PHARMACOMETRICS
PHARMACOMETRICS THE SCIENCE OF QUANTITATIVE PHARMACOLOGY
Edited by
Ene I. Ette Anoixis Corporation
Paul J. Williams University of the Pacific and Anoixis Corporation
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright © 2007 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, 201-748-6011, fax 201-748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at 877-762-2974, outside the United States at 317-572-3993 or fax 317-572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Wiley Bicentennial Logo: Richard J. Pacifico Library of Congress Cataloging-in-Publication Data: Pharmacometrics : the science of quantitative pharmacology / [edited by] Ene I. Ette, Paul J. Williams. p. ; cm. Includes bibliographical references. ISBN 978-0-471-67783-3 1. Pharmacology. 2. Pharmacokinetics. I. Ette, Ene I. II. Williams, Paul J. [DNLM: 1. Chemistry, Pharmaceutical–methods. 2. Drug Evaluation–methods. 3. Models, Theoretical. 4. Pharmacoepidemiology–methods. 5. Pharmacokinetics. 6. Technology, pharmaceutical–methods. 7. Drug Development. 8. Pharmacometrics. QV 744 P5363 2006] RS187.P4553 2006 615′. 1–dc22 2006016629 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1
To my wife, Esther, who supports, comforts, and inspires and is always there for me. E. I. E. To my wife, Debbie, who supports, comforts, and inspires. P. J. W.
CONTENTS
CONTRIBUTORS
xi
PREFACE
xv
ACKNOWLEDGMENTS
xix
1. Pharmacometrics: Impacting Drug Development and Pharmacotherapy
1
Paul J. Williams and Ene I. Ette
PART I
GENERAL PRINCIPLES
2. General Principles of Programming: Computer and Statistical
25
Sastry S. Isukapalli and Amit Roy
3. Validation of Software for Pharmacometric Analysis
53
Gary L. Wolk
4. Linear, Generalized Linear, and Nonlinear Mixed Effects Models
103
Farkad Ezzet and José C. Pinheiro
5. Bayesian Hierarchical Modeling with Markov Chain Monte Carlo Methods
137
Stephen B. Duffull, Lena E. Friberg, and Chantaratsamon Dansirikul
6. Estimating the Dynamics of Drug Regimen Compliance
165
Ene I. Ette and Alaa Ahmad
7. Graphical Displays for Modeling Population Data
183
E. Niclas Jonsson, Mats O. Karlsson, and Peter A. Milligan
8. The Epistemology of Pharmacometrics
223
Paul J. Williams, Yong Ho Kim, and Ene I. Ette
9. Data Imputation
245
Ene I. Ette, Hui-May Chu, and Alaa Ahmad
PART II POPULATION PHARMACOKINETIC BASIS OF PHARMACOMETRICS 10. Population Pharmacokinetic Estimation Methods
265
Ene I. Ette, Paul J. Williams, and Alaa Ahmad vii
viii
CONTENTS
11. Timing and Efficiency in Population Pharmacokinetic/ Pharmacodynamic Data Analysis Projects
287
Siv Jönsson and E. Niclas Jonsson
12. Designing Population Pharmacokinetic Studies for Efficient Parameter Estimation
303
Ene I. Ette and Amit Roy
13. Population Models for Drug Absorption and Enterohepatic Recycling
345
Olivier Pétricoul, Valérie Cosson, Eliane Fuseau, and Mathilde Marchand
14. Pharmacometric Knowledge Discovery from Clinical Trial Data Sets
383
Ene I. Ette
15. Resampling Techniques and Their Application to Pharmacometrics
401
Paul J. Williams and Yong Ho Kim
16. Population Modeling Approach in Bioequivalence Assessment
421
Chuanpu Hu and Mark E. Sale
PART III PHARMACOKINETICS / PHARMACODYNAMICS RELATIONSHIP: BIOMARKERS AND PHARMACOGENOMICS, PK/PD MODELS FOR CONTINUOUS DATA, AND PK/PD MODELS FOR OUTCOMES DATA 17. Biomarkers in Drug Development and Pharmacometric Modeling
457
Paul J. Williams and Ene I. Ette
18. Analysis of Gene Expression Data
473
Daniel Brazeau and Murali Ramanathan
19. Pharmacogenomics and Pharmacokinetic/Pharmacodynamic Modeling
509
Jin Y. Jin and William J. Jusko
20. Empirical Pharmacokinetic/Pharmacodynamic Models
529
James A. Uchizono and James R. Lane
21. Developing Models of Disease Progression
547
Diane R. Mould
22. Mechanistic Pharmacokinetic/Pharmacodynamic Models I
583
Varun Garg and Ariya Khunvichai
23. Mechanistic Pharmacokinetic/Pharmacodynamic Models II
607
Donald E. Mager and William J. Jusko
24. PK/PD Analysis of Binary (Logistic) Outcome Data
633
Jill Fiedler-Kelly
25. Population Pharmacokinetic/Pharmacodynamic Modeling of Ordered Categorical Longitudinal Data Ene I. Ette, Amit Roy, and Partha Nandy
655
CONTENTS
26. Transition Models in Pharmacodynamics
ix
689
Ene I. Ette
27. Mixed Effects Modeling Analysis of Count Data
699
Christopher J. Godfrey
28. Mixture Modeling with NONMEM V
723
Bill Frame
PART IV
CLINICAL TRIAL DESIGNS
29. Designs for First-Time-in-Human Studies in Nononcology Indications
761
Hui-May Chu, Jiuhong Zha, Amit Roy, and Ene I. Ette
30. Design of Phase 1 Studies in Oncology
781
Brigitte Tranchand, René Bruno, and Gilles Freyer
31. Design and Analysis of Clinical Exposure: Response Trials
803
David Hermann, Raymond Miller, Matthew Hutmacher, Wayne Ewy, and Kenneth Kowalski
PART V
PHARMACOMETRIC KNOWLEDGE CREATION
32. Pharmacometric/Pharmacodynamic Knowledge Creation: Toward Characterizing an Unexplored Region of the Response Surface
829
Ene I. Ette and Hui-May Chu
33. Clinical Trial Simulation: Theory
851
Peter L. Bonate
34. Modeling and Simulation: Planning and Execution
873
Paul J. Williams and James R. Lane
35. Clinical Trial Simulation: Efficacy Trials
881
Matthew M. Riggs, Christopher J. Godfrey, and Marc R. Gastanguay
PART VI PHARMACOMETRIC SERVICE AND COMMUNICATION 36. Engineering a Pharmacometrics Enterprise
903
Thaddeus H. Grasela and Charles W. Dement
37. Communicating Pharmacometric Analysis Outcome
925
Ene I. Ette and Leonard C. Onyiah
PART VII
SPECIFIC APPLICATION EXAMPLES
38. Pharmacometrics Applications in Population Exposure–Response Data for New Drug Development and Evaluation He Sun and Emmanuel O. Fadiran
937
x
CONTENTS
39. Pharmacometrics in Pharmacotherapy and Drug Development: Pediatric Application
955
Edmund V. Capparelli and Paul J. Williams
40. Pharmacometric Methods for Assessing Drug-Induced QT and QTc Prolongations for Non-antiarrhythmic Drugs
977
He Sun
41. Using Pharmacometrics in the Development of Therapeutic Biological Agents
993
Diane R. Mould
42. Analysis of Quantic Pharmacokinetic Study: Robust Estimation of Tissue-to-Plasma Ratio
1035
Hui-May Chu and Ene I. Ette
43. Physiologically Based Pharmacokinetic Modeling: Inhalation, Ingestion, and Dermal Absorption
1069
Sastry S. Isukapalli, Amit Roy, and Panos G. Georgopoulos
44. Modeling of Metabolite Pharmacokinetics in a Large Pharmacokinetic Data Set: An Application
1107
Valérie Cosson, Karin Jorga, and Eliane Fuseau
45. Characterizing Nonlinear Pharmacokinetics: An Example Scenario for a Therapeutic Protein
1137
Stuart Friedrich
46. Development, Evaluation, and Applications of in Vitro/in Vivo Correlations: A Regulatory Perspective
1157
Patrick J. Marroum
47. The Confluence of Pharmacometric Knowledge Discovery and Creation in the Characterization of Drug Safety
1175
Hui-May Chu and Ene I. Ette
INDEX
1197
CONTRIBUTORS
Alaa Ahmad, Clinical Pharmacology, Vertex Pharmaceuticals, 130 Waverly St., Cambridge, MA 02139 [
[email protected]] Peter L. Bonate, Genzyme Corporation, Pharmacokinetics, 4545 Horizon Hill Blvd., San Antonio, TX 78229 [
[email protected]] Daniel Brazeau, Department of Pharmaceutical Sciences, 517 Cooke Hall, State University of New York at Buffalo, Buffalo, NY 14260 [
[email protected]] René Bruno, Pharsight Corporation, 84 Chemin des Grives, 13013 Marseille, France [
[email protected]] Edmund V. Capparelli, Pediatric Pharmacology Research Unit, School of Medicine, University of California—San Diego, 4094 4th Avenue, San Diego, CA 92103 and Trials by Design, 1918 Verdi Ct., Stockton, CA 95207 [
[email protected]] Hui-May Chu, Clinical Pharmacology, Vertex Pharmaceuticals, 130 Waverly St., Cambridge, MA 02139 [
[email protected]] Valérie Cosson, Clinical Pharmacokinetics Modeling and Simulation, Psychiatry, GSK Spa, Via Fleming 4, 37135 Verona, Italy [
[email protected]] and Hoffman—La Roche Ltd., PDMP, 663/2130, CH-4070 Basel, Switzerland [
[email protected]] Charles W. Dement, 260 Jacobs Management Center, University at Buffalo–SUNY, Buffalo, NY 14260 Chantaratsamon Dansirikul, School of Pharmacy, University of Queensland, Brisbane 4072, Australia [
[email protected]] and Department of Pharmaceutical Biosciences, Uppsala University, Box 591, SE-751 24 Uppsala, Sweden Stephen B. Duffull, School of Pharmacy, University of Queensland, Brisbane 4072, Australia [
[email protected]] and School of Pharmacy, University of Otago, PO Box 913, Dunedin, New Zealand [stephen.duffull@stonebow. otago.ac.nz] Ene I. Ette, Clinical Pharmacology, Vertex Pharmaceuticals, 130 Waverly St., Cambridge, MA 02139 and Anoixis Corp., 214 N. Main St., Natick, MA 01760 [
[email protected]] Wayne Ewy, Pfizer, PGR&D, 2800 Plymouth Road, Ann Arbor, MI 48105 [wayne.ewy@pfizer.com] xi
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CONTRIBUTORS
Farkad Ezzet, Pharsight Corporation, 87 Lisa Drive, Chatham, NJ 07928 Emmanuel O. Fadiran, Division of Clinical Pharmacology 2, OCP, FDA, 10903 New Hampshire Avenue, Building 21, Silver Springs, MD 20993-0002 [
[email protected]] Jill Fiedler-Kelly, Cognigen Corporation, 395 S Youngs Rd., Williamsville, NY 14221 [
[email protected]] Bill Frame, C.R.T., 5216 Pratt Rd., Ann Arbor, MI 48103 [framebill@ameritech. net] Gilles Freyer, Ciblage Thérapeutique en Oncologie, Service Oncologie Médicale, EA 3738, CH Lyon-Sud, 69495 Pierre-Bénite Cedex, France [Gilles.Freyer@ chu-lyon.fr] Lena E. Friberg, School of Pharmacy, University of Queensland, Brisbane 4072, Australia [
[email protected]] and Department of Pharmaceutical Biosciences, Uppsala University, Box 591, SE-751 24 Uppsala, Sweden Stuart Friedrich, Global PK/PD and Trial Simulations, Eli Lilly Canada Inc., 3650 Danforth Ave., Toronto, ON, MIN 2E8 Canada [
[email protected]] Eliane Fuseau, EMF Consulting, Aix en Provence Cedex 4, France [
[email protected]] Varun Garg, Clinical Pharmacology, Vertex Pharmaceuticals, 130 Waverly St., Cambridge, MA 02139 [
[email protected]] Marc R. Gastonguay, Metrum Research Group LLC, 2 Tunxis Road, Suite 112, Tariffville, CT 06081 [
[email protected]] Panos G. Georgopoulos, Computational Chemodynamics Laboratory, Environmental and Occupational Health Sciences Institute, 70 Frelinghuysen Road, Piscataway, NJ 08854 [
[email protected]] Christopher J. Godfrey, Clinical Pharmacology, Vertex Pharmaceuticals, 130 Waverly St., Cambridge, MA 02139 and Anoixis Corp., 214 N. Main St., Natick, MA 01760 [
[email protected]] Thaddeus H. Grasela, Cognigen Corporation, 395 S Youngs Rd, Williamsville, NY 14221 [
[email protected]] David Hermann, deCODE Genetics, 1032 Karl Greimel Drive, Brighton, MI 48116 [
[email protected]] Chuanpu Hu, Biostatistics, Sanofi-Aventis, 9 Great Valley Parkway, Malvern, PA 19355-1304 [Chuanpu.Hu@sanofi-aventis.com] Matthew Hutmacher, Pfizer, PGR&D, 2800 Plymouth Road, Ann Arbor, MI 48105 [matt.hutmach er@pfizer.com] Sastry S. Isukapalli, Computational Chemodynamics Laboratory, Environmental and Occupational Health Sciences Institute, 70 Frelinghuysen Road, Piscataway, NJ 08854 [ssi@fidelio.rutgers.edu]
CONTRIBUTORS
xiii
Jin Y. Jin, Department of Pharmaceutical Sciences, School of Pharmacy, 519 Hochstetter Hall, State University of New York at Buffalo, Buffalo, NY 14260 Siv Jönsson, Clinical Pharmacology, AstraZeneca R&D Södertälje, SE-151 85 Södertälje, Sweden [
[email protected]] E. Niclas Jonsson, Hoffmann-La Roche Ltd., PDMP Modelling and Simulation, Grenzacherstr 124, Bldg. 15/1.052, CH-4070 Basel, Switzerland [niclas.jonsson@ roche.com] Karin Jorga, Hoffmann-La Roche Ltd., PDMP Clinical Pharmacology, Grenzacherstrasse 124, Bldg. 15/1.081A, CH-4070 Basel, Switzerland [
[email protected]] William J. Jusko, Department of Pharmaceutical Sciences, School of Pharmacy, 519 Hochstetter Hall, State University of New York at Buffalo, Buffalo, NY 14260 [
[email protected]] Mats O. Karlsson, Division of Pharmacokinetics and Drug Therapy, Department of Pharmaceutical Biosciences, Uppsala University, Box 591, SE-751 24 Uppsala, Sweden [
[email protected]] Ariya Khunvichai, Clinical Pharmacology, Vertex Pharmaceuticals, 130 Waverly St., Cambridge, MA 02139 [
[email protected]] Yong Ho Kim, Clinical Pharmacokinetics, Five Moore Drive, Sanders Bldg. 17.2245 PO Box 13398, Research Triangle Park, NC 27709 [joseph.y.
[email protected]] and Clinical Pharmacokinetics, GlaxoSmithKline, Raleigh, NC [
[email protected]] Kenneth Kowalski, Pfizer, PGR&D, 2800 Plymouth Road, Ann Arbor, MI 48105 [ken.kowalski@pfizer.com] James R. Lane, Department of Pharmacy, Skaggs School of Pharmacy and Pharmaceutical Sciences, University of California San Diego, 200 West Arbor Drive, San Diego, CA 92103-8765 [
[email protected]] Donald E. Mager Department of Pharmaceutical Sciences, School of Pharmacy, 519 Hochstetter Hall, State University of New York at Buffalo, Buffalo, NY 14260 [
[email protected]] Mathilde Marchland, EMF Consulting, 425 rue Rene Descartes, BP 02, 13545 Aix-en-Provence Cedex 4, France [
[email protected]] Patrick J. Marroum, Office of Clinical Pharmacology, CDER, FDA, 10903 New Hampshire Avenue, Building 21, Silver Spring, MD 20993 [patrick.marroum@ fda.hhs.gov] Raymond Miller, Pfizer, PGR&D, 2800 Plymouth Road, Ann Arbor, MI 48105 [raymond.miller@pfizer.com] Peter A. Milligan, Pfizer, Ramsgate Road, Sandwich, Kent, CT13 9NJ, UK [peter.a.milligan@pfizer.com]
xiv
CONTRIBUTORS
Diane R. Mould, Projections Research, Inc., 535 Springview Lane, Phoenixville, PA 19460 [
[email protected]] Partha Nandy, Johnson & Johnson Pharmaceutical Research and Development, 1125 Trenton-Hourborton Road, Titusville, NJ 08560 [
[email protected]] Leonard C. Onyiah, Engineering and Computer Center, Department of Statistics and Computer Networking, St. Cloud State University, 720 4th Avenue South, St. Cloud, MN 56301 [
[email protected]] Olivier Pétricoul, EMF Consulting, 425 rue Rene Descartes, BP 02, 13545 Aix-enProvence Cedex 4, France [
[email protected]] José Pinheiro, Biostatistics, Novartis Pharmaceuticals Corporation, One Health Plaza, 419/2115, East Hanover, NJ 07936 [
[email protected]] Murali Ramanathan, Pharmaceutical Sciences and Neurology, 543 Cooke Hall, State University of New York, Buffalo, NY 14260 Amit Roy, Strategic Modeling and Simulation, Bristol-Myers Squibb, Route 206 and Provinceline Road, Princeton, NJ 08540 [
[email protected]] Matthew M. Riggs, Metrum Research Group LLC, 2 Tunxis Road, Suite 112, Tariffville, CT 06081 [
[email protected]] Mark E. Sale, Next Level Solutions LLC, 1013 Dickinson Circle, Raleigh, NC 27614 [
[email protected],
[email protected]] He Sun, SunTech Research Institute, 1 Research Court, Suite 450-54, Rockville, MD 20850 [
[email protected],
[email protected]] Brigitte Tranchand, Ciblage Thérapeutique en Oncologie, Faculté de Médecine, EA3738, Lyon-Sud, BP12, 69921 Oullins Cedex, France [Brigitte.Tranchand@ adm.univ-lyon1.fr] James A. Uchizono, Department of Pharmaceutics and Medicinal Chemistry, Thomas J. Long School of Pharmacy, University of the Pacific, Stockton, CA 95211 [juchizono@pacific.edu] Paul J. Williams, Thomas J. Long School of Pharmacy and Health Sciences, University of the Pacific, Stockton, CA 95211 and Anoixis Corp., 1918 Verdi Ct., Stockton CA 95207 [pwilliams@pacific.edu,
[email protected]] Gary L. Wolk, 1215 South Kihei Rd., Kihei, HI 96753 [
[email protected]] Jiuhong Zha, Biopharmacentical Sciences, Astellas Pharma, US Inc., Chicago [
[email protected]]
PREFACE
The subspecialty of population pharmacokinetics was introduced into clinical pharmacology / pharmacy in the late 1970s as a method for analyzing observational data collected during patient drug therapy in order to estimate patient-based pharmacokinetic parameters. It later became the basis for dosage individualization and rational pharmacotherapy. The population pharmacokinetics method (i.e., the population approach) was later extended to the characterization of the relationship between pharmacokinetics and pharmacodynamics, and into the discipline of pharmacometrics. Pharmacometrics is the science of interpreting and describing pharmacology in a quantitative fashion. Vast amounts of data are generated during clinical trials and patient care, and it is the responsibility of the pharmacometrician to extract the knowledge embedded in the data for rational drug development and pharmacotherapy. He/she is also responsible for providing that knowledge for decision making in patient care and the drug development process. With the publication of the Guidance for Industry: Population Pharmacokinetics by the Food and Drug Administration (the advent of population pharmacokinetics/pharmacodynamics-based clinical trial simulation) and recently the FDA Critical Path Initiative—The Critical Path to New Medical Products, the assimilation of pharmacometrics as an applied science in drug development and evaluation is increasing. Although a great deal has been written in the journal literature on population pharmacokinetics, population pharmacokinetics/pharmacodynamics, and pharmacometrics in general, there is no major reference textbook that pulls all these facets of knowledge together in one volume for pharmacometricians in academia, regulatory agencies, or industry and graduate students/postdoctoral fellows who work/research in this subject area. It is for this purpose that this book is written. Although no book can be complete in itself, what we have endeavored to assemble are contributors and an array of topics that we believe provide the reader with the knowledge base necessary to perform pharmacometric analysis, to interpret the results of the analysis, and to be able to communicate the same effectively to impact mission-critical decision making. The book is divided into seven sections—general principles, population pharmacokinetic basis of pharmacometrics, pharmacokinetics/pharmacodynamics relationship, clinical trial designs, pharmacometric knowledge creation, pharmacometric service and communication, and specific application examples. In the introductory chapter, the history of the development of pharmacometrics is traced and its application to drug development, evaluation, and pharmacotherapy is delineated. This is followed by Part I on general principles that addresses topics such as the general principles of programming, which is a must for every pharmacometrician, pharmacometric analysis software validation—a subject that has become prominent in last few years, linear and nonlinear mixed effects xv
xvi
PREFACE
modeling to provide the reader with the background knowledge on these topics and thus setting the pace for the remainder of the book, estimation of the dynamics of compliance, which is important for having a complete picture of a study outcome, graphical display of population data—a sine qua non for informative pharmacometric data analysis, the epistemology of pharmacometrics, which provides a pathway for performing a pharmacometric analysis, and data imputation. Data analysis without the proper handling of missing data may result in biased parameter estimates. The chapter on data imputation covers the various aspects of “missingness” and includes an example of how to handle left censored data—a challenge with most pharmacokinetic data sets. In Part II of the book various aspects of population pharmacokinetics, pharmacometric knowledge discovery, and resampling techniques used in pharmacometric data analysis are covered. Thus, various aspects of the informative design and analysis of population pharmacokinetic studies are addressed together with population pharmacokinetics estimation methods. The chapter on pharmacometric knowledge discovery deals with the integrated approach for discovering knowledge from clinical trial data sets and communicating the same for optimal pharmacotherapy and knowledge/model-based drug development. Part III, which is on the pharmacokinetics–pharmacodynamics relationship, deals with biomarkers and surrogates in drug development, gene expression analysis, integration of pharmacogenomics into pharmacokinetics/pharmacodynamics, empirical and mechanistic PK/PD models, outcome models, and disease progression models that are needed for understanding disease progression as the basis for building models that can be used in clinical trial simulation. Part IV builds on the knowledge gained from the previous sections to provide the basis for designing clinical trials. The section opens with a chapter on the design of first-time-in-human (FTIH) studies for nononcology indications. The literature is filled with a discussion of the design of FTIH oncology studies, but very little has been written on the design of FTIH studies for nononcology indications. A comprehensive overview of different FTIH study designs is provided with an evaluation of the designs that provide the reader with the knowledge needed for choosing an appropriate study design. A comprehensive coverage of the design of Phase 1 and phase 2a oncology studies is provided in another chapter; the section closes with a chapter on the design of dose – response studies. Part V addresses pharmacometric knowledge creation, which entails the application of pharmacometric methodologies to the characterization of an unexplored region of the response surface. It is the process of building upon current understanding of data that is already acquired by generating more data (information) that can be translated into knowledge. Thus, the section opens with a chapter on knowledge creation, followed by the theory of clinical trial simulation and the basics of clinical trial simulation, and ends with a chapter on the simulation of efficacy trials. Parts VI and VII discuss what a pharmacometric service is all about, how to communicate the results of a pharmacometric analysis, and specific examples ranging from applications in a regulatory setting, characterization of QT interval prolongation, pharmacometrics in biologics development, pharmacometrics in pediatric pharmacotherapy, application of pharmacometric principles to the analysis of preclinical data, physiologically based pharmacokinetic modeling, characterizing metabolic and nonlinear pharmacokinetics, in vitro in vivo correlation, and the
PREFACE
xvii
application of pharmacometric knowledge discovery and creation to the characterization of drug safety. What makes this book unique is not just the presentation of theory in an easy to comprehend fashion, but the fact that for a majority of the chapters there are application examples with codes in NONMEM, S-Plus, WinNonlin, or Matlab. The majority of the codes are for NONMEM and S-Plus. Thus, the reader is able to reproduce the examples in his/her spare time and gain an understanding of both the theory and principles of pharmacometrics covered in a particular chapter. A reader friendly approach was taken in the writing of this book. Although there are many contributors to the book, we have tried as much as possible to unify the style of presentation to aid the reader’s understanding of the subject matter covered in each chapter. Emphasis has been placed on drug development because of the need to apply pharmacometrics in drug development to increase productivity. Examples have been provided for the application of pharmacometrics in pharmacotherapy and drug evaluation to show how pharmacometric principles have been applied in these areas with great benefit. In the writing of this text, the reader’s knowledge of pharmacokinetics, pharmacodynamics, and statistics is assumed. If not, the reader is referred to Applied Pharmacokinetics by Shargel and Yu, Pharmacokinetics by Gibaldi and Perrier, Pharmacokinetics and Pharmacodynamics by Gabrielson and Weiner, and statistics from standard textbooks. Finally, this book is written for the graduate students or postdoctoral fellows who want to specialize in pharmacometrics; and for pharmaceutical scientists, clinical pharmacologists/pharmacists, and statisticians in academia, regulatory bodies, and the pharmaceutical industry who are in pharmacometrics or are interested in developing their skill set in the subject. Ene I. Ette Paul J. Williams
ACKNOWLEDGMENTS
This book is the result of many hands and minds. None of us is as smart as all of us; therefore we acknowledge the contributions of the chapter authors who withstood bullyragging as this work was put together. Furthermore, the contributions of our parents over the long haul of our lives must be recognized. We thank Esther and the children, and Debbie, who have been patient not only through the process of writing and editing this work but for a lifetime. In addition, we are thankful to Jonathan Rose, Wiley commissioning editor for pharmaceutical sciences books, and Rosalyn Farkas, production editor at Wiley, for their patience and cooperation. Finally and most importantly, we recognize the work of the Father, Son, and Holy Spirit who gave us the idea and provided the energy to complete this work and to whom we are eternally indebted. E. I. E. P. J. W.
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CHAPTER 1
Pharmacometrics: Impacting Drug Development and Pharmacotherapy PAUL J. WILLIAMS and ENE I. ETTE
1.1
INTRODUCTION
Drug development continues to be expensive, time consuming, and inefficient, while pharmacotherapy is often practiced at suboptimal levels of performance (1–3). This trend has not waned despite the fact that massive amounts of drug data are obtained each year. Within these massive amounts of data, knowledge that would improve drug development and pharmacotherapy lays hidden and undiscovered. The application of pharmacometric (PM) principles and models to drug development and pharmacotherapy will significantly improve both (4, 5). Furthermore, with drug utilization review, generic competition, managed care organization bidding, and therapeutic substitution, there is increasing pressure for the drug development industry to deliver high-value therapeutic agents. The Food and Drug Administration (FDA) has expressed its concern about the rising cost and stagnation of drug development in the white paper Challenge and Opportunity on the Critical Path to New Products published in March of 2004 (3). In this document the FDA states: “Not enough applied scientific work has been done to create new tools to get fundamentally better answers about how the safety and effectiveness of new products can be demonstrated in faster time frames, with more certainty, and at lower costs. . . . A new product development toolkit—containing powerful new scientific and technical methods such as animal or computer-based predictive models, biomarkers for safety and effectiveness, and new clinical evaluation techniques—is urgently needed to improve predictability and efficiency along the critical path from laboratory concept to commercial product. We need superior product development science to address these challenges.” In the critical path document, the FDA states that the three main areas of the path that need to be addressed are tools for assessing safety, tools for demonstrating medical utility, and lastly tools for characterization and manufacturing. Pharmacometrics can be applied to and can impact the first two areas, thus positively impacting the critical path.
Pharmacometrics: The Science of Quantitative Pharmacology Edited by Ene I. Ette and Paul J. Williams Copyright © 2007 John Wiley & Sons, Inc.
1
2
PHARMACOMETRICS: IMPACTING DRUG DEVELOPMENT AND PHARMACOTHERAPY
For impacting safety, the FDA has noted opportunities to better define the importance of the QT interval, for improved extrapolation of in vitro and animal data to humans, and for use of extant clinical data to help construct models to screen candidates early in drug development (e.g., liver toxicity). Pharmacometrics can have a role in developing better links for all of these models. For demonstrating medical utility, the FDA has highlighted the importance of model-based drug development in which pharmacostatistical models of drug efficacy and safety are developed from preclinical and available clinical data. The FDA goes on to say that “Systematic application of this concept to drug development has the potential to significantly improve it. FDA scientists use and are collaborating with others in the refinement of quantitative clinical trial modeling using simulation software to improve trial design and to predict outcomes.” The pivotal role of pharmacometrics on the critical path is obvious. Drug development could be improved by planning to develop and apply PM models along with novel pathways to approval, improved project management, and improved program development. Recent advances in computational speed, novel models, stochastic simulation methods, real-time data collection, and novel biomarkers all portend improvements in drug development. Dosing strategy and patient selection continue to be the most easily manipulated parts of a patient’s therapy. Optimal dosing often depends on patient size, sex, and renal function or liver function. All too often, the impact of these covariates on a PM parameter is unstudied and therefore cannot be incorporated into any therapeutic strategy. PM model development and application will improve both drug development and support rational pharmacotherapy.
1.2
PHARMACOMETRICS DEFINED
Pharmacometrics is the science of developing and applying mathematical and statistical methods to characterize, understand, and predict a drug’s pharmacokinetic, pharmacodynamic, and biomarker–outcomes behavior (6). Pharmacometrics lives at the intersection of pharmacokinetic (PK) models, pharmacodynamic (PD) models, pharmacodynamic-biomarker–outcomes link models, data visualization (often by employing informative modern graphical methods), statistics, stochastic simulation, and computer programming. Through pharmacometrics one can quantify the uncertainty of information about model behavior and rationalize knowledge-driven decision making in the drug development process. Pharmacometrics is dependent on knowledge discovery, the application of informative graphics, understanding of biomarkers/surrogate endpoints, and knowledge creation (7–10). When applied to drug development, pharmacometrics often involves the development or estimation of pharmacokinetic, pharmacodynamic, pharmcodynamic– outcomes linking, and disease progression models. These models can be linked and applied to competing study designs to aid in understanding the impact of varying dosing strategies, patient selection criteria, differing statistical methods, and different study endpoints. In the realm of pharmacotherapy, pharmacometrics can be employed to customize patient drug therapy through therapeutic drug monitoring and improved population dosing strategies. To contextualize the role of pharmacometrics in drug development and pharmacotherapy, it is important to examine
HISTORY OF PHARMACOMETRICS
3
the history of pharmacometrics. The growth of pharmacometrics informs much on its content and utility.
1.3 1.3.1
HISTORY OF PHARMACOMETRICS Pharmacokinetics
Pharmacometrics begins with pharmacokinetics. As far back as 1847, Buchanan understood that the brain content of anesthetics determined the depth of narcosis and depended on the arterial concentration, which in turn was related to the strength of the inhaled mixture (11). Interestingly, Buchanan pointed out that rate of recovery was related to the distribution of ether in the body. Though there was pharmacokinetic (PK) work done earlier, the term pharmacokinetics was first introduced by F. H. Dost in 1953 in his text, Der Blutspeigel-Kinetic der Knozentrationsablaufe in der Kreislauffussigkeit (12). The first use in the English language occurred in 1961 when Nelson published his “Kinetics of Drug Absorption, Distribution, Metabolism, and Excretion” (13). The exact word pharmacokinetics was not used in this publication. In their classic work, the German scientists Michaelis and Menton published their equation describing enzyme kinetics in 1913 (14). This equation is still used today to describe the kinetics of drugs such as phenytoin. Widmark and Tandberg (15) published the equations for the one-compartment model in 1924 and in that same year Haggard (16) published his work on the uptake, distribution, and elimination of diethyl ether. In 1934 Dominguez and Pomerene (17) introduced the concept of volume of distribution, which was defined as “the hypothetical volume of body fluid dissolving the substance at the same concentration as the plasma. In 1937 Teorrel (18) published a seminal paper that is now considered the foundation of modern pharmacokinetics. This paper was the first physiologically based PK model, which included a five-compartment model. Bioavailability was introduced as a term in 1945 by Oser and colleagues (19), while Lapp (20) in France was working on excretions kinetics. Polyexponential curve fitting was introduced by Perl in 1960 (21). The use of analog computers for curve fitting and simulation was introduced in 1960 by two groups of researchers (22, 23). The great growth period for pharmacokinetics was from 1961 to 1972, starting with the landmark works of Wagner and Nelson (24). In 1962 the first symposium with the title pharmacokinetics, “Pharmacokinetik und Arzniemitteldosireung,” was held. Clinical pharmacokinetics began to be recognized in the 1970s, especially in two papers by Gibaldi and Levy, “Pharmacokinetics in Clinical Practice,” in the Journal of the American Medical Association in 1976 (25). Of further importance that same year was a paper by Koup et al. (26) on a system for the monitoring and dosing of theophylline based on pharmacokinetic principles. Rational drug therapy is based on the assumption of a causal relationship between exposure and response. There pharmacokinetics has great utility when linked to pharmacodynamics and the examination of pharmacodynamics is of paramount importance.
4
1.3.2
PHARMACOMETRICS: IMPACTING DRUG DEVELOPMENT AND PHARMACOTHERAPY
Pharmacodynamics
In 1848 Dungilson (27) stated that pharmacodynamics was “a division of pharmacology which considers the effects and uses of medicines.” This definition has been refined and restricted over the centuries to a more useful definition, where “pharmacokinetics is what the body does to the drug; pharmacodynamics is what the drug does to the body” (28, 29). More specifically, pharmacodynamics was best defined by Derendorf et al. (28) as “a broad term that is intended to include all of the pharmacological actions, pathophysiological effects and therapeutic responses both beneficial or adverse of active drug ingredient, therapeutic moiety, and/or its metabolite(s) on various systems of the body from subcellular effects to clinical outcomes.” Pharmacodynamics most often involves mathematical models, which relate some concentration (serum, blood, urine) to a physiologic effect (blood pressure, liver function tests) and clinical outcome (survival, adverse effect). The pharmacodynamic (PD) models have been described as fixed, linear, log-linear, Emax, sigmoid Emax, and indirect PD response (29–31). The indirect PD response model has been a particularly significant contribution to PD modeling (30, 31). It has great utility because it is more mechanistic than the other models, does not assume symmetry of the onset and offset, and incorporates the impact of time in addition to drug concentration, thus accounting for a delay in onset and offset of the effect. For these models the maximum response occurs later than the time of occurrence of the maximum plasma concentration because the drug causes incremental inhibition or stimulation as long as the concentration is “high enough.” After the response reaches the maximum, the return to baseline is a function of the dynamic model parameters and drug elimination. Thus, there is a response that lasts beyond the presence of effective drug levels because of the time needed for the system to regain equilibrium. Whenever possible, these mechanistic models should be employed for PD modeling and several dose levels should be employed for accurate determination of the PD parameters, taking into consideration the resolution in exposure between doses. The dependent variables in these PD models are either biomarkers, surrogate endpoints, or clinical endpoints. It is important to differentiate between these and to understand their relative importance and utility.
1.3.3
Biomarkers
The importance of biomarkers has been noted in recent years and is evidenced by the formation of The Biomarkers Definitions Working Group (BDWG) (32). According to the BDWG, a biomarker is a “characteristic that is objectively measured and evaluated as an indicator of normal biological processes, pathogenic process or pharmacologic responses to a therapeutic intervention.” Biomarkers cannot serve as penultimate clinical endpoints in confirming clinical trials; however, there is usually considered to be some link between a biomarker based on prior therapeutic experience, well understood physiology or pathophysiology, along with knowledge of the drug mechanism. Biomarkers often have the advantage of changing in drug therapy prior to the clinical endpoint that will ultimately be employed to determine drug effect, thus providing evidence early in clinical drug development of potential efficacy or safety.
HISTORY OF PHARMACOMETRICS
5
A surrogate endpoint is “a biomarker that is intended to substitute for a clinical endpoint. A surrogate endpoint is expected to predict clinical benefit, harm, lack of benefit, or lack of harm based on epidemiologic, therapeutic, pathophysiologic or other scientific evidence” (32). Surrogate endpoints are a subset of biomarkers such as viral load or blood pressure. All surrogate endpoints are biomarkers. However, few biomarkers will ever become surrogate endpoints. Biomarkers are reclassified as surrogate endpoints when a preponderance of evidence indicates that changes in the biomarker correlate strongly with the desired clinical endpoint. A clinical endpoint is “a characteristic or variable that reflects how a patient feels, functions or survives. It is a distinct measurement or analysis of disease characteristics observed in a study or a clinical trial that reflect the effect of a therapeutic intervention. Clinical endpoints are the most credible characteristics used in the assessment of the benefits and risks of a therapeutic intervention in randomized clinical trials.” There can be problems with using clinical endpoints as the final measure of patient response because a large patient sample size may be needed to determine drug effect or the modification in the clinical endpoint for a drug may not be detectable for several years after the initiation of therapy. There are several ways in which the discovery and utilization of biomarkers can provide insight into the drug development process and patient care. Biomarkers can identify patients at risk for a disease, predict patient response, predict the occurrence of toxicity, and predict exposure to the drug. Given these uses, biomarkers can also provide a basis for selecting lead compounds for development and can contribute knowledge about clinical pharmacology. Therefore, biomarkers have the potential to be one of the pivotal factors in drug development—from drug target discovery through preclinical development to clinical development to regulatory approval and labeling information, by way of pharmacokinetic/pharmacodynamic–outcomes modeling with clinical trial simulations. 1.3.4
PK/PD Link Modeling
PK/PD modeling provides the seamless integration of PK and PD models to arrive at an enlightened understanding of the dose–exposure–response relationship. PK/PD modeling can be done either sequentially or simultaneously (33, 34). Sequential models estimate the pharmacokinetics first and fix the PK parameters, generating concentrations corresponding to some PD measurement. Thus, the pharmacodynamics is conditioned on the PK data or on the estimates of the PK parameters. Simultaneous PK/PD modeling fits all the PK and PD data at once and the PK and PD parameters are considered to be jointly distributed. When simultaneous modeling is done, the flow of information is bidirectional. Both of these approaches appear to provide similar results (33, 35). However, it is important to note that PD measurements are usually less precise than PK measurements and using sequential PK and PD modeling may be the preferred approach in most instances. PK and PD can be linked directly through a measured concentration that is directly linked to an effect site. The direct link model does not work well when there is a temporal relationship between a measured concentration and effect, as when hysteresis is present. When this is the case, an indirect link between the measured concentration and effect must be accounted for in the model. This has been done in
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PHARMACOMETRICS: IMPACTING DRUG DEVELOPMENT AND PHARMACOTHERAPY
general by the construction of an effect compartment, where a hypothetical effect compartment is linked to a PK compartment. Here the effect compartment is very small and thus has negligible impact on mass balance with a concentration time course in the effect compartment. The effect is related to the concentration in the effect compartment, which has a different time course than the compartment where drug concentrations are actually measured. In addition to the effect compartment approach to account for temporal concentration–effect relationships, the indirect response concept has found great utility. PK and PD have been linked by many models, sometimes mechanistic and at other times empirical. These models are especially useful in better understanding the dose strategy and response, especially when applied by stochastic simulation. The population approach can be applied to multiple types of data—for example, both intensely and sparsely sampled data and preclinical to Phase 4 clinical data— and therefore has found great utility when applied to PK/PD modeling.
1.3.5
Emergence of Pharmacometrics
The term pharmacometrics first appeared in the literature in 1982 in the Journal of Pharmacokinetics and Biopharmaceutics (36). At that time, the journal made a commitment to a regular column dealing with the emerging discipline of pharmacometrics, which was defined as “the design, modeling, and analysis of experiments involving complex dynamic systems in the field of pharmacokinetics and biopharmaceutics . . . concerning primarily data analysis problems with such models.” They went on to say that problems with study design, determination of model identifiability, estimation, and hypothesis testing would be addressed along with identifying the importance of graphical methods. Since this time, the importance of pharmacometrics in optimizing pharmacotherapy and drug development has been recognized, and several graduate programs have been established that emphasize pharmacometrics (37). Pharmacometrics is therefore the science of developing and applying mathematical and statistical methods to (a) characterize, understand, and predict a drug’s pharmacokinetic and pharmacodynamic behavior; (b) quantify uncertainty of information about that behavior; and (c) rationalize data-driven decision making in the drug development process and pharmacotherapy. In effect, pharmacometrics is the science of quantitative pharmacology.
1.3.6
Population Modeling
A major development in pharmacometrics was the application of population methods to the estimation of PM parameters (38). With the advent of population approaches, one could now obtain estimates of PM parameters from sparse data from large databases and also obtain improved estimates of the random effects (variances) in the parameters of interest. These models first found great applicability by taking massive amounts of data obtained during therapeutic drug monitoring (TDM) from which typical values and variability of PK parameters were obtained. The parameters once estimated were applied to TDM to estimate initial doses and, using Bayesian algorithms, to estimate a patient’s individual PK parameters to optimize dosing strategies. Population methods have become widely accepted to the
HISTORY OF PHARMACOMETRICS
7
extent that a Guidance for Industry has been issued by the United States Food and Drug Administration (FDA) on population pharmacokinetics. Population methods are applied to pharmacokinetics, pharmacodynamics, and models linking biomarkers to clinical outcomes (39). 1.3.7
Stochastic Simulation
Stochastic simulation was another step forward in the arena of pharmacometrics. Simulation had been widely used in the aerospace industry, engineering, and econometrics prior to its application in pharmacometrics. Simulation of clinical trials first appeared in the clinical pharmacology literature in 1971 (40) but has only recently gained momentum as a useful tool for examining the power, efficiency, robustness, and informativeness of complex clinical trial structure (41). A major impetus promoting the use of clinical trial simulation was presented in a publication by Hale et al. (41), who demonstrated the utility of simulating a clinical trial on the construction of a pivotal study targeting regulatory approval. The FDA has shown interest in clinical trial simulation to the extent that it has said: “Simulation is a useful tool to provide convincing objective evidence of the merits of a proposed study design and analysis. Simulating a planned study offers a potentially useful tool for evaluating and understanding the consequences of different study designs” (39). While we often think of clinical trial simulation as a way for the drug sponsor to determine optimal study structure, it is also a way for the FDA to determine the acceptability of a proposed study protocol. Simulation serves as a tool not only to evaluate the value of a study structure but also to communicate the logical implications of a PM model, such as the logical implication of competing dosing strategies for labeling. The use and role of a simulated Phase 3 safety and efficacy study is still under discussion as confirmatory evidence at the FDA; however, a simulation of this type can serve as supportive evidence for regulatory review (4, 5). It is likely that at some time in the future knowledge of a disease’s pathophysiology plus knowledge of drug behavior and action will be applied to a group of virtual patients as the pivotal Phase 3 study for approval by a clinical trial simulation. Stochastic simulation should result in more powerful, efficient, robust, and informative clinical trials; therefore, more can be learned, and confirming efficacy will be more certain as stochastic simulation is applied to the drug development process. 1.3.8
Learn–Confirm–Learn Process
Drug development has traditionally been empirical and proceeded sequentially from preclinical through clinical Phases 1 to 3. Sheiner (42) first proposed a major paradigm shift in drug development away from an empirical approach to the learn–confirm approach based on Box’s inductive versus deductive cycles (43). Williams et al. (6, 44) and Ette et al. (45) have since revised this process to the learn–confirm–learn approach because of their emphasis on the fact that learning continues throughout the entire drug development process. The learn–confirm– learn process contends that drug development ought to consist of alternate cycles of learning from experience and then confirming what has been learned but that one never proposes a protocol where learning ceases.
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PHARMACOMETRICS: IMPACTING DRUG DEVELOPMENT AND PHARMACOTHERAPY
In the past, Phases 1 and 2a have been considered the learning phases of drug development because the primary objectives are to determine the tolerated doses and the doses producing the desired therapeutic effect. Phase 2 has targeted how to use the drug in the target patient population, determining the dose strategy and proof of concept. Phase 3 has focused on confirming efficacy and demonstrating a low incidence of adverse events, where if the ratio of benefit to risk is acceptable then the drug is approved. An encouraging outcome in these early cycles results in investment in the costly Phase 2b and 3 studies. However, even in the confirming stages of drug development, one ought to continue to be interested in learning even though confirming is the primary objective of a study; that is, all studies should incorporate an opportunity for learning in the protocol. Therefore, the process has been renamed “learn–confirm–learn”. Learning and confirming have quite different goals in the process of drug development. When a trial structure optimizes confirming, it most often imposes some restrictions on learning; for example, patient enrollment criteria are limited, thus limiting one’s ability to learn about the agent in a variety of populations. For example, many protocols limit enrollment to patients with creatinine clearances above a certain number (e.g., 50 mL/min). If this is done, one cannot learn how to use such a drug in patients with compromised renal function. Empirical commercial drug development has in general focused on confirming because it provides the necessary knowledge for regulatory approval, addressing the primary issue of efficacy. The downside of the focus on confirming is that it has led to a lack of learning, which can result in a dysfunctional drug development process and less than optimal pharmacotherapy postapproval. PM modeling focuses on learning, where the focus is on building a model that relates dosing strategy, exposure, patient type, prognostic variables, and more to outcomes. Here the three-dimensional response surface is built (42) (see Section 1.3.9.2). PM models are built to define the response surface to increase the signalto-noise ratio, which will be discussed shortly. The entire drug development process is an exercise of the learn–confirm–learn paradigm. 1.3.9
Exposure–Response Relationship
The importance of elucidating the exposure–response relationship must be emphasized. When the term exposure is used, one is usually referring to dose or variables related to concentration such as area under the concentration–time curve (AUC), maximum concentration (Cmax), minimum concentration (Cmin), or average concentration (Cave) in some biological specimen such as serum, urine, cerebral spinal fluid, or sputum. It is worth noting that dose is a very weak surrogate of exposure, especially where there is no proportionality between dose and AUC or Cmax. Response is a measure of the effect of a drug either therapeutic or adverse, such as blood pressure, cardiac index, blood sugar, survival, liver function, or renal function. 1.3.9.1 Regulatory Perspective The FDA document, Guidance for Industry: Exposure–Response Relationships— Study Design, Data Analysis, and Regulatory Applications, has commented extensively on the exposure–response relationship (46). It states: “Exposure–response information is at the heart of any determination of the safety and effectiveness of
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9
drugs. . . . In most cases, however, it is important to develop information on the population exposure–response relationships for favorable and unfavorable effects and information on how, and whether, exposure can be adjusted for various subsets of the population.” The FDA recognizes the value of exposure–response knowledge to support the drug development process and to support the determination of safety and efficacy. In this document it stated that “dose–response studies can, in some cases, be particularly convincing and can include elements of consistency that, depending on the size of the study and outcome, can allow reliance on a single clinical efficacy study as evidence of effectiveness.” The exposure–response relationship was further refined in the defining of the response surface. 1.3.9.2 Response Surface A significant development of the exposure–response concept was the proposing of the response surface. Sheiner (42) first proposed the pharmacological response surface as a philosophical framework for development of PM models. The response surface can be thought of as three dimensional: on one axis are the input variables (dose, concurrent therapies, etc.); on the second axis are the important ways that patients can differ from one another that affect the benefit to toxicity ratio; and the final axis represents the benefit to toxicity ratio. Sheiner stated: “the real surface is neither static, nor is all the information about the patient conveyed by his/her initial prognostic status, nor are exact predictions possible. A realistically useful response . . . must include the elements of variability, uncertainty and time . . .” Thus, the primary goal of the response model is to define the complex relationship between the input profile and dose magnitude when comparing beneficial and harmful pharmacological effects and how this relationship varies between patients. For rational drug use and drug development, the response surface must be mapped. PM models, once developed and validated, allow extrapolation beyond the immediate study subjects to allow application to other patients from whom the model was not derived. These predictive models permit the evaluation of outcomes of competing dosing strategies in patients who have not received the drug and therefore aid in constructing future pivotal studies. One important aspect of PM models employed in mapping the response surface is that they increase the signal-to-noise ratio in a data set because they translate some of the noise into signal. This is important because when we are converting information (data) into knowledge, the knowledge is proportional to the signal-to-noise ratio. 1.3.10
PM Knowledge Discovery
It is our experience that most drug development programs are data rich and knowledge poor. This occurs when data are collected but all of the knowledge hidden in the data set is not extracted. In reality, huge amounts of data are generated from modern clinical trials, observational studies, and clinical practice, but at the same time there is an acute widening gap between data collection, knowledge, and comprehension. PM knowledge discovery applies 13 comprehensive and interwoven steps to PM model development and communication and relies heavily on modern statistical techniques, modern informative graphical applications, and population modeling (8, 9) (see Chapter 14). The more that is known about a drug the better will be its application to direct patient care, and the more powerful and efficient
10
PHARMACOMETRICS: IMPACTING DRUG DEVELOPMENT AND PHARMACOTHERAPY
will be the development program. To this end, PM knowledge discovery is the best approach to extracting knowledge from data and has been defined and applied to PM model development. 1.3.11
PM Knowledge Creation
Most often, knowledge discovery provides the foundation for knowledge creation and is simply the initial step in the application of PM knowledge (10). The discovered knowledge can be used to synthesize new data or knowledge, or to supplement existing data. PM knowledge creation has something in common with knowledge discovery its intent to understand and better define the response surface. Data supplementation deals with the use of models on available data to generate supplemental data that would be used to characterize a targeted unexplored segment of the response surface (47). 1.3.12
Model Appropriateness
Model appropriateness brought a new epistemology to PM model estimation and development (48) (see Chapter 8). The pivotal event in establishing model appropriateness is stating the intended use of the model. The entire process requires the stating of the intended use of the model, classifying the model as either descriptive or predictive, evaluating the model, and validating the model if the model is to be used for predictive purposes. Descriptive models are not intended to be applied to any external population—that is, their sole purpose is to gain knowledge about the drug in the population studied. Predictive models are intended to be applied to subjects from whom the model was not derived or estimated. Predictive models require a higher degree of correspondence to the external universe than descriptive models and therefore require validation. Under the epistemology of model appropriateness, the purpose for which the model is developed has a significant impact on the modeling process. In the current modeling climate, insufficient consideration is given to the purpose or intended use of the model and little attention is given to whether the model is descriptive or predictive. Model appropriateness is a paradigm that ought to be applied to the model development and estimation process and it provides the framework for appropriate use of PM models.
1.4
PIVOTAL ROLE OF PHARMACOMETRICS IN DRUG DEVELOPMENT
Drug development has become protracted and expensive over the last several decades, with the average length of clinical development being over 7–12 years, the number of studies averaging 66, and a cost of $0.802–1.7 billion per approved agent (1–4). The process has been empirical—driven by identifying all the items needed for registration of an agent, constructing a checkbox for each item, and executing the studies so that each box is checked, with a consequent fulfillment of each requirement. The numbers above indicate that this empirical, “it has always been done this way” approach does not work well and novel approaches need to be applied. The learn–confirm–learn paradigm should be applied to all drug
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development programs, and modeling should follow the epistemology of model appropriateness. To expedite drug development while maintaining patient safety, new technologies and approaches to discovery, improved project and development approaches, portfolio review, application of sound science, novel study structures, and pharmacometrically guided development programs will need to emerge (49). The use of pharmacometrics to define the dose exposure–response relationship has been successful in improving drug development and pharmacotherapy. Of pivotal importance here is the learn–confirm–learn paradigm, which has been previously mentioned as one of the significant proposals in the evolution of pharmacometrics. While pharmacometrics can be an important tool to expedite drug development, it will also play a key role in determining the optimal dose at the time of approval (new drug application approval). Going to market with the optimal dose is not as straightforward as one may expect. A recent retrospective study noted that of 499 approved drugs between 1980 and 1999, one in five had a dosage change postapproval and 80% of these changes were a decrease in dose (50). This study concluded that current drug development frequently does not capture completely the dose information needed for safe pharmacotherapy. To address this, Cross et al. (50) suggested that improved PK and PD information be gathered early in Phase 2 of drug development. Finally, if drug doses are higher than need be during development and adverse events are related to dose, this may result in an increased frequency of adverse events resulting in an increased study dropout rate and therefore a decrease in study power. Finding the optimal dose is one of the primary goals of clinical development, because changing a dose based on patient characteristics can easily be done. Simplified dosing strategies are often sought by the drug sponsor because it results in ease of use by the practitioner and the patient. Often a sponsor wants a “one dose fits all” approach, which may not result in optimized dosing. Often several levels of dose stratification result in surprisingly improved dosing strategies (e.g., elderly versus young). Novel study structures, such as the enrichment trial, fusion, and adaptive design studies, will result in more efficient drug development. Enrichment studies attempt to choose subjects who are likely to respond. Study groups can be “enriched” by enrolling only subjects with response markers in a specific range or by enrolling only subject types demonstrating a good response during a short pretest phase. In enrichment trials the exposure relationship can be studied efficiently, but it is difficult to know how to extrapolate the quantitative relationship (exposure–response) from an enrichment study to the general population. The advantage of the adaptive design study is that it emphasizes study of the drug in the region of useful doses, thus minimizing the number of subjects in regions where the drug is not effective. For adaptive designs, an exposure–response model is used and continuously updated as each subject’s response is observed. The updated model is used to generate the probability of allocation of each new subject to a treatment arm, favoring the allocation to those arms with the better accumulated outcomes to date, with new subjects randomly allocated to arms on the basis of these frequencies. A treatment arm is dropped from the remainder of the study when its allocation probability drops below a specified threshold. The efficiency of this study design is that as few subjects as necessary are studied to determine that
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PHARMACOMETRICS: IMPACTING DRUG DEVELOPMENT AND PHARMACOTHERAPY
one dose level is less useful than another. This approach can decrease study duration and numbers of subject in a clinical study. Adaptive design works best when patient accrual rates are slow. 1.4.1
Preclinical Development
Drug discovery has focused on identifying the most potent lead compound for a specified target. However, many drugs have failed due to poor pharmacokinetic or biopharmaceutical properties such as a short half-life or poor bioavailability. In today’s economic environment such failures can no longer be afforded. It has become recognized that the “best drug” is one that balances potency, good pharmacokinetic–biopharmaceutical properties, good pharmacodynamics, safety, and low cost of manufacturing. It is important to deal with these issues prior to testing in humans. Optimized preclinical development can be a tremendous aid to the design of early clinical studies. This optimization will include a thorough study of preclinical safety by combining traditional toxicology studies with novel methods in toxicoproteomics, toxicogenomics, and metabolomics. These new “-omics” will lead to novel biomarkers to predict toxicology and efficacy. Preclinical development should play an important role in defining the exposure– response (both efficacy and toxicity) relationships, which is a primary role for preclinical pharmacometrics. It is essential to determine the absorption, distribution, metabolism, and elimination during toxicokinetic studies in order to understand the comparison of these across species. It has been demonstrated that by combining preclinical exposure–response data (the steepness of the curve is important here), preclinical pharmacokinetics, and novel approaches to scale up to humans (10, 51) (see also Chapters 29 and 30), Phase 1 can be expedited. This can be done by choosing higher first time in human doses or more rapid escalation (if the dose–response curve is rather flat), resulting in fewer dosing cycles and thus less time, energy, and finances expended on Phase 1, without sacrificing safety. The development of physiologically and pathophysiologically based PM models (PBPM models) during preclinical development deserves attention. These models have the potential to provide accurate and nearly complete characterization of the PK and concentration–effect relationship and quantification of the potency of a drug (52–56). PBPM testing is best executed when the chemistry, biochemistry, metabolism, and exposure response of the drug are well known in addition to the relative physiology of the animals used in preclinical trials versus the parallel human physiology. To utilize PBPM modeling one must define the physiology, pathophysiology, biochemistry, and exposure–response relationships. To execute this type of modeling, some of the physiological variables that often need to be defined include blood flow to various organs such as liver, kidney, and effect organs. The biochemical–pharmacological parameters of a model that often need to be defined are Km and Vmax for the various enzymes that catalyze the metabolism of the drug and/or metabolites; tissue to blood concentration ratios; the distribution of the drug and/or metabolites of interest, for example, protein binding; and the clearance for various organs, for example, liver versus kidney. Exposure–response variables that are associated with a positive response or an adverse event need to be identified such as area under the concentration–time curve (AUC) or maximum concentra-
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tion (Cmax) or nadir concentration (Cmin). The exposure response may be related to the parent compound or to a metabolite and may be a concentration-based variable in plasma or within a specific organ or tumor. Many of these parameters can be estimated in vitro, such as enzyme kinetic parameters and protein binding, and physiologic parameters can be obtained from the literature, such as blood flow rates and organ volumes (56). PBPM modeling enabled the evaluation of the pharmacometrics of capecitabine for determination of the optimal dosing strategy in humans (56). Capecitabine is a prodrug that is converted in three steps to 5-fluorouracil (5-FU). A multicompartmental model was developed to describe the pharmacometrics of capecitabine, two metabolites, and 5-FU. The PBPM model is shown in Figure 1.1. The model included five compartments, all in some way related to either efficacy or adverse event. The parameters included in the model were Km and Vmax for each of the enzymes that catalyze capecitabine to 5-FU; tissue to blood ratio of capecitabine and the metabolites in gastrointestinal (GI), liver, and tumor tissue; protein binding; blood flow rate to liver, GI, and tumor tissue; and urinary clearance of unbound capecitabine and its metabolites. Enzyme activities (liver, breast, and colorectal tumors) and protein binding parameters were derived from in vitro experiments. Physiologic parameters were obtained from the literature. From the model, the 5-FU AUC values in breast and colorectal tumors were simulated at doses from 829 to 1255 mg/m2. The 5-FU AUC in tumor increased in a nonlinear manner relative to the increases in capecitabine dose. The model indicated that, for capecitabine, the 5-FU exposure in the tumors was much greater than in blood, resulting in a relatively low systemic exposure. The simulated blood
Capecitabine
Dose
Carboxylesterase [liver]
5′-DFCR
KA, TLAG
Cytidine deaminase [liver, tumors]
5′-DFUR dThdPase [liver, tumors] DPD [liver]
5-FU * FBAL
V1 CL1 V2 CL2 V3 CL3
* Intermediate metabolites: FUH2, FUPA FIGURE 1.1 Metabolic pathway of capecitabine and its representation by a PK model. Abbreviations: Tissues with high enzyme activites are shown in square brackets; 5′-DFCR = 5′deoxy-5-flurocytidine; 5′-DFUR = 5′deoxy-5-flurouridine; dThdPase = thymidine phosphorylase; DPD = dihydropyrimidine dehydrogenase; FBAL = a-fluoro-b-alanine; FUH2 = dihydro-5-fluorouracil; FUPA = 5-fluoro-ureido-propionic acid. Dose = capecitabine dose (mg); KA = first-order absorption rate constant (L/h); TLAG = lagtime (h); CL1 = apparent 5′-DFUR clearance (L/h); V1 = apparent 5′-DFUR volume (L); CL2 = apparent 5-FU clearance (L/h); V2 = apparent 5-FU volume (V); CL3 = apparent FBAL clearance (L/h); V3 = apparent FBAL volume (L). (From Blesch et al. (56); used with permission.)
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PHARMACOMETRICS: IMPACTING DRUG DEVELOPMENT AND PHARMACOTHERAPY
AUC values were consistent with clinical observations, indicating that the model was able to describe known clinical data. Once the model was developed, a murine xenograft was done and the PK, blood, and tissue binding of capecitabine and its metabolites were measured in vivo and integrated into the PBPM model. Large interspecies differences in tissue distribution and metabolic activity were observed. The predicted blood and tissue concentration profiles of 5-FU in the xenograft were compared to those in humans after simulated oral administration of several levels of capecitabine doses. The 5-FU AUCs in blood and xenograft tumor tissues were lower than those in humans for all capecitabine doses administered. At their effective oral doses of capecitabine (0.0944 mmol/kg, the clinical effective dose for humans; 0.44 mmol/kg, the effective dose for human cancer xenograft) similar 5-FU AUC values were observed in humans and human cancer xenograft models. The results of this study strongly supported the fact that a clinically effective dose can be extrapolated from xenograft models to a corresponding effect dose in humans when thoughtful approaches to the development and application of PBPM modeling is executed. Preclinical PM modeling should be done on a real-time basis so that modeling has been completed prior to planning and protocol development for Phase 1. Biomarkers need to be identified and investigated in preclinical studies, especially those that may predict future safety problems. Sometimes the lowering of blood pressure or the prolongation of the corrected QT interval may give one a “heads up” to potential toxicities or dose-related toxicities that may occur during clinical development. When a thorough job is done during preclinical development, then transition to clinical development can be done efficiently and with confidence. 1.4.2
Clinical Development
Clinical development continues with the application of the learn–confirm–learn paradigm applied to drug development. Scale up to the first-time-in-human (FTIH) study is best done by the application of sound PM methods as described by several authors (10, 51, 56). 1.4.2.1 Phase 1 Studies Phase 1 studies are executed to identify well tolerated doses and, in some cases, the maximum tolerated dose, to study the single and multiple dose pharmacokinetics, and to gain an initial knowledge of the exposure–response relationship. In addition to the above, one sometimes does Phase 1 studies to determine food effect and gender on pharmacokinetics, drug–drug interactions, and pharmacokinetics in special populations such as those with impaired renal or hepatic function or pediatric or geriatric patients. Here one has learned about the dose–exposure–response relationship from preclinical studies, has been guided by that preclinical knowledge, and is confirming or revising what was learned. Both traditional two-stage and population PK methods have been applied to Phase 1 model development with good results. The population approach can provide valuable information that is otherwise not available by the standard two-stage approach. Phase 1 studies are most often conducted in healthy volunteers unless the anticipated toxicity of the drug is severe or the drug is being applied to a life-threatening condition for which no other treatment is available.
PIVOTAL ROLE OF PHARMACOMETRICS IN DRUG DEVELOPMENT
15
In Phase 1, the approach to the FTIH study is critical in determining how much time is expended in this part of development. The central issue here is: “What should the first dose be and how rapidly does escalation occur?” If the very first dose it too high, then an adverse event will occur; if it is too low, then unnecessary time will be expended on low-dose testing. The application of preclinical findings becomes important. A promising approach has been the combining of allometry and mixed effect modeling with stochastic simulation to extrapolate preclinical models and knowledge to humans (10, 51). Applying sound PM methods has been and will be of great value in bringing efficiency to Phase 1 studies and for discovering knowledge that was previously hidden in most Phase 1 data sets. In situations where the maximum tolerated dose (MTD) is sought and defined in healthy volunteers, it should be redefined in patients at some later stage of development if possible (57, 58). In addition to the FTIH studies, the effects of food, drug–drug interactions, and special populations need to be studied. Coadminstration of drugs has been demonstrated to both increase and decrease bioavailability of some agents with the subsequent lack of efficacy or appearance of toxicity. Further details on the design and conduct of food effect studies can be found in Chapter 29. Drug–drug interaction studies have become increasingly important as the number of agents prescribed to patients continues to increase. In one instance, a prominent drug was withdrawn from the market after adverse events were reported, which were due to interactions with other agents. It is important to obtain information for some subpopulations, such as pediatric patients, those with renal impairment, and the elderly, so that group-specific dosing guidelines can be developed. These special studies can be executed with either traditional PK studies or more efficiently by applying population techniques (39) (see Chapters 12 and 39). The need to study subpopulations strongly supports implementing the learn–confirm–learn paradigm. These issues are addressed in Chapter 14. As the development process nears the end of Phase 1, it becomes crucial to extract all knowledge from existing data. PM models should be developed, linking drug exposure to pharmacodynamics (response). These models are applied, often by stochastic simulation, to optimize the structure and designs of Phase 2 studies. Real-time data collection is helpful here so that PM models may be estimated prior to data set closure and then applied to evaluation of competing Phase 2a study designs (39, 48, 59, 60). In this way, efficient and powerful Phase 2 programs can be constructed. 1.4.2.2 Phase 2 Studies Phase 2 studies should focus on both learning and confirming. Historically, Phase 2a has had as its primary goal to demonstrate “proof of concept” that the drug is capable of being effective. It has been a common practice to administer the maximum tolerated dose (MTD) in Phase 2a and this dose may be on the flat part of the efficacy curve. If this is the case, lower doses may have been equally effective and less toxic. This dose is then carried forward into Phase 2b and eventually Phase 3. In Phase 3 the drug will likely be demonstrated to be effective and without significant adverse effects. The result will be NDA approval at the MTD. Therefore, doses may be lowered because “a lower dose is quite adequate for treatment and less expensive” in the opinion of the prescriber or “a lower safer dose may be
16
PHARMACOMETRICS: IMPACTING DRUG DEVELOPMENT AND PHARMACOTHERAPY
needed.” The former may be enacted by practitioners without a change in labeling and the latter would come at the directive of the FDA. The former can be quite costly in terms of gross revenues for the manufacturer because an increase in cost per unit after marketing is in general not a viable alternative. Phase 2a should have learning as its primary focus to define the optimal dose, thus improving the drug development process; while Phase 2b studies should focus on confirming. Phase 2a is the time during development to learn about efficacy; to confirm or modify what was learned in Phase 1 about safety, efficacy, and drug effect on biomarkers; and to refine the dose–PK/PD-biomarkers–surrogate–outcomes relationships. The knowledge discovered in Phase 2a provides information for the later larger trials that will be designed to prove efficacy. The sample sizes are small in Phase 2 and the patients are often the “healthiest” to minimize disease-related variability. With this in mind, the Phase 2a study should be designed to give a first glimpse to the following issues (48): (a) Does the drug work? (b) How does the drug work? (c) What is the dose–response relationship? (d) Is there a difference in any of the pharmacology in subgroups? A very valuable practice here is to power these studies by setting a at a more liberal level of 0.10–0.20 when evaluating efficacy. Addressing these issues will require paying attention to important design points such as number and level of doses studied, timing of endpoint observations, number of subjects at each dosing level, and duration of the study. Furthermore, a well designed Phase 2a trial with 150–200 subjects will usually provide more information and is less costly than several smaller studies, even when these are later combined (48). A well designed study here will usually depend on stochastic simulation of competing study designs. In the end, many of the analyses will be population dose–pharmacokinetics/ pharmacodynamics–response models. In Phase 2 the proof of concept study provides scientifically sound evidence supporting the postulated effect of the new drug, where the effect may be the relevant pharmacological action or a change in disease biomarkers, established surrogate endpoints, or clinical outcomes that may be beneficial and/or toxic in nature. The proof of concept is often used for go/no-go decisions and is therefore one of the most critical steps in the drug development process. Biomarkers play an important role in Phase 2 studies. These are covered in Chapter 20 in detail. Biomarkers are most important in early efficacy and toxicity studies when clinical endpoints take too long to become observable. After approval, biomarkers may prove useful in monitoring the course of pharmacotherapy in individual patients. Prior to advancing to Phase 2b, all the knowledge hidden in the Phase 1 and Phase 2a data ought to be discovered. Then clinical trial simulation (knowledge creation) should be applied to construct Phase 2b. In Phase 2b the knowledge discovered in all previous phases is confirmed, and learning more about the drug in a larger patient population continues. In this phase of development, strong supportive evidence is generated so that if an accelerated approval is sought the knowledge and data generated could be enough to obviate the need for two Phase 3 confirming studies. Attention should be given to informatively designing Phase 2b studies to meet the confirming study objectives and allow learning that will enhance a further characterization of the response surface. Pharmacokinetics enables the refinement and further development of PK/PD models
SUMMARY
17
for dosage optimization (see Chapter 29). In Phase 2b sparse sampling is adequate; this data may be concatenated with previously collected data. The concatenation of these data with previously collected data and the estimation of individual PK or PD parameters via post hoc Bayesian algorithms may be useful for explaining individual treatment failures, toxicities, or positive responses to a drug. The PM models estimated from all previous data and available at the end of Phase 2b are important for constructing the pivotal Phase 3 program through knowledge creation. 1.4.2.3 Phase 3 Phase 3 is the pivotal phase for registration of a drug, where usually two large randomized, controlled trials for establishing efficacy and safety are required. The PM models from all previous studies are crucial for the determination of the dose(s), patient population selection, study duration, number of patients, and so on for Phase 3. In some cases a single pivotal study may be acceptable to the regulatory agency provided there is good supportive science (which may be good PM models) and confirmatory evidence supporting efficacy and safety (6, 7). In Phase 3 it is still advisable to proceed with sparse collection of PK and PD variables. These data can further support registration, may provide explanations for clinical trial success or failure, and are inexpensive to obtain when compared with the cost of enrolling patients. 1.4.2.4 Phase 4 Phase 4 studies are sometimes required by regulatory agencies. This can happen if the regulatory agency is interested in further characterizing safety, exploring new treatment indications, broadening label claims, exploring new drug combinations, or examining dosing in some special subpopulations (e.g., pediatric patients).
1.5
PHARMACOMETRICS AND REGULATORY AGENCIES
The FDA has promoted the role of pharmacometrics in the drug approval process through its approach to review of applications and by publishing its “guidances.” The FDA has gained expertise in pharmacometrics from self-training within and by recruitment of new highly skilled personnel. The value of pharmacometrics continues to be evaluated at the FDA.
1.6
SUMMARY
Pharmacometrics is playing a major role in improving drug development and therapeutics. Improvements in drug development must come through creating and using novel pathways to approval and application of sound scientific principles, partly by applying mechanistic PM models. It is difficult to imagine a more efficient, powerful, and informative drug development process without the expansion of the role of pharmacometrics. Pharmacotherapy is also in great need of improved dosing strategy selection for the avoidance of adverse events and the improvement in efficacy. This will come through the development of pragmatic PM models that provide knowledge
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PHARMACOMETRICS: IMPACTING DRUG DEVELOPMENT AND PHARMACOTHERAPY
about drug behavior and how the drug can be optimally used. As more pragmatic PM models are developed, optimal dosing strategies can be implemented. The acceptance of pharmacometrics in drug use and development cannot, therefore, be overemphasized.
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PART I
GENERAL PRINCIPLES
CHAPTER 2
General Principles of Programming: Computer and Statistical SASTRY S. ISUKAPALLI and AMIT ROY
2.1
INTRODUCTION
Although pharmacometricians are often involved in the development, modification, and use of computer code and programs, formal training in these skills is often neglected. Computer programming skills are acquired in an ad hoc approach, in which the minimal necessary knowledge to devise and code an algorithm is gained to solve the scientific problem at hand. This is not unexpected, as the scientific problem is of primary interest, and programming is simply a means to an end. While the ad hoc approach to acquiring the necessary programming skills may have been adequate in the past, the need for sophistication in computer programming is increasing along with the complexity of computational problems being addressed by pharmacometricians. The programming approach that may appear to be expedient is often not the most efficient with respect to overall productivity. Additional effort in the initial stages of a project can save time and improve accuracy and overall quality of code in subsequent stages. Although there are usually multiple ways in which a scientific programming problem can be addressed, adhering to standard programming approaches is an important step in development of high-quality programs. Standardization facilitates consistency and faster code reviews, and, more importantly, it helps a reviewer identify commonly occurring mistakes. The aim of this chapter is to provide an overview of generally applicable good programming practices that could benefit pharmacometricians with regard to improving the quality and transparency of code, as well as increasing overall productivity. A set of techniques and practices is provided here that will be useful in writing better computer programs. The involvement of pharmacometricians with programming ranges from relatively simple code to complex, software development projects. Likewise, programming skills of pharmacometricians range from
Pharmacometrics: The Science of Quantitative Pharmacology Edited by Ene I. Ette and Paul J. Williams Copyright © 2007 John Wiley & Sons, Inc.
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GENERAL PRINCIPLES OF PROGRAMMING: COMPUTER AND STATISTICAL
novice to proficient, with proficiency usually gained from long experience. Rather than attempt to cover all aspects of programming in detail, this chapter covers the basics of writing good code and provides the reader with references to additional resources that provide more detail on other aspects of programming and software development.
