J Appl Physiol 113: 355–367, 2012.
First published May 17, 2012; doi:10.1152/japplphysiol.00369.2012.
Comparison of various approaches to calculating the optimal hematocrit
in vertebrates
Heiko Stark and Stefan Schuster
Department of Bioinformatics, Friedrich-Schiller University, Jena, Germany
Submitted 23 March 2012; accepted in final form 15 May 2012
optimal hematocrit theory; blood viscosity; Einstein’s equation;
Fåhraeus-Lindqvist effect; adaptation to high altitude
cells (erythrocytes) transport
oxygen. In evolutionary history, oxygen-transporting molecules such as heme are as old as the Bilateria (31, 36).
Erythrocytes belong to the groundplan of the Craniota (108).
Moreover, erythrocytes also transport purine nucleotides
from organs with a nucleotide surplus to organs in which
purines are required (cf. Refs. 55, 85). Obviously, it is
physiologically favorable when a maximum amount of oxygen and nucleotides is transported at given sizes of the
arteries. This can be phrased as an optimality principle.
Such principles are often used in biology, based on Darwinian evolutionary theory. For example, the optimal streamlining of the body shape of different species (shark, barracuda/tuna, and dolphin) has been computed (9). Other examples are the optimal number of letters of nucleotides (4)
in the genome (99), the optimal amount of ATP produced in
metabolic pathways (14, 103, 107), and the optimal leaf
sizes in relation to the environment (72).
IN HIGHER ANIMALS, RED BLOOD
Address for reprint requests and other correspondence: H. Stark, Lehrstuhl
für Bioinformatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2,
07743 Jena, Germany (e-mail: [email protected]).
http://www.jappl.org
The oxygen flow in animals can change by a change in the
erythrocyte-to-blood volume ratio. This ratio is called hematocrit
(cf. Ref. 26). In healthy humans, it amounts to ⬃40%. Note that
there is a difference in most hematological parameters of men vs.
woman; for example, the hematocrit amounts to 45.8 ⫾ 2.7 vs.
40.0 ⫾ 2.4% (27). The sex-related variation in parameters of the
blood is explained by the difference in age distribution of red
blood cells due to menstruation and the subsequent difference in
mechanical properties of blood of premenopausal women and
men (27). The hematocrit values differ considerably across the
animal kingdom (4, 5, 8, 18 –23, 29, 32, 45– 47, 52, 64, 66, 68, 71,
82– 84, 101, 102, 104 –106, 110, 111). The values of several
species are given in Table 1.
The question arises as to whether the hematocrit values
given above are optimal. If the hematocrit is too low, too few
erythrocytes are present to transport oxygen. If it is too high,
the blood is very viscous and cannot flow quickly, so that
oxygen supply to the tissues is again reduced. Within hemorheology (hydrodynamics of the blood), a theoretical framework
called “optimal hematocrit theory” has been established (6, 15,
41, 70, 104, 110). That an optimum has been found in evolution is supported by the observation that the hematocrit values
are quite robust against temperature changes (1, 44) and are
nearly the same in active sportsmen and control persons (88).
On the other hand, there is some dependency on water supply
and on altitude (see below). A general problem in optimality
considerations in biology is that usually several optimality
principles are relevant simultaneously, for which a trade-off
must be found. Here, we focus on the criterion of maximum
oxygen transport velocity, leaving other criteria for future
studies.
These considerations are very important, since oxygen transport is an important factor for physical performance. They are
especially relevant in medicine (e.g., blood transfusion, dialysis, transplants, stent and shunt implants), sports, and mountaineering. For example, the nonalcoholic fatty liver disease
correlates with higher hematocrit levels (58). Regrettably,
some sportsmen use blood doping by infusing additional erythrocytes (3, 7, 35, 49, 67, 90).
It is an interesting question whether optimal hematocrit
theory allows one to calculate hematocrit values that are in
agreement with the observed values in various vertebrate
species. In this paper, we first briefly review previous approaches in that theory. Then we check which empirical or
theoretically derived formulas describing the dependence of
viscosity on concentration in a suspension lead to the best
agreement between the theoretical and observed values. We
consider both spatially homogeneous and heterogeneous distributions of erythrocytes in the blood. By discussing the
results, we critically assess the power and limitations of optimal hematocrit theory.
8750-7587/12 Copyright © 2012 the American Physiological Society
355
Downloaded from http://jap.physiology.org/ by 10.220.33.4 on July 28, 2017
Stark H, Schuster S. Comparison of various approaches to calculating
the optimal hematocrit in vertebrates. J Appl Physiol 113: 355–367, 2012.
First published May 17, 2012; doi:10.1152/japplphysiol.00369.2012.—An
interesting problem in hemorheology is to calculate that volume
fraction of erythrocytes (hematocrit) that is optimal for transporting a maximum amount of oxygen. If the hematocrit is too low, too
few erythrocytes are present to transport oxygen. If it is too high,
the blood is very viscous and cannot flow quickly, so that oxygen
supply to the tissues is again reduced. These considerations are
very important, since oxygen transport is an important factor for
physical performance. Here, we derive theoretical optimal values
of hematocrit in vertebrates and collect, from the literature, experimentally observed values for 57 animal species. It is an interesting
question whether optimal hematocrit theory allows one to calculate
hematocrit values that are in agreement with the observed values in
various vertebrate species. For this, we first briefly review previous
approaches in that theory. Then we check which empirical or
theoretically derived formulas describing the dependence of viscosity on concentration in a suspension lead to the best agreement
between the theoretical and observed values. We consider both
spatially homogeneous and heterogeneous distributions of erythrocytes in the blood and also possible extensions, like the influence
of defective erythrocytes and cases where some substances are
transported in the plasma. By discussing the results, we critically
assess the power and limitations of optimal hematocrit theory. One
of our goals is to provide a systematic overview of different
approaches in optimal hematocrit theory.
356
Optimal Hematocrit in Vertebrates
Table 1. Literature values of the hematocrit in 57 vertebrate
species
Animal
Hematocrit, %
Sample
Size, no.
Mixed dog breeds
Mongolian gerbil
Leopard seal
Mole
Hedgehog
Tasmanian devil
Crabeater seal
Mixed rat breeds
Women
Men
Goat
Mixed rabbit breeds
Lemur
Pronghorn antelope
Lion
Elephant
Cat
Gorilla
Pig
Orangutan
Killer whale
Chimpanzee
Galago
Sheep
Alpaca
Shrew
Horse
Cow
Vicuña
Camel
Llama
Tiger
Armadillo
Reference No.
9
1
2
3
5
12
6
5
5
5
5
1
2
6
12
48
14
21
19
40
36
3
12
5
6
16
13
4
2
7
5
15
15
2
22
1
3
1
8
8
9
10
3
11
12
10
1
2
1
3
3
3
2
4
Guard and Murrish (32)
Bartels et al. (5)
Shaffer et al. (93)
Dhindsa et al. (20)
Ohta et al. (68)
Nemeth et al. (66)
Usami et al. (102)
Ohta et al. (68)
Guard and Murrish (32)
Bartels et al. (5a)
Bartels et al. (5a)
Bartels et al. (5)
Guard and Murrish (32)
Ohta et al. (68)
Nemeth et al. (66)
Ohta et al. (68)
Guard and Murrish (32)
Usami et al. (102)
Weng et al. (106)
Kameneva et al. (50)
Kameneva et al. (50)
Yamaguchi et al. (111)
Usami et al. (102)
Ohta et al. (68)
Kato (52)
Dhindsa et al. (21)
Dhindsa et al. (21)
Dhindsa et al. (21)
Parer et al. (71)
Dhindsa et al. (21)
Riegel et al. (82)
Usami et al. (102)
Ohta et al. (68)
Riegel et al. (83)
Weng et al. (106)
Riegel et al. (83)
Dhindsa et al. (20)
Riegel et al. (83)
Dhindsa et al. (19)
Dhindsa et al. (19)
Usami et al. (102)
Weng et al. (106)
Yamaguchi et al. (111)
Bartels et al. (5a)
Weng et al. (106)
Weng et al. (106)
Yamaguchi et al. (111)
Yamaguchi et al. (111)
Riegel et al. (82)
Yamaguchi et al. (111)
Banchero et al. (4)
Banchero et al. (4)
Parer et al. (71)
Dhindsa et al. (18)
8
13
8
11
not given
2
11
12
not given
Guard and Murrish (32)
Guard and Murrish (32)
Munday and Blane (64)
Guard and Murrish (32)
Isaacks and Harkness (47)
Guard and Murrish (32)
Guard and Murrish (32)
Guard and Murrish (32)
Isaacks and Harkness (47)
Aves (birds)
Blue-eyed shag
Gentoo penguin
Pigeon
Adelie penguin
Turkey
Chinstrap penguin
South polar skua
Giant petrel
Ostrich
55.9
52.6
52.5
47.8
47.0
47.0
45.5
44.9
42.6
Table 1.—Continued
Continued
Mixed chicken
breeds
Mallard duck
Guinea fowl
Quail
Pea fowl
Pheasant
Hematocrit, %
Sample
Size, no.
Reference No.
