BIOENGINEERING: THE FUTURE OF POROMECHANICS ?

J.M. Huyghe 1 (Member, ASCE), Y Schroder 2, F.P.T. Baaijens 3

ABSTRACTThere 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 tothe design of adequate solutions. Several examples are covered. Multiporosity models usually appliedto fractured rocks in the field of petroleum engineering are applied in its finite deformation form tocardiovascular desease, the number one killer in the US. A swelling model developed in the contextof intervertebral disc mechanics, is presently assessed by petroleum engineers in the context of borehole stability in shale formations. The ionisation of shales, clays, hydrogels and tissues result in anintimate coupling between electrical and mechanical behaviour. The authors make a plea in favour ofa much closer collaboration between geomechanics and bioengineering, which are both involved intoionised porous structures imbibed with electrolyte solutions, in which interfacial phenomena are oftendetermining the macroscopic behaviour.

Keywords: coronary heart disease, stroke, hernia, mixture, porous media, swelling

INTRODUCTIONSince 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 ofcivil engineering on present day life expectancy for world citizens is dramatic. Today, morethan half of the medical research is done by technically trained scientists. The NetherlandsFund for Fundamental Research into Matter - traditionally the research fund of physicists tofinance atomic physics research - has decided to redirect a good deal of its funds to the interfacebetween 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 humangenetic code has now called upon the uncovering of the mechanisms through which the geneticinformation is translated into the macroscopic human body. As all the functional constituentsare present already at the level of microsized cell, many of the challenges lie at the nano- andmicroscale 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, wefind health problems are typically one or two orders of magnitude more relevant to society than

1Corresponding author: Department of Biomedical Engineering, Eindhoven University of Technology, Eind-hoven, The Netherlands, email: j.m.r.huyghe@tue.nl, fax:+31-40-2447355

2Department of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands3Department of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

FIG. 1: Relevance for society for a number of issues typically associated with poromechan-ics. The last 3 figures represent the more readily identifiable costs for medical care, workerscompensation payments and time lost from work. It does not include costs associated withlost personal income due to acquired physical limitation resulting from a back problem andlost 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.org2003)

engineering problems ( fig. 1). Business associated with cardiovascular disease is in the ordera 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. Evenon the level of a single cell, the number of molecules involved are so high that if any modellingis going to be successful, it will necessarily involve continuum mechanics. We discuss oneapplication on the microporescale, blood perfusion, and one application on the nanoporescaleand the mechanics of swelling.

CARDIOVASCULAR DISEASEWilson, Aifantis and many other have been successful in using methods of multiporosity

porous media mechanics on fractured rocks (Wilson and Aifantis 1982) . Multiporosity theoryhas applications even much closer to us. Coronary artery disease is believed to be number onecause of mortality in our world. The coronary vascular system is nothing but a pore structureinside a deforming solid which we call the heart muscle. The muscle is subject to large de-formations. The pore structure is highly organised to deliver every heart cell from its wastematerials and supply every cell of its nutrients. The way this is done is through a multiporositystructure (fig. 3). Since many centuries these porosities have names : arteries, arterioles, cap-illaries, venules and veins. They have their characteristic pore size, their characteristic bloodvelocity, their characteristic wall properties and their characteristic pathology. Every doctorin the world even knows a quantity proper to multiporosity flow : perfusion. It is the flow offluid 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 evidencethat blood perfusion is affected by the deformation of the heart muscle (Spaan 1985). There

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FIG. 2: Left : Corrosion cast of the coronary arterial tree of a canine heart (cast is courtesyof dr. P. Santens, University of Gent). Right : Simulated blood pressure distribution acrossa section ����� of the heart wall. During diastole most of the arteriovenous pressure dropis located in the arterioles. During systole the deeper endocardial layers of the heart wall aresubjecting 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 modelledusing a multiporosity, finite deformation approach (fig. 3). The essential difference betweenthe perfusion model (Huyghe et al. 1989; Huyghe et al. 1989) and the classical multiporositymodels of fractured rocks is the coupling between the interporosity flow and the intraporosityflow (Fig. 3): � �������

˜

��˜ � � � �� ������������

˜ ��� ������˜

� (1)

The multiporosity model was implemented in 2D, 3D and axisymmetric finite elements (Vankanet 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 sameequations were derived twice : using mixture theory (Vankan et al. 1996) and using aver-aging theorems (Huyghe and van Campen 1995a; Huyghe and van Campen 1995b). A micro-macro 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 operationalmethods to reconstruct the 3D microgeometry post mortem.

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FIG. 3: A dense tree structure can be modelled as a multiporosity continuum. Each generationof branches - the stem, larger branches, intermediate branches, small branches and twigs - areone porosity. The interface between two subsequent porosities, is plotted on this graph. Thenormal on this interface, pointing from large to smaller branches is directed upwards on aver-age. A gradient of chemical potential from small towards larger branches, therefore, inducesan upward flux of sap. Conversely, a gradient in chemical potential between the upper branchesand lower branches induces a flux of sap from larger branches to smaller branches. This ex-ample illustrates the coupling factor

�˜

�between on the one hand the interporosity flow and

intraporosity flow in the Darcy-type equation (1). This coupling is not accounted for in Wilsonand Aifantis (1982)

SWELLINGSwelling 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 governingequations, as derived by Molenaar et al. (1998), are given in the special case of infinitesimalquadriphasic mechanics of compressible charged porous media. Finite deformation is dealtwith elsewhere (Huyghe and Janssen 1997; van Loon et al. 2003). Specific constitutive mate-rial behavior is considered and corresponding relations for the osmotic pressure and electricalpotential at equilibrium are given. We consider a porous solid saturated with a monovalentionic solution. The current volume fraction of the solid is � , the current volume fraction ofthe ionic solution is � � . The corresponding initial volume fractions are � � , � �� . The solutionis 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� �� . The unstrained volume change �� of constituent � � ����� � is :

� � � ��� �� � (2)

in which � � is an intrinsic reference density such that � �� ��� ��� ���� �� � � � � �� .

