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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
Math Biosci. Author manuscript; available in PMC 2017 September 22.
Published in final edited form as:
PMCID: PMC5609728

Oxygen Diffusion from Capillary Layers with Concurrent Flow


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.

1. Introduction

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., [610]), 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.

2. The Model

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

Fig. 1
Geometry of the problem.

where P is the oxygen tension (partial pressure of oxygen) in the tissue, P1 and P2 are the oxygen tensions in the capillary blood, D is the diffusion coefficient for oxygen in the tissue, S and Sb are the solubility coefficients of oxygen in the tissue and in the blood, respectively, g is the oxygen consumption rate in the tissue, Qi is the capillary blood flow rate, Ni is the oxygen-binding capacity of hemoglobin, ψ is the oxygen-saturated fraction of hemoglobin (oxygen dissociation curve), and Ji is the capillary-tissue diffusive flux per unit length of the capillary. The assumptions that led to Eqs. (1) and (2) were discussed in [1].

The boundary conditions are formulated as follows:


where Γi 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


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


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]


where k=1k2. The quantities I and P0 are




The functions P1(z) and P2(z) satisfy ordinary differential equations




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: Q1 ≥ 0, Q2 ≥ 0. The intracapillary oxygen tensions are specified at the capillary entrances, which imposes the boundary conditions


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.

3. Form of Presentation of Results

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 = (4abπr2/2) 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


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


If 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


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


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


and the mean oxygen tension for the volume,


Equations (20) and (22) yield


These quantities will be used in the following presentation.

4. Computational Procedures

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 xi/2a = ±i/2Mxy and yi/2b = j/Mxy, where i,j = 0, 1, 2, …, Mxy. Since the functions (9)–(11) are independent of parameters of a particular problem (except for the ratio b/a, which is taken to be unity in the present work), it is only necessary to calculate these functions once. Then the solution of the set of two nonlinear ordinary differential equations (14) is obtained using the Runge-Kutta numerical method. Equation (12) is used to calculate P0(z), and then Eq. (7) is used to calculate P(x, y, z) at the points xi, yj, zk = kL/Mz, where k = 0, 1, …,Mz. The numbers Mxy, and MZ were chosen so as to provide the necessary precision in the calculation of frequency distributions F(P).

5. Parameters of The Model

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.

Table 1
Parameters of the Model

The values of the oxygen consumption rate in Table 1 were converted from the values g[similar, equals]0.3–1.2 (cm3 O2)/(100 g tissue)(min) using the conversion factor 1 (cm3 O2)/(100 g tissue)(min)[similar, equals]1.75 × 10−4 (cm3 O2)/(cm3 tissue)(sec), assuming that the mass of 1 cm3 of wet tissue is 1.05 g [26]. The range for the oxygen-binding capacity of blood was chosen according to the relationship N = 0.50H, where H is the blood hematocrit (volume concentration of red blood cells in blood). This relationship can be derived from Hiiffner's equation N = 1.34CHb, where CHb (g/cm3) is the concentration of hemoglobin in blood, taking into account that for normal blood H[similar, equals]0.4 and CHb = 0.15 g/cm3 [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 rt = √8 a/√π. The considered “standard” value of the parameter a corresponds to the Krogh cylinder radius rt = 48 μm, or to the capillary density n = 140 flowing capillaries per mm2 (it should be noted that the number of flowing capillaries is, generally, smaller than the number of capillaries anatomically present [2]). The range a = 10–45 μm corresponds to rt [similar, equals] 16–72 μm, or to the capillary density n[similar, equals]60–1250 flowing capillaries per mm2.

Fig. 2
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/πr2. The low “standard” value for the inlet capillary oxygen tension Pa 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]:


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).

6. Dimensional Analysis of The Problem

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




are certain characteristic values of the corresponding parameters.

Dimensional analysis of the problem suggests [29] that the dimensionless characteristics P/Pa, P1/Pa, P2/Pa, together with properly scaled frequency distributions introduced in Sec. 3, can be presented as functions of the parameters (25) and (26). The parameters (25) are significant for both identical and nonidentical capillary layers, whereas the parameters (26) equal unity in the case of identical layers. Therefore, the parameters (26) may be regarded as characteristics of the asymmetry (heterogeneity) of the problem.

