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Mathematical Biosciences 167 (2000) 123±143
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Calculations of intracapillary oxygen tension distributions in
muscle
C.D. Eggleton *, A. Vadapalli, T.K. Roy, A.S. Popel 1
Department of Biomedical Engineering and Center for Computational Medicine and Biology, The Johns Hopkins
University School of Medicine, Baltimore, MD 21205, USA
Received 1 September 1998; received in revised form 15 June 2000; accepted 14 July 2000

Abstract
Characterizing the resistances to O2 transport from the erythrocyte to the mitochondrion is important to
understanding potential transport limitations. A mathematical model is developed to accurately determine
the e€ects of erythrocyte spacing (hematocrit), velocity, and capillary radius on the mass transfer coecient. Parameters of the hamster cheek pouch retractor muscle are used in the calculations, since signi®cant
amounts of experimental physiological data and mathematical modeling are available for this muscle.
Capillary hematocrit was found to have a large e€ect on the PO2 distribution and the intracapillary mass
transfer coecient per unit capillary area, kcap , increased by a factor of 3.7 from the lowest (H ˆ 0:25) to the
highest (H ˆ 0:55) capillary hematocrits considered. Erythrocyte velocity had a relatively minor e€ect, with
only a 2.7% increase in the mass transfer coecient as the velocity was increased from 5 to 25 times the
observed velocity in resting muscle. The capillary radius is varied by up to two standard deviations of the
experimental measurements, resulting in variations in kcap that are 0:

The boundary condition at the edge of the tissue cylinder is

oP 
ˆ 0:
or 

…18†

rˆrt

Initial conditions in the computational domain are determined by the oxygen tension given by the
Krogh solution at the arterial end of the capillary. The boundary conditions for PO2 at the end
caps of the computational domain are developed by assuming that the domain is moving at a
constant velocity through a cylinder whose PO2 distribution is given by the Krogh solution. This
leads to the following axial boundary conditions at the leading and trailing edges, respectively


oP
Ltot
oPk

Dp ap
t; z ˆ 
; r ˆ Dp ap
; 0 6 r 6 rp ;
oz
2
oz


oP
Ltot
oPk
; r ˆ Dw aw
; rp 6 r 6 rw ;
t; z ˆ 
Dw aw
2
oz
oz



…19†
oP
Ltot
oPk
t; z ˆ 
; r ˆ Di ai
; rw 6 r 6 ri ;
Di a i
oz
2
oz




oP
Ltot
oSMb oPk
Dt a t

t; z ˆ 
; r ˆ  Dt at ‡ DMb NMb
; ri 6 r 6 rt ;
oz
2
oPk
oz
where oPk =oz is the local gradient in PO2 for the Krogh solution.
The model equations were solved using the ®nite element package PDEFlex (PDESolutions
Inc., Fremont, CA). The core erythrocyte tension, Pc , was assumed to remain constant and no ¯ux
boundary conditions for PO2 were imposed in the axial direction when using the Clark approximation Eq. (9) for oxygen ¯ux density at the erythrocyte±plasma interface. Fully time dependent
simulations are performed with the full equations for oxygen transport within the erythrocyte.

3. Model parameters
The parameters used in this study were chosen to represent working hamster retractor muscle
and are given in Table 1. Most of the parameters speci®c to this muscle were taken from Ellsworth

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C.D. Eggleton et al. / Mathematical Biosciences 167 (2000) 123±143