2.2
PHARMACOMETRIC PROGRAMMING TASKS
Change is a dominant factor in scientific programming; hence, a scientific program needs to be easily readable and easily modifiable. In this sense, a scientific computer program is analogous to a scientific document, in that it should provide context, be readable, and contain appropriate references. Furthermore, a well designed program will often be useful far beyond what the original programmer intended, because it will be easily readable, modifiable, and expandable. There is extensive literature on basic programming techniques for scientists and engineers, but a majority of the literature focusing on programming practices is over three decades old (1–4). Many modern books dealing with programming are often focused on highlighting the features of a language, or advanced techniques involving specific programming platforms or approaches. Recently, there has been increased attention on good practices in software design (5, 6). Pharmacometricians are often involved in programming tasks that span a wide range of complexity, ranging from writing a few lines of code to writing scripts and programs. These programming tasks can be classified according to a variety of attributes as shown in Table 2.1. Moreover, pharmacometricians may also be the domain experts on a software development team, providing guidance or input to other programmers. Therefore, much of the programming tasks demanded of a pharmacometric scientist involve writing not full programs from scratch but customizations of existing code or minor modifications to existing modules in order to create a program. One example of systems where a model can be developed without much programming is ADAPT II (7), which provides templates of Fortran subroutines. In ADAPT II, the scientist is required to specify the model by adding code to existing templates of subroutines, in order to create a complete program. These subroutines can then be compiled and linked to other compiled code (object files) to create a stand-alone executable. Another example is the specification of models in NONMEM (8, 9). In NONMEM, the model is specified by a control file, which is then processed, to produce Fortran code that is compiled and linked to other object files to create an executable file. Although sophisticated programming skills are not necessary to develop models using these programs, some of the concepts described in this chapter will be useful in scripting even these relatively simple programs. More extensive programming is often required in writing scripts or programs for software packages such as S-Plus (10, 11) or Matlab (12, 13). These two modern software packages are increasingly used by pharmacometricians: Matlab as a programming environment for numerical simulations and S-Plus as a programming environment for statistical data analysis. It must be noted that there is a considerable overlap between the roles of these two packages, and both provide strong graphical capabilities. Although the princi-
PHARMACOMETRIC PROGRAMMING TASKS
27
TABLE 2.1 Examples of Different Types of Classifications Found in the Scientific Programming Spacea Programming experience Scientific experience Programming role Problem/model complexity Randomness Software project complexity Complexity of the tools Programming approach Program dependencies Program interfaces Documentation complexity Quality assurance level Extensibility and modularity a
Novice to professional programmer Key scientist to programming support staff Use/apply others’ code, review code, develop new software modules Linear models, algebraic equations, ordinary and partial differential equations Deterministic models, simple error models, stochastic systems Individual, local group, distributed group, production versus prototype versions Spreadsheet based, predefined modules (e.g., NONMEM) Procedural, object-oriented, visual, symbolic, pipeline-based, event-based External databases, external web services, other programs; used as module in other programs Command line, noninteractive, distributed, web-based, embedded into other programs (spreadsheets) Simple commenting/memos, detailed documentation published as reports Error checks, automated tests, reproducing results, internal/ external review Single run models versus multiple run, cluster-based simulations
Though the space spans a wide range, the general programming principles are applicable throughout.
ples of good programming practice described in this chapter are generally applicable for a variety of programming environments, they will be mainly illustrated using examples of Matlab code. Although it is possible to use much of the functionality of Matlab through the graphical user interface (GUI) or interactive commands, the full features of these systems can be utilized only through scripts. Furthermore, there are many advantages to writing scripts. First, scripts provide a record of the commands executed and facilitate the reproducibility of the results. Second, scripts provide a means for automating repetitive tasks and relieve the tedium and errors that commonly occur in performing repetitive tasks with a GUI interface, especially for computationally intensive tasks that have long waiting times between user input steps. Third, once a set of scripts that accomplish common tasks have been developed for a given project, they can often be modified for subsequent projects with a much smaller time investment. Some pharmacometricians may be involved in complex software projects, such as the development of software for ADMET (absorption, distribution, metabolism, excretion, and toxicity) predictions or software tools that can be used by other scientists. Examples of such tools include Perl-speaks-NONMEM (14) or Xpose (15). Such tasks often require a diverse set of programming skills and strong programming practices.
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2.3
GENERAL PRINCIPLES OF PROGRAMMING: COMPUTER AND STATISTICAL
OVERVIEW OF SCIENTIFIC PROGRAMMING METHODOLOGY
The programming paradigms applicable to scientific programming have often followed the developments in the field of software engineering. Some of the major paradigms applicable for scientific programming are briefly described in Table 2.2. 2.3.1
Scientific Program Development Steps and General Guidelines
Introductory programming books often provide resources for learning a programming language and programming syntax, for utilizing the development environment, and for compilation, execution, debugging, and optimizing of programs. All these techniques are directly applicable to scientific programming. Furthermore, there are a few additional points that a scientific programmer has to be aware of: (a) change is the dominant factor in scientific programming; (b) quality assurance is more important in scientific programming than in regular programming because it is often difficult to distinguish program errors or bugs from bad science; and (c) it is often very difficult to notice errors in the results. A scientific program may start as a script for solving a specific problem and may find use in related areas. Sometimes, the program finds use in a much broader context. Some of the uses of the program can be (a) as one step in a sequence of steps involving multiple programs (i.e., in the form of a “pipeline”), (b) as a script that is invoked by another script, (c) as a function that is invoked by other functions or scripts, (d) as a program wrapped around a Monte Carlo type simulation or a parameter estimation module, (e) as a module wrapped around a graphical user
TABLE 2.2 Overview of Some of the Main Programming Paradigms and Approaches Applicable to Scientific Programming Procedural programming Flow-driven programming Event-driven programming Object-oriented programming Design patterns-based programming Symbolic programming Visual programming Pipeline programming Collaborative programming Parallel/distributed programming Web-based programming
Modules or procedures are used as computational blocks Execution of code follows a well defined order Execution of code depends on the events such as user clicks Objects, interfaces, and methods are used as computational blocks Utilizing standard solutions to software design problems Calculations are performed in a symbolic manner (e.g., Maple) Assembling of “blocks” visually to form full programs Output of one program is used as input of another (pipeline) Deals with advantages (and issues) of multiprogrammer projects Deals with utilization of multiple machines Programming focusing on web-based interfaces
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interface, (f) as a web-enabled program, and (g) as a program that is run multiple times on a distributed machine cluster. Therefore, it is prudent to follow good programming practices for all levels of programming tasks. Schneider (16) recommends that a beginning programmer should concentrate on semantics and program characteristics of a programming language, and not just on the syntax. The concerns for programming style should be cultivated from the very beginning, and care must be taken to avoid the common mistake of initially writing beginning programs quickly with the idea of coming back later and then refining them. This prevents bad coding habits from ever developing. The programmer should also become familiar with and follow formal processes for debugging, program testing, and verification, as well as for documentation. Seeley (17) argues that following programming practices is more productive than simply using the latest tools. Computer programming tasks in recent times have evolved from writing new code and modules to correctly linking existing modules. The majority of effort involved in solving a scientific programming problem is in identifying the appropriate design for the solution, and in identifying relevant existing modules; the linking of the modules becomes a simple task once the design is completed. The following set of objectives with respect to the quality of scientific programs is recommended in this chapter: (a) program correctness, (b) reproducibility of results, (c) program readability (critical for code reviews), (d) maintainability (bug fixing and minor changes to the program), (e) ease of configuration change (e.g., parameter values and the constants used in the program), (f) portability and extensibility (ability to run the program on different systems and ability to link the program with other programs), and (g) performance (speed and disk space requirements). The general steps involved in the development of a scientific program are common to programming tasks across a wide range of scales, from simple programs developed by an individual to complex software development involving a large group. However, implementation of individual steps varies depending on the type of problem solved, the scale of the project, and the level of quality testing. These main steps are: 1. Mathematical Formulation of the Scientific Problem. In this step the scientific problem is formulated in mathematical terms and may involve reviewing the literature, identifying the appropriate mathematical model, and identifying sources for model parameter values. 2. Algorithm Design. Here, the problem has to be addressed from a computational framework viewpoint. Issues such as selection of a model solution scheme (e.g., choice of a differential equation solver, choices of appropriate modules for random number generation, etc.) are addressed at this stage. 3. Design and Documentation of the Computer Code. Here, the program is designed in a top–down approach. Interactions between the main program and individual modules are defined at this stage, along with brief documentation of the functionality of each module. At this stage, the program does not have much code—only definitions of the functions and parameters. The body of the functions is mostly empty at this stage. Changes to the program and the interactions between the modules can easily be made at this stage.
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4. Design of Test Cases. Representative test cases are identified and documented so that when the actual code is written, it can readily be tested. A brief review of the test cases is also done at this stage. 5. Program Implementation. At this stage, a programmer can focus on individual modules. Typically, a programmer should develop simple, “unit tests” for individual modules at this stage. For modules with very few lines of code, these unit tests may be very simple, but in general writing unit tests is a good practice. As the complexity of the module increases (e.g., for the main module of a pharmacokinetic model), unit tests could involve calculating steady-state estimates with zero input doses (where many target tissue concentrations should reach zero) and very high input doses. Also, simple mass-balance tests can also be added at this stage. For example, when simulating systems involving multiple chemicals and reactions among them, an inert test chemical can also be introduced into the simulation and simple mass balances can be used for testing. The unit tests should be designed in a manner that facilitates easy debugging, so by definition they should be simple. 6. Program Verification and Correction. At this stage, the programmer runs the code, fixes errors, and runs the test cases. If there are subtle errors, an interactive debugger can be used for stepping through the program. For complex simulations (e.g., those that run for several hours), programs can be monitored through log statements. 7. Program Refinement and Optimization. At this stage, the program is refined and optimized. Feature enhancements, performance improvements, improvements in usability, and so on are common at this stage. Adequate documentation and representative test cases are critical for developing good scientific programs. The documentation can be in the form of references (e.g., for assumptions used, mathematical models, and references for parameter values). Furthermore, when the programs are likely to be used by other scientists in a group, following a set of guidelines used in the group (or developing a set of guidelines if none exist) is a good step.
2.3.2 Tools for Numerical and Statistical Programming: Matlab, S-Plus, and Open Source Alternatives The principles and practices discussed here are general in nature and are applicable to a wide range of scientific programming problems. They are also independent of the programming language and approach used. Specific examples are provided using Matlab, which is a programming environment for numerical simulations. These examples can also be readily applied to S-Plus, a widely used programming environment for statistical data analysis; however, it must be noted that there is a considerable overlap between the functionality of Matlab and S-Plus. Matlab is a high-level scientific scripting language and an integrated development environment with interactive tools for visualization and several toolboxes addressing different computing areas such as statistics, database connectivity, and data mining. A pharmacometrician using Matlab may have to purchase Matlab toolboxes, such as the Statistics Toolbox, in addition to the basic Matlab license;
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therefore, some individuals may find the cost of Matlab high. Fortunately, free, open source alternatives to Matlab exist: Octave (www.octave.org) is a high-level language, primarily intended for numerical computations, and Scilab (www.scilab. org) is a scientific software package for numerical computations. Both Octave and Scilab are similar to Matlab, and like Matlab, they both have large sets of toolboxes: Octave toolboxes are available in the form of the octave-forge package, while loosely coupled toolboxes are available for Scilab. S-Plus is a statistical data programming language environment that follows the approach of programming with data. It is scalable and handles massive data sets and provides integrated tools for advanced analytics such as data mining. It also provides some advanced modules relevant to pharmacometricians, for example +SeqTrialTM for designing, monitoring, and analyzing clinical trials using group sequential methods. S-Plus license fees may also be an issue for some individuals. Free, open source alternatives to S-Plus include R (www.r-project.org) and Omega project (www.omegahat.org). R is very closely related to S-Plus, as both are based on the S software from Bell Labs; in fact, a majority of R code can run unchanged in SPlus. A large set of modules for R are available at the Comprehensive R Archive Network (CRAN, which is part of the R Project). The use of free, open source tools is suggested for pharmacometricians who may not have licenses for commercial software. However, the expenses associated with the licenses may not be significant for many organizations. A pharmacometrican can utilize the similarities between the proprietary and open source tools by developing the skills using the free tools and, if needed, transition to the proprietary versions later on. One of the consequences of rapid advances in computer technology is that users are not constrained by the programming language or environment they use. In fact, many interfaces for invoking one language from another exist. For example, S-Plus can operate with SAS (www.sas.com) data sets. The Omega Project provides an R–Matlab interface (currently, an early release status) that facilitates a bidirectional interface between the R and Matlab languages that allows users of either language to invoke functions in the other language using the syntax of their choice. Matlab also provides interfaces to directly invoke functions in Fortran, C, C+ +, and Java. 2.3.3
Scientific Programming Resources
An overview of computational problem solving techniques for beginners can be found in Dijkstra (2) and Dromey (4). Several introductory textbooks on algorithm design are available freely (18–21). Textbooks based on specific programming environments and languages are useful in learning programming techniques—for example, for Matlab (12, 13, 22) or S-Plus (11, 23). Some of the books for advanced programming techniques are also freely available, for example, for object-oriented design (24), parallel computing (25, 26), user interface design (27), and agiledevelopment (28). One of the best ways to learn good programming skills is to read code from experts in the field. Often, reading and understanding code from an experienced programmer within an organization is also recommended, because it provides the novice programmer familiarity with the coding styles and approaches used in the
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organization. Some of the approaches for beginning programmers include (a) reproducing the results from a working program, as this involves becoming familiar with the inputs and outputs used, getting familiar with the operating system and the programming environment, executing the program, and optionally postprocessing of the program outputs; (b) studying the code using “code browsers”; and (c) running the program in an interactive debugger and stepping through the code. Programming productivity can be substantially increased by utilizing available toolkits, libraries, development environments, and relevant programming approaches (29). Some of the productivity-improving features are available in integrated systems such as Matlab and S-Plus. For other features, or for programming in other languages, a programmer can use either specialized integrated development environments (IDEs) or general purpose toolkits. Though a detailed discussion of the available toolkits is beyond the scope of this chapter, some of the widely used general purpose tools include text editors such as XEmacs (www.xemacs.org) and ViM (www.vim.org); general purpose IDEs such as Eclipse (www.eclipse.org); debugging tools such as the GNU Debugger, gdb (www.gnu.org/software/gdb); code profiling tools such as the GNU Profiler, gprof (www.gnu.org/software/ binutils); code browsing and publishing tools such as Glimmer (glimmer.sourceforge.net); version control systems such as Concurrent Versions System (CVS) (www.nongnu.org/cvs); (30); and defect tracking systems such as bugzilla (www. bugzilla.org). Likewise, there is an large set of available libraries for general purpose scientific and statistical programming (31–34). Language-dependent libraries also exist, for example, Matlab libraries (35, 36) and S-Plus/R code (37). A scientific programmer must be aware of available libraries and toolkits and must be familiar with the general tools and approaches for effective computer programming. An awareness of these tools and approaches will help in pursuing the corresponding features in the programming language of choice. For example, a Matlab programmer familiar with the notion of code browsing can either use the general purpose tool Glimmer (glimmer.sourceforge.net) or search for the feature in the Matlab Repository (35) and arrive at the Matlab code browsing tool, M2HTML (38). An awareness of features and utilities one can realistically expect in a programming environment will enable a programmer to seek similar features, often successfully, even in totally new programming environments. As an example, the Matlab IDE provides a majority of such features, and in some cases an auxiliary tool may be needed. 1. Enhanced editing ability, consisting of syntax-based code coloring/highlighting and automatic completion of variable names, facilitates faster coding as well as early detection of simple syntax errors (e.g., unbalanced parentheses, quotes). Some editors and IDEs also support an “outline mode” for navigating large blocks of code. 2. Code “beautifying” tools enhance code readability via automatic indentation and line wrapping, as well as format code from diverse sources in a consistent manner. 3. Code browsing tools provide effective navigation of large blocks of code spanning multiple files and thus are valuable for reviewing or studying programs written by others. Often, the code browsing tools allow publishing of the code
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4.
5.
6.
7.
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in a hyperlinked format (typically as a set of HTML files), which can be then viewed through a regular browser. The M2HTML tool (38) provides this functionality for Matlab. Interactive debugging environments allow for tracing code execution, inspecting variables, and arbitrarily setting breakpoints inside the program. These allow for rapid location of errors. The Matlab IDE provides both a visual debugger as well as a command line debugger via the “dbstop” command. Code profiling tools provide a summary report on the code execution, including time spent in different blocks of code, thus helping in optimizing the code. The Matlab IDE provides a profiler tool as well as the “profile” command. Tools for periodic saving of program state provide value by (a) allowing an interrupted program to restart from a prior valid state and (b) allowing the user to monitor program progress by using the intermediate outputs. This is especially useful in the context of computationally demanding simulations that may run for days to weeks, because errors can be detected early by analyzing the intermediate outputs, and erroneous model runs can be stopped. Likewise, computational time is not lost when a correct model run is interrupted due to unavoidable problems. The Matlab system provides a “save” command to save the entire workspace or a set of objects. Revision control tools allow easier management of source code changes in a transparent and efficient manner. Using these tools, a programmer (or a group of programmers) can easily track code changes, obtain a summary of changes from one version to another, and revert to any version based on either a version number or a date. Matlab provides an interface to several version control systems, for example, via the “cvs” command for CVS (30) and the “sourcesafe” command for SourceSafe (39).
2.4 GOOD PROGRAMMING PRACTICES: BASIC SYNTAX, CODING CONVENTIONS, AND CONSTRUCTS The practices listed here are applicable to all aspects of scientific programming, including small segments of code or complete scripts, as well as modules that form a large software project. 2.4.1
Use Meaningful Names for Program Variables
Giving meaningful names to program variables is one of the simplest ways of enhancing the readability of code. The names of variables in legacy code are often cryptic, because the length of variable names was constrained in older programming languages (a maximum of 8 characters is allowed in Fortran 77). These constraints have been removed for all practical purposes in most of the current programming languages (such as Fortran 90, C, Java, Matlab, and S-Plus), which allow variable names that can be as long as 32 or even 256 characters. Very short names (such as “t” to indicate simulation time) are easy to type, but (a) they are not informative about the nature and context of the variable, and (b)
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(A) Well Named and Formatted Code
% Initialize compartment concentrations to zero FOR iCmpt = 1:N_COMPARTMENTS FOR jChem = 1:N_CHEMICALS % Initialize concentration of chemical j in compartment i conc_cmpt(iCmpt, jChem) = 0; END % End of jChem loop END % End of iCmpt loop (B) Poorly Named and Formatted Code
FOR i = 1:N FOR j = 1:M c(ii, jj) = 0; END END FIGURE 2.1 Code Block 1—impact of variable naming and code formatting on program readability.
they are likely to be misinterpreted or inadvertently redefined in another part of the program. It is preferable to use a meaningful name such as “sim_time” (note that in this scenario, the variable name “time” may not be appropriate because it may conflict with a system command, a reserved word, or an inbuilt function). Likewise, very long variable names should also be avoided, because (a) they are tedious to type and can lead to inaccuracies and lengthy statements, and (b) it is difficult to distinguish between long variable names that differ by only a few characters at the trailing part. Short variable names are convenient and appropriate to hold temporary or intermediate values, such as counters in conditional loops. However, it is recommended that meaningful names be used even for counters. The code in Code Block 1 (Figure 2.1) provides examples of descriptive names for constants, variables, and temporary counters in conditional loops. Although this is a trivial example, the benefits of using descriptive counter variable names increase as the number of statements and nesting levels in the conditional block increase. 2.4.2
Use Consistent Naming Conventions for Program Variables
Many modern programming languages are also case-sensitive, and this feature can be used to advantage in communicating the type and context of a variable name. A convention often followed by Matlab programmers is to use all uppercase names for program constants. The case of a variable name can also be used to distinguish the context of the variable (local versus global) and variable type (scalar or vector versus matrix). In statistical programming, case is often used to distinguish between data set and column/variable names, thereby improving program readability. Some generally used naming conventions are presented in Refs. 40–42. Consistent naming of variables is important for understanding the context of the variable and for writing reusable code. Some commonly used naming conventions are:
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1. Use all uppercase names for constants and global variables. 2. Prefix global variables with an identifier, for example, GL_NUM_ COMPARTMENTS. 3. Use readable variable names either via underscores (num_compartments) or via mixed case naming (NumCompartments). A related issue is the definition and initialization of constants. A commonly followed approach is to place definitions of constants together at the top of a unit of code (script or a module), so that the constants can readily be located. This also enables the programmer to identify at a glance the constants that have been defined. 2.4.3
Follow Organizational Conventions for Code Formatting
Proper formatting of code, such as indentation, wrapping of long lines, and splitting long formulas into shorter formulas, significantly enhances the readability of code, similar to a well formatted document. It also makes it easy to comment out or delete blocks of code. Many programming environments and modern, general purpose text editors have features for “beautifying the code.” This includes appropriate automatic indentations and line wrapping (e.g., a two-space indentation for each nested conditional block). A consistent format not only helps in the readability but also highlights potential problems (e.g., a spurious “END” statement will alter the indentation in a visible manner). 2.4.4
Provide and Maintain Informative Comments
Providing comments that explain the purpose and logic of blocks of code is one of the most simple and effective ways of improving program readability. The main variables used should be commented along with major processing blocks (e.g., comments of the type “initializing the system” or “calculating derivatives”). Likewise, the end of loop constructs (“for,” “while,” and “if-then-else” blocks) should have a short, informative comment that mentions the conditional block that is being ended. This facilitates readability, especially for code that has several nested conditional statements. The major exception is for loops that consist of just one or two statements inside. In general, comments should provide the context of a statement or a block of statements (i.e., why something is done) instead of just a literal translation of the statements themselves. For example, while commenting a break statement, indicating both the innermost loop (e.g., “exiting i_comp compartment loop”) and the significance of the statement (e.g., “convergence reached” or “completed all dose inputs”) is more informative than simply stating “exiting compartment loop.” It is very important to ensure that the comments and code are always consistent, as wrong comments can cause more harm than no comments. However, it is often the case that a good comment turns into a bad one because of changes in the code without updates to the comment. A good practice is to review and update comments whenever the code is changed significantly.
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Avoid Segments of Commented Out Code
When extensive changes are to be made to an existing program, some programmers often tend to comment out working code and add new code, with the idea that the changes can easily be reversed. However, the commented out code often ends up staying in the program long after the code changes are finalized. Furthermore, programmers may add additional comments to describe why the code block was commented. This can lead to even more lines of difficult to follow programs cluttered with noninformative lines of text. The preferred method of revising code is to employ version control utilities such as CVS (30), which enable programmers to keep track of changes while maintaining the coherence of the code. The use of version control is briefly mentioned in Section 2.10.4. 2.4.6
Provide Documentation and References Along with the Code
It is essential to provide references for additional documentation when the code requires extensive documentation (e.g., statements involving a complex formula). This could also be in the form of an electronic document provided along with the code. Relying solely on comments to provide details can lead to comments overshadowing the code. Furthermore, text comments are limited in the type of information they can convey. For example, the documentation of a pharmacokinetic model can include the model schematic (a graphic), along with the model equations (mathematical objects), and additional references; such a document is significantly more useful than large chunks of text-based comments in the code.
2.5 GOOD PROGRAMMING PRACTICES: RELEVANT MATHEMATICAL CONCEPTS Some of the basic mathematical requisites for scientific programming include understanding of (a) rules of operator precedence, (b) machine precision, (c) equality and inequality issues, and (d) potential for overflow/underflow of numbers. Related concepts such as relative and absolute differences are also important for scientific programming. 2.5.1
Operator Precedence
Operator precedence deals with the order in which different operations in a mathematical expression are evaluated. For example, in most programming languages multiplication has a higher precedence than addition. Understanding operator precedence is especially important when writing complex mathematical expressions, because it is a source for subtle errors. Using parentheses for grouping terms is a good technique, as it improves readability as well as reduces potential errors that creep in due to operator precedence issues. 2.5.2
Machine Precision Issues
These issues arise due to limitations in machine representation of numbers and fractions in terms of a limited number of computer bits (e.g., a decimal fraction
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1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
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x = 0.00000000000000000001; % x correctly set to 1E-20 y = 5 - 4. 9999999999999999; % y (1E-16) set to zero w1 = (x == y); % evaluates to zero (false); Potential Error w2 = (x < y); % evaluates to zero (false); Definite Error w3 = (x > y); % evaluates to one (true); Definite Error w4 = y * 1E16; % evaluates to zero (actual value 1); Error w5 = x/y; % evaluates to inf (actual value 1E-4); Error w6 = y/y; % evaluates to NaN (actual value 1); Error c1 = (abs(x-y) < eps); % true; Equal within precision limit (x < eps & y < eps); % true, so avoid inequality comparisons
FIGURE 2.2 Code Block 2—a Matlab example highlighting common machine precision issues encountered.
such as 1/10 cannot be represented adequately with a limited number of bits in binary format1). Therefore, very small errors (“round-off” errors) are introduced, and these can sometimes accumulate over the course of a long simulation. In some programming languages, the problem is exacerbated by the choice of the variable type: for example, in Fortran, a “real” number is less precise than a “double precision” number. Therefore, depending on the problem and the choice of the variable type, the numerical errors can vary significantly. In scientific programming, such errors are sometimes encountered in the solution of systems of differential equations that are solved by numerical integration over a large number of time steps, with the accuracy of the solution highly dependent on the integration time step size and the duration of the simulation. 2.5.3 Equality and Inequality Issues These issues arise due to the limitations imposed by machine precision. Very often, two quantities that should be identical will not pass the equality test because of the different ways in which they are computed. Sometime inequalities are also impacted. As an example, the Matlab statements in Code Block 2 (Figure 2.2) will produce unexpected errors (the statements were tested on Matlab Version 7.01; interactive; default setting of single precision). When Statements 1 and 2 are used to calculate the values of two small numbers, “x” and “y”, one of them (“y”) is incorrectly rounded off to zero, whereas the smaller of the two (“x”) still retains nonzero value. Thereafter, all subsequent comparisons of “x” and “y” lead to unexpected and incorrect results. For example, the results of comparison in Statements 4 and 5 are definitely wrong, whereas the comparison in Statement 3 may or may not lead to an incorrect conclusion. There are some techniques to overcome these issues. For example, Matlab provides a variable “eps” that indicates the smallest floating point increment possible in a given precision. Statements 9 and 10 use “eps” in the context of comparing the difference of two numbers, as well as in deciding whether inequality comparisons can be made, and produce more predictable results. 1
For example, 0.1 represented in binary becomes 0.00011001100110011 . . . with the 0011 recurring; in general, only fractional numbers that can be represented in the form p/q, where q is an integer power of 2, can be expressed exactly, with a finite number of bits.
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Overflow/Underflow Problems
These problems can sometimes arise in numerical calculations due to limitations of machine representation of very large or very small numbers. Overflow errors occur when the number to be represented is larger than what the computer can handle; thus, the number gets assigned a value of “Inf” (infinity). Likewise, underflow errors occur when the number to be represented is smaller than what the computer can handle; thus, the number gets rounded to zero. These issues lead to functions returning the values of “NaN” (Not a Number, or invalid number), for example, when two large numbers (infinity) are subtracted, or when two very small numbers (zero) are divided by each other (see Statements 6 to 8). Such issues may appear pedantic, but in scientific programming, very small numbers often result due to the small time steps in numerical solution of differential equations and are sensitive to the choice of units used to represent different quantities in the simulation. 2.5.5
Absolute and Relative Differences
These need to be used appropriately when the convergence of numerical simulations is to be evaluated, for example, to estimate steady-state values or to estimate the quality of a numerical approximation. This often involves a combination of relative and absolute difference criteria. Absolute difference refers to the magnitude of the difference between two values, whereas relative difference deals with the ratio of the difference to the actual value. When the values to be compared are very small (but substantially more than the machine precision), absolute differences are recommended to judge convergence. Likewise, when the values are very large, relative differences are useful in evaluating convergence. It must be noted that there are several exceptions to these recommendations, and the choice of the criteria depends on the problem at hand. A scheme that employs both the absolute and relative error criteria will provide a more robust means for evaluating convergence.
2.6 GOOD PROGRAMMING PRACTICES: REDUCING PROGRAMMING ERRORS 2.6.1
Explicitly Check for Errors Such as Division by Zero
Many programming languages handle numerical exceptions such as division by zero or square root of a negative number by aborting the program execution. However, some modern languages such as Matlab allow for computation to proceed despite such errors (see discussion on NaN and Inf in Section 2.5.4). Depending on the simulation and the programming environment setting, the following scenarios are possible: (a) the program aborts execution with an error indicating file name and line number of the offending code; this is common in simple programs written in C or Java; (b) the program continues execution and some of the variables will have infinite or nonnumber values; this happens often in Matlab and Fortran; (c) the program suspends execution at the first instance of an exceptional situation; this is common in interactive debugging environments when global error checks are enforced (e.g., in Matlab, one can set the option to interrupt when a NaN or Inf is encountered using the command “dbstop if naninf”); (d) the module makes a log
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entry for the error condition and skips to the next iteration of the function program; this is common when multiple simulations need to be performed in one program, and erroneous simulations can be identified from the log files. 2.6.2 Avoid False Robustness in the Programs Some programs are designed to be robust despite minor errors in the inputs and program state. Examples include web browsers, which are designed to do the best possible job in spite of errors. Such an approach should be avoided in scientific programming, because the robustness of the program comes at a cost: correctness. In a scientific program, it is often advisable for the program to fail noticeably when spurious conditions are encountered. For example, in Code Block 3 (Figure 2.3), the code in lines 1–3 does not perform any error check, the code in lines 5–8 “compensates” for errors in another module, whereas the code in lines 10–14 alerts the user when there is an error in the program. Though the right approach for error handling and alerting is often dependent on the situation, code that alerts when spurious conditions are encountered is preferable, unless otherwise dictated by the situation. One practice for easy error detection is initializing variables to NaNs at the time of definition (or when error conditions are encountered). At any stage of computation, one can see if there are any invalid computations that are performed. Another practice for reducing errors is through explicit checks of function arguments. This is very essential, for example, in web-based business programs, where fraud is of high concern. In scientific programs, accuracy is of high concern, especially when subtle errors can result in seemingly reasonable, but wrong answers. 2.6.3 Check for Unused Variables In languages such as Java and C, it is not possible to use a variable without declaring a type for it. So, the Matlab example in Figure 2.4 would not run properly in those languages. In this code, there is an error on line 11 in Code Block 4, a
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
% A code block with no error checking conc = calculate_tissue_concentration(param1, param2); return conc; % returns conc values without error checks % A falsely robust code block conc = calculate_tissue_concentration(param1, param2); if (conc < 0), conc = 0; end % fix negative concentrations return conc; % compensates for errors in another module
% A fragile, but more correct code block conc = calculate_tissue_concentration(param1, param2); if (conc < 0), error(’Negative concentration encountered’); end 13. % Above check raises an alert when an error is encountered 14. return conc; FIGURE 2.3
Code Block 3—the role of appropriate error checks in scientific programs.
40 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
GENERAL PRINCIPLES OF PROGRAMMING: COMPUTER AND STATISTICAL
function result = get_blood_air_PC(scenario) % Returns Blood-Air Partition Coefficient based on scenario % Inputs: scenario: 1 => default, 2 => updated % Outputs: blood_air_PC (Partition Coefficient) % References: J. Doe, J.Pharm 2004, X. Y. Doe, J. Pharm.,2005 % Author: An Employee, Organization, Inc. PC_BLOOD_AIR_DEFAULT = 0.2; % J. Doe, J. Pharm., 2004 PC_BLOOD_AIR_UPDATED = 0.22; % X. Zmith, J. Pharm., 2005 PC_blood_air = PC_BLOOD_AIR_DEFAULT; if (scenario == 2) PC_blod_air = PC_BLOOD_AIR_UPDATED; end result = PC_blood_air; % return PC for blood compartment
% Output from mlint mlint('get_blood_air_PC') L 11 (C 3-13): The value assigned here to variable 'PC_blod_air' is never used
FIGURE 2.4 Code Block 4—subtle errors in function definitions that can be identified only via auxiliary tools.
misspelled variable name. Some programming languages such as Java and C will produce compilation errors with similar code (compilation errors in which the variable PC_blod_air is not declared). However, this set of statements is valid in languages such as Matlab, and, therefore, no error will be reported. As a consequence of the error, the program will always use the default value. Furthermore, if the error is minor (e.g., the values of default and updated partition coefficients differ very little), the error will become very difficult to track. Fortunately, there are tools to identify such errors. For example, Matlab has a command for comprehensive code checking called “mlint.” It must be noted that mlint is ideal for analyzing function files and is not as effective with script files in dealing with unused variables, since the purpose of a script may be just to initialize a set of variables to be used by another script. Code Block 4 also shows the output of mlint used on the code in lines 1–13. Matlab also provides “lint” report generation on all the files in a folder through a GUI. The report for the entire folder can be saved as an html file and easily reviewed. 2.6.4
Use “Catch-All” Statement Blocks in Conditional Constructs
When using a conditional statement such as “if-then-else” or “switch” statements, it is prudent to use a “catch-all” statement that addresses the unhandled cases. This may not seem important in the beginning, but it increases the robustness of the program when the code is used for different cases, because the program alerts the user when such errors occur. For example, when the statements in Code Block 5 (Figure 2.5) are executed for the set of chemicals handled by the program, the partition coefficient is assigned properly. However, if the “otherwise” portion of the “switch” statement is not used, the program would have produced silent errors when the code is used for new chemicals, by leaving the partition coefficient set to
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switch (chemical_name) case {’cl2’, ’chcl3’, ’chloroform’, ’ccl4’, ’tce’} pc = estimate_pc_voc (chemical); % partition coefficients of volatile organics case {’hg’, ’cd’, ’as’, ’As’} pc = estimate_pc_voc (metal); % partition coefficients of metals otherwise error([’Chemical ’ chemical_name ’ is not yet supported’]); end % The corresponding “if” statement will have several lines of % equality comparisons of the type if (chemical_name == ’cl2’ | chemical_name == ’chcl3’ | ... chemical_name == ’chloroform’) FIGURE 2.5 Code Block 5—use of the appropriate structured programming construct: “switch” versus “if.”
an unknown value. By including a “catch-all” statement, a noticeable error will be triggered upfront whenever the code is used for chemicals it does not handle.
2.7 GOOD PROGRAMMING PRACTICES: BASICS OF SCRIPT AND PROGRAM DESIGN 2.7.1
Avoid Monolithic Blocks of Code
In many programming languages, it is possible to write large, monolithic blocks of code. However, it is cumbersome to maintain and debug large blocks of code that require scrolling through several screens to be viewed in their entirety. One commonly used alternative to writing large blocks of code is to split large code blocks into multiple files, with each file tested individually and linked as a sequence of command files (also known as “including the files”). Though programs written in this fashion may appear to be “modular,” they are in fact similar to single files with one large block of code. 2.7.2
Write Modular Code
The main aspect of modular code is that changes in one module do not alter the behavior of other modules. However, when multiple script files are included in the same module, they share the same “name space” (i.e., they all can access the same set of program variables). Thus, a minor change in one file, such as assigning a value to a variable, can have unforeseen consequences in other files. However, when modular code (via subroutines and functions) is used, changes internal to the function will have no consequences upon other functions. As long as the function parameters and return values are consistent, significant changes to the internals of the functions can be made, without affecting other modules. This reduces the chances of subtle, intractable errors.
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In order to write maintainable code, programmers should break larger pieces of code into shorter functions (subroutines and procedures in some languages) that are small enough to be understood easily, have a well defined set of input arguments, and return corresponding outputs. This approach, though obvious, needs to be emphasized as it makes isolated small pieces of code easier to understand without having to understand the whole program at once. Likewise, once the small functions are tested, they can be assumed to work unchanged despite changes in other, unrelated functions. Modularization via functions enables code reuse and avoids repetition of code. This approach is also superior to that of cut-and-paste of existing code into new code. For example, if there is an existing pharmacokinetic model for Drug A, and a similar model is needed for Drug B, it is preferable to modularize common functions and rewrite just the components that need to be changed, such as the initialization of the parameters. Otherwise, the programmer will inherit an additional task of keeping the two programs synchronized. When the code is modularized, bug fixes or improvements to a module are instantly reflected in all the programs that use that module. Modularization can significantly increase productivity, because modules that are used frequently are likely to be tested more often and improved in terms of accuracy and performance. When modularization is carried to an extreme, it can lead to overengineering and also unreadable code. Programmers should exercise their judgment in deciding what level of modularization is appropriate for a specific set of problems. 2.7.3
Utilize Existing Modules and Libraries to the Fullest
A corollary to using modular programming is the use of modules developed by others. Most modern computer programming languages sport a wide range of modules in the form of libraries or toolboxes for solving a variety of problems. The type of problems they solve varies in scope: from sorting, searching, solution of linear equations or differential equations, random number generation, and plotting, among many others. Thus, a significant amount of computer programming can benefit from the “component model of programming,” where the problem is often posed as finding appropriate modules from an existing toolbox and linking them to solve a specific problem. Despite the availability of well tested standard modules, some programmers tend to write new code to solve standard problems: for example, a module to solve an ordinary differential equation or a module to generate random numbers. It is recommended to perform a simple search to identify any existing modules before embarking on writing new ones, thus avoiding the problem of “reinventing the wheel.” The main exception to this practice is the situation where license restrictions or organization policies necessitate developing new code to solve a standard problem. 2.7.4
Use Structured Programming
Structured programming is an approach in which a program consists of subsections, each with a single point of entry. Structured programming facilitates a “top–down” approach to program design, whereby the large scale structure of a program is mapped out in terms of smaller operations, which can be independently imple-
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mented and tested. Structural programming is achieved by using hierarchical conditional constructs, such as “if-then-else,” “switch,” “for,” and “while” for creating conditional branches of execution. This approach shuns the indiscriminate use of “goto” statements, which allows program control to jump to any line in the code identified by a line number or label and can make it difficult to follow the logic of a program. The “goto” statement is sometimes used to direct program control when a program exception or error occurs. Alternative constructs that can be used in structured programming include (a) the “return” statement, which returns control to the end of the current function; (b) the “break” statement, which terminates the inner most loop; and (c) the “continue” statement, which returns to the next iteration of the innermost loop. 2.7.5
Use Appropriate Structured Programming Constructs
The choice of the structured programming construct used should convey the logic involved in a given operation. This is important because most of the constructs can be expressed in terms of other constructs: for example, a “for” loop can be written as either a “while” or “unless” loop. Some of the guidelines for the appropriate constructs to use are as follows: Use “for” construct when the number of loop iterations is known beforehand. Likewise, use “while” construct when the number of loop iterations is not known beforehand. Cases include reading data from a file or from user input line by line until the end is encountered. Though this can be achieved by using a “for” loop with a conditional “break” statement, the “while” statement conveys the logic clearly. Some special cases require using “do-while” (when the first statement has to be executed before the conditional). Use “switch” construct instead of multiple, nested “if-then-else” statements, especially when all the conditionals are treated at the same level. The resulting code is usually easier to read and follow. However, when different types of conditionals are tested, multiple, nested “if-then-else” constructs are preferable. An example for using “switch” versus “if” statements is shown in Code Block 5 (Figure 2.5). 2.7.6 Use Data Structures Appropriate to the Problem Under Consideration A programmer should select the appropriate program types that properly define the computational problem. For example, if the number of compartments is a userdefined construct (i.e., the program is designed for solving systems of equations for a multicompartment PBPK model), the number of compartments becomes a parameter. However, if a PBPK model is for a specific implementation, a constant should be used to describe the number of compartments. Likewise, depending on the situation, a matrix may be more appropriate than a set of one-dimensional arrays. Similar choices have to be made with respect to selecting complex data structures versus default types provided by the programming language.
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2.8 GOOD PROGRAMMING PRACTICES: MODULAR CODE DESIGN FOR FUNCTIONS 2.8.1 Restrict Use of Global Variables Global variables are variables that are active at all stages of the code, while the scope of local variables is restricted to the function in which they are defined. Global variables should be used with care to avoid inadvertently setting a value in one module of code that could affect computations in another module. Use local variables as far as possible. Global variables are most appropriately used to define constants that do not change during the execution of the program (e.g., molecular weight of a chemical). They are a convenient means of passing values through more than one level of module hierarchy. An example with Matlab code for a differential equation model is given in Code Block 6 (Figure 2.6), in which global variables are used to pass values to the derivative function (which is not called directly from the code block where the constants are defined). The constants could have been defined in the derivative code, but defining them earlier is more efficient as the statements in the derivative code are executed repeatedly. 2.8.2 Pass Information Through Function Parameters and Not Through Global Variables One of the advantages of global variables is that they are accessible from all components of the program. This also means that keeping track of the global variables becomes very difficult. A change in the values of a global variable in one function may trigger a difficult to notice change in another function. Therefore, passing information via function parameters is much more robust than passing information (i) y = solveMyODE(’f’, x0, x1, y0, dt, eps1, 23); % Above function call is not informative (ii) y = solveMyODE(struct(’func’, ’f’, ’xinit’, x0, ’xend’, x1, ... ’yinit’, y0, ’tstep’, dt, ... ’relerror’, eps1, ’ODEMethod’, 23)); (iii) config.function = ’f’; config.xinit = x0; config.xend = x1; config.tstep = dt; config.relerror = eps1; % ODE (Ordinary Differential Equation) solver option config.ODEMethod = 23; y = solveMyODE(config); % passing config object to ODE solver config.ODEMethod = 13; % change just the ODE solver y = solveMyODE(config); % invocation with modified options FIGURE 2.6 Code Block 6—passing large sets of function parameters through custom data structures.
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through global variables. This approach also allows for easier definition of test cases, since the only changes in the function state will be caused by changes in function parameters. The main exception to this practice is when the same information needs to be passed through a nested set of functions. In PBPK modeling in Matlab, such a scenario is often encountered when a main program invokes a subprogram that invokes a differential equation solver. 2.8.3
Avoid Too Few or Too Many Function Parameters
A function with too few parameters is usually less flexible. However, a function with too many parameters is a good candidate for further modularization into multiple functions. The extra inertia in having to provide a large set of parameters to invoke a function will lead to an underused function; often, a programmer will use a simpler alternative. 2.8.4 Write Functions Using a Variable Number of Arguments with Reasonable Defaults Many programming languages support defining functions that operate with a variable number of arguments, with a common example being the “print” function. A function that accepts a variable number of arguments along with reasonable defaults can provide great flexibility and functionality. The function will be easy to invoke, because it does not require a large set of parameters; but at the same time, it will be flexible enough for advanced users of the function. Matlab provides the feature of variable number of function arguments and function outputs through the constructs “varargin” and “varargout.” This feature is often encountered in Matlab in the solution of differential equations: a novice can use the solver with default options and still get a reasonable solution, whereas an expert can tune the function performance by providing advanced options. Of course, there is an additional overhead involved in writing functions that handle a variable number of parameters, including checking whether required parameters are provided, what optional parameters are provided, and what parameters need to be set to default values. However, the code for handling such tasks is similar from function to function, and a well designed, flexible function usually is worth the additional code required. 2.8.5
Use “Try-Catch” Type Exception Handling
One of the most powerful features of modern programming languages is exception handling. However, it is significantly underused in scientific programming. Blocks of code that use exception handling consist of two parts—the normal program (also called as the “try” block) and the errors/exception block (also called as the “catch” block). The main advantage is that the exceptional conditions can be handled in one location without “cluttering” the code for main program flow. The exceptions from the lower level functions (e.g., a square root function) can be propagated to the higher level invoking function, which can then handle the error condition appropriately. Thus, the writer of the lower level function need not focus on how to handle the error condition and can simply focus on alerting the caller function about the error conditions. This enhances the modularization of the code as well as allows more code reuse.
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GENERAL PRINCIPLES OF PROGRAMMING: COMPUTER AND STATISTICAL
Use Custom Data Structures with a Hierarchical Design
Most modern programming languages support user-defined, complex data structures and objects, and that feature can be utilized in writing clearer code. For example, in Matlab, instead of using variables such as PBPK_Human_chloroform_Vmax, PBPK_ Human_chloroform_Km and PBPK_Human_chloroform_PC_blood_lung, one can write compact, easily readable code, by defining the variables as constituents of a custom data structure, as follows: pbpk.human.chloroform.vmax, pbpk.human. choloroform.Km and pbpk.chloroform.PC.blood_lung. Therefore, the parameters can be used in the most appropriate manner depending on the context. For example, in case of a pharmacokinetic module for chloroform, the parameters can be passed as param = pbpk.human.choloroform. Now, the parameters Vmax and Km, can be accessed as param.Vmax and param.Km. Likewise, all the partition coefficients can be accessed as param.PC. This approach provides flexibility in parameter assignment and parameter passing and improves readability. 2.8.7 Use Informative, Custom Data Structures for Function Parameter Passing A function that takes a structure that has informative field names is significantly more readable than a function that takes a large number of parameters. For example, in Code Block 6 (Figure 2.6), a function call of the form shown in (i) is typically used and is not informative. However, by using data structures for parameter passing, as in (ii), the context of the parameters becomes clearer. The parameter passing approach in (iii) is similar to that used in (ii), with the added advantage that the data structures for parameter passing can also be reused. Functions designed in such a manner can also be easily extended to include more parameters without requiring changes in the intermediate calling functions; that is, changes in a function invoked via intermediate functions will not impact the intermediate functions. 2.9 GOOD PROGRAMMING PRACTICES: WRITING EXTENSIBLE AND NONINTERACTIVE PROGRAMS Often, a numerical model has to be run for different combinations of parameter values. Examples include performing a large number of Monte Carlo simulations with a model to estimate the range of uncertainties in model outputs or distributions of outputs for a study population. Likewise, parameter estimation techniques, such as the Bayesian Markov chain Monte Carlo (MCMC) (43, 44) technique, also involve running the full model with varying sets of parameters. Therefore, a program should be designed upfront in a manner that facilitates noninteractive (automated) runs. This aspect is critical in software testing (45, 46). The notion of running programs in a noninteractive mode is common in the area of server-based computing using operating systems such as UNIX and Linux. In contrast, PC-based computing has been predominantly interactive. The advantage of server-based systems is that a user can connect to the server, submit one or more “jobs” for execution, monitor the progress of the simulations for a period of time, set the job status to “background,” and disconnect from the server. In such systems,
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several users can use one server simultaneously and do not need to stay connected for the duration of the simulations. In case the users need to maintain an interactive session, some advanced tools, such as the GNU screen utility (www.gnu.org/ software/screen), provide the feature of a “virtual interactive session” that the users can disconnect from and reconnect to as needed. These tools can be contrasted with the current techniques in PC-based computing, such as remote desktop (www.rdesktop.org) or virtual network computing (VNC; www.tightvnc.com), where only one user can effectively be connected to the server at a given time, and there are no easy means for automating connections to multiple machines. The advantage of noninteractive programs has become more pronounced with the advent of powerful but inexpensive computing clusters. Typically, a user has access to several tens to hundreds of machines in a cluster. Thus, the ability to run a program in a noninteractive or detached mode without continuous monitoring is very useful. Furthermore, since the user’s main computer (typically a desktop computer) is not occupied with multiple connections to the server, one can submit large running jobs to the server without affecting the desktop machine. 2.9.1
Provide a Usable Interface to the Model
An intuitive user interface, either command-line or graphical, is an important factor affecting model usability. Some of the relevant aspects include providing appropriate user input prompts, warnings, and diagnostics, when erroneous conditions are encountered, and user input validation and correction (e.g., re-prompting the user when an input error such as entering a text string when a numerical input is expected). This is complementary to the ability to run the model in an automated, noninteractive mode. Ideally, a program should be designed to operate in both interactive and noninteractive modes. A common approach for designing such programs involves running the model in an interactive mode when no command-line parameters are provided, and running it in an automated mode when the required parameters are passed via command line or through an input file. 2.9.2
Write Programs with Standard Formats for Inputs and Outputs
When the model uses a standard format for model inputs and outputs, it becomes easily extensible in the form of a link in a long chain of models. It also makes it easier for writing scripts to generate reports or plots from model outputs, to aggregate multiple model runs and perform additional analysis, and even to run multiple simulations based on other resources (e.g., using a database of chemistry parameters as an input to the model). Traditionally, the input and output formats are quite variable, and often a programmer would decide on the format based on the flow of the model. Some of the widely used general purpose formats include CSV (comma separated values, supported by most spreadsheet software) and XML (eXtensible Markup Language; www.xml.org). An effective approach is to utilize object storage features of the programming environment. For example, both Matlab and S-Plus provide an option to directly save a set of variables into an object file. These objects can be retrieved later by simply loading the object file. One of the advantages of using the programming environment features to save objects is that, in addition to model inputs and outputs, extensive “metadata”
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related to the inputs and outputs of a model can also be saved and later retrieved. The metadata can include (a) time when a simulation is run, (b) configuration options, (c) machine and folder paths for the simulation, and (d) the script that was run to produce the outputs. It facilitates easier reproducibility of results. 2.9.3
Write Easily Relocatable Programs
Relocatable programs are programs that can be run in isolation, without affecting earlier model runs. This requires that locations of input, output, and configuration files not contain absolute folder paths. This is important when multiple runs of the model are performed—each model run can be performed in a different folder and model runs from one simulation will not affect other runs. 2.9.4 Provide Ability to Uniquely Identify Results from Multiple Model Runs When a model is used for performing multiple simulations, the “management” of simulations becomes a major issue. At a minimum, the simulations should be set up such that the outputs from different model runs can easily be identified. The simulation setup should allow a subset of model runs to be repeated without needing to repeat all the model runs. This is often achieved by providing a separate, appropriately named folder for storing inputs, outputs, and configuration files for each model run, and aggregating the model runs at the end of the simulation. Such an approach allows the model runs to be performed on a distributed cluster of machines.
2.10 GOOD PRACTICES: RELEVANT SOFTWARE ENGINEERING CONCEPTS 2.10.1
Follow Appropriate Directory Structures
Standardized directory structures for source code, documentation files, final executables, configuration files, and model inputs/outputs allow tracking the software development process and also help in easily integrating multiple, independently developed modules. Standardized directory structures allow easy detection of conflicts in the names of functions, scripts, or configuration files. 2.10.2
Utilize Available Libraries and System Tools
The advantage of using existing tools and libraries is that the programmer need not actively maintain or refine them. The modern programming experience often involves taking advantage of a diverse set of libraries, programs, and tools. Most languages provide interfaces to link modules from other languages (e.g., Fortran/C, Matlab/C, Matlab/Java), which can be utilized to link modules written in practically any language. The overhead involved in understanding new libraries, tools, and language interfaces pays off very quickly. Sometimes, the simple approach of using multiple programs in a “pipeline” is also effective. A common example is where a program’s output is used to automatically generate plots and
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reports using preexisting templates. When a programmer has flexibility and initiative in using multiple tools, there is an increased chance of using the right set of tools for a given task, within common constraints such as cost, as well as organization guidelines. License issues also play a major role, since there may be different types of restrictions that arise when using commercial (redistribution issues, code confidentiality, etc.) as well as freely available code (which may contain clauses that affect the derived code). 2.10.3
Use Appropriate Module from a Library
One problem with the availability of a large set of “standard” libraries is that at times it is possible to use the wrong module for a given task. A common situation involves the solution of differential equations. Some systems of differential equations contain derivatives that vary over wide scales, and these are known as “stiff” systems of differential equations. Therefore, a stiff differential equation solver should be used in these cases; otherwise, substantial numerical errors or convergence problems will result. 2.10.4
Use Software Revision Control Tools
Revision control helps in identifying changes to documents or code by incrementing an associated number or letter code, termed the “revision level” or simply “revision.” Most modern revision control systems such as CVS (www.nongnu.org/cvs) (30) provide facilities to track changes based on user, time, or version number. For a group project, such systems are very critical. Even for a single programmer, such systems are essential because they provide some means of being able to reproduce a set of source files that satisfied some set of conditions in the past. These systems are vastly superior to ad hoc approaches for document control such as manual backups of directories. A further advantage of a revision control system is that the programmer has the flexibility to experiment with code changes without having to worry about manually managing extensive changes. 2.10.5 Embed Simple Testing into the Model: Simple Mass-Balance Checks Embedding simple error checks into the model improves its robustness and extensibility. This can be done either at the mathematical model development stage (e.g., incorporating mass-balance checks in a pharmacokinetic model) or at the software implementation level (e.g., incorporating alerts whenever a negative concentration or a negative flow rate is encountered). The mass-balance type checks are valuable in the sense that they can highlight errors in both the mathematical model as well as the software implementation. The overhead associated with incorporating such error checks is warranted because of the benefits provided. 2.10.6
Utilize Test Cases and Peer Review in Program Design
Proper code testing and peer review of code and test cases are critical for scientific programming (46–48). One of the recent advances in software design methodologies includes the approach of “extreme programming” (XP) (6, 49). The XP approach
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advocates the notion of software design by contract, where test cases are designed first and the code is written later (50). A major emphasis is also placed on designing the code such that the costs associated with code changes are lowered. Software testing ranges from testing of an individual module (unit testing) (51) to the established discipline of formal software testing (45, 46). For an individual programmer, Humphrey (5) presents an insight into most major aspects of software development and relevant practices, including test-driven program design and reviews of design, code, and test cases. 2.11
SUMMARY
This chapter provides an overview of generally applicable good programming practices relevant to pharmacometricians. The guidelines provided here can be useful in developing correct, robust, and easily maintainable and extensible programs. These guidelines are targeted toward novice and intermediate programmers and may also provide some relevant tips to experienced programmers. Although sophisticated programming skills are not necessary to develop many pharmacometric programs, the concepts described in this chapter can be useful in writing even relatively simple scripts and programs. The main focus deals with aspects of basic coding style, design issues, and tools that can be quickly used in improving the programming process. The style aspects focus on readability and standardization, which facilitate effective code reviews; the design aspects focus on structured programming and modular function design; and the software engineering discussion focuses on test design and program extensibility. Overall, the development of scientific computer programs is addressed from the perspective of writing scientific documents: they should provide context, be readable, and contain appropriate references. Another aspect addressed in this chapter is that computer programming tasks in recent times have evolved from writing new code and modules to correctly linking existing modules. Programming productivity can be increased substantially by utilizing available toolkits, libraries, development environments, and relevant programming approaches. Once a good design or approach is employed, and relevant existing modules are identified, the linking of the modules to solve a pharmacometric programming problem becomes a more straightforward task. ACKNOWLEDGMENTS The good programming practices described here reflect experience of the authors at the USEPA funded Center for Exposure and Risk Modeling (CERM), and the environmental bioinformatics and Computational Toxicology Center (ebCTC).
REFERENCES 1. J. M. Yohe, An overview of programming practices. ACM Comput Surv 6:221–245 (1974). 2. E. W. Dijkstra, A Discipline of Programming. Prentice-Hall, Englewood Cliffs, NJ, 1976.
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3. B. W. Kernighan and P. J. Plauger, The Elements of Programming Style. McGraw-Hill, New York, 1978. 4. R. G. Dromey, How to Solve It by Computer. Prentice-Hall International, Englewood Cliffs, NJ, 1982. 5. W. S. Humphrey, PSP: A Self-improvement Process for Software Engineers. AddisonWesley, Boston, 2005. 6. S. W. Ambler, Agile Modeling: Effective Practices for eXtreme Programming and the Unified Process. Wiley, Hoboken, NJ, 2002. 7. D. Z. D’Argenio and A. Schumitzky, ADAPT II User’s Guide: Pharmacokinetic/Pharmacodynamic Systems Analysis Software. Biomedical Simulations Resource (BMSR), Los Angeles, 1997. 8. M. de Hoog, R. C. Schoemaker, J. N. van den Anker, and A. A. Vinks, NONMEM and NPEM2 population modeling: a comparison using tobramycin data in neonates. Ther Drug Monit 24:359–365 (2002). 9. B. Frame, R. Miller, and R. L. Lalonde, Evaluation of mixture modeling with count data using NONMEM. Pharmacokinet Pharmacodynam 30:167–183 (2003). 10. S. P. Millard and A. Krause, Applied Statistics in the Pharmaceutical Industry: With Case Studies Using S-Plus. Springer, New York, 2001. 11. W. N. Venables and B. D. Ripley, Modern Applied Statistics with S-PLUS. Springer, New York, 1999. 12. S. J. Chapman, MATLAB Programming for Engineers. Brooks/Cole-Thomson Learning, Pacific Grove, CA, 2002. 13. S. R. Otto and J. P. Denier, An Introduction to Programming and Numerical Methods in MATLAB. Springer, New York, 2005. 14. L. Lindbom, J. Ribbing, and E. N. Jonsson, Perl-speaks-NONMEM (PsN)—a Perl module for NONMEM related programming. Comput Methods Programs Biomed 75: 85–94 (2004). 15. E. N. Jonsson and M. O. Karlsson, Xpose—an S-PLUS based population pharmacokinetic/pharmacodynamic model building aid for NONMEM. Comput Methods Programs Biomed 58:51–64 (1999). 16. G. M. Schneider, The introductory programming course in computer science: ten principles, in Papers of the SIGCSE/CSA Technical Symposium on Computer Science Education. ACM Press, Detroit, MI, 1978. 17. D. M. Seeley, Coding smart: people vs. tools. Queue 1:33–40 (2003). 18. R. L. Read, How to Be a Programmer: A Short, Comprehensive, and Personal Summary. Available freely from http://samizdat.mines.edu/howto/ (2002). 19. H. Abelson, G. J. Sussman, and J. Sussman, Structure and Interpretation of Computer Programs. MIT Press, Cambridge, MA, 1996. 20. M. Felleisen, How to Design Programs: An Introduction to Programming and Computing. MIT Press, Cambridge, MA, 2001. 21. A. Downey, How to Think Like a Computer Scientist: Learning with Python. Green Tea Press, Wellesley, MA, 2002. 22. W. Gander and J. Hrebícek, Solving Problems in Scientific Computing Using Maple and MATLAB. Springer, New York, 2004. 23. M. J. Crawley, Statistical Computing: An Introduction to Data Analysis Using S-Plus. Wiley, Hoboken, NJ, 2002. 24. O. M. Nierstrasz and D. C. Tsichritzis, Object-Oriented Software Composition. Prentice Hall, London, 1995.
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25. I. Foster, Designing and Building Parallel Programs: Concepts and Tools for Parallel Software Engineering. Addison-Wesley, Reading, MA, 1995. 26. W. H. Press, Numerical Recipes in Fortran 90: The Art of Parallel Scientific Computing. Volume 2 of FORTRAN Numerical Recipes. Cambridge University Press, New York, 1996. 27. J. Spolsky, User Interface Design for Programmers. http://joelonsoftware.com/ uibook/chapters/Fog0000000057.html (2006). 28. K. Ka Iok Tong, Essential Skills for Agile Development. http://www.agileskills. org/ (2006). 29. M. Giorgio, Survey of existing programming aids. SIGPLAN Not 11:38–41 (1976). 30. K. Fogel, Open Source Development with CVS. Coriolis Group Books, Scottsdale, AZ, 1999. 31. NIST, Guide to Available Mathematical Software. http://gams.nist.gov/ (2005). 32. GNU, GSL–GNU Scientific Library. http://www.gnu.org/software/gsl/ (2005). 33. StatLib, StatLib—Data, Software and News from the Statistics Community. http:// lib.stat.cmu.edu/ (2005). 34. Netlib, Netlib—A Collection of Mathematical Software, Papers, and Databases. http:// www.netlib.org/ (2005). 35. MathWorks, The MATLAB Central File Exchange. http://www.mathworks.com/ matlabcentral/fileexchange/ (2005). 36. OctaveForge, GNU Octave Repository. http://octave.sourceforge.net/ (2005). 37. CRAN, CRAN—The Comprehensive R Archive Network. http://cran.r-project. org/ (2005). 38. G. Flandin, M2HTML: Documentation System for Matlab in HTML. http://www. artefact.tk/software/matlab/m2html/ (2005). 39. L. C. Whipple and T. Roche, Essential SourceSafe. Hentzenwerke Publishing, Whitefish Bay, WI, 2001. 40. Wikipedia, Programming Style—Appropriate Variable Names. http://en.wikipedia. org/wiki/Coding_standard#Appropriate_variable_names (2005). 41. MathWorks, Programming (Version 7), 2005. 42. S. McConnell, Code Complete. Microsoft Press, Redmond, WA, 2004. 43. F. Jonsson and G. Johanson, The Bayesian population approach to physiological toxicokinetic–toxicodynamic models—an example using the MCSim software. Toxicol Lett 138:143–150 (2003). 44. W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, Markov Chain Monte Carlo in Practice. Chapman & Hall, Boca Raton, FL, 1998. 45. R. Patton, Software Testing. Sams, Indianapolis, IN, 2001. 46. C. Kaner, J. L. Falk, and H. Q. Nguyen, Testing Computer Software. Van Nostrand Reinhold, New York, 2002. 47. K. E. Wiegers, Peer Reviews in Software: A Practical Guide. Addison-Wesley, Boston, MA, 2002. 48. W. S. Humphrey, A Discipline for Software Engineering. Addison-Wesley, Reading, MA, 1995. 49. J. Newkirk and R. C. Martin, Extreme Programming in Practice. Addison-Wesley, Boston, MA, 2001. 50. K. Beck, Test-Driven Development: By Example. Addison-Wesley, Boston, MA, 2003. 51. P. Hamill, Unit Test Frameworks. O’Reilly, Sebastopol, CA, 2004.
CHAPTER 3
Validation of Software for Pharmacometric Analysis GARY L. WOLK
3.1
INTRODUCTION
The development, installation, and utilization of software for pharmacometric studies require the pharmacometrician to interact with at least three different organizational entities. Management must first be convinced of the need, and the appropriate expense must be justified, for implementing the tools regarded as necessary by the pharmacometrician to perform a successful analysis. Next, there is the interaction with the local suppliers of technology, the information technology (IT) group. This interaction is critical to determining the timeliness and the success of the implementation process. Finally (and perhaps most important) is the interaction of the pharmacometrician with the regulatory group responsible for the software validation process. The responsibilities of the scientist will vary, depending on the organizational size. If the pharmacometrician is employed by a small or startup pharmaceutical or biotechnology firm, it is plausible that the pharmacometrician may be filling all three of these roles—clinical developer/manager, information technologist, and regulatory specialist. In this instance, interdepartmental delays become nonexistent, but the burden on the pharmacometrician is immense. For scientists working in medium size institutions, there is probably a specialist available from each area, but the burdens on each group tend to be immense since the company is more than likely in a “growth” mode. Finally, in a large corporate environment, the scientist is confronted by the possibility of dealing with a less personal, highly specialized IT or regulatory organization or, possibly, organizations that have been specifically devoted to business segments such as clinical development. In a sense, this last condition is the closing of the business organizational loop where one person is responsible for the entire process to a set of organizations that is entirely focused on the success of this particular part of the pharmaceutical realization process.
Pharmacometrics: The Science of Quantitative Pharmacology Edited by Ene I. Ette and Paul J. Williams Copyright © 2007 John Wiley & Sons, Inc.
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This chapter outlines the software implementation and validation process, to an extent that the pharmacometrician could in fact establish the quality assurance infrastructure, implement the hardware and software, and validate the implementation independent of dedicated internal IT or regulatory resources. Though such an approach is not recommended, the purpose of this chapter is to give the scientist a clear understanding of what is required in order to be successful in such an endeavor. First, we review the concepts behind software quality assurance, testing, and validation. We review the process from the historical perspective of how other industries have faced these quality assurance issues, the role of independent organizations, and finally the role that federal regulatory agencies have played and how each of these has impacted the validation process in the pharmaceutical industry (1, 2). The rule-making efforts of the US FDA in the last 5 years, in particular, the 21CFR11 guidance (3, 4), is discussed in the context of this historical perspective. We also note the critical issues that face pharmacometricians in executing their scientific methodology: obtaining/finding data, creating/defining models in software, creating/finding results, and reproducing analysis. We then outline the basic methodology for software validation: quality assurance practices (corporate policy, standard operating practices, validation processes), technology practices (assuring the proper infrastructure, influencing and participating in the IT process), and the process for making “buy or build” decisions. Often the decision is to buy and then build on to the software base. This is particularly true of software tools that allow the pharmacometrician to either automate existing software processes or design variations on existing algorithmic routines offered by the commercial tool. The validation process is outlined from writing user requirements specification to testing and validating specific analysis using estimation methods. This is followed with brief examples of validation approaches for some commonly encountered software, such as S-Plus®, SAS®, WinNonlin®, and NONMEM®.
3.2
SOFTWARE DEVELOPMENT AND IMPLEMENTATION: BACKGROUND
In the late 1980s at AT&T Bell Laboratories, it came as quite a shock to be told that the “quality” of our work needed to be addressed. The scientific staff was insulted and the nontechnical managers who implemented “quality improvement” programs based on the Japanese models of the time were without a clue as to why there would be such resentment. It took several years for all to realize that, indeed, the quality of business practices that surrounded R&D efforts needed improvement, not necessarily the quality of the technical effort. The processes surrounding R&D—documentation of work, sharing of information, the need to avoid duplicate effort—soon were understood to be significant areas of improvement that both technical staff and nontechnical managers could work together on to improve the overall nature of the business. The manufacturing division, the former Western Electric, had indeed been a center of quality improvement and statistical process control 20 years earlier. Telephony transmission and switching systems, by their nature of being large, complex, engineered entities, had always been subject to high levels of review and quality assurance.
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What remained was to take the appropriate pieces of the quality assurance world and implement them in such a way that scientists were able to work in a “structured” framework, while at the same time assuring that the creativity of the scientists was not stifled. In the ensuing years, many industries began to adapt the guiding principles of ISO 9000 (5), using the actual certification process as a way both to identify and improve business processes and to leverage certification as a marketing tool. In parallel with these industrywide quality improvement efforts, the software industry had recognized the need to identify processes and standards that assured the quality of commercial software. ANSI (6) and IEEE (7) have been issuing practice standards and definitions for many years in an effort to unify quality assurance methodology in the software development industry. Furthermore, the software industry recognized early on that establishment of quality principles prior to the initiation of a development effort reduced the cost of repairing faulty software later in the process (8). In the early 1990s, the implementation of the Clean Air Act, along with major changes to other environmental laws and regulations, produced a tremendous effort in the field of data acquisition and analysis, which clearly needed to be aided by advances in the information sciences. Hence the Environmental Protection Agency (EPA) issued guidance in late 1990, the EPA Good Automated Laboratory Practices (GALPs) (9), that were the first effort by a regulatory agency to assure the proper use of IT in the acquisition and analysis of regulated data. The Food and Drug Administration (FDA) had previously taken the position that most good laboratory practice (GLP) and good clinical practice (GCP) processes involving information systems were covered by existing regulation (10). Although a guide to inspection of computerized equipment in drug processing (11) and a technical reference (12) on software development activities were issued in the 1980s, the major FDA guidance on the use of electronic records and systems was not issued until 1999. Once that guidance, 21CFR11, was issued (13), an entire industry arose to attempt to explain, implement, and modify the guidance (7). In general, there are certain basic quality assurance principles that can be invoked that will satisfy the spirit, if not the fine detail, of most regulatory requirements: 1. Document the processes used to generate, accept, analyze, store, and archive data and analytical results. 2. Document the physical and logical security of hardware and software systems used on regulated data. 3. Document the installation and testing of hardware and software used on regulated data. 4. Document that the system design achieves the intended purpose/use. 5. Document performance, both initial and ongoing, of the software system. 6. Document training and education backgrounds of the users and providers of the systems. 7. Document that the business practices are in place to operate, backup, and recover (including disaster recovery) regulated software systems. Each of these issues focuses on documentation. The purpose of the validation process and the generation of process standards (or standard operating
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procedures—SOPs) are to establish a documentation framework a priori, rather than de facto with regard to the installation and use of key software. All of the processes listed above occur within disciplined scientific organizations. The validation methodology is used to demonstrate the structure of these processes for the purpose of both internal and external review.
3.3
METHODS
A successful validation strategy is aided by several elements including: 1. A corporate policy on quality assurance/validation. 2. Existing, corporatewide SOP infrastructure and pharmacometric specific SOPs. 3. Definition of the validation process. 4. Understanding the user requirements generation process. 5. Identifying the system specification for a particular implementation. 6. Understanding the current information technology infrastructure/ organization. 7. Recognizing the constraints of “building” versus “buying.” We discuss each of these in turn. 3.3.1
Corporate Policy
In the case of industries that decide to pursue ISO 9000 certifications, the role of management is well defined (5). The standard clearly states that management will define and document its policy, objective, and commitment to quality. The burden of implementing, explaining, and maintaining the quality plan is clearly on corporate management. A similar approach needs to be undertaken in approaching validation of regulated systems. A clear corporate policy document should exist, which: 1. Establishes a working group to define and maintain policy and objectives regarding validation of software systems. 2. Ensures that employees are trained and retrained on the policy. 3. Empowers the resources necessary to carry out the policy. In the absence of support from the highest levels of corporate management, it is unlikely that the competing priorities of clinical development, information technology, and regulatory affairs will somehow “align” to enable the success of a validation project. 3.3.2
Establishment of SOP Infrastructure
The first priority of a regulatory group should be the establishment of “SOP on SOPs”—that is, how they are to be created, reviewed, implemented, and revised. If a policy applies across corporate groups including information technology,
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pharmacokinetics, data management, biostatistics, regulatory affairs, quality assurance, and materials management, then the pharmacometrician will have model SOPs as well as the collegial support necessary to implement a procedural structure that may be used to clarify workflow and serve as a training tool for new scientists. In particular, the process by which preclinical and clinical pharmacokinetic/ pharmacodynamic (PK/PD) data is received, identified, and analyzed (at least for initial parameters such as AUC and t½) should be documented in a series of SOPs. Furthermore, the manner in which such data and analysis should be stored for latter retrieval is also a key consideration for optimal efficiency in the drug development process. The process by which PK summaries and reports are approved and released to other groups must also be documented in order to prevent misunderstandings. The procedure for use of randomization codes by the PK group must clearly be documented by an SOP, consistent with the needs and requirements of data management, biostatistics, and regulatory groups. Table 3.1 shows a plausible sample of SOPs that could be written to encompass the activities of both clinical and preclinical PK and PD analysis. It should be noted that many of the chapter titles in this text could also serve as the basis of clinical pharmacology SOPs! Given the current desire of management to be able to “mine data” and “see trends across studies” and the availability of PK/PD repository systems (which we discuss more fully later), the first and foremost operating procedure requirement in pharmacometrics is the definition of key metadata that describes the process flow. Metadata, from the information science perspective, is simply information that describes data: that is, where the data goes, what the data is, and what possible
TABLE 3.1 SOP # PKPD001 PKPD002 PKPD003 PKPD004 PKPD005 PKPD006 PKPD007 PKPD008 PKPD009 PKPD010 PKPD011 PKPD012
Standard Operating Procedures for Clinical Pharmacology SOP Title Training requirements for pharmacologists and toxicologists Definition of nomenclature: project, study, indication, NCE ID, etc. Review and approval process for PK summaries and related reports Standards for PK data analysis: basic parameters to be obtained Use of blinded data PK analysis standards, data preparation, statistical analysis, expected output for clinical bioavailability studies PK analysis standards, data preparation, statistical analysis, expected output for clinical bioequivalence studies PK analysis standards, data preparation, statistical analysis, expected output for human dose proportionality studies PK analysis standards, data preparation, statistical analysis, expected output for drug interaction studies PK analysis standards, data preparation, statistical analysis, expected output for clinical renal studies PK analysis standards, data preparation, statistical analysis, expected output for first dose in human studies PK analysis standards, data preparation, statistical analysis, expected output for compartmental study types
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value it may have. For example, having consistent definitions and values for the words “Portfolio,” “Project,” “Protocol,” “Study,” “Study Design,” “Study Type,” “Indication,” “Method,” “Period,” “Phase,” and “Relative Nominal Time” can lead to a dramatic increase in the ability to find and leverage information within clinical development. Unfortunately, there are enough clinical data management, repository, and laboratory information management systems (LIMSs) available to completely confuse the end user as to how the corporate metadata matches a software vendor’s definition. Nevertheless, a group of scientists who have established procedural definitions of such metadata a priori have built a common ground that can serve as a basis for leveraged information management. 3.3.3
Definition of the Validation Process
In general, the validation process should also be defined by several SOPs, originating in either the regulatory or information technology groups. Table 3.2 shows a sample list of IT or Quality Assurance SOPs appropriate to the task. The validation process generally will consist of the following: 1. Validation Project Plan is a summary document identifying software, hardware, and related systems involved in a specific validation effort. The document explains the approach that will be employed, the responsible parties, and the expectations of those parties for each task involved in the validation. 2. User requirements specification is the responsibility of those end users who have identified the need for the system. It must adequately define the functional requirements of the system/software so that the end users can satisfy the stated business requirement.
TABLE 3.2 Standard Operating Procedures for Information Technology or Quality Assurance SOP #
SOP Title
QA001 QA002 QA003
Format, functionality, and maintenance of standard operating procedures Membership and purpose for the software validation standards committee Validation process: validation planning, user requirements, and system specifications Installation qualification protocol format and requirements for software Operational qualification protocol format and requirements for software Performance qualification protocol format and requirements for software Change control procedures Deviation procedures Reporting “out-of-specification” events Physical security procedures Logical security procedures Backup and recovery procedures Hardware installation qualification procedures Software development life cycle practices and procedures
QA004 QA005 QA006 QA007 QA008 QA009 IT001 IT002 IT003 IT004 IT005
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3. Systems specifications provide all the information needed for the technical implementation of the system. This includes hardware, networking connections, and backup requirements as well as all information needed to install, operate, and qualify the performance of the system. It generally includes all of the qualification protocol documents (installation, operation, and performance qualifications) created during the validation process. 4. Completion and change control is the closure of the process and a methodology for maintenance activities. 3.3.4
Understanding the User Requirements Specification (URS) Process
This process is usually the first exposure of the pharmacometrician to the validation process. It is a difficult first step, where the scientist must document the use of a tool (which is of obvious utility from the perspective of the pharmacometrician) to an audience that may not have a good understanding of the clinical development process. The major point is that the process can be quite generic for many of the software tools utilized in PK/PD analysis. For example, the user requirements specification for implementing S-Plus, SAS, or Graphpad Prism® could all be essentially the same document. Similarly, NONMEM, WinNonlin, WinNonMix®, Kinetica®, and ADAPT II would contain the same basic set of user requirements. Specific capabilities that would be used for a particular software tool would need to be identified, but the basic form of the requirements is the same. Once again, having a sound basis set of SOPs that actually describes the acquisition, analysis, and reporting requirements for clinical data will enable the pharmacometrician to crossreference the particular software capabilities with the technical (business) process. For systems such as PK/PD repositories, a broader view is needed. Such systems by definition are intended to exchange data with other systems and integrate with analytical tools such as those described earlier. In this case, the pharmacometrician needs to have a well established process in place and be able to document how such a repository system will be implemented to either augment or replace current manual processes. 3.3.5
Identify the System Specification for a Particular Implementation
The selection of platforms (i.e., UNIX versus Microsoft Windows Server) is primarily within the realm of system specification rather than user specification. Nevertheless, it is useful for the end user to consider early on which tools are preferred and which platform will be used or whether several platforms might be utilized (depending on business requirements). We discuss this issue further when the interaction with the IT group is reviewed. The pharmacometrician must be able to specify key system requirements with regard to recovery of data and archiving. Furthermore, the end user needs to participate heavily in the definition of the operational qualification protocol, since it is this protocol that will determine if the software is meeting the basic user requirements that have been recorded in the URS. Finally, the performance qualification is the responsibility of the pharmacometrician, since this testing will determine whether the software system is functioning within the business/technical needs of the end user.
