Articles in PresS. J Appl Physiol (May 17, 2012). doi:10.1152/japplphysiol.00369.2012
1
1
Comparison of various approaches to calculating
2
the optimal hematocrit in vertebrates
3
Heiko Stark* and Stefan Schuster
4
5
Friedrich-Schiller-University
7
Department of Bioinformatics
8
Ernst-Abbe-Platz 2
9
07743 Jena
10
Germany
11
12
Abbreviated title: Optimal hematocrit in vertebrates.
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14
Word count: 5970
15
16
*
17
Dr. Heiko Stark
18
Lehrstuhl für Bioinformatik
19
Friedrich-Schiller-Universität Jena
20
Ernst-Abbe-Platz 2
21
07743 Jena
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Germany
23
E-mail: [email protected]
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Phone: ++49-3641-949584
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Fax: ++49-3641-946452
Corresponding author:
Copyright © 2012 by the American Physiological Society.
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2
26
Abstract
An interesting problem in hemorheology is to calculate that volume fraction of
28
erythrocytes (hematocrit) that is optimal for transporting a maximum amount of
29
oxygen. If the hematocrit is too low, too few erythrocytes are present to transport
30
oxygen. If it is too high, the blood is very viscous and cannot flow quickly, so that
31
oxygen supply to the tissues is again reduced. These considerations are very
32
important since oxygen transport is an important factor for physical performance.
33
Here, we derive theoretical optimal values of hematocrit in vertebrates and collect,
34
from the literature, experimentally observed values for 57 animal species. It is an
35
interesting question whether optimal hematocrit theory allows one to calculate
36
hematocrit values that are in agreement with the observed values in various
37
vertebrate species. For this, we first briefly review previous approaches in that theory.
38
Then, we check which empirical or theoretically derived formulas describing the
39
dependence of viscosity on concentration in a suspension lead to the best agreement
40
between the theoretical and observed values. We consider both spatially
41
homogeneous and heterogeneous distributions of erythrocytes in the blood, and also
42
possible extensions like the influence of defective erythrocytes and cases where
43
some substances are transported in the plasma. By discussing the results, we
44
critically assess the power and limitations of optimal hematocrit theory. One of our
45
goals is to provide a systematic overview of different approaches in optimal
46
hematocrit theory.
47
Keywords
48
Optimal hematocrit theory, blood viscosity, Einstein's equation, Fåhræus-Lindqvist
49
effect, adaptation to high altitude
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27
3
50
a
Proportionality factor for substances transported in the plasma
b
Proportionality factor for substances transported in erythrocytes
D
Tube diameter
η
Viscosity of fluid (viscosity of suspension)
η0
Viscosity of fluid solvent (so-called embedding fluid)
ηrel (D,φ)
Viscosity relationship between D and φ
J
Total flow across the blood vessel (Hagen-Poiseuille law)
Joxygen
Total oxygen flow across the blood vessel
l
Tube length
Δp
Pressure difference
φ
Volume fraction of particles in the suspension
φm
Maximum volume fraction (maximal packing density)
φopt
Optimal volume fraction of particles in the suspension
φint
Volume fraction of intact particles in the suspension
φdef
Volume fraction of defective particles in the suspension
v
Fluid velocity
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51
Glossary
4
52
1. Introduction
In higher animals, red blood cells (erythrocytes) transport oxygen. In evolutionary
54
history, oxygen transporting molecules such as heme are as old as the Bilateria (31,
55
36). Erythrocytes belong to the groundplan of the Craniota (108). Moreover,
56
erythrocytes also transport purine nucleotides from organs with a nucleotide surplus
57
to organs in which purines are required (cf. 55, 85). Obviously, it is physiologically
58
favourable when a maximum amount of oxygen and nucleotides is transported at
59
given sizes of the arteries. This can be phrased as an optimality principle. Such
60
principles are often used in biology, based on Darwinian evolutionary theory. For
61
example, the optimal streamlining of the body shape of different species (shark,
62
barracuda/tuna and dolphin) has been computed (8, 93). Other examples are the
63
optimal number of letters of nucleotides (4) in the genome (99), the optimal amount
64
of ATP produced in metabolic pathways (14, 103, 107), and the optimal leaf sizes in
65
relation to the environment (72).
66
The oxygen flow in animals can change by a change in the erythrocyte to blood
67
volume ratio. This ratio is called hematocrit (cf. 26). In healthy humans, it amounts to
68
about 40 %. Note that there is a difference in most hematological parameters of men
69
versus woman; for example, the hematocrit amounts to 45.8 ± 2.7 % vs.
70
40.0 ± 2.4 %, respectively (27). The gender-related variation in parameters of the
71
blood is explained by the difference in age distribution of red blood cells due to
72
menstruation and the subsequent difference in mechanical properties of blood of pre-
73
menopausal women and men (27). The hematocrit values differ considerably across
74
the animal kingdom (4, 5, 7, 18–23, 29, 32, 45–47, 52, 64, 66, 68, 71, 82–84, 101,
75
102, 104–106, 110, 111). The values of several species are given in Table 1.
76
The question arises whether the hematocrit values given above are optimal. If the
77
hematocrit is too low, too few erythrocytes are present to transport oxygen. If it is too
78
high, the blood is very viscous and cannot flow quickly, so that oxygen supply to the
79
tissues is again reduced. Within hemorheology (hydrodynamics of the blood), a
80
theoretical framework called 'optimal hematocrit theory' has been established (6, 15,
81
41, 70, 104, 110). That an optimum has been found in evolution is supported by the
82
observation that the hematocrit values are quite robust against temperature changes
83
(1, 44) and are nearly the same in active sportsmen and control persons (88). On the
84
other hand, there is some dependency on water supply and on altitude (see below). A
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53
5
85
general problem in optimality considerations in biology is that usually several
86
optimality principles are relevant simultaneously, for which a trade-off must be found.
87
Here, we focus on the criterion of maximum oxygen transport velocity, leaving other
88
criteria for future studies.
89
These considerations are very important since oxygen transport is an important
90
factor for physical performance. They are especially relevant in medicine (e.g. blood
91
transfusion,
92
mountaineering. For example, the non-alcoholic fatty liver disease (NAFLD)
93
correlates with higher hematocrit levels (58). Regrettably, some sportsmen use blood
94
doping by infusing additional erythrocytes (3, 12, 35, 49, 67, 90).
95
It is an interesting question whether optimal hematocrit theory allows one to calculate
96
hematocrit values that are in agreement with the observed values in various
97
vertebrate species. In this paper, we first briefly review previous approaches in that
98
theory. Then, we check which empirical or theoretically derived formulas describing
99
the dependence of viscosity on concentration in a suspension lead to the best
100
agreement between the theoretical and observed values. We consider both spatially
101
homogeneous and heterogeneous distributions of erythrocytes in the blood. By
102
discussing the results, we critically assess the power and limitations of optimal
103
hematocrit theory.
104
2. Material & Methods
dialysis,
transplants,
stent
and
shunt
implants),
sports,
and
First, we briefly review some fundamentals from rheology needed for the subsequent
106
calculations.
107
2.1.The Hagen-Poiseuille law
108
A central parameter characterizing the flow behaviour of a liquid is its viscosity, η.
109
Consider a blood vessel (e.g. an artery) with circular cross-section. Let R and l
110
denote the radius and length of the tubular vessel. The blood flow is driven by a
111
pressure difference, Δp. A basic assumption in hydrodynamics, which is well justified
112
by experiment, is that the velocity of the liquid very near to a rigid surface is zero (or
113
is the same as that of the surface if that is moving). This is because a thin layer of
114
molecules of the fluid are attached to the surface. Moreover, for symmetry reasons,
115
we can assume that the velocity of a liquid being in a stationary flow in a circular tube
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105
6
116
forms a profile with rotational symmetry. Thus, it depends only on the radial
117
coordinate r (with 0 < r < R): v = v(r). This is justified as long as the flow is non-
118
turbulent, i.e., for low velocities. To calculate this velocity profile, we can use the
119
equilibrium of forces between driving pressure and resistance due to the viscous
120
properties of the liquid. It is a well-known result from fluid mechanics that the velocity
121
obeys the eq. (1) (33, 75–77, 98).
v=
122
R2 − r 2
Δp
4ηl
(1)
This is a parabolic velocity profile, in which the highest velocity is reached in the
124
middle of the blood vessel (Fig. 1a). It is of interest to know the total flow across the
125
blood vessel. To calculate this, we need to integrate over the cross-section, A, of the
126
tube:
J=
127
R
R2 − r 2
R2 − r 2
ΔpdA= Δp 
2πrdr
4ηl
4ηl
0
J=
128
πΔp  R 2 r 2 r 4  R πΔp  R 4 R 4 
−  |0 =
−



