Analysis of transcapillary exchange and intraluminal transport in the microocclusion of single capillaries

MICROVASCULAR

RESEARCH

25, 156-175 (1983)

Analysis of Transcapillary Exchange and lntraluminal Transport in the Microocclusion of Single Capillaries THOMAS

R. BLAKE* AND JOSEPH F. GROSS

Department of Chemical Engineering, University of Arizona, Tucson, Arizona 85721 Received June 29. 1981 A theoretical model, appropriate for data interpretation, is presented for the transcapillary fluid exchange and associated intraluminal hydrodynamic and solute transport occurring in the microocclusion of single capillaries. This analysis describes the spatial and temporal behavior of the intraluminal flow. A comparison of the theory with in vivo data suggests good qualitative agreement with that data and, further, the specification of filtration parameters in the model leads to good quantitative agreement regarding the displacement histories of erythrocytes. The model includes both natural and tagged colloidal osmotic influences and can, therefore, be used to represent experiments wherein the concentration of a dyed colloid is measured and related to fluid filtration. The relative merits of measuring colloidal concentration or erythrocyte displacement are discussed within the context of the model.

INTRODUCTION Fluid exchange between capillaries and tissue in the microvasculature is essential to the function of the cardiovascular system. Starling (1896) postulated that this exchange is influenced by the relative magnitudes of hydrostatic and protein osmotic pressures in the capillaries and in the tissue. The Starling hypothesis has been partially verified through fluid exchange experiments on single organs (cf. Pappenheimer and Soto-Rivera, 1948; Levine et al., 1967; however, most of the experimental evidence for the hypothesis has been obtained through microocclusion techniques applied to single capillaries. With the microocclusion experiment, pioneered by Landis (1927) and modified by Intaglietta and Zweifach (1966), Zweifach and Intaglietta (1968), and Michel et al. (1974), quantitative data on the filtration coefficient and the driving forces causing filtration can be obtained. The experimental technique consists of transilluminating a surgically exposed tissue in which a single capillary is microoccluded. High-speed microcinematography or television is used to record time histories of either the erythrocyte displacement or the optical density of a dyed colloid. Then the erythrocyte motion or the colloidal concentration is used to deduce the filtration of fluid from the occluded capillary. The microocclusion experiment is an indirect technique for establishing the filtration parameters since an analytical model of fluid filtration from the occluded * Adjunct Professor of Physiology and Chemical Engineering, Permanent Address, Professor, Mechanical Engineering Department, University of Massachusetts, Amherst, Massachusetts 01003. 156 0026.2862/83/020156-20$03.00/O Copyright Q 1983 by Academic Press, Inc. All rights of reproduction in any form reserved Printed in U.S.A.

ANALYSIS

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157

capillary is required for the interpretation of the data. In the present article we discuss some of the hydrodynamic and transport aspects of the microocclusion experiment and present an analytical model which may be useful for the interpretation of such experiments. There have been several analytical models of the microocclusion experiment. Zweifach and Intaglietta (1968) assumed that the conservation of fluid between individual erythrocytes was the major consideration in the interpretation of the data. Subsequently Lee et al. (1971) formulated a model for fluid filtration from the occluded capillary which provided a transient description of erythrocyte displacement and included the decreasing filtration which is caused by increasing intraluminal protein concentration. Some investigators, e.g., Michel et al. (1974), cannulate the capillary for the purpose of regulating the hydrostatic and osmotic pressures in the capillary during the microocclusion experiment; Levich and Michel (1977) discuss a model for such an experimental protocol, where the concentration history of an injected albumin solution is analyzed and is related to the filtration coefficient for the capillary. The models of Lee et al. (1971) and Levick and Michel (1971) are appropriate models for capillaries wherein the filtration parameters do not change along the length of the capillary. In the former analysis these parameters must be constant along the capillary between erythrocytes which comprise the boundaries of the fluid volume under consideration. In the latter analysis the parameters must be constant throughout the length of the capillary. Further, neither analysis can be applied to an experimental protocol wherein both natural and tagged colloid are present. Such a situation will occur when the tagged colloid is introduced into and is thoroughly mixed in the cardiovascular system prior to an in viva microocclusion experiment. The model presented herein provides an analytical basis for the interpretation of in viva microocclusion experiments and, in particular, it should be useful in describing phenomena observed in capillaries where there are axial variations of filtration parameters. It includes the influences of both natural and tagged colloids upon the colloidal osmotic pressures and permits one: to relate filtration parameters to both the displacement of erythrocytes and to the concentration of the tagged colloid. The model is an extension of the analysis of Blake and Schneyer (1974) and Blake and Gross (1976) and is based upon a system of one dimensional time dependent transport equations describing the flow field in the occluded capillary. BOUNDARY VALUE PROBLEM FOR FLOW IN THE OCCLUDED CAPILLARY Our motivation is to delineate the intraluminal hydrodynamic and solute transport phenomena occurring in the microocclusion of single capillaries. We introduce several simplifying assumptions which permit a tractable analysis and which seem to be justified by observations of the microvasculature. For example, the presence of erythrocytes is neglected in most of our theory. It is expected that the plug motion of the erythrocytes is coincident with the local plasma displacement. Therefore, with the very low intraluminal plasma velocities in the microoccluded capillary, the dynamics of the erythrocytes are approximately that of the plasma.

