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16 Polarographic Oxygen Tension Measurements in Microstructures of the Living Tissue

Downloaded by UNIV OF IOWA on September 2, 2016 | http://pubs.acs.org Publication Date: June 1, 1973 | doi: 10.1021/ba-1973-0118.ch016

A Digital Computer Study on the Oxygen Tension Histogram H E R M A N N METZGER Department of Physiology, Johannes Gutenberg-University, Mainz, West Germany

An alternative to the Krogh capillary-tissue model for describing oxygen tension distribution in the microstructure of the organism is developed. The new three-dimensional network model covers the inhomogeneities in capillary blood flow in a relation of 27:1. Oxygen tension distribution (steady-state case) of the capillary network is compared with that of the con- and counter-current flow systems. This study was performed to obtain information about the dependence of the oxygen tension frequency distribution pattern on capillary arrangement. Theoretical oxygen tension frequency distributions are compared with experimental results from rat cortex. The frequently observed low oxygen tension values are probably caused by low flow velocities in some branches of the capillary network.

Qome typical concepts that are well known i n chemical engineering ^ systems theory can be applied to elucidate processes i n biological systems. Biological systems are characterized b y several transport phe nomena such as diffusion, convection, and chemical reactions which occur in capillaries and tissue. In this study a simulation of such processes for various arrangements of capillaries was performed to attain a better understanding of oxygen supply conditions i n the brain cortex. Oxygen tension values were calcu lated for steady-state conditions by numerical solution of partial differen tial equations for different capillary-tissue systems. The analysis helps to explain the experimental results obtained from the microstructure of rat 328 Reneau; Chemical Engineering in Medicine Advances in Chemistry; American Chemical Society: Washington, DC, 1973.

16.

METZGER

Oxygen Tension in Mycrostructures

329

or cat brain cortex with small oxygen microsensors. In particular, an explanation has been sought for the frequently observed low oxygen tension values.


Geometrical Model of Capillary-Tissue

System

The first mathematical treatment of oxygen exchange between capil laries and tissue was made by Krogh and Erlang ( 1 ). They assumed a parallel, equidistant arrangement for capillaries in which blood flows concurrently at a constant velocity while oxygen diffuses into the tissue and is consumed. This geometric arrangement has been the basis for numerous theoretical calculations of oxygen tension distribution values in the blood stream and tissue (2,3,4,5). T o simulate the histological complexity of capillary-tissue arrange ments i n a better way and to develop an alternative concept to the Krogh model, a three-dimensional network system is suggested here, providing for differences in directions and flow velocities within the branches of the network (Figure 1). A new inhomogeneously perfused tissue model is considered for a tissue cube with 3 χ 3 χ 3 capillaries. Arterial inputs and venous outputs are located at opposite corners of the tissue cube. As a simplify ing condition, it is assumed that the hydrodynamic resistances of all the capillaries are equal. Furthermore, KirchofFs laws are assumed to be valid, and capillary blood flow rates and directions can be calculated theoretically. For symmetry, the analyzed tissue cube can be divided into 24 parts, where only one part, a tetrahedron, had to be considered i n the final analysis (thick fines at the left-hand corner, lower part). Blood flow directions at the inputs and outputs as well as in the tetrahedron are marked by arrows. This study is an extension of a series of papers for the two-dimen sional case (6,7,8). Results from the three-dimensional case approximate the physiological system more accurately and compare better with experi mental results. The Oxygen Tension Histogram

Concept

Using small oxygen tension microsensors (tip diameter = 1 /mi) enabled oxygen tension values to be measured in the microstructure of different organs. As the small oxygen sensor is moved forward from the surface into deeper layers of the tissue, tremendous oxygen tension dif ferences are encountered between capillaries and the oxygen consuming structures i n the cell (9,10,11).

Reneau; Chemical Engineering in Medicine Advances in Chemistry; American Chemical Society: Washington, DC, 1973.


330

C H E M I C A L ENGINEERING I N M E D I C I N E

Figure 1.

