An examination of the contribution of red cell spacing to the uniformity of oxygen flux at the capillary wall

MICROVASCULAR

RESEARCH

27, 273-285 (1984)

An Examination of the Contribution of Red Celi Spacing to the Uniformity of Oxygen Flux at the Capilbry Wall WILLIAM Departments

of Chemical

J. FEDERSPIEL AND INGRID H. SARELIUS’ Engineering and Radiation Biology and Biophysics, Rochester, Rochester, New York 14642 Received

University

of

October 4, 1982

It is generally assumed that capillary blood is homogeneous for 0, supply and that red cells can provide a constant, uniform flux of 0, out of the capillary regardless of the spacing between cells. Using a simplified model of red cells moving through a capillary in skeletal musde, an approximate analysis is developed to study the effect of red cell spacing on the ability of erythrocytes to provide a constant, uniform flux of O2 at the capillary wall. The results suggest the existence of a critical red cell separation distance above which the flux of 0, at the capillary wall between red cells cannot remain uniform and the capillary blood is no longer homogeneous for O2 supply. In resting muscle the predicted critical separation distance is greater than four cell lengths. During maximal 0, consumption, the critical separation distance predicted by the model is one cell length. These predictions agree closely with in vivo observations of red cell spacing. The total red cell flux through a capillary is determined not only by red cell spacing (hematocrit) but also by erythrocyte velocity; a simple example is given which suggests that changes in each of these variables are not equivalent in maintaining a constant and uniform flux of 0, at the capillary wall.

INTRODUCTION It is well established that in capillaries, erythrocytes flow in single file with plasma gaps of variable distance separating the cells (Krogh, 1918; Johnson, 1971; Klitzman and Duling, 1979). Further, the spacing between erythrocytes varies. This is indicated by changes in capillary hematocrit (which reflects the number of cells per unit length and hence red cell spacing). Thus capillary hematocrit increases during functional hyperemia (Klitzman and Duling, 1979) and maximal vasodilation (Sarelius et al., 1981)and is decreased by tissue hyperoxia (Klitzman and Duling, 1979). These changes in hematocrit, and hence cell spacing, are consistent with the accepted role of capillary blood in supplying oxygen to surrounding tissues. An important assumption is implicit in the majority of both experimental and theoretical investigations of capillary supply of oxygen to tissue (for example, Popel, 1982; Tenney, 1974; Grunewald and Sowa, 1978; Bourdeau-Martini et al., 1974; Granger and Nyhof, 1982). This assumption is that capillary blood is ’ To whom reprint requests should be addressed at: Dept. of Radiation Biology and Biophysics, University of Rochester, 601 Elmwood Avenue, Rochester, N.Y. 14642. 273 0026-2862/8d $3.00 Copyright 0 1934 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in U.S.A.

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homogeneous with respect to oxygen supply. That is, while it is recognized that an axial PO2 gradient in the capillary will exist as oxygen is consumed, and while it is also recognized that diffusion and convection of oxygen may occur within the capillary, it is assumed that capillary blood is homogeneous and thus is effectively equivalent for oxygen supply at any location within the capillary. In addition it is assumed that changes in blood flow will lead to parallel but uniform changes in oxygen supply; clearly this is only necessarily true if the assumption that capillary blood is a homogeneous oxygen source is valid. The idea that PO2 gradients between cells due to cell spacing might influence oxygen delivered from capillary blood was raised in a recent study (Homer et al., 1981) but the key question has not been addressed. That is, can capillary blood, with its characteristic red cell spacing, be treated us homogeneous with respect to oxygen supply? In this paper we present a simple model which specifically addresses this question. For capillary blood to be viewed as homogeneous, it must be able to supply a constant flux of oxygen at the capillary wall. Note that it does not necessarily follow from this that the O2 flux through the capillary wall in vivo will be constant. Tissue heterogeneities in the transport or consumption of O2 may create a nonuniform flux of O2 out of the capillary; however, the capillary blood may still be regarded as homogeneous if it has the potential to provide a constant and uniform flux of O2 at the capillary wall. For our model we have set constant oxygen flux at the wall as the limiting boundary condition and determined what cell spacings would be consistent with this in different physiological states. Our model uses experimentally derived data from striated muscle to predict the limits for cell spacing over which the boundary condition is satisfied. The spacings predicted by the model are consistent with in vivo observations of cells per unit length in striated muscle (Sarelius et al., 1980; Sarelius, personal communication). We conclude that in striated muscle, capillary blood may be assumed to be homogeneous in studies of the interaction between capillary oxygen supply and heterogeneous tissue oxygen consumption. THE MODEL We have chosen a simplified model to represent red cells inside a capillary. In Fig. lA, the red cells are represented by cylinders of finite length L inside a cylindrical capillary of radius R. The cells are separated by a distance 4 and nearly fill the luminal cross section, so that the plasma gap of width b between the red cells and the capillary wall is small with respect to the capillary radius. The implications of a finite width red cell-capillary wall gap will be discussed later. Our model is thus consistent with the most widely utilized capillary models (e.g., Aroesty and Gross, 1970; Hellums, 1977; Homer et al., 1981). The red cells move with a velocity U,,, through the capillary; but since for simplicity we have chosen a frame of reference moving with the red cells, the capillary wall is moving with a velocity U,,, to the left in Fig. IA. The movement of the capillary wall and the pressure gradient inside the capillary create fluid motion in the plasma gaps. A schematic of the fluid motion (for a frame of reference moving with the red cells) appears in Fig. 1B. The streamlines and velocity profile are qualitatively drawn based on results for spheres in tube flow (Wang and Skalak, 1969).

