A Model for Oxygen Transport from Blood in Microvessels to Tissue

Abstract

Oxygen is vital for cellular energetics and metabolism in the human body. Blood transports oxygen to the tissues with hemoglobin in red blood cells playing a key role in the transportation of oxygen. To account for the Fåhraeus and Fåhraeus–Lindqvist effects, we use Haynes marginal zone concept, which subdivides each microvessel into a cell free layer surrounding a core region of uniform red blood cells concentration. The marginal zone concept is used to develop a steady state model for the transport of oxygen from blood to tissue where chemical reaction of oxygen occurs to produce energy. The approach is based on fundamentals of fluid flow and mass transfer in the two zones while accounting for the role of hemoglobin in the transport process and including mass transfer and chemical reaction in the tissue to produce energy using the Krogh cylinder concept. In contrast to transport modeling of solutes such as glucose, the present model includes the key role of hemoglobin in the transport of oxygen from blood to tissue. The model is analytical and provides analytical expressions for the oxygen level profiles in the blood cell free layer, the core zone, and the Krogh cylinder. The results are found to agree with published results in the literature for oxygen transport from blood in capillary size microvessel to its Krogh tissue cylinder. The model is not restricted to transport from capillaries and includes transport of oxygen from microvessels to tissue in general. Extensions of the model include further investigations in the case where changes in the blood microvessel or red blood cells occur due to pathological conditions.

1. Introduction

Flow in microvessels of diameter less than about 500 μm has a larger discharge hematocrit than the tube hematocrit (hematocrit ratio less than one) (Fåhraeus effect) and a decreasing apparent viscosity as capillary diameter decreases (Fåhraeus–Lindqvist effect) [1,2,3,4,5,6]. Both effects are explained by Haynes marginal concept [7]. The model divides the capillary side into a cell-free annular layer and a core region containing plasma and the red blood cells. A review of the marginal zone concept can be found in [8,9,10,11]. Shear induced migration and elastic stress induced models are considered in [12,13,14,15] using the concept introduced in [16,17], and [18,19,20,21,22], respectively. Particulate (mesoscale) models are discussed in [23,24], and viscoelastic models are reviewed in [23].

Krogh mass transport of oxygen in tissue [25], review and extensions of the Krogh model including mass transport in the capillary surrounded by the Krogh tissue cylinder can be found in [3,23,26,27,28,29] and the references therein. Several numerical models considered oxygen transport in capillaries while accounting for the particulate nature of RBCs, assuming different shapes of RBCs: spherical, cylindrical, and parachute form [30,31,32,33,34] with different frames of reference. A simplified analytical model is also provided in [34]. Computational models are introduced for the transport of oxygen (i) based on a one-dimensional (1D) model for microvascular network and a 3D model for porous tissue in [35] and (ii) using a computational method based on incomplete boundary conditions to model microvascular networks in [36].

The model of Fournier [3] for oxygen transport accounts for oxygen transport in capillaries using a uniform velocity profile while accounting for transport of oxygen in plasma and the role of hemoglobin in the transport process, with Krogh model used for transport by diffusion and oxygen consumption in the tissue side. On the other hand, the particulate models account for mass transport in microvessels with each cross-sectional area containing no more than one RBC. In a microvessel, there is a cell-free layer in which no RBCs are present, with a core region where RBCs concentrate. This segregation is considered in the present analysis of oxygen transport. The present model accounts for mass transport in both zones while including the role of hemoglobin in transport of oxygen in the core region. In addition, non-uniformity of the velocity profiles in the two zones and transport of oxygen in the tissue side are included in the present analysis of oxygen transport from blood to tissue. In contrast to a previously published model for solute transport (such as glucose) from blood to tissue [37], the present model includes the key role of hemoglobin in the transport of oxygen.

2. Governing Equations

2.1. Blood Flow Model

The model used is based on the Haynes marginal theory [3,7,11]. The annular zone is the cell-free layer, and the core zone contains plasma and red blood cells (Figure 1).

The main results are summarized, and results are needed for the mass transfer model of oxygen from blood to the surrounding tissue Krogh cylinder.

The hydrodynamic model is a continuum model using the no-slip boundary condition at the capillary wall (r = R), finite shear stress at the axial location (r = 0), and continuity of the velocity and the shear stress at the interface between the annular layer and the core region (r = r_i).

The model yields the following velocity profiles in the fully developed flow region: v_c in the core region, v_a in the annular layer, and v_i at the interface

$${\mathrm{v}}_{\mathrm{c}}={\mathrm{A}}_{\mathrm{c}}\left({\mathrm{r}}^2-{\mathrm{r}}_{\mathrm{i}}^2\right)+{\mathrm{v}}_{\mathrm{i}}; {\mathrm{v}}_{\mathrm{i}}={\mathrm{A}}_{\mathrm{a}}\left({\mathrm{r}}_{\mathrm{i}}^2-{\mathrm{R}}^2\right); {\mathrm{v}}_{\mathrm{a}}={\mathrm{A}}_{\mathrm{a}}\left({\mathrm{r}}^2-{\mathrm{R}}^2\right)$$

(1)

where

$${\mathrm{A}}_{\mathrm{c}}=\frac{-{\mathrm{P}}_{\mathrm{g}\mathrm{r}}}{4\hspace{0.33em}{\mathsf{\mu }}_{\mathrm{c}}}; {\mathrm{A}}_{\mathrm{a}}=\frac{-{\mathrm{P}}_{\mathrm{g}\mathrm{r}}}{4\hspace{0.33em}{\mathsf{\mu }}_{\mathrm{a}}}$$

(2)

P_gr is the pressure gradient and μ_c and μ_a denote the core region and annular layer viscosities, respectively. Other relations include the following relation for the volumetric flow rate Q, and the average velocity v^*

$$\mathrm{Q}={\mathrm{v}}^{\ast }\mathsf{\pi }{\mathrm{R}}^2=\frac{\mathsf{\pi }{\mathrm{P}}_{\mathrm{g}\mathrm{r}}}{8}\left[\frac{{\mathrm{r}}_{\mathrm{i}}^4}{{\mathsf{\mu }}_{\mathrm{c}}}+\frac{{\mathrm{R}}^4-{\mathrm{r}}_{\mathrm{i}}^4}{{\mathsf{\mu }}_{\mathrm{a}}}\right]$$

