Home | About | Journals | Submit | Contact Us | Français |

**|**HHS Author Manuscripts**|**PMC5609725

Formats

Article sections

- Abstract
- 1. Introduction
- 2. Mathematical Model
- 3. Parameters of The Model
- 4. Method of Solution
- 5. Results And Discussion
- 6. Nomenclature
- References

Authors

Related links

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

Published in final edited form as:

PMCID: PMC5609725

NIHMSID: NIHMS878399

Aleksander S. Popel, Department of Mechanical Engineering, University of Houston, Houston, Texas 77004;

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

The effect of the surface oxygen tension on the oxygen-tension distribution in the tissue is considered. A mathematical formulation of the problem is presented, and the solutions are obtained numerically using a finite-element method. It is shown that the oxygen tension at the surface of the tissue may significantly affect the oxygen-tension distribution in layers of tissue situated within several intercapillary distances below the surface.

Measurements of tissue oxygen tension (*P*O_{2}) with oxygen microelectrodes are most valuable when combined with intravital microscopy, but are limited to the vicinity of the tissue surface superfused with a solution containing a certain percentage of oxygen [1–6]. Variation of the solution *P*O_{2} may cause variation of *P*O_{2} in the surface layers of the tissue accompanied by variation of the capillary velocities [2, 3, 5–7], the capillary hematocrit [7], the density of flowing capillaries [3, 8], and the inlet capillary *P*O_{2} [2]. Therefore, oxygen transport in surface layers of tissue is a complex process that may involve many factors. A careful analysis of these factors is needed in order to separate regulatory adjustments of the microcirculation from “passive” effects of diffusion and convection. Obviously, the Krogh cylinder model [9], widely used for estimates of oxygen delivery in deep layers of tissue, generally cannot be applied to the surface layers. In the present paper a mathematical model of oxygen transport in the surface layers of tissue is formulated, and the effect of the surface *P*O_{2} on the tissue *P*O_{2} is analyzed while such variables as capillary velocities, capillary hematocrit, intercapillary distances, and oxygen consumption rate remain constant. In subsequent papers the effect of variation of these parameters on the oxygen transport in surface layers of tissue will be examined.

Consider a slab of tissue bounded by plane surfaces − ∞ < *x* < ∞, 0 ≤ *y* ≤ *H*, 0 ≤ *z* ≤ *L*, as shown in Figure 1. The capillaries are parallel to the surface of the tissue and form a square lattice. The intercapillary distances equal 2*d*, and the distance between the tissue surface and the row of capillaries closest to the surface is *d*_{1}.

Steady-state oxygen transport in the tissue is governed by the diffusion equation

$$D\alpha {\nabla}^{2}P-M=0,$$

(1)

where *D* is the oxygen diffusion coefficient, *α* is the solubility coefficient, *P* is the tissue oxygen tension, and *M* is the oxygen consumption rate per unit volume of tissue.

It has been shown that the oxygen transport equation for the capillary lumen can be written in the form [10, 11]

$$\begin{array}{cc}\pi {r}^{2}{\upsilon}_{i}\frac{\partial}{\partial z}[{\alpha}_{b}{P}_{i}+{C}_{i}\psi ({P}_{i})]=-{J}_{i},& i=1,2,\dots ,\end{array}$$

(2)

where the left-hand side is a derivative of the flux of oxygen through a cross-section of the capillary (see Appendix). In Equation (2)
*r* is the capillary radius, *υ _{i}* is the velocity of blood in the

The oxygen tensions *P*(*x, y*, *z*) and *P _{i}*(

$$\begin{array}{ccc}P={P}_{s}& \text{at}& y=0\end{array}.$$

(3)

At the lower tissue surface the no-flux boundary condition is imposed:

$$\begin{array}{ccc}\frac{\partial P}{\partial y}=0& \text{at}& y=H\end{array}.$$

(4)

