Calculations of intracapillary oxygen tension distributions in
muscle
C.D. Eggleton
*, A. Vadapalli, T.K. Roy, A.S. Popel
1Department 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 O2transport from the erythrocyte to the mitochondrion is important to understanding potential transport limitations. A mathematical model is developed to accurately determine the eects 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 eect on the PO2 distribution and the intracapillary mass transfer coecient per unit capillary area,kcap, increased by a factor of 3.7 from the lowest (H0:25) to the highest (H0:55) capillary hematocrits considered. Erythrocyte velocity had a relatively minor eect, with only a 2.7% increase in the mass transfer coecient 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 inkcap that are <15% at the reference case. The mag-nitude of these changes increases with hematocrit. An equation to approximate the dependence of the mass transfer coecient on hematocrit is developed for use in simulations of O2 transport from a capillary network. Ó 2000 Elsevier Science Inc. All rights reserved.
Keywords:Computational model; Oxygen transport; Microcirculation; Striated muscle
1. Introduction
Oxygen is required by cells for the sustained production of ATP. The resistances to oxygen transport from erythrocytes to mitochondria are investigated using a mathematical model
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*
Corresponding author. Present address: Department of Mechanical Engineering, UMBC, 1000 Hill top Circle, Baltimore, MD 21250, USA. Tel.: +1-410 455 3334; fax: +1-410 455 1052.
E-mail address:eggleton@umbc.edu (C.D. Eggleton).
1
Author for reprint requests.
designed to assess tissue oxygenation and to identify the extent to which dierent factors aect oxygen transport in the microcirculation. We brie¯y review recent advances in the modeling of oxygen transport relevant to the present work. The reader is referred to general reviews on oxygen transport [1±3].
Oxygen transport in the peripheral circulation is known to occur from arterioles, capillaries, and venules acting as sources and sinks [4]. The fact that oxygen transport occurs on a scale larger than that of individual capillary segments means that simulations should be carried out at the microvascular network level in order to understand how heterogeneities in ¯ow and consumption interact to generate observed PO2 distributions. The goal of this work is to calculate the single capillary mass transfer coecients for subsequent use in these network models.
Identifying oxygen transport resistances and their primary determinants is crucial to an un-derstanding of the oxygen transport process and to developing methods of enhancing oxygen availability. This paper describes the construction and use of a mathematical model at the single capillary level that characterizes each region and leads to estimates of the component transport resistances along the pathway for oxygen. The model utilizes experimentally determined values for required parameters in order to make quantitative comparisons of measured quantities.
The pathway for oxygen transport to tissue begins at the erythrocyte. Erythrocytes in the pe-ripheral circulation release oxygen by diusion into the surrounding plasma with a ¯ux related to the local PO2 gradient. The oxygen then diuses through the plasma, across the capillary wall, across the interstitium, into muscle cells and to the mitochondria, where most of the oxygen is consumed in the process of oxidative phosphorylation resulting in the production of ATP. The drops in PO2as oxygen traverses these dierent regions allow us to de®ne transport resistanceskÿ1 (inverse mass transfer coecients) for each region relating the local ¯uxjto the PO2 gradientDP
across the region according to
jkDP: 1
Erythrocytes contain most of their oxygen in the form of oxyhemoglobin, which diuses toward the erythrocyte membrane and dissociates to release free oxygen which then diuses out of the cell.
The eect of oxyhemoglobin dissociation kinetics on oxygen transport out of the erythrocyte can be represented by a mass transfer coecient of the form shown in Eq. (1). Under physi-ological conditions, the erythrocyte contains oxyhemoglobin in equilibrium with free oxygen except in a layer near the membrane. The coupled transport equations for oxygen and oxy-hemoglobin in the entire erythrocyte volume are represented. We also utilize an expression for the ¯ux through the non-equilibrium boundary layer derived by Clark et al. [5] based on the equations for one-dimensional transport through a slab as a function of the PO2 on each side for comparison.
capillary and a constant PO2boundary condition at the capillary wall. They included the eect of oxyhemoglobin dissociation kinetics and oxygen transport through plasma, although the eects of erythrocyte movement and plasma convection were not considered; PO2 gradients between ad-jacent erythrocytes were neglected, permitting the use of periodic boundary conditions. They calculated the capillary mass transfer coecient as a function of capillary hematocrit and found the capillary transport resistance to decrease with decreasing cell separation. The mass transfer coecient relating the ¯ux and the PO2 drop between the erythrocyte and the capillary wall was not found to vary greatly over a wide range of erythrocyte PO2 or hemoglobin saturation.