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3.3.6
VALIDATION OF SOFTWARE FOR PHARMACOMETRIC ANALYSIS
Information Technology Infrastructure and Organization
The interaction between clinical pharmacology and the IT resources, whether internal or externally by contract, is of paramount importance when considering the productivity and analytical capability of the pharmacometrician. Once the pharmacometrician has clearly stated her/his software needs, based on the URS, the actual definition of the overall systems that will be used to service the needs of the pharmacometrician must be decided. The clinical pharmacology area is one that is subject to the same regulatory demands of other clinical areas such as biostatistics or clinical data entry and validation, yet it is a discipline that utilizes scientific methodologies that are closer in reality to discovery and preclinical drug development. That is, modeling of data, attempting to establish the validity of a hypothesis based on accumulated data and prior scientific knowledge, is the process employed. While some variables and covariates may be well defined and understood, in many cases, especially in population-dependent studies, it is the “expert system” of the scientist’s experience and ability that unveils the critical issues surrounding pharmacokinetic, pharmacodynamic, or toxicity effects. To this extent, clinical pharmacology, while considered a development activity, is more closely akin to a discovery process. In general, drug discovery areas such as medicinal chemistry, target identification and structure, or preclinical assay development are not subject to the regulatory information system requirements that clinical pharmacology must follow. Hence, the IT support structures normally associated with areas such as clinical development, which more typically involve electronic document control or clinical database management, need to be imbued with a technical understanding of the work of the pharmacometrician. Ideally, organizations should strive to identify a pharmacometrician, or other members of the scientific staff, with an interest in IT. Such individuals would not be “lost” to PK/PD research but rather would become an invaluable asset in communicating the specific needs of clinical research. Nevertheless, the usual situation is one where a computer engineer needs to be educated as to the needs of the pharmacometrician. If, in fact, that engineer is not devoted entirely to the clinical area, the probability of a successful interaction will decrease dramatically. It is improbable that an IT individual can successfully support business software dealing with human resources, purchasing, and customer relationship management while at the same time understanding the needs of the pharmacometrician to create a model (perhaps by generating new code to do so), automate the analysis of a large number of studies, and then generate a PK report using completely separate tools. The ultimate goal of the IT staff assigned to the clinical development groups must be customer service. In order to increase the throughput and accuracy of the pharmaceutical realization process, the clinical development area must be given the IT resources and attention necessary to determine the efficacy and safety of the subject chemical entity. If such resources are available, they must be encouraged to serve as advocates for the pharmacologists they support to IT management and corporate management. Once again, it is up to the pharmacometrician to establish a relationship with the IT support structure that encourages this attitude on the part of the IT support personnel. The clinical pharmacology group should at the least identify an individual within their organization as the liaison with IT. That individual should be included in meetings held within IT regarding policy,
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infrastructure, and resources so that there is a clear source of information regarding IT infrastructure. In smaller company situations, where the pharmacometrician may in fact be the IT support person, clearly the ability to influence and participate in the IT support process is critical. 3.3.7
The Buy or Build (or Buy and Build) Decision
Most clinical organizations take for granted that their key capabilities lie in PK/PD rather than in software development. Nevertheless, some of the most utilitarian tools used in pharmacometrics have been written by pharmacologists. While many of these have arisen from university endeavors, several commercial packages began as “skunks works,” projects that have evolved into private companies that provide valuable tools. The point is that it is probable that, within current organizations, there are individuals who are certain that a better mousetrap is within reach. Furthermore, most commercially available tools have specifically enabled programming and automation tools (such as WinNonlin) or interfaces (such as S-Plus) where custom development is not only possible but more than likely would have a positive impact on the drug development process. The issue to consider when going down the “home-brewed” or automation road is that the software development process, as discussed in Chapter 2, must be well documented before the development process begins. Just as in the documentation of the clinical drug evaluation processes with SOPs, the clinical group must become familiar with and document (via SOPs) the software development life cycle process (SDLC) as it will be implemented within the group; or possibly, if in a large company, it is plausible that the IT group already has established SOPs for SDLC. The starting point for commercial and internal development is exactly the same—the user requirements document. In addition, a functional requirements document should be written, outlining the details of how the specific functionality of the software (i.e., subroutines used, function of a dropdown menu, a panel of buttons, or what to type in as a command to execute some specific task) needs to be written. Unit test plans—how the person or persons generating the code will determine if well defined subunits of code are working—must also be generated. Finally, installation of the code (or the macros, if a commercial tool is being automated) and operational and performance qualification should be performed in the same manner as any commercial application would be. Another important consideration is that some type of source code control system must be identified and employed so that the history of the software development process, as well as any change control process after installation, may be documented. Alternatively, if the decision is made to buy only commercially available software, or only commercially developed add-ons or automation scripts, then the pharmacometrician needs to participate in the key processes used to evaluate the vendor. The occurrence of key quality failures in widely used software has been previously documented (14). Therefore, the pharmacometrician should be intimately involved in the vendor audit process. If the vendor is not performing the quality assurance procedures just outlined for internal development, the cost (both in quality and accuracy of future work) will be in jeopardy. As discussed later in the section on validation documentation, the ability to state what the vendor’s quality processes are will mitigate the need to perform functional software testing at the same level that has already been executed by the vendor’s quality assurance group.
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3.4
VALIDATION PROCESS
The primary issues surrounding the documentation of the validation process and the role of the pharmacologist are now addressed. The main areas of concern to the end user are: 1. 2. 3. 4.
What documentation needs to be created? What is the order or priority of the document generation process? Who is responsible for various sections of the documentation? What are the content, format, and future (i.e., what are the maintenance requirements.) of the documents?
We also note that the approach of quantity over quality of documentation is preferred by many organizations. This course of action will lead to a general disillusionment with the validation process and should be avoided at all costs. A good installation qualification should fit on one or two sides of a page (three with boilerplate, if that is unavoidable). A user requirements specification (URS) may be no more than a paragraph. While it is plausible that a URS may turn out to be several hundred pages for an internally developed repository system, that would be an exception rather than the rule. If an FDA inspector arrives with a method to determine the mass of your documentation, rather than with a desire to view the processes that such paperwork documents, it will be time to find other sources of advice on validation. 3.4.1
User Requirements Specification (URS)
The key document to be generated solely by the pharmacometrician is the user requirements specification. The URS simply states the purpose of the software. It is quite worthwhile to note what the URS is not. For example, the business process (or scientific process) that is being addressed by the software should have already been addressed in the SOPs relevant to the department and should not appear in the URS. That is, describing what you do and the generic manner in which you do it is the fodder for a good set of SOPs, not the requirements that outline a particular tool that you wish to use. Furthermore, the system requirements—software/ hardware availability, user access, recovery of data—are not elements of the user requirements. In many cases these elements should be covered by SOPs of the IT group or in a separate systems requirements document. The document containing these requirements (see Section 3.4.2) is a document generated by the IT engineer in collaboration with the pharmacometrician. What the URS should contain are the features and functionality of the software tool that are required by the pharmacometrician to accomplish the business/scientific objectives at hand. As an example, Table 3.3 shows some generic user specifications that might be included in the URS for a statistical package. Note that the URS is generic; it could fit SAS as well as S-Plus or GraphPad. One need not list all the features and functionality of the package being implemented, but the key features that one will use (and therefore test) must be included.
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TABLE 3.3 Possible Elements of a User Requirements Specification Requirement Data input formats Data output formats Other data I/O Data manipulation Reporting Statistics Graphics Automation or customization
3.4.2
Description Must be able to import .xls, .txt, SAS transport files, etc. Must be able to export to .xls, .txt, SAS transport files, etc. ODBC or JDBC connectivity Ability to subset, merge, transpose, or filter (using multiple criteria) data Ability to integrate output into word processing software Descriptive, hypothesis testing, multivariate, nonparametric, etc. Charts, plots, and user designed graphs Standard or vendor-designed programming, macro or automation language
System Specification
The generation of this document requires the interaction of the pharmacometrician with the IT group. This document reflects additions to practices already defined in IT and clinical SOPs. That is, IT should already have (see Section 3.3) SOPs that set forth: 1. 2. 3. 4. 5.
Data/system backup procedures for validated systems. User access (Logical Security) for validated systems access. Physical access (Security) for validated systems. Disaster recovery plans for validated systems. Installation requirements for hardware and operating systems used for validated systems.
The main purpose of this document is to reflect the input of the IT professional regarding the system requirements, usually as documented by the vendor of the software. This clearly involves assuring that the information technologist has become familiar with the vendor’s installation procedure and requirements. The selection or identification of hardware cannot proceed until the IT professional has ensured that the appropriate processor, disk space, and communication interfaces exist as required by the vendor. Furthermore, it is plausible that the software system will need software interfaces (such as database connectivity), which require additional resources. This document might reflect the “coupling” of validated systems (i.e., obtaining clinical data from a clinical database for PK/PD analysis). The document also might reflect the use of a hardware system (i.e., a user workstation) already validated for use as the target system of the new application. The primary responsibility for generation of this document lies with the IT group. Since these are the experts at systems implementation, it behooves the pharmacometrician to engage these resources early and to try as best as possible to understand the constraints, both technical and political, under which these colleagues may be operating.
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3.4.3
VALIDATION OF SOFTWARE FOR PHARMACOMETRIC ANALYSIS
Validation Plan
As discussed earlier, the validation plan is a document that should be well defined by existing SOPs written within either the Quality Assurance Organization or the IT group. Unless the project is quite unique, the validation plan should follow the same general course. The user requirements and system specifications documents will impact this, of course, but the generation of this document, which also should be quite brief, should be straightforward. The key elements of the validation plan are given in Table 3.4. Once again, sections of the validation plan regarding security, access, and so on may be better covered in SOPs that are resident with the IT group, rather than being specified for each validation. 3.4.4 Installation Qualification (IQ), Operational Qualification (OQ), and Performance Qualification (PQ) One analogy used to describe the function of these three processes is the installation, operation, and performance of an overhead projector. The IQ involves receiving the projector from the vendor, unpacking it according to the vendor’s instructions, setting it on a cart or table (consistent with the vendor’s requirements regarding how strong a table or cart), putting together the projector arm and head, plugging the projector into an electric socket, and turning on the power. Assuming the projector comes on, following the vendor’s recommended shut-down procedure (i.e., making sure the cooling fan stays on for some fixed time after the bulb has been turned off) successfully would imply a successful IQ. The OQ would then involve turning on the projector, taking a standard, widely used transparency, placing it on the glass, adjusting the height, distance, and focus of the projector and projector head, and so on until a satisfactory image is obtained on an acceptable image surface (i.e., a screen). Finally, the PQ would require the same type of process as in the OQ and that could be successfully performed on the end user’s specific viewgraphs, be they color, black and white, multiple levels, or partially blocked. At any stage in these processes, there needs to be an ordered set of steps and tests that verify the successful execution of the intended actions. This is referred to as the test script (or test plan). For each document that describes one of these qualification processes, the test script is the main functional part of the document. The “boilerplate,” describing the project, referencing the validation plan, and documenting who is executing the qualification, could be as small as a single page (or even paragraph).
TABLE 3.4 Key Elements of a Validation Plan Overview of the system Definition of the system: user requirements, system requirements, and software description Organization and responsibilities of the validation team (usually the end user, and the information technology and quality assurance members) Outline of timeframe for implementation Documentation: URS, SRS, IQ, OQ, PQ, change control, acceptance
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The IQ test script clearly needs to be generated by the individual who is responsible for the installation. Usually this will be an information technologist. Note that this individual must be familiar with the software (i.e., the individual has read the installation instructions and warnings provided by the vendor) and will have to document the steps that will be followed. It is highly recommended that while this area is not the specific responsibility of the end user, the pharmacometrician would be wise to become familiar with the installation process. Vendors often provide a good deal of related information that the information technologist either misses or does not understand the analytical implications of, and it is best if the end user asks as many questions as possible before the IQ is generated. The OQ test script may be written by either the information technologist or the pharmacometrician. The ideal situation is for this to be a collaborative effort. One highly positive result of such a collaboration is that the OQ test can turn into the best “software training” experience that both individuals will have for the particular software involved. The need to actually read the vendor’s user manual in order to generate meaningful test scripts can lead to an unanticipated benefit of identifying software capabilities that were previously unknown. There is one school of thought that claims that all of the features, functionality, buttons, menus, and so on of a particular software package must be exercised in order to successfully test the operation of the software. In general, this is extreme. Almost all of the software that is purchased has been quality assured by the vendor. Assuming that the software vendor has been audited (or that the customary use of the software by industry and regulatory agencies is widespread and it is generally agreed that the software is of high quality) and there is documented vendor evidence of functional testing, the OQ can generally be executed based on recommended tests provided by the vendor, in addition to statistical testing provided by standards organizations (15). In Section 3.5 some specific examples are outlined. The OQ also needs to test some of the system specification requirements. These include security (i.e., authorized users can access the software, unauthorized users cannot), recovery (the software can be reinstalled and critical data recovered from original media or backup systems in the event of either accidental or disasterrelated events), and boundary tests (e.g., maximum users allowed, maximum data set size). Finally, the PQ will execute some of the same tests performed in the OQ, but using the particular functionality (noncompartmental and/or compartmental models, statistical tests, graphics, integrations, fitting) of the software that is particular to the uses of the pharmacometrician. These tests should be performed on actual data or at least data that is indicative of that analyzed during the PK/PD analysis. As in any well designed scientific investigation, this will involve the use of estimation (perhaps using other tools), boundary testing, and calibration with data standards in order that the pharmacometrician is confident that the result is “reasonable.” Clearly, this is the domain of the scientist. The IT and quality assurance resources may be available to help with execution of a performance qualification, but ultimately the design responsibility for these tests lies with the pharmacometrician. If a particular type of analysis is common (i.e., bioavailability–bioequivalence–drug interaction), often the vendor or provider (i.e., for software originating in academic venues) of the software has a canonical example for the particular type of analysis. This may
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be used as a model for the PQ testing, with both the vendor’s and the end user’s data being used to validate the algorithmic approach and result. In the case of new types of analysis that are developed after the software has been qualified, it is incumbent upon the scientist to follow a similar process of estimation or validation in order to document the validity of the approach. This can be documented in a separate SOP (particularly if the approach becomes widely used within the organization) rather than “requalifying” the software. A change control document can be issued to indicate to the quality assurance group that “new” functionality is being employed within the software package. During the next “requalification” or upgrade of the software package, the new analytical approach can be integrated into a revised PQ. 3.5
INFORMATIVE EXAMPLES
The outlines of typical test scripts for an IQ, OQ, and PQ appear in Appendixes 3.1–3.3. Note that the “boilerplate” for these documents will be determined to a great extent by the quality assurance group. The main points to note are that each script provides a general outline of what will be tested, a statement as to responsible parties, and then a sequence of test steps that must be followed, verified, and documented as to anomalies or unexpected results. In some steps figures are called for. These are location specific and have not been reproduced here. If there are unexplained events that cannot be corrected and documented during the test, it may be necessary to regenerate the test script (maintaining the original test data as an appendix to the validation documents) and retest. We now discuss useful starting points for operational qualification scripts for various PK/PD analysis tools. ADAPT II There are several sample tests provided by D’Argenio and Schumitzky (16). The Fortran compiler is a key software subsystem for both ADAPT II and NONMEM. In this regard it is best to have a separate qualification for the installation of the compiler, followed by careful review of the expected output provided in Ref. 16. Older Fortran f77 compilers may show discrepancies that can only be resolved by implementing the most current versions of the f77 compiler. NONMEM For the operational qualification, a careful review of the parameters discussed in Section 2.9 of the NONMEM Users Guide—Part III (17) should be performed. These values should be identified and set during the IQ and tested properly during the OQ. The specific examples provided for NONMEM’s PREDPP, NM-TRAN, and associated library subroutines are highly recommended as a starting point for the OQ. The Phenobarbital and Theophylline data files provided with the software (18) offer even more extensive testing appropriate (with modification) for a PQ. The output is well documented and individuals may seek to modify or parameterize the examples for their needs. S-Plus The validate( ) function (19) is particularly appropriate for use during the OQ. As with NONMEM, the system settings and systemwide user parameter files (20) should be identified and implemented during the IQ. As with any statistical package, it is highly recommended that appropriate statistical analyses from standards organizations (15) be utilized as appropriate to the organization.
INFORMATIVE EXAMPLES
67
SAS The SAS Institute support organization has recently published resources for both validation (21) and actual IQ/OQ guidance (22). This should certainly be reviewed as a plausible starting point for the OQ. The same advice regarding analyses from standards organizations (15) applies. Please note that both S-Plus and SAS provide a wide range of capabilities for model creation, data analysis, presentation, and interfacing to databases and other software. It is incumbent on the user community to identify, at least initially, the capabilities that will be utilized in the user requirements documentation. Such software-specific capability should then be appropriately tested in the OQ. WinNonlin WinNonlin comes with a well documented set of exercises (23) that can be used as the basis of an OQ. These exercises, as well as several additional tests, can also be obtained as an automated test package (24). This is quite useful if several installations of the product are being validated on independent workstations, or if it is anticipated that frequent requalification (due to product updates or releases) will be needed. There is a significant initial investment of time that must be made in order to learn and utilize the automated package. There may also be issues surrounding whether automated test software in itself must be qualified. Nevertheless, for those organizations willing to invest the effort, such testing is without a doubt more rigorous (and quite rapid) once implemented. As with other tools, WinNonlin provides the capability to create new model strategies with user-generated code as well as the ability to highly automate software functionality (25). As with other tools, the ability to write software for new modeling strategies adds the requirement that a SDLC process be in place for the pharmacometrician to adhere to. Other PK/PD Software Kinetica, WinNonMix, and Trial Simulator® are examples of other software tools that may be utilized within PK/PD organizations. Each of these products provide example tests (26–28) that may form the basis of the OQ. In many circumstances, it will be difficult to anticipate the full range of use of some tools. Nevertheless, the vendor documentation generally provides a wide range of examples of functionality, which can be incorporated into an OQ. Repository Systems Several software systems (29–31) (PKS®, EP2®, SAS Drug Development) have been released in the last several years, which enable the pharmacometrician to store PK/PD data, analyze such data in several ways, and then perform various reporting tasks (including data/results mining) across a wide variety of projects, studies, and so on. While each of these products have virtues and weaknesses, the fundamental issue that must be addressed by the clinical pharmacology community prior to considering the use of a specific system is: How does this software fit our current processes? Many times the need to answer this question leads to a major effort to define just what the current processes are! These systems require a high degree of organizational discipline and structure around the concept of metadata. That is, what data do we use to describe the models, data, analysis results, and reporting variables that are critical to our organization? The important point to recognize here is that software systems such as “repository” systems are considered “enterprise” software. The implementation is not customizable to an individual’s requirements or a department’s needs. The architecture of the software is the vendor’s “impression” of how a clinical pharmacology effort
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VALIDATION OF SOFTWARE FOR PHARMACOMETRIC ANALYSIS
may be organized. This “impression” may have no connection to your current processes; it may, especially if one’s organization was the model used when the software was architected, reflect your processes exactly. Under any circumstances, the implementation of such a package requires a large effort to identify processes, especially processes between groups such as data management, biostatistics, quality assurance, and clinical pharmacology, before considering individual software systems. Recent analyses of enterprise software have characterized this effort as the “organizational capital” (32) that must be expended in addition to the resources for “capital equipment and software expense.” Once a system is chosen, the implementation team needs to recognize that the fundamental way they work will be changed. The rewards may be tremendous, but the road to implementation may be long and arduous.
3.6
SUMMARY
This chapter is a brief attempt to aid the pharmacometrician in understanding how “quality” standards need to be applied to research and development activities involving software tools. Specifically, the needs of the ethical pharmaceutical industry are addressed, but one could argue that the ability to document such activities is critical in any industry. In the chapters that follow, several specific analytical approaches to numerous problems in clinical pharmacology are discussed. If the software tools utilized in these creative and important analytical methodologies are properly installed, validated, and supported, the quality and throughput of the pharmaceutical realization process will be assured, for both the development teams and the regulatory agencies involved in the process of ethical drug discovery and development.
REFERENCES 1. T. Stokes, R. C. Branning, K. G. Chapman, H. J. Hambloch, and A. J. Trill, Good Computer Validation Practices: Common Sense Implementation. Interpharm Press, Buffalo Grove, IL, 1994. 2. R. Chamberlain, Computer Systems Validation for the Pharmaceutical and Medical Device Industries. Alaren Press, Libertyville, IL, 1994. 3. FDA, http://www.fda.gov/ora/compliance_ref/part11/. 4. FDA: Help Us Plot Act Two for Part 11. http://www.bio-itworld.com/news/051904_ report5155.html. 5. ISO, http://www.iso.org/iso/en/iso9000–14000/iso9000/iso9000index.html. 6. ANSI, Document ANSI/IEEE 730-2002, Software Quality Assurance Plans; also see www.ansi.org. 7. IEEE, Document 1061–1998, IEEE Standard for a Software Quality Metrics Methodology; also see http://standards.ieee.org/catalog/olis/se.html. 8. G. J. Myers, Software Reliability: Principles and Practices. Wiley, Hoboken, NJ, 1976. 9. US EPA, Good Automated Laboratory Practices: Recommendations for Ensuring Data Integrity in Automated Laboratory Operations with Implementation Guidance. US Environmental Protection Agency, Research Triangle Park, NC, 1990.
REFERENCES
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10. FDA, 21CFR Part 312: New Drug Product Regulations; Final Rule. Department of Health and Human Services, Washington, DC; Fed Reg 3/19/1987, Part VII. 11. FDA, Guide to Inspection of Computerized Systems in Drug Processing. US Government Printing Office, Washington, DC, 1983-381-166:2001, 1983. 12. FDA, Technical Report on Software Development Activities: Reference Materials and Training Aids for Investigators, US Government Printing Office, Washington, DC, July 1987. 13. An internet search for “21 CFR 11” or “21CFR11” using www.google.com retrieved 9550 results at the time of this writing. 14. B. D. McCullough and B. Wilson, On the accuracy of statistical procedures in Microsoft Excel 2000 and Excel XP. Comput Statistics Data Analysis 40:713–721 (2002). 15. NIST Statistical Reference Datasets, www.itl.nist.gov/div898/strd. 16. D. Z. D’Argenio and Alan Schumitzky, ADAPT II Pharmacokinetic/Pharmacodynamic Systems Analysis Software User’s Guide to Release 4. The Biomedical Simulations Resource, University of Southern California, Los Angeles, Mar. 1997, Chapters 2, 5, and 6. 17. A. J. Boeckmann, S. L. Beal, and L. B. Sheiner, NONMEM Users Guide—Part III. NONMEM Project Group, University of California at San Francisco, May 1999, pp. 25–30. 18. A. J. Boeckmann, S. L. Beal, and L. B. Sheiner, NONMEM Users Guide—Part III. NONMEM Project Group, University of California at San Francisco, May 1999, pp. 102–104. 19. S-Plus 6 for Windows Programmer’s Guide. Insightful Corporation, Seattle, WA, July 2001, Chapter 24. 20. S-Plus 5 for Unix Installation and Maintenance Guide. Data Analysis Products Division, Insightful Corporation, Seattle, WA, May 1999, Chapters 4 and 7. 21. SAS Corp., http://support.sas.com/rnd/migration/planning/validation/. 22. SAS Corp, http://support.sas.com/rnd/migration/resources/SAS_IQOQ_SOP. pdf. 23. WinNonlin® Getting Started Guide Version 4.1. Pharsight Corporation, Mountain View, CA, 2003, Chapter 3. 24. Pharsight, http://www.pharsight.com/products/winnonlin/wnl_validation.php. 25. WinNonlin® Users Guide Version 4.1. Pharsight Corporation, Mountain View, CA, 2003, Chapters 11, 12, 19, and 20. 26. Kinetica® R4.0 User Manual V4.0. InnaPhase Corp., Philadelphia, PA, 2001, pp. 256–301. 27. WinNonMix® Getting Started Guide Version 2.0.1. Pharsight Corporation, Mountain View, CA, 2000, Chapter 2. 28. Pharsight® Trial SimulatorTM Getting Started Guide Version 2.1.2. Pharsight Corporation, Mountain View, CA, 2001, Chapter 4. 29. Pharsight Corp, http://www.pharsight.com/products/pks_suite/. 30. Innaphase Corp, http://www.innaphase.com/products_epseries.html. 31. SAS Corp, http://www.sas.com/industry/pharma/develop/index.html. 32. Federal Reserve Bank of Minneapolis Research Department Staff Report 291, http://research.mpls.frb.fed.us/research/sr/sr291.pdf.
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APPENDIX 3.1
SAMPLE INSTALLATION QUALIFICATION
Document Number: IQ00-00001 Document Location: QA Document Owner: Technolgist, Info
Install Qualification: S-Plus 6.0.4
Test Version: 1.0 Reference Document(s) #: SV00-00001
Purpose (This defines the purpose for the current test(s). It may refer to previous tests and/or documents.) This test is conducted to qualify the installation of software components used for S-Plus 6.0.4. These are components of the S-Plus 6.0.4 software and are used to verify that a client interface may access S-Plus 6.0.4. These tests will also be used to establish a baseline for future testing. Scope (This defines the scope of the test(s). It is a written description of the scope and restrictions of the test.) Testing is done to prove the S-Plus 6.0.4 software has been correctly installed. This test does not prove the Installation Process; instead, it proves that the end result of the process was successful based on software functionality. Validating the result of the installation implicitly proves the success of the process. Test Requirements Testing is done to prove the following: 1. Verify the S-Plus 6.0.4 installation 1.1. Verify the my_server_name server is started 1.2. Verify the /home/splus6 directory and permissions 1.3. Verify the S-Plus scripts have been copied to my_server_name: /usr/local/bin 1.4. Verify the file and directory listing for the /home/splus6 directory 1.5. Verify that /usr/local/bin/Splus and /usr/local/bin/Splus invoke S-Plus 6.0.4 2. Verify Terminal/HOST Client/Server Interface 2.1. Verify UNIX Server login from Telnet client (terminal) 2.1.1. Verify Security 2.1.1.1. Bad Username 2.1.1.2. Good Username, Bad Password 2.2. Verify Logout from Telnet client (terminal) 2.3. Verify UNIX Server login from X-windows client 2.3.1. Verify Security 2.3.1.1. Bad Username 2.3.1.2. Good Username, Bad Password 2.4. Verify Logout from X-windows client Test Prerequisites (A list of requirements necessary to run the test. These can include environment reqs (e.g., NT, with MS Office loaded), tester reqs (e.g., tester is trained in operating MS Office), software reqs (e.g., test assumes xyz software to have already been loaded), or other reqs (e.g., paper documents for scanning).) The following conditions must be met before testing:
SAMPLE INSTALLATION QUALIFICATION • • •
71
The environment is ready to test. Tester is trained in basic usage of UNIX and S-Plus 6.0.4. Testing files are prepared and put in place.
Test Instructions (Gives any special instructions to the tester. Tester is assumed to be qualified to execute test.) For each test condition in the Testing Table, the Tester must initial each graybar section when completed regardless of success or nonsuccess. If the test condition has been met and Expected Result is the same as the actual result (the result of executing the test condition), then the test is successful and must be marked as OK in the OK column. If the test condition has not been met, or the Expected Results are not exactly the same as the actual results, then the Tester must stop, report the deviation in the Comments column, and report the occurrence to the Test Coordinator. At that time, the Test Coordinator will make a judgment on whether or not the test can be continued. In the event that the deviation is considered acceptable and that the test can continue, the Test Coordinator must log the event, any workarounds necessary, and initial the Tester’s comments (this may be done on the script if there is room). In the event that the deviation is not acceptable, then the test must stop. Test Tables Test tables show: 1. 2. 3. 4. 5. 6.
Line Number: Allows reference for tracking anomalies and errors. Test Condition: Defines the test. Expected Results: Defines what should happen. Any deviation is an error. OK: Were the expected results met? Initials: Tester proof of execution. Comments: Allows information about the test condition.
Signoffs (Signoffs for Document Owner and Author with printed name and date spaces.)
Author (By signing this the author of this document agrees that the document is complete and accurate.) Printed Name
Signed Name
Responsible
Date: MM/DD/YYYY
Author
Owner (By signing this the owner of this document agrees that the document is complete and accurate.) Printed Name
Signed Name
Responsible Owner
Date: MM/DD/YYYY
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Test Condition
Verify the home directory by typing pwd.
2.
System responds with user’s home directory, /home/”username”
Login prompt and shell prompt observed following successful login as in Figure 1.
Type cd /home Type ls –al | grep splus
The output is: drwxr-x— 12 splus clinical 1024 Mar 23 11:06 splus Indicating the splus directory exists and is accessible (r-x) by the group clinical
Type cd /usr/local/bin Type ls –l
The output is: -rwxr-x— 1 root clinical 4655 Sep 5 2004 Splus -rwxr-x— 1 root clinical 4655 Mar 22 14:32 Splus
Type cd /home/splus6 to return to the /home/splus6 directory. Type ls –lR | pg
Compare the output of this command to Figure 2. ALL of the files listed in Figure 2 must be in the output. There will be files output by the command that are not displayed in Figure 2.
1.4. Verify the file and directory listing for the /home/splus6 directory
4.
1.3. Verify the S-Plus scripts have been copied to my_server_name: /usr/local/bin
3.
1.2. Verify the /home/splus6 directory and permissions
As a valid my_server_name user and member of the clinical group, open a Telnet client and log on to my_server_name server.
1.
5.
Expected Results
1.1. Verify the my_server_name server is started and accessible
1. Verify the S-Plus 6.0.4 Installation
#
OK
Init
Comments
73
The operating system command prompt should return after the first command. The second command should produce the same result as Step 6. Type q() to exit S-Plus.
Type q() at the > prompt. Type /usr/local/bin/Splus
7.
9.
8.
As a valid my_server_name user and member of the pharmaco group, open a Telnet client and log on to my_server_name server using a good user name and an invalid password.
2.1.1.2. Good username, bad password
As a valid my_server_name user and member of the pharmaco group, open a Telnet client and log on to my_server_name server using a bad user name, good password.
2.1.1.1. Bad username
2.1.1. Verify Security
Login is rejected as in Figure 5.
Login is rejected as in Figure 4.
2.1. Verify UNIX Server login from Telnet client (terminal)
2. Verify Terminal/HOST Client/Server Interface
The output file should be similar to Figure 3. There should be no indication of errors. The line: “Working data will be in /home/wolk/MySwork” should indicate the current “username” instead of “wolk.”
Type cd $HOME Type /usr/local/bin/Splus
6.
1.5. Verify that /usr/local/bin/Splus and /usr/local/bin/Splus invoke S-Plus 6.0.4
74
As a valid my_server_name user and member of the pharmaco group, open a Telnet client and log on to my_server_name server using a good user name and a valid password. Then type exit.
Login is successful as in Figure 1. Telnet client screen is as in Figure 6 and keyboard is unresponsive following exit. Logout is successful.
Login to configured X-windows client using a valid username / invalid password.
2.3.1.2. Good username, bad password
Configure X-windows client to connect to my_server_name server (using eXodusPowerPC, select “Connections/ Connection Manager/Sample XDM Session/Edit” Change “Mode” to “Query” set “Host” to “my_server_name” and “Title” as “my_server_name IQ”; Click connect using an invalidusername.
13.
Login to configured X-windows client as a valid my_server_name user and member of the pharmaco group, log on to my_server_name server using a good user name and a valid password. Then select Mouse Button 3; Click on Exit.
2.4. Verify Logout
12.
11.
2.3.1.1. Bad username
2.3.1. Verify Security
Valid login produces X-windows, Openwindows environment similar to Figure 9. Selecting Mouse button 3 yields window similar to that in Figure 10. Selecting exit yields confirmation screen similar to Figure 11. Clicking on Exit causes windows environment to disappear.
Login is not allowed as in Figure 8.
Login is not allowed as in Figure 7. If using the Common Desktop Environment instead of Openwindows, the windows may appear differently than in the figures below, but the functionality should be the same.
2.3. Verify UNIX Server login from X-windows client
10.
2.2. Verify Logout
SAMPLE OPERATION QUALIFICATION
75
Signoffs (Signoffs for Document Tester and Test Coordinator with printed name and date spaces)
Tester (By signing this the tester of this document agrees that the test has been completed and is accurate.) Printed Name
Signed Name
Responsible
Date: MM/DD/YYYY
Tester
Test Coordinator (By signing this the tester of this document agrees that the test has been completed and is accurate.) Printed Name
Signed Name
Responsible
Date: MM/DD/YYYY
Test Coordinator
APPENDIX 3.2
SAMPLE OPERATION QUALIFICATION
Document Number: OQ00–00001 Document Location: IS Validation
Operation Qualification: S-Plus 5.1
Document Owner: Metrician, Pharmaco
Appendix A
Test Version: 1.0 Reference Document(s) #: SV0000001
Purpose (This defines the purpose for the current test(s). It may refer to previous tests and/or documents.) This test is conducted to qualify the operation of S-Plus 5.1. These tests verify the proper operation of the S-Plus 5.1 software as well as the security of the software as far as user access to the data and executable programs. It also tests the backup and recovery of data and executables. These tests will also be used to establish a baseline for future operational testing. Scope (This defines the scope of the test(s). It is a written description of the scope and restrictions of the test.) Testing is done to prove the S-Plus 5.1 is operational in accordance with the manufacturer’s criteria. Testing is also done to prove that data and executable software access is limited to authorized users and only up to the number of available licenses. Testing is also done to verify the backup and restoration of selected data and executable files.
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VALIDATION OF SOFTWARE FOR PHARMACOMETRIC ANALYSIS
Test Requirements Testing is done to prove the following: 1. Verify the S-Plus 5.1 Operation (these tests are from the S-Plus 2000 Programmer’s Guide, Chapter 25) 1.1. Execute validate(); the complete validation test suite 1.2. Verify the validate function code 1.3. Execute the anova test suite in verbose mode to demonstrate an individual test 1.4. Execute the regress test suite in verbose mode, return and examine the Boolean result 2. Verify the S-Plus Data Files May only be accessed by Authorized Users 2.1. Verify /home/splus directory is not accessible to unauthorized users 2.2. Verify the /home/splus and /home/“user” subdirectories are (not) writeable by (unauthorized) authorized users 2.3. Verify the /home/splus and /home/“user” subdirectories are readable by the authorized group 3. Verify the S-Plus program may be started only by authorized users 3.1. Verify S-Plus or S-Plus 5 may be started only by authorized users 3.2. Verify that the license limit may not be exceeded by authorized users. 4. Verify that a tape backup of data and executable files may be selectively restored Test Prerequisites (A list of requirements necessary to run the test. These can include environment reqs (e.g., NT, with MS Office loaded), tester reqs (e.g., tester is trained in operating MS Office), software reqs (e.g., test assumes xyz software to have already been loaded), or other reqs (e.g., paper documents for scanning).) The following conditions must be met before testing: • The environment is ready to test. • Tester is trained in basic usage of UNIX and S-Plus 5.1. • Testing files are prepared and put in place. Test Instructions (Gives any special instructions to the tester. Tester is assumed to be qualified to execute test.) For each test condition in the Testing Table, the Tester must initial each graybar section when completed regardless of success or nonsuccess. If the test condition has been met and Expected Result is the same as the actual result (the result of executing the test condition), then the test is successful and must be marked as OK in the OK column. If the test condition has not been met, or the Expected Results are not exactly the same as the actual results, then the Tester must stop, report the deviation in the Comments column, and report the occurrence to the Test Coordinator. At that time, the Test Coordinator will make a judgment on
SAMPLE OPERATION QUALIFICATION
77
whether or not the test can be continued. In the event that the deviation is considered acceptable and that the test can continue, the Test Coordinator must log the event, any workarounds necessary, and initial the Tester’s comments (this may be done on the script if there is room). In the event that the deviation is not acceptable, then the test must stop. Test Tables Test tables show: 1. 2. 3. 4. 5. 6.
Line Number: Allows reference for tracking anomalies and errors. Test Condition: Defines the test. Expected Results: Defines what should happen. Any deviation is an error. OK: Were the expected results met? Initials: Tester proof of execution. Comments: Allows information about the test condition.
Signoffs (Signoffs for Document Owner and Author with printed name and date spaces.)
Author (By signing this the author of this document agrees that the document is complete and accurate) Printed Name
Signed Name
Responsible
Date: MM/DD/YYYY
Author
Owner (By signing this the owner of this document agrees that the document is complete and accurate) Printed Name
Signed Name
Responsible Owner
Date: MM/DD/YYYY
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Test Condition
Expected Results
OK
Type: /usr/local/bin/Splus
At the > prompt, type: validate()
2.
3.
4.
At the > prompt, type: sink(“/home/username/validate_code”) where username is the user name observed in the pwd command of Step 1. At the > prompt, type: validate
1.2. Verify the validate function code
As a valid S-Plus user (Section 7.2 of Operation Qualification) open a Telnet or X-windows client and log on to my_server_name server. Verify the current directory is /home/”username” by typing pwd.
1.
System responds with output of Figure 2.
System responds (the test may take a few minutes) with the results depicted in Figure 1.
S : Copyright Lucent Technologies, Inc. Version 5.1 Release 1 for Sun SPARC, SunOS 8 : 2009 Working data will be in /home/myname/MySwork > Except username is user of Step 1. Instead of “myname”
System responds with: License Warning : S-Plus license expires Thu Apr 20 23:59:59 2009 S-Plus: Copyright © 1988, 1999 Insightful, Inc.
Successful login to my_server_name server pwd command returns: /home/”username” (for ksh or sh) /export/home/”username” (for csh) where username is the tester’s login name
1.1. Execute validate(); the complete validation test suite
1. Verify the S-Plus 5.1 Operation (these tests are from the S-Plus 2000 Programmer’s Guide, Chapter 25)
#
Init
Comments
79
System responds with output of Figure 3.
System responds with output of Figure 4.
System responds with the Boolean result “T” for True: [1] T >
In the first Telnet session, type the following: sink(“/home/username /regress_test”) where username is the user name observed in the pwd command of Step 1. At the > prompt, type: Regrspass < -validate(“regress”, verbose = T) In the second Telnet session type: pg regress_test
At the > prompt, type: sink() Regrspass
7.
1.4. Execute the regress test suite in verbose mode, return and examine the Boolean result
In the first Telnet session, type the following: sink(“/home/username /annova_test”) where username is the user name observed in the pwd command of Step 1. At the > prompt, type: validate(“anova”, verbose = T) In the second Telnet session type: pg anova_test
6.
5.
1.3. Execute the anova test suite in verbose mode to demonstrate an individual test
Open a second Telnet session as in Step 1 and Login to that session. In the second session type: pg validate_code
80
Type: cd /home/splus
9.
Access is denied, message is: ksh: /home/splus: permission denied or for csh: /home/splus: Permission denied
Successful login to my_server_name pwd command returns: /home/“username” where username is the tester’s login name
Type: mkdir /home/splus/test_dir
Type: mkdir /home/wolk/test_dir
Type: exit
As a valid S-Plus user (Section 7.2 of Operation Qualification), open a Telnet session and log on to my_server_name.
10.
11.
12.
13.
Successful login to my_server_name
Successful logoff from my_server_ name server
System responds with: mkdir: Failed to make directory “/home/wolk/test_dir”; Permission denied
System responds with: mkdir: Failed to make directory “/home/splus/test_dir”; Permission denied
2.2. Verify the /home/splus and /home/“user” subdirectories are (not) writeable by (unauthorized) authorized users
As a valid my_server_name user who is NOT listed as a S-Plus user (Section 7.2 ofOperation Qualification), open a Telnet session and log on to my_server_ name (The password for the “nmtest” user may be obtained from the test coordinator). Verify the current directory is /home/“username” by typing pwd.
8.
2.1. Verify the /home/splus directory is not accessible to unauthorized users
2. Verify the S-Plus Data Files May only be accessed by Authorized Users
81
Type: pwd
Type: mkdir /home/splus/test_dir
Type: mkdir $HOME/test_dir ls –l $HOME | grep test_dir
Type: rmdir $HOME/test_dir ls –l $HOME | grep test_dir
Type: mkdir /home/wolk/test_dir
14.
15.
16.
17.
18.
System responds with: mkdir: Failed to make directory “/home/wolk/test_dir”; Permission denied (Group members directories are NOT writeable by other group members.)
The ls –l command returns nothing after the directory has been removed.
A new directory named test_dir has been written by the current user to their HOME directory, as demonstrated by the response of the ls –l command: drwxr-xr-x 2 wolk pharmaco 512 Mar 20 09:13 test_dir (Note that the user who logged on in step 13 should be listed instead of “wolk”)
NOTE THAT AUTHORIZED USERS HAVE READ AND EXECUTE BUT NOT WRITE PERMISSION in /home/splus
System responds with: mkdir: Failed to make directory “/home/splus/test_dir”; Permission denied
pwd command returns: /home/“username”
82
Type: cd /home/wolk pwd
Type: cat README
20.
21.
File contents of the README file appear as in Figure 6. Group members files are readable by other group members unless read permission is specifically removed.
Output of the pwd command is: /export/home/wolk (csh) /home/splus (ksh or sh)
File contents of the Copyright file appear as in Figure 5.
Type: /usr/local/bin/Splus5 The S-Plus command was tested in Step 2. That step may be repeated here if so desired.
Type exit
22.
23.
Successful logoff from my_server_name.
System responds with: License Warning : S-Plus license expires Thu Apr 20 23:59:59 2000 S-Plus : Copyright (c) 1988, 1999 MathSoft, Inc. S : Copyright Lucent Technologies, Inc. Version 5.1 Release 1 for Sun SPARC, SunOS 5.5 : 1999 Working data will be in /home/wolk/MySwork > Except username is user of Step 1. Instead of “wolk.”
3.1. Verify S-Plus or S-Plus 5 may be started only by authorized users
3. Verify the S-Plus program may be started only by authorized users
Type: cd /home/splus cat Copyright
19.
2.3 Verify the /home/splus and /home/“user” subdirectories are readable by authorized group
83
Type: /usr/local/bin/Splus
Type: /usr/local/bin/Splus5
Type exit
25.
26.
27.
Successful logoff from my_server_name.
System responds with, for kshell: ksh: /usr/local/bin/Splus: cannot execute for c-shell: /usr/local/bin/Splus: Permission denied
System responds with, for kshell: ksh: /usr/local/bin/Splus: cannot execute for c-shell: /usr/local/bin/Splus: Permission denied
Successful login to my_server_name pwd command returns: /home/“username” where username is the tester’s login name.
29.
28.
Type exit
Repeat Steps 1 and 2 above. For the remainder of this test, a second valid user of S-Plus must be utilized. CONTACT THE TEST COORDINATOR IN ORDER TO CONTINUE THIS TEST! Have the test coordinator repeat Steps 1 and 2 above.
Successful logoff from my_server_name.
System output should resemble that in Figure 7. The “username” should be the tester’s username. The test coordinator is not allowed to execute S-Plus due to the license limit: Terminating S Session: No S-Plus licenses available
3.2 Verify that the license limit may not be exceeded by authorized users.
As a valid my_server_name user who is NOT listed as a S-Plus use r (Section 7.2 ofOperation Qualification), open a Telnet session and log on to my_server_name (The password for the “nmtest” user may be obtained from the test coordinator). Verify the current directory is /home/“username” by typing pwd.
24.
84
Login to my_server_name as kroot (this testmust be executed by a System Administrator or an Application Administrator with System Administrator Privilege).
Rename the file /usr/local/bin/Splus5 to Splus5_orig; Repeat Step 22.
Obtain a backup tape of the my_server_name server. Restore the file/usr/local/bin/Splus5 to it’s original location. Repeat Step 22.
30.
31.
32.
Except username is “kroot” instead of “wolk”
System response is: License Warning : S-Plus license expires Thu Apr 20 23:59:59 2000 S-Plus : Copyright © 1988, 1999 MathSoft, Inc. S : Copyright Lucent Technologies, Inc. Version 5.1 Release 1 for Sun SPARC, SunOS 5.5 : 1999 Working data will be in /home/wolk/MySwork >
An error message occurs, the executable shell script is not found.
Login successful.
4. Verify that a tape backup of data and executable files may be selectively restored
85
Delete the restored version; rename the original version to S-Plus 5. Verify that the directory /home/kroot exists and that the file /home/kroot/ MySwork exists within this directory. Type: ls –l /home/kroot | grep MySwork
Type: rm -r /home/kroot/MySwork ls –l /home/kroot/MySwork | grep MySwork
Repeat Step 22.
Repeat Step 34. Obtain a backup tape of the my_server_name /home/kroot directory. Restore the directory /home/kroot/MySwork to it’s original location. Repeat Step 22.
Type exit.
33.
34.
35.
36.
37.
Successful logoff from my_server_name.
S-Plus generates the message in Figure 9, which since the MySwork directory was restored, does not have the “Creating data. . .” message of Figure 8.
S-Plus generates the message in Figure 8, which, because the MySwork directory was deleted in Step 34, includes: Creating data directory for chapter /home/kroot/MySwork
Output of ls –l /home/kroot | grep MySwork Is empty.
Output of ls –l /home/kroot | grep MySwork: drwxr-xr-x 3 root other 512 Sep 5 1999 MySwork
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VALIDATION OF SOFTWARE FOR PHARMACOMETRIC ANALYSIS
Signoffs (Signoffs for Document Tester and Test Coordinator with printed name and date spaces:)
Tester (By signing this the tester of this document agrees that the test has been completed and is accurate) Printed Name
Signed Name
Responsible
Date: MM/DD/YYYY
Tester
Tester (By signing this the tester of this document agrees that the test has been completed and is accurate) Printed Name
Signed Name
Responsible
Date: MM/DD/YYYY
Tester
Test Coordinator (By signing this the tester of this document agrees that the test has been completed and is accurate) Printed Name
Signed Name
Responsible
Date: MM/DD/YYYY
Test Coordinator
APPENDIX 3.3
SAMPLE PERFORMANCE QUALIFICATION
Document Number: PQ00–00001 Document Location: Quality Assurance Document Owner: Metrician, Pharmaco
Performance Qualification: S-Plus 5.1
Test Version: 1.0 Reference Document(s) #: SV00-00001
Appendix A
Purpose (This defines the purpose for the current test(s). It may refer to previous tests and/or documents.) This test is conducted to qualify the performance of S-Plus 5.1. These tests verify the proper operation of specific, commonly used features of the S-Plus system software. Scope (This defines the scope of the test(s). It is a written description of the scope and restrictions of the test.) Testing is done to prove that specific, commonly used features of S-Plus 5.1 are operational in accordance with the end users needs. The data sets are obtained from the National Institute of Standards and Technology (Ref. 6) and academic reference texts (Ref. 7)
SAMPLE PERFORMANCE QUALIFICATION
87
Test Requirements Testing is done to prove the following: 1. Verify that S-Plus 5.1 performs the NIST StRD Analysis of Variance calculations to within 3 signficant digits 1.1. Test ANOVA with dataset SiRstv, that is, low degree of stiffness, low replicates per cell 1.2. Test ANOVA with dataset AtmWtAg, that is, average degree of stiffness, low replicates per cell 1.3. Test ANOVA with dataset SmLs06, that is, average degree of stiffness, high replicates per cell 2. Verify that S-Plus 5.1 performs the NIST StRD Linear Regression calculations to within 3 signficant digits 2.1. Test Linear Regression with dataset Norris, Low difficulty linear 2.2. Test Linear Regression with dataset NoInt1, Average difficulty linear 2.3. Test Linear Regression with dataset Filip, High difficulty polynomial 3. Verify that S-Plus 5.1 performs the NIST StRD Non-linear Regression calculations to within 3 signficant digits 3.1. Test Nonlinear Regression with dataset Misra1a, Lower difficulty exponential 3.2. Test Nonlinear Regression with dataset Kirby2, Average difficulty rational 3.3. Test Nonlinear Regression with dataset MGH09, Higher difficulty rational 4. Verify that S-Plus 5.1 performs a General Additive Model with Gaussian error Distribution and identity link problem correctly to 3 significant digits
Test Prerequisites (A list of requirements necessary to run the test. These can include environment reqs (e.g., NT, with MS Office loaded), tester reqs (e.g., tester is trained in operating MS Office), software reqs (e.g., test assumes xyz software to have already been loaded), or other reqs (e.g., paper documents for scanning).) The following conditions must be met before testing: • The environment is ready to test. • Tester is trained in basic usage of UNIX and S-Plus 5.1. • Testing files are prepared and put in place. Test Instructions (Gives any special instructions to the tester. Tester is assumed to be qualified to execute test.) For each test condition in the Testing Table, the Tester must initial each graybar section when completed regardless of success or nonsuccess. If the test condition has been met and Expected Result is the same as the actual result (the result of executing the test condition), then the test is successful and must be marked as OK in the OK column. If the test condition has not been met, or the Expected Results are not exactly the same as the actual results, then the Tester must
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VALIDATION OF SOFTWARE FOR PHARMACOMETRIC ANALYSIS
stop, report the deviation in the Comments column, and report the occurrence to the Test Coordinator. At that time, the Test Coordinator will make a judgment on whether or not the test can be continued. In the event that the deviation is considered acceptable and that the test can continue, the Test Coordinator must log the event, any workarounds necessary, and initial the Tester’s comments (this may be done on the script if there is room). In the event that the deviation is not acceptable, then the test must stop. Test Tables Test tables show: 1. 2. 3. 4. 5. 6.
Line Number: Allows reference for tracking anomalies and errors. Test Condition: Defines the test. Expected Results: Defines what should happen. Any deviation is an error. OK: Were the expected results met? Initials: Tester proof of execution. Comments: Allows information about the test condition.
Signoffs (Signoffs for Document Owner and Author with printed name and date spaces.)
Author (By signing this the author of this document agrees that the document is complete and accurate) Printed Name
Signed Name
Responsible
Date: MM/DD/YYYY
Author
Owner (By signing this the owner of this document agrees that the document is complete and accurate) Printed Name
Signed Name
Responsible Owner
Date: MM/DD/YYYY
89
Test Condition
Expected Results
OK
As a valid S-Plus user (Section 7.2 of Performance Qualification) open a Telnet or X-windows client and log on to my_server_name server. Verify the current directory is /home/“username” by typing pwd.
Create a subdirectory in the /home/“username”. Type the following: mkdir splus_pq cd splus_pq pwd
Type the following: Splus CHAPTER
Type the following: mkdir datasests cp ∼username/splus_pq/ datasets/* datasets cd datasets ls -og
1.
2.
3.
4.
System responds with: total 476 -rw-r–r– 1 683 May 26 15:04 AtmWtAg.csv -rw-r–r– 1 3594 May 26 15:04 Filip.csv -rw-r–r– 1 1922 May 26 15:04 Kirby2.csv -rw-r–r– 1 130 May 26 15:04 MGH09.csv -rw-r–r– 1 255 May 26 15:04 Misra1a.csv -rw-r–r– 1 166 May 26 15:04 Misra1a1.csv -rw-r–r– 1 80 May 26 15:04 NoInt1.csv -rw-r–r– 1 388 May 26 15:04 Norris.csv -rw-r–r– 1 292 May 26 15:04 SiRstv.csv -rw-r–r– 1 216126 May 26 15:04 SmLs06.csv -rw-r–r– 1 182 May 26 15:04 Wampler1.csv
System responds with: Creating data directory for chapter. Splus5 chapter splus_pq initialized.
System responds to pwd command with: /home/“username”/splus_pq
Successful login to my_server_name server pwd command returns: /home/“username” (for ksh or sh) /export/home/“username” (for csh) where username is the tester’s login name.
1.1. Test ANOVA with dataset SiRstv, that is, low degree of stiffness, low replicates per cell
1. Verify that S-Plus 5.1 performs the NIST StRD Analysis of Variance calculations to within 3 signficant digits
#
Init
Comments
90
Type the following: cd .. Splus
Type: SiRstv < - importData(file = “datasets/SiRstv.csv”) instr1 < -factor(SiRstv[,“Instrument”]) resist1 < -SiRstv[,“Resistance”] SiRstv.anova.1 |t|) (Intercept) −0.2623 0.2328 −1.1267 0.2677 x 1.0021 0.0004 2331.6058 0.0000 Residual standard error: 0.8848 on 34 degrees of freedom Multiple R-squared: 1 F-statistic: 5436000 on 1 and 34 degrees of freedom, the p-value is 0
2.1. Test Linear Regression with dataset Norris, Low difficulty linear
2. Verify that S-Plus 5.1 performs the NIST StRD Linear Regression calculations to within 3 signficant digits
Review the information in Figure 1.3a.
19.
94
Review the information in Figure 2.1a.
Review the results of Step 22 and Step 23 against the certified results of Figure 2.1c.
Review the data set in the problem of Figure 2.1d against the data imported in Step 22. (Type Norris at the S-Plus prompt to see the data set.)
25.
26.
27.
The data imported matches the data in Figure 2.1d
Results match within 3 significant figures.
Figure 2.1a is the specific model information relevant to the Norris data set. Initial the INIT column if read and understood. Figure 2.1b is the data set information. It is provided for reference.
Figure 2.0a is the general background information for the NIST StRD Linear Regression test. Initial the INIT column if read and understood. Figure 2.0b is a summary of the data sets referred to in Figure 2.0a.
System responds with: Analysis of Variance Table Response: y Terms added sequentially (first to last) Df Sum of Sq Mean Sq F Value Pr(F) x 1 4255954 4255954 5436386 0 Residuals 34 27 1
28.
Type: NoInt1 < - importData(file = “datasets/NoInt1.csv”) NoInt1.lm.1 < -lm(y∼-1+x, data = NoInt1) summary(NoInt1.lm.1)
System responds with: Call: lm(formula = y ∼ −1 + x, data = NoInt1) Residuals: Min 1Q Median 3Q Max −5.207 −2.521 0.1653 2.851 5.537 Coefficients: Value Std. Error t value Pr(>|t|) x 2.0744 0.0165 125.5000 0.0000
2.2. Test Linear Regression with dataset NoInt1, Average difficulty linear
Review the information in Figure 2.0a.
Type the following: anova(Norris.lm.1)
24.
23.
95
Review the information in Figure 2.2a.
Review the results of Step 28 and Step 29 against the certified results of Figure 2.2c.
Review the data set in the problem summary of Figure 2.2d against the data imported in Step 28. (Type NoInt1 at the S-Plus prompt to see the data set.)
30.
31.
32.
The data imported matches the data in Figure 2.2d
Results match within 3 significant figures.
Figure 2.2a is the specific model information relevant to the NoInt1 data set. Initial the INIT column if read and understood. Figure 2.2b is the dataset information. It is provided for reference.
System responds with: Analysis of Variance Table Response: y Terms added sequentially (first to last) Df Sum of Sq Mean Sq F Value Pr(F) x 1 200457.7 200457.7 15750.25 0 Residuals 10 127.3 12.7
33.
Type: Filip < - importData(file = “data sets/Filip.csv”) Filip.lm.1 < -lm(y∼poly(x,10), data = Filip) attach(Filip) poly.transform(poly(x,10), coef(Filip.lm.1))
System responds with: x∧0 x∧1 x∧2 x∧3 x∧4 −1467.49 −2772.18 −2316.371 −1127.974 −354.4782 x∧5 x∧6 x∧ 7 x∧8 −75.1242 −10.87532 −1.062215 −0.06701912 x∧9 x∧10 −0.002467811 −4.029625e-05
2.3. Test Linear Regression with data set Filip, High difficulty polynomial
Type the following: anova(NoInt1.lm.1)
29.
Residual standard error: 3.568 on 10 degrees of freedom Multiple R-Squared: 0.9994 F-statistic: 15750 on 1 and 10 degrees of freedom, the p-value is 0
96
35.
34.
Type the following: anova(Filip.lm.1)
Type: summary(Filip.lm.1)
Value 0.8496 0.4614 −0.0868 −0.0827 0.0967 0.0175 −0.0617 0.0067 0.0340 −0.0155 −0.0150
Std. Error 0.0004 0.0033 0.0033 0.0033 0.0033 0.0033 0.0033 0.0033 0.0033 0.0033 0.0033
t value Pr(> |t|) 2297.8525 0.0000 137.8103 0.0000 −25.9256 0.0000 −24.6980 0.0000 28.8958 0.0000 5.2127 0.0000 −18.4251 0.0000 1.9912 0.0503 10.1625 0.0000 −4.6397 0.0000 −4.4942 0.0000
System responds with: Analysis of Variance Table Response: y Terms added sequentially (first to last) Df Sum of Sq Mean Sq F Value Pr(F) poly(x, 10) 10 0.2423916 0.02423916 2162.44 Residuals 71 0.0007959 0.00001121
ALL VALUES ARE ZERO.
Correlation of Coefficients
Residual standard error: 0.003348 on 71 degrees of freedom Multiple R-Squared: 0.9967 F-statistic: 2162 on 10 and 71 degrees of freedom, the p-value is 0
(Intercept) poly(x, 10)1 poly(x, 10)2 poly(x, 10)3 poly(x, 10)4 poly(x, 10)5 poly(x, 10)6 poly(x, 10)7 poly(x, 10)8 poly(x, 10)9 poly(x, 10)10
Coefficients:
System responds with: Call: lm(formula = y ∼ poly(x, 10), data = Filip) Residuals: Min 1Q Median 3Q −0.008804 −0.002176 4.502e−05 0.002029 Max 0.007096
0
97
Review the information in Figure 2.3a.
Review the results of Step 33, Step 34 and Step 35 against the certified results of Figure 2.3c.
Review the data set in the problem of Figure 2.3d against the data imported in Step 33. (Type Filip at the S-Plus prompt to see the data set.)
37.
38.
39.
The data imported matches the data in Figure 2.3d.
Results match within 3 significant figures.
Figure 2.3a is the specific model information relevant to the Filip data set. Initial the INIT column if read and understood. Figure 2.3b is the data set information. It is provided for reference.
System responds with: NULL
40.
Type: Misra1a < - importData(file = “data sets/Misra1a1.csv”) The following is a single command: Misra1a.nls.1 < -nls(y∼b1*(1-exp (-b2*x)),data = Misra1a,start = list (b1 = 250,b2 = 0.0005)) summary (Misra1a.nls.1)
Correlation of Parameter Estimates: b1 b2 −0.999
Residual standard error: 0.101879 on 12 degrees of freedom
System responds with: Formula: y ∼ b1 * (1 − exp(- b2 * x)) Parameters: Value Std. Error t value b1 2.38942e+02 2.70701e+00 88.2680 b2 5.50156e-04 7.26687e-06 75.7075
3.1. Test Nonlinear Regression with data set Misra1a, Lower difficulty exponential
3. Verify that S-Plus 5.1 performs the NIST StRD Nonlinear Regression calculations to within 3 signficant digits
Type the following: detach(2,Filip)
36.
98
Review the information in Figure 3.1a.
Review the results of Step 40 against the certified results of Figure 3.1c.
Review the data set in the problem summary of Figure 3.1d against the data imported in Step 40. (Type Misra1a at the S-Plus prompt to see the data set.)
42.
43.
44.
45.
Type: Kirby2 < - importData(file = “data sets/Kirby2.csv”) The following is a single command: Kirby2.nls.1 < nls(y∼(b1+b2*x+b3*x∧2)/ (1+b4*x+b5*x∧2),data = Kirby2,start = list(b1 = 1.5,b2 = −0.15,b3 = 0.0025, b4 = −0.0015,b5 = 0.00002)) summary(Kirby2.nls.1)
t value 19.0304 −33.8189 62.0242 −29.2578 107.6260 Residual standard error: 0.163545 on 146 degrees of freedom
System responds with: Formula: y ∼ (b1 + b2 * x + b3 x∧2)/(1 + b4 * x + b5 * x∧2) Parameters: Value Std. Error b1 1.67448e+00 8.79894e−02 b2 −1.39272e−01 4.11818e−03 b3 2.59610e−03 4.18562e−05 b4 −1.72421e−03 5.89314e−05 b5 2.16647e−05 2.01297e−07
*
The data imported matches the data in Figure 3.1d.
Results match within 3 significant figures.
Figure 3.1a is the specific model information relevant to the Misra1a data set. Initial the INIT column if read and understood. Figure 3.1b is the data set information. It is provided for reference.
Figure 3.0a is the general background information for the NIST StRD Nonlinear Regression test. Initial the INIT column if read and understood. Figure 3.0b is a summary of the data sets referred to in Figure 3.0a.
3.2. Test Nonlinear Regression with data set Kirby2, Average difficulty rational
Review the information in Figure 3.0a.
41.
99
Review the results of Step 45 against the certified results of Figure 3.2c.
Review the data set in the problem summary of Figure 3.2d against the data imported in Step 45. (Type Kirby2 at the S-Plus prompt to see the data set.)
47.
48.
The data imported matches the data in Figure 3.2d.
Results match within 3 significant figures.
Figure 3.2a is the specific model information relevant to the Kirby2 data set. Initial the INIT column if read and understood. Figure 3.2b is the data set information. It is provided for reference.
49.
Type: MGH09 < - importData (file = “data sets/MGH09.csv”) The following is a single command: MGH09.nls.1 < nls(y∼(b1*(x∧2++b2*x))/ (x∧2+x*b3+b4),data = MGH09, start = list(b1 = 0.25,b2 = 0.39, b3 = 0.415,b4 = 0.39)) summary (MGH09.nls.1)
System responds with: Formula: y ∼ (b1 * (x∧2 + + b2 * x))/ (x∧2 + x * b3 + b4) Parameters: Value Std. Error t value b1 0.192800 0.0114361 16.858900 b2 0.191355 0.1963730 0.974443 b3 0.123030 0.0808503 1.521700 b4 0.136101 0.0900403 1.511560
3.3. Test Nonlinear Regression with data set MGH09, Higher difficulty rational
Review the information in Figure 3.2a.
46.
Correlation of Parameter Estimates: b1 b2 b3 b4 b2 −0.896 b3 0.803 −0.974 b4 0.569 −0.793 0.903 b5 0.847 −0.984 0.962 0.756
100
Review the results of Step 49 against the certified results of Figure 3.3c.
Review the data set in the problem summary of Figure 3.3d against the data imported in Step 49. (Type MGH09 at the S-Plus prompt to see the data set.)
51.
52.
0.4400
The data imported matches the data in Figure 3.3d.
Results match within 3 significant figures.
Figure 3.3a is the specific model information relevant to the MGH09 data set. Initial the INIT column if read and understood. Figure 3.3b is the data set information. It is provided for reference.
0.5250 0.9890
Parameter Estimates: b2 b3
53.
Type: stack < -data.frame(cbind (stack.x,stack.loss)) (note: stack.x and stack.loss are S-Plus built-in data sets) The following is a single command: names(stack) < c(“AirFlow”,“waterTemp”, “AcidConc”,“Loss”)
System responds with: Call: gam(formula = Loss ∼ s(AirFlow) + s(waterTemp) + s(AcidConc), data = stack, control = gam.control(bf.maxit = 50)) Degrees of Freedom: 21 total; 8.00097 Residual Residual Deviance: 67.79171
4. Verify that S-Plus 5.1 performs a General Additive Model with Gaussian error Distribution and identity link problem correctly to 3 significant digits
Review the information in Figure 3.3a.
50.
Correlation of b1 b2 −0.7440 b3 0.0885 b4 −0.7640
Residual standard error: 0.00662792 on 7 degrees of reedom
101
Type: q()
Type exit.
57.
Review the results of Steps 53 and 54 against the results of Figure 4 (from Ref. 7)
55.
56.
summary(stack.gam.1)
54.
The following is a single command: stack.gam.1 < -gam (Loss ∼ s(AirFlow)+s (waterTemp)+s(AcidConc), control = gam.control(bf.maxit = 50),data = stack) stack.gam.1
Successful logoff from my_server_name.
Successful exit from S-Plus.
Results match within 3 significant figures.
System responds with: Call: gam(formula = Loss ∼ s(AirFlow) + s(waterTemp) +s(AcidConc), data = stack, control = gam.control(bf.maxit = 50)) Deviance Residuals: Min 1Q Median 3Q Max −3.089759 −1.604992 0.2439517 0.876497 1 3.967667 (Dispersion Parameter for Gaussian family taken to be 8.472936) Null Deviance: 2069.238 on 20 degrees of freedom Residual Deviance: 67.79171 on 8.00097 degrees of freedom Number of Local Scoring Iterations: 1 DF for Terms and F-values for Nonparametric Effects Df Npar Df Npar F Pr(F) (Intercept) 1 s(AirFlow) 1 3 0.934405 0.4676402 s(waterTemp) 1 3 3.171167 0.0851828 s(AcidConc) 1 3 0.975555 0.4507614
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VALIDATION OF SOFTWARE FOR PHARMACOMETRIC ANALYSIS
Signoffs (Signoffs for Document Tester and Test Coordinator with printed name and date spaces:)
Tester (By signing this the tester of this document agrees that the test has been completed and is accurate) Printed Name
Signed Name
Responsible
Date: MM/DD/YYYY
Tester
Test Coordinator (By signing this the tester of this document agrees that the test has been completed and is accurate) Printed Name
Signed Name
Responsible Test Coordinator
Date: MM/DD/YYYY
CHAPTER 4
Linear, Generalized Linear, and Nonlinear Mixed Effects Models FARKAD EZZET and JOSÉ C. PINHEIRO
4.1
INTRODUCTION
Biopharmaceutical research often involves the collection of repeated measures on experimental units (such as patients or healthy volunteers) in the form of longitudinal data and/or multilevel hierarchical data. Responses collected on the same experimental unit are typically correlated and, as a result, classical modeling methods that assume independent observations do not lead to valid inferences. Mixed effects models, which allow some or all of the parameters to vary with experimental unit through the inclusion of random effects, can flexibly account for the within-unit correlation often observed with repeated measures and provide proper inference. This chapter discusses the use of mixed effects models to analyze biopharmaceutical data, more specifically pharmacokinetic (PK) and pharmacodynamic (PD) data. Different types of PK and PD data are considered to illustrate the use of the three most important classes of mixed effects models: linear, nonlinear, and generalized linear. Linear mixed effects (LME) models express the response variable as a linear function of both the fixed effects and the random effects, with an additive withinunit error, see Laird and Wase (1) or Searle et al. (2) for a good review of methodology. The frequentist approach to LME models is generally likelihood-based, with restricted maximum likelihood (REML) being the preferred method of estimation (3). Nonlinear mixed effects (NLME) models extend LME models by allowing the response to be expressed as a nonlinear function of the parameters plus a withinunit error term. Much of this work in biopharmaceutical research began in the 1970s, pioneered by Sheiner and Beal (4). Exact likelihood estimation is generally not feasible, as the marginal distribution of the response cannot be expressed in closed form. Approximate likelihood methods are used instead, with different degrees of accuracy and computational intensity having been proposed in the
Pharmacometrics: The Science of Quantitative Pharmacology Edited by Ene I. Ette and Paul J. Williams Copyright © 2007 John Wiley & Sons, Inc.
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LINEAR, GENERALIZED LINEAR, AND NONLINEAR MIXED EFFECTS MODELS
literature; see Davidian and Giltinan (5) for a good review of some of these methodologies. A more detailed account of the theory and application of LME and NLME models, especially under S-Plus (6) can be found in work by Pinheiro and Bates (7). Research to produce computationally efficient and accurate approximate likelihood methods for NLME models is still quite active. Generalized linear mixed models (GLMMs) provide another type of extension of LME models aimed at non-Gaussian responses, such as binary and count data. In these models, conditional on the random effects, the responses are assumed independent and with distribution in the exponential family (e.g., binomial and Poisson) (8). As with NLME models, exact likelihood methods are not available for GLMMs because they do not allow closed form expressions for the marginal distribution of the responses. Quasilikelihood (9) and approximate likelihood methods have been proposed instead for these models. Mixed effects models under a Bayesian framework have been widely studied and used with the use of Markov chain Monte Carlo methods (10). These methods have gained particular popularity as complex problems became easily formulated using the WinBUGS software (11). See Congdon (12) for an extensive coverage of topics and examples and implementation in WinBUGS. In this chapter we investigate and illustrate the use of LME and NLME models, as well as GLMMs using algorithms implemented in the S-Plus functions lme, nlme, and glme, respectively. We attempt to demonstrate that, even under fairly complex hierarchical, correlated data structures, the existing algorithms are capable of properly estimating the underlying parameters (fixed effects and variance–covariance components), thus providing reliable unbiased inference. We begin by considering a simple PK dose proportionality (DP) study in which subjects receive an experimental drug to evaluate if the increase in exposure is proportional to dose. We examine the problem in two ways: (a) using an exposure metric, for example, area under the concentration–time curve (AUC), which leads to an LME model; and (b) using the concentration data directly, which requires the use of an NLME model. Concentration data are simulated using different hierarchical random effects structures. We then extend the DP example to include a clinical response and explore a pharmacokinetic/pharmacodynamic (PK/PD) NLME model. Collapsing the clinical response into a binary measure allows the illustration of GLMMs. Common features among the three different classes of models and their implementation within the S-Plus environment come into light during the analysis of the examples: in particular, the syntax for defining the fixed and random effects in the models, as well as methods for extracting estimates from fitted objects. All data sets discussed in this chapter are fictitious: that is, they are generated by simulation. The reader is encouraged to experiment with the code provided in Appendix 4.1 to explore alternative scenarios.
4.2
PHARMACOKINETIC DOSE PROPORTIONALITY PROBLEM
Consider a dose proportionality study in which each subject is to receive a number of doses, usually two or more, of an experimental drug to evaluate if exposure increases proportionally with dose. We adopt a crossover design and, to keep things simple, assume that issues related to carryover, period, and sequence effects, as well
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as subject by dose interactions (13), are of no concern in this example. The S-Plus function sim.dp.mult, included in Appendix 4.1, generates drug concentrations (C) at times (t) following drug administration, C(t), according to the single dose oral one-compartment PK model: C( t ) =
F ⋅ dose ⋅ Ka − ke t [e − e − kat ] V( Ka − Ke )
(4.1)
Typically, two types of error are recognized: (a) measurement level error resulting from error in concentrations due to assays, time of measurement, and so on and (b) subject level error represented in the model by random effects, accounting for deviations in the PK parameters between subjects, that is, in absorption (Ka), elimination (Ke), and/or volume (V). V is usually expressed as V/F when the fraction of dose absorbed (F) is unknown. Formally, we may express C(t) as Cij(t) = f(qi, dosej, t) [1 + eij(t)], where Cij(t) and f(qi, dosej, t) are the measured and predicted concentrations for the ith subject at the jth dose at time t, respectively, and qi is the vector of PK parameters for the ith subject. Here, the intersubject variability in the PK parameters is assumed proportional. For instance, volume for the ith subject is defined as Vi = V exp(hi,V), where the random effects hi,V are independently distributed as N(0, cv·V). The prefix cv denotes coefficient of variation for V. The measurement error is assumed multiplicative, with the eij(t) independently distributed as N(0, cv·e). The functional form of f is determined by the type of PK model being considered; for the DP example we assume the one-compartment model described above. A third possible source of variation, accounting for deviations in the PK parameters within subject from period to period, often referred to as interoccasion (IO) variability, may also be incorporated in the PK model. For example, we may define Vij = V exp(hi,V + h′ij,V), where hi,V as before represents the intersubject random effect while h′ij,V represents the interoccasion random effect within subject, assumed independently distributed as N(0, cv·occ·V). The S-Plus data frame dp1 is generated by calling the function sim.dp.mult assuming strict dose proportionality and no IO variability, as illustrated below. Figure 4.1 shows a trellis display of the corresponding concentration–time profiles. dose 1 pack a day, male Not a smoker, female ≤1 pack a day, female >1 pack a day, female
Cumulative Probability 0.32 0.45 0.55 0.90 0.97 1.00
Now a random variate U with value 0.85 is simulated. Since 0.55 ≤ 0.85 < 0.90, the simulated value is a female who does not smoke.
33.7
SOFTWARE
Many different software packages exist for simulating clinical trials or PK/PD studies. Which one to use usually depends on the user. Some software is specifically designed to simulate PK/PD outcomes within the context of a clinical trial design, such as the Pharsight Trial Simulator. Then there are more general simulation languages with graphical user interfaces (GUIs), such as Simulink (Mathworks, Natick, MA). While GUIs are useful, they are often limited to what the developers include in the software, sometimes leaving the user caught between a rock and an upgrade. Programs such as NONMEM (Globomax LLC, Hanover, MD), WinNonlin (Pharsight Corp., Mountain View, CA), or ADAPT II (University of Southern California Biomedical Simulations Resource) were designed for model fitting and development, but can be used to simulate data. However, they must be used in conjunction with another program, like SAS (SAS Institute, Cary, NC) or S-Plus (Insightful Corp., Seattle, WA) as they cannot simulate data within the context of a clinical trial (i.e., they are dependent on the inputs supplied by the user and cannot simulate those inputs). Lastly, there are programming languages. While the most difficult to use, they are also the most flexible. These include Matlab (Mathworks, Natick, MA), Gauss (Aptech Systems, Maple Valley, WA), the IML procedure within SAS, and S-Plus. A common combination of software used to simulate clinical trials is the use of NONMEM and SAS or NONMEM and S-Plus, allowing the user to take advantage of NONMEM’s large library of PK models (see Figure 33.4).
