Pergamon M&hi. Comput. Modelling Vol. 22, No. 9, pp. 99-111, 1995

Copyright@1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved

0895-7177/95 $9.50 + 0.00

0895.7177(95)00171-9

An Analytical Model for Oxygen Transport in Tissue Capillaries

in a Hyperbaric Environment with First Order Metabolic Consumption

M. SHARAN~ , M. P. SINGH AND B. SZNGHS Centre for Atmospheric Sciences, Indian Institute of Technology, Delhi

Hauz Khas, New Delhi-110016, India

Abstract-A mathematical model for oxygen transport in the systemic capillary and the sur- rounding tissue is described. The model takes into account the molecular diffusion in both axial and radial directions in the capillary and tissue, the convective effect of the blood, and the saturation of haemoglobin with oxygen in the blood. The first order metabolic consumption rate of 02 in the tissue is considered. The nonlinear 02 dissociation curve (ODC) is approximated by a linear func- tion to simulate the conditions of hyperbaric environment. Its slope is determined by evaluating the derivative of ODC at the arterial PO2. The resulting system of partial differential equations is solved analytically by the method of eigenfunction expansion. It is shown that 02 is transported in the first one-fifth part of the tissue by diffusion (radial and axial), while, in the remaining part of the tissue, radial diffusion is stabilized. The accumulation of 02 in the tissue is found to be larger with the first order metabolic rate in comparison to the zero order metabolic rate.

Keywords-Mathematical model, Oxygen transport, Hyperbaric environment, Eigenfunction ex- pansion, Systemic capillaries, Blood, Tissue, Partial differential equations.

1. INTRODUCTION

The transport of oxygen from the lungs to the systemic capillaries is accomplished by a process of bulk flow as oxygenated blood is carried to the tissue. Once blood reaches the systemic capillaries, oxygen dissociates from haemoglobin, diffuses through the red cell membrane into the plasma and from there into the tissue. The delivery of 02 to the cells depends upon the

molecular diffusion, convection, and chemical combination of haemoglobin with OZ. 02 assumes a much more significant role in a hyperbaric environment [l]. Breathing compressed air at high pressure results in the diffusion of a large amount of 0s and other gases in the blood and leads to a large amount of 02 in the blood plasma. The dissolved 02 can diffuse freely in the tissue and fulfils the normal requirements of the tissue. As a result, depending upon the activity of the tissue, the combined 0s with haemoglobin may not be needed, or if it is needed, much less will be required. A high excess partial pressure of 02, even in a short time, may cause accumulation of a substantial amount of 0s in the tissue, leading to adverse effects on the cells which is known as oxygen poisoning. The symptoms indicating 0s poisoning are convulsions, coma, etc.

The first simple mathematical model for oxygen exchange was formulated by Krogh [2]. The model is based on the concept that oxygen from a capillary diffuses only into a tissue cylinder concentric with the capillary, so that the 0s flux at the external surface of the cylinder vanishes.

$Author to whom all correspondence should be addressed. #Current address: Department of Mathematics, Delhi College of Engineering, Kashmeri Gate Delhi-110006, India.

Typeset by .A@-TEX

MM ZZ?P.H 99

100 M. SHARAN et al.

This model has been extended by several investigators [3-51. Blum [S] has taken into account

the capillary wall resistance to 02 transport and has solved analytically the steady state problem

for zero and first order consumption of 02 in the tissue. However, Salathe and Wang [7] and

Fletcher and Schubert [S] have pointed out that the solution obtained by Blum was not correct.

In these studies, the flow conditions in the capillary have not been considered.

Keeping this in view, Reneau et al. [9,10) have developed a numerical model for the transport

of 02 in the capillary and tissue by taking into account the intracapillary diffusion, convective

effect of the blood, and axial diffusion in the tissue. Salathe et al. [ll] have analyzed the 02

transport in the tissue by assuming radial mixing in the capillary. Gutierrez [12] has emphasized

the importance of nonequilibrium chemical kinetics on the delivery of 02 to the tissue. Recently,

Singh et al. 1131 and Sharan et d. [5,14] have described mathematical models of 02 and COz

transport in the capillary and tissue. The governing equations with physiologically relevant

boundary conditions have been solved analytically, as well as numerically. It has been pointed

out that the toxic effects on the cells may be due to both the so-called “Oxygen poisoning” and

the accumulation of COz. In these studies, the 02 was assumed to be consumed with zero order

metabolic rate. In the present study, we describe a mathematical model of oxygen transport in the systemic

capillaries and its surrounding tissue in a hyperbaric environment. The model takes into account

the molecular diffusion in both axial and radial directions in the capillary and tissue, the convec-

tive effect of the blood, the saturation of haemoglobin with 0s in the blood, and the Hurst order

metabolic consumption rate in the tissue.

