We all are porous media

Jacques M. Huyghe, R. van Loon and F.T.P. BaaijensDepartment of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, The NetherlandsPatricia M. van KemenadePersonal Care Institute, Philips Research, Eindhoven, The NetherlandsTheo H. SmitDepartment of clinical physics and informatics, Vrije Universiteit Medical Centre, Amsterdam, The Netherlands

Today,the focus of physical scientists is shifting more to biology than ever before. Because there is no biologicaltissue that is not a porous medium, the porous media community should be very alert to this shift of attention.The impact that porous media mechanics had on geosciences in the past centuries, may very well be reiteratedin the field of biology - probably in an amplified form. The characteristic pore size in most biological appli-cations is close to the molecular level and hence below the Debye-Hueckel scale. Not only pressure gradientsand concentration gradients, but electrical gradients as well are intimately linked to fluid flow, ion flow anddeformation. To give a feeling for the subject, applications are covered in this presentation : blood perfusion,the skin barrier, the mechanosensory mechanics in bone and the swelling of tissues and cells.

1 INTRODUCTION

More than half of the medical research in the Nether-lands is done by technically trained scientists. TheNetherlands Fund for Fundamental Research intoMatter - traditionally the research fund of physiciststo finance atomic physics research - has decided toredirect a good deal of its funds to the interface be-tween physics and biology. While the 20th centurywas the century of the physical sciences, the 21stcentury is expected to become the century of biol-ogy. The uncovering of the human genetic code hasnow called upon the uncovering of the mechanismsthrough which the genetic information is translatedinto the macroscopic human body. As all the func-tional constituents are present already at the levelof microsized cell, many of the challenges lie at thenano- and microscale and require therefore advancedtechnical skills for both measuring and modelling.What is the role of porous media mechanics in all ofthis ? Even on the level of a single cell, the number ofmolecules involved are so high that if any modellingis going to be successful, it will necessarily involvecontinuum mechanics. We will discuss one applica-tion on the microporescale, blood perfusion, and threeapplications on the nanoporescale, the skin barrier,bone remodelling and the biomechanics of swelling.

2 BLOOD PERFUSION

Figure 1: Corrosion cast of the coronary arterial treeof a canine heart (cast is courtesy of dr. P. Santens,University of Gent)

Wilson, Aifantis and many other have been suc-cessful in using methods of multiporosity porous me-dia mechanics on fractured rocks (Wilson and Aifan-tis 1982) . Multiporosity theory has applications even

1

Figure 2: Simulated blood pressure distribution acrossa section ����� of the heart wall. During diastole mostof the arteriovenous pressure drop is located in the ar-terioles. During systole the deeper endocardial layersof the heart wall are subjecting the coronary systemto high pressure because of the strong muscle con-traction.

much closer to us. Coronary artery disease is be-lieved to be number one cause of mortality in ourworld. The coronary vascular system is nothing buta pore structure inside a deforming solid which wecall the heart muscle. The muscle is subject to largedeformations. The pore structure is highly organisedto deliver every heart cell from its waste materialsand supply every cell of its nutrients. The way thisis done is through a multiporosity structure (fig. 1).Since many centuries these porosities have names : ar-teries, arterioles, capillaries, venules and veins. Theyhave their characteristic pore size, their characteris-tic blood velocity, their characteristic wall propertiesand their characteristic pathology. Every doctor in theworld even knows a quantity proper to multiporos-ity flow : perfusion. It is the flow of fluid from oneporosity 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 de-formation of the heart muscle (Spaan 1985). Thereis evidence that blood perfusion affects the deforma-tion, both metabolically as mechanically (Olsen et al.1981). These findings demonstrate that the coronary

system should be modelled using a multiporosity, fi-nite deformation approach (fig. 2).

The multiporosity model was implemented in 2D,3D and axisymmetric finite elements (Vankan et al.1997). Animal experiments were undertaken to verifythe concepts (van Donkelaar et al. 2001). Compari-son of experiment and model clearly demonstrate thecapabilities. The same equations were derived twice :using mixture theory (Vankan et al. 1996) and usingaveraging theorems (Huyghe and van Campen 1995a;Huyghe and van Campen 1995b). A micro-macrotransformation was derived to compute hydraulic per-meabilities from the microstructure (Huyghe et al.1989; Huyghe et al. 1989; Vankan et al. 1997). Whileat the time of the research, the microstructure washard to recontruct, today micro-computed tomogra-phy are operational methods to reconstruct the 3D mi-crogeometry post mortem.