The tissue PO2within a few microns from the capillary lumen undergoes oscillations caused by passing erythrocytes and plasma gaps. The greater the blood velocity, i.e. the higher the frequency of the passing erythrocytes, the smaller the penetration of these oscillations into the tissue [3]. Tsai and Intaglietta [8] numerically simulated the time-dependent transport of oxygen in capillaries containing equally-spaced cylindrical erythrocytes by applying moving boundary conditions at the capillary wall to simulate erythrocyte movement and calculated the PO2 in the tissue. The moving boundary conditions at the capillary wall were used to represent intracapillary PO2 dis-tribution. The plasma gaps were assumed to make no contribution to oxygen transport and no plasma sleeve was considered.
Other mathematical models used to calculate the intracapillary mass transfer coecient have imposed boundary conditions at the capillary wall. Sheth and Hellums [9] simulated oxygen transport in hemoglobin layers con®gured to match the microcirculatory system with a constant ¯ux boundary condition at the capillary wall. A constant ¯ux boundary condition at the capillary wall was also used by Baxley and Hellums [10] with an axisymmetric model that treated blood as a hemoglobin solution and a suspension of discrete erythrocytes. Constant ¯ux boundary condi-tions were also used by Federspiel and Sarelius [11]. A constant PO2boundary condition was used by Federspiel and Popel [7] who modeled erythrocytes as spheres and Wang and Popel [12] who used realistic erythrocyte shapes. Groebe [25] used both constant PO2and constant ¯ux boundary conditions at a `tissue' boundary several microns away from the blood-vessel interface and considered the eects of erythrocyte velocity. Since the boundary condition at the capillary wall is neither constant PO2nor constant ¯ux, it is dicult to specify boundary conditions that are valid under physiological conditions. The conditions at the capillary wall can be determined from a model that simultaneously solves the equations governing the intracapillary and extracapillary regions. Natural boundary conditions that match PO2 at interface of the capillary wall and the tissue were modeled by Knisely et al. [14] who treated blood as a continuum ¯uid. Here we have formulated a model of intracapillary transport processes that explicitly includes oxygen transport within the erythrocyte, plasma sleeve and plasma gap.
The calculations of Federspiel and Popel [7] show that the intracapillary mass transfer co-ecient is a weak function of erythrocyte PO2; we con®rm the results in the present work and also show that the mass transfer coecient is a weak function of extracapillary parameters. Therefore, calculated intracapillary mass transfer coecients can be utilized in capillary network models, enabling calculations of tissue PO2 without detailed consideration of intracapillary events.
2. Mathematical model
A schematic for the model is shown in Fig. 1. The equations are written in the frame of ref-erence of a single erythrocyte, with the capillary wall, interstitial ¯uid, and tissue regions moving relative to the erythrocyte and its surrounding plasma. Plasma convection is not considered, since its eect on oxygen transport was shown to be small [15].
2.1. Intracapillary transport
The capillary lumen is assumed to contain plasma and equally-spaced erythrocytes modeled as cylinders containing hemoglobin. In the model, oxygen leaves the erythrocyte either through the lateral surface (into the plasma sleeve to the capillary wall) or through the basal surfaces (into the plasma gap between erythrocytes and then to the capillary wall).
