A mathematical model of water flux through aortic tissue

Bulletin of Mathematical Biology, Vol. 41, pp. 79-90 Pergamon Press Ltd. 1979. Printed in Great Britain © Society for Mathematical Biology

A M A T H E M A T I C A L M O D E L OF WATER FLUX T H R O U G H AORTIC TISSUE

[] DOUGLAS E. KENYON D e p a r t m e n t of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A.

Water flux in porcine aortic segments produced by the sudden application of a hydrostatic pressure gradient has been described in a recent paper by Harrison and Massaro (1976). A mathematical model is developed here to explain the results obtained when pressure is applied to either covered or uncovered samples. The model predicts that the rate of exudation in both instances should be substantially identical for a period of time ~0.2z, where z is the consolidation time. The consolidation time is proportional to the hydraulic resistance to liquid flow, and inversely proportional to the compressive stiffness of the artery. The existence of a time-dependent water flux in an artery in vivo during periodic pressurization is predicted by the mathematical model if the resistance to water flow at the endothelium is not excessive. The pore pressure within the bulk of the media is predicted to pulsate in a highly unexpected fashion. These predictions follow naturally from the fact that the consolidation phenomenon in large arteries, as determined by the compression tests of Harrison and Massaro, is of long duration, much longer than the period of a heartbeat. Pressure gradients in vivo in interstitial fluid are then confined to a very small fraction of the total arterial wall thickness. A potential for plasma "sloshing" across the endothelial junctions exists. The convective flux of water across an endothelial layer may therefore be of a pulsatile character in normal arteries in vivo.

1. Introduction. The significance of water transport in the artery wall rests in part in the validity o f the filtration theory o f atherogenesis in which arterial wall thickening is attributed to filtration and entrapment of plasma-borne constituents (Wilens, 1951). Also of interest to atherogenesis are the forces acting on the endothelium and their relationship to endothelial permeability. One should include here both hemodynamic shear stress and pressure gradients across the endothelium. The latter is normally thought to be due entirely to the time mean blood pressure, but theory developed below indicates otherwise. Generalizations which are likely to affect water transport in vivo are described. 79

80

DOUGLAS

E. K E N Y O N

2. Mathematical Model ,/or Compression of Aortic Tissue. Harrison (1975) applied a step change in pressure to segments of fresh swine descending thoracic aortae in a permeability cell. The tissues were restrained at the adventitial surface. The results are shown in Figures 1 and 2, replotted .on log-log paper for reasons which will become apparent shortly. Figure 1 gives the measured water flux as a function of time for uncovered samples and Figure 2 gives the same quantities for samples FLUX

FLUX

( ~ L cm "2 HR -1)

(FLL Cm-2 HR -I) HARRISON AND MASSARO

o

ATHEROSCLEROSIS

HARRISON (1975)

2_._41976

10

o o

e.

o

o °• o, .~,

,o v

W= K f n t es.._[J n o -.91 o

W=Kt

o

•

o o oo o

n

tes._t o

n -.56

- .50

•

- ,46

-.45

o

-.52

- .52

•

-1.55 p

10

T I M E (HR)

Figure 1

T I M E (HR)

Figure 2

which had been covered by a plastic sheet over the endothelium prior to the application of the pressure gradient. The flux in the covered and uncovered tissues for times t < 8 hours is practically indistinguishable. The steady state flux in the uncovered samples reached a roughly constant value at this time. The flow in the covered tissue appeared to be reducing still further at 8 hours but the values are very small. Harrison describes several factors contributing to water flow in live tissue, including smooth muscle cell metabolism and desorption. A simplified mathematical model for the time-dependent water flux in covered and uncovered tissues can be made. The theory as originally described for the consolidation of water-saturated soils (e.g., "Biot, 1941) will be used here. The application of the theory to other biological tissues has been made (e.g., Fatt, 1968; McCutchen, 1962). The bulk stress is viewed as a composite of a pore fluid pressure and an effective or skeleton stress in the elastic-like Structure o f the matrix. In one-dimensional compression, the interstitial fluid pressure and the matrix compressive stress share in supporting the applied compressive stress. The governing equations for flow and deformation in the tissue are three in number: (l) a