(3)

The apparent viscosity is provided by

$$\frac{{\mathsf{\mu }}_{\mathrm{a}}}{{\mathsf{\mu }}_{\mathrm{a}\mathrm{p}\mathrm{p}}}=1+{\mathsf{\sigma }}^4\left[\frac{{\mathsf{\mu }}_{\mathrm{a}}}{{\mathsf{\mu }}_{\mathrm{c}}}-1\right]; {\mathsf{\mu }}_{\mathrm{a}\mathrm{p}\mathrm{p}}=\frac{\mathsf{\pi }\hspace{0.33em}{\mathrm{P}}_{\mathrm{g}\mathrm{r}}\hspace{0.33em}{\mathrm{R}}^4}{8\hspace{0.33em}\mathrm{Q}}$$

(4)

with the following relations reported in [3]

$$\frac{{\mathsf{\mu }}_{\mathrm{a}}}{{\mathsf{\mu }}_{\mathrm{c}}}=1-\mathsf{\alpha }{\mathrm{H}}_{\mathrm{C}}$$

(5)

$$\mathsf{\alpha }=0.070\hspace{0.33em}\exp \left[2.49\hspace{0.33em}{\mathrm{H}}_{\mathrm{C}}+\frac{1107}{\mathrm{T}}\hspace{0.33em}\exp \left(-1.69\hspace{0.33em}{\mathrm{H}}_{\mathrm{C}}\right)\right]$$

(6)

where H_C is the core region hematocrit and temperature T is in Kelvin.

The required values of H_C and the discharge hematocrit H_D can be obtained using the following relations

$$\frac{{\mathrm{H}}_{\mathrm{T}}}{{\mathrm{H}}_{\mathrm{C}}}={\mathsf{\sigma }}^2$$

(7)

$$\frac{{\mathrm{H}}_{\mathrm{T}}}{{\mathrm{H}}_{\mathrm{D}}}={\mathsf{\sigma }}^2\left[1+\frac{{\left(1-{\mathsf{\sigma }}^2\right)}^2}{{\mathsf{\sigma }}^2\left(2-2{\mathsf{\sigma }}^2+{\mathsf{\sigma }}^2\frac{{\mathsf{\mu }}_{\mathrm{a}}}{{\mathsf{\mu }}_{\mathrm{c}}}\right)}\right]$$

(8)

where H_T is the tube hematocrit and σ denotes r_i/R.

2.2. Mass Transfer Model

Oxygen is mainly transported by hemoglobin Hb with n molecules bound to Hb. At equilibrium, Hill’s equation is used to provide hemoglobin saturation Y as a function of partial pressure of oxygen, p_O2 [3], and Hill’s constant n

$$\mathrm{Y}=\frac{{\mathrm{p}}_{{\mathrm{O}}_2}^{\mathrm{n}}}{{\mathrm{p}}_{50}^{\mathrm{n}}+{\mathrm{p}}_{{\mathrm{O}}_2}^{\mathrm{n}}}$$

(9)

where, at a partial pressure of oxygen equal to p₅₀, the binding sites of hemoglobin are half filled.

The governing equations for dissolved oxygen balance in the core region plasma is given by

$${\mathrm{v}}_{\mathrm{c}}\frac{\partial {\mathrm{c}}_{\mathrm{c}}}{\partial \mathrm{z}}={\mathrm{D}}_{\mathrm{c}}\left[\frac{1}{\mathrm{r}}\frac{\partial }{\partial \mathrm{r}}\left(\mathrm{r}\frac{\partial {\mathrm{c}}_{\mathrm{c}}}{\partial \mathrm{r}}\right)+\frac{{\partial }^2{\mathrm{c}}_{\mathrm{c}}}{\partial {\mathrm{z}}^2}\right]-{\mathrm{R}}_{\mathrm{H}\mathrm{b}\mathrm{O}}$$

(10)

where c_c and D_c are the free oxygen concentration in the core and the diffusion coefficient in the core region, respectively, and R is the rate of conversion to oxygenated hemoglobin.

Neglecting diffusional effects as in [3], the conservation equation of oxygenated hemoglobin can be written as

$${\mathrm{v}}_{\mathrm{c}}\frac{\partial {\mathrm{c}}_{\mathrm{H}\mathrm{b}\mathrm{O}}}{\partial \mathrm{z}}={\mathrm{R}}_{\mathrm{H}\mathrm{b}\mathrm{O}}$$

(11)

where c_HbO is the hemoglobin-bound oxygen concentration.

Adding Equations (10) and (11) and using the chain rule yields [3]

$${\mathrm{v}}_{\mathrm{c}}\left(1+\frac{\partial {\mathrm{c}}_{\mathrm{H}\mathrm{b}\mathrm{O}}}{\partial {\mathrm{c}}_{\mathrm{c}}}\right)\frac{\partial {\mathrm{c}}_{\mathrm{c}}}{\partial \mathrm{z}}={\mathrm{D}}_{\mathrm{c}}\left[\frac{1}{\mathrm{r}}\frac{\partial }{\partial \mathrm{r}}\left(\mathrm{r}\frac{\partial {\mathrm{c}}_{\mathrm{c}}}{\partial \mathrm{r}}\right)+\frac{{\partial }^2{\mathrm{c}}_{\mathrm{c}}}{\partial {\mathrm{z}}^2}\right], \mathrm{m}=\frac{\partial {\mathrm{c}}_{\mathrm{H}\mathrm{b}\mathrm{O}}}{\partial {\mathrm{c}}_{\mathrm{c}}}$$