33.8 APPLICATION OF M&S IN DRUG DEVELOPMENT AND REGULATORY REVIEW Much has already been written on the role of M&S in drug development (8, 10, 22–24). All of these reviews present examples of simulation in drug development. However, the examples reported in these reviews are often brief, lacking detail or insight. Rather than reprint what has already been reviewed, it was thought a more informative approach would be to present two case studies, one from the pharma-
APPLICATION OF M&S IN DRUG DEVELOPMENT AND REGULATORY REVIEW
Analyze Output
Simulate Dosing Inputs
Simulate Covariates
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Simulate PK-PD via NONMEM
Merge NONMEM Output with SAS Input
Simulate Clinical Design Features
Done within SAS or S-Plus
Done within SAS or S-Plus
FIGURE 33.4 Schematic illustrating how NONMEM and a more generalized software program, like SAS (SAS Institute, Cary, NC) or S-Plus (Insightful Corp., Seattle, WA), can be used to interact and simulate clinical trials.
ceutical industry and one from the regulatory point of view, in sufficient detail so as to illustrate the methodology. Darbepoetin alfa (Aranesp®) is in the class of recombinant human erythropoietin proteins that has greater in vivo potency than recombinant human erythropoietin (r-HuEPO) through the addition of N-linked sialic acid side chains on the amino acid backbone of the protein. Darbepoetin alfa stimulates the production of red blood cells (RBCs) for the treatment of chemotherapy-induced anemia and in patients with chronic renal failure. The recommended dose is different between the indications. In patients with chemotherapy-induced anemia, the recommended starting dose is 2.25 mg/kg administered as a weekly subcutaneous (SC) injection with the weekly dose adjusted to maintain a target hemoglobin (Hgb). The dose should be increased to 4.5 mg/kg if the Hgb increase is less than 1.0 g/dL after 6 weeks of treatment. The dose should be reduced 25% if either the Hgb exceeds 12 g/dL or the increase in Hgb is more than 2.0 g/dL. Dosing should be withheld until the Hgb falls to at least 12.0 g/dL if the Hgb exceeds 13.0 g/dL, at which point dosing should be reinitiated at ∼25% below the previous dose. Alternative dosing strategies have been proposed to take advantage of the longer half-life of darbepoetin alfa compared to r-HuEPO. One regimen of 3 mg/kg every 2 weeks was shown to be equally efficacious as 40,000 U r-HuEPO once weekly. Dosing a patient based on their body weight may add an additional layer of complexity to the dosing regimen that might not be necessary. Hence, Jumbe et al. (25) used CTS to determine whether a fixed dose of darbepoetin alfa (200 mg every 2 weeks) has the same outcome as a weight-based dose of darbepoetin alfa (3 mg/kg every 2 weeks). Data was pooled from three clinical trials (547 patients) studying the use of darbepoetin alfa in the treatment of chemotherapy-induced anemia. Serial PK and PD (Hgb concentrations) measurements were collected throughout the studies and merged into a single database along with patient-relative covariates. A population PK/PD model was developed that simultaneously modeled darbepoetin alfa
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concentration–time profiles, as well as Hgb–time profiles (see Figure 33.5). The CTS was then developed encompassing the following elements: • Model parameters and their associated between-subject variance estimates from the PK/PD model were fixed to their final values. • The patient population demographics (body weight and baseline Hgb concentrations) were defined based on observed baseline values observed across the three studies. The following study design elements were incorporated in the simulation: • Dosing was every other week for 12 weeks by SC administration based on either weight or fixed dose. • A transfusion was simulated if Hgb declined below 8.0 g/dL, whereby data from these patients were censored for the next 4 weeks. • Dosing was withheld if Hgb was ≥14.0 g/dL in women or ≥15.0 g/dL in men. • Censoring was randomly implemented to coincide with the censoring rates in the clinical trials. • Other protocol elements, such as definition of response or sampling for PD analysis every 2 weeks, were incorporated. Five thousand subjects per treatment arm were simulated. Summary statistics were used to define the mean Hgb concentration, along with its associated variabil-
Subcutaneous Dosing Compartment (F)
⎛ E C ⎞ k in ⎜1 + max ⎟ C + EC50 ⎠ ⎝
kt
kt
RBC Production
Lag Compartment (Lag-time, ka) RBC Production
kt
kt
kout
Hemoglobin
Pharmacodynamic Model
Central Compartment (CL, V)
Pharmacokinetic Model
FIGURE 33.5 Schematic of darbepoetin alfa PK/PD model. Darbepoetin alfa concentrations were modeled using a one-compartment model with absorption and lag time. Hgb concentrations were modeled using an indirect response model where darbepoetin alfa concentrations in the central compartment stimulate an increase in RBC production at the precursor stage via an Emax model. RBCs then mature at a constant rate (kt) and manifest as a change in Hgb. Sampling compartments are denoted with lines with solid circles as an arrowhead.
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ity, at each sampling point, which were then compared across treatment arms and to actual clinical data collected from 33 patients who were dosed subcutaneously every other week at the dose of 3 mg/kg darbepoetin alfa. The results are presented in Figure 33.6. No difference was observed in mean change from baseline in Hgb concentrations or in their associated variability over time between any of the groups. The proportions of subjects who were declared positive responders in the observed, simulated weight-based, and simulated fixed-dose treatment groups were 60%, 77%, and 76%, respectively. No statistical difference was observed between these percentages based on their overlapping confidence intervals. Lastly, the percentages of patients requiring transfusion in the weightbased and fixed-dose treatment groups was 21% and 22%, respectively, compared to 16% in the actual observed patients. The higher transfusion rates in the simulation treatment groups were suspected to be due to the objectivity of having a transfusion in clinical practice compared to the yes/no condition defined in the simulation. In summary, these results indicated that dosing per body weight would be an unnecessary complexity and that dosing could proceed based on a fixed dosing regimen, as long as dosing still proceeded based on individual titration to target Hgb values. Although the results of this analysis have not resulted in a change in the Dosage and Administration section in the Aranesp® product label, based on the results of the simulation, US Oncology, one of the largest oncology consortium in the United States, decided in 2003 to implement a fixed dosing regimen of 200 mg every other week provided Hgb concentrations are maintained at target levels. Thames et al. (26) did a retrospective chart review of US Oncology’s practice in 333 patients dosed under these guidelines (174 were previously treated with epoetin alfa and 156 were darbepoetin alfa naive) and found that a “darbepoetin alfa starting dosage of 200 mg every 2 weeks is effective in both naive patients and in those switched from epoetin
FIGURE 33.6 Scatterplot of mean Hgb change from baseline over time in the simulated treatment arms and in 33 patients who were dosed once every other week subcutaneously with 3 mg/mL darbepoetin alfa. Data are reported as the mean ± standard deviation. (Reprinted with permission from Ref. 25.)
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alfa.” They also found that transfusion rates were about 15% in both the switched and naive treatment groups, which was very near the 16% observed when dosing was done on a weight basis. This simulation illustrates the complexity involved in designing and implementing CTS and illustrates the practical benefit of using CTS in this case—a reduction in the complexity of the dosing regimen. M&S is not limited to use by the pharmaceutical industry. Frequently, regulatory authorities are using it, sometimes with far-reaching results. Recently, Genta, Inc. (Berkeley Heights, NJ) submitted to the Food and Drug Administration (FDA) a New Drug Application (NDA) for their oligonucleotide Genasense for use in combination with dacarbazine (DTIC) in the first line treatment of patients with advanced melanoma. Genta submitted results from a single, randomized Phase 3 study of DTIC versus DTIC plus Genasense in 771 patients. The primary endpoint was survival, which was not statistically significant (p = 0.18). However, the secondary endpoint of progression-free survival (PFS) did show a benefit from 49 days with DTIC to 74 days with DTIC plus Genasense (p = 0.0003) using a last-observation carried forward approach to handling censored data. Despite failing to meet their primary endpoint, Genta, Inc. was requesting approval of DTIC in combination with Genasense based on the secondary endpoint of PFS. In May 2004, the Oncologic Drug Advisory Committee (ODAC) met to discuss Genta’s application. Presented at the meeting were the results of a simulation performed by the FDA. The protocol defined disease-free progression as the date on which a scan or measurement was made, not the date of office visit. If patients in one group are assessed at a later date than patents in the other group, the documented date of disease-free progression would be longer in the former group, even if the two groups were equal. Because of a peculiarity in the study design due to drug administration, patients in two treatment groups were not assessed at the same time after the start of the trial. Assessment of patients in the combination arm were slightly delayed compared to patients in the control arm. Using Monte Carlo simulation, the FDA showed that the statistically significant results produced by the sponsor could be an artifact of the trial design and not due to any real drug effect. This simulation is important for a number of reasons. First, it represents the first real use of simulation by the FDA to question the results from a clinical study. Second, it is informative to listen to a comment made by one of the ODAC members after the FDA presented the results of their simulation. The member stated: “I am sure the 11 or so patients out there still in remission will be disturbed to know that modeling suggests that they shouldn’t be there.” Clearly, the simulation carried no weight with the physician. This comment highlights a particular problem with simulation in general and that is the credibility gap between modeling and simulation. To most, developing models is one thing, using them is another. The use of simulation in making decisions requires putting one’s faith in the model and the assumptions of the simulation. When the 16 members of ODAC were asked whether they believed the observed difference in progression-free survival was real, six members voted “yes.” When members were asked whether the difference in response rate and PFS for the combination of DTIC and Genasense versus DTIC alone provided substantial evidence of effectiveness that outweighed its potential for increased toxicity in chemotherapy naive patients with metastatic melanoma, only three members voted “yes.” Although the conclusion that members rejected Genta’s findings because
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of the results of the simulation cannot be definitely made (in fact, their responses would tend to indicate otherwise), it cannot be denied that the committee did in fact reject the sponsor’s conclusions and rejected their NDA.
33.9
SUMMARY
CTS sounds daunting. To think that you can simulate a process as complicated as a clinical trial simply sounds crazy. I personally believe that CTS has suffered because of the use of this phrase. But, simulation is nothing more than applied modeling. The principles involved are the same principles that have guided Monte Carlo simulations for the last 50 years. First, a model is needed. Second, the sources of variability in the model parameters must be understood, as does how those parameters are correlated. Third, once the system is defined, an input design must be defined. The process is simulated and the outputs are examined for averages, as well as extrema. There are no black boxes in this process—no smoke and mirrors as I have heard some people call it. The processes involved are the same as those used in other fields, including aerospace, manufacturing, and business. Many software packages aim to simplify the process by making the inner workings more consistent and credible across users, but it is still nevertheless important for the user to understand how these programs work and what to do when they cannot do what is needed. It remains to be seen what impact CTS will have on drug development—whether it will become an integral part of the process or will become a specialized tool to be utilized on a case-by-case basis. For the former to occur, M&S must gain greater exposure, not among pharmacokineticists and pharmacometricians, but among others impacted by its use like clinicians, project managers, and clinical research associates. While one likes to be recognized by peers, presenting the results of an analysis at meetings geared toward other pharmaceutical scientists, like the American Association of Pharmaceutical Scientists (AAPS), will not necessarily advance the cause of M&S in drug development. Pharmacokineticists must be willing to present their results at more clinically oriented meetings, like the American Society of Clinical Oncology, or more general research oriented programs like the Drug Information Association. Presenting at AAPS is like preaching to the choir; pharmacokineticists are aware of the methodology and want to implement it in their job, but in order to do so must convince the senior leadership in an organization. Only by making them aware of the methodology will M&S be accepted as an integral part of the drug development process.
REFERENCES 1. J. A. DiMasi, R. W. Hansen, and H. G. Grabowski, The price of innovation: new estimates of drug development costs. J Health Econ 835:1–35 (2003). 2. S. Arlington, S. Barnett, S. Hughes, and J. Palo, Pharma 2010: The Threshold of Innovation. International Business Consulting Services, London, UK, 2003. 3. P. Van Buskirk, Saving corporations millions: the benefits of modeling. PCAI, July/ August, 38 (2000).
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4. M. D. Hale, Using population pharmacokinetics for planning randomized concentration-controlled trial with a binary response, in The Population Approach: Measuring and Managing Variability in Response, Concentration, and Dose, L. Aarons, L. Balant, M. Danhof, M. Gex-Fabry, and U. Gundert-Remy (Eds.). European Commission, Belgium, 1997, pp. 227–235. 5. W. Kang and M. Weiss, Kinetic analysis of saturable myocardial uptake of idarubicin in rat heart: effect of doxorubicin and hypothermia. Pharm Res 20:58–63 (2003). 6. S. M. Eppler, D. L. Combs, T. D. Henry, J. J. Lopez, S. G. Ellis, J.-H. Yi, B. H. Annex, E. R. McCluskey, and T. F. Zioncheck, A target mediated model to describe the pharmacokinetics and hemodynamic effects of recombinant human vascular endothelial growth factor in humans. Clin Pharmacol Ther 72:20–32 (2002). 7. M. Weiss and W. Kang, Inotropic effect of digoxin in humans: mechanistic pharmacokinetic/pharmacodynamic model based on slow receptor binding. Pharm Res 21:231–236 (2004). 8. P. L. Bonate, Clinical trial simulation in drug development. Pharm Res 17:252–256 (2000). 9. A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin, Bayesian Data Analysis. Chapman & Hall, London, 1995. 10. N. H. G. Holford, H. C. Kimko, J. P. R. Monteleone, and C. C. Peck, Simulation of clinical trials. Annu Rev Pharmacol Toxicol 40:209–234 (2000). 11. J. Urquhart, Variable patient compliance as a source of variability in drug response, in Variability in Drug Response, G. T. Tucker (Ed.). Elsevier Science, Amsterdam, 1999, pp. 189–198. 12. P. Girard, T. Blaschke, H. Kastrissios, and L. B. Sheiner, A Markov mixed effect regression model for drug compliance. Stat Med 17:2313–2333 (1998). 13. H. Sun, E. O. Fadiran, E. I. Ette, and L. J. Lesko, Regulatory perspectives on clinical trial simulations, in Pharmacokinetics in Drug Development: Clinical Study Design and Analysis, P. L. Bonate and D. Howard (Eds.). AAPS Press, Alexandria, VA, 2004. 14. S. M. Ross, Simulation. Harcourt/Academic Press, San Diego, CA, 1997. 15. B. D. McCullough, Assessing the reliability of statistical software: Part II. Am Stat 53:149–159 (1999). 16. B. D. McCullough and B. Wilson, On the accuracy of statistical procedures in Excel 97. Comput Stat Data Anal 31:27–37 (1999). 17. B. D. McCullough, Assessing the reliability of statistical software: Part I. Am Stat 52:358– 366 (1998). 18. D. E. Knuth, The Art of Computer Programming: Seminumerical Algorithms. AddisonWesley, Reading, MA, 1981. 19. S. K. Park and K. W. Miller, Random number generators: good ones are hard to find. Commun ACM 31:1192–1201 (1988). 20. A. M. Law and W. D. Kelton, Simulation Modeling and Analysis. McGraw-Hill, New York, 2000. 21. G. E. P. Box and M. E. Muller, A note on the generation of random normal variates. Ann Math Stat 29:160–161 (1958). 22. M. Gauthier, Clinical trial simulation. Appl Clin Trials 6:22–25 (1997). 23. M. Hale, W. R. Gillespie, S. Gupta, N. Tuk, and N. Holford, Clinical trial simulation: streamlining your drug development process. Appl Clin Trials 5:35–40 (1996). 24. M. Sale, Modelling and simulation in drug development, promise and reality. Drug Discovery World Spring:47–50 (2001).
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25. N. Jumbe, B. Yao, R. Rovetti, G. Rossi, and A. C. Heatherington, Clinical trial simulation of a 200-microgram fixed dose of darbepoetin alfa in chemotherapy-induced anemia. Oncology 16(Suppl 11):37–44 (2002). 26. W. A. Thames, S. L. Smith, A. C. Scheifele, B. Yao, S. A. Giffin, and J. L. Alley, Evaluation of the US Oncology Network’s recommended guidelines for therapeutic substitution with darbepoetin alfa 200 mcg every 2 weeks in both naive patients and patients switched from epoetin alfa. Pharmacotherapy 24:313–323 (2004).
CHAPTER 34
Modeling and Simulation: Planning and Execution PAUL J. WILLIAMS and JAMES R. LANE
34.1
INTRODUCTION
In recent years modeling and simulation methods have been increasingly used to construct both clinical and preclinical programs and individual studies (1). Simulation has been used to identify optimal study architecture for clinical and preclinical pharmacokinetic (PK), pharmacodynamic (PD), and scale-ups for first-time-inhuman (FTIH) studies. Simulation has also been used to create data for communication and graphics, so that the meaning of research can be understood by individuals not involved in pharmacometrics (2). It is conceivable that in the not too distant future as a greater understanding of drug action is realized, late phase clinical development may be minimal or become unnecessary. At that point in time several learning trials will be executed and the results of these trials will be applied to virtual patients to determine the outcomes of drug administration. Modeling and simulation are especially useful when several critical issues concerning study structure need to be addressed simultaneously. These issues may include dropout rates (especially if related to dose), deviations from protocols by the subject, deviations from protocol by the practitioner, and nested levels of random effects. Such issues have not been addressed when approaches to study structure were based on prior experience, intuition, and empiricism. Modeling and simulation are a team effort. Several disciplines must be involved in the entire exercise to eventually simulate studies, where there is “buy in” from all stakeholders involved in the drug development process. Prior to a modeling and simulation exercise, appropriate PK, PD, physiology, pathophysiology, and future marketing strategies must be identified; all knowledge must be discovered from all available data (see Chapter 14). It has been pointed out that modeling and simulation should be guided by clarity, completeness, and parsimony.
Pharmacometrics: The Science of Quantitative Pharmacology Edited by Ene I. Ette and Paul J. Williams Copyright © 2007 John Wiley & Sons, Inc.
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Clarity. The report of the simulation should be understandable in terms of scope and conclusions by intended users such as those responsible for committing resources to a clinical trial (see Chapter 37). Completeness. The assumptions, methods, and critical results should be described in sufficient detail to be reproduced by an independent team. Parsimony. The complexity of the models and simulation procedures should be no more complex than necessary to meet the objectives of the simulation project. Program codes sufficient to generate models, simulate trials, and perform replication and simulation project level analyses should be retained but there is no need to store simulated trials and analysis results that can be reproduced from these codes (3). When guided by these principles, simulation in drug development will be streamlined and effective. The remainder of this chapter deals with the elements of the overall process for an overall project. These include simulation project assessment, project planning, project execution, project reporting, and project utilization.
34.2 34.2.1
EXECUTION OF THE SIMULATION EXERCISE Simulation Project Evaluation
For simulation to be considered in the drug development scheme, someone involved in the development of a therapeutic agent must realize that it may contribute significantly to the successful development of the agent, by increasing either the probability of approval or the profitability. Simulation can contribute to approval by providing the knowledge necessary for the design of studies that can meet regulatory requirements and studies that increase the likelihood of success by being more powerful. Profitability can be increased by aiding the proper positioning of the agent, shortening timelines to approval, decreasing the number of studies needed for approval, or decreasing individual study cost. Once the potential value of executing a simulation has been recognized, the feasibility of the project must be determined. To evaluate the feasibility of a project one must assess the current state of knowledge for the agent, evaluate the timeline, and propose a budget. The investigators brochure and all other relevant studies (both clinical and preclinical) must be obtained along with the current proposed study protocol. These must be reviewed and then discussed with those currently involved in the development process. Of note, it is important to involve those who will eventually be marketing the drug to ask what properties of the agent would improve marketability. Pivotal issues to be addressed when determining feasibility are: 1. Is the project really needed in light of prior knowledge? 2. Will the simulation project add value to the drug? 3. Can the simulation project be executed without compromising current timelines?
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4. If the current timeline is compromised, will the simulation add value to the overall drug development process to a such a degree that it is worthwhile? 5. Are the necessary human and monetary resources available for the project? 6. Do existing data and models provide adequate background to make a simulation worthwhile? 34.2.1.1 Project Necessity The necessity of the project must be evaluated. This can only be done after review of all documents, the study plan, and current data, and discussion with the current development leaders for the agent. It may be that sufficient knowledge has already been created for the project to proceed. It is important to address the trade-offs of doing versus not doing the simulation project. If, for example, a Phase 2 study has already documented efficacy, the dosing strategy, optimal patient selection for Phase 3, and Phase 3 would then be executed as a formality and for assessment of adverse events. In that case a modeling and simulation may not be necessary. 34.2.1.2 Project Completion Without Compromising the Timeline Very often models have not been developed for use in the ensuing study, and therefore for the simulation project, models must be developed. Developing and validating pharmacometric models can be time consuming. If the simulation was not originally proposed in the development process, the timeline for approval may be compromised. Any delay in development will necessitate that the role of the simulation be scrutinized. However, the delay in the timeline may be defensible, based on an increased certainty of outcomes of the proposed study, the generation of other knowledge that could support registration, or aiding in supporting a go/nogo decision. A strategy should always be in place to execute the needed models as early in the development process as possible. Real-time data collection should be done whenever possible. If blinding is a problem, real-time data collection and assembly should be done offsite. Furthermore, it is possible to develop pharmacometric models for two purposes; one for regulatory approval and the other for use in a simulation exercise. Thus, one may obtain representative data prior to locking of the data set, especially if there is real-time data collection. These data could be used to expeditiously develop and validate a model for the next stage of development. 34.2.1.3 Project Resources Prior to execution of the project, a serious look at resources required for the simulation project execution must be evaluated. Resources mean more than simply the financial budget. It includes people with the requisite skill set, experience, and knowledge base to execute the project. One must also address whether the necessary computational facilities and softwares are available. If personnel, expertise, or computational facilities are inadequate yet funding is available, then outsourcing the project to a contract research organization (CRO) would be an option. When working with a CRO one must ensure that the contract is very specific regarding the deliverables and timeline. Templates provided to the CRO are of great value when working outside one’s own organization. Finally, these contracts need to be executed in a timely manner to ensure projects are completed on time.
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34.2.2
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Project Plan
34.2.2.1 The End User Planning begins with the consumer of the simulation outcome in mind. The pharmacometrician has to be conscious of who the end user of the simulation will be and be certain to include the individual who will be making critical decisions, such as a go/no-go decision on the project. At this point the end user of the simulation drives the process; therefore, listening is a crucial quality the pharmacometrician must possess. It is important to play back to the end user what they have said to be certain that there is an understanding of all aspects of the project. The pharmacometrician should not ask questions until the end user has finished making his/her point. 34.2.2.2 Project Purpose When planning a simulation project, the purpose and intended use of the simulation must be clearly stated. It is the purpose that drives the modeling and simulation process. The clearly stated purpose and intended use of the simulation outcome provide bases for all decision and actions related to the project. The purpose and intended use must be agreed to by all stakeholders in the simulation project. 34.2.2.3 Project Team The purpose and intended use of the model will determine who the simulation team members are. The team members should cross several disciplines and all pivotal stakeholders need to be identified. The decision maker for future projects and development should be included in the team because it would be extremely frustrating to execute a simulation and, in the end, have it rejected by the end user or by the project decision maker. After the end user and decision makers have been identified, content experts must be added to the team. This would most often include disease experts, statisticians, PM experts (PK and PD experts), computer programmers, and pharmacologists. Individuals from regulatory and marketing are often available for the purpose of identifying what kinds of knowledge would be valuable to gain approval or to optimize the product’s position in the marketplace. If there are health-care related cost issues, a member with a background in pharmacoeconomics would be of value. The team should include a simulation team facilitator who should be a simulation expert. The facilitator’s role is to oversee the project, conduct expert interviews, gather unbiased assessments, evaluate the models, direct the execution of the project, assign each team member a specific role, and communicate the results. Once the team has been identified, planning of the project can begin. 34.2.2.4 The Project Plan Clinical trials always have detailed plans and protocols that describe the objectives, hypotheses, assumptions, data collection methods, data analysis methods, and so on. In like manner, the simulation plan describes a simulation process that is agreed to by the simulation team. The plan describes the work to be done, records to be maintained, and reports to be written. This plan should be written in enough detail so that another researcher could pick it up and execute the simulation with corresponding results. The plan must be critiqued and modified if needed. The preparation of the plan also provides an opportunity to evaluate objectives, assumptions, methods, and goals of the project.
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The overall objectives of the simulation and how the simulation will be used must be clearly stated in the plan. These will drive model selection and the approach to the execution of the simulation. If a model must be developed, the model building process must be described, data retrieved, and the model report generated. This process is described in detail in Chapters 8, 14, and 15. The assumptions of the simulation must be clearly stated. These include the structure of the PK, PD, outcomes, and covariate influence models with the stated values of each parameter. For the execution model, assumptions are made concerning deviations from the protocol, missing data, and patient compliance with the prescribed treatment regimen. The information foundation of the assumptions should be stated; the assumptions are either data and model based, theoretically justified, opinions of domain experts, or conjecture. Premises of greater certainty will often remain unchanged during the simulation process but those of less certainty may be varied to evaluate the final trial for robustness. The final data analysis approaches for the simulated data must be specified. Software that will be employed should be stated. Standard operating procedures should be referenced. The design of the simulation study must be stated. For each study the number of replications that will be performed, the factors (e.g., number of subjects, dose enrollment strategies, dropout rates, compliance) and to what degree these factors can be varied, how the robustness of the design will be assessed, and the required informativeness of the study design must be stated. The impact of varying joint factors must also be considered. The number of replications will vary depending on whether only typical outcomes or also atypical outcomes are of interest. Those studies and simulations where atypical or fringe outcomes are of interest will require more replications. An example of a fringe outcome would be the 5th and/or 95th percentiles of a biomarker. There are three distinct features for each simulation model that must be addressed in the simulation plan. The first are the input–output (IO) models that describe the PK/PD–outcomes models. The inputs here are the rates of drug administration and the outputs are things such as drug concentrations or biomarkers. These IO models should have stochastic elements as part of the model such as between-subject variability and residual variability. It is of primary importance here that the complete probability distribution of the outputs be described in the planning. IO models may be mechanistic or empirical. Mechanistic models attempt to portray the model at the physiological or biochemical level while empirical models simply describe the IO model. Mechanistic models are preferred for simulation as they are more likely to be extrapolated to other studies or drugs in the future. It is important to incorporate this I/O model parameter uncertainty in the simulation of clinical trials. In order to implement parameter or model uncertainty in the simulation model, the typical values (mean values) of model parameters are usually defined as random variables (usually normally distributed), where the variance of the distribution is defined as standard error squared. The limits of the distribution can be defined at the discretion of the pharmacometrician. For a normal distribution, for example, this would be q ± 2 SE, where q is the parameter. This would include 95% of the simulated distribution. When the simulation is performed, each replicate will have different typical starting values for the system parameters. The
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output can then be combined for all replicates and the outcomes of interest studied over a more representative range of variability. The covariate distribution models, which describe the characteristics of the population (weight, height, sex, race, etc.), must be determined and used for the creation of the study population. The virtual subjects are drawn from a probability distribution that can be one of many types (normal, lognormal, binomial, uniform) but that needs to be described in the study plan. For assignments to sex one must account for what proportion of patients will be female versus male. Furthermore, when creating this population the joint distribution of variables such as height and weight or sex and size must be accounted for. This then leads to the execution model. The importance of the execution model cannot be overemphasized. The execution model describes how the study is carried out and deviations from the protocol. There are deviations from the protocol that are done by the patient such as refusal to enter the study, dropouts, and patient noncompliance. Other deviations are due to practitioner behavior such as missing data, wrong recording of data, or improper preparation of doses. It must be decided whether the deviations are completely at random or if there is some influencing factor that may result in protocol deviations. For example, would patients experiencing adverse events have a greater tendency to drop out of the study? As a part of planning, each member of the simulation team must be aware of his/her individual assignments, the deliverables, and timelines. The team members’ responsibilities must be stated explicitly and the simulation team facilitator must assure that all members fulfill their responsibilities in a timely manner. One often overlooked but important part of the plan is for an evaluation of the simulation results when compared to the final study. This must be planned for. Simulation performance criteria must be stated prior to the completion of the final study. Report templates are often included in the plan. These can be useful as they ensure that important outputs from the simulation project are generated and reported. These templates can include due dates to ensure timely delivery of required inputs and outputs for the simulation project. It is also import to note who will generate and receive the information from these report forms. Often, these templates can be used in future studies either without modification or with only slight modification. 34.2.3
Execution of the Project Plan
A simulation project is an iterative process. Therefore, the team must meet periodically once the simulation is being executed. The simulation is initiated per the simulation protocol. However, some things may change during the execution of the simulation, such as underlying assumptions, or the simulation may show problems that need to be addressed by the team. There may need to be changes in the simulation protocol or the simulation may show flaws in the end study design that should be addressed expeditiously. All affected members of the team should be informed of any proposed changes, especially the end user and decision makers. If there are any changes in the simulation assumptions, then all stakeholders should be notified.
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A report of the results of the simulation must be generated. The written report will have the data, assumptions, deviations from the simulation plan with justifications, results, conclusions, recommendations, program scripts, simulation outputs, and supporting literature. The initial draft should be reviewed by the simulation team prior to release to the clinical development team. The final document should be archived.
34.2.4
Applying the Results of the Simulation
The results of the simulation are reported most importantly to the end users and decision makers. The written report is circulated and a team meeting scheduled. The recommendations and reasoning behind any recommendations and options to the recommendations are made concerning the structure of the simulated study. It is important to keep the report and reporting as simple as possible; any presentation with an excessive amount of verbiage or slides is likely to be ignored. The end users and decisions makers will for the most part be interested only in the bottom line of the simulation results. Very often at this point, additional simulations are requested by the clinical development team.
34.3 34.3.1
MISCELLANEOUS POINTS TO CONSIDER Written Standard Operating Procedures (SOPs)
For organizations that frequently engage in modeling and simulation projects, SOPs are of great value. These should be coupled with templates, practice guidelines, policies, and other similar documents. These documents will help in providing structure to the modeling and simulation process and will expedite the process when repeated. These documents should be reviewed and updated on occasion.
34.3.2
The Problem of Blinded Studies and Data Access
For studies where blinding is a necessity, access to the data is seldom available prior to locking of the database. This causes a problem because the time between locking the database and the writing of the next protocol is not usually sufficient to complete a modeling and simulation project. A solution to this problem is off-site real-time data collection combined with off-site modeling and possibly simulation. An example would be the execution of a Phase 2a study, the results of which will eventually be used to design a Phase 2b study. One may need only a portion of the Phase 2a results (say, only 80% of the Phase 2a data, which would be added to Phase 1 PK/PD data) to develop a PK/PD–outcomes model. Thus, the modeling could begin prior to the data lock. A model would be available at the time of the data lock at the end of Phase 2a. The simulation team could also be selected prior to the data lock. Thus, the simulation could begin in a very timely manner after the data lock and be ready as the protocol is written for the Phase 2b. This type of a workaround must be documented and transparent.
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SUMMARY
A well planned simulation project increases the likelihood of providing meaningful and timely simulation results that will enhance the design and improve the efficiency, robustness, power, and informativeness of preclinical and clinical studies. An increase in the efficiency and power of clinical trials should reduce the number of studies and time needed to complete the drug development process with the resultant reduction in cost of pharmacotherapy to the consumer.
REFERENCES 1. N. Holford, H. Kimko, J. Monteleone, and C. Peck, Simlation of clinical trials. Annu Rev Pharmacol Toxicol 40:209–234 (2000). 2. R. Gieschke and J.-L. Steimer, Pharmacometrics: modeling and simulation tools to improve decision making in clinical drug development. Eur J Drug Metab Pharmacokinet 25(1):49–58 (2000). 3. N. H. G. Holford, M. Hale, H. C. Ko, J.-L. Steimer, L. B. Sheiner, and C. C. Peck, Simulation in Drug Development: Good Practices. Available at cdds.georgetown,edu/ research/sddgp723.html (1999).
CHAPTER 35
Clinical Trial Simulation: Efficacy Trials MATTHEW M. RIGGS, CHRISTOPHER J. GODFREY, and MARC R. GASTONGUAY
35.1
INTRODUCTION
Expanding on the model-based drug development concept proposed by Sheiner (1), the US FDA in its March 2004 document Challenge and Opportunity on the Critical Path to New Medical Products advocates “using simulation software to improve trial design and to predict outcomes.” Simulation of a Phase 2b/3 efficacy trial generally occurs during Phase 2 development following proof of concept and dose ranging. However, to say that simulation begins at this point in development is an injustice, or if it is true, a shortcoming of the clinical and scientific development team. Simulation of an efficacy trial should occur as part of a continuum, where accumulated preclinical and clinical information is used to make informative decisions for each next step in development. In this case, simulation will inform the efficacy trial design, conduct, analysis, and interpretation. In its April 2003 Guidance, Exposure–Response Relationships—Study Design, Data Analysis, and Regulatory Applications (http://www.fda.gov/cder/ guidance/5341fnl.pdf), the FDA advocates a critical role for exposure–response evaluation in decreasing the uncertainty of drug development. To further emphasize the importance of exposure–response modeling and simulation prior to embarking on large-scale efficacy trials, the FDA Clinical Pharmacology Subcommittee for Pharmacentical Science has suggested early sponsor–FDA meetings (at the end of Phase 2a development) to discuss exposure–response issues (Advisory Meeting Nov. 17/18 2003, http://www.fda.gov/cder/audiences/acspage/acslist1.htm). From this stimulus to learn more about drug effects earlier, a wealth of information should be available for consideration at the point of designing an efficacy trial. Included are preclinical and clinical investigations pertaining to conceptual efficacy and safety endpoints, pharmaceutical and manufacturing considerations for drug product formulation, in vitro and in vivo metabolism characterization, and, in many instances, clinical pharmacology studies exploring potential metabolic interactions and special populations. Efficacy and safety information may include biomarker responses that ideally have been quantified through exposure–response modeling. Pharmacometrics: The Science of Quantitative Pharmacology Edited by Ene I. Ette and Paul J. Williams Copyright © 2007 John Wiley & Sons, Inc.
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The degree of translational relevance of the biomarker(s) to the actual clinical endpoint will be based on prior experience. Information pertaining to adverse event profiles will be emerging from Phase 2 and contextualized through Phase 1 special population and drug–drug interaction (DDI) studies. Additionally, rigorous understanding of “external” information such as disease progression models and comparator profiles will further shape the focus of efficacy trials. Altogether, the clinical profiles, in conjunction with the nonclinical considerations, should provide a therapeutic range in which to characterize the expected clinical efficacy. Modeling and simulation can offer considerable insight into the design of studies to target this range and ultimately confirm the clinical outcomes. Considerations for performing these simulations are the focus of this chapter. A simulation of a hypothetical efficacy trial for a zidovudine analog in HIV patients is provided as an application and example of many, although not all, of these considerations.
35.2
SIMULATION PLANNING
To begin, all information pertaining to the clinical efficacy trial should be assembled and categorized as possible inputs (known information) or outputs (information that needs to be known) for the trial simulation. This information gathering should involve subject matter experts and key stakeholders of the drug’s development (e.g., clinical pharmacology, clinical, statistics, regulatory, operations, and commercial leads). Additional invitees may include other research members (e.g., biology, pharmacology, biopharmaceutics, outcomes research) as determined by the complexity of the input factors (e.g., a thoroughly mechanistic pharmacokinetic/pharmacodynamic (PK/PD) model may benefit from biologist and pharmacologist insight) and output responses (e.g., outcomes research to help define inclusion of clinical utility functions). As part of this start-up meeting, the identified input factors and output responses should also be assigned a level of precision, either how well it is known (input) or how well it needs to be known (output). Defining this level of uncertainty may be done qualitatively at first, whereas a quantified level of precision will need to be developed for those factors entering into the simulation. The sections that follow provide examples of both model-based and trial-based input factors and output responses that should be considered for efficacy trial simulations. 35.2.1
Model-Based Input Factors
35.2.1.1 Pharmacokinetic (PK) Model PK models describe the continuous drug concentration–time course resulting from an administered dose. By doing so for each individual either through (a) intensive collections and standard two-stage PK analyses, or (b) sparse sample collections and population analyses, these continuous descriptions provide a less discrete and so often more informative measure of drug exposure than does dose alone. In concert, mechanistic PK/PD models are being developed more frequently and more congruently during drug development to describe exposure–response relationships (2, 3).
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These models often incorporate intermediate biomarker responses. Consequently, trial simulations driven by PK models, rather than more traditional dose–response relationships, will enable more detailed simulations. For example, exposure differences due to interactions, inclusion of special populations, or from dosing regimen or formulation changes may be explored with the PK models driving PD responses. This will place additional emphasis on the modeler to develop reliable PK models using Phase 1 and 2 data that translate into the patient population. Appropriate consideration of covariates, as discussed later, will be an important part of this development. An additional component of the PK model that may warrant consideration in the simulation is the relative bioavailability of the drug formulation to be used in the efficacy trial. Formulation changes may occur at this point in development, where a suboptimal formulation for commercial-scale manufacture may have been used in previous studies. If such changes have occurred between the dose-ranging study and current design, the relative bioavailability between formulations (and associated 90% CIs) may also be considered for simulation evaluation. A sensitivity analysis (see Section 35.3.1) may be conducted to evaluate whether this effect will be influential on the simulation and/or if additional data may be required to provide acceptable precision. 35.2.1.2 Exposure–Response (ER or PK/PD) Model PK/PD models are becoming less empirical and more mechanistic in nature due to the increasing reliance on biomarkers and the collection of this information earlier and earlier in drug development. The questions being probed by these models through simulation are becoming more focused, including evaluation of exposure differences (4) or even for multiple drugs (2, 5). The former case allows for in silico exploration of formulation (e.g., IR vs. CR) and regimen (e.g., QD vs. BID), or to evaluate effects of exposure differences in special populations or due to DDI. The latter case affords us the opportunity to compare among competing candidates within a development program, or to compare to other available agents. Implicit to the promising roles of mechanistic PK/PD models is a link to clinical outcomes. A few such examples have been presented, including a model linking blood glucose concentrations and glycosylated hemoglobin (HbA1c) (6), where HbA1c has been linked with progression of nephropathy (7) and retinopathy (8). For atherosclerosis, high serum concentrations of LDL-C have been demonstrated to be a major risk factor for coronary heart disease (CHD) and therapeutic LDL-C lowering (e.g., with HMG CoA reductase inhibitors) has been linked to reduced risk of major coronary events (9). In addition to exposure–response models for clinical efficacy, the simulation also may include models for safety markers. These may include more immediate or direct effects, such as a drug affecting the QT/QTc interval (10). Although less frequent, longer term effects such as changes in liver function likely are not well defined at this point in development. As known, or potentially expected, such effects may be considered with longitudinal mixture models (11). Taken together, the ER models for efficacy and safety will define the therapeutic window, where at the point of simulating an efficacy trial, an acceptable separation should exist. However, if multiple markers are being used to determine this separation, there may be a desire to “weigh” some markers more than others. For
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example, one marker may be more of a tolerance issue (e.g., moderate incidence of transient nausea upon initiation of therapy) and another a more serious but with a much lower incidence (e.g., edema). Both may warrant inclusion in the simulation but may be assigned different levels of influence through the use of weights. The use of utility functions has been reported to provide this relative balancing (12), which in addition to safety and efficacy parameters, can include any other pertinent input factors as well (e.g., study cost and/or duration). 35.2.1.3 Baseline Distribution Model Appropriate definition of baseline values for biomarkers or endpoints is critical in setting initial simulation conditions. Both the mean and range of baseline values can have important effects of the projected outcome. This is particularly important for mechanistic models, since the degree of effect is often defined as a function of baseline values. For example, in a typical “indirect” model describing the rate of change in the measured endpoint, R, the assumption may be made that the input rate (kin) is equal to the output rate using a first-order rate constant (kout) and the baseline endpoint measure (Rbase): kin = koutRbase (13). Therefore, baseline models should be evaluated to ensure an appropriate distribution of baseline values is being generated for the patient population of interest. These values also need to be checked against specific inclusion or exclusion criteria for the efficacy study and adjusted as needed. 35.2.1.4 Longitudinal Effect (Disease Progression) Model The longitudinal course of the targeted disease is of particular interest in many chronic ailments where conditions worsen over time, including Alzheimer’s disease, Parkinson’s disease, osteoporosis, diabetic nephropathy, and respiratory disease (14). Alternative models have been developed for other disease states where cyclical effects may be observed, such as depression (15) or myelosuppression (5). Included in longitudinal models are descriptions for the natural progression of disease as well as nonpharmacological intervention (e.g., placebo treatment). For simulation of an efficacy trial, the mechanism by which the drug affects progression must be considered (e.g., symptomatic or protective). This mechanism may or may not be the same as a comparator agent, which itself may be useful during the simulation. Altogether, the progression model may be used within the simulation to evaluate when clinical endpoints will be measured and the optimal duration of the trial. For example, it may take more or less time (trial duration) than originally anticipated to show separation between treatments depending on the rate of change in the disease progression relative to that affected by standard of care therapy. Detailed discussion of structural considerations for disease progression models is provided in Chapter 21 and elsewhere (14). 35.2.1.5 Covariate Models Both a drug’s PK response and pharmacological response may be influenced by various patient characteristics. Additionally, the progression or extent of disease may be affected by comorbidities. Covariate models attempt to account for and quantify the influence of these factors. For example, a covariate model would be used in simulating PK differences between males and females. Likewise, for a disease progression model of atherosclerosis, or for the overall evaluation of
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antihyperlipidemic efficacy, the presence of diabetes raises an individual’s cardiovascular risk and results in varying treatment goals for therapy (9). Thus, diabetes may be included as a variable for a lipid efficacy trial simulation where diabetics are part of the study population. A potential limit at this point in development is that many covariates are still being identified and explored, so the “true” effect is not yet known. Consequently, the precision of the estimated covariate effect may be relatively low. Although it would be advisable to limit the number of covariates included at this stage to those of direct clinical relevance, it is recommended that a “full” model approach (16) be employed. The “full” model would include all covariates of interest with associated mean estimates and precision (e.g., confidence interval (CI) calculated using asymptotic standard errors or bootstrap replication procedures). Collinear covariates should be used with caution as they may affect the precision of the estimates (17). Using PK differences between males and females as an example, suppose the sex difference was considered “not statistically significant” from Phase 1 and 2 data. However, it may be that the evaluated effect was not “powered” appropriately to rule out a clinically significant difference from the available data. For example, if the resulting 90% CI for the covariate parameter estimate was not well defined (e.g., outside [0.8, 1.25]), then a “no effect” conclusion may not be the most appropriate assumption at this juncture. Therefore, if differences between males and females are of interest to the trial or program outcome, retention of this covariate parameter in the model would be advised. A sensitivity analysis (see Section 35.3.1) assessing the influence of a PK sex effect may be conducted. This would determine whether the point estimate of the effect influences the simulation outcome and how its precision may lend to overall uncertainty in the results. Covariates are incorporated into the simulation as distributions that are either simulated stochastically or resampled from an existing database (18). Correlation between covariates is handled during stochastic simulations using multivariate distributions with appropriate variance–covariance structure. Alternatively, covariates resampled from a sufficiently large existing database carry all relevant covariates from an individual into a simulated individual and so capture inherent correlation. Regardless of the method, the simulated outputs for covariates need to be checked to ensure that they reflect the expected trial population and are consistent with trial inclusion and exclusion criteria. 35.2.1.6 Compliance Model Failure to account for nonadherence to study drug administration schedules will lead to biased and imprecise trial simulation outcome measures (19). Models to assimilate compliance often involve a hierarchical Markov model, where the probability for an individual to take a scheduled dose is conditional on whether this individual had taken the previous dose (20, 21). The model may also contain covariates as predictors of compliance. For example, compliance has been shown to be affected by dosing frequency, where an increased frequency (e.g., three times daily vs. once daily) has been associated with worse compliance (22, 23). Alternatively, the consequence of missing a once-a-day dose may have more significant impact on efficacy. PK/PD-based simulations play an important role in understanding the balance of these situations.
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In addition to the Markov model, compliance may be modeled using a more simplified model as a mixture (fraction) of patients who are either compliant or noncompliant (all-or-none) (24). Or, similar to drawing covariate distributions from databases of representative populations, a nonmodel-based option for compliance would be to draw from prior compliance data collected from a representative patient population. 35.2.1.7 Study Retention (Dropout) Models Subjects will drop out of trials for either random (ignorable) reasons or perhaps for a reason attributable to their disease, trial conditions, or other nonignorable factor. Both conditions are important to consider for efficacy trial simulation. In the former case, subjects who drop out (are missing) at random will result in a decrease in total sample size and may affect the study power. In the latter case, nonrandom dropout is considered to be nonignorable in that the reason for dropout is informative to the trial outcome and may bias the results. In the seminal paper by Sheiner (25), an example of nonrandom dropout is presented for an analgesic trial, where those subjects not achieving adequate pain relief were more likely to drop out (i.e., to take rescue medication). Numerous methods exist for handling dropout, including a recent example by Hu and Sale (26). The reader is referred to a published tutorial (27) for guidance on evaluating the most appropriate method for a particular situation. 35.2.2
Trial-Based Input Factors
Although some of the trial-based input factors will be fixed (e.g., if a design must be set a certain way due to unwavering logistics), many of the trial-based factors will be variables for which the simulation will attempt to find an appropriate combination to achieve the trials objectives. These variables are the “what ifs” of the efficacy trial simulation. An attempt to provide a thorough, albeit not all inclusive, list and brief description of trial-based factors to consider for efficacy trial simulation is provided below. Elemental considerations may be broad comparisons, such as parallel group versus randomized crossover designs. Simulation also may assist in assigning the trial’s primary endpoint, where the simulated probabilities of a successful trial for several clinically meaningful outcomes could be used to determine the most appropriate primary endpoint. Simulation evaluation often considers many numerical factors of a trial design. These include the total number of subjects, the proportion of subjects allocated to the treatment groups, and the number of treatments included (where the range of treatment that is most informative already has been defined by the dose-ranging study). Included in these components may be evaluations to explore effects within specific subpopulations, the inclusion of, and effect in, specific strata within treatment groups, or the impact of other inclusion or exclusion criteria. As discussed earlier, study duration and the number and timing of endpoint measures may also be considered through the trial simulation. Many of these trial-dependent factors are ultimately evaluated relative to their influence on the statistical power of the study. Additionally, the study data analysis method(s) to be employed may also be a consideration for the simulation.
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For example, in the analgesic example cited above (25), a comparison was made between an analysis using the last observation carried forward (LOCF) method and the proposed mixed effects maximum likelihood method. Although this was a retrospective analysis, similar contrasts could be included in the trial’s simulation to ascertain the most appropriate analytical methodology to include in the study design (protocol). Other analysis factors for consideration include appropriate correction of variability, where such sources may include differences between sites or regional differences. 35.2.3
Output Responses
The output responses can be categorized as either measurements directly from the trial (e.g., endpoints) or measures of how well the trial is expected to perform (power or probability of success). Either these responses may be evaluated for how well they can be defined for a given study design, or vice versa, they can be defined as needing to be known to a given level of precision, and the study design consequently optimized for this goal. The focal clinical endpoints (need to know) are likely to fall into the latter, while more peripheral endpoints (nice to know) are likely to be assessed as the former. The simulated output responses provide the measures for which to optimize the design and analysis of the study. In the case of the model-based input factors, their effects are often used to determine how useful (informative) the model(s) and associated parameters are for the output responses and, importantly, which model assumptions may be most critical to the validity of the simulation. Evaluation of the trial-based input factors is often an iterative process, where various scenarios (e.g., ranges of subject numbers, cohorts, strata, duration) are simulated and the most favorable is carried forward for study planning. The definition of “most favorable” often involves several response factors in addition to the probability of trial success (or failure). Trial costs or other “utility costs” from utility functions may be included in this decision. For example, a larger parallel group study may cost more but finish more quickly than a crossover design. Depending on several other factors, such as within-subject versus between-subject PK and response (efficacy and safety) variances, relative drug effect on disease progression rate, and dropout rates, the designs may have a different probability of success. Therefore, criteria for the cost to benefit ratio for each design would need to be developed and applied to the simulation results. Ideally, such decision trees are formed prospectively, at least in concept, to enable decisive application of the simulation results. 35.2.4
Simulation Team Review of Model and Assumptions
As important as the initial simulation start-up meeting, regular contact with team members must be maintained throughout the trial simulation to ensure continued clarity of the goals, to develop understanding and agreement of the underlying assumptions, and to foster interest, involvement, and ultimately informed application of the simulation results. A second simulation team meeting should ensue once the initial input factors and output responses are identified and initial estimates with accompanying uncertainties are defined. This review should occur prior to
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extensive coding, and certainly before beginning the simulation execution. During this review session, all assumptions considered for the simulation should be detailed and agreed upon. A general schematic of the inputs and outputs may be beneficial, with indicators of key inputs, outputs, and assumptions. Once all goals, strengths, benefits, and limitations are agreed upon, the simulations may begin; during which the team is to be provided with regular status updates.
35.3 35.3.1
SIMULATION EXECUTION AND INTERPRETATION Sensitivity Analysis (SA)
Once the simulation conditions have been properly defined, the simulations are now ready to execute. One important consideration in interpreting simulation results is the sensitivity of these results to underlying assumptions or uncertainty about the simulation model and parameters. Trial simulation outputs should be viewed relative to their sensitivity to model parameter uncertainty. There are two general methods, local and global, for performing SA. Local SA involves repeated groups of simulations, where in each group a fixedpoint perturbation of one parameter is used for the simulations. Trial simulation output metrics are then calculated for each group and the impact on outcome is considered relative to this range of parameter estimates. For example, the degree of sensitivity may be considered by the rate of change in response relative to the unit change in the parameter. This process is then repeated for each parameter of interest. Limitations of the local sensitivity approach are that it only reflects sensitivity to uncertainty in one parameter or assumption at a time. It is therefore inefficient and conclusions about sensitivity are conditional on assumptions of all other parameters. Global SA is based on simulations where results are conditioned on uncertainty distributions across all parameters. Uncertainty is quantitatively defined for all parameters (models) through the use of appropriate distribution models (28) or using distributions from prior reports or models. The latter method, which does not require an assumed model parameter probability distribution function, may include use of fuzzy set theory (29) or the use of bootstrapped estimates from previous estimations. Monte Carlo methods are required to simulate from the uncertainty distributions at the intertrial level. This usually requires one set of simulations with a large number of replicates. The number of trial replicates is discussed in Chapter 33, where this number may need to be further increased for the global SA. One benefit of global versus local SA is that by running only one set of simulations, the sensitivity of simulation outcome(s) to simulation parameter uncertainty (assumptions) can be viewed over a continuous range of parameter uncertainty because the sensitivity to uncertainty is incorporated in all model parameters simultaneously. Examples of simulation with uncertainty are available for physiologically based PK models (29, 30) and clinical trial simulations (31, 32). In practice, a blend of local and global SA approaches may be employed. Quantitative uncertainty in the form of standard errors may be available only for some model inputs. Multiple executions of the simulation to characterize the global SA of the quantified parameters may be undertaken conditional on a series of fixed-
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point perturbations of another parameter. This “semiglobal” SA would determine whether the findings about global sensitivity change given the uncertainty in the locally perturbed parameter. 35.3.2 Simulation Team Review of Sensitivity Analysis and Impact on Assumptions The SA will identify which parameters are most influential on the simulation outcome and consequently which assumptions are either reasonable or in doubt. This provides another decision point for the overall trial simulation, and so a meeting with the simulation team should be convened. This meeting should cover the impact of the SA on the trial (and/or program) planning and design. As an example discussion point, if the SA reveals previously unexpected, and possibly undesirable, response ranges in a certain patient subpopulation, then the team would discuss whether (a) to proceed with the current design, given the risk of unexpected or undesirable outcomes, which has now been quantified; (b) the model assumptions for that subpopulation are appropriate; (c) more data be collected to reduce the uncertainty surrounding this patient population before executing the trial; (d) the current trial design requires modification to make it more robust to potential differences in this subpopulation; or (e) this subpopulation be excluded altogether if they are not of interest in the current trial. The SA may also provide rationale for simplifying a model. This may occur if the outcome is shown to be robustly tolerant to wide ranges of parameter uncertainty, which may allow for the removal of some parameters and thus lead to a more parsimonious simulation model. Conversely, the SA may reveal that insufficient information currently exists to define a precise or reliable range of trial outcomes. In this latter case, either more time may be required to obtain additional informative experimental data and thus reduce the uncertainty to an acceptable range, or separate sets of plausible assumptions may need to be considered and subsequently tested for their own sensitivity. Such decisions need buy-in from the subject matter experts and should be considered in the full context of the development program. 35.3.3
Simulation Recommendations
Following discussion and acceptance of the SA results, including both model-based and trial-based input factor adjustments, the efficacy trial simulations may proceed as planned. For each possible trial design, the appropriate input factors and output responses are simulated and results are compared to determine the most appropriate design. As discussed previously, this final decision likely will not only be based on a specific p-value or trial power, but will also include valuations based on trial duration, monetary cost, or information gained or lost toward continuing development goals (e.g., an overall measure of clinical utility). 35.4
EFFICACY TRIAL SIMULATION EXAMPLE
A simulation of a hypothetical efficacy trial for a zidovudine analog (ZDVA) in HIV patients was completed to evaluate the probability of a successful Phase 3 trial if Phase 2b was skipped, given the Phase 2a results and prior knowledge from
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CLINICAL TRIAL SIMULATION: EFFICACY TRIALS
a marketed competitor. The goal of this simulation was to evaluate the probability of trial success under a predefined trial design while examining the sensitivity of this probability estimate to underlying assumptions. Included in this example is an assessment of the effect of uncertainty (from none through varying degrees) on the trial outcome. 35.4.1
Model-Based Inputs
Input factors, as defined in Table 35.1 and presented schematically in Figure 35.1, included an adherence model, PK and PK/PD models, placebo response and disease progression models, and both random and nonignorable dropout. The PK/PD model was used to simulate patient survival times (or censored events) for each individual as a function of drug exposure, which was defined as the average steadystate drug plasma concentration (Cavg). Cavg was modeled as a function of clearance (CL), which depended on hepatic disease (HEP), weight (WT), and methadone use (METH). Dosing compliance/adherence was modeled with a bimodal distribution (high and low compliance groups). A hazard function for dropout due to inefficacy included terms for placebo dropout, effect of CD4+ count, and effect of ZDV. Random dropout was also simulated. The full model is provided in the chapter appendix. “Uncertainty” distributions were defined for all parameters including typical PK, PD parameters, covariate effects, and interindividual and residual variance parameters. These distributions were derived from the variance–covariance matrix of the estimates obtained from a prior analysis, and from a review of prior knowledge and published results. Separate simulations were performed with no, low, moderate, or high degrees of uncertainty to explore the effect of uncertainty on the probability of trial success using a global SA. Uncertainty, expressed as %CV for each parameter, was set for the fixed model effects at 0% (none), 10% (low), 35% (moderate), or 50% (high).
TABLE 35.1 Description of Simulation Model Parameters Used for the Zidovudine Analog Efficacy Trial Simulation Name CD4 CD4PT CD4SLP CLBASE HAZP HAZR HEPCL HICOMP IC50 LOWCOMP
Description Mean CD4+ count CD4+ breakpoint for beneficial effect Effect of CD4 count on hazard Typical clearance Placebo risk for dropout (hazard) Random dropout hazard Effect of hepatic disease on CL Mean high compliance Drug potency Mean low compliance
Name
Description
METHCL PrFEM
Methadone effect on CL Probability of female
PrHEP
Probability of hepatic disease
PrHICOMP PrLOWCOM
Probability of high compliance Probability of low compliance
PrMETH SEXCL
Probability of methadone use Sex effect on CL
WTFEM WTMALE ZDVSL
Mean weight for females Mean weight for males Maximum drug effect on hazard
EFFICACY TRIAL SIMULATION EXAMPLE
Dosing /Adherence Model
Placebo Response Model
PK Model (covariates)
Disease Progression Model
891
+
PD Model (covariates) + Disease Progression Model (covariates)
FIGURE 35.1
Survival Analysis Success = significant difference from placebo
Schematic of zidovudine analog efficacy trial simulation components.
Uncertainty for random effects was defined as high for all evaluations. The high level of uncertainty for random effects represented a “worst case” scenario, but also was representative of the greater uncertainty in these parameters at this stage in development relative to the fixed effects parameters. In an actual trial simulation, prior estimates of these uncertainties could be used, and possibly inflated to accommodate the additional parameter uncertainty associated with extrapolation to a new trial scenario or population. To illustrate a local SA, the parameter describing the maximum drug effect on the hazard parameter (ZDVSL) was fixed at values ranging from 0.25 to 1.0. 35.4.2
Trial-Based Inputs
Two thousand patients were to be randomly assigned to placebo, ZDVA 500 mg or ZDVA 1500 mg, daily. Follow-up visits were planned every 28 days for 2 years. The number of “survivors” was defined as the number of patients remaining in the study at each observation time. 35.4.3 Simulation Execution and Analysis Simulations with and without varying degrees of uncertainty were performed, as described above. Simulation with uncertainty can be implemented in a variety of programs with Monte Carlo simulation capabilities. In this example, simulations were carried out using S-Plus® (Insightful, Seattle, WA) and NONMEM® (GloboMax LLC, Ellicott City, MD). PROC PHREG in SAS® (SAS Institute Inc., Cary, NC) was used for survival analyses. Local regression plots were created with S-Plus and the LOCFIT library. 35.4.4 Simulation Results and Recommendation The effect of the degree of uncertainty on trial success (power) is provided in Table 35.2. A total of 500 replicate simulations were performed for each extreme of uncertainty (none and high). Fewer (n = 100) replicates were performed for the intermediate levels (low and medium), based both on practical computational limitations and an observation of relative stability in the results after 100 replications. More (n = 5000) replicates were performed for graphical clarity in the local regression plots. The results show that failure to account for uncertainty in the model would result in an overestimate of the trial power and thus a falsely optimistic design.
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CLINICAL TRIAL SIMULATION: EFFICACY TRIALS
TABLE 35.2 Effect of Uncertainty Level on the Estimate of Trial Power Using a Global Sensitivity Analysis for the Zidovudine Analog Efficacy Trial Simulation Uncertainty None Low (10%)b Medium (35%)b High (50%)b
Successful Trialsa (%) 94.4% 93% 86% 82.8%
(n = 500) (n = 100) (n = 100) (n = 500)
a
Results reflect n simulated trials of 2000 patients. Uncertainty in fixed effect parameters is approximate %CV. Uncertainty in random effect parameters was high for all simulations.
b
FIGURE 35.2 Results of global SAs from the zidovudine analog efficacy trial simulation. Parameters displayed are for drug clearance (CLBASE), drug potency (IC50), placebo risk on the dropout hazard (HAZP), and maximum drug effect (ZDVSL) on the dropout hazard. The trial power (probability of success) is plotted relative to the parameter value for each of the 5000 replicate simulations. The solid lines indicate the local logistic regression (locfit) and the dotted lines provide the 95% confidence interval for the locfit.
A global SA for the simulated outcome, probability of a successful differentiation from placebo, was not sensitive to uncertainties in CL and IC50 parameters (Figure 35.2). Simulation conclusions were sensitive to assumptions about ZDVSL and HAZP parameter values (Figures 35.2), and a wide range of possible outcomes was evident.
EFFICACY TRIAL SIMULATION EXAMPLE
893
0
0.2
Pr(Success) 0.4 0.6 0.8
1
In other words, at the hypothetical doses explored (500 and 1500 mg), uncertainties about exposure (CL) and potency (IC50) were not important contributors to the uncertainty of response, whereas uncertainties about the maximal effect (ZDVSL) and placebo dropout did contribute notably to the trial power. An additional benefit of the global SA is that it provided the interdependence of the ZDVSL and HAZP parameters on the trial outcome (Figure 35.3), illustrating a benefit of global versus local SA. The results of a local SA using the ZDVSL parameter are provided in Table 35.3. These results are similar to the global SA, but are conditioned on the fixed estimates of the other parameters. As shown with both global and local SA, the probability of a successful trial was less than 80% across a large range of uncertainty in the parameters (Figures 35.2 and 35.3 and Table 35.3). Given prespecified criteria of ≥80% power, the
0.0 25 0.0 2 0.0 15 HA Z
2 P
0.0 1 0.0 05
1
ZD
1.5 VSL
0 .5
FIGURE 35.3 Results of the global SA from the zidovudine analog efficacy trial simulation displaying the multidimensional effect of parameter uncertainty on trial power (probability of success). Parameters displayed are placebo risk on the dropout hazard (HAZP) and maximum drug effect (ZDVSL) on the dropout hazard. TABLE 35.3 Effect of Single Parameter (ZDVSLa) Uncertainty on the Estimate of Trial Power Using a Local SA for the Zidovudine Analog Efficacy Trial Simulation Fixed Value of ZDVSL 0.25 0.5 0.735 1.0
Successful Trialsb (%) 30.6% 70.4% 93.0% 99.0%
a ZDVSL represents the maximum drug effect on the dropout hazard. b Results reflect 500 simulated trials of 2000 patients.
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CLINICAL TRIAL SIMULATION: EFFICACY TRIALS
simulation results represented an unacceptable risk of proceeding with the current design, given the current state of knowledge. More robust trial designs might have proved to be impractical. Therefore, before proceeding with the current design, it would be recommended to refine estimates of these influential model components. The overall recommendation from this hypothetical example would be to run the Phase 2b trial, rather than skipping directly into Phase 3.
35.5
SUMMARY
Returning to the FDA March 2004 document Challenge and Opportunity on the Critical Path to New Medical Products, “the current medical product development path is becoming increasingly challenging, inefficient, and costly. A new product development toolkit—containing powerful new scientific and technical methods such as . . . computer-based predictive models . . . is urgently needed to improve predictability and efficiency along the critical path from laboratory concept to commercial product.” Efficacy trial simulation satisfies this initiative by offering a tangible benefit to drug development through an a priori in silico evaluation of trial design performance. Consistent development and application of this approach to program planning will lead to more informed study designs and more focused outcomes, while minimizing costs and time. To appropriately apply this approach, development and management teams must champion its benefit and provide dedicated resources, as well as commit project team time to correctly interweave simulation into the development process.
ACKNOWLEDGMENT The authors wish to acknowledge William R. Gillespie, Stuart L. Beal, and GloboMax LLC for their contributions to the efficacy trial simulation example.
REFERENCES 1. L. B. Sheiner, Learning versus confirming in clinical drug development. Clin Pharmacol Ther 61:275–291 (1997). 2. M. O. Karlsson, T. Anehall, L. E. Friberg, A. Henningsson, C. Kloft, M. Sandstrom, and R. Xie, Pharmacokinetic/pharmacodynamic modelling in oncological drug development. Basic Clin Pharmacol Toxicol 96:206–211 (2005). 3. D. R. Huntjens, M. Danhof, and O. E. la Pasqua, Pharmacokinetic-pharmacodynamic correlations and biomarkers in the development of COX-2 inhibitors. Rheumatology 44:846–859 (2005). 4. S. Chabaud, P. Girard, P. Nony, and J. P. Boissel, Clinical trial simulation using therapeutic effect modeling: application to ivabradine efficacy in patients with angina pectoris. J Pharmacokinet Pharmacodyn 29:339–363 (2002). 5. L. E. Friberg, A. Henningsson, H. Maas, L. Nguyen, and M. O. Karlsson, Model of chemotherapy-induced myelosuppression with parameter consistency across drugs. J Clin Oncol 20:4713–4721 (2002).
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6. T. M. Post, W. de Winter, J. DeJongh, I. Moules, R. Urquhart, D. Eckland, and M. Danhof, Treatment efficacy of combination-therapy based on a mechanistic characterisation of disease processes in type 2 diabetes mellitus over a two-year period. Abstracts of the Annual Meeting of the Population Approach Group in Europe. PAGE 14:Abstr 757 (2005). 7. P. Hovind, P. Rossing, L. Tarnow, U. M. Smidt, and H. H. Parving, Progression of diabetic nephropathy. Kidney Int 59:702–709 (2001). 8. O. Brinchmann-Hansen, K. Dahl-Jorgensen, L. Sandvik, and K. F. Hanssen, Blood glucose concentrations and progression of diabetic retinopathy: the seven year results of the Oslo study. BMJ 304:19–22 (1992). 9. S. M. Grundy, J. I. Cleeman, C. N. Merz, H. B. Brewer, Jr., L. T. Clark, D. B. Hunninghake, R. C. Pasternak, S. C. Smith, Jr., and N. J. Stone, Implications of recent clinical trials for the National Cholesterol Education Program Adult Treatment Panel III guidelines. Arterioscler Thromb Vasc Biol 24:e149–e161 (2004). 10. D. M. Jonker, L. A. Kenna, D. Leishman, R. Wallis, P. A. Milligan, and E. N. Jonsson, A pharmacokinetic-pharmacodynamic model for the quantitative prediction of dofetilide clinical QT prolongation from human ether-a-go-go-related gene current inhibition data. Clin Pharmacol Ther 77:572–582 (2005). 11. K. G. Kowalski, L. McFadyen, M. M. Hutmacher, B. Frame, and R. Miller, A two-part mixture model for longitudinal adverse event severity data. J Pharmacokinet Pharmacodyn 30:315–336 (2003). 12. G. Graham, S. Gupta, and L. Aarons, Determination of an optimal dosage regimen using a Bayesian decision analysis of efficacy and adverse effect data. J Pharmacokinet Pharmacodyn 29:67–88 (2002). 13. N. L. Dayneka, V. Garg, and W. J. Jusko, Comparison of four basic models of indirect pharmacodynamic responses. J Pharmacokinet Biopharm 21:457–478 (1993). 14. P. L. Chan and N. H. Holford, Drug treatment effects on disease progression. Annu Rev Pharmacol Toxicol 41:625–659 (2001). 15. N. Holford, J. Li, L. Benincosa, and M. Birath, Population disease progress models for the time course of HAMD score in depressed patients receiving placebo in antidepressant clinical trials. Abstracts of the Annual Meeting of the Population Approach Group in Europe. PAGE 11:Abstr 311 (2002). 16. F. E. Harrell, Regression Modeling Strategies: With Applications to Linear Models, Logistic Regression, and Survival Analysis. Springer-Verlag, New York, 2001, pp. 79–83. 17. J. Ribbing and E. N. Jonsson, Power, selection bias and predictive performance of the population pharmacokinetic covariate model. J Pharmacokinet Pharmacodyn 31:109–134 (2004). 18. D. R. Mould, Defining covariate distribution models for clinical trial simulation, in Simulation for Defining Clinical Trials, H. Kimko and S. Duffull (Eds.). Marcel Dekker, New York, 2001, pp. 31–54. 19. R. R. Bies, M. R. Gastonguay, K. C. Coley, P. D. Kroboth, and B. G. Pollock, Evaluating the consistency of pharmacotherapy exposure by use of state-of-the-art techniques. Am J Geriatr Psychiatry 10:696–705 (2002). 20. P. Girard, T. F. Blaschke, H. Kastrissios, and L. B. Sheiner, A Markov mixed effect regression model for drug compliance. Stat Med 17:2313–2333 (1998). 21. P. Girard, L. B. Sheiner, H. Kastrissios, and T. F. Blaschke, Do we need full compliance data for population pharmacokinetic analysis? J Pharmacokinet Biopharm 24:265–282 (1996).
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22. J. A. Cramer, M. M. Amonkar, A. Hebborn, and R. Altman, Compliance and persistence with bisphosphonate dosing regimens among women with postmenopausal osteoporosis. Curr Med Res Opin 21:1453–1460 (2005). 23. J. J. McNabb, D. P. Nicolau, J. A. Stoner, and J. Ross, Patterns of adherence to antiretroviral medications: the value of electronic monitoring. AIDS 17:1763–1767 (2003). 24. A. J. O’Malley and S. L. Normand, Likelihood methods for treatment noncompliance and subsequent nonresponse in randomized trials. Biometrics 61:325–334 (2005). 25. L. B. Sheiner, A new approach to the analysis of analgesic drug trials, illustrated with bromfenac data. Clin Pharmacol Ther 56:309–322 (1994). 26. C. Hu and M. E. Sale, A joint model for nonlinear longitudinal data with informative dropout. J Pharmacokinet Pharmacodyn 30:83–103 (2003). 27. J. W. Hogan, J. Roy, and C. Korkontzelou, Handling drop-out in longitudinal studies. Stat Med 23:1455–1497 (2004). 28. A. M. Law and W. D. Kelton, Selecting input probability distributions, in Simulation Modeling and Analysis, 3rd ed. McGraw-Hill, New York, 2000, pp. 292–401. 29. I. Nestorov, I. Gueorguieva, H. M. Jones, B. Houston, and M. Rowland, Incorporating measures of variability and uncertainty into the prediction of in vivo hepatic clearance from in vitro data. Drug Metab Dispos 30:276–282 (2002). 30. D. Farrar, B. Allen, K. Crump, and A. Shipp, Evaluation of uncertainty in input parameters to pharmacokinetic models and the resulting uncertainty in output. Toxicol Lett 49:371–385 (1989). 31. W. R. Gillespie, M. Sale, and M. R. Gastonguay, Modeling uncertainty in clinical trial simulation: a Bayesian approach. Clin Pharmacol Ther 65:Abstr, PIII-21 (1999). 32. H. Kraiczi and M. Frisen, Effect of uncertainty about population parameters on pharmacodynamics-based prediction of clinical trial power. Contemp Clin Trials 26:118– 130 (2005).
APPENDIX 35.1
NONMEM CODE FOR EFFICACY TRIAL SIMULATION
The code does not include parameters for uncertainty. Parameter ZDVSL noted in chapter text is termed BZDV in the NONMEM code. $PRED ;FIRST RANDOM SOURCE IS FOR ETAS IF (ICALL.LE.1) RETURN ;IF BEGINNING OF RUN OR PROBLEM Y=0 ;2ND RANDOM SOURCE (UNIFORM) IF (ICALL.EQ.4) THEN ; FOR SIMULATION OF SURVIVAL TIMES GIVEN HAZARD FUNCTION CALL RANDOM (2,R) ;RANDOM SEED FROM UNIFORM (0,1) X=R ENDIF ;2ND RANDOM SOURCE (UNIFORM) IF (ICALL.EQ.4) THEN ; FOR SIMULATION OF SURVIVAL TIMES GIVEN HAZARD FUNCTION CALL RANDOM (2,R) ;RANDOM SEED FROM UNIFORM (0,1) X2=R ENDIF
NONMEM CODE FOR EFFICACY TRIAL SIMULATION
897
;NOW TAKE CARE OF PATIENT DEMOGRAPHICS ;FOR PROBABILITIES, USE TRANSFORMATION PR=EXP(THETA)/(1+EXP(THETA)) TO CONSTRAIN 0-1 IF (ICALL.EQ.4.AND.NEWIND.NE.2) THEN ;FOR $SIM AND 1ST RECORD OF EACH IND ;EACH RECORD IN THIS DATA SET IS THE ;FIRST RECORD FOR THAT INDIVIDUAL ;ONLY 1 RECORD PER INDIVIDUAL ;2ND RANDOM SOURCE (UNIFORM) - FOR HEPATIC DISEASE (3% OF PATIENTS) CALL RANDOM (2,R) RH=R HEP=0 PRHD=EXP(THETA(10))/(1+EXP(THETA(10))) ;PROBABILITY OF HEPATIC DISEASE - THIS WILL HAVE UNCERTAINTY IF (RH.GT.(1-PRHD)) HEP=1 ;HEPATIC DISEASE ;2ND RANDOM SOURCE (UNIFORM) - FOR SEX (79% MALE, 21% FEMALE) CALL RANDOM (2,R) RS=R SEX=0 ;MALES PRFE=EXP(THETA(11))/(1+EXP(THETA(11))) IF (RS.GT.(1-PRFE)) SEX=1 ;FEMALES ;2ND RANDOM SOURCE (UNIFORM) - FOR METHADONE USE (4.4% OF PATIENTS) CALL RANDOM (2,R) RM=R METH=0 PRME=EXP(THETA(12))/(1+EXP(THETA(12))) ;PROBABILITY OF METHADONE USE IS 0.044 IF (RM.GT.(1-PRME)) METH=1 ;ON METHADONE ;3RD RANDOM SOURCE (NORMAL) - FOR WEIGHT (NORMAL WITH MEAN OF 73 FOR MALES, ; 60 FOR FEMALES CV=20%, RANGE=40-100) WT=1 DO WHILE (WT.LT.40.OR.WT.GT.100) CALL SIMETA (ETA) LBWT=40 UBWT=100 RGWT=UBWT-LBWT WT1=LBWT+RGWT*EXP(THETA(13))/(1+EXP(THETA(13))) WT2=LBWT+RGWT*EXP(THETA(14))/(1+EXP(THETA(14))) TVWT=(WT1*(1-SEX) + WT2*SEX) WT=TVWT*(1 + ETA(7)) ENDDO
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CLINICAL TRIAL SIMULATION: EFFICACY TRIALS
;1ST RANDOM SOURCE (NORMAL) - FOR CD4+ COUNT (NORMAL WITH MEAN OF 348, SD=90, MAX=500) CD4=0 CD=501 DO WHILE (CD.LE.0.OR.CD.GT.500) CALL SIMETA (ETA) LBCD=0 UBCD=500 RGCD=UBCD-LBCD TVCD=LBCD+RGCD*EXP(THETA(15))/(1+EXP(THETA(15))) CD=TVCD+ETA(6) CDPT=EXP(THETA(16)) ; BREAKPOINT FOR SIGNIFICANT CD4 EFFECT ON EFFICACY IF (CD.GE.CDPT) CD4=1 ENDDO ;2ND RANDOM SOURCE (UNIFORM) - FOR DOSE ASSIGNMENT CALL RANDOM (2,R) RD=R DOSE=0 IF (RD.GE.0.33.AND.RD.LT.0.67) DOSE=500*28 IF (RD.GE.0.67) DOSE=1500*28 ; DOSE FOR 4 WEEKS ;USE 2ND RANDOM SOURCE FOR COMPLIANCE MODEL - COMPLIANCE IS RANDOM ;COMPLIANCE IS MODELED AS A BINOMIAL DISTRIBUTION WITH 90% HI COMPLIANCE MEAN=80%, SD=10 ;AND 10% LOW COMPLIANCE COMPL=40%, SD=5 (MIN=0, MAX=100%) ;CALL 2ND RANDOM SOURCE (UNIFORM) - FOR TYPE OF COMP (90% HIGH, 10% LOW) CALL RANDOM (2,R) RC=R CTYP=0 ;HIGH COMPLIANCE PRHI=EXP(THETA(17))/(1+EXP(THETA(17))) IF (RC.GT.PRHI) CTYP=1 ;LOW COMPLIANCE COMP=1.1 DO WHILE (COMP.LT.0.OR.COMP.GT.1) CALL SIMETA (ETA) CMP1=EXP(THETA(18))/(1+EXP(THETA(18))) CMP2=EXP(THETA(19))/(1+EXP(THETA(19))) COMP=(CMP1+ETA(8))*(1-CTYP) + (CMP2+ETA(9))*CTYP ENDDO ADOS=DOSE*COMP ;ACTUAL DOSE=ASSIGNED DOSE*COMPLIANCE ENDIF ; CL=1.3+/-0.3 L/KG/HR CL IN L/4 WEEKS = CL L/KG/HR*24HRS/DAY *28DAYS*WT
NONMEM CODE FOR EFFICACY TRIAL SIMULATION
899
; USE EXP MODEL AND 23% INTERINDIVIDUAL CV INSTEAD TVCL=EXP(THETA(1)) CL1 = TVCL *EXP(ETA(1)) CL2=(EXP(THETA(2)))**HEP CL3=(EXP(THETA(3)))**SEX CL4=(EXP(THETA(4)))**METH CL = CL1*WT*24*28*CL2*CL3*CL4 CAVG = ADOS/CL ;(AVERAGE CSS) TVIC=EXP(THETA(5)) IC50 = TVIC*EXP(ETA(2)) TVHP=EXP(THETA(6)) HP = TVHP*EXP(ETA(3)) TVBC=EXP(THETA(7)) BCD4 = TVBC*EXP(ETA(4)) TVZD=EXP(THETA(8)) BZDV = TVZD*EXP(ETA(5))
;HAZARD MODEL FOR BASELINE CD4 EFFECT ON EFFICACY HAZ = HP*EXP(-BCD4*CD4 + (-BZDV*CAVG/(CAVG+IC50)))
;IN THIS CASE, THE HAZARD FUNCTION IS CONSTANT OVER TIME FOR EACH INDIVIDUAL ;THE OBSERVATION IS THE EVENT TIME ;X IS RANDOM NUMBER DRAWN FROM A UNIFORM (0,1) DISTRIBUTION CTIME=24 ;24 MONTHS OBS1=LOG(1/(1-X))/(HAZ) CEN1=1 ;FLAG INDICATING THAT THE EVENT DID HAPPEN BEFORE CENSORING TIME IF (OBS1.GT.CTIME) THEN CEN1=0 ;EVENT HAPPENS AFTER CENSORING TIME OBS1=CTIME ENDIF ;RANDOM DROPOUT EVENT: Pr(T15% improvement in score) increased with dose. The results demonstrated that the probability of reaching the highest score for the PD endpoint was not significantly higher at 60 mg when compared with the 30 mg dose. Overall, it appeared that the population response reached a plateau at approximately 30 mg; therefore, administration of doses greater than 30 mg did not provide additional benefit. This conclusion was consistent with the findings resulting from the efficacy data reported in the Phase 3 trials. Based on the pooled data from the three Phase 3 studies, there was a trend toward dose-dependent increases in the frequency of drug-related adverse events, particularly dyspepsia, myalgia, and back pain, although the overall absolute incidence was between 0.5% and 10%. This was dependent on the event and drug dose. Given the results of the pharmacometric analysis, a starting dose of 30 mg was recommended for patients with the functional disorder who are otherwise healthy. With regard to special populations, systemic Enhibitor exposure was approximately twofold higher for the parent drug and three- to fourfold higher for the major metabolite in subjects with mild and moderate renal impairment following administration of the 30 mg dose. The increased systemic Enhibitor exposure in this population was associated with significant increase in the incidence of drug-related
DRUG ENHIBITOR
941
1
Probability of Achieving the Efficay Score
0.9 0.8 0.7 0.6 0.5 0.4 0.3
Efficacy Score = 5
0.2
Efficacy Score = 3 0.1 0 0
10
20
30
40
50
60
70
80
Dose (mg)
FIGURE 38.1 Estimated probability of achieving a given score of efficacy measure for Enhibitor. In this plot, the efficacy measure is the probability of achieving a desired clinical efficacy score (or percentage of patients who achieve the desired efficacy score). Two representative curves are displayed—one for when the desired efficacy score is 5 (solid line), and the other when the desired efficacy score is 3 (dashed line). When the desired clinical efficacy score is 3 (i.e., clinically satisfactory), the probability of clinically achieving this score plateaued at 20 mg of dose (the probability is 93%). Increasing dose will not further increase this probability. If the desired efficacy score is 5 (clinically very satisfactory), the probability of achieving this score plateaued somewhere beyond the 75 mg dose, but the added benefit from 30 mg to 60 mg is relatively small (increased from 60% to 66%). Therefore, doses greater than 30 mg would not likely add significant clinical benefit.
adverse events. Due to the increased incidence of adverse events in moderate renal impairment subjects, the exposure of Enhibitor was not investigated in patients with severe renal impairment. It should be noted that all the pivotal Phase 3 trials for Enhibitor excluded patients with clinically significant renal failure. 38.2.5
Results
Based on the E-R data analysis, there were no significant differences (p > 0.05) in response between 30 and 60 mg doses. Both 30 and 60 mg doses were statistically significantly better than placebo. There did not seem to be any additional benefit with the 60 mg dose compared to the 30 mg dose. Based on the increased Enhibitor exposure of parent drug and the metabolite in patients with moderate renal impairment, and the associated increase in incidence of adverse events, a dose of 15 mg was recommended in this patient population.