M O D E L O F WATER FLUX T H R O U G H AORTIC TISSUE

81

force balance in the tissue as a whole; (2) a law for interstitial fluid flow as a function of the pore pressure gradient; and (3) a volume flux conservation equation (Kenyon, 1976a). These equations combine together with the result that the volume change A in the tissue satisfies a diffusion equation (4). Using the symbols aij for the Cartesian components of the effective or skeleton stress in the matrix, P as the interstitial liquid pressure, Ui as the displacement vector of the tissue under load, Wz as the filtration velocity, 2 and G as the incremental moduli of elasticity of the skeleton, and #/K as the hydraulic resistance, the governing equations are as follows: c~P Ox~ aP

~x~

~xi -

Oxi

-

Oa0 Oxj

(1)

~- w ,

(2)

~c

Wi+

OA

(3)

=0

~2A (4)

c3t -- Dx (~xi c~xi ,

where

,,-

{ev, aa, + o t

+ #x, )

D~- to(2+ 2G) // gUi a =- c~xi

(5)

The important property of the diffusion system is the pore volume diffusivity Dx (of order 10 -6 cm2/sec) and it governs the manner in which transients in water flow are established in a fashion analogous to, e.g., the transfer of heat in a conductor. In writing (1)-(3), one is assuming negligible effects due to inertia and that the interstitial fluid and the walls of the skeletal members are incompressible. Also, skeleton viscoelasticity is neglected and the matrix is assumed to be elastically and hydraulically isotropic and homogeneous, assumptions to be discussed later. Finally, the

82

DOUGLAS E. KENYON

permeability ~c/# and the stiffness 2 and G are assumed to be constant during compression. The experiments by Harrison mentioned above are illustrated schematically in Figure 3 (top). The one-dimensional nature of the experiment

t

l

se°l

\ I

I ~

uncovered fissure pressure (theory)

f

y/h

7 '' p°r°us grid L~ "~" filtrate

coveredtissue pressure (theory)

y/h ,2

old/ o

p/p.

~

, 1

ol/-- / o

~"" p/p~

,

igure 3

greatly simplifies the governing equations. Thus, one seeks to determine the tissue displacement U( y, t) and the pore or interstitial fluid pressure P(y, t) as functions of position and time. For an uncovered tissue, the boundary conditions are, for a step change in applied pressure PAH(t), P(O, t)=O; U(0, t ) = 0 ;

P(h, t)= PAH(t)

(6)

0U (h. t ) = 0 . 0y

(7)

Thus, the surfaces of the tissue are assumed to drain freely and the tissue is restrained at y = 0 . The tissue experiences no contact or effective stress at the upper surface y = h. For the covered tissue the boundary conditions are P(O, t)=O;

~P - - (h, t)=O ~y

(8)

MODEI~ OF WATER FLUX THROUGH AORTIC TISSUE

U(0, t ) = 0 ;

8U P(h, t ) - (2+2G) ~ - (h, t)=PAH(t ). oy

83

(9)

Thus, the upper surface is impermeable to liquid flow and the applied stress PA is shared by the liquid pressure and the skeleton stress at the tissue surface, inside the cover, in a proportion which depends on time. The governing equations in either case are

8U

P - (2+2G) ~ 7 =PA 8P

(~2p

8t -- D~ 8y 2 .

(10)

The second equation of (10), a result unique to one-dimensional loading, follows from the first equation of (10) and (4) above. The first equation of (10) is the integrated form of (1"). The solutions are obtained easily by standard techniques (e.g., Laplace transform). The general solution can be expressed in terms o f a similarity variable y/6 and a consolidation time z, defined as

b2=_D~t h2

=

D/c.

