Substrate concentrations in tissue surrounding single capillaries

Substrate Concentrations in Tissue Surrounding Single Capillaries ERIC P. SALATHB ANDTSENG-CHAN Cenier for the Application of Mathematics, Lehigh Uniwrsi@, Bethlehem, Penmylwnia Receiwd

WANG 18015

10 Ju& 1979; revised 26 October 1979

ABSTRACT The mathematical analysis of a model for substrate concentration in tissue is given. The model consists of a single capillary from which substrate diffuses into a surrounding cylinder of tissue. The equations governing this model were derived by Blum, who also attempted to obtain their solution. Blum’s method of solution was incorrect, and he failed to obtain meaningful results. The correct illustrated by numerical examples.

analysis is presented here, and the results

INTRODUCTION In 1960, a paper by Jacob Blum [l] on the mathematical analysis of a model for substrate concentration in tissue appeared in the American Journal of Physiology. The model, due originally to Krogh [5], consists of a single capillary from which substrate diffuses into a surrounding cylinder of tissue. Unfortunately, Blum’s solution is incorrect, and his results are not meaningful. The error in Blum’s analysis has never been realized, and the paper is widely quoted in the literature, including a number of textbooks on microcirculatory physiology ([6], [7], f or example). Lih [6] devoted an appendix of his book to a detailed explanation of Blum’s solution, and Fletcher [3], in an appendix to a paper on unsteady substrate transport, appears to have recently obtained the same solution independently. Since the problem describes an interesting physiological process, it is important to provide the correct analysis, and this is done in the present paper. A brief description of Blum’s analysis is given in the appendix, where it is shown why his solution is wrong. Numerical examples are presented for a particular choice of physiological data. No examples were given by Blum, presumably because of the contradictory nature of his results. Since the appearance of Blum’s paper, a considerable amount of attention has been given to a variety of related problems. A summary of some of this work can be found MATHEMATICAL

BIOSCIENCES

49:235-247

QElsevier North Holland, Inc., 1980 52 Vanderbilt Ave., New York, NY 10017.

235

(1980)

002S-5564/80/040235

+ 13$02.25

236

ERIC P. SALATHk

AND

TSENG-CHAN

WANG

in the book by Middleman [7], and in the proceedings of a symposium on oxygen transport to tissue [2]. A more recent reveiw has been given by Fletcher [4]. ANALYSIS The model consists of a single capillary of radius R, and length L surrounded by a concentric cylinder of tissue of radius R,. The tissue cylinder is bounded at both ends by planes normal to the capillary. Blood containing substrate at concentration C, enters the arterial end of the capillary with flow velocity u. As the substrate is convected through the capillary, it diffuses into the surrounding tissue, where it is consumed at a constant rate M. The permeability of the capillary endothelium to the substrate is denoted by H, and it is assumed that no substrate crosses the boundary of the tissue cylinder. The concentration of substrate in the capillary, C,, is assumed to be a function only of distance t along the capillary, while the concentration in the tissue, c,, varies not only with t but also with radial distance r from the capillary axis. The diffusivities of substrate in the tissue in the radial and axial directions may differ, and are denoted by D, and D,, respectively. A detailed discussion of the model was given by Blum [ 11. In terms of the nondimensional variables C= C,/C,, c = c,/C,, r= f/R,, and z = Z/L, the governing equations are

i a ac --r-+f2a4f r ar

Lo ar

%O az

ac ar

’

7

O
02

az2

ar

r=l,

O
z=o, 1,

R
< 1,

(2)

R
= -h[C(z)-c(R,z)],

o-sz<

1,

r-R

Pg=;l,_,.

O
where R=&/R,, e=m (R,/L), MO= R:M/D,C,,, h=HR,/D,, and fi= vR,R,/2D,L. The solution to Eqs. (l)-(4) will be obtained in terms of C(z), and this solution will be substituted into Eq. (5) to obtain a single equation for C(z).

