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HANDBOOK OF HEMORHEOLOGY AND HEMODYNAMICS

Biomedical and Health Research Volume 69 Recently published in this series: Vol. 68. Vol. 67. Vol. 66. Vol. 65. Vol. 64.

Vol. 63. Vol. 62.

Vol. 61. Vol. 60. Vol. 59. Vol. 58. Vol. 57. Vol. 56. Vol. 55. Vol. 54. Vol. 53. Vol. 52. Vol. 51. Vol. 50. Vol. 49. Vol. 48. Vol. 47. Vol. 46.

J.-F. Stoltz (Ed.), Mechanobiology: Cartilage and Chondrocyte – Volume 4 R.J. Schwartzman, Differential Diagnosis in Neurology H. Strasser (Ed.), Traditional Rating of Noise Versus Physiological Costs of Sound Exposures to the Hearing T. Silverstone, Eating Disorders and Obesity: How Drugs Can Help S. Eberhardt, C. Stoklossa and J.-M. Graf von der Schulenberg (Eds.), EUROMET 2004: The Influence of Economic Evaluation Studies on Health Care Decision-Making – A European Survey M. Parveen and S. Kumar (Eds.), Recent Trends in the Acetylcholinesterase System I.G. Farreras, C. Hannaway and V.A. Harden (Eds.), Mind, Brain, Body, and Behavior – Foundations of Neuroscience and Behavioral Research at the National Institutes of Health J.-F. Stoltz (Ed.), Mechanobiology: Cartilage and Chondrocyte – Volume 3 J.-M. Graf von der Schulenburg and M. Blanke (Eds.), Rationing of Medical Services in Europe: An Empirical Study – A European Survey M. Wolman and R. Manor, Doctors’ Errors and Mistakes of Medicine: Must Health Care Deteriorate? S. Holm and M. Jonas (Eds.), Engaging the World: The Use of Empirical Research in Bioethics and the Regulation of Biotechnology A. Nosikov and C. Gudex (Eds.), EUROHIS: Developing Common Instruments for Health Surveys P. Chauvin and the Europromed Working Group (Eds.), Prevention and Health Promotion for the Excluded and the Destitute in Europe J. Matsoukas and T. Mavromoustakos (Eds.), Drug Discovery and Design: Medical Aspects I.M. Shapiro, B.D. Boyan and H.C. Anderson (Eds.), The Growth Plate C. Huttin (Ed.), Patient Charges and Decision Making Behaviours of Consumers and Physicians J.-F. Stoltz (Ed.), Mechanobiology: Cartilage and Chondrocyte, Vol. 2 G. Lebeer (Ed.), Ethical Function in Hospital Ethics Committees R. Busse, M. Wismar and P.C. Berman (Eds.), The European Union and Health Services T. Reilly (Ed.), Musculoskeletal Disorders in Health-Related Occupations H. ten Have and R. Janssens (Eds.), Palliative Care in Europe – Concepts and Policies H. Aldskogius and J. Fraher (Eds.), Glial Interfaces in the Nervous System – Role in Repair and Plasticity I. Philp (Ed.), Family Care of Older People in Europe

ISSN 0929-6743

Handbook of Hemorheology and Hemodynamics

Edited by

Oguz K. Baskurt, M.D., Ph.D. Professor and Chairman, Department of Physiology, Akdeniz University Faculty of Medicine, Antalya, Turkey

Max R. Hardeman, Ph.D. Clinical Biochemist, Laboratory for Clinical Hemorheology, Department of Physiology, Academic Medical Center, Amsterdam, The Netherlands

Michael W. Rampling, Ph.D. Honorary Senior Lecturer, Department of Bioengineering, Imperial College, South Kensington, London, UK

and

Herbert J. Meiselman, Sc.D. Professor and Vice-Chairman, Department of Physiology and Biophysics, University of Southern California, Keck School of Medicine, Los Angeles, CA, USA

Amsterdam • Berlin • Oxford • Tokyo • Washington, DC

© 2007 The authors. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without prior written permission from the publisher. ISBN 978-1-58603-771-0 Library of Congress Control Number: 2007931478 Publisher IOS Press Nieuwe Hemweg 6B 1013 BG Amsterdam Netherlands fax: +31 20 687 0019 e-mail: [email protected] Distributor in the UK and Ireland Gazelle Books Services Ltd. White Cross Mills Hightown Lancaster LA1 4XS United Kingdom fax: +44 1524 63232 e-mail: [email protected]

Distributor in the USA and Canada IOS Press, Inc. 4502 Rachael Manor Drive Fairfax, VA 22032 USA fax: +1 703 323 3668 e-mail: [email protected]

LEGAL NOTICE The publisher is not responsible for the use which might be made of the following information. PRINTED IN THE NETHERLANDS

Handbook of Hemorheology and Hemodynamics O.K. Baskurt et al. (Eds.) IOS Press, 2007 © 2007 The authors. All rights reserved.

v

Foreword The appearance of this Handbook is a timely event, as it is 20 years since the publication of the “Handbook of Bioengineering” [1] that dealt mainly with basic aspects of hemodynamics and hemorheology. Also, in the 1980s and 1990s a number of books were published that focused on clinical aspects of blood rheology [2–5]. In selecting topics for the present handbook the editors have attempted to provide a general overview of both basic science and clinical hemorheology and hemodynamics. Hemorheology and hemodynamics are closely related, the former dealing with all aspects of the flow and interactions of the individual blood cells mostly studied in vitro, the latter with the in vivo relationships among vessel architecture, driving pressure, flow rate and shear stress. The linkage between the in vitro and in vivo research described in the book will be of interest to both basic science and clinical investigators. With respect to hemorheology, the new book successfully updates developments and advances in the flow properties of human blood cells (microrheology). Furthermore, in the chapters on cell mechanics, these flow properties are related to events occurring at the level of the bonds between the interacting corpuscles (platelets and white cells as well as red cells), and between the corpuscles and the vessel wall (molecular rheology). A welcome feature of the handbook is that it includes a chapter on comparative hemorheology, showing that the rheological properties of red cells vary widely among the animal species, thus shedding light on the process of adaptation to a specific environment or lifestyle, and a chapter on neonatal and fetal blood rheology showing the considerable adaptation processes in play at birth and in infancy and childhood. Also dealt with in some depth are the effects of diseases on the mechanical and adhesive properties of red cells and the underlying molecular mechanisms, particularly those found in malaria. A related subject, the damage sustained by red cells due to flow-induced mechanical trauma, is also presented. With respect to hemodynamics, it is evident in the chapters of section III of the handbook that the field has advanced significantly in the last 30 years, particularly with respect to our understanding of microcirculatory blood flow using novel experimental techniques, the latter being the subject of a separate chapter. The handbook closes with chapters on clinical states associated with abnormal blood rheology, including a chapter on the yet controversial subject of rheological therapy. The editors of the handbook have each been active in the fields of bio- and hemorheology for many years, and have published extensively. They have successfully achieved their objective to publish a well-written and well-edited handbook that will be valuable for researchers and students in the field. Shu Chien, MD, PhD Harry L. Goldsmith, PhD [1] R. Skalak and S. Chien, Eds., Handbook of Bioengineering, McGraw-Hill, New York, 1987. [2] S. Chien, J. Dormandy, E. Ernst and A. Matrai, Eds., Clinical Hemorheology, Martinus Nijhoff Publ., Dordrecht, 1987.

vi

[3] G.D.O. Lowe, Ed. Clinical Blood Rheology, CRC Press, Boca Raton, FL, 1988. [4] A.M. Ehrly, Therapeutic Hemorheology, Springer-Verlag, New York, 1991. [5] J.F. Stoltz, M. Singh and P. Riha, Hemorheology in Practice, IOS Press, Amsterdam, 1999.

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Preface The fields of hemorheology and hemodynamics are active and expanding areas of research, yet no combined reviews or “handbooks” have been published within the past 20 years. It was thus felt appropriate to attempt such a task. An outlined proposal was submitted to IOS Press and, after cordial telephone and email exchanges, was approved for publication. In planning for this book, the editors realized that it would be impossible to cover in detail the entire field of hemorheology and hemodynamics, and hence that it would be necessary to limit its scope. It was therefore decided to primarily focus on the macro-and micro rheological behavior of blood and its formed elements, on interactions between the formed elements and blood vessel walls, and on the microvascular aspects of hemodynamics; areas such as cardiac hemodynamics and theory for pulsatile flow in large vessels were omitted. Since many aspects of hemorheology and hemodynamics can be affected by disease or a wide variety of clinical states, these areas were deemed relevant as were the hyperviscosity syndromes and therapy for disturbed blood rheology. In addition, discussions of methods in hemorheology and hemodynamics were included to provide a practical framework for studies in these areas. In the search for authors needed to prepare each section, the global nature of the fields was recognized: Australia, Austria, France, Germany, Hungary, Italy, Netherlands, Singapore, South Korea, Turkey, United Kingdom and the United States of America are represented. We sincerely thank every contributor for writing their section, for allowing editorial corrections/modifications, and for accepting “helpful” criticism without threatening the editors with hostile actions. This book would not have been possible without their contributions. We also wish to thank Ms. Rosalinda B. Wenby for her valuable editorial assistance. We hope that we have been successful in reaching our objectives for this book, and that it will be of value to researchers and clinical scientists engaged in hemorheology and hemodynamic studies. In particular, we trust that the book will serve to foster greater cooperative efforts between these fields. It is notable that the field of hemorheology was, in large part, prompted by direct observations of RBC aggregation, “blood sludging”, white cell effects, and disturbed flow in the human retinal microcirculation. These observations have lead to in vitro studies of blood viscosity, cell rheology and aggregation, and blood flow in small tubes and in more complex geometries; studies aimed at understanding relations between in vitro and in vivo hemorheologic and hemodynamic phenomenon are also ongoing. Further collaborative efforts will be of mutual value, and will hopefully lead to improved health for normal individuals and for those with various clinical conditions. O.K. Baskurt M.R. Hardeman M.W. Rampling H.J. Meiselman

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Contents Foreword Shu Chien and Harry L. Goldsmith Preface O.K. Baskurt, M.R. Hardeman, M.W. Rampling and H.J. Meiselman

v vii

I. History of Hemorheology I. History of Hemorheology Michael W. Rampling

3

II. Hemorheology 1. Basic Aspects of Hemorheology Giles R. Cokelet and Herbert J. Meiselman

21

2. Compositional Properties of Blood Michael W. Rampling

34

3. Blood Rheology a. Macro- and Micro-Rheological Properties of Blood Giles R. Cokelet and Herbert J. Meiselman b. Viscoelasticity of Human Blood George B. Thurston and Nancy M. Henderson 4. Cell Mechanics a. Mechanical and Adhesive Properties of Healthy and Diseased Red Blood Cells Brian M. Cooke and Chwee T. Lim

45 72

91

b. Red Blood Cell Aggregation Björn Neu and Herbert J. Meiselman

114

c. Mechanical Properties of Leukocytes and Their Effects on the Circulation Roger Tran-Son-Tay and Gerard B. Nash

137

d. Adhesion of Circulating Leukocytes and Platelets to the Vessel Wall Susan L. Cranmer and Gerard B. Nash

153

5. Pathophysiology of Blood Rheology a. Mechanisms of Blood Rheology Alterations Oguz K. Baskurt

170

b. Hemorheology of the Fetus and Neonate Otwin Linderkamp

191

c. Mechanical Trauma to Blood Marina V. Kameneva and James F. Antaki

206

x

d. Hemorheological Considerations in Stored Blood Transfusion James P. Isbister

228

6. Methods in Hemorheology Max R. Hardeman, Peter.T. Goedhart and Sehyun Shin

242

7. Comparative Hemorheology Ursula Windberger and Oguz K. Baskurt

267

III. Hemodynamics 1. Basic Principles of Hemodynamics Timothy W. Secomb and Axel R. Pries

289

2. Blood Rheology Aspects of the Microcirculation Herbert H. Lipowsky

307

3. In Vivo Hemorheology Oguz K. Baskurt and Herbert J. Meiselman

322

4. Endothelium and Hemorheology Tommaso Gori and Sandro Forconi

339

5. Methods in Hemodynamics Sehyun Shin, Hideyuki Niimi, Max R. Hardeman and Peter.T. Goedhart

351

IV. Clinical Aspects of Hemorheology 1. Hyperviscosity: Clinical Disorders James P. Isbister

371

2. Clinical Significance of Hemorheological Alterations Kalman Toth, Gabor Kesmarky and Tamas Alexy

392

3. Treatment in Clinical Hemorheology: A Current Overview Michel R. Boisseau, Katalin Koltai, Zsolt Pecsvarady and Kalman Toth

433

A Note on the Editors

445

Subject Index

447

Author Index

455

I. History of Hemorheology


Handbook of Hemorheology and Hemodynamics O.K. Baskurt et al. (Eds.) IOS Press, 2007 © 2007 IOS Press. All rights reserved.

3

History of Hemorheology Michael W. RAMPLING1 School of Medicine, Imperial College, South Kensington, London SW7 2AZ, UK.

Introduction If the rate of appearance of publications in the field can be taken as a criterion, hemorheology can be considered as coming of age in the fairly recent past - perhaps forty or so years ago. This relative lateness is due largely to the previous lack of measuring equipment with the required sophistication; a particular problem being the complex nature of blood viscosity and the need for adequate viscometers capable of measuring it. Nevertheless the ease of availability of blood, its dramatic color and its obvious connection to well being have made it a subject of study since ancient times. What is more, many of those ancient studies were of physical properties of blood that have direct hemorheological relevance. So it could be said that hemorheology is one of the oldest of clinical research areas.

1. Ancient history It quickly becomes clear to anyone who has left a tube of blood undisturbed for a significant period that it will separate out into different phases. The very fact that, in these circumstances, movement of blood components is taking place makes it a hemorheological phenomenon, and the phenomenon was well known to the ancients, and probably the first hemorheological phenomenon to be studied. They found that such settled blood generally formed four layers, so the idea developed that blood was composed of four fluids. The top layer, the first of the fluids, became known as the yellow bile or cholera, and is now known to be serum, i.e. the fluid that separates from the blood clot. The next layer was called the mucus or phlegma. This is now referred to as the buffy coat of leukocytes, platelets and fibrin that settles on top of the third layer called the blood or sanguinis. The third layer is actually the packed red cells. Finally, at the bottom is the last layer, i.e. the black bile or melancholia; this is really the red cells that have failed to oxygenate after the blood collection [1]. This separation, and the fact that the rate of separation of the layers is increased in association with disease, was known to the ancients Greeks and they developed it into a diagnostic tool dependent on the relative proportions of the four layers. It would have been a small step for the ancients to have moved from the ideas presented above to an interest in the rate at which the blood layers settle out. Strangely, it was many centuries before proper study of this took place. The person to give it comprehensive investigation and to allow its development into a proper 1

Corresponding author: School of Medicine, Imperial College, South Kensington, London SW7 2AZ, UK; E mail: [email protected]

4

M.W. Rampling / History of Hemorheology

quantitative diagnostic tool was Robin Fåhraeus in the early twentieth century. It became known as the Erythrocyte Sedimentation Rate Test, and we shall return to it later in this chapter.

2. Middle Ages Following on from the idea of the humours in the blood, and their disturbance in disease, it was not a great conceptual jump to the idea that the physician might do something about the imbalance with the aim of alleviating the pathology. The method for doing that was blood letting, which was to become a common therapeutic tool. The bleeding could be induced grossly by lancing a major vessel or, perhaps more subtly, by leaches. As might be expected the efficacy of the treatment was not high and, indeed, often resulted in death of patients already weakened by the disease from which they were suffering. Nevertheless the concept in principle is very close to that of a hemorheological technique that was developed in the twentieth century, i.e. hemodilution, to reduce blood viscosity and assist blood flow [2].

3. Age of Reason The discovery by William Harvey, in the early seventeenth century, of the circulation of the blood was one of the greatest physiological discoveries of all time (Figure 1). Prior to Harvey’s insight, it had been believed that blood ebbed and flowed in the veins and arteries spreading “vital spirit” to all the tissues, and the massive flow from the heart was considered to be involved in the replenishment of the blood that was consumed in the process.[3]. Harvey’s opus magnus, usually referred to as “De Motu Cordis”, was published in 1628 [4]. His insight was based primarily on the observations of the one-way valves in the veins and in and around the heart, the different pressures in the veins and arteries and the effects of ligations on blood flow. They enabled him to hypothesize the idea of blood circulating from the heart through the arteries and returning to the heart via the veins. This was a huge physiological insight and is obviously of great hemorheological significance as, for the first time, the importance of flowing blood became exposed and so the rheological properties of blood became relevant. Harvey’s discovery stands as one of the pivotal points in the history of hemorheology. It is also interesting that, after the discovery, the raison d’etre of blood letting gradually changed from being to readjust the humours to being, rather, to thin the blood and so make blood flow easier. Harvey’s insight can be seen to have been especially acute as he did not have access to microscopy and so, though he could see the arteries and the veins, he could not see the microcirculatory vessels. However, his evidence for a complete circulatory system was so compelling that he assumed that there were “anastomosis of these vessels or pores in the flesh and solid parts permeable to the blood” [5]. It was left to Malpighi in 1660 to be the first to see the microcirculation [6] and to prove microscopically the connection of the network of small vessels between the arterial and venous sides of the circulation.


5

Figure 1. William Harvey (National Portrait Gallery, London)

Figure 2. Antoni van Leeuvenhoek (Whipple Museum of the History of Science, University of Cambridge)

A major problem at this time was that the true nature of blood was still very poorly understood. It was largely looked on as a simple liquid, but that view began to change after Malpighi had seen the red cells, even though he mistook them for fat globules. It was left to Anthoni van Leeuwenhoek (Figure 2) to give the first accurate description of the red cell in 1674 [7]. He was able to see them flowing in the microcirculation and

6


so was the first to confirm Harvey’s postulate about the circulation. He observed the deformation of red cells as they negotiated the capillaries, and the extent of their deformability seems to have amazed him, as he commented in one of his letters that they could elongate “up to three times their original dimensions without break-up”, but he realized that such deformability was necessary for them to negotiate the minute blood capillaries. He also observed red cell aggregation and noticed that it increased during pregnancy and during infections, and postulated that red cells can lose their deformability in certain circumstances and thus cause disease. Perhaps most astonishing of all was his estimate of the diameter of the red cell which he put at 1/3000inch – or 8.5μm [8]. He was clearly one of the founders of the field of hemorheology, and a scientific giant. From now on it was realized that blood had a particulate nature, i.e. it was composed in part of “corpuscles”, and to be complex and an interesting fluid to study. For example, red cell aggregation was investigated by a number to physicians, John Hunter in particular. In the mid 1700’s he reported [9] that “in all inflammatory dispositions …. the red globules become less uniformly dispersed, and their attraction to one another becomes stronger”. So not only did he see the aggregation but he realized that the degree of aggregation increased in association with many pathologies. This is, of course, an area of particular interest to hemorheologists today. However still the white blood cells had not been seen, and their discovery fell to one of Hunter’s students, William Hewson, in 1770 [10]. He also investigated the lymphatics and explained the relationship between the lymph glands and the lymphocytes.

4. Jean Leonard Marie Poiseuille After Harvey’s discovery of the circulation of the blood, physicians began to consider what were the factors that determined the flow of blood in and to different organs. Some invoked nervous influences, some the inherent properties of the walls of the blood vessels and some the flow properties of the blood. A particular advocate of the last, after many microscopic investigations of blood flow in various animals, was Jean Leonard Marie Poiseuille (Figure 3). In order to get clearer views of the factors determining flow in the circulation he simplified the problem and investigated flow in glass tubes. He found that blood was too difficult to use, or to get consistent results from, so ended by performing most of his studies on simpler liquids such as water and alcohol. He published the results of these painstaking studies in the 1840’s [e.g. 11]. Subsequently, Hagenbach took Poiseuille’s data and derived the relationship between the factors determining the rate of flow of a simple liquid through cylindrical tubes that is usually referred to as Poiseuille’s Law [12]. It is summarized in the equation:F/t = (Spr4)/(8Kl) where F/t is the fluid flow rate, p is the driving pressure across the tube, r and l are respectively the radius and length of the tube and K is the viscosity of the fluid. It is of interest that Hagen independently found the same law [13], so the law is sometimes given the name Hagen-Poiseuille. We now know that this equation only applies strictly to Newtonian liquids, but the principles are applicable even to non-Newtonian liquids such as blood. It is also interesting that the difficulties that Poiseuille had encountered in using blood in his ex vivo studies were partly due to its non-Newtonianism, and this in turn means that, though he did not realize it, he was probably the first to react to the


7

non-Newtonian character of blood. Poiseuille’s breakthrough investigations may be considered the real beginning of the science of hemorheology.

Figure 3. Jean Leonard Marie Poiseuille as portrayed on the Poiseuille Medal. The Medal is awarded by the International Society of Biorheology for services to Biorheology. (Prof Shu Chien)

5. Robin Fåhraeus and John Harkness Earlier in this chapter mention was made of the settling of blood and its relevance to the Ancients in showing up the four humours of the blood. There was a renewed interest in the phenomenon once red cell aggregation had been seen under the microscope and it became realized that there was an association between increased aggregation and the rate of settling. Hewson for example showed that defibrinated blood aggregated less and its settling rate was reduced compared to whole blood. It was also known that the blood even of healthy horses aggregated to a remarkable degree and the settling rate was very high. However, a clear view of what was happening here awaited the appearance of Robin Fåhraeus [1] (Figure 4). His interest in the area would appear to have been stimulated by his observations, while working at the Karolinska Sjukhuset in Stockholm in 1917, on the blood from pregnant women, which was already known to settle more quickly than that from healthy non-pregnant women [14]. His publications started soon after [1] and it was his detailed study of the factors affecting the settling of the blood and its variation in sickness and in health that led to the development of the Erythrocyte Sedimentation Rate (ESR) Test as a valuable quantitative diagnostic tool. As old as this test is, and as simple as it is, it is still one of the most common diagnostic tests done world wide. What is more, newer

8


methodologies for measuring the ESR are still being developed and papers on the topic have never stopped being published.

