A compartmental model for oxygen transport in brain microcirculation

Annals of Biomedical Engineering, Vol. 17, pp. 13-38, 1989 Printed in the USA. All rights reserved.

0090-6964/89 $3.00 + .00 Copyright 9 1989 Pergamon Press plc

A Compartmental Model for Oxygen Transport in Brain Microcirculation Makhili Sharan,* M.D. Jones, Jr., **t R.C. Koehler, t R.J. T r a y s t m a n , t A.S. Popel* *Department of Biomedical Engineering School of Medicine Johns Hopkins University Baltimore, MD **Departments of Pediatrics (Eudowood Neonatal Pulmonary Division), and Gynecology and Obstetrics Johns Hopkins University 1-Anesthesiology/Critical Care Medicine Johns Hopkins University

(Received 10/6/87; Revised 7/1/88)

A compartmental model is formulated f o r oxygen transport in the cerebrovascular bed o f the brain. The model considers the arteriolar, capillary and venular vessels. The vascular bed is represented as a series o f compartments on the basis o f blood vessel diameter. The formulation takes into account such parameters as hematocrit, vascular diameter, blood viscosity, blood flow, metabolic rate, the nonlinear oxygen dissociation curve, arterial P02, Pso (oxygen tension at 50% hemoglobin saturation with 02) and carbon monoxide concentration. The countercurrent diffusional exchange between paired arterioles and venules is incorporated into the model. The model predicts significant longitudinal P02 gradients in the precapillary vessels. However, gradients o f hemoglobin saturation with oxygen remain fairly small. The longitudinal P02 gradients in the postcapillary vessels are f o u n d to be very small. The effect o f the following variables on tissue P02 is studied: blood flow, P02 in the arterial blood, hematocrit, Pso, concentration o f carbon monoxide, metabolic rate, arterial diameter, and the number o f perfused capillaries. The qualitative features o f P02 distribution in the vascular network are not altered with moderate variation o f these parameters. Finally, the various types o f hypoxia, namely hypoxic, anemic and carbon monoxide hypoxia, are discussed in light o f the above sensitivity analysis. K e y w o r d s - O x y g e n transport, Microcirculation, Cerebral circulation, Carbon monoxide, Mathematical model, Computer simulation.

INTRODUCTION

Oxygen transport in the brain is dependent on the oxygen-carrying capacity of the blood, the rate of local blood flow, the diffusion conditions within the brain, the kiAcknowledgments-The authors wish to thank Ms. Brenda Pope for the typing of the manuscript. This work was carried out while one of the authors (M.S.) was on leave from the Indian Institute of Technology, Delhi. Supported in part by NIH grant NS 20020. Address correspondence to Dr. Aleksander S. Popel, Department of Biomedical Engineering, School of Medicine, Johns Hopkins University, Baltimore, MD 21205.

13

14

M . Sharan et al.

netics of oxygen release from hemoglobin, hemoglobin oxygen affinity, 02 consumption of the tissue and the partial pressure of oxygen (PO2) in the blood entering the capillaries. Alterations in these parameters in physiologic and pathologic conditions influence the ~capillary-tissue O2 exchange. The classical view of oxygen transport maintains that oxygen diffuses exclusively through the walls of the capillaries and the precapillary vessels do not participate in the exchange. However, Davies and Bronk (4) found that oxygen could diffuse across the walls of pial vessels. Later, Duling and Berne (5) demonstrated precapillary losses of 02 by measuring the periarteriolar PO2 in several tissues. It was also shown that the hemoglobin saturation with oxygen (SO2) decreases as blood proceeds towards the capillaries (7). These measurements were made in tissues with a relatively low bIood flow rate. Duling et al. (6) also demonstrated precapillary losses o f oxygen in the brain, where the blood flow rate is several times higher, by measuring the perivascular PO2 in the vicinity of the pial vessels in the cat. However, the losses are smaller than those measured in other tissues. Ivanov et al. (13) measured directly the PO2 at the surface of arterioles, capillaries and venules of rabbit brain cortex and demonstrated that a considerable amount of oxygen diffuses through the walls of the precapillary vessels. Thus, the arterial blood enters the capillaries with a lower PO2 and thereby the capillary-tissue 02 exchange is altered. Mathematical models also suggest that the observed longitudinal oxygen gradients are qualitatively consistent with diffusive mechanism of 02 transport from the arterioles (28,30). Hence, the arterioles and, possibly, venules can play a significant role in the exchange of gases between the blood and tissue. However, it is not known how the 02 lost from the arterioles is distributed or whether this phenomenon is universal in different tissues and in different physiologic and pathologic conditions. Further, little is known about the participation of the venules in 02 transport and, finally, there is no mathematical model that is in quantitative agreement with experimental data. The venous PO2 is commonly considered the indicator of tissue and end-capillary PO2. However, Gutierrez (10) has shown that the kinetics of 02 uptake by hemoglobin may play an important role in the delivery of 02 to the tissue and, as a result of nonequilibrium chemical reaction, the venous PO2 can be greater than the end-capillary PO2. The predicted effect is more pronounced during hypoxia. Pittman and Duling (27) reported that the PO2 in the venular blood may rise as the blood flows from the smaller to larger vessels. The mechanism of countercurrent diffusional exchange between paired arterioles and venules running side by side parallel to each other has been implicated. Sharan and Popel (33) examined theoretically the problem of countercurrent diffusional exchange between a pair of arteriolar and venular vessels and found the rise in the venular PO2 to be small in normal conditions. However, under some conditions (e.g., ischemia)the rise in the venular PO2 can be significant. Another possible mechanism of countercurrent exchange is that the oxygen that diffuses from arterioles is picked by the capillaries running in the vicinity of the arterioles; the capillaries in turn unload the oxygen directly into the venules by convection. The above arguments suggest that the consideration of venous PO2 as an accurate measure of the end-capillary PO2 and tissue PO2 in whole organ experiments might be questionable. A number of mathematical models have been developed to describe the transport of oxygen and other gases in the brain. Reneau, Bruley and their co-workers (2) extended the classical model of Krogh by including the radial and axial diffusion in

Model of Oxygen Transport

15

the capillary, nonlinear oxygen dissociation curve as well as the axial diffusion in the tissue. Other geometrical models, different from Krogh's, of the cerebral capillary network have been described in the literature. Metzger (21) represented the network by a three dimensional square mesh surrounded by uniform tissue. Ivanov et aL (14) studied the distribution of O2 in a microvascular unit that consists of a neuron surrounded by several capillaries; the arteriolar and venular networks were not included. In the present study, we formulate a compartmental model of oxygen transport in the cerebrovascular bed of the brain which considers the arteriolar, capillary, and venular vessels. Parameters such as hematocrit, vascular diameter, Pso, and carbon monoxide concentration are included in the formulation to allow analysis of experimental data in various types of hypoxia (12,17-19). The countercurrent diffusional exchange between the paired arterioles and venules is incorporated. In this report, we present the results of the sensitivity analysis that reveal to what extent various parameters of the model affect the distribution of PO2 in the vascular network. The effect of the above parameters under conditions of hypoxic, anemic and carbon monoxide hypoxia is examined. In a subsequent report, the model is applied to the analysis of experimental data on oxygen transport and cerebral blood flow in the lamb and sheep in hypoxic, anemic, and carbon monoxide hypoxia and after changes in oxyhemoglobin affinity. This analysis should provide a better understanding of the mechanisms controlling tissue oxygen availability. MATHEMATICAL FORMULATION Blood flows through the vascular bed from the arterioles through the capillaries to the venules. We represent the bed as a series of vascular compartments (Fig. 1) sur-

L .... t _%o_Jl ....