SUBSTRATE

CONCENTRATIONS

A solution to Eq. (1) satisfying the eigenfunction expansion Mor2 c(r,z)

=

4

the boundary

conditions

Bessel functions

--

of order n, a,,a2,An,Bn

and pn = EM. Using the boundary condition

c(r,z)=$

g-lnr-$+lnR

(2) gives

+B, 1

( + nz,

(3) is given by

+a,+a,lnr

where K, and Z,, are the modified are constants,

237

IN TISSUE

B,[K,(p,)Z~(p,r)+Zl(p,)Ko(p,r)lcosn~z,

(6)

where the constant (r, has been written as B,- (Mo/2)(R2/2 -In R) for convenience, and B,, = &/Z,(p,). These constants are determined by substituting the solution into the remaining boundary condition, Eq. (4). Writing

P,[K,(P,YI(P,R)un = [K,(P,JZO(P,R) u, =

this substitution B,+

From that

Zd~n)K,(~nR)lv + Z,(PXO(P,J 11,

(7)

yields z,

this equation

B,(u~-

;u.)cosnlrr=C(z)+

and the orthogonality

f).

$(R-

properties

of cosn~z

it follows

B,=g(R-$)+fC(z)dz, B, =

2 o,-(f/h)u,

1C(z)COSn7rzdz. s IJ

(8)

Equation (6), with B,, Bn given by Eq. (8), expresses c(r,z) explicitly in terms of the capillary concentration C(z). Substituting this solution into Eq.

238

ERIC P. SALATHE

(Sj yields an integrodifferential

equation

AND TSENG-CHAN

WANG

for C(z):

s‘C(S)cosnd)dS. (R-i 1+ 2 pjvny$) (coSnTz)

p$

0

“_,

Integrating with respect to z and applying the boundary condition C(O)= 1 yields a Fredholm integral equation of the second kind with separable kernel, for the function

C(z):

(9) Multiplying

Eq. (9) by cosmrz

a,,,=b,,,+

and integrating

5

e-a,,

from z =0 to z = 1 gives

m=l,2,3

,...,

(10)

n-1 where a,=

1+ $(R-

b,,,=i’[

/0

i)z]cosmvrzdz

0,

=

1 --

MO

pm%’

R-f (

)

m=2i,

i=l2

m=2i-1,

i= 1,2 9...?

>

0, =

(“1

‘C(z)cosm7rzdz

4%

>

,.**,

m=n

or m+n

m+n

odd.

(12)

euen, (13)

P(D.--kun)(n2-m2)7r2’

Equation (10) represents an infinite set of linear algebraic equations for the truncated or infinite set of unknown constants a,. The corresponding

SUBSTRATE

CONCENTRATIONS

239

IN TISSUE

“reduced” system of equations, N

&,,=b,,,+

x

n-l

m=1,2 ,..., N,

e,,,,,&

(14)

can readily be solved for the N unknowns ci,, and it can be shown that the &,, converge to the desired constants a,,, as N-co. Therefore, a solution of any desired accuracy can be obtained by choosing N sufficiently large. Once the a, have been determined, C(z) and C(T,Z) are found from 2a,u, 5 n-1 an+“iU”)

C(z)=l+~(R-;)z+

slnm7z

(15)

and c(r,z) =

Z(R- +)+p(z)dz

+Z ‘“; 1WNdw-) n-l

v,-

+ Zl(p,)Ko(p,r)lcosn~z, (16)

-u h ”

where

s

‘C(z)dz=l+$’

0

is used in Eq. (16). NUMERICAL

RESULTS

AND DISCUSSION

Numerical examples will be presented for a 110 pm diameter Krogh cylinder surrounding a 10 pm diameter capillary 1000 pm long. The axial and radial tissue diffusivities are both 900 pm’/sec, and the capillary blood flow velocity is 400 pm/set. To determine the normalized concentrations C(z) and c(r,z), it is necessary only to specify the ratio M/CA, and not both quantities separately. The examples correspond to M/CA =0.002 set-i.

ERIC P. SALATHk

240

AND TSENG-CHAN

WANG

The results are illustrated for a range of capillary permeabilities, including an infinitely permeable capillary endothelium. In this case, which was not considered by Blum, Eq. (4) must be replaced by the condition C(z)= c(R,z). It can be shown that the corresponding solution is obtained directly from the above solutions for finite h simply by setting l/h =O. Figure 1 shows the substrate concentration in the capillary for various values of H. In all cases, C = 0.4 at z = 1, since the amount of substrate consummed is independent of the capillary permeability. For the larger values of H, the concentration decreases more rapidly at the arterial end. The more permeable capillary permits a rapid entry of substrate into the capillary at the arterial end, and this substrate reaches other regions of the tissue by axial diffusion. The limiting case H = co is approximated fairly closely by the solution for finite H when H > 50 ym/sec. Figures 2 and 3 show the tissue concentration, as a function of axial position, at the capillary wall and at the outer edge of the Krogh cylinder, respectively. The concentration decreases with decreasing H, since less substrate enters the tissue when the endothelium is less permeable. At a value slightly less than H = 2 ,nm/sec the substrate concentration becomes negative over some portion of the tissue. Since this is physically impossible, 1.0

