BIOENGINEERING: THE FUTURE OF POROMECHANICS ? J.M. Huyghe 1 (Member, ASCE), Y Schr¨oder 2 , F.P.T. Baaijens 3 ABSTRACT There is no human tissue which is not a porous medium. Accurate quantitative understanding of the poromechanics of these tissues provide a powerful tool to understand the etiology of diseases and to the design of adequate solutions. Several examples are covered. Multiporosity models usually applied to fractured rocks in the field of petroleum engineering are applied in its finite deformation form to cardiovascular desease, the number one killer in the US. A swelling model developed in the context of intervertebral disc mechanics, is presently assessed by petroleum engineers in the context of bore hole stability in shale formations. The ionisation of shales, clays, hydrogels and tissues result in an intimate coupling between electrical and mechanical behaviour. The authors make a plea in favour of a much closer collaboration between geomechanics and bioengineering, which are both involved into ionised porous structures imbibed with electrolyte solutions, in which interfacial phenomena are often determining the macroscopic behaviour.
Keywords: coronary heart disease, stroke, hernia, mixture, porous media, swelling INTRODUCTION Since many centuries civil engineers have played a mayor role in human health care. Even if we only consider the dramatic improvement of sewage systems in the cities, the impact of civil engineering on present day life expectancy for world citizens is dramatic. Today, more than half of the medical research is done by technically trained scientists. The Netherlands Fund for Fundamental Research into Matter - traditionally the research fund of physicists to finance atomic physics research - has decided to redirect a good deal of its funds to the interface between physics and biology. While the 20th century was the century of the physical sciences, the 21st century is expected to become the century of biology. The uncovering of the human genetic code has now called upon the uncovering of the mechanisms through which the genetic information is translated into the macroscopic human body. As all the functional constituents are present already at the level of microsized cell, many of the challenges lie at the nano- and microscale and require therefore advanced technical skills for both measuring and modelling. The relevance of work done for health issues is paramount. When translated in numbers, we find health problems are typically one or two orders of magnitude more relevant to society than 1
Corresponding author: Department of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands, email:
[email protected], fax:+31-40-2447355 2 Department of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands 3 Department of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands
FIG. 1: Relevance for society for a number of issues typically associated with poromechanics. The last 3 figures represent the more readily identifiable costs for medical care, workers compensation payments and time lost from work. It does not include costs associated with lost personal income due to acquired physical limitation resulting from a back problem and lost employer productivity due to employee medical absence. Sources : (1) (van Oort 1997), (2) (BOAT 1994), (3) (www.nof.org/osteoporosis/stats.htm 2004), (4)(www.amaricanheart.org 2003)
engineering problems ( fig. 1). Business associated with cardiovascular disease is in the order a half trillion dollar for US only. What is the role of porous media mechanics in all of this ? Most biology involves fluid flow, convection of solutes, deformation of a solid skeleton. Even on the level of a single cell, the number of molecules involved are so high that if any modelling is going to be successful, it will necessarily involve continuum mechanics. We discuss one application on the microporescale, blood perfusion, and one application on the nanoporescale and the mechanics of swelling. CARDIOVASCULAR DISEASE Wilson, Aifantis and many other have been successful in using methods of multiporosity porous media mechanics on fractured rocks (Wilson and Aifantis 1982) . Multiporosity theory has applications even much closer to us. Coronary artery disease is believed to be number one cause of mortality in our world. The coronary vascular system is nothing but a pore structure inside a deforming solid which we call the heart muscle. The muscle is subject to large deformations. The pore structure is highly organised to deliver every heart cell from its waste materials and supply every cell of its nutrients. The way this is done is through a multiporosity structure (fig. 3). Since many centuries these porosities have names : arteries, arterioles, capillaries, venules and veins. They have their characteristic pore size, their characteristic blood velocity, their characteristic wall properties and their characteristic pathology. Every doctor in the world even knows a quantity proper to multiporosity flow : perfusion. It is the flow of fluid from one porosity to another, measured per unit volume tissue. Flow has a dimension and as it is defined per unit volume, perfusion has a dimension of . There is evidence that blood perfusion is affected by the deformation of the heart muscle (Spaan 1985). There
2
FIG. 2: Left : Corrosion cast of the coronary arterial tree of a canine heart (cast is courtesy of dr. P. Santens, University of Gent). Right : Simulated blood pressure distribution across a section of the heart wall. During diastole most of the arteriovenous pressure drop is located in the arterioles. During systole the deeper endocardial layers of the heart wall are subjecting the coronary system to high pressure because of the strong muscle contraction.