J--"

.... r---

I,l I P

c

i

c

II Ii i I

M.

Pt

Jl

l;v Q.

t v: 1.......

Pv

FIGURE 1. Compartmental model of a brain microcirculatory unit with diffusional 02 exchange between tissue and all the vascular elements and with the convective 02 transport along the circulation. Also, there is countercurrent diffusional exchange between the paired arterioles and venules. The arterioles ( a O , a l , . . . , a m ) and venules (vO, v l , . . . , vm) are classified into (2m + 2) compartments on the basis of vessel diameter. P denotes oxygen tension in the vascular and tissue compartments, E are the diffusion conductances between the vascular and tissue compartments, and M is the rate of tissue oxygen consumption.

16

M . S h a r a n et al.

rounded by a tissue compartment, similar to model (30). Each compartment contains vessels of the same diameter arranged parallel to each other. Each of the vascular compartments exchanges 02 with the tissue compartment. In order to examine the possible effect of diffusional O2 shunting from arteriole to venule on tissue PO2, we assume that the large venules run in proximity with and parallel to large arterioles. As the blood flows through the vascular bed, oxygen is transported by convection in the vascular compartments and by free diffusion from the blood to the surrounding tissue where it is utilized. Also, there is diffusional interaction between some arteriolar and venular compartments at the same branch level in the process of countercurrent exchange, as will be described below. It is assumed that the vascular bed contains the following compartments: rn + 1 arteriolar, m + 1 venular, and one capillary. The mass balance principle in each of the vascular compartments leads to the following equations in steady state: Q(C,, -

Q(Ca(k-l)

C~o) - E , , o ( P a o - P t ) - F a o ( P a o - Poo) = O,

-- Cak) -- E , t k ( P a ~ -- P t ) - F a k ( P ~ k -- Poe) = O; k = 1,2 . . . . . m

Q ( C a m - C~) - E ~ ( P c - P , ) Q(C~

-

Cvm ) -

Q ( Co~k+I) -

Evm(Pvm

--

Pt)

-- Fvm(evm

Co~) - E ~ k ( P o k - I t ) k = m-

l, m -

=

o, -

- Fok(Pok

2 .....

1,0.

Pam)

=

O,

- Pae) = O;

(1)

where Cai and Pai are, respectively, the oxygen content and oxygen partial pressure in ith arteriolar compartment, Q is the blood flow rate, E~i and F~i are the diffusion conductances or exchange coefficients for the ith arteriolar compartment, P t is the partial pressure of oxygen in the tissue, and Ca is the 02 content in the blood entering the vascular network with the partial pressure P,. The subscripts "v" and "c" refer to the corresponding quantities in the venular and capillary c o m p a r t m e n t s , respectively. The oxygen tension in each compartment represents the spatially averaged quantity. In each of Eqs. (1), the first term represents the transport of oxygen by convection, the second term gives the transport of oxygen from the blood to the surrounding tissue, while the last term is due to countercurrent diffusional exchange between the paired arterioles and venules. Oxygen is carried by the blood in two forms: physically dissolved in plasma, and in combination with the hemoglobin inside the red blood cells. In arterial blood most of oxygen, about 98%, is bound to hemoglobin. We recognize that nonequilibrium kinetics of the oxygen-hemoglobin chemical reaction may be important, especially under hypoxic conditions; the reaction is also affected by the presence of carbon monoxide. However, since this work is a first attempt to investigate the transport interactions between different parts of the microcirculation in the brain, for the sake of simplicity we assume that the chemical reaction between oxygen and hemoglobin is in equilibrium. The oxygen content in the blood flowing through the ith vascular compartment is calculated as: G = ~P~ + 5 H S (Pg)

(2)

Model of Oxygen Transport

17

in which c~ is the solubility of 0 2 in the blood, P~ is the partial pressure of 0 2 in the ith compartment, H is the blood hematocrit, S is the fractional saturation of hemoglobin with O2, and/3 is the oxygen-carrying capacity per unit volume of red blood cells. When a fraction of hemoglobin molecules is occupied by carbon monoxide, CO, the coefficient/3 can be expressed as:

/3 -

1.34[Hb] Ho

(1 -

sco)

(3)

where [Hb] is the total hemoglobin concentration in grams per unit volume of blood under normal conditions, H ~ is the blood hematocrit under normal conditions and SCO is the fractional saturation of hemoglobin with carbon monoxide. In Eq. (3), it was assumed that variations in hematocrit from the arterial to the venous blood are negligibly small. In recent experiments studying the cerebral blood flow and oxygen transport during anemic hypoxia (12), part of the hemoglobin was replaced by methemoglobin. In this case, only the hemoglobin available for O2 and CO binding, [Hb], should be accounted for in Eq. (3). In the tissue compartment, the rate of accumulation of O2 is balanced by the net transport of O2 by diffusion from the vascular compartments and the utilization of oxygen in the metabolic processes. The mass balance in the tissue compartment is expressed by the following equation in the steady state:

~Eai(P.i i=0

- P~) + E c ( P ~ - P t ) + ~ ] E ~ i ( P v , - P t ) - M V t = 0

(4)

i=0

where M is the metabolic rate, and Vt is the volume of tissue compartment. In the tissue, the consumption of O2 is commonly represented by the nonlinear function based on the Michaelis-Menten kinetics. However, it has been shown that the metabolic consumption of Oz in the brain remains constant down to low levels of oxygen tension (23) and, therefore, M is taken to be constant (i.e., zero order chemical kinetics). Equations (1) and (4) constitute a system of algebraic equations. This system contains several parameters such as saturation function, S, blood flow, Q, and the diffusion conduOances, E and F, for the vascular compartments. The procedures of computation of these parameters are explained below. OXYHEMOGLOBIN DISSOCIATION CURVE The relationship between the fractional saturation of hemoglobin with oxygen, SO2, and oxygen tension, PO2, is known as the oxyhemoglobin dissociation curve (ODC). The aim of the present study is to formulate a model capable of analyzing oxygen transport in the sheep brain during hypoxic, anemic, and carbon monoxide hypoxia. The function representing the ODC should depend on Pso, pH, PCO2 and the carboxyhemoglobin concentration in order to enable us to simulate the experimental hypoxic conditions. Also, the mathematical relation should be capable o f approximating the lower saturations of hemoglobin with O: under hypoxia. In most studies, the two-parameter Hill equation is used to represent ODC in the mathematical modeling of 02 transport. The effect of pH, PCO2 and CO can be incorporated

18

M. Sharan et al.

through the manipulation o f Ps0. However, the Hill equation does not accurately describe the lower saturations which are encountered in hypoxia. Also, the relations presented in the literature for the ODC are not convenient for mathematical modeling of O2 transport in the presence of carbon monoxide. Thus, here we describe an algorithm that allows an accurate representation of ODC at low saturations. We use the Adair equation: a l p + 2a2 P2 q- 3 a 3 P 3 + 4a4 P4 S = 411 + a l P + a2 P2 + a3 P3 + a4 p 4 ]

(5)

in which S is the fractional saturation of hemoglobin with 02, P is the partial pressure of 02, and ai are the Adair parameters. The parameters ai are expressed in terms of Ps0 for the sheep blood (31): log10 al = -0.4948 loga0 Pso - 1.117 lOgl0 a2 = 0.7473 logl0 Pso - 5.207

(6)

logloa4 = --3.955 log~oPso + 0.0238. The parameter a3 was assumed to be zero based on the study (35) in which the curve is accurately described by three parameters for human blood. Ps0 has been expressed in terms of P C O 2 and pH (31): logl0Pso = (0.1902 x 1 0 - 2 p C 0 2 -- 0.3916)pH - 0.0126 P C O 2 q- 4.4527.