(11 )

The pressure profiles obtained from the theory are illustrated in the lower half of Figure 3. It is seen that for sufficiently small time (t=0.05r), the pressure profiles in the two cases are indistinguishable due to the small depth of the consolidation boundary layer formed at the p o r o u s grid. Consequently, pressure gradient and the resulting water flux relative to the tissue occur only within the boundary layer near the bottom of the tissue. There is no filtration near the top of the tissue and for tl~is reason the presence o f a cover on the top makes no difference in the flux o f water squeezed out at the bottom. For larger time t ~ z , the pressure in the uncovered tissue approaches a linear profile and the pressure and flux in the covered tissue gradually falls to zero everywhere. Of more interest here is the flux of water W(O,t) at the restraining surface y = 0 . The results are most conveniently expressed in nondimensional form by comparing the flux W(O,t) to the steady state flux Wo~ obtained in the uncovered sample. The solution for the exudation ratio W(0, O/Woo is a function only of b/h, i.e., the depth of penetration 6 of

84

DOUGLASE. KENYON

the consolidation front as a fraction of the tissue thickness h. Since ( 3 / h ) 2 = t/r, the results can also be expressed in terms of time as a fraction of the consolidation time. The theoretical results are plotted in Figure 4 for the covered and 10

W FLUX Wa°

£

w _(__~V

COVERED

I ,I

.01

i 1

TIME i / r

Figure 4 uncovered tissues. Notice that there is identical flux from the covered and uncovered tissues for a substantial time, t < 0 . 2 r . The time at which the measured fluxes in covered and uncovered samples begin to depart in Harrison's experiment was roughly 8 hr, so that z = 40 hr. A lower bound for the theoretical value of -c can be estimated from (5) and (11). The permeability estimated by Harrison and Massaro (1976) is

#

<7

x 10 -13

cm4/dyne-sec.

They do not give an estimate of stiffness. However, Tucker et al. (1969) estimated a dynamic elastic modulus E' for compression of a canine descending thoracic aorta. Since these tests were conducted at a high frequency where wring-out is insensible, the steady state modulus of elasticity is only 2(1 +v)/3 of the values given, assuming an isotropic tissue as an approximation. Here, v is Poisson's ratio and is probably closer to zero than ½ if measured in equilibrium. Also, the tissue was pre-stretched in an unspecified manner and, therefore, the measured Young's modulus E' 5 x 106 dynes/cm 2 is an upper bound for the true steady state stiffness in radial compression of unstretched tissue of the kind tested by Massaro.

MODEL OF WATER FLUX THROUGH

A O R T I C TISSUE

Thus, 2 + 2 G = E ( 1 - v ) / ( l + v ) ( 1 - 2 v ) ~ E < 5 x 1 0 6 d y n e s / c m bound for r for a 2 m m thickness of tissue is, therefore,

"c=h2#/K()~ + 2G)>~3 hr.

2.

A

85

lower

(12)

While it is tempting to speculate on the effects of leaks on the permeability to/# and the effects of prestress, anisotropy, and viscoelasticity on the steady state stiffness E, there is no way to determine the magnitudes of these corrections. Rather, of more importance here is the result that with the exception of a single experiment in both the series of covered and uncovered samples, the remaining data shown in Figures 1 and 2 plot as reasonably good straight lines with slopes nested around n = - 0 . 5 0 , the theoretical value (Figure 4). It may therefore be assumed that consolidation in large artery walls is a very long term process, of order hours, and certainly much longer than the duration of a single heartbeat. The implication of this is discussed below.