At the capillary-tissue interface the oxygen tension is continuous, and the oxygen flux from the capillary, *J*, is equal to the total oxygen flux into the tissue:

$$\begin{array}{ccc}P={P}_{i},& {J}_{i}=-D\alpha {\int}_{{\Gamma}_{i}}\frac{\partial P}{\partial n}d{\Gamma}_{i}& \text{on}{\Gamma}_{i}\end{array}$$

(5)

where is Γ* _{i}* is the intersection of the

$$\begin{array}{ccc}{P}_{i}={P}_{ai}& \text{at}& z=0\end{array}.$$

(6)

It is realized that Equation (2) might be an oversimplification in view of recent analyses by Hellums [12], who pointed to a potential intracapillary radial resistance to diffusion of oxygen, and by Fletcher [13], who took into account nonlinear kinetics of oxygen binding by hemoglobin. Both factors could be incorporated into the theory; however, it is felt at this point that excessive details might cloud the main issue under consideration, i.e. the effect of the tissue surface *P*O_{2}.

Replacement of the boundary condition (3) above with

$$\begin{array}{ccc}\frac{\partial P}{\partial y}=0& \text{at}& y=0\end{array}$$

corresponds to a sealed tissue surface. This boundary condition will be used to compare oxygen distributions in the case of a sealed boundary and in the case of specified *P*O_{2} at the surface.

Assuming that capillary oxygen distributions in horizontal rows are identical, the spatial domain can be reduced to 0 ≤ *x* ≤ *d*, 0 ≤ *y* ≤ *H*, 0 ≤ *z* ≤ *L* (Figure 1, darkened area), with the no-flux boundary condition, *P*/*x* = 0, on the tissue portion of the planes *x* = 0 and *x* = *d*.

If all second derivatives in the Laplace operator in (1) are of importance, the described mathematical formulation is incomplete without some kind of boundary conditions on the tissue portion of the planes *z* = 0 and *z* = *L*. In the present work the so-called axial diffusion and, associated with it, the second derivative in (1) in the direction parallel to the capillary axes are neglected (the effect of axial diffusion is discussed in [14, 15]); thus Equation (1) reduces to the form

$$D\alpha (\frac{{\partial}^{2}P}{\partial {x}^{2}}+\frac{{\partial}^{2}P}{\partial {y}^{2}})-M=0.$$

(7)

In this case the mathematical formulation of the problem is complete and no additional boundary conditions are required.

This study is not aimed at a quantitative description of the *P*O_{2} distribution in any particular tissue under realistic experimental conditions, but rather at an investigation of the effect of surface *P*O_{2} variation under the condition that other parameters remain constant. Such a consideration is hardly applicable to physiological situations, since it is well known that in reality a change of the surface *P*O_{2} induces changes in such parameters as capillary velocity, capillary hematocrit, intercapillary distances, etc. However, this idealized case is a necessary step toward more detailed modeling.

The following parameters were chosen for simulation: *D* = 1.6 × 10^{−5} cm^{2}/sec; *α* = 3 × 10^{−5} (cm^{3} O_{2})/(cm^{3} tissue)(mmHg); *α _{b}* = 2.9 × 10

$$\psi (p)=hp\frac{{(1+hP)}^{3}+m{(hP)}^{3}}{{(1+hP)}^{4}+m{(hP)}^{4}}$$

(8)

with *h* = 0.01278, *m* = 124 [17].

Equations (7) and (2) were rewritten in the form of a single parabolic nonlinear partial differential equation

$$U(S)\frac{\partial S}{\partial z}=\lambda (\frac{{\partial}^{2}S}{\partial {x}^{2}}+\frac{{\partial}^{2}S}{\partial {y}^{2}})-Q(S),$$

(9)

where

$$S=\{\begin{array}{cc}{P}_{i}& \text{in the}i\text{th capillary},\\ P& \text{in the tissue},\end{array}$$

$$U=\{\begin{array}{cc}\pi {r}^{2}{\upsilon}_{i}{\alpha}_{b}(1+{C}_{i}{\alpha}_{b}^{-1}d\psi /d{P}_{i})& \text{in the}i\text{th capillary},\\ 0& \text{in the tissue},\end{array}$$

$$\lambda =\{\begin{array}{cc}{\alpha}_{b}{D}_{b}& \text{in the capillaries},\\ \alpha D& \text{in the tissue},\end{array}$$

$$Q=\{\begin{array}{cc}0& \text{in the capillaries},\\ M& \text{in the tissue}.\end{array}$$