(7)

In order to introduce the dependence of the saturation function on carbon monoxide, we compute oxygen pressure, Tso, at 50% saturation of hemoglobin with O2 in reference to the remaining binding sites of hemoglobin available for O2 in the presence of CO according to the theory developed by Collier (3): 1-SCOf-1(0.5+ Ts0 - 1 + S C ~

~-)

(8)

Here, f is the relation between SO2 and P 0 2 , as defined by the Adair equation. Ts0 is computed from Eq. (8) by inverting Eq. (5) for the fractional saturation 0.5 (1 + SCO) numerically. The Pso in Eq. (6) is replaced by Ts0, to compute the parameters ai of the Adair equation in the presence of CO, and the complete ODC is computed from Eq. (5) using the modified values of the parameters. Thus, Eq. (5) with modified parameters is used in the calculation of S in Eq. (2). Blood flow

Let qa and qv be the arterial and venous hydraulic pressures. Since the vascular compartments are arranged in series, the total blood flow, Q, is given by: Q=

(qa - q v ) R

i

(9)

Model o f Oxygen Transport

19

where m

R =

Rai + R c + ~_j R vi i=0

(10)

i=0

is the total vascular resistance, and Ri is the resistance of ith compartment. We assume that vessels in each of the vascular compartments are arranged parallel to each other. Total resistance in the compartment is given by: 128Li (11)

R i -- 7rNiDi4 IZpl~r(Di,H)

in which Ni is the number o f vessels in the ith compartment, Li is a characteristic length of each of the vessels, Di is the vessel diameter,/~ap is the apparent viscosity o f blood in the microvessel, #p/is the plasma viscosity and/~r = IZ~p/lZpl is the relative apparent viscosity that depends on the vessel diameter and hematocrit. Equation (11) becomes the classical Hagen-Poiseuille equation for fully developed laminar pipe flow of a Newtonian fluid when #r = 1. The low Reynolds numbers characterizing the microcirculation make it possible to neglect inlet effects and consequently justify the assumption of fully developed flow. The particulate nature of blood in the microvessels plays an important role in determining the flow resistance. For computing the resistance (Eq. 11) in each of the vessels, we calculate the relative apparent viscosity, t~r, from the equation presented in (24). Diffusion Conductances f o r the Noncapillary Vessels

A mathematical model for the oxygen transport between paired arterioles and venules was formulated in (33) based on an earlier model (28) in which 02 diffusion in the extra-arteriolar space was described by a linear transport equation: 1 v~P,

-

w

(P,

-

(12)

P = ) = o.

Here, 1~ is a phenomenological parameter, referred to as the penetration length, and P~ is the tissue PO2 far from the arteriole. We use this model for calculating the diffusion conductances. Briefly, we consider a pair o f circular parallel unbranched arteriolar and venular vessels of diameters Da and Do at a distance b apart, surrounded by tissue. Oxygen diffuses from the arteriole to the tissue where it is utilized and it can also diffuse from the arteriole through the tissue to the venule. The flux of oxygen from the arteriole is expressed as the sum of these amounts: J.

Dt c~t

- ~(G-

~)+~(G-

~).

(13)

Here, P~, Po, and Pt are the oxygen tensions in arteriole, venule, and tissue, respectively; O t and at are the diffusion and solubility coefficients for O2 in the tissue; ea

20

M. Sharan et al.

and f~ are coefficients that depend only on the vascular geometry (i.e., on D~, Do, and b) and on the penetration length, lt. Similarly, the total O2 flux from the venule, Jo is expressed as:

J~ Ot O~t

--

.fo(P~ -- P~) + e~(Pv - Pt)

(14)

where fu and eo are geometrical coefficients. The coefficients ea, f~, eo, and fo are computed numerically for each set of parameters D., Do, b, and It (33). The diffusion conductances for the ith arteriolar compartment are calculated from: Ea i = NiLgOto:teai;

Fag = NiLiOto~tfai

(15)

where Ng is the number of vessels in the ith compartment, and Lg is the characteristic length of each of the vessels. The diffusion conductances for the ith venular compartment are calculated with the help of Eq. (15) by replacing ea by eo and fa by fu- It has been shown (33) that the value of ea computed from Eq. (13) without countercurrent exchange (e.g., when b ~ oo) approaches that computed from the model (28). D i f f u s i o n C o n d u c t a n c e f o r the Capillary C o m p a r t m e n t

The diffusion conductance for the capillary compartment is calculated differently from that of the other vascular compartments. We model the capillaries as a compartment with exchange properties equivalent to a large number of spatially averaged Krogh tissue cylinders. The transport of O2 in a tissue cylinder of radius rt surrounding a capillary of radius rc is governed by:

Dte~ t -1 -d- r - - dpt - M = 0 rdr dr

(16)

in which Pt is the partial pressure of O2 in the tissue and M is the metabolic rate. We assume that the axial diffusion of 02 in the tissue is negligible in comparison to the radial diffusion. Equation (16) is subject to the following boundary conditions:

dPt

--

dr

=0atr=rt

(17)

and

Pclr=rc = P, lr=.c

(18)

where Pc is the partial pressure of 0 2 in the capillary. Integrating Eq. (16) and using the boundary conditions (17) and (18), we obtain:

Model of Oxygen Transport

21

pt(r,z) =pc(z) + ~

= (r2r~

r 2In

r)

(19)

where z is the coordinate along the length of the capillary. Let Pt be the average partial pressure of 02 in the tissue cylinder of length L defined by:

Pt-

1 lo lr/

zc(rtZ - r~)L

27rrpt(r,z) drdz.

(20)

Similarly, the average partial pressure of 0 2 in the capillary Pc is defined by:

lloL

Pc = ~

Pc(Z) dz.

(21)

Taking the average of Eq. (19) and using Eqs. (20) and (21), we obtain:

Pt = Pc + ~

M

(3r[_r [

4r 4 l n r t ) r 2 - r-----~ r~ "

(22)

The diffusive flux for the compartmental model can be represented by the macroscopic transport equation: de = Ne(Pc - Pt)

(23)

where N is the total number of capillaries and e is the diffusion conductance. If capillary blood is the only source of 02 for the surrounding tissue, total flux of oxygen from the capillary to the tissue is:

(24)

Jc = NM~r(rt 2 - r2~) L.