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CAPILLARY

IN

275

CAPILLARIES

WALL

J

P

t

X

i

PLASMA -f?-

CAPILLARY

WALL

FIG. 1. Schematic of red cells in a capillary. (A) Cylindrical capillary of radius R containing a cylindrical red cell of length L. The red cell is separated from the capillary wall by a uniform small plasma gap of width b (b << R), and is separated from its neighboring red cell by a distance e. The cylindrical coordinate system (X, p) is placed in the center of the inter-red cell gap and moves with the red cell. (B) Qualitative streamlines (upper portion) and velocity profile (lower portion) for a frame of reference moving with the red cells.

The transport equation for 02, assuming constant plasma density, O2 diffusivity, and 0, solubility, is

aPoz

-

at

+ V . VP02 = DV2P02,

where dPO,/dt is the time rate of change of the O2 tension in a differential control volume, V is the fluid velocity vector field, VP02 is the gradient of O2 tension, D is the diffusion coefficient, and V2P02 is the Laplacian of the O2 tension. The term V . VP02 represents the transport of O2 by bulk fluid motion (convection), and the term DV2P02 describes the transport of O2 by molecular diffusion. Several simplifications of the transport equation (Eq. (1)) can be made. Aroesty and Gross (1970) modeled the effect of the fluid convection between red cells on the transport of diffusible species through the plasma. Their results indicate that the convection in the region between red cells negligibly affects the O2 transported by molecular diffusion. To a first approximation then, the convective term in Eq. (1) is negligible with respect to the diffusion term: V * VPO, << D V2P02. Another major simplification of the transport equation (Eq. (1)) can be made for intracapillary O2 transport. The independent variable 1 and the operator V2

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are made dimensionless using the following scaling: t* = t/e V*2

= R2 V2,

where 8 is a characteristic capillary transit time and R is the capillary radius. Substituting these definitions in Eq. (1) (without the convective term) results in the following equation: R2 aP02 --= De at*

v*2p02

If we choose appropriate values for R, D, and 8 to calculate an upper-bound value for R2/D8 (R = 3 pm, D = 2.0 x 10e5 cm2 set-‘, and 8 = l/2 set), we find that R2/D13 is about 0.01. This indicates that the relative magnitude of the time derivative term is small, and to a first approximation the term can be neglected in the transport equation. The transport is of a quasi-steady nature; i.e., it adjusts rapidly to any changes in the environment. Therefore, the equation for the transport of O2 between red cells reduces to V2P02 = 0.

(2)

The O2 transport equation requires a set of boundary conditions. The first boundary condition is a specified O2 tension POzc on the surface of the red cells. POzc changes with time as the red cells unload O2 but because of the quasisteadiness of the transport, P02c can be regarded as a constant. We assume that P02, for two adjacent red cells is the same. In actuality, a finite difference in pO2, exists between red cells in vivo, but the difference is minimized by hemoglobin buffering. The other boundary condition is specified at the capillary wall. Since the assumption being tested is that red cells can deliver a constant and uniform flux of O2 at the wall, we require that A, (the flux of O2 at the capillary wall) is a constant. A, is calculated from values for tissue consumption, capillary density, and capillary radius (see Parameters). Our choice of the flux boundary condition limits the model to addressing only the specific question raised in the introduction; addressing more general questions about intracapillary O2 transport would require a more extensive treatment of both boundary conditions. Consider the cylindrical coordinate system (X, p) centered in the plasma gap between the two red cells (Fig. 1A). The location of the ends of the red cells are at X = e/2 and X = - C/2; the capillary wall corresponds to p = R. Assuming that the red cells nearly fill the luminal cross section (i.e., b << R) and that there is angular symmetry, the transport equation for O2 in the inter-red cell gap, with boundary conditions, is +-

atp=R:cuD-=

apo2

a2P0, ax2

-A

ap w atX = *e/2 : PO2 = P02,,

(3)