Here *D _{b}* is the oxygen diffusion coefficient in blood. It has been shown that the results are practically independent of

The previously formulated boundary conditions can be rewritten in the form

$$\begin{array}{ccc}S={P}_{s}& \text{at}& y=0,\end{array}$$

(10)

$$\begin{array}{ccccc}\frac{\partial S}{\partial x}=0& \text{at}& x=0& \text{and}& x=d,\end{array}$$

(11)

$$\begin{array}{ccc}\frac{\partial S}{\partial y}=0& \text{at}& y=H\end{array}.$$

(12)

The distribution of *S* at *z* = 0 (“initial” condition) was determined by solving Equation (7) at *z* = 0.

Equation (9) was solved numerically using a computer program IMSL TWODEPEP implementing a finite-element scheme with triangular elements. The numerical scheme was tested on a number of geometries, including the simplest case of identical capillaries forming a square lattice as well as more complex geometries studied by the author [19, 16]. Figure 2 shows a comparison of intracapillary *P*O_{2} obtained by application of the finite element scheme to equation (9) with boundary conditions (11), (12) and *S*/*y* = 0 at *y* = 0 corresponding to a sealed tissue surface, with the “exact” solution governed by the conservation-of-mass relationship (FicK's principle):

$$\pi {r}^{2}\upsilon \frac{d}{dz}[{\alpha}_{b}P+C\psi (P)]=-\pi ({R}^{2}-{r}^{2}){M}_{0},$$

(13)

$$P(0)={P}_{a},$$

(14)

where *R* = 2*d*/√*π* is the effective “Krogh cylinder” radius. In all cases considered to test the numerical scheme, the discrepancy between the solutions obtained with the finite-element technique and the solutions of (13) (or, for more complex geometries, solutions obtained by the author [16, 18]) did not exceed 1 mmHg.

It will be seen now how the diffusive interaction between the surface and the tissue affects the oxygen distribution in the surface layers of tissue. For convenience the results are presented in terms of dimensionless coordinates

$$\begin{array}{ccc}x\ast =x/d,& y\ast =y/d,& z\ast =z/L.\end{array}$$

(15)

Figure 3 demonstrates variation of intracapillary oxygen tensions along the capillary length for *P _{s}* = 40 mmHg. The dashed line depicts the intracapillary

Distribution of intracapillary oxygen tensions *P*_{1–4} when *P*_{s} = 40 mmHg; *P*_{K} is the intracapillary oxygen tension when the tissue surface is sealed.

Panels 1–3: oxygen-tension distribution in the planes *z** = 0, 0.5, and 1, respectively, for *P*_{s} = 40 mmHg. Panel 4: oxygen-tension distribution in the case of zero capillary flow. Numbers on the isobars designate oxygen tension in mmHg.

$$P=\{\begin{array}{ccc}{P}_{s}{(1-\frac{y}{l})}^{2}& \text{for}& 0\le y\le l,\\ 0& \text{for}& l\le y\le H,\end{array}$$

(16)

where
$l={(2D\alpha {P}_{s}{M}_{0}^{-1})}^{1/2}$ is the penetration depth. Comparison of the last two panels shows that even though the gradients between the capillary closest to the surface and the surrounding tissue are very small, the *P*O_{2} distribution near the surface is significantly different from the distribution at zero blood flow.

Figures 7–9 show intracapillary and tissue *P*O_{2} distributions in the case of a lower surface *P*O_{2} (20 mmHg). In this case intracapillary oxygen tensions in all four capillary layers, *P*_{1}–*P*_{4}, are below the values of *P _{K}*. Both Figures 8 and and99 demonstrate that the issue

Intracapillary oxygen-tension distribution for *P*_{s} = 20 mmHg (*P*_{1}_{–4}) and for a sealed tissue (*P*_{K}).