Table 1
Values of parameters used in calculations for hamster retracter muscle. When values for this speci®c muscle were not
available, parameters were chosen as discussed in [18]
Parameter
Vrbc
Lrbc
vrbc
rp
rw ÿ r p
ri ÿ r w
NA …c; f †
Drbc
Dp
Dw
Di
Dt
arbc
ap
aw

ai
at
kÿ
n
P50
NHb
DHb
Mb
P50
NMb
DMb
Mc

Units
3

cm
cm
cm sÿ1
lm

lm
lm
Capillaries per mm2
cm2 sÿ1
cm2 sÿ1
cm2 sÿ1
cm2 sÿ1
cm2 sÿ1
ml O2 mlÿ1 Torrÿ1
ml O2 mlÿ1 Torrÿ1
ml O2 mlÿ1 Torrÿ1
ml O2 mlÿ1 Torrÿ1
ml O2 mlÿ1 Torrÿ1
sÿ1
±
Torr
mol cmÿ3
cm2 sÿ1
Torr
mol cmÿ3

cm2 sÿ1
ml O2 sÿ1

Value

Notes
ÿ11

6:93  10
[20]
8:16  10ÿ4 [19]
9:35  10ÿ3 [17]
1:8  0:175 [28]
0.3 [29]
0.35 [29]
1435 [17]
9:5  10ÿ6 [5]
2:18  10ÿ5 [30]
8:73  10ÿ6 [31]
2:18  10ÿ5 [30]

2:41  10ÿ5 [32]
3:38  10ÿ5 [33]
2:82  10ÿ5 [34]
3:89  10ÿ5 [35]
2:82  10ÿ5 [33]
3:89  10ÿ5 [35]
44 [5]
2.2 [17]
29.3 [17]
2:03  10ÿ5 [5]
1:44  10ÿ7 [5]
5.3 [36]
4:0  10ÿ7 [23]
6:0  10ÿ7 [37]
1:62  10ÿ3 [17]

See text

Human
Human

Dp
Hamster retractor
Human
Human
at
ap
Frog sartorius
Human
Hill coecient

et al. [17]; exceptions are noted below. Values for most other parameters were taken from [18]
where their selection is explained in detail.
The dimensions of the erythrocyte are calculated from observed values of the lineal density,
LD ˆ 632 cells cmÿ1 [17], and observed erythrocyte length Lrbc ; these values yield the length of the
plasma gap Lp from Eq. (4). Microscopic observations of single capillaries [19] provided average
erythrocyte length; since this value was found to depend on hamster age, we used an interpolated
value, Lrbc ˆ 8:16  10ÿ4 cm, for 34 day old hamsters, the average age of hamsters considered in
the study of Ellsworth et al. [17]. The speci®ed erythrocyte volume, Vrbc ˆ 6:93  10ÿ11 cm3 [20], is
used to calculate erythrocyte radius, rrbc , from Eq. (2). Capillary hematocrit is then calculated
from Eq. (3) using rp ˆ 1:8 lm [17], which was the average of the mean radii observed for arteriolar and venular capillaries.

In the frame of reference of a single erythrocyte, the capillary wall, interstitial ¯uid, and tissue
regions move with the erythrocyte velocity, vrbc , relative to the erythrocyte and its surrounding
plasma. We used a value of vrbc ˆ 9:35  10ÿ3 cm sÿ1 [17], which was the average of the mean
velocities observed in arteriolar and venular capillaries in resting muscle, and used factors from 5
to 25 to assess the e€ect of velocity variations.

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131

In the tissue region, we used the working muscle consumption Mc estimated by Ellsworth et al.
[17] as 10 times the resting muscle consumption of 0.89 ml O2 100 gÿ1 minÿ1 measured by
_ 2 max , estimated
Sullivan and Pittman [21]. This value falls below the maximum consumption, VO
for this muscle as 21 times the resting rate based on the mitochondrial volume density [22]. The
value of NMb ˆ 4  10ÿ7 mol cmÿ3 in hamster retractor muscle has been measured by Meng et al.
[23]. A resting value of NA …c; f † ˆ 1435 capillaries per mm2 was calculated based on in vivo
microscopic intercapillary distances in [17]. Capillary recruitment was not considered since it has
been shown to be small for animals of this size [24].