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where (Y is the solubility of O2 in plasma. The latter boundary condition does not strictly apply in the region p > R - b; however, for the analysis here, that region is considered to be small with respect to the capillary radius. We introduce the following dimensionless variables: r = p/R x = XfL P = POJPOzc. Substituting these definitions in Eq. (3) leads to the dimensionless form of the O2 transport equation for the inter-red cell plasma gap:

atr=l:$=

-F

atx = +t/2L:P

= 1.

(4)

F is the normalized flux of O2 at the capillary wall and is related to the dimensional flux by AR aDPo2c’

F=---“---

Equation (4) with associated boundary conditions describes the transport of O2 in the plasma gap between the two red cells. Since the effect of convective transport is small, the transport is a balance of radial diffusion to the capillary wall and axial diffusion between the adjacent red cells. Equation (4) is solved readily by separation of variables; the solution expresses the O2 tension, normalized by P&c, at any position between the two red cells. PARAMETERS All the information we need to apply the model is not available for a single muscle preparation in one animal species. Consequently, we have chosen representative values for the necessary parameters from several sources. In addition, reported values for the parameters of interest may vary widely, for example, due to differences in physiological state, species, or experimental conditions. A critical parameter is the O2 consumption (VO,) of a muscle preparation. An average resting VOz for dog gastrocnemius-plantaris muscle reported by Chapler er al. (1979) is 0.5 ml 02/100 g tissue/min. Resting i/O, in cat soleus and gracilis muscle is 0.25 and 0.32 ml OJlOO g tissuelmin, respectively (Bockman et al., 1980). Honig et al. (1971) reported an average VOz of 0.25 ml 02/100 g tissue/ min for resting dog gracilis. Honig and Frierson (1980) measured VOZmaxin dog gracilis and reported a value of 16 ml OJlOO g tissue/min. For dog gastrocnemius and extensor digitorum longus, Maxwell et al. (1977) give values for i/OZmaxof 16 and 20 ml 02/100 g tissue/min, respectively. From these values we choose a range of VO, from 0.2 to 20 ml 02/100 g tissue/min.

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Relating VO, to the O2 flux through a single capillary requires capillary density (CD) values. An average anatomical CD for several dog muscles is 900 capillaries/ mm* of tissue as reported by Maxwell et al. (1980). Assuming a capillary recruitment factor of 2, we have chosen 450 capillaries per square millimeter of tissue for a resting CD value. To compute the average flux of O2 at a single capillary wall, we assume that the capillaries are uniformly spaced in the muscle tissue; i.e., each capillary supplies its own cylinder of surrounding tissue (Krogh cylinder). This is an idealization of the capillary arrangement inside a muscle tissue, but is needed to relate muscle e0, to O2 flux at a single capillary. The cross-sectional area, A,, of the Krogh cylinder surrounding a single capillary is given by AKr = l/CD

- ?rR2

The flux of O2 through the capillary wall is then calculated using the following equation:

where k converts the units of A, to moles cm-* set-‘. Capillary diameter is most often assumed, but actual measurements have been reported for various muscle preparations: rabbit tenuissimus, 5.4 ,um (Myrhage and Hudlicka, 1978); hamster cremaster, 5.1 pm (Klitzman and Duling, 1979); hamster cremaster, 5.7 pm (Sarelius et al., 1981). We choose a representative value of capillary radius equal to 2.5 pm. The length of a red cell inside a capillary is approximately the diameter of the unstressed red cell biconcave disk. (According to Bagge et al., 1980, cells merely fold over the longitudinal axis of the capillaries they enter.) This is confirmed by data from Klitzman and Duling (1979), who give a value of 5.6 pm for the length of a hamster red cell inside a capillary; this value agrees with values reported for the diameter of an unstressed hamster red cell (Altman and Dittmer, 1971). We choose a red cell length between 6 and 7 ,um, which is a range representative of dog red cells (Altman and Dittmer, 1971). Red cell intracapillary separation distances have not been reported extensively in the literature. For hamster cremaster (Sarelius et al., 1980), red cell separation is two to three cell lengths at rest. With maximal dilation, the separation distance decreases to one cell length (Sarelius, unpublished observations). Also, from data reported for hamster cremaster (Klitzman and Duling, 1979), a red cell separation distance of about four cell lengths at rest can be calculated. From Klitzman and Duling’s measurements obtained with the muscle stimulated at 1 Hz contractions, we calculated a red cell separation’distance of less than two cell lengths. In capillaries, POZ is substantially lower than systemic PO* (Duling and Berne, 1971; Duling and Pittman, 1975). Model studies (Pope1 and Gross, 1979) suggest that the PO* of red cells entering capillaries. is between 20 and 50 mm Hg. End capillary PO, will be much lower depending on O2 demand. We chose a representative red cell PO* along the capillary to be 20 mm Hg. The parameters chosen from the available experimental data are summarized in Tables 1 and 2.

