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- Abstract
- 1. Introduction
- 2. The Model
- 3. Form of Presentation of Results
- 4. Computational Procedures
- 5. Parameters of The Model
- 6. Dimensional Analysis of The Problem
- 7. Symmetric Case: Identical Capillary Layers. Comparison with The Krogh Cylinder Model
- 8. Asymmetric Case: Nonidentical Capillary Layers
- 9. Conclusions
- References

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Math Biosci. Author manuscript; available in PMC 2017 September 22.

Published in final edited form as:

PMCID: PMC5609728

NIHMSID: NIHMS878395

Aleksander S. Popel, Department of Chemical Engineering, University of Arizona, Tucson, Arizona 85721 and Department of Physiology, College of Medicine, University of Arizona, Tucson, Arizona 85724;

The publisher's final edited version of this article is available at Math Biosci

Oxygen transport from capillary layers with concurrent flow is considered for symmetric and asymmetric distributions of oxygen concentration between the layers. The analysis is based on the solution previously obtained by the author [1]. Solutions for the symmetric case are shown to be very close to the corresponding solutions of the Krogh cylinder model. Asymmetry in oxygen distribution is introduced systematically by considering different velocities of blood in the alternate capillary layers, different inlet capillary oxygen tensions, and different capillary hematocrits. It is shown that increase of the degree of asymmetry leads to diminution of the mean oxygen tension.

Recent studies have demonstrated that the Krogh cylinder model [2, 3] does not provide an adequate description of oxygen transfer between capillary blood vessels and the surrounding tissue [4, 5]. It has been shown that significant heterogeneity exists in tissue perfusion at the capillary level—in particular, heterogeneity of the capillary geometry (e.g., [6–10]), of capillary flow [11, 12], and of capillary hematrocrit [13, 14]. Besides, the results of the experimental studies of the precapillary oxygen transport [15, 16] suggest that the degree of oxygenation of the blood entering the capillaries may also be heterogeneous.

Therefore, it appears to be important to study the mechanisms whereby these factors affect the oxygen transport characteristics. Significant progress in this direction has already been achieved in numerical simulation analyses by Metzger [17, 18], Grunewald [19], and Grunewald and Sowa [20].

In the present work an analytical solution recently developed by the author [1] is utilized and applied to a simple case of heterogeneous capillary flow represented by alternate layers of capillaries with concurrent flow. Parameters pertaining to capillaries in adjacent layers may be different, which causes asymmetry in the oxygen distribution.

In the preceding paper [1] a model of the capillary-tissue diffusional exchange was developed for a rather general geometry of parallel capillaries, and under a number of assumptions the solution of the three-dimensional diffusion problem was reduced to the solution of a set of ordinary differential equations. In the present paper a particular case of the problem is considered with the capillaries arranged in alternate parallel layers. Figure 1 shows the capillary geometry in a cross-section perpendicular to the capillaries, and an elementary microvascular unit supplied by two capillaries pertaining to different layers. Introducing a rectangular coordinate system (*x*, *y*, *z*) as shown in Figure 1, we write the equations governing oxygen distribution in the tissue and in the capillaries, respectively, in the form

$$DS(\frac{{\partial}^{2}P}{\partial {x}^{2}}+\frac{{\partial}^{2}P}{\partial {y}^{2}})-g=0,$$

(1)

$${Q}_{i}{S}_{b}\frac{d}{dz}[{P}_{i}+{N}_{i}{S}_{b}^{-1}\mathrm{\psi}({P}_{i})]=-{J}_{i},\phantom{\rule{0.3em}{0ex}}i=1,2,$$

(2)

where *P* is the oxygen tension (partial pressure of oxygen) in the tissue, *P*_{1} and *P*_{2} are the oxygen tensions in the capillary blood, *D* is the diffusion coefficient for oxygen in the tissue, *S* and *S _{b}* are the solubility coefficients of oxygen in the tissue and in the blood, respectively,

The boundary conditions are formulated as follows:

$$\frac{\partial P}{\partial x}=0\phantom{\rule{0.2em}{0ex}}\text{at}\phantom{\rule{0.2em}{0ex}}x=-a,\phantom{\rule{0.2em}{0ex}}r\le y\le 2b,\phantom{\rule{0.2em}{0ex}}0\le z\le L\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=a,\phantom{\rule{0.2em}{0ex}}0\le y\le 2b-r,\phantom{\rule{0.2em}{0ex}}0\le z\le L;$$