Figure 4. Shu Chien before a portrait of Robin Fåhraeus. Professor Chien was the first recipient of the Fåhraeus Medal that is awarded by the European Society for Clinical Haemorheology for services to Clinical Hemorheology. (Prof Shu Chien)

Fåhraeus made many other contributions to hemorheology, but an area of especial importance resulted from his studies of blood flow in very small diameter glass tubes. These seem to have been stimulated by observations of in vivo flow in microvessels. One observation that particularly impressed him was the flow of blood through the microvessels of the mesentery of the horse. Here, the red cells do not separate at all but flow as a continuous string held together by the forces of aggregation that produce rouleaux, constraining them to flow along the centre of the vessel. This was very different from the granular nature of the flow of human blood that he observed in the nail bed capillaries.[1]. In his subsequent studies he was able to show that the apparent viscosity of blood decreases as the diameter of the tube through which it flows decreases until the diameter became significantly smaller that the red cell when the apparent viscosity starts to rise again, sharply. This is now known as the FåhraeusLindqvist Effect [15]. It is due to the finite size of the cellular compartment of the blood and it becomes significant only when the diameter of the tube through which the blood is flowing is of a similar order of size to that of the cells. In particular it is caused by the “cell reduced “layer near the wall of the tube. The ESR Test that Fåhraeus had done so much to pioneer has the great advantage of simplicity and cheapness – and this is why it is still so widely used, especially in the


9

third world. Nevertheless, it has significant problems, e.g. the necessity for hematocrit correction and the frequency of anomalous results. However, it had been known for a considerable period that the elevated ESR is associated with increase in the degree of rouleaux formation in the blood, and that this in turn was associated with elevated levels of large plasma proteins, especially fibrinogen [1]. So, because the main determinants of plasma viscosity are the large plasma proteins, it was to be expected that there should be some association between plasma viscosity and pathology. Many studies had been made of the relationship of plasma viscosity to individual diseases, but one of the first papers comprehensively to investigate its potential as an alternative to the ESR as a non-specific diagnostic tool to follow the progress of pathology was produced in 1946 by Harkness et al [16]. However the adoption of this new concept in clinical practice was hampered by the lack of a decent plasma viscometer for the clinical setting. This was finally solved when Harkness described such a capillary viscometer in 1963 [17]. It was to go into commercial production and there was a period, especially in England, when plasma viscosity was regularly used as an alternative to the ESR. However, with time more specific biochemical tests for the large proteins that influence plasma viscosity (e.g. fibrinogen) have superseded it and the commercial production of the Harkness Viscometer has ceased. Now, in the developed world at least, plasma viscosity is only used for rare clinical cases, e.g. macroglobulinemia.

6. The Other Pioneers of the Earlier Twentieth Century As was discussed above, Poiseuille was frustrated in his studies of fluid flow in cylindrical tubes. He had wanted to investigate blood but found it too difficult and so produced his ground-breaking research results using simple liquids such as water and alcohol. Nevertheless work on blood did continue, and in the early part of the twentieth century the dependency of the apparent viscosity of blood on the diameter of the tube through which it flows and on its rate of flow was found, as was the fact that it possessed shear-thinning properties [15]. What had been discovered was that blood possessed non-Newtonian characteristics, and that was a property that was going to bedevil the field for a considerable time to come. The problem was that the measurement techniques that were then available disallowed real comparisons of viscometric data to be made between laboratories. In view of these complexities a remarkably brave and forward-thinking study was performed by Whittaker and Winton and published in 1933 [18]. They perfused the isolated hind leg of a dog with blood of various hematocrits, and at various driving pressures and compared the flow rates with those of physiological saline to allow the apparent in vivo viscosity of flowing blood to be calculated. They were able to show that the apparent viscosity was considerably less than that measured on a sample of the same blood by an Oswald-type viscometer. Their data also indicated that there was a logarithmic relationship between viscosity and hematocrit and that, at the driving pressures and flow rates used, the viscosity was almost constant, i.e. the blood behaved as a Newtonian liquid in this in vivo experiment. Although it is now possible to criticize aspects of the investigation, it was nevertheless a major study of in vivo hemorheology. It is a great shame that such studies have been few and far between in the history of hemorheology. Only in the recent past have significant studies in this direction started to appear in anything like adequate numbers. We shall return to this area later.

10


The major development that stimulated the interest in hemorheology that was to burgeon in the second half of the twentieth century was that of viscometers capable of measuring non-Newtonian viscosity in a standardized manner that was reproducible in laboratories across the world. These were originally developed for use on complex liquids of commercial interest, e.g. paints, molten chocolate etc. One of the most notable was the Weissenberg Rheogoniometer [19]. AL Copley, amongst others, made a huge contribution to hemorheology by modifying the instrument to allow its use on blood [20]. For the first time, data from separate laboratories became truly capable of comparison. There followed an inundation of physicists and engineers studying the basic viscometric properties of blood – great names like GW Scott Blair, RL Swank, RE Wells, EW Merrill, but also RL Whitmore who brought much of this work together in a hugely influential book in 1968 [21]. Leopold Dintenfass was another of these early ‘hemo-viscometrists’. He developed his own advanced viscometer, and with it made a huge range of seminal studies in the area. However, perhaps his greatest contribution to the field was his first book [22]. This was much more clinically based than Whitmore’s had been and covered not only basic rheology but also investigations of hemorheological changes associated with a vast range of clinical conditions – the 66 pages of references bear testimony to that fact! A.L. Copley has already been mentioned as one of the investigators who helped bring modern viscometry into hemorheology and he did considerable work of merit afterwards. Even before that, since the early 1940’s, he had been very active in the field, using a rolling ball viscometer of his own design, and had made some of the earliest studies of blood viscosity, yield stress, thixotropy etc [23]. However, perhaps his major contribution to the subject of hemorheology was his tireless promotion of it in everyway possible. He was involved in the establishment of the two journals that promote all things hemorheological, i.e. Biorheology, first published in 1962, and Clinical Hemorheology, first published in 1980. He was also heavily involved in the organizing of the First International Conference on Hemorheology which was held in Reykjavik in 1966. This led ultimately to the establishment of the International Society of Biorheology and its first International Congress was held in Lyon in 1972. These Congresses have since recurred every 3 years. Eventually it was realized that there was a need for a predominantly clinical forum for hemorheologists and so the biennial European Conference on Clinical Hemorheology was born in Nancy in 1979. The final stage in the maturation of the subject was the founding of the International Society of Clinical Hemorheology in 1993 at the First International Conference on Clinical Hemorheology held in Vienna. Such events, together with the arrival of relatively easy to operate commercial viscometers capable of dealing with the vagaries of blood viscosity e.g. from the Wells Brookfield and Contraves companies, were responsible for sucking considerable numbers of clinicians into hemorheological research. Thus the glory days of clinical hemorheology began in the late 1960’s and ran through to the late 1980’s. It was a time of rapid accumulation of research data, facilitated by considerable drug company interest and, therefore, funding. At that time there were considerable numbers of investigators in the US, but they tended to concentrate on the more theoretical aspects of the subject, and groups were active elsewhere, especially in Japan, but the centre of clinical hemorheology was Europe. There were very active groups in Britain and Italy but the majority of the groups were in France and Germany. The number in France was sufficient that for a time they actually had 3 separate societies of clinical hemorheology! The result of all of this activity was the discovery of viscometric abnormalities in a vast array of clinical conditions [24].


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7. The Glory Days of Hemorheology In the early days viscosity of blood was the main area of study. However, eventually workers in the field began to be interested in the mechanical properties of the red cell, and its deformability in particular. This was triggered by the realization that blood viscosity could be affected by stiffened cells [25], and that the rate of gaseous exchange by the erythrocyte depends on its flexibility and the stirring of its internal hemoglobin as it negotiates the capillaries [26]. The interest was also stimulated by the considerable study of sickle cell anemia at the time – this being a condition where erythrocytes have difficulty negotiating the microcirculation because they become stiffened when deoxygenated [27]. Sickle cell anemia is, of course, an extreme case where reduced red cell deformability becomes a hemodynamic embarrassment; nevertheless it was considered that less extreme conditions could still have circulatory consequences. The result was an interest in the study of erythrocyte deformability in its own right. To begin with, the methodologies for measuring red cell deformability were complex and really only found in high powered research laboratories. Such techniques were based on glass micropipettes, of similar dimensions to the red cell, into which whole red cells were sucked and the aspiration pressure used as an index of the cellular deformability. Alternatively, using even smaller pipettes, part of the red cell membrane was sucked into the pipette and from its radius of curvature and the aspiration pressure an estimate of the detailed mechanical properties of the membrane could be made [28]. These methods were too complex for the routine clinical laboratory. However, simpler methodologies gradually began to be developed, but the most common was based on the use of micropore filters, which were produced by the Nucleopore Company. These filters had pores of a similar size to red cells or small blood capillaries, and simple filtration methods gave indices of erythrocyte deformability [29]. However there was still a problem; the methods were generally manufactured in each laboratory and interlaboratory comparison not adequate. An important breakthrough took place in 1985 when Dormandy et al described the St George’s Filtrometer [30]. This simplified the methodology and became a commercially available product, which appeared subsequently in many laboratories and expanded the rate of presentation of results, in a conforming way. Nevertheless there were still many problems, not least the facts that the presence of white cells tended to make the data inconclusive and the anticoagulant used could affect results. Dormandy was largely responsible for organising frequent workshops, in London, to try to standardize methodology. Ultimately this all led to standardization being imposed from outside in the shape of the International Committee for Standardization in Haematology [31]. Interestingly the influence of the white cell on microfiltration studies was at first seen only as an inconvenience and various methods, both practical and theoretical were developed to try to remove their influence on micropore filtration measurements of erythrocyte deformability. However, eventually the realization that the leukocytes could affect micropore flow reawakened interest in their influence on flow in the microcirculation in vivo. This was remarkable since, as early as the 1930’s, reports were appearing showing that the white cells are stiffer than the red and are capable of plugging microvessel flow [32]. The effects of the leukocytes on micropore flow sparked a burst of activity looking at the mechanics of the white cell using both micropipette and micropore technology. It is now clear that the white cells have deformabilities some three orders of magnitude less that the red and so, even though they are present in healthy blood only in concentrations of about 1 to every 1000 red

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cells, nevertheless they are roughly equally influential on microvascular flow to the red mass [33, 34]. Once the white cell had become an object of particular interest to hemorheologists, factors of rheological significance, other than its deformability, began to be appear, in particular its ability to adhere to the wall of blood vessels. This is a continuing active area of research [35]. Later it was discovered that the red cell too could under certain circumstance adhere to the walls of blood vessels. An example of this is found in patients suffering from Diabetes mellitus. This is an area especially studied by JeanLuc Wautier, who has shown that the accumulation of RAGE (receptors for advanced glycation end-products) in the membranes of the red cells is probably responsible. Such adhesion can be expected to influence hemodynamics in a deleterious way [36]. One of the factors that pushed the development of clinical hemorheology in the period between the 60’s to 80’s was the commercial development of a number of hemorheologically based pharmaceutical products. Names such as dextran, pentoxyfylline (Trental) and Troxerutine immediately spring to mind. These products were all reputed to have beneficial hemorheological effects of one sort or another. Appropriate preparations of dextran were used to induce hemodilution and improve blood fluidity, and were also thought to reduce red cell aggregation [24]. Pentoxyphylline was reputed to improve red cell [37] and white cell mechanics [38] and Troxerutine to reduce red cell aggregation [39]. There were also more physical techniques appearing in the field with potential hemorheological effects, such as hemodilution, that needed investigation [2, 40. 41]. All of this brought considerable funding into the field to support, at least in part, its research groups, finance its scientific meetings and to fund clinical trials. It had a huge catalytic effect on activity in the area. It is unfortunate that much of this support has now evaporated. In the early days of the modern developments in hemorheology the studies were devoted entirely to the adult, but it was inevitable that the neonate would eventually also be studied and this has shown that fascinating differences exist between the neonate and the adult [42, 43]. Furthermore, once in utero blood sampling techniques had been perfected, the hemorheology of the fetus became the object of study and showed even more extraordinary differences from the adult [44], and has influenced attitudes to in utero blood transfusion. Furthermore, as studies of clinical conditions unique to the neonate expand so it is increasingly being realized that a number of them have hemorheological associations, e.g. neonatal hyperviscosity syndrome [45]. It is interesting, when one looks at science from an historical point of view, to see how topics come into and out of fashion. The study of rouleaux formation was high on the agenda in the 70’s but then interest waned. However, there has been a considerable resurgence of interest in the fairly recent past [46]. This seems to have been triggered by the advent of a new hypothesis to explain the cause of this unusually weak form of cellular aggregation. Previously, the cross-bridging hypothesis had been generally accepted but recently a very different one based on the depletion layer idea has been proposed [47]. In an effort to decide between the hypotheses investigators have been using enzymes to alter the surface properties of the red cell and then to study the effects on the aggregation phenomenon [48, 49]. The conflict between the hypotheses is still on-going. However, the interest in rouleaux formation has spawned a new concept; that of coating poly(ethylene glycol), and related compounds, onto the red cell surface to alter its rouleaux forming potential in a regulated manner [50]. This has two important potential uses, first as a means of altering aggregation tendencies by varying only one parameter in hemodynamics studies thus making their analysis easier and


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potentially allowing the hemodynamics influence of rouleaux formation to be determined once and for all, second it offers possible therapeutic benefit. Professor Baskurt’s group in Antalya, and their collaborators, have taken full advantage of the former use. They have investigated the effects of red cell aggregation on in vivo hemodynamics, where the aggregation level is altered by coating the cells with poloxamer copolymers, thus the only hemorheological parameter that they vary in their studies is the inherent aggregability of the red cells [51, 52, 53, 54]. Another area on which some work had been done in the past but which had fallen out of favor is that of athletic performance and hemorheology. In the past few years papers in this area have reappeared in earnest. The group from which most recent papers have come is that of Brun [55, 56, 57], but other groups are increasingly getting involved [58, 59]. A unique event in hemorheology took place in 1985 when Dintenfass et al [60] reported on what is believed to have been the first and, so far, the only hemorheological experiment to take place on a space flight. The study was devoted to the question of whether red cell aggregation is affected by zero gravity conditions. The flight took place on a Space Shuttle in early 1985 and the general conclusion seems to have been that the lack of gravity had little, if any, effect on the morphology of the aggregates. An aspect of modern hemorheology has been the steady introduction of newer and more sophisticated methodologies. A particularly good example is the use of optical trapping to study the forces and mechanism involved in red cell aggregation formation [61]. This technology allows the aggregating red cells to be individually manipulated during the aggregation process and the minute forces involved estimated so that the process can be followed in detail. One of the newer instruments on the market, also based on laser technology, is the LORCA (Laser-assisted Optical Rotational Cell Analyser), which allows for rapid automated analysis of red cell deformability and aggregation [62]. Yet another development has been the use of ultrasound backscattering to estimate red cell aggregation [63] It is always good when new groups become involved in hemorheology. So the recent interest from veterinary research groups is to be welcomed. This interest seems to have been triggered by the realization that the significance of hemorheology to human health could extend to other mammals, especially those of economic and environmental importance, e.g. race horses and endangered species in zoos [64]. However hemorheological studies on animals other than the human are not new. In the early 1970’s Shu Chien’s (Figure 4) group reported on a comprehensive study of five different mammalian and four different non-mammalian species [65] and in 1992 Johnn et al published a comprehensive hemorheological comparison of 31 different mammalian species [66]. Furthermore, increasingly workers are using animal models to study in vivo influences of hemorheological changes and the effects of therapeutic regimen [67, 68]. One of the most common of these animal models is the rat for reasons of cheapness, size and relatively similar hemorheology to that of the human adult [69, 70] A continuing problem with our field in the view of the author is that relatively few studies have attempted directly to relate hemorheology to hemodynamics in vivo or in vitro. Too often hemorheological changes ‘found in the viscometer’ are assumed, on the basis of logic, to be of hemodynamic significance. However, what is needed is hemodynamic experiments that prove these hypotheses. Some studies in this direction have, of course, been done. A classic, which has been discussed above, is that of Whittaker and Winton [20] reported as long ago as 1933. Others have artificially

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altered hemorheological characteristics and studied their effects on hemodynamics both in vitro [71, 72] and in vivo [73, 74, 75, 76, 77, 78]. However this is a very complex area and even today the data is still confusing. For example it is still not clear what influence rouleaux formation has on in vivo hemodynamics. However, the work coming out of Professor Baskurt’s laboratory in Antalya, referred to above [51, 52, 53, 54], is beginning to clarify the situation. There is a crying need for more research groups to become involved in in vivo studies. So the history of hemorheology is a very long one, with a great surge in work in the field between the 1960’s and 1980’s. Since then the areas of particular interest have changed but clinical hemorheology is still very active. An indication of the change and wellness of the subject can be gleaned from looking at the in house journal Clinical Hemorheology. In 1985 only one volume was printed and it contained 11 papers from the UK, 8 from Germany, 7 from France, 4 from the USA, Italy and Austria and 1 each from Japan, Denmark and Greece, i.e. 41 papers in total. This shows the major influence of Western Europe, followed by the US and some contribution from Japan. In 2003 two volumes of the journal were published, but just to take figures from volume 2: there were 6 papers from Germany, 4 from France, 3 from China, 2 each from the USA, Turkey and Hungary and 1 each from Argentina, Egypt, Japan, Switzerland, Portugal, Italy and Poland, i.e. 26 papers. This indicates two things; first, work in the field continues at a similar rate to that in the 80’s and second that it is no longer centered in Western Europe but rather has widened to include most regions of the world. This is also bringing with it newer attitudes and interests, for example the study of hemorheological effects of traditional Chinese pharmaceuticals [79, 80]. So clinical hemorheology is still very much alive and kicking and long may it continue to be so.

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[45] J.H. Drew, R.L. Guaran, M. Cichello and J.B. Hobbs. Neonatal whole blood hyperviscosity: The important factor influencing later neurologic function is the viscosity and not the polycythaemia, Clin. Hemorheol. 17 (1997), 67-72. [46] M.W. Rampling, H.J. Meiselman, B. Neu and O.K. Baskurt, Influence of cell-specific factors on red cell aggregation, Biorheology 41 (2004), 79-90. [47] H..J. Meiselman, Red blood cell role in RBC aggregation – 1963 -1993 and beyond, Clin. Hemorheol. 13 (1993), 575-592. [48] Z. Wen, W. Yao, L. Xie, Z. Yan, K. Chen, W. Ka and D. Sun, Influence of neuraminidase on the characteristics of microrheology of red blood cells, Clin. Hemorheol. 23 (2000), 51-58. [49] N.Z. Ertan, M.W. Rampling, M.J. Pearson, C.P. Winlove, P.M. Gribbon and D. Denison, Time dependent effects of trypsin treatment of human red cells, Clin. Hemorheol. 13 (1993), 473-480. [50] J.K. Armstrong, H.J. Meiselman, R.B. Wenby, R.B. and T.C. Fisher, Modulation of red blood cell aggregation and blood viscosity by the covalent attachment of pluronic copolymers, Biorheology 38 (2001), 239-247. [51] O. Yalcin, M. Uyuklu, J.K. Armstrong, H.J. Meiselman and O.K. Baskurt, Graded alterations of RBC aggregation influence in vivo blood flow resistance. Am. J. Physiol. 287 (2004), H2644-H2650. [52] O.K. Baskurt, O. Yalcin, S. Ozdem, J.K. Armstrong and H.J. Meiselman, Modulation of endothelial nitric oxide synthase expression by red blood cell aggregation, Am. J. Physiol. 286 (2004), H222H229. [53] O.Yalcin, H.J. Meiselman, J.K. Armstrong and O.K. Baskurt, Effect of enhanced red blood cell aggregation on blood flow resistance in an isolated-perfused guinea pig heart preparation, Biorheology 42 (2005) 511-520. [54] O. Yalcin, F. Aydin, P. Ulker, M. Uyuklu, F. Gungor, J.K. Armstrong, H.J. Meiselman and O.K. Baskurt, Effects of red blood cell aggregation on myocardial haematocrit gradient using two approaches to increase aggregation, Am. J. Physiol. 290 (2006) H765-H771. [55] E. Varlet-Marie, A. Gaudard, J-F Monnier, J-P Micallef, J. Mercier, F. Bressolle and J-F Brun, Reduction of red blood cell aggregability during submaximal exercise: relationship with fibrinogen levels, Clin. Hemorheol. 28 (2003), 139-150. [56] E. Varlet-Marie, A. Gaudard, J. Mercier, F. Bressolle and J-F Brun, Is the feeling of heavy legs in overtrained athletes related to impaired haemorheology? Clin. Hemoreheol. 28 (2003). 151-160. [57] S. Khaled, J-F Brun, J-P Micallef, L. Bardet, L. Cassanas, G. Monnier and J.F. Orseetti. Serum zinc and blood rheology in sportsmen (football players), Clin. Hemorheol. 17(1997), 47-58. [58] A.V. Muravyov, S.V. Draygin, N.N. Eremin and A.A. Muravyov, The microrheological behaviour of young and old red blood cells in athletes, Clin. Hemorheol. 26 (2002), 183-188. [59] A. Temiz. O. Yalcin, H. Resmi and O.K. Baskurt, Can white blood cell activation be one of the major factors that affect hemorheological parameters during and after exercise, Clin. Hemorheol. 26 (2002), 189-194. [60] L. Dintenfass, P.D. Osman and H. Jedrzejczyk, First haemorheological experiment on NASA space shuttle ‘Discovery’ STS 51-C: aggregation of red cells, Clin. Hemorheol. 5 (1985) 917-936. [61] P.J.H. Bronkhorst, J. Grimbergen, G.J. Brakenhoff, R.M. Heethaar and J.J. Sixma, The mechanism of red cell aggregation investigated by means of direct manipulation using multiple optical trapping,. Br. J. Haematol. 96 (1997), 256-258. [62] R. Hardeman, P.T. Goedhart, J.G.G. Dobbe and K.P.LLetterina, Rotational Cell Analyzer L.O.R.C.A.) I A new instrument of laser-assisted optical measurement of various structural hemorheological parameters, Clin. Hemorheol. 14 (1994), 605-618. [63] M. Boynard, J.C. Lelievre and G. Guillet, Aggregation of red blood cells studied by ultrasound backscattering, Biorheology, 24 (1987) 451-461. [64] U. Wendberger, R. Plasenzotti and Th. Voracek, The fluidity of the blood in African elephants (Loxodonta africana), Clin. Hemorehol. 33 (2005), 321-326 [65] S. Chien, S. Usami, R.J. Dellenback and C.A. Bryant, Comparative hemorheology - hemorheological implications of species differences in blood rheology, Biorheology 8 (1971), 35-58. [66] H. Johnn, C. Phipps, S. Gascoyne, C. Hawkey and M.W. Rampling, A comparison of the viscometric properties of the blood of a wide range of mammals, Clin. Hemorheol. 12 (1992), 639-647. [67] H. Baumler, B. Neu, R. Mitlohner, R. Georgieva, H.J. Meiselman and H. Kiesewetter, Electrophoretic and aggregation behavior of bovine, horse and human red blood cells in plasma and in polymer solutions, Biorheology, 38 (2001), 39-51. [68] O.K. Baskurt and H.J. Meiselman, Susceptibility of equine erythrocytes to oxidant–induced rheological alterations, Am. J. Vet. Res. 10 (1999), 1301-1306. [69] X. Nie, Z-Y. Wen, Z-Y Yan, L. Huang , D. Sun and B. Cheng, Effects of morphine on rheological properties of rat red blood cells, Clin. Hemorheol. 22 (2000), 189-196.