3. Mathematical Model for an Artery in vivo. The existence of a consolidation time long compared to the duration of a single heartbeat implies that volume change in the artery wall during normal pulsatile pressurization is negligible, or at least almost negligible. The fact is that the diffusion equation (4) for interstitial pore volume predicts that volume change is confined to thin boundary layers of thickness 6 = x / D x / J ; where f is the frequency of a heart beating. For D~ ~ 1 0 - 6 c m 2 / s e c and . f ~ l Hz, then 6 ~ 1 0 # m , a small fraction of the total arterial wall thickness, h. Pressure gradients, as in the case of compression (Figure 3), are then confined to the boundary layers at the inside and outside of the vessel. What is perhaps more surprising is the excursion of the pore pressure produced in the bulk of the tube wall. A detailed theoretical examination of the pore pressure produced in a thin-walled, isotropic elastic porous tube has been made (Kenyon, 1976b). The "deep" tissue liquid pressure within the tube wall for thin boundary layers is such that liquid is "sucked" into the wall during a brief period of tube pressurization from within. Conversely, liquid will be expelled from the tube wall during a brief period of depressurization. These results are explained by the manner in which the stress in the "deep" tissue of the tube wall not involved in consolidation is a composite of a liquid stress and a skeletal stress. Whefi the skeleton is incrementally isotropic and homogenous, it is impossible to produce a change in the isotropic stress in the skeleton if it is prevented from undergoing volume

86

DOUGLAS E. KENYON

change. In a dynamic loading, this is achieved in the bulk of the tube wall due to the miniscule depth of penetration of the consolidation boundary layer during the period between reversals of applied pulsatile pressure. Figure 5 shows the theoretical pressure distributions in the idealized

a. Average pressure

PA 1

i

R 2h

diastole

y/h

b. Pulsatile Dressure

p'

PX y/h

R 2h

systole

Figure 5

tube in the case where c~, the depth of penetration of the consolidation boundary layer, is much less than the tube thickness h, which in turn is small compared to the neutral radius R of the tube. The pressure in the interstitial spaces is the sum of a time average pressure /5 and a pulsatile part P'. The pressure applied by the blood at the intima, or inner wall, is idealized as P A + P ~ sin cot for simplicity. The position y = 0 corresponds to the lumenal surface and y = h corresponds to the adventitial capillary bed, in which the pressure is assumed negligible in comparison to the applied pressure at the lumen. The time mean pressure 15 is linearly varying across the wall thickness and contributes to the time mean filtration rate FV according to Darcy's Law (2):

W~- ~c PA # h

(13)

M O D E L O F WATER FLUX T H R O U G H AORTIC TISSUE

87

The pulsatile pressure P' is a sinusoidal function of time and a n exponentially decaying function of the similarity variable y/6. The pulsatile pressure produced in the deep tissue region, Figure 5b, is +-P'AR/2h. The pulsatile pressure gradient across the boundary layer (5 is therefore of order P'A R/2h6. The accompanying pulsatile filtration rate at the lumen y--0 is therefore given approximately as P~ R [W'I= ~ 6 2h"

(14)

Since c ~ h and physiological pressures in humans are PAN100torr, P~ ~ 2 0 t o r r , we see that the magnitude of the pulsatile filtration IW'I is much greater than the magnitude of the steady filtration We are therefore led to the following deduction. The theoretical calculations for time mean and pulsatile pore pressure in an arterial wall a t normal heartbeat rates are such that the pulsatile filtration at the lumen, a purely sinusoidal function of time in this simplified analysis, is much greater than the time average filtration rate. Unlikely though it may seem, the theory predicts that plasma "sloshes" back and forth across the endothelium, and a little beyond, while a much more sedentary flow (due to the mean filtration rate ITV)is expected in the deep tissue regions. In addition to these peculiarities in the plasma filtration process associated with the non-quasi-steady character of the interstitial pressure, there is also an unusual force on the endothelial layer itself. For example , during "retrograde" flux from the intima to the lumen in diastole, there is a "lift" force tending to pull the endothelium (of thickness he) off the basement membrane of order R he P~ 2h x ~- x (endothelial cell area).

The factor he~6 is a very rough measure of the fraction of the pulsatile pressure drop which exists across the endothetium, assuming the resistance of the latter to flow is the same as the bulk tissue. This is discussed below. For the present, it is important to bring attention to this force level in regard to endothelial cell durability since it is several orders of magnitude larger than a typical physiological shear stress due to the flowing blood.

