MICROVASCULAR

RESEARCH

A Theory

42, 113-116

for Fluid Transport through Connective Tissue T. R.

Department

of Mechanical

(1991)

BLAKE

AND

Engineering,

E.

Univemity Received

HATZIMANOLAKIS of Massachusetts,

July

Interstitial

II,

Amherst,

Massachusetts

01003

1989

INTRODUCTION A theoretical continuum description of the fluid and solid mechanics of the interstitium is used to model fluid transport through the extracellular space of connective tissue. The equations for conservation of mass, and Darcy’s relation, are added to the balance of forces in Blake (1989) to yield a theory for both transient and steady flows. This theory is more complex than that of Salathe (1977), since it includes the influence of the ultrastructure. However, it is simpler than that of Mow et al. (1980) for articular cartilage, since the concentration of polysaccharides in connective tissue is dilute and inertial effects can be neglected. CONTINUUM

MODEL

In Blake (1989), analogous to Wiederhielm (1979), the interstitial space in the connective tissue is conceptualized as a multiphase environment, where one phase is a “polysaccharide” phase of water-imbibed polysaccharides entangled with collagen and elastin. A second phase is the “free fluid” phase. For the present study, this free fluid phase is absent, and proteins, electrolytes, and nutrients are neglected. The stress within the polysaccharide phase (Scholander, 1971; Laurent, 1972) is the sum of a hydrostatic pressure of the imbibed fluid, p; an osmotic pressure of the polysaccharides, rTT,; and an elastic stress of the polysaccharide entanglement with the collagen and elastin, 7ij. Within the context of the continuum theories for multiphase mixtures (Bird, et al., 1987; Blake, 1989), the conservation of momentum for the polysaccharide phase, neglecting inertia, is O

=

b,-

CP

+

Tfl)S,

+

Tjj},

(1)

I

where 7ij is composed of a trace component Tkk and a deviatoric component r$ such that rii = 8;j$rkk + 7: and & = 0. The volume change of the polysaccharide phase is due solely to the imbibition 113 0026.2862/91 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in U.S.A.

114

BRIEF

COMMUNICATION

of water, and, consequently, the density of the polysaccharide pn and the density of the imbibed water pp are related according to (Blake, 1989) (2)

These densities are defined by the mass of the the polysaccharide phase, and consequently are substance, e.g., if p” is the density of water then responds to some reference state. Conservation requires that

substance in a unit volume of not the densities of the pure pp s p”. The subscript e corof mass of the polysaccharide

(3) where ug is the velocity of the polysaccharide. written as

The conservation

of fluid mass is

(4) where uf is the velocity of the fluid. In accord with Levick (1987), the velocity of the imbibed fluid z$ is defined, relative to the velocity of the polysaccharide up by a Darcy relationship ,f.euUa=1

-k

dp ’ aXi

(5)

where kp is a hydraulic conductivity, affected by the densities of glycosaminoglycans, proteoglycans, and collagen. Also, experimental observations on the osmotic pressure of the polysaccharides (Wiederhielm et al., 1976) indicate that 7rIT, = r,(p,), and measurements of swelling pressure in tissue (Guyton, 1972; Guyton et al., 1971) suggest that the trace of the polysaccharide mechanical stress is a function of the density of the imbibed water rkk = rkk(@). A constitutive equation for the deviatoric stress must be defined. The simplest expression for $ would involve a dependence on the polysaccharide strain (Fung et al., 1966; Bird et al., 1987). In the special case of small perturbations from an equilibrium state, the stress deviator can be written 7: = pe(auq'/axj + a2$/ax, - $3, auf/a&), where pp is the shear modulus and us’ is the polysaccharide displacement from equilibrium (Fung et al., 1966). The Eqs. (l-5) become analogous to those of Biot (1956) for flow through an elastic matrix. Specifically, a perturbation from equilibrium implies: pa = pae + p:, pp = pPe + pb, p = pe + p’, ug = ug’, and r$ = u?‘, where p; 10m4 cm, the properties of the modified Bessel function cause the transient p’ to decay within a few radii of the neighborhood of r,. Of course in a truly three-dimensional environment (cf. Blake and Gross, 1982) the elliptical solution of Eq. (6) can be important and, together with the parabolic solution, may yield a spatial decay which is different from that for cylindrical geometry. Jackson and James (1982) measure kpe. This is obtained indirectly from a laboratory rig, where a pressure differential causes solvent to flow through a chamber containing hyaluronic acid. Since collagen and elastin are not in the solution of hyaluronic: A, = {(dm,/dp,)p,},, p, = 0. The time scale for steady flow in that cell can be estimated from the transient solution of Eq. (6): p”L2/(6pS,kp,A,) = 4 x lo4 set, where L = 0.3 cm is the axial length of the test cell, pae = .Ol g/cm3, (d~~/dp~)~ = 2 x lo6 dynes-cm/g (Wiederhielm et al., 1976), and kP = 2 x lo-” cm4/dynes-set (Levick, 1987). This estimate of the characteristic time is not consistent with the observation by Jackson and James that steady-state occurs approximately 3 hr after the imposition of the pressure differential for 1% concentration hyaluronic solutions. If the present analysis is applied to lower concentrations of hyaluronic, the characteristic time is found to be relatively insensitive to concentration. This behavior is occasioned by the nonlinear nature of (dm,/dp,), and kpe, and appears to differ from the observations of Jackson and James. The qualitative nature of the steady flow conditions in the cell of Jackson and James can be assessed from the solution of Eq. (6). For example, the axial displacement uy’ of the polysaccharide relative to its equilibrium position is uy’ = -Ap’(x, -