CapiUary network system

From any oxygen sensor experiment i n the animal tissue, a high data output was obtained, and hundreds of oxygen tension values were regis tered. To understand better the results, the frequency distribution of oxygen tension values (histograms) from a small tissue volume has been illustrated by numerous authors (e.g., 12,13,14,15). Steady-state condi tions of oxygen supply were usually assumed, resulting in an almost constant oxygen tension distribution independent of time (for method see Figure 2 ). The oxygen tension profiles measured with oxygen sensors were analyzed; a classification of 1 m m H g was chosen. Results are from differ ent experiments i n a small volume of the occipital cortex of the anesthe tized albino rat: E x p A : 898 measuring points. E x p B : 904 measuring points. E x p C : 840 measuring points. E x p D : 374 measuring points [according to ( 1 5 ) ] . Questions arise as to what extent the experimental steady-state histo grams depend upon the different physiological parameters—e.g., as oxy gen consumption rate, local blood flow velocity, capillary length and diameter, and capillary arrangement. Here we make a first step toward providing an overall agreement between experimental and theoretical results. Oxygen tension values were calculated for different capillary-tissue arrangements, and the results were illustrated by histograms. A systematic investigation was performed by varying various physiological parameters and observing the histogram pattern.



16.

METZGER


331

ο ι

Figure 2. Typical oxygen tension profiles in the occipital rat cortex registered with an oxygen sensor of about 1 μπι in diameter (according to (11) and (15)) (upper part). Steps of the micromanipulator are 10 μm (marked by thin lines). Frequency of oxygen tension values (H) (lower part).


332


Derivation

of Capillary-Tissue

Equations

In developing equations to describe oxygen transport in blood and tissue, several assumptions were used. The most important are sum marized as follows:


List of Assumptions 1. Oxygen is transported i n tissue by diffusion only and in capillaries by convection only. 2. Chemical reactions involving oxygen are assumed to take place at steady-state conditions and are described by H i l l s law in the capillaries and by Michaelis and Menten's law i n the tissue. 3. Oxygen consumption i n the tissue is assumed to be homogeneous. 4. F l o w velocity in the capillaries is calculated according to K i r c h hofFs laws and is constant over the capillary cross section. 5. The diffusion resistance of the capillary wall is small compared with that of the tissue and can be neglected. 6. Oxygen concentration is continuous at the capillary-tissue inter face, and transport across the interface can be described by Fick's first law. 7. The analyzed tissue area continues in the same way to infinity; the considered tissue volume is representative for the whole organ. 8. The differences of blood flow rate within the different capillaries are 27:11:4:3:1. A mathematical description of the capillary-tissue model, based on assumptions 1-8, is possible and can be described by two coupled non linear differential equations. Derivation of the Equations. The mass balance for oxygen i n the tissue and capillaries is given by the equation of continuity for an ele ment with volume V and surface F :

(1)

where V is the tissue volume, Q(c)

is the oxygen consumption i n terms

of a chemical reaction per unit time and volume, and / is the oxygen flow through the surface element d F . The first integral describes the temporal change of the oxygen quantity i n volume V . The second integral sum marizes the convection of oxygen at the volume surface F . The third integral represents the oxygen consumption in V . B y applying Gauss's theorem and using differentiation according to the volume, we obtain the following partial differential equation


16.

METZGER

333

Oxygen Tension in Mycrostructures -

f

t

= d i v j + Q(c)

(2)

This differential equation is valid for both capillaries and tissue, but with different values. In the capillaries, Q(c) = 0; for the tissue we obtain


Q(0 = A ^

(3)

( A — maximal respiration volume, Κ = kinetic rate constant). Only steady-state conditions are analyzed (dc/dt = 0). Furthermore, the validity of Fick's first law is assumed. j

=

— D grad c

(4)

( D is the diffusion coefficient and is independent of position i n the volume element. ) W e get

Equation 5 w i l l be transformed into the dimensionless form

If

+ %

~

+

ciTTc -

A G

0

(6)

with AG = AV/Dc

a

and CH = K/c

(7)

a

(Partial pressure values are calculated by dividing the concentration c by the solubility coefficient a). To obtain a differential equation for c? in the capillaries, we refer to Equation 1. For a stationary case and Q(c) = 0, we get

f

jdF

= 0

(8)

F The integration is arrived at by a surface element of thickness ds: Qv[c'(s + ds) -

c'(s)] -

dXdsD

+ (9)