3 SKIN BARRIER

Figure 3: Hydraulic permeability profile across theskin (van Kemenade et al. ).

As we are sitting in the room we all are porous me-dia in contact with air. As the chemical potential of thevapour in the air is lower than the chemical potentialof the water in the body, we continually loose waterthrough evaporation. This process bears the name oftransepidermal water loss. If we were to put a bathof water at 37

Centigrade next to us - with same

contact area with air as our body, we would find theloss of fluid from the bath to be many times morethan from our body. So, there is a barrier that protectus against desiccation. Minute bruises to our skin re-duces the barrier very significantly, underscoring therole of the upper micrometers of the horny layer of theepidermis in maintaining the barrier. Because of inter-est in the skin barrier from the industry, we undertook

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the challenge to model transepidermal water loss us-ing Biot’s theory coupled to an air vapour diffusionmodel. The finite element model was compared to invivo data obtained from volunteers. A sharp drop inhydraulic permeability (fig. 3) and a sharp increase incapillary effects towards the skin surface explains thebarrier function satisfactorily..

4 BONE MECHANOSENSING

Figure 4: Longitudinal section of a femur. The spon-gious bone orients its porous structure according tothe local stress field (Williams et al. 1989)

Bone is known to adapt its pore structure to lo-cal mechanical load (fig. 4). Load-related alignmentand mass of bone are the foundations of the Lawof Bone Transformation (Wolff’s Law), formulatedby Julius Wolff in 1892. Computer simulations con-firmed his postulate, that bone adapts its form accord-ing to rules of mathematical design (Huiskes et al.2000). However, the cellular mechanism that under-lies the typical secondary bone structure is as yet un-explained. Bone remodelling involves groups of cellsof 3 different types, which collaborate in so-calledbasic multi-cellular units: osteoblasts, osteoclasts andosteocytes Basic multicellular units proceed by tun-

Figure 5: Schematic view of an osteon. The vascularporosity contains osteoclasts (cells destroying bonemass) and osteoblasts (cells constructing bone mass).In the surrounding porous bone the osteocytes reside.They are interconnected by the lacuno-canalicularnetwork (nano pore size).

nelling, during which osteoclasts excavate a canal thatis partly refilled by osteoblasts, thus forming an os-teon (fig. 5). The osteocytes reside in lacunae in-side the bone matrix and thus have a good positionfor mechano-sensing. With their long slender protru-sions that run through canaliculi they form a three-dimensional network that reaches to the bone sur-face, which allows them to signal (pre-) osteoclastsand (pre-) osteoblasts. In order to respond to me-chanical stimuli, bone necessarily needs to sense themechanical load it adapts to. The most accepted hy-pothesis on how bone senses its own stress field, in-volves fluid flow induced by mechanical load (Cowinet al. 1991). In other words it involves a consolida-tion process, similar to the consolidation of geomate-rials. Burger and Klein Nulend (Burger and Nulend1999) have demonstrated through several experimen-tal studies in vitro that osteocytes react to fluid flow.As an early response they release nitric oxide (NO)and prostaglandins (PG) E2 and I2, followed by ex-pression of the enzyme cyclo-oxygenase 2 (COX-2),which allows for continued release of prostaglandins.NO, PGE2 and COX-2 play a crucial role in the induc-tion of bone formation (Chambers et al. 1999), whileNO and PGE2 also inhibit osteoclasts. Large tissuestrains thus conceivably lead to osteoblast recruit-

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Figure 6: Local fluid content change in an osteon dur-ing walking as computed by Biot‘s theory. In front ofthe cutting cone appears an area of volumetric expan-sion that effectively inhibits the exchange of fluid be-tween the bone matrix and the vascular porosity (Smitet al. ).

ment. Conversely, decreased stimulation of osteocytesmay activate osteoclasts, possibly by the mechanismof cell death. As osteocytes are likely stimulated bystrain-induced fluid flow, my colleagues of the FreeUniversity of Amsterdam and myself aimed to deter-mine the local pattern of fluid flow at a remodellingsite (Smit et al. ).Cortical bone has two systems of interconnectedchannels. The largest of these is the vascular poros-ity consisting of Haversian and Volkmann’s canals,with a diameter of some 50 � m, which contains a.o.blood vessels and nerves. The smaller system consistsof the canaliculi and lacunae. The canaliculi are at the

