Interstitial plasma protein concentration profiles and transport mechanisms

MICROVASCULAR

RESEARCH

Interstitial

42, 209-216 (1991)

Plasma Protein Concentration Transport Mechanisms

Profiles

and

D. G. TAYLOR, J. L. BERT,* AND B. D. BOWEN* Department of Chemical Engineering, University of Ottawa, 161 Louis Pasteur, Ottawa, Ontario, Kl N 6N5, Canada; and *Department of Chemical Engineering, University of British Columbia, 2216 Main Mall, Vancouver, British Columbia, Canada V6T lW5 Received January 10, 1991

INTRODUCTION Until recently, it was standardly assumed that the interstitial compartment could be treated as a well-mixed space, devoid of pressure or concentration gradients. However, advances in experimental techniques, such as microfluorometry, electron microscopy, and digital image analysis, have led to a number of experimental studies challenging that view (see, for example, [ 1,3,11]). Further, it has been proposed [7] that measured interstitial plasma protein distribution profiles be used to identify the principal mechanisms governing their transport within the interstitial space. In an earlier paper [lo], we used a distributed model of interstitial transport to investigate fluid and plasma protein exchange within a model tissue representing mesentery. The study suggested that interstitial plasma protein distribution data alone contain insufficient information to identify the principle interstitial transport mechanisms. We substantiate that conclusion here with an example drawn from the literature. MODEL DESCRIPTION Figure 1 is a schematic diagram of the model mesentery. The tissue segment is bounded left and right by arteriolar and venular capillaries, respectively. The upper and lower surfaces of the tissue consist of identical mesothelial layers, bathed by the same well-mixed peritoneal fluid compartment. The model tissue is therefore assumed symmetric about the x axis. It is also assumed, for simplicity, that variations in the z direction can be ignored, reducing the system to two dimensions. A detailed description of the mathematical model used to simulate fluid and plasma protein transport within the segment of interstitial space is found in two earlier papers [9,10]. Further study (see [S]) revealed that the two-dimensional 209 lw26-2862/91 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in U.S.A.

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Peritoneal fluid

-WrJ

Meeolhellal

boundary

I

Peritoneal fluid

FIG. 1. A schematic diagram of the model tissue [lo].

tissue could be described adequately by a one-dimensional model in which variations in the y direction are neglected. In that case, the interstitial resistances to fluid and plasma protein transport are lumped together with the resistances of the mesothelial layer itself. We present only the final differential equations and boundary conditions for this simplified one-dimensional system. Fluid transport within the interstitium is described by a modified version of Darcy’s Law that accounts for the effects of both hydrostatic and colloid osmotic pressure gradients. If jz is the fluid flux in the x direction, then .o lw = -p

d(P’ - II’) dx a

(1)

The peritoneum serves as a source/sink for fluid and plasma proteins in the system. A steady-state material balance on a differential volume of interstitium gives -- HKo&(P’ & - II’) + h’lnes_ - 0, 2

(2)

where JF, the net fluid flux crossing each of the mesothelial layers, is described by the Starling equation: ,y = gyp* - pmes_ gyII* _ IIm-)]. (3) The interstitial colloid osmotic pressure, II’, is related to the local concentration of plasma proteins within the accessible volume, C’, by II’ = Al . C’ + A2 . (Cl)” + A3 - (Cl)‘.

(4)

Plasma protein transport within the interstitium occurs via convection and diffusion. In addition, plasma protein exchange occurs between the interstitium and the peritoneum. A steady-state balance on interstitial plasma proteins contained within a differential volume of interstitium yields

(5) The first and second terms contained within the brackets represent the net convective and net diffusive transport of plasma proteins in the tissue, respectively, while J? is the net flux of plasma proteins across each mesothelial layer, which is given by the nonlinear flux equation

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,y

= (1 _ g=s),y

1

C’ - C”““exp( - Pemes) 1 - exp( - Pemes) ’ [

(6)

where Pemes= (1 - +TjF D mes Convective transport is governed, in part, by the degree of convective hindrance, characterized by 5 which lies between 0 and 1. From Eq. (5), if 5 is zero, all plasma protein transport is assumed to occur by diffusion. A nonzero value of 5, on the other hand, implies that both convective and diffusive protein transport are possible. Boundary conditions are needed to complete the problem specifications. For example, a fluid balance at the arteriolar boundary gives