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38.2.6
PHARMACOMETRICS APPLICATIONS IN POPULATION EXPOSURE–RESPONSE DATA
Conclusions
The sponsor had requested approval for only a 60 mg dose. Based on the E-R analysis of the data from Phase 2 and Phase 3 studies, a lower starting dose of 30 mg in patients with the functional disorder who are otherwise healthy was recommended. Based on the evidence of increased Enhibitor exposure and adverse events in renal impairment patients, it was concluded that dosing in patients with moderate renal impairment should not exceed 15 mg and the drug should be contraindicated in patients with severe renal impairment.
38.3 38.3.1
DRUG BOTANI Objective
The objective of this analysis was to determine the lowest effective dose of a botanical drug product, Botani, indicated for a chronic disorder. The drug was to be given chronically over many years. The sponsor requested approval of a dose that is similar or better than a marketed comparator, Compara. 38.3.2
Data Resource
Botani and Compara were both administered once daily for 8 weeks. There was a run-in period and follow-up phase. Data from Phase 2 dose–response studies were available. The E-R relationship was characterized in two Phase 2 clinical trials with the daily doses of 0, 3, 7.5, 15, 30, 60, 120, and 240 mg. Percent change of plasma R from baseline (ΔR%) was used as a PD clinical endpoint. 38.3.3
Data Transfer
Data was transferred from SAS transport file to ASCII format using StatTransfer. Final data sets for the consequent analysis were saved in S-Plus data structure. 38.3.4
Data Analysis and Results
Data analyses included data visualization, nonparametric statistical analysis on observations (data from study 1 only), and parametric analysis with nonlinear mixed effects modeling. 38.3.4.1 Data Visualization Various plots were generated to check ΔR% versus treatment time relationship, ΔR% versus dose relationship, variability, relative potency between Botani and Compara, and the effect of covariates on the response variable. Botani exposure– response plots indicated that plasma R gradually drops and essentially reaches plateau over the 6 week treatment time. The profiles of ΔR% over time showed similar pattern for both Botani and Compara (Figure 38.2). After approximately 4 weeks of treatment time, percent change of endpoint (ΔR%) reaches 86–90% of the maximum effect for a given dose. This was observed for all dose levels studied (3–240 mg). It did not appear that any covariates (age, sex, baseline R, etc.) had any
943
DRUG BOTANI
0
28
56
120 mg
240 mg
15 mg
30 mg
84
11
Female Male Compara Missing PLACEBO Botani
14 Na
40 20 0 -
60 mg
40 20 0 -
R (%) Placebo
3 mg
7.5 mg
40 20 0 0
28
56
84 112 140
0
28
56
84 112 140
Day
FIGURE 38.2 Plots of ΔR% over time by dose and treatment for study 1 (a comparative study of Botani and Compara). Data show treatment effect over time and visual comparisons on relative potency of Botani and Compara. At all doses, the ΔR% gradually increases and reaches steady states. When the drug is withdrawn, R slowly returns to the baseline level. The new drug, Botani, produced more R% reduction than Compara as shown at the 30 mg dose. In addition, there is a clear dose–response relationship for Botani.
effect on ΔR%. Botani produced up to approximately 4× R lowering effect when compared with Compara, in both studies (Figure 38.3). As shown in Figure 38.3, the maximum effect for Botani was approximated at the 30 mg dose. 38.3.4.2 Data Analyses Botani’s dose–response and concentration–response analyses were conducted. In the analyses, no effort was made on checking per protocol (PPT) data, but the focus was on the intention-to-treat (ITT) data. Analyses focused on the dose–response data in study 2 and study 1 data were used for external model validation (see Chapter 8 for a detailed discussion on model validation). Data from other studies were not modeled. Since concentration–effect data from study 2 did not appear to offer any advantage when compared with the dose–response data, it was examined without detailed modeling analyses. Statistical comparisons between R% reductions at a given Botani dose versus 4× Compara dose resulted in the 90% CI of the mean ratios of the two treatments falling within 80–125%, indicating that the mean observations or parameters for the two drugs at doses 4× different were equivalent. Some statistical analysis results of Botani’s E-R data are presented in Tables 38.1. The ΔR% reduction at week 6 was not significantly different from that at week 4. Results of mean (and SD) of R% reduction of study 1 and study 2 were
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PHARMACOMETRICS APPLICATIONS IN POPULATION EXPOSURE–RESPONSE DATA
FIGURE 38.3 Plot of ΔR% versus dose by treatment. Data show visual comparisons on relative potency of Botani and Compara at all dose levels. In all doses, the new drug Botani produces a greater ΔR% than Compara. The dose–response relationship is clearly demonstrated. The data points are observed ΔR%; the lines are links of the mean values at various doses.
TABLE 38.1 Mean ΔR% at Various Dose Levels for Botani at Week 6 Botani Dose (mg/day)
Mean ΔR% at Week 4
Mean ΔR% at Week 6
Study 1 Observed ΔR% at Week 6 (Meanand SD)
Study 2 Observed ΔR% at Week 6 (Meanand SD)
Approximate Minimum ΔR% at Week 6 in About 85% of Patients
0 3 7.5 15 30 60
−1.38 −36.67 −39.16 −42.55 −47.39 −53.09
−1.99 −38.32 −40.93 −44.47 −49.53 −55.9
−0.598 (7.18) −35.23 (8.81) −41.57 (9.27) −44.63 (7.16) −49.43 (17.0) −54.26 (11.9)
−1.318 (6.58) — — −41.6 (9.94) −49.95 (10.67) −52.21 (9.86)
— −26 −28 −32 −39 −43
very consistent. Also, at the 15 mg dose, R% reduction is more than 30% in 85% of subjects, yielding a clinically significant therapeutic outcome. 38.3.4.3 Nonlinear Mixed Effects Modeling Analyses The objective of this analysis was to integrate all of the above information for making a final recommendation on optimal therapeutic dose of Botani. Nonlinear mixed effects modeling analyses were conducted only on dose–response data from study 2 because of its completeness at multiple dose levels and larger number of subjects. A total of 374 subjects with 1816 observations were included in the data analyses. An inhibitory effect Emax model describes the response–time relationship (at a given
DRUG BOTANI
945
dose) and response–dose relationship (at a given time point) very well. However, concentration–response data did not offer any advantage over dose–response data in these modeling analyses. The nonlinear mixed effects analyses were conducted using Pharsight WinNonMix software. Model-building criteria for adding covariate effects were based on objective function change by more than 30 units (p < 0.001) and the examination of diagnostic plots. An Inhibitory Maximum Effect function available in WinNonMix was used for both time effect and dose effect on ΔR%. The FOCE BLOCK method was applied. Intersubject variability was modeled as lognormal distribution and residual error was modeled unweighted. The final nonlinear mixed effects model was (applicable to the dose range 3–240 mg only) Max observed ΔR% = Baseline (day 0) + Treatment time effect + Dose effect + Drug effect Baseline at day 0 = − 1.99% ΔR% over time = Baseline − Emax ⋅ (1 − exp( − K ⋅ Time))
(38.1)
where Emax is a regression parameter to describe the maximum effect at time infinity and is drug and dose dependent: K = 0.1657 − 0.00789 ⋅ ( trt-1) Emax = − 39.94 − 38.40 ⋅ Dose (16.43 + Dose) + 10.03 ⋅ ( trt-1) trt = 1 if drug is Botani; trt=2 if drug is Compara
(38.2)
The model excellently predicted individual ΔR% over time as indicated in a set of goodness-of-fit diagnostic plots (not shown) for study 1. The predicted mean values have less than ±4% error. Post hoc individual prediction for all 374 subjects was excellent (see the appendix for details and individual predictions). The c2 statistics confirmed that there were significant treatment effects, dose effects, and time effects on ΔR% profiles for Botani and Compara. Based on the model, it was concluded that: (a) Botani offered superior potency to Compara; for example, 30 mg Botani produced approximately equal degree of R% lowering effect as 120 mg Compara; (b) at the same dose level (e.g., 30 mg), Botani would produce an additional 20% R lowering effect than Compara (e.g., 49.95% drop in R for Botani versus 37.87% drop in R for Compara); and (c) at least 4 weeks of treatment time are needed to approximate the corresponding maximum effect for all doses. 38.3.4.4 Results E-R analyses indicated the following. Significant reductions in plasma R were seen within 1 week of therapy and most of the total effect was achieved by 4 weeks. The extent of R% lowering was dose-related and was in the range of −35% to −60% for 30–60 mg doses. Although the sponsor had proposed to market 30, 60, 120, and 240 mg doses, the E-R data showed that doses lower than the 30 mg also reduced plasma R levels significantly. A 7.5 mg dose reduced the plasma R concentrations by a mean of 40% and a 3 mg dose, the lowest dose studied, reduced plasma R by approximately a mean of 35%. In addition, prediction based on nonlinear mixed effects modeling
946
PHARMACOMETRICS APPLICATIONS IN POPULATION EXPOSURE–RESPONSE DATA
0.03
0.02
0.01
0.00 –120
–100
–80
–60
–40 ΔP%
–20
0
FIGURE 38.4 Simulated population distribution plots of ΔR% for doses 3, 15, and 30 mg. Data are from the NONMEM model prediction. If ΔR% = 20% is assumed to be a clinical effective benefit marker, the increase in the number of patients who will benefit from the treatment is small when the dose was increased from 3 to 30 mg.
indicated that the approximate minimum ΔR% at 6 weeks in about 85% of patients who took the 3 mg, 7.5 mg, 15 mg, and 30 mg doses were −25%, −30%, −35%, and −40%, respectively. The efficacy measure appeared to plateau at the 120 mg dose with little additional benefit achieved when titrated from 3 to 30 mg (Figure 38.4). This absence of additional benefit was coincident with greater risk for myopathy and rhabdomyolysis observed at higher doses. 38.3.5
Conclusions
Although the sponsor had proposed to market 30, 60, 120, and 240 mg doses, the E-R data indicated that doses lower than the 30 mg were efficacious. Considering the potential toxicity at higher doses and the increased risk of greater exposure in the patient subgroup (e.g., renally impaired), it was recommended that the optimal starting dose be lower than 30 mg (5 and 15 mg doses). The safe use of drug Botani would require the availability of low-dosage strengths (e.g., for patients treated with other interacting drugs, patients with renal impairment, or the elderly).
38.4
DRUG AMICID
This Investigational New Drug (IND) application for this drug was initially submitted in July 2001. Drug Amicid is a water-soluble hormone antagonist. The product is formulated as a lyophilized powder, reconstituted with sterile water and mannitol for subcutaneous and intramuscular injection, or with sterile 5% glucose solution for intravenous infusion. Drug Amicid is relatively safe. The single dose tested was up to 360 mg without adverse effects. There was a very low ADR rate. The sponsor conducted several Phase 2 trials with various subcutaneous injection volumes, dose, dosing interval, and length of durations. With these data, the sponsor
DRUG AMICID
947
conducted comprehensive modeling and simulation work to design the optimal dose and dose regimen for their planned Phase 3 trials, including monthly and trimonthly dose regimens. Sponsor requested an end of Phase 2a (EOP2A) meeting to receive the FDA’s input on the modeling & stimulation (M&S) performed, and the Phase 3 clinical program for the drug. Given the amount of trough concentration data available, the data were subdivided into 1 ng/mL bins. The corresponding percentage of subjects within each concentration bin who met a clinical efficacy measure criterion was calculated. The clinical group desired the efficacy rate to be greater than 90%. Figure 38.5 is a composite plot of such data. It indicates that if subjects maintained Amicid trough concentrations within 5–6 ng/mL, the clinical success rate would be about 92%. Since controlling mean trough drug concentrations in a population was more practical, it was necessary to incorporate variability, factoring in the concentrations and, with the above criterion, the percentage of subjects who had Amicid trough concentrations lower than 5 ng/mL at various mean Amicid trough concentrations were generated and plotted (see Figure 38.6). From the figure it can be observed that when the mean concentration was 10 ng/mL, the percentage of subjects with trough concentrations less than 5 ng/mL was less than 1%, and the 10 ng/mL average concentration appears to be the inflection point on the curve. When the mean plasma drug concentration was less than 10 ng/mL, the percentage of subjects who
10
Percent Success Rate
90 1 to 1.5 month 1.5 to 3 month 3 to 6 month 6 to last day
80
70
60
50
0
5
10
15
20
25
Average Amicid Conc. at each bin (ng/mL)
FIGURE 38.5 Percent success rate versus drug concentration. Data were generated by binning the concentrations (see text for explanation). Different symbols show data from different treatment times. Tolerance was not observed. If all subjects have trough Amicid concentration of 5 ng/mL, the success rate would be about 92%. Due to concentration variability in a population, a mean concentration in a population of 7.5 ng/mL corresponds to a success rate ranging from 70% to 97% (mean 94%, as shown in the darker square), and a success rate of 92–97% (mean 96.4%, the lighter square) for a corresponding mean concentration of 9.5 ng/mL.
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PHARMACOMETRICS APPLICATIONS IN POPULATION EXPOSURE–RESPONSE DATA
10
77 7 777 7 7 6 77 76
9
6
8 % Amicid 5 ng/mL
6
7 6
7 6
7 77
7 2 76722
7
7 7
2
6 7
2
6
72 7
7
7 2 6 2
272
6
7
7
7
2
77
7
5
2 2 22 2
77
7 7
7
4
7
7
2
22
22 2 2227 2
3
222 7
2
2
2
2 2 2
1
14
2 2
12 12
14
12
2 14
11
0 0.
2.
5.
7.
12
1 1
10.
14 12 12
1 11 1
1 1
12.
1
1
15.
17.
20.
Mean Amicid Concentration
FIGURE 38.6 The percentage of subjects who have trough Amicid concentrations lower than 5 ng/mL at various mean Amicid concentrations. When mean Amicid concentration is greater than 10 ng/mL, the percentage of subjects whose Amicid concentration is less than 5 ng/mL is about 2–4%. However, a further increase in the mean concentration will not offer much added benefit to reduce the percentage of subjects with trough Amicid concentrations less than 5 ng/mL.
would not be able to maintain Amicid trough concentrations greater than 5 ng/mL increased in an almost linear manner. Furthermore, an increase in mean concentration beyond 10 ng/mL would not offer any substantial improvement in decreasing the fraction of subjects with plasma Amicid concentration lower than 5 ng/mL. Based on the above information, trial design was simulated using the sponsor’s PK model with the criterion that the mean drug concentration remains on or slightly above 9.5 ng/mL. Furthermore, various dose–dosage regimen combinations were simulated to recommend the optimal dose–dosage regimen combination. The sponsor accepted the Agency’s recommendation for the dose–dosage regime combination to study in their Phase 3 clinical trials.
38.5
DISCUSSION
According to the ICH-E4 guideline (12) and certain FDA guidances (12), a knowledge of the relationships among dose, drug concentration, and clinical response (desirable and undesirable effects) can (a) guide the selection of an appropriate starting dose, (b) guide the selection of an adjusted dose for patient subgroups, and (c) identify a dose beyond which increases would be unlikely to provide added benefit or would produce unacceptable side effects.
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949
For a pharmaceutical product that is prescribed to a patient, the ideal goal is for the patient to attain maximum efficacy (i.e., maximum benefit) with minimal adverse events (i.e., limited risk). These examples illustrate how E-R relationships can guide the selection of not only an efficacious dose but an optimal dose as well. Using traditional approaches for assessing E-R relationships and population PK/PD methodologies, optimal doses for Drug Enhibitor and Drug Botani were chosen from several available efficacious doses. Without such information, higher than necessary doses may have been approved with associated greater risks of adverse events. In the case of drug Amicid, an understanding of the E-R relationship led to the recommendation of the right dose to be studied in the Phase 3 clinical program. Dose finding in Phase 3 is also important for dosage optimization (14).
38.6
SUMMARY
In conclusion, these examples taken from two recently submitted NDAs to the FDA and an EOP2A meeting between FDA and a sponsor demonstrate that E-R information can significantly contribute to a better understanding of optimal doses and dosage regimens. It is also important that dose finding not only occur in Phase 2 but that it continues into Phase 3 in order to optimize dosing.
REFERENCES 1. Innovation or Stagnation: Challenge and Opportunity on the Critical Path to New Medical Products. FDA, Rockville, MD, 2004. http://www.fda.gov/oc/initiatives/ critcalpath/whitepaper.html.
2. 3. 4. 5.
6. 7. 8.
9.
10.
11.
PDUFA of 1992. http:/www.fda.gov/oc/pdufa. FDAMA of 1997. http://www.fda.gov/cder/fdama. CDER Annual Report. http://www.fda.gov/cder/reports/rtn/2001/rtn2001.htm. J. L. Lesko, M. Rowland, C. C. Peck, and T. F. Blasche, Optimizing the science of drug development: opportunities for better candidate selection and accelerated evaluation in humans. J Clin Pharmacol 40:803–814 (2000). C. C. Peck, Drug development: improving the process. Food Drug Law J 52:163–167 (1997). P. Roland, The contribution of clinical pharmacology surrogates and models to drug development—a critical appraisal. Br J Clin Pharmacol 44:219–225 (1997). H. Derendorf, L. J. Lesko, P. Chaikin, W. A. Colburn, P. Lee, R. Miller, R. Powell, G. Rhodes, D. Stanski, and J. Venitz, Pharmacokinetic/pharmacodynamic modeling in drug research and development. J Clin Pharmacol 40:1399–1418 (2000). J. S. Cohen, Dose discrepancies between the PDR and the medical literature, and their possible role in the high incidence of dose-related adverse drug events. Arch Intern Med 161:958–964 (2001). J. Cross, H. Lee, A. Westelinc, J. Nelson, C. Grudzinska, and C. Peck, Postmarketing drug dosage changes of 499 FDA-approved new molecular entities, 1980–1999. Pharmacoepidemiol Drug Safety 11:439–446 (2002). E. R. Heerdink, J. Urquhart, and H. G. Leufkens, Changes in prescribed drug doses after market introduction. Pharmacoepidemiol Drug Safety 11:447–453 (2002).
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12. http://www.fda.gov/cder/guidance/index.htm: ICH E4: Dose–Response Information to Support Drug Registration (1994). Exposure–Response Relationships: Study Design, Data Analysis and Regulatory Applications (2003). Providing Clinical Evidence of Effectiveness for Human Drug and Biologic Products (1998). ICH E9: Guidance on Statistical Principles for Clinical Trials (1998). 13. R. Temple, Dose–response and registration of new drugs, in Dose–Response Relationships in Clinical Pharmacology, L. Lasagna, S. Eri, and C. A. Naranjo (Eds.). Elsevier, Amsterdam, 1989, pp. 145–167. 14. L. B. Sheiner, Learning vs. confirming in clinical drug development. Clin Pharmacol Ther 61:275–291 (1997).
APPENDIX 38.1 WINNONMIX NONLINEAR MIXED-EFFECTS ESTIMATION PROGRAM (V2.0.1) Core Version 28NOV2000 Listing of Input Commands: MODEL DNAMES ID~1 OBDAY~2 PD~3 WEEK~4 TRT~5 DOSE~6 TRT2~7 DAY~8 METHOD 1 /R MINIMIZATION 0 /STEP=1 DINCREMENT 0.001 NPOINTS 100 ITERATIONS 100 CONVERGENCE FUNC /TOL=0.0001 SUBJECT ID XNAME DAY YNAME RP STDERR 0 MIXEFFECTS BASE=BASE1_0 ;Baseline term in equ 38.1 K=(K0_1+K0_2*(TRT2-1))*EXP(K0_ETA0) ;K term in equ 38.1 E=(E0_0+E0_1*DOSE/(E0_2+DOSE)+E0_3*(TRT2-1))*EXP(E0_ETA0);Emax term in equ 38.2 END INITIAL 1: (-2) (0.19) (0.002) (-42) (-35) (15) (8) END NOBOUND VFUNCTION IDENTITY OMEGA K0_ETA0 E0_ETA0 END Command Parsing Completed. Data Input Finished.
APPENDIX 38.1
951
Ordinary Least Square Estimates of Fixed Effects
EMAX_0 -2.2878E+00
EC50_0 3.9733E+00
EC50_1 -4.1081E-01
E0_0 -5.2616E+01
E0_1 -2.2337E-01
E0_2 9.3982E+00
Initial Estimates of Random Effects (For All ID)
EC50_ETA0 0.0000E+00
E0_ETA0 0.0000E+00
Initial Estimates of Covariance Parameters
SIGMA^2: Covariance Matrix of Random Effects: EC50_ETA0 E0_ETA0 EC50_ETA0 1.3998E+00 E0_ETA0 7.6006E-02
5.6588E+01
Computation of Initial Estimates Completed.
REML Estimation Iteration History
Iteration 1 EMAX_0 -1.9804E+00
Objective Criterion 21711.9269 1.0000 EC50_0 EC50_1 E0_0 4.2012E+00 1.3577E-01 -5.1956E+01
E0_1 -2.7583E-01
E0_2 8.8429E+00
Iteration 2 EMAX_0 -1.9804E+00
Objective Criterion 21711.9269 0.0000 EC50_0 EC50_1 E0_0 4.2012E+00 1.3577E-01 -5.1956E+01
E0_1 -2.7583E-01
E0_2 8.8429E+00
Convergence Achieved. Model Estimation Completed.
Class Level Information
Class
Levels Values
ID
374 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ……373 374
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PHARMACOMETRICS APPLICATIONS IN POPULATION EXPOSURE–RESPONSE DATA
Model Fitting Information Description
Value
Number of Subjects Total Observations Minimum Objective Function Value REML Log Likelihood Akaike’s Information Criterion (AIC) Schwarz’s Bayesian Criterion (SBC) -2 * REML Log Likelihood
374 1816 21711.9269 -6635.7629 13287.5257 13331.5344 13271.5257
Solution For Fixed Effects Parameter EMAX_0 EC50_0 EC50_1 Estimate -1.9804E+00 4.2012E+00 1.3577E-01 StdError 4.3267E-01 2.8473E-01 4.6437E-01
E0_0 -5.1956E+01 1.1187E+00
E0_1 -2.7583E-01 2.1160E-02
Parameter E0_2 Estimate 8.8429E+00 StdError 1.3235E+00 Covariance Parameter Estimates Parameter SIGMA^2
Estimate 6.1175E+01
StdError 6.3153E+00
Variance/Covariance Of Fixed Effects:
EMAX_0 EC50_0 EC50_1 E0_0 E0_1 E0_2
E0_2
EMAX_0 1.8720E-01 -5.4909E-03 -3.4663E-02 1.1176E-02 1.2942E-04 2.6031E-02
EC50_0
EC50_1
E0_0
E0_1
8.1072E-02 -7.3546E-02 -1.3797E-01 -2.4125E-04 1.3654E-01
2.1564E-01 1.3743E-01 -2.1655E-04 -3.3834E-01
1.2515E+00 -1.3464E-02 -6.9148E-01
4.4774E-04 -3.4679E-03
EC50_1
E0_0
E0_1
1.0000E+00 2.6456E-01 -2.2039E-02 -5.5050E-01
1.0000E+00 -5.6878E-01 -4.6703E-01
1.0000E+00 -1.2383E-01
E0_2 1.7517E+00
Correlation Of Fixed Effects: EMAX_0 EC50_0 EMAX_0 1.0000E+00 EC50_0 -4.4571E-02 1.0000E+00 EC50_1 -1.7252E-01 -5.5624E-01 E0_0 2.3091E-02 -4.3316E-01 E0_1 1.4137E-02 -4.0042E-02 E0_2 4.5458E-02 3.6231E-01
E0_2
E0_2 1.0000E+00
APPENDIX 38.1
Variance/Covariance Of Random Effects: EC50_ETA0 E0_ETA0 EC50_ETA0 2.6183E-01 E0_ETA0 2.0195E-02 Standard Error of Variance/Covariance Of Random Effects: EC50_ETA0 E0_ETA0 EC50_ETA0 6.0387E-02 E0_ETA0 2.9767E-03 Correlation Of Random Effects: EC50_ETA0 E0_ETA0 EC50_ETA0 1.0000E+00 E0_ETA0 1.0000E+00 Variance/Covariance Of Individual Estimates:
EMAX EC50 E0
EMAX 0.0000E+00 0.0000E+00 0.0000E+00
EC50
E0
7.7233E+01 6.6950E+00
7.6264E+01
Solution For Random Effects ID 1 2 37 38
EC50_ETA0 -4.3592E-01 -1.0799E+00 -2.2513E-01 -5.8753E-01
E0_ETA0 1.4015E-01 2.9815E-01 -2.8100E-02 8.0395E-02
Individual Parameter Estimates ID 1 2 373 374
EMAX -1.9804E+00 -1.9804E+00 -1.9804E+00 -1.9804E+00
EC50 2.8046E+00 1.4730E+00 3.7591E+00 5.1892E+00
E0 -4.9600E+01 -5.8090E+01 -4.0335E+01 -3.3586E+01
Estimates Of Secondary Parameters ID 1 2 373 374
EMAX-E0 4.7620E+01 5.6110E+01 1.2392E+00 1.0547E+00
Program Completed (total time used: 00:02:48.45). Normal Ending.
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CHAPTER 39
Pharmacometrics in Pharmacotherapy and Drug Development: Pediatric Application EDMUND V. CAPPARELLI and PAUL J. WILLIAMS
39.1
INTRODUCTION
The use of medication by children is widespread, with 20% of school-age children receiving one or more prescription drugs each year. Given the extensive use and altered pharmacokinetics resulting in a wide range of drug exposures, it is not surprising that recent history is replete with tragic consequences of drug administration to pediatric patients when pediatric clinical pharmacology information is lacking. To promote healthy children, it is of paramount importance that optimal dosing strategies be determined. The development of dosing strategies in this population is, in general, more complex than adults because of the diversity in the pediatric population, which is much greater than adults. For example, this patient population can range from several hundred grams to over a hundred kilograms and there is diversity of maturation where organs of elimination have varying degrees of functionality at different ages. Many biomarkers also have age-dependent ranges and therefore can exhibit significant within-subject maturational changes even over the course of a study. Size, age, ongoing growth, altered disease progression, and maturation are only several of the factors that impact dosing strategy in pediatric patients. Comprehensive and well defined development of pharmacokinetic (PK), pharmacodynamic (PD), and outcomes models are a necessity for the generation of optimal pediatric therapy. By the mid-1960s the problem of administering inappropriate drug doses to pediatric patients had been clearly documented. In 1959 reports of the gray baby syndrome in neonates were published documenting the toxicity of chloramphenicol when adult doses were “miniaturized” to infants without consideration of maturation differences. The resulting deaths occurred because neonates have immature glucuronyl transferase activity necessary for the biotransformation of chloramphenicol and therefore accumulation occurred. When chloramphenicol Pharmacometrics: The Science of Quantitative Pharmacology Edited by Ene I. Ette and Paul J. Williams Copyright © 2007 John Wiley & Sons, Inc.
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PHARMACOMETRICS IN PHARMACOTHERAPY AND DRUG DEVELOPMENT
accumulated in the neonate’s bloodstream, it caused hypotension, cyanosis, and often death. Many of the laws defining the US Food and Drug Administration (FDA) activities have been developed as a result of therapeutic misadventures in infants and children. The 1938 Amendment to the Federal Food and Drug Act was enacted as a direct result of the distribution of a drug, sulfanilamide elixir, that killed 107 children due to the use of diethylene glycol as a solvent. Passage of the Kefauver–Harris FDA Amendment in 1962 was prompted by the tragic malformations seen in babies who were exposed to thalidomide in utero. Even with these additional safety regulations in place, infant deaths occurred from gasping syndrome due to benzyl alcohol use as a preservative and the limited ability of infants to metabolize and eliminate this “nontoxic” preservative (1). Beyond PK changes, the growth and development that occur during childhood can be adversely affected. Recently, it has been recognized that use of systemic steroids in infants with respiratory distress results in stunted head growth and cerebral palsy (2). The clinical pharmacology of drugs administered to children cannot be extrapolated from adult data on absorption, metabolism, and excretion alone. Although this concept is now generally accepted, the special characteristics of the pharmacology of drugs in children continues to be underappreciated. Although the mantra for the National Institute for Childhood Diseases (NICHD) rightly proposes that children are not miniature adults, they are not Martians either. As the knowledge of pediatric pharmacology has increased, some extrapolations can be made with a reasonable level of certainty. With this in mind, the FDA has recently published the draft of Guidance for Industry: General Considerations for Pediatric Pharmacokinetic Studies for Drugs and Biological Products (3). For ethical and logistic reasons, pharmacometric studies in children almost always occur after the characterization of a drug in adults and thus pediatric trials can utilize existing knowledge, including the use of preclinical data, in their design. Distribution characteristics and elimination pathways show commonality across ages but their relative importance may differ. Therefore, a drug that is a substrate for a particular drug metabolizing enzyme isoform in adults will remain a substrate for that isoform in children. However, in extreme cases, pathway switches may occur such that the primary route of elimination in adults may be undeveloped in infants to such an extent that an alternative pathway predominates. Caffeine is an example of this, where it is primarily renally eliminated in infants due to the almost completely undeveloped CYP 1A2 biotransformation pathway. However, use of the growing knowledge of the developmental pattern of the physiologic processes that are important for a drug’s disposition can be used to predict PK behavior in pediatric populations. While there is a high degree of similarity across age groups relative to enzyme presence and activity, one important exception exists—the CYP 3A system. In adults, the CYP 3A4 isoform is responsible for metabolism of more drugs than any other enzyme but this isoform is essentially absent in infants with the primary isoform in this family being CYP 3A7, which is not found in adults. While there is overall homology in substrates for these two enzymes, there are differences that may lead to unexpected developmental changes in the pharmacokinetics for drugs that are CYP 3A substrates.
INTRODUCTION
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In pediatrics, the maturation changes in pharmacodynamics can be of a greater magnitude than PK differences, yet pediatric-specific PD models remain infrequently and poorly defined. Potential maturational PD differences can be pronounced depending on the physiologic or pathophysiologic system involved. This is especially true for systems that undergo extensive maturation after birth and for the least developed infants, those born premature. The immune and central nervous systems are particularly prone to altered pharmacodynamics due to their extensive postnatal development. GABA pathways, which produce inhibitory responses in adults, do not fully develop until 10 years of age and may be involved in excitatory pathways in preterm infants, which has been clinically manifested by clonic responses to lorazepam. In older infants paradoxical excitatory responses to nonsedating antihistamines may also be a manifestation of a similar phenomena. The immune system response to potential pathogens is grossly underdeveloped in infants and acceptable anti-infective PD targets in adults may not be adequate for infants. It also puts infants at greater risk for therapies whose primary adverse effects include immunosuppression. A further example of significant postnatal development is the autonomic nervous system that in the first year of life impacts the pharmacodynamics of drugs that affect the cardiovascular and gastrointestinal systems. The exposure–response surface (see Chapters 8 and 32) is rarely mapped in children, thus limiting extrapolation of therapy from adults to children. Models linking biomarkers to patient outcomes—outcomes link models—are virtually absent in pediatric pharmacometrics. There is no reason to believe that PK/PD-biomarkersurrogate-outcomes linkage in children should uniformly parallel adults; therefore, this is an area where there is a great need for further model development. Population methods can play an important role for filling in these critical pieces to determine optimal pediatric therapy. Developmental differences, disease presentation, disease progression, and comorbidities also need to be considered when determining pediatric pharmacotherapy. Even when the mechanism of action and PD response surface may be similar between pediatric and adult populations, differences in therapy may be indicated based on disease progression. For example, hypertension rarely presents as primary finding in children but most frequently as secondary to renal disease or other processes, which frequently impact the pharmacologic goals of therapy. HIV infection and AIDS will result in a 50% 2-year mortality in untreated infants yet typically takes 10 years in adults to wear down the immune system to the point at which opportunistic infections and AIDS take hold. Thus, therapeutic targets must account for these differences especially if these therapies will be used for chronic conditions. The application of pharmacometrics is the only feasible manner to get optimal drug use for pediatric patients. This is accomplished primarily through the estimation of differing levels of covariate influence on pharmacokinetics and pharmacodynamics, and thus dosing strategies. The impact of covariates on pharmacokinetics and pharmacodynamics is more important in children than in adults because of the large range of influential covariates such as weight and age in children (4, 5). In pediatrics, dosing is most often based on patient size such as body weight, body surface area (BSA), weight bands, or age. The goal of these dosing strategies is to generate near identical exposures (peak concentrations, area under the concentration–time curve, time above a threshold, or trough concentrations) across age–weight groups
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adjusted for known PD differences. This goal is complicated by limitations in formulations, where dose may be dictated by solid oral dosage form strengths. There is a need to balance appropriate exposure for the population against overly complex pediatric regimens that are likely to result in abandonment or dosing errors.
39.2
REGULATORY CLIMATE
In 1962, the Kefauver–Harris Drug Amendment was passed to ensure drug efficacy and greater drug safety. For the first time, drug manufacturers were required to prove to the FDA the effectiveness of their products in the treated population before marketing them. In addition, the FDA was given closer control over investigational drug studies, FDA inspectors were granted access to additional company records, and manufacturers had to demonstrate the efficacy of already approved products. There was a conservative climate following the Kefauver–Harris Drug Amendment that led to the avoidance of the study of drugs in pediatric populations. This avoidance led to a lack of pediatric labeling for greater than 90% of drugs in certain pediatric populations. The lack of labeling studies for pediatric patients was seen as posing significant health risks to children. Initially, the FDA implemented largely voluntary measures through the Pediatric Rule to encourage the study of drugs in children and to enhance pediatric labeling in the early 1990s. These measures failed. Therefore, in 1997 the Food and Drug Administration Modernization Act (FDAMA) included incentives to study drugs in children through FDAMA and the subsequent Best Pharmaceuticals for Children’s Act (BPCA) by adding six months of attached exclusivity to any existing exclusivity. This legislation had a potent effect on increasing drug studies done in children and in a report to Congress it was stated that “the pediatric exclusivity provision has done more to generate clinical studies and useful prescribing information for the pediatric population than any other regulatory or legislative provision to date.” In 2003 Congress enacted the Pediatric Research Equity Act (PREA) to further promote drug study in the pediatric population. Here the FDA was given the authority to require studies for the registration of a new drug when deemed necessary. Therefore, now the FDA has both a “carrot” (BPCA) and a “stick” (PREA) to encourage the study of drugs in pediatrics.
39.3 39.3.1
OBSTACLES OF PM RESEARCH IN PEDIATRICS Gaining Permission for Participation
The first hurdle in executing a pediatric pharmacometric (PM) study is ethically obtaining informed consent from the patient. This requires informed consent from at least one and sometimes both parents or a legal guardian. For older pediatric patients, assents are usually required from the study participants themselves. In addition to the complicated logistical issues of getting this consent, if the legitimate goals of the research are not presented well to the parents, it may result in concern that their child or infant is being used as a guinea pig. This compounds the overall
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reluctance to allow participation of their child in a study, even if the parents would likely participate in similar studies themselves. Many PM studies require close collaboration between the study site and pediatric subspecialists to develop the trust needed to gain subject enrollment. Even when the parents are agreeable, the potential pediatric participant may decline to participate. In dealing with a vulnerable population, pediatric PM studies must justify design based on some degree of direct benefit. So, as with adult cancer chemotherapy PM studies, where healthy volunteers are not utilized, almost all pediatric PM studies are performed in children with disease or who are at risk for the disease that the drug is being used to treat. Furthermore, the child or parent cannot be coerced or bribed with reimbursement to participate in studies and some pediatric centers’ investigational review boards (IRBs) do not allow any monetary or other inducements at all. In situations where direct reimbursement is a barrier to participation, other financial barriers such as parking fees, taxi rides, delivery of supplies, or overnight accommodations can and should be removed. 39.3.2
Study Design Issues
Some study designs are very difficult to utilize or implement in pediatric patients. The need for direct clinical benefit precludes use of “healthy” pediatric volunteers. Crossover designed studies, which are ideal for assessing drug interactions, are extremely difficult to perform and are seldom executed in pediatric PM studies. Even single-dose PM studies, for drugs with very long half-lives, are challenging because is it difficult to get children to participate for a duration long enough to complete the sampling. It is very difficult and disruptive for parents to coordinate their family’s schedules to accommodate bringing infants and children back to a PM study center on successive days. Thus, there is a large drop-off in participation when the sampling duration increases from 8 to 12 or 24 hours and beyond. Population approaches to estimation and development of PM models are advantageous in this setting because several samples may be obtained during the first dosing interval and several more may be obtained at steady state. Here the patient can participate for the first several hours after the first dose and return at a convenient time within a flexible time window after steady state is achieved for additional sampling, thus increasing patient retention. It should be further noted that the population approach can often be applied to evaluate drug–drug interactions, thus obviating the need for crossover studies, although randomization is needed to establish causality. 39.3.3
Pediatric PM Studies Are Time Consuming
Almost every aspect of pediatric PM study execution and analysis requires greater resources to perform and the overall study duration is longer than a similar study in adults. Tasks that require greater time include locating study centers, identifying suitable subjects for participation, obtaining informed consent, and coordinating data quality assurance schedules. Rarely can sites coordinate multiple study participants to synchronize their visits together into a weekend or two. Pediatric PM studies collect more variable data than studies in adults; thus, the data clean-up can be time consuming as well. With more than 200-fold range in possible weight (0.5 kg to >100 kg), standard data filters can become problematic. Transciption and
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recording errors of weight in pounds rather than kilograms may not be grossly apparent. Rarely is the same single milligram dose used across all subjects. The multiple doses used provide the opportunity for errors in recording and transcribing dose information. In addition, the normal range for many laboratory values is age dependent, so any filter for toxicity grades must account for different values at different ages. 39.3.4
Collinearity
Collinearity refers to a situation where, in the same data set, some of the covariates are highly correlated with others. A high degree of collinearity between covariates important for PM studies has been demonstrated for at least weight, age, body surface area, height, and creatinine clearance (6). This is demonstrated in Figure 39.1. Many standard laboratory test values are age dependent, so that wide covariate PM univariate screens that include serum creatinine, uric acid, alkaline phosphatase, lactate dehydrogenase, bilirubin, or albumin may identify associations that reflect age or size misspecification. There is also collinearity for many PD biomarkers and surrogate endpoints with age; CD4+ cells, blood pressure, and absolute neutrophil count are just a few examples. Formulation differences may also confound age effects. Conditions where disease progression is a prominent feature, especially if it affects drug elimination, will complicate assessment of age and size effects with other covariates. The implications of collinearity in covariates on the modelbuilding process for nonlinear mixed effects modeling have been investigated by 0
20 40 60 80 100
0
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10
15 2.0 1.5
BSA
1.0 0.5 0.0
100 80 60 40 20 0
W TKG
170 120
HTCM
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AGE
5 0
SCR
0.0 0.5 1.0 1.5 2.0
20
70
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1.35 1.10 0.85 0.60 0.35 0.10
0.100.350.600.851.101.35
FIGURE 39.1 Collinearity of various demographic variables in pediatric patients. Of special importance are the collinearities of body surface area (BSA), weight (WTKG), height (HTCM), and age.
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Bonate (7). He states that covariates showing a high degree of correlation (r > 0.50) when included in a model at the same time may indicate that one or both do not improve the model, even when in fact both should be included. Thus, collinearity is a potential source for spurious covariate exclusion in a pediatric PM model. It has to be pointed out that although the Bonate term r = 0.5 represents high correlation, r ≥ 0.75 is generally regarded as high correlation and an r value between 0.5 and less than 0.75 as moderate correlation. 39.3.5
Cost of PM Studies in Pediatrics
In contrast to adult studies, where the cost–benefit ratio is seldom an issue that impacts study design, it can play a significant role in pediatric study design. Singledose intensive PM studies may cost less than multiple-dose studies and have the additional benefit of rapidly generating data. For drugs where additional exclusivity is being sought from performing a pediatric PM study per the BPCA, there is a race to complete the study and analysis prior to patent expiration. Thus, in some settings the pediatric PM study completion time may be the most important design consideration. However, single-dose studies are not as powerful for determining covariate effects in PM models because of the increased subject homogeneity and smaller number of subjects. Thus, traditional single-dose studies are more likely to oversimplify dosing across age groups. There is the further issue of whether a single-dose study can predict steady-state pharmacometrics, particularly for pharmacokinetics. For drugs where single-dose pharmacokinetics can predict steadystate pharmacokinetics and can be characterized within an 8–12 hour interval, the cost of collecting and analyzing the additional samples needed for standard noncompartmental analysis (NCA) may be less than enrolling additional subjects for a population analysis. However, the knowledge generated is uniformly less with a NCA. Most BPCA-inspired pediatric drug programs also require a larger safety and drug effect study. This may provide the opportunity to rapidly collect data for pediatric population PK/PD evaluations. 39.3.6
Sampling How Much and What
The quantity of sample for adults is rarely an issue, in contrast to pediatric patients where it is almost always an issue. This is especially true when it comes to obtaining blood, plasma, or serum because of the limited quantity that most pediatric patients possess and can safely be collected for purposes of the study. Even in PM studies of larger children, where blood volume is not a safety issue, it is important to minimize unnecessary blood collection volume for both ethical and recruitment considerations. Not only does the total blood volume need to be minimized but also the number of venipunctures. There is much greater subject and parent acceptance of six PK samples drawn over an 8 hour dose interval through placement of an indwelling catheter than three PK samples, each collected by direct venipuncture, drawn at PK optimized collection times over 24 hours. Liquid chromatography tandem mass spectometry (LC MS-MS) is a good method because one can use a small sample size due to the sensitivity and specificity of the assay method. A specific assay may minimize the need for baseline samples, further reducing the quantity of samples needed.
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It is a good idea to use an analytical laboratory that specializes in or has extensive experience with pediatric samples. Many adult contract laboratories have default sample volume requirements for standard hematology and chemistry evaluations that are greater than may be needed for the PK sampling of the study. Also, since many clinical chemistry values are age dependent, using a pediatrics-specific laboratory will prevent artificial labeling of many values as outside the normal range, when they are simply outside the normal range for adults but normal for infants or children. A further recommendation in pediatric PM studies is the use of a local anesthetic, such as Emla cream, whenever possible. Not only will this reduce patient discomfort but it will also ameliorate anxiety from both potential subjects and parents, thus reducing reluctance to study participation. Use of local anesthetics requires planning as application is require well in advance of the venipuncture. Pediatric PM studies should be done in a pediatrics friendly environment with age-appropriate supplies (toys, games, etc.) and age-specific activities available. The phlebotomy team should be experienced and dedicated to pediatric venipuncture so that prior to obtaining a specimen the patient is properly assessed. One common practice that is helpful in pediatric sample collection is the use of two individuals, one to obtain the sample and one to distract, during the phlebotomy procedure. A population analysis approach allows for the use of opportunistic blood sampling for pediatric PM studies. Opportunistic samples can be obtained when there is blood, plasma, or serum “left over” from other procedures or processes. This requires use of a very sensitive assay and can be helpful in hospitalized pediatric patients. This is particularly pertinent for infants where sampling is especially limited and for drugs with long half-lives where potentially reduced accuracy in collection times will not adversely affect PM model development. Another source for PK information to consider is urine, which may be useful for renally eliminated drugs. In newborns, use of pulp derived diapers can provide reasonably accurate collection of urine for determination of renal clearance. Urine collection can provide valuable mass balance information, which may provide additional stability to a PM model. However, use of adhesive bags for urine collection can lead to maceration of the skin and therefore should be employed with caution. Finally, a sampling and study strategy that recognizes that some collection will be incomplete or sample will be lost is best. The quantification of metabolites can also be useful in pediatric PM studies. The metabolite models can actually provide improved information concerning the parent compound as well as providing mechanistic information concerning age effects on drug disposition. When metabolites are active, then improved PD and outcomes models can be developed, increasing model applicability for proposed dosing strategies. For population pediatric PK studies, Jones et al. (8) have addressed many of the sampling issues by executing Monte Carlo simulations. In these simulation analyses, the authors assumed that a single sample was obtained from each subject in a crosssectional design. The ability to estimate both a one- and two-compartment model was investigated to evaluate timing and number of samples needed for accurate and precise estimation of parameters including random effects parameters. The timing of these samples was determined using the informative (profile) block randomized design (see Chapter 12). Specifically, the informative times obtained with ADAPT II and the concentration–time profile were divided into three sampling blocks and subjects were randomly sampled within each block. Several sampling schemes were
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investigated for both the one- and two-compartment models. Both single-dose and multiple-dose study impacts were also evaluated. For the one-compartment model, the sampling window blocks were 0.06–0.10, 0.1–2.9, and 2.9–5.4 hours. For instance, with sampling scheme A, samples were collected in a proportion of 3 : 4 : 3 for the three blocks, while sampling scheme B maintained 20 samples for the simulations with 50 or 100 subjects and 10 samples for the simulations with 20 and 30 subjects at the middle region sampling interval (be reminded that only one sample per subject was observed in the simulation). The detail of the sampling scheme is presented in Tables 39.1 and 39.2. The true parameters were those used in the simulation and therefore the accuracy and precision of the estimates of the parameters could be quantified. Both the degree of bias and precision of estimates relative to “true” values were of interest and were computed. One hundred replicate simulations were done for each scenario and the percent prediction error (%PE) and the percent root mean squared precision error (%RMSE) were estimated for the parameters for bias and precision, respectively. This investigation showed that the sampling strategy impacted the ability to accurately estimate PK parameters. For a one-compartment model, an N of 50 estimated the typical values of CL, V, and between-subject random effect for CL (wCL) with little bias and good precision. TABLE 39.1 Sampling Scheme A* Number of Experimental Units 20 30 50 60 70 80 90 100
Number of Samples in Sampling Block 1
Number of Samples in Sampling Block 2
Number of Samples in Sampling Block 3
6 9 15 18 21 24 27 30
8 12 20 24 28 32 36 40
6 9 15 18 21 24 27 30
*Source: From Jones et al. (8), used with permission from Taylor and Francis Group LLC, www.taylorandfracis.com.
TABLE 39.2 Sampling Scheme B* Number of Experimental Units 20 30 50 60 70 80 90 100
Number of Samples in Sampling Block 1
Number of Samples in Sampling Block 2
Number of Samples in Sampling Block 3
5 10 15 20 25 30 35 40
10 10 20 20 20 20 20 20
5 10 15 20 25 30 35 40
*Source: From Jones et al. (8), used with permission from Taylor and Francis Group LLC., www.taylorandfracis.com.
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The estimate of the between-subject random effect for V (wV) lacked precision. Neither sampling scheme (scheme A versus scheme B) appeared to perform better than the other. When the residual variability was increased, bias in the estimation of V increased and the precision of wCL and wV worsened. For the two-compartment model, an N of 80 resulted in accurate estimates of CL, V1, V2, and wCL, and precise estimates of CL, V2, and wCL. All estimates of intercompartmental clearance (Q) were inaccurate and imprecise regardless of sample size and the precision of V1 and wV were poor regardless of sample size. This study demonstrated that for typical drugs, when an informative profile block randomized design is used, single sample cross-sectional designs can be constructed to adequately estimate CL and V and their respective variability for populations of 50 subjects for one-compartment drugs and 80 subjects for two-compartment drugs. For drugs with high residual variability, there was worsened accuracy and precision of parameter estimates. For each drug with its own set of PK parameters, the exact informative sample times will be different. If within-subject longitudinal samples are added to a study, the accuracy and precision of all parameters would be expected to improve over the purely cross-sectional studies. Given the ability of these cross-sectional studies to estimate pivotal parameters (CL and V) with accuracy, dosing strategies can be proposed based on their PK parameter estimates. This type of cross-sectional study could be very useful when there are severe restrictions in obtaining samples as is often the case in pediatric studies.
39.4 39.4.1
DIFFERENCES BETWEEN ADULT AND PEDIATRIC PATIENTS Differences in Pharmacokinetics
39.4.1.1 Absorption Absorption patterns in pediatric patients are very different from adults and one must be aware of these differences when conducting PM research. For example, absorption can be age dependent because infants have less gastric acid production, which influences agents where acid is needed for absorption such as azole antifungals and atazanvir. Infants also produce less lipase than adults and older children; this enzyme may be necessary for the absorption of some drugs. More recently, it has been noted that active transporters in the gut limit absorption of some compounds. The degree of transporter expression may exhibit age dependency. Although overall GI transit time may be shortened, gastric emptying is delayed, and one may see slow absorption in very young infants (9). Absorption may also be altered due to age-specific formulations with liquid formulations often administered to younger children and infants. For drugs with absorption sensitive to food intake, the ability to fast prior to dosing or take with prescribed food is more difficult to accomplish in infants and younger children. 39.4.1.2 Distribution Changes in body composition seen throughout infancy and childhood have an impact on the apparent volume of distribution (V). Newborn infants have higher total body water on a L/kg basis than older populations; therefore, drugs distributed to water have higher V values in newborns when compared to older children
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or adults. For example, aminoglycoside antibiotics are highly polar and thus are distributed primarily to extracellular fluid and have double the V value in newborns when compared to adults. Body fat content may also be altered especially in preterm infants who have very little fat, reducing V for lipophilic agents. In neonates, albumin and alpha-1-acid glycoprotein concentrations are low and albumin has a specific “fetal” structure with different binding characteristics compared to “adult” albumin. Infants also have higher concentrations of endogenous compounds that compete for drug binding sites on protein and in tissue. Overall in neonates, highly bound drugs will have a higher free-fraction of drug in plasma, leading to proportionally greater distribution into tissues. Thus, the V will on average be greater. These differences in binding are important to consider in interpreting pediatric PM results. Doses that achieve similar total concentration–time area under the curve (AUC) profiles for adults and infants may be associated with higher free-drug AUCs in infants, possibly resulting in increased toxicity. 39.4.1.3 Metabolism Metabolism in pediatric patients can be quite different from adults. In the very young infant, drug uptake by the liver is decreased due to reduced transport proteins. The biliary excretion of antibiotics with dual routes of elimination suggests that hepatic transport maturation is even slower than glomerular filtration or renal transport maturation. Overall, mixed function oxidases are present at 30–50% of adult activity, while individual enzymes may be less than 5% of adult activity. In particular, isoenzymes of CYP 2C9 and 1A2 have greatly reduced activity in neonates; however, there is a rapid increase in 2C9 activity in the first weeks of life. After birth, Phase I and II enzymes have a programmed order of expression, which is different for each isoenzyme. Some isoenzymes increase in days, others over weeks, and still others over months. Infant hepatic mass is two to three times adult mass on a weight basis. Thus, overall enzyme capacity is higher on a weight basis. A study from St. Jude’s Children’s Hospital correlated clearance with liver size by scan (10). Both hepatic enzyme activity and hepatic blood flow impact liver metabolic clearance. The influence of each component depends on the substrate in question. While liver size may be a surrogate for total liver enzymatic activity, much less is known about development changes in hepatic blood flow. However, given changes in developmental cardiac output, it is likely that hepatic blood flow and clearance of highly extracted drugs correlate more closely with body surface area than body weight. 39.4.1.4 Renal Excretion At birth, renal blood flow is 12 mL/min and the kidneys receive only 5–6% of cardiac output compared to 15–25% in adults (normalized to body surface area). Glomerular filtration rate (GFR) is directly proportional to gestational age beyond 34 weeks. GFR increases rapidly over the first few weeks of life, with smaller increases throughout the first year of life. Preterm infants have reduced GFR but exhibit more rapid development in the postnatal period than would have occurred in utero if delivered at term. The peripartum use of steroids to promote lung development during preterm labor may expedite maturation of renal function as well. Tubular secretion is less mature when compared to GFR and increases twofold over the first week of life and tenfold over the first year of life. Renal function
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parallels increases in body weight and this collinearity can be problematic when one is attempting to include both covariates in population PK models concurrently. Clinical serum creatinine measurements, which are used to evaluate renal function in adults, have limited precision in estimating renal function in infants and young children due to low granularity in reported results, typically given only with one-tenth of mg/dL precision. Normal serum creatinine values are 0.2 mg/dL, and because of the reporting units this encompasses true concentrations from 0.1501 to 0.2499 mg/dL or a 66% range difference. Additional analysis difficulties can be encountered with renal function development in longitudinal PM studies. While serial assessments at various stages of development can provide information-rich data, the continued growth and maturation result in large within-subject variability in “base” models. This must be recognized and can cause computational difficulties using standard model-building approaches. The first few weeks of life are also characterized by large inter-subject variability resulting from maturation differences among infants. 39.4.2
Differences in Pharmacodynamics
Ongoing growth and development affects PD analyses. When attempting to employ a biomarker or surrogate endpoint, one must be aware that tests or procedures that are easily applied to adults often cannot be used in infants or children. For example, only limited pulmonary function evaluations can be performed in infants because they cannot execute the standard test. Even though newborns feel pain, it is difficult to get assessments of pain in children and pain scales are difficult to compare across age categories. As a substitute for pain scales, physiologic changes such as blood pressure, catecholamine release, and heart rate variability can be employed, but even these are age dependent and may be affected by concomitant therapies used in infants and young children. Although CD4+ cell count is accepted as a valid surrogate endpoint for HIV disease, its appropriate use as a PD marker in pediatric studies in HIV-infected children is not clear. The CD4+ count is higher in infants compared to older children and does not stabilize until around 5 years of age. Therefore, while “successful” HIV therapy in adults will typically be associated with increases in CD4+ cells, immunologic “success” in infants may be manifested by a smaller drop in CD4+ cells or change in the CD4+/CD8+ cell ratio.
39.5 39.5.1
COVARIATE IMPACT IN PEDIATRIC PHARMACOMETRICS Size as a Covariate
Size is a critical element for understanding, analyzing, and applying PM principles to pediatrics. Weight can range more than 200-fold between premature infants and adolescents, and it correlates with age and other factors that may impact drug disposition. However, many physiologic covariates that affect drug clearance such as renal function do not scale directly to weight. Body surface area (BSA) has been found empirically to correlate more closely with the clearance of many drugs rather than weight. This most likely occurs because physiologic processes are slower
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in larger individuals than in smaller ones. BSA can be calculated on the basis of several equations (11–13). One problem with BSA is that it is must be estimated from height and weight and is prone to calculation errors. Allometric scaling of PM parameters is the preferred approach during childhood. A major advantage of allometric size adjustment is that it is a mechanistic approach that is based on dispersion theory (14, 15). This theory suggests that clearance parameters should be scaled by WT0.75 and volume parameters WT1.0. Use of standardized allometric exponents, besides being mechanistically appropriate, facilitates comparison among models. This approach does not take maturation or age effects into consideration, which must be evaluated separately from the size effects. It also structurally suggests half-life will increase with size and age, which is commonly seen after maturation of CL processes during infancy. Recently, it has been suggested that, based on an evaluation of a large number of xenobiotics across species, the allometric exponent value differs based on route of elimination: 0.67 for clearance is for drugs eliminated mainly by biotransformation and 0.75 is more appropriate for drugs eliminated by the kidney (16). However, the added complexity of a different exponent remains to be justified for assessing size across the age continuum within a species. It is important to recognize that dosing strategies based on allometry can match AUCs across age groups; however, the peak–trough differences would still exist (with identical dose intervals) and would be greater in smaller (younger) patient populations. This may have importance for drugs whose PD response is linked through a threshold or a peak concentration effect. It is best to include variables of size into PM covariate models prior to incorporating other covariates for useful models. 39.5.2
Age as a Covariate
Age comes in multiple structures for pediatric populations. Infant age may be defined in terms of postnatal and gestational ages, while for adolescents biological age (Tanner scores) may be more appropriate than chronological age. When attempting to evaluate age as an influential covariate, for infants, if data is dense, one should evaluate the impact of postnatal age and gestational age at birth as two separate covariates, thus recognizing that maturation occurs at different rates in in utero versus postnatal environments. Metabolic and excretory functions increase more rapidly following birth than in utero and thus composite measures of infant age such as postconception age are less accurate in this setting. The impact of age can have a nonlinear relationship to PM model parameters and therefore graphics, particularly nonparametric smooths, and generalized additive models should be used to determine the relationship between age and the PM parameter of interest. During an analysis, it must be remembered that age is a dynamic covariate and subjects can have large increases in drug clearance in sequential PK evaluations. While much of the power from population PM analyses in infants can be derived from this mixture of longitudinal with cross-sectional patient evaluations, models without interoccasion variability components will output average parameters within a subject in the posthoc estimates from the base model. Graphics from these outputs will have the true maturation processes blunted and this must be taken into account during the modeling process.
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39.5.3
PHARMACOMETRICS IN PHARMACOTHERAPY AND DRUG DEVELOPMENT
Creatinine Clearance as a Covariate
The impact of creatinine clearance (CLCr) as a predictor variable for pediatric PM parameters is often low even for renally eliminated drugs due to the inclusion of size (height) in the calculation of renal function and its normalization to adult size. Although the original pediatric renal studies by Schwartz (6) show excellent correlation between measured and estimated CLCr from serum creatinine, others have noted these equations as not predictive in some pediatric subpopulations. The poor precision for the clinical assay of serum creatinine partially accounts for this lack of predictability. However, many pediatric PM analyses have demonstrated the reciprocal of serum creatinine is a powerful covariate for predicting the clearance of renally eliminated drugs, even in infants. 39.5.4
Drug Interactions
Drug interaction screens do not determine cause; therefore, caution is needed in interpreting the meaning of a drug interaction covariate effect on pediatric PM parameters. For example, inotropes have been included in models as significant predictors of clearance in infants; however, they likely identify a characteristic of the severity of underlying illness of the subpopulation rather than a true drug interaction. While pediatric patients may receive a different scope of potentially interacting drugs, the expected qualitative effects are typically similar to adults. Enzyme inducers and inhibitors have been identified as covariates for predicting clearance in pediatrics. Some drug interactions only suspected in adults have been documented in pediatric populations such as dapsone and rifabutin. Development may impact the capacity of an enzyme to be induced so there may be important quantitative differences between adults and children. It is also common for standard pediatric dosing to result in a different exposure to the interacting drug than is seen in adults. If the drug interaction is concentration dependent, the altered drug exposure of the inducer or inhibitor may affect the magnitude of the drug interaction in pediatric patients. 39.5.5
Other Covariates
Other covariates that have been identified as important in pediatric PM studies include the level of metabolism in the gastrointestinal tract, ECMO that may be a marker of hypoperfusion, nutrition, and genetics–genomics. Ethnic differences may also exist in pediatric populations while known gender differences in adults are likely absent or greatly reduced. 39.6
POPULATION MODELING IN PEDIATRICS
Population modeling has great utility in pediatric patient populations because less intense and opportunistic sampling can be executed to estimate population parameters such as typical values for clearance. The population approach in pharmacometrics allows for variable dosing regimens, variable sample collections, the use of unbalanced data, the study of a broad spectrum of patients, and a screen for drug–drug interactions, provides estimates of individual drug exposure for explor-
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atory analysis of efficacy and toxicity, and allows pooling of data across studies. It is especially useful for incorporating significant covariates into PM models, assessing complex PM models, modeling at steady state, and comparing formulations, and pharmacokinetics–pharmacodynamics–outcomes links models can be generated. The stochastic elements of population models are more accurately estimated than for traditional standard two-stage models (17). 39.6.1
Sampling Strategies in Population Modeling
Of particular interest is the utility of population modeling when sampling is limited, as is often the case in pediatric studies. For example, etoposide toxicity and efficacy have been related to exposure. It was not reasonable to execute an intense sampling PK study in the pediatric population, therefore a limited sampling strategy was proposed and done. In each subject only two samples were collected—one at about 3 hours postdose and another 5.5 hours postdose. The approach was shown to be able to estimate PK parameters that had little bias (18). Optimal sampling strategies for pediatric population studies can be determined by using Monte Carlo simulation. Here one constructs plausible data sets that would be obtained under several competing study structures by varying study characteristics such as number of subjects, number of samples per subject, missing data, mistimed sampling, and mislabeled sampling. From the basic model, several hundred plausible data sets are generated by the simulation software and then models are estimated for each data set. These competing study structures are then compared for power, efficiency, robustness, and informativeness. Once the PM models are estimated, then dosing strategies can be proposed based on important covariates. These dosing strategies can be assessed by Monte Carlo strategies prior to a future pediatric patient ever receiving a dose. A drawback to population modeling is that it can be logistically difficult to perform, often a great deal of education is needed at the sites where the data are collected, and quality assurance of the data can be difficult. 39.6.2
When to Incorporate Size in a Population PM Model
A frequently addressed question is whether size should be incorporated into the population model first before other covariates are tested for inclusion, because it is well known that size impacts pharmacokinetics especially in pediatric patients. A further and related question is: Should one fix the allometric exponent or should one estimate the exponent? The advantage of estimating the allometric exponent is that a statistically superior model may be found. However, this is at the expense of two degrees of freedom from studies where the total information may be limited. Fixing the allometric exponents requires making additional assumptions about the relationship of size to the parameter but it increases the utility of the resulting model when comparing to other models. In addition, situations where the allometric exponent deviates significantly from the expected are often a reflection of developmental differences among the study population, with changing exponent values depending on the age group included in the analysis. One problem with allometric scaling is that it is difficult for those not familiar with the concept to grasp the meaning of clearance as L/h·kg0.75 and how it may
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affect dosing. Some advocate presenting the parameters at the median or mean weight of the study population. While this is preferable to the raw exponents, the concept of normalizing clearance to size is so ingrained that many pediatric clinicians will then divide the value by the typical weight clearance in L/h·kg. To prevent this mistake, one should also present the model predicted clearance for the smallest and largest subjects (this can be done graphically) to provide a clearer representation of clearance across the population.
39.7
CLINICAL TRIAL SIMULATION
One of the most potent applications of pharmacometrics is the informative construction of clinical trials by using clinical trial simulation (CTS). Population PM models are of great value when used in CTS because estimates of typical parameters along with parameter variability can be incorporated. There are three basic types of models needed to execute a CTS: an input–output model, a covariate model, and an execution model. These are described in detail in Chapter 34 of this book. Clinical trial simulation can improve pediatric study structure by examining the impact of many important factors such as dropouts, choosing varying endpoints, and deviations from protocol. Pediatric PM models find great utility when applied to CTS.
39.8
AN INFORMATIVE EXAMPLE
We present a pediatric population PK (PPK) model development example to illustrate the impact that the model development approach to scaling parameters by size can have on pediatric PPK analyses; a typical pediatric study is included. It is intuitive that patient size will affect PK parameters such as clearance, apparent volume, and intercompartmental clearance; and that the range of patient size in most pediatric PPK data sets is large. Thus, it is expected that in most pediatric PPK studies subject size will affect multiple PK parameters. However, because there are complex interactions between covariates and parameters in pediatric populations, there are also intrinsic pitfalls of stepwise forward covariate inclusion. Selection of significant covariates via backward elimination has appeal in nonlinear model building; however, it requires knowledge of the relationship between the covariate and model parameters (linear vs. nonlinear impact) and can encounter numerical difficulties with complex models and limited volume of data often available from pediatric studies. Thus, there is a need for PK analysis of pediatric data to treat size as a “special” covariate. Specifically, it is important to incorporate it into the model, in a mechanistically appropriate manner, prior to evaluations of other covariates. The current example is drawn from results of a PK study that was designed to evaluate the pharmacokinetics of cyclosporine in stable pediatric transplant patients receiving chronic oral dosing. Since many of the subjects had evaluations from two separate formulations, a secondary objective of the study was to evaluate the relative absorption characteristics of the two formulations. The study included 32 children and adolescents, a typical size for a pediatric Phase 1–2 study. This modest number of subjects in a pediatric PK study is common but reduces the power to
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include extensive covariates in a final PK model. Therefore, it is essential to limit the scope of covariates that will be evaluated to those related to study objectives and those that are expected to impact dosing. In this study, there was relatively intensive cyclosporine PK samplings: eight samples per subject were collected over a dose interval of the primary formulation. Many subjects had an additional, limited, three-sample PK evaluation performed a few months after the primary PK evaluation. In addition, a subset of subjects had previously participated in another separate cyclosporine PK study utilizing a different formulation. Their PK data from this prior study were included in the analysis. The average subject age in this study was 13.7 years with a range of 3–21 years. Like most pediatric trials, there was a large range in overall subject size that reflected true differences in absolute liver and kidney organ mass and function. The weight range encompassed nearly a tenfold range (from 12.2 to 121 kg) and BSA averaged 1.32 m2 (range 0.53–2.39 m2). This study did not represent the entire pediatric continuum, as no infants below 3 years of age were included. The PK analysis issues related to size would have been magnified further if infants had been included, because not only the range of size would have been expanded but also the diversity of hepatic and renal elimination maturation issues would have been encountered. For the development of this model, the a priori level of decrease in the minimized objective function (MOF) for retaining covariates in the model was set at 6.6 (p < 0.01) for a single degree of freedom in hierarchical models. A two-compartment model greatly improved the fit of the data as indicated by a greater than 100 point decrease in the MOF. The first-order conditional estimates algorithm was utilized after a log transformation of concentration data. The most commonly used approach to population PK model building, developed in adults, starts with a base model that contains no covariates for PK parameters and adds covariates one at a time, assessing the impact on the model by changes in the MOF. Weight or other metrics of subject size are added with size covariates to each PK parameter separately. This approach is not optimal for pediatric PK modeling, and utilizing it in this cyclosporine example resulted in no inclusion of size covariates for any PK parameter in the final model. To illustrate this point, models were developed starting with a base model devoid of size covariates and adding individual covariates of BSA or weight to CL, V2, or V3 one at a time. From this approach, no covariates resulted in a reduction in the MOF by 6.6 and only one decreased the MOF by more than 4, weight added to V3 (decreased the MOF by 4.5). These results occurred despite the clear post hoc graphical suggestion of size effects on CL (see Figure 39.2). However, when size covariates were applied to CL, V2, V3, and intercompartmental clearance (Q) simultaneously, the objective function was significantly reduced by over 25 (p < 0.001 on 4 df). The univariate screening approach would have become even more confusing if many covariates were evaluated in this step due to multiple collinearities between size, age, and serum creatinine. Age as a covariate had a similar impact to WT and BSA on PK parameters and in some instances was the most powerful covariate. Although simultaneous inclusion of size on all parameters improved the model, some of the size covariates did not remain statistically significant for all parameters if evaluated by stepwise backward elimination. While mechanistically this is not a plausible reflection of the true nature of cyclosporine’s pediatric PK disposition, it is not totally unexpected that all of these covariates do not all reach “statistical
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CSA CL/F (L/h /m 2 )
40
30
20
10 0.50
0.90
1.30
1.70
2.10
2.50
BSA Formulation 1
Formulation 2
FIGURE 39.2 Relationship between BSA and total clearance demonstrated by graphical presentation from post hoc estimates of a base model without size. Note that BSA was not a significant covariate in the univariate screen when included in isolation size effects on parameters despite the obvious relationship between BSA and clearance.
significance” based on the modest study size and the limited information on some PK parameters from the sampling design. Removing individual size effects would result in a model with peculiar characteristics. It makes mechanistic sense that all of the PK parameters of CL, V2, V3, and Q would be expected to increase with increasing size; that is, the smallest patient in this study (12.2 kg) would not be expected to have the same value for any of these four parameters as the largest patient (121.0 kg). While a model developed through pruning size effects from some PK parameters would be statistically correct, the usefulness would be severely limited. The complexity of a two-compartment model for this data, although clearly justified by the goodness-of-fit plots and MOF (MOF for one-compartment model was 100 points more than for a two-compartment model), may have limited the ability of the univariate screening to detect size covariates. However, the one-byone addition of size covariates to PK parameters to a one-compartment model also failed to decrease the objective function by 4 in any permutation. A preferable approach that was used in this analysis was to start with allometrically scaled parameters, clearance and volume terms scaled by WT0.75 and WT1.0, respectively, before assessment of other covariates. The underlying assumption that these size relationships exist is plausible and results in a base model that has the same degrees of freedom as a base model without the size assumption. In the cyclosporine analysis, allometric scaling accounted for much of the apparent pediatric PK “age” effects. However, even after scaling CL by WT0.75 in the final model, children less than 12 years of age were associated with a 26% higher clearance. Formulation had a similar magnitude impact on F (24%). Different formulation F was associated with a drop in the objective function of 12. The final model included the covariance estimation between CL and V2 as well as interoccasion variability on F. The final model is presented in Table 39.3.
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TABLE 39.3 Complete Description of the Final Irreducible Model Valuea
Parameter
0.75
SEE
CL V2 KA F1 Q V3
1.86 · WT · AGE12 1.58 · WT 0.65 1 + 0.24 · FORM1 1.48 · WT0.75 7.75 · WT
0.22 (AGE12: 0.09) 0.35 0.07 0.13 0.21 2.41
Intersubject Variability CL V2 IOV—F1 Residual Error
23% 65% 31% 33%
13% 36% 18% 14%
a
Where AGE12 = 1.27 if ≤12 or 1 if >12 and where FORM1 = 1 if formulation 1 and 0 for formulation 2.