2. MATHEMATICAL DESCRIPTION

When the blood flows through the systemic capillary, the dissolved oxygen diffuses into the

tissue where it is utilized by the cells during the metabolic process. The 0s diffusing from

plasma to the surrounding tissue is replenished by the 02 released from the oxyhaemoglobin

present inside the red blood cells.

The transport of 02 in the blood flowing through the capillary depends upon the molecular

diffusion, convection, and release of 02 from the oxyhaemoglobin. Taking into account these

transport mechanisms, the mass balance principle leads to the following equation 1151

g = V.(DVc),

where c is the 02 concentration in the blood, D is the diffusion coefficient of 02 in the blood,

s is the total oxygen content in the blood, and the derivative 6 is given by

D if_ + u.v, E = dt

in which U is the velocity vector.

Oxygen is carried by the blood in two forms:

(i) as physically dissolved in plasma, and

(ii) in combination with the haemoglobin.

It is recognised that the chemical kinetics of 02 with haemoglobin in the blood plays an important role in the 02 delivery to the tissue [16]. For the sake of simplicity, we assume that the chemical

reaction between haemoglobin and 0s is in equilibrium. The total 02 content in the blood is

defined by

s=CIP+N$, (3)

where Cr is the oxygen solubility coefficient in the blood, N is the oxygen carrying capacity of

blood, P is the partial pressure of 02, and _1cI is the fractional saturation of haemoglobin with Oz.

Analytical Model for Oxygen Transport 101

The transport of oxygen in the tissue depends upon the molecular diffusion and 02 consumption during the metabolic process, and is governed by

g = V.(D’ Vc’) - g(c’), (4)

where c’ is the 02 concentration in the tissue, D’ is the diffusion coefficient of 02 in the tissue, and g(c’) is a function of the 02 concentration in the tissue, indicating the utilization of 02 by

the tissue cells.

We consider the channel version (Figure 1) of an idealized Krogh capillary tissue cylinder

as the geometrical representation of the capillary beds. A layer of blood is surrounded by a symmetrically placed layer of tissue. The arterial blood enters the systemic capillary at z = 0. The blood flows with a uniform speed w. The 02 is transported in the capillary and tissue by the molecular diffusion in the radial, as well as axial directions. In tissue, the rate of metabolic consumption is assumed to be of the form

g(c’) = go + 91 c’, (5)

where go and gi are constants.

A$Jor;$l

z=o

Lethol corner Vulnerable corner

*axVenous _c--- capillary L _ ___?zJ blood

=Yv

--------I Figure 1. Sketch of the mathematical model.

Under steady state conditions, equations (1) and (4) become

D d2P zG+Dzg=Vg, O<x<Rl and Ouzel, (6)

D, a2Pl 5 dz2 + 0; g = $ + g1 P’, RI < x < R2 and 0 < t < L, (7)

where

V(P) = u(I + 4(P)),

in which $ is proportional to the slope of the 02 dissociation curve (ODC):

(8)

b(P) = -$ $. (9)

Here, RI and Rz denote the radii of the capillary and tissue, respectively, L is the length of the capillary, P is the partial pressure of 02 in the blood, and D, and D, are the diffusion coefficients in the x and .Z directions. A symbol with primes denotes the corresponding quantity in the tissue.

102 bf. SHARAN et al.

Equations (6) and (7) are to be solved with respect to the following boundary and interface

conditions:

(1)

(11)

Boundary Conditions:

(i) Due to symmetry,

at 2 = 0, ap 0 z= 7 O<zIL. (104

(ii) Since the tissue layer surrounding the capillary layer is closed, the 02 cannot penetrate through the walls of the tissue, and accordingly, we have:

dP’ atx=Ra, -=O,

dX OIZIL,

at z = 0, dP’ o z= 7 RllXLR2, and

at z = L, dP’ o -7$y= 7 RI < x 5 Rz.