(12)

where D_c is estimated using Maxwell equation [3,38]

$$\frac{{\mathrm{D}}_{\mathrm{c}}}{{\mathrm{D}}_{\mathrm{a}}}=\frac{2{\mathrm{D}}_{\mathrm{a}}+{\mathrm{D}}_{\mathrm{R}\mathrm{B}\mathrm{C}}-2{\mathrm{H}}_{\mathrm{c}}\left({\mathrm{D}}_{\mathrm{a}}-{\mathrm{D}}_{\mathrm{R}\mathrm{B}\mathrm{C}}\right)}{2{\mathrm{D}}_{\mathrm{a}}+{\mathrm{D}}_{\mathrm{R}\mathrm{B}\mathrm{C}}+{\mathrm{H}}_{\mathrm{c}}\left({\mathrm{D}}_{\mathrm{a}}-{\mathrm{D}}_{\mathrm{R}\mathrm{B}\mathrm{C}}\right)}\hspace{0.33em}$$

(13)

We can also express equations using partial pressure ${\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{c}}$ = c_c/α_a, where α_a denotes oxygen solubility, to get

$${\mathrm{v}}_{\mathrm{c}}\left(1+\mathrm{m}\right)\frac{\partial {\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{c}}}{\partial \mathrm{z}}={\mathrm{D}}_{\mathrm{c}}\left[\frac{1}{\mathrm{r}}\frac{\partial }{\partial \mathrm{r}}\left(\mathrm{r}\frac{\partial {\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{c}}}{\partial \mathrm{r}}\right)+\frac{{\partial }^2{\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{c}}}{\partial {\mathrm{z}}^2}\right]$$

(14)

Using Hill’s equation yields [3]

$$\mathrm{m}=\mathrm{n}\hspace{0.33em}{\mathrm{p}}_{50}^{\mathrm{n}}\hspace{0.33em}\mathrm{H}\hspace{0.33em}{\mathrm{c}}_{\mathrm{c},\mathrm{s}\mathrm{a}\mathrm{t}}\frac{{\mathrm{p}}_{{\mathrm{O}}_2}^{\mathrm{n}-1}}{{\left({\mathrm{p}}_{50}^{\mathrm{n}}+{\mathrm{p}}_{{\mathrm{O}}_2}^{\mathrm{n}}\right)}^2}, \mathrm{H}={\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{c}}/{\mathrm{c}}_{\mathrm{c}}=1/{\mathsf{\alpha }}_{\mathrm{a}}$$

(15)

where n and p₅₀ are equal to 2.34 and 26 mm Hg, respectively, and c_c,sat is the maximum concentration for oxygen (with four oxygen molecules of oxygen bound and a maximum concentration of hemoglobin in blood of 2200 μM) is 8800 μM [3].

The mass transfer of dissolved oxygen in the annular zone (cell-free layer) is governed by

$${\mathrm{v}}_{\mathrm{a}}\frac{\partial {\mathrm{c}}_{\mathrm{a}}}{\partial \mathrm{z}}={\mathrm{D}}_{\mathrm{a}}\left[\frac{1}{\mathrm{r}}\frac{\partial }{\partial \mathrm{r}}\left(\mathrm{r}\frac{\partial {\mathrm{c}}_{\mathrm{a}}}{\partial \mathrm{r}}\right)+\frac{{\partial }^2{\mathrm{c}}_{\mathrm{a}}}{\partial {\mathrm{z}}^2}\right]$$

(16)

where c_a and D_a denote the dissolved oxygen concentration and diffusion coefficient in the annular zone, respectively. Dissolved oxygen levels can be expressed in terms of oxygen partial pressure using Henry’s law [3]. The above equation yields

$${\mathrm{v}}_{\mathrm{a}}\frac{\partial {\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{a}}}{\partial \mathrm{z}}={\mathrm{D}}_{\mathrm{a}}\left[\frac{1}{\mathrm{r}}\frac{\partial }{\partial \mathrm{r}}\left(\mathrm{r}\frac{\partial {\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{a}}}{\partial \mathrm{r}}\right)+\frac{{\partial }^2{\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{a}}}{\partial {\mathrm{z}}^2}\right]$$

(17)

In the tissue side, the partial pressure profile in the Krogh model where mass transfer is limited to the Krogh tissue cylinder of radius r_T surrounding the microvessel [3,23,25,26,27,28]. The governing equation is

$${\mathrm{D}}_{\mathrm{T}}\left[\frac{1}{\mathrm{r}}\frac{\partial }{\partial \mathrm{r}}\left(\mathrm{r}\frac{\partial {\mathrm{c}}_{\mathrm{T}}}{\partial \mathrm{r}}\right)+\frac{{\partial }^2{\mathrm{c}}_{\mathrm{T}}}{\partial {\mathrm{z}}^2}\right]-\mathsf{\Gamma }=0$$

(18)

where c_T and D_T are the oxygen dissolved concentration and diffusion coefficient in the tissue side, respectively, and Γ is the rate of consumption of oxygen. Dropping the diffusional term in the z-direction and setting the flux to zero at r_T, to exclude mass transfer across the different Krogh tissue cylinders, yields the Krogh-Erlang solution [25,27]

$${\mathrm{c}}_{\mathrm{T}}(\mathrm{r},\mathrm{z})-{\mathrm{c}}_{\mathrm{T}}(\mathrm{R},\mathrm{z})=\frac{\mathsf{\Gamma }}{4{\mathrm{D}}_{\mathrm{T}}}\left[{\mathrm{r}}^2-{\mathrm{R}}^2\right]-\frac{\mathsf{\Gamma }\hspace{0.33em}{\mathrm{r}}_{\mathrm{T}}^2}{2{\mathrm{D}}_{\mathrm{T}}}\ln \left(\frac{\mathrm{r}}{\mathrm{R}}\right)$$

(19)

This gives the following in terms of oxygen partial pressure

$${\mathrm{p}}_{\mathrm{T}}(\mathrm{r},\mathrm{z})-{\mathrm{p}}_{\mathrm{T}}(\mathrm{R},\mathrm{z})=\frac{\mathsf{\Gamma }}{4{\mathsf{\alpha }}_{\mathrm{T}}{\mathrm{D}}_{\mathrm{T}}}\left[{\mathrm{r}}^2-{\mathrm{R}}^2\right]-\frac{\mathsf{\Gamma }\hspace{0.33em}{\mathrm{r}}_{\mathrm{T}}^2}{2{\mathsf{\alpha }}_{\mathrm{T}}{\mathrm{D}}_{\mathrm{T}}}\ln \left(\frac{\mathrm{r}}{\mathrm{R}}\right)$$

(20)

where α_T denotes oxygen solubility in tissue.