The erythrocyte and its surrounding plasma are treated as stationary relative to the moving capillary wall, interstitial ¯uid, and tissue regions. Erythrocytes are assumed to be cylindrical, leading to the following expression for erythrocyte volume,Vrbc, in terms of erythrocyte radius,rrbc, and erythrocyte length, Lrbc
Vrbcpr2
rbcLrbc: 2
The capillary (or tube) hematocrit, Hcap, can be expressed in terms of these parameters and the length of the plasma gap, Lp
Hcap
rrbc2 Lrbc
r2
p LrbcLp
; 3
whererp is the inner radius of the capillary. The lineal density, LD, of erythrocytes is de®ned as the number of cells per unit capillary length and can be calculated from
LD LrbcLpÿ 1
4
and the total length of the tissue cylinder is given by
The equations that govern oxygen transport within the erythrocyte were developed by Clark et al. [5] and Federspiel and Popel [7] and account for diusion of free oxygen and oxygen bound to hemoglobin, as well as dissociation/association. The equation for free oxygen is
arbc
wherePis the oxygen tension,Sthe hemoglobin saturation, Drbc the diusion coecient of free O2 inside the erythrocyte, arbc the intraerythrocyte O2 solubility coecient, kÿ is the
oxyhemo-globin dissociation rate constant,P50the PO2 corresponding to 50% hemoglobin (Hb) saturation,
n the Hill coecient, and NHb is the hememonomer concentration inside the erythrocyte. The boundary conditions for P and S are the continuity of PO2 at the erythrocyte membrane, and
oS=on0. The latter boundary condition represents no-¯ux of oxyhemoglobin through the
erythrocyte membrane, wheren is the erythrocyte surface normal. We will compare the solution governed by Eqs. (6) and (7) to an approximate analytic solution for intraerythrocyte transport given by Clark et al. [5].
In their solution, the local dimensional oxygen ¯ux density at the erythrocyte surface can be represented as
j Pc;P q Pc;P DrbcarbckÿP50NHb1=2; 8
wherePcis the PO2in the erythrocyte core,P the PO2 at the surface of the erythrocyte, andqis a dimensionless ¯ux density. The dimensionless ¯ux density of oxygen,q, leaving the erythrocyte is given by
The saturationScS Pcis determined from Hill's equation for the equilibrium oxyhemoglobin dissociation curve
S P=P50
n
1 P=P50n
; 10
wherenis the Hill coecient.
j ÿDpap oP
on; 11
where n is the surface normal, Dp the diusion coecient of free O2 in plasma, and ap is the
plasma O2 solubility coecient.
The oxygen transport equation in the plasma re¯ects the free diusion of oxygen and plasma movement with local erythrocyte velocity. Neglecting velocity gradients in the plasma, mass transport is governed by
The capillary wall and interstitium are modeled as annular regions of ®nite thickness with appropriate transport properties. Oxygen consumption in these regions is not considered, since they occupy only a small volume compared to the tissue region. The muscle ®bers are assumed to contain myoglobin and to consume oxygen at a constant rate.
Inside the capillary wall,
where Dw is the diusion coecient of free O2 in the capillary wall, aw the capillary wall O2
solubility coecient andvrbc is the erythrocyte velocity. In the interstitial ¯uid layer,
whereDiis the diusion coecient of free O2in the interstitial ¯uid, andaiis the interstitial ¯uid
O2 solubility coecient.
The radius of the tissue cylinder,rt, is related to the capillary density, NA c;f, through
pr2
t NA c;f
ÿ1
15
(here the common stereological notation for capillary density,NA c;f, is used). We assume that myoglobin and oxygen are in local chemical equilibrium
SMb
P P PMb
50
; 16
o
where the terms represent free oxygen and oxygen bound to myoglobin,Dt is the diusion co-ecient of free oxygen in the tissue,DMbthe diusion coecient of oxygen bound to myoglobin in the tissue,atthe tissue O2 solubility coecient,NMb the tissue concentration of Mb, andoSMb=oP
is derived from Eq. (16).M Prepresents a concentration-dependent consumption rate. Here, we use a zero-order chemical kinetics model of consumption, such that
M P 0 for P 0;
M0 for P >0:
The boundary condition at the edge of the tissue cylinder is
oP
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
Dpap
whereoPk=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 approx-imation 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
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, LD632 cells cmÿ1[17], and observed erythrocyte lengthLrbc; 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,Lrbc8:1610ÿ4 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,Vrbc6:9310ÿ11 cm3[20], is used to calculate erythrocyte radius, rrbc, from Eq. (2). Capillary hematocrit is then calculated from Eq. (3) using rp1:8 lm [17], which was the average of the mean radii observed for ar-teriolar 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 vrbc9:3510ÿ3 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 eect of velocity variations.
