BULLETIN OF MATHEMATICAL BIOPHYSICS VOLUME 30~ 1 9 6 8

LINEAR BULK DIFFUSION

INTO HETEROGENEOUS

TISSUE

9 B. A. HILLS* D e p a r t m e n t of Chemical Engineering, The University of Adelaide, Adelaide, Australia

In this paper an expression is derived which describes the transient overall uptake of an inert solute by a section of tissue excised with parallel faces and placed upon an impermeable base. The approach diverges from the conventional analyses for perfused tissue (Morales and Smith, Bull. Math. Biophysics, 6, 125-141, 1944; 7, 47-99, 1945) because the extravaseular zone is regarded as a heterogeneous diffusion medium. Account for this is taken by regarding tissue as effectively composed of two phases--a continuous (extracellular) phase similar to water, and a dispersed phase comprising cells of irregular profile. In both phases the relevant mode of uptake is taken as bulk diffusion rather than surface permeation, thus emphasizing the influence of the internal geometry of the tissue upon its overall exchange response. Introduction. B l o o d perfusion h a s been widely a c c e p t e d as t h e r a t e - l i m i t i n g process for t h e e x c h a n g e of i n e r t s u b s t a n c e s b e t w e e n blood a n d e x t r a v a s e u l a r tissue (Jones, 1951; K e t y , 1951). H o w e v e r , m o s t o f t h e facets o f blood-tissue e x c h a n g e w o u l d s e e m to be equally c o m p a t i b l e w i t h a simple diffusion-contrib u t i n g m o d e l if tissue were r e g a r d e d as h e t e r o g e n e o u s on the m i c r o scale (Hills, 1966). T h e v i t a l f a c t o r in appraising such parallel e x p l a n a t i o n s , a n d in deciding w h e t h e r or n o t t h e r a t e - l i m i t i n g c o n t r i b u t i o n afforded b y diffusion is significant, h a s b e e n t h e d e t e r m i n a t i o n o f diffusion t i m e s in e x t r a v a s c u l a r tissue. H o w ever, such e s t i m a t i o n s are influenced g r e a t l y b y t h e values one uses for diffusion coefficients in cellular a n d extracellular m a t e r i a l , a n d b y w h e t h e r one p r e s u m e s *Presen~ address: Division of Biological and Medical Sciences, Brown University, Providence Rhode Island. 4---~.M.~. 47

48

B.A. HILLS

that these values are equal. The latter assumption has been implied b y the equation for transient radial diffusion used in such estimations b y K e t y (1951), Roughton (1951), and others. I f one dispenses with this assumption, then it would not be realistic to substitute into transient equations values of diffusion coefficients determined on macro-sections b y steady-state methods, e.g. those of Krogh (1918). Evidence for heterogeneous permeability is provided b y Dick (1959) who quotes a range of 1.5 • 10 .8 - 5.0 • 10 -1~ cm2/sec for water in the cytoplasm of various biological cells compared with 10 -5 cm2/sec for the self-diffusion coefficient of water. Diffusion coefficients in water should be very similar to those for the same solute in extracellular fluid. :Feniehel and Horowitz (1963) have determined values ranging from 3 • 10 -1~ to 3 • 10 -8 cm2/sec for ten inert compounds in the relatively uniform and isolated fibers of a frog's sartorious muscle. These include urea for which compound Perry (1950) quotes a value of 1.4 x 10 -5 cm2/sec as the diffusion coefficient in water. However, the greater geometric complexity of mammalian skeletal muscle, and of most tissues in their undisturbed state, render them far less inducive to mathematical description of their overall uptake response. Hence the extraction of cellular diffusion coefficients from experimental data becomes considerably more difficult than extraction from the isolated fibers of the frog's sartorius muscle. Since cells are too small to provide continuous sections large enough for steady-state determinations of their transmission properties, one is forced to use a transient method, and hence a mathematical model, for interpretation of data. Many models have been proposed for the transfer of cations in which linear responses are postulated between any isolated pair of adjacent compartments in series. The many series, parallel and combined models are ideally suited to analysis using Laplace transforms, as shown b y the general solution given b y Segre (1965). However, the assignment of a linear response to an isolated diffusion stage of uptake implies membrane limitation. While this would seem perfectly reasonable for cations, experimental evidence indicates that it is a gross approximation for inert substances. Fenichel and Horowitz (1963) have shown that bulk diffusion, rather than surface permeation, was the relevant mode of uptake of such solutes b y the isolated fibers of the frog's sartorious muscle. Thus any quantitative description of the exchange of inert substances in tissue should be more valid if it can take account of: (a) dissimilar diffusion coefficients for the same solute in cellular and extracellular material; (b) the geometric relationship between these two "phases." While the microgeometry of the dispersed (cellular) phase is fixed by the