is evidence that blood perfusion affects the deformation, both metabolically as mechanically (Olsen et al. 1981). These findings demonstrate that the coronary system should be modelled using a multiporosity, finite deformation approach (fig. 3). The essential difference between the perfusion model (Huyghe et al. 1989; Huyghe et al. 1989) and the classical multiporosity models of fractured rocks is the coupling between the interporosity flow and the intraporosity flow (Fig. 3):
˜ ˜ ˜ ˜
(1)
The multiporosity model was implemented in 2D, 3D and axisymmetric finite elements (Vankan et al. 1997). Animal experiments were undertaken to verify the concepts (van Donkelaar et al. 2001). Comparison of experiment and model clearly demonstrate the capabilities. The same equations were derived twice : using mixture theory (Vankan et al. 1996) and using averaging theorems (Huyghe and van Campen 1995a; Huyghe and van Campen 1995b). A micromacro transformation was derived to compute hydraulic permeabilities from the microstructure (Huyghe et al. 1989; Huyghe et al. 1989; Vankan et al. 1997). While at the time of the research, the microstructure was hard to recontruct, today micro-computed tomography are operational methods to reconstruct the 3D microgeometry post mortem. 3
FIG. 3: A dense tree structure can be modelled as a multiporosity continuum. Each generation of branches - the stem, larger branches, intermediate branches, small branches and twigs - are one porosity. The interface between two subsequent porosities, is plotted on this graph. The normal on this interface, pointing from large to smaller branches is directed upwards on average. A gradient of chemical potential from small towards larger branches, therefore, induces an upward flux of sap. Conversely, a gradient in chemical potential between the upper branches and lower branches induces a flux of sap from larger branches to smaller branches. This exbetween on the one hand the interporosity flow and ample illustrates the coupling factor ˜ intraporosity flow in the Darcy-type equation (1). This coupling is not accounted for in Wilson and Aifantis (1982)
SWELLING Swelling is a symptom of disease since antiquity. It is associated with ionisation of a porous medium. This is true for clays as well as for living tissues. In this section the governing equations, as derived by Molenaar et al. (1998), are given in the special case of infinitesimal quadriphasic mechanics of compressible charged porous media. Finite deformation is dealt with elsewhere (Huyghe and Janssen 1997; van Loon et al. 2003). Specific constitutive material behavior is considered and corresponding relations for the osmotic pressure and electrical potential at equilibrium are given. We consider a porous solid saturated with a monovalent fraction of ionic solution. The current volume fraction of the solid is , the current volume the ionic solution is . The corresponding initial volume fractions are , . The solution is a molecular mixture of water (w), cations (+) and anions (-). The partial densities of water, cations and anions are in the current state , and , and the in the initial state , and is : . The unstrained volume change of constituent
in which is an intrinsic reference density such that BALANCE LAWS The momentum balance, neglecting inertia, reads:
!
4
(2)
.
(3)
with
the Cauchy stress tensor. Mass balance of constituent
requires:
!
(4)
with the unstrained volume flux of constituent with respect to the solid. In this paper denotes all constituents, i.e. electrically charged solid (s), water (w), cations (+) and anions (-); , all constituents except the charged solid matrix. As the mixture is assumed to be fully saturated,the following relation for the saturation condition holds:
in which is the solid velocity,
and
with , and tively.
!
(5)
are the Biot coefficients. Electroneutrality requires:
!
(6)
Faraday’s constant, the valence and molar volume of constituent respec-
CONSTITUTIVE BEHAVIOR The fluxes obey a coupled form of Darcy’s, Fick’s and Ohm’s law:
for
(7)
with a positive definite symmetric permeability matrix. In relationship (7) the electro chemical potentials, are defined as:
(8)
with and the electrical potential and the energy function respectively. The stress appearing in the momentum balance (3) is the partial derivative energy func " of " the # # , with " the with respect to the infinitesimal strain tensor tion ! displacement vector of the solid :
(9)
The energy function is the sum of the poroelastic strain energy and a mixing energy:
/.1032
%$
#%&
(')
(' ,+- ) 64 5 #%465 1 475 # 1 . 0 . 0
& # * $ &
#
(10)
The form (10) of assumes linear isotropic elasticity and Donnan osmosis as the swelling mechanism of the shale. ) is the shear modulus, ' is Poisson’s ratio, is the volumetric 5
modulus of the solution, is the reference potential of constituent , . is gas constant, 0 the absolute temperature and 2 the osmotic coefficient. Given this expression for the energy function, the electrochemical potentials (8) take the form:
.1032 .1 0 475 #
.1 0 475 #
considering that
%$
#
(11) (12) (13)
(14)
The differential form of eqs. (11-13) is:
in which
#
&
(15)
(16)
The numerical implementation of the above equations is done along the lines of (van Loon et al. 2003). The displacements, the pressures, the electrochemical potentials of water, cations and anions, and the electrical potentials are the degrees of freedom.
!