4. Results
4.1. Calculated quantities
The intracapillary mass transfer coecient per unit area of capillary wall, kcap , is de®ned in
terms of the volume averaged partial pressure of oxygen within the erythrocyte, Pc , and the PO2
and ¯ux density at the capillary wall, Pp and Jp , kcap , respectively:
Jp
…20†
kcap ˆ 
…Pc ÿ Pp †2prp Ltot
with
Jp ˆ

Lÿ1
tot

Z

‡Ltot =2

Jp …z† dz;

ÿ1
Pc ˆ 2pVrbc

Pp ˆ Lÿ1
tot

…21†

ÿLtot =2

Z

Z

‡Ltot =2
ÿLtot =2

Z

rrbc

P …r; z†r dr dz;

…22†

0

‡Ltot =2

Pp …z† dz:

…23†

ÿLtot =2

The local mass transfer coecient, kcap …z†, is de®ned using the local PO2 and ¯ux density at the
capillary wall (Pp …z† and Jp …z†), and the volume averaged partial pressure of oxygen within the
erythrocyte, Pc ,
Jp …z†
kcap …z† ˆ 
:
…Pc ÿ Pp …z††2prp Ltot

…24†

4.2. Reference case
The parameters given above were used in the model to determine the transient PO2 pro®les and
the transport resistances in each of the regions using the full equations for oxygen transport within
the erythrocyte. Initially the PO2 within the erythrocyte is set to 50 Torr and in equilibrium with
oxygen bound to hemoglobin. The initial oxygen tension outside the erythrocyte corresponds to

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C.D. Eggleton et al. / Mathematical Biosciences 167 (2000) 123±143

that for the steady-state two-dimensional case for the Krogh model. Consumption drives oxygen
from the erythrocyte into the tissue. The simulation is performed for one second of dimensional
time during which the erythrocyte would move 467 lm at vrbc , i.e. approximately the capillary
length in this muscle [17]. Fig. 2 shows the PO2 at the following points (indicated in Fig. 1); the
center of the erythrocyte, Pc ; a point in the plasma gap, Pg ; the point on the midplane of the
erythrocyte on the outer edge of the capillary wall, Pw ; the point on the midplane of the erythrocyte on the outer edge of the interstitial ¯uid, Pi ; and the point on the midplane of the erythrocyte on the outer edge of the tissue wall, Pt . Transients from the initial conditions decay in the
®rst 0.01 s, after which ¯ux density out of the erythrocyte matches consumption in the tissue and
the PO2 decreases uniformly over the entire domain. The following characteristics of the reference
case that are presented correspond to a dimensional time of one half second. Fig. 3 shows the PO2
pro®les as a function of r from the axis of the capillary to the edge of the tissue rt in units of rp , the
inner capillary radius. The PO2 continues to fall through the plasma sleeve, the capillary wall, and
the interstitial ¯uid region before reaching the tissue, where consumption lowers the PO2 even
further. The changes in slope in each region are due to di€erences in the solubility and di€usivity
of oxygen.
The two RBC boundary pro®les show the PO2 variation in the radial direction in a cross
section just adjacent to the erythrocyte, at the leading and trailing edge caps of the cylinder.
The PO2 values at r ˆ 0 are lower than the erythrocyte core PO2 due to intraerythrocyte resis-

Fig. 2. Transient PO2 pro®les for reference case. PO2 values at di€erent points in the computational domain as
illustrated in Fig. 1, Pc , at the center of the erythrocyte; Pg , in the plasma gap; Pw , at the centerline on the outer edge of
capillary wall; Pi , at the centerline on the outer edge of the interstitial ¯uid; Pt , at the centerline on the outer edge of
tissue cylinder.