\dn )u

\dn )r

\dn /1_


334


This equation is already adapted to our three-dimensional model. In cluded are the additional assumptions that the capillary cross section Q is a square one (Q = cP) and that c' can only change within the capillary in the capillary directions. The total concentration i n the capillary interior d is composed of a few physically dissolved molecules (c) and chemically bounded molecules

c

' =

1

3

m

4

x

m

c

( 1 0 )

with s , the saturation of oxyhemoglobin and c b, the hemoglobin concen tration. The saturation of hemoglobin depends on the oxygen concentra tion or tension ρ i n the plasma and can be described b y Hill's law H

T


+

Kyy 1+Kpy

st_

100

K

J

(K and γ are constants which are calculated from the actual dissociation curve.) Furthermore, / = c'v was placed i n the capillary and Fick's first law applied to the areas bordering the tissue, the magnitude of which d X ds ν is the velocity of blood i n the different capillaries, which is calculated in the inhomogeneous model according to KirchofFs laws. In the con- and countercurrent systems, the capillaries are perfused at the same velocity. The four differential quotients (dc/dn) at the areas bordering the tissue are taken i n their normal directions. After dividing by d X ds and after ds -> o, we obtain:

S = I f ê ) „ fê). (Î), (£).] +

+

+

2 1

with the expression

™ - jrrhv

(13)

The constants AK and K' are defined as AK

,S V α10

= jdv 1.34 c b γΚ'

(14)

H

and (15) (p is equal to the arterial oxygen tension. ) a


16.

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335


Equations 5 and 12 are two coupled nonlinear differential equations that completely describe the oxygen concentration i n the three-dimen sional model, provided that the influence of the capillary membrane as a diffusion barrier can be neglected. Boundary Conditions. The equations for the blood-tissue interface are


C/ blood =

(16)

C/ tissue

7 =èV

dc drti

k / blood

W

àU I tissue

The initial capillary point (arterial input) is = 1.0

c

a

(18)

A t the boundaries i n the model, it is assumed that the oxygen tension field continues i n the same way to infinity with the boundaries as sym metry lines. If, for example, the concentration of the boundary point with the coordinates x , y , z is to be calculated, concentrations at the points with the coordinates x , y — h, z or x , y , z — h had to be con sidered. The following equations are valid 0

0

0

0

c(x o,

y ο -

c(x ο,

y

h,

ο, ζ ο

-

0

0

0

ζ ο) = c(x ο, = c(x ,

h)

y

0

+ Λ, ζ ) 0

0

y ο,

0

0

ζο

+

h)

(10) (20)

Similar equations have been derived for a l l the points at the boundaries of the same tissue block as well as the tetrahedron (Figure 3 ) . ( A repre sentative tissue and a capillary point with the corresponding neighboring points necessary for calculation are marked i n Figure 3. Only the twodimensional case of the problem is shown to demonstrate the principle of calculation at the boundary Unes. ) Special Equations. A t special points of the network system a mixing of blood from two or more capillaries occurs and can be described b y the equation 6

Σ

v=l

6

c» i» = c

Σ

v=l

*v

(21)

with c the concentration of the v capillary with a blood flow rate t„ and the mean concentration c at the mixing point. v

th



336


Figure 3. Basis area of the tetrahedron (thick lines: capillaries; thin lines: mesh system for calculation; arrows: blood flow input and output)

I V

The values i are calculated according to Kirchhoffs laws. A t the nodes the equation v

6

Σ

v=l

tv = 0

(22)

is valid. F o r each mesh we arrive with the equation 4

Σ *V

=

0

() 23

μ=1 The hydrodynamic resistances Wn of the capillaries are assumed to be equal. Transformation into Difference Form. T h e differential equations have been transformed into difference form and solved iteratively. T h e


16.

METZGER

337


function c(x,y,z) was developed into a Taylors series around the point Xo> y] X l

[c(i,m + 1, nY "» + c(l,m - l , n ) v


c(Z,m,n - 1 )

(,,)

( y )

+ c(Z,m,n + l ) * " " " +

(24)

- 4c(Z,ra,n) ] (,,)

Z,m,n are coordinates i n capillary direction and perpendicular to the quadratic sides; ν is the iteration number. (b) Tissue equation: c(i,i,*) "

Polarographic Oxygen Tension Measurements in ... - ACS Publications

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