(iii) At the exit of the capillary, we assume the nondiffusional flux condition:

(lob)

(16c)

God)

at z = L, dP 0 T&= 7 0 5 x 5 RI. (1Oe)

(iv) At the entry of the capillary, the partial pressure of 02 in the blood is the same as in the arterial blood, i.e.,

at 2 = 0, P = Part, 0 Ix 5 RI. (lOf>

Interface Conditions: We assume that the resistance offered by the capillary and tissue interface to msss transfer is negligibly small. Accordingly, the pressure and the flux across the interface are continuous.

(i) P(Ri, 2) = P’(Ri, z), 0 I z IL, and (ila)

(ii) D,Ci g(R~,z) = D;C; ~(RIJ), O<z<L. W)

Since the saturation function for the ODC is nonlinear, equations (6), (7) with the condi- tions (lo), (11) form a nonlinear system of partial differential equations. When the partial pressure of 02 is increased under high pressure, the amount of dissolved oxygen increases. In this case, the tissue would use the dissolved oxygen before calling upon that combined with the haemoglobin. As a result, depending upon the activity of the tissues, combined oxygen may not be needed, or if it is needed, much less will be required [l]. The ODC is almost linear for higher range of PO2 [17], which is relevant to hyperbaric conditions. Thus, we approximate the saturation function 1c, by

$(P) = h p + k2, (12)

where kl and kz are constants. Slope ICI represents the deviation of the saturation curve from the line of zero slope (the horizontal line). In the computation, the value of ICI is difficult to assign. Sharan et al. [5] and Singh et al. [13] have chosen kl arbitrarily. In the present analysis, we calculate ICI by evaluating the slope of the ODC at the inlet PO2; i.e.,

Analytical Model for Oxygen Transport 103

Many forms of the equation can be used to represent the ODC (see, e.g., [12,18]). In the present

model, we use the Hill equation

G(P) = Plp50)n

1 + (pIp5o)” ’ 04

where PRO is the partial pressure of 0s at 50% saturation and n is the Hill parameter.

By approximating the saturation $J by a linear function, the equations (6) and (7) become

linear and are amenable to find an analytical solution.

NONDIMENSIONALIZATION.

By introducing the dimensionless variables

X X* = -

R2’ z* = t

L' p* = $,

c T* = ;,

c and suppressing the stars, the governing equations (6) and (7) become

a2P 2 @P m+CY -@.= vi!!z

e a2 O<x<R and O<z<l,

and

~+P2~=ET+~, R<x<l and O<zcl,

where

(15)

(16)

(17)

and R = Rl/R2 are dimensionless constants, and P, is the characteristic partial pressure of 02.

The corresponding boundary conditions (lOa)-( 10f) and interface conditions (lla)-( llb) be-

(I) Boundary Conditions:

dP 0 z= 7

dT 0 z= 7

dT 0 z=,

dT 0 z= 1

dP 0 dz=,

P = PO,

in which

x = 0, O<zSl,

x= 1, OSZll,

z = 0, R<xIl,

z= 1, RSxll,

z = 1, O<xIR,

z = 0, OL:xlR,

Po2$. c

(II) Interface Conditions:

P(R 2) = T(R, 21, Olzll,

dp=6aT dx dx’

and

(184

(18b)

(18~)

(184

(18e)

(18f)

(1%

WI

* = Da ci --. Dz Cl (20)

104 M. SHARAN et al.

3. ANALYSIS

A solution to equation (16) in the tissue region satisfying the boundary conditions (18b)-(18d) is given by

T(x,z) = g A, cosh{p,(l -x)} cosm~z - ;, (21) m=O

where

pm = &+m2P2n2, A’

A, = m cash pm ’

m = 0, 1,2, . . . , and

in which AL are unknown coefficients.

A solution of equation (15) in the capillary region satisfying boundary conditions (18a), (18e), and (18f) is obtained by the method of separation of variables in the form

P(x, z) = PO + f) B, eaz cosh(X,z) sin(b,x), (22) n=O

where B,, n=0,1,2 ,..., are the unknown coefficients,

n = 0, 1,2, . . . ) (23)

are the eigenvalues, and the constants b,, n = 0, 1,2,. . . , are the roots of the transcendental equation

tanb, = -k; a

It can be shown that all the roots of equation (24) accounted for in the solution, we can assume b, > 0.

satisfy the condition

a > 0. (24

will be real. Since -b, has already been We further notice that each of b, > 0 will

(2n+l)7r <b 2 n

< (2n+2)7r 2 ’

n = 0, 1,2, . . . .