(3)

$$\frac{\partial P}{\partial y}=0\phantom{\rule{0.2em}{0ex}}\text{at}\phantom{\rule{0.2em}{0ex}}-a+r\le x\le a,\phantom{\rule{0.2em}{0ex}}y=0,\phantom{\rule{0.2em}{0ex}}0\le z\le L\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-a\le x\le a-r,\phantom{\rule{0.2em}{0ex}}y=2b,\phantom{\rule{0.2em}{0ex}}0\le z\le L;$$

(4)

$$P={P}_{i}\phantom{\rule{0.2em}{0ex}}\text{on}\phantom{\rule{0.2em}{0ex}}{\mathrm{\Gamma}}_{i},\phantom{\rule{0.2em}{0ex}}i=1,2,$$

(5)

$${J}_{i}=-DS\int \frac{\partial P}{\partial n}d{\mathrm{\Gamma}}_{i},\phantom{\rule{0.2em}{0ex}}i=1,2,$$

(6)

where Γ* _{i}* is a quarter of the cylindrical surface of the capillary situated inside the parallelepiped shown in Fig. 1.

An approximate solution to this problem obtained in [1] can be written in the form

$$P(x,y,z)=\frac{g{x}^{2}}{2DS}+\frac{g{a}^{2}}{2\pi DSK}I(\xi ,\eta )-\frac{{J}_{1}}{8\pi DS}ln[{(\xi +1)}^{2}+{\eta}^{2}]-\frac{{J}_{2}}{8\pi DS}ln[{(\xi -\frac{1}{k})}^{2}+{\eta}^{2}]+{P}_{0},$$

(7)

where *K*(*k*) is the complete elliptic integral of the first kind [21], and the elliptic modulus *k* is a solution of the transcendental equation

$$\frac{K(\sqrt{1-{k}^{2}})}{K(k)}=\frac{2b}{a},\phantom{\rule{0.2em}{0ex}}0<k<1.$$

(8)

For *a* = *b* the solution of (8) is *k* = 0.1716.

The variables *ξ* and *η* are expressed in terms of *X* = *xK*/*a* and *Y* = *yK*/*a* using the Jacobian elliptic functions [21]

$$\xi =\frac{\text{sn}(X,k)\text{dn}(Y,{k}^{\prime})}{{\text{cn}}^{2}(Y,{k}^{\prime})+{k}^{2}{\text{sn}}^{2}(X,k){\text{sn}}^{2}(Y,{k}^{\prime})},$$

(9)

$$\eta =\frac{\text{cn}(X,k)\text{dn}(X,k)\text{sn}(Y,{k}^{\prime})\text{cn}(Y,{k}^{\prime})}{{\text{cn}}^{2}(Y,{k}^{\prime})+{k}^{2}{\text{sn}}^{2}(X,k){\text{sn}}^{2}(Y,{k}^{\prime})},$$

(10)

where
${k}^{\prime}=\sqrt{1-{k}^{2}}$. The quantities *I* and *P*_{0} are

$$I(\xi ,\eta )={\int}_{0}^{1}\frac{ln\left\{\right[{(\frac{\sqrt{1-{k}^{\prime 2}{t}^{2}}}{k}-\xi )}^{2}+{\eta}^{2}\left]\phantom{\rule{0.2em}{0ex}}\right[{(\frac{\sqrt{1-{k}^{\prime 2}{t}^{2}}}{k}+\xi )}^{2}+{\eta}^{2}\left]\right\}}{\sqrt{(1-{t}^{2})(1-{k}^{\prime 2}{t}^{2})}}\mathit{\text{dt}},$$

(11)

$${P}_{0}=\frac{q}{2\pi}ln\frac{rK\sqrt{{k}^{\prime}(1+k)}}{a\sqrt{2}}+\frac{1}{2ln\mu}[{P}_{1}ln\frac{rK{k}^{\prime}}{a\sqrt{2(1+k)}}+{P}_{2}ln\frac{rK{k}^{\prime}\sqrt{k}}{a\sqrt{2(1+k)}}],$$

(12)

where

$$q=\frac{8gab}{D{S}_{b}},\phantom{\rule{0.2em}{0ex}}v=\frac{S}{{S}_{b}},\phantom{\rule{0.2em}{0ex}}\mu =\frac{rK{k}^{\prime}{k}^{1/4}{2}^{1/2}}{a\sqrt{1+k}}.$$