2ηl  2
4
2ηl  2
4 
J=
129
with J =
πΔpR 4
8ηl
V
t
(2)
(3)
(4)
130
Eq. (4) is called the Hagen-Poiseuille law (cf. 26). In a stationary, non-turbulent flow
131
across a cylindrical tube, the total flux of liquid is proportional to the 4th power of the
132
tube radius.
133
2.2. Dependence of viscosity on hematocrit
134
Dilute suspensions
135
The question is now how the viscosity depends on the hematocrit, φ. Blood can be
136
considered as a suspension (rather than a solution) because erythrocytes are much
137
larger than molecules. The solid phase (erythrocytes) is dispersed throughout the
138
fluid phase (plasma) through mechanical agitation. Note that other blood cells such
139
as leukocytes and thrombocytes contribute a negligible volume (< 1%) in comparison
140
to erythrocytes (27). For suspensions of spheric particles with low concentration (but
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123
7
141
not considering blood at that time), Albert Einstein (24, 25) derived the formula:
η = η0 (1 + 2.5φ )
142
(5)
143
(Fig. 2a) with η = viscosity of suspension, η0 = viscosity of fluid solvent (so-called
144
embedding fluid, e.g. water), φ = volume fraction of particles in the suspension (i.e.,
145
hematocrit in the case of blood).
146
Suspensions with higher concentrations
147
Einstein’s formula has been extended in various ways to cope with higher
148
concentrations. For example, the formula of Saitô (86) reads
150
151
152
153
154
(6)
while the formula of Gillespie (28) reads
η = η0
1+ φ / 2
(1 − φ)
(7)
2
and the formulas of Quemeda (80) and Krieger & Dougherty (55) reads
η = η0
η = η0
1
(1 − φ / φm )
2
1
(1 − φ / φm )
2.5φm
(8)
(9)
155
respectively, where φm is the maximum volume fraction possible (Fig. 2a). For
156
example, spheres can be packed with a maximum volume fraction of π / √18 ≈ 74 %
157
(43). The formulas (6), (7) and (9) and, with some restriction, formula (8), have the
158
properties that they converge to Einstein’s formula in the case of low φ (as shown
159
below) and obviously diverge in the limit φ → 1 and φ → φm, respectively. In this limit,
160
the suspension becomes practically solid and cannot flow any longer, so that the
161
viscosity becomes infinite. A maximum volume fraction less than 1 can easily be
162
included in Saitô's and Gillespie’s formulas as well by replacing φ by φ/φm.
163
The case of low φ can be treated by a Taylor expansion up to the linear term. In the
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149

2.5φ 
η = η0 1+

 (1 − φ ) 
8
164
case of the Saitô equation, this reads:
η ( φ )  η0 +
165


∂η
2.5
|φ= 0 φ = η0 1+
φ  = η0 (1+ 2.5φ )
2


∂φ
 ( 0 − 1) 
(10)
166
This shows that the formula is consistent with Einstein's equation for low φ.
167
The proof for the case of the Gillespie equation is as follows:
η ( φ )  η0 +
168

− (5 + 0) 
∂η
|φ=0 φ = η0 1 +
φ  = η0 (1 + 2.5φ )
 2 ( 0 − 1)3 
∂φ


(11)
For the case of Quemada's formula, we can write a Taylor series, as φ / φm is then a
170
small parameter,
171
η ( φ )  η0 +

∂η
2
|φ=0 φ = η0 1 +
 (1 − 0 / φ )2+1 φ
∂φ
m
m



2
φ  = η0 1 +

 φm


φ |
= η0 (1 + 2.5φ )
φ =0.8
 m
172
(12)
173
Thus, Quemada's formula is consistency with Einstein’s formula only for a maximum
174
volume fraction of 80 %.
175
For the Krieger-Dougherty formula, we obtain the more general result:
176
177
η ( φ )  η0 +

2.5φm
∂η
|φ=0 φ = η0  1 +
2.5φm+1

∂φ
φm
 (1 − 0 / φm )

φ  = η0 (1 + 2.5φ )


(13)
In hemorheology, sometimes exponential functions are used:
η = η0 e pφ+q
178
(14)
179
(Fig. 2b) as has first been proposed by Arrhenius (2). However, they have the
180
disadvantage not to diverge in the limit φ → φm. To be consistent with Einstein’s
181
formula in the case of low φ, the parameters must be chosen as follows: p = 2.5,
182
q = 0, as can again be shown by a Taylor series.
183
η ( φ )  η0 +
∂η
|φ=0 φ = η0 (1 + e 0∗2.5+0 0 ) φ = η0 (1 + 2.5φ )
∂φ
(15)
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169
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184
Nearly exponential curves with p values of about 2.5 have indeed been measured in
185
experiments (15, 110).
186
The question arises what the theoretical basis for the exponential function is. One
187
possible explanation is that it is a simple function showing the appropriate
188
monotonicity properties. Another explanation would be that it is the solution of a
189
differential equation
190
an increase in the volume fraction is proportional to the volume fraction.
191
To comply with a divergence at high volume fractions, Mooney (62) proposed the
192
following formula:
(16)
(Fig. 2b). At low φ, we can write:
η ( φ )  η0 +
195
∂η
|φ=0
∂φ