P[ @*i lI1q=o= L;Y[P’],=,- part- aart. ([II’]x=o- IF)),

(8)

while a protein balance at this same boundary yields

- ([j%C’],,, - n’D.,[ $j=,) = (1 - d”‘)(V”l,=0)[

1

[C’],,, - @“exp( - Pea”) ’ 1 - exp( - Pea”)

(9)

Equations similar to (8) and (9) can be written for the venular boundary at x = L. Note that the right-hand sides of these two equations represent the net flux of material crossing the capillary wall from the interstitium to the circulation. Solution of Eqs. (1) through (5), subject to the attendant boundary conditions, yields the distributions of interstitial plasma protein concentration and interstitial fluid pressure within the model tissue. This information can then be compared to available experimental data. RESULTS AND DISCUSSION As mentioned at the beginning of the paper, the objective of this study is to demonstrate the fact that interstitial plasma protein concentration profiles contain insufficient information, in and of themselves, to draw definitive conclusions regarding the relative importance of interstitial plasma protein convection versus diffusion within mesentery. To date, there is little experimental data in the literature describing the distribution of native plasma proteins within the interstitia of specific tissues. However, Friedman and Witte [3] and, more recently, Barber and Nearing [l] have measured the interstitial plasma protein concentration profile in segments of rat mesentery. In particular, the geometry of the system investigated by Friedman and Witte is analogous to the model tissue. This data set, therefore, represents a useful one for investigating the relationships between interstitial concentration profiles and interstitial plasma protein transport mechanisms using the existing model.

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O.OO1

1.0 DINENSIONLES3 DISTANCE FIG. 2. The model predictions of c’ assuming 4 is zero (solid line) and assuming 4 is 0.2 (dotted line) are compared here to the mean concentration profile determined by Friedman and Witte [3] (solid dots). The maxima and minima associated with the experimental determination of interstitial plasma protein concentration distribution in rat mesentery are indicated by the error bars.

The experimental procedure adopted by Friedman and Witte yielded the total plasma protein concentration, C t, which is related to the concentration within the plasma protein distribution volume, C’, by the equation

C’ = Iz’ 1 - .scl.

(10)

The results of the experimental study are shown in Fig. 2. Note that the data exhibit considerable scatter that the authors identify as “noise” associated with the procedure. There are four principal components within the model tissue that impact on exchange: the three permeable boundaries and the interstitium. Model parameters associated with each of these were adjusted using a trial and error method to obtain a reasonable fit between the model predictions of C and the measured concentration profile shown in Fig. 2. Assuming the parameter values listed in Table 1, simulations were performed for two different cases: one in which it was assumed that interstitial plasma protein transport occurs by diffusion alone (i.e., 5 = 0) and one in which both convection and diffusion take place (i.e., 5 = 0.2). Hence, the only system parameter to differ between the two simulations was the convective hindrance, 5, which, according to Eq. (5), determines the degree of plasma protein convection within the tissue. In each case, the model equations were solved using the finite element method, the solution yielding the model’s prediction of the spatial distribution of the interstitial plasma proteins. (Details of the procedure can be found elsewhere [S].) The results of the simulations are also presented in Fig. 2. Some comment regarding the choice of parameter values is warranted. LF’ and LT are somewhat higher than, but of the same order of magnitude as, experimental values reported for mesentery [4]. Since measurements of single

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TABLE 1 VALUES

OF MODEL

PARAMETERS ASSUMED IN THE SIMULATIONS

Parameter

Value”

2.8 x 10’ dyne-cm/g 2.1 x IO6 dyne-cm2/g2 1.2 x 10’ dyne-cm3/g3 0.06 gm/cm3 0.03 gm/cm’ 0.06 gm/cm3 1.0 X lo-’ cm*/sec 2.4 x lo-’ cm/set 3.6 x lo-’ cm/set 3.6 x lo-@ cm/set 3.0 X 10m3cm 3.1 x lo-*’ cm4/(dyne - set) 3.0 X IO-* cm 4 x 10m9cm3/(dyne - set) 8.0 x 10m9cm3/(dyne - set) 6.0 x 10e9 cm’/(dyne - set)