158

BLAKE

AND

GROSS

The neglect of the erythrocytes implies that the analysis will include higher estimates for both the plasma volume and the axial diffusion of colloids. The latter will be neglected anyway, and the former involves an error of the order of 10%. dependent, of course, on the local hematocrit. Consequently we do not expect that the exclusion of the erythrocytes will affect the major conclusions of this analysis. However, in a later section, we present an analysis which includes the influence of erythrocytes through the hematocrit and we discuss the implication of that extension of the theory. We now define the remaining assumptions and the mathematical statement of analysis. The capillary is a straight cylindrical tube, with a constant diameter, connecting two well-mixed reservoirs. The high and low pressure reservoirs correspond, respectively, to the arteriolar and venular vessels. Again, the presence of erythrocytes is neglected and we assume that the blood is an incompressible, homogeneous, viscous fluid wherein the colloidal molecules are a dilute solute. We assume that both a natural colloid and a tagged colloid are present and are completely mixed in the arteriolar reservoir. The wall of the tube is permeable to the fluid, but impermeable to the solute. We consider that the x axis coincides with the axis of the capillary (cf. Fig. 1) and we assume that the flow properties are a function of only this axial distance X, and of time, t. With this assumption, we are necessarily dealing with a flow where the solute is well mixed and the axial velocity is an average measure of the axial flow. In the present study we will examine the time dependent flow between the open arteriolar end of the capillary (X = I) and the occlusion (X = 0). This occlusion, a plane surface normal to the axis of the capillary, is an idealization of the microneedle used in the actual experiment; cf. Zweifach and Intaglietta (1968). We will assume that this plane surface is suddenly imposed at t = 0 upon a steady capillary flow. The conservation of mass, momentum, and solute concentration are determined by the following differential equations (Blake and Schneyer, 1974) for velocity u, pressure p, and solute concentrations c and T, where these concentrations are for, respectively, the natural and tagged colloids:

p!F+4j=o ax

a) FIG. la. Schematic and venular reservoirs.

diagram of capillary

d

’

X=p

as a straight cylindrical

tube connecting

the arteriolar

ANALYSISOF TRANSCAPILLARYEXCHANGE

159

x: 0 x=-P ” Vendor reservoir

, Arterialor I

reservoir

1

I

I

b)

X.1

L

I I

Location

i

0: occlusion

FIG. lb. Section of idealized capillary illustrating the one dimensional nature of the geometry in the present model.

CIP --zo$+) ax d2

’

(2)

W (3b) In these equations p is the density of the fluid, d is the diameter of the capillary, j is the fluid mass flux through a unit area of the semipermeable capillary wall, p is the viscosity of the fluid, D is the molecular diffusion coefficient, and f is a constant in the assumed relationship between the local wall shear stress and the local fluid velocity. We neglect fluid inertia in (2) and retain (&/I%) in (3), implying that the time scale associated with a change in concentration is expected to be large compared to the time scale for hydrodynamic change of comparable import. The local fluid mass flux across the capillary wall is related to the local driving forces by the mathematical statement of Starling’s hypothesis j = pW){P

- (7r]C + 7r2c7 ~ (z-,E + 7r2C2)- P,(x)}.

(4)

In this equation k(x) is the filtration coefficient, r,c + GT~C*, with V, and 7~~being constants, is the colloidal osmotic pressure produced by the presence of the natural colloid, with corresponding definitions for the tagged colloid, and P,(x) represents the tissue hydrostatic and colloidal osmotic pressures. The dependence of K(x) and P,(x) upon the axial distance along the capillary is explicitly indicated by the use of the argument notation. Because of the one dimensional aspect of the present analysis, Starling’s hypothesis occurs, through (1) and (4), in the system of differential equations, rather than as a boundary condition. There are explicit boundary conditions for that system of equations which apply to the pressure, velocity, and solute concentration in the occluded capillary flow. Specifically we have PC/, d = p, 9 c(l, t) = c,, ?(I, t) = c,, u(0, t) = 0,

(5)

160

BLAKE AND GROSS

E (0, t) = 0, E (0, t) = 0. That is, the hydrostatic pressure and solute concentration at the arteriolar end of the capillary (x = I) are equal to the respective arteriolar values. Further, the fluid velocity and the solute flux are both zero at the occlusion (x = 0). The occlusion is instantaneously impressed upon a steady preocclusion flow with some distribution of solute. Since hypothetically there can be no exchange of fluid during the imposition of the occluding needle, cf. Blake and Schneyer (1974), the solute concentration in the preocclusion flow provides an initial condition for the system of Eqs. (l)-(4). c(x, 0) = c(x), qx, 0) = S(x).

(6)

The boundary value problem, defined in (l)-(6) for the postocclusion flow, represents a difficult mathematical study. Fortunately, the flow environment in capillaries permits approximations to that system of equations which simplify the problem. In the next section we introduce important restrictions on the order of magnitude of the terms in those equations.

ORDER OF MAGNITUDE APPROXIMATIONS OCCLUDED CAPILLARY

FOR FLOW IN

Again the above boundary value problem (l)-(6) is rather intractable. In Blake and Schneyer (1974) and in Blake and Gross (1976) that boundary value problem for only a single solute is solved using matched asymptotic expansions based upon the smallness of various parameters in the governing equations. For our present analysis we shall introduce similar order of magnitude concepts, but in a heuristic fashion, to eliminate certain terms in Eqs. (l)-(6). From such an evaluation we will recover the lowest order terms in the more formal analyses of Blake and Schneyer (1974) and Blake and Gross (1976). Further, this more informal approach permits a statement of the physicochemical approximations without the use of the mathematics of matched asymptotic expansions. For the establishment of any order of magnitude criteria we must use specific data on the microvasculature and consequently our conclusions must be restricted to particular organs. In the present case we consider two microvascular environments: the rat intestinal muscle (Gore et al., 1976) and the rabbit omentum (Lee et al., 1971). In these environments we will show for the intraluminal flow in the microoccluded capillary that the following approximations can be applied to (l)-(6): (i) The colloidal concentration c(5) in the occluded capillary exhibits a relatively small change from that in the arteriolar vessel: c = c, + A, where A << c,.

