The influence of space-distributed parameters on the calculation of substrate and gas exchange in microvascular units

The Influence of Space-Distributed Parameters on the Calculation of Substrate and Gas Exchange in Microvascular

Units

HERMANN METZGER Department of Physiology, Medical School Hannover, 3 Hannover, Karl- Wiechert-Allee 9, West Germany Communicated

by W. Reichardt

ABSTRACT Capillary-tissue exchange processes are mathematically analyzed as space-distributed parameter systems: capillary flow and the consumption rate of the tissue are introduced as space-dependent for the calculation. Presentation of the results as orthogonal isoconcentration and flux lines as well as histogram plots of the concentration values is discussed, with special regard to the oxygen transport to brain and kidney tissue. It is concluded that the space dependence of physiological parameters has a tremendous influence on the calculation of substrate and gas-exchange processes and has to be considered in all mathematical investigations of microcirculatory units.

INTRODUCTION Substrate and gas-exchange processes between capillary blood and the metabolic active tissue of different organs such as brain cortex or kidney cortex have been mathematically dealt with on the basis of homogeneity of the various physiological and morphological parameters. Parallel and equidistant capillaries perfused with the same flow velocity and direction of the capillary blood as well as tissue areas of equal and space-independent consumption rates of oxygen and glucose molecules are characteristic assumptions of the calculations performed by many authors [l-5]. In order to simulate capillary-tissue exchange processes in a more realistic way, systems with space-dependent parameters (inhomogeneous systems) are here analyzed. For purposes of comparison with the homogeneous capillary-tissue systems of substrate and gas exchange, the models have to be deterministic. Extended versions of previously reported capillary-tissue arrangements are used for calculation [6-lo]. The analysis is performed to prove the influence of spatial dependence of capillary flow and consumpMATHEMATICAL

0 American

Elsevier

BIOSCIENCES Publishing

30, 3145

Company,

(1976)

Inc., 1976

31

HERMANN

32

METZGER

tion rate on substrate and gas-concentration distributions in capillaries and tissue. Well-known parameters and results from the physiological literature have been used to simulate the exchange processes in brain and kidney tissue as closely as possible. Despite this, a number of assumptions have had to be made for the appropriate representation of physiological results which have been published as relative values or in a form which was not adequate for the model calculations presented in this paper. Nevertheless, the assumptions are reasonable and can be replaced in the computer programs if more accurate experimental results become available. In the following, microvascular units for brain and kidney cortex as well as cell arrangements are extracted from the histological literature. Mathematical equations describing substrate and gas-exchange processes will be derived; the equations are presented for oxygen transport to tissue only, but similar equations can be derived for carbon dioxide and glucose as well as electrolyte transport [ 1l-121. MORPHOLOGICAL EVIDENCE FOR SPATIAL DEPENDENCE OF PARAMETERS Homogeneity of capillary flow is one of the most important assumptions for the calculation of capillary-tissue exchange processes, as the shape of the microvascular unit is coaxial and can be treated mathematically by introducing cylindrical coordinates. A second important assumption for all model calculations is homogeneity (position independence) of consumption rates of oxygen and glucose molecules within the tissue. Both assumptions are analyzed, and their influence on predictions of oxygen concentration values in capillaries and microareas of the tissue is investigated. Brain cortex and kidney cortex are considered as typical examples for substrate exchange processes within microvascular units. SPACE

DISTRIBUTION

OF CAPILLARY

FL0 W IN THE KIDNEY

CORTEX

Histological investigations of kidney cortex have demonstrated a capillary network system existing around the proximal and distal tubulus. Capillaries form a branched mesh system with differences in flow velocity as well as in flow direction. The basic microvascular unit of the kidney consists of branched capillaries with distributed, input points for the arterial blood. Each capillary network surrounds at least one tubulus in order to guarantee substrate exchange on a minimal diffusion path. SPACE

DISTRIBUTION

OF CONSUMPTION

RATE

IN THE BRAIN

CORTEX

Like the kidney cortex, the brain cortical tissue has a high metabolic rate for aerobic glycolysis. As in kidney cortex, capillary mesh systems are the

SPACE-DISTRIBUTED

PARAMETER

SYSTEMS

33

dominant architecture in many areas of the cortex. Capillaries surrounding cells in the form of single loops provide for optimal functioning of the cells [13]. Recently, it was pointed out that a distribution of cell density is to be observed within the rat brain cortex: a decrease of cell density exists between the arterial and the venous end of the capillary mesh system [14]. Consequently, a decrease of consumption rate of oxygen and glucose molecules has to be assumed for theoretical analysis. DEVELOPMENT MODELS