Eliminating M from Eqs. (22), (23) and (24), we can express the diffusion conductance for the capillary compartment as: 4~rNLDtat(1

Ec = Ne =

2 - 1 -

-

w 2)

(25)

3 - w2 In(l/w) WE

2

where w = rc/rt is the ratio of capillary radius to tissue cylinder radius. Severns and Adams (32) derived an expression for the diffusion conductance for a compartmental model by considering the unsteady transport of a substrate in the Krogh model. A step rise in the partial pressure was used at the capillary-tissue interface and the first term in the series solution of the governing equation was retained for calculating the diffusion conductance. The advantage of Eq. (25) compared to that of Severns and Adams is that it becomes an exact solution of the corresponding Krogh model when capillaries are the only source of 02 in the tissue.

22

M . Sharan et al. O0

03

d O~ 0 0

0"~

e.i

X

t.~ ...2. c-

0 0

CO

X t.O

O 0

0

0 0

0 0')

d

X r CO

II

• p-

0

d

II

3 r

0 0 0

S 0

',,,0

d

• r O~

L.O X

d

3 CO

E e~

E

0 0

0 0

U.I ..I

CD 0 0

0 0"~

,:5

LO 03

r

e.i

• O3 r

X t..O

X 1.0

k(N 0

t.O

E 0

=__

Model of Oxygen Transport

23

Eqs. (1) and (4) together with the relations (2), (5) and (9), form a nonlinear system of algebraic equations. The system is solved numerically using the NewtonRaphson method (29). PARAMETERS OF THE MODEL

In order to calculate the distributions of P O 2 in the vascular and tissue compartments in any particular tissue, the values of the parameters such as the diffusion coefficients, solubility coefficients, geometrical parameters pertaining to the morphology of the vascular bed, metabolic rate, blood viscosity, etc . . . . must be specified. Keeping in mind application of the model to the cerebral circulation in sheep, we will choose the corresponding parameters relevant for the sheep brain. However, only a few of the necessary parameters for the sheep brain have been determined experimentally. The morphological parameters such as number of vessels, average length of each of the vessels and blood vessel diameters for this vascular bed, and, in fact, for cerebral circulation in any species are not available. Moskalenko et al. (22) reported a set of morphological parameters for the vascular bed of the dog brain with a weight of 80 g. The total cerebral blood flow based on these morphological data was calculated at 57.2 ml/100 g/min, which approximates the known experimental results. However, t h e flow and pressure distributions computed from the data in each vascular order are not consistent with available measurements (11,16,20). For example, the velocity in a vessel of 120/~m diameter comes out to be 22.5 cm/s which is more than five times the velocity reported in (16). Also, about half of the pressure drop occurred in vessels of this size. We have assembled a set of morphological parameters (Table 1) for the vascular bed of the cerebral part of sheep brain. The weight of the cerebral part is taken as 75 g (total brain is 100 g) with a blood flow rate of 70 ml/100 g/min (18). The number of vessels in a given order for arterioles are calculated from velocities in the cat pial arteries (16): vi = 0.34Di + 0.309

(26)

where vi is the blood velocity in m m / s and D / i s the vessel diameter in t~m. These velocity measurements are consistent with those reported for the cerebral microvessels of rats (20). Diameter of the venules is assumed to be 50~ larger than the diameter of the corresponding arterioles, and the numbers of arterioles and venules are assumed to be equal. A capillary velocity of 0.6 mm/s is assumed from measurements in cortical capillaries of rats (15). The average capillary length is chosen to be 0.6 mm, which results in capillary transit time of 1 sec. We have also chosen: arterial pressure q, = 60 m m H g at the inlet of 120/~m diameter arterioles, venous pressure qv = 10 mmHg at the outlet of 180 ~tm diameter venules (11), and plasma viscosity t~pt = 1.28 centipoise. The average length of noncapillary vessels is assumed to be proportional to vessel diameter and the coefficient of proportionality is calculated by satisfying the pressure-flow relationship in the network. The pressure distribution computed from Table 1 is consistent with the microvascular pressure measurements in the pial circulation in the rat (11). Also, the intercapillary distances computed from the present data are close to the one based on morphological studies in the cat (25) and rat (34). Thus, the set of morphological data

24

M. Sharan et al.

in Table 1 which is used in the present model is consistent with flow and pressure measurements and also with the stereological measurements in the brain reported in the literature. The diffusion and solubility coefficients for 02 have been determined for other tissues (1): Dt = 1.5 • 10 -5 cm2/s; o~ = elt =3 • 10 -5 m l / m l / m m H g . The penetration length, It, used in the calculation of the diffusion conductances was chosen to be 100/~m, similar to the model of oxygen transport in arteriolar networks (28). For the sheep [Hb] = 0.11g/(ml blood); H ~ = 0.3; the metabolic rate M = 4.8 ml O2/100 g / m i n (18). POE = 95 m m H g is taken in the arterial blood in normal conditions. For calculating the diffusion conductances in noncapillary vessels, the distance, b, between the surfaces of the paired arteriole and venule in the first and second order vessels is taken as 50/zm while in the vessels of remaining orders it was assigned a large value (b = 1000 #m) to provide the diffusion conductance parameters without countercurrent exchange (33). There is at least one report (8) that suggests the arterioles and venules in the human brain are not paired; however, as will be seen from results below, the model predicts a very small amount of diffusional countercurrent exchange. Thus, the exact value assigned to b is not essential for the results. The value of the Krogh tissue cylinder radius, rt, is calculated f r o m the volume of the cerebral part of the brain, the number of capillaries, and the capillary length. For the normal conditions, we chose p H = 7.4, PCO2 = 40 m m H g , H = 0.3 and S C O = 0. The value of Ps0 computed from Eq. (7) is 41.1 m m H g . For the morphological data (Table 1), the total number of vascular compartments is 11 and rn = 4. RESULTS A N D D I S C U S S I O N In the figures, " I N " corresponds to the network inlet. " A I " to "AS" correspond to arteriolar compartments, " C A P " designates the capillary compartment, "V5" to " V I " the venular compartments and " T " denotes the tissue compartment. The total blood flow computed f r o m Eqs. (9)-(11) using the relative viscosity obtained from (24) is 0.875 cm3/s which is equivalent to 70 ml/100 g / m i n in the cerebral circulation o f sheep (18). The distribution of PO2 and SO2 in the vascular compartments and PO2 in the tissue compartment is presented in Fig. 2. It shows that the precapillary gradients o f PO2 are present in all arterioles. The longitudinal PO 2 gradient increases in the successive branch levels o f the arteriolar network. The saturation is almost constant in the first four orders o f arterioles and a significant fall in the saturation and PO2 occurs in the capillaries. The PO2 and the corresponding saturation are almost constant in the blood flowing through the venular compartments. The venous PO2 is 40.4 m m H g which is close to the reported experimental value 39 m m H g (17). It is found that 32~ of the total PO2 drop between the inlet and outlet of the vascular network occurs at the precapillary level with the remaining 68 o70 in the capillaries. For the saturation in absolute terms, a 5~ drop occurs in the arterioles and a 39~ drop in the capillaries. In order to examine the possible contribution of countercurrent exchange between the paired arterioles and venules on the venous PO2, we assumed that the arteriolar and venular vessels are paired at the level of the first two vascular orders. As a standard case, the distance between the surfaces of the paired vessels was chosen as b = 50 ~m. However, the results are not affected when b is varied between 25 and 1000

Model of Oxygen Transport

25

100

90

80 "I"

E

~E

70

60 84

~,~

50-

O O 0 " I

40-

I

l

\

\ \\ \

30-

20 IN

, A1

i A2

i A3

i M

i A5

I

I

I

I

I

|

CAP

V5

V4

V3

V2

Vl

BRANCH LEVEL FIGURE 2. D i s t r i b u t i o n of PO2 and SO2 in t h e v a s c u l a r n e t w o r k , n SO2; + PO2. Here and in subs e q u e n t figures, " I N " c o r r e s p o n d s to the network inlet, " ' A 1 " t o " ' A 5 " and " ' V I " t o " ' V S " correspond to arteriolar and venular compartments respectively, "CAP" designates the capillary compartment, and " T " the tissue c o m p a r t m e n t .