ANALYSISOF TRANSCAPILLARYEXCHANGE

161

(ii) The pressure gradient is negligible with the pressure in the occluded capillary being equal to that in the arteriole: c3P ax=-

P, 1

Or

P-P,.

(iii) The diffusion of colloids is much less significant than the convection of colloids and, further, this convection is dominated by the axial change in velocity (divergence) in the capillary

The first of these approximations (i) may be deduced from the balance of hydrostatic and osmotic pressures contributing to filtration of fluid from the occluded capillary; viz the quantity {P - (T,C + rr2c2) - (iY,T + if2E2) - P,} in (4). When this bracketed term becomes identically zero through an increase in colloidal concentration associated with fluid filtration, that filtration will cease and the colloidal concentration will have some value higher than that in the arteriolar vessel. It is noted that the nonlinearity of 7r,c + rrZc2 is important in this balance. The largest possible departure from the arteriolar concentration will occur with P = P, and without tagged colloid. The colloidal concentration (c, + A) coinciding with that balance of hydrostatic and osmotic pressures is obtained from r,(c, + A) + TT*(C,+ A)’ = P, - P,. If we use 7r, = 2.72 cm H,O/g/lOO ml, 7r2 = 0.326 cm H20/(g/100 ml)*, c, = 5.8 g/100 ml from Lee et al. (1971) and we also use the value of P, - P, = 36.78 cm HZ0 from Fig. 5 in that same reference we find that c, + A = 7.2 g/ ml, or A/c, = 0.26. Alternatively, the analysis of Blake and Schneyer (1974), using the same colloidal representation, leads to a slightly different value of P, - P, = 33.6 cm (cf. Fig. 2 of the latter reference) and suggests that c, + A = 6.8 g/ml or A/c, = 0.16. These two estimates give the largest possible variation of colloidal concentration from the arteriolar value and, further, that variation will occur in the limit as the occlusion experiment is ending: A/c, + 0.20. For earlier times in the experiment A/c, is much less than this value. From a mathematical view A << c, and this establishes the approximation (i). The second of the above approximations (ii) may be deduced from an order of magnitude assessment of (l), (2), and (4) together with c := c, from (i). The mass flux of fluid into the occluded capillary is equal to the filtration of fluid through the wall of capillary. This balance, obtained from (I) and (4), can be approximated by pu, $

- prd lK{P,

- [T,c, + &I

- P,}

or ua - ; K{P, - [7r,c, + 7T,cf] - P,},

(7)

162

BLAKE AND GROSS

where U, is the axial fluid velocity into the capillary from the arteriolar vessel during the occlusion. Now

or with (7) we obtain [?T,c, + Tr*c;] - P,}.

Ap = P - P, -

In Table 1 we list the values of v, and Ap obtained from (7) and (B), together with the individual parameters corresponding to the analyses of the microocclusion experiment in Gore et al. (1976), Lee et al. (1971), and Blake and Schneyer (1974). It is clear from the last two lines in Table 1 that P, >> Ap and that P = P,,. which establishes approximation (ii). The approximation (iii) is established from (i) and from (7). First we have from (9

and second we have from (7) au -Liz-< cv ‘ax 1 ac “ax--

0 4+iPa - [TIC, + ~dl

Au, - A+{P,, 1

- P,),

- [~T,c, + ~,c:l - P,}.

In Table 2 we list these quantities, again for the analysis of the microocclusion experiments in Gore er al. (1976), Lee et al. (1971), and Blake and Schneyer (1974). In Table 2, the first line measuring the diffusive transport of the colloid TABLE 1 FILTRATION PARAMETERSIN ANALYSIS OF MICROOCCLUSIONEXPERIMENT

Parameter

Gore et a/. (lY76) Rat intestinal muscle

Lee et al. (1971) Rabbit omentum

Blake and Schneyer (1974) Rabbit omentum

lo-<

10m6

1.9 x IOmh

25 20

31 5.8 27

33 5.8 27

IO-’ 5 x lo-”

1.4 lo-’ 8 x 10m4

1.0 IO-’ 8 x 10m4

16

16

16

lo-” 4 x lo-’

1om6 s x 1om4

P, (cm I&O)

25

4 (cm H,O)

-0.5

37 -2.5

K (cm/set cm H,O) P,-P, (cm H,O) c, (g/100 ml) P(c,) (cm WX A (g/100 ml) 1 (cm) d (cm) f D (cm’isec) u, (cm/set)

x lo-’

x lo-’

IO o 6x 33 -3.0

IO4 x lo-’

ANALYSIS

OF TRANSCAPILLARY

163

EXCHANGE

TABLE 2 ESTIMATE OF CONVECTIVE AND DIFFUSIONAL TRANSPORT IN ANALYSIS OF MICROCICCLLJSION EXPERIMENT. NOTE A,c, u,, U ARE FROM TABLE 1 WITH (A,c,) FOR GORE ET AL. (1976) FROM BLAKE AND SCHNEYER (1974). TRANSWRT Is IN gisec/lOO ml