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133

Fig. 3. Radial PO2 pro®les for reference case at the middle of the capillary. PO2 values at di€erent cross sections
(C, erythrocyte centerline; LE, leading edge erythrocyte cap; TE, trailing edge erythrocyte cap; LE gap, leading edge of
the domain boundary in the plasma gap; TE gap, trailing edge of the domain boundary in the plasma gap); in the
model domain as a function of normalized radial position r=rp , where r ranges from 0 to rt . The vertical dotted lines
from left to right indicate the positions of the erythrocyte lateral surface, inner capillary wall, outer capillary wall, outer
interstitial space, and outer tissue edge, respectively.

tance, and the PO2 at the leading edge of the erythrocyte is seen to be slightly lower than the
trailing edge pro®le.
The two midgap pro®les represent the left and right domain boundaries. Although there is a
large di€erence between the PO2 in the plasma adjacent to the erythrocyte as compared to the gap,
the di€erences between the pro®les become minimal about halfway into the tissue. This is further
illustrated in Fig. 4, which shows the PO2 pro®les in the axial direction at the inner capillary wall,
r ˆ rp , the outer capillary wall, r ˆ rw , the interstitial ¯uid, r ˆ ri , and at the outer edge of the
tissue cylinder, r ˆ rt . Although large axial PO2 gradients exist at the capillary wall, these have
largely disappeared by the time oxygen reaches the edge of the tissue cylinder; only a small
variation in PO2 remains. Note the absence of large radial PO2 gradients in the plasma gap region
and the existence of an erythrocyte `zone of in¯uence' in the regions of the domain close to the
erythrocyte.
The radial ¯ux density calculated with the local PO2 gradient in each of these regions is shown
in Fig. 5. The ¯ux density is minimal in the region of the plasma gaps but increases as the erythrocyte edges are approached. The slight asymmetry in the ¯ux density is due to erythrocyte
movement, with higher ¯uxes at the leading edge where the erythrocyte is unloading to a
lower PO2 .

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C.D. Eggleton et al. / Mathematical Biosciences 167 (2000) 123±143

Fig. 4. Axial PO2 pro®les for reference case at the middle of the capillary. PO2 values at di€erent radial distances (IC,
inner capillary wall; OC, outer capillary wall; OI, outer interstitial space; OT, outer tissue edge) in the model domain as
a function of normalized axial position z=rp , where z ranges from ÿLtot =2 to ‡Ltot =2. The vertical dotted lines indicate
the trailing and leading edges of the erythrocyte.

4.3. E€ect of hematocrit
By keeping all other parameters constant and varying only lineal density it was possible to
isolate the e€ect of changing Lp . Fig. 6 shows the e€ect of varying hematocrit on the radial PO2
pro®les at t ˆ 0:5 s, with the case above shown for reference. As the hematocrit decreases the
gradient in PO2 from the lateral edge of the erythrocyte to the inner edge of the tissue increases.
The detrimental e€ect on tissue edge PO2 with increasing cell spacing is clearly evident.
This e€ect also occurs in the plasma gap (Fig. 7(a) and (b)), with the plasma PO2 at the centerline decreasing as cell spacing increases. The tissue edge PO2 compared with the previous ®gure
shows only minor variations between the regions adjacent to the plasma gap and adjacent to the
erythrocyte at the same hematocrit.
The PO2 pro®le in the axial direction at the capillary wall is shown as a function of hematocrit
in Fig. 8. The curves extend to di€erent axial positions, re¯ecting the increase in the length of the
plasma gap, since distances are normalized by rp which was kept constant. Again, there is a
signi®cant variation in PO2 at the capillary wall, with increasing asymmetry apparent at lower
values of hematocrit. Average tissue edge PO2 decreases with increasing cell spacing. At the hematocrits considered the PO2 is nearly uniform along the tissue edge.
Increases in cell spacing are also re¯ected in the asymmetry of the radial ¯ux density distribution in the axial direction along the capillary wall (Fig. 9). The higher ¯uxes at lower hematocrit
are due to increased domain size with the same value of tissue consumption. The ¯ux density
decreases rapidly with increasing distance from the erythrocyte in all cases.