0.089 0.68 2.942 x lo4 dyne/cm’ 0.0 dyne/cm’ 2.000 x lo4 dyne/cm’ 0.8

0.59 0.75 0.87 o-o.2 y Rationale for parameters given in [lo] and text.

vessel conductivities are subject to wide variations between capillaries within the same vascular bed [6], these values are not considered unreasonable. The blood capillary reflection coefficients, meanwhile, are somewhat lower than reported in the literature for rat mesentery. But again, the values of aart and u-‘“” lie within the range published for single capillaries in frog mesentery, for example [6]. The value for p assumed in each case lies within the range of values reported for mesentery [5], while the capillary hydrostatic pressures lie within the overall range of microvascular pressures measured in mesentery [6]. Most importantly, it is emphasized that these and all other model parameters, with the exception of 4, remained unchanged from one simulation to the next. We now turn to the model predictions. In both simulations, fluid and plasma proteins enter the interstitial space from the two vascular compartments and leave the interstitium via the mesothelium. The majority of the exchange across the mesothelial layers occurs in the vicinity of the blood vessels. Hence, the concentration of plasma proteins decreases, in each case, as one moves toward the central regions of the tissue. Clearly, when 5 is zero, all interstitial plasma protein transport is by diffusion alone. However, when 4 is 0.2, there is substantial convective transport of plasma proteins within the interstitium. For example, the average

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ratio of convection to diffusion within the interstitial spaces adjacent to the arteriolar and venular boundaries is 0.99 and 0.80, respectively. It is noted further that, in regions within the interstitium, convective and diffusive transport of proteins are in opposite directions. It is interesting to note that neither of the model predictions agrees with Friedman and Witte’s interpretation of the data. These authors assumed that fluid and plasma proteins entered the interstitium across the arteriolar wall, some of the proteins then crossing the mesothelium to cause the local gradient in concentration near that vessel. However, in contrast to the two model predictions described above, they further assumed that fluid is reabsorbed by the venular vessel. The plasma protein gradient near that vessel is largely attributed to a combination of diffusion and the presence of a sink for plasma proteins (i.e., the mesothelium). This interpretation therefore represents a third possible explanation for the same concentration profile! It is clear from Fig. 2 that reasonable agreement between the experimental data of Friedman and Witte and our model predictions is possible assuming drastically different interstitial plasma protein transport mechanisms. Contrary to the assumptions made in some past analyses [11,7], it is evident from this example that, without better information regarding the transport properties of the mesothelial layer (such as its hydraulic conductivity, permeability, and sieving characteristics), one cannot draw conclusions regarding interstitial plasma protein transport mechanisms from concentration profiles in mesentery. Further, if in fact the mesothelial transport properties are similar to those assumed in the simulations (i.e., a highly conductive layer offering only moderate sieving of plasma proteins), then the concentration profiles in mesentery are largely determined by exchange across this boundary and are not greatly affected by interstitial plasma protein transport mechanisms. CONCLUDING

REMARKS

The model study presented here demonstrates that steady-state interstitial plasma protein concentration profiles alone yield insufficient information to determine the principal mechanisms of plasma protein transport within the interstitium. In caseswhere plasma protein transport is predominantly convective, the profiles can be virtually indistinguishable from those in which plasma protein transport is purely diffusive. These profiles are strongly influenced by the transport properties of the confining boundaries, for example. Further information about the boundary exchange characteristics, as well as other system parameters, is therefore needed to allow a mechanistic interpretation of measured plasma protein distribution data within the interstitia of mesentery. Other experimental techniques, such as photobleaching [2], may provide greater insight into the relative importance of plasma protein convection and diffusion within the interstitia of tissue.

