Or
I
Z
FEB 28 1963
THE RHEOLOGY OF HUMAN BLOOD
L.ISRAR(
by
Giles R. Cokelet
B. S., California Institute of Technology, 1957
M. S., California Institute of Technology, 1958
Submitted in Partial Fulfillment of the Requirements
for the Degree of Doctor of Science
at the
Massachusetts Institute of Technology
January 1963
Signature Redacted
Signature of the Author:
Department of Chemical En
eering
Certified by:
E. W. Merrill, Thesis Supervisor
Certified by:.
E. R. Gilliland, Thesis Supervisor
Accepted by:
G. C. Williams, Chairman
Dept. Committee on Graduate Theses
THE RHEOLOGY OF HUMAN BLOOD
by
Giles R. Cokelet
Submitted to the Department of Chemical Engineering on January
21, 1963, in partial fulfillment of the requirements for the degree of
Doctor of Science.
ABSTRACT
This thesis reports the results of a rheological study of human
blood obtained from healthy individuals.
The rheological properties
of blood, as a bulk material, were determined in the shear rate region
from zero to 100 inverse seconds, with particular attention at the
shear rate region of zero to 10 inverse seconds.
The effects of
changes in red cell volume fraction (hematocrit), temperature, plasma
composition, anticoagulant, and red cell and blood age were investigated.
The GDM viscometer, a concentric cylinder, Couette - type viscometer,
was employed to make the viscometric measurements.
The "roughness" of the viscometer surfaces was found to be
important in making viscometric measurements on blood.
.In
addition,
migration of the red cells away from at least one viscometer wall at
shear rates below about 1 sec
was detected.
The possibility of an
error in calculation of the shear rate due to the slight variation in the
shear stress across the viscometer gap was considered.
An under-
standing of these effects, which are revealed in part by time
dependence of the shear stress at constant shear rate, is shown to be
essential to the correct interpretation of the data.
ii
The relationships between shear stress, shear rate, and
hematocrit, developed by Casson for his model suspensions, were
found to be good means of correlating blood data in the low shear rate
region.
The limits of applicability of the Casson equations were found
to be from zero shear rate up to a value which increased in magnitude
as the hematocrit decreased; at a hematocrit of 45%, the upper limit
usually was about 1 inverse second. However, these relationships can
not be used to determine fundamental properties of the red cell.
It was established that blood has a yield stress which, in normal
blood, is dependent only on fibrinogen, of all the plasma proteins, for
its formation.
Free calcium ions are not essential for the formation
of a yield stress in blood. It was also discovered that hemoglobin
(as from lysis), and lipids in the plasma have important roles
contributing to the yield stress yet to be determined in adequate detail.
With respect to blood containing normal concentrations of fibrinogen,
the interrelationships of yield stress and hematocrit, and of the
temperature effect on rheological parameters in general, were
extensively investigated.
Edward W. Merrill,
Associate Professor of Chemical
Engineering
Thesis Supervisors;
Edwin R. Gilliland,
Professor of Chemical Engineering
iii
Department of Chemical Engineering
Massachusetts Institute of Technology
Cambridge 39, Massachusetts
January 18, 1963
Professor Philip Franklin
Secretary of the Faculty
Massachusetts Institute of Technology
Cambridge 39, Massachusetts
Dear Sir:
The thesis entitled "The Rheology of Human Blood" is herewith
submitted in partial fulfillment of the requirements for the degree of
Doctor of Science.
Respectfully submitted,
Giles R. Cokelet
iv
ACKNOWLEDGMENTS
For having suggested the subject of this work, and for his patience
and encouragement during the ups and downs of this study, special
thanks are due to Professor E. W. Merrill.
Without his overall guidance,
and continuous aid, this project would not have progressed as well as
it has.
Mr. P. J. Gilinson, Jr., and Mr. C. R. Dauwalter, of the
Instrumentation Laboratory, Massachusetts Institute of Technology,
were continuous sources of information about the GDM Viscometer.
Their interest in improving this vital instrument was indispensible.
The Instrumentation Laboratory, under the direction of Professor
C. S. Draper, supplied the GDM Viscometer.
Dr. A. Britten, of the Massachusetts General Hospital, Boston,
not only arranged for the supply of blood used in this work, but also
patiently contributed his ideas and medical knowledge.
Hyunkook Shin, Karin Ippen, and Bill Margetts supplied not only
their labors, but also their ideas and humor.
All their contributions
were essential to the progress of this study.
Mr. Jerry Pelletier performed the protein analyses.
Thanks too to Sally Drew for her contributions as a draughtsman,
typist, and humour equilibrator.
This investigation was supported by PHS Research Grant H6423
from the National Heart Institute, Public Health Service.
v
TABLE OF CONTENTS
Section
Page
SUMMARY..........
II
III
IV
..................
1
15
INTRODUCTION. ................................
A.
Background and Objectives. ...................
15
B.
Composition and Properties of Human Blood.. .
17
(1)
The red cells. ..........................
17
(2)
The white cells .........................
24
(3)
The platelets ................
28
(4)
Blood plasma ................
(5)
Coagulation and aggregation .........
32
35
C.
Proposed Model. ..........................
D.
Results of Previous Investigators ..........
PROCEDURE ..................
.
42
The GDM Viscometer ...................
B.
The Merrill - Brookfield Viscometer
C.
Preparation of Blood Samples. ..............
(1)
Obtaining blood samples ..............
(2)
Preparation of samples ...........
DISCUSSION OF RESULTS.........
48
. . . . . .
51
51
52
. . .
.
0
...
54
Whole Blood ........................
(1)
54
Derivation of vis'cometer equations
. .
54
Assuming constant fluid viscosity in
the viscometer gapp. . . . .. .. .. . . .
54
(b)
The Krieger - Elrod equation. . . . . . .
57
(c)
The Vand wall effect . . . .
58
(a)
(2)
37
42
... s... .....
A.
A.
29
......
Time effects .....................
vi
. ...
. . . .
.
.
.
67
TABLE OF CONTENTS (Cont)
Page
Section
Time effects at constant bob
rotational speeds .................
67
(b)
Time effect upon stopping the
viscometer bob ..................
81
Correlation of shear stress - shear
rate data. . . . . . . . . . . . . . . . . . . . . .
96
(a)
The low shear rate region . . . . . . . .
96
(b)
The high shear rate region. . . . . . . .
105
The yield stress. . . . . . . . . . . . . . . . . .
112
.
.
(b)
Effect of hematocrit . . . . . . . . . . . .
116
(c)
Effect of temperature . . . . . . . . . . .
118
(d)
Variation with source . . . . . . . . . . .
122
.
.
.
112
Effects of physical factors on blood
rheological properties . . . . . . . . . . ...
122
(a)
Hematocrit . . . . . . . . . . . . . . . . . .
122
(b)
Temperature . . . . . . . . . . . . . . ..
123
(c)
Sample age . . . . . . ......
. . . . .
134
(d)
Centrifugation . . . . . . . . . . . . . . ...
138
(a)
Anticoagulants . . . . . . . . . . . . . . . .
138
(b)
Plasma protein content . . . . . . . . . .
142
(c)
Plasma lipid content . . . . . . . . . . . .
146
(d)
Plasma hemoglobin content . . . . . . .
150
.
.
.
.
Red Cell Suspensions..........
(1)
........
151
Red cells suspended in saline . . . . . . . . .
.
B.
138
Effects of chemical factors on blood
rheological properties. . . . . . . . . . . . .
.
(6)
Method of determination..........
.
(5)
(a)
.
(4)
.
.
(3)
(a)
vii
151
TABLE OF CONTENTS (Cont)
Section
Page
(2)
154
Red cells suspended in a-globulin
saline solutions .
155
Red cells suspended in y -globulin
saline solutions .
158
Red cells suspended in fibrinogensaline solutions.. ...........
158
.
(3)
Red cells suspended in albumin-saline
solutions. ............................
.
(4)
(5)
Plasma. . . . . . . . . . .
161
.
C.
.
. . . ... . . . . .
CONCLUSIONS . .
. . . . . .
163
.
V
*
. . ... . . . . . .
VI
RECOMMENDATIONS.....
166
APPENDIX
B.
The Krieger
-
Elrod Enu ation.. . ... . ..
169
Derivation
. . . . . . . . . . . . . . . . .
169
(2)
Application . . . . . . . . . . . . . . . . .
175
.
(1)
.
A.
180
C.
Location of Data and Calculations . . . . .
191
D.
Literature Citations . . . ... . . . . . .. . . .
192
E.
Nomenclature . . . .. . .
198
.
.
.
Use of Theoretical Equations to Correlate
Blood Data in the Shear Rate Range 2 to
20 sec-1. . . . . . . . . . . . . . . . . . . . .
viii
.
. . . . . . . . . . .
LIST OF TABLES
Table No.
2-1
Page
Title
. . . .
30
Rate of torque decay at constant viscometer
rotational speed. ....................
. . . .
75
4-2
Yield stresses of kaolin suspensions . . . . . .
. . . .
88
4-3
Casson constants and rouleaux axial ratios
calculated from equation (4-25) from data of
figures (4-20) and (4-21) . . . . . . . . . . . . .
103
.
4-1
.
Main constituents of blood plasma and
representative normal concentrations.....
Experimentally determined yield stresses
of blood . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
115
Effect of temperature on the rheological
properties of blood . . . . . . . . .. . . . . . . . .
. . . .
124
Effect -of high temperatures on the rheological
properties of blood . . . . . . . . . . . . . . . . .
. . . .
128
Effect of high temperature on the rheological
properties of blood. .. . . . . . . . . . . . . . . .
. . . .
130
.
4-4
.
4-5
.
4-6
.
4-7
Effect of temperature on a cold-agglutinating
.
4-8
b lo o d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
131
Effect of storing blood containing ACD at 400C on
the rheological properties of blood . . . . . . . . . . .
135
Rheological properties of suspensions of different
aged red cells in plasma . . . . . . . . . . . . . . . . . .
137
The effect of centrifuging on the rheological
properties of blood . . . . . . . . . . . . . . . . . . . . .
139
The effect of anticoagulants on the rheological
properties of blood. . . . . . . . . . . . . . . . . . . . . .
141
A-1
Shear stress - shear rate data for a blood sample
. .
179
B-I
-1
Viscosity of blood at shear rates between 4 sec1
1
.
and 20 sec- . . . . . . . . . . . . . . . . . . . . . . . . . .
183
Red cell volume fraction at closest packing for
suspensions of Table B-1. . . . . . . . . . . . . . . . . .
187
4-9
4-10
4-11
4-12
B-2
ix
LIST OF FIGURES
Page
Title
Figure No.
2-1
The Human Red Cell. ........................
2-2
The Fibrinogen Molecule in The Dry State
32
2-3
Model Red Cell Aggregate .....................
35
2-4
Comparison of The Data of Dintenfass with
The Data of This Thesis ................
41
3-1.
Overall View of The GDM Viscometer ........
42a
3-2
Schematic Diagram of the GDM Viscometer .
3-3
Detailed Schematic Diagram of the GDM
Viscom eter ........................
.
45
Schematic Diagram of The Grooved Viscometer
Surfaces ...........................
.
47
3-4
.18
.
. .
43
3-5
The Merrill - Brookfield Viscometer......... .... 49
4-1
Comparison of Shear Rates Calculated From
Viscometric Data by the Krieger - Elrod
Equation and the Conventional Equation .. .......
4-2
59
Comparison of the Viscometer Cylindrical
Surface - Suspension Interface When The
Cylindrical Surface is (a) Smooth, and
(b) Rough on a Scale Greater Than The
Particle Size ........................
4-3
4-4
62
Effect of Viscometer Wall Roughness on the
Apparent Rheological Properties of Blood ....
64
Typical Torque - Time Curves for Human
Blood, at Constant Viscometer Rotational
Speed. .........
4-5
.............................
Time Required To Reach Torque
-
Time
Curve Peak versus Viscometer Bob
Rotational Speed, for 3 Blood Samples .......
4-6
4-7
68
.
69
Photograph of Hyperlipidemic Blood and Plasma,
and Normal Blood Plasma ...................
71
Hyperlipidemic Blood in The Viscometer,
0.5 Minutes After Stirring Stopped, Bob
Speed Is 0.2 RPM. ..........................
72
x
LIST OF FIGURES (Cont)
Figure No.
4-8
Page
Title
Hyperlipidemic Blood in The Viscometer,
5.5 Minutes After Stirring Stopped, Bob
Speed Is 0.2 RPM. ...................
. .
73
Sedimentation of Red Cells Normal to a
Shear Field. ..........................
76
4-10
Sedimentation Rate of Hyperlipidemic Blood
78
4-11
Torque - Time Curve for Blood at Constant
Viscometer Rotational Speed ...........
82
4-12
Shear Stress - Shear Rate Data for Human
Blood, Using Extrapolated, Peak, and Steady
State Torque Values .....................
4-13
4-14
.
4-9
Obtained on Stopping the Viscometer Bob
Rotation .........................
. .
84
. .
85
Torque Decay Curve Obtained for Kaolin
Viscometer ...........................
Torque Decay Curve for 4% Kaolin Suspension
.
in the GDM Viscometer, Rotation Stopped.
Comparison of Yield Stress Data for Kaolin
Suspensions as Determined in the Merrill
Brookfield and GDM Viscometers .........
.
87
. .
89
. .
91
-
4-16
83
Torque - Time Curve for Human Blood,
Suspension in the Merrill - Brookfield
4-15
. .
4-17
Torque - Time Curves for Blood, GDM
4-18
Torque - Time Curves for Blood, GDM
Viscometer Rotation Stopped ...........
92
Viscometer Rotation Stopped ............
4-19
4-20
4-21
4-22
Casson Plot of Data for Blood of Various
Hematocrits ..........................
Test of Casson Equation using Human Blood
Data. ............................
99
. . .
100
Test of Casson Equation using Human Blood
Data. ...................................
102
Shear Stress - Shear Rate Data for Blood,
Red Cells in Serum, Plasma, and Serum ......
108
xi
LIST OF FIGURES (Cont)
Figure No.
4-23
4-24
Title
Page
Shear Stress - Shear Rate Behavior of Blood
and Red Cells Suspended in Albuminated
Saline. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
Shear Stress - Shear Rate Behavior of Red
Cell - Serum and Red Cell - Saline
4-25
Suspensions. . . . . . . . . . . . . . . . . . . . . . . ..
111
Cube Root of Yield Stress versus Hematocrit
for 5 Different Normal Bloods . . . . . ... . . . . .
117
4-.26
Casson Plots for a Typical Normal Human
Blood, at Three Temperatures and Four
Hematocrit Levels . . . . . . . .. . . . . . . . .. .. 120
4-27
Viscosity (log scale) versus Reciprocal
Absolute Temperature, Determined from
Table (4-5) ...............................
125
Relative Viscosity (log scale) versus Temperature, Computed from Figure (4-27) ............
127
4-28
4-29
4-30
4-31
4-32
Effect of High Temperatures on The
Rheological Properties of Blood ..........
.
129
Effect of Temperature and Hematocrit on
The Slope of The Casson Plot for a Blood. ....
130
Comparison of Anticoagulated Blood Samples
with Native Blood.............
. ..
.....
140
Rheological Properties of Red Cells Suspended'.
in Plasma, Serum, and a Plasma - Serum
M ixture ..........................
.
4-33
Rheological Data for a High Lipid Content Blood
4-34
Effect of Hemoglobin on the Rheological
...........
Properties of Blood ............
4-36
4-37
148
152
Rheological Properties of Red Cell - Saline
.......
Suspensions, Effect of Hematocrit.
153
Rheological Properties of Red Cell
Albuminated Saline Suspensions ..............
156
Rheology of Red Cell - a--globulin saline
Suspensions ........................
157
-
4-35
145
xii
LIST OF FIGURES (Cont)
Figure No.
Title
Page
Rheology of Red Cell - y -globulin Saline
Suspensions. ..............................
159
Rheology of Red Cell - Fibrinogen Saline
Suspensions.........................
160
4-40
Effect of Temperature of Plasma Viscosity. .....
162
A-i
Diagram for Evaluating d w1 /d Ti for The
Krieger - Elrod Equation.................
176
4-38
4-39
A-2
Diagram for Evaluating d 2 1n w /(d In T1)2
for The Krieger - Elrod Equation . ......
B-1
3-2
B-3
B-4
. ...
177
Specific Viscosity Divided By Hematocrit
versus Hematocrit For Red Cells Suspended
in Plasm a................
...........
181
Test of The Mooney Equation with Red Cell
Suspension Data .....................
185
Test of Brinkman's Equation with Data for
Red Cell - Plasma Suspensions. ................
186
Test of Simha's Equation with Data for Red
Cell - Plasma Suspensions. ..............
xiii
.
188
I.
SUMMARY
Introduction
The objective of this study was to investigate the rheological properties of human blood, obtained from donors in good health, near and at
zero shear rate.
This particular shear rate range had not been pre-
viously investigated because of a lack of sufficiently sensitive viscometers
and because of the almost exclusive use of capillary viscometers by those
interested in blood flow.
Interest was focused on the very low shear rate region because it is
in this region that the interparticle forces become important in comparison to the hydromechanical forces.
The nature and cause of the inter-
particle forces were to be studied.
Blood consists of several types of cells suspended in a complex
solution (plasma) of inorganic salts and organic compounds.
The red
cells occupy about 45% of the blood volume while the other particles together occupy less than 1% of the blood volume.
The red cells are bi-
concave discs, 8 p in diameter and 2 g in maximum thickness; they are
easily elastically deformed.
From the work of Fahraeus (23), it is
known that the presence of certain of the plasma proteins has a profound
influence on the ability of the red cells to aggregate in stationary blood.
Procedure
Blood samples and red cell suspensions were studied in the GDM
viscometer, which is a concentric cylinder, Couette type instrument
(Figure 3-3).
The unique features of this particular viscometer are
that the stationary outer cylinder, the "cup", is mounted on an air bearing, and that the torque is measured by a "torque-to-balance" system
1
which permits torques to be measured with a precision of 0. 0001 dyne
cm or 0. 1%, whichever is larger, in the torque range from 0. 0100 to
1999 dyne cm.
Because of the dimensions of the viscometer cylinders,
2
this corresponds to shear stresses from 0. 00036 to 74 dynes/cm .
The
inner cylinder, the "bob", is rotated at speeds from 0. 01 to 100 rpm,
which corresponds to shear rates in the viscometer gap from 0. 01 to
100 sec .
The viscometer bob, constructed of coin silver, is hollow
and attached to a hollow shaft down which passes a tube; water at a
chosen constant temperature is circulated through the shaft and bob at
a rate of about 2 liters per minute.
of lucite.
The viscometer cup is constructed
Because of the high thermal conductivity of the bob, the high
heat capacity of the bob, and the poor thermal conductivity of the lucite
cup, the fluid in the viscometer gap has been calculated to be within
0. 05*C of the temperature of the water passed through the bob.
constant temperature water is maintained within
temperature.
The
0. 01*C of the chosen
A stationary guard ring, which penetrates the surface of
the fluid in the viscometer, prevents the mechanical transfer of a torque
from the rotating viscometer bob to the viscometer cup by any surfactant
layer which might form at the liquid-gas interface.
Two sets of viscometer cylinder surfa'ces were used: (1) a smooth
surfaced set, and (2) a rough surfaced set consisting of cylinders vertically grooved with 720 equilateral-triangular cuts 66 microns deep.
the viscometer surfaces, rough or smooth, were "siliconized".
All
The vis-
cometer gap (1. 5 mm) is large in comparison with the red cell size.
Blood samples were obtained from donors in good health at the Blood
Bank of the Massachusetts General Hospital, Boston.
were collected by routine blood bank procedure:
2
Most of the samples
ACD solution was mixed
with the blood to prevent coagulation and to permit blood storage at 4*C.
Other samples were collected without addition of anticoagulant and with
the addition of other anticoagulants.
The volume fraction of the blood
occupied by the red cells was varied by combining various portions of
Red cells were also suspended in
centrifuged red cells and plasma.
isotonic saline containing plasma proteins and protein fractions.
Results and Discussion
(a)
Before discussing the rheological behavior of blood, it is
essential that certain phenomena which might lead to an erroneous
interpretation of the experimental data first be presented.
(1)
Time Effects at Constant Viscometer Rotational Speed.
A recorder continuously traces out the torque reading of
the viscometer as a function of time.
When the viscometer contains whole
human blood, and the bob is rotated at a constant angular speed, the torquetime curve takes one of two forms.
1 sec
If the shear rate is greater than about
the torque rapidly climbs to a value which is constant thereafter
(upper diagram of Figure 4-4);-if the shear rate is less than about 1 sec,
the torque initially rises to a maximum and then decays (lower diagram of
Figure 4-4).
The time necessary to reach the torque maximum increases
as the shear rate (bob rotational speed) decreases, and the rate of torque
decay immediately after the peak, expressed as dyne-cm per minute,
appears to be independent of the shear rate, but varies with the nature
of the viscometer surfaces and with the blood donor.
This behavior has
been observed without exception in all blood samples from donors in
normal health, and the shear rate at and below which the time effect is
first observable has always been at about 1 - 4 sec-4
From the observation that viscous homogeneous fluids show the same
behavior as the initial portion of the torque-time curves for blood, it has
3
been concluded that this behavior is the transient period during which
the blood is attaining its steady state flow pattern in the annular viscometer gap.
The subsequent torque decay period, is explained by the
mechanism of a developing layer of cell-free blood plasma at one or
both cylindrical surfaces.
This layer, which acts as a lubricant, de-
velops only at shear rates below about 1 sec'
and grows in thickness
with time until the torque has decayed to a steady value.
This argument
requires that the red cells of the blood receed from the viscometer walls.
Visual evidence of this mechanism was obtained with the fortunate discovery of a blood donor whose blood plasma contained about 8% fat. The
fat concentration was sufficiently high to cause the blood plasma to be
opaque and milky white, instead of the usual clear, straw -colored, fluid.
This blood was placed in the rough surfaced viscometer and stirred by
raising and lowering the rotating bob, which was rotating at a rate of
0. 2 rpm.
Its appearance about 1/2 minute after stirring was stopped
was normal (red in color) , but thereafter it became milky white in
color as time progressed.
This color change was caused by the develop-
ment of a plasma layer at the outer viscometer cylinder wall.
The blood
was again stirred, returning to a normal color, but the viscometer bob
was not rotated - the blood did not whiten with time.
Clearly, the de-
velopment of the plasma layer is induced by the flow of the blood.
Further
experiments with this blood sample showed that the plasma layer did not
develop at shear rates greater than 1 sec~I and that the layer developed
only when a torque decay was simultaneously observed in the torque-time
curve.
From this interpretation of the torque time curves, it is quite obvious
that the correct torque value to be associated with a particular shear rate
is close to the torque-time curve peak, if the peak occurs shortly after
4
starting the fluid motion.
However, in the case of low shear rates,
where the peak occurs several minutes after start up, some correction
must be made to the peak value in order to correct for the plasma layer
which has been developing in this time; a linear extrapolation to time
zero, of the torque-time curve after the peak would be one such attempted
correction.
Such an extrapolation procedure was used at the lowest shear
rates in this study.
No torque decay (at constant viscometer rotational speed) was found
for suspensions of red cells in isotonic saline, or in isotonic saline containing a plasma protein or protein fraction, with one very important
exception.
The exception was red cells suspended in saline containing
the protein fibrinogen.
This strongly suggests that the migration of the
red cells in the viscometer at low shear rates is dependent on, or coincident with, the ability of the red cells to aggregate into rouleaux at low
shear rates, which is known to depend on fibrinogen.
The cause of the
migration cannot lie in the deformation either of the red cells or the
rouleaux, but may be due to a Magnus effect, or due to the iritercellular
attractive force between red cells.
(2)
The Effect of Viscometer Surface Roughness
As a consequence of the geometric hinderance of a smooth
wall, a suspension occupying the space immediately next to the wall does
not contain the same volume fraction of particle material as the bulk
fluid.
A model of this situation in the viscometer could consist of thin
layers of the suspension suspending medium at the smooth walls and
uniform suspension in the rest of the viscometer gap. Such a situation
would result in lower torque values being recorded at each viscometer
bob speed than would be recorded if no wall layers existed.
5
The rough surfaced set of viscometer cylinders was prepared with a
"roughness" which is large compared to the red cell size.
Both the rough
surfaced and the smooth surfaced viscometer surfaces were used to determine the rheological properties of a blood sample.
The difference was
found to be significant and, usingthe equation derived by Vand (64) for the
model of the smooth surfaced situation, the wall layer thickness was
calculated to be 1 to 3 microns, in good agreement with the expected
thickness considering the red cell dimensions.
This wall effect was
verified by similar tests on several blood samples, and also by the use
of sand-coated viscometer surfaces.
The effect was not due to cali-
bration errors, as Newtonian fluids of viscosities ranging from 1 cp to
500 cp were found to have the same viscosity in both the smooth and
rough surfaced viscometers.
(3)
Calculation of the Shear Rate
The commonly used equations for relating the rotational speed
of a concentric cylinder viscometer to the shear rate in the viscometer gap
assume that at a particular rotational speed the viscosity of the fluid in the
gap is constant across the gap.
This assumption is valid for Newtonian
fluids, but, because of the slight variation in shear stress across the gap,
is not exactly correct for non-Newtonian fluids (although the error due to
the assumption is generally small) . Krieger and Elrod (42) derived an
equation which permits the calculation of the shear rate from the experimental data without making any assumption about the properties of the
fluid in the viscometer gap.
time consuming.
Application of their equation, however, is
It was found that it usually was not necessary to use
the Krieger-Elrod equation to calculate the exact shear rate for blood
samples from healthy donors if the hematocrit (red cell volume fraction)
was below about 45%.
At higher hematocrits, use of the usual equation
6
led to low shear rate values, and the Krieger-Elrod equation proved to
be of value.
(b)
The Rheological Properties of Blood
The considerations discussed in the previous section were
essential to the study of and interpretation of the rheological properties
of blood, as will become clear.
(1)
Yield Stress
The yield stress of blood has never been recorded in the.
It was experimentally determined in this study in the follow
-
literature.
ing manner: after the viscometer had been in operation at some constant
shear rate the rotation of the inner cylinder was stopped and torque-time
curves, such as shown in Figure (4-13)., were obtained.
If the fluid in
the viscometer was water, plasma, or red cells suspended in albuminated
saline, the torque decayed to zero in a few seconds, as indicated by the
dashed line.
When blood is in the viscometer, the torque initially decays
at the same rate as in the case with water, until a certain value, T
reached below which the torque decays much slower.
really two exponential curves.
is
This curve is
It has been found empirically, using
suspensions having known yield values, that the point where the transition
from one exponential curve to the other first takes place corresponds to
the fluid yield stress.
For blood, the agreement between this independent determination of
the yield stress and the value obtained by extrapolation of the low shear
rate data is always within a few percent.
Analysis of the torque-time curves obtained after the viscometer
rotation is stopped shows that the curves are qualitatively in agreement
7
with the following hypothesis.
The three dimensional network giving
stationary blood its yield stress is supposed to contain rouleaux, and
the average length of the rouleaux is assumed to be a function of the
shear rate to which the blood was subjected immediately before becoming stationary (higher shear rates causing shorter rouleaux).
The dependence of yield stress on hematocrit was found to be correlated by an empirical expression proposed by Norton (51) for clay
suspensions:
y
where Tr
c
)
T 3 = a(c - c
is the yield stress, "a" is a constant, and "c" is the hematocrit.
The constant "c
"
is the red cell concentration below which blood cannot
have a yield stress because it is not geometrically possible to construct
a 3-dimensional network throughout the blood with the amount of red cells
available.
Figure (4-25) shows the yield stress-hematocrit data for 5
different bloods.
In the case of blood samples of hematocrit less than about 40%, the
yield stress appears to be independent to temperature in the range from
10*C to 37*C.
At higher hematocrits, the yield stress decreases slightly
as the temperature increases.
It has been hypothesized that, for blood
samples of hematocrit below 40%, the product of the linkage density and
the average link strength of the 3-dimensional network in stationary blood
must be independent of temperature, but that the members of the product
vary with temperature (link density decreases and link strength increases
with temperature increase).
At higher hematocrits, the red cell density
is becoming so high that the nature of the structure giving the blood a
yield stress is different from that of the lower hematocrit blood.
8
(2)
Correlation of Shear Rate - Shear Stress Data
The ability of the red cells in human blood to aggregate when
the blood is stationary was first extensively studied by Fahraeus (23).
In
normal health, the red cells, which are disks with concave faces, aggregate by joining together at their faces to form rod-like rouleaux (having
as their diameter the diameter of one red cell).
These rouleaux are
flexible and easily broken down when the blood is caused to flow.
At
sufficiently low shear rates, the rouleaux, because of their asymmetry,
will become aligned along the fluid streamlines and will act as though
they were rigid straight rods if the streamlines are straight.
Because
of their frailness, the rouleaux will decrease in length as the shear rate
increases, until at high enough shear rates the red cells exist only as
individuals.
To a remarkable extent this behavior of blood is identical
to that of a model suspension proposed by Casson (10):
mutually
attractive particles are suspended in a Newtonian medium; these particles aggregate at low shear rates to form rigid, rod-like aggregates
whose length varies inversely with the shear rate.' For this model,
Casson found that the relationship between the axial ratio J (length to
diameter ratio) of the aggregates and the shear rate - was
9FA
2
a p _1
S48
provided J was very much greater than unity (FA is the cohesive force
between the particles forming the aggregates, dynes/cm2,
fl is the
suspending medium viscosity, and "a" is a constant whose value depends
on the orientation of the aggregates with respect to the fluid streamlines).