Since the bn9s are positive and real, the Xn7s will also be positive and real.

3.1. Estimation of Series Coefficients

Applying the interface conditions (19a) and (19b), we obtain

PO + 2 B, cosh(X,R) eaz sin(b,z) = 2 A, n=O m=O

cash {p,(l - R)} cosm~.z - ;,

and

2 B, X, sinh(X,R) eaz sin&z) = -S 2 A, pm sinh {pm( 1 - R)} COSTMZ.

n=O m=O

(25)

(26)

(27)

Analytical Model for Oxygen Transport 105

Multiplying (26) and (27) by cos k’l~z and integrating from z = 0 to z = 1, we get

1

Ao = cosh{po(l - R))

1

(28)

Ak = coshba(l -RI) n=O 2 B, cosh(LR) d,, ,

1

,

k= 1,2,..., (29)

Ao = -26 pe sinh {pc(l - R)) n=O 2 I?, A, sinh( X,R) d,, , and (30)

1

Ak = -6~k sinh{pk(l -R)} n=O 2 B, A, sinh(X,R) d,, , k = 1,2, s . . . (31)

The relations (28) to (31) yield

2 B, {YO cosh(X,R) + A, sinh(X,R)}d,, = -2ro n=O

, and (32)

9 B, {rk cosh(X,R) + A, sinh(X,R)} dnk = 0, (33) a==0

in which rk = 6 pk tanh (&( 1 - R)} , k = 0, 1,2,. . . , and

1

d,, = 2 s

eat sin(b,z) cos(&rz) dz, k-:0,1,2 ,..., n==0,1,2 ,.... (34) 0

Equations (32) and (33) form an infinite set of nonhomogeneous algebraic equations for the

infinite set of unknown constants B,, n = 0, 1,2,. . . . Once the B,‘s are known, another set of

infinite constants A,, n = O,l, 2,. . . , is computed from equations (28) and (29). The truncated

system of equations corresponding to (32) and (33) for finite N, becomes

k l?, (70 cosh(X,R) + A, sinh(X,R)}d,, = -2~0 n=o ( >

: + PO , and (35)

N

c kn {Q cosh(X,R) -I- A, sinh(X,R)} d,, = 0; k-1,2 ,..., N. (36) n=O

This system (equations (35) and (36)) can be solved for the (N + 1) unknowns & and the

approximate coefficients & converge to B, with the desired accuracy as N -+ co. Once @,,

n = 0, 1,2, . . ) co, are known, another set of constants, a,, n = 0, 1,2,. . . , N, can be computed

with the desired accuracy for sufficiently large N, after truncating equations (28) and (29) for

finite N. Finally, the solution in the tissue, as well as in the capillary can be obtained from

relations (21) and (22). The convergence of the solution is tested by the interface condition (19a);

i.e.,

IPN(R, z) - zV(R, z)l < 6.

The error decreases as N increases.

(37)

3.2. Particular Cases

(i) When 71 = 0, i.e., the first order metabolic consumption. For 11 = 0, the solution (21) in the

tissue region takes the form

T(z, z) = 2 A, cash {p,(l - 5)) cos m;lrz . (38) m=O

106 M. SHAFIAN et al.

The form of the solution in the capillary region remains unchanged with 71. As discussed earlier, the constants A,,, and B,, m = 0, 1,2,. . . , can be computed by solving the equations obtained from equations (28)-(32) by putting Q = 0.

(ii) When < -+ 0, the zero order metabolic consumption or constant metabolism. When < -+ 0,

the term independent of z in the series (21) becomes

which can be derived directly from equation (16) by putting < = 0.

The solution (21) in the tissue region takes the form

(40) T(z,z)=A:,+q ;- ( )

x + 5 A, cosh{p,(l - z)} cos rnrz, m=l

where pm = m p 7r. Rewritting the constant

A;=Ao-r] Z-R , ( )

the solution (40) becomes

T(x, Z) = v f - x - $ + R) + AC, + 2 A, cash {p,(l - x)} cos mx2. (41) m=l

Thus, by taking [ = 0, the solution of the system will be the same as obtained earlier [5,13].

The solution (22) in the capillary region does not change when c = 0. It may be noticed that

equations (29) and (31) for the coefficients remain unchanged, whereas equations (28) and (30)

become

A0 = PO + f 2 B, cosh(X,R) d,, , n=O

and (42)

6v(R-1) = $BnX, sinh(X,R) d,, . (43) n=O

As discussed earlier, the system of equations (33) and (43) can be solved for the constants B,. Once B, are known, the coefficients A, are computed from (29) and (42).