  2.5
2.5 ∗ 0
φ = η0 1 + 
+
2
  1 − 0 / φm (1 − 0 / φm ) φm


2.5 ∗ 0
 1− 0 / φ
m
e




 φ = η0 (1 + 2.5φ )



(17)
196
197
Formula based on experimental data
198
Pries (79) has compiled own and literature data to a description of relative apparent
199
blood viscosity as a function of tube diameter and hematocrit. The combined data
200
base comprises measurements at high shear rates in tubes with diameters ranging
201
from 3.3 to 1,978 µm at hematocrit values of up to 90 %. It includes also the
202
Fåhræus-Lindqvist effect (see below), which implies a significant decrease of
203
apparent blood viscosity in tubes with diameters ranging between ~500 and 50 µm.
204
This is important because human blood vessels exhibit diameter variations over four
205
orders of magnitude ranging from ~3 cm in the large systemic vessels down to 3 µm
206
in skeletal muscle capillaries (27). The hematocrit-viscosity relationship (φ and ηrel) is
207
described by the steepness B(D) and curvature C(D) in relation to the tube diameter
208
D:
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2.5φ
1 − φ / φm
η = η0 e
193
194
∂η
= const.φ , which means that the increase in viscosity upon
∂φ
10
(
ηrel = 1 + B ( D ) ∗ (1 − φ )
209
C( D)
)
−1
(18)
210
(Fig. 2c). The parameter C describes the curvature of the relationship between
211
relative apparent blood viscosity and hematocrit. Its dependence on D had been
212
fitted by the empirical equation:


1
1
 −1 +
C ( D) =
+
12
D
D12
1 + 11 
1 + 11
10
10

213


 ( 0.8 + e−0.075∗D )



(19)
After solving the relative viscosity eq. (18) (with a measured hematocrit value of
215
45 %) for B (D)
(
ηrel0.45 ( D ) = 1 + B ( D ) ∗ (1 − 0.45 )
216
217
C( D)
)
−1
(20)
one obtains:
ηrel0.45 ( D ) − 1
B ( D) =
218
(1 − 0.45)
C( D)
(21)
−1
219
With substitution of ηrel0.45 (D) from a fit of experimental data with a hematocrit of
220
45 %
ηrel0.45 ( D ) = 3.2 − 2.44e −0.06∗D
221
222
223
224
225
0.645
+ 220e −1.3∗D
(22)
into the equation (21) results in:
B ( D) =
2.2 − 2.44e−0.06∗D
0.645
(
+ 220e −1.3∗D
W+( −1 +W ) 0.8 + e−0.075∗D
−1 + 0.55
with W =
)
1
D12
1 + 11
10
(23)
Substitution of the eqs. (19) and (23) into eq. (18) results in:
ηrel ( D,φ ) = 1 +
(
2.2 − 2.44e −0.06D
(
)

W+ −1 +W ) 0.8 + e −0.075D 
+ 220e −1.3D  −1 + (1 − φ ) (



−0.075D
W+( −1 +W ) 0.8 + e
−1 + 0.55
0.645
)
(
)
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214
11
(24)
226
227
2.3. The case of inhomogeneous concentration
Above, we have assumed that the concentration of particles is the same everywhere
229
in one given artery. However, when a suspension is flowing, the concentration is
230
becoming inhomogeneous. The mechanistic explanation is that the velocity of the
231
fluid is different at different distances from the artery wall, so that each particle is
232
influenced by different velocities. Different velocities cause different pressures
233
(Bernoulli effect), so that a force acting on the particles in the direction perpendicular
234
to the artery axis results. This force is directed such that the particles are driven
235
towards the centre of the artery because the pressure is the lower the higher the
236
velocity is.
237
An explanation arising from irreversible thermodynamics is the Principle of minimum
238
entropy production (30). It says that non-equilibrium systems that are not too far from
239
thermodynamic equilibrium tend, at constant boundary conditions, to a state at which
240
entropy production is minimal. The flow of a liquid is an irreversible process because
241
mechanical energy is permanently converted into heat due to the inner friction of the
242
liquid. In other words, the flow is producing entropy. If the particles are enriched in the
243
centre, the regions where the velocity gradient is high (low) show a low (high)
244
viscosity (Fig. 1b). Thus, the total entropy production due to inner friction is lower
245
than if the viscosity were the same everywhere. For the case of flows of suspensions,
246
this phenomenon is called Fåhræus-Lindqvist effect (cf. 25).
247
2.4. The optimality principle
248
To phrase the optimality principle in a general way, we take into account that the
249
substance of interest, for example, purine nucleotides, may not only be transported
250
by erythrocytes but also in the plasma. This case is also relevant for insects feeding
251
on blood proteins, which occur both in erythrocytes and in the plasma (16) and for
252
oxygen transport in organisms without red blood cells, for example, arthropods.
253
The optimality principle can be written as a maximization of the flow of the substance
254
of interest:
255
maximize
J substance =  a (1 − φ ) + bφ J ( η ( φ ) )
(25)
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228
12
256
subject to the side constraint
0 ≤ φ ≤1
257
(26)
J is the blood flow as calculated by the Hagen-Poiseuille law (eq. 4). This depends on
259
the viscosity, which in turn depends on hematocrit. Constraint (26) is of importance
260
because otherwise unrealistic values of φ>1 could be obtained.
261
φ is a factor in front of J (formula 25) because the amount of transported substance is
262
proportional to the number of erythrocytes. The proportionality constant is here
263
denoted by a, although it is not really relevant because it does not affect the
264
optimum. The same optimality principle is relevant for the transport of purine
265
nucleotides.
266
Substitution of the Hagen-Poiseuille law (4) into eq. (25) results in:
 πΔpR 4 
J substance =  a (1 − φ ) + bφ 

 8η ( φ ) l 
267
(27)
268
It can be seen that φ enters the equation at least twice: once in the numerator and
269
once in the denominator. This can lead to the occurrence of an optimum.
270
3. Results
271
In Section 3, we focus on the case of oxygen transport. Since the transport of
272
oxygen in the blood plasma can be neglected in mammals, we can put the parameter
273
a in eq. (27) equal to zero. Interestingly, the remaining parameter b then does not
274
affect the optimum because it is a proportionality factor. In the general case, the ratio
275
between the substance concentrations in the erythrocytes and plasma needs to be
276
known.
277
278
279
3.1. Einstein’s formula
We now try to substitute the formula (5) into the eq. (27) for the oxygen flow:
J oxygen =
πΔpR 4 bφ
8η0 (1+ 2.5φ ) l
(28)
280
We now look for a maximum of Joxygen with respect to varying φ. It becomes clear that
281
all the factors such as Δp, R4, l etc. do not play any role. In the Hagen-Poiseuille law,
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258
13
282
it is only important for our purpose here that the flow is proportional to 1 / η. The
283
function
φ
1+ 2.5φ
284
(29)
is monotonic increasing in the form of a saturation function (Fig. 3a). So, it would lead
286
to the erroneous result that φ should have its maximum value, 100 %. Then,
287
however, the blood would consist of erythrocytes only. The reason for this failure of
288
the calculation is that Einstein’s equation cannot be applied to our case because
289
blood is not a dilute suspension, that is, the concentration of erythrocytes is not low
290
enough to allow usage of this equation.
291
292
3.2. Saitô' formula
Now we substitute Saitô's formula in the equation for the oxygen flow:
J oxygen =
293
const.φ