LINEAR BULK DIFFUSION

49

histological structure of the particular tissue, experimentally one has a choice of outer boundaries for the continuous phase (extracellular fluid). Selection of structurally simple sections has the advantage of minimizing mathematical complexity. Thus, a suitable form of tissue sample which can be readily obtained experimentally (Hills, 1967) is a thin parallel-faced slab placed upon an impermeable base for mechanical support. I f the exposed face of the section is large relative to the area of the sides, then the effect of the latter m a y be ignored. Thus, the relevant case for response analysis is that of linear bulk diffusion into a continuous fluid in which a less permeable phase is distributed as particles of irregular profile.

The "isolated"cellularphase.

Before attempting to express the overall uptake response of the tissue model postulated and the mutual interaction between the phases, it is essential to describe the mode of uptake b y the cellular phase as though this were isolated from the rest of the system. While geometric complexity renders any exact description an impossible task, the following general observations serve as an approximation: (a) Although there are m a n y microirregularities seen in any histological section, there is a repetitive pattern such that the distribution of the cellular phase over maeroregions (say 1 mm 3) is remarkably uniform. (b) There is similar uniformity in cellular contact, and in the area of cellular material available for mass transfer with extracellular fluid. Thus, if V is the volume of cellular material per cell, A is the surface area of contact between the phases per cell, and r is the effective cell radius, then (Ar/V) is a dimensionless parameter indicative of distribution of the cellular phase. Such "shape" factors can be obtained very easily for geometric forms whose boundaries can be expressed in simple mathematical terms, e.g., a sphere, annulus, cylinder, etc.--in fact all the shapes of a uniform medium for which one can obtain exact solutions for uptake b y bulk diffusion. The solutions for the net uptake b y all such perfect shapes m a y be expressed in the general form:

G= VSP[1- ~ R,.exp (-a2nDt)].

(1)

?t=l

Here (7 is the net uptake of inert gas b y the shape of volume V at a time (t) following a step change in external gas tension (p) defined b y p = 0 for t < 0 to p = P for t /> 0. S is the solubility, D is the diffusion coefficient, R , is a dimensionless geometric term determined b y boundary conditions and a, is the n th root, real and positive, of an auxiliary equation peculiar to the particular shape. Expressions for R . and a. are given in Table 1 for various perfect

shapes.

50

B.A.

HILLS

v

,~

O

~0

II

mm,~ v/ ~V/

-r .

I

I

r

r162

oo

(.9 oo r

v

o~

--s

I

"-g

o~

~.~ o ~

LINEAR

BULK

DIFFUSION

51

It has been shown that (a2/al) 2 and (ral) give smooth curves when plotted against the corresponding values of (At~ V) for a sphere, an infinite cylinder, a parallel-faced slab and annuli of various relative dimensions (Hills, 1967). Thus, an approximation to the uptake response of a particular biological cell of irregular profile m a y be obtained b y interpolation. In practice it has been shown b y Fenichel and Horowitz (1963) and Hills (1967) that only the first two terms of the infinite series contained in equation (1) represent capacity with any appreciable delay with r e s p e c t to external changes. Moreover, the capacity represented b y the root terms, proportional to R~, decreases rapidly as n increases. Hence, for the case of inert substances diffusing into cells, some approximation is introduced b y writing equation (1) as: G = VScP[1 - R1 exp ( - a~Dt) - R 2 exp ( - a~Dt)],

(2)

where Sc is the solubility in cellular material, the assumption being that the small cellular capacity represented b y (VScP. ~ , ~ s R , ) responds so rapidly that it may be regarded as effectively equilibrated with extracellular fluid. However, an overall uptake of form identical to equation (2) m a y be obtained from two compartments parallel to each other yet each in a series with the same source if: (a) their capacities for the solute are VS~PRI and VScPR2, and (b) each displays a linear response with respect to the same source of solute, where the time constants are (anD) and (a~D) respectively. Thus if p~ and P2 are the mean tensions which can be attributed to these hypothetical compartments, (ep~/~t) = a~D(p - p~),

(3)

(ep2/~t) = a~D(p - P2).