Reference

Transport

Lee rt ul. (1971) Rabbit omentum

Blake and Schneyer (1974) Rabbit omentum

0.014

0.01

-

0.07

0.06

2.3

0.29

0.35

Gore el ul. (1976) Rat intestinal muscle

0.01 dc Au “iii -2 I au c’,U‘, (‘--dx 1

is at least one order of magnitude smaller than the third line which measures the contribution of c(&/&) to convective transport of the colloid. Further, the second line, being less than the third line, shows the dominance of c(&/&) relative to u(&/&) upon the convective transport. Again, as we noted with approximation (i), the values of A used herein are only achieved in the limit as the occlusion experiment is ending. Consequently the transport in the first and second lines in this Table 2 are much less than the values indicated and approximation (iii) is established. SOLUTION OF APPROXIMATE BOUNDARY VALUE PROBLEM FOR FLOW IN OCCLUDED CAPILLARY With the approximations @-(iii) we can combine and write (1) and (4) as

au yqP(, iii+

- P,(x) - (~T,c,, +

~T~c:) - (n, + 2xr2c,)

- (+7,-E,+ ST& - (5, + 2?i*Z,) ii} = 0,

A (9)

where we retain only terms which are linear in A(&). Further, (3a) and (3b) become

(loa)

and the corresponding boundary (5) and initial (6) condition are u(0, t) = 0,

(11)

A(x, 0) = A(x) = 0, A(x, 0) = iI(x) = 0,

(12)

164

BLAKE

AND

GROSS

where again, A(x, t) is the change in solute concentration, c(x, t) = c, + A(x, t), from that in the arteriolar vessel. With the approximations (i)-(iii) the boundary conditions on A and A in (5) need not be included in the solution. We combine (9) with (10a) and (lob) to obtain two equations for A and A, aA --c

{P, - P,(x) - (rlc,

aA -at

{P, - P,(x) - (rr,c, + &)

at

+ qcf)

- (7~~+ 27~+,)A

- (T, + 2r,c,) A

- (%,E, + 5Fj2c$)- (5, + 25?*Z,)ii} = 0,

(13b)

and as one would expect from (9) and (10) the multipliers of c&F,) in the second of the terms in (13a) and (13b) are the same. This means that ii(;)

= it(;)

and upon integration and the use of the initial conditions (12) we obtain A -=c,

A T,’

(14)

Thus we need to solve only one of Eqs. (13), say, the second, for (A/F,). With (14) that equation is written as

(15) Integration of (15) yields d(x, t), the increase of the tagged colloidal concentration from that in the arteriole:

li - = [[P, - P,(x) - (~,C, + 4) I, - (5,-E, + %j2z;)}/{Ta(;ji, + 2%*Za) + c,(7r + 27T$,)]]

Since the concentration of the tagged colloidal concentration is E(x, t) = T, + &Xl t)

(16)

ANALYSIS

OF TRANSCAPILLARY

165

EXCHANGE

wehavethetimehistoryof thatcolloidata location x in theoccluded capillary: -E= T, t Z,[[{P, - P,(x) - (T,C, t a$?;) (17)

- (%,Z, + ?i2~~)}/{~a(~, + 25Yj2Fa) + c,(Tr, + 27Qc,)}]] . {{I

- exp - @(L(;i,

+ 2?r3,) + cLn, + 2%,)]]}}.

Naturally, from (14) and (16) we have an equation for A/c, and, consequently, we also have a definition of c(x, t) which is analogous to (17). Further, from (14) and (16) we can integrate (9) to establish the fluid velocity in the occluded capillary

(J= - ; fy[Pa I

- Pr(5)- (7T,c,+ Tr,c:, - (;ii,Z, + ;ir&)]] 4K(5)

--ph

4. 11

+ T,(;ii, + 2%i,C,)} + 25TTT2c,)

(18)

Since Eqs. (17) and (18) are the result of approximations (i)-(iii) it is appropriate to inquire about the range of validity in time of those equations. We show in the Appendix, for microvascular environments such as the rabbit omentum or the rat intestinal muscle, that, indeed, (17) and (18) with approximations (i)-(iii) are valid for the time scales of microocclusion experiments in those environments. DISCUSSION OF APPROXIMATE SOLUTlON AND RELATIONSHIP ERYTHROCYTE DISPLACEMENT IN MICROOCCLUSION EXPERIMENT

TO

If the erythrocytes merely move with the plasma then the above equation (18) for the fluid velocity in the microoccluded capillary can be used to correlate typical data. Specifically, Zweifach and Intaglietta (1968), Lee et al. (1971), Michel et al. (1974), and Gore et al. (1976) measure time histories of erythrocyte displacements during the microocclusion experiment. In most of those experiments (cf. our discussion below) the filtration coefficient K(x) and the tissue pressure P,(X) are constant along a significant length of the capillary between the occlusion, x = 0, and the arteriolar vessel, x = 1. An exception to this is the rat intestinal muscle which is studied by Gore et al. (1976). However, in that case if the relative displacements of erythrocytes, which are initially close together, are measured, then the filtration coefficient and the tissue pressure between those erythrocytes can be approximately constant. Case when Filtration Parameters Are Constant along Capillary Consider the case where there is not any tagged colloid, and where the filtration coefficient K(x) and the tissue pressure P,(X) are constants. For that environment we integrate (18), yielding an explicit expression for the fluid (erythrocyte) velocity; the magnitude of the velocity increases in a linear fashion with x, the distance between the fluid element and the occluding needle. This has been discussed by Blake and Schneyer (1974) and Blake and Gross (1976) and is consistent with the experimental observations of Zweifach and Intaglietta (1968). Further, the

BLAKEANDGROSS

166

displacement of a fluid element or an erythrocyte is related to that fluid velocity by the equation dx = u = -Y{Po dt

-

fw

- (?r,c, + ?72c:) - P,)

4K --p,c,

+ 2?Qcf>t .