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[70] A. Temiz, O.K. Baskurt, C. Pekcetin, F. Kandemir and A. Gure, Leukocyte activation, oxidant stress and red cell properties after acute, exhausting exercise in rats, Clin. Hemorheol. 22(2000), 253-260. [71] W. Reinke, W. Gaehtgens and P. C. Johnson, Blood viscosity in small tubes: effect of shear rate, aggregation, and sedimentation, Am. J. Physiol. 253 (1987), H540-H547. [72] A.A. Palmer and W. Jedrzejczyk,. The influence of rouleaux on the resistance to flow through capillary channels at various shear rates, Biorheology 12 (1975), 265-270. [73] S. Simchon, K-M Jan and S. Chien, Influence of reduced red cell deformability on regional blood flow, Am. J. Physiol. 253 (1987), H898- H 903. [74] M. Cabel, H.J. Meiselman, A.S. Popel and P.C. Johnson, Contribution of red cell aggregation to venous vascular resistance in skeletal muscle, Am. J. Physiol. 272 (1997), H1020-H1032. [75] R. Greene, J.M.B. Hughes, I.D. Iliff and G.F. Pineo, Red cell flexibility and pressure-flow relations in isolated lungs, J. Appl. Physiol. 34 (1973), 169-175. [76] G.A. Pantely, L.J. Swenson, C.H. Tamblyn, G.Y.F. Seaman, C.G. Anselone, W.B. Johnson and I.D. Bristow, Increased vascular resistance due to a reduction in red cell deformability in the isolated hind limb of swine, Microvasc. Res. 35 (1988,) 86-100. [77] M.P. Doyle, W.R. Galey and B.R. Walker, Reduced erythrocyte deformability alters pulmonary haemodynamics, J. Appl. Physiol. 67 (1989), 2593-2599. [78] H.H. Lipowsky and J.C. Firrell, Microvascular hemodynamics during systemic hemodilution and hemoconcentration, Am. J. Physiol. 250 (1986), H908-H922. [79] Yu Zhui, Ou-Yang Jing-Ping, Liu Yongming, Wei Lei, Tu Shuzheng, Yang Hailu, Zheng Hanqiao and Yan Xiaohomg, Experimental study of the antiatherogenesis effect of Chinese medicine angelica and its mechanisms, Clin. Hemorheol. 22 (2000), 305-310. [80] F. Liao and B.Li, Inhibition of shear–induced platelet aggregation by Chinese herbal medicines, Clin. Hemorheol. 17 (1997), 315-3180.


II. Hemorheology



21

Basic Aspects of Hemorheology Giles R. COKELETa,1 and Herbert J. MEISELMANb Department of Chemical and Biological Engineering, Montana State University, Bozeman, MT 59717, USA and bDepartment of Physiology and Biophysics, Keck School of Medicine, University of Southern California, Los Angeles, CA, 90033, USA. a

Introduction In analyzing blood flows, one is generally interested in how the blood responds to forces (e.g., pressure gradients, shear stresses). The general fluid mechanical procedure used to predict how a fluid flows in response to forces involves three steps: (1) Consideration of all the forces being exerted on an infinitesimally small volume of fluid. This is done by use of the physical principle known as the conservation of momentum, and results in equations which relate the forces to velocity gradients. (2) Introduction of rheological (“constitutive”) equations which are specific to the fluid being analyzed. These equations indicate how the fluid responds to forces, and relate the forces to the resulting velocity gradients. The rheological equations contain fluid specific characteristics (e.g., apparent viscosity as a function of shear rate). (3) Substitution of the rheological equations into the conservation of momentum equations, and integration of the resultant differential equations to obtain macroscopic relationships, such as between flow rates and pressure gradients. In this chapter, this procedure will be illustrated for specific applications to blood.

1. The Continuum Model All fluids consist of particles (molecules, cells, etc.). Analyzing a fluid flow by keeping track of each and every particle in the fluid is an impossible task. To reduce the analytical work to a reasonable level, we use a model of the fluid which ignores the particulate nature of the fluid and replaces the real fluid with a model fluid whose properties vary smoothly and continuously in space. This is the continuum model. Of course, we must be aware of when this model does not adequately represent the real fluid. The limits of applicability of the continuum model are determined empirically. One type of limit is determined by measuring a property of the fluid (e.g., density), using smaller and smaller fluid sample volumes, until it is determined that the fluid properties no longer are independent of sample size. Alternatively, one can determine the macroscopic characteristics of the flow (e.g., pressure drop versus flow 1 Corresponding author: Department of Chemical and Biological Engineering, Montana State University, Bozeman, MT, USA; E mail: [email protected]

22

G.R. Cokelet and H.J. Meiselman / Basic Aspects of Hemorheology

rate in a tube) in smaller and smaller flow channels, until the measured flow properties do not obey the predicted relationship derived by the fluid mechanical procedure described in the Introduction. The limit of the applicability of the continuum model for suspensions is generally taken to be 20-30 times the particle major dimension; this limit for blood is discussed in Chapter II.3.a.

2. Rheological Concepts Since the rheological equations for the response of a specific fluid to forces involves fluid velocities, a coordinate system must be defined so that we can measure distances and velocities. Here, we consider coordinate systems that do not move relative to the local earth’s surface. Three types of coordinate systems are useful in looking at various fluid flows: (1) a rectangular system (2) a cylindrical system (3) a spherical system. Which coordinate system is used for a particular flow situation depends on the flow boundary shape and the directions of the non-zero velocity components. Thus, for laminar, steady flow between parallel plates where only one velocity component is non-zero, one uses the rectangular system. For flows in cylindrical tubes or between concentric cylinders viscometers, one uses the cylindrical coordinate system, while for flow in a cone-and-plate viscometer, one uses the spherical system. After selecting an appropriate coordinate system, one then proceeds with the fluid mechanical procedure described in the Introduction. This will be illustrated for an analysis in the rectangular coordinate system. To obtain the force-response relationships, one considers an infinitesimally small volume of the fluid, whose shape is defined by the chosen coordinate system and where the faces of the small volume must all be perpendicular to the coordinate directions. In the rectangular coordinate system, such an elemental volume would be a cube or rectangular parallelepiped, whose edges are oriented parallel to the coordinate axes, and which have lengths of įx in the x-direction, įy, and įz in the y and z directions. The forces acting on this elemental volume of fluid are then considered. In Figure 1, a force acting on the front face of the elemental volume is shown; it is not generally perpendicular to the face on which it is acting. This force, which is a vector, is broken into three components that are aligned with the coordinate system axes as shown. Forces Fx and Fz act in the plane of the face, and when divided by the area of the face are called shear stresses. Fy is perpendicular to the face, and when divided by the area of the face is known as a normal stress, or pressure. On the top face, another vector force per face area has been decomposed into its three stress components. Each stress is identified by two subscripts, the first of which indicates the face that the stress is operating on, with the face identified by the coordinate system axis perpendicular to the face, and the second identifies the stress direction. Thus, Ĳzx is the stress operating on the face which is perpendicular to the z-direction, and which operates in the x-direction.


23

Figure 1. An elemental volume of fluid, subjected to some forces. The bold force vector acting on the front face is shown decomposed into its three components, Fx, Fy and Fz, acting on that face.

In general, each face of the elemental volume has three stresses acting on it, and so the elemental fluid volume has a total of 18 stresses on it; there is also a gravitational or body force, but its effect is generally considered negligible when studying blood flow. If we imagine that the elemental volume shrinks to a point, so that we can evaluate the stresses at a point, we will have nine different stresses: three normal stresses with one for each coordinate direction, and six shear stresses, two in each plane perpendicular to a coordinate direction. By a mathematical argument, shear stresses which differ only in the order of their subscripts (e.g., Ĳxy and Ĳyx) are equal. Consequently, at any point in the continuum fluid, only three normal and three shear stresses are independent and need to be specified. Having defined the forces acting at a point in the fluid in terms of stresses, we now have to define the fluid’s response to these stresses. Any motion of an infinitesimal fluid volume can be decomposed into one or more basic motions: (1) Pure translation (i.e., movement relative to the coordinate system, without rotation, shape change or volume change) (2) Pure rotation (3) Angular distortion (i.e., a shape change without a volume change) (4) Volumetric distortion. Of these motions, only two are caused by stresses: a volumetric distortion caused by normal stresses, and an angular distortion arising from shear stresses. Since blood is considered to be an incompressible fluid, only the shear stresses need to be considered. One means of measuring the rates of angular distortion is to measure the rates of changes of the angles between adjacent faces of the infinitesimal fluid volume. This is illustrated, for the two-dimensional case, in Figure 2, where a measure of the rate of angular distortion could be the rate of change of the angle Ȗ. Since Ȗ = 1800 – Į – ȕ, the time rate of change of Ȗ is given by

wJ wt

wD wE wt wt

24


where “t” denotes time. We now relate these angular changes to the appropriate velocity components.

Figure 2. Angular distortion of a 2-dimensional fluid element due to shear stresses.

In Figure 2, the lower left corner of the volume moves with velocity components ux and uy. However, the lower right corner may move with slightly different velocities:

ux

wux dx and uy wuy dx . wx wx

Thus, in the time increment when the volume moves from the lower to the upper position, the volume edge originally parallel to the x-axis will become oriented in a new direction, at an angle dĮ relative to the coordinate axis x. The relative displacement, dy, in the y-direction is

(uy

wuy dx ) (uy ) wx

wuy dx . wx

For small angles, the tangent of the angle is equal to the angle in radians, so

dD

dy dx

wuy dx wx dx

By similar reasoning dE .

§ wu · ¨ ¸ © wx ¹

wux dJ . Therefore, wy dt

(

. wux wuy ) { J ; wy wx

where J is defined as the shear rate in the x – y plane. Shrinking the elemental volume to a point, there are three possible angular distortions, one in each coordinate plane through the point. For a rectangular coordinate system, these are

25

G.R. Cokelet and H.J. Meiselman / Basic Aspects of Hemorheology .

Jxy .

Jxz .

Jyz

wux wuy wy wx

(1a)

wux wuz wz wx

(1b)

wuy wuz . wz wy

(1c)

For cylindrical coordinates, the shear rates are .

Jrx .

Jr\

.

Jx\

wur wux wx wr

1 wur wu\ u\ r w\ r wr 1 wux wu\ . r w\ wx

(2a)

(2b)

(2c)

For spherical coordinates .

JrI

. r\

J

.

JI\

uI ) 1 wur r r wr r wI w(

u\ w( ) 1 wur r r r sin I w\ wr

sin I r

w(

u\ ) 1 wuI sin I wI r sin I w\

(3a)

(3b)

(3c)

In many situations, where the flow is steady, laminar flow of an incompressible fluid, these equations are greatly simplified because there is only one non-zero velocity component: ux in flow between parallel plates and in cylindrical tubes, u\ in a concentric cylinder viscometer, and uI in a cone-and-plate viscometer. Thus, for these simple flows:

26


.

Jxy

Jrx

dux dy

dux dr

. u\ Jr\ du\

dr

.

JrI

r

uI ) r r dr

parallel plates

(4a)

cylindrical tube

(4b)

concentric cylinders viscometer

(4c)

cone-and-plate viscometer

(4d)

d(

All the other shear rates for these simple flows are zero. Now that we have measures of the rates of deformation (i.e., shear rates) at a point in any fluid, we must now introduce the unique flow (i.e., rheological) properties of the fluid into our analysis in order to incorporate the shear stresses which cause the deformations. These flow properties are expressed in the constitutive or rheological fluid equations.

3. Constitutive Equations The rheological properties of fluids are measured in viscometers specially designed so that only one velocity component exists and does not vary in the direction of flow. In addition, except for tube viscometers, the fluid space in the viscometer is small enough so that all of the fluid is subjected to approximately the same shear rate. General details of viscometer operation can be found in Van Wazer, Lyons, et al. [1], and special details of determining blood’s rheological properties can be found in Cokelet [2] and Meiselman and Cokelet [3]. The rheological properties of a fluid may be divided into two classes: timeindependent and time-dependent. Since the time-dependent behavior of blood is discussed in Chapter II.3.b. and because blood’s rheological properties are timeindependent in almost all in-vivo and in-vitro flows, we restrict the discussion here to time-independent rheological properties. The relationship between the shear stress and the shear rate must be determined experimentally. To be most useful, this relationship must be expressed as a mathematical equation. Some such equations are the result of curve-fitting experimental data, while others are based on a model of the fluid. Some useful empirical relationships are given here:

W

.

K J

for Newtonian fluids, where K is the viscosity and a constant at all shear rates of interest.

27

G.R. Cokelet and H.J. Meiselman / Basic Aspects of Hemorheology .

W

a J b

W

aJ

for W t b . Here “a” and “b” are constants, with “b” being the minimum shear stress needed to cause flow, known as the yield stress. This is the Bingham fluid model.

n

where “a” and “n” are constants. This is the power law fluid model

It is common to define an “apparent viscosity” by the relationshipKa

{

W . Except J

for Newtonian fluids, it is a function of shear rate. Three equations, based on models of suspensions, are: Here Ko is the suspending fluid Newtonian viscosity, “k” is a constant dependent on particle shape, and H is the volume fraction of the suspension occupied by particles. This equation is applicable for suspensions having a low volume fraction of particles. Einstein showed that k=2.5 for spherical particles.

Ka Ko(1 kH )

W 1/ 2

aJ

1/ 2

b

1/ 2

where “a” and “b” are constants. It is known as the Casson [4] equation; at very low shear rates, b is the yield shear stress. However, for blood, the experimental data can not be fit over all shear rates with only one set of constants “a” and “b”, whereas fairly good fit is possible by applying the equation over several shear rate ranges and thereby obtaining several sets of constants.

Ka Ko (1 0.5kH )2

where

k

k o k f J r1 / 2 1 J r1 / 2

and

Jr {

J Jc

with

k 0, kf , and Jc being constants. This equation, known as the Quemada [5-7] equation, accurately fits blood data over a very wide range of shear rates. The use of the Casson equation to represent blood’s rheological properties is discussed further in Chapter II.3.a. 4. Applications Given the angular distortion or shear rate equations (1a – 4d) and a suitable constitutive equation for a specific fluid, it is now possible to derive an equation using easily measured macroscopic properties of a viscometric fluid flow (e.g., pressure drop versus flow rate in tube viscometers) to determine the coefficients in the constitutive equation.

28


This will be illustrated for the Newtonian fluid model that includes the Einstein type equation above. 4.1. Tube Viscometer For steady, uniform, laminar flow of an incompressible fluid in a cylindrical tube, the appropriate shear rate equation is equation (4b) above:

Jrx

wux wr

(5)

And the constitutive equation for a Newtonian fluid is:

Wrx

KJrx

(6)

Substitution of equation (5) into equation (6) gives:

Wrx

K

wux wr

(7)

Inspection of this equation indicates: 1) for conditions at the centerline where r equals 0, Wrx 0 based on symmetry of the velocity profile about r=0; 2) at the wall, Wrx Wo , which is not zero because, with ux=0 at r=r0, there must be a velocity gradient there. Consequently, we see that Wrx is a function of r, with the relationship between Wrx and r obtained by the application of the principle of the conservation of momentum.

Figure 3. A small fluid shell for the case of flow though a cylindrical tube.

Figure 3 shows a very small “sleeve” of fluid. From equation (7) we see that there is a non-zero shear stress wherever the velocity gradient is non-zero. Consequently, there is a shear stress, Wrx , on the inner cylindrical surface of the elemental volume, and also one on the outer surface, Wrx

wWrx dr , which may differ from Wrx because the wr

velocity gradient may be different on each of the two cylindrical surfaces. Likewise,

29


the pressure at axial positions x and x+dx may be different. Because the flow is uniform, (i.e., the velocity does not change in the flow direction), the momentum added to the elemental volume by flow into the volume at axial position x is balanced by the flow out of the volume at the axial position x+dx. Consequently, the x-component of the momentum equation is:

2SrdrP 2Srdr ( P

wP dx) 2SrdxWrx (2S (r dr )dx)(Wrx dWrx dr ) dr wx

0.

Simplification reduces this equation to

wP wx

dWrx dr

(8)

w 2P If this equation is differentiated with respect to x, we see that 0 and wx 2 wP a constant. Use of the r-component of the momentum equation shows therefore wx wP dP which is a constant that P does not vary in the r direction, and thus wx dx everywhere. If equation (8) is now integrated, we obtain:

dP r dx

Wrx kI

(9)

where kI is a constant of integration. By symmetry of the velocity profile, r=0, so kI=0. Combining equations (7) and (9) yields:

dP r dx

K

Wrx

0 at

dux dr

Separation of variables and integration gives:

dP r 2 dx 2

Kux kI .

At r=r0, ux=0, so the velocity profile is given by:

dP r 0 2 r 2 ( ) 2 dx 2

Kux

which is the well-known parabolic velocity profile. To obtain the total flow rate, Q, one must integrate the equation from r=0 to r=r0: 4

Q

³ 2Sr (dr )ux

dP Sro dx 8K

(10)

30


Equation 10 is, of course, the Hagen-Poiseuille Equation. The Hagen-Poiseuille Equation allows us to determine if experimental data for pressure gradient and corresponding flow rate are fit by the Newtonian fluid flow

dP 'P , which is equal to , versus Q must be linear with dx 'x 8K . Note that to test if the continuum the slope of the line fitting the data equal to r04

equation. That is, a plot of

model was appropriate for fluid flow in a tube of radius r0, one must obtain data with several tube sizes to see if the calculated viscosity is a constant and independent of tube radius. It not, the continuum model has failed for this fluid flowing in the tested tube; this matter is discussed further for the case of blood in Chapter II.3.a. One could, in a manner similar to the above, derive equations analogous to equation (10) for the other constitutive equations. Fortunately, this is not necessary, because the Mooney-Rabinowitz-Weissenberg equation [8, 9] is generally valid for fluids whose rheological properties are not time dependent:

J0

2U (3

d ln U ) d ln W 0

where U = the average velocity divided by the diameter and

J0

= the shear rate at the

tube wall; W o is obtained from equation (9). Thus, by plotting ln U versus ln W 0 , one can obtain the wall shear rate for the corresponding wall shear stress. Note that all of the above equations are only valid for laminar fluid flow. The determinant for laminar tube flow for a Newtonian fluid is embodied in the Reynolds number

Re {

2ro u

K

d 2100 , where u is the average fluid velocity; Re values > 2100

indicate non-laminar flow. 4.2. Concentric Cylinders Viscometer For steady, uniform, laminar flow of an incompressible Newtonian fluid in a concentric cylinders viscometer, equation (9) and the Newtonian fluid constitutive equation give:

Wr\

K (

wu\ u\ ) wr r

u\ ) r Kr wr w(

(11)

With this type of viscometer, one usually measures: 1) the rotational rate of one of the cylindrical surfaces while the other remains stationary; 2) the total torque exerted by the fluid on one of the cylindrical surfaces. The total torque, T, for the case where the torque is measured on the rotating inner cylindrical surface is: T 2SriLiWi where the subscript “i” refers to the inner surface, and Li is the length of the torque measuring surface, usually taken as the fluid gap length. Thus, using equation (11),

31


T

§ u\ · ¨ w( ) ¸ 2Sri 2 Li (Kri¨ r ¸ ) ¨ wr ¸ ¸ ¨ ¹i ©

§ u\ · ¨ w( ) ¸ 2SKri 3 Li¨ r ¸ ¨ wr ¸ ¸ ¨ ¹i ©

If the gap width is very small, it can be assumed, within instrumental precision, that the derivative in this equation is a constant across the gap and equal to with ro being the outer surface radius. In this case, T terms of the angular rotational speed,

T

2SKLi (u\ )i

(u\ / r ) i , ri ro

ri 2 , or, in ( ri r o )

Zi :

2SKLiri 3Zi ( ri r o )

Thus, one can use the measured torque and rotational speed to calculate viscosity. It is necessary to test to see whether the viscometer gap is small enough so that the small gap approximation is valid; this testing is achieved by using various size gaps. If not, one must construct the viscometer with a smaller gap, or utilize equation (11) and the conservation of momentum equation to determine the velocity profile in the gap. Fortunately, Krieger and Elrod [10] developed a method, somewhat analogous to the Mooney-Rabinowitz-Weissenberg equation, for obtaining the shear rate at the rotating surface of a concentric cylinder viscometer, even for non-Newtonian fluids and non-small gaps. For a rotating inner surface, the Krieger-Elrod Equation is

Ji

º (ln s ) 4 d 4Zi d ln Zi (ln s ) 2 d 2Zi 1 ln ...» s « 2 4 ln s ¬ 45Zi (d ln Wi ) 3Zi (d ln Wi ) d ln Wi ¼

Zi ª

where s = ro/ri. Solution of this equation requires graphical determination of the derivatives. For blood, a precision of 1% or better requires three terms of this series for shear rates of 10 s-1 or less, whereas shear rates above 10 s–1 require only two terms. The shear stress on the inside surface can be calculated from the measured torque from the equation:

Wi

T (2Sri 2 Li )

The criteria for laminar fluid flow is again embodied in a Reynolds number for a Newtonian fluid, Re {

ri Z (ro ri ) U

K

.