OF

SPATIALLY-DEPENDENT-PARAMETER

For the derivation of deterministic models taking into account distribution of capillary flow as well as the tissue consumption propriate mathematical models are described below. Substrate concentration distributions are analyzed for these systems. SPATIALLY

DEPENDENT

CAPILLARY-FLOW

the space rate, apand gas-

MODEL

For the presented mathematical analysis histological capillary-tissue systems had to be transformed into an idealized form suitable for the organ under consideration. In developing an appropriate number of capillaries per unit tissue volume for the system, the number of capillary input and output points and the type of branching of the capillaries have to be defined. In order to obtain a microvascular unit which is representative for a large part of the brain or kidney, it is helpful to introduce the condition of symmetry of the arrangement of arterial sources and venous sinks of the supplying blood. The development of an appropriate capillary system will be described for the two-dimensional case, as the complexity of the threedimensional systems makes it difficult to understand. Nevertheless, for calculation of substrate transport to tissue the three-dimensional case had to be considered. (a) Two-Dimensional Case. If a small part of a microvascular system is analyzed, we arrive at the simplest case: the square one-mesh system. Only two sources and sinks are connected by capillaries. Both input points are to be found either on the same side or arranged at opposite corners of the square. If both input points are on the same side, no difference in hydrostatic potential exists and no capillary flow occurs in the horizontal direction. If sources and sinks are alternatly located at opposite corners of the quadratic tissue, the same flow is to be found within all capillaries (Fig. 1). The two-mesh system has additional capillaries in the middle of the area. It is easy to see that no capillary flow exists within the central capillaries, as the same hydrostatic potential exists half way between each source and sink. Consequently, the two-mesh system can be reduced to a one-mesh system if all hydrodynamic resistances of the capillaries are equal.

HERMANN

34

FIG.

1.

Two-dimensional

case

of

spatially

dependent

capillary

METZGER

flow

systems:

capillaries form square meshes with different locations of inputs (arterioles) and outputs (venoles) of blood. Arrows mark blood flow directions. Horizontal and vertical arrows represent capillary flow; diagonal arrows, input and output flows. Systems are part of an infinite system; the boundary of the organ is neglected (for further explanation see text).

For the three-mesh system differences of capillary flow are calculated by applying Kirchhoff’s laws. If sources and sinks are arranged at opposite corners in the three-mesh system, the flows are in a ratio of 5 : 3 : 1. The three-mesh system is the simplest form of inhomogeneously perfused capillary system, as it includes differences of flow as well as flow directions. The three-mesh system was used for further analysis of space-distributed capillary flow systems. If the square form of the tissue element under consideration is replaced by a rectangular unit, a concurrent and a countercurrent system result. In this case, no flow is allowed in the horizontal direction (Fig. 1). In this particular example, the horizontal side length corresponds to the side length of the mesh system, and the vertical side length is chosen as 3 times the horizontal one.

(b) Three-Dimensional Case. For purposes of comparison with experimental results obtained by means of oxygen microsensors in brain cortex analysis has to be [7-81 as well as kidney cortex [15-161, mathematical performed for three-dimensional space. Previously reported results for the two-dimensional case [6] have a restricted validity, as diffusion processes in microcirculatory units occur in three dimensions. Transformation of the three-mesh system from the two- to the threedimensional case results in a tissue cube with side lengths of 3 X3X 3

SPACE-DISTRIBUTED

PARAMETER

35

SYSTEMS

capillaries; input and output points are located at opposite corners of the cube. A capillary-flow ratio of 27 : 1 is found if all hydrodynamic resistances of the capillaries are equal (Fig. 2). For further variation of the inhomogeneity of capillary flow, hydrodynamic resistances of the different capillaries can be varied: in one example, hydrodynamic resistances of the central capillaries were increased in such a way that only one-fourth of the original flow occured within these capillaries; in another example, capillary-flow ratios between the rapid perfused capillaries and the connecting horizontal capillaries were changed from 27 : 11 : 4 : 3 : I to 27 : 7 : 5. 25 : 3 : 1; the second example has a more homogeneous flow distribution than the first one. The question arises to what extent substrate and gas concentrations in capillaries and tissue are influenced by the prescribed flow variations.

concurrent

FIG. 2.

t

system

Three-dimensional

case

three-mesh

of the position-independent

system

capillary

flow

(con-

current) system and position-dependent capillary flow (mesh) system. The three-mesh system is used for analysis. It consists of a tissue cube with 3 x 3 x 3 capillaries. Input and output points are located at opposite tetrahedron has to be used for numerical

SPATIALLY

DEPENDENT

TISSUE

corners of the cube. calculations.