(13)

The functions *P*_{1}(*z*) and *P*_{2}(*z*) satisfy ordinary differential equations

$$[1+\mathrm{\Phi}({P}_{i})]\frac{d{P}_{i}}{dz}=-\frac{DL}{{Q}_{i}}[q\mp \frac{\pi v({P}_{1}-{P}_{2})}{ln\mu}],\phantom{\rule{0.2em}{0ex}}i=1,2,$$

(14)

where

$$\mathrm{\Phi}(P)=N{S}_{b}^{-1}\frac{d\mathrm{\psi}}{dP}$$

(15)

is proportional to the slope of the oxygen dissociation curve, and *z** = *z/L* is a dimensionless coordinate.

In the present work the concurrent flow in the capillary layers is considered: *Q*_{1} ≥ 0, *Q*_{2} ≥ 0. The intracapillary oxygen tensions are specified at the capillary entrances, which imposes the boundary conditions

$${P}_{1}(0)={P}_{a1},\phantom{\rule{0.2em}{0ex}}{P}_{2}(0)={P}_{a2}.$$

(16)

It should be remembered that the applicability of the solution is limited to cases where the simplifications made in the derivation can be justified. The most important of these simplifications are the assumption of a constant oxygen consumption rate, and the neglect of oxygen diffusion in the tissue in the direction parallel to the capillary axes.

In this section a convenient form of presentation of the oxygen tension distributions inside the volume is suggested for the description of both symmetric and asymmetric cases.

Let *V* be the volume of the parallelepiped without the capillaries (Fig. 1), *V* = (4*ab* – *πr*^{2}/2) *L*. The frequency distribution of oxygen tension in the volume *V* is defined in the following way. If *dV* is the volume of the region in which the tissue oxygen tension lies between *P* and *P* + *dP*, then

$$\frac{dV}{V}=F(P)dP,$$

(17)

where *F*(*P*) is the frequency distribution of oxygen tension in the volume. It follows from (17) that

$${\int}_{0}^{\infty}F(P)dP=1.$$

(18)

If *dσ* is the area in the plane of the volume *V* with coordinate *z* = const, in which the oxygen tension lies between *P* and *P* + *dP*, then

$$\frac{d\sigma}{\sigma}=f(z,P)dP,$$

(19)

where *σ* is the area of the cross-section of the volume *V* at *z* = const, and *f*(*z*, *P*) is the frequency distribution of oxygen tension at the cross section. From the definition (19) it follows that

$${\int}_{0}^{\infty}f(z,P)dP=1,\phantom{\rule{0.3em}{0ex}}\frac{1}{L}{\int}_{0}^{L}f(z,P)dz=F(P).$$

(20)

Using the frequency distribution functions, one can define the mean oxygen tension for the cross section,

$${P}_{m}(z)={\int}_{0}^{\infty}Pf(z,P)dP,$$

(21)

and the mean oxygen tension for the volume,

$${\overline{P}}_{m}={\int}_{0}^{\infty}PF(P)dP.$$

(22)

$${\overline{P}}_{m}=\frac{1}{L}{\int}_{0}^{L}{P}_{m}(z)dz.$$

(23)

These quantities will be used in the following presentation.

The computations have been done in the following order. First, the functions *ξ*(*x*, *y*), *η*(*x*, *y*), and *I*(*ξ*, *η*) given by equations (9)–(11) are calculated at the points *x _{i}*/2

Parameters of the model are chosen for simulation of oxygen transport in resting mammalian skeletal muscle. Table 1 shows the parameters together with appropriate references. It should be noted that experimental data obtained for different muscles under different experimental conditions vary within broad limits. Therefore, the purpose of the table is not to give a comprehensive review of the experimental data for resting mammalian skeletal muscle, but rather to present some typical numbers pertinent to the problem. Table 1 indicates that some of the parameters are fixed, whereas for others the range of variation is shown. As will be explained in the subsequent section, in which the solution of the problem will be presented in a proper dimensionless form, variation of one dimensional parameter may be in a certain sense equivalent to variation of other parameters; for example, variation of the capillary velocity may have the same effect as variation of the capillary length. Thus, we limit the analysis to variation of the parameters shown in Table 1.