φ 
η0 1+ 2.5

(1 − φ) 

(30)
294
As the function is continuous and differentiable (Fig. 3a), we can find the maximum
295
by differentiation.
∂J oxygen
296
297
298
299
300
∂φ
  2.5
2.5φ
 −φ 
+
2
const.   1 − φ (1 − φ )
=

2
η0 
 2.5φ 
1 +


 1− φ 




+
1 − 2φ − 1.5φ2
=0
1+1.5φ2



1
=0
 2.5φ  
1 +

 1 − φ  

(31)
(32)
Equating the numerator with zero leads to a quadratic equation, which has two zeros:
φ1,2 =
1
−2 ± 10
3
(
Only the positive solution is relevant. We obtain
)
(33)
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285
14
φopt =
301
−2 + 10
≈ 0.387
3
(34)
302
Thus, an optimal hematocrit of about 39 % is obtained. This is already quite near to
303
the experimentally determined value of 40 %.
304
305
3.3. Gillespie’s formula
Now we substitute Gillespie’s formula:
J oxygen =
306
const.φ (1 − φ )
2
η0 (1+ φ / 2 )
(35)
It can be seen that this formula has a maximum because it is zero for φ = 0 and φ = 1
308
and positive in between (Fig. 3a). As the function is continuous and differentiable, we
309
can find the maximum by differentiation.
∂J oxygen
310
∂φ
2
2 1

1 − φ ) − 2φ (1 − φ )  (1 + φ / 2 ) − φ (1 − φ ) 

const.  (
2 =0
=


2
η0 
(1+ φ / 2 )



(36)
311
The numerator involves the factor (1 - φ). Therefore, φ = 1 is a solution, but certainly
312
not a maximum. The remainder of the numerator gives
313
(1 − φ − 2φ )(1 + φ / 2 ) − φ (1 − φ ) / 2 = 0
314
1 + φ / 2 − 3φ − 3φ2 / 2 − φ / 2 + φ2 / 2 = 0
315
1 − 3φ − φ 2 = 0
316
317
318
319
(37)
(38)
(39)
This quadratic equation has two zeros:
3
9 4
3
13
φ1,2 = − ±
+ =− ±
2
4 4
2
2
(40)
Only the positive solution is relevant. We obtain
φopt =
−3 + 13
≈ 0.303
2
(41)
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307
15
320
This is fairly close to the experimentally observed value of 40 %, but less close than
321
the result from Saitô's formula.
322
323
3.4. Krieger-Dougherty formula
Now we try the Krieger-Dougherty formula:
J oxygen =
324
const. φ (1 − φ / φm )
2.5φm
(42)
η0
325
This has a maximum as well because it is zero for φ = 0 and φ = φm (Fig. 3a).
326
Differentiation gives
327
328
∂φ
=
const. 
2.5φ
2.5φ −1
(1 − φ / φm ) m − 2.5φm φ (1 − φ / φm ) m / φm  = 0

η0 

(43)
We can divide by the term in parentheses to the power of 2.5 φm -1 and obtain:
329

φ
1 −
 φm

 = 2.5φ

(44)
330
 2.5φm +1 
1= φ

 φm

(45)
331
 φm

φ= 

 1+ 2.5φm 
(46)
332
The solution strongly depends on the packing density of blood. It can be assumed
333
that this is the range from the packing density of spheres, π / √18, and the maximum
334
value of 100 %. The latter can, as an extreme case, nearly be reached when blood is
335
centrifuged.
336
In the extreme case where φm = 1, the optimal value would be
337
338
339
φopt =
2
≈ 0.286
7
(47)
and for the sphere packing, where φm = π / √18, we obtain
φ1,opt =
2π
6 2 + 5π
≈ 0.260
(48)
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∂J oxygen
16
340
The other solution is a minimum.
341
Generally, when φm < 1, then φopt would be smaller than the value given in Eq. (47).
342
In any case, it is smaller than the result based on Gillespie’s formula.
343
3.5. A surprisingly simple solution
344
A solution that is very near to the experimental value can be derived as follows. The
345
factor 2.5 in Einstein’s formula would give the desired result of 40 % just by dividing 1
346
by 2.5. This results indeed from the formula proposed by Arrhenius (2):
η = η0 e 2.5φ
347
Substituting this formula into the equation for the oxygen flow gives
J oxygen ∝ φe −2.5φ
349
(50)
350
This formula has a maximum, since it is zero for φ = 0, tends to zero for φ → ∞ and is
351
positive in between (Fig. 3b). The question is whether the maximum lies at a value
352
smaller than 1. Differentiation gives
∂J oxygen
353
∂φ
= e −2.5φ − 2.5φe −2.5φ = 0
1 − 2.5φ = 0
354
φopt = 0.4
355
(51)
(52)
(53)
356
This is a surprisingly simple and excellent solution. The derivation does not take into
357
account the criterion that η should diverge for φ → 1 (or even earlier, when the
358
maximal packing density φm of cells is reached).
359
The more complex formula (16) of Mooney leads to results less consistent with reality
360
(see Fig. 3b and Tab. 2).
361
362
363
3.6. Pries formula
Now we try the Pries formula:
J oxygen =
const. φ
η0 ηrel ( D,φ )
(54)
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348
(49)
17
364
For values of D > 1,000 µm we obtain an optimal hematocrit of
φopt ≈ 0.392
366
Note that the Pries formula has an asymptotic behaviour such that the viscosity does
367
not practically depend on D anymore for D values above 1,000 µm.
368
The good agreement of the calculated optimal value with the real hematocrit is not
369
surprising, since the Pries formula is based on experimental data. Interesting is the
370
region below 1,000 µm, were we obtain optimal values from 0.392 up to 1 (see
371
Fig. 4). The Fåhræus-Lindqvist effect, which occurs mainly at tube diameters of less
372
than 300 µm, leads to an increase in the optimal hematocrit (see Discussion for
373
further explanation).
374
3.7. Possible extensions of the theory
375
An approximative solution in the spatially inhomogeneous case
376
Now we include the Fåhræus-Lindqvist effect. The extreme situation is reached when
377
all particles are shifted to the middle axis and compressed there so that a quasi-solid
378
cylinder occurs (Figs. 1c and 5). This has indeed been discussed for the case of
379
blood for sufficiently wide vessels (26, 37, 38, 78). The cylinder (with radius R0) can
380
have a certain packing density, φ*, e.g. the maximal packing density, φm. This
381
cylinder then moves as a whole without velocity gradients in its interior. Outside of it,
382
the pure liquid (water, or blood serum in our case) is flowing.
383
The value φ is now an averaged, effective quantity. It is related to the packing density
384
φ* in the moving cylinder by the following formula, which corresponds to the cross-
385
section area of the blood vessel:
386
387
388
389
πR02 φ* = πR 2 φ
(55)
This allows us to calculate the radius of the cylinder:
R0 =
φ 2
R
φ*
(56)
We use the formula (1) and substitute r within formula (56)
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365
18
φ