When the inner surface rotates, Taylor

ro , resulting in incorrect shear stresses. For ro ri either surface rotating, turbulence will occur if Re t 50,000 . vortices will form if Re t 41.3

32


4.3. Cone-and-Plate Viscometer This viscometer employs: 1) a cone with an angle measured from the cone axis to the cone surface of slightly less than 90o; 2) a flat plate. The cone is situated so that its axis is perpendicular to the flat plate and its apex is just slightly separated from the flat plate; in some situations, a small portion of the apex is removed to allow proper separation with the “virtual” apex at the plate surface. Usually, the cone is rotated at selectable constant rates and the resultant torque is measured on the cone; the fluid is in the gap between the cone and the plate. For steady, laminar uniform fluid flow, application of the momentum equation results in:

1 dW TZ W TZ dT

2 cot T

Note that it is assumed that uT

rf (T ) , where f (T ) is only a function of T ;

this assumed velocity satisfies all the boundary conditions. Integration and use of the boundary condition on the flat plate surface gives

W T\

(W T\ ) S ( 2

1 ) sin 2 T

From this equation, it can be seen that the shear stress, as a function of the angle T , varies by 0.12% if the angle in the fluid (i.e., between the cone surface and the flat plate), T L , is 2.0o. Since in most cases the liquid angle is less than 2.0o, one may assume that the shear stress is constant. Thus, the measured torque T is

T

2 ³ 2Sr W T\ dr

where the integration is from r=0 to r=R, the radius of the cone. Rearrangement gives

W T\

3T . 2SR 3

The corresponding shear rate, with the use of .

J T\

Z TL

, where

Z

T L d 2 o , becomes (from equation 4d):

is the rotational rate of the cone.

If the rotational velocity is too high, the fluid flow will have vortices. The criterion for the onset of this condition is again a Reynolds number, defined as Re

R 2ZU

P

. For

a 4o liquid angle, secondary flows are present if Re is greater than 1100; the criterion for onset of vortices formation depends on the fluid angle [11].


33

References J.R. Van Wazer, J.W. Lyons, K.Y. Kim and R.E. Colwell, Viscosity and flow measurement, John Wiley, New York, 1963. [2] G.R. Cokelet, The rheology of human blood, In: Biomechanics: Its foundations and objectives, Y.C. Fung, N. Perrone and M. Anliker, Eds. Prentice Hall, Englewood Cliffs, NJ, 1972. [3] H.J. Meiselman and G.R. Cokelet, Blood Rheology: Instrumentation and Techniques, In: Flow: Its measurement and control in science and industry, R.D. Dowdell, Ed., Instrument Society of America, Pittsburgh, PA, 1974, Vol. 1, Part 3. [4] N. Casson, A flow equation for pigment-oil suspensions of printing ink type, In: Rheology of disperse systems, C. C. Mills, Ed., Pergamon, New York, 1959, pp 84-102. [5] D. Quemada, Rheology of concentrated suspensions and minimum energy dissipation principle, I. Viscosity-concentration relationship, Rheol. Acta. 16 (1977), 82-94. [6] D. Quemada, Rheology of concentrated dispersed systems. II. A model for non-Newtonian shear viscosity in steady flows, Rheol. Acta. 17 (1978), 632-642. [7] D. Quemada, Rheology of concentrated disperse systems. III. General features of the proposed nonNewtonian model. Comparison with experimental data, Rheol. Acta, 17 (1978), 643-653. [8] B. Rabinowitsch, Ueber die viskositaet und elastizitaet von solen, A. Physik. Chemie, A145 (1929), 126. [9] M. Mooney, Explicit formulas for slip and fluidity, J. Rheol. 2 (1931), 210-221. [10] I.M. Krieger and H. Elrod, Direct determination of the flow curves of non-Newtonian Fluids. II. Shearing rate in the concentric cylinder viscometer. J. Appl. Physiol. 24 (1953),134-144. [11] M.E. Fewell and J. D. Hellums, The secondary flow of Newtonian fluids in cone-and-plate viscometers. J. Rheol. 21(1977), 535-565. [1]

See Also: H.A. Barnes, J.F. Hutton and K. Walters, An Introduction to Rheology, Elsevier Science, Amsterdam, 1989. R.G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press, Oxford, 1998. C.W. Macosko, Rheology: Principals, Measurements and Applications, Wiley-VCH, New York, 1994. A.Y. Malkin and A.I. Isayev, Rheology: Concepts, Methods and Applications, ChemTec Publishing, Norwich, NY, 2005. T.G. Mezger, The Rheology Handbook, Vincent Verlag, Hanover, 2002. F.A. Morrison, Understanding Rheology, Oxford University Press, Oxford, 2001.

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Compositional Properties of Blood Michael W. RAMPLING1 School of Medicine, Imperial College, South Kensington, London SW7 2AZ, UK

Introduction The function of blood is to feed all the tissues of the body with vital materials and to remove waste. To do this in the human adult it has to traverse the complicated vascular network, which varies in diameter from some 3 cm down to about 5 μm. Furthermore, the blood must circulate above a limiting rate if it is to do its work effectively enough to keep the organism healthy. This rate of circulation is determined by the driving pressure generated by the heart, by the geometrical resistance offered by the vasculature and by the flow properties of the blood. These flow properties are the concern of the hemorheologist and they are dependent on the composition of the blood and the properties of its constituents; hence, knowledge of them is vital to any understanding of hemorheology. This chapter gives an overview of the composition of normal adult human blood and some indication of the ways in which it can be altered in diseased states. There is also discussion of the normal changes in blood composition that take place as the fetus develops through to the neonatal period. Finally, there is a brief review of the variations that occur in other mammals, emphasizing the similarities and the great differences that exist compared with the human adult.

1. Human Adult In the healthy adult human, a little less than half the blood volume is made up of the cellular compartment and the rest is plasma. The plasma is a mixture of many metabolites, proteins and lipoproteins suspended in a complex salt solution. The cellular compartment is also complex, consisting mainly of erythrocytes, or red blood cells, but also of white cells, or leukocytes, and platelets, or thrombocytes. The red cells are essentially all the same in that each is little more than a membranous sack containing a very concentrated hemoglobin solution. Their prime role is to facilitate the transport of oxygen and carbon dioxide around the circulation. The white cells on the other hand are made up of a number of distinct varieties, and they all have complicated interiors, containing organelles, a nucleus, fibres etc suspended in a viscous cytoplasm. They are involved in defense against infection. The platelets are essentially all similar, are much smaller than the other blood cells, 1

Corresponding author: School of Medicine, Imperial College, South Kensington, London SW7 2AZ, UK; E mail: [email protected]

M.W. Rampling / Compositional Properties of Blood

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are anucleate and have relatively complex contents consisting of vacuoles and fibres suspended in their viscous cytoplasm. They have a role to play in hemostasis. 1.1. Plasma The liquid phase in which the cellular components of blood are suspended is the plasma. It is a complex solution of materials ranging in size from a few tens to millions of Daltons. These solutes make up some 8 to 9% of plasma by weight, the rest being water. 1.1.1. Ions The smallest of the solutes are the ions of the dissolved and dissociated inorganic salts, with molecular weights of a few tens of Daltons and they make up some 1% of the plasma by weight. There are many different ions present in blood, e.g. Na+, Ca++, K+, Cl-, HCO3- and PO4--. However, by far the most concentrated of the cations in Na+ (derived largely from dissociated NaCl) and so it is the most potent from an osmotic point of view. For this reason amongst others, it is necessary that its concentration be tightly controlled. There are potent physiological mechanisms to ensure this, so that in the healthy human the concentration is maintained in the normal range of 135 to 145mM [1]. From a hemorheological standpoint this is very important, because if its concentration rises or falls outside this range, red cells shrink or swell, with this volume change having a profound influence on their mechanical properties and, hence, on their effect on blood viscosity [2]. The other particularly important ion from a hemorheological point of view is the anion HCO3-. It has a normal range of 24 to 30mM [1]. Its importance lies in it being one of the factors involved in controlling blood pH and in maintaining it in the very narrow normal range of 7.35 to 7.45 which is vital to normal bodily function [1]. Again, if the pH deviates far from the normal range, it has deleterious effects on the mechanical properties of red cells and hence on their viscometric effects [3]. 1.1.2. Metabolic Molecules The metabolic molecules generally have molecular weights of a few hundreds of Daltons (e.g., glucose, urea and amino acids) and make up about 1% of the plasma by weight. Their concentrations are generally somewhat less well controlled than the salts mentioned above so, for example, the normal ranges for glucose and urea are 0.7 to 1.0g/l and 80 to 250mg/l (urea nitrogen) respectively [1]. Generally, they have relatively minor hemorheological effect and so no further discussion will be made of them here. 1.1.3. Proteins The plasma proteins are all very large, having molecular weights ranging from a few tens of thousands to millions of Daltons, and make up about 7% of plasma by weight. They are generally subdivided according to their electrophoretic mobility into albumin, Į, ȕ and Ȗ groups. However, this approach masks the wide variety of different proteins that exist in the plasma (e.g., there are carriers of lipids, metals and other factors, many immunoglobulins, clotting factors and fibrinogen). They are necessary to the carriage of many vital materials, to defense against infection, to

36


hemostasis, etc. [1], and are hemorheologically important for two reasons. First, because of their relatively high concentration, their large size and often asymmetrical shapes, they have a large effect on the viscosity of plasma; the viscosity of whole blood is dependent in part on the viscosity of the plasma. The normal range for the viscosity of human plasma is 1.25r0.10 mPa.s at 37oC, while the viscosity of water at the same temperature is 0.69 mPa.s; the difference between the viscosity of water and plasma is due almost entirely to the plasma proteins [4]. In health the concentrations of these proteins are reasonably well controlled but they can vary enormously in diseased states and correspondingly cause plasma viscosity to vary widely [5]. The second reason that they are of hemorheological significance is that some of them cause red cells to stick loosely together in characteristic face-to-face aggregates, like piles of coins, called rouleaux. From a hemorheological point of view rouleaux formation is important because it causes the viscosity of blood to be very dependent on the shear rate to which it is exposed [4]. Thus the viscosity of normal blood is high at low shear rate, but steadily falls as the shear rate increases and the shear forces increasingly disperse the rouleaux (see Chapter II.3.a). This phenomenon is known as shear thinning. There is no doubt that the protein with the greatest effect in this respect is fibrinogen, and while its normal range in health is usually taken as 3.5r0.4 g/l [6], it can rise substantially in association with a wide variety of diseases and hence greatly affect the shear thinning properties of blood [4]. Other large plasma proteins also have rouleaugenic effects (e.g., D2-macroglobulin, IgG and IgM) but they are not as potent as fibrinogen [7]. 1.2. Formed Elements 1.2.1. Erythrocytes In the past the relative volume of the blood that was made up of cells was measured centrifugally. The blood sample was spun in a uniform cylindrical tube, with the ratio of the length of the cell column to that of the cells plus plasma termed the hematocrit. Of course, strictly speaking, it was not a true hematocrit (i.e., volume fraction of only red cells) as the spun cell column would contain leukocytes and platelets as well as red cells. Nowadays, red cell parameters are commonly determined electronically and the hematocrit is obtained by multiplying the measured red cell count and the mean cell volume. This gives a “true” hematocrit (i.e., not contaminated by other cells). The convention now is to call the electronic value the hematocrit and the centrifugal one the packed cell volume (PCV). However in practice, for normal blood, it makes little difference as the red cells vastly outnumber the leukocytes and the platelets are much smaller and less numerous than the red cells. The normal range for the hematocrit differs between men and women and is 40 to 50% and 36 to 46% respectively [8]. By comparison, the leukocytes and platelets together comprise only about 1%. This high concentration of red cells is the main reason that they are hemorheologically of enormous importance. The other reasons are the various physical properties of the red cells. Firstly, they have an unusual shape since they are biconcave discs, about 8.5 Pm in diameter and about 2.4 Pm thick, and so have the capacity to align with the direction of flow. Secondly, they are extremely flexible and so will deform and extend under shear forces. Thirdly, they have the tendency to adhere together loosely, as rouleaux, under the influence of plasma proteins (especially fibrinogen) as


37

mentioned above. Fourthly, they contain a hemoglobin solution of high concentration (normal range 32 to 36 g/dl [8]) which has an effect on the speed with which they can deform under shear forces. These properties act together to give blood a viscosity very substantially in excess of that of plasma alone, and to endow it with prominent shear thinning properties. The relationship between blood viscosity and hematocrit is approximately of the form: log (viscosity) = k1 + k2(hematocrit) where k1 and k2 are shear rate dependent. Typical values for the viscosity of healthy blood at shear rates of 0.277 s-1 and 128.5 s-1 are 39r4 and 4.3r0.2 mPa.s respectively for females and 48r6 and 4.7r0.2 mPa.s respectively for males [6]. The gender differences are, of course, due to the lower hematocrit that is normal for females. The above arguments should make it clear that the red cell mass has a huge influence on the viscosity of normal whole blood, and hence on blood flow in large vessels. However, blood must also pass through the microcirculation where vessels may be as small as a few Pm in diameter. Here the cells are of a size similar to the vessels, so the concept of blood viscosity is hardly relevant because the blood cannot be approximated to a uniform solution or suspension. When such flow is observed under a microscope, a bolus of plasma followed by one or two deformed cells followed by a further bolus of plasma, etc. is seen in microvessels. Under these circumstances, the factors determining resistance to flow are the viscosity of the plasma, the deformability and concentration of the red cells as well as other cells (see below), and the forces involved in holding the red cell rouleaux together. Of course another factor is the size of the red cells. The mean red cell volume (MCV) for healthy adults is 83 to 101 fl and is gender independent [8]. However, this range can be misleading, as it is the average MCV across the healthy adult population. The reason that this is misleading is that, within an individual, the variation in size of their red cell population is far greater than this range; this is because young cells released to the circulation from the bone marrow have a large volume, as much as 120 fl, but subsequently progressively lose membrane, contents and size and deformability as they age. Finally, they are destroyed, after about 120 days in the circulation, when their volume may be as little as 60 fl. A recent area of study involving the red cell concerns cell-specific effects on erythrocyte aggregation. It is now clear that individual red cells vary in their response to rouleaugenic molecules. So, for example, young red cells, newly released from the bone marrow, respond much less than do older cells. Furthermore, red cells from different healthy individuals differ in their rouleaugenic response to molecules such as fibrinogen. The reasons, which are presumably determined by the cell membrane, are not yet clear, but the consequences are hemorheologically significant [9]. 1.2.2. Leukocytes As implied above, the leukocytes have very little role to play in determining the viscosity of normal whole blood because their volume concentration is so much smaller than that of the red cells. However, they do play a major role as a determinant of flow in the microcirculation. There are two reasons for this. First, their internal contents are a complex array of nucleus, organelles, fibres, cytoplasm,

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etc, which results in them having viscoelastic characteristics (i.e., viscosity, elasticity) much greater than those associated with the red cells. Second, they generally have somewhat larger volumes than the red cells. There are several varieties of leukocytes (i.e., monocytes, granulocytes and lymphocytes), all of which are larger than the red cell: the largest (the monocyte) has a mean volume of about 230fl, while for the smallest (the lymphocyte) it is about 120fl. It is now thought that, in a healthy individual, the white cell population confers an overall resistance to microcirculatory flow similar to that of the whole red cell mass. It is interesting to note the number concentration of the leukocytes is normally 4-8 x109 per liter while that for the red cells is about 5x1012 per liter (i.e., some 3-orders of magnitude difference), thus implying the individual leukocyte has a flow resistance, in small vessel flow, some 3-orders of magnitude greater than that of the red cell; the latter point has been proven by micropipette studies [10]. 1.2.3. Platelets From a hemorheological point of view the platelets are of little importance, even though they have relatively complex internal contents with considerable viscosity. The reason for this finding is that they are much smaller than either the erythrocytes or leukocytes, having diameters of the order of 2-3 Pm; their overall volume in normal blood is even less than that associated with the leukocytes. The consequence is that they neither influence whole blood viscosity nor microvascular resistance to any significant degree. Their primary role is to participate in the hemostatic mechanism, and as such they do have a major part to play in what is essentially the solidification of blood as it clots [1].

2. Human Juvenile It has long been known that the blood parameters change as the child ages from birth through to adulthood. In the relatively recent past, in utero blood extraction techniques have been developed and have allowed the substantial hematological changes that take place during fetal development to be evaluated. 2.1. The Fetus Data have now been published on hematological values in the fetus from 18 to 35 weeks of gestation [11]. They show that, over this period, there are progressive increases in hematocrit from 32% to 48% and in plasma protein concentration from 25 g/l to 43 g/l. The most remarkable change over the same time period is the increase in fibrinogen concentration; at 18 weeks the average value is 0.4 g/l but in many of the samples studied no clottable material (i.e., fibrinogen) was extractable at all, while at 35 weeks it had risen to about 2 g/l. One inevitable hemorheological consequence of these changes in the concentrations of fibrinogen and other plasma proteins is that the plasma viscosity rises from 0.9 to 1.05 mPa.s over the same period. Another is that the viscosity of the blood rises very substantially. Thus at a high shear rate (128.5 s-1) it increases, on average, from 2.5 to 3.7 mPa.s, while at a low shear rate (0.277s-1) it rises from about 3 to 14 mPa.s. What is clear from these data is that at 18 weeks there is virtually no shear thinning associated with the blood


39

of the fetus (i.e. it has almost Newtonian characteristics); this flow behavior is due largely to the extremely low levels of fibrinogen and other rouleaugenic proteins in the blood. There are also more subtle changes with potentially hemorheological consequences. One is the large size of the red cells of the very early fetus. At week 24 the MCV is 135 fl and, although it decreases with gestational age, even at term it is still about 119 fl and thus larger than that of the adult [12]. This, of course, has potential consequences to the resistance to flow in the microcirculation. There are also subtle differences in the erythrocyte membrane composition compared to that of the adult, with these differences affecting the mechanical properties of the membrane [13] and the innate ability of the fetal erythrocytes to form rouleaux. The hemoglobin at this stage is predominately HbF, rather than the adult form HbA1, but its intracellular concentration (i.e., MCHC) changes little from 24 weeks of gestation to term, being about 31g/dl and therefore a little less than that of the adult [12]. Another significant difference in the fetus is that the fibrinogen molecules are more strongly sialinated than in the adult [14], and although the level of sialination steadily decreases towards term, it is still higher at term than that of the adult. This increased sialination hinders its ability to induce rouleaux formation, and thus fetal blood overall exhibits less rouleaux formation than the adult, the blood shows much less shear thinning, and the overall viscosity is low compared with the adult. This raises a potentially important question to those involved with in utero transfusion: is it reasonable to transfuse adult blood which has such different hemorheological properties from that of the fetus? 2.2. At Birth A comprehensive study has been made of the hematological/hemorheological characteristics of the blood of the pre-term babies, as early as 24 weeks, and of the term baby [15]. As with the fetal data discussed above, the results show a steady increase in most hematological parameters with gestational age. Thus, hematocrit rises on average from 45 to 50 % with large variations between individuals, plasma protein concentration changes from 44 to 55 g/l, with increases in IgG, D2macroglobulin and fibrinogen from 2.1 to 2.6 g/l. Comparing these values with those obtained in utero at a corresponding gestational age, it is clear that the ex utero values are higher; indicating a degree of hemoconcentration takes place during birth. The result of the aforementioned changes is that the viscosities of both plasma and blood, at both high and low shear rates, increase over the same period: the shear thinning properties become more pronounced, with the magnitude of the changes being greater than found in the in utero study. Low shear rate (0.277 s-1) blood viscosity increases 3-fold from 12 to 40 mPa.s while the high shear rate (128.5 s-1) value increases by 20% from 3.7 to 4.5 mPa.s and plasma viscosity rises by 20% from 0.96 to 1.15 mPa.s. The alteration in the high shear rate viscosity can be explained almost entirely by changes in hematocrit and plasma viscosity over the period, whereas this explanation is not valid at low shear rate; here there are effects due to altered rouleaux formation secondary to cell-specific factors [9] and to the alteration in fibrinogen sialination that accompanies changes in gestational period.

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2.3. Childhood At birth the red cells are still relatively large compared to the adult and still contain predominantly HbF. Over the following 6 to 12 months the hemoglobin of the red cell population is progressively replaced by HbA and the red cells acquire a similar size to those of adults. There are no hematological differences between males and females in pre-pubescent children, but post puberty the red cell parameters slowly diverge as the male red cell count increases beyond that of the female to give the normal differences found in adult humans described above.

3. Changes of Hemorheological Significance Associated With Pathology Because of space limitations only a review of the general principles will be given here. 3.1. Plasma Proteins In healthy blood it is the plasma protein concentration, especially of the larger proteins, that is mainly responsible for the elevation of plasma viscosity above that of water, so conditions associated with dramatic changes in plasma protein composition can be expected to change the hemorheological properties of blood. Two obvious examples are Waldenstrom’s macroglobulinaemia and multiple myeloma. In these conditions the concentration of large macroglobulins increases massively and has a very large effect on plasma viscosity, often with up to a 20-fold increase in the viscometric effect of these proteins. Furthermore, many of the proteins in these conditions are abnormal and highly rouleaugenic. The consequent large increase in the strength and size of rouleaux leads to markedly increased low shear rate blood viscosity. These effects have deleterious effects on blood flow in all vessels of the circulation, especially those in the microcirculation [5]. As discussed above, the plasma protein with the greatest rouleaugenic effect is fibrinogen, and hence the relevance of plasma fibrinogen concentration increases in association with many clinical conditions. Examples are diabetes mellitus, hypertension, pregnancy, post surgical trauma, infection, etc. This means that the strength of rouleaux formation and the shear thinning of blood is frequently elevated in association with these conditions [16]. 3.2. Erythrocytes. It should by now be clear that the red cell is the most prominent hematological factor influencing hemorheology. Hence, diseases characterized by alterations in this area are generally associated with hemorheological abnormalities. The most obvious is anemia due to any cause, where the lowering of the whole blood hemoglobin concentration inevitably reduces the viscous effect of the red cell mass. The result is that, though anemia itself can have serious physiological effects, it does not bring hemorheological embarrassment. What makes anemia more of a hemorheological problem is that it is often complicated by associated changes in the individual red cells. For example, in iron deficiency anemia the cells become smaller than normal,


41

while in folic acid- or vitamin B12-deficiency anemia the cells increase in size. In thalassemia, the cells are again often smaller than normal and they are also less deformable. Such changes may have negative influences on microvascular flow. However, most significant of all in this respect is sickle cell anemia where the cells sickle in low oxygen conditions, become quite rigid and have very substantial problems negotiating the microcirculation. In addition, the repeated sickle-unsickle cycles during flow through the body eventually leads to their membranes becoming damaged and irreversibly rigid. As a consequence, the resistance to blood flow in large vessels is lower than normal due to the low hematocrit, while in the microcirculation it can be considerably higher, even leading to stasis . At the other end of the scale is the problem of increased red cell concentration. There are various pathologies that lead to this; examples are respiratory disease where the resultant hypoxia stimulates red cell production and polycythemia consequent to malignant changes in the hemopoietic tissue [17]. Inevitably, they lead to substantial increases in blood viscosity. Such changes can also occur in the neonate, in that excessive production of red cells can result from intrauterine hypoxia. However, the effects can also result from passive in utero events such as transfusion between mother and foetus, between twin and twin, or as a result of delayed clamping of the cord [18]. All of these phenomena cause the blood viscosity to increase considerably and in extreme cases can be very deleterious to blood flow. 3.3. Leukocytes As mentioned above, leukocytes in the healthy individual are not of sufficient concentration to affect whole blood viscosity, whereas they do have a significant resistive effect at the microcirculatory level due to their relative stiffness and enhanced viscoelastic characteristics compared to the erythrocyte. Nevertheless, there are conditions in which the leukocyte count can become extremely high: the most prominent are the classic leukemias. It is not unusual for the leukocyte count to increase by an order of magnitude or even more. Under these circumstances they are relevant to the viscosity of the whole blood and, because of their large volume concentration, relative inflexibility, irregular shape, and viscoelasticity, they can cause the viscosity to rise enormously and so affect large vessel flow. However, equally important is that now their resistive effect at the microcirculatory level far exceeds that of the red cell mass, especially as the deformability of leukemic leukocytes is less even than that of normal leukocytes [19]: all of these factors have very deleterious effects on tissue perfusion with obvious consequences [20].