40.5
30.9
38.4
33.7
33.5
29.0
24.0
not given
5
31
4
not given
not given
not given
Isaacks et al. (46)
Ohta et al. (68)
Guard and Murrish (32)
Isaacks et al. (46)
Isaacks and Harkness (47)
Isaacks and Harkness (47)
Isaacks and Harkness (47)
Crocodilia (crocodiles)
Estuarine crocodile
19.2 (⬃25°C)
11
Wells et al. (105)
Testudina (turtles) and Serpentes (snakes)
Green sea turtle
Grass snake
Loggerhead sea
turtle
38.9 (⬃25°C)
37.0 (21–24°C)
2
6
Isaacks and Harkness (47)
Munday and Blane (64)
27.2 (⬃25°C)
2
Isaacks and Harkness (47)
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. (110)
Weathers (104)
Weathers (104)
Osteichthyes (bony fishes)
Yellowfin tuna
Rainbow trout
Brill and Jones (8); Bushnell
35.0 (25°C)
not given and Jones (13)
30.1–37.0 (12°C)
24
Gingerich et al. (29)
Tetens and Christensen
23.0 (15°C)
7
(101)
Glossary
a
b
D
␩
␩0
␩rel (D,␸)
J
Joxygen
l
⌬P
␸
␸m
␸opt
␸int
␸def
v
Proportionality factor for substances transported in the plasma
Proportionality factor for substances transported in erythrocytes
Tube diameter
Viscosity of fluid (viscosity of suspension)
Viscosity of fluid solvent (so-called embedding
fluid)
Viscosity relationship between D and ␸
Total flow across the blood vessel (HagenPoiseuille law)
Total oxygen flow across the blood vessel
Tube length
Pressure difference
Volume fraction of particles in the suspension
Maximum volume fraction (maximal packing
density)
Optimal volume fraction of particles in the
suspension
Volume fraction of intact particles in the suspension
Volume fraction of defective particles in the
suspension
Fluid velocity
MATERIALS AND METHODS
First, we briefly review some fundamentals from rheology needed
for the subsequent calculations.
J Appl Physiol • doi:10.1152/japplphysiol.00369.2012 • www.jappl.org
Downloaded from http://jap.physiology.org/ by 10.220.33.4 on July 28, 2017
Human
63.2
53.0
52.6
46.0
52.1
45.9
44.1
49.6
49.2
47.2
47.0
47.0
46.5
45.5
44.5
44.8
44.4
44.0
44.0
40.0
45.3
43.0
28.9
42.9
39.8
42.8
42.5
42.7
42.5
42.1
38.7
35.9
42.0
42.0
41.0
40.9
40.1
39.8
39.4
38.4
37.0
37.0
36.0
35.5
35.0
34.0
34.0
34.0
22.1
30.0
26.8 (3,420 m)
24.4 (0 m)
29.8
29
Stark H et al.
Animal
Mammalia (mammals)
Weddell seal
Kangaroo
Beluga whale
•
Optimal Hematocrit in Vertebrates
•
357
Stark H et al.
冋
The Hagen-Poiseuille Law
A central parameter characterizing the flow behavior of a liquid is its
viscosity, ␩. Consider a blood vessel (e.g., an artery) with circular cross
section. Let R and l denote the radius and length of the tubular vessel,
respectively. The blood flow is driven by a pressure difference, ⌬P. A
basic assumption in hydrodynamics, which is well justified by experiment, is that the velocity of the liquid very near to a rigid surface is zero
(or is the same as that of the surface if that is moving). This is because a
thin layer of molecules of the fluid are attached to the surface. Moreover,
for symmetry reasons, we can assume that the velocity of a liquid being
in a stationary flow in a circular tube forms a profile with rotational
symmetry. Thus it depends only on the radial coordinate r (with 0 ⬍ r ⬍
R): v ⫽ v(r). This is justified as long as the flow is nonturbulent, i.e., for
low velocities. To calculate this velocity profile, we can use the equilibrium of forces between driving pressure and resistance due to the viscous
properties of the liquid. It is a well-known result from fluid mechanics
that the velocity obeys Eq. 1 (33, 75–77, 98).
R ⫺r
2
v⫽
2
⌬P
(1)
This is a parabolic velocity profile, in which the highest velocity is
reached in the middle of the blood vessel (Fig. 1, left). It is of interest
to know the total flow across the blood vessel. To calculate this, we
need to integrate over the cross-section, A, of the tube:
J⫽
兰
R2 ⫺ r2
4␩l
J⫽
R
⌬PdA ⫽ ⌬P
冉
R2 ⫺ r2
冊冏
4␩l
0
␲⌬P R2r2
2␩l
兰
2
⫺
r4
4
J⫽
R
⫽
0
␲⌬PR
2␲rdr with J ⫽
冉
␲⌬P R4
2␩l
2
⫺
R4
4
冊
V
t
(2)
(3)
8␩l
Equation 4 is called the Hagen-Poiseuille law (cf. Ref. 26). In a
stationary, nonturbulent flow across a cylindrical tube, the total flux of
liquid is proportional to the fourth power of the tube radius.
Dependence of Viscosity on Hematocrit
Dilute suspensions. The question is now how the viscosity depends
on the hematocrit, ␸. Blood can be considered as a suspension (rather
than a solution) because erythrocytes are much larger than molecules.
The solid phase (erythrocytes) is dispersed throughout the fluid phase
(plasma) through mechanical agitation. Note that other blood cells,
such as leukocytes and thrombocytes, contribute a negligible volume
(⬍1%) compared with erythrocytes (27). For suspensions of spheric
particles with low concentration (but not considering blood at that
time), Albert Einstein (24, 25) derived the formula:
␩ ⫽ ␩0共1 ⫹ 2.5␸兲
(5)
(Fig. 2A) with ␩ ⫽ viscosity of suspension, ␩0 ⫽ viscosity of fluid
solvent (so-called embedding fluid, e.g., water), and ␸ ⫽ volume fraction
of particles in the suspension (i.e., hematocrit in the case of blood).
Suspensions with higher concentrations. Einstein’s formula has
been extended in various ways to cope with higher concentrations. For
example, the formula of Saitô (86) reads
共 1 ⫺ ␸兲
册
(6)
while the formula of Gillespie (28) reads
␩ ⫽ ␩0
1⫹␸⁄2
(7)
共 1 ⫺ ␸兲 2
and the formulas of Quemeda (80) and Krieger and Dougherty (55)
read
␩ ⫽ ␩0
␩ ⫽ ␩0
1
(8)
共 1 ⫺ ␸ ⁄ ␸ m兲 2
1
(9)
共1 ⫺ ␸ ⁄ ␸m兲2.5␸m
respectively, where ␸m is the maximum volume fraction possible (Fig.
2A). For example, spheres can be packed with a maximum volume
fraction of ␲/18 ⯝ 74% (43). Formulas 6, 7, and 9 and, with some
restriction, formula 8, have the properties that they converge to
Einstein’s formula in the case of low ␸ (as shown below) and
obviously diverge in the limit ␸ ¡ 1 and ␸ ¡ ␸m, respectively. In this
limit, the suspension becomes practically solid and cannot flow any
longer, so that the viscosity becomes infinite. A maximum volume
fraction ⬍1 can easily be included in Saitô’s and Gillespie’s formulas,
as well by replacing ␸ by ␸/␸m.
The case of low ␸ can be treated by a Taylor expansion up to the
linear term. In the case of the Saitô equation, this reads:
␩共 ␸兲 ⯝ ␩0 ⫹
⭸␩
⭸␸
4
(4)
2.5␸
冏
冋
␸ ⫽ ␩0 1 ⫹
␸⫽0
2.5
共 1 ⫺ 0兲 2
册
␸ ⫽ ␩0共1 ⫹ 2.5␸兲
(10)
This shows that the formula is consistent with Einstein’s equation for
low ␸.
The proof for the case of the Gillespie equation is as follows:
␩共 ␸兲 ⯝ ␩0 ⫹
⭸␩
⭸␸
冏
冋
␸ ⫽ ␩0 1 ⫹
␸⫽0
⫺ 共 5 ⫹ 0兲
2共 1 ⫺ 0兲 3
册
␸ ⫽ ␩0共1 ⫹ 2.5␸兲
(11)
For the case of Quemada’s formula, we can write a Taylor series, as
␸/␸m is then a small parameter,
␩共 ␸兲 ⯝ ␩0 ⫹
⭸␩
⭸␸
冏
冋
␸ ⫽ ␩0 1 ⫹
␸⫽0
冉
⫽ ␩0 1 ⫹
2
␸m
2
共1 ⫺ 0 ⁄ ␸m兲2⫹1␸m
␸
冊冏
␸m⫽0.8
␸
册
(12)
⫽ ␩0共1 ⫹ 2.5␸兲
Thus Quemada’s formula is consistent with Einstein’s formula only
for a maximum volume fraction of 80%.
For the Krieger-Dougherty formula, we obtain the more general
result:
␩共 ␸兲 ⯝ ␩0 ⫹
⭸␩
⭸␸
冏
冋
␸ ⫽ ␩0 1 ⫹
␸⫽0
2.5␸m
␸
册
共1 ⫺ 0 ⁄ ␸m兲2.5␸m⫹1␸m
⫽ ␩0共1 ⫹ 2.5␸兲
(13)
Fig. 1. Left: parabolic velocity profile in the case of homogeneous concentration. Middle: oblate velocity profile in the case where particles are concentrated in
the middle (Fåhraeus-Lindqvist effect). Right: extreme situation with a blood thread moving in the middle of the vessel.
J Appl Physiol • doi:10.1152/japplphysiol.00369.2012 • www.jappl.org
Downloaded from http://jap.physiology.org/ by 10.220.33.4 on July 28, 2017
4␩l
␩ ⫽ ␩0 1 ⫹
358
Optimal Hematocrit in Vertebrates
•
Stark H et al.
The question arises what the theoretical basis for the exponential
function is. One possible explanation is that it is a simple function
showing the appropriate monotonicity properties. Another explanation would be that it is the solution of a differential equation
⳵␩/⳵␸ ⫽ const. ␸, which means that the increase in viscosity upon
an increase in the volume fraction is proportional to the volume
fraction.