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]
In the tissue region, we used the working muscle consumptionMcestimated by Ellsworth et al. [17] as 10 times the resting muscle consumption of 0.89 ml O2 100 gÿ1 minÿ
1
measured by Sullivan and Pittman [21]. This value falls below the maximum consumption,VO_ 2 max, estimated for this muscle as 21 times the resting rate based on the mitochondrial volume density [22]. The value ofNMb 410ÿ7 mol cmÿ3in hamster retractor muscle has been measured by Meng et al. [23]. A resting value of NA c;f 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 coecient 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:
kcap
The local mass transfer coecient,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,
kcap z Jp z
PcÿPp z2prpLtot: 24
4.2. Reference case
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 ery-throcyte on the outer edge of the interstitial ¯uid, Pi; and the point on the midplane of the ery-throcyte 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 PO2decreases 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 ofrfrom the axis of the capillary to the edge of the tissuertin units ofrp, the inner capillary radius. The PO2continues 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 dierences in the solubility and diusivity 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 r0 are lower than the erythrocyte core PO2 due to intraerythrocyte
resis-Fig. 2. Transient PO2 pro®les for reference case. PO2 values at dierent 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
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 dierence between the PO2in the plasma adjacent to the erythrocyte as compared to the gap, the dierences between the pro®les become minimal about halfway into the tissue. This is further illustrated in Fig. 4, which shows the PO2pro®les in the axial direction at the inner capillary wall,
rrp, the outer capillary wall, rrw, the interstitial ¯uid,r ri, and at the outer edge of the tissue cylinder, rrt. 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 PO2remains. Note the absence of large radial PO2gradients 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 ery-throcyte edges are approached. The slight asymmetry in the ¯ux density is due to eryery-throcyte movement, with higher ¯uxes at the leading edge where the erythrocyte is unloading to a lower PO2.
Fig. 3. Radial PO2 pro®les for reference case at the middle of the capillary. PO2 values at dierent 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 positionr=rp, whererranges from 0 tort. The vertical dotted lines
4.3. Eect of hematocrit
By keeping all other parameters constant and varying only lineal density it was possible to isolate the eect of changing Lp. Fig. 6 shows the eect of varying hematocrit on the radial PO2 pro®les at t0:5 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 eect on tissue edge PO2 with increasing cell spacing is clearly evident.
This eect also occurs in the plasma gap (Fig. 7(a) and (b)), with the plasma PO2 at the cen-terline decreasing as cell spacing increases. The tissue edge PO2compared 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 dierent 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 he-matocrits 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 distri-bution 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.
Fig. 4. Axial PO2pro®les for reference case at the middle of the capillary. PO2 values at dierent 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 positionz=rp, wherezranges fromÿLtot=2 toLtot=2. The vertical dotted lines indicate
Values of the intracapillary mass transfer coecient per unit capillary wall area,kcap, given by Eq. (20), as a function of hematocrit are shown in Fig. 10(a). The mass transfer coecient,kcap, fell by a factor of 2.3 at the lowest hematocrit (H 0:25) and rose by a factor of 1.6 at the highest hematocrit (H 0:55) when compared to the reference case (H 0:43). A single calculation was done at the reference hematocrit at the velocity and consumption for resting conditions and the calculated kcap diered by only 7.6% from that for working conditions. Simulations were calcu-lated for the same parameters with no-¯ux boundary conditions at the end caps of the compu-tational domain and the resulting mass transfer coecients diered 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 coecient with hematocrit,H, is ®t to a quadratic function
kcap1:21ÿ4:38H23:6H2 10ÿ6 ml O2 sÿ1 Torrÿ1 cmÿ2; 25
whereHis hematocrit given as volume fraction. Note that the intercept value atH 0 does not correspond to cell-free plasma.
One of the main objectives of this study is to determine the eect of the boundary conditions at the capillary wall on the calculated values of the capillary mass transfer coecient. 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
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 dierent 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 positionz=rp, wherezranges fromÿLtot=2 to
¯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 0:25 and 1.15 times higher at
H 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 ar-ti®cial constant wall PO2 boundary condition is imposed. Importantly, the calculated intravas-cular 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,Pc40 Torr, and natural boundary con-ditions at the wall. These predictions show the same trend with cell spacing, but overpredict the mass transfer coecient when compared to the full equations for oxygen transport within the erythrocyte. The mass transfer coecient 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.