4. Other Mammals Until fairly recently the study of hemorheology has concentrated on the human, and other mammals have largely been ignored. The situation is now beginning to change as investigators look to animal models for hemorheological studies not possible in the human, as veterinary workers begin to realize that the hemorheological effects in pathology found in the human may also exist in mammals of economic importance, and as investigators realize that the study of mammalian hemorheology may offer new insights into human hemorheology, especially in pathological states. Thus

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fascinating data are beginning to emerge showing that some hematological/ hemorheological parameters seem to be held remarkably constant across the whole class of mammals, while others vary widely. 4.1. Plasma Proteins Comparative data on variations across the mammalian kingdom of the concentrations of the different plasma proteins are difficult to locate, but data from Johnn, et al. [21] indicate that they are likely to vary significantly between mammals. In their study, 31 different mammalian types were investigated with results indicating that the plasma viscosity varied widely: at one extreme were the New Zealand rabbit and the cheetah with values of 1.1 mPa.s and 1.2 mPa.s respectively, and at the other the chimpanzee and goat each with a value of 1.6 mPa.s. These variations were only partly explained by variations in the concentrations of fibrinogen, because while the rabbit had one of the lowest fibrinogens at 2.4 g/l, that of the cheetah was relatively high at 3.8 g/l. Similarly, that of the goat was high at 5.4 g/l while that of the chimpanzee was fairly average at 3.0 g/l. It is notable that these fibrinogen levels are all overshadowed by the level in the black buck which was recorded as 7.0 g/l. Plasma protein data on nine different domestic mammals from Windberger, et al. [22] are useful since they show that total plasma protein concentrations varied from as low as 43 g/l in the mouse to as high as 72 g/l in the cat and in cattle. However, they found only a partial correlation between plasma protein concentration and plasma viscosity. 4.2. Erythrocytes The most important finding here is the remarkable constancy of the hematocrit and whole blood hemoglobin across mammals. In the study by Johnn, et al. [21] mentioned above, they found the lowest hematocrit in the goat at 31% and the highest in the wallaby at 53%. However, these are extremes and when the values for the 31 mammals studied were averaged the mean and standard deviation were 40r5%, indicating the narrowness of spread across the class as a whole. Whole blood hemoglobin shows an even narrower range judging by data from Gascoyne and Hawkey [23] who show a mean and SD of 14.8r2.1 g/l. This seems remarkable considering the large variation in size, heart rate, life style, etc. between the animals. On the other hand, they found enormous differences in MCV, with the goat having the smallest values at 22 fl and the Asian elephant had 6-fold greater value of 120 fl! Windberger, et al. [24] have shown that for the African elephant the MCV is even larger at 138 fl. In spite of this, the mean cell hemoglobin concentration is remarkably conserved. Gascoyne and Hawkey [23] found a coefficient of variation of only 8.5% (34.6r3.0, mean±SD). In view of the relative constancy of hematocrit and whole blood hemoglobin concentration, this narrow range of hemoglobin means that the red cell count varies by a similar order of magnitude to that of the MCV across the mammals. There is no obvious connection between the MCV and the size, heart rate, life styles, etc. of the mammals studied. Johnn, et al. [21] measured the high and low shear rate (128.5 and 0.222 s –1 respectively) blood viscosity characteristics of the 31 mammals they investigated, and found that high shear rate values varied by some 2-fold, with this variation


43

almost totally explained by the variations in hematocrits and plasma viscosities between the mammals. On the other hand, the low shear rate values varied by some 18-fold and this could only be partly explained by the hematocrit and plasma viscosity differences. The other factor of great relevance here is the variation in the inherent rouleaugenicity of the red cells from the various mammals. Thus, for example, in sheep and cattle, rouleaux formation is non-existent, while in the horse and elephant it is very increased such that if it occurred in a human it would be considered pathological. The reasons for the differences may be partly due to different levels of fibrinogen in the plasma of these animals, but it is now becoming increasingly clear that there must be cell specific factors involved as well [9]. That is, in the non-rouleaux forming animals such as sheep, some peculiarity of the membrane structure inhibits rouleaux formation. This has been shown by Kaibara [25] who found that bovine red cells would not form rouleaux in their own plasma, but if they were treated with trypsin to remove significant surface protein they would then form intense rouleaux. Furthermore, in animals where rouleaux formation is massive such as the horse, it is to be expected that membrane factors will be found to be facilitate aggregate formation. Further study in this area is clearly warranted. 4.3. Breed Variation The data discussed above are clear in showing large differences of some hemorheologically important variables between mammals. What is also of interest is to what extent these variable values are conserved across breeds within an individual mammal species. Such studies are rare; indeed, the only one that the author has found is a study of nine different breeds of dog [26]. Very substantial differences in whole blood viscosity were found in this study, so high shear rate (128.5 s-1) values varied by 50% between the highest and the lowest, while the low shear rate (0.277 s-1) values varied by 140%. Of the breeds, it was the more athletic, that undergo brief periods of intense anaerobic exercise, which had the highest values (e.g., greyhounds and deerhounds), while the golden retriever and Springer spaniel were at the other end of the range. However, the differences could be explained entirely by the different hematocrits between the dogs, with such other potential factors such as cellular parameters and plasma composition not being significantly different between the breeds. In this context, an interesting study was done by Anwar and Rampling [27] who compared the hematological/ hemorheological parameters of five different ethnic groups of humans. The only significant difference found between the groups was that Caucasians had significantly lower concentrations of plasma proteins and hence lowered plasma viscosity.

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S.I. Fox, Heart and circulation, Chapt. 13, In: Human Physiology, Times Mirror, Dubuque, 1996, pp. 342-385. H.J. Meiselman, E.W. Merrill, E.R. Gilliland, G.A. Pelletier and E.W. Saltzman, Influence of plasma osmolarity on the rheology of human blood, J. Appl. Physiol. 22 (1967), 772-781. P.W. Rand, W.H. Austin, E. Lacombe and N. Barker, pH and blood viscosity, J. Appl. Physiol. 25 (1968), 550-559. G.D.O. Lowe and J.C. Barbenel, Plasma and blood viscosity, In: Clinical Blood Rheology, Vol I, G.D.O. Lowe, Ed., CRC Press, Boca Raton, (1988), pp 11- 44.

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[22]

[23] [24] [25] [26] [27]


T. Somer, Rheology of paraproteinaemias and plasma hyperviscosity Syndrome, Bailliere’s Clin. Haematol. 1 (1987), 695-723. M.W. Rampling, J.R. Brown, R.J. Robinson, M.D. Feher, S. Cholerton and P.S. Sever, The short term effects of abstention from tobacco by cigarette smokers on blood viscosity and related parameters, Clin. Hemorheol. 11 (1991) 441-446 M.W. Rampling, Red cell aggregation and yield stress, In: Clinical Blood Rheology, Vol I, G.D.O. Lowe, Ed., CRC Press, Boca Raton, (1988), 45-64. J.V. Dacie and S.M. Lewis, Practical Haematology, Churchill Livingstone, London (1995) pp. 1217. M.W. Rampling, H.J. Meiselman, B. Neu and O.K. Baskurt, Influence of cell-specific factors on red cell aggregation, Biorheology, 41 (2004), 91-112. S. Chien, White blood cell rheology, In: Clinical Blood Rheology, Vol I, G.D.O. Lowe, Ed., CRC Press, Boca Raton, (1988), 87-109. C.R. Welch, M.W. Rampling, M.A. Anwar, D.G. Talbert and C.H. Rodeck, Gestational reference ranges for fetal haemorheological parameters, Clin. Hemorheol. 14(1994), 93-103. D.G. Nathan and F.A. Oskai, Hematology of Infancy and Childhood, Vol 1, WB Saunders Co, Philadelphia (1990) p 30. O. Linderkamp, G.B. Nash, P.Y.K. Wu & H.J. Meiselman, Deformability and intrinsic material properties of neonatal red blood cells, Blood 67(1986), 1244-1250. J.L. Francis and D.J. Armstrong, Sialic acid and enzymic desialination of cord blood fibrinogen, Haemostasis 11 (1982), 223-228. M.A. Anwar, M.W. Rampling, S. Bignal and R.P.A.Rivers, The variation with gestational age of the properties of the blood of the new-born, Brit. J. Haematol. 86 (1994), 163-168. T. Somer and H.J. Meiselman, Disorders of blood viscosity, Ann. Med. 25 (1993), 31-39. T.C. Pearson and G. Wetherley–Mein, The course and complications of ideopathic erythrocytosis, Clin. Lab. Haematol. 1 (1979), 189-196. E.A. Letsky, Fetal and neonatal transfusion, Brit. Med. J., 300(1990), 862-866 M.A. Lichtman and G.A. Kearney, The filterability of normal and leukemic leukocytes, Blood Cells 2 (1976), 491-501. M.W. Rampling, Hyperviscosity as a complication in a variety of disorders, Semin. Thrombos. Haemostas. 29 (2003), 459-465. H. Johnn, C. Phipps, S. Gascoyne, C. Hawkey and M.W. Rampling, A comparison of the viscometric properties of the blood from a wide range of mammals, Clin. Hemorheol. 12 (1992), 639-647. U. Windberger, A Bartolovitsch, R. Plasenzotti, K.J. Korak and G. Heinze, Whole blood viscosity, plasma viscosity and erythrocyte aggregation in nine mammalian species: reference values and comparison data, Exp. Physiol. 88 (2003), 431-440. S.G. Gascoyne and C. Hawkey, Patterns of variation in vertebrate haematology, Clin. Hemorheol. 12 (1992), 627-637. U. Windberger, R. Plasenzotti and Th. Voracek, The fluidity of blood in African elephants (Loxodonta africana), Clin. Hemorheol. 33 (2005), 321-326. M. Kaibara, Rheological behaviors of bovine blood forming artificial rouleaux, Biorheology 20 (1983), 583-592. A.R. Bodey and M.W.Rampling, A comparative study of the haemorheology of various breeds of dog, Clin. Hemorheol. 18 (1998), 291-298. M. Anwar and M.W. Rampling, Comparative haemorheology of five healthy, ethnically diverse groups: results of a pilot study, Clin. Hemorheol. 14 (1994), 697-707.

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Macro- and Micro-Rheological Properties of Blood b,

Giles R. COKELETa and Herbert J. MEISELMAN 1 Department of Chemical and Biological Engineering, Montana State University, Bozeman, MT 59717, USA and bDepartment of Physiology and Biophysics, Keck School of Medicine, University of Southern California, Los Angeles, CA, 90033, USA. a

Introduction In order to quantitatively understand the conditions of blood flow through various in vitro and in vivo geometries, the flow properties of blood must be experimentally determined. In this chapter, we initially consider the rheological behavior of blood under conditions where the blood is treated as a homogenous fluid and thus where the formed elements (e.g., red blood cells, white blood cells) are tacitly ignored. This approach is then modified in order to consider flows where the blood cell characteristic dimensions approach those of the geometries in which the flow takes place. The former approach yields the macro-rheological properties of blood while the latter yields micro-rheological characteristics; in general, data obtained in geometries of 200 μm or less are considered micro-rheological. It is of interest to note that the study of blood rheology dates, at least, to the work of Poiseuille who attempted to derive an equation for blood flow in tubes. However, due to experimental difficulties associated with blood coagulation he was unsuccessful with these attempts, and thereafter turned to simpler fluids such as water and oil to develop his well-known equation [1].

1. Special Problems in Measuring Blood Rheology Prior to discussing specific rheological properties, it is of value to discuss some difficulties and potential artifacts associated with rheological measurements on blood and related fluids. Unless otherwise stated, it is the authors’ belief that all data presented herein are unaffected by such experimental problems. There are three properties of blood, and related fluids, which can lead to difficulties in measuring rheological properties: (1) The plasma proteins are surfactants and form a protein layer or film at fluid-air interfaces; (2) The erythrocytes are denser than plasma; (3) At low shear rates, the erythrocytes aggregate. The surfactant layer formed by plasma proteins at the fluid-air interface has mechanical strength and is thus a semi-rigid film. Consequently, the pressure drop 1 Corresponding author: Department of Physiology and Biophysics, Keck School of Medicine, University of Southern California, Los Angeles, CA, 90033, USA; E mail: [email protected]

46

G.R. Cokelet and H.J. Meiselman / Macro- and Micro-Rheological Properties of Blood

across such interfaces, as in a capillary viscometer, is not zero. Likewise, the surfactant layer at an interface can transmit significant torque to or cause drag on the torquemeasuring element in rotational viscometers. In general, this effect is a significant problem only for plasma, serum and RBC-plasma or RBC-serum suspensions at very low hematocrits; at normal or high hematocrits, the additional torque due to the film is only a small percent of the total. An unfortunate result of this additional torque is the artifact of non-Newtonian behavior for plasma or serum (Figure 1); data obtained at high shear rates and thus high shear stresses yields the true Newtonian viscosity. 7

PLASMA VISCOSITY (mPa.s)

6

DONOR A DONOR B

5 4 3 2 1 0

0

500

1000

1500

2000

-1

SHEAR RATE (SEC )

Figure 1. Viscosity-shear rate data for human plasma which shows protein film artifacts. These results were obtained in a cone-plate viscometer in which the film added extra drag at the radius of the cone. The apparent non-Newtonian behavior is the result of this added drag; dashed lines indicate viscosity values obtained by extrapolation of viscosity to infinite shear rate (i.e., to the reciprocal of shear rate = zero).

Procedures to minimize or eliminate the influence of the protein layer depend on the type of viscometer used for the measurements. In the case of capillary viscometers, one can use fluid filled pressure transducers that connect to the fluid reservoirs below the fluid-air interfaces, or can obtain pressure drop and flow rate data using two capillaries of the same diameter but significantly different lengths. In the latter case, one subtracts the lengths and pressure drops for the two capillaries, at a selected flow rate, to obtain a pressure drop free of the effects of the surfactant interfaces. In the case of rotational viscometers, one useful approach is to insert a “guard” ring through the interface, thereby isolating the torque-sensing element from the film-associated drag. Note, however, that the guard ring must rotate at the same angular velocity as the torque-measuring element of the viscometer, and is thus not useful for cone-plate viscometers in which the cone both rotates and senses torque. Alternatively, one can use two bobs that are of different lengths or two cones that are of different radii; in this method, a subtraction technique will eliminate the surfactant effects if the interface is the same for the two bobs or cones.


47

Red cell sedimentation is slow for individual cells, since the density difference between the cell (1.096 gm/ml) and the plasma (1.025 gm/ml) is small. Consequently, the sedimentation rate of an individual erythrocyte in plasma is only about 0.13 μm/minute or 0.08 mm/hr. If the suspending medium is serum and thus does not promote RBC aggregation, and if the RBC volume fraction H (i.e., hematocrit) is 15% or greater, the presence of neighboring red cells causes the settling to be hindered and hence slower. For hindered settling free of wall effects, the sedimentation rate (U, μm/min) is given by the equation where H is hematocrit [2]: U = 0.13 (1 – H)4 40 50 uM ID 100 uM ID 500 uM ID

HEMATOCRIT INCREASE (%)

35 30 25 20 15 10 5 0

0

5

10

15

20

25

30

DISTANCE SETTLED (uM)

Figure 2. Calculated increase of hematocrit of the settled RBC phase for non-flowing blood in horizontal tubes. Horizontal axis is the distance RBC settle from the top inside wall of the tube. Minimal effects are noted for large tubes whereas large increases can occur in smaller tubes.

RBC aggregation, as seen in plasma at stasis or slow flow, greatly increases the sedimentation rate since the settling rate of particles is dependent on the square of the particle size. A typical, steady, hindered settling rate for H=40% normal human blood is about 200 μm/min or about 10 mm/hr. Usually, sedimentation in viscometers is not a significant problem if the wall shear rate is above about several inverse seconds. However, at lower shear rates, the effect of erythrocyte sedimentation must be considered, especially for horizontal tube and cone-and-plate viscometers. Figure 2 presents calculated percent increases of hematocrit versus settled distance in 50, 100 and 500 μm tubes. It is obvious that minor settling has almost no effect in large tubes whereas even a 10 μm settling (i.e., about one RBC diameter) markedly elevates hematocrit in smaller tubes. Erythrocyte aggregation also causes another effect termed red cell syneresis. This phenomenon is the inward movement of RBC as they aggregate, leaving layers of cellpoor plasma at the walls of viscometers of all types [3]. Although syneresis is usually considered to be a slow process (e.g., separation of fluid from a gel), the effects of

48


syneresis are seen very soon after starting the viscometer. This has led to conflicting data for blood’s rheological properties for shear rates below about 2 s-1. For example, consider the events that occur when using a concentric cylinder viscometer filled with well-mixed and well-stirred blood: upon starting the viscometer at a rotational speed corresponding to a steady shear rate of about 2 s-1 or less, the torque-time record of the viscometer first rises from zero to a peak, then decays in an exponential manner with time, finally reaching an essentially steady value after approximately 30 minutes. This time-dependent torque response is due to the startup of the fluid motion at very short times, syneresis of the erythrocytes away from the viscometer surfaces at all times, and finally, at very long times, sedimentation effects. Note that essentially all investigators agree that using the final steady-state value of torque is incorrect: it is the result of fully completed syneresis plus a variable degree of sedimentation: its use results in shear stress and viscosity values that are too low. 1000

RELATIVE VISCOSITY

PEAK TORQUE EXTRAPOLATE

100

10 0.001

0.01

0.1

1

10

-1

SHEAR RATE (SEC )

Figure 3. Viscosity-shear rate results for normal human blood at a hematocrit of 40%. Two different methods were used for assigning the correct torque at low shear rates. For shear rates below 1 s-1 the use of peak torque underestimates viscosity, with the difference becoming greater with decreasing shear. Peak torque values from [4, 5] and extrapolated to zero time from [6, 7].

Two different methods have been proposed for assigning the correct torque at shear rates below about 2 s-1: Chien’s approach uses the peak torque value to calculate the apparent viscosity of blood [4, 5], while Merrill and co-workers extrapolate the exponentially decaying torque data back to zero time and use that torque value to calculate apparent viscosity [6, 7]. The rationale for the latter procedure is that during the time of the exponential torque decay, only the syneresis effect is a significant cause for the decreasing torque behavior; extrapolation back to zero time assumes that the syneresis effect is the same in nature even in the earlier, pre-peak time period. Figure 3 shows a comparison of the results of the two methods for a normal blood at a 40% hematocrit, where it can be seen that the two methods diverge at shear rates less than 1 s-1; results are usually coincident at higher shear.


49

2. Macro-Rheological Behavior of Normal Human Blood Literature reports describing the macrorheological behavior of human blood are numerous [e.g., 8-16] and may be consulted for additional details; reviews of the development of hemorheology [17] and of the molecular biology aspects of hemorheology [18] also exist. In general, it is agreed that normal human blood or reconstituted RBC-plasma suspensions are non-Newtonian fluids without measurable normal stresses. In addition, for the usual time scales encountered in macrorheology studies, rheological properties are history and time independent (i.e., no evidence of thixotropy or rheopexy). That is, although RBC disaggregation and RBC deformation occur during establishing viscometric flow and do require small but finite periods of time, their time scales are relatively short and thus do not affect steady shear results. 50

SALINE PLASMA

VISCOSITY (mPa.s)

40

30

20 RIGID CELLS 10

0 0.1

1

10

100

1000

10

4

-1

SHEAR RATE (SEC )

Figure 4. Typical viscosity-shear rate results for three RBC suspensions all at 40% hematocrit: 1) RBC in plasma; 2) RBC in saline; 3) chemically-fixed rigid RBC in saline. RBC aggregation increases viscosity at lower shear rates for the RBC in plasma suspension, with cell deformation occurring at higher shear rates. RBC-plasma and RBC-saline data are not coincident at high shear due to different suspending phase viscosity.