In the present study the last two parameters in (25), v and m, are kept constant. Variation of dimensional parameters υ, g, Pa, N, and a (Table 1) affects the first five dimensionless groups (25). Once the solution of the problem has been obtained, e.g., for the case L = 0.04 cm, υ = 0.04 cm/sec, the same solution will apply to L = 0.08 cm, υ = 0.08 cm/sec, or L = 0.1 cm, υ = 0.1 cm/sec, because the quantities (25) and (26) remain constant despite these changes.

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.

7. Symmetric Case: Identical Capillary Layers. Comparison with The Krogh Cylinder Model

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


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]:


where the prime refers to the Krogh model: P′(z) is the intracapillary oxygen tension; Pmin(z) is the minimum value of the tissue oxygen tension at a cross-section z = const; and Pm, is the mean oxygen tension


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) = Pa.

Figure 3 shows the intracapillary oxygen tension P1 = P2, the minimum tissue oxygen tension Pmin(z) in a plane z = const, and the mean oxygen tension Pm defined in (21), versus the dimensionless coordinate z* = z/L. The calculated quantities P′ and Pm given by (29) and (31) coincide with P1 = P2 and Pm, respectively, within the plotting accuracy. The largest difference, of about 0.5 mmHg, is between Pmin and Pm, which is due to the difference in geometry of the two models. Indeed, the minimum oxygen tension Pmin is reached at the points most distant from the capillaries: x = a, y = 0 and x = −a, y = 2a; these points are separated by the distance 2a from the axes of the capillaries. In the case of the Krogh cylinder model, the minimum values of the oxygen tension are reached at the distance rt = √8 a/√π [similar, equals] 1.6a from the capillary axis.

Fig. 3
Distribution of oxygen tensions along the length of the capillary.

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

Fig. 4
Tissue oxygen tension profiles.

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 P1 = P2 the function P(x*, y*, z*) can be expressed in the form


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, Pa, υ, N, and a in the case of identical layers. This should provide a reference for assessment of the results pertaining to asymmetric distributions. Figure 5 depicts the frequency distributions F(P) for different values of the oxygen consumption rate g. All unspecified parameters assume the “standard” values from Table 1. Vertical arrows indicate the mean values Pm. It is instructive to present the results of calculations in histogram form, since this form is frequently used in experimental studies. The histograms in Fig. 5, depicted by dotted lines, are defined by

Fig. 5
Frequency distributions of tissue oxygen tension for different values of the oxygen consumption rate.

for 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, Pυ = P1(1) = P2(1); the mean oxygen tension Pm; and the minimum oxygen tension Pmin, defined as

Fig. 6
The mean tissue oxygen tension Pm, the minimum tissue oxygen tension Pmin, and the intravascular oxygen tension at the venous end of the capillary, Pυ, versus the oxygen consumption rate.

as functions of the oxygen consumption rate. The circles indicate the corresponding values Pυ, P¯m and P¯min obtained within the framework of the Krogh cylinder model. Clearly, the values Pυ and P¯min practically coincide with Pυ and Pm, respectively; the values of P¯min are slightly higher than Pmin.

Figures 7 and and88 demonstrate the effect of the inlet capillary oxygen tension on the frequency distributions F(P) and on the parameters Pυ, Pm, and Pmin. Analogous results demonstrating the effect of variation of the capillary velocity υ are presented in Figs. 9 and and10.10. Finally, Figs. 11 and and1212 show the variation of Pυ, Pm, and Pmin caused by variation of the oxygen-binding capacity N and of the geometrical parameter 2a.

Fig. 7
Frequency distributions of tissue oxygen tension for different values of the inlet capillary oxygen tension.
Fig. 8
Oxygen tensions as functions of the inlet capillary oxygen tension.
Fig. 9
Frequency distributions of tissue oxygen tension for different values of the capillary velocity.
Fig. 10
Oxygen tensions as functions of the capillary velocity.
Fig. 11
Oxygen tensions as functions of the oxygen-binding capacity.
Fig. 12
Oxygen tensions as functions of the geometric parameter.

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.