116

BRIEF COMMUNICATION

,5)x,/2,%,, where Ap’ is the small pressure differential caused by raising the constant head reservoir in the experiment, and x1 is the axial distance measured from the upstream end of the cell. Hence, even at low flows, the distribution of polysaccharide is nonuniform. ACKNOWLEDGMENTS This research is supported by NIH HL 17421 to the Department of Physiology, College of Medicine, University of Arizona, with a subcontract to the Department of Mechanical Engineering at the University of Massachusetts, Amherst.

REFERENCES BIOT, M. A. (1956). General solutions of the equations of elasticity and consolidation for a porous material. J. Applied Mech. 78, 91-96. BIRD, R. B., CURTISS, C. F., ARMSTRONG, R. C., AND HASSAGER,0. (1987). “Dynamics of Polymer Liquids,” Vols. I and II, Wiley, New York. BLAKE, T. R. (1989). A theoretical view of interstitial fluid pressure-volume measurements. Microvasc. Rex 37, 178-187. BLAKE, T. R., AND GROSS, J. F. (1982). Analysis of coupled intra and extraluminal flows for single and multiple capillaries. Math. Biosci. 59, 173-206. FUNG, Y. C., ZWEIFACH, B. W., AND INTAGLIETTA, M. (1966). Elastic environment of the capillary bed. Circ. Res. 19, 44-461. GUYTON, A. C. (1972). Compliance of the interstitial space and the measurement of tissue pressure. PfIusers Arch. 336(Suppl.), Sl-S20. GUYTON, A. C., GRANGER, H. J., AND TAYLOR, A. E. (1971). Interstitial fluid pressure. Physiol. Rev.

51 (3), 527-563. G. W., AND JAMES, D. F. (1982). The hydrodynamic resistance of hyaluronic acid and its contribution to tissue permeability. Biorheology 19, 317-330. LAURENT, T. C. (1972). The ultrastructure and physical-chemical properties of interstitial connective tissue. Pflugers Arch. 336 (Suppl.), S21-S42. LEVICK, J. R. (1987). Flow through interstitium and other fibrous matrices. Quart. J. Exp. Physiol. JACKSON,

72, 402-438. V. C., KUEI, S. C., LAI, W. M., AND ARMSTRONG, C. G. (1980). Biphasic creep and stress relaxation of articular cartilage in compression: Theory and experiments. J. Biomech. Eng. 102,

Mow,

73-84. M. P. (1976). Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents. Rev. Geophys. Space Phys. 14 (2), 227-241. SALATHE, E. P. (1977). An analysis of interstitial fluid pressure in the web of the bat wing. Am. J. Physiol. 232, H297-H304. SCHOLANDER, P. R. (1971). State of water in osmotic processes. Microvasc. Res. 3, 215-232. WIEDERHIELM, C. A. (1979). Dynamics of capillary fluid exchange: A nonlinear computer simulation. RICE, J. R., AND CLEARY,

Microvasc. WIEDERHIELM,

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C. A., Fox, J. R., LEE, D. R. (1976). Ground substance mucopolysaccharides and plasma proteins: Their role in capillary and water balance. Am. J. Physiol. 230, 1121-1125. WIEDERHIELM, C. A., AND WESTON, B. V. (1973). Microvascular, lymphatic and tissue pressures in the unanesthetized mammal. Am. J. Physiol. 225, 992-996.

A theory for fluid transport through interstitial connective tissue.

MICROVASCULAR RESEARCH A Theory 42, 113-116 for Fluid Transport through Connective Tissue T. R. Department of Mechanical (1991) BLAKE AND En...
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