Figure 4 presents typical viscosity-shear rate results for three RBC suspensions, all at a hematocrit of 40%: 1) RBC in plasma; 2) RBC in isotonic buffer; 3) RBC made extremely rigid by fixation in glutaraldehyde and suspended in isotonic buffer. The observed hematocrit of the rigid cell suspension was 66% which, due to poor packing of rigid cells, corresponds to a true hematocrit of 40% [19]. Dealing first with the RBC-plasma suspension, the feature most notable is the strong effect of shear rate on viscosity and hence the marked non-Newtonian flow behavior. Over the shear rates shown (0.5 to 1,500 s-1), there is an 11-fold decrease in blood viscosity, with essentially Newtonian behavior at or above 1,000 s-1; the exact shear rate necessary for Newtonian flow is hematocrit dependent, with higher shear needed at higher hematocrits. In contrast, the RBC-buffer suspension is much less shear dependent, with only a 2-fold

50


decrease over the range of shear rate and Newtonian flow expected prior to 1,000 s-1. Since normal RBC in plasma undergo reversible aggregation (i.e., rouleaux formation, see chapter II.4.b) while those in buffer do not, RBC aggregation is the primary determinate of low shear blood viscosity. With increasing shear forces, the RBC aggregates in the plasma suspension become progressively dispersed and the viscosities approach each other. Note that since plasma has a higher viscosity than buffer, the two curves in Figure 4 will not coincide at higher shear but would do so if suspending viscosities were equalized [9, 11]. The data for the rigid, non-deformable RBC suspension in Figure 4 demonstrates the role of RBC deformability as a determinant of blood rheology. Since unlike normal cells, rigid RBC are unable to deform in response to shear forces, increases of viscosity at high shear are ascribed to decreases of RBC deformability. Dintenfass [20], and subsequently other investigators, have utilized this behavior at higher shear (i.e., > 200 s-1 or sufficient to disperse RBC aggregates) to assess RBC rigidity via a calculated “Tk” parameter: Tk = [(Kr0.4 – 1) / Kr0.4 ] / H where Kr is relative viscosity (i.e., suspension viscosity divided by suspending medium viscosity) and H is hematocrit. In overview, normal human blood behaves as a shear thinning non-Newtonian fluid, with this behavior generally ascribed to two processes [14, 15]: 1) RBC aggregates formed at stasis or low shear are dispersed by increasing fluid shear forces, with complete dispersion achieved in the range of 80-120 s-1; 2) RBC undergo deformation and alignment with flow at medium to high shear rates. Thus, as shown in Figure 4, increases of viscosity at low shear usually implies enhanced RBC aggregation while increases at high shear are ascribed to decreased RBC deformability. 2.1. Hematocrit Effects Not surprisingly, the volume fraction of particles in a suspension affects its rheological behavior. In the case of normal blood, this volume fraction is represented by the volume fraction of RBC (i.e., hematocrit); unless aggregated, platelets are too few in number and volume to affect viscosity, and white cells only influence viscosity when their volume fraction is abnormally elevated [21]. Figure 5A presents blood viscosity data for hematocrits of 20, 30, 40 and 50% over a 200-fold range of shear rate (i.e., 0.1 to 20 s-1), and Figure 5B presents viscosity at selected shear rates for the four hematocrits. It is obvious from these figures that hematocrit affects both viscosity values and the degree of non-Newtonian behavior. At the lowest shear rate shown, a hematocrit change from 20 to 50% increases viscosity by about 9-fold whereas at the highest shear the same hematocrit change causes a 3-fold increase. Deviation from Newtonian behavior is also hematocrit dependent: at 20% hematocrit there is a 6-fold decrease in viscosity over 0.1 to 20 s-1 whereas at 50% the decrease is about 20-fold.

G.R. Cokelet and H.J. Meiselman / Macro- and Micro-Rheological Properties of Blood 120

20% 30% 40% 50%

VISCOSITY (mPa.s)

100

80

60

40

20

0

0.1

1

10

100

-1

SHEAR RATE (SEC )

Figure 5A. Effect of hematocrit on viscosity-shear rate data for normal human RBC suspended in autologous plasma. Note that hematocrit affects both viscosity and the degree of non-Newtonian behavior.

120 -1

0.11 SEC 0.51 SEC -1

100

VISCOSITY (mPa.s)

-1

2.0 SEC 20 SEC -1

80

60

40

20

0 0

10

20

30

40

50

60

HEMATOCRIT (%)

Figure 5B. Hematocrit-viscosity data at four different shear rates for normal human RBC suspended in plasma. The effects of hematocrit changes are greater at lower rates of shear. Data are from Figure 5A.

51

52


2.2. Yield Shear Stress The reversible RBC aggregation observed in normal blood is similar to that for particles found in many other suspensions (e.g., clay), thus prompting Merrill and coworkers to consider the existence of a yield shear stress for human blood [22, 23]. These investigators utilized a data-fitting model developed by Casson [24] for pigmentoil suspensions used as printing inks: W1/2 = aJ1/2 + Wy1/2 where W is shear stress, J is shear rate and Wy is yield shear stress. Figure 6A presents a so-called “Casson Plot” for the experimental data shown in Figure 5A. It is clear that for each hematocrit the data can be fitted by linear regression (r>0.99), and extrapolation to zero shear rate yields the yield shear stress. As expected [22], the yield stress is strongly hematocrit dependent, with the relationship well-fitted by a power law equation (Figure 6B). At constant hematocrit, the yield stress increases with the strength of RBC aggregation [23], and has been used by some investigators as an index to red cell aggregation in disease and for cells in various polymer solutions [26, 27].

1/2

SQUARE ROOT SHEAR STRESS (mPa )

14 20% 30% 40% 50%

12 10 8 6 4 2 0

0

1

2

3

4

5

SQUARE ROOT SHEAR RATE (SEC -1/2)

Figure 6A. Shear rate-shear stress data for normal human RBC in plasma. Note that the square root of shear rate and shear stress are on linear axes (“Casson” plot). Straight lines obtained via linear regression (r>0.99) with their intercepts being the square root of yield shear stress. Data are from Figure 5A.

Using the Casson Equation to determine yield stress requires shear rate-shear stress data at very low rates of shear (e.g., 20 s-1 and lower) in order to obtain a linear fit in the region near zero shear. Application of the equation to data obtained at higher shear rates is possible, but extrapolation to zero shear will result in incorrect values of yield stress. Although the data shown in Figures 5 and 6 were obtained with a rotational


53

viscometer, direct determinations of yield stress in a tube viscometer (i.e., residual pressure drop at zero flow) are consistent with the results presented herein [27]. Note that both the Casson extrapolation method and the tube viscometer data were obtained going from low shear to zero shear; measurements made starting from zero shear are fraught with artifacts, including RBC settling or syneresis in tubes (Figure 2). 10

YIELD STRESS (mPa)

8

6

4

2

0

15

20

25

30

35

40

45

50

55

HEMATOCRIT (%)

Figure 6B. Yield shear stress results obtained from Figure 6A plotted versus hematocrit. Curved line through points obtained using power law regression (exponent = 3.1, r = 0.999) in agreement with [22].

2.3. Cellular Determinants of Low Shear Viscosity Low shear viscometry is a technique commonly used to assess RBC aggregation [9, 12], and its use follows from the classical description of the factors that affect the low and high shear rate rheological behavior of RBC suspensions [9]: at constant hematocrit and temperature, low shear blood viscosity is primarily determined by RBC aggregation while high shear viscosity is dependent on RBC deformability (Figure 4). It is thus tacitly assumed that comparisons of blood viscosity at low shear will reflect relative degrees of red cell aggregation. Experimental findings now exist that challenge the unique contribution of aggregation to low shear viscosity. One approach is the use of chemical agents that produce a dose-dependent, constant volume change of RBC shape from the normal biconcave morphology either to a crenated, spiky, spherical form or to a mono-concave form [28]. Coincident with such dose-related shape changes are progressive decreases of RBC deformability: rather than causing complete cell rigidity (e.g., Figure 4), the cells are less deformable at low stress levels yet deform normally at higher stress levels [29]. Figure 7 presents rheologic data for crenated RBC, suspended in buffer, at increasing levels of the shape-altering agent (DNP, di-nitrophenol) and hence at greater extents of morphology changes. Microscopic examination of these suspensions indicated absolutely no evidence of RBC aggregation, yet there are marked, dose-

54


dependent increases of low shear viscosity: 5 mM DNP causes a 2-fold increase at 37.5 s-1 and a 15% increase at 150 s-1, whereas at 1,500 s-1 the data are identical with control. Based solely on the usual interpretation of such data [9, 11], DNP would be considered to be an agent causing aggregation that, with sufficient shear stress, is dispersed: in fact, no aggregation occurs and increasing shear serves to deform the cells, eventually resulting in a normal deformed appearance [29]. Similar observations have been reported for RBC treated with hydrogen peroxide or very low levels of glutaraldehyde, or heated to 48 qC: either unaltered or very slight changes of aggregation in plasma yet markedly increased low shear viscosity [30]. 35 CONTROL 1 mM DNP 2 mM DNP 5 mM DNP

VISCOSITY (mPa.s)

30 25 20 15 10 5 0

10

100 -1 SHEAR RATE (SEC )

1000

Figure 7. Viscosity-shear rate data for human RBC re-suspended at 60% hematocrit in saline containing increasing concentrations of the shape-altering chemical di-nitrophenol (DNP). RBC do not aggregate in DNP-saline solutions but deformability at low shear stresses is reduced; normal deformation occurs at high shear stress [29].

2.4. Constitutive Equations The relationships between shear stress and shear rate for blood must be determined experimentally and expressed as mathematical equations, usually referred to as constitutive equations. Given the complex macro-rheological behavior of blood described above, it is not surprising that a single equation fails to completely describe the effects of various rheological variables (e.g., hematocrit, shear rate). Thus, several approaches to defining these equations exist, with some the result of curve-fitting experimental data and others based on a particular rheological model. Chapter II.1 (Basic Aspects of Rheology) presents constitutive equations of two types. One type results from using empirical relationships and includes Newtonian fluids, the Bingham fluid model and the power-law fluid model. All of these use the ratio of shear stress to shear rate to define an “apparent viscosity” for blood that, with the exception of Newtonian fluids, is a function of shear rate. The other type results from the use of models of suspensions and includes an Einstein-model approach based


55

on the volume fraction of the suspension occupied by particles (e.g., hematocrit), the Casson Equation detailed above in section 2.2, and an equation developed by Quemada. While the Einstein model is only useful for suspensions having low volume fractions of particles and the Casson Equation is really only applicable at very low shear rates, the one by Quemada accurately fits blood data over a very wide range of shear. Several other approaches to constitutive equations for blood have been published (e.g., [31-36]), a comparison of constitutive equations has been presented by Easthope and Brooks [37], and a microstructure-based constitutive equation, developed by Owens and co-workers, has been applied to the pulsatile flow of blood [38].

3. Micro-Rheological Properties of Human Blood As the characteristic dimension of a flow channel approaches the size of the particles in a suspension, one should expect that the simple continuum model of the suspension will fail to be applicable. Often, this limit of the applicability of the continuum model begins to manifest itself at characteristic channel dimensions that are about 30 times the particle diameter: in the case of blood with a characteristic RBC dimension of 8 μm, an apparent failure occurs at about 300 microns. This was demonstrated by Fåhraeus and Lindqvist [39], who found that the apparent viscosity of blood was a function of tube diameter, for diameters of 300 microns and less, when they flowed constant-hematocrit blood from a well-stirred reservoir through a tube. Subsequent studies by others showed that the hematocrit of the blood flowing from the tube is equal to that of the blood in the feed reservoir, as long as the tube diameter is 20 microns or greater [40, 41]. This means that, at steady state, there is no accumulation or loss of erythrocytes from the blood as it flows through the tube. However, Fåhraeus earlier had shown that the average hematocrit of the blood in the tube was less than that of the blood in the feed reservoir [42]. The practical consequence of these findings is that the local hematocrit across the cross section of the tube is not uniform, with the hematocrit highest near the center of the tube where the velocity is highest. All of these factors are discussed in more detail in the material below. 3.1. The Fåhraeus Effect The finding that, for blood flowing steadily in tubes with diameters of less than 300 microns, the average hematocrit of the blood in the tube is less than the hematocrit of the blood in the reservoir feeding the tube is known as the Fåhraeus Effect. This effect is generated in the concentration entrance length of the tube, in which erythrocytes move towards the central region of the tube as they flow downstream. This entrance length is estimated to be about the distance that the blood travels in a quarter of a second for blood where red blood cell aggregation is negligible and the vessel diameter is greater than about 20 microns. In a survey of all the data available at the time, Goldsmith et al. [43] compiled the information about the Fåhraeus Effect, and their results are shown in Figure 8.

56


D, μm Figure 8. Relative tube hematocrit as HT/HD (i.e., tube /discharge hematocrit ratio) versus tube diameter D in microns. The shaded area contains all the data in the literature as of 1988. The data points are the original data of Fåhraeus [39]. All flow rates are sufficiently high to prevent red cell aggregation, and the feed hematocrits were 40 - 45%. HT is the average hematocrit of the blood in the tube, and HD is the hematocrit of the blood leaving the vessel which, at steady state, also equals the hematocrit of the blood actually entering the tube. Reproduced with permission from [43].

In capillaries of diameters below about five microns, the red cells flow down the tube in single file and the erythrocyte tends to fill the lumen [44]. As a consequence, HT/HD approaches unity as the capillary diameter approaches 2.7 microns, the minimum diameter tube in which a human erythrocyte can fit without breaking its membrane. A detailed graph of the literature data, for three feed reservoir hematocrits, is presented in Pries et al. [45]. Erythrocytes are not the only blood cells that show the Fåhraeus Effect. White cells and platelets also have such an effect, but it is qualitatively different. For white blood cells, the radial distribution depends on the blood flow rate: at high flow rates they tend to be in the central regions of the flow, whereas at low flow rates, when erythrocytes can aggregate, the white cells are pushed to the wall [46, 47]. For the platelets, they are always found in higher concentrations near the tube wall, and so, at low flow rates, have their concentration in the tube as high as 1.3 - 1.4 times their concentration in the blood leaving the tube, and about a 1.1 times greater at high blood flow rates [43]. The general qualitative rule for blood flow in tubes is that the largest particles are in the central region of the flow, and the smallest particles are near the wall. At high shear rates (i.e., flow rates), the red cells and white cells prefer to be in the faster flowing regions of the tube, but at low flow rates, where red cell aggregation occurs, the red cell aggregates are the biggest particles in the flow, and so they prefer to be in the central region, displacing the white cells towards the perimeter of the flow. For details on the mechanisms of particle radial migration, see Goldsmith [48, 49].


57

3.2. The Fåhraeus -Lindqvist Effect The finding that for small tubes with diameters below about 300 microns and for faster flow rates which do not allow appreciable erythrocyte aggregation, the effective viscosity of the blood depends on tube diameter is known as the Fåhraeus-Lindqvist Effect. The effective viscosity is determined from the Hagen-Poiseuille equation using the measured pressure drop, the flow rate, and tube dimensions. In Figure 9 the relative effective viscosity of the blood (i.e., effective viscosity divided by the plasma viscosity) for such relatively fast flows is indicated by the shaded area, which includes all literature data. The filled squares are the original data of Fåhraeus and Lindqvist [39]. Also shown are data for slow flows, where red cell aggregation can occur.

Figure 9. The Fåhraeus-Lindqvist Effect. The shaded area contains all the published data for normal human blood flowing at high rates, with the filled squares being the original data of Fåhraeus and Lindqvist [39]. Top upward arrow indicates asymmetric syneresis and bottom downward arrow indicates axisymmetric syneresis; vertical dashed line indicates critical diameter for RBC entry (see text for other curves). Reproduced with permission from [43].

In Figure 9 the top curve from Reinke et al. [50] indicates the relative effective viscosity of blood for very slow flow in horizontal tubes where red cell aggregation and sedimentation occur; average wall shear rates were in the range of 1.6 to 4.0 s-1. At steady state, one anticipates that red cells have settled on the bottom of the horizontal tube, causing a high hematocrit there and consequently a very low flow rate, while above this more concentrated hematocrit region, the flow is faster because the

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hematocrit will be lower than in the settled region. The average hematocrit in the tube will thus be higher than that of the blood flowing form the tube. The bottom two curves in this figure are for blood flowing slowly through vertically oriented tubes, with curve P indicating the minimum relative effective viscosity for slow flow in tubes, and curve D similarly indicating the minimum slow flow relative effective viscosity for blood to which dextran 250 kDa has been added to the plasma to enhance erythrocyte aggregation. With the vertical tube orientation, syneresis makes the red cells move into the core of the flow, leaving an appreciable plasma wall region [51], thereby resulting in greatly reduced resistance to flow. When dextran 250 is added to the plasma, the erythrocyte aggregation is enhanced, and the red cell aggregation is “tighter”, making the plasma wall layer even larger and reducing the flow resistance even further. A detailed graph of the literature data for the Fåhraeus -Lindqvist Effect is in Pries et al. [45]. Considering the Fåhraeus and Fåhraeus-Lindqvist Effects together, one asks whether or not the Fåhraeus-Lindqvist Effect is the result of the Fåhraeus Effect alone. This question has been approached in two ways: (1) In the first, a model of the blood flow is constructed with a wall layer of plasma and a core containing the red cells and plasma. The velocity profile calculated from this model is not parabolic due to the wall layer, but the average velocity of the blood must equal the measured flow rate, and the average hematocrit in the tube must match the measured Fåhraeus Effect. Using the measured Fåhraeus-Lindqvist Effect and the Fåhraeus Effect, the width of the plasma layer can be calculated from this model. Typically, this wall layer thickness is about 0.5 - 3 microns. In reality, the wall layer thickness varies in the axial direction [51], and so this thickness must be considered to be an average width. Accurate measurements of this wall layer thickness requires high speed optical recording of the flow. From such recordings, this average wall layer thickness is found to be comparable to that calculated with the model. This implies that the continuum model of blood can be used to predict these flows. (2) The other technique uses the Mooney-Rabinowitz-Weissenberg (MRW) method (see Basic Aspects of Rheology, chapter II.1) to analyze the flow data from small tubes. This method requires that all data, regardless of the tube diameter, must fall on one curve when plotted as the wall shear stress versus (8U/D), where U is the average flow velocity and D is the diameter of the tube. Consequently, if one so plots the data for flow in a small tube where the Fåhraeus and Fåhraeus-Lindqvist Effects occur, and compares those data with data for blood flow in a large tube with the same average hematocrit in the larger tube as in the small tube, all the data should fall on one curve. This approach was used to analyze data obtained for blood flow in a 29 micron diameter tube and the corresponding data from flow in an 811 micron tube, and the results are reproduced in Figure 10.


59

W w (dynes/cm2)

100

10

1

0.1 1

100

10 U

1000

(sec-1)

Figure 10. Use of the MRW method to analyze blood flow data of 29 and 811 micron diameter tubes, with the average hematocrit of the blood in the tubes being equal. Reproduced with permission from [52].

In Figure 10, the solid curves represent the data from the 811 micron diameter tube, and the points represent the data obtained from the 29 micron tube, with the average hematocrits in the two tubes being equal. The fact that the two data sets for a given tube hematocrit define the same curve indicates that the MRW method was satisfied, and that the effective viscosity of the blood in the smaller tube can be predicted from the macroscopic rheological properties of the blood, if one uses the correct, measured average tube hematocrit. Some evidence indicates that this is true even for tubes with diameters down to about 20 microns [53]. While this makes it easy to predict the pressure drop – flow rate relationship for blood flow in tubes greater than 20 microns in diameter using macroscopic rheological data, the fact that the MRW method is satisfied is just a coincidence. In reality, there is a non-uniform red cell distribution in such blood flows due to the wall layer of plasma, and this violates one of the conditions imposed in the derivation of the MRW method. Nevertheless, from a practical viewpoint, this finding also supports the concept that one can predict blood flow behavior in small tubes or vessels from the macroscopic rheological properties of the blood.

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The famous Whittaker-Winton paper [54] reported that the in-vivo effective viscosity of blood, as measured in the hindlimb of the dog, is about half that found with a glass capillary viscometer. As these authors suggested, this difference in effective viscosities is due to the Fåhraeus-Lindqvist Effect, as can be demonstrated with a very simple model of the hindlimb vascular network which utilizes the Fåhraeus-Lindqvist Effect. 3.3. Flow in Networks of Vessels The average hematocrits of blood flowing in in-vivo networks of small vessels in the microcirculation have been measured by optical densitometry. While these measurements have an inherent uncertainty in them because the method is calibrated with blood in circular artificial tubes and the cross sections of in-vivo vessels are not circular, they nevertheless are a good indication of the hematocrits in such vessels. Such measurements have been made by Lipowsky et al. [55] and Kanzow et al. [56], as well as by others. For vessels which are smaller than about 30 microns in diameter, one can determine vessel average hematocrits from video records of the flow using tracer fluorescent red blood cells [57]. Figure 11 shows some data for in-vivo vessel average hematocrits as a function of vessel diameter.

Figure 11: The ratio of the vessel average hematocrit to the systemic hematocrit versus vessel apparent diameter, for arterial, capillary and venous vessels in the rat mesenteric microcirculation. The curves are for the Fåhraeus Effect for two blood conditions. Adapted from Kanzow et al. [56].


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There are a number of characteristics of Figure 11 that are striking. First, the data in the larger vessels are not as scattered as the data for the vessels where the diameter approaches that of capillaries. For the capillaries, the distribution of observed hematocrits is very much dispersed. The solid curve on the graph is the Fåhraeus Effect as determined for suspensions of red cells in saline at room temperature, and the dashed curve represents the Fåhraeus Effect for red blood cells suspended in plasma at a temperature of 37°C. Consequently, the second outstanding feature of this graph is that the Fåhraeus Effect can not alone explain the very low hematocrits seen in many of the capillaries and neighboring vessels. In some of the vessels, the hematocrit ratio can exceed unity, as shown by Lipowsky et al. [58] and by Desjardins and Duling [59]. In the later paper, the data are reported only for vessels in which the flow was very fast, and all those data are clustered about a hematocrit ratio of unity regardless of the vessel diameter. These characteristics of the vessel hematocrit distribution must be explained by some mechanism which has not yet been considered here. This mechanism is basically due to the Fåhraeus Effect, and the way red blood cells are distributed at an arterialtype bifurcation from the inflow vessel into each of the two downstream vessels. This is illustrated in Figure 12.

Figure 12: Blood flow through a series of small diameter arterial-type bifurcations, showing an idealized hematocrit distribution at various axial positions. Reproduced with permission from [53].