CONSUMPTION

By symmetry,

only

the

RATES

As described in the preceding chapter, tissue consumption rates have to be considered as spatially dependent. As a first guess, tissue oxygen consumption is increased within the input area of the capillary bed and lowered within the output area. An eightfold increase of the oxygen consumption rate is chosen for the input area, following Opitz and Schneider [17], who calculated the oxygen tension at the center of a brain cell assuming a consumption rate of 0.8 ml/g min. Areas with extremely high metabolic activity have been placed in the square meshes in the vicinity of the input of the mesh system (Fig. 3). The oxygen consumption rate of the rest of the tissue area in the neighborhood of the venule is adjusted in such a way that total consumption is kept constant; furthermore, the arteriovenous oxygen

HERMANN

36

METZGER

difference as well as the blood flow corresponded to values taken from the literature [7-91. These assumptions are necessary in order to obtain a rapidly converging procedure in the digital-computer calculations. The results of the calculations presented here are a first step towards analyzing the influence of space-distributed parameters on theoretical predictions of concentration values for substrate and gas exchange within microvascular systems.

FIG. 3.

Two-dimensional

case of spatially

dependent

consumption

rates. Three-mesh

system is considered for analysis. Tissue areas with high metabolic activity are marked by high mitochondria densities (see points). These areas are located near the input points of the arterial blood. The venous end (output points) is characterized by low consumption rates.

DIFFERENTIAL ADDITIONAL CALCULATING LIST

EQUATIONS AND RELATIONSHIPS FOR CONCENTRATION DISTRIBUTIONS

OF CONDITIONS

FOR THE CALCULATION

The equations for substrate and conditions necessary for an orderly writing digital computer programs. sion of previously derived method briefly:

gas exchange are based on a number of performance of the calculation and for These conditions are an extended verof calculation; they will be summarized

(1) Oxygen is transported in tissue by diffusion and in capillaries by convection only. (2) Chemical reactions are assumed in steady-state conditions and described by Hill’s law in the capillaries and by Michaelis and Menton’s law in the tissue. The reaction constant is between 1 and 10 mm Hg.

SPACE-DISTRIBUTED

PARAMETER

37

SYSTEMS

(3) The flow velocity in the capillaries is calculated according to Kirchhoff’s laws. There is no velocity gradient within the cross-section of the capillary, and the shape of the cross-section is that of a square. (4) The diffusion and solubility coefficients in plasma and tissue are almost equal and are assumed constant for capillaries as well as tissue. (5) Differences in capillary flow are calculated according to the hydrodynamic resistances of the capillary meshes. In the first example, the resistances have been assumed equal. In the second example, the central capillaries have an increase in resistance equal to four times that of the other capillaries. The third example of the mesh system is characterized by an increase in resistance of the second-order capillaries which run perpendicular to the main capillaries between each source and sink. (6) Oxygen consumption rates are assumed space-dependent: high consumption rates are localized in the input area, low rates in the output region. An increase of consumption is assumed to be established by a high concentration of mitochondria. (7) The analyzed tissue volume is only a small part of the whole organ. The influence of the organ surface is neglected; calculated results are valid for tissue volumes less than 100-200 microns only. (8) For calculation of oxygen and carbon dioxide transport, the diffusion resistances of capillary wall and cell membrane are neglected. If glucose and electrolyte transport are analyzed, appropriate permeability coefficients have to be introduced. Furthermore, active transport influences exchange processes and has to be considered in the analysis. Based on the above assumptions, change processes is derived. MATHEMATICAL

EQUATIONS

a mathematical

OF EXCHANGE

description

of the ex-

PROCESSES

The mathematical equations are presented in dimensionless form. Dimensionless numbers characterize capillary and tissue qualities as well as nonlinearities of the steady-state chemical reactions in blood and tissue. Tissue equation:

Capillary

equation:

ac, ““:;r-y)([$I.+[ $I.+[ $I,+[ $1,). _= as

HERMANN

38 Michaelis

and Menton’s

law:

&&

Q(c2)= Derivative

of 0, dissociation

2

curve: m-l

f(c,)=

Capillary

parameter

c’

.