The following boundary conditions express the continuity of the partial pressures of oxygen

$$\mathrm{at} \mathrm{r}=0{,}_{ }\frac{\partial {\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{c}}}{\partial \mathrm{r}} \mathrm{is} \mathrm{finite}$$

(21)

$$\mathrm{at} \mathrm{r}={\mathrm{r}}_{\mathrm{i}}, {\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{a}}={\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{c}}$$

(22)

$$\mathrm{at} \mathrm{r}=\mathrm{R}{,}_{ }{\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{a}}={\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{T}}$$

(23)

$$\mathrm{at} \mathrm{r}=\mathrm{R}{,}_{ }-{\mathrm{D}}_{\mathrm{a}}{\mathsf{\alpha }}_{\mathrm{a}}\frac{\partial {\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{a}}}{\partial \mathrm{r}}=-{\mathrm{D}}_{\mathrm{T}}{\mathsf{\alpha }}_{\mathrm{T}}\frac{\partial {\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{T}}}{\partial \mathrm{r}}$$

(24)

3. Solution

We seek the following solution

$${\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{c}}={\mathrm{p}}_{{\mathrm{O}}_2,0}+{\mathsf{\phi }}_{\mathrm{c}}-{\mathrm{b}}_{\mathrm{c}}\mathrm{z}; {\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{a}}={\mathrm{p}}_{{\mathrm{O}}_2,0}+{\mathsf{\phi }}_{\mathrm{a}}-{\mathrm{b}}_{\mathrm{a}}\mathrm{z}; {\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{T}}={\mathrm{p}}_{{\mathrm{O}}_2,0}+{\mathsf{\phi }}_{\mathrm{T}}-{\mathrm{b}}_{\mathrm{T}}\mathrm{z}$$

(25)

where oxygen pressures are linear function of z, and φ_c, φ_a, and φ_T are functions of r. Continuity at the interfaces requires equality of the slopes: b_c = b_a = b_T = b, with subscripts dropped. In Equation (25), φ simply represents the deviation of oxygen partial pressures from the average oxygen partial pressure in blood ${\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{b}}(\mathrm{z})={\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{b}}(0)-\mathrm{b}\hspace{0.33em}\mathrm{z}$. Substituting the relevant expressions for oxygen partial pressure in Equation (31) into Equations (14) and (17) yields the following ordinary differential equations

$$-{\mathrm{v}}_{\mathrm{c}}\left(1+\mathrm{m}\right)\mathrm{b}={\mathrm{D}}_{\mathrm{c}}\frac{1}{\mathrm{r}}\frac{\mathrm{d}}{\mathrm{d}\mathrm{r}}\left(\mathrm{r}\frac{\mathrm{d}{\mathsf{\phi }}_{\mathrm{c}}}{\mathrm{d}\mathrm{r}}\right)$$

(26)

$$-{\mathrm{v}}_{\mathrm{a}}\mathrm{b}={\mathrm{D}}_{\mathrm{a}}\frac{1}{\mathrm{r}}\frac{\mathrm{d}}{\mathrm{d}\mathrm{r}}\left(\mathrm{r}\frac{\mathrm{d}{\mathsf{\phi }}_{\mathrm{a}}}{\mathrm{d}\mathrm{r}}\right)$$

(27)

subject to the following boundary conditions

$$\mathrm{at} \mathrm{r}=0{,}_{ }{\mathsf{\phi }}_{\mathrm{c}} \mathrm{is} \mathrm{finite}$$

(28)

$$\mathrm{at} \mathrm{r}={\mathrm{r}}_{\mathrm{i}}{,}_{ }{\mathsf{\phi }}_{\mathrm{c}}={\mathsf{\phi }}_{\mathrm{a}}={\mathsf{\phi }}_{\mathrm{i}}$$

(29)

$$\mathrm{at} \mathrm{r}=\mathrm{R}{,}_{ }-{\mathrm{D}}_{\mathrm{a}}{\mathsf{\alpha }}_{\mathrm{a}}\frac{\mathrm{d}{\mathsf{\phi }}_{\mathrm{a}}}{\mathrm{d}\mathrm{r}}=-{\mathrm{D}}_{\mathrm{T}}{\mathsf{\alpha }}_{\mathrm{T}}\frac{\mathrm{d}{\mathsf{\phi }}_{\mathrm{T}}}{\mathrm{d}\mathrm{r}}$$

(30)

Integrating Equations (26) and (27) using boundary conditions (28) and (29) yields

$${\mathsf{\phi }}_{\mathrm{c}}-{\mathsf{\phi }}_{\mathrm{i}}=-\frac{\left(1+{\mathrm{m}}_{\mathrm{a}\mathrm{v}}\right)\mathrm{b}}{{\mathrm{D}}_{\mathrm{c}}}\left[\frac{{\mathrm{A}}_{\mathrm{c}}}{16}\left({\mathrm{r}}^4-{\mathrm{r}}_{\mathrm{i}}^4\right)+\left({\mathrm{v}}_{\mathrm{i}}-{\mathrm{A}}_{\mathrm{c}}{\mathrm{r}}_{\mathrm{i}}^2\right)\frac{{\mathrm{r}}^2-{\mathrm{r}}_{\mathrm{i}}^2}{4}\right]$$

(31)

$${\mathsf{\phi }}_{\mathrm{a}}-{\mathsf{\phi }}_{\mathrm{i}}=-\frac{\mathrm{b}{\mathrm{A}}_{\mathrm{a}}}{{\mathrm{D}}_{\mathrm{a}}}\left(\frac{{\mathrm{r}}^4-{\mathrm{r}}_{\mathrm{i}}^4}{16}-{\mathrm{R}}^2\frac{{\mathrm{r}}^2-{\mathrm{r}}_{\mathrm{i}}^2}{4}\right)+\mathrm{K}\ln \frac{\mathrm{r}}{{\mathrm{r}}_{\mathrm{i}}}\hspace{0.33em}$$