R2 −  R2 
 φ*  Δp
v=
4ηl
390
391
392
393
After integration over the entire tube radius we obtain:
J=
πΔpR 4 ( φ − φ*)
4ηlφ*
πΔpR 4 
φ
1 − 
4ηl  φ* 
(58)
With the formula (25) for the optimality principle we can write:
J oxygen =
∂J oxygen
395
=
∂φ
const. φ 
φ
1 − 
η0  φ* 
=−

1
φ
φ+ 1 −  = 0
φ*
 φ* 
2φ
=1
φ*
396
φopt =
397
φ*
2
(59)
(60)
(61)
(62)
398
The maximal packing density depends on the shape of the particles. Let us assume
399
that the erythrocytes are deformed approximately into spheres, so that the maximal
400
packing density is π / √18. This yields
φopt = 37 %
401
(63)
402
which is very near to the experimental value of 40 %. If it were assumed that the
403
erythrocytes are compressed completely (like in the centrifuge), φ* would be 100 %,
404
so that φopt = 50 % would be obtained.
405
Influence of defective erythrocytes
406
As mentioned above, the observed hematocrit in humans differs between women and
407
men. Typical values are 40.0 ± 2.4 % and 45.8 ± 2.7 %, respectively (27, 50, 51). A
408
straightforward explanation is menstruation. This causes a periodic outflow of
409
erythrocytes including defective ones in women. Since they are replaced by new
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394
(57)
19
410
erythrocytes, the percentage of defective erythrocytes is lower in women than in
411
men. This explanation is supported by the observation that the hematocrit in women
412
increases after the menopause (10).
413
To compute the optimal value in the presence of defective erythrocytes, eq. (27) (with
414
a = 0) needs to be modified. Let φint and φdef denote the hematocrit values
415
corresponding to intact and defective erythrocytes, respectively. Then the numerator
416
in eq. (27), which describes oxygen transport, should involve φint only. In contrast, the
417
demoninator, which corresponds to the viscous flow, should involve the sum φint +
418
φdef, so that eq. (27) should be extended as:
J oxygen
(64)
420
This equation can now be used to compute the optimal values based on the different
421
formulas given in Table 2. However, this is beyond the scope of this paper and will be
422
done in a sequel paper. Here, we show, by way of example, the calculation for the
423
Arrhenius equation. φdef is assumed to be a given parameter determined by
424
physiological properties.
J oxygen ∝ φint e
425
426
(65)
The optimum is computed to be
∂J oxygen
427
( +φdef )
−2.5 φint
∂φint
=e
( +φdef )
−2.5 φint
− 2.5φint e
428
1 − 2.5φint = 0
429
φint, opt = 0.4
( +φdef )
−2.5 φint
=0
(66)
(67)
(68)
430
It can be seen that the optimal total hematocrit is, in the case of the Arrhenius
431
equation, simply the sum of the ideal optimal hematocrit (in the case where no
432
defective erythrocytes would occur) and the volume fraction of defective erythrocytes:
433
434
φtot, opt = 0.4 + φdef
(69)
Case where some substance is transported in the plasma
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419