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135

Fig. 5. Axial pro®les of the radial ¯ux density for reference case at the middle of the capillary. Radial ¯ux density …j†
values at di€erent radial distances (IC, inner capillary wall; OC, outer capillay wall; OI, outer interstitial space; OT,
outer tissue edge) in the model domain as a function of normalized axial position z=rp , where z ranges from ÿLtot =2 to
‡Ltot =2. The vertical dotted lines indicate the trailing and leading edges of the erythrocyte.

Values of the intracapillary mass transfer coecient per unit capillary wall area, kcap , given by
Eq. (20), as a function of hematocrit are shown in Fig. 10(a). The mass transfer coecient, kcap ,
fell by a factor of 2.3 at the lowest hematocrit (H ˆ 0:25) and rose by a factor of 1.6 at the highest
hematocrit (H ˆ 0:55) when compared to the reference case (H ˆ 0:43). A single calculation was
done at the reference hematocrit at the velocity and consumption for resting conditions and the
calculated kcap di€ered by only 7.6% from that for working conditions. Simulations were calculated for the same parameters with no-¯ux boundary conditions at the end caps of the computational domain and the resulting mass transfer coecients di€ered by less than 3% with the
moving boundary conditions over the range of hematocrits considered.
To facilitate applications of these results to calculations of oxygen transport in capillary networks,
the dependence of the mass transfer coecient with hematocrit, H , is ®t to a quadratic function
kcap ˆ 1:21 ÿ 4:38H ‡ 23:6H 2 …10ÿ6 ml O2 sÿ1 Torrÿ1 cmÿ2 †;

…25†

where H is hematocrit given as volume fraction. Note that the intercept value at H ˆ 0 does not
correspond to cell-free plasma.
One of the main objectives of this study is to determine the e€ect of the boundary conditions at
the capillary wall on the calculated values of the capillary mass transfer coecient. A number of
previous studies set a constant PO2 value at the wall [7,12,18], and Groebe and Thews [13] showed
that the results were dependent on the type of the boundary condition: constant PO2 or constant

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C.D. Eggleton et al. / Mathematical Biosciences 167 (2000) 123±143

Fig. 6. Midcell radial PO2 pro®le at the middle of the capillary as a function of hematocrit (H ˆ 25%, 35%, 43%, 55%).
PO2 values at z ˆ 0 as a function of normalized radial position r=rp , where r ranges from 0 to rt . The vertical dotted
lines from left to right indicate the positions of the erythrocyte lateral surface, the inner capillary wall, outer capillary
wall, outer interstitial space, and outer tissue edge, respectively.

¯ux. Thus in Fig. 10(a) we also present the results of solving the full equations with a constant
wall PO2 (chosen at 10 Torr). The calculated values are signi®cantly higher than those obtained
with the natural boundary conditions (2 times higher at H ˆ 0:25 and 1.15 times higher at
H ˆ 0:55). The reason for this large underestimation of the transport resistance (1=kcap ) is that
each erythrocyte is surrounded by a region of elevated PO2 , and the PO2 gradients in the vicinity
of an erythrocyte are smaller for the natural boundary conditions than in the case when an arti®cial constant wall PO2 boundary condition is imposed. Importantly, the calculated intravascular resistance when the full equations and tissue boundary conditions are used is even higher
than the previously reported values. In some studies [18,25] the approximation introduced by
Clark et al. [5] was used instead of the full equations. This approximation given by Eq. (9)
simpli®es the calculations signi®cantly, but its validity has only been established for a limited set
of conditions [5]. Therefore, in Fig. 10(a) we compare this approximation with the full solution for
a wide range of capillary hematocrits. In this case, the calculations were done at steady state, with
a preset value of the erythrocyte core oxygen tension, Pc ˆ 40 Torr, and natural boundary conditions at the wall. These predictions show the same trend with cell spacing, but overpredict the
mass transfer coecient when compared to the full equations for oxygen transport within the
erythrocyte. The mass transfer coecient was overpredicted by a factor of 1.34 at the lowest
hematocrit where the ¯ux density was largest, and by 2.32 at the highest hematocrit where the ¯ux
density was lowest.