4. RESULTS AND DISCUSSION

The study is undertaken to analyze 02 transport in the systemic circulation in a hyperbaric

environment, by incorporating the axial diffusion in the capillary as well as in tissue and the first

order metabolic consumption.

In order to compute the distribution of PO0 in the capillary (equation (22)) and tissue (equa-

tion (21)), the values of parameters such as diffusion coefficients, solubility coefficients, metabolic

rate, speed of the blood, 02 carrying capacity of the blood, the dimensions of the capillary tissue cylinder, etc., must be specified. The values of the parameters taken for the computation are given in Table 1. The values of the parameters RI, Rz, v, D,, DL, N, Cl, and Ci are taken

from [9,19]. The length L = 0.02 cm is chosen on the basis of the model predictions [14], which is

close to the value 0.018cm given by Hunt and Bruley [19]. The rate of 02 consumption gi = 0.5

Analytical Model for Oxygen Transport 107

Table 1. Values of the parameters used in the computation.

Parameter Value Units

D, 1.2 x 10-S cm2/sec

DZ 1.95 x 10-d cm2/sec

0: 1.7 x 10-s cm2/sec

0: 5.1 x 10-S cm2/sec

V 0.03 cm/set

Rl 3 x 10-d cm

R2 3 x 10-s cm

L 2 x 10-s cm

91 0.5 set-’

Cl 1.527 x 1O-g mole/(cm3 mm Hg)

c; 1.295 x 10-e mole/(cm3 mm Hg)

N 9.1099 x 10-s mole/cm3

0.4 0.6

Axial distance (2) Figure 2. Axial variation of PO2 in the capillary at zr = 0.01 (-0-B) and tissue (-) at z = 1.0. Here cx2 = 0.3656, f12 = 0.0675, V, = 1.192, 6 = 1.2, < = 0.2647,

n = 0. The values of the parameters are given in Table 1.

is chosen from [6]. The diffusion coefficients in the axial direction are chosen in order to get the

physiologically known value of PO2 in venous blood under normal conditions.

The amount of 02 decreases in the capillary-tissue system as the blood flows from the arterial

to the venous end (Figure 2). Comparison of concentration profiles in the capillary and tissue

shows that the decrease of PO2 in the capillary is rapid and significant radial diffusion of 02 takes

place in the tissue close to the arterial end (i.e., about 20% of the capillary length). Gradually, its contribution stabilizes to a constant value.

A larger axial drop of 02 tension in the capillary is seen in the arterial end in the first 20% of

the tissue and, subsequently, axial fall of PO2 is the same in the capillary as well as in the tissue.

Thus, 02 is transported in the first one-fifth part of the tissue by diffusion (radial and axial) while,

in the remaining part of the tissue, radial diffusion stabilizes. The maximum concentration of 02

occurs at the entry point of the tissue on the arterial side. Thus, owing to excessive accumulation

108 M. SHARAN et al.

of 02 in the tissue, the toxic effects, if any, will be first felt (realized) in the region close to the

arterial end of the tissue.

Figure 3 represents the axial variation of POz in the capillary and tissue with the zero and

first order metabolism. For zero order metabolism, the partial pressure of 02 in the capillary

and tissue is obtained from equations (22) and (41), in which the value of constant metabolic consumption rate go is taken from [9]. Equations (22) and (38) are used for computing the 0s

partial pressure in capillary and tissue region, for first order metabolic consumption in which

the value of the rate of metabolic consumption is taken from [6]. The values of the remaining

parameters are the same in both the cases, as given in Table 1. Figure 3 shows that the PO2 in

the cells is lower with constant metabolism, in comparison to the first order metabolism. The

02 accumulation in the tissue becomes larger with the first order metabolic rate, in comparison

to the zero order metabolic rate.

1.0

Axial distance (2)

Figure 3. Oxygen partial pressure along length of the capillary at 2 = 0.01 (-0-B) and tissue at z = 1.0 (-) with zero (i.e., constant) and first order metabolism. For first order < = 0.2647 and q = 0, and for constant metabolism < = 0 and 17 = 0.152.