πΔpR 4
=  a (1 − φint ) + bφint  

 8η ( φint + φdef ) l 
20
435
As mentioned in Section 2, the substance of interest may not only be transported by
436
erythrocytes but also in the plasma. To analyse this case, non-zero values of the
437
parameter a should be used. Again, the different formulas given in Table 2 can be
438
used. This will be done in a sequel paper. Here, we only use, by way of example,
439
the Arrhenius equation. We have
J substance ∝  a (1 − φ ) + bφ  e −2.5φ
440
441
(70)
The optimum is computed to be
443
[ −1+ b] − 2.5 a (1 − φ ) + bφ = 0
φopt =
444
445
446
7a − 2b
5a − 5b
(71)
(72)
(73)
In the special case a = 0, this leads again to Eq. (53).
4. Discussion
447
Here, we have derived theoretical optimal values of hematocrit in vertebrates and
448
collected, from the literature, experimentally observed values for 57 animal species.
449
The theoretical values are based on different formulas for the dependence of
450
viscosity on the volume fraction of suspended particles taken from the literature. We
451
have shown that relatively simple approaches based on Newtonian fluid dynamics
452
provide results that are consistent with experimentally observed values. Although
453
blood is, of course, a non-Newtonian fluid, the simplification to Newtonian properties
454
appears to be justified here.
455
Similar considerations are certainly of interest also in the suction of nectar by insects
456
(cf. 15, 39, 51, 52, 71, 72) and in technological applications with respect to
457
suspensions other than blood. This concerns, for example, the question which
458
concentration a cement-water suspension should have to enable a maximum
459
pumping flow of cement along tubes (65, 85).
460
We started with trying Einstein’s formula for dilute suspensions. Although this has not
461
led to realistic results, we included it into the paper to show the step-wise way of
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442
∂J substance
= [ − a + b ] e −2.5φ − 2.5  a (1 − φ ) + bφ  e −2.5φ = 0
∂φ
21
finding good descriptions in science.
463
The observed values differ considerably among species, ranging from 19 %
464
(estuarine crocodylus) to 63 % (weddell seal). Also the theoretical values differ in
465
about that range. The deviations between theoretical and observed values may
466
provide an indication of the extent to which other optimality criteria are relevant as
467
well, so that a trade-off must have been found in evolution, as discussed earlier in the
468
case of optimal stoichiometries of metabolic pathways (107). As for the weddell seal
469
and beluga (white) whale, the rather large hematocrit value may be due to an
470
additional criterion relevant for diving animals. Since they have to store as much
471
oxygen as possible before diving, the storage capacity of the blood for oxygen needs
472
to be particularly high. However, this does not appear to be a consistent feature for
473
all diving animals since the killer and beluga whales show hematocrit values below
474
50 % (Table 1). Also for birds, the values differ considerably. This might be due to a
475
different activity, although Optimal Hematocrit Theory would not predict any
476
dependency on activity. It may be assumed that again additional criteria are relevant.
477
The exponential function (Arrhenius' formula) and Saitô's formula lead to the
478
surprisingly simple solutions of 40 % and 39 %, respectively, which excellently match
479
the observed values in humans, chimpanzee, gorilla, rabbit, cat, pig, and several
480
other species. Mooney's equation leads to a value of 23 % (assuming a maximal
481
packing density of 1), which is in perfect agreement with the observed value in the
482
rainbow trout. The Krieger-Dougherty and Gillespie formulas yield about 30 %,
483
matching with llama, tiger, armadillo, pea fowl and quail.
484
The question arises why more complex formulas such as the Gillespie and Krieger-
485
Dougherty equations do not give such good results for the species with high
486
hematocrit as the simple exponential function, while they may be relevant for species
487
with lower hematocrit.
488
A possible answer is that blood is a very complicated fluid involving a lot of effects,
489
while most of the formulas used here for the dependence of viscosity on volume
490
fraction had been derived for other types of suspensions such as plastic globules or
491
cement (60, 61, 63, 69, 81, 85). Erythrocytes are not usually spheres. In humans and
492
many animals they are biconcave disks, while in camels and llamas, they are
493
elliptical (97). They aggregate to each other, orient and are deformed in flowing
494
blood. Brinkman et al. (9) found that erythrocyte aggregation could be divided into
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462
22
four types: no rouleaux formation type (ox, sheep, and goat), slight rouleaux
496
formation type (rabbit), moderate type (human, pig, dog, cat, and rat), and excessive
497
type (horse) (cf. 66). Due to all these effects, blood is a non-Newtonian fluid. That
498
means that viscosity depends on the velocity gradient.
499
Interestingly, for complex situations or processes, simple formulas sometimes lead to
500
better results than complicated ones, even if the simple ones do not have a firm
501
theoretical basis. This is, in fact, the essence of modelling, since a model is a
502
simplified representation of some aspect of reality. Different models can be built for
503
the same process, and it is decided by the practical application which one works
504
best. As mentioned above, in hemorheology, many effects play a role such the
505
dependence on flow velocity, diameter etc. Here, we have simplified things
506
considerably to concentrate on the essential properties. In spite of these
507
simplifications, realistic results can be derived in optimal hematocrit theory.
508
Red blood cells of camels and llamas are elliptical, as contrasted to the disc-shaped
509
biconcave red cells of other mammals (97). The resistance to flow which is offered by
510
a suspension of asymmetric particles, such as blood cells, must depend upon their
511
orientation with respect to the direction of flow. It is possible that elliptical cells might
512
orient in the direction of flow more easily than discoidal cells, but with the methods
513
used in this study we did not find differences in viscosity that could be attributed to
514
the difference in shape. The rationale behind the study was that the elliptical cells of
515
camel and llama might be of advantage in their respective environments, desert and
516
high altitude (97). The camel is confronted with intense heat and potential
517
dehydration with hemoconcentration, and the llama encounters low oxygen
518
pressures where a high red cell concentration is an advantage.
519
It is interesting to discuss the physiological advantages of the emergence of
520
erythrocytes during evolution. One advantage over oxygen-transporting molecules
521
dissolved in the plasma is the Fåhræus-Lindqvist effect, which allows the blood cells
522
to concentrate in the centre of the vessels and, thus, to decrease effective viscosity.
523
Moreover, the kidneys can filter blood cells much more easily than heme molecules
524
(11). A further advantage is that the interaction of heme with other molecules is
525
avoided.
526
A phenomenon worth being discussed here is blood doping (12, 49, 88, 90). First of
527
all, we stress that we are against such a practice for legal and medical reasons.
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495
23
528
Nevertheless, it is of academic interest to discuss why this doping is efficient (within
529
certain limits) in spite of the assumed optimality property of hematocrit in the
530
physiological standard situation. This assumption is supported by the observation
531
that the hematocrit is nearly the same in active sportsmen and control persons (88).
532
Böning et al. (12) called the effect of blood doping the Hematocrit Paradox.
533
Blood doping (induced erythrocythemia) is the intravenous infusion of more
534
concentrated blood to produce an increase in the blood’s oxygen carrying capacity.
535
This can increase the hemoglobin level and hematocrit by up to 20 %. Does this