C.D. Eggleton et al. / Mathematical Biosciences 167 (2000) 123±143

137

Fig. 7. Midgap radial PO2 pro®le at middle of the capillary as a function of hematocrit (H ˆ 25%, 35%, 43%, 55%). PO2
values at the leading (a) and trailing (b) edges of the domain as a function of normalized radial position r=rp , where r
ranges from 0 to rt . The vertical dotted lines from left to right indicate the positions of the erythrocyte lateral surface,
the inner capillary wall, outer capillary wall, outer interstitial space, and outer tissue edge, respectively.

The e€ect of ¯ux density and PO2 variation on the local intracapillary mass transfer coecient
is shown in Fig. 10(b). Here, the local mass transfer coecient given by Eq. (24) is normalized by
average mass transfer coecient de®ned by Eq. (20) at the same hematocrit. For the reference
case, H ˆ 0:43, the local mass transfer coecient along the capillary wall decreased by a factor of
16 at the leading edge gap and increased by a factor of 3.77 at the eyrthrocyte midpoint with
respect to the average value kcap .
4.4. E€ect of velocity
The e€ect of erythrocyte velocity, vrbc , was investigated by altering the velocity from the reference case. The velocity in the reference case is ®ve times that observed in resting muscle, and
hereby designated v0 . Simulations were done for v0 , 2v0 , and 5v0 (25 times the observed resting
velocity). Fig. 11 demonstrates that the mass transfer coecient per unit area varies with velocity,
but that variations are less than 2.7% when the velocity increases ®ve-fold.
4.5. E€ect of capillary radius
The e€ect of capillary radius on the mass transfer coecient is evaluated in this section. The
average of the standard deviations of measured capillary radii on the arteriolar and venular sides
[17], 0:175 lm, is used here to determine the variation in the capillary radius, rp . At the reference
hematocrit, simulations are done for radial increases/decreases of 1 and 2 standard deviations.
The plasma sleeve width between the lateral surface of erythrocyte and endothelium, 0:16 lm, is

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C.D. Eggleton et al. / Mathematical Biosciences 167 (2000) 123±143

Fig. 8. Inner capillary wall (axial) PO2 pro®le at the middle of the capillary as a function of hematocrit (H ˆ 25%, 35%,
43%, 55%). PO2 values at r ˆ rp as a function of normalized axial position z=rp , where z ranges from ÿLtot =2 to ‡Ltot =2.
The vertical dotted lines indicate the trailing and leading edges of the erythrocyte.

held ®xed as the capillary radius is varied. Given the capillary radius, the cylindrical cell radius is
determined, and the length is found from the ®xed volume of the cell from Eq. (2). The length of
the plasma gap is calculated from the speci®ed hematocrit from Eq. (3) and the length of the tissue
cylinder is determined from Eq. (5). The resulting calculated mass transfer coecients, kcap , at the
reference hematocrit (H ˆ 0:43) are shown in Fig. 12. The mass transfer coecient (compared to
rp ˆ 1:8 lm) decreased to 0.96 and 0.95 of its reference value for radial decreases of 1 and 2
standard deviations, respectively. An increase to 1.06 and 1.14 of its reference value was calculated for a radial increase of 1 and 2 standard deviations, respectively. The e€ect of capillary
radius is also determined at the lower hematocrit of H ˆ 0:25. The resulting calculated mass
transfer coecients, kcap , at the lower hematocrit (H ˆ 0:25) are shown in Fig. 12. The mass
transfer coecient per unit area decreases (compared to rp ˆ 1:8 lm) to 0.98 and 0.99 of its
reference value for radial decreases of 1 and 2 standard deviations, respectively, and increases to
1.04 and 1.10 of its reference value for radial increases of 1 and 2 standard deviations, respectively.