(4)

If ~ is the true volume fraction of the extracellular space, the effective capacities for the solute per unit volume of tissue are: C 1 = (1 - iz)R1Sc for the hypothetical compartment representing the first root term for bulk diffusion in cells, C 2 = (1 - ~)R2S c for the hypothetical compartment representing the second root term for bulk diffusion in cells, C e = t~Se + (1 - R 1 - R2)(1 - i~)Sc for the effective extracellular zone. where S e is the solubility of the solute in extracellular fluid. Hence, capacities m a y be expressed relative to that of the effective extracellular region b y C~ (1 - I~)R~Sc fll = C--~ = /~Se + (1 - R 1 - R2)(1 - F)Sc

(5)

52

B.A. HILLS

for the first hypothetical cellular compartment, and b y f12 = ~ S

(I -- /~)R2Sc e + (1 - R 1 R2)(1 - / ~ ) S c

(8)

for the second hypothetical cellular compartment. The extraceUular phase. Since cells are very small relative to the dimensions of tissue sections of the order of 0.5 to 3.0 mm thickness needed for experimental work (Hills, 1967), it m a y be assumed that: (a) The fraction of cross-sectional area of tissue occupied b y cellular material is effectively constant in all planes parallel to the prepared surfaces of the section, and there is negligible convolution of the diffusion p a t h w a y in the continuous phase. Thus incorporation of the compartmental concept of the cellular phase gives the model illustrated in Figure 1. (b) The cellular phase is evenly distributed, the constants ill, f12, a~D c and a~D c being the same at all depths in the section.

Figure 1. The overall model showing linear bulk diffusion in the extracellular phase with "side" bulk diffusion, at all depths, into the less permeable cellular phase---represented by two sets of an infinite number of parallel compartments. Each set represents one root term of the expression for bulk diffusion into a homogeneous cellular phase of the relevant geometric distribution In view of the values of the order of 104 for the ratio of extracellular to cellular diffusion coefficients, there should be negligible error in ignoring increased permeation of the section contributed b y intercellular diffusion. This simplifies the analysis of the model given in Figure 1 b y permitting the omission of transfer in the x direction within the cellular zones. Thus molecules of solute reach the surface of the cellular material at each tissue depth b y diffusion from the surface through the extracellular fluid only, that is, linear bulk diffusion

LINEAR BULK DIFFUSION

58

through the continuous phase with the complication of exchange between t h e phases at all intermediate depths. The overall response. The expression for linear bulk diffusion in a homogeneous medium has been derived by taking a mass balance for the solute at any point, Carslaw and Jaeger (1959) quote the result as D

to~p(x, t) tox2

=

(x,t),

(7)

where p(x, t) is the point tension at depth x and time t. However, in the two-phase model (Fig. 1), there is greater capacity for solute at each microregion by virtue of the presence of the cellular phase although no greater area of transmission through the continuous phase exists relative to its volume. Hence, repetition of the mass balance for a point at depth x gives De . a2p(x, t) tOp(x, t) ax~ = a---i-- + ~

topi(x, t) tot + ~

top2(x, $) a----/--'

(s)

where D e is the diffusion coefficient for the solute in the extracellular phase and p i ( x , t) and p~.(x, t) are the mean tensions of the solute at depth (x) and time it) in the first and second of the hypothetical compartments invoked to simulate bulk diffusion in the isolated cellular phase. Writing equations (3) and (4) for the cellular phase at depth x, aPi(X, t) = a~Dc[p(x ' t) - p i ( x , t)], tot