/~m. The model predicts that only a small amount of oxygen diffuses from the arterioles to venules.

Sensitivity Analysis The model contains various parameters such as the penetration length, It, blood flow rate, Q, metabolic rate, M, oxygen affinity, Pso, hematocrit, H ~ carboxyhemoglobin saturation, (SCO), and the POz in the entering blood. We will now study the effect of individual parameters on POz distribution in the brain. At the present time, only a few experimental measurements of precapiUary gradients are available. The values of the penetration length, It, in Eq. 12 have not been determined experimentally (26). In fact, it is not clear whether the model of Popel and Gross (28) can adequately describe the experimental data. In our calculations, the penetration length was chosen as 100/zm, similar to (28). Because of this uncertainty, we analyze the sensitivity of our results to this parameter. Figure 3 shows the PO2 distribution in the vascular and tissue compartments with the variation of penetration length from 10 #m to 150/zm. The longitudinal PO2 gradients in the pre-

26

3,1. Sharan et aL 100 90 80

70 "~ "I-

SO

E ..EE

s0

n

40

u

D

"

=

n

30 20-

100

IN

I

I

|

t

At

h2

A3

M

f

~

I

I

I

I

I

I

CAP

V5

V~

V3

V2

Vl

BRANCH LEVEL FIGURE 3. Distribution of PO2 in the vascular network for various values of penetration length. n 150/Lm; + 100/r o 50/~m; A 25/=m; x 10/~m.

capillary vessels increase a s I t decreases; for example, the POz in the A5 vessel is reduced from 78.9 m m H g to 65.1 m m H g with a fifteenfold decrease in It, and the tissue PO2 is increased by 4 mrnHg from 29.1 m m H g to 33.2 mmHg. The venular PO2 is not affected by It. Figure 4 shows PO2 distribution in the vascular and tissue compartments with elevation and reduction o f blood flow. Blood flow is increased by 100o70 and reduced by 25~ and 50~ from the control conditions. The model predicts an increase of precapillary 02 losses with the reduction in blood flow when the vessel diameters remain at the control level. It is found that the tissue PO2 becomes a/most zero (0.2 mmHg) when the blood flow is reduced by 50~ from its control. With 50o/o reduction in the blood flow, the change in the venous PO2 due to countercurrent exchange increases to 5~ from 0.507o in the control. The regulatory changes in the microcirculation usually involve changes in arteriolar diameters as welt as changes in blood flow rate. In our computations, the arteriolar diameters are increased by 20o70, 50~ and 100o7o, or reduced by 15~ from the control value. We found that for a constant flow, the arteriolar dilation or constriction does not affect the distribution o f PO2. For example, with I00o70 dilation, the tissue PO2 changed from 29.4 to 29.7 mmHg, and PO~ in the terminal arterioles changed from 77.6 to 74.7 mmHg. However, if the blood flow rate is allowed to

Model of Oxygen Transport

27

lOO 90

80

\

70

"i" E

80

c

s

c

c

o, ~x

50-

\

40

I

I

I

I

t,xxx\

30

\

'it IN

A1

\

L

1

I

\

\

|

I

A2

A3

A4

I

l

I

I

I

I

I

A5

CAP

V5

V4

V3

V2

Vl

BRANCH

x

LEVEL

FIGURE 4. Distribution of P O 2 in the vascular network for blood flow elevated or reduced with respect to the control level. [] 2Q; + Q; o 0 . 7 5 Q ; n 0.SQ.

change with the dilation of the arterioles (in accordance with Eqs. 9 to 11), then the PO2 gradients decrease (Fig. 5). Also, the P O 2 level in the venous blood and tissue increases. The amount of oxygen carried by the blood (Eq. 2), and the blood flow rate (Eqs. 9 and 11) depend upon the systematic hematocrit, which can change significantly in anemia and polycythemia (12). The PO2 distributions predicted by the model are shown in Fig. 6 for hematocrit in the range 20~ to 40~ for constant and for variable blood flow. For the constant flow, the precapillary PO2 gradients increase as the hematocrit decreases. P O 2 in the venous blood and tissue are decreased by 10 m m H g as the hematocrit falls from 30~ to 20%. When the hematocrit rises from 30% to 40%, the venous and tissue P O 2 increase by 6 mmHg. Increase in the blood hematocrit causes the viscosity of blood to increase and the blood flow rate to decrease. When the blood flow rate is allowed to change with the hematocrit in accordance with Eqs. (9)-(11), the difference between P O 2 at constant flow and at control level in a compartment for a given hematocrit is diminished. The venous and tissue PO2 decrease by 6 mmHg when the hematocrit falls from 30% to 20%. The corresponding increase is 2 mmHg with the rise in hematocrit from 30% to 40%. Two factors affect the delivery of oxygen to the tissue: (a) for constant flow, the

28

M . Sharan et al. 100gO80. 70-

~

60

~.

40-

20100

i

i

I

I

I

A1

A2

A3

/~

~

I

I

C/~P V5

I

i

I

]

V4

V~

V2

V1

T

BRANCH LEVEL FIGURE 5. Distribution o f PO2 in t h e v a s c u l a r n e t w o r k w i t h t h e variation in arteriolar diameter. 0 . 8 5 D o ; + De; o 1.2Da; ~ 1.5D~; x 2D~.

10090"

80-

70-

~-~ "r

E

60-

.~E

50-

(:L

40-

\\ \, 20

10

0

i IN

A1

~

A3

A4

A5

CAP

V5

V4

V3

V2

Vl

BRANCH LEVEL FIGURE 6. PO 2 distribution in t h e n e t w o r k for t h e h e m a t o c r i t in t h e range 0 . 2 to 0 , 4 f o r c o n s t a n t and variable f l o w . + H = 0 . 3 in normal c o n d i t i o n s ; H = 0 , 2 w i t h Q c o n s t a n t ; A H = 0 , 2 w i t h Q varied; o H = 0 . 4 w i t h Q c o n s t a n t ; • H = 0 . 4 w i t h Q varied.