For shorter aggregates, not being able to determine the J - -k relationship, Casson assumed that over short ranges of P, the relationship was
9
(S-i)
+p
*
J=a
where a and 1
are constants. Using this later equation, he then found
the following relationship between the shear stress and the shear rate
of his model suspension:
1
T2
_
1
2 +b
(S-2)
where
S =2
(S -3)
-1
(1-c au
ac -1
b
aa-1
1
(-c4/
In these equations, "c" is the volume fraction of the suspension occupied
by the particles..
The Casson suspension must fulfill certain conditions. First, over
1
1.
a certain shear rate range, a plot of T2 versus PP should be linear. Also,
from equation (S-3), a plot of ln s versus ln(1-c) should also be linear
with a slope equal to [-(aa -1) /2].
Having thus determined the quantity
(aa ), equation (S-4) indicates that a plot of b versus 1/(l-c)(aa -1) /2
should be a straight line, with a slope equal to the negative of the intercept.
The data of Figure (4-19) indicates that at low shear rates (below
1 sec I) the first condition of a Casson suspension is fulfilled by human
blood. Figure (4-20) shows a plot of ln s versus ln(1-c)for three blood
samples: from this graph the value of the quantity (aa -1)/2 has been evaluated
as 1.19 for two bloods at 19*C and 1.09 for one sample at 25*C. Using these
data, a plot of b versus 1/(1-c)(aa -1) /2 was prepared - Figure (4-21).
10
The slopes and intercepts of these curves are shown: the intercepts are
approximately the negatives of the sl6pes.
On one point does blood not
fulfill all of the graphical properties of the Casson model: the lines
should all pass through the point [ 1/(1-c) (aa-1) /2
1. 0, b = 0 in
Figure (4-21); instead they pass slightly away from this point.
This is
because blood has a critical concentration of red cells below which it
does not have a yield value.
It has been found that a decreases and P increases as the temperature
increases.
The range of applicability of the Casson equations is from zero shear
rate up to a limiting value, which decreases -in value as the hematocrit
-1 at a hematocrit of
increases. The limiting shear rate is about 1 sec
about 45% in normal blood.
Because of approximations and assumptions in the derivation of the
Casson equations, they cannot be used to calculate any fundamental
properties of human blood.
The fact that blood, in the low shear rate range, obeys the Casson
relationships is not sufficient prcof in itself that blood behaves in detail
like the Casson model suspension.
(3)
Effects of Blood Constituents on the Rheological Properties
of Blood
(a)
In the normal plasma protein concentration ranges found
in blood from healthy individuals, only the protein fibrinogen seems able
to cause blood to have a yield stress.
This is shown clearly in Figure
(4-32), which shows data for red cells suspended in plasma, in serum
made from the same plasma, and in a mixture consisting of equal volumes
of the plasma and the serum.
In these suspension solutions, only the
11
fibrinogen concentration varies, the other plasma constituent concentrations remaining constant.
The red cell
have no yield stress, and the red cell
-
-
serum suspension was found to
plasma
-
serum mixture sus-
pension had a yield stress about one quarter that of the red cell - plasma
suspension (blood).
The yield stress is not directly proportional to the
plasma fibrinogen concentration although higher fibrinogen concentrations
do cause higher yield stresses.
The red cell migration in the viscometer
(at a constant rotational speed of 1 rpm or less) occured only in those
red cell - protein containing saline suspensions which contained fibrinogen.
The speed of migration increases as the fibrinogen concentration increases;
blood samples showing high yield stress also show high migration speed.
Free calcium ions do not play a role in causing the intercellular red
cell force.
(b) The red cell properties change with the red cell age, but
these changes do not affect the rheological properties of the blood.
(c) The lipid content of the blood may influence the blood
properties, especially at the higher lipid concentrations where the yield
stress and apparent viscosity at a given shear rate increase.with increase
in lipid content.
(d) Hemoglobin in plasma increases the yield stress and
apparent viscosity of blood.
This may prove to be of great importance
in open-heart surgical procedures with the "heart-lung" machine, in
which the hemoglobin concentration continually rises with time.
(4)
Effect of Temperature on the Flow Properties of Blood
In the temperature range of 100C to 37*C, changes in the
rheological properties of blood from healthy donors were reversible.
shear rates above about 20 sec
,
At
the temperature dependence of blood
12
is the same as that of water, while at lower shear rates the temperature
dependence decreases as the shear rate decreases.
This is a consequence
of the temperature independence, or near independence, of the blood yield
stress.
Blood plasma, which is Newtonian, has the same temperature dependence as water.
When blood is held at temperatures a few degrees above 37*C (98.6 0 F)
irreversible changes occur, as indicated by changes in the flow properties.
It is tentatively postulated that this irreversibility is due to an instability
of the protein fibrinogen.
Several abnormal bloods, known as cold agglutinating bloods, were
found to undergo irreversible changes at temperatures a few degrees
below 37*C.
All of the irreversible changes referred to were characterized
by higher viscosities and higher yield stresses, when the blood was retested at 37*C, after the thermal treatment (heating or cooling).
Conclusions and Recommendations
A procedure for obtaining meaningful low shear rate data for blood
has been developed, and the Casson equations have been found to be
satisfactory correlative means, though imperfect in their fine detail.
A model for blood, similar to the Casson model suspension, seems
appropriate.
The role of fibrinogen in causing the red cell attractive force is
beginning to become clear.
The studies described herein point to the
need for research into the details of the fibrinogen effect, such as the
competitive sorption on the red cell surface by the other plasma proteins,
the role of lipids in the plasma and on the red cell, the fibrinogen effects
13
under conditions of disease, and so on.
Many other effects noted in this
thesis, such as the red cell migration at low shear rates and the irreversible effects of temperature changes, remain to be investigated in detail.
The use of low shear rate viscometry, as developed here with a
sensitive concentric cylinder type instrument, can be used as a tool for
the study of medical and biological problems.
It offers the advantages
of requiring small samples, using non-destructive testing, and giving
results rapidly.
Hopefully this technique will find wider use in the
future.
14
II.
A.
INTRODUCTION
Background and Objectives
Blood, which is a suspension of several types of deformable particles
in a complex aqueous solution, has been a subject of medical interest for
a long time.
Today, the interest in blood is not just clinical in nature,
but is also aimed at understanding the properties of the individual constituents of blood.
On obtaining a knowledge of the forces governing the
behavior of individuals, the inter -relationships of the parts can be better
understood.
Studies of the rheological properties of blood, and of parts of blood,
offer one means of investigating the interactions of the members of this
important suspension.
Since there are many diseases in which circula-
tion difficulties arise, and many of these diseases are marked by abnormalities in size, shape and/or concentration of one or more of the
constituents of the blood, it is important to see what effect these abnormalities have on the flow properties of human blood.
However, before
an understanding of the abnormal can be obtained, it is essential to
understand the normal.
Considering the ease with which the importance of blood flow studies
can be ascertained,. it is not surprising to find that the flow of blood has
been investigated for quite some time; indeed, the French physician
Poiseuille seems to have been the first to have made such an investigation.
With few exceptions, until very recently, these experimental
studies have been conducted in capillary tube viscometers, probably because of the gross physical similarity between the blood vessels and
capillary tubes, and because of the ease with which one can make and
use such viscometers.
While such studies will permit one to determine
15
pressure drop-volumetric flow rate data, they will not permit one to
gain an insight into the properties of the blood.
-
If one is interested in investigating the interconstituent forces re
sponsible for the flow properties of blood, one must make his studies at
low shear rates.
In order to do this in a capillary viscometer, with any
degree of precision, one must use very small capillaries.
using small capillaries presents two major objections:
However,
(1) the blood
flowing through the capillary is not subjected to a shear rate which is
even approximately uniform, and (2) in sufficiently small capillaries
(less than about 0..3 mm in diameter) the influence of the tube walls becomes large.
Consequently, if one is interested in the flow properties
of blood at low shear rates, another type of investigative instrument
must be considered.
An instrument which overcomes the objections to the use of capillary
viscometers is the Couette-type viscometer, which physically is two
concentric cylinders with a gap between them.
The fluid to be studied
is placed in the gap, one cylinder is rotated at constant speeds, and the
torque transmitted through the fluid from the rotating cylinder is measured
at the other cylinder, which is stationary.
The shear stress and shear rate
are almost constant across the viscometer gap, and the approach to constant conditions is determined by the dimensions of the cylinders and the
gap.
In addition, the gap can be made large enough to eliminate the in-
fluence of viscometer dimensions on the measured properties of the fluid
being tested.
Such an instrument, capable of making measurements at
shear rates as low as 0. 01 sec,
has been developed at M. I. T. , and has
come to be known as the GDM Viscometer.
16
Considering the almost complete lack of data on the flow properties
of human blood at shear rates below about 50 sec
, the objectives of
this thesis have been to make a study of the rheological properties of
normal human blood at shear rates below about 50 sec
In order to
more fully understand the causes of these properties, the studies have
extended into investigations of suspensions and solutions which are a
combination of several of the constituents of blood.
In addition, pre-
liminary studies of abnormal bloods were undertaken, and indicate the
potential usefulness of this type of investigation for understanding the
causes of circulatory difficulties.
B.
Composition and Properties of Human Blood
Human blood is a complex suspension of three general types of
The three types of particles are the
particles in a continuous medium.
erythrocytes (red cells), the leukocytes (white cells). and the platelets.
The continuous medium, known as plasma, is in itself a complex solution of inorganic salts and organic macromolecules in water.
The
importance of each of these parts of the blood merits a brief discussion
of each of them,
1.
The Red Cells*
The normal human red cell, when observed in stationary blood,
has the shape of a biconcave disk with a mean maximum diameter of
about eight microns, a maximum thickness of about two microns, and a
minimum thickness of about one micron.
cubic microns.
Its average volume is 87 ( 5)
The shape of the erythrocyte is shown in Figure (2-1).
*Most of the information contained in this section is discussed in references (53) and (34).
17
While this is its shape when it is viewed in stationary blood, it is very
flexible and is deformed into almost every shape during its circulation
This flexibility arises because the red cell is essen-
through the body.
tially a thin membrane container filled with a solution.
Figurp (2-1)
The human red cell
In health, the human male has about 5. 4 million red cells per cubic
millimeter of blood,Z~18
while the human female has about 4. 6 million per
cubic millimeter.
This means that normally, in a man, the red cells
occupy about 47%o of the blood by volume, and, in a woman, about 42%
of the blood volume (the percentage of the blood volume occupied by the
red cells is referred to as the blood "hematocrit" in medical language,
and does not include the volume of the blood occupied by the whiecells
or platelets).
When a human has a low hematocrit, he is said to be
anemic;- anemia can be caused by (1) blood loss due to excessive bleeding,
(2) lack of functioning of bone marrow (the source of red cells) such as
that due to excessive exposure to X-rays or benzene compounds, (3).
failure of the red cells to mature such as in pernicious anemia, and
(4) destruction of red cells such as occurs in severe sickle cell anemia.
The other extreme in hematocrit, that of a high hematocrit, is called
polycythemia; this condition develops in humans who live at high altitudes
(physiologic polycythemia) as well as in persons suffering from a tumerous condition of the red cell forming organs (erythremia).
Under the
conditions of these diseases, the hematocrit may be anywhere from about
10 percent to 70 or 75 percent.
The red cell is a "living" body in the sense that metabolism does
go on in it, and that it does age.
It has generally been found that the
red cell survives for 110 to 120 days in the human.
During this life
time, the red cell is reported to undergo changes in its dimensions,
shape, volume, density, osmotic fragility, metabolism and composition.
At the physiological pH of 7. 2, the red cells have a net negative
charge.
This has been determined by electrophoretic measurements.
Attempts to find the isoelectric point of the cells have lead to conflicting conclusions:
some workers report isoelectric points between pH
values of 3. 6 and 4. 7, while other investigators have not been able to
find an isoelectric point.
The problem is complicated by the inability
of the red cell to remain intact at pHs which differ very much from 7. 2.
The source of the cells net negative charge has been attributed to the
free phosphate radicals of the phospholipids, and to carboxyl groups of
other substances, all-which are constituents of the red cell membrane.
Some workers have found that quartz particles, of about the same size
as the red cell, when coated with the protein albumin, have the same
mobility as red cells; these workers therefore concluded that a surface
layer of albumin is important in giving the red cell its negative charge.
Since the mobility of cells from the different serological groups are not
19
the same (the mobility of A and B cells being about 17% less than that of
0 cells), it is not surprising that cells coated with antibodies have lower
mobilities than uncoated cells.
In spite of the fact that red cells have a net negative charge, the
cells will stick to each other if the potential difference between the cells
and the surrounding medium becomes less than a critical potential. Thus,
cells can be made to stick to each other (agglutinate) either by lowering
the cell potential or by raising the critical potential.
Red cells suspended
in sucrose solutions or in weak concentrations of electrolytes lose their
negative charge and agglutinate.
On the other hand, cells treated with
antibody agglutinate because the critical potential is raised.
This later
type of agglutination occurs when bloods of the incompatible serological
groups are mixed, and the cells are believed to be held together by an
antibody -antigen interaction.
Unlike the case with agglutinated cells, which are randomly joined
together rather strongly, the red cells in stationary blood will aggregate,
flat face to flat face, to form "poker chip" stacks of red cells, called
rouleaux.
The red cells in rouleaux are only weakly held together and
when the blood is set into motion the rouleaux decrease in length, or
break up completely.
breaking.
The rouleaux are not rigid, but can bend without
The formation of these rouleaux was carefully studied by
Fahraeus, (24), whose work showed that the plasma protein albumin
prevented rouleaux formation, while the globulins aided rouleaux formation to some extent, but fibrinogen greatly enhanced the formation of
rouleaux.
In stationary blood, these rouleaux may contain only a few
red cells or may contain up to about 30 red cells.
Quite obviously the membrane of the red cell strongly determines
the physical and chemical properties of the erythrocytes.
20
Estimates of
the thickness of the membrane range from 50
ductivity measurements) to about 5000
R (by
R
(by electrical con-
birefringence measure-
ments). - The electron microscope has yielded estimated thicknesses
throughout this entire range.
The membrane thickness does not appear
to be uniform in all areas of the cell.
Most information on the cell
membrane has been obtained with membrane material prepared by
causing red cells to haemolyze (lose the red cell contents) and washing
the remaining cell "ghosts", called stroma.
Unfortunately, the methods
of causing red cell haemolyzation and of washing the stroma have a large
effect on the results of investigations on the stroma.
However, work
preformed on stroma indicates that it makes up about 10 volume percent
In chemical composition, the stroma appears to be
of the original cell.
about 90% protein and 10% lipid, although the proteins and fats seem to be
combined by Ca
and Mg
ions.
The proteins include hemoglobin and
several protein fractions, which depend both in number and activity on
the analytical method used to obtain them.
The main lipid constituents
are cholesterol, phospholipid, cerebroside and neutral fat.
While there is some agreement as to the chemical constituents, oY
the membrane, the architectural arrangement of the constituents is not
agreed upon.
The classical picture of the cell membrane is of a double
molecular layer of lipid coated on both the inner and outer surfaces with
protein; the proteins are associated through Ca
groups of the lipids.
ions with the polar
Some workers believe that an incomplete albumin
layer forms the outer surface.
Others, working with the electron
microscope, concluded that the red cell outer surface is covered with
plaques, 50
R thick,
protein in nature, held together by lipids.
What-
ever the spatial arrangement within the membrane, the membrane seems
to have some elastic force maintaining the red cell shape since changes
21
in the cell shape, brought about by changes in the suspending medium
(such as changes in pH, osmotic pressure, temperature, and pressure)
are reversible.
The red cell membrane permits the diffusion of low molecular
weight substances into and out of the cell.
However, this process is not
one of simple diffusion since the red cell, at equilibrium with blood
plasma at 37*C (human body temperature), has a sodium ion concentration
about one tenth that of the plasma and a potassium ion concentration about
30 times that of plasma.
Red cells which have been cold stored have a
distribution of ions more like the plasma, but upon raising the blood
temperature, the usual ion distribution is restored.
Hence, some meta-
bolic process seems responsible for the active rejection of one ion from
the cell and the active inclusion of another ion.
The rate of permeation
through the membrane is very high for water: red cells burst when placed
in distilled water for 2.4 seconds (volume on bursting being about 160% of
the original volume - a volume closely corresponding to that of a sphere
with a surface area equal to that of the red cell).
The passage of chloride
and bicarbonate ions through the red cell membrane occurs by simple
diffusion and is very rapid: the time required for the ion concentration
change to reach 50% of the total change that is obtained when the red cell
environment is changed is about 0. 1 second at 38"C.
The important fuel,
glucose, also enters the cell rapidly, but its mechanism of diffusion
appears to be one which utilizes specific sites on the membrane surface.
Other substances, present in blood plasma, also have rapid transport
rates through the membrane.
The membrane of the red cell appears to flicker.
While this motion
might be due to the Brownian motion of molecules in the cell environment,
22
some experiments have indicated that the flicker is related to the metabolic activity of the cell.
The cause of flicker has not been demonstrated
to be solely due to either Brownian motion or a metabolic process, and
both sources probably contribute an appreciable fraction of the motion.
The contents of the red cell interior are dissolved in a water solution and include substances which cannot diffuse through the membrane,
These non-diffuseable substances are
as well as diffuseable materials.
responsible for many of the more important functions of blood: haemoglobin, which permits the blood to transport large quantities of oxygen
and which accounts for about 70% of the blood buffering action, and carbonic anhydrase, which acts as a catalyst for converting carbon dioxide
into carbonic acid.
Haemoglobin is a spherodial molecule composed of
four haem units combined with the protein globin, which has a molecular
weight of 66,700 and forms crystals which consist of alternate layers of
haemoglobin and bound water.
The concentration of haemoglobin in the
red cell is high, about 34 gm/100 ml of cells, a concentration sufficiently
high to make the physical state of the haemoglobin somewhere between
that of a liquid and a crystal.
In fact, it has been proposed that the red
cell changes its shape fron a biconcave disk to a half melon shape during
circulation through the body of a person suffering with sickle cell anemia
because his type of haemoglobin forms long crystals (or gells) in a high
carbon dioxide environment.
cell.
These gells cause the distortion of the red
As implied above, haemoglobin exists in several types, some of
which are responsible for red cell diseases.
The remaining main con-
stituents of the red cell interior, and their concentrations are:
Reduced Glutathione
1. 1 gm/ 1H 2 0
Chloride Ion
73 m. eq. /1.
H20
Bicarbonate Ion
25 m. eq. /1.
H2 0
23
Phosphate Ion
0. 04 gm/1. H20
Ester Phosphate
0. 69 gm as P
Sodium Ion
15 m. eq./1. H2 0
Potassium Ion
150 m. eq. /1.
Water
69 wt.
/
1. H2 0
%
H2 0
These species represent only the more abundant constituents of the cell
interior, many other entities being present only in very small trace
amounts.
The red cell is physically relatively rugged while it is in its normal
natural environment.
However, when placed in other media, it easily
looses constituents both from its membrane and its interior..
Suchlosses
often dramatically alter the physical properties of the cell membrane, and
the cell shape.
If the changes in environment are large enough, the cell
will rupture, spilling its contents out and leaving behind its "ghost", the
stroma.
The change in environment needed to rupture the cell may be
relatively small.
This short discussion of the red cell mentions only those features
of the red cell which might have some influence on the rheological and
flow properties of blood.
It has been concerned with normal red cells,
and has not touched upon variations in cell shape, construction, or behavior which occur in some human diseases.
2.
The White Cells'
The leukocytes, or white cells, are of five types:
(1) three
types of polymorphonuclear cells known as the neutrophils, the eosinophils,
and the basophils, (2) the monocytes, and (3) the lymphocytes.
cells have discrete nucleoli, cytoplasm, and mitochondria.
24
All these
The white cell concentration in an adult usually is about 7000 per
cubic millimeter of blood, of which about 63% are neutrophils, 1. 6%
are eosinophils, 0. 4% are basophils, 5. 0% are monocytes, and the remaining 30% are lymphocytes.
In children, the normal white cell count
is higher and the cell type distribution is different.
The concentration
and distribution of white cells can vary widely and rapidly, even in
healthy persons.
Extremely hard exercise, even for a very short period
of time, can increase the neutrophil concentration 6 or 7 fold; taking a
very deep breath can cause the neutrophil concentration to increase by
Damage to tissue will cause a rise in the white cell concentration
50%.
in blood, and certain diseases can cause a great increase in one particular type of white cell, e. g., whooping cough, which may cause the
lymphocyte concentration to rise from the usual 2100 per cubic millimeter up to 100, 000 or more per cubic millimeter.
The various forms
of leukemia may be characterized by high increases in the while cells,
with or without a distribution change, or by the production of mutant
cells.
As generally described*, the shape of all of the white cells is that
of an easily deformed sphere: the polymorphonuclear cells being 10 to
12 microns in diameter, monocytes 12 to 15 microns in diameter, the
small lymphocytes 8 microns and the large lymphocytes 13 microns in
diameter.
An idea of the flexibility of these cells is obtained when it is
realized that not only do these cells squeeze through capillaries, which
have a smaller diameter than the white cells, but they also squeeze
through the pores of the blood vessels.
See, for example, reference (34).
25
While agreeing that the white
cells are extemely flexible, Tullis (62) says that it is probable that a white
cell is never globular, and detection of a rounded edge can be accepted as
sufficient evidence that the cell is dead.
He pictures the white cell as an
irregularly shaped, gelatinous body.
The length of the life spans of the white cells are not known.
This
lack of knowledge arises because the white cells are not restricted to
the circulatory system, as the red cells are, but rather, use the circulation system as -a mode of transportation from the bone marrow and
lymphogenous organs (where they are produced) to the areas of the body
where they are needed to overcome infectious agents.
From studies on
people who have been subjected to gamma rays, which cause destruction
of white cell producing material, it. is estimated that the polymorphonuclear white cells have a life span of perhaps 8 to 12 days (of which time
only a small fraction may be spent in the blood).
The life span of mono-
cytes is completely unknown, because of their greater mobility through
the parts of the body.
Since the production of lymphocytes in a day is
several times greater than the number of lymphocytes in the blood
stream at any time, it has been estimated that the lymphocyte life span
is well under 24 hours, some estimates being as low as 4 hours.
The white cell membrane is even more elastic than that of the red
cell; the white cell volume may increase to 1000
shape changes occurring.
3 without irreversible
Thus, when the white cell ingests foreign
particles (phagocytosis), it can increase its volume considerably, and
become more sphere-like as it ingests more and more particles.
This
may explain why one sees, in motion pictures of the microcirculation
(29),spherical white cells rolling along the blood vessel walls: these
cells are extended because of ingested material.
26
The "stickiness" of white cells is often referred to, but Tullis (62)
says that the white cells, when in the body (invivo)do not adhere together, even when highly concentrated.
When out of the body (in vitro),
clumping is usual; this clumping generally is due to cell damage done
during cell removal from the body, and in subsequent handling.
Unlike
red cells which have a hydrophobic surface, the white cell is hydrophilic
and will stick to wettable surfaces.
Consequently, when attempting to
collect white cells, one must be sure that only unwettable surfaces are
used.
At least two types of in vitro white cell clumping are known.
One
form, called agglutination, is irreversible and shown by dead or dying
cells.
It occurs from exposure to wettable surfaces, mismatched blood,
and other causes of white cell death.
The other form of clumping is
reversible, but will lead to white cell death if continued for an hour or
more.
It is brought about by those agents which also cause the for-
mation of red cell rouleaux.
The anion permeability of the white cell membrane is similar to
that of the red cell membrane.
Potassium and sodium ions also pene-
trate the white cell membrane.
Determinations indicate that the water
permeability is high, but less than that for the red cell.
The interior of a white cell is more complex than that of a red cell.
It contains cytoplasm, often containing globulin-holding bodies, a nucleous, nucleoli, granules, and ingested particles.
According to Endes
and Herget (21) the inorganic chemical composition of a leucocyte
interior is:
113 millimoles /liter
Potassium
22
Calcium
2
27
"
Sodium
70 millimoles /liter
Chloride
Inorganic Phosphorus
Bicarbonate
10
1.8
The white cells contain a high concentration of various enzymes.
As might be expected, the white cell is very active metabolically.
Buckley (62) has found that the oxygen consumption of a single white cell
is 10 to 12 x 10
3.
-9
3
mm /minute.
The utilization of glucose is also high.
The Platelets
The'third type of blood cell, the platelet, is an incomplete cell,
lacking both a nucleus and fine structure.
It has long been held that the
platelets are fragments of megakaryocytes, cells which are developed
from the same primitive cell (myeloblast) as the white cells.
The
platelets are small, one to three microns in size, and normally ate
present in the blood in a concentration of about 400, 000 per cubic millimeter.
Their life span is about four days.
The classical picture of a platelet as a small disk is not correct.
With phase contrast microscopy, it has been established that the platelets
have many small fibrils projecting from their surface.
In young platelets,
the formation and destruction of these fibrils is reversible and dependent
on the dissolved CO2 and 02 content of the platelet environment; high
CO2 - low 02 contents result in fibril formation, while
low CO2 - high 02
environments result in fibril destruction.
These observations, reported
by Tullis (63), were made in vitro, and have not been made in vivo,
although there is no reason to suppose that platelet fibrils do not exist
in vivo,
The visualization of these fibrils has permitted a reasonable
explanation for several features of fibrinogen clot retraction, e, g, , the
effect of gas phase composition on clot retraction, when the clot forms
in blood which is exposed to oxygen and carbon dioxide mixtures.
28
The finding that the platelet fibrils decrease in number and size as
the platelet ages is also consistent with the evidence which suggests that
the "stickiness" of platelets is a sign of platelet youth; the older a
platelet, the less the tendency for the platelet to agglutinate with other
platelets.
Tests on stored platelets also show that the destruction and
reformation of fibrils, caused by changes in the dissolved gas content
of the environment, becomes less reversible as time passes.
However,
age does not seem to affect the ability of the platelet to play its role in
the mechanisms which lead to clot formation: only its role in clot retraction seems to be a function of age.
Like the other particles of the blood, platelets have a negative net
charge.
4.
Blood Plasma*
By centrifugation, the cellular particles of blood can be sepa-
rated from the suspending medium, which is known as blood plasma.
Blood plasma is usually a clear, slightly straw -colored fluid whose
specific gravity is generally about 1. 03.
Its pH is 7.46.
Plasma is an aqueous solution of a seemingly infinite variety of
organic and inorganic substances.
Table (2-1) is a list of the more
abundant substances, and representative normal concentrations of these
substances.
The proteins are large molecules composed of various amino acids.
Some, such as albumin and fibrinogen, have been fractionated from
*Plasma composition data obtained from reference (16).
is the source of the data on the proteins.
29
Reference (56)
TABLE 2-1
Main constituents of blood plasma and representative normal
concentrations
6. 8 weight
%
Proteins
Albumin
a -globulins
P -globulins
y -globulins
Fibrinogen
3. 5 w eight o
0. 83
0.89
0.70
0.49
Other Organic Substances (in mg/100 ml plasma)
-Sugar
Urea
Cholesterol
123
22
107 - 320 (194 average)
Inorganic Ions (mg/100 ml plasma)
Na 4
311 - 334
13. 7 - 19.5
C++
9.2 - 11.2
Mg ++
C1
1.22 - 2.43
so4O4
HC 3
352 - 373
22. 1
13.3
170
30
Others, such as the globulins, are
plasma in a relatively pure form.
really protein fractions which are separated from plasma by a separation procedure because the proteins making up the fraction have some
common property, such as electrical charge at a given pH, density, or
solubility.
And some others are detected by a biological property, but
have not been isolated because of their low concentration in blood plasma
and their similarity to other proteins.
The molecular weights of the species making up the plasma protein
fractions ranges from 44,000 to about one million, but the colloid molecular weight of whole serum (plasma with the fibrinogen removed)
about 90,000.
is
This is because the most abundant protein, albumin, has
The p-globulin molecular weight is
a molecular weight of about 69,000.
93,000, y- globulin 160,000, and fibrinogen 340,000.
Determination of protein size and shape is complicated by the fact
that the proteins are hydrated when in aqueous solution.
With the ex-
ception of fibrinogen, the more abundant proteins are generally considered to be roughly spherical in shape.
Albumin has been estimated
to be a prolate ellipsoid of axial ratio of about 4.2 to 3.3, an oblate
ellipsoid of 5.4 to 4.0 axial ratio (depending on the assumed degree of
hydration), and also as a prism 145 R long, 50
R wide,
and 22 A thick.
The y -globulin molecule has been estimated to be a prolate ellipsoid
235
R long and
44
A
in diameter.
The shape and size of fibrinogen, in
the dry state, as determined by Hall and Slayter (36), by using the
electron microscope, is unusual for the proteins and is shown in Figure
At physiological pH, all of the proteins have a negative net charge.
The order of electrophoretic mobility is albumin > a
31
-
globulins
4-75 *k z5 A
Figure (2-2)
> a 2 - globulins >
The fibrinogen molecule in the dry state (36)
p -globulins > fibrinogen > y - globulin.
In addition,
the proteins act as surfactants and will orient themselves at gas -liquid
interfaces to form surfactant films, which may lead to experimental
artifacts in viscometric experiments (40).
Upon reaching an interface,
the proteins irreversibly change their spatial configuration (denature),
thereby changing their physical and chemical properties because of the,
exposing of new constituent parts to the environment, as well as changing their size and shape.
5.