(1+ U&q

and distribution

function:

fi (X?Y>Z>. Kirchhoff’s

laws:

Tissue parameter

and distribution

function:

&$

AG(x,y,z)=

(1

I

2

f2 (X,YJ>.

Mixing points: 6 T z

6 v”=

v=l

Boundary

x

conditions:

a,c,

Q(3=D2(3,

Arterial

c,v,.

v-1

input: c,= 1.0.

Tissue margin:

ac2=o.

-x&y

=

a2c2,

METZGER

SPACE-DISTRIBUTED

PARAMETER

PHYSIOLOGICAL

MORPHOLOGICAL

USED

AND

39

SYSTEMS PARAMETERS

FOR CALCULATION

n = maximal consumption

rate of 0, molecules = 0.1 ml/g min,

PO= arterial oxygen tension = 100 mm Hg,

(Y,= solubility

coefficient

of 0, in plasma = 0.024 ml/ml

atm,

(Ye= solubility

coefficient

of 0, in tissue = 0.0225 ml/ml

atm,

D, = diffusion

coefficient

of 0, in plasma = 2.4 X 10-s cm’/sec,

D, = diffusion

coefficient

of 0, in tissue = 2 x 10e5 cm2/sec,

C Hb =

hemoglobin

concentration

m = Hill’s exponent K= Hill’s constant

As = arteriovenous

of capillary blood = 33 wt%,

of the 0, dissociation of the 0, dissociation

curve = 4.04, curve = 0.8575 x 10W4,

oxygen difference = 53%,

Pk = critical oxygen tension of the mitochondria

= 1 mm Hg,

d= side length of the square capillary = 6 microns, I = length of the capillary mesh system = 60 microns, c, = initial concentration W = relati-Je capillary

of the mesh system = 1.O,

resistance = 1.O,

0 = capillary flow velocity. As will be seen from the list above, the diffusion and solubility coefficients of blood and tissue are almost equal. Concentration values can be replaced at each point of the system by oxygen-tension values; our results are presented as relative values and can be interpreted as relative oxygentension values. For further references on the parameters which have been obtained from literature or by our own experiments, see [7]. RESULTS

AND DISCUSSION

Numerical solutions of coupled differential equations with boundary conditions have been obtained for each point of the tissue as well as the capillaries. A graphical representation of the results can be presented in different ways: ISO-CONCENTRA

TION AND

FLUX

LINES

A system of orthogonal trajectories is formed by lines of the same concentration values and the path of the molecules perpendicular to the concentration lines. Sources and sinks of the molecules can easily be detected in these graphs, so that tissue areas with insufficient supply (anoxic and hypoglycemic areas) are made visible (Fig. 4).

40

HERMANN

METZGER

=-

_*

FIG. 4.

Iso-concentration

Iso-concentration

AK=0.0525.

lines

and

of oxygen

As is seen,

flux lines for the bottom (or O,-isobars)

considerable

shunt

are

diffusion

shown. occurs

area

of the tissue

Parameters: between

cube.

AG=0.91; the

different

capillaries.

The capillary mesh system shows the specific phenomenon of shunt diffusion: molecules diffuse from capillaries with high concentration values to those with low ones via the tissue. Shunt diffusion is most typically observed in capillary meshes with short capillaries and high capillary perfusion. Despite high metabolic activity of the tissue, a bypass for the molecules between the arterial and venous ends of the capillary bed exists, and venous concentration remains high. No shunt diffusion can be expected in the case of the homogeneously-perfused-cylinder model, as each capillary is analyzed separately. Computer calculations have been performed for many combinations of the parameters AC and AK displaying shunt diffusion within capillary meshes. As AC is proportional to the product of A and I*, capillary length strongly influences shunt diffusion. From morphological data on the kidney

SPACE-DISTRIBUTED

PARAMETER

41

SYSTEMS

cortex it is evident that shunt diffusion has to be a basic phenomenon in the organ. Furthermore, shunt diffusion can be found within some parts of the brain cortex. HISTOGRAM

PLOT

For oxygen and glucose concentrations as well as for hydrogen, sodium, and potassium ions, point measurements by means of microelectrodes can be performed in cells and tissues. Histogram plots have been published for oxygen-tension values in the different organs. Results from kidney and brain are summarized for illustration in Table 1. The mean value, median (50% value), and module (maximum-frequency value) have been calculated. As is seen from Table 1, left-shifted histograms have a low module, a higher median, and a mean value somewhat higher still. The question arises whether experimental results can be reasonably explained on the basis of theoretically derived histograms of space-distributed parameter models (Fig. 5). A systematic variation of the parameters AC and AK has been performed to answer this question. Both parameters influence the left-shifted histogram of the mesh system tremendously, as is seen from the graphical 17 “, 12

th

P (modul) 02

P HISTOGRAMM 02

HI%)

AG=O91 AK = 0 0525

0-

CH=OOl

7

BN=L

06

BK=15L -P _I

02

lmedlanl

L-

_

532

P a02 I

pvo2

-.I

4

l-hhpwh++ 25

50

FIG. 5. Histogram plot for the same set of parameters of the histogram is seen. Mean value, median, mode, marked.