H = 50 ,,m/sec H = 10 ,m/sec H = 2 iim/sec

0.6

I

0

0

I 0.2

I

I

1

0.4

I 0.6

I

I 0.8

I

I 1.0

z

FIG.1. Normalized capillary concentration values of the permeability H.

as a function of distance, for various

SUBSTRATE

CONCENTRATIONS

IN TISSUE

241

0.8

H = 50 ~~m/sec H = 10 umfsec 0.6

0

0.2

0.4

0.6

0.8

1.0

FIN. 2. Nonnalked tissue concentration at’the capillary wail as a function of axial position, for various values of the permeability H. 1.0

F 0.8

H = 10 umlsec 0.6

0

0.2

0.4

0.6

0.8

1.0

FIG. 3. Norrnaiized tissue concentration at ke outer edge of the tissue as a function of axial position, for various values of the permeability H.

ERIC P. SALATHk

242

AND TSENG-CHAN

WANG

the solutions obtained are not valid for such values of H. Below some critical value of capillary permeability, it is impossible for sufficient substrate to enter the capillary to be consumed at the uniform rate M. A new solution must be obtained in which regions with zero substrate concentration appear in the tissue. An approximate analysis of this type has been given recently for oxygen transport to tissue by Salathe and Beaudet [8]. The concentration profiles show zero slope at z = 0 and z = 1, in accordance with the boundary conditions (3). An exception occurs at r= R for H = 00, since c( R, z) = C(z), and z = 0,1, r = R, are singular points for this solution. 1.0

\

L =

0.8

H

50

um/sec

-

H = 10

,m/sec

0.6

C(r,z)

H = 2 umlsec 0.4

0.2

0

0

-

I R

0.2

I

I

I

0.4

I 0.6

,

I 0.8

1

J 1.0

r

FIG. 4. Normalized concentration as a function of radial position at I ~0, for various values of the permeability H.

SUBSTRATE CONCENTRATIONS 1.0

-

0.8

-

243

IN TISSUE

....

-.-.-.-.-.-.-.-._.-.-.-------a--

0.6

------

-

Ch-,z)

.--.- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.4 -

0.2

-

I

o0

R

0.2

I

I 0.4

1

=

1 0.6

I

I 0.8

,

] 1.0

FIG. 5. Normal&d concentration as a function of radial position at I =0.3. H-~,-~-~-~-H~50pm/sec,_______W=10pm/sec;~~~~~H~2~m/sec. Figures

4-6

show the concentration

profiles

as a function

of radial

position at z = 0, z = 0.3, and z = 1.0. These profiles are rather flat, as a result of the large tissue diffusivity. Not only does large diffusivity permit radial diffusion into the tissue from the capillary to occur at low concentration gradients, but it also permits substrate delivery by axial diffusion from the arterial end. When the diffusivity is low, the radial gradients are steeper. However, each section of the capillary then supplies substrate mainly to the tissue surrounding it at that location, since axial diffusion is of less significance. The result is an approximately linear profile in the capillary. For larger values of diffusivity, marked nonlinearity of the capillary concentra-

ERIC P. SALATHE

244

AND TSENG-CHAN

WANG

1.0

0.8

0.6

C(r,z)

0.4

H = 2

um/SeC

0.2

-

0

I1

I1

R

0.2

0.4

I, 0.6

I

I]

1.0 r FIG. 6. Normalized concentration as a function of radial position at t = 1, for various values of the permeability H. 0

0.8

tion profile occurs as a result of axial diffusion. This effect is illustrated in Fig. 7, which shows the substrate concentration in the capillary for various values of diffusivity when H = 50 pm/set. It is clear from this figure that axial diffusion can have a significant effect on the concentration profiles in the capillary and surrounding tissue. Since the solution is obtained as an infinite series, it must be truncated after a finite number of terms for numerical computations. In addition, the truncation of the infinite set of algebraic equations at N terms, leading to Eq. (14), means that the N numbers ri, only approximate the correct values a,. Therefore, the accuracy of the solution should be examined. Since all the governing equations except Eq. (4) are satisfied exactly, the error can be