The one-by-one addition of all covariates to PK parameters in the model, treating size just like any other covariate, would have resulted in a model that would not make physiologic–pharmacologic sense. It is often difficult to tease out independent effects of the covariates from pediatric data. Incorporation of prior knowledge of the pediatric physiology, development, disease presentation, and disease progression is often needed to provide important guidance in model development. As one would expect, age and size are moderately correlated in this data set, the R2 was 0.57 between age and weight and 0.68 between age and BSA. If one were also to include time since transplant, serum creatinine, or other age-dependent laboratory measures that vary with size or age, a typical univariate screen could select any of these potential associations over weight, allometric scaling, or BSA. Thus, a model suggesting that clearance and absolute dose (in mg) be solely a function of time since organ received could be the best statistical model yet a harmful model in application outside the study population. It would grossly underdose a 17 year old with new transplant and possibly overdose a very young transplant recipient. In the standard one-by-one covariate addition model-building paradigm, most of these covariates will drop out at the multivariate step, but “cluttering” this stage with many confounded variables can greatly affect the approach taken in testing the covariate in a multivariate step (categorical, linear, or nonlinear) and one may miss the covariate that has a true causal influence. The inclusion of many covariates due to collinearities with size can also change the order of evaluation in the forward selection process or result in computation difficulties due to the large number of factors if one assesses independence using a backward elimination selection approach. Size, weight, or BSA is intuitively related to CL and V and therefore should be included as prior knowledge in all pediatric population PK modeling exercises.
39.9
SUMMARY
Population PM methodologies represent a powerful approach to generating clinical pharmacology data in infants and children. In the last 15 years there has
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been a great increase in the use of this method to analyze PK data. Its potential role in pediatric PD model and disease model development remains to be cultivated. In pediatric PM trials, it is important that they are designed to ask the right questions. Pediatric PM data are often limited and assumptions may be required for the analysis. Thus, it is essential that pediatric expertise be sought to assist in study design and analysis. Useful models should be mechanistically relevant and, at a minimum, should account for size and assess developmental changes to be useful. With the increased performance of pediatric population PM studies, opportunities exist to learn more about pharmacologic ontogeny and build better developmental models. These models may ultimately enhance the safety and effective use of drugs in this important population. REFERENCES 1. J. L. Hiller, G. I. Benda, M. Rahatzad, J. R. Allen, D. H. Culver, C. V. Carlson, and J. W. Reynolds, Benzyl alcohol toxicity: Impact on mortality on intraventrucular hemmorrhage among very low birth weight infants. Pediatrics 77:500–506 (1986). 2. T. M. O’Shea, J. M. Kothadia, K. L. Klinepeter, D. J. Goldstein, B. G. Jackson, R. G. Weaver, and R. G. Dillard, Randomized placebo-controlled trial of a 42 day tapering course of dexamethasone to reduce the duration of ventilator dependency in very low birth weight infants: outcome of study participants at 1-year adjusted age. Pediatrics 104:15–21 (1999). 3. Department of Health and Human Services, Guidance for Industry: General Considerations for Pediatric Pharmacokinetic Studies for Drugs and Biological Products. U.S. Food and Drug Administration, Rockville, MD, 1998. 4. P. Rajagopalan and M. R. Gastonaguay, Population pharmacokinetics of ciprofloxacin in pediatric patients. J Clin Pharamacol 43:698–710 (2003). 5. E. Chatelut, A. V. Boddy, and B. Peng, Population pharmacokinetics of carboplatin in children. Clin Pharmacol Ther 59:436–443 (1996). 6. G. J. Schwartz, G. B. Haycock, C. M. Edelmann, and A. Spitzer, A simple estimate of glomerular filtration rate in children derived from body length and plasma creatinine. Pediatrics 58:259–263 (1976). 7. P. Bonate, The effect of collinearity on parameter estimates in nonlinear mixed effect models. Pharm Res 16:709–717 (1999). 8. C. D. Jones, H. Sun, and E. I. Ette, Designing cross-sectional population pharmacokinetic studies: Implications for pediatric and animal studies. Clin Research Reg Affairs 13:133–165 (1996). 9. G. Kearns, P. K. Robinson, J. Wilson, D. Wilson-Costello, G. R. Knight, R. M. Ward, and J. van den Anker, Cisapride disposition in neonates and infants: in vivo reflection of cytochrome P450 3A4 ontogeny. Clin Pharmacol Therap 74:312–325 (2003). 10. D. J. Murry, W. R. Crom, W. E. Reddick, R. Bhargava, and W. E. Evans, Liver volume as a determinant of drug clearance in children and adolescents. Drug Metab Dispos 23:1110–1116 (1995). 11. D. Dubois and E. Dubois, A formula to estimate the approximate surface area if height and weight be known. Arch Int Med 17863–17871 (1916). 12. E. A. Gehan and S. L. George, Estimation of human body surface area from height and weight. Cancer 54:225–235 (1970).
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13. R. D. Mosteller, Simplified calculation of body-surface area. N Eng J Med 317:1098 (1987). 14. G. B. West, J. H. Brown, and B. J. Enquist, The fourth dimension of life: fractal geometry and allometric scaling of organisms. Science 284:1677–1679 (1999). 15. G. B. West, J. H. Brown, and B. J. Enquist, A general model for the origin of allometric scaling laws in biology. Science 276:122–126 (1997). 16. T. M. Hu and W. L. Hayton, Allometric scaling of xenobiotic clearance: uncertainty versus universality. AAPS Pharm Sci 3:E29 (2001). 17. L. B. Sheiner and S. L. Beal, Pharmacokinetic parameter estimates from several least squares procedures: superiority of extended least squares. J Pharmacokinet Biopharm 13(2):185–201 (1985). 18. J. C. Panetta, M. Wilkinson, C. H. Pui, and M. V. Relling, Limited and optimal sampling strategies for etoposide and etoposide catechol in children with leukemia. J Pharmacokinet Pharmacodyn 29(2):171–188 (2002).
CHAPTER 40
Pharmacometric Methods for Assessing Drug-Induced QT and QTc Prolongations for Non-antiarrhythmic Drugs HE SUN
40.1
INTRODUCTION
The electrocardiogram has been intensely studied. Figure 40.1 demonstrates the waves of a heartbeat as recorded on the electrocardiogram (ECG) rhythm strip. They are labeled, according to well accepted practice, with the letters P, Q, R, S, and T. This chapter primarily addresses the QT segment, as recorded on the ECG rhythm strip, which includes the time interval (measured in milliseconds, ms) from the beginning of ventricular depolarization, the Q wave, to the end of the T wave, at which point cardiac repolarization is complete. QT prolongation refers to lengthening of a normal QT interval. While the extent of QT prolongation is acknowledged as an imperfect biomarker for proarrhythmic risk, there is a quantitative relationship between QT prolongation and the risk of torsades de pointes (TdP), especially for drugs that cause substantial prolongation of the QT interval (1). Because of the QT’s inverse relationship to heart rate (HR), the measured QT interval is routinely corrected by various formulas that relate the QT to the HR, known as the QTc interval. Although it is not clear whether arrhythmia development is more closely related to an increase in the absolute QT interval or QTc, most drugs that have caused TdP clearly increase both the absolute QT and the QTc. In pharmacometric analysis, QTc is used as the biomarker of choice for drug-induced QT change assessment. Several new developments on QT/QTc prolongation assessment have occurred recently. The most important includes the publication of the new International Conference of Harmonisation (ICH), Step 4 guidance, issued on May 12, 2005 entitled The Clinical Evaluation of QT/QTc Interval Prolongation and Proarrhythmic Potential for Non-Antiarrhythmic Drugs. It contains relevant information obtained from the ICH Steering Committee, as well as information discussed at the October 2003 Food and Drug Administration (FDA) and Drug Information Association (DIA) Pharmacometrics: The Science of Quantitative Pharmacology Edited by Ene I. Ette and Paul J. Williams Copyright © 2007 John Wiley & Sons, Inc.
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R
P S-T
Q
T
S Q-T
FIGURE 40.1 ECG rhythm strip. The extension of the QT interval is termed QT prolongation. The ending point of the T wave sometimes is difficult to determine, which contributes to the variability of QT measurements.
Meeting, “ECGs in Clinical Trials: The New Regulatory Realities,” and provides recommendations to sponsors concerning the design, conduct, analysis, and interpretation of clinical studies to assess the potential of a drug to delay cardiac repolarization. It includes testing the effects of new agents on the QT/QTc interval and the collection of cardiovascular adverse events (AEs). Readers are encouraged to access this document for detailed information. One key point in the E14 guidance for a pharmacometrician who is conducting QTc data analysis is that “a negative thorough QT/QTc study is one in which the upper bound of the 95% one-sided confidence interval for the largest time-matched mean effect of the drug on the QTc interval excludes 10 ms. This definition is chosen to provide reasonable assurance that the mean effect of the study drug on the QT/ QTc interval is not greater than around 5 ms. When the largest time-matched difference exceeds the threshold, the study is termed ‘positive QT trial.’ A positive study influences the evaluations carried out during later stages of drug development, but does not imply that the drug is pro-arrhythmic.” Additional statistical guidance can also be obtained from the PhRMA Working Group paper released in 2003 (2). The FDA will consider substantial QT/QTc interval prolongation, with or without documented arrhythmias, as grounds for nonapproval or discontinuation of clinical development, or require the sponsor to include relevant information in the product label. If it is a feature shared by other drugs of the therapeutic class, it may require a study to compare the extent and incidence of any QT/QTc interval prolongation effects to other drugs in the same class with concurrent positive control groups in the trials (see later discussions on the use of positive control). Documented experiences from the literature where CYP450 metabolism resulted in QTc prolongation and torsades de pointes have prompted regulatory actions such as label warnings or market withdrawal. ICH, FDA, and industry will benefit by working together in order to clarify outstanding issues on QT/QTc interval prolongation. Additional QT/QTc interval prolongation guidances or methodologies will be issued as ICH, FDA, and industry collaborate on this topic. As the FDA continues to provide more guidance and additional data from thorough QT/QTc clinical studies are submitted to the FDA, additional modifications of new product labels can be expected.
CORRECTION OF THE QT INTERVAL FOR HEART RATE
40.2
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CORRECTION OF THE QT INTERVAL FOR HEART RATE
Heart rate or RR interval (defined as 60 divided by the heart rate) correction plays a very important role in the analysis of QT/QTc data. Baseline corrections depend heavily on the clinical assumptions that the baseline data represent the subjects’ physiological condition in drug-free conditions, and that the QT/HR relationship stays the same before and after drug administration, whether or not the heart rate changes. With this assumption, QT values are corrected for HR in order to summarize and compare QTc values across different subjects, trials, or conditions. The objective of correcting the QT interval for HR or RR is to obtain a corrected QT interval that is statistically independent of the HR or RR interval. Figure 40.2 shows the dependence of QT on HR. In order to eliminate the dependence of QT on heart rate, numerous HR or RR correction formulas have been proposed in the ECG literature, reflecting the variety of statistical models that have been fit to the data. The reader should be aware that there is no best QT interval correction method for heart rate, but there are some practical methods. The most popular corrections are the Bazett (3) and Fridericia (4) formulas. Both are based on the simple power model QTc = QT/RRb; that is, calculation of the QTc is equal to the observed QT in milliseconds divided by the term of a root of the RR interval in milliseconds. Bazett’s method uses b = 0.5, and Fridericia’s correction uses b = 0.333. Both can produce similar or different QTc intervals for the same QT, depending on the HR value. For RR values less than 1 second (i.e., HR greater than 60 bpm), the square root function is smaller than the cube root function and the Bazett-corrected QT interval (QTcB) will be larger than the Fridericia-corrected QT interval (QTcF). Thus, at high heart rates, QTcB is much larger than QTcF and the Bazett formula may “overcorrect” QT interval. Thus, when a drug increases the heart rate substantially but does not truly prolong the QT interval, the use of Bazett’s formula can inflate the probability of concluding a positive QT/QTc signal when such a signal
550 manual mechine 450
350
250 20
40
60
80
100
Heart Rate (beats per min)
FIGURE 40.2 Plot of baseline QT versus heart rate. Two methods of QT interval measurement are presented—the manual read and machine read. QT interval is clearly depended on the heart rate. Also, machine read QT interval is longer than manual read QT interval.
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does not exist. Similarly, Fridericia’s formula is said to “somewhat undercorrect” at low heart rates. In most QTc data submitted in a New Drug Application (NDA), with the same data set, the QTc prolongation assessment endpoint of maximum mean QTc change is always larger for QTcB than for QTcF. This mainly is due to the fact that the mean HR is normally greater than 60 bpm. In fact, Bazett’s and Fridericia’s formulas are just specific cases for the data available. In a new study, linear or log-linear models can also be used to derive an empirical population- or subject-specific correction based on the observed QT/RR baseline (predrug) data, including ECGs from all subjects in the study. For example, a pharmacometrician can fit log(QT) versus b log(RR) to drug-free data from all patients in all periods to obtain an estimate of b, then calculate QTc by applying QTc = QT/RRb to all treatment arm/period data. This will generate population-corrected QTc data (QTcP). Similarly, if log(QT) versus b log(RR) is fit to baseline data from individual subjects for individual estimates of b, then calculating QTc by applying QTc = QT/RRb to the individual subject at treatment will generate the subject-specific corrected or individual corrected QTc (QTcI). In most cases, population b values are in the range of 0.22–0.6, and individual b values range from 0.1 to 0.8. Subject-specific corrections may be preferable to population-based correction formulas in studies with multiple ECG recordings per subject at baseline (5), but their effective use depends on an adequate range of heart rates in the baseline data for each subject. If subjects are in resting condition during the experiment, their heart rates do not usually vary much. Therefore, an individually derived correction based on a narrow range of heart rate pretreatment may not be the most accurate correction available to correct for QT measurements and can lead to false conclusions. Also, considering the large interbeat variability for QT interval (see later for variability), the amount of baseline data needed to generate QTcI is substantially large. These problems clearly have prevented the broad use of the individual correction method and are why QTcI data in NDA submissions were rarely seen. An important reminder is that the effectiveness of any correction formula for a particular set of QT/RR data from a population or an individual should be examined graphically. If the correction is adequate, QTc will be statistically independent of HR/RR. See the example plots in Figure 40.3 for examining correction effect on QT intervals. In the four panels of plots for different correction methods applied to this data, QTcF provides the best HR correction since the slope of QTcF versus HR is most close to zero (i.e., QTcF is independent of HR). However, a slope of zero in a population may not be enough for an adequate correction. No single correction formula will work for every data set, and therefore understanding the limitations of each correction is critical (6). For most populations, Fridericia’s formula is generally simple to apply and provides an acceptable correction and is the one the FDA normally uses in new drug evaluations. See ICH 14 guidance (7) for further discussion. Although the Fridericia correction is often designated as the primary correction, it is prudent to present the results using several corrections. The FDA encourages sponsors to include population and individual empirical corrections derived from baseline data, in addition to the Fridericia method. When statistical results differ because various corrections were applied to the same data set, which is often seen, discrepancies should be explained and can usually be traced to the range of heart rates prior to and after drug administration, a change
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FIGURE 40.3 Heart rate corrected QT versus heart rate. Four methods are applied to the same data set. From the resulting slopes, QTcI (individual correction method) provided the best correction. The Bazett method yields the worst correction.
in heart rate, a change in the QT/RR relationship, or differences between subjects in individual QT/RR relationships. It is recommended that pre- and postdose RR intervals (or heart rates) as well as uncorrected QT intervals also be examined and analyzed in order to understand the relationships between the variables.
40.3 DATA ANALYSIS CONSIDERATIONS IN STUDY DESIGN 40.3.1 Study Type and Number of Subjects The ICH 14 guidance has clearly indicated that time-matched baseline and placebo correction should be used in QT prolongation assessment. Although the choice of a crossover or parallel group, single or multiple dose, study design is usually determined by the primary objectives of the study, the pharmacokinetic (PK) and pharmacodynamic (PD) properties of the compound (e.g., half-life of the drug, expected margin, delay, or accumulation of QTc prolongation), and the current knowledge of the drug’s safety and tolerance profile, a single-dose crossover design is easier for time-matched correction for each individual subject. Also, as normal healthy volunteers are generally to be employed in the design of thorough QTc
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trials (TQT), the application of crossover designs is of better utility in improving the accuracy and precision of the relevant prolongation measurements. Whenever possible and appropriate, the four basic components—the baseline, placebo, positive control, and multiple dose levels of the drug—will assist in evaluating QT/QTc prolongation. Here, baseline data is used for HR correction; placebo control should be employed as QTc is known to show diurnal variation and in order to provide a reference basis for an assessment and potential claim of no-effect; a concurrent positive control group or arm is strongly encouraged and most of time is required; and finally, one or more dose levels that represent the worst scenarios of clinical drug exposure should be included. The most common procedure of a TQT trial has this order: baseline day for all subjects, positive control treatment for all subjects, then crossover design for placebo and treatment drug, or crossover design for placebo, and two or more dose levels of the tested drug. There are other designs that have a separate parallel group for placebo control, or sequential parallel designs, or dose escalation design. Table 40.1 presents a list of various designs seen for recent non-antiarrhythmic drugs. Due to the large inter- and intrasubject variability in QT/QTc (see later for variability discussion), the sample size for crossover study design would typically be about 60–80 subjects (typically 65 subjects) exposed per treatment. This is to ensure
TABLE 40.1 List of Various Study Designs of Nine Recently Approved Non-antiarrhythmic Drugs
Doses Useda
Trial Designb
Number of Subjects
Baseline Collection Period (Data Were Used for Heart Rate Corrections)
Drug 1
SD
XO
68
Drug 2
SD
XO
44
Drug 3
SD
XO
58
Drug 4
MD
PL
40
Drug 5
MD
ESC
85
Drug 6
MD
ESC
25
Drug 7.1
SD
XO
48
Drug 7.2 Drug 8
SD SD
XO XO
61 90
Drug 9
MD
PL
76
On day 1: multiple points until 8 h postdosing On day 1: only 1 h before dosing On day 1: at 30, 15, and 0 min before dosing On day 1: multiple points until 8 h postdosing On day 1: multiple points until 24 h postdosing On day 1: multiple points until 24 h postdosing On day 1: multiple points until 24 h postdosing Right before dosing On day 1: multiple points until 12 h postdosing On day 1: multiple points until 4 h postdosing
Drug
a b
SD, single dose; MD, multiple dose. XO, crossover design; ESC, dose escalation design; PL, parallel design.
Number of Replicate QT Measures Per Time Point 1 3 6 3 3 3 1 1 10 3
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adequate study power Thus, for parallel study with four treatment arms, 260 subjects would be required, whereas for crossover design, only 65 volunteers (allow for a few dropouts) would need to be exposed to each treatment and washout period. Again, variability is the key determinant of study sample size. 40.3.2
Number and Timing of ECG Recordings
One of the most significant clarifications that evolved from ICH E14 was the move from time-averaged to time-matched analysis (7). A problem with the time-averaged analysis is that the maximum effect of drug on QT interval is diluted, and therefore in the absence of a separate analysis of maximum mean QT change, a false-negative conclusion may be obtained. The advantage of the time-matched analysis is that each time point on treatment (active drug, placebo, or positive control) is compared with the baseline values for the corresponding time point (i.e., to calculate the timematched ΔQTc). The ΔQTc value at each time point is calculated using the average of the replicated QTc values taken at that time point for each individual. This is the change of QTc from baseline at that time. The parameter of interest is the difference between the ΔQTc on the active treatment (or positive control) and the placebo at the same time (i.e., the ΔΔQTc). When the same baseline data is used, this is equivalent to subtracting the QTc value of placebo from the active treatment or positive control. This approach requires the measurement at baseline and during treatment (placebo, active control) to be the same. The potential problem with this requirement is that if there is a missing value at baseline, or placebo, the data at treatment arm will be useless. No imputation methods to adjust such a missing value have been used or published so far. The optimal number and timing of ECG recordings in a TQT clinical trial is an area of active research and depends mainly on the endpoint being evaluated, the PK properties of the drug, and the stage of development. The optimal timing to cover a range of concentrations for PK/PD analyses is important. In general, it is recommended to use the time schedule that was normally used for the PK study, or to seek statistical guidance to select the optimal number and timing of ECGs for the objectives of a study, taking into consideration the subject population, endpoint, statistical model, sample size, and cost effectiveness. Normally, 12 time points over the concentration–time profile, in which several points are near the time of maximum drug concentration, are recommended. Replicated ECGs are needed to avoid the bias in outcome parameters. Malik and Camm (8) recommend that it would be “worthwhile to consider recording 3 to 5 replicate ECGs at each time point within a 2 to 5 minute period.” This is because QT interval is not an absolute constant and it is measured with error. The clinical assumption is that even under stable conditions, an individual’s true QT/QTc interval can vary largely within a minute. How much of the variability is under stable conditions depends on the natural biological variability and the measurement error. The number of replicates and the number of subjects per trial jointly influence the outcome of a TQT trial. Sun et al. (9) evaluated QT interval variabilities in its chaotic (beat-to-beat), circadian (across a few minutes/hours), and occasion (across days/weeks) domains and provided a reference for clinical trial designs for TQT prolongation assessment. In the study, QTcB, QTcF, and QTcI data from 57 normal young healthy male subjects, with 6 replicated QTs
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at each of 3 protocol time points within 10 minute intervals, for 6 cycle days one week apart were used. Standard deviations (SDs) were indices for variability. The impact of inter- and intrasubject variabilities on sample size was estimated statistically and/or via simulations. The overall mean QTcB, QTcF, and QTcI were 391 ± 15.35, 386 ± 14.62, and 381 ± 14.72, respectively, showing that the intersubject variability is independent of correction methods. With QTcF, the average chaotic SD is 14.71 ms (range 9–41). The lowest circadian SD in 20 min, based on the means of 6 replicates per time, was 6.69 ms (range 2.78–11.8). The lowest between-period SD, using the means of all 18 measures for each day, is 7.1 ms (range 1.52–25). These components of variability increased nonlinearly as the number of replicated QTs decreased. Data are presented in Table 40.2. Sun et al. (10) conducted a bootstrap resampling simulation trial. From 80 subjects, 10, 20, 30, 40, or 60 subjects were randomly drawn from drug-free baseline data. Each subject provided two sets of 1, 2, 3, 4, 5, or 6 replicated QT values 10 min apart. The ΔΔQTc values were calculated. Since the true ΔΔQTc was best believed to be zero, any trials with ΔΔQTc greater than zero were considered false-positive results. Figure 40.4 shows the large intersubject variability within a few minutes, and in 30 minute periods. Figure 40.5 shows the distribution of the SD and the high–low limits of 1000 bootstrap resampling analysis results. When the primary objective of a study is to estimate change in QTc at a specific point in time, say, Tmax, the use of replicate ECGs can reduce uncertainty. Agin et al. (11) report that for time-matched or within-day changes from baseline, the use of triplicate ECGs instead of single ECGs reduced the within-subject standard deviation as estimated by two different methods: the standard deviation of the observed changes from baseline was 14.7 to 9.2 ms and from 13.5 to 8.1 ms for model-based estimates. Using replicates in this case can substantially decrease the sample size necessary to estimate a response with a desirable precision or test a hypothesis with a prespecified power. With the known large intrasubject chaotic, circadian, and day-to-day variability in QT intervals, Sun et al. (10) independently investigated the frequency of QT replications and the corresponding limit of QT prolongation assessment. Baseline only QT data were obtained from several prospectively designed TQT trials in normal young healthy male or female subjects (N ≥ 40 in all trials), with up to 6 replicated QT values at multiple time points of a day (intervals ranged from a few minutes to a few hours) for 2–6 treatment periods. One to six jackknife randomly resampled QT values were drawn from each of the TABLE 40.2 Intersubject Variability as a Function of the Number of Replicate QT Measurements Number of Replicates 1 2 3 4 5 6
Mean of WithinSubject SDs for All 57 Subjects
SD of the WithinSubject SDs for All 57 Subjects
6.82 5.04 4.06 3.91 3.79 3.08
4.31 2.96 2.3 2.1 1.98 1.85
Ranges of SD for All 57 Subjects 1.52–12.5 0.5–11.16 0.19–10.9 1.0–12.25 0.8–9.8 0.44–8.35
DATA ANALYSIS CONSIDERATIONS IN STUDY DESIGN -30 s ubject: 101 period: 1
-10
s ubject: 101 period: 2
-30 s ubject: 101 period: 3
-10
s ubject: 101 period: 4
-30 s ubject: 101 period: 5
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-10
s ubject: 101 period: 6
400
390
380 QTcF 370
360
350
340 -30
-10
-30
-10
-30
-10
Minutes
FIGURE 40.4 Example of large intersubject variability in QTcF for an individual. ECGs were recorded six times, 1 minute apart, and repeated three times in 30 min and for six different time periods one week apart.
Bias and high-low limits of experimental Means 0 1 2 3 4 5 6 7 10
No.Sub: 10
No.Sub: 20
0 1 2 3 4 5 6 7 No.Sub: 30
No.Sub: 40
No.Sub: 60
8 6 4
SD
2 0
0 1 2 3 4 5 6 7
0 1 2 3 4 5 6 7
0 1 2 3 4 5 6 7
No.Replications
FIGURE 40.5 Relationship between the number of replicates per time point versus potential experimental bias.
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time points and the means, SDs, and changes of means from time to time were determined. The inter- and/or intrasubject QT variabilities and the lowest limit of reliable QT replicates were estimated from 1000 jackknife data sets. Results show that with 1–6 replicates per time point, the intrasubject SD nonlinearly decreased from 16 to 7.82 ms, the time to time QT changes were never zero, and the natural existing QT change ranged from 9.51 to 4.35 ms and maximized from 12 to 49 ms. The intersubject variability is less affected by the frequency of sample replicates. It was discovered that increasing the number of replicates to more than 3 is not needed, as shown in Figure 40.4. 40.3.3
Baseline Days
Since time-matched analysis will be used, the baseline ECGs should be recorded at the same time of day as ECGs collected during active treatment. This will provide insight into diurnal and food effect on the QT/QTc interval. It is important to note that due to the high intrasubject variability, the replicated baseline alone may show false-positive or false-negative QTc changes. Figure 40.6 is an analysis result by Lee et al. (12) with data from the same group of subjects who provided drug–free QT values on consecutive days. 40.3.4
Choice of Endpoint
Many endpoints can be considered for analysis of QT/QTc intervals. The choice should be dictated by the primary objective of the study as well as by the different 6
14
6
14
PATNO: 21
PATNO: 22
PATNO: 23
PATNO: 24
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PATNO: 16
PATNO: 17
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PATNO: 20
PATNO: 7
PATNO: 8
PATNO: 9
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PATNO: 12
460
380 460
380 PATNO: 1
PATNO: 2
PATNO: 3
PATNO: 4
PATNO: 5
PATNO: 6
460
380 6
14
6
14
6
14
TIME
FIGURE 40.6 Example to show the diurnal and food effect on the QT/QTc interval over an 18 h period.
DATA ANALYSIS CONSIDERATIONS IN STUDY DESIGN
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analyses performed to achieve that objective. It is advisable to mention that other approaches have been used to derive these endpoints. Among the various endpoints, the FDA/ICH concept paper gives greater weight to time-matched QT/QTc interval changes and the maximum mean change (and 90% CI) in the QT/QTc interval for the analysis of central tendency. It is important to point out that the commonly used upper threshold of CI 450 ms Regimen
N(A)
N(B)
N Outliers TRT A (%)
N Outliers TRT B
M M3 M5 P P3 P5 S10 S30
3537 — — 3423 — — 4957 3060
1751 782 750 1667 1563 751 — —
19 (0.54) — — 5 (0.15) — — 68 (1.37) 181 (5.92)
31 (1.77) 24 (3.07) 21 (2.80) 1 (0.06) 10 (0.64) 3 (0.40) — —
TABLE 40.4 Presenting Outliners: The Change of QTcF from Baseline >30 ms and > 60 ms >30 and 60 ms
TRT A
TRT B
TRT A
TRT B
23 (3.6) — — 2 (0.3) — — 128 (10.8) 239 (21.3)
18 53 43 0 24 43
0 (0.0) — — 0 (0.0) — — 0 (0.0) 3 (0.3)
0 (0.0) 1 (0.3) 0 (0.0) 0 (0.0) 0 (0.0) 0 (0.0) — —
(5.9) (18.5) (15.6) (0.0) (8.4) (15.6) — —
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40.3.7
ASSESSING DRUG-INDUCED QT AND QTc PROLONGATIONS FOR NON-ANTIARRHYTHMIC DRUGS
Automation of Data Analysis
As many new drugs are subjected to a TQT study, the FDA will receive more and more information from sponsors. In order to capture all the data in submissions in a single location for future reference, and for future data analysis, a database with an analytical tool—the Qtech—was developed in the Office of Clinical Pharmacology and Biopharmaceuticals within the FDA (15). The QTech is a Visual Basic
FIGURE 40.7
The QTech platform for QT data analysis automation.
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Application that works in combination with S-Plus, MS Access, MS Excel, and MS Word to provide the users with many functionalities such as generating powerful graphs and reports in addition to storing, loading, querying, and analyzing data in the database. (Minimum Software requirements are S-Plus 6.0, Access 2002 with Service Pack 3 (SP), Excel 2002 with SP3, and Word 2002 with SP3). The platform is presented in Figure 40.7. There are two important features in QTech: (a) loading data into the database and (b) data querying and analysis. By selecting one of the tasks, the user can go directly to the task at hand. New improvement of the software is to include a drop-down menu for more functions and a link to S-Plus for fast loading of data and analysis. Industry and academia are encouraged to use this FDA in-house software as a template to develop other types of automation for the QT assessment world.
40.4
SUMMARY
This chapter focuses on pharmacometric methods used in QT data analysis. Pharmacometric issues in the design and analysis of thorough QT clinical trials (TQT), and the methods used to assess QT prolongation potential are described. Analysis methods discussed are patterned after the new International Conference on Harmonisation (ICH) guidance for clinical QTc assessment and are illustrated with data from real clinical trials. Issues in the interpretation of data are also discussed.
REFERENCES 1. S. G. Prior, Risk stratification in long QT syndrome. N Engl J Med 348:1866–1874 (2003). 2. QT/QTc Prolongation: Report from the PhRMA Statistics Expert Team, 14 August 2003, Investigating drug-induced QT and QTc prolongation in the clinic: statistical design and analysis considerations. Drug Inf J 39:243–266 (2003). 3. H. C. Bazett, An analysis of time relations of electrocardiograms. Heart 7:353–367 (1920). 4. L. S. Fridericia, Die Systolendauer im Elecktrokardiogramm bei normalen Menschen und bei Herzkranken. Ada Med Scand 53:469–486 (1920). 5. M. Malik, Problems of heart rate correction in the assessment of drug-induced QT interval prolongation. J Cardiovasc Electrophysiol 12:411–420 (2001). 6. C. Funck-Brentano and P. Jaillon, Rate-corrected QT interval: techniques and limitations. Am J Cardiol 72:17B–22B (1993). 7. International Conference on Harmonisation (ICH) E14, Guidance for Industry, E9 Statistical Principles for Clinical Trials. www.fda.gov/cder/guidance/ (1998). 8. M. Malik and A. J. Camm, Evaluation of drug-induced QT interval prolongation. Drug Saf 24:323–351 (2001). 9. H. Sun, P. Chen, L. Kenna, and P. Lee, The chaotic QT interval variabilities on risk assessment trial designs. Clin Pharmacol Ther 75(2):P55 (2004). 10. H. Sun, P. Chen, J. Hunt, and H. Malinowski, Frequency of QT recording and the reliability of QT prolongation assessments. Clin Pharmacol Ther 75(2):P48 (2004).
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11. M. Agin, D. Kazierad, R. Abel, R. Anziano, W. Billing, G. Layton, J. Mancuso, D. Stricter, W. Xu, R. Blum, and D. Jorkasky, Assessing QT variability in healthy volunteers. Presented at ACCP Annual Meeting, September 2003. 12. S. H. Lee, H. Sun, P. Chen, S. Doddapaneni, J. Hunt, and H. Malinowski, Sensitivity/reliability of the time-matched baseline subtraction method in assessment of QTc interval prolongation. Clin Pharmacol Ther 75(2):P56 (2004). 13. L. A. Kenna, A. Parekh, V. Jarugula, D. J. Chatterjee, H. Sun, M. J. Kim, S. Ortiz, J. P. Hunt, and H. Malinowski, Experience evaluating QT prolongation data. Clin Pharmacol Ther 75(2):P7 (2004) 14. C. Chuang-Stein, Safety analysis in controlled clinical trials. Drug Inf J 32:1363S–1372S (1998). 15. Quantitative Tools for Electrocardiogram Computation Harmonization (QTech), FDA, OCPB, Rockville, MD, 2004.
CHAPTER 41
Using Pharmacometrics in the Development of Therapeutic Biological Agents DIANE R. MOULD
41.1 41.1.1
PHARMACOKINETICS OF THERAPEUTIC PROTEINS Background
Overall, there are some profound differences between the pharmacokinetic (PK) behavior of biologics and small molecules. Table 41.1 summarizes the major differences between these two broad classes of molecules. When evaluating the PK behavior of any protein, it is important to understand the biology and the pharmacology of the system that the therapeutic biologic is acting on in order to anticipate the expected covariates and behavior of the drug. Initially, the structure of the protein that is being developed can provide some information as to the likely clearance and also can suggest feasible routes of administration. Table 41.2 presents the relationships between the molecular weight (MW), bioavailability, and clearance of biologics. In the development of any therapeutic biological agent, the ability to anticipate the PK and pharmacodynamic (PD) behavior of the agent is helpful. However, the PK behavior of proteins is quite distinct from the behavior of small molecules and the drug development path for biologics is not as standardized as it is for small molecules. Table 41.3 gives a general overview of some characteristics of proteins that can help the analyst anticipate the likely PK behavior of a novel biological agent before it goes into clinical testing. 41.1.2
Protein Structure
The architect Louis Sullivan first coined the phrase “form follows function.” In the biological setting, however, the reverse is true: function follows form. During any PK or PD assessment of therapeutic proteins, the size and structure of the molecule Pharmacometrics: The Science of Quantitative Pharmacology Edited by Ene I. Ette and Paul J. Williams Copyright © 2007 John Wiley & Sons, Inc.
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TABLE 41.1 Overview of Pharmacokinetic Differences Between Chemical Entities and Proteins Chemical Entities
Therapeutic Biologics
Pharmacokinetics usually independent of pharmacodynamics Usually linear pharmacokinetics Metabolic breakdown Renal clearance often important Free concentrations useful (“coverage”) Binding implies distribution Tissue penetration often good PK drug interactions possible PD drug interactions rare
Pharmacokinetics often dependent on pharmacodynamics Often nonlinear pharmacokinetics Proteolytic breakdown Renal clearance uncommon if MW higher than 50 kD Free concentrations may cause problems (immunogenicity) Binding implies clearance Usually poor tissue penetration PK drug interactions rare PD drug interactions possible
TABLE 41.2 Relationships Between Molecular Weight, Bioavailabilitys, and Clearance MW < 20 kD Good bioavailability when given SC Can be given via inhalation Usually very fast clearance (minutes to hours)
20 kD < MW < 50 kD
MW > 50 kD
Adequate bioavailability when given SC Inhalation possible Usually moderate clearance (hours)
Poor bioavailability when given SC Not suitable for inhalation Moderate to slow clearance (hours to days)
needs to be taken into account in order to develop appropriate models for both the pharmacokinetics and pharmacodynamics of a biological agent. Proteins are characterized by their primary, secondary, tertiary, and quaternary structures. The primary structure is the sequence of the amino acids in the polypeptide chain that makes up the protein. Secondary structure refers to the first folding of the amino acid chain and reflects, for example, disulfide bonds. Tertiary structure (a monomer) is the final folded configuration of the protein that is controlled by the primary and secondary structures and is thermodynamically driven by the relative hydrophobicity of the component amino acids in the structure. Quaternary structure refers to the functional association of several polypeptides (monomers). For example, the final structure of hemoglobin consists of four associated monomers. Any change in the primary structure of a protein often results in changes to all the higher level structure. Protein structures must be characterized and controlled during the production process. To give some specific examples of how physical structure plays a role in the pharmacokinetics of a therapeutic protein, first consider a monoclonal antibody binding fragment (Fab), which is the active binding region of an antibody, and Fab fragments, which have been extensively investigated as potential therapeutic agents but because they are rapidly cleared, their potential as therapeutic agents is limited. Reengineering the structure by replacing a single disulfide bridge between Fab arms with a thioether bridge increased the mean residence time of the fragment
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Human enzymes, Epogen, factor VIII,
Example
a
Yes
Consider modificationa
Porcine insulin vaccines
Yes
Linear, fast, constant NAB
Protein construct, “fusion protein”
No
Linear, slow, constant N-NAB
Yes
No
Linear, slow, constant NAB
Yes
Glucocereb rosidase, Ceridase, and Cerizyme
Yes
Nonlinear, fast, constant N-NAB
No
No
Fab fragment, singlechain antibody
Yes
Nonlinear, fast, constant NAB
No
No
Refers to modifications such as PEGylation or hyperglycosylation, which alters pharmacological behavior.
Fuzeon
Linear, fast, constant N-NAB
Clearance
No
No
Yes
No
No
Yes
Functional Fc present
No
No
No
No
Changes receptor number
No
No
Yes
Receptormediated clearance
No
No
Yes
Yes
Human structure
No
Answer
Question
GCSF, TPO
Yes
Nonlinear, fast, variable N-NAB
No
Yes
Yes
Yes
TABLE 41.3 Lookup Table of Protein Characteristics and Associated PK Behavior
OKT3
Yes
Nonlinear, fast, variable NAB
No
Yes
Yes
No
“Coating antibody,” Enberel, Clenoliximab
No
Nonlinear, slow, constant N-NAB
Yes
No
Yes
Yes
Chimeric coating antibody
Yes
Nonlinear, slow, constant NAB
Yes
No
Yes
No
Herceptin, Campath
No
Nonlinear, slow, variable N-NAB
Yes
Yes
Yes
Yes
Chimeric lytic antibody
Yes
Nonlinear, slow, variable NAB
Yes
Yes
Yes
No
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USING PHARMACOMETRICS IN THE DEVELOPMENT OF THERAPEUTIC BIOLOGICAL AGENTS
in normal mice by threefold (1). A second example of a structural change resulting in altered pharmacokinetics is Tenecteplase, a fibrinolytic protein developed from human tissue plasminogen activator (alteplase) for the treatment of acute myocardial infarction. Specific mutations at three sites in the original alteplase molecule resulted in 15-fold higher fibrin specificity, 80-fold reduced binding affinity to the physiological plasminogen activator inhibitor PAI-1, and sixfold increase in the half-life (2). Clearly, the structure of the protein is important to both the pharmacokinetics and pharmacodynamics of any biological therapeutic agent, emphasizing the need to characterize the protein structure at all levels and control it adequately during manufacturing. Some of the basic aspects of protein production and engineered structural changes are presented in the following sections. 41.1.2.1 Production of Therapeutic Proteins Bacteria and Chinese hamster ovary (CHO) cell lines are commonly employed for the production of recombinant therapeutic proteins. The process given below is used in the development and production of therapeutic antibodies, which constitute a majority of the proteins being used clinically today. In 1975, Kohler and Milstein (3) presented a method for preparing murine cell cultures that would produce antibodies targeted against a specific antigen. Producing mouse antibodies to selected antigens is very easy to do, and murine antibodies have been shown to have clinical utility, although all murine antibodies have been associated with the formation of human anti-murine antibodies (HAMA). This method ultimately led to the development of Orthoclone muromonab-CD3 (OKT3®), the first monocloncal antibody approved for use in humans. OKT3 was a murine IgG2 antibody that binds and modulates the CD3 receptor site on cytotoxic T-lymphocytes, interfering with antigen recognition and preventing cellular proliferation. OKT3 is used for the treatment of acute rejection in renal transplantation. However, it has since been determined that antibodies that have murine structure are more immunogenic than ones that have been engineered to have human structure. This is because foreign proteins are recognized as such and consequently elicit an antibody response against them (e.g., HAMA) or more broadly human anti-globulin antibody (HAGA). In general, HAMA response is polyclonal, with increased levels of IgM and IgG that are directed against the mouse-specific determinant, the isotype (the heavy chain or FC portion), the binding region (F(ab′)2), and the “idiotype” of mouse immunoglobins (4). The development of neutralizing antibodies directed against the foreign protein restricts their usefulness for several reasons: the development of HAMA can result in anaphylaxis or other related adverse events (e.g., fever, chills, serum sickness, anemia, leukopenia, arthralgia, rash), and HAMA forms a new, and fast, route of elimination that removes the therapeutic antibody from circulation, which reduces the circulating therapeutic protein concentrations and consequently its beneficial effect (5). Because of the HAMA response, murine antibodies are not suitable for treatment durations exceeding 10 days and cannot be administered at some later time to a patient who was previously exposed because of the potential for anaphylaxis. 41.1.2.2 Humanization Structural changes in proteins can result in substantially different PK behavior. Protein engineering using recombinant DNA technology has provided a partial
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solution to the problems associated with HAGA by developing methods to construct chimeric genes that fuse rodent exons for the monoclonal antibody variable regions with human exons encoding the heavy and light chain constant (Fc) regions of the antibodies (6, 7). These constructs produce antibodies with human effector functions due to the human Fc portion and, at the same time, theoretically reduce the likelihood of HAGA response to a major part of the protein. The transfer of murine binding regions into human frameworks transfers the ability to recognize the antigen but provides a slightly less immunogenic framework. In addition to reducing immunogenicity, the use of the human constant (Fc) region is theoretically associated with improved effector function of the therapeutic antibody (e.g., complement protein and antibody-dependent cell cytotoxicity (ADCC)) as well as decreased clearance due to the improved ability of the antibody to take advantage of the body’s tendency to conserve antibodies through protective mechanisms such as Brambell receptors. In summary, the more “human” the antibody structure, the less immunogenic the agent is, the longer the half-life, and the greater the likelihood of utilizing ADCC. The benefits of humanization are not completely straightforward. Many features of immunoglobulin sequences are conserved between species and thus there is no concept of an immunoglobulin sequence appearing to be completely murine. Therefore, the humanization of antibodies does not automatically preclude the development of HAGA (8) because the variable regions of the antibody are still murine and therefore chimeric and humanized proteins may still develop HAGA. In addition, there is considerable homology between many murine and human variable region sequences, making the development of HAGA difficult to predict. 41.1.2.3 PEGylation One of the drawbacks of biological agents is the need for parenteral routes of administration. In some cases, these agents are administered as subcutaneous injections, which reduces the number of clinic visits required for treatment. However, many of these agents require frequent dosing because of their short half-life. The discomfort associated with frequent injections can negatively impact patient compliance and there are other issues associated with the disposal of used syringes. The concept of modifying therapeutic molecules through the covalent attachment of poly(ethylene glycol) (PEG) moieties (PEGylation) was first introduced by Abuchowski et al. (9) in 1977. This approach has been shown to be effective in decreasing the clearance of therapeutic protein agents, as well as reducing the incidence of neutralizing antibody formation, the mechanism of which is described below. PEGylation reduces renal and hepatic clearance and, for some products, effectively increasing the circulating half-life of the agent. PEGylation also results in a more sustained absorption after subcutaneous administration as well as restricted distribution. These PK changes usually result in more constant plasma concentrations, which can be maintained near the desired target levels with less frequent dosing. Additionally, PEG modification may decrease adverse effects caused by the large variations in peak-to-trough plasma drug concentrations associated with intermittent administration and by “covering” the foreign protein (resulting in PEG-induced steric hindrance) can prevent immune recognition and reduce the immunogenicity
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as compared to the unmodified protein (10, 11). However, it should be noted that for all covalently PEGylated and successfully marketed PEGylated agents, the weight of the added PEG was at least as great as the weight of the protein being modified. Modification of a protein by PEGylation also causes changes to the observed PD properties due to altered protein structure and hydrophilicity, which in turn results in different binding properties of the native protein. In general, the binding capacity of PEGylated proteins is reduced as compared to the native protein. Because the size, geometry, and attachment site of the PEG moiety play pivotal roles in observed changes of these properties, therapeutically optimized PEGylated agents must be individually designed. 41.1.2.4 Hyperglycosylation Perhaps the best known example of modification of a protein resulting in altered pharmacokinetics is darbepoetin alfa (Aranesp®) (12). Darbepoetin alfa is a hyperglycosylated analog of recombinant human erythropoietin. The addition of sialic acid residues to erythropoietin resulted in a substantial prolongation of circulating half-life. The terminal half-life of darbepoetin alfa was two to three times longer and the clearance was approximately four times slower than epoetin (13). Similar results have been reported when “glycoengineering” was applied to thrombopoietin and leptin (14). In addition, when l-asparaginase was conjugated with colominic acid (polysialic acid), the immunogenicity was reduced (15). Antibody titers appeared highest for the native enzyme and were generally reduced as the degree of polysialylation increased. In addition, the half-lives of these preparations were three- to fourfold greater than that of the native enzyme. 41.1.3
Clearance
For proteins, structure has an impact on the clearance. For example, the half-life of an intact IgG molecule is 23 days, while for an intact Fc fragment the half-life is 10–20 days (16). Similarly, the binding regions of antibodies (F(ab′)2 fragments) are cleared very rapidly (17). As mentioned previously, the PK behavior can be altered not only by changes in the amino acid sequence but also by changes in the pattern of glycosylation on the protein (18). Consequently, structural changes can alter the PK and the PD behavior of the drug. Therapeutic proteins can undergo several routes of elimination: renal, hepatic, receptor mediated, and HAGA directed. Not all proteins undergo clearance through all possible routes. Again, the type of elimination is partly dependent on the structure of the protein, its molecular weight, and immunogenicity. In addition, the role that receptor-mediated clearance plays in the overall clearance of a protein depends on the functionality of that protein. An excellent example of characterization of different routes of clearance of a biologic agent is a report of the pharmacokinetics of SB-251353, a low molecular weight protein that is a truncated form of the human CXC chemokine growth-related gene product beta (19). The pharmacology of this agent was studied in the mouse. Primarily, the clearance appears to be mediated by its pharmacologic receptor, CXCR2, through endocytosis with subsequent lysosomal degradation. SB-251353 is eliminated via renal and hepatic routes as well. Microscopic autoradiography showed uptake into renal proximal tubule epithelial
PHARMACOKINETICS OF THERAPEUTIC PROTEINS
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cells with limited excretion of SB-251353 in the urine ( 16 kD were primarily absorbed via the lymphatic system. An interesting aspect of this mechanism of absorption is the link between the pharmacokinetics and PD activity of some drugs. A study of the pharmacokinetics and pharmacodynamics of G-CSF (46) found no correlation between Cmax and
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Saturation of Brambell Receptor Clearance
Clearance
Saturable Receptor Mediated Clearance
Organ and Macrophage Based Clearance
Concentration
FIGURE 41.2 Clearance of antibodies is dependent on concentration. At low concentrations, therapeutic proteins may exhibit concentration-dependent clearance, with the clearance decreasing as concentrations increase. For most biological agents that display this behavior, the clearance will decrease to a particular level (the nadir in the curve) and then remain constant. However, for antibodies, a second form of saturation can occur, where the Brambell receptors become saturated, resulting in an apparent increase in clearance as concentrations increase.
increases in neutrophil count, but there was a negative correlation between AUC and neutrophils. The measured G-CSF concentrations reflect the fraction of drug that escapes clearance in the lymphatic system prior to entering the circulation (somewhat analogous to first pass metabolism), and with continued dosing the relative bioavailability of G-CSF would be expected to decrease. 41.1.4.2 Relationship Between Molecular Weight and Bioavailable Fraction When administered via SC, intramuscular, or inhalation routes, the bioavailability of therapeutic proteins is variable and the fraction absorbed is dependent on the molecular weight of the protein (47). Interferon alpha, which is a relatively low molecular weight protein (19 kD), has good bioavailability following SC administration (80%), whereas most therapeutic monoclonal antibodies have bioavailability of approximately 20–60% following SC administration. The reason that proteins generally exhibit low and variable bioavailability is not presently understood, although several mechanisms such as degradation, aggregation, or metabolism at the site of injection have all been proposed (48, 49); however, there is little evidence supporting these theories. 41.1.5
Volume of Distribution
For large proteins, transfer across cell membranes is limited due to size and hydrophilicity of the molecule. There is some evidence that Brambell receptors may play a
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TABLE 41.4 Pharmacokinetic Parameters for Several Marketed Therapeutic Monoclonal Antibodies and Derivatives Generic Name
Brand Name
Abciximab Basiliximab Bevacizumab Cetuximab Daclizumab Etanercept
Repro Simulect Avastin Erbitux Zenapax Enberel
Gemtuzumab Infliximab Trastuzumab
Myelotarg Remicade Herceptin
Type Fab fragment Chimeric IgG1 Humanized IgG1 Humanized IgG1 Humanized IgG1 Fusion protein linked to the Fc portion of human IgG1 Humanized IgG4 Chimeric IgG1 Humanized IgG1
Target Antigen
Vss (L)
GP IIb/IIIa CD25 VEGF EGF receptor CD25 TNF
8 9 — 4.4 6 —
CD33 TNF Her2-neu
20 3 4
role in facilitating crossing cell membranes for macromolecules that have functional Fcg binding (40), but the extent of this activity appears to be limited. Karanikas et al. (50) demonstrated that there is little cellular penetration of monoclonal antibodies, even in cells carrying target receptors. Lin et al. (51) evaluated the distribution of a recombinant humanized IgG1 monoclonal antibody (MAb) directed against vascular endothelia growth factor (VEGF) in rabbits (51). These findings showed that, as expected, serum concentrations of the MAb were 10 times higher than the highest tissue concentration. Furthermore, after 24 hours, evaluable autoradiography was limited due to the recycling of the labeled amino acids by the body, making assessments of tissue distribution difficult. Consequently, most high molecular weight therapeutic proteins (MW > 50 kD) appear to have a distributional volume on the order of 0.1 L/kg, which is approximately equal to the extracellular fluid volume. The volumes of distribution of several marketed therapeutic monoclonal antibodies are provided in Table 41.4. The values for these parameters were taken from the Physician’s Desk Reference (52), which includes labeling information for each agent. Lower molecular weight proteins (MW > 50 kD) generally have a slightly higher volume of distribution, with the volumes ranging from 0.2 to 0.8 L/kg.
41.2 EVALUATING PHARMACOKINETICS USING MODEL-BASED ANALYSIS 41.2.1
Pharmacokinetic Models
In general, the pharmacokinetics of most therapeutic proteins can be described using either a one- or a two-compartment model. This behavior is dependent in part on the route of administration and the molecular weight of the protein. In general, most proteins display nonlinear clearance or there are parallel linear and nonlinear routes of clearance. Absorption following SC administration is often not straightforward to describe, with many analysts using parallel routes of uptake or modeling the absorption as a slow first-order process.
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41.2.1.1 Absorption Models Following SC administration, absorption is generally variable and is often described using parallel routes of uptake (53, 54), although simple first-order models have been used successfully (55). However, the absorption of a biological agent usually follows a complex process. Several simple alternative functions are provided below that have been used to describe the absorption kinetics of various biologics. There are many other functions that can be tested during the model development process. Inverse Gaussian Input Function The inverse Gaussian input function is described as follows: Input (T ) = Dose ⋅ F ⋅
(T − MAT )2 ⎞ MAT ⎛ exp − ⎜⎝ 2 ⋅ NV 2 ⋅ MAT ⋅ T ⎟⎠ 2π ⋅ NV 2 ⋅ T 3
(41.1)
In this equation, Dose is the administered dose, F is fraction absorbed, MAT is the mean input time or mean absorption time, NV2 is the normalized variance of the Gaussian density function, and T is the modulus time following administration of a dose. Spline Input Function A cubic spline function can be used to reproduce the input function described by the inverse Gaussian function. This function is somewhat simpler to code than the inverse Gaussian but is also an empirical function. The spline function is Input = ⎢⎣ A ⋅ T 3 − B ⋅ T 2 + C ⋅ T ⎥⎦
(41.2)
In this equation, T is the relative time postdose, and A, B, and C are the coefficients of the cubic spline. Unlike the inverse Gaussian function, the spline input function does not rely explicitly on dose and can be evaluated both as an explicit input function and as an infusion rate. Biexponential Input Function A biexponential (Bateman) function is sometimes useful because it is also capable of producing an input profile similar to the inverse Gaussian function and can describe nonlinear absorption processes. The representation for this input function is given as Input = K a 0 ⋅ {exp [ − A ⋅ (T − Alag )] − exp [ − B ⋅ (T − Alag )]}
(41.3)
In this equation, Ka0 is the basic absorption rate, T is the modulus time postdose, A and B are coefficients of the exponential input curves, and Alag is the lag time before absorption begins. The input is 0 at all modulus times less than Alag. Surface Input Function One additional function can be considered when investigating absorption models that might describe a nonstandard absorption process. This is a simple surface function. This input function is represented as follows: Input = Ka 0 ⋅ [1 − Alpha ⋅ (T − Alag )]
(41.4)
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In this equation, Ka0 is the basic absorption rate, T is again the modulus time postdose, Alag is the lag time prior to the onset of absorption, and Alpha is a coefficient defining the input function curvature. 41.2.1.2 Clearance Models As might be expected, many biological agents exhibit nonlinear behavior. In some cases, the clearance can be described using parallel linear and nonlinear mechanisms of clearance (19, 53, 56). In order to be able to develop this model, data must be available from a wide range of doses. However, depending on the mechanism of action, the mechanism of clearance, and other aspects such as patient covariates, clearance for some proteins has been described using wholly linear or nonlinear mechanisms. In addition, time- or receptor-dependent clearance mechanisms have also been utilized to explain time-dependent changes in clearance due to the effect of repeated administration of therapeutic proteins. The most common form of clearance is represented as ⎛ V ⋅ Concentration ⎞ ClearanceTotal = ⎜ max + CLLinear ⎝ K m + Concentration ⎟⎠
(41.5)
In this equation, ClearanceTotal is the sum of nonlinear and linear clearance. Vmax is the maximum nonlinear clearance, Km is the concentration required to achieve halfmaximal nonlinear clearance, and Concentration is the drug concentration. CLLinear is the linear component of clearance. 41.2.2
Potential Covariates
During the development of many therapeutic proteins, many of the standard Phase 1 pharmacology studies such as those conducted in special populations (e.g., elderly subjects, renally impaired subjects) and drug interaction studies can be omitted. This is done for numerous reasons, including the fact that administration of proteinbased agents can cause the formation of antibodies, which could potentially affect that subject’s treatment should they later develop a disease requiring therapy using a similar biologic. Furthermore, in many cases, normal volunteers do not have high levels of receptors for these agents because they do not have the disease, so the basic PK behavior of a biologic agent cannot always be translated between a normal volunteer and a patient. Finally, patients are a more heterogeneous group than normal volunteers, making accrual of appropriate patients for such studies difficult, and if the drug has shown efficacy it may be inappropriate to conduct single-dose studies in these patients. These factors limit the number of studies that are conducted during the development of a biological agent and place a greater importance on the use of population PK modeling to assess the effects of covariates on the PK behavior of these agents. A discussion of the potential covariates that should be considered during a population-based evaluation is given in the following sections. 41.2.2.1 Number of Receptors For therapeutic proteins that are cleared by receptor-mediated binding, the number of receptors is usually one of the major covariates. Receptor density or receptorpositive cell count has been identified numerous times as a covariate for cytokines
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and peptide hormones as well as antibodies (49). In some cases, information on receptor density or number of receptor-positive cells is not available. In these cases, treatment duration can sometimes be used to account for changing (usually decreasing) receptor density over time with the consequent decrease in clearance as treatment with the biologic progresses. An example of this is a population PK evaluation of alemtuzumab (Campath®) (57), which found that the concentration–time profile was best described by a two-compartment model with nonlinear elimination. Campath is cleared by binding to CD52+ cells, which was a strong covariate on the maximum velocity for clearance (Vmax). During treatment, the number of CD52+ cells is markedly decreased, resulting in decreased clearance of this agent. 41.2.2.2 Patient Characteristics Patient characteristics can have a profound effect on the pharmacokinetics of therapeutic proteins. For patients undergoing immunosuppressive therapy or those who have a disease that compromises their immune response, antibody formation against the biologic agent is often blunted, delayed, or of lower frequency. However, such patients can and do form antibodies against biologics. As mentioned previously, cancer patients who are treated with asparaginase can develop antibodies against this protein, although PEGylated asparaginase appears to ameliorate the immune response, prolonging the duration of effective treatment with this agent (35). For proteins that undergo receptor-mediated clearance, the stage of the disease can be a predictive covariate when receptor number is missing. Presumably, patients with more advanced disease would have a tendency to have a greater number of receptor-positive cells. In a categorical sense, patient disease state or disease status may be useful as an explanatory covariate when receptor number is not available. The impact of disease type has been reported for infliximab when used to treat Crohn’s disease (58). Other disease-related covariates would include ascites and pleural effusion. These comorbid conditions would be expected to increase the volume of distribution of proteins, thus lowering the measured concentrations in the serum. In one case (59), ascites was found to be a weak covariate of clearance as well. 41.2.2.3 Body Weight As might be expected based on their mechanism of clearance, many monoclonal antibodies have demonstrated a strong relationship between weight and PK behavior (56). This relationship is often true for volume of distribution, since the distribution of most biologics is limited to extracellular fluid volume and one might expect that a patient with a high body weight would have a correspondingly larger extracellular fluid volume. Patient body weight can also be correlated to clearance in situations where receptor-mediated clearance is not predominant. The use of interspecies scaling has been shown to have some utility for scaling from animal to human exposure (60), although care should be taken to account for binding specificity and immunogenicity when attempting to scale the pharmacokinetics of human proteins evaluated in animal models. 41.2.2.4 Drug Interactions The mixed function oxidases and cytochrome P450 enzyme systems do not play a role in the clearance of macromolecules. Nor do large proteins interact with transporter proteins such as P-gp, despite the fact that one site of clearance is the
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intestines. Consequently, formal drug interaction studies are not often conducted for biologics. For example, there are several marketed therapeutic monoclonal antibodies (e.g., daclizumab, Zenapax®) that did not conduct formal drug interaction studies, although there are some marketed biologics that have reported drug interaction studies (61–63) with the expected negative outcomes. In general, reported drug interactions with biological agents and chemical agents are largely PD in nature (64, 65). Although proteins are not cleared by cytochrome enzymes or mixed function oxidases, and would therefore not be expected to alter the pharmacokinetics of other medications, there are potential mechanisms of interaction between proteins and concomitant medications, which could affect patient exposure to the protein. For example, steroids, which alter macrophage cell trafficking (66, 67), could potentially alter the clearance of large therapeutic proteins. An aspect of concomitant medications that should be considered during a population PK evaluation is previous treatment with other related biologic agents or others that have been derived from similar processes. For example, if a patient has developed antibodies to an agent that was derived from a prokaryotic cell line (e.g., E. coli), the patient may also be cross-reactive with a second protein that was derived from E. coli . 41.2.2.5 Liver Function Proteins are not cleared by hepatic enzyme systems. However, liver size and function, spleen function, and macrophage function would be expected to account for variability in observed clearance of large therapeutic proteins. For instance, Sewell et al. (68) demonstrated that Kupffer cell function is decreased in aged rats, which is a function of age as much as it is of liver function. Standard measures of liver function such as alanine transferase may not provide relevant information and are rarely identified as a covariate even with small molecules. However, patients with advanced liver disease such as cirrhosis may have reduced clearance due to poor liver function and reduced liver blood flow. 41.2.2.6 Renal Function Creatinine clearance is not usually a covariate for protein clearance. However, the kidney does form one site of clearance of proteins. Glomerular functionality and renal blood flow might be expected to have some impact on the clearance of low molecular weight proteins. For example, dose reductions are recommended in end stage renal disease patients receiving PEGylated interferon (52). However, accounts of reduced clearance in anuric patients are rare, suggesting that alternate routes of clearance are used in situations where renal function is nonexistent. 41.2.2.7 Age As mentioned previously in Section 41.2.2.5, age can be a covariate for biologics. The effects of age on the PK properties of a protein appear to be due to changes in endothelial and macrophage function and to a lesser extent to changes in organ blood flow. Alterations of the immune response in elderly patients have been associated with increased amounts of memory and alloreactive T-cells, as well as altered cytokine responses (69), which can impact both on the pharmacokinetics and pharmacodynamics of a protein therapeutic agent.
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41.2.2.8 Sex After accounting for weight, patient sex is not usually identified as a covariate. However, as mentioned previously, there have been reports of differential expression of neutralizing antibodies between the two sexes (36). In addition, gender is occasionally identified. In one example, a population PK evaluation of cetuximab was performed (70). The covariates evaluated included demographic data (age, weight, height, body surface area, sex, and race), hepatic and renal function, cancer type, concurrent therapy, EGFr status, clinical response, and presence of a skin rash. A two-compartment model with saturable elimination was used to describe the concentration–time data. The volume of the central compartment was found to have a 27% reduction in the typical value of the central volume in females as compared to males and the typical value of Vmax showed a 26% reduction in females, giving a maximal clearance from the saturable pathway of 0.059 L/h in males and 0.043 L/h in females. No other covariates were found to have a significant impact on the pharmacokinetics of cetuximab. In addition, enfuvirtide was found to have a 20% lower clearance in females than males, even after adjusting for body weight (52). 41.2.2.9 Race Given the complexities of the pharmacology and pharmacokinetics of therapeutic proteins, the effect of race would not be expected to be important. There is no known difference between racial characteristics that would cause additional PK variability. Attempts have been made to examine the effect of race as a covariate, but it has only rarely been identified once patient weight or sex and other covariates (particularly those related to disease) were taken into account. Such was the case for cetuximab (70).
41.3 PHARMACODYNAMICS OF THERAPEUTIC PROTEINS: BACKGROUND The PD response–time profiles for most of the biological agents on the market follow some variant of the basic indirect effect PD models described by Dayneka et al. (71). There are four basic indirect response models. The applicability of these models can be readily identified from the characteristic lag between plasma concentrations and measured response. In all four models described (71), there is a zero-order input rate constant of formation for the marker (Ksyn) and a first-order degradation rate constant (Kdeg). Prior to administration of a drug, the ratio of Ksyn to Kdeg determines the baseline level of the biomarker. An administered drug can act on either the rate of synthesis or rate of degradation and can be either inhibitory or stimulatory. The resulting change in biomarker levels depends on the site and type of action of the drug. For instance, an agent that acts to stimulate the synthesis rate constant would result in an increase in biomarker level. If that agent acted to inhibit the rate of synthesis, then the biomarker level would be expected to fall transiently. Once the effect of the drug wears off, the biomarker would be expected to return to the baseline level. The differential equation for an indirect system at the baseline state is given as
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dBiomarker = ksyn − kdeg ⋅ A (1) dt
(41.6)
In this equation, Biomarker is the PD biomarker being described, ksyn is the synthesis rate constant of that biomarker, kdeg is the degradation rate constant for the biomarker, and A(1) is the amount of biomarker. However, when drug is administered, the effect of that drug (Eff written below as a nonlinear stimulatory effect) can be added to the system (here it is added to the synthesis rate constant): Eff =
Emax ⋅ C p EC50 + C p
(41.7)
dBiomarker = ksyn ⋅ (1 + Eff ) − kdeg ⋅ A (1) dt
In this equation, Eff is the drug effect, Emax is the maximum effect of the drug, EC50 is the concentration of drug required to attain half-maximal effect, Cp is the concentration of the drug, Biomarker is the PD biomarker being described, ksyn is the synthesis rate constant of that biomarker, kdeg is the degradation rate constant for the biomarker, and A(1) is the amount of biomarker. If the delay between measured drug levels and response is very long, additional effect compartments can be added to allow the model to describe a longer lag period. A schematic diagram of an indirect effect model with the additional effect compartment for a precursor added is provided in Figure 41.3. When the delay between concentration and response is protracted, additional effect compartments can be added to help describe the delay. These additional compartments also provide new places at which the drug effect can be evaluated. The equations for this new schematic are given below. Note that Ksyn, the rate of synthesis of the biomarker, has now become a first-order process and that the effect of drug has to be included for both the biomarker and the precursor pool. Eff =
Emax ⋅ C p EC50 + C p
d Precursor = k0 − ksyn ⋅ A ( Precursor ) ⋅ (1 + Eff ) dT dBiomarker = ksyn ⋅ A ( Precursor ) ⋅ (1 + Eff ) − kdeg ⋅ A ( Biomarker ) dt
K0
Precursor Pool
Inhibition or Stimulation
Ksyn
Pharmacodynamic Biomarker
Inhibition or Stimulation
(41.8)
Kdeg
Inhibition or Stimulation
FIGURE 41.3 Multiple compartment indirect effect model.
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In this equation, Eff is the drug effect, Emax is the maximum effect of drug, EC50 is the concentration of drug required to attain half-maximal effect, Cp is the concentration of drug, Precursor is the predecessor for the biomarker, K0 is the rate constant of formation of the precursor, Biomarker is the PD biomarker being described, Ksyn is the synthesis rate constant of that biomarker, Kdeg is the degradation rate constant for the biomarker, and A(1) is the amount of biomarker. This schematic provides new sites for adding a drug effect, such as on the rate of formation of the precursor pool (K0). An additional aspect of this model is that it provides a basic mechanistic model for tolerance. If one assumes an agent works to stimulate Ksyn, then the drug cannot have an effect when the amount in the precursor pool is depleted. Therefore, the duration of action of a drug can be dependent on the amount of precursor components that are available. There are many variants of this basic model, including the variation published by Movin-Osswald and Hammarlund-Udenaes (72) in 1995 and then by Sharma et al. (73) in 1998 in a slightly modified form. Developing a PD model for a biological agent must be done on a case-by-case basis. The analyst must develop an understanding of the complex pharmacology that underlies the mechanisms of these agents. However, it should be noted that the basic PD behavior of many pharmacologically related proteins can often be described using similar models. For that reason, the PD behavior of several broad classes of therapeutic proteins may be broken down by the type of protein.
41.4 41.4.1
SPECIFIC PROTEINS Cytokines
Cytokines form a family of proteins including interleukins and lymphokines that are released by cells in the immune system and act as intercellular mediators in immune response. Cytokines are produced by various cell populations, although they are predominantly produced by helper T cells and macrophages. Cytokines that are secreted from lymphocytes are referred to as lymphokines, whereas those secreted by monocytes or macrophages are referred to as monokines. Many lymphokines are also referred to as interleukins (ILs), because they are also capable of affecting leukocyte cellular responses. Several different broad classes of cytokines are produced by the body, the largest of which stimulates immune cell proliferation and differentiation. The largest class includes IL-1, which activates T cells; IL-2, which stimulates proliferation of antigenactivated T and B cells; IL-4, IL-5, and IL-6, which all stimulate proliferation and differentiation of B cells; interferon gamma (IFN-γ), which activates macrophages; and IL-3, IL-7, granulocyte-macrophage colony-stimulating factor (GM-CSF), and granulocyte colony-stimulating factor (G-CSF), all of which stimulate hematopoiesis. There are not many examples of cytokines that have been approved as therapeutic agents. In part, this lack is due to the pleiotropic effects of these agents (74), which makes the overall effect difficult to predict, and because administration of these agents will often mediate increased immunological response. In general, the administration of interleukins results in elevated blood cell counts, particularly white cells. However, because they help to increase blood cells and also induce
SPECIFIC PROTEINS
1013
immune response, interleukins have been found to be useful in the treatment of advanced cancers. Interleukin-2 (Proleukin®, aldesleukin, IL-2) was approved for treatment of metastatic renal cell carcinoma in 1992, and then later for the treatment of metastatic melanoma in 1998. The pharmacokinetics and pharmacodynamics of recombinant interleukin-2 (IL2) in patients with human immunodeficiency virus (HIV) infection have been evaluated (75). Patients were administered IL-2 either by continuous infusion or by SC injection for 5 days over multiple cycles. Following repeated injection, soluble IL-2 receptors were substantially but transiently increased. A dose-dependent decrease in area under the concentration–time curve (AUC) between days 1 and 5 was attributed to a receptor-mediated change in clearance. Concentrations were described using an unusual model that employed an indirect stimulatory PD model to link the time-dependent changes of the pharmacokinetics with the change in IL-2 receptor density following repeated administration. 41.4.2
Interferons
Interferons, which were discovered in the 1950s as a result of their antiviral activity (76), are pleiotropic agents exhibiting a wide variety of effects including antiviral, antiproliferative, hematopoietic, and immunomodulatory activities (77, 78). Interferons are sometimes considered to be cytokines because of their role in cellular and humoral immune responses. Interferons are generally stimulatory proteins that exert their activity through interactions with cell surface receptors, inducing cellular processes and enhancing specific gene translation (79). Interferons also regulate the expression of unique antiviral proteins such as MX protein, which alters microtubule formation and mitosis, and 2′-5′-oligoadenylate synthetase (2,5-OAS), which induces the destruction of viral RNA. A general schematic diagram of the mechanism of action of interferons on cellular protein production is presented in Figure 41.4. The pharmacodynamics of interferon alpha using MX protein as the biomarker have been described using a simple indirect effect stimulatory model (55). Although they predominantly exhibit stimulatory activity, interferons can inhibit general cellular protein synthesis, including the synthesis of cytochrome P450 enzymes, making interferon one of the few biological agents that have the potential for causing “classic” drug interactions. The mechanism by which IFN-α exerts antitumor activity is unclear, particularly in hematological cancers. In melanoma and renal cell carcinoma, antitumor effects may be mediated by augmented immune responses including activation of natural killer lymphocytes and enhanced expression of cell surface antigens (e.g., MHC I and II). However, these mechanisms have not been decisively proved. 41.4.3
Growth Factors
Growth factors are proteins that bind to receptors on the cell surface, with the primary result of activating cellular proliferation and/or differentiation. Many growth factors are quite versatile, stimulating cellular division, maturation, and margination in numerous different cell types; while other factors are specific to
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IFN
FIGURE 41.4 Mechanism of action of interferon on cellular processes. Interferon (IFN) binds to a cell surface receptor, which, through a series of cellular processes, enhances the formation of RNA and subsequently increases protein formation when the RNA binds to ribosomes (hexagons). The proteins (cylinder) can either remain in the cell or be excreted.
a particular cell type. Growth factors are not the only agents that exhibit hematopoietic activity; cytokines are also capable of modulating cell growth and maturation. At present, there are several hematopoietic factors that have been approved for clinical use. These approved agents include granulocyte growth and stimulation factor (G-CSF, Filgrastim, Neupogen®) and its PEG-modified variant (Neulasta®), granulocyte-macrophage colony-stimulating factor (GM-CSF), stem cell factor (SCF), and erythropoietin (EPO) and its hyper-glycosylated variant (Aranesp®). A simplified diagram of the hematopoiesis is provided in Figure 41.5. The growth factors and cytokines that influence each pathway are provided in the figure. One aspect that must always be considered in the development of any PD model for a biological agent is the presence of endogenous factors that will also influence the measured biomarker activity. This is particularly true for growth factors. For example, patients with neutropenia following chemotherapy will have elevated endogenous levels of G-CSF. The same is true of an anemic patient, whose endogenous EPO levels will be elevated. The pharmacodynamics of growth factors are complex, but they do have some common characteristics between classes. 41.4.4
G-CSF
G-CSF is a protein that regulates the production of neutrophils by stimulating neutrophil progenitor proliferation (80, 81), differentiation (82), and selected endcell functional activation (83). G-CSF has little effect on the production of other hematopoietic cells. Endogenously, G-CSF is produced by monocytes, fibroblasts, and endothelial cells. As a low molecular weight protein (MW < 20 kD), G-CSF is subject to clearance by glomerular filtration and there is good evidence that
SPECIFIC PROTEINS
1015
IL-7
T Progenitor
IL-2 IL4 IL-7
T Helper
Thymocyte IL-3 IL-3 IL-5 IL-6
IL-3 IL-7
Lymphoid Stem Cell
T Killer
B Progenitor
IL-3
B Cell IL-3
Pluripotent Stem Cell
Macrophage
Monocyte
Stromal Cell GM-CSF M-CSF
Granulocyte Monocyte Progenitor IL-3 GM-CSF
IL-3, IL-6 GM-CSF
GM-CSF G-CSF
Neutrophil IL-5 GM-CSF
IL-3 GM-CSF
Eosinophil IL-4 GM-CSF
Myeliod Stem Cell
IL-3 IL-11 GM-CSF EPO, TPO IL-3 GM-CSF EPO
Basophil IL-6 GM-CSF EPO, TPO
Platelets
Megakaryocyte erythrocyte EPO
Erythroid Progenitor Reticulocyte
FIGURE 41.5 Schematic diagram of hematopoiesis. The cytokines and other factors that control cell formation and maturation are provided for reference.
endogenous concentrations are also cleared by binding to a receptor on the surface pluripotent stem cells as well as neutrophils (84). The first binding effects the differentiation of the stem cells toward eventual maturation to neutrophils. The latter receptor interaction appears to play a critical role in hemostasis, increasing clearance of G-CSF from both endogenous and exogenous sources when cell counts are high as a means of controlling neutrophil count by a feedback mechanism (31). Because G-CSF works to increase circulating neutrophils, the clearance of G-CSF varies over the course of treatment and is dependent on individual PD response (85). There is a strong PK–PD interaction with this agent and, therefore, a physiological limitation to the PD activity of G-CSF. Wahlby et al. (86) have demonstrated the importance of using time-dependent covariates and this is particularly relevant with biological agents. Another aspect of the pharmacological response to G-CSF administration is the unusual pattern of induced changes in neutrophil count over time. Immediately following the first administration, G-CSF dose-independently induces neutropenia
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and causes substantial downregulation of its own receptor (CD114) on neutrophils (87). This G-CSF–CD114 interaction dose-independently induces degranulation of neutrophils, which results in increased levels of gelatinase B, an enzyme that precipitates neutropenia and subsequent neutrophilia. The gelatinase B release into plasma may also contribute to mobilization of neutrophils or stem cells into peripheral circulation. The PK–PD relationship for G-CSF following IV and SC administration was well characterized in healthy volunteers (53). The PK model was a two-compartment PK model with bisegmental absorption from the site of SC administration, parallel first-order and saturable elimination pathways, and an indirect effect PD model describing the time course of neutrophils. A sigmoidal Emax model was applied for the stimulation of the neutrophil input rate. In addition, a time-variant scaling factor for absolute neutrophil count (ANC) observations was introduced to account for the early transient depression of ANC. A simple indirect effect stimulatory model adequately describes the time course of neutrophils following G-CSF administration. However, G-CSF is commonly administered following chemotherapy to treat the associated neutropenia. There is a substantial lag time between the administration of chemotherapeutic agents and the nadir ANC value that can be described more accurately using the “cell transit” PD model (88) than a simple indirect effect model. The cell transit PD model utilizes a gamma distribution to provide a semiphysiological description of cell maturation. The schematic diagram for this model is provided in Figure 41.6. Despite its apparent complexity, this model is relatively easy to use. There is only one transit rate constant “Kt” that is used to describe the transfer of cells from one compartment to the next. Ksyn and Kdeg are the synthesis and degradation rate constants, respectively. Chemotherapy is assumed to act on the synthesis rate constant in an inhibitory fashion. The effect of G-CSF can also be added to this model, making it a better model for comparing the efficacy of G-CSF and other variants in a clinically relevant system. 41.4.5
EPO
Unlike G-CSF, erythropoietin (EPO) is a pleiotropic agent with multiple actions and different sites of activity. The various sites that EPO can affect in the hemaKsyn
Feedback
Stem Cell
Kt
CFU
Kt
Early Cells
Kt
Mature Cells
Kt
Circulate Blood Cells Kdeg
FIGURE 41.6 Schematic diagram of “cell transit” model. In this model, pluripotent stem cells are assumed to be produced on a first-order process, and differentiation (i.e., passing though each stage or compartment of cell growth) occurs over a fixed transit time (Kt). Mature cells are lost via a first-order process (Kdeg). The model can also allow for a negative feedback by which mature white cells diminish the formation of new stem cells.