This equation can be integrated to give the time history of fluid displacement and, hence, the displacement of an erythrocyte which moves with the local fluid velocity. If we designate x0 as the location of the fluid element (or erythrocyte) at t = 0, we have X

- = exp x0 [I

P, - (7r,c, + qcf) - P, ?r,c, + 2a,c; I 4K(n,c, + 2?T&)t d

II

and if we consider two fluid elements (or erythrocytes) which are initially at x0 and xh the history of their relative distance (x - x’) is given by x - x’

= exp x0 - x(, [-I

P, - (?r,c, + 7r,cf> - P, 7rlC, + 25Tp5 I

II

- 1 - exp - F(7i,ca + 2n&)t . 1 From the former of these (19a) we conclude that if two or more erythrocytes are observed, where these erythrocytes move with the local fluid velocity, the normalized histories of the erythrocyte locations will be identical. In Fig. 2 we show measurements by Intaglietta (1980) of two erythrocyte positions during a microocclusion experiment in the rabbit omentum; these time

ov

TIME

(sec.)

FIG. 2. Comparison between present model and measurements of erythrocyte trajectories during the microocclusion of a single capillary in the rabbit omentum (Intaglietta, 1980). Instantaneous erythrocyte location is normalized by initial erythrocyte distance from the occluding needle. Two different ervthrocvtes are indicated bv the svmbols.

ANALYSIS

OF TRANSCAPILLARY

EXCHANGE

167

histories are normalized by the original location of the two erythrocytes. Those trajectories are identical and, therefore, suggest that the above equation (19a) is consistent with the experimental measurements. That equation with a specification of the filtration coefficient and pressure parameter, as indicated on the figure, provides an excellent qualitative and quantitative description of the measured time histories. Conversely, the identical nature of the two measured histories, through (19a), suggests that the filtration coefficient and tissue pressure, in that occluded capillary, are constant. The second of the above equations (19b) has been used by Blake and Schneyer (1974) to analyze the measurements of Lee et al. (1971). That analysis of a microocclusion experiment in the rabbit omentum showed good agreement between (19b) and the data. Consequently, it suggests that both the filtration coefficient and the tissue pressure are also constant in the latter microocclusion experiments. The representation of that data for a specific case is shown in Fig. 3. In that figure we also indicate the influence of a parametric variation in the driving pressure for filtration with a fixed value of K. Case when Filtration

Parameters

Change along Capillary

In the measurements for the rabbit omentum, e.g., Zweifach and Intaglietta (1968), and Lee et al. (1971), different values for the filtration coefficient and the driving forces for capillary filtration are observed, dependent upon the location of the capillaries in the microvascular bed. For example, the filtration coefficients in capillaries on the venous side of the omentum are larger than those for capillaries in the arterial section of that bed. While such gradients are on a spatial scale larger than that considered herein, gradients of capillary filtration parameters also exist on the length scale of the axial distance along the occluded capillary. This is evident in the measurements by Gore et al. (1976) of erythrocyte motion during microocclusion experiments in the rat intestinal muscle. Those measurements indicate dramatic axial gradients of the filtration coefficient K(x). A perusal of the theoretical expression (18) for the fluid velocity (and for erythrocytes which

FIG.

rocytes

3. Comparison between present model and measurements of relative positions of two erythduring microocclusion of single capillary in the rabbit omentum (Lee ef al., 1971).

168

BLAKE

AND

GROSS

move with that velocity) suggests that the inversion of such erythrocyte velocity measurements through (18) to evaluate K(x) would be difficult. Gore ef al. (1976) avoid this difficulty by measuring relative erythrocyte trajectories for erythrocytes which are initially close together. For such experimental protocol we must integrate (18), for distances from the occlusion (x) which are large compared to the capillary diameter, to establish the history of fluid element or erythrocyte location. We shall proceed in a manner analogous to that used to derive (19b) and we shall introduce some approximations which will permit us to obtain a tractable relationship like (19b), but for the case where the filtration parameters change along the capillary. We consider two fluid elements (or erythrocytes) which are initially at x0 and ~6. The differential equation describing their relative distance (x - x’) is obtained from (18).

4W)

exp - t -+,(T,

+ 27~2~0)+ C,(i?, + 2Tr,Z,)}

When the following inequalities hold,

11 d(.

(21)

namely, that the erythrocytes are initially close together and are not displaced very far, then the filtration coefficient K(x) and the tissue pressure P,(X) are essentially constant in the length through which the two fluid elements or erythrocytes are displaced. With these approximations we obtain the differential equation for the relative displacement of the two erythrocytes, A$ - y) = -(x - x~)~ * {P, - P,(x;,) - (?-r,c, + 7r,c;, - (%,T, + %&)} . exp -t

fJy&T,

[ which upon integration gives

x- x’

1 ,

P, - P,(xh) - (T,C, + ?T,c;> ~ (E,T, + ;ii&) c,(n-, + 2T,C,) + Z,(?i, + 2?i,E,)

[r

~ = exp xg - x;l .{1

+ 2T,C,) + E&Y, + 2;jj2EJ}

(224

-..,-,[!$a

{c,(T, + 27~~~~)+ E,(?r, + 2?j2F,)}

III

.