Coagulation and Aggregation
Two properties of whole blood are especially important in any
rheological study of blood: (1) the coagulation process, and (2) the
aggregation of red cells in stationary or very slowly moving blood. Each
of these will be discussed separately.
(a)
Blood hemostasis is the process by which the body seals off
severed blood vessels so that a person does not bleed to death. It is a very
complicated mechanism, parts of which are still not fully understood,
which consists of the following steps:
32
(1)
damage of the blood vessel wall,
(2) contraction of the blood vessel,
(3) adhesion of platelets to the damaged vessel wall
of a temporary platelet clot,
(4) and formation of a fibrin clot from fibrinogen,
(5)
shrinkage of the fibrin clot to form a dense
structure.
(6) relaxation of the blood vessel.
The third and fourth steps are important in any blood study since they
will occur, with any pretense as an excuse, unless special precautions
are taken.
The third step, platelet adhesion, will occur on any wettable surface.
The platelets, upon sticking to the foreign surface, burst and spill their
contents out into the surrounding blood.
These spilled contents cause
the neighboring platelets to stick together and on the foreign surface.
Also, the spilled platelet contents help set off the fourth step of hemostasis: the coagulation step.
Clearly, in any study of blood, it is
important to prevent platelet adhesion.
This can be done by siliconizing
all surfaces which may come into contact with the blood.
Step four of hemostasis is known as coagulation and involves the
conversion of the protein prothrombin into the enzyme thrombin, so
that the thrombin can cause the polymerization of fibrinogen into fibrin
fibers (43).
Many reactions occur simultaneously in this process, some
aiding the coagulation process and others combatting it so that only that
blood in the vicinity of the wound will clot.
Obviously, one does not
desire blood coagulation to occur during a rheological study of the fluid.
Coagulation will occur, at 37"C, within several minutes after the blood
33
is removed from the body even if all surfaces which come in contact
with the blood are siliconized.
If the blood sample is chilled, coagulation
can be prevented for perhaps half an hour.
One way to prevent coagu-
lation for longer times is to add to the blood sample a small amount of
the natural anticoagulant heparin, an electrolytic polysaccharide.
Heparin works as an anticoagulant by blocking the formation of thrombin.
Several other substances can be added to the blood to prevent coagulation:
oxalate ions and citrate ions for example.
These substances are effective
because they form very weak complexes with calcium ions, an essential
ingredient in the coagulation mechanism, thereby blocking the coagulation
process by preventing one of its essential steps.
Complexing calcium ion.
also prevents the formation of platelet clots on foreign surfaces, since
calcium ion also seems to be important in that process.
Of course, the
effect of anticoagulants on the rheological properties of blood must be
determined.
(b) The ability of the red cells to aggregate in stationary,
normal human blood was first extensively studied by Fahraeus (23, 24);
It was found, by observation of whole blood drops, that the red cells
aggregate with their flat sides together to form the analogy of a stack of
identical coins.
These aggregates, called rouleaux, normally fall apart
easily when fluid motion is induced.
In illness, these rouleaux often are
much longer than normal and the rouleaux may themselves aggregate together in clumps which do not fall apart easily in plasma currents.
Fahraeus found that albumin acted to prevent aggregation and fibrinogen
acted to enhance aggregation, with the globulins being weak aggregation
aids.
The variation in plasma protein concentration which occurs in
certain diseases and other clinical situations will therefore change the
degree of aggregation of the red cells in stationary or slowly flowing
34
blood, and this change in aggregation forms the basis of the explanations
for several clinical tests, such as the red cell sedimentation rate (24,32),
and the guttadiaphot test (26).
Normally, in stationary healthy blood, the
rouleaux may contain about eight to thirty red cells, with the average
being about fifteen red cells.
The rouleaux are flexible and will bend
when flowing around obstructions.
Unlike the case with coagulation, it is not desired that red cell
aggregation be prevented when making rheological tests on blood, since
the reversible formation and destruction of rouleaux does occur in
normal healthy blood (26).
Indeed, this ability of the red cells to aggre-
gate can have a dominant role in establishing the rheological properties
of whole blood at low shear rates.
C.
Proposed Model
The model used to interpret the rheological behavior of human blood,
fron a healthy donor, in the low shear rate region consists of mutually
attractive, flexible, disc-like particles (red cells) suspended in a
Newtonian fluid of a slightly lower density than the particles (plasma).
Because of the interparticle attractive force hypothesized to exist, the
aggregates are peculiar in that they are formed only by the face - to
face joining together of the disc-like particles (Figure 2-3).
axial
ratio
Figure (2-3)
Model red cell aggregate
35
J
1_
8g
-
particles will reversibly aggregate at very low shear rates, but the
The length of these aggregates, called rouleaux, varies in an inverse
fashion with the shear rate.
The maximum length at a particular shear
rate is determined by the stress exerted on the aggregate by the suspending media; when the stress at the center of the aggregate (the point of
maximum stress) just exceeds the cohesive force holding two particles
together, the aggregate will fall apart into two fragments of equal length.
Because of the high collision rate of particles, even in very dilute
suspensions, aggregates of less than half the maximum length will be
rare, and almost all of the aggregates will have lengths between J
and 1/2 Jma, where J
length at a given shear rate.
is the maximum permissible aggregate
The value of Jax
will depend on the
orientation of the aggregate in the shear field.
The rouleaux are not rigid, but slightly flexible as evidenced by
their ability to bend when flowing past obstacles, as observed under the
microscope by causing blood to flow between two microscope slides between which were trapped small stationary air bubbles.
The rouleaux, and indeed the individual particles themselves, are
large enough so that the effects of Brownian motion are negligible* Thus,
the rouleaux, being anisometric, will tend to rotate in a laminar, uniform
shear rate field with a variable velocity and a period which increases as
the shear rate decreases and the rouleaux axial ratio increases, and
which depends on the orientation of the rouleaux, (9) , (39).
In dilute
suspensions the tendency is for the anisometric particles to align parallel
to or perpendicular to the direction of flow.
In the concentration region
*Brownian motion may be the cause of the flickering of the red cell
membrane. This membrane flicker may aid red cells in attaining their
face-to-face orientation in the rouleaux.
36
of interest here, not only are there hydromechanical forces to consider,
but also mechanical interactions, which tend also to align the anisometric
particles along shear planes.
And as the concentration increases, the
mechanical interactions dominate.
It has also been observed with a
microscope, that when blood flows between microscope slides, the
rouleaux do align themselves with their length parallel to the direction
of flow.
Because of this tendency for the rouleaux to align, any 3-dimensional
network formed when blood is caused to become stationary after previously having been under a low shear rate field will differ from a network formed from blood which previously had been in a sufficiently high
shear rate field so that no aggregates, or only very small aggregates,
could exist.
In the first case the network will be analogous to a bunch of
fibers which have been combed more or less parallel to each other, while
the second case will be analogous to a wad of fibers which are randomly
interwoven.
In addition, the length of the rouleaux formed while the blood
was flowing (and hence the shear rate) will affect the nature of the network,
since the rate of aggregation of cells in stationary blood has been microscopically observed to be very slow.
The mechanical properties of such
networks would therefore depend on the prior history of the material.
Consequently, blood, by this model, would be expected to have a yield
stress which would vary with the blood's prior history, predominantly
shear rate, and the direction of stress.
D.
Results of Previous Investigators
Until recently, most of the rheological investigations related to
blood were made in capillary viscometers, although Brundage (8) used
a concentric cylinder viscometer, and Copley, Krchma, and Whitney(14)
37
used a falling ball viscometer.
by Bayliss (1) (2).
This work has been reviewed in detail
The main conclusions to be drawn from these works
are that blood is not a Newtonian fluid, but rather its viscosity decreases
as the shear rate increases.
In addition, Fahraeus and Lindqvist (25)
first pointed out that below a capillary diameter of about 0.3 mm the
rheological properties were a function of the tube radius.
Attempts to
explain this abnormality generally use the idea of a plasma layer at the
tube wall, created by the "axial drift" of red cells away from the tube
wall.(25) (58) (38) (15), or the idea that the flow of blood through a tube
is analogous to alternating tubes of sheared and unsheared fluid (the
"sigma" effect) (38) (20).
The results of investigations aimed at determining the relationship
between the hematocrit and the apparent viscosity of blood have led to
conflicting correlations.
Nygaard, Wilder and Berkson, (52)
,
apparently
using red cells in serum, found a linear relationship between viscosity
and hematocrit in the hematocrit range of 15 to 50%. Brundage (8)
found
that the viscosity increased at a faster rate than the hematocrit and at a
constant hematocrit was proportional to the plasma viscosity.
Bingham
arid Roepke (4) found that the reciprocal of the viscosity was a linear
function of the hematocrit.
Haynes (38)
found that for hematocrits below
10%, the viscosity was a linear function of hematocrit, but above 10% it
was an exponential function of the hematocrit.
Obviously, there is not
general agreement on the effect of hematocrit on blood viscosity.
It is generally agreed that the viscosity of blood decreases at a
slightly faster rate than that of water as the temperature increases (2).
Several erroneous ideas have gotten into the literature.
Two of
these ideas are (1) that removal of the fibrinogen from blood does not
38
affect the rheological properties of blood (2),
be effectively replaced with saline (37).
and (2) that the plasma can
The first view seems to be based
on the work of Bingham and Roepke (3), who showed that fibrinogen did
not contribute very much to the viscosity of plasma.
The fallacy of this
view was demonstrated byWells, Merrill and Gabelnick (69), who showed
that the viscosity of blood at a given shear rate was much higher than that
for red cells in saline. Reference to the difference in behavior between
red cells -suspended in plasma and red cells suspended in serum has not
yet appeared in print.
Another serious error which has been reported in the literature is
the apparent finding that blood plasma is non-Newtonian (11, 68).
How-
ever, as has been shown by Joly (40), and Merrill, et al (47), this error
arises from an experimental artifact caused by the formation of a layer
of denatured plasma proteins at the liquid-air interface formed in the
viscometers.
This layer has some mechanical strength and if the effect
of this layer on the viscometric measurements is not eliminated, erroneous conclusions can be drawn from experimental data.
The experimental work discussed above was all done at shear rates
above about 60 - 100 sec
,
at shear rates below 10 sec
and until very recently, no work was reported
-1
.
In addition, it was mainly done in capillary
viscometers; data from such instruments are difficult to interpret, not
only because of wall effects or sigma effects, but also because the fluid
in the tube is not being subjected to a uniform shear rate.
The rotational
viscometers (cone - in - cone, plate - and - cone, and concentric cylinder)
offer several advantages:
the fluid being tested in them is subjected to an
essentially uniform shear rate, and, in some forms, the viscometers are
capable of measuring very small shear stresses at very low shear rates.
39
These instruments are now being used by a few investigators (69,
1%, 19, 12).
Dintenfass, (18.
9) has used a cone - in
-
cone viscometer
to make measurements down to a shear rate of 0.006 sec,.
conclusions are that:
17,
His general
(1) at shear rates above about 8 sec- 1 the viscosity
of blood is constant, (2) at about 8 sec-
the rheological nature of blood
changes drastically, and (3) at shear rates below about 8 sec
the
logarithm of the viscosity is a linear function of the logarithm of the
shear rate.
While the data of Dintenfass are in the same range as the
data reported in this thesis, none of Dintenfass' three general blood
characteristics were found (Figure 2-4).
Dintenfass does not report
any surface or time effects, as is reported herein.
It is believed that
these effects, especially the time effects and possibly sedimentation
effects, not considered by Dintenfass, may account for the divergence
of his data and the data reported here.
He does not report an experi-
mentally determined yield stress for blood.
The work of Dintenfass
supplies the only known data, besides that reported here, on the rheological properties of blood at shear rates below 1 sec
40
I
I
FIGURE
2-4
Comparison of data of Dintenfass
with data of tis tnee S
100
------
Dinten rass
Temp,
o
= 3600
This t hesis,
a = 44 .6%
Temp.
S44.6%
10
pN
0
Temp
*
log
secm
1
0.1
10.000
100
10
log r. , op.
41
37,,0O0
III.
A.
PROCEDURE
The GDM Viscometer
The, viscometer employed to determine the rheological properties
of blood was developed by adapting a very sensitive torque measuring
device so that it could be used to measure the torque transmitted by a
viscous fluid from the rotating "bob" to the stationary "cup" of a conventional concentric cylinder Couette viscometer.
The necessary torque
measuring device had been developed earlier at the Instrumentation
Laboratory for the testing of parts used in the guidance system of the
Polaris missile.
The adaptation was performed by P. J. Gilinson, Jr.,
C. R. Dauwalter, who were responsible for the development of the
original torque measuring device, and E. W. Merrill; hence, the "GDM"
designation of the machine which resulted from their efforts.
A photograph of the actual viscometer is presented as Figure (3-1),
together with a labelled silhouette of the equipment.
A schematic diagram,
illustrating the method of measuring torques, is presented as Figure (3-2).
As shown in this figure,, the outer cylinder of the viscometer, the "cup"
(C) is mounted on the top plate (D) of a frictionless air bearing (G).
The
plates (D) and (F) provide vertical positioning and the cylindrical surface
around shaft (E) provides horizontal positioning of the bearing.
The
shaft (Q), an extension of shaft (E), has attached to itself two metal
rotors, (R) and (S), which can rotate in the gaps of two electromagnets,
(L) and (K), which are called the microsyn voltage signal generator and
the microsyn torque generator respectively.
The voltage signal generated
in (L), which is proportional to the angular displacement of the rotor (R)
from its null position, is fed into a voltage amplifier (J), which, in turn,
feeds into a current amplifier (M).
42
A feedback current from (M) is
Elba&,
C=X=
1I
4k
I
*~;
~
of
-Vl
time
Srecorder'
77-
II
si1-
INSTRUMENTATION LAhUA
CRY
M.I.T.
Datet
JAN 8 1963
print No.
3) 64
FIGURE
3-2
Schematic diagram of'the GDM viscometer
A
D
air
G
F
Eag
Q
IR
K
43
supplied to the microsyn torque generator (K), causing a torque to be
applied on the shaft (Q).
This torque is applied in the direction opposite
to that of the torque causing the shaft (Q) to originally rotate away from
its rest position.
Thus, when a torque is applied to the viscometer
"cup" (C) by the rotation of the viscometer inner cylinder, "bob" (B),
in a viscous fluid contained between the cylinders, shaft (Q) rotates until
the torques exerted by the microsyns on the shaft (Q) exactly balance the
viscometer torque.
The magnitude of this torque is read on a calibrated
meter (N), which measures the feedback current sent to the microsyn
torque generator (K).
Details of the concentric cylinder viscometer itself are shown
schematically in Figure (3-3).
made of coin silver and hollow.
which passes a tube (4).
The inner cylinder, the "bob" (1), is
It is attached to a hollow shaft (3) down
This arrangement permits water from- a constant
temperature bath to be circulated through the bob.
The outer cylinder of
the viscometer is a cup (10) constructed of lucite.
A cylindrical guard
ring (2) penetrates through the liquid
gas interface formed when the
viscometer is filled with liquid; this prevents the transmission of a
torque from the rotating bob to the stationary cup by any surfactant layer
which might form at the liquid-gas interface.
The bob is positioned
concentrically within the cup by adaptor (8) and a shield (9), which
rests on the air bearing frame (14).
The cup is positioned by studs
which are part of the floating table (15) of the air bearing.
The bob is
rotated by means of a gear train which is activated by a constant speed
synchronous motor; by the proper selection of gears, various bob
rotational speeds can be obtained.
In its present form, the bob can be rotated at speeds from 0.01 rpm
up to 100 rpm.
Torques ranging from 0.0100 to 1999 dyne-cm can be
44
FIGURE
3-3
Detailed schematic diagram of the GDM viscometer
1
Silver rotor, the "bob"
2 Guard ring
4
Hollow drive shaft and tube
Water lead-in and -out tubes
3
5
Assembly housing
6
Sealed bearings (4)
7
Plate attachment to housing
5 to bring to correct vertical
elevation
8
Adapter
9
Lucite shield
16
13
-011
-.'Ile
1 IK
_
_Ij
10
Lucite cup
11
Upper sealing plug for
shaft 3
12
Cross bar in 11 engaging
with slot 13
Slotted motor shaft from
Brookfield Variable Speed
Drive
it/
12
KJ
13
5
14
Circular rim of air bearing
15
16
Air bearing table
Brookfield motor
a
2
1'
elf
-w
10
15
45
measured with an uncertainty of
0.0001 dyne-cm.
The cup movement,
away from its null position, necessary to permit measurement of a
torque, is generally about 0.002* of arc, or less.
The constant temper-
ature water, whose temperature varies no more-than
0.01*C, is sucked
through the bob at a rate of about 2 liters per minute.
Because of the
high thermal conductivity of the bob, the high heat capacity of the bob, and
the poor thermal conductivity of the cup, it has been calculated that the
fluid in the viscometer gap is maintained within 0.05*C of the temperature
of the water sucked through the bob.
Three sets of cylindrical surfaces were utilized in this work: all
bobs were constructed of coin silver and all cups were made of lucite
(polymethyl methacrylate).
Two of these sets were used primarily for
-
testing human blood, and required about 8.5m1 of fluid to fill the vis
cometer. 'The third set was used to test smaller samples of fluids; it
required only about 1 ml. of fluid. The dimensions of the cylinders are:
o. d. of inner cylinder
i. d. of outer cylinder
length of inner cylinder
A
B
C
2.224 cm
2. 574 cm
2. 896 cm
2. 224 cm
2. 438 cm
2. 917 cm
1. 397 cm
1. 537 cm
1. 669 cm
The surfaces of set A were smooth, while those of sets B and C were
vertically grooved (the dimensions given above for sets B and C are those
of the smooth surfaced cylinders from which the grooved surfaces were
made).
Figure (3-4) is a schematic diagram of the grooved surfaces:
each cylinder was vertically grooved with 720 cuts, 66 microns deep,
by use of a 60* broaching tool.
Unless otherwise indicated, all surfaces
were siliconized, using the Clay Adams product, "Siliclad".
46
The outside
&
,
/. 0 -;
K:
1~-~
Viscometer "bob"
surface
"
cup
"
Viscometer
I
surface
FIGURE 3-4
Schematic diagram of the grooved viscometer surfaces
47
of the viscometer cup is rubbed with a small amount of "Statnul"*, a
conducting fluid which causes any static charge, which may have been
built up by the handling of the cup during assembly of the viscometer,
to be rapidly dissipated.
A more detailed description of the air-bearing, and the torque
measuring system, may be found in references (30) and (31).
B.
The Merrill
-
Brookfield Viscometer
The Merrill
-
Brookfield viscometer is a concentric cylinder vis-
cometer which uses standard Brookfield drive heads, manufactured by
the Brookfield Engineering Corporation, to which is attached a special
rotor.
The rotor is suspended and rotated in a special stationary well
(the rotor - well unit was designed by Professor E. W. Merrill of
M. I. T.).
The viscometer is shown schematically in Figure (3-5).
In
use the viscometer was mounted vertically, by clamping the well, in an
air conditioned room whose temperature was 22"C.
The unique feature
-
of this viscometer is its large surface area, which enables the vis
cometer to be used to measure small shear stresses accurately.
Two Brookfield drives were utilized:
(1) a drive which could be run
at 8 rotational speeds (0. 1, 0. 2, 0. 5, 1, 2, 10, and 20 rpm) and (2)
another which had six drive speeds (0. 3, 0. 6, 1. 5, 3, 6, and 12 rpm).
The two Brookfield drive units had torsion wires for different shear
stress regions.
The torsion wire deflection scales of the Brookfield
units were marked in arbitrary divisions.
*"Statnul" is a product sold commercially by Daystrom, Inc., Newark 12,
New Jersey.
48
FIGURE 3-5
The Merrill - Brookfield viscometer
Standard Brookfield
iscometer Drive
Head
A0
a
In
Z.pacer
Suspended Rotor.
Wetted Rotor Area
= 57.6 in 2
Well
m1-0
I
o"
+-
2.6005''
,
i
.
49
Fluid
From the dimensions of the rotor and well, the relationship between
the rotational speed of the rotor and the shear rate in the viscometer gap
can be determined:
?= 2. 10 (rpm)
where
y
-1
is the shear rate in sec 1 .
This relationship is derived in the
same manner as that for the GDM viscometer in section IV-A-1-a.
To
obtain the relationship between the shear stress and the scale deflection
of the torsion wire of the Brookfield units, the scales were calibrated by
putting Newtonian fluids of known viscosity (water, molasses - water
mixtures) in the viscometer and determining the scale deflection - shear
stress relationship.
These relationships were
8 = 0. 0616 (Scale Reading)
for the 8 speed unit, and
T6 = 0. 0048 (Scale Reading)
for the 6 speed unit (T is the shear stress in dynes/cm 2
The viscometer was filled to the level indicated in Figure (3-5), the
excess fluid overflowing over the inner weir of the well when the rotor
was lowered into place.
Measurements were made starting at the highest
rotational speed and going to the lowest speed.
After making the measure-
ment at the lowest speed, the highest speed measurement was repeated to
see if any change had occured in the fluid during the measurements.
The yield stress of a fluid was determined in this viscometer by
rotating the rotor at the lowest speed until a steady shear stress reading
was obtained and then the rotation was stopped.
If the fluid had a yield
stress, the shear stress reading did not return to zero, but to a finite
value; this steady value at zero shear rate was taken as the yield stress.
50
This viscometer was used only to verify the yield stresses of
suspensions determined in the GDM viscometer.
Because of the large
volume required - 60 ml. - it is not suitable for the testing of blood.
C.
Preparation of Blood Samples
1.
Obtaining Blood Samples
Blood samples were collected by two procedures, depending
on the size of the sample.
Large samples (one or one-half pint) of blood were obtained from
donors in good health at the blood bank of the Massachusetts General
Hospital, Boston.
The blood was drawn by venepuncture and collected
in plastic bags (Fenwal
cedure.
Code No. JA-2C) by routine blood bank pro-
The plastic bags come prefilled with a standard ACD solution,
which for a bag for 500 ml of blood is of the following composition:
Citric Acid:
0. 6
Dextrose:
1. 83 gm
Sodium Citrate:
1. 65 gm
gm
Total ACD solution volume
75 ml
Blood which was not immediately used was refrigerated at 4 *C..
Small blood samples (about 10 - 15 ml) were collected by venepuncture from the donor with siliconized needle (size 20) and siliconized
syringe.
Samples which were not anticoagulated were transferred
immediately to the siliconized viscometer or else chilled immediately
to about 40C.
Anticoagulated samples were prepared by transferring
10 ml of the blood in the syringe to a siliconized glass tube containing
*Fenwal blood collection units are manufactured by Fenwal Laboratories,
Morton Grove, Illinois.
51
200 units of heparin, 1.34 gm sodium oxalate, 3.8 gm trisodium citrate,
or 3.8 gm ACD, all of which were either in solution or solid form.
Samples not used immediately were refrigerated at 4"C.
2.
Preparation of Samples
To obtain blood samples of various red cell volume fractions
(hematocrit), an anticoagulated original blood sample (as obtained from
the donor) was centrifuged at 3000 g and 40 C for 10 minutes.
This sedi-
mented all cells, -leaving above the suspending medium, plasma.
The
white cells, occupying less than 1% of the blood volume, sedimented at
a slower rate than the red cells and were left on top of the red cell pack;
the white cell layer and the very top layer of red cells were removed
from the centrifuged blood by use of a syringe and large bore needle.
The remaining plasma and red cells were separated by decanting off
the plasma.
Various proportions of red cells and plasma were combined
to get samples of the desired hematocrits.
Hematocrits were determined by filling small glass capillaries
(diameter about 1 mm, length about 5 mm) with the sample, sealing the
bottom of the tube with clay, and centrifuging at 17,000 g for at least 15
minutes; the ratio of the red cell pack length to the total length of the
sample was taken as the hematocrit.
Generally two capillaries were
filled for each sample and the hematocrits determined with the two
capillaries usually agreed within 0.5 hematocrit units.
Serum was prepared from blood by either allowing native blood to
clot naturally for half an hour and removing the clot and cells by centrifuging, or by adding thrombin to ACD containing blood to initiate clotting.
After removal of cells and clot, the serum was kept at 37*C for at least
12 hours to allow the remaining thrombin in the serum to be destroyed.
52
Suspensions of red cells in media other than plasma were prepared
by centrifuging an ACD blood sample, removing plasma and white cells
with a syringe and large bore needle, washing the remaining red cells
several times in the final suspension medium, and finally suspending the
washed cells in a fresh portion of the final suspending medium.
The
washing procedure involved adding to the red cell pack an equal volume
of the final suspending medium, suspending the cells with gentle agitation
by inverting the sample several times, and again centrifuging for 10
minutes at 3000 g (4C).
The supernatant solution was removed by the
use of a syringe and needle, and the process repeated the desired number
of times.
Details as to the composition of the various suspending media
are given where suspensions of red cells in such media are discussed.
53
IV.
A.
DISCUSSION OF RESULTS
Whole Blood
When determining the rheological properties of a suspension near
limiting quantities, such as very low shear rates and very low shear
stresses, it is essential that the physical arrangement for the measurement of these properties be closely looked at.
Under limiting conditions,
small causes may lead to large errors in interpretation of experimental'
data, whereas the same small causes may have a negligible effect at
higher conditions.
Several such factors, discussed in detail, and found
significant, in the following sections, must be considered when interpretating low shear rate rheological measurements of blood.
1.
Derivation of viscometer equations
(a) Assuming constant fluid viscosity in the viscometer gap.
The viscometer under consideration consists of an inner
cylinder of radius r1 rotating at a constant angular speed 62, and of an
outer cylinder whose inside radius is r 2 and upon which a restraining
torque G (per unit height of the inner cylinder) is exerted to prevent
movement of the outer cylinder.
The space between the cylinders is oc-
cupied by the fluid whose rheological properties are being investigated.
At a radius r (r1 5 r < r2 ), the shear stress and the restraining
torque are related (at steady state):
=0
T (27rr)r-G = I
dO
T
G
27rr 2
(4-1)
This equation is valid for any type of fluid provided that the largest unit
of matter in the fluid is small in comparison to the viscometer gap size,
and that the flow is laminar.
54
The rate of shear in the fluid occupying the viscometer gap is
= r d(u/r)
r4-w
(4-2)
dr
dr
Considering a fluid whose properties are independent of time and viscometer gap size (equivalent to the condition stated above that the largest
material unit in the fluid is small compared to the viscometer gap size),
we can write
(4-3)
f r)
From Equations (4-1), (4-2), and (4-3), and using the boundary conditions
appropriate for the GDM viscometer, i. e.,
at r
r
r
u
r
r2
u
0
o
one can obtain
r1
2
dr
f(T
(4-4)
r2T
2
For a Newtonian fluid
(4-5)
T
where 77 is a constant, the fluid viscosity, and therefore
f(T)= 1
for a Newtonian fluid.
(4-6)
)
Substitution of (4-6) in (4-4) and integration of the
result gives
which upon use of (4-1) becomes
~1
[1
1 0 Ir4
27j21r
55
J
1]
r~
or
G
4 ro
r7_'
2
2
r2-r1(47
_r7 2
ss1
(2
so that, using (4-1), (4-5), and (4-7)
r
2 1
r ~ ILr
r2
2
2
2 1
2 -r1
(4-8)
-
-2w
From Equations (4-1) and (4-8), the shear stress and shear rate at any
point in the viscometer gap canbe calculated from experimentally determined quantities, provided, of course, that the fluid in the vicometer is
Newtonian, or that the fluid viscosity does not vary significantly across
the gap, the flow is laminar, and the fluid can be considered a continuum.
For a point at the outer cylinder surfaces (r = r 2 ), the equations for the
shear stress and shear rate are as follows:
viscometer set A
T2(dyne/cm2
~Ui~t.
2
2 (sec- )
cm)
_ T(ne
28. 77
0. 7463 N (rpm)
viscometer set B
=
T
2
27. 26
2
1. 026 N
(4-9)
viscometer set C
T
6.19
2
=
1. 05N
The validity of these equations was checked by putting three standard
oils (K, D, and H oils obtained from the National Bureau of Standards)
56
in each of the viscometers: the difference between the experimentally
determined fluid viscosity and the viscosity reported by the National
This close agreement indi-
Bureau of Standards was less than 0. 5%.
cates the absence of any "end effects".
(b)
The Krieger-Elrod equation.
The equations for calculating the shear stress and shear
rate from experimentally determined quantities, which were presented
in the previous section, ar~very convenient to use.
However, they are
based on the assumption that the fluid viscosity does not vary across the
viscometer gap.
Since blood is known to be non-Newtonian, and since
the shear stress varies across the viscometer gap by about 28% in the
larger vicometers, the use of equations (4-9) for blood is open to some
question.
For the case of laminar flow of a continuum in a concentric cylinder
viscometer, whose inner cylinder was stationary and outer cylinder rotated, Krieger and Elrod (42) developed an equation for the shear rate at
the surface of the outer moving cylinder making only one assumption; this
assumption was that a functional relationship existed between the shear
rate and shear stress:
y,
=
f(r)
The method of Krieger and Elrod can also be applied to the viscometer
used in this work (see Appendix A).
The resultant expression for the
shear rate at the surface of the rotating inner cylinder is
w
rh
111+1
d2 w
2
dInw 1
(ln s)4
45 W1
57
+ (In s)
4
1
4
d
1
(d ln T1 )4
+
(4-10)
when s = r2/r
If the term jin s (d in w 1 /d in T1)] is less than 0.2, only the firsttwoterms
of the series need be used and the error caused by this procedure willbe
less than 1%.