(9)

ap2(x, t) = a~Dc[p(x ' t) - p2(x, t)], tot

(10)

and

where D c is the diffusion coefficient for the solute in the cellular phase. Substituting for api(x, t)/tot and top2(x, t)/tot according to equaAnalysis. tions (9) and (10) respectively, equation (S) gives

a2p(x, t) De

9-

-

tox2

+ flla~Dc[Pi(x, t) - p(x, t)] + fl2a~Dc[P2(x, t) - p(x, t)]

ap(x, t) at

(11)

Initial conditions are p ( x , t) = p l ( x , t) = p2(x, t) = o

(12)

54

B.A. HILLS Boundary conditions are

fort ~< 0 a n d 0 < x < L.

@(x, t) @l(X, t) ~p~(x, t) ~---7-= ~----~--- = ~--7--- = o

(13)

p ( x , t) = p l ( x , t) = p~(x, t) = p

(14)

at x = L, and

at x ~< 0 for t > 0, that is, a step change from 0 to P in the tension at the exposed face (x = 0) at t = 0, while the remote face (x = L) is impermeable. Assume

n=O

Pl(x,O

=

P + ~ bn(t) sin [(2n + 1)Trx] n:o ~-LJ'

(16)

+ 1)~-x] ~Z7 J"

(17)

p2(x,t) = P +

~ Cn(t) sin [(2n "

n=o

Equations (15) and (16) satisfy conditions (13) and (14) automatically. Substituting equations (15) through (17) in (9) through (11), and equating coefficients of sin [.(2n + 1)TrX] ~-L" J: we find _De[(2n + 1)~-]2

2L

J an + flxa~Dc(b" - an) + fi2a~Dc(c" - an) = ~d (an)'

(18)

d a~Dc(an - bn) = ~(b,~),

(19)

d cn) = ~(cn).

(20)

a~Dc(an

-

-

Assume a,~(t) = an(0)'exp ()it), bn(t) = bn(0).exp (At) and ca(t) = cn(O)'exp (At), when substitution in equations (18) through (20) gives:

(A-I- De[ (2n h'l)Tr]2 + Dc(flla~ + fl2a~))an(O) - Dc(flla~bn(O) + fl2a~cn(O))=O, -a~Dc.an(O ) + (~ + a~Dc)bn(O) = O, -a~Dc.an(O ) + (A + a~Dc)c,~(O) = O.

55

LINEAR BULK DIFFUSION

Hence,

a~Dc.an(O) . 2 ~ b~ = (2t + aiDe)

Cn

=

a~Dc'a,(O) 2 ' (~ + a2Dc)

and an satisfies (2+ De[!2nhl'~r]

+ Dc([31a~+ [32a2)- D2c['(~t+a2Dc)+ (2t "-~"~Dc)]) an(O) =

O,

which is a solution only if )~ satisfies the expression

D~[B~(a + ~D~) + ~(a

+ ~ D o ) ] = 0.

(21)

This is a cubic equation for t. Let the roots be I~, I~2, t~a. Then there are three solutions a.~(o); b.do) = ~Dca.~(O)/(a.~ + ~D~); e.~(0) = ~D~a~(O)/(a.~ + ~ D ~ a.~(0); b.~(0) = ~D~a.~.(O)/(a~ + ~Do); c.~(0) = ~Dca.~.(O)/(a.~ + ~D~); a.~(0); b.~(0) = ~D~a.~(O)/(a~ + ~D~); c~(0) = ~Doan~(O)/(,~.~ + ~D~).

Hence, (22) an(t) = anl(O)-exp (2nlt) + a~2(O).exp (2n2t) + ana(O).exp ()~3t); a~.(O).exp (,~=#) a~s(O)9ex__p(,~nst)] ; ( 2 3 ) b.(t) = a ~ D c [ a"l(O)'-ex-p (2~1t) L (~.1 + a~Dc) + (An2+ a~Dc) + (A~a+ a~Dc) J cn(t) =- a ~ D c [ a ~ )'exp (2tnlt)-~-~-~c) a,9.(O).exp (An2t) a~3(O).exp (An3t)]. (24) +

(y.2 + .~no)