It may be noted from the graph that for the first order metabolic rate, the PO2 is minimum in

the capillary tissue system at the point (z = 1, z = 1) located on the tissue periphery near the

venous end of the capillary. The PO2 at this point decreases further with zero order metabolic

rate. In this case sufficient 02 may not be available to the cells near the venous end of the

capillary. Thus, the cells may be affected more adversely with zero order metabolic rate in

comparison to first order metabolism. Although the analogy is not strictly applicable, Reneau et

al. [9] have suggested the existence of a lethal corner at the venous end, in their Krogh cylinder

model, where the supply of oxygen is least.

Figure 4 shows the axial variation of PO2 in the capillary-tissue system with the variation

of PO2 in the arterial blood. Since we are mainly interested in the hyperbaric studies, the

computations have been done by taking an arterial 02 partial pressure of more than 100 mmHg.

The PO2 increases in the capillary and tissue as the PO2 in the arterial blood increases. For a

fixed value of PO2 at the entry, the amount of 02 decreases continuously as the blood flows from

the arterial to the venous end of the capillary. The fall becomes more rapid as the entry PO2 increases. The difference in the levels of PO2 between the axis of the capillary and the periphery

Analytical Model for Oxygen nansport 109

of the tissue increases with the increase in arterial PO2, and the net supply of 0s by the capillary is found to increase. This can be explained from the 0s dissociation curve as follows. There are

two factors operating here: (a) the PO2 gradients between the capillary and tissue, and (b) the slope of the 02 dissociation curve. When the capillary PO2 is high, a large difference between the capillary PO2 and tissue PO2 determines the flux of 0s from the capillary to the tissue, but the small slope of the ODC indicates that the amount of 0s bound to haemoglobin does not change signi~cantly. On the other hand, at lower PO2, the 0s tension gradients between the capillary and tissue regions become smaller and the slope of the ODC becomes steeper. A small variation in the PO2 causes a release of a large amount of 02 from the haemoglobin. In an earlier study,

Sharan et al. /5] arterial PO2 and

found that the net supply of oxygen by the capillary was independent of the it was due to constant metabolic rate.

0.4 0.6

Axial distance Df

0.8

Figure 4. The variation of oxygen partial pressure along the length of the capillary at z = 0.01 (-D-U-) and along tissue at z = 1.0 (-), with PO2 in the arterial blood. The values of the parameters are given in Table 1.

The model depends on various dimensionless parameters, such as 02, ,B2, 6, and V,; a2 is the axial diffusion parameter in the capillary and includes the effect due to the diffusion coefficient of 0s in the blood in the axial and radial directions, and due to the radius of the tissue cylinder and its length; p2 is the corresponding axial diffusion parameter in the tissue. The convective parameter V, includes the effect due to blood flow, the 02 carrying capacity of the blood, the radial diffusion coefficient of 0s in the blood, and the radius of the tissue cylinder and its length. The parameter b depends on the diffusivity and solubility properties of two different mediums. The effect of an individual parameter on the delivery of 02 to tissue can be examined in a similar way as was done in [5].

Figure 5 shows that the concentration of 02 in the capillary and tissue decreases as the meta- bolic rate parameter < increases. This is due to the fact that a large amount of 0s is consumed by the cells when the metabolic rate is increased. The difference in the level of PO2 between the venous blood and the tissue periphery decreases as < decreases. The accumulation of 0s at the arterial end increases with the decrease in 5. For E = 0.053, corresponding to gi = 0.1, the difference in PO2 between the arterial and venous blood is insignificant, and the 02 accumulation

110 M. SHARAN et al.

20’ I I I I I I I I I I

0 0.2 0.4 0.6 0.8 1.0

Axial distance(z)

Figure 5. The variation of oxygen partial pressure along the length of the capillary at I = 0.01 (-O-C-) and along tissue at z = 1.0 (-), with the metabolic rate parameter < = gr @/Da, where Rz = 3 x 10m3 cm and Dj, = 1.7 x 10e5 cm2/s. The values of the parameters are given in Table 1.

in the tissue is of the same order as the arterial POs. This can be explained by the fact that, in this case, 02 transported from plasma gets instantaneously replenished from the haemoglobin.