5. Discussion
A mathematical model has been used to predict the e€ect of hematocrit, erythrocyte velocity
and capillary radius on the mass transfer coecient. A feature of this model is that appropriate

C.D. Eggleton et al. / Mathematical Biosciences 167 (2000) 123±143

139

Fig. 9. Axial pro®les of the radial ¯ux density at middle of the capillary as a function of hematocrit (H ˆ 25%, 35%,
43%, 55%). Radial ¯ux density …j† values at r ˆ rp as a function of normalized axial position z=rp , where z ranges from
ÿLtot =2 to ‡Ltot =2. The vertical dotted lines indicate the trailing and leading edges of the erythrocyte.

matching boundary conditions are used at the capillary wall, while most previous models speci®ed
either constant PO2 or constant ¯ux boundary conditions. The non-uniformity of these quantities
at the capillary wall induce large variations in the pro®le of the local mass transfer coecient,
pointing out the importance of calculating conditions at the capillary wall rather than specifying
them. For all cases considered here, the local mass transfer coecient along the capillary wall,
kcap …z†, showed variation. It was largest in the section above the erythrocyte (where the PO2 was
largest) and smallest in the plasma gaps.
This study shows that the mass transfer coecient changes substantially with hematocrit. From
the lowest hematocrit (H ˆ 0:25) to the largest (H ˆ 0:55), the mass transfer coecient per unit of
capillary wall area increased by a factor of 3.7 with ®xed velocity and capillary radius. Using the
approximation of Clark et al. [5] consistently overpredicts the value of the mass transfer coecient, with the di€erence diminishing with hematocrit. In addition, the results show only a 2.7%
increase in the intracapillary mass transfer coecient due to a ®ve fold increase of erythrocyte
velocity. This is in qualitative agreement with the results of Groebe and Thews [13], who found
that erythrocyte movement at a velocity of 0.4 cm/s could enhance oxygen transport by up to 20%
compared to the stationary case because erythrocyte movement improved the uniformity of oxygen tension and oxygen ¯ux. The capillary radius was decreased and increased up to two
standard deviations to determine its e€ect on the mass transfer coecient and the variations of
kcap did not exceed 15% compared to the reference case. Erythrocyte shape, velocity and capillary

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C.D. Eggleton et al. / Mathematical Biosciences 167 (2000) 123±143

Fig. 10. Intracapillary mass transfer coecient: (a) per unit area capillary wall kcap as a function of hematocrit;
(b) normalized local capillary wall mass transfer coecient.

C.D. Eggleton et al. / Mathematical Biosciences 167 (2000) 123±143

141

Fig. 11. Intracapillary mass transfer coecient per unit area of capillary wall, kcap , as a function of the reference
velocity, where v0 is ®ve times the velocity observed in resting muscle.

Fig. 12. Capillary wall mass transfer coecient, kcap , as a function of capillary radius.

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C.D. Eggleton et al. / Mathematical Biosciences 167 (2000) 123±143

radius are related in a complex manner. A more physiologically realistic model would require the
calculation of the hamster erythrocyte deformation as a function of velocity and radius, as was
done by Secomb et al. [26] for the human erythrocyte. Wang and Popel [12] have shown that
realistic shape changes with velocity with parameters corresponding to human erythrocytes can
lead to a 25% decrease in the mass transfer coecient.
A number of important questions have been answered by using the mathematical models developed here in conjunction with available physiological data. A model was developed to more
accurately determine the e€ect of erythrocyte spacing (hematocrit) and movement on the mass
transfer coecient. Intracapillary processes were considered in detail to avoid the arbitrary
speci®cation of a boundary condition at the capillary wall. Calculations done for hamster retractor muscle yielded values of mass transfer coecients as a function of hematocrit, velocity and
capillary radius. Capillary hematocrit was found to have a dominant e€ect on the PO2 distribution
and the intracapillary mass transfer coecient, whereas velocity and capillary radius had relatively minor e€ects. The calculated values of the mass transfer coecients are being used in
simulation of O2 transport from realistic capillary networks [27].

Acknowledgements
This work was supported by NIH grants HL18292 and HL52864.

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