(22b)

169

ANALYSIS OF TRANSCAPILLARY EXCHANGE

Equation (22), subject to (21), permits the evaluation of K(x) and p,(x) at x = xl, based upon the local and relative erythrocyte displacement. We now apply (22b) together with (21) and Z, = 0 to selected data like that of Gore (1982a). Again, Gore and his coworkers performed microocclusionexperimentson single capillaries in the rat intestinal muscle and measured erythrocyte displacement histories. Our purpose is to demonstrate the applicability of the present analytical technique to such data where significant changes in K(x) occur in the capillary. Indeed we shall conclude that while the spacing of the erythrocyte pairs x0 - x6 is small it is not small enough to permit a completely satisfactory evaluation of K(x) from all of the erythrocyte trajectories for this selected data. In Fig. 4 we show measurements (Gore, 1982b) obtained in the manner of Gore (1982a), of the normalized relative distance between erythrocytes (X - x’)/

(x0 - x6) for two different erythrocyte pairs (x;, x0), in the same capillary. The occluding needle was placed near the venular end of the capillary and it was convenient to measure distances from the arteriole rather than from the occluding needle. That is (cf. Fig. l), Gore and his coworkers define the locations of the erythrocyte pairs as (I - x;, 1 - x0). However, since only the relative distance between the erythrocytes enters (22b) any definition of the location is satisfactory. The theoretical expression (22b), with Eu = 0, is fit to the data in Fig. 4 by

selecting K(xh) and P,(xA) for the two experimental histories. That theory is shown as the solid curves in Fig. 4; in this particular capillary I’,(&) is the same for both of the observed histories but K(xA) is different. The agreement, in both a qualitative and quantitative sense, between theory and experiment is quite

I

Measurements

I

1.00

FIG. 4. Comparison between present model and measurements, cf. Gore (1982b), of relative displacement histories of two pairs of erythrocytes during microocclusion of a single capillary in the rat intestinal muscle. The length of the capillary is I + I,, T 302 pm.

170

BLAKE AND GROSS

good. However, we note that the application of (22b) to such data is predicated on the basis of inequality (21). We now a posteriori estimate the magnitude of the terms K(xh) and (&l&),&x - xi) and assess whether or not inequality (21) is satisfied. We note that (x - x;l) = (x0 - x;) = 50 F and that the distance separating the two pairs is -100 p: O.O415(pm/sec . cm H,O) - O.l204@m/sec . cm H,O) 100 pm = -8.0 x 10m4(set . cm HzO)-‘. Therefore the product

This is much smaller than K(xA) = O.l204(pm/sec * cm H,O) and therefore (21) is satisfied for the curve (1 - xA)/(f + I,) = 0.88 in Fig. 4. But the product (dK/dx),,,(x - x;) = O.O40(pm/sec . cm HzO) is not much smaller than K(xh) = O.O415(Fm/sec . cm H,O) and consequently (21) is not satisfied for the other curve in Fig. 4. Consequently, that analytical result must be interpreted with some caution. This example illustrates that the present theory provides both a conceptual structural to evaluate the filtration parameters K(x), P, - P,(x) and a criteria for assessing the applicability of that structure to particular experimental environments. Most importantly, for the experimentalist, the theory indicates when the erythrocyte pairs (x,, x6) are spaced too far apart to be useful in the diagnosis of the filtration parameters. DISCUSSION OF APPROXIMATE SOLUTION AND MEASUREMENT TAGGED COLLOID IN MICROOCCLUSION EXPERIMENT

OF

In the solution for transport in the microoccluded capillary (17)-(18) and in the expression for relative erythrocyte displacement (22) we have included the presence of both natural and tagged colloid. Our motivation is twofold. First, there are a few experiments (cf. Levick and Michel, 1977), wherein tagged colloid has been used as a tracer and optically monitored during the microocclusion of capillaries. With such measurements (17) can be used to deduce the filtration parameters from the data. Our second reason is to demonstrate, within the context of the specific morphological and physiological properties of the microvasculature in Tables 1 and 2, that the concentration history of tagged colloid can be quite useful in delineating the filtration parameters. This application is especially valuable when the filtration parameters vary along the capillary length. Specifically consider the theoretical expression for the tagged solute concentration in (17); the local tissue pressure P,(x) is measured by the large time asymptote of that expression, while the local filtration coefficient K(x) determines the characteristic time for achieving that asymptote. We have shown in the previous section of this article that (22b) can be similarly used to deduce the filtration parameters, K(x), P,(x), from the erythrocyte motion

ANALYSIS

OF TRANSCAPILLARY

EXCHANGE

171

during the microocclusion experiment. However, for a capillary in which there are axial gradients of the filtration parameters the application of (22) is restricted by the inequalities (21). In a practical sense, as we indicated by a specific example in that section, one might not find two adjacent erythrocytes which are initially close enough (x0 - ~6) such that the filtration parameters are constant in the length through which the two fluid elements or erythrocytes are displaced. The interpretation of the tagged colloid concentration through application of (17) does not suffer from such limitations. Another advantage of introducing a tagged colloid and measuring the concentration of that colloid is that there is some uncertainty about the actual mechanisms associated with the motion of erythrocytes in the microoccluded capillary. Both Intaglietta (1974) and Gore (1982a) suggest that the accuracy of the technique is too dependent upon the interpretation of erythrocyte motion. Gore (1982a) has noted that the motion of erythrocytes in some occlusion experiments could be caused by displacement of a compliant capillary wall without any filtration of fluid from the capillary. For some microvascular environments, Gore (1982a) comments that such displacement of the capillary wall cannot be measured within the resolving power of a light microscope. While such compliant behavior during the microocclusion experiment has not been documented, its possibility does obscure the meaning of some of the measurements. The introduction of a tagged colloid and the monitoring of the colloid concentration would provide an empirical technique which is less affected by such spurious behavior. If there is no filtration of fluid from the capillary then the concentration of colloid will not increase. Thus, whether one is assessing filtration parameters in capillaries with constant K and P, or in capillaries with axial variations in K and P, it appears that the monitoring of a tagged colloid offers some significant advantages relative to the monitoring of erythrocyte motion. INFLUENCE