BALANCE LAWSThe momentum balance, neglecting inertia, reads:

���� � � ! � (3)

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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 fullysaturated,the following relation for the saturation condition holds:� ���� �� � � ���� � �

�� ��� ��� �� � ��� � ! � (5)

in which�� is the solid velocity, � and

are the Biot coefficients. Electroneutrality requires:

� � � � ��� ��

� �� � � ��� � ! � (6)

with , � and�� Faraday’s constant, the valence and molar volume of constituent respec-

tively.

CONSTITUTIVE BEHAVIORThe 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 of the energy func-tion

�with respect to the infinitesimal strain tensor � � ����! �� �" � �� �"�# � # , with

�" thedisplacement vector of the solid : � � ���� � (9)

The energy function is the sum of the poroelastic strain energy and a mixing energy:

� � � � �%$ � � � � � ��� �

� � � �

�� ��� ��� �� #%& � � �('�) � �(' � � & # �*$ & �

� ) �,+-� � � � � � ��� �

� � � �/.1032 � �� � � � �� � #%465 � � � .10 � �� � 475 � �� � � # � .10 � �� � 465 � �� � � # (10)

The form (10) of�

assumes linear isotropic elasticity and Donnan osmosis as the swellingmechanism of the shale. ) is the shear modulus, ' is Poisson’s ratio,

� � is the volumetric

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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 energyfunction, the electrochemical potentials (8) take the form:

� � � � � � � � � .1032 � �� � � �� � �� � � �

# (11)

� � � � �� � � � .10� � 475 � � � � # � � �� ��� � (12)

� � � � �� � � � .10� � 475 � � � � # � � �� � � � (13)

considering that � � � � �%$ � � � � � � ��� �

� (14)

The differential form of eqs. (11-13) is:���� � � � �� � � � # � �� � � � ��� �

� � � � ���� � � � ����� � (15)

in which � � � � & � � � � � � � (16)

The numerical implementation of the above equations is done along the lines of (van Loonet al. 2003). The displacements, the pressures, the electrochemical potentials of water, cationsand anions, and the electrical potentials are the degrees of freedom.

� ��� � � ���˜

�˜

��� � � � ��� ��� � � � �� � ��� � � � � ��� ��� � �������� � � � ˜ � ˜ ��� ��� ˜ �

�� � � � � � � �!� � � � � � � � � ���� ��� � �#"

˜

� � ˜ � � � �%$ � ˜ & � � ˜ ')(�� � ��� ��� � ���

� ��� � � �*� � ���� ��� � � � ��� � �

� ��� � � � ��� ��� � �#+

˜� � � � � ��� � � ˜ �

� �,.-�/ $10243 05 (76%8 � ')9�1: ' 8 0243 05�;=<?> � �

,!-A@ 02CB#3 05 � <?>� � � �

, -4D˜ $ 9 � � (FE & D ˜ � <?> � � �HG , - 02 D 6 B JI� � B 02 D�KML <N>

˜ � �

,.O 3 05 B�PQB 0RS< � �,1- PUTB 02V3UW 05�<?> �

,.- 02CB#3 05 � <?> � ˜ �,.- D

˜ $ LYX : �LYX:!Z � � � � Z� ( <?>� ˜ & �%G , - $ � � B 02 " � ( B 02 D ˜ KAL <?> � ˜ ' � $ � � G ( , - $ � � B 02 " � Z ( B 02

D˜

KAL <?>� ˜ � �

, - D˜ $\[ � � [ �Z ( <?>

(17)

Continuity of the displacements and the electrochemical potentials is enforced both withinthe elements as between element. As the displacement are continuous, but non-differentiable

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across the element interfaces, the strains, unstrained volumes and concentrations are discon-tinuous across the element boundaries. From eqs. (11), (12) and (13), one infers that thecontinuity of the electrochemical potentials is incompatible with the continuity of pressure andelectrical potential, if the unstrained volumes are discontinuous. Hence, it is necessary notto enforce continuity of the pressure and the electrical potential. Fig 5 shows the deformedmesh of swelling simulation of an intervertebral disc after excision. The swelling responseis observed experimentally and indicates that in the in vivo state (before excision) swellingpressures tensions the disc fibers even in its unloaded state. This prestressing might have aprotecting function against hernia (fissuring of the disc).

FIG. 4: Chemical potential of the fluid as a function of time during swelling of a one dimen-sional ionised medium. The solution from a 3D finite element code (van Loon et al. 2003) iscompared to the analytical solution (van Meerveld et al. 2003).

CONCLUSIONSThe 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 mate-rials have a structure that is far more sophisticated. Hence, cooperation with many more disci-plines is mandatory in order to cover the whole range expertise needed to address all issues. Itis vital that the community of porous media experts contribute their share to understanding ofour own bodies and those of other species.

ACKNOWLEDGEMENTThe authors acknowledge financial support from the Technology Foundation STW, the

technological branch of the Netherlands Organisation of Scientific Research NWO and theministry of Economic Affairs (grant MGN 3759). Part of the research is done in the context

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FIG. 5: Free swelling of an intervertebral disc in physiological salt solution, after excision fromits 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 thescale of the geometry

of EURODISC, a project funded by the Quality of Life Key action ’The Aging Population’ ofThe European Union.

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