Model of Oxygen Transport

29

decrease in hematocrit reduces the 02 content in the blood and causes a decrease of the tissue P O 2 and (b) for constant 02 content, the decrease in hematocrit leads to an increase o f blood flow rate resulting in an increase o f tissue PO2 (Fig. 4). With these opposing tendencies, the model shows that the tissue PO2 decreases as the hematocrit decreases (Fig. 6). Thus, the model predicts the dominant role of 02 content over blood flow alterations due to viscosity alone on the resultant tissue PO2. During hypoxic hypoxia the arterial PO2 is reduced. Figure 7a shows that for a constant flow the fractional fall in precapillary PO2 is diminished with reduced arterial PO2. There are two factors operating here: (a) the PO2 gradient between the intravascular and tissue regions and (b) the slope of the oxygen dissociation curve. When the intravascular POE in high, a large difference between the intravascular PO2 and the tissue PO2 determines the flux of 02 from the vessel to the tissue, but the small slope of ODC indicates that the amount o f oxygen bound to hemoglobin does not change significantly. On the other hand, at lower PO2, the 02 tension gradients between the intravascular and tissue regions become smaller and the slope of ODC becomes steeper. A small variation in the PO2 causes a release o f a large amount of O2 from the hemoglobin. The venous PO 2 and tissue PO2 decrease as the arterial PO2 is reduced. The fall in tissue POE is only about 2 mmHg when the arterial POE is reduced from 95 to 80 mmHg, whereas the corresponding fall is 13.8 m m H g when the arterial PO2 is

90'

'~176 l 80 " 70

"I"

60

E ~E

50 r

12_

40 2~ ,

30-

\ \

20-

X\, \, \ \ \\

)6"~X

10-

X

"" \\

0 iN

l

I

I

l

I

I

I

l

I

[

I

A1

A2

A3

A4

AS

CAP

V5

V4

V3

V2

Vl

BRANCH LEVEL

FIGURE 7a. Longitudinal distribution of PO2 in the vascular network for various values of inlet PO2 for constant flow. + 9 5 mm Hg; [] 8 0 mm Hg; 6 0 m m Hg; & 50 mm Hg; x 41 m m Hg.

30

M. Sharan et al. 10o 9o I

80 7o r

6O

d O3

5O F 403020100

IN

I

I

I

1

I

I

I

I

I

I

I

A1

A2

A3

A4

A5

CAP

V5

V4

V3

V2

Vl

BRANCH LEVEL FIGURE 7b. Distribution of SO2 in the network. The symbols are same as in Fig. 7a.

reduced from 50 mmHg to 41 mmHg. Thus, for lower PO2, a small change in arterial POz leads to significant variation in the tissue PO2, whereas the tissue PO2 is practically independent of the arterial P02 level if the latter is sufficiently high. Figure 7b represents the distribution of SOz in the vascular compartments with the reduction in the arterial PO2. It shows that the total fall in oxygen saturation (the difference between the inlet and outlet SO2 values) remains constant. This is a consequence of the conservation of mass because the flow and the metabolic uptake of 02 remain constant. The precapillary gradients of SOz are not significantly affected by the PO2 in the arterial blood. However, the precapillary gradients of PO2 decrease (Fig. 7a) as the arterial PO2 is reduced. Experimental studies on the effects of shifts in ODC provide evidence of the importance of cerebral tissue PO2 in regulation of cerebral blood flow in normal and hypoxic conditions (19). The position of ODC is described by Pso which can change with pH and P C 0 2 . In vivo, P50 can also be altered by exchange transfusion. Figure 8a exhibits the PO2 distribution in the vascular and tissue compartments with variation in Pso for constant flow. It shows that precapillary PO2 gradients increase with the decrease in Pso. The reason is that the ODC becomes steeper with decreased Pso, and the reduction in SOz causes smaller change in intravascular PO2 with reduced P50. As a result, the gradient between the intravascular PO2 and the tissue PO2 increases with decrease in Pso. Also, the venous and tissue

Model of Oxygen Transport

31

'il 7O

6o

3O 2O \\\I

10 0 IN

i AI

I /,2

i ~

i Nl

i /~5

i CAP

i V5

l V4

i V'3

i V2

i VI

BRANCH LEVEL FIGURE 8a. L o n g i t u d i n a l distribution of PO 2 in t h e v a s c u l a r n e t w o r k f o r v a r i o u s values of P5o f o r c o n s t a n t f l o w , [] 5 5 m m Hg; + 4 0 m m Hg; o 3 0 m m Hg; ~ 2 5 m m Hg; x 2 0 m m Hg,

60

50.

40-IE

o. W

I-

20-

10-

0 - ' I0

i

I

i

i

30

50

Pso

(mm

i

70

Hg)

FIGURE 8b. Variation of tissue P 0 2 w i t h P5o f o r different values of blood f l o w rate. [] 0 . 7 5 Q ; + Q; o 1 . 5 Q ; ~ 2Q.

32

M. Sharan et al. 100,

90 80 70 60 v

ae r

50

I

40-

2010-

0

IN

I

I

I

I

I

I

I

I

I

|

A1

A2

k3

M

A5

CAP

V5

V4

V3

V2

Vl

BRANCH LEVEL FIGURE 8c. Distribution o f SO2 in vascular network with Pso for constant f l o w . D 55 m m Hg; + 40 m m Hg; o 3 0 m m Hg; A 2 0 m m Hg.

P O 2 decrease as Pso decreases because in this case less oxygen is released from the hemoglobin. The decrease in v e n o u s P O 2 with decrease in P50 has been observed experimentally, (17) and (19). It is known (19) that increase in Pso reduces the cerebral blood flow. Figure 8b shows the effect of simultaneous variation of Ps0 and blood flow on tissue P O 2. It shows that tissue P O 2 increases as the Ps0 and the blood flow rate increase. However, in vivo, these two parameters have opposite effects on tissue POz because the cerebral blood flow decreases as Pso increases. Figure 8c shows the variations o f oxygen saturation in the vascular compartments with Ps0. As Pso increases, SO2 decreases not only in the vascular compartments, but also in the arterial blood entering the vascular bed. The total fall in oxygen saturation is almost constant with Ps0 (i.e., the reduction in the amount o f SO2 in the arterial blood due t o increase in Ps0 is the same as the corresponding reduction in the venous blood). Also, the precapillary gradients of SO2 are not significantly affected with the variation in Ps0. These results follow from the assumption that the blood flow and metabolic rate are constant. Carbon monoxide exposure causes a leftward shift in the ODC, thus increasing oxygen affinity. Figure 9a represents the P O E distribution as the carboxyhemoglobin saturation ( S C O ) changes from 0~ to 48070. For constant flow, the precapillary

Model of Oxygen Transport

33

70-

"I" E ~E

O050-

\\ '\

~

\ \

20-

\

10\\\; 0

IN

I

I

l

I

I

AI

A2

A3

A4

A5

I

CAP

1

I

F

I

I

V5

V4

V3

V2

Vl

\1

BRANCH LEVEL FIGURE 9a. Longitudinal distribution o f PO2 in the v a s c u l a r n e t w o r k f o r d i f f e r e n t levels o f hemoglobin saturation w i t h carbon m o n o x i d e f o r c o n s t a n t f l o w . + 0 % ; o 1 0 % ; ~ 2 5 % ; • 4 0 % ; v 4 8 % .

100

90. 807060-

o

403020-

I

I

I

I

I

I

I

I

r

I

A1

A2

A3

A4

A5

CAP

V5

V4

V3

V2

Vl

BRANCH LEVEL

FIGURE 9b. Distribution of SO2A in the vascular network for the same parameters as in Fig. 9a. SO2A is the hemoglobin saturation with 02 in reference to the remaining available binding sites of hemoglobin for 02.