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TABLE 1 O2 TRANSPORTPARAMETER+-RESTING MUSCLE CONSUMPTION

O2 solubility 0, diffusion coefficient capillary radius red ceil length red cell separation red cell surface 0, tension 0, consumption capillary density O2 flux at capillary wall Normalized OZ flux through capillary wall

1.23 x 10m9mole O2 cm-’ mm Hg-’

2.0 x 10-j cm’ sect’ 2.5 x 10m4cm 6.0-7.0 x lo-” cm three to four cell lengths

20 mm Hg. 0.2 ml 0, (100 g))’ mm’ 450 capillaries mm-* tissue 2.09 x lo-” mole 0, cm-’ set-’

0.01-0.02

RESULTS Figure 2A shows the mean O2 tension relative to red cell PO* in the inter-red cell plasma gap for a ratio of capillary radius to red cell length (R/L) of 2.5/6.0. The mean PO2 in the gap is plotted as a function of the normalized distance from the red cell. The distance in Fig. 2A is normalized so that the red cell corresponds to an abscissa value of zero and the center of the plasma gap to an abscissa value of unity. The solid curve represents the axial profile of mean O2 tension obtained during resting O2 consumption conditions (Table 1) with the red cells separated by a distance of three cell lengths. Under these conditions, the mean POZ in the plasma gap differs from the red cell POZ by less than 20% at any point in the plasma gap. However, if the O2 consumption increases to VO 2max(Table 2), the mean PO? profile changes significantly (dashed lines in Fig. 2A). For example, if the red cells remain separated by a distance of three cell lengths, the mean PO* quickly decreases to zero away from the red cell. Under these conditions the flux of O2 through the capillary wall cannot remain uniform since large anoxic regions exist in the capillary. To maintain a uniform flux of O2 at the capillary wall, the mean PO* must be elevated to eliminate anoxic loci. Figure 2A shows that if the separation distance between red cells is reduced to one cell length, the mean POZ is substantially elevated. Since the radial PO, profile at the center of the gap (see Fig. 2B) also remains above zero, the flux of O2 through the capillary wall is uniform between the two red cells. These observations show the existence of a critical cell separation distance S, . When two adjacent red cells are separated by a distance greater than S,, a constant, uniform flux of Q2 cannot be maintained at the capillary wall between the two red cells, and the capillary blood is not homogeneous for O2 supply. TABLE 2 PARAMETERSTHAT CHANGE WITH MAXIMAL 0, CONSUMPTION

e +Qmax CD ‘4,

one cell length 20 ml 0, (100 g))’ min-’ !900capillaries mm-’ tissue 1.05 X loo9 mole 0, cm-’ sect’

F

OS-O.6

280

FEDERSPIEL AND SARELIUS 1.0

I 0.6 0 P E 6 0.6 z d z B 0.4 5 2 p 0.2

I

0.0 ( I

0.2

0.4 0.6 0.6 NORUAUZED AXIAL POSITION

o.oI 0.0

0.2

0.6 0.6 0.4 NORUALIZED RADIAL POSITION

1.0

1 .o

FIG. 2. (A) Mean O2 tension in the inter-red cell plasma gap. The mean O2 tension is expressed relative to red cell PO*, and is plotted as a function of the normalized axial distance from the red cell (i.e., red cell = 0, center of gap = 1). The mean Oz tension is obtained by averaging the Oz tension over the cross-sectional area of the capillary at any axial position. Parameters are R/L = 2.5/&O; C = one, two, three cell lengths; rest conditions F = 0.015; and maximal exercise F = 0.52. (B) Normalized Oz tension radial profiles for C = 1 and F = 0.52. The curves correspond to different axial positions between the center of the gap (1) and the red cell (0).