(32)

where m_av is an average value for m. Using boundary condition (30) provides a relationship between the two remaining unknowns b and K

$$\mathrm{K}=-\frac{\mathrm{b}{\mathrm{A}}_{\mathrm{a}}}{{\mathrm{D}}_{\mathrm{a}}}\frac{{\mathrm{R}}^4}{4}-\frac{\mathsf{\Gamma }}{{\mathsf{\alpha }}_{\mathrm{a}}{\mathrm{D}}_{\mathrm{a}}}\frac{{\mathrm{r}}_{\mathrm{T}}^2-{\mathrm{R}}^2}{2}\hspace{0.33em}$$

(33)

The constant b needed to find the drop in the partial pressure of oxygen per unit change of length in the axial direction, and a mass balance on oxygen is performed in the annular zone between z and z + Δz

$$\begin{gathered} {\displaystyle {\int }_{{\mathrm{r}}_{\mathrm{i}}}^{\mathrm{R}}2\mathsf{\pi }}\mathrm{r}{\mathrm{v}}_{\mathrm{a}}\left[{\left.\left({\mathrm{c}}_0+{\mathsf{\alpha }}_{\mathrm{a}}{\mathsf{\phi }}_{\mathrm{a}}-{\mathsf{\alpha }}_{\mathrm{a}}\mathrm{b}\mathrm{z}\right)\right|}_{\mathrm{z}}-{\left.\left({\mathrm{c}}_0+{\mathsf{\alpha }}_{\mathrm{a}}{\mathsf{\phi }}_{\mathrm{a}}-{\mathsf{\alpha }}_{\mathrm{a}}\mathrm{b}\mathrm{z}\right)\right|}_{\mathrm{z}+\mathsf{\Delta }\mathrm{z}}\right]\mathrm{d}\mathrm{r} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-2\mathsf{\pi }{\mathrm{r}}_{\mathrm{i}}\mathsf{\Delta }\mathrm{z}{\mathrm{D}}_{\mathrm{c}}{\mathsf{\alpha }}_{\mathrm{a}}{\left.\frac{\mathrm{d}{\mathsf{\phi }}_{\mathrm{c}}}{\mathrm{d}\mathrm{r}}\right|}_{\mathrm{r}={\mathrm{r}}_{{}_{\mathrm{i}}}}-\mathsf{\pi }\left[{\mathrm{r}}_{\mathrm{T}}^2-{\mathrm{R}}^2\right]\mathsf{\Delta }\mathrm{z}\hspace{0.33em}\mathsf{\Gamma }=0 \end{gathered}$$

(34)

Using Equations (31) and (32) yields after substitution into Equation (34), integration and rearrangement

$$\mathrm{b}=\frac{\left({\mathrm{r}}_{\mathrm{T}}^2-{\mathrm{R}}^2\right)\mathsf{\Gamma }/{\mathsf{\alpha }}_{\mathrm{a}}}{\left({\mathrm{R}}^2-{\mathrm{r}}_{\mathrm{i}}^2\right){\overline{\mathrm{v}}}_{\mathrm{a}}+\left(1+{\mathrm{m}}_{\mathrm{a}\mathrm{v}}\right)\left[\frac{{\mathrm{A}}_{\mathrm{c}}}{2}{\mathrm{r}}_{\mathrm{i}}^4+\left({\mathrm{v}}_{\mathrm{i}}-{\mathrm{A}}_{\mathrm{c}}{\mathrm{r}}_{\mathrm{i}}^2\right){\mathrm{r}}_{\mathrm{i}}^2\right]}$$

(35)

In order to determine φ_i, we apply the overall mass balance to the annular zone between 0 and z

$$\begin{gathered} {\displaystyle {\int }_{{\mathrm{r}}_{\mathrm{i}}}^{\mathrm{R}}2\mathsf{\pi }}\mathrm{r}{\mathrm{v}}_{\mathrm{a}}{\mathrm{c}}_0\mathrm{d}\mathrm{r}-{\displaystyle {\int }_{{\mathrm{r}}_{\mathrm{i}}}^{\mathrm{R}}2\mathsf{\pi }}\mathrm{r}{\mathrm{v}}_{\mathrm{a}}\left({\mathrm{c}}_0+{\mathsf{\alpha }}_{\mathrm{a}}{\mathsf{\phi }}_{\mathrm{a}}-{\mathsf{\alpha }}_{\mathrm{a}}\mathrm{b}\mathrm{z}\right)\mathrm{d}\mathrm{r} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-2\mathsf{\pi }{\mathrm{r}}_{\mathrm{i}}\mathrm{z}{\mathrm{D}}_{\mathrm{c}}{\mathsf{\alpha }}_{\mathrm{a}}{\left.\frac{\mathrm{d}{\mathsf{\phi }}_{\mathrm{c}}}{\mathrm{d}\mathrm{r}}\right|}_{\mathrm{r}={\mathrm{r}}_{{}_{\mathrm{i}}}}-\mathsf{\pi }\left[{\mathrm{r}}_{\mathrm{T}}^2-{\mathrm{R}}^2\right]\mathrm{z}\hspace{0.33em}\mathsf{\Gamma }=0 \end{gathered}$$

(36)

Using Equation (34) reduces the above equation to

$${\displaystyle {\int }_{{\mathrm{r}}_{\mathrm{i}}}^{\mathrm{R}}\mathrm{r}{\mathrm{v}}_{\mathrm{a}}{\mathsf{\phi }}_{\mathrm{a}}\mathrm{d}\mathrm{r}}=0$$

(37)

Substituting for the expressions of v_a and φ_a given in Equations (1) and (32), and integrating gives

$${\mathsf{\phi }}_{\mathrm{i}}=4\frac{{\mathrm{f}}_{\mathsf{\chi }=\mathsf{\lambda }}-{\mathrm{f}}_{\mathsf{\chi }=1}}{{\left({\mathsf{\lambda }}^2-1\right)}^2}$$