536
falsify our above calculations? There are at least three factors to be considered.
1. The addition of erythrocytes increases blood volume. As the blood vessels are
538
rather elastic, they can be dilated, so as to take up the increased volume. This,
539
however, increases the parameter R (artery radius) in our calculations.
540
2. We assumed the pressure difference, Δp, to be constant. However, the heart does
541
not produce the same pressure under all conditions. If blood viscosity and, thus, the
542
resistance against heart contraction increase, the heart tries to pump harder, to
543
reach the same blood flow velocity. However, this is possible only over a small
544
range of viscosity above the optimal one. Above that range, the increased viscosity
545
indeed reduces oxygen transport. And even in the range where increasing the
546
hematocrit seems to be beneficial, the heart is more stressed. There is a
547
hyperviscosity syndrome, which can lead to heart failure in the long term.
548
3. We cannot completely be sure that the hematocrit found in humans or other animals
549
is really optimal. One has to be careful that no circular reasoning is used. Such a
550
reasoning would be to ask whether the hematocrit is optimal, then try various
551
calculations until one of them gives the experimental value and then say the answer
552
is positive - the hematocrit is optimal. It might be that the hematocrit is slightly below
553
the optimum in order to give the organism the chance to realize a better oxygen
554
flow under special circumstances. For example, at high altitudes, where oxygen
555
pressure is lower, the hematocrit indeed increases. The hematocrit also changes in
556
camels during longer dry seasons and after drinking much water (95).
557
4. Böning et al. (12) mention the following additional potential factors: augmented
558
diffusion capacity for oxygen in lungs and tissues because of the enlarged red cell
559
surface area, increased buffer capacity, vasoconstriction, reduced damage by
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537
24
560
radicals, and even perhaps placebo effects. They suggested that blood doping has
561
multifactorial effects not restricted to the increase in blood oxygen content.
In summary, blood doping is dangerous (and anyway illegal). One of the risks in the
563
hyperviscosity syndrome is blood clotting inside of the blood vessels due to higher
564
density of erythrocytes.
565
It is an empirical fact that hematocrit increases at higher altitudes. In the case of
566
humans, this can be easily observed in mountaineering (39, 89, 100); for the llama,
567
see Table 1. At first sight, this is intuitively understandable because the lower oxygen
568
pressure should be compensated. One of the most documented physiological
569
adaptations to a reduced O2 uptake is the increased release of erythropoietin, which
570
causes an increase in red blood cell mass (91, 100). On the other hand, it may seem
571
to contradict optimal hematocrit theory because a higher hematocrit makes the blood
572
more viscous. The paradox can be resolved by the observation that the increase in
573
hematocrit at higher altitudes is partly due to a decrease in the amount of blood
574
plasma (cf. 89, 96). This is first caused by dehydration and later by a shift of
575
intravasal fluid to the interstitial space. Thus, the total blood volume decreases, while
576
in the above calculations, blood volume was considered constant. A lower volume
577
makes it easier for the heart to drive blood circulation, so that an increase in viscosity
578
can be tolerated.
579
There are a number of additional effects. It has been shown that the native
580
highlander is characterised by a larger pulmonary diffusion capacity (17) and
581
adaptations in the structural and metabolic organisation of skeletal muscle that result
582
in a tighter coupling between ATP hydrolysis and oxidative phosphorylation (42).
583
From the above reasoning about blood doping and adaptation to high altitudes, it
584
may be concluded that the observed hematocrit values are slightly suboptimal.
585
Suboptimal states were also discussed in the optimal kinetics of oxygen binding to
586
hemoglobin (109) and in the optimal degree of redundancy in metabolism (56, 96).
587
This allows to increase the hematocrit under stress conditions and, thus, to reach a
588
better performance. This is worth being studied further.
589
At the end of the Results section, we have outlined several interesting extensions of
590
the theory. One extension concerns the consideration of spatially heterogeneous
591
distributions of erythrocytes. Here, we have modelled the Fåhræus-Lindqvist effect by
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562
25
considering the extreme case where all particles move in the middle of the blood
593
vessel as a quasi-solid cylinder. This leads to higher optimal hematocrit values than
594
in spatially homogeneous suspensions. This can be explained as follows: The
595
cylinder moves as a whole without velocity gradients in its interior. Thus, a high
596
erythrocyte concentration can be reached without inner friction. At the boundary of
597
the cylinder, a steep velocity gradient occurs because there, viscosity is lower. A
598
similar reasoning can be applied to spatially heterogeneous suspensions in general.
599
To simulate continuous spatial concentration gradients, variational calculus (13, 34,
600
48, 59, 80, 94) can be used (not done here). Both the velocity and the concentration
601
would then be written as functions of the radial coordinate r. From the Principle of
602
minimum entropy production (30), a minimization principle can be written, from which
603
the optimal velocity and concentration profiles can be computed by variational
604
calculus. Averaging the concentration over the cross-section leads to the optimal
605
hematocrit.
606
The second potential extension concerns the difference in the observed hematocrit
607
values of men vs. women. A straightforward explanation is the different percentage of
608
defective erythrocytes in the two genders due to menstruation. A similar effect has
609
been discussed for the case of blood doping by erythropoietin - the percentage of
610
young red blood cells with good functional properties increases. This gender
611
difference can be included in the equations relatively easily.
612
The third generalization concerns the case where the substance of interest is not
613
only transported by erythrocytes but also in the plasma. We have considered this in
614
the equations from the outset, but focussed for most calculations on the case where
615
the substance (e.g. oxygen) is only transported by erythrocytes. The general case
616
requires an additional parameter: the ratio between the substance concentrations in
617
the erythrocytes and plasma.
618
A promising calculation is the following. Based on the theory presented here, the
619
additional pumping power of the heart necessary to achieve the same oxygen
620
transport in the case of high hematocrit values (larger than 40%) can be calculated.
621
Optimal hematocrit theory can be extended also in other ways. For example, non-
622
Newtonian fluid mechanics can be used (16), although Newtonian approaches lead
623
to very good results, as shown here. Moreover, in very small capillaries such as
624
sinusoids in the liver, the theory based on viscous flow is not valid because just one
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592
26
625
erythrocyte can pass at a time. Then, the discrete, corpuscular nature of cells needs
626
to be considered.
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27
627
Acknowledgements
628
We thank Jörn Behre for stimulating discussions and for an idea leading to Eq. (63),
629
and Nadja Schilling for valuable comments on the manuscript. Financial support by
630
the University of Jena and the Virtual Liver Network funded by the German Ministry
631
for Education and Research (BMBF) is gratefully acknowledged.
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28
632
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Goats to Temperature, 0° to 40° C. Journal of Animal Science 17: 326-335, 1958.
635