If [in s(d in o/d in
Tr))
is greater than 0. 2 but less than
1, one additional term must be used in order to keep the error below 1%.
A comparison between the shear rate - shear stress data of a sample
of human blood calculated from the experimental information by use of
Equation (4-10) and by use of equations of the same form as Equations
(4-9) but derived for a point onthesurface of the inner cylinder, is made
in Figure (4-1).
In both cases the shear stress was calculated from the
equation.
1
G
2,rr 1
Details of these calculations are presented in Appendix A.
The data are
plotted as the square root of the shear stress versus the square root of
the shear rate for reasons which will be explained later.
should be noted from Figure (4-1):
Two things
(1) the error introduced by the use
of Equations (4-9) to calculate the shear rate at a point in the viscometer
gap can be appreciable if blood is the fluid filling the viscometer, and
(2) the correction introduced by use of the Krieger-Elrod equation does
not change the general qualitative features of a plot of the square root of
the shear stress versus the square root of the shear rate for human blood.
(c)
The Vand wall effect.
Both Equations (4-9) and (4-10) are based on the supposition
that the shear rate of the fluid in the viscometer gap is solely a function of
the shear stress.
In the case of blood, or a fairly concentrated suspen-
sion of any type of particle, this is not strictly true since the particle
58
FIGURE
Comparison of shear rates calculated from viscometric data by
the Krieger - Elrod equation and the conventional equation
1.0
CD
C
C
El
.5
0Calculated assuming constant
viscosity across viscometer gap
DO
0
0 0-woo
j, Calculated from Krieger
equation
Hematocrit
=
47.4
Temperature = 24.8
%
CD
4-1
0
C
Viscometer set B
0
0
1
2
1/2, (sec-l/2)
3
-
Elrod
concentration adjacent to the cylinder walls is lower than in the bulk of the
fluid.
This variation in particle concentration arises because the pres-
ence of the solid wall physically prevents the particles from occupying
the space next to the wall in the same manner as they do in the bulk of
the suspension.
For example, red cell centers can never touch the space
adjacent to the solid walls; hence, the space adjacent to a wall is occupied
by a red cell - deficient layer of blood.
Because the shear rate - shear
stress relationship for blood appears to be a strong function of the red
cell concentration, it is evident that the shear rate of blood in the viscometer gap is not solely a function of the shear stress, but is also a
function of the red cell concentration, or equivalently, the radial position of the point of interest in the viscometer gap.
Because of the red cell deficient layer of blood next to the solid
boundaries of the viscometer gap, the fluid in this region will have a
lower viscosity than the bulk fluid, and this will give rise to an apparent
slippage velocity at the wall.
Of course, at very low particle concentra-
tions, the wall slippage effect will disappear.
Vand (65) considered the problem of apparent wall slippage in capillary and Couette viscometers by considering the fluid flow as two phase
flow (a wall layer of one fluid and a bulk fluid core or layer).
He derived
the expression relating the viscosity of the bulk fluid (ij) with the wall
layer fluid viscosity (77) and the viscosity calculated from experimental
data ignoring the wall effect (y ):
x
- -1
H (L
-)1
(4-11)
where, for the capillary viscometer
H
x (
1- p
)4
rw
60
(4-12)
and for the Couette viscometer (with rotating bob)
(-)1-Df
H
x
=
2(4-13)
1+ 2D
r -r2
in which D is the thickness of the wall layer.
In deriving these equations,
it is necessary to assume that both the wall layer liquid and the bulk liquid
have constant viscosities.
This is a poor assumption in the case of a cap-
illary viscometer filled with a non-Newtonian fluid, since the shear stress
varies from zero at the capillary center to its highest value at the capillary wall.
However, in the case of the Couette viscometer, this assump-
tion will not lead to a large error if the shear stress (and hence viscosity)
does not vary too greatly across the viscometer gap.
The division of the
flow into simple two phase flow is, of course, a simplification.
The above. theory would not be valid if the solid boundaries were
rough on the scale of the size of the particles.
In such a "rough wall"
situation, if the "roughness" were great enough, the wall-slippage effect
would not be seen.
This is illustrated in Figure (4-2). If the viscometer
surface is relatively smooth, as shown schematically in Figure (4-2(a)),
the wall-slippage effect may be noted.
However, if the surface is grossly
rough, as shown in Figure (4-2(b)), the particles and suspending medium
which are in the "cracks" of the surface will remain at rest relative to
the surface, whether the surface itself moves, or remains stationary.
If the "cracks" in the surface are numerous enough, the bulk suspension
in the viscometer gap will "see" cylindrical viscometer walls of itself
(shown by the dashed line in Figure (4-2(b), -and there then will not be
any cause for a wall-slippage effect.
61
/
P.0
/
(b)
EP
i
(b)
1,0
C)
(a)
Figure 4-2
Comparison of the viscometer cylindrical surface - suspension interface when the cylindrical surface is (a) smooth
and (b) rough on a scale greater than the particle size.
The GDM viscometer set B, which consists of vertically grooved
cylindrical surfaces (see Figure (3-4)), fits the conditions of the rough surface described in the last paragraph, if used with human blood.
Conse-
quently, if shear stress - shear rate data are obtained for a blood sample
by using the rough cylindrical surfaces (set B) and the smooth cylindrical
surfaces (set A), the magnitude of H, inEquation (4-11) can be determined;
this calculation is based on the use of the rough surfaced viscometer
data to calculate rj, the smooth surfaced viscometer data to calculate Itj
and the viscosity of the blood plasma as Tr
0
The "thickness" of the wall
layer, D, can then be calculated from Equation (4-13).
62
Figure (4-3) shows, on a plot of the square root of the shear stress
versus the square root of the shear rate, data obtained on a blood sample
by use of the smooth surfaced and rough surfaced viscometers.
The data
points were calculated by use of both equations (4-9) and the Krieger-Elrod
Equation (4-10); in this case the difference between the results of the two
methods of calculation was not greater than about 0.25%.
To insure that
the difference between the two sets of data obtained with the two different
sets of viscometer surfaces was not due to an instrument calibration error, Newtonian fluids with viscosities varying from about 1. 1 centipciise
to about 500 centipoise were tested in both viscometers: the data obtained
from both viscometers for any fluid agreed, with the greatest difference
between the two experimentally determined viscosities being 0. 4% of the
viscosity.
The difference in the two curves for blood in Figure (4-3) must be
attributed to the surface effect which has been discussed above.
Making
use of Equation (4-11) the following values of H were calculated from
Figure (4- 3):
H
49 sec
1.010
25
1.016
9
1.019
1
1.018
0.04
1.006
These values of Hx correspond to wall layer thicknesses of between 1 and
3.microns, as determined from Equation (4-13). When one recalls the
fact that the red cell is roughly disc shaped with a thickness of 2 microns
and a flat face diameter of 8 microns, this wall layer thickness seems
reasonable.
63
FIGURE
4-3
Effect of viscometer wall roughness on the
apparent rheological properties of blood
1.5
* Blood, rough surfaced viscometer
A Blood, smooth surfaced viscometer
o Red cells in albumin - saline, rough
surfaced viscometer
ARed cells in albumin - saline, smooth.
surfaced viscometer
1.0
Temperature
C
=
25
C
4.0.5
Area of inset
H
0-3
Woo
0 -5
00..1.0
00
0
>
7-5
2
2
3
56
1/2, (sec-l/2
7
1.0
8
Also shown in Figure (4-3) are data for a suspension of red cells in an
isotonic saline solution containing 3. 5 weight percent human (crystallized)
serum albumin.
The data for this supension, obtained with the two viscoAt first, this seemes to contradict the
meters, fall on a single curve.
conclusions reached in the previous paragraph, but if Equation (4-11)
is
rearranged to give
H
17X
+
0
(4-14)
*-
q 1+ (H -1) 7
x
0
and it is assumed that H
= 1.01 and use is made of the experimentally
determined value of 77 (1.01 cp at 25*C), Equation (4-14) becomes
0
L02(4-15)
T
*
77~
1.01 + 0.01-)
x
The largest value of 7) determined from the data shown in Figure (4-3)
for red cells suspended in albuminated saline, is 6 centipoise; using
this value for
tj
in Equation (4-15) gives
Tx 1 /2 = 0. 98 71/2
This represents the largest difference to be expected between r 1/2 and
71/2 since the suspension viscosity decreases below 6 cp as the shear
rate increases.
Consequently, it is not surprising that the data obtained
for a red cell suspension in albuminated saline, using both the smooth
and rough surfaced viscometers, should all fit on one curve.
Several groups of investigators have attemped to detect the wall
Sweeny and
Geckler (61) made a suspension of 124 micron glass beads in a glycerol
-
slippage effect in the concentric cylinder type viscometer.
water - zinc bromide solution (40 volume percent beads), and made measurements at 3 shear rates in a concentric cylinder viscometer which had
3 different size bobs.
These authors claim that no wall slippage effect
65
was detectable, or else it was smaller than the experimental variation
in their results.
Considerable doubt is cast upon the validity of their
results since their data indicates that their suspensions have a maximum viscosity at a shear rate of about 100 sec',
at both higher and lower shear rates.
with lower viscosities
This behavior is not consistent
with the generally found behavior of particle suspensions of this type.
Evenson, Whitmore, and Ward (22) suspended spheres of p'lymethyl
methacrylate (38 to 388 microns in diameter) in a water -glycerol - lead
nitrate solution.
Tests on these suspensions in a Couette - type visco-
meter with gaps of 2, 3, and 4 mm showed no wall slippage effect.
How-
ever, these results are not surprising when it is realized that the maximum volume concentration of spheres was only 20%; at such low concentra-
tions the particles are separated by large enough distances to make the
nature of the viscometer wall surfaces of very little or no importance.
The' results summed in Figure (4-3) are typical of the results obtained
with several normal bloods.
Tests were also made with smooth cylindri-
cal surfaces, and surfaces coated with washed sand particles (0. 0029
0. 0041 inches in diameter); in the case of the sand-coated surfaces, the
effective viscometer dimensions were determined by making torque - bob
rotational speed measurements on several Newtonian fluids of known viscosity and by utilizing Equation (4-7).
(In the tests with the Newtonian
fluids, tests were made with one surface smooth, of known radius, and
one sand-coated surface, and then the tests were repeated for both surfaces sand-coated; this procedure permitted the radii of each sanded
surface to be calculated.)
In all cases, the results indicated a wall-slip-
page effect on the smooth surfaces, and the "thickness" of the wall layer
was always about 1-3 microns.
66
2.
Time effects
(a)
Time effects at constant bob rotational speeds
A recorder attached to the viscometer continuously traces
out the torque reading of the viscometer as a function of time.
Whenthe
viscometer contains whole blood, and the bob is rotated at a constant angular speed, the torque - time curve takes one of the two forms shown in
Figure (4-4): (1) if the shear rate is greater than about 1 sec
-1
,
the
torque rapidly climbs, after the bob is started rotating, to a value at which
it remains constant thereafter (upper diagram - Figure (4-4)); (2) if the
shear rate is less than about 1 sec
, the torque initially rises, upon
starting bob rotation, to a maximum value and then decays with time
(lower diagram - Figure (4-4)).
less than about 1 sec
,
For the cases where the shear rate is
the torque will decay, after having reached its
maximum value, until a steady value is attained.
If the blood is one whose
red cells sediment rapidly, the torque will slowly rise from this steady
state 20 - 30 minutes after bob rotation was first started; this is due to
the increase in hematocrit which occurs in the bottom of the viscometer
as the cells settle downward.
If the blood in the viscometer is sufficiently
stirred, by raising and lowering the stillrotating bob, the torque - time
curve is reproduced when the stirring is stopped.
The time necessary to reach the maximum torque value increases
as the shear rate (bob rotational speed) decreases, as is illustrated in
Figure (4-5) for several different blood samples.
Even though the data
in this figure represent the behavior of "normal" bloods (from donors in
good health), the great variation among bloods is apparent.
Because of
the complexity of the rheological properties of blood, and the complexity
of the mechanism by which steady state blood flow is attained in the viscometer, no quantitative importance can be attached to Figure (4-5). However,
67
Ii1
I
I
11
11
)
I
I
IM
Full Scale Torque 50 dyne-cm
Upper Curve 10.2 sec-1
Lower Curve 20.4 sec-1
t
t4
'1
Id1
I I
I
"Mt
4ft
TflT
Full Scale Torque - 3 dyne-cm ; 0.01 sec-1
Figure 4-4. Typical torque-time curves for human blood; upper diagram for shear rates
greater than 1 sec-1, lower diagram for shear rates less than 1 serTF
68
4-5
FIGURE
Time requir ed to reach torque - time
curve peak vers us viscometer bob rotational
speed, for 3 blood samples
1.0
0.7
--
)(
0
C
Sample
m 8264, W3
Sample
M
Sample
K 4370
8264, N2
0.2
x 10-4-
rpm
0.1
0
0.07-
X0
0.05I
0.021-
EM
0.01
1.1
0
20
10
8.6 x Peak time,
69
,
(minutes)
because viscous Newtonian fluids, such as aqueous glucose solutions also
show the same qualitative behavior, it is hypothesized that the initial
portion of the time-torque curve, illustrated in the lower diagram in
Figure (4-4) by the portion of the curve preceding the maximum torque
value, is the transient period during which the blood is accelerating to
its steady state flow pattern.
The subsequent torque decay period is explained by the mechanism
of a developing layer of blood plasma at one or both viscometer cylindrical surfaces.
This layer, which acts as a lubricant at the viscometer
surfaces, develops only at shear rates below about 1 sec
1
and it grows
in thickness with time until the torque has decayed to a steady value.
-
This argument requires that the red cells of the blood recede from the vis
cometer walls at low shear rates.
Visualevidence of this mechanism
was obtained withthe fortunate discovery of a blood donor whose blood
plasma contained about 8% fat.
The fat concentration was sufficiently
high to cause the blood plasma to be opaque and milky white, instead of
the usual clear, straw-colored fluid.
In Figure (4-6), the right tube con-
tains a sample of this blood with the red cells settled down in the lower
part of the tube, and the milky plasma in the upper part of the tube.
tube of normal blood plasma is shown on the left.
A
The middle tube shows
a sample of the high fat blood with its red cells in the process of settling
down (this particular blood had a very high sedimentation rate).
This
blood was placed in the rough-surfaced viscometer and stirred by raising and lowering the rotating bob, which was rotating at a speed of 0.2 rpm.
Upon stopping the stirring, the blood appeared to be uniformly red, as
-
shown in Figure (4-7), which is a color photograph of the blood in the vis
cometer taken about 0.5 minutes after stirring was stopped.
Continued
bob rotation resulted in the development of a growing whiteness in the
70
FIGup
4-6
Photograph of hyperlipidemic blood and
plasma, and normal blood plasma
71
Hyperlipidemic blood in the
after stirring stopped, bob
72
viscometer, 0-5 minutes
rpm
speed is 0.2
Hyperlipidemic blood in the viscometer, 5.5 minutes
after stirring stopped, bob speed in 0.2 rpm
73
apparent blood color, as is shown in Figure (4-8), which was taken about
5 minutes after Figure (4-7).
This whiteness can only be interpreted as
being due to the presence of a plasma layer at the outer viscometer surThe blood was again stirred, but this time the bob was not rotated;
face.
the blood did not show a plasma layer at the outer wall of the viscometer
gap 5-1/2 minutes after the stirring was stopped.
Clearly then, this de-
velopment of a plasma layer is induced by the flow of the blood.
Further
experiments with this blood sample showed that the plasma layer did not
-1
develop at shear rates greater than 1 sec , and that the layer developed
only when a torque decay was simultaneously observed inthe torque - time
curve.
The use of smooth surfaced viscometer surfaces duplicated these
observations.
The magnitude of the rate of torque decay immediately after the maximum torque is reached is a function of blood sample donor, hematocrit, and
shear rate.
For a particular donor, the effects of hematocrit and shear rate
are qualitatively the same as those found for bloods fron other healthy donors.
It has been observed that the rate of torque decay is zero for blood
samples whose hematocrit is below about 10% and above about 55%; the
maximum rate of torque decay occurs at about 35 - 40% hematocrit, as
is illustrated by the data in Table (4-1).
It might be suspected that the
torque decay found at low shear rates might be due to sedimentation of
the red cells in the blood, especially since the sedimentation rate of red
cells also is low at very low and high hematocrits, with the maximum
sedimentation rate also at 35 - 40% hematocrit.
If the sedimentation of
the red cells is the. explanation for the torque decay, then since the torque
does decrease with time, there must be a decrease in the concentration
of red cells in the blood (in the viscometer gap) with time.
74
Granting
that red cells are leaving the viscometer gap due to sedimentation out
the bottom of the viscometer gap, and also entering the top of the gap at
the same time, it is necessary to determine the net change with time in
the number of red cells in the viscometer gap.
Figure (4-9) illustrates
how the red cell sedimentation rate varies when blood is subjected to a
shear stress normal to the direction of the gravitational force causing
the red cells to settle.
Two effects kre to be noted: (1) the initial period
of slow settling decreases in length as the normal stress is increased,
and (2) the maximum settling rate.decreases as the normal stress inci-eases.
In the limit, then, since the blood above the viscometer gap
TABLE 4-1
Rate of torque decay at constant viscometer rotation speed.
Rate of Torque- Decay (dyne cm/min)
Hematocrit
Bob
Rotational
Speed
10.2%
1 9.2% 21.6%
27.5% 33.2%
39.10% 46.0%
0. 5 RPM
0.000
0.030
0.026
0.061
0.067
0.048
0.042
0. 2 RPM
0.000
0.030
0.045
0.046
0.064
0.069
0.057
0. 1 RPM
0.000
0.038
0.047
0.057
0.065
0.062
-----
Blood Sample
Temperature
Rough Surfaces
M-8264
=
25. 0* C
Viscometer
is essentially stationary (and could have a long ''induction" period of
little or no settling), and when the blood in the viscometer gap is at a
low shear rate (having no "induction" period and the maximum possible
sedimentation rate), the maximum possible rate of torque decay due to
settling of the red cells would be
75
i
FIGURE
4-9
Sedimentation of red cells normal to a shear field
66.2
Sample
Oqi
K 1184
Rough hamster bob
Smooth large cup
66.o
0%
eN
*sNO
65.8
0%
'0
%G
\
C)
-3
O*z
..
.1-I
C)
'0
'0
+~
65.6
'0
0
0
'*0tpm0
**0
65.4
rpm = 2
65.2I
65.0'
0
rpm =
0. 1
20
10
Time, (minutes)
I---
30
dT
"dO/max
_ (maximum sedimentation rate)
bob length
(Torque)
Returning again to the high lipid blood shown in Figures (4-6) through (4-8)
the maximum sedimentation rate at 25 0 C was found to be 1. 11 mm per
minute, Figure (4-10), which becomes 1. 7 mm per minute at 37 0 C if one
applies a temperature correction to the sedimentation rate (66).
For the
experiment recorded in Figures (4-7) and (4-8), the peak torque value was
9. 2 dyne cm and the initial rate of torque decay was 1. 4 dyne cm per minute; the blood was at 37. 0*C.
Therefore, the maximum torque decay
possible, due to red cell sedimentation is
(
)min
dO max
1.7 min
X 9. 2 dyne cm= 0.
53
dne cm
mm
29.17mn
Thus, the maximum decay rate to be expected, due to red cell sedimentation, is about 1/3 that experimentally found.
Since the extreme situation
considered above would be visually detectable if it occurred, and it never
has been seen, it is apparent-that red cell sedimentation does not account
for the torque decay phenomena being discussed.
It might be argued that there was sufficient torque transmitted from
the bottom of the bob to the bottom of the cup of the viscometer to permit
the torque decay to be explained on the basis of the sedimentation of red
cells away from the bottom of the viscometer bob.
Such an argument
would require the torque transmitted between the bottom of the viscometer
bob and the cup to be appreciable.
This particular "end effect" was dem-
onstrated to be negligible by (1) putting National Bureau of Standards "K"
oil in the viscometer and determining the oil viscosity with the bob in its
normal position, and raised 0.0417 inches and 0.0614 inches above its
normal position (viscosity at 25*C is 33 cp), (2) repeating the experiment
77
I
:'IGURE
4-l0
Sedimentation rate of hyperlipidemic blood
~0
79
1.11 mm/min
78
0
0
\0
0
77
\0
co
4,
4e
76
K
C' .0
Sample: M 5855, type 0+
Temperature = 25.0 0C
75
1
0
10
20
I
30
I
I
50
Time, (minutes)
I
60
I
70
I
80
withmolasses at 25*C (viscosity =.2400 cp), and (3) repeatingthe experiments with blood at- 37*C and 2.05 sec
.
In all cases, the data indicated
that no "end effect" was detectable.
The torque decay is therefore clearly due to the migration of the red
cells of blood away from one or both viscometer walls at low shear rates.
Goldsmith and Mason (33) have reported that when suspensions are passed
through circular tubes at particle Reynolds Numbers below 106, the
particles willmigrate towards the region of lowest shear rate if they are
deformable spheres or threads; rigid spheres, disks, or rods do not migrate at all.
These authors attribute the migration to a force which arises
when a particle is deformed in a variable shear rate region.
Thus a
sphere suspended in a fluid inthe gap of a Couette viscometer ideally
would not migrate because of the (almost) uniform shear rate inthe viscometer gap.
According to this theory, if the red cells in blood could be
prevented from aggregating at low shear rates, no migration might be detected because of the very smallvariations in shear rate across the viscometer gap; rouleaux might migrate because of their greater deformability.
The experimental evidence partially supports this viewpoint:
the red cells
in human blood do migrate at low shear rates (when rouleaux can form),
but red cells suspended in an isotonic saline solution containing 3. 5%
human serum albumin (which prevents any red cell aggregation, as has
been verified by microscopic investigation) have not been found to migrate
(as evidenced by the lack of any torque decay phenomena).
However the
hypothesis of Goldsmith and Mason requires migration to the region of
lowest shear rate.
In the GDM viscometer, this region of lowest shear
rate is at the surface of the outer cylinder.
Since with both the rough and
smooth surfaced viscometers, the red cells were found to migrate away
79
from the outer wall, it must be that Goldsmith and Mason's theory does not
apply to the situation of human blood in a Couette -type vis cometer.
It has been reported by Segre and Silberberg (59) that in a suspension
flowing through a verticaltube at particle Reynold's numbers between 10
-3
and 6 X 10-2 rigid spheres.of polymethyl methacrylate not only migrate
away from the tube wall, but also simultaneously migrate away from the tube
axis.
Thus, the spheres come to occupy an annular space in the tube. Segre,
and Silberberg attributed the migration of the spheres away from the tube
wall to the Magnus effect,. but offe red no explanation for the migration away
from the tube axis.
S. G. Mason has reported (46) that he has duplicated the
w ork of Segre and Silberbe rg and found the ir findings to be corre ct.
It appears, therefore, that there are physical forces which could explain
how red cells in blood recede from one or both walls of a concentric -cylinder
viscometer atlow shear rates.
Notest has been devised whichwould per-
mit one to determine if the red cells recedefrom the inner cylinder wall, but
there is no doubt that the ability to migrate at low shear rates is related to the
ability of the red cells to aggregate:
red cells suspended in albumin-saline
solutions do not aggregate or migrate; red cells suspended in aIand y - globulin - saline solutions, where some smallaggregation may occur, do not migrate; but, red cells suspended in fibrinogen - saline solutions, and in plasma,
where aggregation can be observed under the microscope, do migrate.
The
migration may arise not only because of the increased deformation of aggregates over individual particles, the Magnus effect, and other purely
physicalforces, but also because of an attractive force between red cells and
red cell aggregates which is similar, but not identical, to the force causing
aggregation.
This attractive force would permit more efficient "packing"
of the cells in blood at low shear rates.
80
The attractive force is not identical
to the force causing aggregation because red cells will normally only aggregate face-to-face; however, both forces may arise because of a fibrinogen
-
cell surface site linkage, with the majority of sites being on the cell faces.
From this interpretation of the torque
-
times curves, it is obvious that
the correcttorque value to be associated with a particular shear rate is close
to the torque - time curve peak, provided the peak occurs shortly after startingthe fluid motion.
However, inthe case of low shear rates, where the peak
occurs several minutes after start up, some correction must be made to the
peak value in order to correct for the plasma layer which has been developing
atone or both viscometer walls in this time.
Since thetorque decay afterthe
peak of the torque - time curve is exponential, as illustrated in Figure (4-11)
an exponential extrapolation to time zero of the torque decay section of the
torque - time curve would be one such attempted correction; this corresponds,
within the limits of experimental error, to a linear extrapolation of the portion of the torque - time curve immediately after the peak totime zero.
Figure (4-12) shows, on a plot of y 1/2 versus
1/2, thepeak, extrapolated,
and "steady state"values for a blood sample
(b)
Time effect upon stopping viscometer bob
Upon stopping the viscometer bob rotation, the torque recorded
bythe viscometer does not drop to zero instantaneously, even though the bob
is stopped essentially instantaneously.
This occurs because it takes some
time for the viscometer cup and torque -summing shaft to return to its null
position, and because the fluid inthe viscometer does not stop flowing instantaneously.
Nevertheless, with a fluid such as water inthe viscometer, the
torque will return to zero within a few seconds afterbob rotation is stopped.
However, with blood intheviscometer, the torque does not return to zero
until a time lapse of about 5 minutes or more has occurred since the viscometer rotationwas stopped; a typical example of this behavior is shown in
81
FIGURE
100 -
4-li
Torque - time curve for blood at
constant viscometer rotational speed
90
70
7
70
4
0
s-
G
0
50
P4
0
4,-
CG
30
S0.01
0.02
0. 1
20~
0
2
rpm
rpm
rprn
4
6
8
Time, e
82
(minutes)
10
12
1.2
FIGUKE
4-12-
Shear stress - shear rate data for human blood,
using extrapolated, peak, and steady state torque values
o.8
Cu
H
CU
Time
Effects
'-
U
C)
o .6
Cu
H
No Ttme
O0.2
0 Extrapolated
-
t =0 v
flues
9Peak
values
0- Minimum (steady state) values
0
0
1
2
.1/2
(e-/
3
'4.
ff1
IIIII Ll 1 1LIL" LL.L' I
I I I I I I I I I I I 11144"444- 1i i I I I I i i i i i i i+
KimI_
III
Li
.......... !.+.f + ! T! i ! 6 11 ! t 1
U
U
U
U
U
F"
I
.1-
-Mr
r=L
11-111- 1-1 .1
. .
I . I I. -
-
f1
4 ~tI
if
I
1-Er
N
'
rA
L1I
M[I.E
!
II
Full Scale Torque - 4 dyne-cm
Initial Rotation
= 0.1 rpm.
Figure 4-13. Torque-time curve for human blood, obtained on stopping
bob rotation.'
84
I
Figure (4-13). When the bob rotation is stopped, the torque initially decays
at a rate which is the same as when the viscometer is filled withwater (dashed
line in Figure (4-13)), until a certaintorque value, T , is reached, below
y
which the torque decays at a much slower rate." This curve is really two
exponential curves, and it has been found empirically that the point where the
-
transition from one exponential curve to the other first takes place corres
ponds to the fluid yield stress
Kaolin suspensions in water were prepared and their yield stress
values were determined in the Merrill-Brookfield viscometer by the
same methods used by Bolger (5).
In the Merrill-Brookfield viscometer,
the torque decay curve was of the form shown in Figure (4-14).
These
same suspensions were then placed in the GDM viscometer (rough surfaces) and it was found that they displayed the same type of torque
-
time
W
0-
YIELD
STRESS
-50BROTATION
STOPPED
Figure 4-14. Torque decay curve obtained for kaolin suspension in the
Merrill-Brookfield viscometer.
85
curves as blood when the viscometer rotation was stopped.
As the Kaolin
concentration was increased,, the length of time required for the torque
By projecting the original torque
-
reading to return to zero increased.
time curves traced out by the viscometer recorder on the wall of a room,
it was possible to obtain time-torque data from the curves.
These data
were then plotted onsemilog paper, Figure (4-15) being one such curve
for a 4% Kaolin suspension.
From such curves, the shear stresses cor-
responding to the torque values where the curves departed from one exponential curve to the next were calculated.
In Table. (4-2), the yield
stresses, as determined in the Merrill-Brookfield viscometer, are compared with the shear stresses of departure from the initial exponential decay curves obtained with the GDM viscometer.
The agreement between
the Merrill-Brookfield yield stress and the GDM viscometer departure
stress is good, considering the inherent errors of the methods used to
obtain these values.
Bolger found for these Kaolin suspensions that the
cube root of the yield stress was linearly related to the volume fraction
of the suspension which was occupied by the Kaolin flocs.
The data of
Table (4-2) are tested in Figure (4-16) for this relationship, the floc
concentrations having been obtained from Bolger's Sc. D. Thesis (5).
The data of Table (4-2) do satisfy the relationship.
It was not feasible to determine the yield stresses of the usual human
blood samples in the Merrill-Brookfield viscometer because of the relatively large sample of this valuable fluid required to fill this viscometer.
However, a sample of "outdated "blood (stored over 3 weeks) was placed
in both the Merrill-Brookfield and the GDM viscometers and its yield
stress determined by the methods discussed above; its yield stress was
found to be 0.108 dyne/cm2 by use of the Merrill-Brookfield viscometer
and 0.110 dyne/cm
by use of the GDM viscometer.
is excellent.