4. Limitations of the Simplified Theory.

There is no data known to the author by which the behavior o f real arteries in vivo compares to the theoretical predictions given here. However, from a theoretical viewpoint,

88

D O U G L A S E. K E N Y O N

there are several ways in which the elementary theory and its provocative consequences are too crude to be complete in detail for real arteries. First of all, the endothelial monolayer is likely to offer more unit resistance to flow than the bulk tissue. An estimate of the equivalent pore diameter d of the bulk tissue can be made using the Carman Kozeny equation; K=

0.0056 ~b3d 2 (1 -qS) 2 '

where ~b is the porosity of mobite interstitial fluid fraction. For tissues, part of the interstitial fluid content is bound and the rest is mobile and a reasonable upper bound for q5 is 0.4, so that the bulk permeability has as an approximate upper bound: K<0.001 d 2. If the viscosity of the mobile interstitial fluid is taken as that of water, then the value of ~c/#~7x 10-13cm4/dyne-sec measured by Harrison corresponds to a bulk pore diameter d > 2 6 0 A . This is comparable to the "large pore" system envisioned for capillary endothelial cells but is much greater than the gaps of so-called "tight junctions" exhibited by normal arterial endothelium (Robertson and Khairallah, 1973). Since the frequency of occurrence of endothelial cell junctions per unit of area is probably less than, perhaps much less than, the frequency of "pores" per unit area in the bulk wall, it stands to reason that the unit resistance to hydraulic flow across a healthy endothelial monolayer with tight junctions is likely to be much higher than that of the bulk tissue. However, the frailty of the monolayer has prevented a direct determination of its contribution to hydraulic resistance. In the limit in which the endothelium offers very large resistance to both the time mean and pulsatile filtration flows, the importance of the sloshing flow relative to the mean flow is reduced, since the pressures PA and P~ R/2h are approximately the same and both are then extinguished across the highly resistive monolayer. In the event that endothelial cells are caused to part slightly at their junctions, however, the pulsatile pressure gradient reduces slightly compared to the reduction in mean pressure gradient. With "open" endothelial junctions, plasma constituents could be swept into the subendothelial spaces during systole, where they might be "trapped" and unable to flow, retrograde, back to the lumen during diastole. Thus, filtration and entrapment are likely to be exaggerated by the plasma sloshing predicted here. Mechanically, the theory assumes that the arterial wall is incrementally

M O D E L O F WATER FLUX T H R O U G H AORTIC TISSUE

89

isotropic and is homogeneous, but the artery is known to be anisotropic (e.g., Patel et al., 1973). This will undoubtedly affect the deep tissue fluid pressures produced by the pulsatile pressures on the lumen. However, the calculations for deep tissue pressure require some knowledge of the axial strain or axial tethering, both of which are somewhat uncertain at this time, but the mechanism described here is not affected in its principle. The only uncertainty lies in magnitude of this effect on the deep tissue pressure. Deep pore pressure is unlikely to be identical to the applied pressure and therefore the large pressure gradient associated with pulsatility will still be dominated by the small boundary layer thickness 3, but perhaps scaled down (or up) by a factor impossible to calculate at present. Another interesting phenomena will occur in the event that the intima is very °°soft". Physically, the intima then behaves as a liquid-like layer which may be thought of as an extension of the blood and it supports no skeleton stress. In this event, the consolidation process occurs in the interior of the artery wall at the border between the soft intima and the first "stiff' circumferential layer, perhaps the internal elastic lamina, if it is intact. However, French (1966) has pointed out that the internal elastic layer fragments with age in larger animals and this is often accompanied by intimal thickening and smooth muscle cell migration from the media to the intima. It is purely speculative, but the existence of a consolidation boundary layer at the internal elastic lamina, with the associated "internal sloshing", could provide a mechanism for the fragmentation of this layer. It is also necessary to remark on the role o f skeleton viscoelasticity in consolidation. Wherever there are large strain rates in the skeleton due to rapid, local volume changes, there is the distinct possibility of an elevated skeleton stress associated with a high speed of, e.g., macromolecular and cellular straining in the skeleton and the skeleton becomes stiffer than it is in low strain rate regions. Skeletal viscoelasticity increases the depth of penetration 6 for a given frequency f from that predicted using a more modest strain rate to predict skeleton stiffness. This will reduce the pulsatile pressure gradient and the associated amplitude of sloshing. There is no data available on the viscoelasticity of an artery intima and the viscoelastic correction cannot be estimated. 5. Conclusions. A theory for an artery viewed as a mixture of an incompressible liquid and an internally incompressible, isotropic elastic matrix or skeleton is used to interpret compression tests of arterial segments. The process of liquid release in compression is a very long one in large arteries. As a consequence, the extrapolation of the theory to arteries in vivo leads to some highly interesting conclusions. (a) There is a pulsatile pressure gradient which is much greater than the