34

M. Sharan et al.

gradients increase substantially with the increase in SCO. There are two factors working here: (a) the increase in the concentration of carbon monoxide causes an increase in the oxygen affinity or decrease of Ps0 and (b) the net amount o f hemoglobin available for oxygen transport in the blood decreases because O2 binding sites of hemoglobin are occupied by CO. The effect of hemoglobin concentration on the PO2 distribution in the vascular and tissue compartments has the same characteristic as that of hematocrit at constant flow. As discussed earlier, precapillary P O 2 gradients increase as Pso or hematocrit decrease. Thus, additional precapillary O2 losses are found (Fig. 9a) as S C O increases. Further, the tissue and v e n o u s P O 2 decrease with increase in S C O because the blood carries less oxygen and also less oxygen is released from hemoglobin. For constant flow, tissue P O 2 approaches zero (0.2 mmHg) with 48% carboxyhemoglobin saturation. Figure 9b represents the fractional hemoglobin saturation with respect to remaining available binding sites of hemoglobin for oxygen ( S O z A ) in the vascular compartments with the variation of carboxyhemoglobin saturation. The fall in saturation in the precapillary vessels is about 5% with 0% H b C O and increases to 6.5% with 48% H b C O . The hemoglobin saturation in the arterioles increases whereas in the capillaries and venules it decreases as the concentration of CO increases due to the leftward shift in the ODC. Figure 10 presents the PO2 distribution in vascular and tissue compartments with 50% elevation and 50% reduction of O2 consumption rate from control level. The model predicts that for constant flow, additional precapillary losses occur with the increase in metabolic rate; the venous and tissue P O 2 a r e decreased. In the present study, the value of the Krogh tissue cylinder radius is calculated from the brain volume, the number of perfused capillaries in the brain, and the average capillary length. For the data in Table 1, the ratio of Krogh tissue cylinder radius to capillary radius is 8.99, similar to a previously used value of 9 (22). However, in reality, the number of perfused capillaries and consequently the ratio R may vary within a certain range. Thus, in the model the number of perfused capillaries in the brain is increased or decreased by 50% from control. The corresponding values of the ratio are 7.34 and 12.72. The alterations in the number of perfused capillaries are associated with changes in blood flow and intercapillary distances. For constant flow, the decrease or increase in the number of perfused capillaries does not significantly affect the P O 2 distribution in the vascular compartments. However, it changes the tissue PO2 significantly. For example, with a 50% increase in the number of perfused capillaries, PO2 changes in the vascular compartments are less than 1.2% and the tissue P O 2 rises from 29.4 to 33.8 mmHg. With a 50% decrease, tissue PO2 dropped from 29.4 to 15.6 mmHg and the changes in arteriolar P O 2 a r e less than 3.4%. On the other hand, the increase in the number of perfused capillaries leads to an increase in the blood flow. We also note that the difference between venous and tissue P O 2 is sensitive to the number of perfused capillaries. To summarize the results of the sensitivity study, we have found that the tissue P O 2 decreases as: the blood flow decreases; PO2 in the arterial blood is reduced; hematocrit decreases; Ps0 decreases; the concentration of carbon monoxide increases; the metabolic rate increases; the arterial diameter decreases; the number of perfused capillaries decreases. Also, the longitudinal PO 2 gradients in the precapillary vessels increase in all cases except when P O 2 in the arterial blood is reduced. Further, the PO 2

Model of Oxygen Transport

35

I

90 80 7O

I

r

r

E

\

"

e

.~ \ \ \ \

1

IN

I

I

I

I

i

[

I

i

A1

~

~

~

~

CAP

~

V4

i

~

i

i

V2

V1

BRANCH LEVEL FIGURE 10.

D i s t r i b u t i o n o f PO 2 in t h e v a s c u l a r n e t w o r k

with constant fow.

for different values of the metabolic

rate

D 1 . 5 M ; + M ; 0 . 5 M .

l o n g i t u d i n a l P O 2 gradients in the postcapillary vessels are found to be small and these gradients are not sensitive to the above parameters. One of the motivations of the present study is to analyze the oxygen transport during hypoxic, anemic, and CO hypoxia. These types of hypoxia can be described in terms of the parameters of the model. However, the quantitative analysis of experimental data (12,16-18) during hypoxia using the present model is not presented here and will be reported elsewhere.

CONCLUSIONS In future studies, the model should be improved in a number of ways. First, the treatment of pre- and postcapillary transport could be i m p r o v e d by taking into account explicitly the geometry and hemodynamics of the capillary bed, instead of using a phenomenological model (28). Second, intravascular transport can play an important role in oxygen exchange (9,10). In the present work the intravascular resistance to oxygen transport is neglected. Finally, detailed morphological data for the sheep brain, or for brain of any species, are also not currently available. Even though the set of data constructed here is consistent with pressure and flow measurements and stereological information available in the literature, it is clear that for the model to be reliable for quantitative predictions, a more detailed knowledge of vascular morphology is required.

36

M . Sharan et al.