To comply with a divergence at high volume fractions, Mooney
(62) proposed the following formula:
␩ ⫽ ␩ 0e
2.5␸
1⫺␸⁄␸m
(16)
(Fig. 2B). At low ␸, we can write:
␩共 ␸兲 ⯝ ␩0 ⫹
⭸␩
⭸␸
⫹
冏
再 冋
冎
兲 册
␸ ⫽ ␩0 1 ⫹
␸⫽0
2.5 ⴱ 0
共 1 ⫺ 0 ⁄ ␸m
2.5
1 ⫺ 0 ⁄ ␸m
(17)
2.5ⴱ0
2
␸m
e 1⫺0⁄␸m ␸ ⫽ ␩0共1 ⫹ 2.5␸兲
␩rel ⫽ 1 ⫹ B共D兲 ⴱ 关共1 ⫺ ␸兲C共D兲 ⫺ 1兴
(18)
(Fig. 2C). The parameter C describes the curvature of the relationship
between relative apparent blood viscosity and hematocrit. Its dependence on D had been fitted by the empirical equation:
1
C共 D兲 ⫽
1⫹
Fig. 2. Plots of viscosity vs. hematocrit based on different formulas. A: Einstein
(solid line), Saito (dotted line), Gillespie (dashed line), Quemada (dotted-dashed
line) and Krieger-Dougherty (dotted-dotted-dashed line; the latter two lines are
superimposed). B: Arrhenius (solid line) and Mooney (dotted line). C: for the Pries
formula, the diameter increases in the direction of arrow (5, 50, and 500 ␮m for
solid lines, and 10, 100, and 1,000 ␮m for dashed lines).
␩共 ␸兲 ⯝ ␩0 ⫹
⭸␸
冏
␸⫽0
␸ ⫽ ␩ 0共 1 ⫹ e
0ⴱ2.5⫹0
冢
(14)
0兲␸ ⫽ ␩0共1 ⫹ 2.5␸兲
1⫹
D12
1011
冣
共0.8 ⫹ e⫺0.075ⴱD兲
␩rel0.45共D兲 ⫽ 1 ⫹ B共D兲 ⴱ 关共1 ⫺ 0.45兲C共D兲 ⫺ 1兴
(20)
␩rel0.45共D兲 ⫺ 1
(21)
共1 ⫺ 0.45兲C共D兲 ⫺ 1
With substitution of ␩rel 0.45 (D) from a fit of experimental data with
a hematocrit of 45%
␩rel0.45共D兲 ⫽ 3.2 ⫺ 2.44e⫺0.06ⴱD
0.645
⫹ 220e⫺13ⴱD
(22)
into Eq. 21 results in:
B共 D兲 ⫽
2.2 ⫺ 2.44e⫺0.06ⴱD
0.645
⫹ 220e⫺1.3ⴱD
with W ⫽
⫺0.075ⴱD
兲
⫺1 ⫹ 0.55W⫹共⫺1⫹W兲共0.8⫹e
(15)
Nearly exponential curves with p values of ⬃2.5 have indeed been
measured in experiments (15, 110).
(19)
one obtains:
B共 D兲 ⫽
(Fig. 2B) as has first been proposed by Arrhenius (2). However, they
have the disadvantage not to diverge in the limit ␸ ¡ ␸m. To be
consistent with Einstein’s formula in the case of low ␸, the parameters
must be chosen as follows: p ⫽ 2.5, q ⫽ 0, as can again be shown by
a Taylor series.
⭸␩
1011
1
⫹ ⫺1 ⫹
After solving the relative viscosity (Eq. 18) (with a measured hematocrit value of 45%) for B (D)
In hemorheology, sometimes exponential functions are used:
␩ ⫽ ␩0e p␸⫹q
D
12
Substitution of Eqs. 19 and 23 into Eq. 18 results in:
J Appl Physiol • doi:10.1152/japplphysiol.00369.2012 • www.jappl.org
1
1⫹
D12
1011
(23)
Downloaded from http://jap.physiology.org/ by 10.220.33.4 on July 28, 2017
Formula based on experimental data. Pries et al. (79) have compiled their own and literature data to a description of relative apparent
blood viscosity as a function of tube diameter and hematocrit. The
combined data base comprises measurements at high shear rates in
tubes with diameters ranging from 3.3 to 1,978 ␮m at hematocrit
values of up to 90%. It includes also the Fåhraeus-Lindqvist effect
(see below), which implies a significant decrease of apparent blood
viscosity in tubes with diameters ranging between ⬃500 and 50 ␮m.
This is important because human blood vessels exhibit diameter variations over four orders of magnitude, ranging from ⬃3 cm in the large
systemic vessels down to 3 ␮m in skeletal muscle capillaries (27). The
hematocrit-viscosity relationship (␸ and ␩rel) is described by the steepness B(D) and curvature C(D) in relation to the tube diameter D:
Optimal Hematocrit in Vertebrates
␩rel共D, ␸兲 ⫽ 1 ⫹
共2.2 ⫺ 2.44e⫺0.06D
•
359
Stark H et al.
⫹ 220e⫺1.3D兲关⫺1 ⫹ 共1 ⫺ ␸兲W⫹共⫺1⫹W兲共0.8⫹e
⫺0.075D
兲
⫺1 ⫹ 0.55W⫹共⫺1⫹W兲共0.8⫹e
0.645
The Case of Inhomogeneous Concentration
兲兴
(24)
the optimum, because it is a proportionality factor. In the
general case, the ratio between the substance concentrations in
the erythrocytes and plasma needs to be known.
Einstein’s Formula
We now try to substitute formula 5 into the Eq. 27 for the
oxygen flow:
Joxygen ⫽
␲⌬PR4b␸
8␩0共1 ⫹ 2.5␸兲l
(28)
We now look for a maximum of Joxygen with respect to varying
␸. It becomes clear that all the factors such as ⌬P, R4, l, etc.,
do not play any role. In the Hagen-Poiseuille law, it is only
important for our purpose here that the flow is proportional to
1/␩. The function
␸
1 ⫹ 2.5␸
(29)
is monotonic increasing in the form of a saturation function
(Fig. 3A). So it would lead to the erroneous result that ␸ should
have its maximum value, 100%. Then, however, the blood
The Optimality Principle
To phrase the optimality principle in a general way, we take into
account that the substance of interest, for example, purine nucleotides,
may not only be transported by erythrocytes, but also in the plasma.
This case is also relevant for insects feeding on blood proteins, which
occur both in erythrocytes and in the plasma (16) and for oxygen
transport in organisms without red blood cells, for example, arthropods.
The optimality principle can be written as a maximization of the
flow of the substance of interest:
maximize Jsubstance ⫽ 关a共1 ⫺ ␸兲 ⫹ b␸兴J关␩共␸兲兴
(25)
subject to the side constraint
0ⱕ␸ⱕ1
(26)
J is the blood flow as calculated by the Hagen-Poiseuille law (Eq. 4).
This depends on the viscosity, which, in turn, depends on hematocrit.
Constraint (26) is of importance because otherwise unrealistic values
of ␸ ⬎ 1 could be obtained.
Substitution of the Hagen-Poiseuille law (Eq. 4) into Eq. 25 results in:
Jsubstance ⫽ 关a共1 ⫺ ␸兲 ⫹ b␸兴
冋
␲⌬PR4
8␩共␸兲l
册
(27)
It can be seen that ␸ enters the equation at least twice: once in the
numerator and once in the denominator. This can lead to the occurrence of an optimum.
RESULTS
In RESULTS, we focus on the case of oxygen transport. Since
the transport of oxygen in the blood plasma can be neglected in
mammals, we can put the parameter a in Eq. 27 equal to zero.
Interestingly, the remaining parameter b then does not affect
Fig. 3. Plot of the oxygen transport flux vs. hematocrit. A: Einstein (solid line),
Saito (dotted line), Gillespie (dashed line), Quemada (dotted-dashed line), and
Krieger-Dougherty (dotted-dotted-dashed line). B: Arrhenius (solid line) and
Mooney (dotted line).
J Appl Physiol • doi:10.1152/japplphysiol.00369.2012 • www.jappl.org
Downloaded from http://jap.physiology.org/ by 10.220.33.4 on July 28, 2017
Above, we have assumed that the concentration of particles is the
same everywhere in one given artery. However, when a suspension is
flowing, the concentration is becoming inhomogeneous. The mechanistic explanation is that the velocity of the fluid is different at
different distances from the artery wall, so that each particle is
influenced by different velocities. Different velocities cause different
pressures (Bernoulli effect), so that a force acting on the particles in
the direction perpendicular to the artery axis results. This force is
directed such that the particles are driven toward the center of the
artery, because the pressure is lower the higher the velocity is.
An explanation arising from irreversible thermodynamics is the
principle of minimum entropy production (30). It says that nonequilibrium systems that are not too far from thermodynamic equilibrium
tend, at constant boundary conditions, to a state at which entropy
production is minimal. The flow of a liquid is an irreversible process,
because mechanical energy is permanently converted into heat due to
the inner friction of the liquid. In other words, the flow is producing
entropy. If the particles are enriched in the center, the regions where
the velocity gradient is high (low) show a low (high) viscosity (Fig. 1,
center). Thus the total entropy production due to inner friction is
lower than if the viscosity were the same everywhere. For the case of
flows of suspensions, this phenomenon is called Fåhraeus-Lindqvist
effect (cf. Ref. 25).
⫺0.075D
360
Optimal Hematocrit in Vertebrates
would consist of erythrocytes only. The reason for this failure
of the calculation is that Einstein’s equation cannot be applied
to our case because blood is not a dilute suspension; that is, the
concentration of erythrocytes is not low enough to allow usage
of this equation.
•
Stark H et al.
The numerator involves the factor (1 ⫺ ␸). Therefore, ␸ ⫽ 1
is a solution, but certainly not a maximum. The remainder of
the numerator gives
共1 ⫺ ␸ ⫺ 2␸兲共1 ⫹ ␸ ⁄ 2兲 ⫺ ␸共1 ⫺ ␸兲 ⁄ 2 ⫽ 0
(37)
1 ⫹ ␸ ⁄ 2 ⫺ 3␸ ⫺ 3␸ ⁄ 2 ⫺ ␸ ⁄ 2 ⫹ ␸ ⁄ 2 ⫽ 0
(38)
1 ⫺ 3␸ ⫺ ␸ ⫽ 0
(39)
2
Saitô’s Formula
2
Now we substitute Saitô’s formula in the equation for the
oxygen flow:
const . ␸
冋
Joxygen ⫽
␩0 1 ⫹ 2.5
␸
共1 ⫺ ␸兲
册
This quadratic equation has two zeros:
3
␸1,2 ⫽ ⫺ ⫾
2
(30)
⫽
␩0
冦
冋
2.5
⫹
1⫺␸
冉
1⫹
2.5␸
共1 ⫺ ␸兲2
2.5␸
1⫺␸
⫹
冉
1 ⫺ 2␸ ⫺ 1.5␸
冊
册
1
2.5␸
1⫺␸
冊冧
(31)
1
3
⫽0
(33)
Only the positive solution is relevant. We obtain
␸opt ⫽
3
⬇ 0.387
(34)
Thus an optimal hematocrit of ⬃39% is obtained. This is
already quite near to the experimentally determined values.