(38)

where λ = R/r_i = 1/σ and

$$\begin{gathered} \mathrm{f}=-\frac{\mathrm{b}{\mathrm{A}}_{\mathrm{a}}{\mathrm{R}}^4}{{\mathrm{D}}_{\mathrm{a}}}{\mathsf{\sigma }}^4\left[\frac{{\mathsf{\chi }}^8}{8}-5{\mathsf{\lambda }}^2\frac{{\mathsf{\chi }}^6}{6}+\left(4{\mathsf{\lambda }}^4+4{\mathsf{\lambda }}^2-1\right)\frac{{\mathsf{\chi }}^4}{4}+\left(-4{\mathsf{\lambda }}^4+{\mathsf{\lambda }}^2\right)\frac{{\mathsf{\chi }}^2}{2}\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\mathrm{K}\left[\frac{{\mathsf{\chi }}^4}{4}\ln \mathsf{\chi }-\frac{{\mathsf{\chi }}^4}{16}-{\mathsf{\lambda }}^2\left(\frac{{\mathsf{\chi }}^2}{2}\ln \mathsf{\chi }-\frac{{\mathsf{\chi }}^2}{4}\right)\right], \mathsf{\chi }=\mathrm{r}/{\mathrm{r}}_{\mathrm{i}} \end{gathered}$$

(39)

4. Computational Procedure and Results

The calculations require expressions for A_a and A_c. The following results are derived from Equations (1)–(3)

$${\mathrm{A}}_{\mathrm{c}}=-\frac{2{\mathrm{v}}^{\ast }}{{\mathrm{R}}^2\left[{\mathsf{\sigma }}^4+\left(1-{\mathsf{\sigma }}^4\right)\frac{{\mathsf{\mu }}_{\mathrm{c}}}{{\mathsf{\mu }}_{\mathrm{a}}}\right]}; {\mathrm{v}}^{\ast }=\frac{\mathrm{Q}}{\mathsf{\pi }{\mathrm{R}}^2}; {\mathrm{A}}_{\mathrm{a}}={\mathrm{A}}_{\mathrm{c}}\frac{{\mathsf{\mu }}_{\mathrm{c}}}{{\mathsf{\mu }}_{\mathrm{a}}}$$

(40)

Using the values σ = 0.9, H_T = 34.2%, previously used in [37] to study glucose transport from blood to tissue, yields: H_C = 42.2%, μ_c/μ_a = 1.947, H_D = 40.0%, and μ_app/μ_a = 1.469 using Equations (7) and (6) at T = 300 K, and (5) and (4) sequentially.

The expression for slope b can be written in a more convenient form as

$$\mathrm{b}=\frac{\left[{\left(\frac{{\mathrm{r}}_{\mathrm{T}}}{\mathrm{R}}\right)}^2-1\right]\frac{\mathsf{\Gamma }}{{\mathsf{\alpha }}_{\mathrm{a}}}}{\left(1-{\mathsf{\sigma }}^2\right){\overline{\mathrm{v}}}_{\mathrm{a}}+\left(1+\overline{\mathrm{m}}\right)\left[\frac{{\mathrm{A}}_{\mathrm{c}}{\mathrm{R}}^2}{2}{\mathsf{\sigma }}^4+\left({\mathrm{v}}_{\mathrm{i}}-{\mathrm{A}}_{\mathrm{c}}{\mathrm{R}}^2{\mathsf{\sigma }}^2\right){\mathsf{\sigma }}^2\right]}$$

(41)

where m_av is calculated using Equation (15) and an average value for ${\mathrm{p}}_{{\mathrm{O}}_2}$ taken as ${\mathrm{p}}_{{\mathrm{O}}_2,0}-\mathrm{b}\mathrm{L}/2$, which obviously requires simple trial and error calculations, by successive substitutions to get m_av. The expressions for ${\overline{\mathrm{v}}}_{\mathrm{a}}$ and v_i are obtained using Equation (1)

$${\overline{\mathrm{v}}}_{\mathrm{a}}=\frac{{\displaystyle {\int }_{{\mathrm{r}}_{\mathrm{i}}}^{\mathrm{R}}2\mathsf{\pi }}\mathrm{r}\mathrm{v}\mathrm{d}\mathrm{r}}{\mathsf{\pi }\left({\mathrm{R}}^2-{\mathrm{r}}_{\mathrm{i}}^2\right)}=\frac{-{\mathrm{A}}_{\mathrm{a}}{\mathrm{R}}^2}{2}\left(1-{\mathsf{\sigma }}^2\right); {\mathrm{v}}_{\mathrm{i}}=-{\mathrm{A}}_{\mathrm{a}}{\mathrm{R}}^2\left(1-{\mathsf{\sigma }}^2\right)$$

(42)

The profiles for the deviations from the pressure at the interface are given by the following expressions:

$$\begin{array}{cc}\hfill {\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{c}}(\mathrm{r},\mathrm{z})-{\mathrm{p}}_{{\mathrm{O}}_2}({\mathrm{r}}_{\mathrm{i}},\mathrm{z})=& {\mathsf{\phi }}_{\mathrm{c}}(\mathrm{r})-{\mathsf{\phi }}_{\mathrm{i}}=\hfill \\ & -\left(1+\overline{\mathrm{m}}\right)\left[\frac{1}{16}\frac{\mathrm{b}{\mathrm{A}}_{\mathrm{c}}{\mathrm{R}}^4}{{\mathrm{D}}_{\mathrm{c}}}\left({\mathsf{\xi }}^4-{\mathsf{\sigma }}^4\right)+\frac{1}{4}\frac{{\mathrm{v}}_{\mathrm{i}}{\mathrm{R}}^2\mathrm{b}}{{\mathrm{D}}_{\mathrm{c}}}\left({\mathsf{\xi }}^2-{\mathsf{\sigma }}^2\right)-\frac{1}{4}\frac{\mathrm{b}{\mathrm{A}}_{\mathrm{c}}{\mathrm{R}}^4}{{\mathrm{D}}_{\mathrm{c}}}\left({\mathsf{\xi }}^2-{\mathsf{\sigma }}^2\right)\right]\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill {\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{a}}(\mathrm{r},\mathrm{z})-{\mathrm{p}}_{{\mathrm{O}}_2}({\mathrm{r}}_{\mathrm{i}},\mathrm{z})=& {\mathsf{\phi }}_{\mathrm{a}}(\mathrm{r})-{\mathsf{\phi }}_{\mathrm{i}}=\hfill \\ & -\frac{\mathrm{b}{\mathrm{A}}_{\mathrm{a}}{\mathrm{R}}^4}{{\mathrm{D}}_{\mathrm{a}}}\left[\frac{1}{16}\left({\mathsf{\xi }}^4-{\mathsf{\sigma }}^4\right)-\frac{1}{4}\left({\mathsf{\xi }}^2-{\mathsf{\sigma }}^2\right)\right]+\mathrm{K}\ln \left(\frac{\mathsf{\xi }}{\mathsf{\sigma }}\right)\hfill \end{array}$$