Fig. 6. Midcell radial PO2pro®le at the middle of the capillary as a function of hematocrit (H25%, 35%, 43%, 55%).
PO2values atz0 as a function of normalized radial positionr=rp, whererranges from 0 tort. The vertical dotted
The eect of ¯ux density and PO2 variation on the local intracapillary mass transfer coecient is shown in Fig. 10(b). Here, the local mass transfer coecient given by Eq. (24) is normalized by average mass transfer coecient de®ned by Eq. (20) at the same hematocrit. For the reference case,H 0:43, the local mass transfer coecient 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 valuekcap.
4.4. Eect of velocity
The eect of erythrocyte velocity, vrbc, was investigated by altering the velocity from the ref-erence case. The velocity in the refref-erence 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 coecient per unit area varies with velocity, but that variations are less than 2.7% when the velocity increases ®ve-fold.
4.5. Eect of capillary radius
The eect of capillary radius on the mass transfer coecient is evaluated in this section. The average of the standard deviations of measured capillary radii on the arteriolar and venular sides [17], 0:175lm, 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
Fig. 7. Midgap radial PO2pro®le at middle of the capillary as a function of hematocrit (H25%, 35%, 43%, 55%). PO2
values at the leading (a) and trailing (b) edges of the domain as a function of normalized radial positionr=rp, wherer
ranges from 0 tort. The vertical dotted lines from left to right indicate the positions of the erythrocyte lateral surface,
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 coecients,kcap, at the reference hematocrit (H 0:43) are shown in Fig. 12. The mass transfer coecient (compared to
rp 1:8lm) 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 calcu-lated for a radial increase of 1 and 2 standard deviations, respectively. The eect of capillary radius is also determined at the lower hematocrit of H 0:25. The resulting calculated mass transfer coecients, kcap, at the lower hematocrit (H 0:25) are shown in Fig. 12. The mass transfer coecient per unit area decreases (compared to rp1:8 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 eect of hematocrit, erythrocyte velocity and capillary radius on the mass transfer coecient. A feature of this model is that appropriate
Fig. 8. Inner capillary wall (axial) PO2pro®le at the middle of the capillary as a function of hematocrit (H25%, 35%,
43%, 55%). PO2values atrrpas a function of normalized axial positionz=rp, wherezranges fromÿLtot=2 toLtot=2.
matching boundary conditions are used at the capillary wall, while most previous models speci®ed either constant PO2or 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 coecient, pointing out the importance of calculating conditions at the capillary wall rather than specifying them. For all cases considered here, the local mass transfer coecient 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 coecient changes substantially with hematocrit. From the lowest hematocrit (H 0:25) to the largest (H 0:55), the mass transfer coecient 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 dierence diminishing with hematocrit. In addition, the results show only a 2.7% increase in the intracapillary mass transfer coecient 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 ox-ygen tension and oxox-ygen ¯ux. The capillary radius was decreased and increased up to two standard deviations to determine its eect on the mass transfer coecient and the variations of
kcapdid not exceed 15% compared to the reference case. Erythrocyte shape, velocity and capillary
Fig. 9. Axial pro®les of the radial ¯ux density at middle of the capillary as a function of hematocrit (H25%, 35%, 43%, 55%). Radial ¯ux density jvalues atrrpas a function of normalized axial positionz=rp, wherezranges from
Fig. 10. Intracapillary mass transfer coecient: (a) per unit area capillary wall kcap as a function of hematocrit;
Fig. 12. Capillary wall mass transfer coecient,kcap, as a function of capillary radius.
Fig. 11. Intracapillary mass transfer coecient per unit area of capillary wall,kcap, as a function of the reference
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 coecient.
A number of important questions have been answered by using the mathematical models de-veloped here in conjunction with available physiological data. A model was dede-veloped to more accurately determine the eect of erythrocyte spacing (hematocrit) and movement on the mass transfer coecient. Intracapillary processes were considered in detail to avoid the arbitrary speci®cation of a boundary condition at the capillary wall. Calculations done for hamster re-tractor muscle yielded values of mass transfer coecients as a function of hematocrit, velocity and capillary radius. Capillary hematocrit was found to have a dominant eect on the PO2distribution and the intracapillary mass transfer coecient, whereas velocity and capillary radius had rela-tively minor eects. The calculated values of the mass transfer coecients 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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