2
Now we substitute Gillespie’s formula:
const . ␸共1 ⫺ ␸兲2
(35)
␩ 0共 1 ⫹ ␸ ⁄ 2 兲
It can be seen that this formula has a maximum because it is
zero for ␸ ⫽ 0 and ␸ ⫽ 1 and positive in between (Fig. 3A).
As the function is continuous and differentiable, we can find
the maximum by differentiation.
⭸ Joxygen
冦
⭸␸
⫽
2
⬇ 0.303
const . ␸共1 ⫺ ␸ ⁄ ␸m兲2.5␸m
␩0
(40)
(41)
const.
(42)
This has a maximum as well, because it is zero for ␸ ⫽ 0 and
␸ ⫽ ␸m (Fig. 3A). Differentiation gives
⭸ Joxygen
⭸␸
⫽
const.
␩0
关共1 ⫺ ␸ ⁄ ␸m兲2.5␸
m
⫺ 2.5␸m␸共1
⫺ ␸ ⁄ ␸m兲2.5␸m⫺1 ⁄ ␸m兴 ⫽ 0
(43)
We can divide by the term in parentheses to the power of 2.5
␸m ⫺ 1 and obtain:
冉
1⫺
␸
␸m
冉
冉
1⫽␸
␸⫽
Gillespie’s Formula
Joxygen ⫽
兹13
Now we try the Krieger-Dougherty formula:
Joxygen ⫽
(32)
共⫺2 ⫾ 兹10兲
⫺2 ⫹ 兹10
3
⫽⫺ ⫾
4
2
⫺3 ⫹ 兹13
␸opt ⫽
⫽0
Equating the numerator with zero leads to a quadratic equation,
which has two zeros:
␸1,2 ⫽
4
Krieger-Dougherty Formula
2
1 ⫹ 1.5␸2
4
⫹
This is fairly close to the experimentally observed values of
about 40%, but less close than the result from Saitô’s formula.
2
1⫹
9
冊
⫽ 2.5␸
冊
冊
2.5␸m ⫹ 1
␸m
␸m
1 ⫹ 2.5␸m
(44)
(45)
(46)
The solution strongly depends on the packing density of blood.
It can be assumed that this is the range from the packing
density of spheres, ␲/18, and the maximum value of 100%.
The latter can, as an extreme case, nearly be reached when
blood is centrifuged.
In the extreme case where ␸m ⫽ 1, the optimal value would be
␸opt ⫽
2
7
⬇ 0.286
(47)
and for the sphere packing, where ␸m ⫽ ␲/18, we obtain
␩0
关共1 ⫺ ␸兲2 ⫺ 2␸共1 ⫺ ␸兲兴共1 ⫺ ␸ ⁄ 2兲 ⫺ ␸共1 ⫺ ␸兲2
共1 ⫹ ␸ ⁄ 2兲2
1
2
冧
(36)
⫽0
␸1,opt ⫽
2␲
6兹2 ⫹ 5␲
The other solution is a minimum.
J Appl Physiol • doi:10.1152/japplphysiol.00369.2012 • www.jappl.org
⬇ 0.260
(48)
Downloaded from http://jap.physiology.org/ by 10.220.33.4 on July 28, 2017
⭸␸
const.
⫺␸
冑
Only the positive solution is relevant. We obtain
As the function is continuous and differentiable (Fig. 3A), we
can find the maximum by differentiation.
⭸ Joxygen
2
Optimal Hematocrit in Vertebrates
•
361
Stark H et al.
Generally, when ␸m ⬍ 1, then ␸opt would be smaller than the
value given in Eq. 47. In any case, it is smaller than the result
based on Gillespie’s formula.
A Surprisingly Simple Solution
A solution that is very near to the experimental value can be
derived as follows. The factor 2.5 in Einstein’s formula would
give the desired result of 40% just by dividing 1 by 2.5. This
results indeed from the formula proposed by Arrhenius (2):
␩ ⫽ ␩0e2.5␸
(49)
Substituting this formula into the equation for the oxygen flow
gives
Joxygen ⬀ ␸e⫺2.5␸
(50)
⭸ Joxygen
⭸␸
⫽ e2.5␸ ⫺ 2.5␸e⫺2.5␸ ⫽ 0
(51)
1 ⫺ 2.5␸ ⫽ 0
(52)
␸opt ⫽ 0.4
(53)
This is a surprisingly simple and excellent solution. The derivation
does not take into account the criterion that ␩ should diverge for
␸ ¡ 1 (or even earlier, when the maximal packing density ␸m of
cells is reached).
The more complex formula 16 of Mooney leads to results
less consistent with reality (see Fig. 3B and Table 2).
Table 2. Optimal hematocrit values calculated by different
formulas
Reference No.
Einstein (24, 25)
Saitô (86)
Gillespie (28)
Quemada (80)
Viscosity Formula
␩ ⫽ ␩0共1 ⫹ 2.5␸兲
冋
␩ ⫽ ␩0 1 ⫹ 2.5
␩ ⫽ ␩0
␩ ⫽ ␩0
␸
共1⫺␸兲
Mooney (62)
Pries et al. (79)
—
38.7
册
1⫹␸⁄2
共1⫺␸兲2
1
24.7 (␸m ⫽ ␲/18)
共1⫺␸⁄␸m兲
2
33.3 (␸m ⫽ 1)
26.0 (␸m ⫽ ␲/18)
28.6 (␸m ⫽ 1)
␩ ⫽ ␩0e2.5␸
␩ ⫽ ␩0e
2.5␸
1⫺␸⁄␸m
See Eq. 24
See text for definitions of terms.
Pries Formula
Now we try the Pries formula:
Joxygen ⫽
const . ␸
␩0␩rel共D, ␸兲
(54)
For values of D ⬎ 1,000 ␮m, we obtain an optimal hematocrit of
␸opt ⬇ 0.392
Note that the Pries formula has an asymptotic behavior such
that the viscosity does not practically depend on D anymore for
D values ⬎1,000 ␮m.
The good agreement of the calculated optimal value with
the real hematocrit is not surprising, since the Pries formula
is based on experimental data. Interesting is the region
⬍1,000 ␮m, where we obtain optimal values from 0.392 up
to 1 (see Fig. 4). The Fåhraeus-Lindqvist effect, which
occurs mainly at tube diameters of ⬍300 ␮m, leads to an
increase in the optimal hematocrit (see DISCUSSION for further
explanation).
Possible Extensions of the Theory
30.3
Krieger and Dougherty
1
␩ ⫽ ␩0
(55)
共1⫺␸⁄␸m兲2.5␸m
Arrhenius (2)
Optimal Hematocrit, %
Fig. 4. Optimal hematocrit in relation to the tube diameter after Pries et al. (79).
40.0
20.7 (␸m ⫽ ␲/18)
23.4
39.2
39.3
39.7
44.5
48.4
60.6
(␸m ⫽ 1)
(D ⫽ 10,000 ␮m)
(D ⫽ 1,000 ␮m)
(D ⫽ 500 ␮m)
(D ⫽ 100 ␮m)
(D ⫽ 50 ␮m)
(D ⫽ 10 ␮m)
An approximative solution in the spatially inhomogeneous
case. Now we include the Fåhraeus-Lindqvist effect. The
extreme situation is reached when all particles are shifted to the
middle axis and compressed there, so that a quasi-solid cylinder occurs (Figs. 1, right, and 5). This has indeed been
discussed for the case of blood for sufficiently wide vessels
(26, 37, 38, 78). The cylinder (with radius R0) can have a
certain packing density, ␸*, e.g., the maximal packing density,
␸m. This cylinder then moves as a whole without velocity
gradients in its interior. Outside of it, the pure liquid (water, or
blood serum in our case) is flowing.
The value ␸ is now an averaged, effective quantity. It is
related to the packing density ␸* in the moving cylinder by the
following formula, which corresponds to the cross-sectional
area of the blood vessel (Fig. 5):
␲R20␸ⴱ ⫽ ␲R2␸
This allows us to calculate the radius of the cylinder:
J Appl Physiol • doi:10.1152/japplphysiol.00369.2012 • www.jappl.org
(55)
Downloaded from http://jap.physiology.org/ by 10.220.33.4 on July 28, 2017
This formula has a maximum, since it is zero for ␸ ⫽ 0,
tends to zero for ␸ ¡ ⬁, and is positive in between (Fig. 3B).
The question is whether the maximum lies at a value smaller
than 1. Differentiation gives
362
Optimal Hematocrit in Vertebrates
•
Stark H et al.
and defective erythrocytes, respectively. Then the numerator in
Eq. 27, which describes oxygen transport, should involve ␸int
only. In contrast, the demoninator, which corresponds to the
viscous flow, should involve the sum ␸int ⫹ ␸def, so that Eq. 27
should be extended as:
Joxygen ⫽ 关a共1 ⫺ ␸int兲 ⫹ b␸int兴
Fig. 5. Extreme situation with a blood thread moving in the middle of the
vessel. See text for definition of terms.