SUBSTRATE CONCENTRATIONS

IN TISSUE

0.8

2700 wn’/sec 0.6

C(z) 0.4

0.2

0 0

0.2

0.4

0.6

0.8

1.0

FIG. 7. Normalized capillary concentratioias a function of distance, for permeability H = 50pm/seeand various values of tissue diffusivity.

determined

by calculating

ac

ar r-R +h[C(z) -

C(O)]

h[ C(z) - C(m)]

*

In the examples shown, 100 terms were used in the expansion, and the error was exceedingly small. Only at z -0, for H= cc, did the error rise to a few percent. By choosing N = 120, the above ratio was 4.7 X lo-* at this singular point, but at z = 0.01 it had already decreased to 6.2 X 10P4. APPENDIX For simplicity, Blum’s analysis will be discussed only for D,= D,, and the notation of the present paper will be used.’ Blum obtained a general

‘In his paper, Blum studied two cases. Case I, which neglected axial diffusion, assumed a consumption rate proportional to concentration (first order kinetics). Case II, wrrespending to the present analysis, assumed a constant consumption rate and included axial diffusion. Only Blum’s case II is considered here.

ERIC P. SALATHE

246 solution

AND TSENG-CHAN

WANG

to Eq. (1) of the form c(r,z)

= y+o,+ozlnr+yr+

5 F,(r)G,(z) n-l

where F,(r) = e,Ja( hr) +f, Y,,( kr), G,,(z) = g, coshbz + h, sinhhz, and e,,f,,g,,,h, are constants. The appearance of the Bessel functions J,,, Y,, instead of the modified Bessel functions I,,, K,,, and of the hyperbolic functions sinh, cash instead of the trigonometric functions sin,cos, distinguish Blum’s solution from the present solution. These differences result from using the opposite sign for the separation constant when solving Eq. (1) by separation of variables, and the choice made by Blum is incorrect for this problem. Blum satisfies the boundary condition (2) by setting (r2= - M,,/2, e, = Yo(EsIR),L= - Jo( p,,R) and determining & from the eigenvalue equation Yo(K,R)J,(PA-Jo(PJ)Y,(P,J=O. choice of e,,f, results in c(R,z)=MoR2/4+~,-(Mo/2)lnR+ yz, so that the tissue concentration is a linear function of z at the capillary wall. Such a result is not consistent with the governing equations, and it is now impossible to satisfy all of the remaining equations. The boundary condition (3) was satisfied by Blum by choosing g, = A,, (1 - coshk)/sinh A and determining h. from This

The tissue concentration c(r,z) is now completely determined, and the capillary concentration C(z) is determined uniquely by Eq. (5) and the initial condition C(0) = 1. No use has been made of Eq. (4) in obtaining these solutions, and there is no way that it can now be satisfied. In fact, substituting the solutions for c(r,z) and C(z) into Eq. (4) shows that it is not satisfied. This work was supported by N.I.H. program and NSF Grant No. ENC. 77-21.542.

Project

Grant PO1 HL-I7421

REFERENCES 1 2

J. J. Blum, Concentration profiles in and around capillaries, Amer. J. Physiol. 198:991-998 (1960). D. F. Bruley and H. I. Bicher (Eds.), Owgen trunrporr to tissue. Pharmacobgv, Mathematical Studies, and Neonatology, Advances in Experimenial Medicine and Biologv, 37B, Plenum, New York, 1973.

SUBSTRATE CONCENTRATIONS

IN TISSUE

247

3 J. E. Fletcher, A model describing the unsteady transport of substrate to tissue from the microcirculation, SIAM J. Appl. Ma& 29~449-480 (1975). 4 J. E. Fletcher, Mathematical modeling of the microcirculation, Math. Biosci. 5

6

38: 159-202 (1978). A. Krogh, Anatomy and Physiology of Cqillaries, Yale U.P., New Haven, 1936. M. Lih, Transport Phenomena in Medicine and Biology, Wiley-Interscience, New York,

1975. S. Middleman, Transport Phenomena in the Cardiowscular System, Wiley-Interscience, New York, 1972. 8 E. P. Salathi: and P. R. Beaudet, The anoxic curve for oxygen transport to tissue during hypoxia, MicrowscuIar Res. 15:357-362 (1978).

I