The values of the oxygen consumption rate in Table 1 were converted from the values *g*0.3–1.2 (cm^{3} O_{2})/(100 g tissue)(min) using the conversion factor 1 (cm^{3} O_{2})/(100 g tissue)(min)1.75 × 10^{−4} (cm^{3} O_{2})/(cm^{3} tissue)(sec), assuming that the mass of 1 cm^{3} of wet tissue is 1.05 g [26]. The range for the oxygen-binding capacity of blood was chosen according to the relationship *N* = 0.50*H*, where *H* is the blood hematocrit (volume concentration of red blood cells in blood). This relationship can be derived from Hiiffner's equation *N* = 1.34*C*_{Hb}, where *C*_{Hb} (g/cm^{3}) is the concentration of hemoglobin in blood, taking into account that for normal blood *H*0.4 and *C*_{Hb} = 0.15 g/cm^{3} [28]. Therefore, the values of *N* in Table 1 approximately correspond to hematocrit *H* = 0.1–0.5 with the “standard” value *H* = 0.2. Recent experimental studies indicate that the hematocrit distribution in the capillaries is heterogeneous [13,14]; the effect of hematocrit heterogeneity on the oxygen transport will be studied.

In the present work the simplest geometrical case, *a* = *b*, is considered. As can be seen in Fig. 2, the shortest intercapillary distance for this geometry is 2√2 *a*. When the capillary layers are identical, i.e., the intracapillary oxygen tensions in both capillary layers are equal to each other, the situation is close to the one described by the Krogh cylinder model [2,3]. In this case each capillary supplies a parallelepiped whose cross-section is shaded in Fig. 2; an effective “Krogh cylinder” can be introduced such that the volume of the cylinder surrounding a given capillary is equal to the volume of the parallelepiped (Fig. 2). Thus, the radius of the cylinder equals *r _{t}* = √8

An area (shaded) supplied by a capillary in the case of identical capillary layers, and an effective Krogh cylinder.

The capillary velocities of blood in Table 1 are proportional to the volumetric blood flow rate: *υ* = *Q*/*πr*^{2}. The low “standard” value for the inlet capillary oxygen tension *P _{a}* have been chosen to reflect the phenomenon of precapillary oxygen losses [15, 16]. However, since it is not clear at the present time to what extent the precapillary oxygen losses are significant in different tissues and under different conditions, the effect of variation of the inlet capillary oxygen tension will be studied over a wide range.

It has been shown that Hill's equation does not provide an accurate description of the oxygen dissociation curve; therefore we used the more complicated Margaria equation, which approximates the experimental data more accurately [28, 3]:

$$\mathrm{\psi}(P)=hP\frac{{(1+hP)}^{3}+m{(hP)}^{3}}{{(1+hP)}^{4}+m{(hP)}^{4}}.$$

(24)

The coefficients *h* and *m* were determined by fitting Eq. (24) to the experimental data for human blood cited in [22], which gave *h* = 0.01278, *m* = 124. It is important for the following analysis that the parameter *m* is dimensionless, whereas *h* has the dimensions of inverse oxygen tension (1/mmHg).

The following independent dimensionless groups can be constructed using the physical parameters characteristic to the problem:

$$\frac{DL}{\pi {r}^{2}\upsilon},\frac{g{a}^{2}}{D{S}_{b}{P}_{a}},\frac{g{a}^{2}h}{D{S}_{b}},N,\frac{r}{a},v=\frac{S}{{S}_{b}},m;$$

(25)

$$\frac{{P}_{a1}}{{P}_{a2}},\phantom{\rule{0.2em}{0ex}}\frac{{\upsilon}_{1}}{{\upsilon}_{2}},\phantom{\rule{0.2em}{0ex}}\frac{{N}_{1}}{{N}_{2}};$$

(26)

where

$$\upsilon =\frac{{\upsilon}_{1}+{\upsilon}_{2}}{2},\phantom{\rule{0.2em}{0ex}}{P}_{a}=\frac{{P}_{a1}+{P}_{a2}}{2},\phantom{\rule{0.2em}{0ex}}N=\frac{{N}_{1}+{N}_{2}}{2}$$

(27)

are certain characteristic values of the corresponding parameters.

Dimensional analysis of the problem suggests [29] that the dimensionless characteristics *P/P _{a}*,

In the present study the last two parameters in (25), *v* and *m*, are kept constant. Variation of dimensional parameters *υ*, *g*, *P _{a}*,

Although the results of calculations will be presented in the dimensional form because that appears to be more convenient for interpretation, the graphs can be easily replotted in a dimensionless form.