8. Asymmetric Case: Nonidentical Capillary Layers

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, Pa1Pa2, which may result, for example, from uneven precapillary oxygen losses. The distributions P1(z*), P2(z*), Pm(z*), and Pmin(z*) shown in Fig. 13 correspond to the boundary conditions Pa1 = 80 mmHg, Pa2 = 40 mmHg. The initial difference of 40 mmHg between the intracapillary oxygen tensions practically disappears at z* = 0.5; therefore, considerable diffusional shunting occurs among the capillaries. Profiles of the tissue oxygen tensions are shown in Fig. 14 in two perpendicular planes; y* = 0 and x* = 0.5. The lower panel of Fig. 15 presents the frequency distributions F(P) for symmetric and asymmetric cases corresponding to the same value (Pa1 + Pa2)/2 = 60 mmHg. It is seen that the shapes of the distributions are somewhat different, and the values of Pm and Pmin are smaller in the case of the asymmetric distribution. Figure 16 further illustrates this effect; it depicts the functions Pυ, Pm, and Pmin versus the parameter Pa1/ Pa2 which characterizes the asymmetry of the problem. These results demonstrate that an increase of Pa1/ Pa2 (increase of the degree of asymmetry provided that Pa1 + Pa2 = const) leads to a reduction of the oxygen tensions Pυ1, Pυ2, Pm, and Pmin.

Fig. 13
Distribution of oxygen tensions along the length of the capillary in the case of different inlet capillary tensions.
Fig. 14
Tissue oxygen tension profiles in the case of different inlet capillary oxygen tensions.
Fig. 15
Frequency distributions of tissue oxygen tension for symmetric and asymmetric cases.
Fig. 16
Oxygen tensions as functions of the inlet-capillary oxygen-tension ratio.

Consider another type of heterogeneity caused by different capillary velocities, υ1 = υ2. Figure 17 shows the distributions P1(z*), P2(z*), Pm(z*), and Pmin(z*) for particular values of υ1, and υ2. Contrary to the case of different inlet capillary oxygen tensions, in this case the difference between P1 and P2 increases as z* increases. The tissue oxygen tension profiles in the planes y* = 0 and x* = 0.5 are presented in Fig. 18. It should be noted that the “slow” capillary is a source of oxygen over its entire length (cf Fig. 14). The upper panel of Fig. 15 shows the frequency distributions of the tissue oxygen tension for the symmetric and asymmetric cases. Clearly, the minimum and the mean oxygen tensions corresponding to the case υ1 = υ2 are shifted to the left with respect to the ones corresponding to the symmetric case. The cumulative effect of asymmetry caused by uneven capillary velocities is shown in Fig. 19. An increase of the velocity ratio υ1/υ2 causes reduction of the mean oxygen tension Pm and of the minimum oxygen tension Pmin, despite conservation of the total blood flow to the microvascular unit.

Fig. 17
Distribution of oxygen tensions along the length of the capillary in the case of different capillary velocities.
Fig. 18
Tissue oxygen tension profiles in the case of different capillary velocities.
Fig. 19
Oxygen tensions as functions of the capillary velocity ratio.

The effect of variation of the last parameter in (26), N1/N2, in most cases can be predicted from the previous calculations. Indeed, it can be shown that Φ(P) [dbl greater-than sign] 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 QiNi = πr2υiNi, proportional to the volume flux of red blood cells, appears in the equations (14). This implies that instead of two independent parameters υ1/υ2 and N1/N2 in (26), only a single parameter υ1N1/υ2N2 is of importance. Therefore, the results shown in Fig. 19 will remain valid if the independent variable υ1/υ2 is replaced by N1/N2 provided that υ1 = υ2 = 0.04 cm/sec, (N1 + N2)/2 = 0.1. Of course, if the solution of the problem includes low values of the capillary oxygen tensions P1 and/or P2, then independent calculations for variation of the parameter N1/ N2 are necessary.

9. Conclusions

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.


geometrical parameters
diffusion coefficient
f(z, P)
frequency distribution, equation (19)
frequency distribution, equation (17)
oxygen consumption rate
parameter in (24)
integral given by equation (11)
capillary-tissue diffusive flux
elliptic modulus
complete elliptic integral of the first kind
capillary length
parameter in (24)
oxygen-binding capacity of blood
oxygen tension
mean oxygen tension defined by (21)
mean oxygen tension defined by (22)
oxygen tension at the arterial end of the capillary
oxygen tension at the venous end of the capillary
oxygen tension given by (12)
dimensionless parameters in (13)
volumetric blood flow rate
capillary radius
S, Sb
solubility coefficients of oxygen
capillary velocity of blood
tissue volume
x, y, z
dimensional coordinates

Greek Symbols

capillary-tissue interface
variable given by (10)
dimensionless parameter defined in (13)
solubility ratio
variable given by (9)
cross-sectional area
dimensionless function defined by (15)
oxygen dissociation curve


i, 1, 2
pertaining to capillaries
mean value
minimum value


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