R0 ⫽
冑
␸
␸ⴱ
R2
(56)
R2 ⫺
␸ⴱ
Joxygen ⬀ ␸inte⫺2.5共␸int⫹␸def兲
⌬P
(57)
⭸ Joxygen
J⫽
␲⌬PR 共␸ ⫺ ␸ⴱ兲
⫽
4␩l␸ⴱ
␲⌬PR
4
4␩l
(66)
冉
1⫺
␸
␸ⴱ
冊
(58)
With formula 25 for the optimality principle we can write:
Joxygen ⫽
⭸ Joxygen
⭸␸
⫽
const . ␸
␩0
1
␸ⴱ
冉
冉
1⫺
␸⫹ 1⫺
2␸
␸ⴱ
⫽1
␸opt ⫽
␸
␸ⴱ
␸
␸ⴱ
冊
冊
(59)
⫽0
(60)
(61)
␸ⴱ
2
(62)
The maximal packing density depends on the shape of the
particles. Let us assume that the erythrocytes are deformed
approximately into spheres, so that the maximal packing density is ␲/18. This yields
␸opt ⫽ 37%
(65)
⫽ e⫺2.5共␸int⫹␸def兲 ⫺ 2.5␸inte⫺2.5共␸int⫹␸def兲 ⫽ 0
⭸ ␸int
After integration over the entire tube radius we obtain:
4
(64)
The optimum is computed to be
R2
4␩l
册
(63)
which is very near to the experimental value of 40%. If it were
assumed that the erythrocytes are compressed completely (like
in the centrifuge), ␸* would be 100%, so that ␸opt ⫽ 50%
would be obtained.
Influence of defective erythrocytes. As mentioned above, the
observed hematocrit in humans differs between women and
men. Typical values are 40.0 ⫾ 2.4% and 45.8 ⫾ 2.7%,
respectively (27, 50, 51). A straightforward explanation is
menstruation. This causes a periodic outflow of erythrocytes,
including defective ones in women. Since they are replaced by
new erythrocytes, the percentage of defective erythrocytes is
lower in women than in men. This explanation is supported by
the observation that the hematocrit in women increases after
menopause (11).
To compute the optimal value in the presence of defective
erythrocytes, Eq. 27 (with a ⫽ 0) needs to be modified. Let ␸int
and ␸def denote the hematocrit values corresponding to intact
1 ⫺ 2.5␸int ⫽ 0
(67)
␸int,opt ⫽ 0.4
(68)
It can be seen that the optimal total hematocrit is, in the case of
the Arrhenius equation, simply the sum of the ideal optimal
hematocrit (in the case where no defective erythrocytes would
occur) and the volume fraction of defective erythrocytes:
␸tot,opt ⫽ 0.4 ⫹ ␸def
(69)
Case where some substance is transported in the plasma. As
mentioned in MATERIALS AND METHODS, the substance of interest
may be transported not only by erythrocytes, but also in the
plasma. To analyze this case, nonzero values of the parameter
a should be used. Again, the different formulas given in Table
2 can be used. Here, we only use, by way of example, the
Arrhenius equation. We have
Jsubstance ⬀ 关a共1 ⫺ ␸兲 ⫹ b␸兴e⫺2.5␸
(70)
The optimum is computed to be
⭸ Jsubstance
⭸␸
⫽ 共⫺a ⫹ b兲e⫺2.5␸
⫺ 2.5关a共1 ⫺ ␸兲 ⫹ b␸兴e⫺2.5␸ ⫽ 0 (71)
关⫺1 ⫹ b兴 ⫺ 2.5关a共1 ⫺ ␸兲 ⫹ b␸兴 ⫽ 0
␸opt ⫽
7a ⫺ 2b
5a ⫺ 5b
(72)
(73)
In the special case a ⫽ 0, this leads again to Eq. 53.
DISCUSSION
Here, we have derived theoretical optimal values of hematocrit in vertebrates and collected, from the literature, experimentally observed values for 57 animal species. The theoretical values are based on different formulas for the dependence
of viscosity on the volume fraction of suspended particles
taken from the literature. We have shown that relatively simple
approaches based on Newtonian fluid dynamics provide results
J Appl Physiol • doi:10.1152/japplphysiol.00369.2012 • www.jappl.org
Downloaded from http://jap.physiology.org/ by 10.220.33.4 on July 28, 2017
v⫽
冉 冊
8␩共␸int ⫹ ␸def兲l
This equation can now be used to compute the optimal values
based on the different formulas given in Table 2. However, this
is beyond the scope of this paper. Here, we show, by way of
example, the calculation for the Arrhenius equation. ␸def is
assumed to be a given parameter determined by physiological
properties.
We use formula 1 and substitute r within formula 56
␸
冋
␲⌬pR4
Optimal Hematocrit in Vertebrates
Stark H et al.
363
Interestingly, for complex situations or processes, simple
formulas sometimes lead to better results than complicated
ones, even if the simple ones do not have a firm theoretical
basis. This is, in fact, the essence of modeling, since a model
is a simplified representation of some aspect of reality. Different models can be built for the same process, and it is decided
by the practical application which one works best. As mentioned above, in hemorheology, many effects play a role, such
the dependence on flow velocity, diameter, etc. Here, we have
simplified things considerably to concentrate on the essential
properties. Despite these simplifications, realistic results can be
derived in optimal hematocrit theory.
Red blood cells of camels and llamas are elliptical, as
contrasted to the disk-shaped biconcave red cells of other
mammals (97). The resistance to flow, which is offered by a
suspension of asymmetric particles, such as blood cells, must
depend on their orientation with respect to the direction of
flow. It is possible that elliptical cells might orient in the
direction of flow more easily than discoidal cells, but, with the
methods used in this study, we did not find differences in
viscosity that could be attributed to the difference in shape. The
rationale behind the study was that the elliptical cells of camel
and llama might be of advantage in their respective environments: desert and high altitude (97). The camel is confronted
with intense heat and potential dehydration with hemoconcentration, and the llama encounters low oxygen pressures where
a high red cell concentration is an advantage.
It is interesting to discuss the physiological advantages of
the emergence of erythrocytes during evolution. One advantage over oxygen-transporting molecules dissolved in the
plasma is the Fåhraeus-Lindqvist effect, which allows the
blood cells to concentrate in the center of the vessels and thus
to decrease effective viscosity. Moreover, the kidneys can filter
blood cells much more easily than heme molecules (12). A
further advantage is that the interaction of heme with other
molecules is avoided.
A phenomenon worth being discussed here is blood doping
(7, 49, 88, 90). First of all, we stress that we are against such
a practice for legal and medical reasons. Nevertheless, it is of
academic interest to discuss whether this doping is efficient
(within certain limits), despite the assumed optimality property
of hematocrit in the physiological standard situation. This
assumption is supported by the observation that the hematocrit
is nearly the same in active sportsmen and control persons (88).
Böning et al. (7) called the effect of blood doping the hematocrit paradox.
Blood doping (induced erythrocythemia) is the intravenous
infusion of more concentrated blood to produce an increase in
the blood’s oxygen-carrying capacity. This can increase the
hemoglobin level and hematocrit by up to 20%. Does this
falsify our above calculations? There are at least four factors to
be considered.
1) The addition of erythrocytes increases blood volume. As
the blood vessels are rather elastic, they can be dilated, so as to
take up the increased volume. This, however, increases the
parameter R (artery radius) in our calculations.
2) We assumed the pressure difference, ⌬P, to be constant.
However, the heart does not produce the same pressure under
all conditions. If blood viscosity and thus the resistance against
heart contraction increase, the heart tries to pump harder, to
reach the same blood flow velocity. However, this is possible
J Appl Physiol • doi:10.1152/japplphysiol.00369.2012 • www.jappl.org
Downloaded from http://jap.physiology.org/ by 10.220.33.4 on July 28, 2017
that are consistent with experimentally observed values. Although blood is, of course, a non-Newtonian fluid, the simplification to Newtonian properties appears to be justified here.
Similar considerations are certainly of interest also in the
suction of nectar by insects (cf. Refs. 15, 39, 51, 52, 71, 72)
and in technological applications with respect to suspensions
other than blood. This concerns, for example, the question which
concentration a cement-water suspension should have to enable a
maximum pumping flow of cement along tubes (65, 85).
We started with trying Einstein’s formula for dilute suspensions. Although this has not led to realistic results, we included
it into the paper to show the step-wise way of finding good
descriptions in science.
The observed values differ considerably among species,
ranging from 19% (estuarine crocodile) to 63% (Weddell seal).
Also, the theoretical values differ in about that range. The
deviations between theoretical and observed values may provide an indication of the extent to which other optimality
criteria are relevant as well, so that a trade-off must have been
found in evolution, as discussed earlier in the case of optimal
stoichiometries of metabolic pathways (107). As for the Weddell seal and beluga (white) whale, the rather large hematocrit
value may be due to an additional criterion relevant for diving
animals. Since they have to store as much oxygen as possible
before diving, the storage capacity of the blood for oxygen
needs to be particularly high. However, this does not appear to
be a consistent feature for all diving animals, since the killer
whale and crabeater seal show hematocrit values ⬍50% (Table
1). Also for birds, the values differ considerably. This might be
due to a different activity, although optimal hematocrit theory
would not predict any dependency on activity. It may be
assumed that, again, additional criteria are relevant.
The exponential function (Arrhenius’ formula) and Saitô’s
formula lead to the surprisingly simple solutions of 40% and
39%, respectively, which excellently match the observed values in humans, chimpanzee, gorilla, rabbit, cat, pig, and several
other species. Mooney’s equation leads to a value of 23%
(assuming a maximal packing density of one), which is in
perfect agreement with the observed value in the rainbow trout.
The Krieger-Dougherty and Gillespie formulas yield ⬃30%,
matching with llama, tiger, armadillo, pea fowl, and quail.
The question arises why more complex formulas, such as the
Gillespie and Krieger-Dougherty equations, do not give such
good results for the species with high hematocrit as the simple
exponential function, while they may be relevant for species
with lower hematocrit.