*P*O_{2} distributions in the tissue were studied within a wide range of variation of the surface *P*O_{2}: 10 to 80 mmHg. In Figure 10 the intracapillary oxygen tensions at the venous end of the capillaries are plotted for different values of *P _{s}*, together with the value of

Oxygen tension at different depths (*y** = 1–6) for *x** = 0, *z** = 1 versus *P*_{s} (solid lines—with capillary flow; dashed lines—no capillary flow).

The results obtained in the present work generally indicate that variation of oxygen tension at the surface of the tissue can significantly alter the oxygen distribution in the surface layers of tissue. Therefore, the use of an appropriate mathematical model, applicable to this geometry, is required for correct interpretation of experimental data on oxygen distributions obtained in the surface layers of exposed tissue.

C | oxygen-binding capacity of blood |

D | oxygen diffusion coefficient |

H | thickness of the tissue |

H_{t} | tube hematocrit |

J | capillary-tissue diffusive flux |

L | length of the capillary |

l | penetration depth |

M | oxygen consumption rate per unit volume of tissue |

P | oxygen tension |

r | capillary radius |

R | Krogh cylinder radius |

υ | velocity of blood |

υ_{pl} | mean velocity of plasma |

υ_{c} | mean velocity of red blood cells |

x, y, z | dimensional coordinates |

x*, y*, z* | dimensionless coordinates |

GREEK SYMBOLS | |

α | oxygen solubility coefficient |

ψ | oxygen saturation fraction of hemoglobin |

SUBSCRIPTS | |

a | pertaining to arterial end of the capillaries |

b | pertaining to blood |

Hb | pertaining to hemoglobin solution |

i = 1, 2, 3, 4 | pertaining to capillaries |

K | pertaining to “Krogh-like” geometry |

pl | pertaining to plasma |

s | pertaining to the surface of the tissue |

The work was supported by the National Institutes of Health Grants HL-23362 and HL-17421.

The author wishes to thank Dr. Bruce M. Klitzman for valuable discussions and comments.

If the hemoglobin solution inside red blood cells and the surrounding plasma are in local thermodynamic equilibrium, and therefore a mixing-cup intracapillary oxygen tension *P* can be introduced, then the oxygen flux through any cross-section of the capillary can be determined from the expression

$$F=\pi {r}^{2}[{\upsilon}_{\text{pl}}(1-H){\alpha}_{\text{pl}}P+{\upsilon}_{\mathrm{c}}{H}_{t}{\alpha}_{\text{Hb}}P+{\upsilon}_{\mathrm{c}}{H}_{t}{C}_{0}\psi (P)]$$

(A1)

where *P* is the mixing-cup oxygen tension in the capillary, *υ*_{pl} is the mean velocity of plasma, such that *πr*^{2}*υ*_{p1},(1 — *H _{t}*) is the volume flux of plasma,

Using the relationships between the mean velocity of blood, *υ*, the discharge hematocrit *H _{D}* [20], and the variables in (Al), we have

$$\upsilon ={\upsilon}_{\text{pl}}(1-{H}_{t})+{\upsilon}_{c}{H}_{t},$$

(A2)

$$\upsilon {H}_{D}={\upsilon}_{c}{H}_{t},$$

(A3)

from which follows

$$\upsilon (1-{H}_{D})={\upsilon}_{\text{pl}}(1-{H}_{t}).$$

(A5)

The relationship (A1) can be rewritten in the form

$$F=\pi {r}^{2}\upsilon [{\alpha}_{b}P+C\psi (P)],$$

(A5)

where *C* = *C*_{0}*H _{D}*, and

$${\alpha}_{b}={\alpha}_{\text{pl}}(1-{H}_{D})+{\alpha}_{\text{Hb}}{H}_{D}.$$

(A6)