In the case of identical capillary layers the relationships (27) yield

$${\upsilon}_{1}={\upsilon}_{2}=\upsilon ,\phantom{\rule{0.2em}{0ex}}{P}_{a1}={P}_{a2}={P}_{a},\phantom{\rule{0.2em}{0ex}}{N}_{1}={N}_{2}=N.$$

(28)

The solutions obtained within the framework of the present model will be compared with the corresponding solutions for the Krogh cylinder model [2, 3, 30]:

$$[1+\mathrm{\Phi}({P}^{\prime})]\frac{d{P}^{\prime}}{dz}=-\frac{g({r}_{t}^{2}-{r}^{2})}{{S}_{b}\upsilon {r}^{2}},$$

(29)

$${P}_{min}^{\prime}(z)={P}^{\prime}(z)-\frac{g{r}_{t}^{2}}{2DS}(ln\frac{{r}_{t}}{r}-\frac{{r}_{t}^{2}-{r}^{2}}{2{r}_{t}^{2}}),$$

(30)

$${P}_{m}^{\prime}={P}^{\prime}(z)-\frac{g{r}_{t}^{2}}{2DS}(\frac{{r}_{t}^{2}}{{r}_{t}^{2}-{r}^{2}}ln\frac{{r}_{t}}{r}-\frac{3{r}_{t}^{2}-{r}^{2}}{4{r}_{t}^{2}}),$$

(31)

$${\overline{P}}_{m}^{\prime}=\frac{1}{L}{\int}_{0}^{L}{P}_{m}^{\prime}(z)dz,$$

(32)

where the prime refers to the Krogh model: *P′*(*z*) is the intracapillary oxygen tension;
${P}_{min}^{\prime}(z)$ is the minimum value of the tissue oxygen tension at a cross-section *z* = const; and
${P}_{m}^{\prime}$, is the mean oxygen tension

$${P}_{m}^{\prime}(z)=\frac{2}{{r}_{t}^{2}-{r}^{2}}{\int}_{r}^{{r}_{t}}\rho {P}^{\prime}(\rho ,z)d\rho ,$$

(33)

where *P′*(*ρ*, *z*) is the tissue oxygen tension and *ρ* is the radial coordinate. The solution of Eq. (29) is subject to the boundary condition *P′*(0) = *P _{a}*.

Figure 3 shows the intracapillary oxygen tension *P*_{1} = *P*_{2}, the minimum tissue oxygen tension *P*_{min}(*z*) in a plane *z* = const, and the mean oxygen tension *P _{m}* defined in (21), versus the dimensionless coordinate

Figure 4 presents the profiles of oxygen tension in two perpendicular planes: *y** = *Q* and *x** = 0.5, where

$$x\ast =x/2a,\phantom{\rule{0.2em}{0ex}}y\ast =y/2a$$

(34)

are dimensionless coordinates. Obviously, in the considered case of identical capillary layers the profiles shown on the two panels of Fig. 4 are identical. The shape of the profiles is independent of *z**, as follows from Eq. (7); indeed, in the case *P*_{1} = *P*_{2} the function *P*(*x**, *y**, *z**) can be expressed in the form

$$P(x\ast ,y\ast ,z\ast )={P}_{0}(z\ast )+G(x\ast ,y\ast ),$$

(35)

where *G* is a function of *x** and *y** but not of *z**.

Prior to considering asymmetric oxygen distributions between the capillary layers, we intend to systematically study the effect of variation of the parameters *g*, *P _{a}*,

Frequency distributions of tissue oxygen tension for different values of the oxygen consumption rate.

$$F(P,\delta )=\frac{1}{\delta}{\int}_{n\delta}^{(n+1)\delta}F(P)dP$$

(36)

for *nδ* ≤ *P* ≤ (*n* + 1)*δ*, *n* = 0, 1, 2, The value of *δ* in Fig. 5 is 5 mmHg. Figure 6 shows the intracapillary oxygen tension at the venous end of the capillary, *P _{υ}* =

The mean tissue oxygen tension _{m}, the minimum tissue oxygen tension _{min}, and the intravascular oxygen tension at the venous end of the capillary, *P*_{υ}, versus the oxygen consumption rate.

$${\overline{P}}_{min}=min{P}_{min}(z\ast )\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}0\le z\ast \le 1,$$