Conversely, when the two red cells are separated by a distance less than S,, the flux of O2 can remain uniform everywhere on the capillary wall between the two red cells, and the particulate nature of the O2 supply is not recognized. We can calculate S, from our solution of the O2 transport equation in the interred cell plasma gap. For a given value of the normalized flux of O2 through the capillary wall ,and a given ratio-R/L, we can calculate the value of the red cell separation that first leads to an anoxic locus between the two red cells. The anoxic-locus first appears at the capillary wall midway between the red cells. This value of the red cell separation is defined as S, (expressed in units of cell

281

RED CELL SPACING IN CAPILLARIES

lengths) and defines an upper-bound red cell separation above which the flux cannot remain uniform. Figure 3A shows the behavior of S, for low values of F. At rest, S, is greater than about four cell lengths for all values of R/L displayed. The behavior of S, for larger values of F is displayed in Fig. 3B. With maximal O2 consumption (Table 2), S, decreases to about one cell length (or slightly greater) for the values of R/L represented in Fig. 3B. DISCUSSION Our analysis shows that red cell spacing significantly affects the ability of red cells to provide a uniform flux of O2 at the capillary wall. For values of parameters 10

0.01

0.02

01

0.1

0.06

0.10

O2 FLUX, F

I 0.2

0.4 NOfiNALIZED

FIG. 3.

0.06

0.04 NORMALIZED

0.6 4

0.11

1.0

FLUX, F

Critical separation distance, S,, calculated from the inter-red cell plasma gap solution. S, is plotted as a function of +e normalized flux of OZ through the capillary wall, F. (A) F = O.Ol0.02 corresponds to resting VO, parameters as listed in Table 1. (B) F = 0.5-0.6 corresponds to maximal VO, parameters as listed in Table 2. The ratio R/L defines a family of curves: a, 2.5/5.0; b, 2.5/6.0 or 3.017.0; c, 2.5/7.0; d, 2.0/7.0.

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representing resting O2 consumption in a muscle preparation (Table l), the critical separation distance is at least four cell lengths (Fig. 3A). If the value of POP, is greater than the value of 20 Tot-r that we used (which may be the case for resting conditions), the value for F decreases (Eq. (5)), and S, would be even larger. Therefore, we can use four cell lengths as a lower-bound estimate of the actual critical separation distance between red cells in capillaries of resting tissue. Since observed in viva red cell spacing in resting tissue is about three to four cell lengths, we can conclude that under resting conditions it is likely that red cells can provide an even, uniform flux of 0, through the capillary wall and that capillary blood is homogeneous for O2 supply. With the onset of exercise, the O2 demand at the. level of a single capillary increases. For values of parameters corresponding to VOZmax (Table 2), the critical separation distance S, decreases to about one cell length (Fig. 3B). Observations of red cell spacing in capillaries of maximally dilated tissue preparations show that the red cell spacing does decrease to the order of one cell length or less (Sarelius, unpublished observations). If red cells were to remain separated by three cell lengths, they could not supply a uniform, high flux of O2 at the capillary wall between them. Instead the O2 flux would necessarily have to vary on a length scale smaller than the separation length of the cells and capillary blood could not be treated as homogeneous for O2 supply. The existence of a critical separation distance has important implications for the interpretation of physiological responses. When tissue metabolism increases, the raised oxygen requirement may be met by either or both of an increase in capillary red cell flux (the number of cells traversing a capillary per unit time) or an increase in O2 extraction. The flux of erythrocytes through a capillary may be increased either by decreasing the red cell spacing or by increasing the red cell velocity. It is generally assumed that the capacity of changes in red cell flux to meet oxygen demand is independent of how the change in flux is achieved (that is, by changing cell velocity or cell spacing, or both). That this assumption need not be correct is illustrated by a simple example. Since we wish to examine components of red cell flux we will evaluate the case where 0, extraction in the capillary remains constant, and changes in metabolism are met entirely by changes in red cell flux. For instance, consider conditions in a muscle undergoing moderate exercise, with a normalized O2 flux at the capillary wall of F = 0.3 and red cells separated by two cell lengths. From Fig. 3B, it can be seen that this cell separation is adequate for a constant uniform O2 flux at the capillary wall, as S, is about two cell lengths. Now let the exercise level increase such that O2 consumption is raised and the O2 flux at the capillary wall is doubled, to F = 0.6. Figure 3B shows that the critical separation distance will decrease to about one cell length. Since we are considering the case where O2 extraction in the capillary remains constant, and since the flux of O2 at the capillary wall has doubled, the red cell flux through the capillary must also double to meet the O2 demand. If the red cell flux doubles by halving red cell spacing from two cell lengths to one cell length, the red cells will be able to deliver a locally uniform flux of O2 out of the capillary. However, if the red cell flux doubles by doubling red cell velocity (leaving red cell spacing at two cell lengths), then the required S, of one cell length at F = 0.6 would not be met, and the O2 flux out of the capillary would