86
Again, the agreement
I
i
FIGURE
4-15
Torque decay curve for 4% kaolin suspension
500
in the GIM viscometer, rotation stopped,
= 1.67 (dyne/cm2)1/3
7 1/3
-
--
200
(D 70
50
30
0
I
I
i
1
2a
3
I
Scale time,
a,
87
(minutes)
I
I
5
6
YiELD
FLOC
,
CONCENTRA TIN COMCEMTR4TION
wr
rR4crIoM
*
VO I U/ME
Y/EO STrESSES DEPARTURE
M-B v/SVMETR
DYNE /CM
2
s rC ss
y
DYNE/CM 2
'/3
C
\ CM
P/ACrO
Y3
(CN
(\3
2
"
KAOlI N
TABLE 4-2
STRESSES OF kAOLIN SUSPENSIONS
/\ct
0.0/
0. 89
0.02
0.160
0.5?
0.55
0.04
0.279
4.7
4.4
/..67
1.61-
0.05
0.330
9.3
9.Z5
2./o
2.12
-
OBr4INED FRoM REFR 5, PAGE 168
0.0289
0.307
0.838
0.8/
I
FIGURE
21-
I
4-iE
Comparison of yield stress data for kaolin
suspensions as determined in the Merrill-Brookfield and GDM viscometer0
rn
H
Cu
Li
a)
Co
%-
eel
H
o Measurement
by GDM
I)Measurement by Merrill--Brookfield
Temperature
=
25.0 "C
0'
0
0.1
0.2
6F
,
Floc volume concentration
0.3
One question remains to be answered:
why is there a difference in
the nature of the torque'- time curves for the same substance in the two
different viscometers?
the torque
is stopped.
-
Figures (4-17) and (4-18) are semilog graphs of
time curves obtained when the GDM viscometer bob rotation
Four curves are shown one each for four initial bob rotat-
ional speeds (before bob motion was stopped).
The two exponential posi-
tions of these curves can be seen, together with a first non-exponential
section immediately after the bob rotation is stopped, and a second nonlinear section between the two exponential portions.
During the period of time represented by the first nonlinear sections
and subsequent first linear sections shown in Figure (4-17), the electromagnetic torque exerted on the torque-summing shaft of the GDM viscometer by the microsyns is greater than the viscous torque exerted on the
viscometer cup by the fluid in the viscometer gap, and the viscous torque
of the damper and the gas in the air-bearing gap.
Consequently, the vis-
cometer cup is rotated towards its null position; this motion is analogous
to the initial rapid torque decay period of the Merrill-Brookfield viscometer (see Figure (4-14)).
In the Merrill-Brookfield viscometer, the torque does not return to
zero when the bob rotation is stopped if the fluid being tested has a yield
value.
Instead, the torque drops to the value corresponding to the fluid
yield stress and remains constant at this value (within the time span of
the experiment, which may be an hour or two).
In the same situation in
the GDM viscometer, the torque decays at a very slow rate after the
torque has dropped to the torque value corresponding to the fluid yield
stress.
In addition, for blood, the rate of torque decay below the yield
torque depends on the shear rate to which the blood was subjected just
before the bob rotation was stopped.
90
10.0
FIGURE
7.0-
4-17-
Torque - time curves for blood,
GDM viscometer rotation stopped
5.0
Rough surfaced viscometer set B
Initial rpm
2.0
A
Full scale torque
o
1.0
10 dyne cm
A
0.5
5 dyne-cm
O
0.2
3 dyne cm
1.0 --
0.7
0-5
E'0.5-
0.2
0.1
T
Time,
,
20
(see.)
0
91
30
FIGURE
0.2
4-18
Torque - time curves for blood,
GDM viscometer rotation stopped
Rough surfaced viscometer set B
Initial
0.1
rpm
Full scale
value
E
0.1
3 dyne cm
V
0.2
3 dyne cm
A
0-5
0
1.0
5 dyne cm
10 dyne cm
S
U
a,
U
k
-.
35 x 105
4.35
dynecm
x .105
5.80 x 10 5
7.25 x 10 5
El
0.07
A
0.05
0
0.02 1
I
0.01
50
100
I
I
200
150
Time,
e, (see)
92
I
300
400
The reason for the difference in behavior of the blood in the two different viscometers lies in the "spring constants" of the two viscometers.
In the Merrill-Brookfield viscometer, the spring constant is 0.10 dyne
cm per radian while for the GDM viscometer the spring constant is
4. 35 K 105 dyne cm per radian (for a torque full scale setting of 3 dyne
cm).
Therefore, in the GDM viscometer, at the yield torque position,
the cup is rotated about 1. 24 X 10-6 radians, or 7. 1 X 10-5 degrees of
arc, away from its null position.
This means that the viscometer cup
surface, in returning to its null position from the yield torque position,
travels a distance of about 0.02 micron, which is much less than the diameter of a red cell.
This represents the distance the cup moves as
the blood in the viscometer gap relaxes.
Presumably, the blood in the
Merrill-Brookfield viscometer is also relaxing, even though the torque
reading appears to be constant.
If it is relaxing at about the same rate
as it does in the GDM viscometer, the movement is so slow that it would
not be detectable in the Merrill-Brookfield viscometer within the time
period of an experimental test.
The viscometer cup could return to its null position by "slipping"
over the structured fluid in the viscometer gap.
However, this does
not seem likely when the rough-surfaced viscometer surfaces are used,
and the cup must move because the fluid in the viscometer gap "creeps".
Considering the blood in this "creeping" situation to consist of rouleaux,
more or less entangled to form a network structure throughout the blood,
we might expect, by analogy to the case of long slipping ropes entangled
in a network structure (27), that the "viscosity" of this network would be
proportional to the length of the rouleaux length raised to the 2.5 power.
Assuming that the network viscosity is proportional to the rouleaux length
raised to some power n, it follows that the torque exerted on the viscometer cup by the blood rouleaux network is given by
93
Tb
KJn -
(4-16)
where K is a constant, J is the rouleaux axial ration (length to width
ratio), and 0 is the angular velocity of the cup.
Neglecting the viscous
torque of the viscometer damper and air-bearing fluid, and making use
of the fact that the electromagnetic torque exerted by the viscometer
microsyns on the torque-summing shaft is proportional to the viscometer
cup displacement from the null position:
T
= k
0
(4-17)
the torque balance on the viscometer cup - torque summing shaft gives:
14 = k, O-K Jn 0
(4-18)
where I is the rotating system moment of inertia.
Since under the
"creeping" motion being considered here for blood, the viscometer
torque reading, Te (and hence angular displacement
4),
is an exponential
function of time (see Figure (4-18)):
T
In
= ln Te-ln ki 1= In (constant)
= ln
+ st
(4-19)
1
where s is the slope of the straight lines in Figure (4-18), and t is time.
Differentiation of Equation (4-19) gives
dT
dt s T e
(4-20)
d 2Te
dt
s2 T
2
e
Substitution of Equations (4-20), using Equation (4-17), converts the differentiation Equation (4-18) to
s2 + KJn
0
(4-21)
and 'therefore:
KJn
5
94
-Is
(4-22)
Since k 1 is of the order of 106 dyne cm, I is about 205 dyne cm sec2
and s is of the order of -10
-2
sec
-1
, the Is term is negligible when com-
pared to kl/s:
k
KJ
1
(4-23)
From the data in Figure (4-18), the following values of KJn were calculated for each curve in the figure:
N. (rpm)
-KJn (dyne cm sec)
0.1
3.6 X 108
0.2
1.6
0.5
0.69
1.0
0.60
where N
is the viscometer bob rotational speed before rotation was
stopped.
From this information, it appears that the axial ratio, J, of
the rouleaux making up the network in the creeping blood decreases as
the shear rate of the sample immediately prior to cessation of viscometer
rotation increases.
This qualitatively agrees with the model of blood
aggregation set forth in this thesis; namely, at low shear rates the red
cells aggregate to form rouleaux, whose length decreases as the shear
rate increases.
From the above discussion, it must be evident that the concept of a
yield stress is not simply the shear stress necessary to initiate material
flow.
Given sufficient time, any material under stress will "flow" to re-
lieve the stress.
Therefore, the idea of a yield stress involves a time
lapse in its definition.
As a convenience to the experimenter, the yield
stress can be defined as the minimum stress needed to cause the material
to "flow" a minimum (perceptible) amount in a convenient length of time.
For some materials, such a stress can be relatively sharply defined for
95
experimental time lapses of minutes or hours, and this stress is called
the yield stress.
3.
Correlation of shear stress
(a)
-
shear rate data
The low shear rate region.
In the low shear rate region, it has been proposed that the
red cells in blood aggregate to form flexible rod like aggregates.
These
aggregates, called rouleaux, have a length which is a function of the shear
rate under which the blood is flowing: their length decreases as the shear
rate increases.
In addition, blood normally contains a rather high con-
centration of cells, so that there is considerable interference between
This complex situation is not amenable totheo-
particles and rouleaux.
retical treatment, but a recent suspension model considered by Casson
(10) seems closest to normal human blood, of all the models considered
in the literature.
Casson's model suspension consists of particles suspended in a
Newtonian medium.
The particles are mutually attractive, so that at
low shear rates the individual particles aggregate.
These aggregates
are allowed to be only rigid rod-like aggregates whose length varies inversely with the suspension shear rate.
For this model, at dilute con-
centrations, Casson found that the relationship between the axial ratio,
J, of the aggregates and the shear rate,
}
, was, for J very much greater
than unity:
J=
9F A
1/2
4 8 q af
(4-24)
where FA is the cohesive force between the particles forming the ag-
-2
gregates, dynes/cm ,
i7
is the suspending medium viscosity, and "a"
0
is a constant whose value depends on the average orientation of the aggregates with respect to the fluid streamlines.
96
For shorter aggregates,
for mathematical simplicity, Casson assumed that over a limited shear
rate range the relationship between the aggregate axial-ratio and the
shear rate was
J =a + p
where a and p are constants.
)1/2
(4-25)
Casson then relates J to the viscosity of
the suspension by determining the rate of energy dissipation due to the
rods and the suspending medium.
For a dilute suspension he finds
7= 77(1-c) + i
a Jc
(4-26)
where fl is the suspension viscosity and "c" is the volume fraction of
Combination of Equations
the suspension occupied by the particles.
(4-25) and (4-26) gives the relationship between shear rate and fluid
To obtain an expression applicable at
viscosity, for dilute suspensions.
higher particle concentrations, use is made of an approximate method
which considers that upon adding to a suspension an additional small
amount of particles, the resulting new suspension can be treated as
dilute suspension of the newly added particles, with the old suspension
being considered as a continuous medium.
The net result of this tech-
nique is an expression relating the shear stress and shear rate for the
suspension:
T
1/2
(4-27)
1/2 + b
-
where
s=
1/2
[
(4-28)
1-c)aa-1
b
a-2
a
a-1
97
2
-1
-1]
(4-29)
This technique, in addition to not being rigorous, does-not -take into account interactions between aggregates due to crowding.
The Casson -suspension must fulfill certain conditions.
a limited shear rate range, a plot of 7
Also, from Equation (28), a
1/ 2'
versus
First, over
1/
y1/2 should be linear.
plot of ln s versus ln(1-c) should be linear,
with a slope equal to [-(aa-1)/2]. From such a plot, it is then possible
to make a plot of "b" versus (1-c)- (aa-1)/2
which, according to Equa-
tion (4-29), should be a straight line whose slope equals the negative
of the line's intercept on the "b" axis.
Several examples showing that normal human blood fulfills the first
condition, namely a linear T 1/2 versus P 1/2 plot, have already been
presented, Figures (4-1), (4-3), (4-12).
Figure (4-19) shows a plot of
T1/2 versus .1/2 for a normal blood, with various red cell volume frac-
tions (hematocrits).
Typically, the shear rate range over which such a
plot is linear increases as the hematocrit decreases,
so that usually a
linear plot is obtained for a blood sample with a hematocrit of 45% only
from zero to about 1 sec
.
The linearity of such plots always extends
from zero shear rate to a high shear rate where nonlinearity first occurs;
this means that the quantity "b" in Equation (4-27) is the square root of
the blood yield stress.
Having established that normal blood fulfills the first Casson model
suspension condition, one takes the slopes of the linear plots just described
and tests them for the second condition, namely that ln s versus In (1-c)
should be a straight line.
In Figure (4-20), the data from Figure (4-19),
as well as data for two other normal bloods, are shown.
In this figure,
the data from two of the bloods seem to fall on the same curve.
These
data fit linear curves, and thus show that blood does satisfy the second
98
FIGUE
1.0
4-19
Caston plot of data for blood of
various hematocrits
Sample M 8264
0
Temperature = 25.0C
00
-0
0.5
ci,
-r----
-0
1.0
-
--
2.0
J7i
,(sec-/2)
3.0
FIGURE
4-20
Test of Casson equation
using human blood data
0.3
0in
0.2
A Whiting
series II
19.060, slope = -1.19
.0 Whiting series II,
19.000, slope = -1.19
D M8264, 2500, slope = - 1.09
0
I
-0.7
-0.5
-0.3
in
(1 -
100
a)
-0.1
0.1
0
Casson model condition.
As will be illustrated later when the effect of
temperature is discussed, the slope of the lines in Figure (4-20) is a
function of temperature, decreasing as the temperature increases.
With Figure (4-20) enabling one to evaluate the quantity (aa -1), the
third test of blood data can be made.
In Figure (4-21), such a test is
made on the data obtained for the three bloods described in Figure (4-20).
The data again fall on a straight line and the slopes of the lines are approximately equal to the negatives of the "b" axis intercepts (not shown,
and well off the plot).
Note that even though the data for two of the
bloods fell on the same line in Figure (4-20), they do not fall on the same
line in Figure (4-21).
On only one condition does normal blood not fulfill all of the conditions of the Casson model: the slopes and negative intercepts of Figure
(4-21) are not equal.
They are not equal because the lines do not pass
through the point (1-c)(a
1)/2
1.00, b = 0.
The reason that the lines
do not pass through that point is that blood, as all suspensions, does not
have a yield stress at very low particle concentrations because at very
low concentrations there are not a sufficient number of particles available to construct a three-dimensional network throughout the fluid.
The
Casson equation does not recognize this, as can be seen from Equation
(4-29), where "b" (the square root of the yield stress) does not become
exactly zero until "c" equals zero.
Unfortunately, Equations (4-24) and (4-25) cannot be combined to
enable one to calculate FAs the cohesive force between the red cells
forming the rouleaux.
To do so, one would have to know that J was
much greater than unity, so that
1/2
A1
(48a) /2
101
(4-30)
1.8
FIGURE 4-21
Test of Casson equation
using human blood data
1.6
0
1.2
1.0
10b
(dyne/cm2)i
0.8
0.6
0.41-
0.2
0
Sample
Intercept
Whiting,III
-0.253
Slope
0.242
A
0.156
-0.163
Whiting, II
19.000
0
0.290
-0.316
M8264
25.000
/
1.0
1.2
Temp.
19.000
0
1.6
1.4
r-
)
102
ad - 1
2-g-
1.8
_________________
2.0
2.2
2.0
This would require a to be zero, which it clearly is not, being calculated
from Figure (4-20) to be 4. 5 in one case and 4. 8 in two other cases (assuming the time average orientation of the rouleaux to be along the fluid
streamlines).
If it is assumed that the average orientation of the blood rouleaux
are along the fluid streamlines, the constant "a" become equal to 1/V.
With this assumption, the values of a and
p
canbe determined, and the
average axial ratio of the rouleaux, as a function of shear rate, can be
calculated.
Table 4-3 lists the values of a, p, 'r 1/2 and the axial ratios
0
of the rouleaux at shear rates of 1 and 0.01 sec~, calculated from Equation (4-25), for the three bloods described in Figures (4-20) and (4-21).
TABLE 4-3
Casson constants and rouleaux axial rations calculated from Equation
(4-25) from data of Figures (4-20) and (4-21).
Whiting Series
M 8264
0
Symbol in Figures
)
a
(
p
(dyne/cm2)1/2
77 1/2 (poise)1/2
J
Whiting Series
4.8
4.8
4.5
0.54
0.82
.0.93
0.13
0.13
0.13
9.0
1
s
1 sec
47.
J0. 01 sec-1
11.1
11.8
68.
77.
The axial ratios calculated for a shear rate of 1 sec
but the values at 0.01 sec-
seem reasonable,
do not. As pointed out by Casson (10), be-
cause of the simplifying assumptions made in deriving the equations to
describe the model suspension, the axial ratios calculated from Equation
(4-25) will be high.
By comparison with the Einstein equation for dilute
103
suspensions of spheres, Casson concluded that suspensions of spheres
would have a values of about 5 (p = 0).
Therefore, Casson model sus-
pensions whose fundamental particles are spherical would be expected
to have values of a of about 5.
Blood seems to fall into this category,
and this seems fairly reasonable considering the red cell shape.
It is interesting to note that the Casson derivation predicts that for
large values of J, the suspension viscosity is proportional to J2 (shown
by combining Equations (4-24) and (4-25), and (4-27)), while the statistical
law of flow for long slipping ropes says the viscosity should be proportional
to J
The fact that the Casson model equations correlate the shear stress
- shear rate data for normal human blood does not conclusively prove
that blood acts in an analogous manner to the Casson model, since fluids
which flow by other mechanisms may have flow properties which can be
ye/.2
_P 1/2 plot
correlated with the Casson equations. A linear r 1/2 versus
does not mean that the suspension acts like the Casson model:
Bolger
(5) found that kaolin suspensions, which are not Casson model suspensions, act as Bingham fluids at high shear rates:
7
=
m-? + n
where "m' and "n" are constants.
For this fluid, at high P values, the
quantity d r1/2/d P 1/2 is equal to ml/2, which means' a 71/2 versus
? 1/2 plot would be linear.
In spite of its inability to describe precisely the physical behavior
of human blood at low shear rates, because of simplifying assumptions
and necessary approximations, the Casson equation is very useful as a
means of correlation.
On the other hand, it appears unreasonable to
attempt to force out of it parameters of fundamental significance.
104
(b)
The high shear rate region
At sufficiently high shear rates, the red cells in blood will
not be able to remain aggregated in the form of rouleaux because the
shear stress on any potential rouleaux will cause the tension at any point
in the rouleaux to exceed the force of cohesion between the red cells.
However, this does not mean that blood at such high shear rates would
be simply a suspension of individual particles.
Mason and Bartok (45) have found that in dilute suspensions of equal,
neutral spheres, under a uniform shear rate, the fraction of the spheres
which are present as shear-induced doublets is given by fd = 8c, where
"c" is the particle volume fraction.
Thus, a 7% (volume fraction) sus-
pension of spheres will have 56% of the spheres in collision doublet form
at any time and any shear rate.
This relationship is not valid at high
particle concentrations because of the formation of triplets, etc. by
collision, and because of the depletion of single particles at high doublet
concentrations (both effects neglected in deriving the doublet relationship).
In addition, attractive forces between particles will add to the steady
state number of doublets in a suspension by increasing the average life
of the doublets.
Consequently, as a result of purely hydrodynamic inter-
actions, the fraction of the red cells present in blood as individuals, even
at high shear rates, is very small; interparticle attractive forces reduce
this fraction even further.
Any relationship aimed at describing the rheological properties of a
suspension at high shear rates would therefore have to take these hydrodynamic aggregations into account.
On the basis of the theoretical equa-
tions proposed for dilute colloidal suspensions of neutral particles, it
has been suggested (28) that this relationship can be represented by a
concentration power series:
105
7
{1 +
] c + k, [712 c2 + k
[
3
c 3 +.
(4-31)
'
where [t7] is the suspension (particle) intrinsic viscosity, and k , k
1
2
etc., .are constants which are functions of the shear rate and the dimensions of the particles.
Vand (64), considering a dilute suspension of
spheres in which hydrodynamic aggregation occurred, found theoretically
that Equation (4-31) was:
7+
2.5 c + 7.349 c2 +..
(4-32)
Mason and Bartok (45) found experimentally that sphere doublets rotated
as though their axial ratio was unity (instead of two) and that the doublets
had a longer life than Vand assumed. On this basis, they proposed that
2
the coefficient of the c term should be 10.05 instead of 7.349. They re.ported that experimentally this coefficient has been found to be 11.7,
12.5, and 12.7.
The difficulties experienced in attempting to find theoretical equations needed to describe the flow properties of dilute suspensions of
neutral spheres at very low shear rates are only minor when compared
to the problems involved for concentrated suspensions of flexible, nonspherical particles, such as blood.
Attempts to solve this more difficult
problem, at least for neutral sphere suspensions, have all lead to results which have a common aspect (see Appendix B): namely, at a particular shear rate, the suspension relative viscosity is a function only
of the particle concentration.
Thus, suspensions of particles in various
suspending media should all have the same relative viscosity at the same
shear rate and particle concentration.
Figure (4-22) shows the shear stress
-
shear rate data for red cells
suspended in plasma (blood) and red cells suspended in serum (plasma
106
FIGUIE
4-22
Shear stress - shear rate
data for blood, red cells
in serum, plasma, and serum
1.0 -
c = 4i2.5 %50
Temperature - 37.0 0C
0.9
Predicted curve for .blood, based on data
W
0
of red cells in serum,
plasma, and serum.
CDD
000
-e
0.5
0.3
400
Serum
0
0
1
2
3
4
5
10
, (sec-)
20
whose fibrinogen has been removed by clotting), both suspensions containing the same volume fraction of red cells; the same data for the two
If the red cells are acting as neutral
suspending media are also shown.
individuals at the upper shear rates shown in this figure, then at the higher
shear rates the relative viscosities of the two suspensions should be equal:
UBlood
_
?7Plasma
7RC-S
Vserum
or, at a. given shear rate
_
Blood
-
Plasma
yserum
RC-S
where the subscript RC-S refers to the suspension of red cells in serum.
The dotted line in Figure (4-22) is the curve predicted by the above equation from the data on plasma, serum, and the suspension of red cells in
serum.
Although it approaches the actual blood curve, it does not co-
incide with the blood curve in the shear rate range of Figure (4-22).
Figure (4-23) is similar to Figure (4-22) except that instead of using
a suspension of red cells in serum, a suspension of red cells in a 3. 5%
albumin - saline solution was used; the albumin prevents any red cell
aggregation, as was confirmed by microscopic investigation of the susAgain, the predicted blood curve does not coincide with the
actual blood curve, even at shear rates of 103 sec
.
pension.
The suspension of red cells in the albumin - saline solution is as
close to a suspension of neutral* red cells as one can get; the red
cells do no aggregate even in the stationary suspension.
Consequently,
*Neutral means no net attraction or repulsion between the particles of
the suspension.
107
FIGURE
4-23
Shear stress - shear rate behavior of blood
and red cells suspended in albuminated saline
-
5
0 Blood, c = 41.7
0 Red cells in albumin - saline solution, c
4
Cu
V Albumin
S
U
=
= saline solution
Plasma
C)
=25.0
10
20
00
-
3
Temperature
C
to
2
0
0
50
1.026
41.7
the fact that the predicted blood curve in Figure (4-23) does not coincide
even at 103 sec
1
with the actual blood curve indicates that interparticle
forces still play an appreciable role at 103 secrheological properties of blood.
in determining the
Higher shear rates are needed in order
to make these forces insignificant enough, compared to the hydrodynamic
forces, for the blood to act like a suspension of neutral particles.
The similarity between Figures (4-22) and (4-23) makes one wonder
if red cells in serum act like a suspension of neutral particles.
To test
this idea, Figure (4-24) presents data on a suspension of red cells in
serum, and a suspension of red cells (from the same blood sample) in
isotonic saline.
In each suspension case, the red cells were removed
from blood by centrifugation, and washed in isotonic saline by mixing the
cell pack with an equal volume of saline and recentrifuging (washing was
repeated once more for serum suspension cells and twice more for saline
suspension cells).
The serum suspension cells were then washed twice,
by the same process, with serum.
Following these treatments, the
cells were suspended in their respective media to a particle volume fraction of 37.3%.
This washing treatment not only insures that no fibrino-
gen is in the final suspensions, but probably also removes some outer
layers of material from the red cell membrane, so that the red cells in
the serum suspension may no longer be identicalto red cells whichwould
be found in serum if one could employ Maxwell demons to remove fibrinogen molecules from blood without altering the blood in any other way.
These suspensions did not show rouleaux formation when examined under
the microscope, although some small aggregates of 2 or 3 cells could be
seen.
Although the two suspensions have very different shear stress-
shear rate curves, the dashed curve in Figure (4-24) shows that the curve
for the red cells in serum can be predicted from the data for the saline
110
FIGUBE
4-24
Shear stress - shear rate behavior of
red cell - serum and red cell - saline suspensions
o
red cells in serum, c
=
37.3
%
5
0 red cells in saline, c =37.3%
- predicted red cells in serum from red cells in saline
4
E-
serum
= 1.69
saline
3
Temperature =25.0
C
43
5
+2
CO
ELEl
*~0
10
20
30
40
Shear rate,
50
t
10
, (sec1
)
0
suspension and the two suspending media, at shear rates above 30 sec
Below 30 sec
1
the predicted curve and actual curve for the red cell
suspension in serum do not agree, probably because the proteins in the
serum alter the interparticle forces between the red cells from the
forces between red cells in saline.
Thus, below about 50 sec -1 , the
effects of the interparticle forces are not negligible, while above about
50 sec
they are negligible compared to the effects of the hydrodynamic
forces (assuming -the red cells suspended in saline to be neutral).
In view of the above paragraph, the discrepancy betweenthe predicted
curves and the actual curves for blood in Figures (4-22) and (4-23), es-1
pecially at shear rates above about 30 sec , arises because the protein
fibrinogen increases the interparticle forces between red cells so much
that the highest shear rates used in this work are insufficient for making
the effects of these forces negligible with respect to the effects of hydodynamic forces.
This influence of fibrinogen is not measured by the
viscosity of plasma since the other proteins contribute a large amount
to the viscosity of plasma.
Consequently, as shown in Appendix B, no
satisfactory theoretical relationships exists to correlate the rheological
properties of blood at the highest shear rates used in this study.
For
interpolation purposes, any of the methods tested in Appendix B can be
used to "correct" data for differences in hematocrit.
4.
The yield stress
(a) Method of determination
An experimental method of determining the yield stress of
a fluid has already been discussed in subsection A-2-b of section IV
(Time Effect Upon Stopping Viscometer Bob Rotation).
This procedure
is time consuming and an alternate method usually was employed to determine the yield stress.
112
The alternate method makes use of the fact that the initial rate of
torque decay upon stopping bob rotation is so high that it appears to be
linear on the torque
-
time curve traced out on the viscometer recorder.
If the viscometer contains a fluid which does not have a yield stress,
the torque - time curve appears linear until the torque drops down to
about 6 - 8 percent of the full scale torque value.
If the fluid in the
viscometer has a yield stress and the torque corresponding to the yield
stress is several times greater than 6 - 8 percent of the full scale torque
value, the point where the torque - time curve departs from its initial
linear section can be easily determined within about 1o of the full scale
torque value.
This point of departure is the same as the departure
point as determined by the lengthy procedure discussed in the earlier
subsection mentioned in the previous paragraph.
The yield stress was determined by the alternate method by stopping the bob rotation when the peak of the torque - time curve was reached.
Because the time required to reach the torque - time curve peak was up
to several minutes at bob rotational speeds of 0.01, 0.02, and 0.05 rpm,
the yield stresses were not determined when the bob rotation was stopped
from these initial speeds.
This was because such a determination would
result in a low yield stress because of the plasma layer which formed at
one or both viscometer walls during the several minutes required to
reach the peak of the torque - time curve.
On the other hand, initial bob
speeds of 1 rpm or greater could not be used because the full scale torque
value at these speeds was so large that the torque corresponding to the
yield stress was less than 10% of the full scale torque value, thereby
not permitting the yield stress to be determined.
Consequently, the
yield stresses were determined with initial bob rotational speeds of
0.05, 0.2, and 0.1 rpm only.
113
Table (4-4) gives representative experimental data on the yield
stresses of blood samples, as determined by the alternate method just
discussed.
If there is any variation in yield stress with the initial bob
rotational speed it is lost in the error of the method used to determine
the yield stress.
One might expect a variation of the yield stress with
bob initial speed, since when the bob is rotating (at low speeds) the
length of the rouleaux in the blood varies inversely with the bob speed
(shear rate).
Assuming that the three-dimensional red cell network in
blood, which gives rise to the yield stress, is composed of the rouleaux
which were present when the blood was being sheared, networks composed of longer rouleaux would be expected at the lower initial bob rotational speeds.
Possibly, this variation in red cell network should give
rise to a variation in yield stress, but this does not seem to be the case,
or else the variation is very small.
It may be that the variation in
rouleaux length with shear rate is not significant in the shear rate range
considered here.
Alternately, the seemingly constant yield stress may
arise because of variations in the viscometer wall plasma layer thickness which cancel the effect of variations in rouleaux length.' Even
though the time available for this wall layer to develop is small, during
the yield stress determination, it may nevertheless be of some small
importance.
When the wall layer is purposely allowed to become large
before making the yield stress determination, a much lower yield stress
is measured.
From this discussion, it is apparent that a "dynamic" yield stress
is being measured and if the network were formed of randomly oriented
rouleaux of randomly varyinglengths the yield stress. might be different.
114
TABLE (4-4)
IN! IAL
-
STRESSE5
YeL-a
(dyne/cn)
DEX TER BLOOD
M 8264 BLOOD
PEED
OF BLOOD
STRESSES
YiELD
EXPERIMEANTALL Y DETER MiNED
0.5
0.054
0.036
0.019
0.059
0.054
0.044
0.044
0.2
0.052
0.030
0.0/7
0.062
0.059
0.051
4.042
0.1I
0.048
0.031
0.019
0.059
0.057
0.053
0.042
HEMATOCRJT
46.0
S9.0
53.2
TEM PERAqTUtE
25.0
25.0
25.0
("c)
)_
_
_
_
_
_
_
_
44.6
44.6
9 .98
_
_
_
_
_
44.6
44.6
2.0
36.8
t.C
_
_
_
_
_
In all the graphs presented in this thesis, the experimentally determined yield stress is indicated on the ? = 0 axis by the same symbol as
used to designate the experimentally determined shear stresses at other
shear rates.