A possible answer is that blood is a very complicated fluid
involving a lot of effects, while most of the formulas used here
for the dependence of viscosity on volume fraction had been
derived for other types of suspensions, such as plastic globules
or cement (60, 61, 63, 69, 81, 85). Erythrocytes are not usually
spheres. In humans and many animals, they are biconcave
disks, while in camels and llamas, they are elliptical (97). They
aggregate to each other, orient, and are deformed in flowing
blood. Brinkman et al. (10) found that erythrocyte aggregation
could be divided into four types: no rouleaux formation type
(ox, sheep, and goat), slight rouleaux formation type (rabbit),
moderate type (human, pig, dog, cat, and rat), and excessive
type (horse) (cf. Ref. 66). Due to all of these effects, blood is
a non-Newtonian fluid. That means that viscosity depends on
the velocity gradient.
•
364
Optimal Hematocrit in Vertebrates
Stark H et al.
ocrit under stress conditions and thus to reach a better performance. This is worth being studied further.
At the end of the RESULTS section, we have outlined several
interesting extensions of the theory. One extension concerns
the consideration of spatially heterogeneous distributions of
erythrocytes. Here, we have modeled the Fåhraeus-Lindqvist
effect by considering the extreme case in which all particles
move in the middle of the blood vessel as a quasi-solid
cylinder. This leads to higher optimal hematocrit values than in
spatially homogeneous suspensions. This can be explained as
follows: the cylinder moves as a whole without velocity gradients in its interior. Thus a high erythrocyte concentration can
be reached without inner friction. At the boundary of the
cylinder, a steep velocity gradient occurs because, there, viscosity is lower. A similar reasoning can be applied to spatially
heterogeneous suspensions in general.
To simulate continuous spatial concentration gradients, variational calculus (34, 48, 59, 80, 94) can be used (not done
here). Both the velocity and the concentration would then be
written as functions of the radial coordinate r. From the
principle of minimum entropy production (30), a minimization
principle can be written, from which the optimal velocity and
concentration profiles can be computed by variational calculus.
Averaging the concentration over the cross section leads to the
optimal hematocrit.
The second potential extension concerns the difference in
the observed hematocrit values of men vs. women. A straightforward explanation is the different percentage of defective
erythrocytes in the two sexes due to menstruation. A similar
effect has been discussed for the case of blood doping by
erythropoietin: the percentage of young red blood cells with
good functional properties increases. This sex difference can
be included in the equations relatively easily.
The third generalization concerns the case in which the
substance of interest is transported not only by erythrocytes,
but also in the plasma. We have considered this in the equations from the outset, but focused for most calculations on the
case in which the substance (e.g., oxygen) is only transported
by erythrocytes. The general case requires an additional parameter: the ratio between the substance concentrations in the
erythrocytes and plasma.
A promising calculation is the following. Based on the theory
presented here, the additional pumping power of the heart necessary to achieve the same oxygen transport in the case of high
hematocrit values (larger than 40%) can be calculated. Optimal
hematocrit theory can be extended also in other ways. For example, non-Newtonian fluid mechanics can be used (16), although
Newtonian approaches lead to very good results, as shown here.
Moreover, in very small capillaries, such as sinusoids in the liver,
the theory based on viscous flow is not valid, because just one
erythrocyte can pass at a time. Then the discrete, corpuscular
nature of cells needs to be considered.
ACKNOWLEDGEMENTS
We thank Jörn Behre for stimulating discussions and for an idea leading to
Eq. 63, and Nadja Schilling for valuable comments on this paper.
GRANTS
Financial support by the University of Jena and the Virtual Liver Network
funded by the German Ministry for Education and Research is gratefully
acknowledged.
J Appl Physiol • doi:10.1152/japplphysiol.00369.2012 • www.jappl.org
Downloaded from http://jap.physiology.org/ by 10.220.33.4 on July 28, 2017
only over a small range of viscosity above the optimal one.
Above that range, the increased viscosity indeed reduces oxygen transport. And even in the range where increasing the
hematocrit seems to be beneficial, the heart is more stressed.
There is a hyperviscosity syndrome, which can lead to heart
failure in the long term.
3) We cannot be completely sure that the hematocrit found
in humans or other animals is really optimal. One has to be
careful that no circular reasoning is used. Such a reasoning
would be to ask whether the hematocrit is optimal, then try
various calculations until one of them gives the experimental
value and then say the answer is positive: the hematocrit is
optimal. It might be that the hematocrit is slightly below the
optimum to give the organism the chance to realize a better
oxygen flow under special circumstances. For example, at high
altitudes, where oxygen pressure is lower, the hematocrit
indeed increases. The hematocrit also changes in camels during
longer dry seasons and after drinking much water (95).
4) Böning et al. (7) mention the following additional potential factors: augmented diffusion capacity for oxygen in lungs
and tissues because of the enlarged red cell surface area,
increased buffer capacity, vasoconstriction, reduced damage by
radicals, and even perhaps placebo effects. They suggested that
blood doping has 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 hyperviscosity syndrome is blood
clotting inside of the blood vessels due to higher density of
erythrocytes.
It is an empirical fact that hematocrit increases at higher
altitudes. In the case of humans, this can be easily observed in
mountaineering (39, 89, 100); for the llama, see Table 1. At
first sight, this is intuitively understandable because the lower
oxygen pressure should be compensated. One of the most
documented physiological adaptations to a reduced O2 uptake
is the increased release of erythropoietin, which causes an
increase in red blood cell mass (91, 100). On the other hand, it
may seem to contradict optimal hematocrit theory, because a
higher hematocrit makes the blood more viscous. The paradox
can be resolved by the observation that the increase in hematocrit at higher altitudes is partly due to a decrease in the
amount of blood plasma (cf. Refs. 89, 96). This is first caused
by dehydration and later by a shift of intravasal fluid to the
interstitial space. Thus the total blood volume decreases, while,
in the above calculations, blood volume was considered constant. A lower volume makes it easier for the heart to drive
blood circulation, so that an increase in viscosity can be
tolerated.
There are a number of additional effects. It has been shown
that the native highlander is characterized by a larger pulmonary diffusion capacity (17) and adaptations in the structural
and metabolic organization of skeletal muscle that result in a
tighter coupling between ATP hydrolysis and oxidative phosphorylation (42).
From the above reasoning about blood doping and adaptation to high altitudes, it may be concluded that the observed
hematocrit values are slightly suboptimal. Suboptimal states
were also discussed in the optimal kinetics of oxygen binding
to hemoglobin (109) and in the optimal degree of redundancy
in metabolism (56, 96). This allows one to increase the hemat-
•
Optimal Hematocrit in Vertebrates
DISCLOSURES
No conflicts of interest, financial or otherwise, are declared by the author(s).
AUTHOR CONTRIBUTIONS
H.S. and S.S. conception and design of research; H.S. and S.S. performed
experiments; H.S. and S.S. analyzed data; H.S. and S.S. interpreted results of
experiments; H.S. and S.S. prepared figures; H.S. and S.S. drafted manuscript;
H.S. and S.S. edited and revised manuscript; H.S. and S.S. approved final
version of manuscript.
REFERENCES
Stark H et al.
365
21. Dhindsa DS, Metcalfe J, Hoversland A. Comparative studies of the
respiratory functions of mammalian blood. IX. Ring-tailed lemur (Lemur
catta) and black lemur (Lemur macaco). Respir Physiol 15: 331–342,
1972.
22. Dhindsa DS, Metcalfe J, McKean T, Thorne T. Comparative studies of
the respiratory functions of mammalian blood. XI. Pronghorn antelope
(Antilocapra americana). Respir Physiol 21: 297–306, 1974.
23. Dhindsa DS, Sedgwick CJ, Metcalfe J. Comparative studies of the
respiratory functions of mammalian blood. VIII. Asian elephant (Elephas
maximus) and African elephant (Loxodonta africana africana). Respir
Physiol 14: 332–342, 1972.
24. Einstein A. Eine neue Bestimmung der Moleküldimensionen. Annalen
der Physik 324: 289 –306, 1906.
25. Einstein A. Berichtigung zu meiner Arbeit: “Eine neue Bestimmung der
Moleküldimensionen”. Annalen der Physik 339: 591–592, 1911.
26. Fåhræus R, Lindqvist T. The viscosity of the blood in narrow capillary
tubes. Am J Physiol 96: 562–568, 1931.
27. Galdi GP, Robertson AM, Rannacher R, Turek S. Hemodynamical
Flows. Basel: Birkhäuser Basel, 2008.
28. Gillespie T. The effect of aggregation and particle size distribution on
the viscosity of Newtonian suspensions. J Colloid Interface Sci 94:
166 –173, 1983.
29. Gingerich WH, Pityer RA, Rach JJ. Estimates of plasma, packed cell
and total blood volume in tissues of the rainbow trout (Salmo gairdneri).
Comp Biochem Physiol A Comp Physiol 87A: 251–256, 1987.
30. Glansdorff P, Prigogine I. Thermodynamic Theory of Structure, Stability and Fluctuations. New York: Wiley-Interscience, 1971.
31. Goodman M, Pedwaydon J, Czelusniak J, Suzuki T, Gotoh T, Moens
L, Shishikura F, Walz D, Vinogradov SN. An evolutionary tree for
invertebrate globin sequences. J Mol Evol 27: 236 –249, 1988.
32. Guard CL, Murrish DE. Effects of temperature on the viscous behavior
of blood from antarctic birds and mammals. Comp Biochem Physiol A
Comp Physiol 52A: 287–290, 1975.
33. Hagen G. Ueber die Bewegung des Wassers in engen cylindrischen
Röhren. Annalen der Physik und Chemie 122: 423–442, 1839.
34. Haines BM, Mazzucato AL. A proof of Einstein’s effective viscosity for
a dilute suspension of spheres. Arxiv preprint arXiv11041102 16802:
1–26, 2011.
35. Harder L, Boshkov L. The optimal hematocrit. Crit Care Clin 26:
335–354, 2010.
36. Hardison RC. A brief history of hemoglobins: plant, animal, protist, and
bacteria. Proc Natl Acad Sci U S A 93: 5675–5679, 1996.