90

DOUGLAS E. KENYON

t i m e m e a n p r e s s u r e g r a d i e n t across the arterial intima. T h e a s s o c i a t e d w a t e r flows are t h e n d o m i n a t e d b y " s l o s h i n g " at the intima, p l a s m a b e i n g " s u c k e d " i n t o the a r t e r y d u r i n g systole a n d expelled b a c k to the l u m e n d u r i n g diastole, p r o v i d e d e n d o t h e l i a l j u n c t i o n s p a r t slightly. A very soft i n t i m a leads to " i n t e r n a l sloshing" at the i n t i m a l elastic lamella. (b) T h e r e is a s u b s t a n t i a l lift force on the e n d o t h e l i a l m o n o l a y e r d u r i n g d i a s t o l e d u e to the m o d e s t d r o p in m e a n p r e s s u r e across the m o n o l a y e r c o m p a r e d to the steep rise in the pulsatile p r e s s u r e across the m o n o l a y e r which exists d u r i n g diastole. T h i s w o r k w a s s u p p o r t e d b y a g r a n t f r o m the N a t i o n a l I n s t i t u t e of H e a l t h , H L 1 4 2 0 9 , with i m p o r t a n t assistance f r o m P r o f e s s o r C. F. D e w e y , Jr. LITERATURE Biot, M. A. 1941. "General Theory of Three-Dimensional Consolidation." J. Appl. Phys., 12, 155 164. Fatt, I. 1968. "Dynamics of Water Transport in the Corneal Stroma." Exp. Eye Res., 7, 402 412. French, J. E. 1966. "Atherosclerosis in Relation to the Structure and Function of the Arterial Intima, with Special Reference to the Endothelium." In International Review o.1 Experimental Pathology, G. W. Richter and M. A. Epstein (Eds.), Vol. 5. New York: Academic Press. Harrison, R. G., Jr. 1975. "Water Flow Across the Isolated Artery Wall." Ph.D. Thesis, University of Wisconsin. . . . . . . . and T. A. Massaro. 1976. "Water Flux through Porcine Aortic Tissue Due to a Hydrostatic Pressure Gradient." Atherosclerosis, 24, 363 367. Kenyon, D. E. 1976a. "The Theory of an Incompressible Solid Fluid Mixture." Archs Ration Mech. Analysis, 62, 131-147. . . . . . . . 1976b. "Transient Filtration in a Porous Elastic Cylinder." J. Appl. Mech., 98, 594~ 598. McCutchen, C. W. 1962. "The Frictional Properties of Animal Joints." Wear, $, 1-17. Patel, D. J., J. S. Janicki, R. N. Vaishnav and J. T. Young. 1973. "Dynamic Anisotropic Viscoelastic Properties of the Aorta in Living Dogs." Circ. Res.i 32, 93 107. Robertson, A. L., Jr. and P. A. Khairallah. 1973. "Arterial Endothelial Permeability and Vascular Disease: the "Trap Door" Effect." Exp. Molec. Path., 18, 241 260. Tucker, W. K., J. S. Janicki, F. Plowman and D. J. Patel. 1969. "A Device to Test Mechanical Properties of Tissues and Transducers." J. Appl. Physiol., 26, 656-658. Wilens, S. L. 1951. "The Experimental Pr:6dtaetion of Lipid Deposition in Excised Arteries." Science, 14, 389-393. RECEIVED 5-10-77 REVISED 9-20-77