2. Arrhenius S. The Viscosity of Solutions. The Biochemical Journal 11: 112-133, 1917.
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637
638
3. Bailey DM, Davies B. Physiological implications of altitude training for endurance
performance at sea level: a review. British Journal of Sports Medicine 31: 183-190,
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640
4. Banchero N, Grover RF, Will JA. Oxygen transport in the llama (Lama glama).
Respiration Physiology 13: 102-115, 1971.
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642
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930
Figures
931
Figure 1: a) Parabolic velocity profile (blue) in the case of homogeneous
932
concentration. b) Oblate velocity profile in the case where particles are concentrated
933
in the middle (Fåhræus-Lindqvist effect). c) Extreme situation with a blood thread
934
moving in the middle of the vessel.
935
Figure 2: Plots of viscosity vs. hematocrit based on different formulas. a) Einstein (
936
), Saito (
937
Arrhenius (
938
the direction of arrow (5 µm, 50 µm, 500 µm for solid lines and 10 µm, 100 µm,
939
1,000 µm for dashed lines).
940
Figure 3: Plot of the oxygen transport flux vs. hematocrit. a) Einstein (
941
Gillespie (
942
Mooney (
943
Figure 4: Optimal hematocrit in relation to the tube diameter after Pries et al. 1992.
944
Figure 5: Extreme situation with a blood thread moving in the middle of the vessel.
) and Mooney (
), Quemada (
), Quemada (
) and Krieger-Dougherty (
). b)
). c) For the Pries formula the diameter increase in
) and Krieger-Dougherty (
), Saito (
). b) Arrhenius (
) and
).
Tables
946
Table 1: Literature values of the hematocrit in 57 vertebrate species.
947
Table 2: Optimal Hematocrit values calculated by different formulas.
),
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945
), Gillespie (
36
948
949
950
Disclosures
No conflicts of interest, financial or otherwise, are declared by the author(s).
Author contributions
S.S. and H.S. conception and design of research; S.S. and H.S. analysed formulas;
952
H.S. and S.S. interpreted results of experiments in literature; H.S. prepared figures
953
and tables; H.S. and S.S. drafted manuscript; H.S. and S.S. edited and revised
954
manuscript; H.S. and S.S. approved final version of manuscript.
955
We (H. Stark & S. Schuster) certify that we have made a direct and substantial
956
contribution to the work reported in the manuscript in all of the following three areas:
957
1) conception and design of the study; 2) data acquisition, and analysis and
958
interpretation of the data; 3) drafting of the manuscript or providing critical revision of
959
the manuscript for intellectual content. We also certify that all authors approved the
960
final version of the manuscript.
Downloaded from http://jap.physiology.org/ by 10.220.33.2 on July 28, 2017
951
Figure 1: a) Parabolic velocity profile (blue) in the case of homogeneous concentration. b)
Oblate velocity profile in the case where particles are concentrated in the middle (FåhræusLindqvist effect). c) Extreme situation with a blood thread moving in the middle of the vessel.
Downloaded from http://jap.physiology.org/ by 10.220.33.2 on July 28, 2017
Downloaded from http://jap.physiology.org/ by 10.220.33.2 on July 28, 2017
Figure 2: Plots of viscosity vs. hematocrit based on different formulas. a) Einstein ( ), Saito
( ), Gillespie (
), Quemada ( ) and Krieger-Dougherty (
). b) Arrhenius ( )
and Mooney ( ). c) For the Pries formula the diameter increase in the direction of arrow
(5 µm, 50 µm, 500 µm for solid lines and 10 µm, 100 µm, 1,000 µm for dashed lines).
Figure 4: Optimal hematocrit in relation to the tube diameter after Pries et al. 1992.
Downloaded from http://jap.physiology.org/ by 10.220.33.2 on July 28, 2017
Figure 3: Plot of the oxygen transport flux vs. hematocrit. a) Einstein ( ), Saito ( ),
Gillespie (
), Quemada ( ) and Krieger-Dougherty (
). b) Arrhenius ( ) and
Mooney ( ).
Figure 5: Extreme situation with a blood thread moving in the middle of the vessel.
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Table 1: Literature values of the hematocrit in 57 vertebrate species.
Animal
Hematocrit (%)
Sample
Size
References
Mammalia (Mammals)
63.2
9
Guard & Murrish, 1975
Kangaroo
53.0
1
Bartels et al., 1966
Beluga whale
52.6
46.0
2
3
Shaffer et al., 1997
Dhindsa et al., 1974
Mixed dog breeds
52.1
45.9
44.1
5
12
6
Ohta et al., 1992
Nemeth et al., 2009
Usami et al., 1969
Mongolian Gerbil
49.6
5
Ohta et al., 1992
Leopard Seal
49.2
5
Guard & Murrish, 1975
Mole
47.2
5
Bartels et al., 1969
Hedgehog
47.0
5
Bartels et al., 1969
Tasmanian devil
47.0
1
Bartels et al., 1966
Crabeater Seal
46.5
2
Guard & Murrish, 1975
Mixed rat breeds
45.5
44.5
6
12
Ohta et al., 1992
Nemeth et al., 2009
Human
44.8
44.4
44.0
44.0
40.0
45.3
48
14
21
19
40
36
Ohta et al., 1992
Guard & Murrish, 1975
Usami et al., 1969
Weng et al., 1996
Kameneva et al., 1998
Kameneva et al., 1998
Goat
43.0
28.9
3
12
Yamaguchi et al., 1987
Usami et al., 1969
Mixed rabbit breeds
42.9
39.8
5
6
Ohta et al., 1992
Kato, 1991
Lemur
42.8
42.5
16
13
Dhindsa et al., 1972
Dhindsa et al., 1972
Pronghorn antelope
42.7
4
Dhindsa et al., 1974
Lion
42.5
2
Parer et al., 1970
Elephant
42.1
38.7
35.9
7
5
15
Dhindsa et al., 1972
Riegel et al., 1967
Usami et al., 1969
Cat
42.0
15
Ohta et al., 1992
Gorilla
42.0
2
Riegel et al., 1966
Pig
41.0
22
Weng et al., 1996
Orang utan
40.9
1
Riegel et al., 1966
Killer whale
40.1
3
Dhindsa et al., 1974
Women
Men
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Weddell Seal
39.8
1
Riegel et al., 1966
Galago
39.4
38.4
8
8
Dhindsa et al., 1982
Dhindsa et al., 1982
Sheep
37.0
37.0
9
10
Usami et al., 1969
Weng et al., 1996
Alpaca
36.0
3
Yamaguchi et al., 1987
Shrew
35.5
11
Bartels et al., 1969
Horse
35.0
12
Weng et al., 1996
Cow
34.0
10
Weng et al., 1996
Vicuña
34.0
1
Yamaguchi et al., 1987
Camel
34.0
22.1
2
1
Yamaguchi et al., 1987
Riegel et al., 1967
Llama
30.0
26.8 (3420 m)
24.4 (0 m)
3
3
3
Yamaguchi et al., 1987
Banchero et al., 1971
Banchero et al., 1971
Tiger
29.8
2
Parer et al., 1970
29
4
Dhindsa et al., 1971
Armadillo
Aves (Birds)
Blue-eyed Shag
55.9
8
Guard & Murrish, 1975
Gentoo Penguin
52.6
13
Guard & Murrish, 1975
Pigeon
52.5
8
Munday & Blane, 1961
Adelie Penguin
47.8
11
Guard & Murrish, 1975
Turkey
47.0
Chinstrap Penguin
47.0
2
Guard & Murrish, 1975
South Polar Skua
45.5
11
Guard & Murrish, 1975
Giant Petrel
44.9
12
Guard & Murrish, 1975
Ostrich
42.6
not given Isaacks & Harkness, 1980
Mixed chicken breeds
40.5
30.9
not given Isaacks et al., 1976 (53)
5
Ohta et al., 1992
Mallard Duck
38.4
31
Guard & Murrish, 1975
Guinea Fowl
33.7
4
Isaacks et al., 1976 (55)
Quail
33.5
not given Isaacks & Harkness, 1980
Pea Fowl
29.0
not given Isaacks & Harkness, 1980
Pheasant
24.0
not given Isaacks & Harkness, 1980
not given Isaacks & Harkness, 1980
Crocodilia (Crocodiles)
Estuarine Crocodile
19.2 (~25°C)
11
Wells et al., 1991
Testudina (Turtles) & Serpentes (Snakes)
Green Sea Turtle
38.9 (~25°C)
2
Isaacks & Harkness, 1980
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Chimpanzee
Grass snake
Loggerhead Sea Turtle
37.0 (21-24°C)
6
Munday & Blane, 1961
27.2 (~25°C)
2
Isaacks & Harkness, 1980
Amphibia (Amphibians)
American Bullfrog
27.2 (22°C)
24.5-28.2 (20°C)
24.4-26.9 (5°C)
6
17
17
Withers et al., 1991
Weathers, 1976
Weathers, 1976
Osteichthyes (Bony Fishes)
Yellowfin Tuna
35.0 (25°C)
Rainbow Trout
30.1-37.0 (12°C)
23.0 (15°C)
not given Brill and Bushnell, 1994
24
7
Gingerich et al., 1987
Tetens and Christensen,
1987
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Table 2: Optimal hematocrit values calculated by different formulas.
Einstein 1906; Einstein 1911
Saitô, 1950
Gillespie 1983
Quemada 1977
Krieger & Dougherty 1959
Mooney 1951
Pries et al. 1992
Optimal hematocrit (%)
η = η0 (1+ 2.5φ )
-

φ 
η = η0  1 + 2.5

(1 − φ ) 

38.7
η = η0
η = η0
η = η0
1+ φ / 2
(1 − φ )
2
30.3
1
2
24.7 (φm=π / √18)
33.3 (φm=1)
2.5φm
26.0 (φm=π / √18)
28.6 (φm=1)
(1 − φ / φm )
1
(1 − φ / φm )
η = η0 e2.5φ
40.0
2.5φ
1 − φ / φm
η = η0e
20.7 (φm=π / √18)
23.4 (φm=1)
See equation (24)
39.2 (D=10,000 µm)
39.3 (D=1,000 µm)
39.7 (D=500 µm)
44.5 (D=100 µm)
48.4 (D=50 µm)
60.6 (D=10 µm)
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Arrhenius 1917
Viscosity formula