37. Haynes RH. Blood flow through narrow tubes. Nature 185: 679 –681,
1960.
38. Haynes RH. Physical basis of the dependence of blood viscosity on tube
radius. Am J Physiol 198: 1193–1200, 1960.
39. Heinicke K, Prommer N, Cajigal J, Viola T, Behn C, Schmidt W.
Long-term exposure to intermittent hypoxia results in increased hemoglobin mass, reduced plasma volume, and elevated erythropoietin plasma
levels in man. Eur J Appl Physiol 88: 535–543, 2003.
40. Heyneman AJ. Optimal sugar concentrations of floral nectars– dependence on sugar intake efficiency and foraging costs. Oecologia (Berl) 60:
198 –213, 1983.
41. Hillman SS, Withers PC, Hedrick MS, Kimmel PB. The effects of
erythrocythemia on blood viscosity, maximal systemic oxygen transport
capacity and maximal rates of oxygen consumption in an amphibian. J
Comp Physiol B 155: 577–581, 1985.
42. Hochachka PW, Stanley C, Matheson GO, McKenzie DC, Allen PS,
Parkhouse WS. Metabolic and work efficiencies during exercise in
Andean natives. J Appl Physiol 70: 1720 –1730, 1991.
43. Hudson DR. Density and packing in an aggregate of mixed spheres. J
Appl Phys 20: 154 –162, 1949.
44. Huggins SE. Blood volume parameters of a poikilothermal animal in
hypo- and hyperthermia. Proc Soc Exp Biol Med 108: 231–234, 1961.
45. Isaacks RE, Harkness DR, Froeman GA, Goldman PH, Adler JL,
Sussman SA, Roth S. Studies on avian erythrocyte metabolism. II.
Relationship between the major phosphorylated metabolic intermediates
and oxygen affinity of whole blood in chick embryos and chicks. Comp
Biochem Physiol 53: 151–156, 1976.
46. Isaacks RE, Harkness DR, Sampsell R, Adler JL, Kim CY, Goldman
PH. Studies on avian erythrocyte metabolism. IV. Relationship between
the major phosphorylated metabolic intermediates and oxygen affinity of
whole blood in adults and embryos in several galliformes. Comp
Biochem Physiol 55: 29 –33, 1976.
J Appl Physiol • doi:10.1152/japplphysiol.00369.2012 • www.jappl.org
Downloaded from http://jap.physiology.org/ by 10.220.33.4 on July 28, 2017
1. Appleman RD, Delouche JC. Behavioral, physiological and biochemical responses of goats to temperature, 0° to 40°C. J Anim Sci 17:
326 –335, 1958.
2. Arrhenius S. The viscosity of solutions. Biochem J 11: 112–133, 1917.
3. Bailey DM, Davies B. Physiological implications of altitude training for
endurance performance at sea level: a review. Br J Sports Med 31:
183–190, 1997.
4. Banchero N, Grover RF, Will JA. Oxygen transport in the llama (Lama
glama). Respir Physiol 13: 102–115, 1971.
5. Bartels H, Riegel K, Kleihauer E, Lang EM. Comparative studies of
the respiratory functions of mammalian blood. II. Marsupialia: great grey
kangaroo and Tasmanian devil. Respir Physiol 1: 145–150, 1966.
5a.Bartels H, Schmelzle R, Ulrich S. Comparative studies of the respiratory functions of mammalian blood. V. Insectivora: shrew, mole and
nonhibernating and hibernating hedgehog. Respir Physiol 7: 278 –286,
1969.
6. Birchard GF. Optimal hematocrit: theory, regulation and implications.
Integr Comp Biol 37: 65–72, 1997.
7. Böning D, Maassen N, Pries A. The hematocrit paradox– how does
blood doping really work? Int J Sports Med 32: 242–246, 2011.
8. Brill R, Jones D. The influence of hematocrit, temperature and shear rate
on the viscosity of blood from a high-energy-demand teleost, the yellowfin tuna Thunnus albacares. J Exp Biol 189: 199 –212, 1994.
9. Brill RW. Selective advantages conferred by the high performance
physiology of tunas, billfishes, and dolphin fish. Comp Biochem Physiol
113: 3–15, 1996.
10. Brinkman R, Zijlstra WG, Jansonius NJ. Quantitative evaluation of
the rate of rouleaux formation of erythrocytes by measuring light reflection (“Syllectometry”). In: Proceedings of the Koninklijke Nederlandse
Academie van Wetenschappen. Series C: Biological and Medical Sciences. 1963, p. 234 –248.
11. Brown E, Hopper J Jr., Hodges JL Jr., Bradley B, Wennesland R,
Yamauchi H. Red cell, plasma, and blood volume in the healthy women
measured by radiochromium cell-labeling and hematocrit. J Clin Invest
41: 2182–2190, 1962.
12. Bunn HF. The use of hemoglobin as a blood substitute. Am J Hematol
42: 112–117, 1993.
13. Bushnell PG, Jones DR. Cardiovascular and respiratory physiology of
tuna: adaptations for support of exceptionally high metabolic rates.
Environ Biol Fishes 40: 303–318, 1994.
14. Christoffersen J, Christoffersen M, Hunding A. Comments on optimization of glycolysis. Biochem Educ 26: 290 –291, 1998.
15. Crowell JW, Smith EE. Determinant of the optimal hematocrit. J Appl
Physiol 22: 501–504, 1967.
16. Daniel TL, Kingsolver JG. Feeding strategy and the mechanics of blood
sucking in insects. J Theor Biol 105: 661–677, 1983.
17. Dempsey JA, Reddan WG, Birnbaum ML, Forster HV, Thoden JS,
Grover RF, Rankin J. Effects of acute through life-long hypoxic
exposure on exercise pulmonary gas exchange. Respir Physiol 13: 62–89,
1971.
18. Dhindsa DS, Hoversland A, Metcalfe J. Comparative studies of the
respiratory functions of mammalian blood. VII. Armadillo (Dasypus
novemcinctus). Respir Physiol 13: 198 –208, 1971.
19. Dhindsa DS, Hoversland A, Metcalfe J. Comparative studies of the
respiratory functions of mammalian blood. XII. Black galago (Galago
crassicaudatus argentatus) and brown galago (Galago crassicaudatus
crassicaudatus). Respir Physiol 47: 313–323, 1982.
20. Dhindsa DS, Metcalfe J, Hoversland A, Hartman R. Comparative
studies of the respiratory functions of mammalian blood. X. Killer whale
(Orcinus orca linnaeus) and beluga whale (Delphinapterus leucas).
Respir Physiol 20: 93–103, 1974.
•
366
Optimal Hematocrit in Vertebrates
Stark H et al.
75. Poiseuille JLM. Recherches expérimentales sur le mouvement des
liquides dans les tubes de très-petits diamètres. I. Influence de la pression
sur la quantité de Iiquide qui traverse les tubes de très-petits diamètres.
Comptes Rendus Académie des Sciences Paris 11: 961–967, 1840.
76. Poiseuille JLM. Recherches expérimentales sur le mouvement des
liquides dans les tubes de très-petits diamètres. II. Influence de la
longueur sur la quantité de liquide qui traverse les tubes de très-petits
diameters. III. Influence du diamètre sur la quantité de liquid. Comptes
Rendus Académie des Sciences Paris 11: 1041–1048, 1840.
77. Poiseuille JLM. Recherches expérimentales sur le mouvement des
liquides dans les tubes de très-petits diamètres. IV. Influence de la
temperature sur la quantité de liquide qui traverse les tubes de très-petits
diameters. Comptes Rendus Académie des Sciences Paris 12: 112–115,
1841.
78. Popel AS, Johnson PC. Microcirculation and hemorheology. Annu Rev
Fluid Mech 37: 43–69, 2005.
79. Pries AR, Neuhaus D, Gaehtgens P. Blood viscosity in tube flow:
dependence on diameter and hematocrit. Am J Physiol Heart Circ
Physiol 263: H1770 –H1778, 1992.
80. Quemada D. Rheology of concentrated disperse systems and minimum
energy dissipation principle. I. Viscosity-concentration relationship.
Rheologica Acta 16: 82–94, 1977.
81. Quemada D. Rheology of concentrated disperse systems. III. General
features of the proposed non-Newtonian model. Comparison with experimental data. Rheologica Acta 17: 643–653, 1978.
82. Riegel K, Bartels H, Buss IO, Wright PG, Kleihauer E, Luck CP,
Parer JT, Metcalfe J. Comparative studies of the respiratory functions
of mammalian blood. IV. Fetal and adult African elephant blood. Respir
Physiol 2: 182–195, 1967.
83. Riegel K, Bartels H, Kleihauer E, Lang EMM, Metcalfe J. Comparative studies of the respiratory functions of mammalian blood. I. Gorilla,
chimpanzee and orangutan. Respir Physiol 1: 138 –144, 1966.
84. Riegel K, Bartels H, Yassin el D, Oufi J, Kleihauer E, Parer JT,
Metcalfe J. Comparative studies of the respiratory functions of mammalian blood. III. Fetal and adult dromedary camel blood. Respir Physiol
2: 173–181, 1967.
85. Roussel N, Lemaître A, Flatt RJ, Coussot P. Steady state flow of
cement suspensions: a micromechanical state of the art. Cement Concrete
Res 40: 77–84, 2010.
86. Saitô N. Concentration dependence of the viscosity of high polymer
solutions. I. J Phys Soc Japan 5: 4 –8, 1950.
87. Salerno C, Giacomello A. Hypoxanthine-guanine exchange by intact
human erythrocytes. Biochemistry 24: 1306 –1319, 1985.
88. Schmidt W, Biermann B, Winchenbach P, Lison S, Böning D. How
valid is the determination of hematocrit values to detect blood manipulations? Int J Sports Med 21: 133–138, 2000.