If the yield stress was not experimentally determined, the
extrapolated yield stress is not marked by a symbol in any graph.
(b)
Effect of hematocrit
As the hematocrit of a blood sample is increased, the
number concentration of rouleaux present at a given shear rate must also
increase since the number of red cells in a rouleaux is determined by the
shear stress.
Consequently, as the hematocrit is increased, the density
of linkages in the red cell network formed in blood upon stopping the
fluid motion must increase and the yield stress will also increase.
This
is what is found experimentally to happen.
As has already been shown in Figure (4-21), the yield stress - hematocrit relationship can be expressed by the relationship,
1/2 = k
2
(4-33)
+ k
where k 1 is about 0.1 to 0.3, k 2 about 1.0 to 1.2, and k3 is roughly equal
to (-k1 ).
This relationship developed from the Casson model.
An empirical equation, proposed by Norton (51) for clay suspensions,
and confirmed for kaolin suspension by Bolger (5):
(T )1/3 = a(c-c
c
(4-34)
has also been found to correlate the yield stress-hematocrit data for
blood of hematocrits below about 50 percent.
(4-25) for five different normal bloods.
This is shown in Figure
In Equation (4-34), the "a" is
a constant which does not vary markedly between bloods, as is shown
by the small variation in slope among the lines in Figure (4-25).
116
The
FIGURE
4-25
Cube root of yield stress versus
hematocrit, for 5 different normal bloods
-
0.5
cm 0.4
3
cu
S2
-
0.3
0.2 -,-
0.1
00
0
20
40
0,
117
(%)
60
quantity cc is the red cell concentration below which the red cell - plasma
suspension cannot have a yield stress, since there are not enough parti-
to give stationary blood a yield stress.
The possibility that red cell
-
cles present with which to build up the three-dimensional networkneeded
plasma suspensions have no yield stress at red cell concentrations below cc has been experimentally confirmed.
In Figure (4-25), there is a six-fold variation in cc.
By the discus-
sion of the above paragraph, this variation in cc means that in some red
cell - plasma suspensions (those with low cc values) more effective use
is made of the cells to make a network.
This more efficient use is that
the cells are organized into longer rouleaux; thus in'two suspensions of
the same concentration one has in one a network composed of a number
of long rouleaux while the other suspension consists of a larger number
of smaller rouleaux, which cannot be randomly spatially arranged to
This variation in rouleaux
give a continuous three-dimensional network.
length arises because of a variation in the cohesive force between red
cells; this variation in cohesive force arises because of variations in
the fibrinogen content of the plasma in which the cells are suspended
(to be discussed in a later section).
The fact that a suspension follows Equation (4-34) is not a sufficient
condition that it will also follow the behavior of the Casson model; this
was demonstrated by Bolger's data on kaolin suspensions.
(c)
Effect of temperature
In Figure (4-26), the effect of temperature onthe rheologic-
al properties of blood is shown.
The intercept on the ordinate of this
figure is the square root of the yield stress.
As indicated, the yield
stress appears to be independent of temperature for red cell - plasma
suspensions with hematocrits below about 40%.
118
With suspensions of
higher hematocrits, there is a small inverse variation in yield stress
with temperature, as shown in Figure (4-26) and also in Table (4-4).
This temperature independence, or near independence, of the yield
stress is an indication that the energy of interaction holding the red cell
network together (in stationary blood) is much greater than the thermal
energy of the blood constituents.
The fact that the yield stress becomes
slightly temperature dependent and also deviates from Equation (4-34),
as shown in Figure (4-25), at a hematocrit above 40% is indicative that
the nature of the red cell network in this particular stationary blood
changes as the hematocrit increases over 40%.
One of the features of the Casson equations, which are satisfied by
normal blood, is that the rouleaux length depends only on the shear rate
and is independent of particle concentration.
At a given shear rate, as
the particle concentration increases, the number density of the rouleaux
increases, and this rouleaux concentration will be directly proportional
to the particle concentration.
However, as the particle concentration
becomes very high, the particles become so packed together that the
nature of the red cell network formed from the rouleaux changes, and
the rheological properties of the individual cells become more important.
The fact that the yield stress is temperature independent at hematocrits below 40% not only implies that Brownian agitation is of negligible importance, but also means that the product of the network link
density and the average link strength remains constant.
This link
strength does not necessarily have to be the red cell cohesive force holding two red cells together in the rouleaux, since the cohesive force
holding two cells together may be a function of what parts of the red cells
are in contact (two cells joined at their flat faces may have a much higher
cohesive force than two cells joined at their rims).
119
FIGURE
4-26
b
Casson plots for a typical normal human blood,
at three temperatures and four hematocrit levels
b
cC05
"00--
a
C"
0
0
02
0)
['3
tC
--c
c
=
=
35.5
49$--
49.8
e
--
=
/-
Dexter blood
--
--
1
2
iv
(sejl)l/2
=
4, Figure 4-25)
21.0 0 C
.a
=
37 . Q
(curve
c == 30.. 0
0
,
0$
CO
3
In order that the linkage density - link strength product remain constant, any change in the number of linkages would have to be inthe opposite
direction of the change in the average linkage strength (fractional changes
would be equal but opposite).
This is not unreasonable since a decrease in
the number of linkages could come about bya decrease in the number of rouleaux, which would mean an increase in the rouleaux length and the cohesive
force holding the cells together in a rouleaux.
This increase in cohesive
force holding the cells in a rouleaux together would most likely be accompaniedby anincrease in link strength since the source of a change in either
bond would also be the same source of a change in the other (changes in
adsorbed protein concentration, red cellmembrane "flicker" etc.). Thus,
the terperature independence of the yield stress means either the average
link strength and link density are independent of temperature, or that they
vary in opposite directions.
Since for suspensions of hematocrit above about 40% the yield stress
is slightly temperature dependent, some, property of the individual red
cell must be temperature dependent.
There is no reason to suspect
that such a property, be it the severity of membrane flicker or the nature
of the adsorbed protein layer on the red cell surface, should become less
temperature dependent at lower hematocrits (at lower hematocrits the environment of the red cells contains a larger amount of plasma constituents per red cell, and so the likelihood of exhausting a plasma constituent
by adsorption or metabolism is less).
Changes in red cell geometry with
temperature, if any, should be unaffected by red cell concentration in
the range considered here.
Consequently, it must be argued that the
link density - link strength product at hematocrits below 40% remains
constant but that the individual members of the product vary with temperature.
121
It is of course possible that at hematocrits above 40% the part of the
individual red cell which governs the red cell network strength is different from the part which is important in suspensions of hematocrits below 40%.
Such a change in network character could account for the tem-
perature behavior being discussed, but it is difficult to visualize such a
mechanism change.
The proposition that the link density - link strength product remains
independent of temperature while the members of the product vary with
temperature does not conflict with the Casson model suspension.
(d)
Variation with source
All of the bloods whose yield value - hematocrit data are
shown in Figure (4-25) came from persons in good health.
From this
figure it is apparent that the yield stress can vary over a large range,
even at the same hematocrit level; the range in Figure (4-25) is from
0.015 to 0.050 dyne/cm2 at a hematocrit of 45%.
be due to variations in red cells or plasma.
This variation could
The effect of the plasma,
especially the fibrinogen content, seems the most important factor and
is discussed in the section dealing with the effect of fibrinogen on the
rheological properties of blood.
5.
Effects of physical factors on blood rheological properties.
(a)
Hematocrit
The effect of red cell concentration on the rheological
properties of blood has been discussed in Section A-3, Theoretical
Correlation of Shear Stress
-
Shear Rate Data, of the Discussion of
Results.
122
(b)
Temperature
In as much as the viscosity of water, the solvent in plasma,
decreases as the temperature is increased, one expects the viscosity of
blood to likewise decrease as the temperature is increased.
In Table
(4-5), the effect of temperature on the rheological properties of a sample
of normal human blood is shown; the blood viscosity at a given shear rate
does decrease with increase in temperature.
Because the hematocrit
of this sample is above 40%, the yield stress is also found to decrease
as the temperature increases.
Figure (4-27) is an Arrhenius plot of the logarithm of the blood viscosity versus the reciprocal of the absolute temperature.
Also shown on
this graph is the viscosity of water multiplied by ten, so that the effect
of temperature on blood at high shear rates can be compared to the effect
of temperature on the viscosity of water.
The similarity in the curve
for water and blood at the highest shear rates indicates that at the highest shear rates, the effect of temperature on the viscosity of blood is
governed predominantly by the effect of temperature on the viscosity of
the suspending medium, plasma.
This agrees with the postulation that
at the higher shear rates used in this work the interparticle forces are
becoming negligible compared to the hydrodynamic forces.
As blood is subjected to lower shear rates, the interparticle forces
become more important and the temperature effect on blood changes, as
shown in Figure (4-27).
Since the yield stress of blood is only slightly
temperature dependent at a hematocrit of 44.6%, the cu-rves in Figure
(4-27) tend to become less steep as the shear rate decreases.
In this
figure, the limiting slope for this blood is shown, based on the yield
value - temperature data, and a limiting horizontal straight line which
123
TABLE (4-5)
REOLOG/CAL
TAE
OF rEMPERATURES ON
EFFECT
PROPERTIES OF BLOOD
7EMPERATURE5
/5.0 *C
1O.0*C
S/EAR.
SHEAR
RATE
STRESS-
Sec-i
/02.6
dyne/cnl
/2.6
3 Z 0 'C
25 0G
-ea .0 *C
VIScosiry
cp
/2.2
/0.7
/0.5
9.31
9.08
8.25
8.05
4.28
51.3
7.25
14.1
6.04
//.8
5.38
10.5
4.80
9.35 3.56
20.5
10.3
3.58
2.16
/7.5
21.1
3.10
1.86
/5./
/3.1
/8./
2.69
1.62
2.40
1.46
//.7
/4.2
5.13
2.05
1.34
0.730
26.1
35.6
1.14
0.6/ 7
22.3
30.1
/.0/
0.553
/9.6
1.03
0.513
0.472
46.1
0.413
40.3
0.369 36.0
0.321
62.5
0.277
53.9
0.258
/0/
150
0.180
0.131
87.7
0.147
0.205 0.206
0.103
0.153
YIELDSTS 0.0609
130
0.0562
SAMPLE : DEXTER
15.9
27.1
6.95-
1.88
1.13
9.19
//.0
0.908 17.7
0.502 24.5
0.703
0.393
/3.7
/9.2
33.0
0.268
26.1
-
0.189
36.9
-
0.128
62.3
-
0.0988 96.3
-
0.0443
0.338
50.3
81.4
0.131 /28
0.0526
6./2
-
-
HEMA70CR1T = 44.67%
Limiting slope curve, constant
r
Limiting slope curvebased on
.
200
.
.
-e
o-o.-103
100
sec
0.25
50
a053
s~c
0'
0
p4
C)
3
Cd
,.3
.0 sec
4)
*14
U,
C
U
C,
*rI
10
Pure water, x 10
5
FIGURE
4-2y
Viscosity (log scale) versus reciprocal absolute
temperature determined from data of Table 4,5
at constant hematocrit of 44.6 (AOD as anticoagulant)
32
I
I
I
33
34
35
x
104
125
-C)
would be obtained if the blood hematocrit were less than about 40%, so
that the yield stress were independent of temperature.
Figure (4-28) is the same data plotted as the blood relative viscosity
(relative to water) versus temperature.
It clearly shows the change in
the viscosity - temperature function of blood as the shear rate is lowered.
Over the temperature range (10*C to 37*C) considered thus far, the
viscosity changes withtemperature are reversible.
of reversibility are not indefinite.
However, the limits
In Table (4-6), are given data for a
sample of blood at different temperatures; the temperature sequence was
from 25* C up to 48* C and then down to 250C again.
Some of these data
are shown in Figure (4-29), from whichfigure it is quite apparent that
some irreversible change has occurred in the blood during the time that
it was at the higher temperatures.
The cause of this change may be two
(1) the destruction of some red cells at the higher temperatures
fold:
would result in the spilling of hemoglobin into the plasma (verified by a
slight darkening of the plasma color during the experiment), and (2) the
denaturation of proteins in the plasma.
Kreuzer (41) reports that the
denaturation of proteins in blood serum was noted by several investigators to occur if the serum was heated to 52 - 57*C.
This temperature
range is higher than the temperatures used in Table (4-6), and Figure
(4-29), but the protein fibrinogen, not present in serum, may be more
temperature sensitive than the other plasma proteins.
Two features regarding the experiment just described should be
noted.
First, the torque decay rate observed when the viscometer bob
was rotating at a constant speed was high at 25*C, became normal at
36*C, and was non-existent at all times afterwards, even when the blood
was returned to 25*C.
Secondly, the yield stress, as recorded in
126
I
I
200 1-
-1
100 t-
0.205 see
-
;k
0
4>)
0
F4
0.513 see
50
I-
10
W
*0
-1
1.03 sec 1
---
-1
0
0
a
U
2.05
see
5.13
sec-
10.3
sec
W
a
F,'
4-)
-1
%
CE
0
I-
-
10
. I
-1
51.3 sec
51-3 see
103
FIOURE
5
sec
i-28
Relative viscosity (log scale) versus temperature,
computed from Figure 4-27
I
10
I
20
30
Temperature,
127
(0 C)
40
TABLE 4-6
EFFECT of
#icu
TEMPERA7URES
ON THE
SPEAR
S//EAR RA T
.sec
-)
103
51.3
1EO1OG/CAL PROPERTIES OF B1000
SrRESSES
25.0 C
36. 0 *C
4. 62
3.49
-2.57
/.91
/.0
2.26
#2.0,0 C
3.27
(dyne/cm)
48.00 c
42.C0
4.10
C
25.0*c
7.04
-
20.5
/.25
0.935
0.876
/.08
10.3
0.740
0.550
0.5/
0.6/5
0.740
5.13
0.435
0.322
0.300
0.34f
0.-23
2.05
0.226
0.168
0.157
O.175
1.03
0.145
0.115
0./1
0-.114
0.136
0.184-
0.513
0.0945
0.088
0.0814
0.0775
0.0866
0.123
0.205
0.064-5
0.0575
0.103
0.0485
0.0390
0.0538
0.04/5
0.0505
0.0389
0.0527
0.028
0.0546
0.0/5
0.013
0.012
YiE0 ST/?ESS
51.000 SAMPLE
A'CMATOCRIT:
A4722/
38.7
%
(ORIGINAL.)
3S.5% (FINAL)
/.07
_-.-
0.305
-
-
0.013
Mo te:
OSrERM/VArIONS
OP 044. ABVE,
CotUMN TO RIer
WERE MACS IN OROER
G.OING
PROM L Fr
1.0
10.3 seC
1
0.7
0.5-
0.2CU
1.03 seC 1
%0.1
---
0.07
0-0 --
- .103 see ..
0.00
FIGURE
%
-
temperatures on the rheological properties of blood
Sample
M 7221
Hematocrit = 38.7
Arrows indicate sequence of experiments-
,Effect of high
4-29
0.02 -
31-.0
32.0
(10
33.0
(0K)G/T),
129
34.o
Table (4-6), was virtually independent of temperature and was not affected by the blood having been subjected to a high temperature.
Both
of these facts indicate that the high temperature resulted in primarily
irreversible changes in the plasma, and not in the red cells.
In an Qffort to more clearly define the high temperature effect on
blood, another sample of blood was, subjected to a temperature of 39.2*C
and then returned to a reference temperature of 25.2*C and finally heated
to 41.20 C and then returned to the reference temperature again.
The
viscometriq measurements made at each temperature are recorded in
Table (4-7).
It appears from this table that some irreversible change
occurs at 39.2* C, but the greater change occurs at 41.2* C.
Inmaking
these measurements, the blood sample properties were determined after
the sample had been at a given temperature about 15 minutes, and the
properties did not seem to be a function of time.
TABLE 4-7
Effect of high temperature on rheological properties of blood
Shear Stresses (dyne/cm 2
25.2"C
37.2 0 C
39.2 0 C
25.2*C
41.2*C
25.2 0 C
0.775
0.555
0.550
-----
0.565
0.901
5.13
0.461
0.330
0.322
-----
0.334
0.539
2.05
0.237
0.170
0.170
0.252
0.172
0.288
1.03
0.164
0.121
0.119
0.122
0.194
0.513
0.113
0.0805
0.0805
0.0861
0.135
0.205
0.0746
0.0542
0.0539
0.0566
0.0898
0.103
0.0588
0.0421
0.0410
0.0436
0.0670
(sec
)
Shear
Rate
10.3
Sample M7,221
Hematocrit:
Note:
39. 6% (original)
39. 2% (final)
130
----
Measurements made in order
of columns from left to right
A lower temperature below which an irreversible change in normal
blood's rheological properties occurs must be below 4*C, if it exists at
all.
Normal blood cooled to 4 C and stored for several days at 4 0 C,
when reheated to 25.0*C showed no change in rheological properties
from the original sample at that temperature.
However, such can not
be said for some abnormal bloods, the so-called cold-agglutinating
bloods.
Table (4-8) is an illustration of such a blood, with the rheo-
logical -properties determined at various temperatures.
From thischart
it is seen that an irreversible change occurs when the blood is cooled
down to 31"C.
TABLE 4-8
Effect of temperature on a cold-agglutinating blood
Shear Stresses (dyne/cm 2
37.0C
35.50C
37.00C
31.00C
37.00C
1.12
0.665
1.14
0.678
1.13
0.668
1.32
0.802
1.17
0.696
4.10
2.05
0.363
0.235
0.369
-----
0.363
0.240
0.403
0.295
0.257
1.03
0.410
0.176
0.180
0.176
0.213
-----
0.131
-----
-----
0.156
0.142
0.205
0.103
0.105
0.0778
-----
0.103
0.0778
0.112
0.0884
0.110
0.0899
Rate
(sec
)
Shear
20.5
10. 3
Sample:
Hammerley
Note:
Sequence of measurements is
from left column to right
Hematocrit = 38.8%
So far this discussion on the effect of temperature on the rheological
properties of blood has been concerned with a sample of constant hematocrit.
The effect of temperature at different hematocrit levels in the
131
same blood is illustrated in Figure (4-26). Particularly interesting is
the low shear rate region-where the data fulfill the Casson model equations.
In Figure (4-30) are shown data illustrating how the slope of the
-Casson plot, s, varies with temperature and hematocrit; generally, the
slope decreases with increase in temperature (at a constant hematocrit),
and the change in s with temperature becomes less as the hematocrit is
lowered.
Consequently, a plot of ln s versus ln(1-c)would show lines of
decreasing negative slope (and decreasing intercept) as the temperature
increased (a plot similar to Figure (4-20)); this means that the Casson
constant a decreases with temperature increases.
Also, a plot of
(1/(1-c)) (aa -1)/2 versus b (similarto Figure (4-21)) would show lines
of increasing slope as temperature was increased; therefore, the Casson
constant P increases as the temperature increases.
Both of these varia-
tions inthe Casson constants are in agreement with the effect of temperature on the rheologicalproperties of blood as set forth in the section
dealing withthe effect of temperature on the yield stress.
The increase
in P with increasing temperature means that the rouleaux length increases'
with temperature; this is because inthe shear rate range where the Casson
model seems applicable, the rouleaux length calculated from the Casson
equations depends predominantly on the term containing P and not the
term containing a.
Since p is also a direct function of the cohesive
force holding the red cells inthe rouleaux together, the increase in p with
temperature means the cohesive force increases with temperature also;
this corresponds to an increase in linkage strength as temperature increases.
This could result in a constant link density - link strength pro-
duct for the three-dimensional network which gives stationary blood its
yield stress.
132
I
i
FIGURE
4-30
Effect of temperature and hematocrit
-
0.40
on the slope of the Casson plot for a blood
Sample:
Dexter
CU
0
0)
0.30
0
CH
0
%
%--= 35-5
0.20
%
c = 30.0
O
o.14
20
30
Temperature, (0c)
133
G
c = 20.3
4o
%.
(c)
Sample age
Blood samples which have had the standard ACD solution
added to them can be stored at 4*C.
This method of storing blood is
standard blood bank practice, although blood so stored more than three
weeks is no longer considered satisfactory for transfusions since if it
is used in a transfusion about 30o of the red cells will be immediately
destroyed.
There is a gradual maturing of the red cells, even at 4 0 C,
which accounts for more cells being destroyed upon transfusion as the
length of storage of the blood increases.
It was the usual practice in this thesis work to store blood samples
at 4*C, and to use such blood as long as its storage time did not exceed
8 days.
To see how the length of blood storage affected the rheological
properties of the blood, the data shown in Table (4-9) were obtained.
Variations in hematocrit slightly complicate the results, but it is apparent that if there is any change in the rheological properties with length
of storage during the first 10 days, it is a slight rise in shear stress at
a given shear rate.
After about 12 days storage, the change in rheologi-
cal properties is unmistakable.
This gradual change could come about
because of changes in the blood constituents, or because the red cells.
sedimented out during storage and remained agglutinated to some extent
even during the viscoroetric testing.
This latter argument does not seem
valid: a blood sample routinely agitated and tested in the viscometer
did not show signs of unusual aggregation when looked at under the microscope.
In an effort to see if the age of the red cell had an influence on the
rheological properties of blood, the red cells in a blood sample were
fractionated, according to age, in a centrifuge.
134
As red cells age, they
TABLE (4-9)
EFFECT
4 0 C OA
P4
(sECt)
____
OF STORING BLOOD CONTA/NING A CD AT
RHEOL06/CAL PROPERTIES OF BLOOD
T~HE
SMEAR
_
STRESSES,
Cdyne
2/ct)
/0/17/62 /0/18/62 /0/19/62 /0/22/62 /0/23/42 /0/24/62 10/25/42 /0/27/62 /0/2962 /0/1J/62 1/1/6Z
zo.5
0.914
0.734
0.697
0.963
0.973
0.973
/.07
1.02
1.06
/.0S
1.09
10.3
0.528
0.391
0.4/3
0.564
0.58/
0.573
0.603
6.6~05-
O.6WS
6.608
0.629
0.268
0.167
0.206
0.135
0.187
0.1/5
0.312
0.294
0.194
0.3/6
0.20/
0.316
0.187
0.31/
0.104
0.308
0.198
0.30g
0.194-
0.330
.205
0.1/4
0.0808
0.0652 0.133
0.123
0.1-32
o.129
0.129
0.128
0,12-8
0.142
0.0679
0.0552
00369 0.081/
0.0804 0.0845
.08/0
0.0777
0,08A
0.0847
0.0508
0.0457
0.0313
0.0591
0.0570
0.0583
-
ao4i4
0.0322
0.OiO6
0.0469
38.8
32.4
36.9
0.194
39.3
0.0595
39.6
1.0846
0.0632
0.0608
o.0491
0.0495
39.3
.4
.0-45C 0.473
-3.O
/017/62
BLOOD DRA WN:
SAMPLE K-2 /96
TEMPERATURE = 370* C
39.0
--
0.0625
0528
B.
38.4
change their physical properties sufficiently to permit them to change
their sedimentation rate.
Borun, Figueroa, and Perry. (6) and Prankerd
(55) found, by tagging cells with radioactive tracers, that if a blood
sample is centrifuged, the age of the cells in the sedimented bed increases
as one goes from the top to the bottom of the red cellbed.
An attempt
was made to use this behavior to prepare red cell fractions, by centrifuging at 3000 g, a normal blood sample.
-
shown below:
The fractionation scheme is
ii
RED CELL/
FROM
ORIGINAL
BLoO D
--
5 AMPLE
B,
where
indicates agitation and suspension. of the red cells in an equal
volume of albuminated saline, and centrifuging.
fractions then ranged from T
(the oldest).
The average age of the
(the youngest) through T 2 , B1 , and B2
Finally, each fraction was washed once and then suspended
in plasma at a hematocrit of about 391/ and the rheological properties were
determined at 25.00C.
Table (4-10) presents this data, together with the
like data for the original blood sample.
No really significant difference
is found between the properties of the various age fractions.
136
TABLE (4-10)
Rheological properties of suspensions of different ages red cells in
plasma
Shear
Rate
(sec
2
Shear Stresses, (dyne/cm)
)
B1
T2
T
Original
B2
(25.00 C)
(25.0*C)
(25.00C)
(25.0 0
102,6
4.41
4.36
4.54
4.61
4.18
51.3
2.37
2.33
2.47
2.53
2.16
20.5
1.17
1.10
1.19
1.23
1.12
10.3
0.688
0.654
0.725
0.725
0.650
5.13
0.402
0.382
0.418
0.435
0.380
2.05
0.205
0.191
0.212
0.222
0.187
1.03
0.121
0.116
0.144
0.146
0.119
0.513
0.0915
0.0792
0.0855
0.101
0.0800
0.205
0.0593
0.0499
0.0610
0.0636
0.0489
0.103
0.0414
0.0355
0.0359
0.0480
0.0368
0.011
0.0095
0.012
0.012
0.012
39.3
38.1
39.7
39.9
39.3
C)
(25.2 C)
Yield
Stress
(dyne/cm
Hemato-
crit (%)
Sample M 7221
All temperatures = 25.0*C (except original sample)
This lack of difference in rheological properties of the suspensions
of red cells of various average ages, as prepared in the experiment
just described, may not be conclusive evidence that the change in red
cell properties with age do not affect the rheological properties of red
cell - plasma suspensions.
The fractionation procedure of Prankerd
(55) was followed exactly with only one exception: human albumin instead
of bovine albumin was used in the albumine - saline suspension solution.
137
The use of this solution may have removed the outer layer of the red
On the other hand, using plasma as a suspending medi-
cell membrane.
um during fractionation probably would not be effective since the fibrinogen present in the plasma would probably cause such large intercellular
attractive forces that the red cells could not sedimentate as individuals.
At this time it would appear most likely that the rheological properties of blood change with time because of changes in the blood plasma
(increase in hemoglobin content, etc.).
(d)
Centrifugation
In the course of preparing red cell
-
plasma suspensions,
it is usual to centrifuge the blood so that the red cells and plasma can
The question arises as to the effect of this centrifuging
be separated.
on the properties of blood.
To see what effect centrifuging had on the rheological properties of
blood, a sample of blood was subjected to viscometric measurements at
25.0*C, centrifuged at 4*C and 5000 g for 20 minutes, rewarmed to 25.0*C.
and agitated, and its rheological properties again determined,
(4-11) shows the results of this experiment.
Table
The effect of centrifuging
under these conditions, which are more drastic than those used in preparing red cell
6.
-
plasma suspensions, is negligible.
Effects of chemical factors on blood rheological properties
(a)
Anticoagulants
Because blood will clot soon after withdrawal from the
body, it is convenient to add an anticoagulant to the blood to prevent clotting.
However, the question arises whether or not the addition of an anticoagulant changes the properties of native blood.
138
TABLE 4-11
The effect of centrifuging on the rheological properties of blood
(25.0*C)
)
Shear Stresses (dyne/cm
Shear Rate
(sec
)
Sample After
Centrifuging
Original Sample
5.64
5.60
51.3
3.20
3.13
20.5
1.65
1.64
10.3
0.994
0.980
5.13
0.601
0.588
2.05
0.322
0.316
1.03
0.212
0.205
0.513
0.160
0.159
0.205
0.112
0.107
0.103
0.0814
0.0814
103
Hematocrit
42. 4%
4'2. 7%
Centrifuging at 5000 g (4*C) for 20 minutes.
To determine the answer to this question, a sample of blood was
drawn from a healthy donor with a siliconized syringe and needle.
A
portion of the native blood was placed in the viscometer and its rheological properties were determined at 19* C.
To 8. 5 ml of the native blood,
20 mg of heparin in 0.2 ml of isotonic saline was added.
To another 8.5
ml portion of the native blood trisodium citrate (45 mg Na 3 C 6 H 5 0 7 - 2H 2 0)
was added, and to a 10 ml sample of native blood ACD (33 mg trisodium
citrate, 12 mg citric acid, 37 mg dextrose) was added.
These anticoagu-
lated blood samples were examined at 19*C in the GDM viscometer and
the results of the investigation are tabulated in Table (4-12) and shown
graphically in a Casson plot in Figure (4-31).
139
FI6UBE
4-31
Comparison of anticoagulated blood
samples with native blood
2.0
cu
cu
El Native blood
=
40.2
%
Heparinized blood, c
0
1.0-
=
36.5
%
Citrated blood, c
Temperature = 19.0 0C
0
0
1
2
34
5
/2,(see-1/2)
6
7
8
9
TABLE 4-12
The effect of anticoagulents on the rheological properties of blood
Shear Stresses, r, dynes/cm 2
Shear
Rate
? _
(sec )
Native
Blood
74.6
5.98
6.39
6.38
5.92
37.3
3. 27
3.48
3.29
3.21
14.9
1.56
1.64
1.51
1.54
7.46
0.920
0.960
0.823
0.887
3.73
0.573
0.595
0.473
0.514
1.49
0.326
0.345
0.248
0.262
0.746
0.250
0.156
0.191
0. 373
0.172
0.108
0.147
0.149
0.123
0.0654
0.108
-----
0.0917
0.0536
-----
40.2%
0.0746
Hematocrit
Heparinized
Blood
Trisodium
Citrated
Blood
..36.5%
ACD
Blood
38. 3%
All temperatures = 19.0*C
Because the native blood sample clotted in the viscometer, its hematocrit
could not be determined.
of blood are reproducible.