SPECIFIC PROTEINS
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topoietic chain are shown in Figure 41.5. Primarily, EPO functions to maintain appropriate oxygenation of cells and its production is regulated via feedback from oxygen pressure detecting cells in the kidneys. A schematic diagram of the process of red blood cell (RBC) production and maturation is provided in Figure 41.7. In the circulatory system, EPO also effects the development of new blood vessels. EPO may also act to facilitate the survival and proliferation of nonerythroid cells as well. In addition to production in the kidneys, EPO is produced in the brain, although this form has a lower molecular weight than the peripherally produced variant (89). In the central nervous system (CNS), EPO plays a critical role in brain function and development. EPO receptors have also been isolated in ovary, oviduct, uterine, and testes cells (90), as well as some tumor cell lines such as breast cancer (91). The function of EPO binding in these alternate cell types has not yet been determined, although the appearance of EPO receptors on cancer cells has been indicated as a poor prognosis factor (92). The use of EPO to treat anemia arising from chemotherapy or from cancer has therefore been called into question (93). Recombinant human EPO has a relatively low molecular weight (MW = 30.4 kD). Because of its low molecular weight, EPO would be expected to undergo both renal and hepatic elimination, although these routes of elimination appear to be relatively minor (94). The pharmacokinetics of EPO have been extensively studied, and in many cases, nonlinear elimination was reported (95) and the clearance was determined to change following phlebotomy or bone marrow ablation in a fashion that is consistent with receptor-mediated clearance. Following SC administration to healthy volunteers, the pharmacokinetics of EPO were best described with a dualabsorption rate model (fast zero-order and slow first-order inputs) with nonlinear disposition (96). Unlike white cells, which are generally assumed to follow first-order kinetics, red cells are attributed as having a lifespan of 120 days in a normal adult (97).
Stem Cells BFU-E
EPO
CFU-E
Erythroblast Reticulocyte
Red Cells
Kidney Oxygen-Sensing EPO Production
EPO
FIGURE 41.7 Mechanism for control of endogenous erythropoietin and red cell production. The regulation of EPO production depends on the oxygenation of the blood as it passes through the kidney. If the oxygen levels are low, the kidney synthesizes additional EPO, which acts to prolong the survival of the blast forming units (BFU-E) and colony forming units (CFU-E), allowing enhanced red blood cell (RBC) production.
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This theory was originally established in the 1960s and still remains in place today. Therefore, a specialized set of indirect PD models for agents that alter the generation of natural cells based on a lifespan concept were developed (98) based on the concept of a fixed lifespan. In this “lifespan model,” mature cells are assumed to be produced at a constant rate and to survive for a fixed interval of time, after which they are lost. Therefore, rate of cell loss must equal the cell production rate at the time those cells were produced (e.g., one “lifespan” ago). This aspect of lifespan-mediated loss requires that the model also track the number of cells produced at a prior interval of time. A stimulatory or inhibitory effect of a therapeutic agent such as EPO then results in a delayed increase in cell count (depending on the number of intervening compartments) but the return to baseline cell count is dependent on the lifespan of the cells. It is interesting to note that the “transit model” and the “lifespan model” both produce a very similar time course of effect if the transit model has sufficient (e.g., at least five) compartments. 41.4.6
Antibodies
When engineered monoclonal antibodies were initially undergoing development as therapeutic agents, it was assumed that the mechanism of activity of these antibodies was directly related to the effector activity of antibodies in vivo. That is, the MAb would bind to its target receptor, which would precipitate antibody-dependent cell cytotoxicity (ADCC) by attracting natural killer (NK) cells. Because many of these early MAbs were targeted against cancer cells, ADCC was considered a desirable mechanism of action, although these early theories have not been borne out. NK cells provide two types of effector function: cell cytotoxicity and lymphokine secretion. In conjunction with antibodies, NK cells can cause cytotoxicity through recognition and lysis of MAb-coated target cells. A proposed schematic for the mechanism of such “lytic” antibodies is presented in Figure 41.8. NK cells also possess a wide range of regulatory receptors that can prevent cytotoxic responses (99) based on the cell surface expression of killer-cell inhibitory receptors (KIRs). Regulation of NK cell effector activity has called into question the extent to which
MAb Add MAb R+
R+
Ab-Ag Binding R-MAB
Fc Binding
Opsonization Decrease in R+ Cells
FIGURE 41.8 Mechanism of action of “lytic” monoclonal antibodies. MAb binds to receptor on antigen presenting cell. The receptor binding attracts phagocytes, resulting in cell destruction.
SPECIFIC PROTEINS
1019
therapeutic MAbs utilize cytotoxic effector mechanisms to provide clinical benefit. It is also unclear whether the concentrations of MAbs following therapeutic dosing are sufficiently high to saturate the substantial numbers of targeted receptors present in many therapeutic indications. Conversely, there is evidence from Fc receptor knockout mice suggesting that, in certain systems, Fc binding is required for clinical activity of therapeutic MAbs (100). Furthermore, anticancer MAbs do not appear to have functional activity when administered as a Fab fragment, which would support the theory that clinical activity of MAbs is linked to effector function. Fc receptors do more than recruit effectors; therefore, part of the requirement for Fc function may be attributable to crosslinking on the target cells, which interferes with cellular function. However, many therapeutic MAbs are currently being investigated for autoimmunity and immunosuppression in therapeutic areas such as rheumatoid arthritis, where the role of ADCC is less appropriate (101). An alternative mechanism of blocking or modulating responses is more desirable than ADCC. A key aspect of this mechanism is that a relatively short exposure to MAb may break the inflammatory cycle and allow the repair process to begin. A schematic diagram of this mechanism is presented in Figure 41.9. MAbs therefore can provide a long-term effect following short-term treatment through a mechanism referred to as “infectious tolerance” (102). The PD behavior of therapeutic MAbs is complex. Fortunately, the activity of these agents can be evaluated by the use of fluorescence activated cell sorting (FACS). Like most therapeutic biologics, the mechanism of action is commonly described using the standard indirect effect models. However, because FACS data can determine the fraction of cell surface receptor bound by the MAb, and because the change in receptor density can be followed using this same method, more mechanistic models have been proposed (29). In these models, the relationship between concentration of antibody and bound receptors can be explicitly described and the bound antibody is then used to drive the indirect effect model. A schematic of this model is provided in Figure 41.10.
Antibody R+ R+
R+ R+ Antigen
Antigen Interaction at Receptor Stimulates More Cell Activation
Activated Cell R+ Cell R+ Antibody Binding Prevents Antigen Interaction At Receptor Antigen
? Unactivated Cell
R+ Cell Stimulate R Shedding
FIGURE 41.9 Mechanism of action of coating monoclonal antibodies. MAb binds to receptor on antigen presenting cell. The receptor binding is “nonproductive”, resulting in a stimulation of receptor loss. Alternatively, when the MAb binds to the receptor, it results in a steric blockage of receptor.
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Kin
Kout R+ Cell Emax[MAb] EC50+[MAb] Kon
Koff
R-MAb
FIGURE 41.10 Example schematic for PK/PD model for therapeutic monoclonal antibodies. The antibody binds (potentially reversibly) to the receptor on the cell surface. The bound antibody–receptor complexes are then what drives the stimulation of loss of receptor-positive (R+) cells.
It should be noted that, unlike small molecules, the action of MAbs is often directly related to the bound concentration of drug rather than the free concentrations, in contrast to the PD behavior of small molecules.
41.5
COVARIATES FOR PHARMACODYNAMIC RESPONSE
When modeling the pharmacodynamics of a biological agent, some consideration needs to be given to the identification of covariates. Because the physiological system that is targeted by the agent is often well characterized, the selection of covariates for investigation can often be limited to those that have a strong likelihood of being identified. For instance, when characterizing the pharmacodynamics of a hematopoietic factor, the time to last treatment with chemotherapy or even the type or number of cycles of treatment might be projected to be covariates. Chemotherapy results in loss of bone marrow function, which in turn should reduce the PD response in a patient. Similarly, concomitant administration of drugs that alter receptor density could also affect the PD response of the investigational agent. PD models often take a long time to converge, so the number of covariates that can be investigated is often limited. Careful selection of the covariates for evaluation is therefore necessary.
41.6 EVALUATING PHARMACODYNAMICS USING MODEL-BASED ANALYSIS Three examples of a NONMEM (Version 5, Globomax LLC, Hanover, MD) control stream and the necessary data format for a commonly employed PK/PD model are
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provided in the following section. The generic code and associated example data file format must be suitably altered for more complex models. 41.6.1
Control Stream
One of the more common PD models used to describe the time course of a biomarker is the simple indirect effect model. In this example, drug concentrations increase the rate of degradation of the biomarker and act to reduce the biomarker concentration. Such a model has been used to characterize the pharmacokinetics/ pharmacodynamics of PEGylated interferon (55) and other proteins. A fragment of an example control stream for this basic PK/PD model is presented and explained in Table 41.5. Explanations for the different commands used are provided as well. TABLE 41.5 Example NONMEM Code 1: Commonly Used Indirect Effect PK/PD Model NONMEM Code
Explanation
$SUBROUTINES ADVAN6 TRANS1 TOL 3
Typically evaluating a PK/PD model requires the use of ADVAN6 (or one of the other ADVANs used to evaluate differential equations). Initially, TOL is set to a low value such as 3 to facilitate convergence.
$MODEL COMP=(CENTRAL,DEFOBS) COMP=(PERIPH) COMP=(EFFECT)
These lines of code define the PK and PD model compartments. Here, the first two compartments are defined for the pharmacokinetics and the third compartment is for the biomarker.
$PK ;Define the PK Parameters CALLFL=-22 TVMX = THETA(1) VMAX = TVMX*EXP(ETA(1)) TVKM = THETA(2) KM=TVKM*EXP(ETA(2)) K23 = THETA(4)*EXP(ETA(4)) K32 = THETA(5)*EXP(ETA(5)) TVV1 = THETA(3) V1 = TVV1*EXP(ETA(3))
This portion of the code defines the parameters for the PK model. The semicolon is a “comment” statement and is not read by NONMEM. Whenever developing a model, adding good commenting is always recommended. This is particularly true when the model is complex and the control stream is long. The present PK model is a two-compartment model with Michaelis–Menton elimination. VMAX and KM define the parameters for elimination, K23 and K32 define the intercompartmental transfer rate constants, and V1 is the central volume of distribution. As with many biologics, this theoretical agent is being administered intravenously.
;DEFINE PD PARAMS TVKDEG=THETA(6) KDEG=TVKDEG TVKSYN=THETA(7) KSYN=TVKSYN*EXP(ETA(8)) TVEMAX=THETA(8) EMAX=TVEMAX*EXP(ETA(6)) TVEC50=THETA(9) EC50=TVEC50*EXP(ETA(7))
This portion of code defines the parameters for the PD model. This is a simple indirect effect stimulatory model. KDEG is the degradation rate constant for the biomarker, KSYN is the formation rate constant of the biomarker, EMAX is the maximal effect of the drug, and EC50 is the concentration at which half-maximal effect occurs.
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TABLE 41.5 Continued NONMEM Code
Explanation
;SCALE COMPARTMENTS S1=V1 F3=KSYN/KDEG
This section of code scales the PK compartments. An important aspect to note here is the use of F3, which is the bioavailability term for the effect compartment. Here, F3 is used to initialize the effect compartment to the baseline value of the biomarker being modeled, which is why it is set to the ratio of KSYN/KDEG. Prior to therapy, the biomarker is presumed to be at some steady-state value, which should be equivalent to the ratio of formation to degradation. F3 is used in conjunction with a special dose item for this compartment to initialize the effect compartment.
$DES ;PK Model CP=A(1)/V1 ;PLASMA CONCS IN UG/L CLMM=(CP*VMAX)/(KM+CP) DADT(1)=-CLMM-K23 *A(1)+K32*A(2) DADT(2)=K23*A(1)-K32*A(2) ;PD Model EFF=EMAX*CP/(EC50+CP) DADT(3)=KIN-KOUT*A(3) *(1+EFF)
This section contains the differential equations that define the pharmacokinetics and the pharmacodynamics of the drug. Here, the effect is stimulatory on KDEG, which will result in a transient reduction of the biomarker.
$ERROR QK=0 OD=0 IF (CMT .EQ. 1) QK=1 IF (CMT .EQ. 3) QD=1 PKY = F * EXP(ERR(1)) + ERR(2) PDY = F+ERR(3) Y=QK*PKY+QD*PDY
This section defines the residual error models for the pharmacokinetics and pharmacodynamics. Note that the PD observations have a simple additive error function, which is different from the PK residual error function. Residual error models must be selected separately for the PK and PD models.
It is important to realize that the parameter estimates obtained for one agent do not usually translate readily to other agents for the PD models. Furthermore, the evaluation of complex sets of differential equations using NONMEM generally requires that the initial estimates for parameters be reasonable since the models can find local minima or may not converge at all if the initial estimates for the parameters are not reasonable. Therefore, when developing a model for a new agent, care should be taken to ensure that the initial estimates are good, and running the model without the estimation step to evaluate the performance of these estimates is critical. A more complex binding type model is defined in Table 41.6. The model defined here is similar to the model shown in Figure 41.10, although it uses scaled binding
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TABLE 41.6 Example NONMEM Code 2: “Cell Transit” Gamma Distribution Model $SUBS ADVAN6 TOL=3 $MODEL COMP=(CENTRAL) COMP=(PERIPH) COMP=(STEM) COMP=(WBC) COMP=(TRANSIT1) COMP=(TRANSIT2) COMP=(TRANSIT3) “FIRST “COMMON/PRCOMG/IDUM1,IDUM2,IMAX,IDUM4,IDUM5 “INTEGER IDUM1,IDUM2,IMAX,IDUM4,IDUM5 “IMAX=50000000 $PK V1 = THETA(4)*EXP(ETA(2)) K10 = THETA(1)*EXP(ETA(1)) K12 = THETA(2) K21 = THETA(3) BASE = THETA(5)*EXP(ETA(3)) IF (BASE .LE. 0) EXIT 1 101 MTT = THETA(6)*EXP(ETA(4)) K = 4/MTT F3 = BASE F4 = BASE F5 = BASE F6 = BASE F7 = BASE POWER = THETA(7) SLOP = THETA(8)*EXP(ETA(5)) $DES ;Pharmacokinetics CP = A(1)/VOF DRUG = SLOP*CP ;****************KINETICS**************************** DADT(1)=A(2)*K21-A(1)*K10-A(1)*K12 DADT(2)=A(1)*K12-A(2)*K21 ;****************DYNAMICS*************************** DADT(3) = K*A(3)*(1-DRUG)*(BASE/A(4))**POWER - K*A(3) ;Pharmacodynamics DADT(4) = K*A(7) - K*A(4) ;WBC DADT(5) = K*A(3) - K*A(5) ;TRANSIT 1 DADT(6) = K*A(5) - K*A(6) ;TRANSIT 2 DADT(7) = K*A(6) - K*A(7) ;TRANSIT 3 $ERROR IF (CMT .EQ. 1) QK=1 IF (CMT .EQ. 4) QD=1 PKY = F * EXP(ERR(1)) + ERR(2) PDY = F * EXP(ERR(3)) + ERR(4) Y=QK*PKY+QD*PDY
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kinetics to define the binding. This control stream has several similarities to the previous example in that a stimulatory indirect effect model is used to describe the change in time course of the biomarker, although in this present example, it is bound drug causing the change in biomarker, not the total drug concentration as was seen previously. This control stream uses a separate compartment for the binding of drug, with limitations set to scale the amount of drug that can bind. Information to develop these models must usually combine information from in vitro studies as well as nonclinical pharmacology studies. Table 41.6 shows the control stream used to implement the model shown in Figure 41.6. This is the “cell transit” model, which is defined by a simple queuing or gamma function. The model is particularly useful for describing the pharmacodynamics of agents that impact cell growth or turnover or have very long lag times. 41.6.2
Database Requirements
An example database for a two-compartment PK model driving a simple indirect effect model is presented in Table 41.7. NONMEM nominally assigns a value of 0 TABLE 41.7 Example Database Format for Indirect Effect Model CID 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3
DATE
TIME
AMT
RATE
DV
CMT
MDV
EVID
7/16/2000 7/31/2000 7/31/2000 8/2/2000 8/2/2000 8/7/2000 8/7/2000 8/7/2000 12/19/1991 1/3/1992 2/3/1992 2/3/1992 2/6/1992 2/10/1992 2/10/1992 2/13/1992 2/17/1992 12/25/1991 1/9/1992 1/27/1992 1/27/1992 1/27/1992 1/27/1992 1/27/1992 1/27/1992 2/2/1992
8:00 13:20 13:20 10:15 12:47 10:45 10:45 13:15 2:00 8:00 15:00 15:15 16:59 8:50 10:05 0:00 8:36 8:00 11:06 8:21 11:30 11:43 14:30 16:30 18:30 9:15
1 3,000 . 30,000 . 30,000 . . 1 . . 7,500 . 7,500 . . . 1 . . 7,500 . . . . .
. 1,500 . 15,000 . 15,000 . . . . . 3,600 . 3,600 . . . . . . 3,600 . . . . .
. . 124 . 700 . 37 729 . 6 8 . 7 . 5 6 6 . 117 176 . 155 81 91 114 169
3 1 3 1 1 1 3 1 3 3 3 1 3 1 3 3 3 3 3 3 1 3 3 3 3 3
1 1 . 1 . 1 . . 1 . . 1 . 1 . . . 1 . . 1 . . . . .
1 1 . 1 . 1 . . 1 . . 1 . 1 . . . 1 . . 1 . . . . .
SUMMARY
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for the initial conditions for each compartment. While the assumption that the initial condition is 0 is acceptable for a PK model, it is not an appropriate initial condition for many PD models. This is particularly true for indirect effect models because the biomarker is usually present at some steady-state level prior to the administration of drug. Once drug is administered, the biomarker will either increase or decrease depending on the activity of the drug. Therefore, it is necessary to use a bioavailability term to initialize the effect compartment and to provide a unit dose into that compartment in the data set. The unit dose record should be read in prior to the other records so that the effect compartment can be initialized. It need be entered only once per individual. The other specialty item for an indirect PK/PD database is the use of the CMT item. This item is set to 1 for the PK observations and to 3 for the PD observations in the example data provided so that the observations are properly associated with their assigned compartments. In this example database, the PK model was assumed to be a two-compartment model. For more information on the use of special data items, see the NONMEM User Guides (103).
41.7
SUMMARY
Developing models that describe the pharmacokinetics and pharmacodynamics of therapeutic proteins is a challenging process. Before getting started, it is always best to develop an understanding of the system that the therapeutic protein will impact. It is also helpful to review available information on the PK behavior of other related proteins. Understanding the design and manufacture of the biological agent is also critical. For instance, in many cases, proteins that have been modified by the addition of such agents as polyethylene glycol will have lower binding affinities to target receptors, but these agents also have a much lower systemic clearance, yielding a net therapeutic benefit. During evaluation of these agents, the usual covariates that are evaluated for chemically based drugs such as creatinine clearance may not be relevant given the size and modification of the biologic. Careful consideration of the protein and its pharmacology is helpful to determine what covariates are likely to be relevant. Similarly, it is not uncommon for the PD activity of a therapeutic protein to have an impact on the pharmacokinetics. This is particularly true for agents that are cleared by binding to a receptor and, through that binding, alter the receptor expression. In most cases, the PD activity of a therapeutic protein does not follow direct effect type behavior. Rather, these agents act at a cellular level and the resulting activity is governed by slower moving processes such as cell or protein turnover. Again, understanding the pharmacology of the therapeutic agent will be helpful in designing a PD model that adequately describes the activity of the drug. The information provided in this chapter is meant to serve only as a basic overview to therapeutic proteins. There are many agents that have been developed that cannot be described using the basic concepts described here. Evaluation of these agents is always a learning process!
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ABBREVIATIONS ADCC ANC CFU CHO EPO Fab FACS FC G-CSF HAGA HAMA IFN IG IL K kD KIR MAb MW NAb NK N-NAb PD PEG PK PMNs R+ RBC rHu SC WBC
Antibody-dependent cell cytotoxicity Absolute neutrophil count Colony forming units Chinese hamster ovary Erythropoietin Fragment antigen binding (binding region of a monoclonal antibody) Fluorescence activated cell sorting Fragment constant (heavy chain region of monoclonal antibody) Granulocyte colony-stimulating factor Human anti-globulin antibody Human anti-murine antibodies Interferon Immunoglobin Interleukin Rate constant Kilodaltons (a molecular weight for proteins) Killer-cell inhibitory receptor Monoclonal antibody Molecular weight Neutralizing antibody Natural killer Nonneutralizing antibody Pharmacodynamic Polyethylene glycol Pharmacokinetic Polymorphonuclear neutrophils Receptor positive Red blood cell Recombinant human Subcutaneous White blood cell
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CHAPTER 42
Analysis of Quantic Pharmacokinetic Study: Robust Estimation of Tissue-to-Plasma Ratio HUI-MAY CHU and ENE I. ETTE
42.1
INTRODUCTION
Preclinical pharmacokinetic (PK) studies provide information useful for supporting efficacy and safety evaluation studies in animals, preclinical and clinical study designs, dosing regimen development, and interpretation of toxicity data. These studies provide PK data that may be useful in dose escalation in healthy volunteers and patients. Toxicokinetics is a major component of toxicology studies. It enables the investigation of the relationship between drug dose and measured concentration, primarily the establishment of the dose proportionality and linearity or nonlinearity in pharmacokinetics. Toxicokinetic and PK research studies are characterized by some uncertainty regarding the process studied and significant variation in the concentration measurements obtained. Variability in PK parameters among homogeneous strains of small laboratory animals has been reported to be between 30% and 50% in some cases (1, 2). In addition to the inherent variability of the biological system, there is the uncertainty associated with the assay and process noise. The number of samples that can be obtained per subject is limited to one sample per subject (especially when destructive sampling is implemented) in most rodent toxicokinetic studies. The fact is that, for small laboratory animals, the periods between successive sampling times are simply not long enough to allow sufficient recovery. A major disadvantage of this sampling scheme is that intraindividual concentration–time profiles are unavailable. This poses a data analysis challenge because the one sample per subject data constitutes the extreme case of sparsely sampled PK data, hence extremely sparse data, with independent observations over time. The situation is complicated when tissue sampling (e.g., in tissue distribution studies) is involved, and the ratio of tissue to plasma concentrations is the object of
Pharmacometrics: The Science of Quantitative Pharmacology Edited by Ene I. Ette and Paul J. Williams Copyright © 2007 John Wiley & Sons, Inc.
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the investigation. Equally, only one tissue sample/subject is obtained in such studies because the animal is usually sacrificed. A solution for analyzing extremely sparse data is to use the nonlinear mixed effects modeling approach. This has been elegantly addressed in the literature (3–6). However, it is common practice that to compute a PK parameter such as area under the concentration–time curve (AUC) some form of data pooling is used. Thus, to compute noncompartmental AUC, actually “composite” AUC, data are averaged at each time point and the parameter is estimated using the trapezoidal rule. This composite approach to the estimation of AUC has problems associated with it. The AUCs and other mean PK parameters are estimated with no measures of uncertainty associated with them. Other measures characterizing the distribution of the parameters are, in general, difficult to obtain. Some ad hoc solutions have been proposed for the estimation of the average (or “typical”) AUC in a population of extremely sparsely sampled subjects (7–11). Pai et al. (12) proposed the use of the bootstrap resampling technique for the estimation of AUC from sparsely sampled populations in toxicology studies.
42.2
ESTIMATION OF TISSUE-TO-PLASMA RATIO
The challenge in the analysis of quantic (one sample/subject) data is further complicated when tissue sampling (e.g., in tissue distribution studies) is involved, and the ratio of tissue to plasma concentrations is the object of the investigation. Equally, only one tissue sample/subject is obtained in such studies because the animal is usually killed. Tissue-to-plasma ratio is commonly determined from the ratio of average concentrations at specified time points. It is not uncommon, in practice, for the ratios to be calculated at selected time points corresponding to peak and trough concentrations, and the variations in the ratios are usually very large. This finding could be attributed in part to the variations in the concentrations and a lack of accounting for the correlation in observations from the biological matrices sampled from each subject. Occasionally, tissue-to-plasma ratio is calculated using area under the concentration curves (AUCs) calculated from mean profiles using the noncompartmental approach. These “composite” AUCs are usually computed from data that are averaged at each time point (naive data averaging approach) using the trapezoidal rule. The tissue-to-plasma ratios computed using either average concentrations at specified time points or composite AUC values are usually reported without regard to the correlation structure in the data, and no measures of dispersion and uncertainty associated with them. In the sections that follow a data set from a quantic PK study is assumed and various approaches used to estimate tissue-to-plasma ratio are discussed as they are applied to the data. Thus, the approaches used in the estimation of tissue-toplasma ratio are presented with their advantages and disadvantages; the importance of examining convergence with resampling algorithms is discussed, as well as the impact of outliers on the performances of the ratio estimation approaches. This is followed with an overall thought on the estimation of tissue-to-plasma ratio with a recommendation on a preferred approach (a nonparametric random sampling
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approach) for the robust estimation of robust tissue-to-plasma ratio in a drug development setting. 42.2.1
Data Set
Since the objective of this chapter is the development of a methodology for estimating robust tissue-to-plasma ratio in a drug development setting, such details as are necessary for understanding the proposed methodology are presented. It is important to note that in drug development, pragmatism, efficiency, and effectiveness are major considerations. A toxicokinetic study was performed to determine the tissue-to-plasma ratio of a drug in development. An oral dose of a drug in development was administered to 18 rats, and each animal was killed at one of six specified time points: 0.5, 1, 2, 4, 6, and 8 hours. Therefore, each animal had only 1 pair of concentrations, 1 each from plasma and tissue, respectively. Table 42.1 shows the data set from the study used in our investigation. The effect of correlation structure in the data set is also of interest. Thus, we investigated the effect of maintaining or breaking the relationship between tissue and plasma concentrations within the same animal, using both paired and unpaired tissue and plasma data to evaluate the effect on the robustness of estimation of tissue-to-plasma ratio. 42.2.2
Approaches for Estimating Tissue-to-Plasma Ratio
We have taken a very practical approach in addressing the computation of tissueto-plasma ratio in a drug development setting. Thus, approaches that are commonly used in practice (i.e., the naive data averaging and ratios of concentrations by time point approaches) for computing tissue-to-plasma ratio were employed in this investigation and compared with our proposed methodology—the random sampling approach. Since our approach is a sampling-based approach, we have included a comparison of the performance of our approach with another sampling-based approach reported in the literature, the PpbB (14), in the estimation of tissueto-plasma ratio. First, the naive data averaging approach is discussed, followed by a discussion of the random sampling approach, and then the PpbB approach.
TABLE 42.1 A Sample Data Set from an Oral Toxicokinetic Study a Time Point (hour) Biological Matrix
0.5
1
2
4
6
8
Plasma Tissue
0.18 9.05
0.13 1.76
0.12 1.26
0.04 0.18
0.00 0.02
0.01 0.42
Plasma Tissue
0.18 5.24
0.14 1.65
0.11 1.67
0.03 0.64
0.05 0.28
0.00 0.07
Plasma Tissue
0.17 2.92
0.18 4.18
0.05 0.74
0.02 0.19
0.02 0.00
0.00 0.10
a
Each cell represents a pair of values from one animal; there are 18 animals in total.
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All methods were implemented in the statistical software, S-Plus Version 6.02 (Insightful, Seattle, WA). 42.2.3 Naive Data Averaging Approach The approach involves computing the average value of the data for each sampling time k: Ck =
1 I ∑ Cik I i =1
for i = 1, . . . , I, where I is the standard number of individual subject data at time point k. The averaging of data across subjects is a common practice owing to the assumption that all concentrations at each time point have been measured under identical conditions. Thus, tissue-to-plasma ratio is estimated independently for each time point using the averaged concentration at each time point. Alternatively, the noncompartmental AUC, actually the composite AUC, can be estimated using the trapezoidal rule. From this point, the use of the term “naive data averaging approach” will be reserved for estimation of AUC. The term “unpaired independent time points approach” will be reserved for use in cases where tissue-to-plasma ratio is calculated at each time point using a measure of central tendency (mean or median) of the measured concentrations without regard to the correlation structure in the observations. The term “paired independent time points” approach will be used when the pairing of observations is taken into account in the calculation of the tissue-toplasma concentration ratio at each time point. 42.2.4 Traditional Naive Data Averaging Approach Incorporating Independent Time Points Approaches The results of tissue-to-plasma ratio values obtained with three approaches: (a) unpaired independent time points approach, (b) paired independent time points approach, and (c) naive data averaging approach are presented herewith. Table 42.2 illustrates ratios obtained across time points by calculating the mean and median for each time point independently for tissue and plasma. There is no measure of variability around each time point, as expected. When zero was returned for plasma concentration (e.g., 6 and 8 hour time points in Table 42.2) because the levels were not quantifiable or below the limit of quantification, the zero divisor of the ratio yielded the result #DIV/0. Such an outcome cannot be interpreted and is usually discarded in the presentation of results with the independent time points approach. Table 42.3 contains summary statistics of paired tissue-to-plasma ratios obtained with the paired independent time points approach. The tissue-to-plasma ratios by time point can vary from 0 to 50.3 across different time points, as shown in the last row of Table 42.3. Table 42.4 contains the tissue-to-plasma AUC ratio (TPAR) derived by calculating the mean AUC values using the naive data averaging approach across time points for both tissue and plasma without regard to the correlation structure in the data. As expected, there is also no measure of variability around TPAR obtained with the naive data averaging method. The code for implementing the naive data averaging approach is in Appendix 42.1.
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TABLE 42.2 Tissue-to-Plasma Ratios Calculated Using the Unpaired Independent Time Points Approacha Time Point (hour)
Tissue/Plasma Ratio
0.5
1
2
4
6
8
Median Mean
30 32
14 16
14 13
10 12
#DIV/0 #DIV/0
#DIV/0 #DIV/0
a #DIV/0 indicates the denominator (plasma concentration) of the ratio is 0 (or below the quantifiable limit—BQL).
TABLE 42.3 Tissue-to-Plasma Ratios Calculated Using the Paired Independent Time Points Approach Ratiosa Time (hours) 0.5 1 2 4 6 8 All time points a
Minimum
Q1
Mean
Q3
Maximum
17 12.2 10.3 5.2 0 32.6
23.4 13.2 12 7.4 1.5 32.6
32.3 16.4 13.4 11.8 2.9 32.6
40 18.5 14.9 15.2 4.4 32.6
50.3 23 16 20.7 5.9 32.6
10
17.4
21.8
50.3
0
Q1 indicates first quartile and Q3 the third quartile.
TABLE 42.4 Tissue-to-Plasma Ratios Calculated from AUC Values Obtained Via the Naive Data Averaging Approach Central Tendency Mean a
Plasma AUCa
Tissue AUC
Tissue-to-Plasma AUC Ratio
0.45
7.62
17.05
AUC indicates area under the curve.
42.2.5
Random Sampling Approach
The random sampling (RS) approach was recently proposed by Chu and Ette (13). To implement the approach, the population sampling pool is first generated, and it comprises a large set of individual pharmacokinetic (PK) profiles based on the empirical data by resampling with replacement. This potential population pool contains M1 copies of PK profiles for each subject to ensure equal opportunity for each subject to be resampled for the next step. Next, M2 copies of the virtual study are drawn from the population pool, and then any function of interest is computed from the virtual study level. Figure 42.1 is a schematic chart illustrating the RS approach. The RS algorithm, therefore, is defined in two phases.
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Individual Sampling Pool for ith Subject
Population Sampling Pool This pool contains M1 copies of PK profiles and associated calculated function of interest for each subject and replicate to ensure equal opportunity for each subject to be resampled for the virtual study step.
Combine All
(1) Randomly resample M1 times with replacement from the available values independently at each unsampled time point. (2) Calculate functions of interest from each profile, for example, AUC.
Study Level Sampling (1) Draw M2 copies of size N (where N is the sample size, total number of animals, in the real study) of functions of interest from the population sampling pool.
Combine All
The Sampling Distributions of Functions of Interest From M2 Virtual Studies
(2) Calculate summary statistics of the function of interest in each virtual study, i.e., quantiles, mean, and median.
(1) Calculate mean, median, and quantiles of parameters such as AUC.
FIGURE 42.1 Schematic chart for random sampling approach.
Phase 1: Setting Up the Population Sampling Pool by Generating Individual Subject Sampling Pools Phase 1 is done by constructing the individual level sampling pool (i.e., the concentration values for the ith subject at rth replicate resampling (C*ir)). The steps to do this are as follows: Step 1. For the ith subject with datum observed at time point w, randomly resample M1 times with replacement from the available values independently at each time point that the subject had no observation. For a subject that has wth time point observation, for example, the concentration values are to be resampled (i.e., plasma and tissue concentrations) at other k time points C*ikr, (where, k = 1, . . ., K, but k ≠ w, and r = 1, . . ., M1) to create a “complete profile” encompassing all sample points, including the observed Ciwr and the resampled vector C*ikr. More specifically, C*i.., M1 replicates of “complete profiles” for the ith subject, can be expressed as the following matrix: ⎡ ci*11 ⎢ Ci*.. = ⎢ . . . ⎢ ⎣ci*1M1
... ...
ciw1 ...
. . . ciwM1
*1 ⎤ ciK ⎥ ... ⎥ ⎥ * 1⎦ . . . ciKM
... ...
Each row represents one profile encompassing all sample/time points. Each column is M1 copy of the same time point. Step 2. Repeat Step 1 of Phase 1 to construct the individual profile pool for each subject. Step 3. Calculate functions of interest from each profile (e.g., AUC, Cmax).
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The population sampling pool of complete profiles is now ready to be sampled for the next phase of virtual study resampling. Phase 2: Generation of Samples at the Study Level Step 1. Draw M2 copies of size N (where N is the sample size, total number of animals, in the real study) of functions of interest from the population sampling pool obtained from Phase 1. Step 2. Calculate the summary statistics (i.e., quantiles, mean, and median) of the function of interest from each virtual study obtained from Step 1 of Phase 2. Step 3. Derive the summary statistics of required parameters across virtual studies with their associated standard deviations. The S-Plus code used in implementing the RS approach is in Appendix 42.2. 42.2.6
Pseudoprofile-Based Bootstrap
The PpbB approach (14) generates estimates of both the distributions of the raw data and the corresponding measures of variability. The term “pseudoprofile” was applied to the information obtained when one sample is obtained per animal but several animals are sampled at each of several times postdose. Bootstrap resampling is performed twice within the PpbB approach to generate PK pseudoprofiles from which the function of interest is estimated. More specifically, the following scheme is adopted for the b1th replicate at each time point: Step 1. Resample with replacement at one concentration, denoted as C*b1(tk), at time tk for k = 1 to K from the respective concentration vectors and keep K concentrations. c*b1(tk), k = 1, . . . , K. Step 2. Construct a pseudoprofile, that is, c*b1 = {c*b1(t1),c*b1(t2), . . . , c*b1(tk−1),c*b1(tk)}. Step 3. Repeat Steps 1 and 2 B1 times to generate a PK pseudoprofile pool Fˆ *, an estimate of the distribution F. Step 4. Calculate a function of interest from each pseudoprofile (i.e., AUC, Cmax). Step 5. Perform B2 times bootstrap resampling with replacement from this empirical distribution Fˆ * with sample size n¯ each, where n¯ is the average number of concentration replicates, and the corresponding parameter for each b2 = 1, . . . ,B2 is estimated. Step 6. Calculate the bootstrap estimates of the mean parameter and its standard deviation. The S-Plus code used in implementing the PpbB approach is in Appendix 42.3. Given the limitations associated with the naive data averaging approaches in estimating the tissue-to-plasma ratio, the RS approach is compared with the PpbB approach in subsequent sections. However, occasional references are made to the naive data averaging and independent time points approaches because of their use in common practice.
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42.3 COMPARISON OF PpbB AND RS APPROACHES 42.3.1 Paired Versus Unpaired Data Figure 42.2 illustrates the results obtained with the RS approach using paired and unpaired data at different replication levels (i.e., M1 equals 10, 100, and 500 to build up the population pool). This was then followed by a calculation of TPAR over M2 (i.e., 50) virtual studies (with N = 18 for each study). Across all three population pool levels (i.e., M1 = 10, 100, and 500), paired observations consistently yielded tighter distributions than unpaired ones. Similar results were obtained with the PpbB approach. If a drug is designed to target a particular tissue, the interest might be in having a minimal target TPAR. In that case, having knowledge of mean TPAR would not be enough. Having knowledge of the distribution of TPAR across virtual studies (i.e., replicates) in terms of the summary statistics (first quartile Q1, mean, median, third quartile Q3) becomes valuable. Thus, knowing that the TPAR is not below a certain cutoff, such as the first quartile of the TPAR distribution, would be important. To provide such an insight, we examined the distribution of TPAR within and between replicates and have provided a summary of the distribution of TPAR across virtual studies. Consequently, Figure 42.3 provides an amplification of the outcomes with the two approaches when the first and third quartiles (Q1 and Q3, respectively) for paired and unpaired data are compared. The quartiles for the unpaired data have a
25
TPAR
20
15
10 10-p
10-u
100-p
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No. of Replications for Each Subject (p = paired, u = unpaired)
FIGURE 42.2 Computation of tissue-to-plasma AUC ratios from paired and unpaired data using the random sampling approach. The line inside the box represents the median, and the box represents the limits of the middle half of the data. The range of the box, from the first quartile (Q1) to the third quartile (Q3), is called the interQuartile range (IQR). The standard span of the data is defined within the range from Q1 − 1.5IQR to Q3 + 1.5IQR. Whiskers, the dotted line, are drawn to the nearest value not beyond the range of the standard span; points beyond (outside values) are drawn individually.
COMPARISON OF PpbB AND RS APPROACHES
Random Sampling
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Pseudoprofile-based Bootstrap
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Q = quartile, p = paired, u = unpaired
FIGURE 42.3 Comparison of the performance of the RS and PpbB approaches when tissue-to-plasma AUC ratios (TPAR) are computed from paired and unpaired data. The comparison is focused on distribution of the first and third quartiles (Q1 and Q3, respectively) of TPARs.
wider spread, with the lower adjacent value of the distribution of Q1 values in the box plot extending beyond that for paired data in both RS and PpbB approaches. Disrupting the correlation structure in the data by unpairing the data yielded more variable results than when the correlation structure in the data was maintained by pairing. Thus, breaking the correlation structure between tissue and plasma observations resulted in a loss of information. Therefore, the rest of the study is focused on the paired scenario only. A tabular comparison of the results obtained with the RS and PpbB approaches is shown in Table 42.5. In addition to the typical fashion of only describing distribution of mean of TPAR, Table 42.5 also includes distributions of quartiles of TPAR in terms of Q1 and Q3 with associated standard errors. The resampling approaches yielded comparable results when the number of replications was at least 600 with mean TPAR around 17, but the RS approach converged faster than the PpbB approach. (See Section 42.3.2 for more details.) 42.3.2
Convergence
Convergence was determined for both RS and PpbB approaches. That is, the number of replications (i.e., the number of times the sampling/resampling has to be repeated) needed for stable estimates of tissue-to-plasma AUC ratio (TPAR) to be obtained was determined for both methods. An empirical approach was used to determine convergence (13).
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14.31 15.01 14.46 14.75 14.57 14.83 14.34 14.49 14.53 14.61 14.84 14.75 14.43
5 10 50 100 200 300 400 500 600 700 800 900 1000
(0.94) (0.95) (1.00) (0.96) (0.99) (1.01) (1.12) (0.89) (1.08) (1.05) (0.99) (1.05) (1.01)
(SE)
16.64 17.51 17.17 17.28 17.15 17.39 17.08 17.16 17.13 17.25 17.41 17.24 17.09
Mean
(SE) (0.88) (0.85) (0.95) (0.82) (0.84) (0.90) (0.95) (0.84) (0.91) (0.97) (0.84) (0.89) (0.87)
Mean
18.64 19.80 19.33 19.38 19.35 19.71 19.28 19.48 19.28 19.53 19.61 19.49 19.41
Mean
Q3
(1.45) (1.29) (1.48) (1.21) (1.26) (1.32) (1.46) (1.36) (1.17) (1.38) (1.20) (1.24) (1.35)
(SE) 5 10 50 100 200 300 400 500 600 700 800 900 1000
Replc 14.43 14.05 14.12 14.45 14.63 14.89 14.28 14.57 14.34 14.90 14.50 14.53
Mean
Q1
(0.42) (0.75) (0.93) (1.15) (0.94) (0.93) (0.85) (0.96) (1.09) (1.02) (0.94) (1.02)
(SE)
15.76 16.71 17.93 17.02 17.16 17.27 17.08 17.30 17.05 17.44 17.12 17.18
Mean
(SE) (0.41) (0.83) (0.68) (0.85) (0.87) (0.80) (0.78) (0.80) (0.88) (0.91) (0.84) (0.78)
Mean
b
17.15 18.70 20.36 19.07 19.20 19.35 19.54 19.76 19.38 19.77 19.32 19.43
Mean
Q3
(1.10) (1.09) (0.95) (1.20) (1.34) (1.16) (1.29) (1.19) (1.33) (1.25) (1.34) (1.17)
(SE)
Pseudoprofile-Based Bootstrapa: Tissue-to-Plasma AUC Ratios
Q1 and Q3 refer to the first and third quartiles of distribution of TPAR; AUC indicates area under the curve; and Repl stands for replication. Replication is M1 (i.e., the number of replicates for each subject in Phase 1 of the RS approach). c Replication represents B1 replicates used in the PpbB approach.
a
Mean
Replb
Q1
Random Sampling Approacha: Tissue-to-Plasma AUC Ratios
TABLE 42.5 Distribution of Tissue-to-Plasma AUC Ratio Parameter Estimates Obtained Using Random Sampling and Pseudoprofile-Based Bootstrap Approaches
COMPARISON OF PpbB AND RS APPROACHES
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To examine the effect of the number of replications (i.e., M1 in RS and B1 in PpbB), a graphical presentation of percentage change (PC) of mean TPAR is shown in the middle panel of Figure 42.4 and Figure 42.5 for the RS and PpbB approaches, respectively. In addition, the PC values of Q1 and Q3 are also plotted in the left and right panels of each figure. The acceptable range for the percentage change is calculated from summary statistics/confidence intervals of PC across all replication levels considered (i.e., from M1 with as little as 5 replications to as high as 1000 replications), and for statistics Q1, mean, and Q3. This range was determined by visual inspection of the convergence graphs with the assumption that, over the range of the replications, the PC trend should be stabilized with limited amount of fluctuations. Therefore, the percentile cutoff range was chosen using a trimming approach, and the range of percentiles 12.5 and 87.5 was found to be appropriate for both sampling approaches and across the three summary statistics. Figure 42.4 shows the convergence trend for the RS approach. For all three statistics (Q1, mean, and Q3) of interest, 100 replications are sufficient. On the other hand, the number of replications needed for the distributions of summary statistics of TPAR with the PpbB is at least 600 replications (see Figure 42.5), owing to the instability in Q1. The range for the RS approach is considerably tighter than that for the PpbB approach. In fact, the range of PC is −1.28% to 1.56% for the RS approach, and −2.26% to 5.10% for the PpbB approach. This finding indicates that there was a larger variability in TPAR estimates obtained with the PpbB approach when compared with that obtained using the RS approach. The uniqueness of the RS approach lies in the population sampling pool, which is populated by generating M1 replications
1st Quartile
Mean
3rd Quartile
10
Percentage Change (%)
5
0
-5
-10 5 10
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700
900
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500
700
900
5 10
100
300
500
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900
No. of Replications for Each Subject
FIGURE 42.4 Convergence trend monitoring using percentage change in summary statistics (Q1, mean, and Q3) of tissue-to-plasma AUC ratio (TPAR) estimates obtained by the random sampling approach.
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ANALYSIS OF QUANTIC PHARMACOKINETIC STUDY
1st Quartile
Mean
3rd Quartile
10
Percentage Change (%)
5
0
-5
-10 10
100
300
500
700
900
10
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500
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900
10
100
300
500
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Number of Bootstraps
FIGURE 42.5 Convergence trend monitoring using percentage change in summary statistics (Q1, mean, and Q3) of distribution of tissue-to-plasma AUC ratio (TPAR) estimates obtained by the PpbB approach.
through resampling concentration–time profiles for each subject (i.e., in this study with N = 18 animals, 100 replications for each animal is equivalent to total of 1800 [= 18 · 100] distinct PK profiles in the population sampling pool). Also, M2 copies of virtual studies are sampled from the population sampling pool to derive a distribution for any function of interest. The code for monitoring convergence is shown in Appendix 42.4. 42.3.3 Outlier Effect on Robustness To investigate the effect of outliers on the robustness with which TPAR was estimated with naive averaging, RS, and PpbB approaches, new data sets were simulated by introducing outlier(s) into the data set. The scenarios we chose can be mapped as a grid (2 × 4 table) (i.e., one or two outliers produced by inflating the higher tissue concentration time points by 10%, 20%, 30%, or 40%). The higher tissue concentration time points were defined as concentrations obtained within 4 hours postdose. These concentrations were randomly chosen in each replication. The outliers were introduced in the region of the concentration–time profile (i.e., around the higher concentrations), where they were likely to produce maximum effect (see S-Plus code in the Appendix 42.5). Figure 42.6 shows the distribution of mean TPAR obtained from simulating 50 replicates (i.e., M2 = 50) of the base data set with the value of one tissue concentration value inflated to create an outlier in each replicate. The effect of one outlier can be measured by how big the distance is from the original mean TPAR value of
OVERALL ASSESSMENT OF TISSUE-TO-PLASMA RATIO ESTIMATION (A) 10%
(B) 20%
(C) 30%
(D) 40%
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20 19 18 17 16 15
TPAR
14
20 19 18 17 16 15 14 Naive
PpbB
RS
Naive
PpbB
RS
Method
FIGURE 42.6 The effect of outlier on distribution of tissue-to-plasma AUC ratios (TPAR) when inflating one concentration by (A) 10%, (B) 20%, (C) 30%, or (D) 40% using the naive averaging (naive), PpbB, and RS approaches.
∼17 (see Table 42.5). The naive averaging approach performed the worst of all three approaches, and PpbB had a wider spread than the RS approach. In Figure 42.6, it appears that the distribution of TPAR estimates obtained with the naive averaging approach was the tightest. It has to considered that by the very nature of the naive averaging approach variability has been eliminated, hence the results. When the scenario for two outliers was considered, Figure 42.7 illustrates the effect when two tissue concentration values were randomly selected to create outliers in each replicate by calculating the PC from mean TPAR of 17 across the three methods, given the four levels of outlier perturbation (i.e., 10%, 20%, 30%, or 40% increase in concentration values). Clearly, the RS approach provides results that are more robust than the other two. The bias in the estimation of TPAR is more prominent with the PpbB and naive averaging approaches than with the RS approach (Figure 42.7).
42.4 OVERALL ASSESSMENT OF TISSUE-TO-PLASMA RATIO ESTIMATION A nonparametric random sampling approach proposed by Chu and Ette (13) for the estimation of robust TPAR was compared with the PpbB and naive averaging approaches. Also, the estimation of tissue-to-plasma ratio using the independent time points approach was examined. It is obvious from Tables 42.2 and 42.3 that estimating tissue-to-plasma ratio independently at various times is a very
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ANALYSIS OF QUANTIC PHARMACOKINETIC STUDY Naive Averaging
PpbB
Random Sampling
20
T P A R P e r c e n t a g e C ha n g e
15
10
5
0
-5
-10 10%
20%
30%
40%
10%
20%
30%
40%
10%
20%
30%
40%
Percentage Increase of Outliers
FIGURE 42.7 The effect of outlier based on the percent increase distribution of tissueto-plasma AUC ratios (TPAR) when inflating two concentrations by 10%, 20%, 30%, or 40% using the naive averaging (left panel), PpbB (middle panel), and RS (right panel) approaches.
unreliable method, since various ratios are obtained at various time points and it is unclear which of the ratios to choose. Also, it is impossible to compute ratios when samples from a particular biological matrix are below the limit of quantification or are unquantifiable. The independent time points approach for calculating tissueto-plasma ratio should, therefore, be avoided. Although the naive data averaging approach for computing AUC provides a single AUC value for drug exposure in each of the two matrices and consequently a single value of TPAR, the correlation in the data structure is unaccounted for and there is no measure of variability or uncertainty around the estimates. With this method, when concentrations are below the limit of quantification, they are usually ignored in the calculation of the mean concentration at the particular time point. The mean concentration is calculated only with available data. Thus, the mean profile obtained in such a situation does not represent the actual mean profile since mean concentrations at each time point are not calculated from an equal number of time points. These drawbacks not withstanding, the approach is better than the independent time points approach. However, both approaches are inferior to the resampling approaches—PpbB and RS. Breaking the correlation structure between tissue and plasma observations results in a loss of information when using any of the resampling approaches. Therefore, it is important to maintain the correlation structure in paired data sets used in estimating TPAR. By doing this, variability in the calculated TPAR is minimized. Although there are similarities in the TPAR estimates produced by the PpbB and RS approaches, the latter converges faster than the former. Convergence is achieved with only 100 replications (i.e., M1 = 100) per subject with the RS approach,
REFERENCES
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while at least 600 bootstrap (i.e., B1 = 600) replications are required for the PpbB approach. In general, 100 replications are adequate in the first phase of the RS approach for robust estimation of TPAR. Also, the acceptable range for TPAR estimates is narrower for the RS approach (Figure 42.4) when compared with the PpbB approach (Figure 42.5). Thus, the PpbB approach requires a larger number of replications to yield robust estimates. The difference lies in the twophase—population and study level—sampling of the RS approach. The tightness of the distribution of estimates obtained with the RS approach can be attributed to the creation of the representative population sample pool for subsequent study level sampling of parameters of interest. This is a unique feature of the RS approach. Also, the estimation of TPAR by the RS approach is not affected by missing data or imbalance in the number of concentrations at each time point over the sampling duration. Individual PK profiles are generated by sampling from available data at each time point across time points. Similarly, the PpbB approach is not affected by missing data or data imbalance. When the effect of outliers on robustness was investigated, the naive data averaging approach performed the worst, while the RS approach performed the best. The edge that the RS approach has over the PpbB approach is, again, owing to the two-phase nature of implementation of the methodology. The robustness of the RS approach lies in the creation of the population sample pool before the study level sampling for the estimation of TPAR. Also, the greater bias obtained with the PpbB approach when compared with the RS approach is probably due to the fact that mean parameter estimates obtained from bootstrap replicates may be influenced by data in the tails of the distribution (15). 42.5
SUMMARY
Traditional approaches used in the estimation of TPAR have been compared with the PpbB approach and the recently proposed RS approach. The traditional approaches—independent time points and naive data averaging approaches—are inferior to the sampling/resampling approaches. The RS approach performed better than the PpbB approach because of its unique algorithm. Also, fewer replications are required for robust estimation of TPAR. The computer intensive methods provide estimates of TPAR with measures of dispersion and uncertainty. The RS approach is the method of choice for obtaining robust estimates of TPAR, when analyzing extremely sparsely sampled data. REFERENCES 1. L. T. Lindstrom and D. S. Birkes, Estimation of population pharmacokinetic parameters using destructively obtained data: a simulation study of the one compartment open model. Drug Metab Rev 15:195–264 (1984). 2. R. D. McArthur, Parameter estimation in a two compartment population pharmacokinetic model with destructive sampling. Math Biosci 91–157 (1988). 3. E. I. Ette EI, A. W. Kelman, C. W. Howie, and B. Whiting, Analysis of animal pharmacokinetic data: performance of the one point per animal design. J Pharmacokinet Biopharm 23(6):551–566 (1995).
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4. E. I. Ette EI, A. W. Kelman, C. W. Howie, and B. Whiting, Efficient experimental design and estimation of population pharmacokinetic parameters in preclinical animal studies. Pharm Res 12(5):729–737 (1995). 5. E. I. Ette, A. W. Kelman, C. W. Howie, and B. Whiting, Inter-animal variability and parameter estimation in preclinical animal pharmacokinetic studies. Clin Res Regul Affairs 11(2):121–139 (1994). 6. E. I. Ette, A. W. Kelman, C. W. Howie, and B. Whiting, Interpretation of simulation studies for efficient estimation of population pharmacokinetic parameters. Ann Pharmacother 27:1034–1039 (1993); and Correction 27:1548 (1993). 7. J. V. Bree, J. Nedelman, J.-L. Steimer, F. Tse, W. Robinson, and W. Niederberger, Application of sparse sampling approaches in rodent toxicokinetics: a prospective view. Drug Info J 28:263–279 (1994). 8. C. D. Jones, H. Sun, and E. I. Ette, Designing cross-sectional pharmacokinetic studies: implications for pediatric and animal studies. Clin Res Regul Affairs 13(3&4):133–165 (1996). 9. A. J. Bailer, Testing for the equality of area under the curves when using destructive measurement techniques. J Pharmacokinet Biopharm 16:303–309 (1988). 10. C. Yeh, Estimation and significant tests of area under the curve derived from incomplete blood sampling. ASA Proc (Biopharm Sec) 74–81 (1990). 11. J. R. Nedelman, E. Gibiansky, and D. T. W. Lau, Applying Bailer’s method for AUC confidence intervals to sparse sampling. Pharm Res 12:124–128 (1995). 12. S. M. Pai, S. H. Fettner, G. Hajian, M. N. Cayen, and V. K. Batra, Characterization of AUCs from sparsely sampled populations in toxicology studies. Pharm Res 13:1283–1290 (1996). 13. H.-M. Chu and E. I. Ette, A random sampling approach for robust estimation of tissue to plasma ratio from extremely sparse data. AAPS J 7(1):E249–E258 (2005). 14. H. Mager and G. Goller, Resampling methods in sparse sampling situations in preclinical pharmacokinetic studies. J Pharm Sci 87:372–378 (1008). 15. E. I. Ette and L. C. Onyiah, Estimating inestimable standard errors in population pharmacokinetic studies: the bootstrap with winsorization. Eur J Drug Metab Pharmacokinet 27:213–224 (2002).
APPENDIX 42.1
CODE FOR NAIVE DATA AVERAGING APPROACH
################################## # calculate AUC ratio by average data at each time point # to construct the AUC curve by average conc. by time point # source data set is named “data” ################################## ## calculate AUC ratio by average data by time point ## to construct the AUC curve by average conc. by time point t_c(0,0.5,1,2,4,6,8) a_apply(data.tissue,2,mean) a_as.vector(a[-1]) b_apply(data.plasma,2,mean) b_as.vector(b[-1])
CODE FOR RANDOM SAMPLING APPROACH
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# Trapezoidal.hmc is a subroutine (see 42.6.6) c1_as.vector(Trapezoidal.hmc(time=t,DV=a)) c2_as.vector(Trapezoidal.hmc(time=t,DV=b)) ## calculate ratio by time point t_c(0.5,1,2,4,6,8) a_rep(NA,length(t)+1) tmp_data.frame(time=a,min=a,Q1=a,median=a,mean=a,Q3=a,max=a,NA=a,s d=a,n=a) for(i in 1:length(t)) { tmp[i,1]_as.character(t[i]) print(t[i]) print(summary(data$ratio[data$Time==t[i]])) b_summary(data$ratio[data$Time==t[i]]) tmp[i,2:(length(b)+1)]_as.vector(b) tmp[i,9]_stdev(data$ratio[data$Time==t[i]]) tmp[i,10]_length(data$ratio[data$Time==t[i]]) } tmp[length(t)+1,1]_”overall” b_summary(data$ratio) tmp[length(t)+1,2:(length(b)+1)]_as.vector(b) tmp[length(t)+1,9]_stdev(data$ratio) tmp[length(t)+1,10]_length(data$ratio)
APPENDIX 42.2
CODE FOR RANDOM SAMPLING APPROACH
################################################################## ###### #### to impute AUC from scarified animals, each rat has one plasma and one #### tissue data point ## to construct the plasma and tissue data sets and associated column names (time points) ## // start tmp_data.frame(rep=seq(1,irep),AUC.p=rep(NA,irep),AUC. l=rep(NA,irep)) t_sort(unique(data$Time)) size.n_length(t) data.plasma_data.frame(iter=c(1,2,3)) data.tissue_data.frame(iter=c(1,2,3)) for(i in 1:size.n) { aa_data$plasma[data$Time==t[i]]
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ANALYSIS OF QUANTIC PHARMACOKINETIC STUDY
data.plasma_cbind( data.plasma,aa) aa_data$tissue[data$Time==t[i]] data.tissue_cbind( data.tissue,aa) } dimnames(data.plasma)[[2]]_c(“iter”,paste(“h”,t,sep=””)) dimnames(data.tissue)[[2]]_c(“iter”,paste(“h”,t,sep=””)) ## // end ## ------------------------------------------------------------### to generate “irep” (M copies of) replication of Psuodoprofile and ### associated AUC for each subject in the data set ## // start irep_1000 tmp.tissue_matrix(rep(NA,irep*size.n),ncol=size.n) tmp.plasma_matrix(rep(NA,irep*size.n),ncol=size.n) #data.plasma_data.plasma[,-c(1,2)] #data.tissue_data.tissue[,-c(1,2)] ## irow*icol (18) is the loop for each subject ## icol is the loop for each time point within each subject ## j is the M-copy loop of calculating AUC for each subject ## the outcome stored in ans and renamed as ans.rep.** ID.n_0 for(irow in 1:nrow(data.tissue)) { for(icol in 3:ncol(data.tissue)) { # for each subject ID.n_ID.n+1 tmp_data.frame(ID=rep(ID.n,irep),rep=seq(1,irep),AUC. p=rep(NA,irep),AUC.l=rep(NA,irep)) tmp.tissue_matrix(rep(NA,irep*(size.n-1)),ncol=(size.n-1)) tmp.plasma_matrix(rep(NA,irep*(size.n-1)),ncol=(size.n-1)) for(i in 1:(size.n-1)) { a_sample(x=c(1:3),size=irep,T) #cat(“\n(“,irow,icol,”) -- with hour index”,i,”\n”) #print(a) #print(data.tissue[a,i+2]) tmp.tissue [,i]_data.tissue[a,i+2] #print(tmp.tissue[,i] )
CODE FOR RANDOM SAMPLING APPROACH
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tmp.plasma[,i]_data.plasma[a,i+2] #print(tmp.plasma[,i] ) } for (j in 1:irep) { tmp$AUC.l[j]_Trapezoidal.hmc(time=t,DV=c(0,tmp.tissue[j,])) tmp$AUC.p[j]_Trapezoidal.hmc(time=t,DV=c(0,tmp.plasma[j,])) #cat(“\n iter”, irep,”AUC of tissue = “,tmp$AUC.l[j],”\n”) } if ( icol*irow==3) {ans_tmp} else {ans_rbind(ans,tmp)} } } ans$ratio_ans$AUC.l/ans$AUC.p summary(ans$ratio) summary(ans$AUC.p) boxplot(ans$ratio,ylab=”ratio”) title(“Distribution of Tissue to Plasma Ratio”,cex=.9) #ans.rep5_ans #ans.rep10_ans #ans.rep100_ans #ans.rep500_ans ## // end ## ------------------------------------------------------------boxplot(ans.rep5$ratio,ans.rep10$ratio,ans.rep100$ratio,ans. rep500$ratio, names=c(“5”,”10”,”100”,”500”), xlab=”Replications for Each Subject”,ylab=”Tissue to Plasma Ratio”) ######################################################### #### part 2 #### to generate M copies of N=18 virual studies data_ans.rep10 irep_50 n_18 #ans_matrix(rep(NA,n*irep),nrow=n) # row is the animal index and column is the replicates ans_matrix(rep(NA,6*irep),nrow=irep) # columns are summary stats and row is the replicates
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ANALYSIS OF QUANTIC PHARMACOKINETIC STUDY
for (i in 1:irep) { a_sample(x=seq(1:nrow(data)),size=n,T) #ans[,i]_data$ratio[a] ans[i,]_as.vector(summary(data$ratio[a])) } dimnames(ans)[[2]]_c(“Min”,”Q1”,”Q2”,”Mean”,”Q3”,”Max”) #ans.rep5.summary_ans #ans.rep10.summary_ans #ans.rep100.summary_ans #ans.rep500.summary_ans par(mfrow=c(2,2)) boxplot(ans.rep5.summary[,3],ans.rep10.summary[,3],ans.rep100. summary[,3],ans.rep500.summary[,3], names=c(“5”,”10”,”100”,”500”), xlab=”replications”,ylab=”Tissue to Plasma Ratio”,ylim=c(10,25)) title(“Distribution of Median \n over 50 rep. of Virtual Study”,cex=.8) boxplot(ans.rep5.summary[,4],ans.rep10.summary[,4],ans.rep100. summary[,4],ans.rep500.summary[,4], names=c(“5”,”10”,”100”,”500”), xlab=”replications”,ylab=”Tissue to Plasma Ratio”,ylim=c(10,25)) title(“Distribution of Mean \n over 50 rep. of Virtual Study”,cex=.8) boxplot(ans.rep5.summary[,2],ans.rep10.summary[,2],ans.rep100. summary[,2],ans.rep500.summary[,2], names=c(“5”,”10”,”100”,”500”), xlab=”replications”,ylab=”Tissue to Plasma Ratio”,ylim=c(10,25)) title(“Distribution of Q1 \n over 50 rep. of Virtual Study”,cex=.8) boxplot(ans.rep5.summary[,5],ans.rep10.summary[,5],ans.rep100. summary[,5],ans.rep500.summary[,5], names=c(“5”,”10”,”100”,”500”), xlab=”replications”,ylab=”Tissue to Plasma Ratio”,ylim=c(10,25)) title(“Distribution of Q3 \n over 50 rep. of Virtual Study”,cex=.8) ################################# #### part 3 #### par.old_par() frame() par(oma=c(0,0,2,0),mar=c(5,5,4,3)+0.1)
CODE FOR PSEUDOPROFILE-BASED BOOTSTRAP
1055
par(mfrow=c(1,1)) par(fig=c(x1=0,x2=0.55,y1=0.45,y2=1)) boxplot(ans.rep5.summary[,3],ans.rep10.summary[,3],ans.rep100. summary[,3],ans.rep500.summary[,3], ans.rep1000.summary[,3],names=c(“”,””,””,””,””),xlab=””,ylab= ””,ylim=c(10,25)) mtext(side=3,”Median”,line=1) par(fig=c(x1=0.45,x2=1,y1=0.45,y2=1),yaxs=”d”) boxplot(ans.rep5.summary[,4],ans.rep10.summary[,4],ans.rep100. summary[,4],ans.rep500.summary[,4], ans.rep1000.summary[,4],names=c(“”,””,””,””,””),xlab=””,ylab= ””,ylim=c(10,25),axes=F) box() #title(“Mean”,cex=.7) mtext(side=3,”Mean”,line=1) par(fig=c(x1=0,x2=0.55,y1=0,y2=0.55),xaxs=”d”,yaxs=”d”) boxplot(ans.rep5.summary[,2],ans.rep10.summary[,2],ans.rep100. summary[,2],ans.rep500.summary[,2], ans.rep1000.summary[,2],names=c(“5”,”10”,”100”,”500”,”1000”), xlab=””,ylab=””,axes=T) box() mtext(side=3,”1st Quartile”,line=1) par(fig=c(x1=0.45,x2=1,y1=0,y2=0.55),xaxs=”d”,yaxs=”d”) boxplot(ans.rep5.summary[,5],ans.rep10.summary[,5],ans.rep100. summary[,5],ans.rep500.summary[,5], ans.rep500.summary[,5],names=c(“5”,”10”,”100”,”500”,”1000”),x lab=””,ylab=””,axes=F) box() mtext(side=3,”3rd Quartile”,line=1) mtext(“Distribution of Summary Stat. Over 50 Replications of Virtual Study”,outer=T) mtext(side=1,”# of Replications for Each Subject”,outer=T,line=-2)
APPENDIX 42.3
CODE FOR PSEUDOPROFILE-BASED BOOTSTRAP
############################################################ ### to generate “irep” (M copies of) replication of Psuodoprofile and ### associated AUC for each subject in the data set ### ### n is the number of animals in the study ### irep == (B1 time loop in step 1 and 2 of the paper) ############################################################
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ANALYSIS OF QUANTIC PHARMACOKINETIC STUDY
b_c(10,50,seq(100,1000,100)) t_sort(unique(data$Time)) size.n_length(t) for (k in 1:length(b)) { irep_b[k] n_18 tmp.tissue_matrix(rep(NA,irep*size.n),ncol=size.n) tmp.plasma_matrix(rep(NA,irep*size.n),ncol=size.n) ## ## ## ##
irow*icol (18) is the loop for each subject icol is the loop for each time point within irow-th subject j is the M-copy loop of calculating AUC for each subject the outcome stored in ans and rename as ans.rep.**
# to sample by time point myStart_proc.time() for(i in 1:size.n) { a_sample(x=c(1:3),size=irep,T) tmp.tissue [,i]_data.tissue[a,i+1] tmp.plasma[,i]_data.plasma[a,i+1] } ## to calcualte AUC ans_data.frame(rep=rep(b[k],irep),AUC.l=rep(NA,irep),AUC. p=rep(NA,irep)) for ( i in 1:irep) { a_as.vector(unlist(tmp.tissue[i,])) ans$AUC.l[i]_Trapezoidal.hmc(time=t,DV=a) a1_as.vector(unlist(tmp.plasma[i,])) ans$AUC.p[i]_Trapezoidal.hmc(time=t,DV=a1) } } # to calculate the paired ratio ans.PpbB.total$ratio.p_ans.PpbB.total$AUC.l/ans.PpbB.total$AUC.p # to calculate the unpaired ratios ans.PpbB.total$ratio.u_rep(NA,nrow(ans.PpbB.total)) for (i in 1:length(b)) #for (i in 1:1)
CODE FOR CONVERGENCE
1057
{ b1_ans.PpbB.total$AUC.p[ans.PpbB.total$rep==b[i]] b2_ans.PpbB.total$AUC.l[ans.PpbB.total$rep==b[i]] b3_sample(x=c(1:length(b1)),size=length(b1),F) #print(b1) #print(b2) #print(b3) ans.PpbB.total$ratio.u[ans.PpbB.total$rep==b[i]]_b2/b1[b3] }
APPENDIX 42.4
CODE FOR CONVERGENCE
#### compare the convergence among different M copies of paired ratios ############################################################ ############### random sampling approach set.seed(565) y_c(5,10,50,seq(100,1000,100)) # y is the # of replication for each subject # columns are summary stats and row is the replicates ans1_matrix(rep(NA,3*length(y)),nrow=length(y)) ans2_matrix(rep(NA,3*length(y)),nrow=length(y)) ans3_matrix(rep(NA,3*length(y)),nrow=length(y)) for(k in 1:length(y)) { if(k==1){data_ans.rep5} if(k==2){data_ans.rep10} if(k>2 ){ data_ans.rep1000[ans.rep1000$rep= 4) amount = opt_amount_initial; else amount = zeros(c.SV.TotNum, 1); % All state variables set to zero end nstages = length(events.timestages) - 1; saved_amounts = []; saved_times = []; saved_mass_balances = []; saved_sv = []; saved_amounts(end+1,:) = amount; % initial values saved_times(end+1,:) = events.timestages(1); % initial time saved_mass_balances(end+1,:) = 0; % initial values -- zeros SVID = c.SV.ID; % state variable IDs odeset(‘RelTol’, 1e-8, ‘AbsTol’, 1e-12, ‘MaxOrder’, 5, ‘BDF’, ‘on’); for istage=1:nstages stage_beg = events.timestages(istage); stage_end = events.timestages(istage+1); E = local_get_event(events, istage); tspan = [stage_beg:events.timeincrements(istage):stage_end]; amount(SVID.GILumen) = amount(SVID.GILumen) + . . . events.bolus_ingestion(istage) * c.F_bioavail_water; amount(SVID.Ingested) = amount(SVID.Ingested) + . . . events.bolus_ingestion(istage); amount(SVID.Feces) = amount(SVID.Feces) + . . . events.bolus_ingestion(istage) * (1 - c.F_bioavail_water); amount(SVID.Blood) = amount(SVID.Blood) + events.bolus_injection(istage); [cur_time_new, amount_new] = ode15s(‘PBPK_voc_deriv’,tspan, amount); saved_amounts = [saved_amounts(1:end-1,:); amount_new]; mass_balances = local_check_mass_balance(c, amount_new); saved_mass_balances = [saved_mass_balances(1:end-1), . . . mass_balances]; % save mass balances saved_times = [saved_times(1:end-1); cur_time_new];
CODE LISTING 2
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% Save the intermediate state variables such as concentrations and rates saved_sv = saved_sv(1:end-1); for i=1:length(tspan) [dummy, optSV] = PBPK_voc_deriv(tspan(i), amount_new(i,:)’); saved_sv = [saved_sv; optSV]; end amount = amount_new(end,:); % Set the state for the next stage end simout.amounts = saved_amounts; % amounts in tissues, etc simout.mass_balances = saved_mass_balances; % mass balance term vs time simout.times = saved_times; % times at which model outputs are saved simout.sv = saved_sv; % state variables of the model simout.PBPK_config = c; simout.event_config = events; simout.concs = [saved_sv.concs]; % main tissues, exhaled, and venous blood function thisevent = local_get_event(events, istage) thisevent.Q.inh = events.rate_inhale(istage); thisevent.Q.DW = events.drinking_water_rate(istage); thisevent.C.inh = events.airconcs(istage); thisevent.C.water = events.waterconcs(istage); thisevent.dermal_contact_media = events.contact_media(istage); thisevent.Q.infusion = events.infusion(istage); function res = local_check_mass_balance(c, orig_amount) SVID = c.SV.ID; % state variable IDs amount = orig_amount’; % Body burden for all compartments (including stratum corneum) Body_burden = sum(amount(1:length(c.CompName),:)); Body_burden = Body_burden + amount(SVID.GILumen,:) + amount(SVID.Blood,:); mass_difference = amount(SVID.Inhaled,:) - amount(SVID.Exhaled,:) . . . + amount(SVID.Ingested,:) - amount(SVID.Feces,:). . . + amount(SVID.Dermal,:) - amount(SVID.Metabolized,:) - Body_burden; input_dose = amount(SVID.Inhaled,:) + . . . amount(SVID.Ingested,:) + amount(SVID.Dermal,:); if (abs(input_dose) > eps) mass_balance_term = mass_difference ./ input_dose; if ( max(abs(mass_difference/input_dose)) > 0.0001 ) error(‘*** mass balance error exceeds 0.01 % ***’); end else mass_balance_term = zeros(1,size(amount,2)); end res = mass_balance_term;
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PHYSIOLOGICALLY BASED PHARMACOKINETIC MODELING
APPENDIX 43.3
CODE LISTING 3
This Matlab code is for the PBPK model configuration for chloroform. This produces a PBPK configuration object based on the individual’s physiological parameters and other model configuration options. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42.
function c = PBPK_chloroform_config(indiv, modelconf) % Provides default configuration values for the PBPK_inhalation model % Structure c holds the complete PBPK Model Configuration % General parameters obtained from Roy et al., Risk Analysis 16(2) % Exception: Q_cardiac from Fisher et al., 1999 % % Usage: x = PBPK_inhalation_default_chloroform_config(indiv, modelconf) % Function Inputs: % indiv: a structure with the following fields % age -- age of the individual % sex -- sex of the individual (‘M’ or ‘F’) % BW -- body weight of the individual % BSA -- body surface area % modelconf: a structure that defines model configuration with fields % DM.nskins -- specifies type of dermal model to use % 1 => one skin model % 2 => two skin model (Stratum Corneum (SC) + Viable Skin) % >2 => distributed skin model with nskins nodes in SC % if this structure is not provided, the default is a one-skin model % % Outputs % c – configuration object for the PBPK model parameters % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % PBPK Model Structure Parameters (e.g. modeled compartments) c.IDNum.Fat = 1; c.CompName{1} = ‘Fat’; % Fat c.IDNum.SP = 2; c.CompName{2} = ‘SP’; % Slowly Perfused Tissue c.IDNum.RP = 3; c.CompName{3} = ‘RP’; % Rapidly Perfused Tissue c.IDNum.Liv = 4; c.CompName{4} = ‘Liv’; % Liver c.IDNum.Gut = 5; c.CompName{5} = ‘Gut’; % Gut c.IDNum.Kid = 6; c.CompName{6} = ‘Kid’; % Kidney c.IDNum.Skin = 7; c.CompName{7} = ‘Skin’; % Skin ID=c.IDNum; % A temporary variable for compartment IDs c.N_Compartments = length(c.CompName); c.Density = 1.0; % Body and tissue density % Set important variables to NaN. Helps in identify initialization errors nanArray = NaN(1,c.N_Compartments); % NaNs corresponding to each compartment c.PC.tissue = nanArray; % Blood partition coefficients for all compartments
CODE LISTING 3
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Chemical-independent physiological parameters (volume ratios, etc) c.AF.QC.M = 15.87; % Allometric factor (AF) for cardiac output (Males) c.AF.QC.F = 17.7; % AF (Females). Units: L/hour/kg. % Percentage of Cardiac Flow Ratios c.QCR(ID.Fat) = 0.05; c.QCR(ID.SP) = 0.156; c.QCR(ID.RP) = 0.26; c.QCR(ID.Liv) = 0.07; c.QCR(ID.Gut) = 0.18; c.QCR(ID.Kid) = 0.25; c.QCR(ID.Skin) = 0.04; c.QCR(ID.SP) = c.QCR(ID.SP) + 1 - sum(c.QCR); % Adjust QCR for SP to sum to 1 c.venous_blood_volume_fraction = 0.01; c.VR(ID.Fat) c.VR(ID.SP) c.VR(ID.RP) c.VR(ID.Liv) c.VR(ID.Gut) c.VR(ID.Kid) c.VR(ID.Skin)
= = = = = = =
0.231; 0.5105; 0.0327; 0.0314; 0.017; 0.0044; 0.1;
% Adjust VR for SP to sum to 1 c.VR(ID.SP) = c.VR(ID.SP) + 1 - sum(c.VR) - c.venous_blood_volume_fraction; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Chemical Specific Information c.Notes = ‘Chloroform PBPK Model for Inhalation Route’; c.Compound = ‘Chloroform’; c.MolecularWeight = 119.4; % Chemical Specific Partition Coefficients for Chloroform c.PC.blood_air = 7.43; % blood_air PC c.PC.tissue(ID.Fat) = 37.7; % fat/blood PC c.PC.tissue(ID.SP) = 1.62; % SP/blood PC c.PC.tissue(ID.RP) = 2.3; c.PC.tissue(ID.Gut) = 2.3; c.PC.tissue(ID.Liv) = 2.3; c.PC.tissue(ID.Kid) = 1.5; c.PC.chem.air_water = 0.163; % Henry’s Law Coefficient from USEPA c.MMK.VMAX_C = 0.26167; % Allometric constant for Vmax c.MMK.KM_C = 0.448 * 1000; % Michaelis-Menten constant in liver (ug/L)
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PHYSIOLOGICALLY BASED PHARMACOKINETIC MODELING
c.Absorption.Gut = 1.0; % Gut-Lumen absorption rate constant (1/hr) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Updating the parameters based on the individual c.indiv = indiv; BW = c.indiv.BW; % simpler variable for some formulae c.V_TISSUE = BW * c.Density * c.VR; % Tissue volumes from volume fractions c.venous_blood_volume = c.venous_blood_volume_fraction * BW * c.Density; c.Q_cardiac = c.AF.QC.(c.indiv.sex)/60 *(BW)^0.75; % Units: L/min -- Fisher c.VMAX = zeros(1,c.N_Compartments); % VMAX initialization (Michelis Menten) c.VMAX(ID.Liv) = c.MMK.VMAX_C * 1000 * BW^0.7; % Units: ug/min c.KM = ones(1,c.N_Compartments) * (-Inf); c.KM(ID.Liv) = c.MMK.KM_C; c.F_bioavail_water = 1; % Oral bioavailability from water c.QC = c.QCR * c.Q_cardiac; % Blood flow rates to each compartment if (nargin < 2) modelconf = []; % default option end c = PBPK_add_dermal_model(c, modelconf); % Add the dermal model % State Variables in the Model c.SV.ID = c.IDNum; % State Variables ID c.SV.Names = c.CompName; c.SV.Names{end+1} = ‘Blood’; % Amount in Blood c.SV.Names{end+1} = ‘GILumen’; c.SV.Names{end+1} = ‘Metabolized’; c.SV.Names{end+1} = ‘Inhaled’; c.SV.Names{end+1} = ‘Exhaled’; c.SV.Names{end+1} = ‘Feces’; c.SV.Names{end+1} = ‘Dermal’; c.SV.Names{end+1} = ‘Ingested’; c.SV.Names{end+1} = ‘Dermal_VS’; c.SV.Names{end+1} = ‘AUCL’; n_comp = c.N_Compartments + c.DM.nskin_comps; for i=length(c.CompName)+1:length(c.SV.Names) thisSVName = c.SV.Names{i}; c.SV.ID.(thisSVName) = n_comp + 1; n_comp = n_comp + 1; end c.SV.TotNum = n_comp; % Total Number of State Variables in this PBPK Model return;
CODE LISTING 3
APPENDIX 43.4
1101
CODE LISTING 4
This Matlab code is for updating the dermal portion of the PBPK model configuration for chloroform, based on the model configuration options. 1. function c = PBPK_add_dermal_model(c_orig, modelconf) 2. % Updates the PBPK model configuration with dermal parameters 3. % 4. % Function Inputs: 5. % c_orig: the configuration of the PBPK model prior to model update 6. % modelconf: a structure that defines model configuration with fields 7. % DM.nskins – specifies type of dermal model to use 8. % 1 => one skin model 9. % 2 => two skin model (Stratum Corneum (SC) + Viable Skin) 10. % >2 => distributed skin model with nskins nodes in SC 11. % if the DM structure is not provided, or if there is no DM.nskins 12. % field, the default is a one-skin model 13. % 14. % Function Outputs: 15. % c: the updated PBPK model configuration 16. % 17. % Function side effects: 18. % Depending on the model configuration, the skin compartment in the 19. % default PBPK model may be replaced by SC, VS, and other compartments 20. 21. nskins = 1; % default number of skin compartments 22. if (isfield(modelconf, ‘DM’) & isfield(modelconf.DM, ‘nskins’)) 23. nskins = modelconf.DM.nskins; 24. if (nskins 1) c = local_update_dermal_parameters(c, nskins); c.K.skin_air = c.K.sc_air; % SC replaces skin c.K.skin_water = c.K.sc_water; end return;
function c = local_update_dermal_parameters(c_orig, nskins) c = c_orig; ID = c.IDNum; % Two-skin compartment c.DM.V_sc = c.DM.L_sc * c.indiv.BSA; c.DM.Vskin = c.V_TISSUE(ID.Skin) * 1000; % in cm^3 % Partition coefficients between multiple chemicals (non-blood) c.PC.chem.octanol_water = 90; % Partition coefficients between multiple compartments (non-blood) c.PCMC.sc_water = (c.PC.chem.octanol_water)^0.71; c.PCMC.sc_air = c.PCMC.sc_water/c.PC.chem.air_water; c.PCMC.vs_water = (c.PCMC.skin_water * c.DM.Vskin - . . . c.PCMC.sc_water * c.DM.V_sc)/(c.DM.Vskin - c.DM.V_sc); c.PCMC.vs_air = c.PCMC.vs_water/c.PC.chem.air_water; c.PCMC.vs_blood = c.PCMC.vs_air/c.PC.blood_air; c.PCMC.vs_sc = c.PCMC.vs_air/c.PCMC.sc_air; %%% Calculate Cleek and Bunge’s B parameter %%%
CODE LISTING 4
92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110.