5. CONCLUSIONS

In this paper, we have developed a mathematical model for the transport of 02 in the blood flowing through the systemic circulation. The model takes into account the molecular diffusion in both axial and radial directions in the capillary and tissue, the convective effect of the blood and the saturation of haemoglobin with 0s in the blood. 0s is assumed to be consumed by first order metabolism. An analytical solution of the resulting system of equations is obtained by linearizing the 0s dissociation curve through a straight line. Its slope is determined by evaluating the derivative of the ODC at the arterial POz. This allows us to incorporate the contribution of combined oxygen. Model predicts that the 02 accumulation in the tissue is larger with the first order metabolic rate in comparison to zero order metabolic rate.

In this study, we have not taken into account the transport of COz, which is important because the carriage of one gas affects the transport of the other. The nonequilibrium kinetics of 0s with haemoglobin in the blood plays an important role in the transport of 02 to the tissue [16]. Also, it is shown experimentally [20] and theoretically [21] that a considerable amount of oxygen diffuses through the walls of the precapillary vessels. Thus, the model can be improved by

taking into account the variable metabolic rate, the interaction of 02 and CO2 in the blood, the nonequilibrium chemical kinetics of the gases in the blood, nonlinear oxygen dissociation curve, and the transport through pre- and post-capillary vessels.

REFERENCES

1.

2.

3. 4. 5.

S. Miles and D.E. Mackay, Under Water Medicine, Adiard Coles, London, (1974). A. Krogh, The supply of oxygen to the tissue and the regulation of the capillary circulation, J. Physiol. 52, 457 (1919). J.E. Fletcher, Mathematical modelling of the microcirculation, Math. Biosci. 38, 159 (1978). AS. Popel, Theory of oxygen transport to tissue, Critical Reviews in Biomed. Eng. 17, 257-321 (1989). M. Sharan, M.P. Singh and I. Sud, Modelling of 02 transport in the systemic circulation in hyperbaric environment, l+ontiers of Med. Biol. Engg. 3, 27-44 (1991).

Analytical Model for Oxygen Transport 111

6. J.J. Blum, Concentration profiles in and around capillaries, Am. J. Physiol. 198, 991 (1960).

7. E.P. Salathe and T.C. Wang, Substrate concentration in tissue surrounding single capillaries, Math. Biosci. 49, 235 (1980).

8. J.E. Fletcher and R.W. Schubert, On the computation of substrate levels in perfused tissues, Math. Biosci. 62, 75 (1982).

9. 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 Chem. Engg. Med. Biol., (Edited by D. Hershey), pp. 135, Plenum, New York, (1967).

10. D.D. Reneau, D.F. Bruley and M.H. Kin&y, A digital simulation of transient oxygen transport in capillary tissue systems (cerebral gray matter), AIChE J. 15, 916 (1969).

11. E.P. Salathe, T.C. Wang and J.F. Gross, Mathematical analysis of oxygen transport to tissue, Math. Biosci. 51, 89 (1980).

12. G. Gutierrez, The rate of oxygen release and its effect on capillary 02 tension, A mathematical analysis, Resp. Physiol. 63, 79 (1986).

13. M.P. Singh, M. Sharan and I. Sud, The process of gas exchange in the systemic circulation in hyperbaric environment, IMA. J. of Mathematics Applied to Med. d Biol. 5, 281 (1988).

14. M. Sharan, M.P. Singh, R.K. Saxena and I. Sud, Numerical simulation of systemic 02 and CO2 exchange in hyperbaric environment, Biosystems 23, 21 (1989).

15. S. Middleman, %msport Phenomena in the Cardiovascular System, Wiley-Intersci., New York, (1972). 16. M. Sharan and S. Selvakumar, The effects of chemical kinetics on the oxygen delivery to tissue, Math.

Biosci. 108, 253-278 (1992). 17. A.C. Guyton, Text Book of Medical Physiology, Saunders, Philadelphia, (1981). 18. M. Sharan, M.P. Singh and A. Aminataei, A mathematical formulation for the computation of oxygen

dissociation curve in the human blood, Biosystems 22, 249 (1989). 19. D.H. Hunt and D.F. Bruley, A nonlinear Monte-Carlo simulation: Oxygen transport in the brain, Simulation

30, 17 (1978). 20. B.R. Duling, W. Kuschinsky and M. Wahl, Measurements of the perivascular PO2 in the vicinity of the pial

vessels of the cat, Pflugers Arch. 383, 29 (1979). 21. M. Sharan, M.D. Jones, R.C. Koehler, R.J. Traystman and A.S. Pope& A compartmental model for oxygen

transport in brain microcirculation, Ann. Biomed. Engg. 17, 13 (1989).