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APPENDIX: NOMENCLATURE Symbol = 123

Description

Units

First, second, and third virial coefficients of colloid osmotic pressure relationship for aggregate plasma protein species. Plasma protein concentration of luminal fluid associated with boundary b. Plasma protein concentration based on total fluid. Local concentration of plasma proteins within the distribution volume fraction (n’). Permeability of membrane boundary to aggregate plasma protein species. Interstitial plasma protein diffusivity. Mesentery thickness. Local solute flux crossing mesothehum. Local total volumetric fluid flux in X, direction. Local fluid flux crossing mesothelium. Local interstitial hydraulic conductivity associated with distribution volume of protein species (assumed = n’p/n’). Local total interstitial hydraulic conductivity. Distance separating arteriolar and venular capillaries. Hydraulic conductance of membrane boundary. Local distribution volume fraction of plasma proteins. Local total mobile fluid volume fraction. Local solid phase volume fraction. Local fluid hydrostatic pressure in plasma protein distribution volume fraction (n’). Local spatial coordinate. Ratio of hydraulic conductivity in distribution volume to total interstitial hydraulic conductivity (K’/p). Local convective hindrance of plasma proteins. Local osmotic pressure of plasma proteins within the distribution volume fraction (n’). Reflection coefficient of membrane boundary b.

F . L’/M M/L’ MfL3 M/L’ LIB LZ/B L M/(L’

. 13)

LIB LIti L“/(F.

0)

L’I(F L L’I(F.

. 0) 8)

F/L’ L

F/L2

Superscripts and Subscripts

art mes ven

arteriolar capillary mesothehum venular capillary

REFERENCES 1.

BARBER, B. J., AND NEARING, R. D. (1990). Spatial distribution of protein in interstitial matrix of rat mesenteric tissue. Am. J. Physiol. 258 (Heart Circ. Physiof. 27), H556-H564. 2. CHARY, S. R., AND JAIN, R. K. (1989). Direct measurement of interstitial convection and diffusion of albumin in normal and neoplastic tissues by fluorescence photobleaching. Proc. Natl. Acad.

Sci. USA 86, 5385-5389.

3.

FRIEDMAN, J. J., AND WITTE, S. (1986). The radial protein concentration distribution in the interstitial space of rat ileal mesentery. Microvasc. Res. 31, 277-278. 4. FRASER, P. A., SMAJE, L. H., AND VERRINDER, V. (1978). Microvascular pressures and filtration coefficients in the cat mesentery. J. Physiol. 283, 439-456. 5. LEVICK, J. R. (1987). Flow through interstitium and other fibrous matrices. Quart. J. Exp. Physiol. 72, 409-438. 6. MICHEL, C. C. (1984). Fluid movements through capillary walls. In “Handbook of Physiology.

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The Cardiovascular System. Microcirculation” (E. Renkin and C. C. Michel, Eds.), pp. 375-409, Am. Physiol. Sot., Bethesda, MD. PAPENFCJSS, H. D., AND WITTE, S. (1987). Perivascular flow and transport of macromolecules: Comparison between intravital microscopic and theoretical studies. In “Microcirculation-An Update, International Congress Series 775” (M. Tsuchiya, M. Asano, Y. Mishima, and M. Oda, Eds.), Vol. 1. Excerpta Medica, Amsterdam. TAYLOR, D. G. (1990). “A Mathematical Model of Interstitial Transport and Microvascular Exchange.” Ph.D. Dissertation, University of British Columbia, Vancouver, Canada. TAYLOR, D. G., BERT, .I. L., AND BOWEN, B. D. (1990). A mathematical mode1 of interstitial transport. I. Theory. Microvasc. Rex 39, 253-278. TAYLOR, D. G., BERT, J. L., AND BOWEN, B. D. (1990). A mathematical mode1 of interstitial transport. II. Microvascular exchange in mesentery. Microvusc. Res. 39, 279-306. WITTE, S., STRASSLE, G., SCHWARZMANN, P., AND PAPENFUSS, H. D. (1987). Transvascular and Update. Inperivascular transport of fluorescent-labelled molecules. In “Microcirculation-An ternational Congress Series 755” (M. Tsuchiya, M. Asano, Y. Mishima, and M. Oda, Eds.), Vol. 1. Excerpta Medica, Amsterdam.