(44)

$${\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{T}}(\mathrm{r},\mathrm{z})-{\mathrm{p}}_{{\mathrm{O}}_2}(\mathrm{R},\mathrm{z})={\mathsf{\phi }}_{\mathrm{T}}(\mathrm{r})-\mathsf{\phi }(\mathrm{R})=\frac{1}{4}\frac{\mathsf{\Gamma }\hspace{0.33em}{\mathrm{R}}^2}{{\mathsf{\alpha }}_{\mathrm{T}}{\mathrm{D}}_{\mathrm{T}}}\left[{\mathsf{\xi }}^2-1\right]-\frac{\mathsf{\Gamma }\hspace{0.33em}{\mathrm{R}}^2}{{\mathsf{\alpha }}_{\mathrm{T}}{\mathrm{D}}_{\mathrm{T}}}\frac{{\mathsf{\xi }}_{\mathrm{T}}^2}{2}\ln \left(\mathsf{\xi }\right)$$

(45)

where K is given by

$$\mathrm{K}=-\frac{1}{4}\frac{\mathrm{b}{\mathrm{A}}_{\mathrm{a}}{\mathrm{R}}^4}{{\mathrm{D}}_{\mathrm{a}}}+\frac{1}{2}\frac{\mathsf{\Gamma }{\mathrm{R}}^2}{{\mathsf{\alpha }}_{\mathrm{a}}{\mathrm{D}}_{\mathrm{a}}}\left(1-{\mathsf{\xi }}_{\mathrm{T}}^2\right)$$

(46)

where ξ_T = r_T / R. The value of ${\mathsf{\phi }}_{\mathrm{i}}$ is used to get the oxygen partial pressure at the two zones interface

$${\mathrm{p}}_{{\mathrm{O}}_2}({\mathrm{r}}_{\mathrm{i}},\mathrm{z})={\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{b}}(\mathrm{z})+{\mathsf{\phi }}_{\mathrm{i}}, {\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{b}}(\mathrm{z})={\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{b}}(0)-\mathrm{b}\mathrm{z}, {\mathsf{\phi }}_{\mathrm{i}}=4\frac{{\mathrm{f}}_{\mathsf{\chi }=\mathsf{\lambda }}-{\mathrm{f}}_{\mathsf{\chi }=1}}{{\left({\mathsf{\lambda }}^2-1\right)}^2}, \mathsf{\lambda }=\mathrm{R}/{\mathrm{r}}_{\mathrm{i}}$$

(47)

where f is given by

$$\begin{gathered} \mathrm{f}\left(\mathsf{\chi }\right)=-\frac{{\mathsf{\sigma }}^4}{16}\frac{\mathrm{b}{\mathrm{A}}_{\mathrm{a}}{\mathrm{R}}^4}{{\mathrm{D}}_{\mathrm{a}}}\left[\frac{{\mathsf{\chi }}^8}{8}-\frac{5{\mathsf{\lambda }}^2{\mathsf{\chi }}^6}{6}+\left(4{\mathsf{\lambda }}^4+4{\mathsf{\lambda }}^2-1\right)\frac{{\mathsf{\chi }}^4}{4}+\left(-4{\mathsf{\lambda }}^4+{\mathsf{\lambda }}^2\right)\frac{{\mathsf{\chi }}^2}{2}\right]+ \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{K}\left[\frac{{\mathsf{\chi }}^4}{4}\left(\ln \mathsf{\chi }-\frac{1}{4}\right)-{\mathsf{\lambda }}^2\frac{{\mathsf{\chi }}^2}{2}\left(\ln \mathsf{\chi }-\frac{1}{2}\right)\right] \end{gathered}$$

(48)

The case of metabolic consumption of oxygen in an islet of Langerhans is considered in [3]. The data, reported in [3], is a capillary blood flow rate of about 7 nL/min for a tissue perfusion rate q_b (capillary blood flow rate over Krogh tissue volume) and about 4 mL/cm³ min. The data used are listed in

Table 1. Data used for oxygen transport from blood flow in a capillary and comparison with (3).

Property	Symbol	Value	Reference
Da	Diffusion coefficient in plasma	2.18 × 10−5 cm2/s	[33]
DRBC	Diffusion coefficient in RBCs	9.5 × 10−6 cm2/s	[33]
DT	Diffusion coefficient in tissue	2.41 × 10−5 cm2/s	[33]
2 R	Capillary diameter	10 μm	Table 5.1 [3]
L	Length	1000 μm	Table 5.1 [3]
Γ	Rate of oxygen consumption in tissue	20 μM/s	Chapter 6 [3]
${\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{b}}(0)$	Average oxygen partial pressure in blood at z = 0	95 mm Hg	Chapter 6 [3]
Q	Volume flow rate of blood	7 nL/min	Chapter 6 [3]
αa	Oxygen solubility in plasma	1.110 μM O2/mm Hg (2.82 × 10−5 mL O2/cm3 mm Hg)	[33]
αT	Oxygen solubility in tissue	1.516 μM O2/mm Hg (3.85 × 10−5 mL O2/cm3 mm Hg)	[33]

included the relevant references.