This implies

$$min({\alpha}_{\text{pl}},{\alpha}_{\text{Hb}})\le {\alpha}_{b}\le max({\alpha}_{\text{pl}},{\alpha}_{\text{Hb}}).$$

(A7)

1. Duling BR, Berne RM. Longitudinal gradients in periarteriolar oxygen tension. A possible mechanism for the participation of oxygen in local regulation of blood flow. Circ Res. 1970;27:669–678. [PubMed]

2. Duling BR. Microvascular responses to alternations in oxygen tension. Circ Res. 1972;31:481–489. [PubMed]

3. Prewitt RL, Johnson PC. The effect of oxygen on arteriolar red cell velocity and capillary density in the rat cremaster muscle. Microvas Res. 1976;12:59–70. [PubMed]

4. Klabunde RE, Johnson PC. Capillary velocity and tissue *P*O_{2} changes during reactive hyperemia in skeletal muscle. Am J Physiol. 1977;233:H379–H383. [PubMed]

5. Tuma RF, Lindbom L, Arfors KE. Dependence of reactive hyperemia in skeletal muscle on oxygen tension. Am J Physiol. 1977;233:H289–294. [PubMed]

6. Gorczynski RJ, Duling BR. Role of oxygen in arteriolar functional vasodilation in hamster striated muscle. Am J Physiol. 1978;235:H505–H515. [PubMed]

7. Klitzman BM, Duling BR. Microvascular hematocrit and red cell flow in resting and contracting striated muscle. Am J Physiol. 1979;237:H481–H490. [PubMed]

8. Klitzman BM. PhD Dissertation. Univ. of Virginia; Charlottesville: 1979. Microvascular determinants of oxygen supply in resting and contracting striated muscle. University Microfilms #8002544.

9. Krogh A. The number and the distribution of capillaries in muscle with the calculation of the oxygen pressure head necessary for supplying the tissue. J Physiol (Lond) 1918/19;52:409–515. [PubMed]

10. Hyman WA. A simplified model of the oxygen supply function of capillary blood flow. Advances Exp Med Biol. 1973;37b:835–841. [PubMed]

11. Stewart RR, Morrazzi CA. Oxygen transport in the human brain— analytical solutions. Advances Exp Med Biol. 1973;37b:843–848. [PubMed]

12. Heliums JD. The resistance to oxygen transport in the capillaries relative to that in the surrounding tissue. Microvas Res. 1977;13:131–136. [PubMed]

13. Fletcher JE. Mathematical modeling of the microcirculation. Math Biosci. 1978;38:159–202.

14. Grunewald WA, Sowa W. Capillary structures and O_{2} supply to tissue. An analysis with digital diffusion model as applied to the skeletal muscle. Rev Physiol Biochem Pharmacol. 1977;77:149–209. [PubMed]

15. Salathé EP, Wang TC, Gross JF. Mathematical analysis of oxygen transport to tissue. Math Biosci. 1980;51:89–115.

16. Popel AS. Oxygen diffusion from capillary layers with concurrent flow. Math Biosci. 1980;50:171–193.

17. Colton CK, Drake RF. Effect of boundary conditions on oxygen transport to blood flowing in a tube. Chem Eng Progr Symp Ser. 1971;67:96–104.

18. Popel AS. Mathematical modeling of convective and diffusive transport in the microcirculation. In: Gross JF, Popel AS, editors. Mathematics of Microcirculation Phenomena. Raven Press; New York: 1980. pp. 63–88.

19. Popel AS. Analysis of capillary-tissue diffusion in multicapillary systems. Math Biosci. 1978;39:187–211.

20. Cokelet GR. Macroscopic rheology and tube flow of human blood. In: Grayson J, Zingg W, editors. Microcirculation 2. Plenum; New York: 1976. pp. 9–32.

PubMed Central Canada is a service of the Canadian Institutes of Health Research (CIHR) working in partnership with the National Research Council's national science library in cooperation with the National Center for Biotechnology Information at the U.S. National Library of Medicine(NCBI/NLM). It includes content provided to the PubMed Central International archive by participating publishers. |