1103
c.DM.BB = c.K.skin_water * sqrt(c.MolecularWeight)/2.6; c.K.sc_water = c.K.skin_water * (c.DM.BB + 1); c.K.sc_air = c.K.sc_water * c.PCMC.sc_air/c.PCMC.sc_water; c.K.vs_sc = c.K.sc_water/(c.PCMC.sc_water * c.DM.BB); % Assign compartment IDs c.IDNum.VS = ID.Skin; c.IDNum = rmfield(c.IDNum, ‘Skin’); c.CompName{c.IDNum.VS} = ‘VS’; c.IDNum.SC = c.IDNum.VS + 1; c.CompName{c.IDNum.SC} = ‘SC’; c.PC.tissue(c.IDNum.VS) = c.PCMC.vs_blood; if (c.DM.distributed == 1) c.DM.D_sc = c.K.sc_water/(c.PCMC.sc_water) * c.DM.L_sc; c.IDNum.SC1 = c.IDNum.SC + 1; c.CompName{c.IDNum.SC1} = ‘InnerMost SC Node’; end
APPENDIX 43.5
CODE LISTING 5
This Matlab code is for calculating the derivatives used in the PBPK model. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
function [d_amount, optSV] = PBPK_voc_deriv(t, amount) % Derivative function defined in a manner the ODE solver in Matlab expects. % Inputs: % t -- current time of the simulation (time variable in the ODE system) % amount -- amount of chemical in each tissue (state variables) % Outputs: d_amount -- derivatives % optSV -- optional state variables such as intermediate concentrations % Global: Configuration variable of the PBPK model global c; % Global PBPK model configuration structure global E; % Global activity event details mc.air = E.C.inh; % media concentration: air; inhalation concentration rate.inh = E.Q.inh; % inhalation rate mc.water = E.C.water; % media concentration: water; ingestion concentration rate.dw = E.Q.DW; % ingestion rate -- drinking water rate.infusion = E.Q.infusion; ID = c.IDNum; % Compartment IDs by name SVID = c.SV.ID; % State Variables ID % Tissue Concentrations C_TISSUE = amount(1:length(c.V_TISSUE))’ ./ c.V_TISSUE;
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PHYSIOLOGICALLY BASED PHARMACOKINETIC MODELING
% Venous Blood Concentration CV_BLOOD = amount(1:length(c.V_TISSUE))’ ./ (c.V_TISSUE .* c.PC.tissue); CV_mixed = (CV_BLOOD*c.QC’)/c.Q_cardiac; % Reroute the venous flow rate from Gut through Liver CV_mixed = CV_mixed + (CV_BLOOD(ID.Liv) - CV_BLOOD(ID.Gut)) * . . . c.QC(ID.Gut)/c.Q_cardiac; venous_blood_concentration = amount(SVID.Blood)/c.venous_blood_volume; C_arterial=(venous_blood_concentration*c.Q_cardiac+mc.air*rate.inh) . . . / (c.Q_cardiac + rate.inh/c.PC.blood_air); mc.exh = C_arterial/c.PC.blood_air; % Exhaled concentration % Rate of metabolism of chemical in each compartment rate_metabolism = c.VMAX .* CV_BLOOD ./ (c.KM + CV_BLOOD); d_amount = (c.QC .* (C_arterial - CV_BLOOD) - rate_metabolism); % Add the flow of venous blood from Gut to Liver to Venous Blood d_amount(ID.Liv) = d_amount(ID.Liv) + c.QC(ID.Gut)*. . . (CV_BLOOD(ID.Gut) - CV_BLOOD(ID.Liv)); d_amount(SVID.Blood)=c.Q_cardiac*(CV_mixed-venous_blood_concentration)+ . . . rate.infusion; d_amount(SVID.Metabolized) = sum(rate_metabolism); d_amount(SVID.AUCL) = C_TISSUE(ID.Liv); d_amount(SVID.Inhaled) = rate.inh * mc.air; d_amount(SVID.Exhaled) = rate.inh * mc.exh; d_amount(SVID.Ingested) = rate.dw*mc.water; rate_gut_absorption = c.Absorption.Gut * amount(SVID.GILumen); d_amount(SVID.GILumen)=d_amount(SVID.Ingested)*c.F_bioavail_water- . . . rate_gut_absorption; d_amount(ID.Gut) = d_amount(ID.Gut) + rate_gut_absorption; d_amount(SVID.Feces) = d_amount(SVID.Ingested)*(1 - c.F_bioavail_water); % Update the derivative for additional processes d_amount = PBPK_add_dermal_derivative(amount, d_amount, c, E); d_amount = d_amount’; optSV.concs.tissue = C_TISSUE; % All tissues optSV.concs.exh = mc.exh; % exhaled breath concentrations optSV.concs.venous_blood = venous_blood_concentration; optSV.rate.metabolism = rate_metabolism;
APPENDIX 43.6
CODE LISTING 6
This Matlab code is for updating the PBPK model derivatives based on the dermal model options.
CODE LISTING 5
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.
1105
functiond_amount=PBPK_add_dermal_derivative(amount,d_amount_orig,c,E) % Updates to the basic PBPK model derivative structure with dermal derivative % Inputs are same as the generic PBPK_update_derivative function % Output: d_amount -- derivative vector updated with the dermal model equations d_amount = d_amount_orig; if (E.dermal_contact_media == 0) mc.dermal = E.C.inh; % dermal/air contact k_skin_media = c.K.skin_air; % Permeability: skin/air pc_skin_media = c.PCMC.skin_air; % Partition coefficient: skin/air if (c.DM.nskins > 1) pc_skin_media = c.PCMC.sc_air; % Partition coefficient: sc/air end else mc.dermal = E.C.water; % dermal/water contact k_skin_media = c.K.skin_water; % Permeability: skin/water pc_skin_media = c.PCMC.skin_water; % Partition coefficient: skin/water if (c.DM.nskins > 1) pc_skin_media = c.PCMC.sc_water; % Partition coefficient: sc/water end end SVID = c.SV.ID; % State variable IDs by name % Tissue Concentrations C_TISSUE = amount(1:length(c.V_TISSUE))’ ./ c.V_TISSUE; switch(c.DM.nskins) case (1) % One compartment Skin = VS C_skin = amount(SVID.Skin)/c.V_TISSUE(SVID.Skin); % Concentration in skin rate_dermal_abs = (k_skin_media * c.indiv.BSA)/1000 * . . . (mc.dermal - C_skin/pc_skin_media); d_amount(SVID.Skin) = d_amount(SVID.Skin) + rate_dermal_abs; d_amount(SVID.Dermal) = rate_dermal_abs; d_amount(SVID.Dermal_VS) = rate_dermal_abs;; case (2) % Two comparments: SC and VS C_sc = amount(SVID.SC)/c.DM.V_sc; % Concentration in SC C_vs = amount(SVID.VS)/c.V_TISSUE(SVID.VS); % Concentration in viable skin rate_dermal_abs = (k_skin_media * c.indiv.BSA)/1000 * . . . (mc.dermal - C_sc/pc_skin_media); rate_vs_abs = (c.K.vs_sc * c.indiv.BSA)/1000 * (C_sc - C_vs/c.PCMC.vs_sc); d_amount(SVID.SC) = rate_dermal_abs - rate_vs_abs; d_amount(SVID.VS) = d_amount(SVID.VS) + rate_vs_abs; d_amount(SVID.Dermal) = rate_dermal_abs; d_amount(SVID.Dermal_VS) = rate_vs_abs; otherwise % Distributed PDE model % Formualte the 1-D PDE in terms of finite differences nn = c.DM.nn_sc; % number of nodes
1106
PHYSIOLOGICALLY BASED PHARMACOKINETIC MODELING
49. dl = c.DM.L_sc/(nn-1); % length of each node 50. 51. % Boundary Conditions 52. C_sc_in = C_TISSUE(SVID.VS)/c.PCMC.vs_sc; % Innermost SC: SC1 53. C_sc_out = mc.dermal * pc_skin_media; % Outermost SC layer 54. ID_sc_out = SVID.SC1 + nn - 1; % ID for outermost SC 55. C_sc = [C_sc_in; amount(SVID.SC1+1:ID_sc_out-1); C_sc_out]; 56. 57. % Central Differnce Forumation across nodes of SC 58. lnodes = C_sc(1:nn-2); % array containing left side nodes 59. mnodes = C_sc(2:nn-1); % middle nodes 60. rnodes = C_sc(3:nn); % right side nodes 61. d_C_sc_nodes = c.DM.D_sc/dl^2 * (lnodes - 2*mnodes + rnodes); 62. 63. % Dermal absorption rates on both sides of SC 64. bsa = c.indiv.BSA; % cm^2 65. rate_dermal_abs = c.DM.D_sc/dl * bsa * (C_sc(nn)-C_sc(nn-1))/1000; 66. rate_vs_abs = c.DM.D_sc/dl * bsa * (C_sc(2) - C_sc(1))/1000; % innermost 67. d_amount(SVID.VS) = d_amount(SVID.VS) + rate_vs_abs; 68. d_amount(SVID.SC) = rate_dermal_abs - rate_vs_abs; % entire SC concentration 69. d_amount(SVID.SC1:SVID.SC1+nn-1) = [0 d_C_sc_nodes’ 0]; % middle nodes 70. d_amount(SVID.Dermal) = rate_dermal_abs; 71. d_amount(SVID.Dermal_VS) = rate_vs_abs; 72. end
CHAPTER 44
Modeling of Metabolite Pharmacokinetics in a Large Pharmacokinetic Data Set: An Application VALÉRIE COSSON, KARIN JORGA, and ELIANE FUSEAU
44.1
INTRODUCTION
When a drug or a prodrug is metabolized to one or more active metabolites, not only the exposure to the parent drug but also the exposure to the active metabolites contribute to the safety and efficacy of that drug/prodrug (1–7). Prodrugs represent an aspect of the biotransformation of parent drug, where only the metabolite is active. Often, the prodrug is not subjected to intense pharmacokinetic (PK) modeling since its concentration declines rapidly; the absorption parameters of the active moiety encompass the transformation of the parent drug to its active form and absorption of the parent. The blood or plasma concentrations of the parent drug and/or its active metabolites (systemic exposure) may provide an important link between drug dose (exposure) and desirable and/or undesirable drug effects (8). For this reason, the modeling of parent drug and metabolite pharmacokinetics, coupled with pharmacodynamic (PD) measurements, offers an essential development tool for prediction and simulation. The simultaneous modeling of parent drug and metabolite allows the evaluation of the impact of organ impairment or of the effects of drug–drug interactions (9–13). The high incidence of adverse events seen in patients with end stage renal disease may, for some drugs, be explained in part by the accumulation of active drug metabolites (1). Any interaction at the site of drug metabolizing enzymes can modify the overall activity of the compound. It is often informative to have the prediction of metabolite concentrations when performing PK/PD modeling (2–7). Delay between the concentration of the parent compound and drug response curve (hysteresis) can be the result of metabolism when the metabolite is more effective than the parent drug.
Pharmacometrics: The Science of Quantitative Pharmacology Edited by Ene I. Ette and Paul J. Williams Copyright © 2007 John Wiley & Sons, Inc.
1107
1108
MODELING OF METABOLITE PHARMACOKINETICS IN A LARGE PHARMACOKINETIC DATA SET
Unless the metabolite has been administered alone (in order to estimate its volume of distribution (14)) or the fraction of parent drug converted to metabolite is known, the modeling of the parent drug and its metabolite requires simplification so that metabolite parameters can be estimated. This is because the rate of conversion of the parent to the metabolite and the distribution volume of the metabolite are structurally not simultaneously identifiable. The ratio of the rate of conversion to the metabolite volume, which is globally identifiable, and apparent elimination rate constant for the metabolite can be estimated and do not constitute an identifiability problem (15–21). The number of metabolites for which parameters can be estimated is not limited. For example, the parent compound, active metabolites, and its conjugate have been modeled together for irinotecan (15). More complicated models can be employed to investigate autoinducible transformations (9, 16, 22).
44.2
THE NELFINAVIR EXAMPLE
Intra- and interindividual variations in protease inhibitor drug exposure can influence the safety and effectiveness of anti-HIV therapy. In the example presented here, in addition to the description of the pharmacokinetics of nelfinavir and its metabolite, it was possible to evaluate the impact of the coadministration of ritonavir. Nelfinavir is the only marketed HIV protease inhibitor that is converted into an active metabolite at plasma levels, which are significant enough to contribute to the overall antiviral activity (23). Nelfinavir distributes largely into tissues and is highly bound to plasma proteins (>98%). The apparent volume of distribution is 2–7 L/kg. Nelfinavir is metabolized in the liver by at least four different cytochrome P450 (CYP) isoenzymes including CYP 3A4, CYP 2C9, CYP 2C19, and CYP 2D6, with CYP 2C19 catalyzing roughly 50% of nelfinavir clearance in normal metabolizers (23–25). CYP 2C19 mediates the formation of the primary metabolite M8 (nelfinavir hydroxy-t-butylamide), which has activity comparable to the parent drug. M8 is subsequently metabolized by CYP 3A4. The majority of nelfinavir and its metabolites (87%) are eliminated in the feces. Urinary excretion accounts for only 1–2%, most of which is unchanged nelfinavir (26). Nelfinavir induces its own metabolism; plasma concentrations decline approximately 40–50% and are stable after 6 days (26, 27). Ritonavir (RTV) is also an inhibitor of HIV proteases, approved for use in combination with nucleoside analog, for the treatment of HIV-1 infected adults, adolescents, and children. It is a potent CYP 3A4 inhibitor and is used at low doses to elevate plasma concentrations of other protease inhibitors being primarily metabolized by CYP 3A4. In combination with saquinavir, this type of interaction has proved favourable (28). The combination with nelfinavir showed much smaller effects on nelfinavir levels, but it appears to change in normal metabolizers the M8/nelfinavir concentration ratio from 0.3 to 0.6. In poor CYP 2C19 metabolizers (∼3–5% of Caucasians and African-Americans, ∼12–20% of Asians), ritonavir addition is not expected to have such an effect on the nelfinavir/M8 ratio (29). In addition, ritonavir induces CYP isoenzymes, so that the full effect of the nelfinavir–ritonavir drug–drug interaction is considered stable after a treatment duration of 10–14 days (30).
THE NELFINAVIR EXAMPLE
44.2.1
1109
Methods
44.2.1.1 Study Design and Data The study was a randomized, stratified, open label, two-arm parallel Phase 4 trial designed to explore the utility of pharmacological parameters as predictors of antiviral response to nelfinavir-containing regimens in patients pretreated with protease inhibitors-sparing regimens. The study population included male or female patients aged 18 years or older with HIV-1 infection who have previously failed only one antiretroviral regimen consisting of nucleoside reverse transcriptase inhibitors (NRTIs), nonnucleoside reverse transcriptase inhibitors (NNRTIs), or at most one protease inhibitor (PI) and had no evidence of PI resistance (genotypic) at screening or had plasma HIV1 RNA greater or equal to 1000 copies/mL at least once in the three months before screening and HIV-1 RNA greater or equal to 1000 copies/mL at screening. Participants in the study were randomized to receive nelfinavir at 1250 mg BID or nelfinavir at 1250 mg BID in combination with ritonavir at 200 mg BID in a 1 : 1 ratio. Both treatments were administered for 48 weeks. In the treatment arm initially receiving nelfinavir as the only PI, ritonavir was added at week 12 if the HIV-1 RNA had not been reduced to 75,000 copies/mL 10,000–75,000 copies/mL 0
η3
(45.4) (45.5)
where V is effective volume of distribution (mL), CL is clearance (mL/h), Ka is the oral absorption rate constant (h−1), BWT is body weight (kg), and LATR is the log10 of the anti-mAb titer. As shown, power functions for V and CL incorporating dose were found to best represent the nonlinear pharmacokinetics in MODEL 1, while the impact of anti-mAb was incorporated as an additive linear increase in CL relative to anti-mAb titers. Proportional intersubject variability was included on V, CL, and Ka. V was found to increase in proportion to body weight. The scalar used to translate model predicted amounts to concentration was set equal to V. A proportional residual error model was found to best represent the data. MODEL 1 was fit to a data set that contained the data from both the Phase 1 and Phase 2 studies. No study impact was found on any model parameters. The basic structure of MODEL 2 is shown in Figure 45.3. MODEL 2 was implemented using ADVAN6 (TRANS1) and the following model differential equations: ∂A1 = − Ka ⋅ A1 ∂t
(45.6)
Dose
Subcutaneous K a*A1 Dose Depot (A1)
Central Compartment (A 2) K e*A2 Elimination of free mAb
E bound
E anti-mAb
Elimination of Elimination of mAb via mAb via antibinding to mAb antibodies target receptor
FIGURE 45.3 MODEL 2 diagram. MODEL 2 was mechanistic in nature and included three elimination routes for mAb, where both Ebound and Eanti-mAb had a nonlinear relationship with concentration of mAb.
AN EXAMPLE
∂A2 = Ka ⋅ A1 ⋅ fu − Ke ⋅ A2 − Eanti − mAb − Ebound ∂t
1143
(45.7)
where Eanti − mAb =
Vmax ⋅ C2 C2 + K m
Ebound = Ke 1 ⋅ Cbound ⋅ V fu =
A2 A2 + Cbound ⋅ V
Cbound =
C2 ⋅ Ctarget,t = 0 K D + C2
C2 =
A2 V
(45.8) (45.9) (45.10) (45.11) (45.12)
In the above equations A1 was the amount of mAb in the subcutaneous dose compartment (mmol), A2 was the amount of mAb not bound to target receptor or free mAb in the central compartment (mmol), C2 was the concentration of free mAb in the central compartment (mmol/mL), Ka was the subcutaneous absorption rate constant (h−1), Ke was the elimination rate constant for free mAb (h−1), fu was the free fraction of mAb (not bound to the target receptor), and KD was the equilibrium dissociation constant for binding between mAb and target receptors (mmol/mL). Included in the model are also terms that represent the elimination due to anti-mAb (Eanti-mAb) and elimination due to binding to the target receptor (Ebound). Elimination via anti-mAb was found to be best represented using the Michaelis–Menten equation (Eq. (45.8)) to capture the dependence of elimination on the concentration of mAb. The elimination via binding to target receptors (Eq. (45.9)) was modeled using a first-order rate constant Ke1 (h−1), where elimination was proportional to the concentration of mAb bound to target receptors (Cbound). Preliminary structures for MODEL 2 (see Appendix 45.2) included differential equations to calculate the concentration of bound and unbound mAb using forward and reverse binding rate constants for the mAb; however, these model equation systems were found to be stiff and computationally resource intensive. In the final model above, an assumption was made to allow a simpler set of model equations. For most mAb, the forward and reverse binding rates to the target receptor are much greater than the rates of absorption or elimination of the mAb, and this was also true for the mAb in this case. So the assumption applied in MODEL 2 was that unbound mAb was in equilibrium with bound mAb at all times. This assumption eliminated the need to include separate equations for bound and free target receptors, and bound mAb in the differential mass balance equations. However, since the elimination of mAb via binding to target receptors was proportional to Cbound, the model required Cbound expressed in terms of C2. This is provided by Eq. (45.11), which is derived by combining the equilibrium binding equation (Eq. (45.13)) with an equation relating the total concentration of target sites to unbound target sites (Eq. (45.14)), where Ctarget,t=0 is the concentration of unbound target sites at time zero.
1144
CHARACTERIZING NONLINEAR PHARMACOKINETICS
KD =
C2 ⋅ Ctarget free Cbound
Ctarget ,bound = Ctarget ,t=0 − Cbound
(45.13) (45.14)
Equations (45.13) and (45.14) assume that the mAb binds stoichiometrically with the target receptor in a 1 : 1 ratio, and Eq. (45.14) assumes that the total number of target receptors remains constant. Since the binding to target receptors was assumed to be in equilibrium at all times, binding was also taken into account when mAb is transferred from the dose compartment to the central compartment, with the central compartment only accounting for free mAb that is in equilibrium with bound mAb. Although not immediately apparent, these model structures preserve the overall mass balance in the model. MODEL 2 also used the following parameter equations:
Vmax =
F1 = 1 150,000
(45.15)
Ka = θ 1 ⋅ eη1
(45.16)
Vmax = 0 for LATR = 0
(45.17)
θ2 ⋅ 10 LATR 150,000 Km =
for LATR > 0
θ3 150,000
(45.18) (45.19)
K e = θ 4 ⋅ e η2
(45.20)
Ke1 = θ 5
(45.21)
V = θ 6 ⋅ BWT ⋅ eη3
(45.22)
where F1 was fixed but used to convert the dose amount from mg to mmol, using the mAb molecular weight of 150,000 daltons. The maximum rate (Vmax, mmol/h) and Michaelis–Menten constant (Km, mmol/mL) for elimination via anti-mAb were divided by 150,000 so that the model q values were expressed in terms of more relevant mass units rather than molar units. Similar to MODEL 1, V was found to increase in proportion to body weight. The scalar used to translate model predicted amounts in mmol to concentrations in mg/mL was set equal to V/150,000. Proportional intersubject variability was included on V, Ke, and Ka. A proportional residual error model was found to best represent the data. MODEL 2 was fit first to the Phase 1 data alone and then to both the Phase 1 and Phase 2 data combined. The population value and intersubject variance for Ka determined from fitting the Phase 1 data were fixed during fitting of the combined Phase 1 and 2 data. KD and Ctarget,t=0 could not be estimated and were fixed to 1 × 10−5 (mmol/mL) and 2 × 10−6 (mmol/mL), respectively, which were expected to represent typical values for a mAb and cell surface target receptor. 45.2.5 Model Comparison and Parameter Values Tables 45.2 and 45.3 list the parameter values for each model, while Figures 45.4 and 45.5 show some diagnostic plots for each model. Overall, both models were
AN EXAMPLE
1145
TABLE 45.2 Final Model Parameters for MODEL 1 Parameter q1 q2 q3 q4 q5 q6 w1 w2 w3 e
Parameter Description Volume of distribution per kg of body weight (mL/kg) Coefficient for dose-dependent clearance term (mL/h) Subcutaneous absorption rate (h−1) Coefficient for anti-mAb-dependent clearance (mL/h) Volume of distribution linearity with respect to dose (power model) Clearance linearity with respect to dose (power model) Intersubject variance of volume of distribution Intersubject variance of dose-dependent clearance term Intersubject variance of subcutaneous absorption Residual error variance Objective function value
Estimate (Value ± SE) 50.0 ± 1.58 29.6 ± 1.05 0.0132 ± 0.000161 0.0244 ± 0.00512 0.156 ± 0.0395 −0.457 ± 0.0243 0.0688 ± 0.0159 0.0977 ± 0.0145 0.00568 ± 0.00230 0.0853 ± 0.0103 −12413
TABLE 45.3 Final Model Parameters for MODEL 2 Parameter q1 q2 q3 q4 q5 q6 w1 w2 w3 e
Parameter Description Subcutaneous absorption rate (h−1) Maximum elimination rate coefficient for antimAb-dependent elimination (mg/h) Michaelis–Menten constant for anti-mAbdependent elimination (mg/mL) Elimination rate constant for free mAb (h−1) Elimination rate constant for mAb bound to target receptors (h−1) Volume of distribution per kg of body weight (mL/kg) Intersubject variance of subcutaneous absorption Intersubject variance of elimination rate constant for free mAb Intersubject variance of volume of distribution Residual error variance Objective function value
Estimate (Value ± SE) 0.0387* 0.418 ± 0.0382 4.87 ± 0.363 0.00132 ± 0.000155 0.0600 ± 0.000831 88.9 ± 6.53 0.0468a 0.172 ± 0.0855 0.0256 ± 0.00415 0.0595 ± 0.00827 −13624
a Subcutaneous absorption parameters were estimated using the Phase 1 data and subsequently fixed when fitting the combined Phase 1 and Phase 2 data.
able to fit the data reasonably well, with MODEL 2 having slightly lower residual error values compared to MODEL 1. When the data were fit using MODEL 1 and MODEL 2 without accounting for anti-mAb effects (q4 set to zero in MODEL 1, q2 set to zero in MODEL 2), the objective function values increased by approximately 1949 and 233 points, respectively, indicating that accounting for anti-mAb-
1146
10^2 10^0
MODEL 2
10^-2 10^-4 10^-3 10^-2 10^-1 10^0 10^1 10^2
10^-4
10^2 10^0
MODEL 1
10^-2 10^-4
Observed mAb Concentration (mcg/mL)
CHARACTERIZING NONLINEAR PHARMACOKINETICS
10^2 10^0
MODEL 2
10^-2 10^-4 10^-3 10^-2 10^-1 10^0 10^1 10^2
10^-4
10^2 10^0
MODEL 1
10^-2 10^-4
Observed mAb Concentration (mcg/mL)
10^-4 10^-3 10^-2 10^-1 10^0 10^1 10^2 Individual Predicted mAb Concentration (μ/mL)
10^-4 10^-3 10^-2 10^-1 10^0 10^1 10^2
Predicted mAb Concentration (μ/mL)
0
0
5
5
MODEL 2
-5
-5
Weighted Residuals
MODEL 1
10^-4 10^-3 10^-2 10^-1 10^0 10^1 10^2
10^-4 10^-3 10^-2 10^-1 10^0 10^1 10^2
Predicted mAb Concentration (μ/mL)
FIGURE 45.4 Goodness-of-fit plots for MODEL 1 and MODEL 2. Plotted data includes all subjects from Phase 1 and Phase 2 studies.
mediated elimination significantly improved the fit to the data. Figure 45.5 includes only those subjects who were anti-mAb positive, and these diagnostic plots suggest that MODEL 2 was able to provide a slightly better fit to anti-mAb-positive subjects; however, both models tended to overpredict the lower concentrations at the end of a dosing interval produced during an anti-mAb response. This is also shown in Figure 45.6, where the lower panels compare the fit of MODEL 1 and MODEL 2 for two subjects in the Phase 1 study who had moderate anti-mAb responses. Both models tended to overpredict the concentrations at later time points during an antimAb response, but MODEL 2 provided a better fit to these low concentrations. Also shown in Figure 45.6 in the upper panels are comparisons of the fit of MODEL 1 and MODEL 2 to the nonlinear profile generated in two sample anti-mAb-negative subjects as a result of binding and elimination through the target receptor. The nonlinear profile in these subjects was more accurately fit using MODEL 2 since CL
10^2
1147
MODEL 2
10^-4
10^-2
10^0
10^2 10^-2
10^0
MODEL 1
10^-4
Observed mAb Concentration (μg/mL)
AN EXAMPLE
10^-4 10^-3 10^-2 10^-1
10^0
10^1
10^2
10^-4 10^-3 10^-2 10^-1
10^0
10^1
10^2
10^0
10^1
10^2
10^0
10^1
10^2
10^2 10^0
MODEL 2
10^-2 10^-4 10^-3 10^-2 10^-1
10^0
10^1
10^2
10^-4
10^2 10^0
MODEL 1
10^-2 10^-4
Predicted mAb Concentration (μg/mL)
Individual Predicted mAb Concentration (μ/mL)
10^-4 10^-3 10^-2 10^-1
4 2
MODEL 2
-4
-2
-2
0
0
2
4
MODEL 1
-4
Weighted Residuals
Predicted mAb Concentration (μ/mL)
10^-4 10^-3 10^-2 10^-1
10^0
10^1
10^2
10^-4 10^-3 10^-2 10^-1
Predicted mAb Concentration (μ/mL)
FIGURE 45.5 Goodness-of-fit plots for MODEL 1 and MODEL 2. Plotted data includes only anti-mAb-positive subjects from Phase 1 and Phase 2 studies.
in MODEL 1 was not dependent on concentration of mAb. Over the 0.05–2.0 mg/kg dose range used in the Phase 2 study, MODEL 1 predicted that V would increase from 31.3 to 55.7 mL/kg and CL would decrease from 116 to 21.6 mL/h. MODEL 1 estimated that the anti-mAb-mediated clearance of mAb would be 7.72 mL/h at an anti-mAb log10 titer of 2.5. The estimated value for Ka was approximately three times higher in MODEL 2 compared to MODEL 1. MODEL 2 estimated the maximum rate nonlinear elimination (Vmax) via anti-mAb to be 0.418 mg/h, with a corresponding Km of 4.87 mg/mL. The estimated Ke and Ke1 in MODEL 2 translate to half-lives of approximately 21.9 days and 11.6 h for elimination of free mAb and mAb bound to target receptors, respectively. The estimated value for V was approximately 1.5–2.8 times higher in MODEL 2 compared to MODEL 1. Proportional residual error expressed as a coefficient of variation was 24.4% for MODEL 2 and 29.2% for MODEL 1.
1148
0.01 0.10 1.00 10.00 100.00
0.01 0.10 1.00 10.00 100.00
Anti-mAb Negative 2
3
4
5
6
0
1
2
3
4
5
6
3
4
5
6
10^0 Anti-mAb Positive 0
1
2
3
4
5
6
10^-4
10^-2
10^0 10^-4
Anti-mAb Negative
10^2
1
10^2
0
10^-2
Concentration of mAb (μg/mL)
CHARACTERIZING NONLINEAR PHARMACOKINETICS
Anti-mAb Positive 0
1
2
Time (week)
FIGURE 45.6 Comparison of individual model predicted versus observed concentrations in four example subjects in the Phase 1 study. The upper panels show the fit to anti-mAbnegative subjects who had nonlinear elimination resulting only from binding to target receptors. The lower panels show the fit to anti-mAb-positive subjects who had moderate anti-mAb responses, and therefore had two routes of nonlinear elimination. Solid lines represent MODEL 1, dashed lines represent MODEL 2, and open circles represent observed data.
45.3
DISCUSSION
Similar to small molecules, the clinical relevance of PK variability and nonlinearity depend on how strong exposure is correlated to efficacy and safety and the width of the therapeutic index. For therapeutics that have a very wide margin of safety and have an efficacy outcome that can be predicted as accurately with an individual’s dose as with an individual’s concentration-related exposure, there may be little value in accurately accounting for PK variability and nonlinearity. However, there are many therapeutics where these two conditions are not true. In these cases, it is important to accurately characterize variability and nonlinearity so that one can better understand how changes in dose, patient characteristics, and disease state may affect safety and efficacy through exposure. The development of antibodies to therapeutic proteins can have direct safety consequences in addition to the impact on pharmacokinetics as addressed in this chapter. Provided the antibody response itself does not have direct safety issues, in some cases it is possible to continue dosing the therapeutic protein in the presence of antibody, provided efficacious levels of the therapeutic protein can be maintained. In this case, it is important to fully understand how the antibody response impacts active concentrations of the therapeutic protein so that it is possible to design dosing regimens that minimize the impact of the antibody response. In the
DISCUSSION
1149
example discussed in this chapter, the antibody response (anti-mAb) reduced mAb exposure, but this occurred in a nonlinear manner with less negative impact at higher exposures relative to lower exposures. In addition to the influence of antimAb, there was an additional nonlinearity observed that was postulated to be the result of mAb binding to the target receptor. The complexity of the modeling that can be done to understand nonlinearity is highly dependent on the amount and quality of data available. In this example, there were anti-mAb data available that were used to model the influence that the anti-mAb response had on mAb clearance. If anti-mAb data were not available, one could use a mixture model in NONMEM, where the population is assumed to be either anti-mAb positive or anti-mAb negative. This method would improve the fit compared to not accounting for anti-mAb elimination at all; however, the fixed effect of anti-mAb titer has a significant impact on elimination and this method would not be able to account for this fixed effect. In this example, there were also no study data regarding the concentration of the target receptor; however, in many cases this is a biomarker that is available as a measure of target receptor occupancy. 45.3.1
MODEL 1 Advantages, Disadvantages, and Limitations
The main advantage of MODEL 1 is that it is a relatively simple model that uses one of the standard NONMEM models in conjunction with parameter equations that incorporate nonlinearity with respect to dose and anti-mAb titer. This model requires fewer parameters and could be used with sparser data sets. The disadvantage of MODEL 1 is that it is more empirical in nature and does not try to account for the observed nonlinearity on a mechanistic basis. The parameter values obtained from MODEL 1 are questionable from a biological standpoint. The estimated half-life based on apparent CL and V is much lower than would be expected for a mAb, and this is because the clearance in MODEL 1 does not distinguish between the normal linear elimination pathway and elimination via binding to the target receptor. What this means is that MODEL 1 would be unable to accurately predict the clearance of the mAb at doses above or below those used in the modeling. The apparent V estimated with MODEL 1 is at or below the plasma volume; therefore, this volume is not an accurate reflection of the true distribution volume. The apparent V was found to increase with increasing dose. It would be more likely that the apparent V would increase with decreasing dose, since more of the mAb would be bound to the target receptor at lower doses and this would cause less of the mAb to be measured, resulting in an increase in the apparent V. 45.3.2
MODEL 2 Advantages, Disadvantages, and Limitations
The main advantage of MODEL 2 is that it attempts to account for the mechanisms that cause the observed nonlinearity. This should allow MODEL 2 to more accurately predict exposures for doses outside the range currently evaluated or for different dosing schedules, and also to directly account for factors that may impact the number of target receptors or their rate of turnover. The parameter values estimated by MODEL 2 are more realistic from a biological standpoint, with an estimated apparent V of 88.9 mL/kg and half-life of 22 days. The apparent V would
1150
CHARACTERIZING NONLINEAR PHARMACOKINETICS
not include binding to target receptors since this is accounted for separately using the equilibrium binding equations. The elimination rate constant (Ke1) for target receptor bound mAb could be considered related to the turnover or recycling rate of the target receptor and was found to have a half-life of 12 h. The main disadvantage of MODEL 2 is that its increased complexity and greater number of parameters required a richer data set and also required assumptions about the initial concentration of target binding sites. The binding affinity (KD) required by MODEL 2 could be reasonably obtained from in vitro binding studies. A limitation of MODEL 2 is that it assumed the number of target binding sites remains constant with treatment and time, whereas in reality it is possible that treatment with the mAb may cause changes in the expression of the target receptor.
45.4
SUMMARY
The example provided in this chapter considers the case of nonlinear pharmacokinetics observed for a protein therapeutic in Phase 1 and Phase 2 studies. The nonlinearity is postulated to be the result of two factors: elimination via anti-mAb antibodies and elimination via binding to target receptors. A comparison is made between fitting the concentration data using an empirical modeling approach versus a mechanistic modeling approach. Both models are able to fit the data reasonably, and the advantages, disadvantages, and limitations of each model are discussed. The best modeling approach to characterize nonlinear pharmacokinetics depends on how much data are available and the intended purpose of the modeling. If the modeling is intended to be used in a predictive manner for future studies that use different dosing regimens and patients with different characteristics, then a more mechanistic approach may lead to more accurate predictions.
REFERENCES 1. T. M. Ludden, Nonlinear pharmacokinetics: clinical implications. Clin Pharmacokinet 20:429–446 (1991). 2. C. A. Van Ginneken and F. G. Russel, Saturable pharmacokinetics in the renal excretion of drugs. Clin Pharmacokinet 16:38–54 (1989). 3. P. Veng-Pedersen, Theorems and implications of a model independent elimination/distribution function decomposition of linear and some nonlinear drug dispositions. I. Derivations and theoretical analysis. J Pharmacokinet Biopharm 12:627–648 (1984). 4. P. Veng-Pederson, Model-independent steady-state plasma level predictions in autonomic nonlinear pharmacokinetics I: Derivation and theoretical analysis. J Pharm Sci 73:761–765 (1984). 5. P. Veng-Pederson, Model-independent method of predicting peak, trough, and mean steady-state levels in multiple intravenous bolus dosing in nonlinear pharmacokinetics. J Pharm Sci 72:1098–1100 (1983). 6. M. T. Smith and T. C. Smith, The unsteady model. An alternative approach to nonlinear pharmacokinetics. Eur J Clin Pharmacol 20:387–398 (1981). 7. C. Pendley, A. Schantz, and C. Wagner, Immunogenicity of therapeutic monoclonal antibodies. Curr Opin Mol Ther 5:172–179 (2003).
MODEL 1 NONMEM CONTROL CODE
1151
8. E. K. Rowinsky, G. H. Schwartz, J. A. Gollob, J. A. Thompson, N. J. Vogelzang, R. Figlin, R. Bukowski, N. Haas, P. Lockbaum, Y. P. Li, R. Arends, K. A. Foon, G. Schwab, and J. Dutcher, Safety, pharmacokinetics, and activity of ABX-EGF, a fully human anti–epidermal growth factor receptor monoclonal antibody in patients with metastatic renal cell cancer. J Clin Oncol 22:3003–3015 (2004). 9. K. R. Lees, H. C. Diener, K. Asplund, and M. Krams, UK-279276, a neutrophil inhibitory glycoprotein, in acute stroke. Tolerability and pharmacokinetics. Stroke 34:1704–1709 (2003). 10. M. Kato, H. Kamiyama, A. Okazaki, K. Kumaki, Y. Kato, and Y. Sugiyama, Mechanism for the nonlinear pharmacokinetics of erythropoietin in rats. J Pharmacol Exp Ther 283:520–527 (1997). 11. D. E. Allison, S. G. Gourlay, E. Koren, R. M. Miller, and J. A. Fox, Pharmacokinetics of rhuMAb CD18, a recombinant humanised monoclonal antibody fragment to CD18, in normal healthy human volunteers. Biodrugs 16:63–70 (2002). 12. R. P. Junghans and C. L. Anderson, The protection receptor for IgG catabolism is the b2-microglobulin-containing neonatal intestinal transport receptor. Proc Natl Acad Sci 96:5512–5516 (1996).
APPENDIX 45.1
MODEL 1 NONMEM CONTROL CODE
$PROB FIT OF SINGLE AND MULTIPLE DOSE DATA RUN=001 $INPUT ID TIME APOS DOSE WGT DV MDV AMT EVID LATR $DATA sdmd.data.final.3.csv IGNORE=C ; Data is in the Excel file. $SUBROUTINE ADVAN2 TRANS2 ;Kineticist: Stuart Friedrich $PK PV=THETA(1)*WGT*DOSE**THETA(5) V=PV*EXP(ETA(1)) IF (LATR .EQ. 0) THEN PCL=THETA(2)*DOSE**THETA(6) ELSE PCL=THETA(2)*DOSE**THETA(6)+THETA(4)*10**LATR ENDIF CL=PCL*EXP(ETA(2)) PKA=THETA(3) KA=PKA*EXP(ETA(3)) S2=V
1152
CHARACTERIZING NONLINEAR PHARMACOKINETICS
$ERROR IPRED=F W=0.001 IF(F.GT.0) W=F IRES=DV-IPRED IWRES=IRES/W Y=F+W*ERR(1) $THETA $THETA $THETA $THETA $THETA $THETA $OMEGA $OMEGA $OMEGA $SIGMA
(20,90,150);1 - Volume (10,40,100);2 - Linear clearance parameter (0.001,0.03,0.1);3 - KA (0,0.01);4 - Clearance parameter due to antibody response (-0.1);5 - Power exponent for change in V with dose (-0.1);6 - Power exponent for change in CL with dose 0.5;1 - Var of V 0.5;2 - Var of CL 0.5;3 - Var of KA 0.5;PROPORTIONAL ERROR
$EST MAXEVAL=5000 PRINT=5 METH=0 POSTHOC $COV $TABLE ID TIME IPRED IWRES APOS LATR DOSE FILE=model_sdmd_1_t1.tb NOPRINT ONEHEADER $TABLE DOSE ID KA V CL ETA1 ETA2 ETA3 FILE=model_sdmd_1_t2.tb NOPRINT ONEHEADER FIRSTONLY
APPENDIX 45.2
MODEL 2 NONMEM CONTROL CODE
Preliminary Evaluated Model $PROB FIT OF SD AND MD DATA RUN=001 $INPUT ID ARM TIME APOS DOSE WGT DV MDV AMT EVID LATR CMT $DATA data.csv IGNORE=C $SUBROUTINE ADVAN8 TRANS1 TOL=3 $MODEL COMP=(COMP1) COMP=(COMP5)
COMP=(COMP2,DEFOBS)
$PK PV=THETA(11)*WGT V=PV*EXP(ETA(11)) PF1=THETA(1)/150000 F1=PF1*EXP(ETA(1))
COMP=(COMP3)
COMP=(COMP4)
MODEL 2 NONMEM CONTROL CODE
PF3=THETA(2)*V F3=PF3*EXP(ETA(2)) PKA=THETA(3) KA=PKA*EXP(ETA(3)) PKF=THETA(4)*1E6 KF=PKF*EXP(ETA(4)) PKR=THETA(5) KR=PKR*EXP(ETA(5)) IF (LATR .EQ. 0) THEN VMX=0 ELSE VMX=(THETA(6)/150000*10**LATR)*EXP(ETA(6)) ENDIF PKM=THETA(7)/150000 KM=PKM*EXP(ETA(7)) PABLG=THETA(8) ABLG=PABLG*EXP(ETA(8)) PKE=THETA(9) KE=PKE*EXP(ETA(9)) PKE1=THETA(10) KE1=PKE1*EXP(ETA(10)) S2=V/150000 $ERROR IPRED=F W=0.001 IF(F.GT.0) W=F IRES=DV-IPRED IWRES=IRES/W Y=F+W*ERR(1) $DES C2=A(2)/V A3=A(3) A5=A(5) IF (T .LE. ABLG) THEN VMAX=0 ELSE VMAX=VMX ENDIF
1153
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CHARACTERIZING NONLINEAR PHARMACOKINETICS
DADT(1)=-KA*A(1);MAB ABSORPTION COMPARTMENT DADT(2)=KA*A(1)-KF*A(3)*A(2)/V+KR*A(4)-KE*A(2)-VMAX*C2/(C2+KM);MAB FREE COMPARTMENT (CENTRAL) DADT(3)=-KF*A(3)*A(2)/V+KR*A(5);TARGET SITES FREE COMPARTMENT DADT(4)=KF*A(3)*A(2)/V-KR*A(4)-KE1*A(4);MAB COMPARTMENT
BOUND
TO
TARGET
DADT(5)=KF*A(3)*A(2)/V-KR*A(5);TARGET SITES BOUND COMPARTMENT $THETA $THETA $THETA $THETA $THETA $THETA $THETA $THETA $THETA $THETA $THETA $OMEGA $OMEGA $OMEGA $OMEGA $OMEGA $OMEGA $OMEGA $OMEGA $OMEGA $OMEGA $OMEGA $SIGMA
(1 FIX);1 - F1 (1 FIX);2 - F3 (0,0.03,0.5);3 - KA (1 FIX);4 - KF (10 FIX);5 - KR (0,0.5);6 - VMAX (0,1);7 - KM (168 FIX);8 - ALAG (0,0.002,0.05);9 - KE (LINEAR ELIM) (0.01 FIX);10 - KE1 (BINDING ELIM) (0,85,150);11 - V/BWT (0 FIX);1 - F1 (0.5);2 - F3 (0.5 FIX);3 - KA (0 FIX);4 - KF (0 FIX);5 - KR (0.5);6 - VMAX (0 FIX);7 - KM (0 FIX);8 - ALAG (0.5);9 - KE (LINEAR ELIM) (0.5);10 - KE1 (BINDING ELIM) (0.5);11 - V/BWT 0.5;PROPORTIONAL ERROR
$EST MAXEVAL=5000 PRINT=5 METH=0 POSTHOC NOABORT $COV
Final Model (Note that theta and omega numbering is not the same as in chapter text.) $PROB FIT OF SINGLE AND MULTIPLE DOSE DATA RUN=001 $INPUT ID TIME APOS DOSE WGT DV MDV AMT EVID LATR $DATA sdmd.data.final.3.csv IGNORE=C ;Data is in the Excel file
MODEL 2 NONMEM CONTROL CODE
1155
$SUBROUTINE ADVAN6 TRANS1 TOL=3 $MODEL COMP=(COMP1,DEFDOSE) COMP=(COMP2,DEFOBS) ;Kineticist: Stuart Friedrich ;Notes: Absorption parameters fixed based on fit of single dose data $PK SIT0=0.000002*EXP(ETA(1)) F1=THETA(1)/150000 PKA=THETA(2) KA=PKA*EXP(ETA(2)) PABLG=THETA(3) ABLG=PABLG*EXP(ETA(3)) IF (LATR .EQ. 0) THEN VMAX=0 ELSE VMAX=(THETA(4)/150000)*(10**LATR)*EXP(ETA(4)) ENDIF PKM=THETA(5)/150000 KM=PKM*EXP(ETA(5)) PKE=THETA(6) KE=PKE*EXP(ETA(6)) PKE1=THETA(7) KE1=PKE1*EXP(ETA(7)) PV=THETA(8)*WGT V=PV*EXP(ETA(8)) S2=V/150000 $ERROR IPRED=F W=0.001 IF(F.GT.0) W=F IRES=DV-IPRED IWRES=IRES/W Y=F+W*ERR(1)
1156
CHARACTERIZING NONLINEAR PHARMACOKINETICS
$DES IF (T .LT. ABLG) THEN VMX=0 ELSE VMX=VMAX ENDIF CBND=A(2)/V*SIT0/(1E-5+A(2)/V) IF (A(2) .EQ. 0) THEN FFRE=1 ELSE FFRE=A(2)/(A(2)+CBND*V) ENDIF DADT(1)=-KA*A(1) DADT(2)=KA*A(1)*FFRE-KE*A(2)-VMX*A(2)/V/(A(2)/V+KM)-KE1*CBND*V $THETA $THETA $THETA $THETA $THETA $THETA $THETA $THETA $OMEGA $OMEGA $OMEGA $OMEGA $OMEGA $OMEGA $OMEGA $OMEGA $SIGMA
(1 FIX);1 - F FIXED FOR MAB (0.0387 FIX);2 - KA (168 FIX);3 - ANTIBODY RESPONSE LAG TIME FROM T=0 ONLY (0,0.4);4 - VMAX OF ANTIBODY RELATED ELIMINATION (0,5);5 - KM OF ANTIBODY RELATED ELIMINATION (0.0005,0.0013,0.005);6 - KE (LINEAR ELIM) (0.002,0.06,0.2);7 - KE1 (ELIM DUE TO BINDING TO TARGET) (70,90,150);8 - V/BWT/F 0 FIX;1 - VAR OF INTIAL TARGET BINDING SITES AT T=0 0.0468 FIX;2 - VAR OF KA 0 FIX;3 - VAR OF ANTIBODY LAG TIME FROM T=0 ONLY 0 FIX;4 - VAR OF VMAX OF ANTIBODY RELATED ELIMINATION 0 FIX;5 - VAR OF KM OF ANTIBODY RELATED ELIMINATION 0.5;6 - VAR OF KE 0 FIX;7 - VAR OF KE1 0.5;8 - VAR OF V/BWT/F 0.5;PROPORTIONAL ERROR
$EST MAXEVAL=5000 PRINT=5 METH=0 POSTHOC $COV $TABLE ID TIME IPRED IWRES APOS LATR CBND VMX FFRE DOSE FILE=model_sdmd_2_t1.tb NOPRINT ONEHEADER $TABLE ID KA KM KE KE1 V DOSE FILE=model_sdmd_2_t2.tb NOPRINT ONEHEADER FIRSTONLY
CHAPTER 46
Development, Evaluation, and Applications of in Vitro/in Vivo Correlations: A Regulatory Perspective PATRICK J. MARROUM
46.1
INTRODUCTION
With the technological advances in the analytical tools and modeling software available to the pharmaceutical scientist, dissolution testing has been used more and more—both by the industry as well as regulatory agencies as a predictor of differences in bioavailability. When drug release from the formulation and its solubilization are the rate-limiting steps, it is possible to predict the resulting plasma concentration–time profile from its in vitro dissolution. In order to achieve this, there should be a well established relationship between the in vitro dissolution of the drug from the formulation and its in vivo bioavailability. In this chapter, the various requirements necessary for establishing an in vitro/in vivo correlation (IVIVC) both in terms of in vitro testing and in vivo modeling are presented. The regulatory requirements in terms of validation are discussed. Since the chapter is focused on the regulatory perspective on IVIVC, emphasis is on practical approaches used in drug development and evaluation and less so theoretical aspects of IVIVC. Finally, applications of IVIVC from both an industrial as well as a regulatory perspective are given in terms of obtaining in vivo bioavalability/ bioequivalence waivers and the setting of clinically meaningful dissolution specifications. In this regard, an example is presented in detail to illustrate the various steps in developing and validating an IVIVC. 46.2 46.2.1
LEVELS OF CORRELATION Level A Correlation
A level A correlation is a point-to-point relationship between in vitro dissolution and the in vivo input rate, as can be seen in Figure 46.1. Such relationships are Pharmacometrics: The Science of Quantitative Pharmacology Edited by Ene I. Ette and Paul J. Williams Copyright © 2007 John Wiley & Sons, Inc.
1157
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DEVELOPMENT, EVALUATION, AND APPLICATIONS OF IN VITRO/IN VIVO CORRELATIONS
120 100 80 % Drug 60 Absorbed 40 20 0 0
20
40 60 80 % Drug Dissolved
100
120
FIGURE 46.1 Level A correlation showing the point-to-point relationship between the fraction of drug absorbed and the fraction of drug dissolved.
12 10 8 MDT 6 In vivo 4 2 0
0
5
10
15
20
25
MDT in vitro
FIGURE 46.2 Level B correlation showing the relationship between the mean in vitro dissolution and the mean in vivo dissolution time.
usually linear, where the in vitro dissolution and the in vivo input curves can be superimposable. Even though nonlinear relationships are uncommon, they can be appropriate since they are useful in predicting the plasma concentration–time profile from in vitro dissolution data (1). 46.2.2
Level B Correlation
In a level B correlation, the mean in vitro dissolution time is compared to either the mean residence time or the mean in vivo dissolution time (Figure 46.2). A level B IVIVC uses the principles of statistical moment analysis. Even though a level B correlation uses all the in vitro and in vivo data, it is not considered a point-topoint correlation. It does not uniquely reflect the actual plasma concentration–time profile because a number of different in vivo profiles will produce similar mean residence times. For this reason, a level B correlation is of little value from a regulatory point of view. 46.2.3
Level C Correlation
A level C correlation establishes a relationship between a dissolution parameter such as the amount of drug dissolved at a certain time and a pharmacokinetic (PK)
CMAX (ng/ml)
DEVELOPMENT OF LEVEL A CORRELATION
500 450 400 350 300 250 200 150 100 50 0
1159
D6 D9
0
20
40
60
80
100
% Dissolved
FIGURE 46.3 Level C Correlation showing the relationship between the amount of drug dissolution at a certain time (for example 6 and 9 hours) and the peak plasma concentration.
parameter of interest such as AUC or Cmax (e.g., see Figure 46.3). Unfortunately, a level C IVIVC does not reflect the complete shape of the plasma concentration– time profile, which is a critical factor in defining the performance of the product. On the other hand, a multiple level C correlation relates one or several PK parameters to the amount of drug dissolved at several time points of the dissolution profile. In general, if one is able to establish a multiple level C correlation, then a level A correlation could be established also and is the preferred correlation to establish.
46.3 46.3.1
DEVELOPMENT OF LEVEL A CORRELATION In Vivo Considerations
Since a level A correlation is the most useful IVIVC both from a regulatory and formulation development point of view, only the development of a level A IVIVC is discussed in this chapter. The following points should be taken into consideration when developing an IVIVC: 1. Sine the PK properties of a drug tend to be somewhat different in animals when compared with humans, only human data is considered from a regulatory point of view. This does not preclude the use of animal data in assessing the performance of pilot formulations. 2. The in vivo PK studies should be large enough to characterize adequately the product under study. In general, the larger the variability in the performance of the formulation, the bigger the study should be (2). 3. The preferred study design is the crossover design since it reduces interstudy variability. Parallel studies as well as data obtained across several studies can be utilized to develop the IVIVC. 4. Inclusion of an immediate release reference in the studies facilitates data analysis since it allows one to better estimate the terminal rate constant for
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DEVELOPMENT, EVALUATION, AND APPLICATIONS OF IN VITRO/IN VIVO CORRELATIONS
each subject and also enables one to normalize the data to a common reference. The reference product could be an intravenous solution, an aqueous oral solution, or an immediate release product. 5. The studies are usually conducted under fasting conditions. However, if there are any tolerability concerns, the studies could be conducted under fed conditions. 46.3.2
Method
The IVIVC should usually be developed with two or more formulations (preferably three formulations) with different release rates. The process involves the following steps: •
•
•
•
Generate in vitro dissolution profiles using an appropriate dissolution methodology that can discriminate among the various formulations. Determine the plasma concentration–time profiles for the tested formulations. Obtain the absorption–time profile for these formulations (fraction of drug absorbed versus time). This can be achieved by the use of appropriate deconvolution techniques. Plot the in vivo absorption profile or the in vivo dissolution profile against the in vitro dissolution profile to determine whether a relationship exists (e.g., see Figure 46.4).
The method described above is called a two-stage procedure (3). An alternative approach is based on a convolution procedure that attempts to model the relationship between in vitro dissolution and plasma concentrations in a single step. The model predicted plasma concentrations are directly compared to the actual plasma concentrations obtained in the studies (4). 46.3.3
Deconvolution Methods
The most commonly used model-dependent deconvolution methods for estimating the apparent in vivo drug absorption following oral administration are the Wagner–Nelson (5) method and the Loo–Riegelman method (6). These methods depend on mass balance and the fraction of drug absorbed for a one-compartment model is expressed as t
( X a )t C + k ∫0 Ct dt Fa( t ) = = ∞ ( X∞ ) k Ct dt
∫
(46.1)
0
where Fa(t) is the fraction of absorbable drug at time t, C is the concentration of drug in the central compartment at time t, and k is the first-order elimination rate constant. For a two- or three-compartment model, the following equation describes the amount of drug absorbed at time t where Vc is the volume of the central compart-
DEVELOPMENT OF LEVEL A CORRELATION
40
60
% Absorbed
60
40
80
20
% Dissolved
100
80
100
120
1161
20
0
0 0
10 20 Time (h)
30
0
5 10 Time (h)
15
120 100 80 % Drug 60 Absorbed 40 20 0 0
25
50 75 % Drug Dissolved
100
125
FIGURE 46.4 Development of a level A correlation, where the fraction of drug dissolved at each time is plotted against the corresponding fraction of drug absorbed at the same time. The top left panel represents the dissolution profiles for three different formulations and the top right panel shows the corresponding percent absorbed plots calculated from the respective in vivo absorption profiles. The bottom panel is a synthesis of the information from the two top panels relating the percent absorbed in vivo to the percent dissolved in vitro.
ment, Ct is the plasma concentration at time t, and k12, k21, k13, k31 are the intercompartmental rate constants, and K10 is the elimination rate constant from the central compartment (7, 8). t t t ( Xa)t = CT + k12 exp− k21 •t ∫ Ct exp− k21 •t dt + k13 exp− k31 •t ∫ Ct exp− k31 •t + k10 ∫ Ct dt 0 0 0 Vc
(46.2)
46.3.4
Convolution-Based IVIVC
In order to be able to develop an IVIVC using a convolution-based approach, the following assumptions should hold true:
1162 • • •
DEVELOPMENT, EVALUATION, AND APPLICATIONS OF IN VITRO/IN VIVO CORRELATIONS
The in vitro release rate approximates the in vivo absorption rate. The PK properties of the drug are linear and time invariant. The pharmacokinetics of the drug administered as IV or immediate release (IR) or drug released from the extended release (ER) formulation are indistinguishable. In others words, once a drug molecule released from the IR or ER formulations is absorbed into the systemic circulation, it behaves just like an intravenously administered one.
In addition, plasma concentrations from an IV dose or from administration of IR formulation such as an oral solution or rapidly dissolving oral formulation are needed to estimate the unit impulse function. If the above conditions are met, then the plasma concentrations are expressed according to the following equation: t
(u) du C ( t ) = ∫ Cδ ( t − u) xrel,vitro ′ 0
(46.3)
where C(t) is the plasma concentration at time t, xrel,vitro is the cumulative amount of drug released in vitro, and x′ is the in vitro release rate obtained by taking the first derivative of x. xrel,vitro can be expressed as any mathematical function that best fits the dissolution profile. Alternatively, x can be estimated by linear interpolation of the mean in vitro dissolution profile. Cd is the unit impulse response, which is the plasma concentration–time course resulting from the instantaneous in vivo release of a unit amount of a drug (9). Cd can be obtained by fitting the IR or the IV plasma concentration–time profile to a polyexponential function. 46.4 EVALUATION OF THE PREDICTABILITY OF THE IVIVC Once an IVIVC has been established, a crucial determination of its applicability is its ability to predict the plasma concentration–time profile accurately and consistently. A relationship between the vitro dissolution and the in vivo absorption rate that is dependent on the release rate of the formulation, as can be seen in Figure 46.5, is usually an indication that a consistent relationship predictive of the in vivo performance does not exist. This is due to the fact that depending on the formulation used one can have a different amount of drug absorbed for the same amount of drug dissolved. On the other hand, a good and consistent relationship would always give you approximately the same slope irrespective of the formulation (whether the slow, fast, or medium formulation is used and whether or not all the data is pooled together). A good illustration of a valid linear level A correlation is presented in Figure 46.6, where the slope of the relationship is the same for each of the individual formulations or for the case where all the formulations are pooled together and treated as one. Since the IVIVC model is going to be used to predict the plasma concentration– time profile, it is imperative to assess the predictive performance of the model via the assessement of the prediction error of the model. Depending on the intended application of the IVIVC and the therapeutic index of the drug, evaluation of the internal or external predictability may be warranted. Evaluation of internal predictability is based on the data that was used to develop the IVIVC. Evaluation of
EVALUATION OF THE PREDICTABILITY OF THE IVIVC
1163
120 100
% Absorbed
80 60 40 20 0
0
20
40
60
80
100
120
% Dissolved
FIGURE 46.5 Poor IVIVC, where the slope of the relationship is dependent on the formulation. Each curve is for a different formulation. 100
100
80 % Absorbed
% Absorbed
80 60 40 20
%
0 0
-20
25
50
75
60 40 20 -20
100
%
0 0
25
% Dissolved
100
100
75
% Absorbed
% Absorbed
75
% Dissolved
100
50 25 0
50
75 50 25 0
0
25
50
75
% Dissolved
100
0 -25
25
50
75 100 125
% Dissolved
FIGURE 46.6 Predictive IVIVC independent of the release rate, where the slope of the relationship is independent of the formulation used. Each plot represents a different formulation.
external predictability involves additional data sets (see the next paragraph) that were not used in the initial development of the IVIVC. If the IVIVC for a non-narrow therapeutic index drug was developed with formulations with three or more release rates, the evaluation of the internal predictability would be sufficient to determine its appropriateness.
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DEVELOPMENT, EVALUATION, AND APPLICATIONS OF IN VITRO/IN VIVO CORRELATIONS
External predictability is warranted in the following situations: • • •
The drug is considered to be a narrow therapeutic index drug. The internal predictability criteria are not met. The IVIVC was developed with two formulations with different release rates.
The data set that is used in the external predictability should ideally be obtained from a formulation with a different release rate. However, it is acceptable to use formulations with similar release rates as those used in the development of the IVIVC. The following represent in decreasing order of preference the types of formulations that can be used to estimate the external prediction errors: • •
•
46.5
Formulations with different release rates A formulation that was made involving a specific manufacturing change (equipment, process, site, etc.) Similar formulations but different lots than the ones used in the IVIVC and the data from a different study than the one used in the development of the IVIVC
APPROACHES TO THE EVALUATION OF PREDICTABILITY
The most common approach to evaluating the predictability of an IVIVC is depicted in Figure 46.7. The procedure involves the conversion of the in vitro dissolution rate into in vivo absorption rate and then, by the use of convolution methods, a prediction of the plasma concentration–time profile. This is represented as Dissolution
➝
IVIC model
Absorption
➝
Plasma profile
PK parameters
The area under the concentration–time curve (AUC) and the peak plasma concentration (Cmax) from the predicted profiles are compared to those obtained from the observed profiles to calculate the percent prediction errors. The absolute prediction errors are calculated as follows:
|(Observed − Predicted)/Observed| × 100 These calculations should be done for each of the formulations used to develop the IVIVC. For internal predictability, an average absolute prediction error of less than 10% for both AUC and Cmax establishes the predictive ability of the IVIVC. In addition, the percent error for each formulation should not exceed 15%. If the above criteria are not met, the IVIVC is declared inconclusive and in this case the evaluation of the external predictability of the IVIVC is required. For external predictability, the percent prediction error should be less than 10% to declare the IVIVC acceptable. A percent prediction error between 10% and 20% is deemed inconclusive, requiring the further evaluation with additional data sets.
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APPLICATIONS OF IVIVC
Cumulative diss
Dissolution rate
120
120 Dissolution rate
80 60 40 20 5
10 15 20 Time (h)
16 14 12 10 8 6 4 2
100 80 60 40 20 0
25
0
10 20 Time (h)
1.2 100
0.8
120
0.6 0.4 0.2 0
30
IVIVC model
1
% absorbed
0
Cp with 1 unit of IV inj
Cp (ng/ml)
0
0
10
20
Time (hours)
30
Absorption rate
% Dissolved
100
100
80 60 40
Observed Predicted
20
80
0
0 20 40 60 80 100 120
60 % dissolved
40 20
0
0
10 20 Time (h)
30
Predicted plasma profiles
0
0
10 20 Time (h)
30
Absorption rate
FIGURE 46.7 Most common approach in evaluating the predictability of an IVIVC—the conversion of in vitro dissolution rate into in vivo absorption rate and the prediction of plasma concentration–time profile by the use of deconvolution methods (see Section 46.5).
A percent prediction error greater than 20% indicates that the IVIVC has a poor predictive ability and thus is considered not useful for any application. Note that the prediction should be made using mean data (mean dissolution profiles as well as population means for the PK parameters) for the following reason: individual dissolution data on the dosage unit that the individual subject was administered is not available. Thus, using average in vitro parameters and individual PK parameters is not appropriate. Since the purpose of the IVIVC is to predict the performance of yet untested formulations, no individual data will be available for such formulations and therefore a decision as to the appropriateness of the in vivo performance of the formulations is best determined on the average performance of these formulations.
46.6 46.6.1
APPLICATIONS OF IVIVC In Vivo Bioavailability Waivers
With a predictive IVIVC, in vitro dissolution would not only be a tool to assure the consistent performance of the formulation from lot to lot but would become a
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surrogate for the in vivo performance of the drug product. The ability to predict the plasma concentration–time profile from in vitro data will reduce the number of studies required to approve and maintain a drug product on the market, and therefore reduce the regulatory burden on the pharmaceutical industry. Once an IVIVC has been established, it is possible to waive the requirements for bioavailability/bioequivalence studies. For example, a biowaiver can be granted for a level 3 process change as defined in SUPAC MR, complete removal or replacement of non release controlling excipient, and level 3 changes in the release controlling excipients (10). If the IVIVC is developed with the highest strength, waivers for changes made with the lowest strengths are possible if these strengths are compositionally proportional or qualitatively the same, the in vitro dissolution profiles are similar, and all the strengths have the same release mechanism (11). However, an IVIVC cannot be used to gain the approval of (a) a new formulation with a different release mechanism, (b) a formulation with a dosage strength higher or lower than the doses that have been shown to be safe and effective in the clinical trials, (c) another sponsor’s oral controlled-release product even with the same release mechanism, and (d) a formulation change involving an excipient that will significantly affect drug absorption. The regultory criteria for granting biowaivers is outlined in the FDA guidance on this topic. Basically, the mean predicted Cmax and AUC from the respective in vitro dissolution profiles should differ from each other by no more than 20% (see Figure 46.8) (12). 46.6.2
Dissolution Specifications
Dissolution Profiles 120
Plasma conc., ng/ml
Cumulative % dissolved
The IVIVC allows one to shift the dissolution criteria from the in vitro side to the in vivo side. The plasma concentration–time profiles that correspond to the lots that are on the upper and lower limits of the dissolution specifications are predicted. Acceptable dissolution specification limits are limits that do not result in more than 20% difference in AUC and Cmax (usually ±10% of the target/bio formulation) (13).
100 80 60 40 20 0
0
5
10
15
20
25
30
Predicted plasma concentrations 120 100 80 60 40 20 0 0 5 10 15 20 25 30
Time (h)
Time (h)
FINAL DISSOLUTION SPECIFICATIONS: Set such that the predicted Cmax and AUC range NMT 20%
FIGURE 46.8 Regulatory criteria for granting a biowaiver using an IVIVC. The upper of the two profiles is for the test formulation, and the lower of the two profiles is for the reference formulation. NMT = no more than.
CASE STUDY
1167
Using the IVIVC to choose clinically meaningful specifications provides several advantages in that (a) it will minimize the release of lots that are different in their in vivo performance, thus optimizing the performance of the product; and (b) in certain cases it will allow wider dissolution specifications.
46.7
CASE STUDY
The following example is presented to illustrate the type of study and data analysis that was undertaken to develop a level A correlation for a once-a-day modified release formulation for metopolol. The in vivo performance of three modified release (MR) formulations with different release rates (fast, medium, and slow formulations) were tested in a fourway crossover single-dose study in healthy volunteers. A fourth other arm of the study included an IR solution of the drug. The individual plasma concentrations are presented in Table 46.1. The mean plasma concentration–time profile for each treatment is shown in Figure 46.9. TABLE 46.1 Individual Concentrations for the Four-Way Crossover Study Plasma Concentrations (ng/mL) Time (hours)
Pt1
Pt2
Pt3
Pt5
Pt6
Pt7
Pt9
ND ND 5.51 17.7 37.6 35.9 42.6 39 34.4 21.1 15.2 6.54 2.39