Prior to clotting, the rheological properties
The ACD blood sample- data are not shown in
Figure (4-31), to avoid confusion of the graph, but Table (4-12) shows
that it falls slightly below the native blood data at high shear rates and
between the heparinized and citrated samples at low shear rates.
Because the native blood hematocrit is unknown, and because of the
differences in hematocrit between the other samples, only the following
conclusions can be drawn from this data:
(1) the addition of heparin or
ACD to native blood does not change the rheological properties of the
141
blood very much, if at all, (2) addition of tiisodium citrate alone seems
to increase the apparent viscosity of blood at the higher shear rates,
and (3) most important of all, native blood, in the region studied, behaves rheologically just as ACD blood does (i. e., obeys the Casson
relationshIp).
It also follows, not only because native flood fits the
Casson plot, but more importantly, because of the data for heparinized
and ACD blood samples, that plasma calcium ions are not necessary
for the formation of rouleaux.
This conclusion is based on the fact
that citrate ions act as a blood anticoagulant because they remove free
calcium ions from the blood by forming weak complexes with them,
while heparin does not affect the calcium ions at all.
There is a decided advantage to using ACD as the anticoagulant for
blood.
Not only does the ACD permit the storage of blood at 4*C, but
heparin, by not removing free calcium ions from the plasma, does not
prevent the platelets in the blood from forming small pin-head size clots,
which make any rheological experiment impossible.
Since calcium ion
is necessary for the formation of platelet clots, citrate ions, by removal
of free calcium ions, prevent this complication from occurring.
(b)
Plasma protein content
The influence of the plasma proteins on the rheological
properties of blood, especially in the very low shear rate region, is
of interest.
In view of the complexity of the plasma protein content,
this influence is probably not simple.
Ideally, one would like to obtain
blood samples from a single source in which the content of just one protein, or protein fraction, varied at a time.
This is not possible, so
one must devise means of adding or subtracting one substance at a time
from or to the blood without changing the blood in any other way.
142
Because of its ability to strongly influence rouleaux formation, the
effect of fibrinogen is especially of interest.
Fortunately, the fibrinogen
content of plasma can be varied, without changing the other protein contents, by mixing various fractions of anticoagulated plasma and serum
formed from the anticoagulated plasma by the addition of thrombin.
Be-
cause residual thrombin in the serum would cause the plasma fibrinogen
to clot when the serum and plasma portions were mixed, it is necessary
to keep the serum at 37*C for 8 hours or more before mixing with the
plasma so that the excess thrombin is destroyed.
The serum and plasma
can then be safely mixed and the mixture will be identical to the plasma
except in fibrinogen content.
Inthe manner just described, red cells suspended in plasma, serum,
and a mixture of equal parts of serum and plasma were investigated.
The red cells'were washed in the suspension medium three times before
finally suspending them in a fresh portion of the medium.
The protein
analyses of the media, before and after the red cells were added, were as
follows
Protein Content
Albumin
Plasma, new
Serum, new
Cells removed
Mixture; new
Cells removed
-
Globulin
1.8
2. 0
2. 0
2.3
1.8
4.0
3. 1
3. 3
3.4
3.8
Percent of Medium
Fibrinogen
0.25
0
0
0.11
0.13
The analysis methods were all direct colorimetric methods, and each
number is the average of two determinations; the greatest difference between two determinations was 0. 3 units and was usually 0. 1 units.
143
The rheological data for the red cells (from the same original blood
sample as the plasma and the serum) suspended in the three media, at
37.0 0 C, are shown in a Casson plot as Figure (4-32).
The yield value s of the
two suspensions which contained the plasma and the serum
plasma mixture
were determined by the torque decay method; no yield stress could be determined by this method for the red cell - serum suspension.
while a torque
-
In addition,
time effect (at a constant viscometer bob speed) was noted
for the red cell - plasma and red cell - serum plasma mixture suspensions,
none was seen forthe red cell - serum suspension.
In view of the protein
analyses, it can be concluded thatthe yield stress is dependent on the fibrinogen content, and does not depend on the concentration of the other proteins
since the red cell - serum mixture has no yield stress.
These comments
apply only to normal blood however, and abnormally high or low concentrations of the other proteins might result in a yield stress dependent on the concentration of a different protein.
These conclusions have been verified by
similar experiments on samples from several blood donors.
The hematocrit of the red cell- plasma suspension is slightly lower
than that of the red cell - serum plasma mixture suspension.
Consequent-
ly, the yield stress of the red cell - plasma suspension at the same hematocrit as the mixture suspension would be slightly higher (4%).
The yield
stresses of these two suspensions are 0.028 dyne/cm 2 and 0.0077 dyne/cm 2
for the red cell - plasma and red cell - mixture suspension respectively;
this is a ratio .of 3.6 while the fibrinogen concentration ratio is 1.9.
Ob-
viously, the yield stress - fibrinogen concentration relationship is not
linear.
Additional detailed work must be done, of the above type of ex-
periment, to define this relationship, which may lead to an understanding of how the fibrinogen causes rouleaux formation, which results in
blood yield stresses.
144
1.0
FIUR -32
Eheolkgical properties of red cells suspended
in plasma, serum, and a plasma - serum mixture
0.8
50 % plasma - 50
c = 43.0 % , 0.13
3 100 %serum,
-
no fibrinogen
% serum
% fibrinogen
e 43.1
%
o
0.6
-
100 %plasma, c = 42.5%
0.25 % fibrinogen
0.4Blood sample
K 5196
Temperature = 37.0'C
Plasma
Serum
7
0.2
0
3
12
/2, (se-1/2
The use of the additive process, that is changing the concentration
of one protein at a time, is of rather limited use, because of practical
difficulties.
Such a process requires a supply of proteins in the solid
form, which is not easy to obtain since proteins are unstable and hydrated.
Even if such a supply is available, getting the solid protein to dissolve
in plasma is a large problem because of slow solution, foam formation,
denaturation, etc.
One exception to these problems seems to be albumin,
which was obtained in solid form, recrystallized twice, from Nutritional
Biochemical Corporation.
It dissolved in plasma rapidly, at room tem-
perature, without foam formation.
Use of this method of varying the
albumin content of plasma and plasma - serum suspension media for red
cells will be valuable in a study such as proposed in the previous paragraph, especially since several investigators believe that the red cell is
partially covered with an outer layer of albumin (54).
content, of normal blood is large and the accumulation of yield stress
-
The variation of the blood yield stress, as a function of the protein
protein content data should give some insight into the red cell - protein
interactions.
The routine accumulation of such data has been started in
the Blood Rheology Laboratory of the Department of Chemical Engineering, M. I. T.
(c)
Plasma lipid content
A group of plasma constituents which might have an influence
on the rheological properties of blood is the lipids.
One possible way of investigating the effect of lipid content on the
flow properties of normal blood is to compare a blood of abnormally high
lipid content with a "normal" blood.
A blood 'sample of extremely high
lipid content was that illustrated earlier in Figures (4-6) through (4-8).
146
While the lipid content of that sample was not determined, the lipid content of a sample drawn from this donor the previous day was:
cholesterol
triglycerides
phospholipids
1,400 mg/100 ml serum
12,000
2,200
(the usual cholesterol level is around 200 mg/100 ml serum).
The flow
properties of this blood at 37.0*C are shown inthe Casson plot, Figure
(4-33). .. The extrapolated value of the square root of the yield stress is
0.313 (dyne/cm2)1/2, which is in good agreement with the value, obtained
by the torque decay method of 0.29, which is known tobe low because the
bob rotation was stopped after the torque - time curve peak (at constant
rotational speed) had beenpassed (this blood showed a large time effect).
The maximum value of the square root of the yield stress recorded for
normal bloods of the same hematocrit is 0.22 (dyne/cm2 1/2, as shown
in Figure (4-25).
However, this high yield stress for this high lipid blood
cannot be so simply attributed to the high lipid content because samples of
this blood obtained on two later occasions also show high fibrinogen and
globulin plasma protein contents:
8/16/62
10/26/62
fibrinogen
globulin
0. 51%
3. 30%
0.46%
3. 55%
albumin
3. 55%
3. 25%
Also shown in Figure (4-33) is data on another sample of blood drawn
from the same donor on October 26, 1962 - about 3 months after the
sample already described.
At this time, the lipid level in the blood had
decreased to the following levels:
cholesterol
triglycerides
895 mg/100 ml serum
1712
phospholipids
875
147
FIGURE
4-33
Theological data for a high
lipid content blood
1.0-
CU
CQ
0.5
WeA
sample 7/31/62, c
=
C
43 -7
%
Temperature = 37.0
0
Sample
0 Hr., c = 31.8
%, 8/16/62
13
Sample
3 Hr -, c = 30-7
%, 8/16/62
o
Sample 9 Hr., c
=
29.3
%, 8/16/62
%
Sample 10/26/62, c'= 41.7
I
0
1
I
I
2
3
(see
,1/2
)
0
4
5
(The plasma protein content is given above).
This represents a large de-
crease and, even considering the difference in hematocrit between this
sample and the previously described one, the change in rheological properties is larger than expected.
For example, assuming that Equation
(4-34)
1/3 = a(c-c C)
is also valid for the two high lipid samples being discussed, one would
predict, upon the basis of the hematocrit and yield stress for the first
sample, a yield stress for the second sample of 0.084 dyne/cm 2, where2
as it was found to be 0.063 dyne/cm2. This difference could be attributed
to the drop in lipid content; however, not knowing the protein analysis of
the first sample, this comparison is not conclusive evidence that the lipid
content of these blood samples had an effect on the rheological properties
of the blood.
Figure (4-33) also shows the behavior of several samples of blood
obtained from the same donor several weeks after the first sample described above.
These samples were taken during a fat tolerance test,
performed at the Massachusetts General Hospital, and collected into ACD.
The first sample, designated "0 HR", was taken just before the person
broke .a fast; the sample designated "3 HR" was taken 3 hours after the
first, and the "9 HR" sample was taken 9 hours after the first. The lipid
contents of these samples, as determined by the Massachusetts General
Hospital, were as follows (in mg/100 ml serum):
cholesterol
0 HR
3 HR
9 HR
triglycerides
455
430
430
485
852
812
149
phospholipids
487
-----
The fact that the 3 hour sample and the zero hour sample have the same
yield stress, inspite of the lower hematocrit of the 3 hour sample, would
seem to indicate that the lipid content does effect the yield stress.
On
the other hand, use of Equation (4-34) and the data of the samples "3 HR"
and " 9 HR" do not predict a reasonable value for cc (14%).
It would ap-
pear that this data is inconclusive due to either experimental error, or
a variation in the samples which is not recorded here.
This variation
could be the introduction of hemoglobin into the plasma from red cells
which break when the blood sample being drawn from the donor is placed
in dry ACD, as it was done here.
The cellbreakage occurs because of a
high salt concentration in local areas of the blood sample before complete mixing is obtained.
The question of what effect the lipid content of blood has on its
rheological properties has not been answered.
It seems likely that at
least very high lipid concentrations may show an effect on the yield stress
of the blood.
(d)
Plasma hemoglobin content
As -indicated in the end of the previous section, it is possible
to cause red cells to rupture during the process of mixing the native
blood and anticoagulant.
Also, during the manipulation of blood samples,.
red cell damage may occur from mechanical causes.
is evidenced by a deepening in color of the plasma.
That such occurs
To determine if libera-
tion of hemoglobin could possibly affect the properties of blood, human
hemoglobin, obtain in solid form from Nutritional Biochemicals Corporation, was added to plasma in the amount of 0.0505 gm hemoglobin per gm
of plasma.
Red cells were suspended in this altered plasma and the pro-
perties of this suspension were compared with those of the originalsample.
150
The data for the suspensions and plasmas are shown in Figure (4-34).
There is a very large discrepancybetweenthe two suspensions which can
only be attributed to the presence of the added hemoglobin.
The viscosity
of the altered blood would be expected to be higher than that of the original
blood, because of the increased plasma viscosity due to the added hemoglobin.
However, this expected increase is shown in Figure (4- 34) by the
dashed curve, and it is apparent that the greater part of the increase in flow
properties is not due to the increased' plasma viscosity.
It appears that
the greater part of the increased blood viscosity is due to the hemoglobin
increasing the intercellular forces between the red cells.
The amount of hemoglobin added in this experiment is many times
the increase in hemoglobin which might be expected from red cell breakage
during -anticoagulant mixing.
Nevertheless, hemoglobin does have a large
effect of the rheologicalproperties and care must be taken to prevent increases in the hemoglobin content of blood during sample collecting and
testing.
B.
Red Cell Suspensions
Solutions of plasma proteins in isotonic saline were prepared, and
red cells were suppended in these solutions.
The mainpurpose of making
and investigating these suspensions was not to gather data on the flow properties of the suspensions, but rather to attempt to determine which plasma
proteins were responsible for giving normal blood its yield stress.
In all the viscometric determinations reported in this section, the
smooth surfaced viscometer Set A was used, unless otherwise noted.
1.
Red cells suspended in saline
A saline solution of the following composition was prepared with
reagent grade chemicals:
151
FIGURE
4
-34'
Effejt of hemoglobin on the rheological
properties of blood
1.2t
1.0Kf841,c=
42.2
K 1184 plus 0.0505 gm
hemoglobin/gm plasma, c
Temperature = 25.7 Oc
Symbols for plasmas
correspond to symbols
for blood samples.
0 Blood
Dashed line is expected
altered blood curve
calculated from
data for the
blood and
plasmas.
E 0.8
Blood
%
O
0
=
42. 3
o.6 I-
~~~0
0..
Plasmnas
o.4
7
0.2
I
0
00
3-1
.1
I
3
2
2
*1/2, (see -1/2)
-
I
I*
I
I
*I
.
iI
I
I
FIGURE
4-35
Eheological properties of red cell
saline suspensions, effect of hematocrit
-
-
1.25
I
I
Bematocrits
o047.1
%
1.0
%
o 38.0
A 27.8%
V 19.0%
So0 -75'
Temperature
=
25.8 0 C
0.56
Go
cc
to
0
0.25
0
1
2
3
4
5
20
10
, (sec 1
)
.0
NaCi
4. 384 gm
3.549
3.403
0. 901
Na 2 HPO4
KH 2 PO4
glucose
diluted with distilled water to make 500 ml of solution.
Red cells were
obtained by centrifuging normal blood, containing ACD anticoagulant, at
3000 g (40C) for 20 minutes. . The plasma and top layer of cells were.removed with a syringe and needle, and the remaining cells were suspended
in an equal volume of the above saline solution.
The suspension was centri-
fugedat 3000 g for 20 minutes, and the supernatant liquid and top layer of
cells removed.
The red cells were again resuspenoed in an equal volume
of saline, maintained for 10 minutes at 37*C, and again centrifuged and
the supernatant fluid removed.
Finally, the red cells were suspended in
the saline at the desired hematocrits.
The rheological properties of red cell-saline suspensions at 25. 80C
are shown in Figure (4-35).
origin.
The curves all extraploate through the graph
No yield stress, or torque decay at constant viscometer rotational
speed, was found experimentally.
Rouleaux formation was not detected
under the microscope.
2.
Red cells suspended in albumin
-
saline solutions
Red cells, obtained from ACD blood and washed by the procedure
given in the previous section, were suspended inthe following solution:
NaCi
KC1
NaHCO3
NaH 2PO4. H2 0
Albumin
154
6. 27 wt. o
0. 0136
0. 0581
0 0117
3.82
The pH of this solution was 7. 1 at 25*C.
The albuminwas twice crystal-
ized human albumin obtained from the Nutritional Biochemicals Corporation.
The albumin dissolved readily with very little foam formation (due
to trapped air in solid albumin) yielding a clear, slightly straw - colored
solution.
The albumin concentration is about that normally found in
human plasma.
The rheological data obtained from several red cell - albuminated
saline suspensions are shown in Figure (4-36).
Again, no yield stresses,
torque decays at constant shear rates, or rouleaux formation were found.
The viscosity of a suspension of this type, at a given hematocrit and shear
rate, is lower than that of the red cell - saline suspensions just discussed;
this is probably due to lower intercellular forces inthe red cell - albuminated saline suspensions.
3.
Red cells suspended in a-globulin -saline solutions.
Because a -globulin, obtained from the Nutritional Biochemicals
Corporation, does not readily dissolve in isotonic saline, the solid aglobulin was first dissolved in a saline solution whose salt content was five
times normal.
The solution was then diluted with distilled water to re-
turn the salt concentration to those given for the albumin solution described in the previous section.
In this fashion, it was possible to pre-
pare solutions containing 0.783% and 0. 308% (by weight) a -globulin solutions.
Red cells, washed once as described previously, were suspended in
these solutions.
The rheological properties of these suspensions are shown in Figure
(4-37).
Again, no yield stresses, torque decays, or rouleaux formation
were detected.
155
Rheological properties of red cell
1.25
1.00
-
4-36
FIGURE
albuminated saline suspensions
1
C~ 0.75
-
--
0
(U
1A
s-1
0.50
H ematocrit
56
0
49
25.4
46
25.4 Oc
0%
j
TeMerature
24.7 'C
c
25.4 'C
42
V2
0.25
V
~
0
,0
1
2
3
4
5
10
i (sec -1
20
FIGURE
Rheology
o.6
of red cefl
-
h-31
a-globulin saline suspensions
0.5
0.3
a-globulin
0
0.783 %
A
0.783
%
22.8
0
0.308
%
30-7
Temperature
0.1
1
2
3
4
5
l0o2
, (see- t
)
0 0
%
%
26.4
%
M.2
Hematocrit
=
25-0 OC
4.
Red cells suspended in y -globulin - saline solutions
Human solid y -globulin was dissolved in saline solution of the
The y
-
same composition as that used to prepare albumin solutions.
globulin, again obtained from Nutritional Biochemicals Corporation, was
2. 5 wt. % and
dissolved in the saline to give two protein concentrations:
1.0 wt. % (slightly above normal plasma concentration).
Previously
washed red cells were suspended in the solutions at a hematocrit of 38%
and the rheological properties were determined in the rough surfaced
viscometer set B.
The data, obtained at 36. 5*C, are shown in Figure
(4-38). Again no yield stresses or torque decays were noted.
5.
Red cells suspended in fibrinogen - saline solutions
Solid fibrinogrin, obtained from Cutter Laboratories and sold
under the trade name "parenogen", was dissolved according to the manufacturers instructions.
The so prepared concentrated fibrinogen solution
was diluted with the saline solution given as the solvent for the albumin
solutions previously discussed.
Fibrinogen conentrations of 0. 3 and
0. 6% were prepared, to cover the usual plasma concentration range,
and washed red cells were suspended in the solutions.
Rheological meas-
urements were made in the rough surfaced Set B viscometer at 25*C; the
data are presented in Figure (4-39) in the form of a Casson plot.
,
These suspensions were found to have yield stresses, both experi-
mentally by the torque decay method and by extrapolation.
A torque de-
cay at constant viscometer rotational speed was found, and it was greater
for the suspension with the higher fibrinogen concentration.
The behavior
of these samples is remarkably the same as would be expected for bloods
of the same hematocrit and fibrinogen content.
158
Rheology of red cell
-
a-globulin saline suspensions
o.6
0.5
o.4
--
0)0
r-globulin.,
c38.0
ar-globulin., c ='37.8
(.3
%
-i'%
SO
2.5
Temperature =36.5 *c
0.1-
>
1
2
3
4
5
302
1.026
,(e
%
0
03
FIGURE
Rheology of red
cell
-
4-39
fibrinogen-saline suspensions
1.0
CU.
CUi
0.5
0.6
% fibrinogen, c
=
43.3
0
0.3
% fibrinogen, c
=
38.5
%
0
%
Ct
o
0
0
2
1
1,/2
(sec1/2
3
This information, together with the data presented for red cells suspended in the other protein - saline solutions, strongly suggests that
-
fibrinogen is almost exclusively responsible for the Casson model
like behavior of blood at low shear rates.
C. Plasma
The viscosity of a plasma sample has been found to be a constant,
independent of the shear rate, in the shear rate range of this work.
This
is illustrated in the following table:
Shear Rate
- l(sec
Shear Stress,
)
27.0*C
20.5
10. 3
5. 13
2.05
1.03
0.513
0. 205
0. 103
dyne/cm
36.20C
2
0.30
0.253
0.152
0.124
0.0764
0.0619
0.0300
0.0247
0.0149
0.0123
0.00750
0.00624
0.00299
0.00247
0.00149
0.00120
Viscosity
1. 45 cp
1.20 cp
Sample source:
Dexter 7/3/62
The temperature dependence of the viscosity of plasma has been
found to be the same as that of the viscosity of water.
demonstrates this on an Arrhenius plot.
Figure (4-40)
The properties of the solvent,
water, therefore govern the change in viscosity of plasma with change in
temperature.
161
I
FIGURE
4-40
Effect of temperature on
plasma viscosity
0
2.0
Dexter
Whiting, II
0
Water
0.8
-
0-
0.7
0.6
34.0
33.0
32.0
T0 4
-1
T
162
V. CONCLUSIONS
1. Normal human blood has a yield stress. Of all the plasma
proteins investigated by any procedure, only the protein fibrinogen is
found to cause this yield stress.
What other molecular elements
participate in the fibrinogen - produced structure are not known.
On
the basis of the work described in this thesis it is possible to state
that:
(i) the yield stress is noi proportional to the fibrinogen con-
centration in the blood plasma, but increases as a higher power of
concentration, and (ii) calcium ions are not needed to act as a "bridge"
between the fibrinogen and the red cells.
At hematocrits below about
35% the yield stress is independent of temperature; above about 35%
hematocrit, the yield stress decreases slightly as the temperature
increases.
2. In the shear rate range near zero, normal blood follows the
relationships derived by Casson for a suspension model which closely
follows the behavior of blood. This model consists of mutually attractive
particles, suspended in a Newtonian fluid, which aggregate at a low shear
rates to form rod-like aggregates, called rouleaux in the case of blood.
The length of these aggregates decreases as the shear rate increases.
The relationships for the Casson model, found to be valid for blood
in the low shear rate range, are:
.1/2
1/2
T
zsy
+ b
In s
b
1/2 ln
0
+ k1 ln (1-c)
=
k2 (1-c) k1 +k3
163
where b = the square root of the yield stress
k , k2, k3 =- constants
and k 3
-k 2 .
The range of applicability of these relationships is
from a shear rate of zero up to an upper limit, the value of which
increases as the hematocrit decreases. As the temperature increases,
the value of k 1 , which is negative, becomes less negative, and k2 , which
is positive, increases.
The equations for the Casson model can not be used to calculate
such quantities as the cohesive force between the red cells in a
rouleaux or the rouleaux axial ratio because the equations contain
assumptions which are not consistent with the observable nature of
blood.
1/3
T 13=
a (c-c
ry
acc
)
3. It has also been found that the empirical equation
correlates the yield stress-hematocrit data of blood for hematocrits
below about 50%.
In this equation, "a" is a constant, as is cc, the
critical particle concentration below which a yield stress can not exist.
4. In the concentric cylinder viscometer, in which the shear rate
is almost constant throughout the fluid, the red cells in blood, at shear
rates below about 1 sec
,
migrate away'from the outer wall of the
viscometer gap, leaving behind a wall layer of plasma.
This migration
may be due to the Magnus effect, or to the intercellular attractive force
of the red cells, but it can not be due to forces arising from the
deformation of the .red cells since such forces would cause migration of
the red cells to the outer viscometer wall.
164
Whatever the migration mechanism is, it requires the aggregation,
or occurs simultaneously with the aggregation, of the red cells since
red cell suspensions in which rouleaux formation does not occur do not
show any evidence of red cell migration. The speed of migration also
is directly related to the fibrinogen concentration in the normal blood
concentration range.
5. Smooth solid surfaces prevent particles in a suspension from
occupying the space next to the wall at the same space concentration as
the bulk concentration.
This effect was found for blood to be equivalent
to the existence of plasma layers, 1 - 3
thick, at the smooth surfaces.
This effect is applicable at all shear rates and was found to cause
appreciably low torque readings in the GDM viscometer when smooth
viscometer surfaces were used.
6. The commonly used equations for concentric cylinder
viscometers relating the viscometer rotational speed to the shear rate
assume that the fluid in the viscometer gap has a uniform viscosity at
each particular viscometer speed. For non-Newtonian fluids, this
assumption may lead to erroneous results.
For blood, in the GDM
viscometer, it was found necessary to use the Krieger-Elrod equation
to calculate the shear rates for samples of high hematocrit (above
about 45%); the Krieger-Elrod equation does not make a by assumption
regarding the rheological character of the fluid in the viscometer.
of the usual equation led to low shear rate values.
165
Use
VI.
RECOMMENDATIONS
The investigation of the effect of fibrinogen on the yield stress of
blood should be extended.
This can be done by preparing a number of
plasma-serum mixtures in which red dells from a single blood sample
can be suspended.
This will enable the fibrinogen - yield stress
relationship to be defined, all other plasma constituent concentrations
being constant.
The addition of albumin to such suspensions will also
define the albumin - yield stress relationship in the presence of
fibrinogen.
In the interest of determining if the albumin and fibrinogen
compete for the same absorption sites on the red cell surface, and the
relative affinity of the sites for the proteins, the time sequence of red
cell environment changes should be varied and changes with time of
the yield stress immediately after the environment changes should be
determined.
The collection of yield stress - plasma composition data should
be continued so that the effects of other plasma constituents on the
rheological properties of blood can be determined.
The variation of red cell surface and its effect on the rheological
properties of blood, including the yield stress, should be looked into.
Two ways of investigating this variable are;
(1) placing red cells from various donors, both healthy and
ill, into one sample of AB plasma.
In this scheme, any difference
between red cell - plasma suspensions would have to be attributed to
differences in red dell surfaces.
Any significant differences would
best be investigated by determining the effects of fibrinogen and
albumin concentrations on the yield stress.
Such an investigation would
yield information on the nature of the difference in the red cell surfaces.
166
(2) washing red cells different numbers of times in saline
and resuspending the washed cells in plasma or fibrinogen-saline
solutions. According to Lovelock (44), the rate of lipid removal
from the red cell with washing is not the same for all of the lipids.
In connection with these investigations, red cell ghosts can be prepared
by various methods and the rheological properties of washed ghosts
suspended in plasma can be studied.
Investigations aimed at determining the source of red cell
migration at low shear rates in blood are of fundamental importance.
It has not been determined if the migration occurs at both walls of
the GDM viscometer, or just at one of them.
If the Magnus effect
explains the migration, the migration should be away from the outer
wall, but not away from the inner wall. To test this question, means
should be devised for rotation of the outer cylinder of the GDM
viscometer while the inner cylinder remains stationary.
Using a
blood sample which has a high fat content, or a dyed plasma, it will be
easy to determine if the red cells migrate away from the outer wall.
If they do, the Magnus effect would have to be discounted as a cause
of red cell migration.
Determination of the state of aggregation of the red cells as a
function of shear rate is most important.
Attempts to determine this
relationship can be pursued in two ways. A microscope has been
mounted on the GDM viscometer so that the fluid in the viscometer gap
can be observed.
Only preliminary use of this device at very low
hernatocrit levels has been made, and the further use of this means
of determining the physical state of blood at low shear rates is to be
encouraged. The study of the sedimentation of red cells in the GDM
167
viscometer gap under various shear rates, combined with data on
the sedimentation of cylinders of fixed axial ratio, may also be useful
in determining the shear rate - aggregation function.
At the present time, the hematocrit of blood samples is measured
by filling a capillary tube with a sample of the blood and centrifuging
the sample at about 17,000 g for about 15 minutes.
Variations in the
hematocrit of a single sample, determined by this method, may vary
by one hematocrit unit. Generally the variation is not this large, but
considering the accuracy of the other measurements being made on
blood, a more satisfactory method of determining the red cell volume
fraction in blood must be found.
If a satisfactory hematocrit measurement can be made, the effect
of red cell size at constant hematocrit can be determined.
A red cell
counter is commercially available.
The cause of the irreversible changes that occur in blood at
temperatures above normal body temperature should be investigated.
Whether the change takes place in the plasma, or the red cells, or
both needs to be determined.
The actual location of the change can
then be pursued.
The effect of changes in the dissolved gas content of blood on the
rheological properties of blood have not been investigated.
Considering
the change in gas content which occurs in the body, this effect should be
determined.
168
APPENDIX A
The Derivation of the Krieger-Elrod Equation*
1.
The derivation of the Krieger-Elrod equation for calculating the
shear rate at the surface of the inner cylinder of a concentric cylinder
viscometer whose outer cylinder is stationary and whose inner cylinder
rotates is as follows:
Consider a concentric cylinder viscometer whose inner cylinder
has radius r1 and a rotational speed .o
radius r 2 = s r.
,
and whose outer cylinder is of
For a fluid which is a continuum, and which is in
laminar flow, the shear rate at any point in the viscometer gap is given
by
? =
r
dlnr
(A-1)
while the shear stress is
T
G2
2ffr
(A-2)
Since G is a constant under steady state conditions
d In T = -2 d(ln r)
(A-3)
and therefore, substituting (A-3) in (A-1)
y
-2 d nr
(A -4)
Integrating this expression, and using the boundary conditions
*The Krieger-Elrod equation, as originally presented, was derived for
the case where the outer cylinder rotated and the inner cylinder was
stationary. The derivation, as presented here, is for the case of the
GDM viscometer - inner cylinder rotates.