(37)

as functions of the oxygen consumption rate. The circles indicate the corresponding values
${P}_{\upsilon}^{\prime}$,
${\overline{P}}_{m}^{\prime}$ and
${\overline{P}}_{min}^{\prime}$ obtained within the framework of the Krogh cylinder model. Clearly, the values
${P}_{\upsilon}^{\prime}$ and
${\overline{P}}_{min}^{\prime}$ practically coincide with *P _{υ}* and

Figures 7 and and88 demonstrate the effect of the inlet capillary oxygen tension on the frequency distributions *F*(*P*) and on the parameters *P _{υ}*,

Frequency distributions of tissue oxygen tension for different values of the inlet capillary oxygen tension.

The results of these calculations give a detailed picture of oxygen diffusion in the case of identical capillary layers. As has been indicated above, the solutions can be easily expressed in a dimensionless form with parameters (25) defining the problem; thus, the obtained solutions are, in fact, applicable to a wider range of parameters than is shown in Table 1. The results of the present model are in good agreement with predictions of the Krogh cylinder model; however, unlike the Krogh model, the present model is suitable for description of diffusional interaction between the capillaries, which is the subject of investigation in the next section.

In the beginning of studies of oxygen transport in heterogeneous microcirculation it is important to investigate simple cases of heterogeneity in order to gain insight in the physical mechanisms of the phenomena. In this section we study the effect of asymmetry of the capillary layers characterized by the parameters defined in (26).

First, consider different inlet capillary oxygen tensions for different capillary layers, *P _{a}*

Distribution of oxygen tensions along the length of the capillary in the case of different inlet capillary tensions.

Consider another type of heterogeneity caused by different capillary velocities, *υ*_{1} = *υ*_{2}. Figure 17 shows the distributions *P*_{1}(*z**), *P*_{2}(*z**), *P _{m}*(

Distribution of oxygen tensions along the length of the capillary in the case of different capillary velocities.

The effect of variation of the last parameter in (26), *N*_{1}/*N*_{2}, in most cases can be predicted from the previous calculations. Indeed, it can be shown that Φ(*P*) 1 for values of *P* exceeding several mmHg, where Φ(*P*) is given by (15); thus the expression [1 + Φ(*P*)] in (14) can be approximated by Φ(*P*). In this case only the product *Q _{i}N_{i}* =

The oxygen diffusion from capillary layers with concurrent blood flow was analyzed. Cases of symmetric and asymmetric oxygen distributions between the capillaries were considered. The analysis has shown that in asymmetric cases the mean oxygen tension and the minimum oxygen tension are lower than in the corresponding symmetric cases. It has been demonstrated that an efficient diffusional shunting of oxygen significantly affects the transport characteristics. The model is a generalization of the Krogh cyUnder model; in the case of identical capillary layers it yields results that practically coincide with the predictions of the Krogh cylinder model. The applicability of the present solution is limited to the cases where the oxygen consumption rate can be assumed constant, and the oxygen diffusion in the tissue in the direction parallel to the capillary axes can be neglected.

The work was supported by the National Institues of Health Grants HL-23362 and HL-17421. The author is indebted to Yin Ho for help in computer programming and to Bernice Anderson for typing the manuscript.

- a,b
- geometrical parameters
- D
- diffusion coefficient
- f(z, P)
- frequency distribution, equation (19)
- F(P)
- frequency distribution, equation (17)
- g
- oxygen consumption rate
- h
- parameter in (24)
- I
- integral given by equation (11)
- J
- capillary-tissue diffusive flux
- k
- elliptic modulus
- K(k)
- complete elliptic integral of the first kind
- L
- capillary length
- m
- parameter in (24)
- N
- oxygen-binding capacity of blood
- P
- oxygen tension
- P
_{m} - mean oxygen tension defined by (21)
_{m}- mean oxygen tension defined by (22)
- P
_{a} - oxygen tension at the arterial end of the capillary
- P
_{υ} - oxygen tension at the venous end of the capillary
- P
_{0} - oxygen tension given by (12)
- q
- dimensionless parameters in (13)
- Q
- volumetric blood flow rate
- r
- capillary radius
- S, S
_{b} - solubility coefficients of oxygen
- υ
- capillary velocity of blood
- V
- tissue volume
- x, y, z
- dimensional coordinates

*i*, 1, 2- pertaining to capillaries
*m*- mean value
- min
- minimum value

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