The solubilities in Table 1 are calculated using the values in mL O₂/cm³ mm Hg and the O₂ molar volume at 36.9 °C in [33]. Using the data in Table 1, and the above–mentioned value of μ_c/μ_a = 1.947 obtained for σ = 0.9 and H_T = 34.2% (H_D = 40.0%), yields the results summarized in

Table 2. Results for oxygen transport using Table 1 for μ_c/μ_a = 1.947 and σ = 0.9. (HD = 40.0%).

Symbol	Value
v*	0.149 cm/s
rT	24 μm
AcR2	−0.224 cm/s
AaR2	−0.436 cm/s
vi	0.0829 cm/s
va	0.0415 cm/s
b	210.88 mm Hg/cm
K	−2.013 mmHg
Dc	1.60 × 10−5 cm2/s
bAaR4/Da	−1.06 mm Hg
bAcR4/Dc	−0.740 mm Hg
ΓR2/(αaDa)	0.207 mm Hg
ΓR2/(αTDT)	0.137 mm Hg
φi	0.0840 mm Hg
φa at R	−0.1555 mm Hg
φT at rT	−1.875 mm Hg
pO2,b at z = L	73.91 mm Hg
mav	12.33

. Using the perfusion rate of about 4 mL/cm³ min used in [3] and the Krogh tissue cylinder volume as a function of R, r_T and L, the outer radius of the Krogh cylinder r_T is found to be about 24 μm.

Using Equations (45) and (46) provides the values for φ (r = 0) and φ (R). Combining Equations (46) and Equation (45) (applied at r = R) gives the following expression for φ_T:

$$\begin{gathered} {\mathsf{\phi }}_{\mathrm{T}}(\mathrm{r})={\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{T}}(\mathrm{r},\mathrm{z})-{\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{b}}(\mathrm{z})={\mathsf{\phi }}_{\mathrm{i}}+\frac{1}{4}\frac{\mathsf{\Gamma }\hspace{0.33em}{\mathrm{R}}^2}{{\mathsf{\alpha }}_{\mathrm{T}}{\mathrm{D}}_{\mathrm{T}}}\left[{\mathsf{\xi }}^2-1\right]-\frac{\mathsf{\Gamma }\hspace{0.33em}{\mathrm{R}}^2}{{\mathsf{\alpha }}_{\mathrm{T}}{\mathrm{D}}_{\mathrm{T}}}\frac{{\mathsf{\xi }}_{\mathrm{T}}^2}{2}\ln \left(\mathsf{\xi }\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{\mathrm{b}{\mathrm{A}}_{\mathrm{a}}{\mathrm{R}}^4}{{\mathrm{D}}_{\mathrm{a}}}\left[\frac{1}{16}\left(1-{\mathsf{\sigma }}^4\right)-\frac{1}{4}\left(1-{\mathsf{\sigma }}^2\right)\right]+\mathrm{K}\ln \left(\frac{1}{\mathsf{\sigma }}\right) \end{gathered}$$

(49)

The results obtained in Table 2 (using the data in Table 1) are used to plot the oxygen partial pressures profiles in Figure 2. As expected, the oxygen partial pressure profiles show a drop in the z direction due to consumption of oxygen in the tissue. The average oxygen partial pressure in blood is higher than the oxygen partial pressure at the outer radius of the Krogh cylinder due to mass transfer resistance to oxygen transport from blood to tissue. The results compare favorably with those in [3], as seen from Figure 2.

The case of anoxia starts when the oxygen pressure at r_T is zero at z = L (critical case):

$$\begin{gathered} {\mathrm{p}}_{{\mathrm{O}}_2}({\mathrm{r}}_{\mathrm{T}})={\mathrm{p}}_{{\mathrm{O}}_2,\mathrm{b}}(0)-\mathrm{b}\mathrm{L}+{\mathsf{\phi }}_{\mathrm{i}}+\frac{1}{4}\frac{\mathsf{\Gamma }\hspace{0.33em}{\mathrm{R}}^2}{{\mathsf{\alpha }}_{\mathrm{T}}{\mathrm{D}}_{\mathrm{T}}}\left[{\mathsf{\xi }}_{\mathrm{T}}^2-1\right]-\frac{\mathsf{\Gamma }\hspace{0.33em}{\mathrm{R}}^2}{{\mathsf{\alpha }}_{\mathrm{T}}{\mathrm{D}}_{\mathrm{T}}}\frac{{\mathsf{\xi }}_{\mathrm{T}}^2}{2}\ln \left({\mathsf{\xi }}_{\mathrm{T}}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{\mathrm{b}{\mathrm{A}}_{\mathrm{a}}{\mathrm{R}}^4}{{\mathrm{D}}_{\mathrm{a}}}\left[\frac{1}{16}\left(1-{\mathsf{\sigma }}^4\right)-\frac{1}{4}\left(1-{\mathsf{\sigma }}^2\right)\right]+\mathrm{K}\ln \left(\frac{1}{\mathsf{\sigma }}\right)=0, {\mathsf{\xi }}_{\mathrm{T}}={\mathrm{r}}_{\mathrm{T}}/\mathrm{R} \end{gathered}$$

(50)

In vitro experiments providing the required data for comparison with the present model are recommended for future investigations.

5. Conclusions

Segregation of red blood cells in microvessels is accounted for in the present model on the basis of fundamentals of fluid flow and mass transfer and including metabolic consumption of oxygen in tissue, using Haynes marginal zone and Krogh cylinder concepts. The key role of hemoglobin is included in mass transfer of oxygen. In addition, velocity gradients in the cell-free zone and the core region inside the microvessel are accounted for in the transport of oxygen model. The final expressions are analytical. The results are found in good agreement with the results of Fournier [3] for oxygen transport from blood in capillary size microvessel to surrounding tissue. Although an average value of m was considered over all the capillary length as in [3], it is possible to subdivide the capillary length into small segments and use an average value for m over each interval for more accuracy. The model is not restricted to transport of oxygen from capillaries and includes transport of oxygen from blood in microvessels to tissue in general.