If the blood coming into the left inflow vessel has traveled a long distance in that vessel, the hematocrit profile across the vessel cross section will be axisymmetric and show a wall layer of plasma and a core of erythrocytes in plasma. This hematocrit distribution is represented by a “top hat” type of distribution, where no erythrocytes are in the wall layer and the hematocrit in the core is constant, regardless of radial position. As the flow approaches the first bifurcation the flow is laminar since the flow has a very low Reynolds number (i.e., inertial forces are extremely small compared to the viscous forces). The dashed line is a separation surface: all the fluid above this curve flows ahead into the continuing downstream vessel and all the fluid below this surface flows into the smaller downstream vessel which is pointed downward. It has been shown that for many flow distributions into the downstream vessels this separation surface is relatively planar [60, 61]. The flow into the smaller vessel contains the wall layer next to it, as well as some of the core region of the inflow vessel. Consequently, the average vessel hematocrit in this vessel will be lower than the average hematocrit in the inflow vessel, and its value will depend on the fraction of the inflow that goes into it. The blood flowing in the continuing main vessel will have to move towards the lower surface of the vessel, as indicated by the streamlines shown in the figure. Because the Reynolds number is so low, the red cells will follow the streamlines, so

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that the hematocrit profile downstream from the first bifurcation will no longer be axisymmetric, and the wall layer at the lower surface of the vessel will be extremely small. At the second bifurcation, the process repeats itself, except that because of the non-axisymmetric hematocrit profile in the upstream vessel, the hematocrit in the smaller downstream vessel is higher than the average hematocrit in the inflow vessel. Of course, what happens at the second bifurcation depends on the orientation of the second bifurcation relative to that of the first bifurcation. If the second bifurcation had the smaller side branch headed upward, the hematocrit in the side vessel would have been very small. At the third bifurcation, with an almost equal flow into the two downstream vessels, one downstream vessel will have a very low hematocrit and the other a very high hematocrit. Therefore, it is possible for the average relative vessel hematocrit, as well as the relative hematocrit of the blood flow from a vessel, to be greater than unity and as low as zero. This is why the earlier cited experimental data have such a wide range of values.

1.0

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0.6

0.4

HD= 20% Diameter= 20 μm *

0.2

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Figure 13: In-vitro data for blood flow through a bifurcation with all vessel diameters equal to 20 microns, with the inflowing blood average hematocrit of 20%, and with the inflowing vessel very long so that the hematocrit profile in the entrance to the bifurcation is axisymmetric. Reproduced with permission from [63].


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Experimental data have been obtained which describe the characteristics of such flows through bifurcations. An example of in-vitro data is shown in Figure 13. This figure is a plot of the fraction of the red cells flowing into the bifurcation that go into one of the downstream vessels, F*, as a function of the fraction of the blood volumetric flow going into the same vessel, Q*. The asterisks indicate data when the downstream vessel examined is directed downward as was the first bifurcation in Figure 13, and the triangles represent data for the flow directions shown in the lower right diagram in the figure. If the fraction of red cells going into a downstream vessel were equal to the fraction of the blood volume going into that vessel, the data would have fallen on the dashed diagonal line. The fact that, within the precision of the measurements, the data all define one curve representing F* as a function of Q* regardless of the flow directions is to be expected, because of the low Reynolds number of the flow and the fact that all the vessels were the same size. As shown by the non-zero intercept of the curve on the F* axis, there is a minimum flow rate into the downstream vessel below which no red cells flow into the branch; the downstream vessel then only receives plasma from the upstream vessel wall layer.

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0.6

0.4

Ȝ 0.08, 0.16, 0.40 0.40, 0.80 1.00

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Figure 14: F* versus Q* for bifurcations with all vessels the same size and axisymmetric inflows. O is the ratio of the red cell maximum diameter to the vessel diameter. The dotted lines represent in vitro data for human blood in tubes of 20, 50 and 100 microns diameter [62]. The solid lines are for model studies using flexible disks suspended in silicone oil [63], and the dashed line is for data from rabbit ear capillaries [64].

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Data for single bifurcations, with axisymmetric inflow, are depicted in Figure 14. As the lines in the figure show, the fraction of the incoming red cells which go into a branch depends strongly on the branch diameter, and the minimum flow fraction required to get any red cells in the branch is also strongly dependent on the O value. Up to this point, data from only single bifurcations with axisymmetric inflows have been described. Enden and Popel [61], in their theoretical study, found that the shape of the dividing surface separating the branch flow from the continuing flow was affected by the value of O . As the branch vessel diameter decreased, the separation surface changed from an almost flat surface when the vessel diameters were about equal to a semi-circular cylindrical surface, budging away from the branch, when the diameter ratio was 0.2 (Q*=0.2). This indicates, at this volume flow split, that as the branch diameter decreases, the average hematocrit of the blood flowing into the branch increases since the amount of core blood from the inflow vessel going into the branch increases and the amount of plasma wall layer decreases.

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Figure 15: The F* vs. Q* plot for two successive bifurcations, with axisymmetric flow into the first bifurcations. The curve represents the data for the first bifurcation. The asterisks represent data at the second bifurcation when Q* at the first bifurcation is 0.50; the diamonds are the data for the second bifurcation when Q* at the first bifurcation is 0.30. Reproduced with permission from [62].


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In vivo blood flow networks consist of multiple series of bifurcations. Since it takes about 10 seconds of flow time after a bifurcation to reestablish an axisymmetric hematocrit profile in a vessel, it is unlikely that a non-axisymmetric profile is altered very much between successive bifurcations. Consequently, the F* versus Q* plot for the second and subsequent bifurcations in a series of bifurcations will depend on what happens at the previous bifurcations. This is illustrated with in-vitro data in Figure 15.

Figure 16: Blood flow through an arterial bifurcation in the rat mesentery. In the upper picture, the flow is occurring in its unperturbed state, whereas in the lower picture, the flow in the right hand vessel of the bifurcation has been reduced by partial occlusion of the vessel downstream from the picture. The arrows indicate the flow directions. Reproduced with permission from [66].

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From these data, it appears that as the fraction of the blood flow which goes into the branch vessel of the first bifurcation increases, the data for the branch of the second bifurcation shift downward. This occurs because with the alternating orientation of the branches, as the blood flow fraction increases at the first bifurcation the hematocrit profile between the vessels is shifted downward in the figure, creating a larger plasma layer at the upper wall of the continuing vessel before the second bifurcation. Of course, the orientation of the second bifurcation relative to the first bifurcation will have a significant effect on red cell distribution at the second bifurcation. This bifurcation effect also occurs in-vivo. An example of this was published in Pries et al. [65] and is shown in Figure 16. This figure shows blood flow through a rat mesenteric arterial-type bifurcation, first in its self-regulating state, and then with flow in the right hand branch retarded. In the latter picture, one can clearly see the non-axisymmetric hematocrit distribution in the branch, with a plasma layer at the upper surface, due to skimming off of the plasma layer that is in the inflow vessel. By adjusting the occlusion extent on the side branch, Pries et al. [61] were able to obtain an F* - Q* plot for this bifurcation. This is shown in Figure 17.

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0.8

0.6

0.4

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Figure 17: The F* - Q* for the bifurcation shown in Figure 16. Reproduced with permission from [66].


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In this figure, the open circles represent data from the right hand branch, and the filled circles are for the continuing vessel. These two sets of data are not independent, since the sum of the two F*s must equal one, as does the sum of the two Q*s. There is some uncertainty in these data, because of the methods of instrument calibration and the assumption that the vessels are circular in cross section [62], but they are nevertheless qualitatively correct. It is clear that this bifurcation red cell distribution effect greatly affects red cell distribution (and oxygen delivery) in the microcirculation. 3.4. Flow in Microvascular Networks of Vessels A considerable amount of in-vivo data on blood flow in the microcirculation has been gathered (e.g., [55, 68]). Other than data on flow direction and pressure drop via micropipette, these data are only qualitative, because the instruments used to measure the data are calibrated using axisymmetric blood flow in circular cross section tubes, whereas in-vivo vessels are not circular in cross section and have varying dimensions and cross-sectional shapes in the axial direction. While in-vitro data are generally accurate, they are not directly applicable to in-vivo blood flow since they generally obtained using blood flow in circular cross section tubes whose diameter does not vary in the axial direction. In spite of these limitations, both in-vivo and in-vitro data have been used to model blood flow in microcirculatory networks of vessels. A very early physical model of the microcirculation was made by A. Fick [69], who constructed his model from glass tubes and rubber connectors; the model consisted of seven levels of vessels in a tree-like trifurcating system with arterial, capillary and venous vessels. The vessels at each level were of different diameters with the smallest vessels being model capillaries. Pressures were measured with manometers at each level of the model. His conclusion from his experimental data was that the major pressure loss occurs in the post-capillary vessels. This erroneous conclusion arises, at least in part, because all his vessels except “capillaries” had turbulent flow, unlike in in-vivo microcirculatory blood flow which is always laminar. While models such as Fick’s are overly simplistic, they can be used to estimate average flow behavior in large networks of vessels. For example, one can model mathematically the Whittaker and Winton experiment [54], in which suspensions of red blood cells in serum were pumped through the isolated hindlimb of the dog, and overall pressure drop and flow rate data were measured. The conclusion from this study was that the effective viscosity of the suspensions, as calculated with the data and Poiseuille’s equation, was about half that of the viscosity of the blood as measured in a tube viscometer. A very simple mathematical model, tree-like in nature, of this experiment can be constructed with the vessel diameter, length and number data for a generic vascular network from the main artery branch to the main venous branch contained in Johnson [70]. From the average hematocrit data for various in-vivo vessels in Lipowsky et al. [55], one can assign to each level of vessel in the model network a vessel hematocrit. Using an appropriate high shear rate apparent viscosity for each vessel, one can then calculate the overall pressure drop for a given overall flow rate in the network. In a separate calculation, one may assume that the hematocrit in each vessel is equal to the systemic hematocrit, and calculate an overall pressure drop at the same flow rate as used in the first calculation. The ratio of the two pressure drops from these calculations is about 0.59, while Whittaker and Winton measured a ratio of 0.52. This simple model calculation strongly supports the idea that the

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Whittaker-Winton Effect is due to the lower average hematocrits found in the smallest vessels of the microcirculation. Although all the necessary data for red cell distribution at bifurcations and pressure drop-flow rate information for non-axisymmetric flow in micro vessels are not available, several investigators have nevertheless constructed mathematical models of blood flow in microvascular networks utilizing the information that is available. A number of different approaches have been used. Pries, et al. [71] used data for network topology and topography, distribution of red cells at bifurcations, effective viscosity as a function of vessel hematocrit, etc., based on in-vivo data, whereas Kiani et al. [72] used the same in-vivo network topology and topography data as Pries et al., but in-vitro data for red cell distribution at bifurcations and the Fåhraeus and Fåhraeus-Lindqvist Effects. There is a significant difference in the method for assigning vessel hematocrit to each vessel: the Pries et al. method calculates discharge hematocrits at all nodes in the network at an instant in time, then assigns to each vessel an average hematocrit based on its discharge hematocrit and the Fåhraeus Effect. The effective viscosity for each vessel is obtained from the Fåhraeus-Lindqvist Effect. The Kiani et al. method keeps track of “slugs” of blood as they travel through the network so that at any instant, the hematocrit varies along a vessel length. The latter method volume averages the hematocrit in a vessel, and based upon this hematocrit an effective viscosity to that vessel. As a consequence of these methods for assigning vessel hematocrits, the Pries et al. method results in steady flows in all vessels whereas the Kiani et al. method shows stationary oscillatory flow in many vessels in a network [72, 73]. Regardless of the details of the computational model, comparisons of predicted vessel parameters to corresponding in-vivo measured parameters are not satisfactory: comparisons of individual vessel hematocrits are very poor and, while somewhat better, comparisons of flow velocities are also not very good [53]. These difficulties arise from two sources: the inaccuracies of the in-vivo data and the inapplicability of the invitro data for the reasons mentioned above. A review of the foundations of modeling the microcirculatory blood flow is given by Popel [74], and in-vivo data are available for several microvascular networks and two rat mesenteric networks on the internet at www.physiology.arizona.edu/people/secomb/network.html. Note, however, that in spite of the limitations of current models, they do duplicate many average in-vivo microcirculatory characteristics [75, 76].

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[40] G.R. Cokelet, Macroscopic rheology and tube flow of human blood. In: Microcirculation, J. Grayson and W. Zingg, Eds., Plenum Press, New York, 1976 ,pp 9-31. [41] A.R. Pries, K.H. Albrecht and P. Gaehtgens, Model studies of phase separation at a capillary orifice, Biorheology 18 (1981), 355-367. [42] R. Fåhraeus , The suspension stability of the blood, Physiol. Rev. IX (1929), 241-274. [43] H.L. Goldsmith, G.R. Cokelet and P. Gaehtgens, Robin Fåhraeus : evolution of his concepts in cardiovascular physiology, Am. J. Physiol. 257 (1989), H1005-H1015. [44] P. Gaehtgens, C. Duehrssen and K.H. Albrecht, Motion, deformation and interactions of blood cells and plasma during flow through narrow capillary tubes, Blood Cells 6 (1980), 799-812. [45] A.R. Pries, D. Neuhaus and P. Gaehtgens, Blood viscosity in tube flow: dependence on diameter and hematocrit, Am. J. Physiol. 263 (1992), H1770-H1778. [46] U. Nobis, A.R. Pries, G.R. Cokelet and P. Gaehtgens, Radial distribution of white cells during blood flow in small tubes, Microvasc. Res. 29 (1985), 295-304. [47] H.L. Goldsmith and S. Spain, Margination of leukocytes in blood flow through small tubes, Microvasc. Res. 27 (1984), 204-222. [48] H.L. Goldsmith, Red cell motions and wall interactions in tube flow, Fed. Proceed. 30 (1971),15781588. [49] H.J. Goldsmith, The microrheology of human erythrocyte suspensions, Appl. Mech., Proceed. of 13th Intern. Congress of Theoret. and App. Mech. E. Becker and G.K. Mikhailov, Eds. Springer-Verlag, Berlin 1973, pp 85-103. [50] W. Reinke, P. Gaehtgens and P.C. Johnson, Blood viscosity in small tubes: effect of shear rate, aggregation and sedimentation, Am. J. Physiol. 253 (1987), H540-H547. [51] G.R. Cokelet and H.L. Goldsmith, Decreased hydrodynamic resistance in the two-phase flow of blood through small vertical tubes at low rates, Circ. Res. 68 (1991), 1-17. [52] J.H. Barbee and G.R. Cokelet, Prediction of blood flow in tubes with diameters as small as 29 microns, Microvasc. Res. 3 (1971), 17-21. [53] G.R. Cokelet, Viscometric, in-vitro and in-vivo blood viscosity relationships: how are they related?, Biorheology 36 (1999), 343-358. [54] S.R.F. Whittaker and F.R. Winton, The apparent viscosity of blood flowing in the isolated hindlimb of the dog, and its variation with corpuscular concentration, J. Physiol. Lond. 78 (1933), 339-369. [55] H.H. Lipowsky, S. Usami and S. Chien, In-vivo measurements of “apparent viscosity” and microvessel hematocrit in the mesentery of the cat, Microvasc. Res. 19 (1980), 297-319. [56] G. Kanzow, A.R. Pries and P. Gaehtgens, Flow dependent hematocrit distribution in microvessel networks, Biblthca anat. 20 (1981), 149-152. [57] I.H. Sarelius and D.R. Duling, Direct measurement of microvessel hematocrit, red cell flux, velocity and transit time, Am. J. Physiol. 243 (1982), H1018-H1026. [58] H.H. Lipowsky, L.E. Cram, W. Justice and M.J. Eppihimer, Effect of erythrocyte deformability on invivo red cell transit time and hematocrit and their correlation with in-vitro filterability, Microvasc. Res. 46 (1993), 43-64. [59] C. Desjardins and B.R. Duling, Microvessel hematocrit: measurement and implications for capillary oxygen transport, Am. J. Physiol. 252 (1987), H494-H503. [60] F. W. Rong and R.T. Carr, Dye studies in flow through branching tubes, Microvasc. Res. 39 (1990), 186-202. [61] E. Enden and A.S. Popel, A numerical study of the shape of the surface separating flow into branches in microvascular bifurcations, Trans. A. S. M .E. 114 (1992), 398-405. [62] G.R. Cokelet, Blood Flow through Arterial Microvascular Bifurcations. In: Microvascular Networks: Experimental and Theoretical Studies. A.S. Popel and P.C. Johnson, Eds., Karger, Basel 1985, pp 155167. [63] B.M. Fenton, R.T. Carr and G.R. Cokelet, Nonuniform red cell distribution in 20 to 100 micron bifurcations, Microvasc. Res. 29 (1985), 103-126. [64] S. Chien, C.D. Tvetenstrand, M.A.F. Epstein and G.W. Schmid-Schhoenbein, Model studies on distributions of white and red blood cells at microvascular bifurcations, Am. J. Physiol. 248 (1985), H568-H576. [65] G.W. Schmid-Schoenbein, R. Skalak, S. Usami and S. Chien, Cell distribution in capillary networks, Microvasc. Res. 19 (1980), 18-44. [66] A.R. Pries, K. Ley, M. Claassen and P. Gaehtgens, Red cell distribution at microvascular bifurcations, Microvasc. Res. 38 (1989), 81-101. [67] G.R. Cokelet and I.H. Sarelius, Perceived vessel lumen and cell-blood ratio: impact on in-vivo blood flow rate determination, Am. J. Physiol. 262 (1992), H1156-H1193. [68] A.R. Pries, T.W. Secomb and P. Gaehtgens, Biophysical aspects of blood flow in the microvasculature, Cardiovasc. Res. 32 (1996), 654-667.


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[69] A. Fick, Ueber den Druck in den Blutkapillaren, Pfluegers Archiv. 42(1888), 482-488. [70] P.C. Johnson, Ed., Peripheral Circulation, Wiley, New York 1978, pp 3. [71] A.R. Pries, T.W. Secomb, P. Gaehtgens and J.F. Gross, Blood flow in microvascular networks, experiments and simulations, Circ. Res. 67 (1990), 826-834. [72] M.F. Kiani, A.R. Pries, L.L. Hsu, I.R. Sarelius and G.R. Cokelet, Fluctuations in microvascular blood flow parameters caused by hemodynamic mechanisms, Am. J. Physiol. 266 (1994), H1822-H1828. [73] R.T. Carr, J.B. Geddes and F. Wu, Oscillations in a simple microvascular network. Ann. Biomed. Eng. 33 (2005), 764-771. [74] A.S. Popel, Network Models of Peripheral Circulation. In: Handbook of Bioengineering. R. Skalak and S. Chien, Eds., McGraw-Hill, New York 1987 chapter 20. [75] A.R. Pries, T.W. Secomb and P. Gaehtgens, Structure and hemodynamics of microvascular networks: heterogeneity and correlations, Am. J. Physiol. 269 (1995), H1713-H1722. [76] A.R. Pries, T.W. Secomb and P. Gaehtgens, Biophysical aspects of blood flow in the microcirculation, Cardiovasc. Res. 32 (1996), 654-67.

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Viscoelasticity of Human Blood George B. THURSTONa,1 and Nancy M. HENDERSONb a University of Texas at Austin, USA b Vilastic Scientific, Inc., USA

1. Introduction and Overview of Blood Viscoelasticity 1.1. Time-Varying Flow of Blood in Vivo The most obvious feature of the circulation is the pulse. Pulsatile flow can be analyzed as containing steady plus harmonic components. The heartbeat of 60-120 beats per minute gives a fundamental Fourier component of 1 to 2 Hz. Because the flow is time varying, pressure-flow relations are a function of both the shear viscosity and the shear elasticity of the blood. The viscoelasticity of blood has a direct effect on the propagation of the pulse throughout the arterial system [1]. Blood flow in vivo covers a wide range of shear rates and varied vascular geometry (smooth wall of uniform diameter, tapered vessels, bifurcations, side branches, stenoses). 1.2. Red Cell Concentration and Deformation Normal human blood contains a high concentration of red blood cells (RBC), which are elastic elements. The maximum theoretical volume concentration of red cells without squeezing and deforming is 58%. Because normal cell concentration is in the range from 30 to 60%, flow cannot occur through the varied geometry of the circulation without elastic cell deformation and orientation and hence storage of elastic energy in the cells. Blood flows only because the RBC are deformable and can be reoriented to slide on the low viscosity plasma. The elastic deformability of cells means that energy can be stored in and recovered from cell deformation. The elastic energy is measurable when flow changes with time. Oscillatory flow is particularly useful for measuring this energy and characterizing viscoelastic properties of blood.

2. Measurement Methods The viscoelastic effects in whole blood are evident from several experimental observations. These effects have been seen in both optical and mechanical measurements. Observations are made in time-varying flow where stored elastic energy is recovered. The first identification of blood viscoelasticity was done using oscillatory flow in a cylindrical tube [2, 3] and in an oscillating Couette geometry [4]. The oscillating cone-plate geometry was used both for measurements and visual 1

Corresponding Author: 1000 Madrone Road, Austin, TX 78746 USA; E mail: [email protected]

G.B. Thurston and N.M. Henderson / Viscoelasticity of Human Blood

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observations of flow-induced cell organization [5, 6]. Stress response to sudden initiation and cessation of flow shows an elastic response [7, 8, 9]. The elastic effects are evident from observations of light reflection from and transmission through blood following sudden cessation of flow [10, 11] and during sinusoidal oscillatory flow [12]. The accuracy and reproducibility of rheological measurements can be affected by the tendency of blood cells to sediment, and by the sample confinement size and geometry. Four confinement geometries have been used to measure the viscoelasticity of blood: cone-plate, plate-plate, concentric cylinder (Couette) and cylindrical tube. The geometry of rotational instruments, such as the cone-plate and plate-plate, can accelerate cell sedimentation during very low shear rotation [13], producing a twophase fluid in the shear plane. A vertically oriented cylindrical tube, with large reservoirs at each end, and the Couette geometry are relatively insensitive to sedimentation during measurements. Because blood viscoelasticity is dependent on the confinement size [14, 15], maintaining a uniform and fixed geometry is important for reproducibility. Therefore, a fixed uniform geometry such as a cylindrical tube is preferable to the variable confinement size of the cone-plate and the user-established gaps of plate-plate and cone-plate geometries. Hence, for the data presented herein, oscillatory flow in a vertically oriented cylindrical tube was utilized (Vilastic-3 Viscoelasticity Analyzer, Vilastic Scientific, Inc., Austin, TX USA). Details of the principle of measurement are given later (see Section 5).

3. Parameters for Shear Flow and Viscoelasticity Consider the action of a steady shearing force on a small cubical volume of blood of height H. The force produces shear stress and shifts the shape of the blood volume to a parallelogram (Figure 1). The displacement, D, of the cube has two components: elastic deformation, E, and slippage, S. The elastic deformation is accompanied by storage of elastic energy within the structure of the blood, while the slippage is associated with a continuous input of viscous energy. When the force is removed, the deformed cube undergoes a partial restoration of shape due to the recovered elastic energy, but the cube remains deformed due to energy loss from slippage. In steady flow, the displacement component continues to increase. Measurements of non-time-varying force and velocity provide no information regarding elastic energy storage. In any time-varying flow, however, the elastic energy component varies with time and may be either increasing or decreasing. Figure 1 can be used to define the parameters: shear stress (F/A), shear rate (V/H), and shear strain (D/H).