75

P ImmHg) 02

used in Fig. 4. Typical left shift and venous concentrations are

42

HERMANN

METZGER

TABLE 1 Comparison of Oxygen Microsensor Results from the Kidney Cortex (7-91 and Brain Cortex [ 15-161 of the Albino Rat (Wistar type). Mean”

Media?

25.8 36.2 35.9 7.8 6.0 8.2 14.4

28.0 29.3 28.7 6.0 6.0 5.0 12.0

Kidney cortex

Brain cortex

‘Results of oxygen-tension measurements;

Modulea

Number of Measurements

25-26 26-27 2627 1-2 ckl l-2 2-3

1047 918 1009 898 904 840 374

values in mm Hg.

01 /AK

FIG. 6.

Mean oxygen-tension values (PO,) as function of the parameters AG and plot shows influence of the various capillary and tissue constants

AK. Three-dimensional on oxygen

supply

representation similar results SPATIALLY

of the tissue, which is represented

by mean oxygen

tension.

of the results in Fig. 6. The mean value is illustrated, have been obtained for the median and mode.

DEPENDENT

CAPILLARY

but

FLOW

The capillary flow distribution was varied according to the prescribed relationships. The percentage flow variation can be replaced by the corresponding variations of the hemoglobin content of the blood, as both, flow and hemoglobin content The space-distribution capillary

flow within

are inversely proportional to the parameter AK. ratio 27 : 11 : 4 : 3 : 1 between the velocities of

the capillaries

of the mesh

system

is an appropriate

SPACE-DISTRIBUTED

relationship

PARAMETER

for approximating

43

SYSTEMS

the situation

within

the kidney.

The form of

the histogram is similar to the experimental results obtained by means of microelectrodes. Nevertheless, it has to be kept in mind that tubular volumes as diffusion spaces and urine flow have not been considered so far in the model

calculation.

In addition,

a decrease

of capillary

flow can be

found between the upper and the deeper layers of the kidney. If these factors are included in the model calculation, volume

of the

experience, is used

capillary

system

have

this is likely to be possible

and

an appropriate

relaxation

to be increased. only if a high-speed factor

the

size

According digital

for the numerical

and

to our computer method

is

applied.’ Differences in capillary flow distribution have not been sufficient for simulation of brain-cortex PO, histograms. In order to improve the simulation,

the inhomogeneity

of the flow

ratio

of 108 : 1. Despite

this,

experimental

results

A

distribution

the correspondence

has been

increased

to a

between

theoretical

and

has been poor (Figs. 7 and 8).

C

B

FIG. 7. Oxygen tension and relative capillary-flow values within the tetrahedron (part of the 3x3 ~3 capillary mesh system). Different degrees of inhomogeneity of capillary flow are to be seen. Flow ratios are the following: case A, 27 : 1 I : 4: 3 : 1, maximum case B, 27 : I I : 4: 3.25 : 0.25, maximum 108 : I; case C, 27 : 7 : 5.25 : 3 : 1, maximum Case B is more inhomogeneous than A; case C is more ratios are less pronounced than in case A. An improvement C is to be seen from the oxygen-tension values.

homogeneous, of the supply

27 : I; 27 : I.

as the internal situation of case

‘The data presented have been calculated with relaxation parameter of about 1.85. Accuracy of the calculation: 5%. Number of iterations: about 70 steps. Number of points of the mathematical grid: 1496. Computer time: less than 60 seconds (CDC 6600 of the Regionales Rechenzentrum fur Niedersachsen in Hannover was used for calculations).

44

HERMANN

i5

I

25

50

is

59

75

FIG. 8. Oxygen-tension histograms for the tetrahedron. Histograms capillaries and tissue. Cases A, B, C, correspond to Fig. 7.