+

(~.~ u

In order to fit the initial conditions [eq. (12)], we require n=o

a~(O).sin [(2n + 1)~rx] 2L J = -P"

Hence, n=o ~ an(O);sin [(2n+.2L'l)rrx]J sin [(2m 2L+1,~rx]. dx

or

89

) = -2PL/(2m + 1),

J

56

B.A. HILLS

that is, am(0) = - 4 P / ( 2 m + 1). Similarly for b=(0) and c=(0) giving 4P

aa~(O) + anu(O) + aaz(O) =

a~D~[

aa(O) L(;~ + ~D~) +

a~D [

a~(0)

an~(0)

.]

(Xnu + (z~Dc) + ()~na + a~n~)J = -an--~(0) ]

an2(O)

a~(O)

(25)

(2n + 1)'

+ =~Dc) +(An= + a~D~) + (A,~a + a~D~)J =

cL(~

4P 9 (26) (2n + 1)' 41) (2n + 1)

(27)

With solution anl(0 ) =

-4PAn~hn~(An~ + a~Dc)(An~ + a~Dc). 2 ~ ~ a~a2Dc(2n + 1 ) ( , ~ -- ,~.2)(,~n~ _ A n a )'

(28)

-4PAns2,~(2n~ + a~D~)(2n~. + a~Dc). -- ~no)(~2 -- ~nl)'

(29)

-4PA~A~:(A~a + a~Dc)(2tn~ + a~Dc)

(30)

a~(o)

= ala2Dc(2n ~ : ~ + 1)(,~

a~(0)

= ala2Dc(2n 2 ~ : + 1)(,~n~ -- ' ~ ) ( ' ~ 3

_

'~.:)

Finally, substitution in equations (15) through (17) gives:--

p(x,t)

= P-

4P ~ s i n [(2n + 1)~rx/2L] ~ ~ ~. " A, ( 2 n + 1) (~:D~) ~=o

[A~2A~3(An~ + a~Dc)(An~ + a~D~).exp (A,~t) + A,~32,~(An2 + a~D~)(2,~2 + a~Dc) + ~nl)~n2(~n3 + a~Dc)()~n3 + a~Dc)'exp (,~n3t)] ~n~ - ~ - ~ ) ( ~ - ~n~) J 4P pl(x, t) = P - a~D----~c.

~ sin [(2n + 1)Trx/L] (2n + 1)

X

9

+

~ -

(31)

)~2)(A~1 -

An3)

(-~.~= L 1 ) - i ~ . ~ = ~

-F

J'

(32)

LINEAR

4P p 2 ( x , t) = P -

"a~-~r

57

DIFFUSION

~ sin [(2n + 1)Trx/2L] (2n + 1)

n=O

[2tn2~tna(Anl + a~De)'exp (2nit) X [

BULK

( ~ n l -- In2)(}~rtl -- an3)

+

~tn3~nl(,~n2 + a~Dc)'exp (,~2t) -~

(}~n2 -- }~n3)(~n2 --

~.1)

]"

(33)

The total uptake of solute (G) by a tissue of area B may now be determined by applying Fick's law to the continuous phase at the exposed surface, that is, dt

x=o

Applying this expression to equation (31), dG 2rrBPSeDe[AnzA~3(A,I + a~Dc)(~x + a2De).exp (hnxt) "-d~ = ~axa2DeL [(An1 -- An2)(Anl - An3 )

"-~

+

(an2 _

~nx)(an2

__ an3)

(34)

2~x~t~z(2n8 + a~Dc)(~tn3 + a2Dc).exp (()tnzt)]

a 2)

]

Thus, according to the two-phase model, the overall uptake of solute by a parallel faced section of tissue can be expressed as the sum of an infinite number of exponential terms. Values of ~t can be expressed in terms of known parameters as the roots of equation (21). Thus + lfir] 2~2~2 /')2. )tn12n22tn3 = De [(2n 2L j x~e~c,

+ alag~Dc(1 2 2 2 + fix + f12);

+ 1)r 2 + a~Dc(1 + fl~) + a~D~(1 + f12). ~,~ + ~ 2 + ~ 8 = De[[(2n2L

(35)

(36) (37)

Discussion. Experimentally the exponential components of the overall response may be extracted by backward projection, a technique whose limitations

58

B.A. HILLS

h a v e b e e n r e v i e w e d b y V a n L i e w (1965). c o m p o n e n t s a r e 8~1, 8~2, 0~3 . . . . t h a t is,

If the response times of these

dG

d-Y = Z l " e x p (-t/On1) + Z 2 " e x p (--t/On2) + Z 3 . e x p (-t/On3) -}- . . . .