89. Schmidt W, Heinicke K, Rojas J, Manuel Gomez J, Serrato M, Mora
M, Wolfarth B, Schmid A, Keul J. Blood volume and hemoglobin mass
in endurance athletes from moderate altitude. Med Sci Sports Exerc 34:
1934 –1940, 2002.
90. Schmidt W, Prommer N. Effects of various training modalities on blood
volume. Scand J Med Sci Sports 18, Suppl 1: 57–69, 2008.
91. Schmidt W, Spielvogel H, Eckardt KU, Quintela A, Peñaloza R.
Effects of chronic hypoxia and exercise on plasma erythropoietin in
high-altitude residents. J Appl Physiol 74: 1874 –1878, 1993.
92. Schmidt W. Effects of intermittent exposure to high altitude on blood
volume and erythropoietic activity. High Alt Med Biol 3: 167–176, 2002.
93. Shaffer SA, Costa DP, Williams TM, Ridgway SH. Diving and
swimming performance of white whales, Delphinapterus leucas: an
assessment of plasma lactate and blood gas levels and respiratory rates.
J Exp Biol 200: 3091–3099, 1997.
94. Siddique MR, Winkle van TL, Wetzel DM. A variational calculus
approach to the suspension flow problem using the minimal force
hypothesis. Bull Math Biol 44: 153–174, 1982.
95. Siebert BD, Macfarlane WV. Dehydration in desert cattle and camels.
Physiol Zool 48: 36 –48, 1975.
96. Stelling J, Sauer U, Szallasi Z, Doyle FJ, Doyle J. Robustness of
cellular functions. Cell 118: 675–685, 2004.
97. Stone HO, Thompson HK, Schmidt-Nielsen K. Influence of erythrocytes on blood viscosity. Am J Physiol 214: 913–918, 1968.
98. Sutera SP, Skalak R. The History of Poiseuille’s Law. Annu Rev Fluid
Mech 25: 1–20, 1993.
99. Szathmary E. Four letters in the genetic alphabet: a frozen evolutionary
optimum? Proc R Soc Lond B Biol Sci 245: 91–99, 1991.
J Appl Physiol • doi:10.1152/japplphysiol.00369.2012 • www.jappl.org
Downloaded from http://jap.physiology.org/ by 10.220.33.4 on July 28, 2017
47. Isaacks RE, Harkness DR. Erythrocyte organic phosphates and hemoglobin function in birds, reptiles, and fishes. Integr Comp Biol 20:
115–129, 1980.
48. Jeffrey DJ, Acrivos A. The rheological properties of suspensions of
rigid particles. AIChE J 22: 417–432, 1976.
49. Jones M, Tunstall Pedoe DS. Blood doping–a literature review. Br J
Sports Med 23: 84 –88, 1989.
50. Kameneva MV, Garrett KO, Watach MJ, Borovetz HS. Red blood
cell aging and risk of cardiovascular diseases. Clin Hemorheol Microcirc
18: 67–74, 1998.
51. Kameneva MV, Watach MJ, Borovetz HS. Gender difference in
rheologic properties of blood and risk of cardiovascular diseases. Clin
Hemorheol Microcirc 21: 357–363, 1999.
52. Kato T. Relation between blood viscosity and hematocrit in rabbits
subjected to repeated treatment with endotoxin. Resuscitation 21: 61–66,
1991.
53. Kim W, Gilet T, Bush JWM. Optimal concentrations in nectar feeding.
Proc Natl Acad Sci U S A 108: 16618 –16621, 2011.
54. Kingsolver JG, Daniel TL. On the mechanics and energetics of nectar
feeding in butterflies. J Theor Biol 76: 167–179, 1979.
55. Krieger IM, Dougherty TJ. A mechanism for non-Newtonian flow in
suspensions of rigid spheres. J Rheol (NY NY) 3: 137–152, 1959.
56. Kuepfer L, Sauer U, Blank LM. Metabolic functions of duplicate genes
in Saccharomyces cerevisiae. Genome Res 15: 1421–1430, 2005.
57. Lalanne M, Willemot J. Adenine and hypoxanthine salvage in erythrocytes of eight mammalian species. Comp Biochem Physiol 66: 367–372,
1980.
58. Li G, Hu H, Shi W, Li Y, Liu L, Chen Y, Hu X, Wang J, Gao J, Yin
D. Elevated hematocrit in nonalcoholic fatty liver disease: A potential
cause for the increased risk of cardiovascular disease? Clin Hemorheol
Microcirc 51: 59 –68, 2012.
59. Lipton R. Variational methods, bounds, and size effects for composites
with highly conducting interface. J Mech Phys Solids 45: 361–384, 1997.
60. Luckham PF, Ukeje M. Effect of particle size distribution on the
rheology of dispersed systems. J Colloid Interface Sci 220: 347–356,
1999.
61. Mendoza CI, Santamaría-Holek I. The rheology of hard sphere suspensions at arbitrary volume fractions: an improved differential viscosity
model. J Chem Phys 130: 44904 –44907, 2009.
62. Mooney M. The viscosity of a concentrated suspension of spherical
particles. J Colloid Sci 6: 162–170, 1951.
63. Mueller S, Llewellin EW, Mader HM. The rheology of suspensions of
solid particles. Proc R Soc Lond A Math Phys Sci 466: 1201–1228, 2009.
64. Munday KA, Blane GF. Cold stress of the mammal, bird and reptile.
Comp Biochem Physiol 2: 8 –21, 1961.
65. Nanthagopalan P, Haist M, Santhanama M, Müller HS. Investigation
on the influence of granular packing on the flow properties of cementitious suspensions. Cement Concrete Compos 30: 763–768, 2008.
66. Nemeth N, Alexy T, Furka A, Baskurt OK, Meiselman HJ, Furka I,
Miko I. Inter-species differences in hematocrit to blood viscosity ratio.
Biorheology 46: 155–165, 2009.
67. Nissenson AR. Optimal hematocrit in patients on dialysis therapy. Am J
Kidney Dis 32: S142–S146, 1998.
68. Ohta K, Gotoh F, Tomita M, Tanahashi N, Kobari M, Shinohara T,
Terayama Y, Mihara B, Takeda H. Animal species differences in
erythrocyte aggregability. Am J Physiol Heart Circ Physiol 262: H1009 –
H1012, 1992.
69. Pal R. Novel viscosity equations for emulsions of two immiscible
liquids. J Rheol (NY NY) 45: 509 –520, 2001.
70. Palenske NM, Saunders DK. Blood viscosity and hematology of American bullfrogs (Rana catesbeiana) at low temperature. J Therm Biol 28:
271–277, 2003.
71. Parer JT, Hoversland A, Metcalfe J. Comparative studies of the
respiratory functions of mammalian blood. VI. Young lion and tiger.
Respir Physiol 10: 30 –37, 1970.
72. Parkhurst DF, Loucks OL. Optimal leaf size in relation to environment.
J Ecol 60: 505–537, 1972.
73. Paul J, Roces F. Fluid intake rates in ants correlate with their feeding
habits. J Insect Physiol 49: 347–57, 2003.
74. Pivnick KA, McNeil JN. Effects of nectar concentration on butterfly
feeding: measured feeding rates for Thymelicus lineola (Lepidoptera:
Hesperiidae) and a general feeding model for adult Lepidoptera. Oecologia (Berl) 66: 226 –237, 1985.
•
Optimal Hematocrit in Vertebrates
100. Tannheimer M, Fusch C, Böning D, Thomas A, Engelhardt M,
Schmidt R. Changes of hematocrit and hemoglobin concentration in the
cold Himalayan environment in dependence on total body fluid. Sleep
Breath 14: 193–199, 2010.
101. Tetens V, Christensen NJ. Beta-adrenergic control of blood oxygen
affinity in acutely hypoxia exposed rainbow trout. J Comp Physiol B 157:
667–675, 1987.
102. Usami S, Chien S, Gregersen MI. Viscometric characteristics of blood
of the elephant, man, dog, sheep, and goat. Am J Physiol 217: 884 –890,
1969.
103. Waddell TG, Repovic P, Melendez-Hevia E, Heinrich R, Montero F.
Optimization of glycolysis: a new look at the efficiency of energy
coupling. Biochem Educ 25: 204 –205, 1997.
104. Weathers WW. Influence of temperature on the optimal hematocrit of
the bullfrog (Rana catesbeiana). J Comp Physiol B 105: 173–184, 1976.
105. Wells RE, Beard L, Grigg G. Blood viscosity and hematocrit in the
estuarine crocodile, Crocodylus porosus. Comp Biochem Physiol 99:
411–414, 1991.
•
Stark H et al.
367
106. Weng X, Cloutier G, Pibarot P, Durand LG. Comparison and
simulation of different levels of erythrocyte aggregation with pig,
horse, sheep, calf, and normal human blood. Biorheology 33: 365–
377, 1996.
107. Werner S, Diekert G, Schuster S. Revisiting the thermodynamic theory
of optimal ATP stoichiometries by analysis of various ATP-producing
metabolic pathways. J Mol Evol 71: 346 –355, 2010.
108. Westheide W, Rieger G. Spezielle Zoologie Teil 2: Wirbel-oder
Schädeltiere. Heidelberg, Germany: Spektrum Akademischer Verlag,
2010.
109. Willford DC, Hill EP, Moores WY. Theoretical analysis of optimal
P50. J Appl Physiol 52: 1043–1048, 1982.
110. Withers PC, Hillman SS, Hedrick MS, Kimmel PB. Optimal hematocrit theory during activity in the bullfrog (Rana catesbeiana). Comp
Biochem Physiol 99: 55–60, 1991.
111. Yamaguchi K, Jürgens KD, Bartels H, Piiper J. Oxygen transfer
properties and dimensions of red blood cells in high-altitude camelids,
dromedary camel and goat. J Comp Physiol B 157: 1–9, 1987.
Downloaded from http://jap.physiology.org/ by 10.220.33.4 on July 28, 2017
J Appl Physiol • doi:10.1152/japplphysiol.00369.2012 • www.jappl.org