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be nonuniform. Thus, the two ways of changing the red cell flux lead to potentially different O2 supply regimes. Our analysis addresses a very specific question: can red cells provide a constant, uniform flux of O2 at the capillary wall regardless of O2 demand and capillary red cell spacing? For this reason we have chosen the boundary condition at the capillary wall to be a constant, uniform O2 flux A,, where A, is calculated from global values of tissue consumption and functional capillary density. It is not clear, either from experimental data or theoretical modeling, which type of boundary condition at the capillary wall should be applied. Consequently, boundary conditions will differ according to the aspect of intracapillary O2 transport being addressed. In general the correct boundary conditions come from matching the intracapillary model with a reasonable tissue model. Our assumption of uniform A, is appropriate for addressing the specific question raised above, provided the values for the critical separation distance are only associated with the ability of the red cells to maintain a uniform flux of O2 at the capillary wall. If the O2 tension at the capillary wall is specified instead, the O2 flux at the wall automatically varies with the passing of a red cell; and though one may study interaction between red cells, the role of red cell spacing in providing a uniform O2 flux out of the capillary cannot be examined. The model investigates nonuniformities in the O2 flux through the capillary wall that occur across a length scale comparable to the red cell separation distance. The boundary condition of constant O2 flux, A,, out of the capillary between red cells does not negate the possibility that small magnitude, large length-scale variations in the O2 flux out of a capillary may occur from the arterial to venous end of the capillary. These variations do not affect the general results of the analysis. For instance if one assumes that the O2 flux out of a capillary slowly decreases from arterial to venous end, the critical separation distances presented in Figs. 3A and B would still predict when red cell spacing may lead to local (i.e., between red cells) nonuniformities in the O2 flux out of the capillary. One may also argue that the flux of O2 out of the region of the capillary wall contiguous to the red cells may be so large that the flux of O2 out of the capillary wall between red cells is small, and red cell spacing is inconsequential. In this view, local nonuniformities in the O2 flux out of the capillary are already present; the intent of our analysis was to address the a-priori belief that despite the large spacing between red cells in capillaries, O2 can leave the capillary in a uniform fashion and capillary blood can be considered as a homogeneous source of OZ. In the formulation of the model the fact that the plasma gap between the red cell and the capillary wall is of a finite width, b, is not considered. Consequently the present work should be seen as a first approximation using the simplest model. Incorporating a finite size red cell-capillary wall gap greatly increases the complexity of the analysis but might be considered as the next step. The presence of such a gap would decrease the critical separation distances computed from our model due to longer diffusion paths, since the red cell is farther from the capillary wall. For this reason the values for the critical separation distance presented in Figs. 3A and B can be considered as upper-bound estimates of the actual critical separation distance.

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Is the critical separation distance really critical in terms of O2 transport in tissue? Obtaining the answer to that question requires extending the present analysis to include a reasonable model of the surrounding tissue. The answer will depend crucially on the distribution of mitochondria around a typical capillary, the O2 consumption rate of the mitochondria, the presence and kinetics of myoglobin O2 buffering, and the scale and frequency of O2 flux variations at the capillary wall. We have limited our analysis to studying the effect that red cell spacing has on the treatment of capillary blood as a homogeneous source of O2; but in doing so, we have suggested a role for red cell spacing-and hence a new role for capillary hematocrit-as an O2 supply variable. ACKNOWLEDGMENTS We thank Dr. Giles R. Cokelet for his critical review of the manuscript and Dr. Alfred Clark, Jr, for his helpful comments during the initial stages of the analysis. Special thanks go to the Word Processing Center in the Department of Radiation Biology and Biophysics for typing the manuscript. This paper is based on work performed partly under Contract DE-AC02-76EV03490 with the U.S. Department of Energy at the University of Rochester Department of Radiation Biology and Biophysics and has been assigned Report No. UR-3490-2222. The work was also supported by NIH Grants GM07136 and HL 18208.

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