E q u a t i o n s (35) t h r o u g h (37) g i v e

(3S)

8~18~28~3 = TIT2T~, OnlOn2 ~- On28n3 -r On3Onl ---- T~T2 + T2Tn(1 + ill) + T1Tn( 1 + f12),

~nl -[- 8n2 -[- %3 = T1 -{- T2 -~- T~(I

TI = I/ fDc,

= I/

Dc

and

(40)

+ fi~ + f12),

where

•

2L

T. = De[(en

+

(39)

]5

For n -- 0, equations (38) through (40) are identical to those derived by Laplace transforms in which it is assumed that the extracellular zone is effectively a fully stirred region of response time given by only the first root of the expression for linear bulk diffusion (Hills, 1967). Equation (34) should enable experimental response constants to be interpreted in terms of the diffusion coefficients of cellular and extracellular phases, their geometric distribution and the external dimensions of the section used.

LITERATURE Carslaw, H. S. and J. C. Jaeger. 1959. Conduction of Heat in Solids, 2nd ed. London: Oxford. Crank, J. 1956. Mathematics of Diffusion. London: Oxford. Dick, D. A . T . 1959. "The R a t e of Diffusion of W a t e r in the Protoplasm of Living Cells." Exp. Cell. Res., 17, 5-12. Feniehel, I. R. and S. B. Horowitz. 1963. "The Transport of Nonelectrolytes in Muscle as a Diffusional Process in Cytoplasm." Acta Physiol. Scand., 60, Suppl. 221. Hills, B . A . 1966. " A Thermodynamic and Kinetic Approach to Decompression Sickness." Adelaide: Libraries Board of S. Australia. 1967. "Diffusion versus Blood Perfusion in Limiting the R a t e of U p t a k e of I n e r t Non-polar Gases b y Skeletal R a b b i t Muscle." Clin. Sci., 33, 67-87. Jones, It. B. 1951. "Gas Exchange and Blood-tissue Perfusion Factors in Various Body Tissues." I n Decompression Sickness, J. F. Fulton, Ed. Philadelphia and London: Saunders, 278-321. K e t y , S . S . 1951. "Theory and Applications of Exchange of Inert Gas at Lungs and Tissues." Pharm. Rev., 3, 1-41. Krogh, A. 1918. "The R a t e of Diffusion of Gases through Animal Tissues, with some Remarks on the Coefficient of Invasion." J. Physiol. (London), 52, 391-415. Morales, M. F. and R. E. Smith. 1945. " A Note on the Physiological Arrangement of Tissues." Bull. Math. Biophysics, 7, 47-51.

LINEAR BULK DIFFUSION

59

Morales, M. F. and R. E. Smith. 1945. "The Physiological Factors Which Govern I n s e r t Gas Exchange." Ibid., 99-106. Perry, J. 1950. Chemical Engineer's Handbook, 3rd ed. New York: McGraw-Hill. Roughton, F. J . W . 1952. "Diffusion and Chemical Reaction Velocity in Cylindrical and Spherical Systems of Physiological I n t e r e s t . " Prec. Roy. Soc., Ser. B, 140, 203229. Segre, G. 1965. "Compartmental Systems and Generating Functions." Bull. Math. Biophys., 27, Special Issue, 49-63. Smith, R. E. and M. F. Morales. 1944. "On the Theory of Blood-tissue Exchanges: I. F u n d a m e n t a l Equations." Bull. Mat. Biophysics, 6, 125-131. Van Liew, H . D . 1962. "Semilogarithmic Plots of D a t a which Reflect a Continuum of Exponential Processes." Science, 138, 682-683. RECEIVED 4-27-67