REFERENCES 1. Altman, P.L.; Dittmer, D.S., editors. Respiration and Circulation. Bethesda, MD: Fed. Am. Soc. Exp. Biol. 1971. 2. Bruley, D.F. Probabilistic solutions and models: oxygen transport in the brain microcirculation. In: Gross, J.F.; Popel, A., eds. Mathematics of Microcirculation Phenomena. New York: Raven Press; 1980; pp. 133-158. 3. Collier, C.R. Oxygen affinity Of human blood in presence of carbon monoxide. J. Appl. Physiol. 40: 487-490; 1976. 4. Davies, P.W.; Bronk, D.W. Oxygen tension in mammalian brain. Fed. Proc. 16:689-692; 1957. 5. Duling, B.R.; Berne, R.M. Longitudinal gradients in periarteriolar oxygen tension. A possible mechanism for the participation of oxygen in local regulation of blood flow. Circ. Res. 27:669-678; 1970. 6. Duling, B.R.; Kuschinsky, W.; Wahl, M. Measurements of the perivascular PO2 in the vicinity of the pial vessels of the cat. Pflugers Arch. 383:29-34; 1979. 7. Duling, B.R.; Pittman, R.N. Oxygen tension: dependent or independent variable in local control of blood flow. Fed. Proc. 34:2012-2019; 1975. 8. Duvernoy, H.M.; Delon, S.; Vannson, J.L. Cortical blood vessels in the human brain. Brain Research Bulletin 7:519-579; 1981. 9. Federspiel, W.J.; Popel, A.S. A theoretical analysis of the effect of the particulate nature of blood on oxygen release in capillaries. Microvasc. Res. 32:164-189; 1986. 10. Gutierrez, G. The rate of oxygen release and its effect on capillary O2 t e n s i o n - a mathematical analysis. Respir. Physiol. 63:79-96; 1986. 11. Harper, S.L.; Bohlen, H.G. Microvascular adaptation in the cerebral cortex of adult spontaneously hypertensive rats. Hypertension 6:408-419; 1984. 12. Hudak, M.L.; Koehler, R.C.; Rosenberg, A.A.; Traystman, R.J.; Jones, M.D., Jr. Effect of hematocrit on cerebral blood flow. Am. J. Physiol. 251:H63-H70; 1986. 13. Ivanov, K.P.; Derry, A.N.; Vovenko, E.P.; Samoilov, M.O.; Semionov, D.G. Direct measurement of oxygen tension at the surface of arterioles, capillaries and venules of the cerebral cortex. Pflugers Arch. 393:118-120; 1982. 14. Ivanov, K.P.; Kislyakov, Y.Y.; Samoilov, M.O. Microcirculation and transport of oxygen to neurons of the brain. Microvasc. Res. 18:434-441; 1979. 15. Kalinina, M.K.; Levkovich, Y.I.; Ivanov, K.P.; Trusova, V.K. Blood flow velocity in brain cortex capillaries (by microfilming). Dokl. Acad. Nauk. SSSR. 226:230-233; 1976. 16. Kobari, M.; Gotoh, F.; Fukuuchi, Y.; Tanaka, K.; Suzuki, N.; Vematsu, D. Blood flow velocity in the pial arteries of cats, with particular reference to the vessel diameter. J. Cereb. Blood Flow Metab. 4:110-114; 1984. 17. Koehler, R.C.; Traystman, R.J.; Rosenberg, A.A.; Hudak, M.L.; Jones, M.D., Jr. Roles of O2hemoglobin affinity on cerebrovascular response to carbon monoxide hypoxia. Am. J. Physiol. 245: H1019-H1023; 1983. 18. Koehler, R.C.; Traystman, R.J.; Zeger, S.; Rogers, M.C.; Jones, M.D., Jr. Comparison of cerebrovascular response to hypoxic and carbon monoxide hypoxia in newborn and adult sheep. J. Cereb. Blood Flow Metab. 4:115-122; 1984. 19. Koehler, R.C.; Traystman, R.J.; Jones, M.D., Jr. Influence of reduced oxyhemoglobin affinity on cerebrovascular response to hypoxic hypoxia. Am. J. Physiol. 251:H756-H763; 1986. 20. Ma, Y.P.; Koo, A.; Kwan, H.C.; Cheng, K.K. On line measurement of the dynamic velocity of erythrocytes in the cerebral microvessels in the rat. Microvasc. Res. 8:1-13; 1974. 21. Metzger, H. The influence of space-distributed parameters on the calculation of substrate and gas exchange in microvascular units. Math. Biosci. 30:31-45; 1976. 22. Moskalenko, Y.E.; Weinstein, G.B.; Demchenko, I.T.; Kislyakov, Y.Y.; Krivchenko, A.I. Biophysical Aspects of Cerebral Circulation. Oxford: Pergamon Press; 1980. 23. Opitz, E.; Schneider, M. The oxygen supply of the brain and the mechanism of deficiency effects. Ergebnisse der Physiologic, Biologischen Chemic, und Experimentallen Pharmakologie, 46:126-260; 1950. 24. Papenfuss, H.D.; Gross, J.F. Mathematical simulation of blood flow in microcirculatory networks. In: Popel, A.S.; Johnson, P.C., eds. Microvascular Networks: Experimental and Theoretical Studies. Basel: Karger; 1986: pp. 168-181. 25. Pawlik, G.; Rackl, A.; Bing, R.J. Quantitative capillary topography and blood flow in the cerebral cortex of cats: an in vivo microscopic study. Brain Research. 208:35-58; 1981.

Model

of Oxygen

Transport

37

26. Pittman, R.N. Oxygen delivery and transport in the microcirculation. In: McDonagh, P.F., ed. Microvascular Perfusion and Transport in Health and Disease. Basel: Karger; 1987: pp. 60-79. 27. Pittman, R.N.; Duling, B.R. The determination of oxyge~a availability in the microcirculation. In: Jobsis, F.F., ed. Oxygen and Physiological Function. Dallas: Professional Information Library; 1977: pp. 133-147. 28. Popel, A.S.; Gross, J.F. Analysis of oxygen diffusion from arteriolar networks. Am. J. Physiol. 237: H681-H689; 1979. 29. Press, W.H.; Flannery, B.P.; Teukolsky, S.A.; Vetterling, W.T. Numerical Recipes. Cambridge: Cambridge University Press; 1986. 30. Roth, ~.C.; Wade, K.'The effects of transmural transport in the microcirculation: a two gas species model. Microvasc. Res. 32:64-83; 1986. 31. Samaja, M.; Gattinoni, L. Oxygen affinity in the blood of sheep. Respir. Physiol. 34:385-392; 1978. 32. Severns, M.L.; Adams, J.M. The relation between Krogh and compartment transport models. J. Theor. Biol. 97:239-249; 1982. 33. Sharan, M.; Popel, A.S. A mathematical model of countercurrent exchange of oxygen between paired arterioles and venules. Math. Biosci. 91:17-34; 1988. 34. Weiss, H.R.; Buchweitz, E.; Murtha, T.J.; Auletta, M. Quantitative regional determination of morphometric indices of the total and perfused capillary network in the rat brain. Circ. Res. 51:494-503; 1982. 35. Winslow, R.M.; Swenberg, M.L.; Berger, R.L.; Shrager, R.I.; Luzzana, M.; Samaja, M.; RossiBernardi, L. Oxygen equilibrium curve of normal blood and its evaluation by Adair equation. 3. Biol. Chem. 252:2331-2337; 1977.

NOMENCLATURE Adair parameters distance between the surfaces of the paired arterioles and venules C = oxygen content D t = diffusion coefficient of 02 in the tissue Oi = vessel diameter in ith c o m p a r t m e n t D~ = arteriolar diameter O v = venular diameter e = geometrical coefficient for the transport f r o m blood to tissue E = diffusion conductance for the vascular c o m p a r t m e n t f = geometrical coefficient for the countercurrent transport F = diffusion conductance for the noncapillary c o m p a r t m e n t s with countercurrent diffusional exchange n blood hematocrit n ~ = blood hematocrit under normal conditions [Hb] = total hemoglobin concentration in the blood under normal conditions J = oxygen flux from the vessel L = characteristic length of the vessel 1t ---- penetration length m+l= number of arteriolar or venular c o m p a r t m e n t s M = metabolic rate N = number of vessels in a c o m p a r t m e n t = partial pressure of 02 in the capillary P P = partial pressure of 02 in the c o m p a r t m e n t = partial pressure of 02 at 50~ hemoglobin saturation with 02 /'50 P~ = tissue PO2 far from the arteriole q = hydraulic pressure = blood flow rate O ai b

=

38

M. Sharan et al.

R SCO S02

--= =

Tso

=

u

= = = = = = =

r

V w Z

radial c o o r d i n a t e resistance s a t u r a t i o n o f h e m o g l o b i n with C O s a t u r a t i o n of h e m o g l o b i n with O2 P O 2 at 50~ s a t u r a t i o n o f h e m o g l o b i n with 0 2 in reference to r e m a i n i n g available b i n d i n g sites o f h e m o g l o b i n for O2 in presence o f C O velocity tissue v o l u m e ratio o f capillary radius to tissue radius axial c o o r d i n a t e solubility o f 02 0 2 carrying capacity, Eq. (3) viscosity

Subscripts: a

ap c

i pl r

t u

= = = = = = = =

arteriole apparent capillary ith compartment plasma relative tissue venule