*
˜
#"
,.-/ 4 2 3 5 ')9 243 5;= $10 0 (76%8 ' 8 0 0 , -4D 1: D 9 (FE & ˜ ˜$ ,.O 3 BPQB ,1- PUT B 2V3UW ., - C 2 B#3 5 5 RS< 5 0 0 0 0 ˜ 0 0 D A K L ," B2 B2 $ 0 ( 0 ˜ ˜ & %G ,- D [ ( ˜ ˜ $\[ Z
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,!-A@ 2CB#3 0 ,- 2 HG ,.- D LYX ˜$ : ˜
' ˜
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5 ?< > 0 D B B 2 DKML JI 0 6 0 LYX Z ( :!Z D KAL ," B2 B2 G( $ 0 ( 0 ˜ Z
(17)
Continuity of the displacements and the electrochemical potentials is enforced both within the elements as between element. As the displacement are continuous, but non-differentiable 6
across the element interfaces, the strains, unstrained volumes and concentrations are discontinuous across the element boundaries. From eqs. (11), (12) and (13), one infers that the continuity of the electrochemical potentials is incompatible with the continuity of pressure and electrical potential, if the unstrained volumes are discontinuous. Hence, it is necessary not to enforce continuity of the pressure and the electrical potential. Fig 5 shows the deformed mesh of swelling simulation of an intervertebral disc after excision. The swelling response is observed experimentally and indicates that in the in vivo state (before excision) swelling pressures tensions the disc fibers even in its unloaded state. This prestressing might have a protecting function against hernia (fissuring of the disc).
FIG. 4: Chemical potential of the fluid as a function of time during swelling of a one dimensional ionised medium. The solution from a 3D finite element code (van Loon et al. 2003) is compared to the analytical solution (van Meerveld et al. 2003).
CONCLUSIONS The application of porous media mechanics in biology is growing field of interest, that presents challenges which in many aspects are similar to those encountered in geomechanics. At small pore scales, electrical effects become very significant. However, the biological materials have a structure that is far more sophisticated. Hence, cooperation with many more disciplines is mandatory in order to cover the whole range expertise needed to address all issues. It is vital that the community of porous media experts contribute their share to understanding of our own bodies and those of other species. ACKNOWLEDGEMENT The authors acknowledge financial support from the Technology Foundation STW, the technological branch of the Netherlands Organisation of Scientific Research NWO and the ministry of Economic Affairs (grant MGN 3759). Part of the research is done in the context 7
FIG. 5: Free swelling of an intervertebral disc in physiological salt solution, after excision from its neighbouring vertebrae. The mesh covers one eight of an axisymmetric intervertebral disc, which is cylindrical in the undeformed state. The scale of the displacements is identical to the scale of the geometry
of EURODISC, a project funded by the Quality of Life Key action ’The Aging Population’ of The European Union. REFERENCES BOAT (1994). “Project briefs: Back pain patient outcomes assessment team.” MEDTEP Update, 1(1). Huyghe, J. M. and Janssen, J. D. (1997). “Quadriphasic mechanics of swelling incompressible porous media.” Int. J. Eng. Sci., 35, 793–802. Huyghe, J. M., Oomens, C. W., and van Campen, D. H. (1989). “Low Reynolds number steady state flow through a branching network of rigid vessels: II. A finite element mixture model.” Biorheology, 26, 73–84. Huyghe, J. M., Oomens, C. W., van Campen, D. H., and Heethaar, R. M. (1989). “Low Reynolds number steady state flow through a branching network of rigid vessels: I. A mixture theory.” Biorheology, 26, 55–71. Huyghe, J. M. and van Campen, D. H. (1995a). “Finite deformation theory of hierarchically arranged porous solids: I. Balance of mass and momentum.” Int. J. Engng Sci., 33(13), 1861–1871. Huyghe, J. M. and van Campen, D. H. (1995b). “Finite deformation theory of hierarchically arranged porous solids: II. Constitutive behaviour.” Int. J. Engng Sci., 33(13), 1873–1886. Molenaar, M., Huyghe, J., and Van den Bogert, P. (1998). “Constitutive modeling of charged porous media.” SPE paper 47332, SPE/ISRM EUROCK’98, Trondheim. 127–132. Olsen, C., Attarian, D., Jones, R., Hill, R., Sink, J., Sink, K., and Wechsler, A. (1981). “The coronary pressure-flow determinants of left ventricular compliance in dogs.” Circulation Research, 49, 856–870. Spaan, J. A. (1985). “Coronary diastolic pressure-flow relation and zero flow pressure explained on the basis of intramyocardial compliance.” Circ. Res., 56(3), 293–309. van Donkelaar, C., Huyghe, J., Vankan, W., and Drost, M. (2001). “Spatial interaction between 8
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