Figure 1. Diagram of a small cubical volume of height, H, in shear. The displacement, D, due to deformation is composed of two parts: an elastic part, E, and a slippage part, S. With constant force F, E remains constant but S continues to increase. When F is removed, E diminishes to zero and S remains.

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Figure 2. Sinusoidal deformation of a cubical volume of fluid. The sinusoidal time varying shear rate and shear stress differ in phase by the angle, I, as shown.

Sinusoidal time varying flow provides a basis for differentiation of the elastic and viscous properties of blood. The time dependences are Shear Rate = J m cos(Z t )

(1)

Shear Stress = W m cos(Z t I)

(2)

S Shear Strain = J m cos(Z t ) 2

(3)

and also,

where Z is the radian frequency (2·S·frequency). Figure 2 shows the cubical volume of blood in oscillatory shear and the time dependence of shear rate and shear stress. For a purely viscous fluid, the phase angle difference (I) between the shear rate and shear stress is 0°. If the fluid is purely elastic then the angle is 90°. For a viscoelastic fluid, such as blood, the angle is between 0° and 90°. The magnitude and phase relation between the stress, strain and shear rate are described by complex numbers. For example Shear Stress = Real Part of (W m e i (Zt I) ) = Real Part of (W * eiZt ) The complex coefficient of viscosity (K*) is given by K*

W* J *

W m e iI J m

(4)

Expanding equation 5 yields K*

W m (cos(I) i sin(I)) J m

K*

Kc iKcc

c W'm iWcm J m

(6)

where K’ is the viscosity and K” is the elasticity. Shear stress relations are W*

c Wcm iWcm

(8)

75


c is the elastic stress. Similarly, the complex where Wcm is the viscous stress and Wcm modulus of rigidity, G*, can be determined from the complex ratio of the shear stress, W*, to the shear strain, J*. G*

c Wc Wcm i m Jm Jm

W* J*

G c i G cc

(9)

where G’ is the storage modulus and G” is the loss modulus. The complex coefficient of viscosity, K*, is related to the complex rigidity by G* i Z K * ; G ' Z Kcc ; G cc

Z Kc

(10)

The three basic parameters: stress, strain and shear rate, are often measured as rms (root mean square) values as opposed to maximum values. The maximum values can be replaced by the product of the rms value times 2 . When utilizing rms values the subscript “m” is removed from the above equations. With sinusoidal oscillatory flow, viscous energy is dissipated while elastic energy is alternately stored and recovered. The energy dissipated per unit volume in each cycle is proportional to the viscosity Ed

§ 2S· Kc J 2 ¨ ¸ © Z ¹

(11)

while the maximum energy per unit volume stored and recovered twice during the cycle is proportional to the elasticity, Es

§1· Kcc J 2 ¨ ¸ © Z¹

(12)

We can show the viscoelastic action for each measurement by an apparent relaxation time that can be obtained by assuming the material behaves as a single Maxwell element. The single element Maxwell model consists of a spring (P) and dashpot (K) in series. For sinusoidal stress and strain the viscosity and elasticity are Kc

Kcc

K 1 Z Tr 2 K Z Tr 1 Z Tr 2

(13)

(14)

where Tr = (K/P) is the relaxation time. From the ratio of Equations. (13) and (14), the Kcc tan(I) . The relaxation time can also be apparent relaxation time is Tr c Z ZK obtained from the stored and dissipated energies indicated in of Equations. (11) and (12), Tr

Kcc Z Kc

Es Ed f

(15)

76


where f is the frequency. Therefore, the relaxation time is a direct indicator of the energy ratio, which depends on how energy is stored and dissipated by the microstructure of the blood. Because of the complexity of the microstructure of blood, there are many modes where elastic energy can be stored and recovered and viscous energy dissipated. The result is a very wide spectrum of relaxation times. This spectrum is sensitive to details of how the red cells are organized and interact, so it is sensitive to the rate of flow, vessel size, etc. The rheology of blood must be studied over a wide range of timings, and one way to do this is to examine its viscoelastic properties for a wide range of frequencies. The Maxwell relaxation time obtained using a single Maxwell element (Equations 13-15) is ultimately determined by the complete relaxation spectrum for many Maxwell elements in parallel. Each has its own characteristic spring (constant Pi) and dashpot (constant Ki) and hence its own relaxation time, Ti= (Ki /Pi) [16, 17]. When the shear rate (and shear stress) is increased, a point will be reached where the Ki and Pi diminish in equal proportions, affecting the longest relaxation times first. Thus, the relaxation spectrum becomes modified. With increasing shear rate further degradation of the longest relaxation process will occur along with degradation of the contribution from shorter relaxation processes. The contribution of the shortest relaxation processes will diminish at the higher shear rates and stresses.

4. Moduli for Blood 4.1. Viscoelasticity Profile

A profile for the viscoelasticity of blood is obtained from measurements near the pulse rate (2 Hz) while progressively increasing the rate of flow in a 1 mm inside diameter (ID) cylindrical tube. For this tube size and frequency, the velocity across the diameter in the cylindrical tube is parabolic (see section 5.). These measurements show the viscosity and elasticity and consequently, how the cellular arrangements are affected by the rate of flow. Figure 3 shows how the viscosity and elasticity change with shear rate at the wall (rms values) over a range comparable with flows in vivo. The 44% hematocrit blood was measured at 22 °C. At the lowest shear rates, near 1 s-1, the blood is nearly in its resting state; in that the rate of flow and internal stresses are small and do not materially disrupt the quiescent state of cell aggregates. With increasing shear rate, the large aggregates begin to break-up into smaller collections of cells. At shear rates above 100 s-1, the disaggregated cells form a layered-like structure where layers of stretched and aligned cells slide on separating plasma layers [18]. The disruption of aggregates and formation of cell layers induced here by oscillatory flow can also occur with steady flow and pulsatile flow [19, 20]. The levels of the viscous and elastic components of the shear stress (Equation 8) for the data in Figure 3 are shown plotted versus the rms value of the shear strain in Figure 4. The dependence of the elastic stress on strain shows that the blood is capable of storing elastic energy with increasing strain. The dashed line adjacent to the elastic stress marks a linear relation between stress and strain. The progressively decreasing ability of the elastic stress to follow this linear relationship is indicative of the degradation of cell aggregates. The maximum elastic shear stress occurs near unit strain. This maximum represents an elastic yield stress, which is the maximum stress the


77

structure can accommodate without major disruption. At strains above the yield value, the aggregates undergo a catastrophic break-up, which is shown by the decreasing ability to store elastic energy. The viscoelasticity profile and elastic yield stress are affected by several factors such as aggregation tendency, cell deformability, hematocrit and plasma composition. [21, 22, 23]. The Maxwell relaxation times for the data in Figure 3 are shown in Figure 5. At low shear rates, the Maxwell relaxation time is nearly constant at 0.027 seconds and diminishes with increasing shear rate. The constant value at low shear rates is indicative of the relaxation characteristics of the aggregates. As the aggregate size diminishes with shear rate, the relaxation time decreases. At high shear rates, the relaxation time reflects the elastic character of the stretched and aligned cells. 100 Viscosity

Viscosity and Elasticity (mPa s)

Elasticity

10

1

0.1

0.01 1

10

100

1000

Shear Rate (1/sec) Figure 3. Viscoelasticity Profile. The viscosity and elasticity for a range of shear rates measured at 2 Hz, 22 °C in a 0.1 cm ID cylindrical tube with length of 6 cm. The blood sample is from a normal donor with a hematocrit of 44%.

4.2. Steady Flow vs. Oscillatory Flow Behavior

Profiles for frequencies from 0.1 Hz to 2 Hz, along with the profile for steady flow (zero frequency), are shown in Figure 6. The difference between steady flow viscosity and oscillatory viscoelasticity is that in steady flow the strain is progressively increasing without limit, whereas in oscillatory flow the strain is limited and reversing periodically. Consequently, the elastic energy stored in steady flow is not recovered and remains hidden. All profiles (steady and oscillatory) tend to merge at high shear rates. This behavior is consistent with the truncation of relaxation processes with increasing shear as postulated in the generalized Maxwell model for blood [17]. The viscoelasticity profile changes with the frequency of the oscillation because of the relationship between the speed of deformation and the characteristic time of the blood structure, the Maxwell relaxation time (Equation 15). At low shear rates, where blood is in its quiescent states, the value of viscosity decreases by 16% and the elasticity decreases by 34% when changing the frequencies from 1 to 2 Hz (Figure 6).


Figure 4. Viscous and Elastic Stress. Measurements were made at 2 Hz at 22°C in a 0.1 cm ID cylindrical tube with length of 6 cm. The blood sample is from a normal donor with a hematocrit of 44%. Results are for viscosity and elasticity data versus shear rate presented in Figure 3.

0.1 Maxwell Relaxation Time

Maxwell Relax. Time (s)

78

0.01

0.001

0.0001 1

10

100

1000

Shear Rate (1/sec) Figure 5. Maxwell relaxation times for 44% hematocrit blood measured in a 0.1 cm ID tube with length of 6 cm. Measurements were made at 2 Hz, 22 °C. Results are for viscosity and elasticity data versus shear rate presented in Figure 3.

79


Figure 6. Viscoelasticity profile for frequencies from 0.1 Hz to 2 Hz, along with the profile under steady flow (zero frequency). Measurements were made at 22 °C. Steady flow measurements were performed with a concentric cylinder Couette viscometer (Contraves LS30) with a gap of 0.5 mm. Oscillatory flow measurements were made with a 0.1 cm ID cylindrical tube with length of 6 cm.


100 Viscosity (Normal) Elasticity (Normal) Viscosity (Hardened) Elasticity (Hardened)

10

1

0.1

0.01 1

10

100

1000

Shear Rate (1/sec) Figure 7. Viscoelastic properties of normal blood (45% hematocrit) and a suspension of heattreated RBC in plasma (45% hematocrit) measured at 2 Hz. Measurements were made in a 0.1 cm ID cylindrical tube with length of 6 cm. The temperature was 22°C.

80


4.3. Deformability

For blood to flow in the circulatory system, the red blood cells must be deformable. With decreased deformability, the pressure required to maintain flow will necessarily increase. In the extreme case, as in sickle cell disease, microvessels may become plugged [24]. The elasticity obtained via a viscoelasticity profile is a direct measure of deformability because it relates directly to the amount of energy storage due to the elastic deformation of the red blood cells (Equation 12). At high shear rates, a decrease in deformability causes an increase in elasticity and Maxwell relaxation times. Deformability of red blood cells can be decreased by treatment with acetaldehyde [23] or heat [25]. Figure 7 compares the viscoelastic profile for normal blood with a suspension of heat-treated RBC in plasma. To prepare the heated treated RBC sample, cells washed with phosphate buffered saline were subjected to a 48 °C bath for 9 minutes and then 22 °C for 1 minute. The cells were re-suspended in autologous plasma [26]. At low shear rates, the viscosity and elasticity are slightly elevated above the normal values. Because of the weaker aggregates in the hardened sample, the elasticity begins to diminish at lower shear rates than in the normal blood sample. This may be due to a decrease in cell-to-cell contact area. At high shear rates, the elevated elasticity of the heated cells can be associated with decreased ability of the cells to conform into layers. As opposed to heat treatment, intense hardening of cells by acetaldehyde causes cells to resist stable layer formation because of their limited ability to conform to neighboring cells and to release trapped plasma. The result is a strong dilatancy, where the decreasing elasticity trend of normal blood at high shear rates is reversed [23]. 4.4. Aggregation

The tendency of red cells in blood to aggregate is affected by the concentration of plasma proteins. In particular, fibrinogen concentration plays a significant role for red cell aggregation in plasma suspensions [27]. The tendency of red blood cells to aggregate can be also be enhanced by the addition of high molecular weight dextrans [28] such as dextran 150 (Mw=150,000) [29] and dextran T100 (Mw=100,000) [30]. Dilution of plasma will lower the concentration of all plasma components, including fibrinogen, resulting in a reduction in aggregation tendency. The effects of this dilution are seen in Figure 8, where normal blood is compared with blood containing plasma diluted by 75% with isotonic saline. At low shear rates, the dilution diminishes the viscosity by 37% and the elasticity by 60%, most likely due to the reduction of aggregate size in the diluted plasma sample. At high shear rates where cells are disaggregated, the values of elasticity are nearly equal and the reduction in viscosity of the diluted plasma sample is due to the overall reduction in plasma viscosity. 4.5. Temperature

While normal body temperature is approximately 37 °C, rheological evaluations of blood are often done near "room" temperatures between 20 and 25 °C. The effects on viscoelasticity are significant and therefore, when comparing samples, care must be taken to match temperatures. Figure 9 shows the viscoelasticity profile for temperatures of 15, 22 and 37 °C. It is seen that the character of the viscosity and

81


elasticity is the same but values are shifted with temperature. The functional form is K=D10-ET. At unit strain (shear rate = 12.56 s-1) for a normal blood at 2 Hz and the temperature in the range of 9 and 42 °C, the equations for the blood sample in Figure 9 are [31]

K c 21.47 10 0.01264T

[mPa.s]

(16)

K cc 4.213 10 0.01022T

[mPa.s]

(17)

[mPa.s]

(18)

K plasma

3.362 10 0.01017T

where T is temperature in °C. While the coefficient, D, is specific to the data in Figure 9, the exponent, E, can be assumed the same for other normal blood samples at this strain. Therefore, it can be used to project normal data acquired at 2 Hz and a strain of 1 (shear rate = 12.56 s-1) to a reference temperature, To KTo KT 10 ETo T

(19)

where K(T) is the viscosity or elasticity at the temperature (T), To is the target temperature and E is experimentally determined at a specific shear rate.


100

10

1

0.1 Viscosity (Normal Plasma) Elasticity (Normal Plasma) Viscosity (Diluted Plasma) Elasticity (Diluted Plasma)

0.01 0.1

1

10

100

1000

Shear Rate (1/sec) Figure 8. Viscosity and elasticity versus shear rate for 44% hematocrit cells in normal plasma and for the same cells in plasma diluted by 75% with isotonic saline. Measurements were made at 2 Hz and 22°C in 0.1 cm ID cylindrical tube with length of 6 cm.

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100

10

1

Viscosity (15°C) Viscosity (22°C) Viscosity (37°C) Elasticity (15°C) Elasticity (22°C) Elasticity (37°C)

0.1

0.01 0.1

1

10

100

1000

Shear Rate (1/sec) Figure 9. Viscoelasticity profile for blood at three temperatures. The frequency of measurement was 2 Hz and the hematocrit was 47%. Measurements were made in a 0.1 cm ID cylindrical tube with length of 6 cm.

4.6. Hematocrit

The viscoelasticity of blood is strongly dependent on hematocrit (i.e., volume fraction red cells). The normal range of hematocrits varies with age and gender and can range from 30% to 60% [32]. Figure 10 shows viscoelasticity profiles for blood with hematocrits of 35%, 45%, 55% and 65%. These samples were derived from a single donor by separating and re-suspending red blood cells in plasma at the desired hematocrit. Measurements were made at 2 Hz and 22 °C. While the character of the curves remains essentially the same for all suspensions, their magnitudes differ significantly with hematocrit. For Figure 10, the functional dependence of viscoelasticity on hematocrit at a shear rate 10 s-1 is given by

K c 1.9736 exp(3.2361 H )

[mPa.s]

(20)

K cc 0.15344 exp(5.240 H )

[mPa.s]

(21)

where H is the hematocrit expressed as a fraction [31]. Similar functional expressions have been determined from statistical analysis of large sample of normals [33, 22]. Because in any study population the hematocrits will vary, researchers have sought ways to account for this variability and its effects on the measured viscoelasticity. One approach is to shift the viscoelasticity data to those values expected if the hematocrit were a standard value such as 45%. This approach requires the determination of shifting functions based on correlations between hematocrit and viscoelasticity of a statistical sampling of normals. These shifting functions are valid only within the


83

normal range of hematocrits [34, 35]. Some have found that the relationship between hematocrit and viscoelasticity for normal blood differs from that of pathological blood [34] and choose to analyze data for pathological blood at the native hematocrit. An additional method is to create a reference set of viscoelasticity data for various hematocrits by the serial dilution of a normal blood sample with plasma and to compare these references with the measured values [36].


100

10

1 Visc. (65%H) Visc. (55%H) Visc. (45%H) Visc. (35%H) Elas. (65%H) Elas. (55%H) Elas. (45%H) Elas. (35%H)

0.1

0.01 0.1

1

10

100

1000

Shear Rate (1/sec) Figure 10. The viscoelasticity profile of blood with hematocrits of 35%, 45%, 55% and 65%. Measurements were made at 2 Hz and 22°C in a 0.1 cm ID cylindrical tube with length of 6 cm.

4.7. Thixotropy

Blood is thixotropic because the flow-induced microstructure requires time to respond to a change in flow. This has been observed by monitoring the effects of time on viscosity in steady flow [37] and on viscoelasticity in oscillatory flow [17]. Light transmission through blood during start-up of flow shows the time for flow-induced microstructure to reach equilibrium [11]. Light transmission following cessation of flow reveals the time for the microstructure to recover and is attributed to recovery of aggregates [10]. The time to reach equilibrium after flow initiation is much shorter than the time for the microstructure to recover after the cessation of flow. The time required for the microstructure to change increases with the magnitude of the stepped increase of flow. At higher shear rates, structural changes occur more rapidly than at lower shear rates. When producing a viscoelasticity profile with increasing steps in shear rate, the flow condition must be maintained long enough to achieve equilibrium before acquiring a data point. For viscoelasticity profiles as presented herein, the blood is subjected to 5 seconds of shear for each data point before data acquisition.

84


4.8. Intrinsic Viscosity

The intrinsic viscosity of blood is a number descriptive of the contribution to the viscosity of a single cell in plasma without interaction with other cells. Such values are determined from measurements of the viscosity while sequentially diluting the cell concentration and extrapolating the viscosity to zero concentration. The intrinsic viscosity is defined as

>Kc@

ª Kc K o º Lim « » c v o0 ¬ K o c v ¼

(22)

where K’ is the viscosity of the solution (blood), Ko is the solvent (plasma) viscosity and cv is the volume concentration of cells. The intrinsic elasticity is outside the range of available measurement techniques [38]. However, from Einstein's viscosity relation for a suspension of rigid spheres, the intrinsic viscosity is 2.5 [39] and for other shapes it is greater than 2.5 [40]. Taylor has shown that the intrinsic viscosity of deformable droplets is below 2.5 [41]. The values for red blood cells in native plasma are below 2 as would be true for deformable objects, while hardened cells are above 3 [38]. Therefore, intrinsic viscosity is an indicator of cell deformability. 4.9. Small Tubes and Boundary Layers

Cell concentration must diminish at the walls of a confining tube because of a boundary layer created by geometric exclusion of cells. It has been shown for cylindrical brass tubes that the thickness of this layer is approximately four cell diameters. With larger tubes of diameters > 1 mm (i.e., > 125 cell diameters), this boundary layer has minimal effect on the measured viscoelasticity [15]. In contrast, for small tubes, the effects become enhanced showing a decrease in viscosity with tube diameter [26]. This is similar to the Fahraeus-Lindqvist effect in steady flow [14]. 4.10. Other Topics Regarding Blood Viscoelasticity

The effects of geometry and plasma environment on the viscoelasticity of blood have been investigated. The influence of flow through tortuous paths such as in porous media provides information about how the viscoelasticity is affected by geometric confinement, which is similar to the tortuosity of arterioles [26]. Additives routinely used in clinical situations such as dextran and ringers also influence blood viscoelasticity [31]. The long-term storage [23] and freezing of blood [42] will alter the viscoelasticity profile as well.

5. Oscillatory Tube Flow with a Viscoelastic Fluid

There are several parameters useful for describing oscillatory flow in cylindrical tubes having the nominal size of small arteries. These parameters start with the pressure gradient in the tube (pressure drop per unit length) and volume flow through it. From these the impedance components (resistance and reactance) can be determined. In the case where the filling fluid is homogeneous and viscoelastic, hydrodynamic theory can

85


be used to relate the flow to the shear stress, shear strain and shear rate at the tube wall. These last parameters then relate to the viscoelasticity of the fluid. This sequence of relations works as well in reverse for calculating pressure-to-flow values. In the case of pulsatile flow, the Fourier components that describe the pulse are additive and can be treated separately by assuming linear relationships. 1000 P' |P"|

P' and |P"| (Pa)

100

10

1 Negative Values

0.1 1.00E-10

1.00E-09

Positive Values

1.00E-08

1.00E-07

3

Volume Flow (m /s) Figure 11. The pressure components of blood (44% H) in a 0.1 cm ID cylindrical tube with length of 6 cm measured at 2 Hz and 22 °C.

To illustrate these relations, consider oscillatory flow in a cylindrical tube with radius of 0.05 cm and length of 6 cm. The tube is filled with normal human blood with 44% hematocrit at room temperature (22 °C). The frequency of oscillation is 2 Hertz. Figure 11 shows the measured pressure-to-flow relation where P' is the component in phase with the volume flow (U) and P" is the component in quadrature with the flow. The negative values are phased 90° behind the flow and positive values are 90° ahead of the flow. For low flow rates the negative values of P” show that the elastic effects of the fluid dominate while at high flow rates the positive values of P” show that the inertial effects due to blood density dominate. Near a volume flow of 3.0E-9 m3/s the elastic effects are equal in size to the inertial effects and pressure is in phase with the flow [43, 44]. This condition is called viscoelastic resonance and occurs when 6 sin I

Y

where I

(23)

cc arctan §¨ K c ·¸ and Y is a dimensionless parameter © K¹

Y

a

UZ K*

(24)

where a is the radius of the tube, U is the density of blood and Z is the radian frequency. The impedance to flow in the tube (Z) is obtained from the ratio of the pressure to flow

86


Pc Pcc i U U

Z

R iX

(25)

where R is the resistance and X is the reactance. These values are plotted in Figure 12. Oscillatory tube flow theory for a viscoelastic fluid serves to relate the impedance to the viscoelastic properties of the fluid and shearing conditions. The density of the fluid is also required. The dimensionless parameter Y serves to separate the fluid motion into two types. If Y

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