OF HEMATOCRIT

In the development of the present analysis we have neglected the influence of erythrocytes upon the fluid and solute transport in the capillary and assumed that in the experiments the erythrocytes merely move with the fluid motion. It is our view that in most microvascular environments these simplifications are appropriate and introduce errors which are small. However, there may be some situations wherein the influence of erythrocytes upon the transport should be included and we now indicate, briefly, how the analysis can be modified to include this influence. This modification is presented for the case where the filtration parameters are constant and also where there are no tagged colloids. The hemotcrit, H, is defined as the volume fraction of erythrocytes in a volume of fluid and erythrocytes. We can use two-phase continuum mixture theory to derive a differential equation for conservation of hematocrit, g

+ ; (Hu) = 0,

(23)

where u is the local velocity of the fluid. Again, we assume that the erythrocytes and the fluid move locally with the same velocity. Further, the differential equations

172

BLAKE

AND

GROSS

(l)-(4) must be modified to account for the diminished volume occupied by the fluid phase. Equation (3a) becomes ; [(l - li)c]

+ ui

[(I - H)c] + (1 - H)c g = (1 - H) D !$

and the remaining equations are modified accordingly. The solution of Eq. (23) and the modified versions of Eqs. (l)-(4) proceeds in a manner identical to that in the previous sections. That solution for the case of uniform initial hematocrit, H,, gives a concentration history identical to that in the previous sections and a velocity history modified by the hematocrit u = - y

(1 - H,){P,

- (?T,c, + ?T,c:, - P,}

4K . exp -7 (7r,c, + 27rz&t . { I

(24)

This velocity is diminished by the presence of erythrocytes. In fact, the erythrocytes tend to decrease the lumen wall area available for filtration of fluid from the capillary. This suggests that the hematocrit will also affect the displacement of erythrocytes. If we consider two erythrocytes which are initially at x0 and ~6, the history of their relative distance (X - x’) is

Again, relative to the case when Ho = 0, cf. Eq. (19b), an increase in hematocrit tends to diminish the relative displacement of adjacent erythrocytes. Now, the asymptotic value of this relative displacement is used to evaluate the driving pressure for filtration: P, - P, = (?T,c, + 7r,c:, -

7r]C, + a?T,& In ~ 1 - H,,

,

t+ x.

Consequently, we see that for a specific value of relative displacement, the driving pressure must increase with an increase in the hematocrit. For example, in the theoretical solution shown in Fig. 3 for the data of Lee et al. (1971), the measured hematocrit was 0.25; if Eq. (25) rather than Eq. (19b) was used to represent that data, the pressure driving force would increase by approximately 2 cm HzO, and the filtration coefficient would decrease by approximately 0.005 x 10e4 cm . /se&m H,O. Again these differences are probably not significant within the context of the many uncertainties, both experimental and theoretical, in the microocclusion experiment. Nevertheless, there may be some situations in which the influence of hematocrit is significant.

ANALYSIS

OF TRANSCAPILLARY

CONCLUDING

EXCHANGE

173

REMARKS

We have presented a theoretical model for the transport and filtration phenomena which occur during the microocclusion of single capillaries. Analytical expressions have been derived which permit one to relate data on erythrocyte displacement or tagged colloidal concentration in the occluded capillary to the capillary filtration coefficient and to the driving forces for fluid filtration. A particular advantage of this model is that, within the context of specific morphological and physiological properties of the microvascular environment, it can be used to interpret data from capillaries wherein variations of the filtration parameters occur along the capillary length. Consequently, the analysis includes criteria for the applicability of different measurements in the deduction of filtration parameters. When these parameters are constant along the capillary length the usual measurement of the relative displacement of erythrocytes is satisfactory. However, when there are axial variations of such filtration parameters the measurement of the concentration of tagged colloid is the most effective means for establishing the filtration parameters. Further, we provide an analytical basis for using relative erythrocyte trajectories even in this latter case if the erythrocytes are sufficiently close together. The model has been applied to the interpretation of erythrocyte trajectories in microocclusion experiments in capillaries of the rabbit mesenteric and omental tissue where the filtration parameters are constant and in the rat intestinal muscle where the filtration coefficient exhibits significant variation along the capillary axis. This model is being applied in collaboration with Gore and his coworkers (cf. Gore, 1982a) in the interpretation of tagged colloid concentration in capillaries of the rat intestinal muscle. APPENDIX: DISCUSSION OF TIME SCALES IN WHICH APPROXIMATE SOLUTION IS VALID In the solutions (17), for the solute concentration, and (18), for the fluid velocity, we have through approximation (iii) neglected the intraluminal diffusion of solute relative to the intraluminal convection of solute. This implies certain restrictions on the applicability of the solutions with regard to the time scale during which Eqs. (17) and (18) provide a valid representation of the intraluminal flow. The major restriction is that at large time the diffusional transport can be as significant as the convective transport and so at large time the approximation, in a mathematical sense, becomes invalid. However, in a practical sense the large time behavior is not significant. This is because most microocclusion experiments with single capillaries are completed on a time scale of l-2 set and we can show that this is typically within the time scale when Eqs. (17) and (18) are valid. There are two possible causes of solute diffusion. The first occurs because of a concentration difference between the arteriolar solute and that in the occluded capillary. Blake and Schneyer (1974) provide a complete analysis, including diffusion of solute, for the case when the filtration parameters are constant. With that analysis or with simple order of magnitude estimates the solute at the