169
1 at r = r
-
T = T ,
7 = T 20
.0
at r = 2
one obtains
2
s
1i
T72
(A-5)
X-dT
2=-
Assuming that a relationship exists between
?
and T of the form
g(T)
differentiating (A-5) with respect to T 1 yields
1
[ g(
1) -
g(r 2 ) I
(A-6)
-
d 1
as follows
=
h(T
)
Define a function h( T)
)
- [g(T
- 2
d 1
dln T
(A-7)
From Equation (A-6)
h(-
1
- g) T d 2)J
(A -8)
- g(s -2T
( g(r)
Also
h(s-2
-2 T ) - g(s 4 T
(A-9)
h(s 4 T ) = [g(s
T 1 ) - g(s -,6
1
etc.
Since s > 1 and g(0)
0
h(s
-2n
T )=g
n=o
170
(A-10)
Series (A-10)
is a slowly convergent one, but its asymptotic value can
be determined by using the Euler-MacLaurin sum formula:
mn'
m
f(n) dn
+
.An)
r[f(0)
+ f(m))
n=o
M
(2k-1)
B2k
Lf(m)
!
+ (2
(2k-)]
- f(O)
k=1
rn-i
+
i+1i
5
i
(2r+1)
x
1)
f(x)
P2r+1
dx
(A-11)
i=0
th
where B = i
Bernoulli number and P = the Bernoulli polynomial.
Krieger and Elrod (42) applied formula (A-11) to the left hand side of
(A-10) and they arrived at the expression
o
co
(s -2nT
h(s - 2n
dn+ - [ h(T7) + h(0)
I
n=o
l[dh(s-2n
12
dn
+
1
dn
3
(A-1 2)
+3''
can be evaluated by making the sub-
stitutions
s -2nT 1
dn-
n=
n0O
h(s -2nT)
720
The integral in Equation (A-12)
T
dy
2y in s
171
and using equation (A-7):
0,
)
h(s-2nT 1
-
dn
1
1
h(y) Au
y In s
C
2 In s
(A-13)
1T
The derivatives in expression (A-12) can be evaluated as follows:
differentiating (A-13)
dw
1
in s
d h(7 1 )
1
dn
(A-14)
dn
1n s
d2 w 1
d2
n2
(A-15)
d
since by definition (A-7)
h( r1 ) =
dw 1
2 d in T
-
Then, using (A-14')
1
dw y
ins
dn
do 1
=-2 dlnn
1
d (0
d2
.dn 2
= - 2 in s
(A-16)
d(ln r 1 ) dn
and
*2
d
=
-2
1
In s
(A-17)
(d ln r 1)2
1
)
dn d(ln
Combining (A-16) and (A-i?)
d2
d2 o
dn2
= 4(n S) 2
di21
(d In
172
T 1)2
2
(A-18)
Substitute (A-18) in (A-15) to obtain
d h(T )
d2 W I
d = 4 1ns
(A-19)
(d 1nT )2
dn
Therefore, the first derivative in the right hand side of series (A-12) is
dh(a -2n
)1'
4 1n s
dn
20
d
)2
2n
(d 1n s
7
(A-20)
Similarly, again starting from (A-13)
1
1
1
in s
)
d3 h(T
dn 3
(A-21)
dn 4
and differentiating (A-18)
3
=
4(n s)
2
dn
______
(A-22)
2
(di 1
) dn
and (A-17)
d3c
d3 w
1n s
1
(A-23)
1
-
(d 1n T )
dn (d 1n T
)
-2
Combining the last two equations
d3____
3
d3
- 8(n s)
41
= - 8(ln s) 3
dn3
d3
1
(d 1n TI) 3
and differentiating
dn 4
173
(A-24)
dn(d 1n T I) 3
Differentiating (A-23) gives
4
d4
d4 o
-2 in s
(d In
dn (d in T ) 3
7)
and therefore
d
d4 W
4
-
(A-25)
16 (In s)
4
dn
(d in T
)
d4cW
Substitution of (A-25) in (A-21) gives
d3 h(Tr)
=
dn3
16 (in s)
(A-26)
1
(d in T 1)
The additional derivatives can be similarly determined.
Since h(7r) and
it derivatives are zero when T, = 0, the final expression for the value of
the shear rate at the inner cylinder surface becomes, after substitution
of (A-26),
(A-20), (A-13) in (A-12) and substitution of the resulting
expression in (A-10):
1'
= g(r1 )
-
2
ins
+
- h(Tr)
1 in1s
s 2 1
3 (d in- T1)2
d4
--I (in s) 3
+ 45
'P =
I
n s
in
W I
d In T7l
d
(n s ) 4
45w
(i
(d in T1)2
+
(d In T
174
)
W11-nd
. .I
(A-27)
This expression, (A-27), is the Krieger-Elrod equation, which permits
the shear rate to be determined at the inner, rotating, cylinder of a
concentric cylinder viscometer provided (1) the flow is laminar, (2) the
fluid can be considered a continuum, and (3) the shear rate is solely a
function of the shear stress.
2.
The application of Equation(A-27) to human blood in the GDM vis-
cometer (cylinder set B) is illustrated below:
The second column of Table (A-1)gives the shear stress at the inner
cylinder wall, calculated from the experimentally measured torques given
in column one, and the third and fourth columns give the corresponding
inner cylinder rotational speed, in revolutions per minute and radians
per second respectively.
plot of In o
From an enlarged version of Figure (A-1), a
versus In T
the values of the slope d in o 1 / d in T 1
were evaluated and recorded in column five of Table(A-1). The second
derivative d2 In w /(d In _ 1)2 was determined from an enlarged version
of Figure (A-2), a plot of the first derivative versus in T 1 and its values
recorded in the sixth column of Table (A-1). By use of the relationship
(d in T )2
1
(d
n T )
In w,
(d
n w 1
d2
d2
+
2
din T
the values of the derivative recorded in column seven of Table (A-1) were
calculated.
Having determined the values of the necessary derivatives,
the value of the shear rate at the inner cylinder surface of the viscometer
was determined from Equation (A-27):
(01
('iKE
In s
1 +Lin sd in o 1
d in 7
175
- (in s)2
+ 3W1
2
d W
(d ln T1) 2
]
I
i
FIGURE A-1
10
4i agrtai
LUr Uva X-LLL6~l
-
1
Elrod equation
/
for the Krieger
w 1~ /U
0
1i
(rad/sec)
0.1
0.01
70
I
1
0,1
10
(dynes/cm2
)
0.001
176
I
I
I
FIGURE
A-2
Diagram for evaluating d2
for the Krieger
13
-
ln c.;/(d ln
Elrod equation
-
15 15
0
11-
9
-
1%
30
I
0
-2.0
-1.0
0.0
ln
1.0
Table (A-1) gives two values of (?1)KE: the first column gives the calculated values for the shear rate using only the first two terms of the series
in Equation (A-27), while the other column gives the values of (fy)KE
using the first three terms of the series.
Use of an addition term in the
series of Equation (A-27) would increase the lowest shear rates by less
than 1% over the values calculated with two terms of the series, while
leaving unchanged the higher shear rates.
gives the values of
The last column of Table (A-i)
calculated by assuming that the fluid viscosity is
constant across the viscometer gap.
A comparison of the shear stress - shear rate data for this sample
of blood is shown in Figure (4-1), which is a plot of the data calculated
here using the Krieger-Elrod equation, and constant viscosity equation
to obtain the shear rate values.
178
TABL.E
SE4AR STRESS
4-1
S/144R fjAtE OArA
d-w,4zz'A., d2K
r_
_
I____
q1____0
dt4A; NAf4)z (Cd9t4)Z 2 TERMS
I_
dyne cm dyne/cmz rev/nin rad/sec
3
FOR
A BLOOD SAMPLE
ci
re- s I
.--
-
sec
sec-'
sec'
sec~
2.32
0.102
0.0/0
o.oa,5
14-.7
- 7?.9
0.145
0.0269
0.0311
0.0124
2.45
0-/08
0.020
0.0209
//.as
-57.5
0.173
0.0474
0.052(
0.0246
2.67
0.117
0.050
0 .o524
7.7
-29.57..54-
3.18
0.140
0.100
o.oos
4.27
-:0.3
3.62
0.15?
0.200
0.0209
3.05
-
S.00
o.220
0.400
0.o4/
2.58
7.o
0.310
1.00
0.oos
00.2
0.5/
2.00
15.5
28.2
0.682
4.00
1.24
96.0
2.OZ
0.159
o.10o
0.1a24
4.27
0.0957
0.291
0.Z747
0.24(
-
1.77
0.205
0.563
0.569
0 .49/
2.1 2
-
0.150
0.372
1.36
0.20?
1.74
-
0.5/5
o.525
2.64
1.37
2.45
1.24
z.i(
0.f/9
1.58
-
0.249
0.134
10.0
/.05
1.1-8
-
0.161
2.13
5.21
12.9
13.0
+.91
12.4-
20.0
2.09
1.42
-
o.oq4 4.03
z5.8
Z5.9
24.6
ROUGH
.V
r
%
0.0933
= 47.
TE MPERATCURE = 2 f.8
W4EAIA-OCRI
4 C
SURPA CED VISCOMETER
S =
.V.
o. 9%G 0.10/ o.061S
/.2i9
. /
cm
12 cm
(sEr a)
= o.o0k99
57.Z4
APPENDIX B
Use of Theoretical Equations to Correlate Blood
Data in The Shear Rate Range 2 to 20 sec
In this section various theoretical rheological relationships
proposed for suspensions of neutral spheres, will be investigated for
possible correlative use with blood.
In view of the fact that all proposed theories applicable to
dilute suspensions of spheres can be expressed in a power series of
the concentration volume fraction of the particles (28)
sp
[
c
it is natural to plot -
+k 1 []
/c versus c.
2 c + k
3 c2 +
---
(B-)
Data for a typical normal blood is
given in Table (B-1) together with the calculated relative viscosities and
7j
Sp /c values.
This information is presented graphically in Figure (B-1)
together with the curves predicted by the first two terms of equation
(B-1), using the constants theoretically calculated by Vand (64), and
Mason and Bartok (45):
sP= 2.5 + 7.35c
C
and
2.5 + 10.05c
These latter theoretical equations are applicable only at low particle
concentrations since the higher order concentration terms have been
neglected.
To illustrate the importance of the higher order terms,
Vand's experiment curve for spheres is also shown in Figure (B-1), for
which the equation is, to three terms;
180
I
it I
FIGURE B-1
101"
T
/
Specific viscosity
divided by conc.
versus conc. for red
cells in plasma
250C
9
~V
/
A
7
-
s
/0//
2.05
4.10
010.3
0 20.5
.....
-7
seesee'
see'
sec-1
Vand
eq.
Mason & Bartok
modified Vand eq.
AM--Experimental
Vand eA., theoret ical
(two terms)
Mao
Bro
2
0
10
20
30
%
a,
181
40
50
= 2.5 + 7.17c + 16.3 c
-
sp
c
From this diagram, two features stand out:
(1) as the shear rate
increases, making the interparticle force effects less important, the
curves approach the theoretical curves for suspensions of neutral
spheres, and (2) the intrinsic viscosity of red cells suspended in
plasma could be 2.5, namely the same as for neutral spheres. It
appears from this figure that Vand's value for the coefficient of the
second term of equation (B-1) is more correct for dilute red cell
suspensions than that value proposed by Mason and Bartok.
The
possibility also exists that at shear rates high enough to make the
effects of interparticle forces negligible, the data for red cells
suspended in plasma may coincide with Vand's experimental curve
for neutral sphere suspensions.
Attempts to obtain a theoretical rheological equation of state for
concentrated suspensions have usually considered a particular particle
concentration as being attained by successive addition of a small
number of particles to a suspension, which is considered as a homogeneous solvent having the rheological properties of the suspension.
Thus, Mooney (jQ) found for a suspension of neutral uniform particles
exp (2.5
(B-2)
where k is a constant, which for suspensions of spheres is between 1.35
and 1.91.
In order to see how closely blood followed Mooney's equation,
equation (B-2) was rearranged to give the relationship
1
2.5c
-
1
ln'rel
k
2.5
182
TABLE
B-1
Viscosity of blood at shear rates between 4 sec-I and 20 sec
1
Viscosities, (centipoise)
10 .2
46.0%
39.0%
33.2%
28.1%
21.6%
19.2%
20-52
7.40
5.79
4.85
4.o4
3.16
2.79
2.142
1.61
10.26
4.1o
9.44
12.71
6-75
8.66
5.41
6.46
4.36
5.15
3-33
3-76
2.86
3.142
2.142
2.21
i.61
1.61
2.05
16.79
11.12
8.15
6.11
4.40
3-54
2-32
1-61
Hematocrits
0
Shear Rates
)
(sec7
Relative Viscosities
0.
Wo
4.59
5.85
3-59
3.01
2.50
1-959
1-730
1-330
1.00
4.18
3.355
2-70
2.064
1-773
1.330
1.00
4.10
7.88
5.37
4-oo
3-19
2.331
1.948
1.370
1.00
2.05
io.41
6.89
5.05
3.79
2.728
2.195
1.439
1.00
3.30
----
20-5P
io.26
20.52
7.80
6.64
6.o6
5.34
/ Hematocrit
3.80
4.44
10.26
10-54
8-15
7.10
6.05
4.93
4-03
3.30
4.10
14.96
11.21
9.04
7.79
6.16
4.94
3.63
2.05
20.46
15-10
12.20
9-93
8.00
6.22
4.30
Specific Viscosity
Temperature = 25.0
Samp1e: m 8264
C
On this basis, Figure (B-2) was prepared by plotting 1/in t 7rel against
1/(2. 5c), using the data of Table (B-1).
From this figure it appears that
Mooney's equation does not satisfactorily define the rheological
equation of state of blood in the higher concentration region. Figure
(B-2) does show thatthe possibility exists that the lower concentration,
range, at high shear rates, may be well represented by Mooney's
equation.
In addition, at higher shear rates it may be that plots of
1/2.5c versus 1/ln ' rel will be linear over a large concentration range.
Using a similar technique, Brinkman (2) and Roscoe (52) developed
similar rheological equations for uniform neutral sphere suspensions of
finite concentration:
..
2 5
(B-4)
(1 - kc)
where for Brinkman's equation k
equation.
1, but is just a constant in Roscoe's
To test this relationship with the blood data, it was converted
to the form
ln 7 re
=
-
2.5 ln (1
-
kc)
(B-5)
Figure (B-3) shows a plot of ln 7rel versus in (1 - c) for the blood data
of Table (B-1).
The data at all the shear rates shown seem to fit straight
lines on this plot, with the slope varying with the shear rate so that as
the shear rate increases the slope seems to approach the theoretical
value. Again, at higher shear rates than used here, the data for blood
may be well described by Brinkman's equation and Roscoe's equation.
with k = 1.0.
184
I
4
I
I
FIGURE B-2
Test of Mooney eq.
with red cell
suspension data
T = 25.000
1
2.9-c
2
2.05 seeA
VAQG
0
4.10 sec l
1
10.3
o 20.5
see"'
se-
1
/
/
Area between dashed lines is
range of theoretically
predicted curves.
0L
0
i
___j
I
2
1/(lm
wrel
185
3
I
I
1
V 052 O 5 sec
10
9
a
4.10 sec
7
6
Y\rel
5
3
20.5 sec
0
4
C;
3
- FIGURE B
Test of Brinc ian's eq.
with data for red cell
plasma susjpension
0
Dashed line is curve
predicted by Brinznari
equation.
1
0.3
.
I
___
.. I
0.7
0.5
(1
....
-
186
c)
_L
0
\
-..
0.9
Simha (_0)
considered the particles in a suspension as being in a
spherical "cage", the diameter of which decreases as the particle
concentration increases.
From his model he found, for concentrated
suspensions of neutral particles
2
JL
54
40 cmax
c
(1
--
(B-6)
3
c
where c max is the particle concentration at closest packing.
In this
case, the value of c max will be taken to be equal to the apparent volume
fraction of the blood occupied by the red cells after sedimenting to a
steady state (over 12 hours) divided into the hematocrit as determined
by centrifuging the suspensions at about 17,000 g for 15 minutes. As
shown in Table (B-2), the value of cmax so determined is a constant
independent of hematocrit.
TABLE B-2
Red Cell Volume Fraction at Closest Packing for
Suspensions of Table B-1
Hematocrit
c(%)
46.0
39.0
33.2
28.1
21.6
19.2
10.2
2
c
cM
maxmax
72.5
75.7
73.7
72.4
75.8
76.0
74.7
c
cma
max
c /c
0.619
0.524
0.446
0.378
0.290
0.258
0.137
3.83
1.41
0.650
0.328
0.130
0.0903
0.0162
Average 74.4
In Figure (B-4) a plot of q rel versus c/(1
-
c/c
)3 is presented
for the data of Tables (B-1) and (B-2), together with the theoretical
curve (equation (B-6)).
The.actual do not resemble the theoretical
187
8
FIGURE
B-11
Test off Simha's equation with data
for red cell - plasma suspensions
2.5sec1
7
61|A
5
H
I-.
A
V
A
o
3
2
0
Dashed line iscurve predicted
by Siniha equation
VA
7'
1
0.
3'
2
I
(-
c/cm)
3
curve.
One might argue that the wrong value had been used for c max
but increasing c ma would lower the slope of the theoretical curve and
shift all the experimental points horizontally to the left; this would not
make the theoretical and experimental curves agree. Lowering the
value of c
similarly would not result in agreement of the theoretical
max
curve with the experimental data. It appears that Simha's theoretical.
equation does not describe the rheological properties of blood.
Thus far the correlative attempts in this appendix have been
concerned with the effect of particle concentration.
Attempts to take'
into account changes in shear rate seem to have been aimed at finding
how changes in shear rate affect the rate of dissipation of energy of
doublets. This rate of energy dissipation of anisometric bodies occurs
because as the shear rate increases, the bodies tend to orient themselves
with their major axis either along or perpendicular to the flow. Having
determined the shear rate - endrgy dissipation relationship, the
intrinsic viscosity of the anisometric particles can be formulated in
terms of the shear rate. This then permits the coefficients in equation
(B-1) to be corrected for-shear rate changes since the coefficients can
be formulated in terms of the intrinsic viscosities of single particles,
doublets, etc, and the interaction coefficients.
The doublet intrinsic viscosity - shear rate relationship has
theoretically been found to be (Q28,
[i]
=[]
=
.O)
{1 - constant v 2
189
+ ---
]
(B-7)
Experimentally, it has been found for polymer solutions that the
relationship is usually of the form (70)
[
=]
7-
9=0
constant
n
(B-8)
There is not, however, general agreement on the form of the shear
rate - intrinsic viscosity relationship.
If equation (B-7) is substituted into the theoretical expressions for
the coefficients of equation, (B-1), then at constant particle concentration one obtains
7
=
k
- k2
2
2
+ --
(B-9)
It would be interesting to see if this equation (assuming higher shear
rate terms to-be negligible) was valid for blood, but data at sufficiently
high shear rates have not been obtained in this thesis work.
In conclusion, it would seem possible that equations (B-1) and (B-4)
hold promise as theoretical correlation means at shear rates above
20.5 sec
,
while equation (B-2) may be applicable at both high con-
centrations and high shear rates. Equation (B-6) does hot seem
applicable at all.
190
APPENDIX C
Location of Data and Calculations
The data and calculations are in the custody of Professor
E. W. Merrill of the Department of Chemical Engineering, and are
physically located in Room 12 - 171.
191
APPENDIX D
Literature Citations
Bayliss, L.E., in "Flow Properties of Blood", A. L. Copley
and G. Stainsby (eds.), Pergamon Press, N.Y., (1958).
2.
Bayliss, L.E., in "Deformation and Flow in Biological Systems",
A. Frey - Wyssling (ed.), North Holland Pub. Co., Amsterdam,
(1952).
3.
Bingham, E. C., and R. R. Roepke, J. Amer. Chem. Soc., 64
1204, (1942).
4.
Bingham, E. C., and R. R. Roepke, J. Gen. Physiol, 28, 131,
(1944).
5.
Bolger, J. C., Sc.D. Thesis, Dept. of Chem. Eng., M.I.T., (1960).
6.
Borun, E. R., Figueroa, W. G., and S. M. Perry, J. Clin.
Invest., 36, 676, (1957).
7.
Brinkman, H. C., J. Chem. Phys., .20, 571, (1952).
8.
Brundage, J. T., Amer. J. Physiol, 110, 659, (1934).
9.
Burgers, J. M., Chapter 3, Second Amsterdam Report on
Viscosity, North Amer. Pub. Co., (1938).
10.
Casson, N., Chapter 5 in "Rheology of Disperse Systems",
C. C. Mill (ed.), Pergamon Press, N. Y., (1959).
11.
Cerny, L. C., Cook, F. B., and C. C. Walker, Amer. J. Physiol.,
202, 1188 (1962).
12.
Charm, S., and G. S. Kurland, Amer. J. Physiol., 203, 417, (1962),
13.
Cokelet, G. R., Merrill, E. W., Gilliland, E. R., Shin, H., Britten,
A., and R. E. Wells, Jr., "The Rheology of Human Blood
Measurement Near and at Zero Shear Rate", presented at
Oct, 1962 meeting of Soc. of Rheology, Baltimore, and accepted
for pub. in Trans. of Soc. Rheol.
14.
Copley, A. L., Krachma, L. C., and M. E. Whitney, J. Gen.
-
1.
Physiol., 2,
49, (1942).
192
15.
Coulter, N. A., Jr., and J. R. Pappenheimer, Amer. J. Physiol.,
159, 401, (1949).
16.
Davson, H., "A Textbook of General Physiology", 2nd ed., Little,
Brown & Co., Boston, (1960).
17.
Dintenfass, L., Angiology, 13, 333, (1962).
18.
Dintenfass, L., Circulation Res., XI, 233, (1962).
19.
Dintenfass, L., Kolloid Z., 180, 160, (1962).
20.
Dix, F. J., and G. W. Scott Blair, J. Appl. Physics, 11, 574,
(1940).
21.
Endres, G., and L. Herget, Z. Biol., 88, 451, (1929).
22.
Eveson, G. F., Whitmore, R. L., and S. G. Ward, Nature
(London), 166, 1074 (1950).
23.
Fahraeus, R., Acta. Med. Scandinav., 55, 1, (1921).
24.
Fahraeus, R., Physiol. Rec., 9, 2, 241, (1929).
25.
Fahraeus, R. and T. Lindqvist, Amer. J. Physiol., 96, 562,
(1931).
26.
Fahraeus, R., Acta. Med. Scandinav., 161, 151, (1958).
27.
Fox, T. G., Gratch, S., and S. Loshaek, in "Rheology",
vol. I, F. R. Eirich (ed.), Academic Press, N.Y., (1956).
28.
Frisch, H. L., and R. Simha, "Viscosity of Colloidal
Suspensions", in "Rheology", vol. I, F. R. Eirich (ed.),
Academic Press, N.Y., (1956).
29.
Fulton, G., "Blood Flow in the Small Vessels", motion picture
produced in Dept. of Biology, Boston University, Boston, Mass.
30.
Gilinson, P. J., Jr., Dauwalter, C. R., and E. W. Merrill,
"A Rotational Viscometer using an A.C. Torque to Balance
Loop and Air Bearing", presented at the Oct, 1962 meeting of
Soc. of Rheology, Baltimore, and accepted for pub. in Trans.
of Soc. Rheol.
193
31.
Gilinson, P. J., Jr., Dauwalter, C. R., and J. A. Scoppettuolo,
"A Multirange Precision Torque Measuring System', Report
R-367, Instrumentation Laboratory, M.I.T., Cambridge,
Mass., July 1962.
32.
Gilligan, D. R., and A. C. Ernstene, Amer. J. Med. Sci.,
187, 552, (1934).
33.
Goldsmith, H. L. and S. G. Mason, Nature, 190, 4781, 1095,
(1961).
34.
Guyton, A. C., "Textbook of Medical Physiology", 2nd.edition,
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35.
Guyton, A. C., "Textbook of Medical Physiology", 2nd edition,
Chapter 13, Philadelphia, W. B. Saunders Co., (1961).
36.
Hall, C. E. and Slayter, H. S., J. Biophys. and Biochem.
Cytology, 5, 11, (1959).
37.
Haynes, R. H., and A. C. Burton, Amer. J. Physiol, 197, 943,
(1959).
38.
Haynes, R. H., Amer. J. Physiol., 198, 1193 (1960).
39.
Jeffery, G. B., Proc. Roy. Soc., A102, 161, (1922).
40.
Joly, M., Biorheology, 1, 15, (1962).
41.
Kreuzer, von F., Helv. Physiol. Acta. 8, 486, '(1950).
42.
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43.
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44.
Lovelock, J. E., Biochem. J., 60, 692, (1955).
45.
Mason, S. G., and W. Bartok, in "Rheology of Disperse
Systems", C. C. Mill (ed.), Pergamon Press, N.Y., (1959).
46.
Mason, S. G., Talk before the 1962 annual meeting of the Soc.
of Rheology, Johns Hopkins Univ., Baltimore, Md., Oct. 1962.
47.
Merrill, E. W., Cokelet, G. R., Gilliland, E. R., Shin, H., and
A. E. Wells, Jr., "Aheology of Human Blood and the Red Cell
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194
Merrill, E. W., Cokelet, G. R., Gilliland, E. R., Shin, H., and
R. E. Wells, Jr., "Human Blood Rheology and Flow in the MicroCirculation - Some New Questions Posed by New Experiments"
sub. to the J. Exptl. Physiol., (1962).
49.
Merrill, E. W., Gilliland, E. R., Cokelet, G. R., Shin, H.,
Britten, A., and R. E. Wells, Jr., "Rheology of Human Blood
as a Homogeneous Substance Near and at Zero Shear Rate
Effect of Temperature and Hematocrit", submitted to
Biophysical J., (1962).
50.
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51.
Norton, F. H., Johnson, A. L., and W. G. Lawrence, J. Amer.
Ceram. Soc., 27, 149, (1944).
52.
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114, 128, (1935).
53.
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55.
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56.
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57.
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58.
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248, (1930).
59.
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60.
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62.
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48.
195
64.
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65.
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66.
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68.
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69.
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196
APPENDIX E
Nomenclature
a
Constant, slope of yield stress - hematocrit correlation
a
Orientation constant in Casson equations, dimensionless
B 1 , B2
Designations for age fractionated red cell groups
b
Constant, intercept (i.
1/21/ dynel/2/cm
/2),
c
Hematocrit (volume fraction of suspension particles)
per cent
c
c
0) of Casson plot ( -r1/2 versus
Critical hematocrit below which blood can not have a
yield stress, per cent
D
Vand wall layer thickness, microns
FA
2
Red cell cohesive forces, dyne/cm.
f
A function
fd
Fraction of suspension particles which are in the form
of doublets, dimensionless
G
Torque per unit length of viscometer bob, dyne
g
Gravitational acceleration, cm/sec
Hx
Vand viscometer constant, dimensionless
h
A function
I
Moment of inertia of the fluid in viscometer and moving
parts of viscometer other than the bob, per unit length
of viscometer bob; dyne sec 2 /cm
J
Rouleaux axial ratio, dimensionless
K
Constant
197
2
ki, k 2 , k 3
Constants
1
Length, cm
m
Constant
N
Rotational speed of viscometer bob before rotation stopped
for yield stress determination, rpm
0
n
Constant
r
Radial distance from viscometer axis, cm
ry
Radius of viscometer bob
r2
Inside radius of viscometer cup
s
Constant
S
Constant, slope of Casson plot (T 1/2 Vs
y1/2),
(dyne sec /cm2 )1/2
s
Viscometer constant r 2 /r 1, dimensionless
T
Temperature, 'C
T
Torque, dyne cm
T1 , T 2
Designations for age fractionated red cell groups
T
Electromagnetic torque
t
Time, sec.
u
Linear velocity, cm/sec
Constant in Casson equation
Constant in Casson equation
Shear rate, sec
198
T
Shear stress, dyne /cm 2
T1
2
Yield stress, dyne/cm
y
Apparent calculated shear stress, dyne/cm 2
7
Viscosity of suspension, centipoise
710
Viscosity of suspension suspending medium, centipoise
71x
Apparent calculated viscosity, centipoise
71]
Intrinsic viscosity, reciprocal concentration
lisp
Specific viscosity, dimensionless
60
Time, sec
'p
Angular displacement of viscometer cup from null
position, radian
Angular velocity, radians/sec
Wi
Angular velocity of viscometer bob, radians/sec
199
BIOGRAPHICAL NOTE
The author was born in New York City on January 7, 1932.
He
attended elementary school in Carle Place, New York, junior high
school in Los Angeles, California, and high schools in Los Angeles
and Monrovia, California.
In September 1950, he entered Pasadena
City College, Pasadena, California, from which he graduated in June
1953.
Transfer as a junior into the California Institute of Technology,
Pasadena, California, was made in September 1953, but studies were
interrupted in January 1954 by action of a local Selective Service Board.
The years 1954 and 1955 were spent in California and Japan as a
member of the U. S. Army.
Returning to Cal. Tech. in January 1956,
the author attained his B.S. degree in Applied Chemistry in June 1957.
The M.S. degree in Chemical Engineering was obtained from Cal. Tech.
in June 1958.
The author entered M.I.T. in February 1960, the intervening years
having been spent with the Dow Chemical Company, Williamsburg,
Virginia.
Work towards the Sc.D. degree began a year later.
During
his graduate work, he was a teaching assistant, instructor, and research
assistant in the Department of Chemical Engineering.
200