SPATIALLY

DEPENDENT

METZGER

CONSUMPTION

are calculated

for

RATES

For better understanding of the experimental results, a spatial dependence of the consumption rate of the oxygen molecules has been introduced. Reasonable agreement is then obtained for the assumptions described in the preceding part. Nevertheless, the assumptions have been only a rough guess. However, further analysis of this kind might help to improve our knowledge of substrate and gas exchange within microvascular units of the brain cortex and possibly stimulate further experimental research. The spatial dependence of physiological parameters is a basic phenomenon well known in the literature but not considered in the model calculations of many authors until now. In particular, the homogeneous cylinder model does not include any of the analyzed factors described here. The analysis demonstrates the strong influence of the spatial dependence of parameters on the calculation of exchange processes. For further theoretical analysis it will be necessary to gather information about the spatial depenFurthermore, diffusion and solubility dence of flow and consumption. coefficients might be position-dependent and will be investigated.

SPACE-DISTRIBUTED

PARAMETER

45

SYSTEMS

REFERENCES 1

A. Krogh, The number and distribution of capillaries in muscles with calculation the oxygen pressure head necessary for supplying the tissue, J. Physiol. (Lo&)

of 52,

2

409415 (1918-19). G. Thews, Uber die mathematische

3

in zylinderformigen Objekten, Acra Biorheor. 10, 105-138 (1953). J. Blum, Concentration profiles in and around capillaries, Am. J. Physiol. 198,

4 5

6

Behandlung

physiologischer

Diffusionsprozesse

991-998 (1960). J. A. Hudson and D. B. Cater, An analysis of factors affecting tissue oxygen tension, Proc. Roy. Sot. (Lond.) B161, 241-214 (1964). D. D. Reneau, D. F. Bruley, and M. H. Knisely, A mathematical simulation of oxygen release, diffusion and consumption in the capillaries and tissue of the human brain, In Chemical Engineering in Medicine and Biology (W. Hershey, Ed.), Plenum, 1967. H. Metzger, Distribution of oxygen partial pressure in a two-dimensional supplied by capillary meshes and concurrent and countercurrent systems, Biosci. 5, 143-154

tissue Moth.

(1969).

7

H. Metzger, Verteilung des 02-Partialdruckes im Mikrobereich des Gehirngewebes: Polarographische Messung und mathematische Anafyse, Habilitationsarbeit, Mainz, 1971.

8

H. Metzger, Polarographic oxygen tension measurements in microstructures of living tissue: a digital computer study on the oxygen tension histogram, Adu. Chem. Ser. 118, 328-342 (1973). H. Metzger, Geometric considerations in modeling oxygen transport processes in

9

tissue, in Oxygen 1973.

Transport IO Tissue (D. F. Bruley

and H. I. Bicher,

Eds.), Plenum,

10

H. Metzger, O,-transport in the homogeneous and inhomogeneous microcirculation, paper presented at the 1st World Congress for Microcirculation, Toronto, 1975.

11

T. A. McCracken, D. F. Bruley, D. D. Reneau, H. I. Bicher, and M. H. Knisely, Systems analysis of transport processes in human brain-Part II (Systems Studies of the Simultaneous Transport of Oxygen, Glucose and Carbon Dioxide in Human Brain), Proc. 1st Pac. Chem. Eng. Congress, Kyoto, 1972.

12

D. D. Reneau, T. Zeuthen, E. Dora, and I. A. Silver, An analysis of ion distribution in brain following anoxia, paper presented at the 2nd Int. Symp. in Oxygen Transport to Tissue, Mainz, 1975.

13

I. Lockard, Existing anatomical parameters and the need for further determinations for various tissue structures, in Oxygen Transport to Tissue (D. F. Bruley and H. I. Bicher, Eds.), Plenum, 1973. J. N. Barker, Blood flow limitation of neuronal density along capillaries of rat brain, Fed. Proc. 31, 1026 (1972).

14 15

16

17

H. Gunther, P. Vaupel, H. Metzger, and G. Thews, Stationare Verteilung der Oz-Drucke im Tumorgewebe (DS-Carcinosarkom): I. Messungen in viva unter Verwendung von Gold-Mikrolektroden, 2. Krebsforsch. 77, 2639 (1972). H. P. Leichtweiss, D. W. Lubbers, Ch. Weiss, H. Baumgartl, and W. Reschke, The oxygen supply of the rat kidney: Measurements of intrarenal %, Pfliigeis Arch. ges. Physiot. 309, 328-349 (1969). E. Opitz and M. Schneider, Uber Mechanismus von Mangelwirkungen,

die Sauerstoffversorgung des Gehirns Ergeb. Physiol. 46, 126260 (1950).

und

den