174

BLAKE AND GROSS

arteriolar end of the microoccluded capillary diffuses on an axial length according to x - (Dry Typically, cf. Lee et al. (1971), microocclusion experiments in the rabbit omentum are conducted for 2 sec. If we use the diffusion coefficient in Table 1 we conclude that diffusion at the arteriolar end of the capillary will include an axial distance of J/? x 10m3cm by the end of 2 set in the microocclusion experiments. This is approximately one capillary diameter and consequently our neglect of solute diffusion for such a microvasculature seems to be justified. Further, even if such experiments were conducted for 10 set this diffusion would still be relatively small. For the case where, say, the filtration parameter K(x) is not constant along the length of the capillary as in the rat intestinal muscle, cf. Gore (1982a), a second mechanism for axial solute diffusion exists. This diffusion is caused by the axial gradients of solute which develop in the capillary during the microocclusion experiment. With (18) we conclude, for a single solute, that axial convection is dV ca ax

- c,f-yPo

- P, - (?T&, + 7T,&}

. exp -

y+Ca,Tl + 27TpzJ)t1, [ while with (17) we conclude, for a single solute, that axial diffusion is

Dd’h D c P, - p, - (r,c, + wl> ax=- a c,(r, + 27T*cJ 4K”(x) c,(7r, + 2?72c,)t d . exp -

y{c,(s,

1

+ 27T*c,)}t .

If we use K(x) = K(I) in the former expression and neglect K”(x) and write K’(x) = K(1)// in this latter expression we find that c,(&/ax) and D (a*A/dx*) become of the same order when l=

y

+(n,

+ 23T,c,)t2.

Then, using the values in Table 1 from Gore et al. (1976), we conclude that this equation is satisfied when t is approximately 10 sec. Consequently, for that microvasculature, diffusional influences are only expected to be significant at a time of the order of 10 set which, again, is longer than typical microocclusion experiments conducted by Gore in the rat intestinal muscle. Finally we note that when the value of K(I) decreases from that reported by Gore et al. (1976) but the other parameters are the same the time during which the solution is valid increases according to the inverse of the square root of K(1).

ANALYSIS

OF TRANSCAPILLARY

EXCHANGE

175

Since the time for “completion” of the solution and the experiment increases according to the inverse of K(1), there are likely to be some combinations of (4K/d)C,(7r, + 27r2C0)and D/l2 for which the present solution is not applicable. ACKNOWLEDGMENTS This research was supported under NIH HL 17421. The authors wish to acknowledge important communications with their colleagues, Professors Marcos Intaglietta of the l.Jniversity of California, San Diego and Robert W. Gore of the University of Arizona, Tucson. Also, the reviews of our initial manuscript provided many important comments which are included in the present article.

REFERENCES BLAKE, T. R., AND GROSS,J. F. (1976). Fluid exchange from a microoccluded capillary with axial variation of filtration parameters. Biorheology 13, 357-366. BLAKE, T. R., AND SCHNEYER,G. P. (1974). On the occluded capillary. Microvnsc. Res. 7, 362-375. GORE, R. W. (1982a). Fluid exchange across single capillaries in rat intestinal muscle. Amer. J. Physiol.

242 (Heart

Circ. Physiol.

ll):H268-H287.

GORE,R. W. (1982b). Measured histories of erythrocyte displacement in microoccluded capillaries of the rat intestinal muscle, private communication. GORE,R. W., SCHOKNECHT, W., ANDBOHLEN,H. G. (1976). Filtration coefficients of single capillaries in rat intestinal muscle. In “Microcirculation” (K. Grayson and W. Zingg, eds.), Vol. 1, pp. 331332. Plenum, New York. INTAGLIETTA,M. (1974). Measurement of fluid exchange between single capillaries and tissue in-vivo. In “Microcirculation” (B. M. Altura and G. Kaley, eds.), Univ. Park Press, Baltimore. INTAGLIETTA,M. (1980). Observed erythrocyte displacements in microoccluded capillary of the rabbit omentum, private communication. INTAGLIETTA,M., AND ZWEIFACH,B. W. (1966). Indirect method for measurement of pressure in blood capillaries. Circ. Res. 19, 199-205. LANDIS, E. M. (1927). Micro-injection studies of capillary permeability. II. Relationships between capillary pressure and the rate at which fluid passes through the walls of single capillaries. Amer. J. Physiol. 82, 217-238. LEE, J. S., SMAJE,L. H., ANDZWEIFACH,B. W. (1971). Fluid movement in occluded single capillaries of rabbit omentum. Circ. Res. 28, 358-371. LEVICK, J. R., AND MICHEL, C. C. (1977). A Densitometric method for determining the filtration coefficients of single capillaries in the frog mesentery. Microvasc. Res. 13, 141-151. LEVINE, 0. R., MELBINS,R. B., SENIOR,R. M., AND FISHMAN,A. P. (1967). Application of Starling’s law of capillary exchange to the lungs. J. Clin. Invest. 46, 934-944. MICHEL, C. C., MASON,J. C., CURRY,F. E., ANDTOOKE,J. E. (1974). A development of the Landis technique for measuring the filtration coefficient of individual capillaries in the frog mesentery. Quat. J. Exp. Physiol.

59, 283-309.

PAPPENHEIMER, J. R., ANDSOTO-RIVERA, A. (1948). Effective osmotic pressure of the plasma proteins and other quantities associated with capillary circulation in hind limbs of cats and dogs. Amer. J. Physiol. 152, 471-491. STARLING,E. H. (1896). On the absorption of fluid from the connective tissue spaces. J. Physio/. 19, 3 12-326.

ZWEIFACH,B. W., ANDINTAGLIETTA,M. (1968). Mechanics of fluid movement across single capillaries in the rabbit. Microvasc. Res. 1, 83-101.