Figure 7: Measured versus simulated streaming po-tential across the shaft of an osteon. The consolida-tion time is in the order of magnitude of one walkingcycle.

submicron level and house the protrusions of the os-teocytes. When bone is loaded, fluids within the solidmatrix sustain a pressure gradient that drives a flow.It is generally assumed that the flow of extra-cellularfluid around osteocytes plays an important role notonly in the nutrition of these cells, but also in thebone’s mechanosensory system. The interaction be-tween the deformation of the bone matrix and the flowof fluid is modeled using Biot’s theory of poroelas-ticity. However, because of the inhomogeneity of thebone matrix and the scale of the porosities, it is notpossible to experimentally determine all the param-eters that are needed for numerical implementation.Smit et al. (Smit et al. ) derive these parameters us-ing composite modelling and experimental data fromliterature. A full set of constants is estimated for alinear isotropic description of cortical bone as a two-level porous medium. An axisymmetric finite elementmodel of the minerised bone surrounding a haversiancanal is constructed. The lacuno-canalicular porosityis modelled as a pore space saturated with compress-ible fluid. The axial load depends on time accordingto the typical loading pattern of a femur during walk-ing. Along the cylindrical part of the tunnel (the clos-ing cone), fluid is pressed out of the bone matrix. Atthe tip (the cutting cone), however, fluid flows intothe bone matrix, because of a local, superficial areaof volumetric expansion. The amplitude of this fluidflow at the cutting cone is about six times lower thanthe flow at the closing cone. At unloading of the bonetissue, the fluid flow pattern is reversed, resulting in afluid outflow at the cutting cone, and an inflow alongthe rest of the osteonic tunnel wall. Inside the bonematrix, the outflow pattern along the closing conedamps out at a depth of some 100 � m. This 100 � mis just about the distance from the deepest osteocytesto the bone surface both in human compact and tra-becular bone, which confirms a role for the transportof nutrients. At the cutting cone, however, a differentphenomenon is observed: here the fluid flow direc-tion is reversed at a depth of some 10 � m, becausethe volumetric expansion is only a superficial phe-nomenon and the deeper layers experience volumet-ric compression (Smit and Burger ). The area of volu-metric expansion in front of the cutting cone not onlyreduces the flow amplitude around the osteocytes lo-cated there, but inhibits the exchange of fluid betweenthe bone matrix and the osteonic lumen as well. Thebone tissue layer immediately in front of the cuttingcone thus appears as an area of local disuse with lackof fluid transport. This is precisely the bone layer thatosteoclasts erode. By contrast, behind the cutting conethe tissue deformation and the canalicular fluid floware increased; there the osteocytes are inhibited fromfurther excavation, while osteoblasts are recruited inorder to refill the tunnel. So by excavating a tunnel,

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osteoclasts create a local area of increased canalicularfluid flow that leads to osteoblast recruitment, and atthe same time create an area of disuse in the loadingdirection that guides their continued resorption activ-ity. This explains the progression of osteon formationalong the principle loading direction (Smit and Burger). From the simulated flow pattern, the time course ofthe streaming potential across the shaft of the osteonis computed (fig. 7) assuming a vanishing streamingcurrent. The hydraulic permeability is chosen so as toreproduce experimentally measured data from the lit-erature (Otter et al. 1992). The above mechanism ex-plains the orientation of the bone structure along theprinciple stress direction and the tuning of bone massaccording to the magnitude of the stress and henceprovides a cellular basis for Wolff’s law (Smit andBurger ). In this mechanism, Biot’s theory is a key el-ement,indicating the strong need for communicationbetween the porous media community and biologists.

5 SWELLING TISSUES AND CELLS

Figure 8: Chemical potential of the fluid as a functionof time during swelling of a one dimensional ionisedmedium. The solution from a 3D finite element code(van Loon et al. ) is compared to the analytical solu-tion (van Meerveld et al. ).

Since antiquity, the phenomenon of swelling of tis-sues has been closely related to health and disease.Biological, synthetic and mineral porous media oftenexhibit swelling or shrinking when in contact withchanging salt concentrations. This phenomenon, ob-served in clays, shales, cartilage and gels, is causedby a combination of electrostatic forces and hydra-tion forces (Lai, Hou, and Mow 1991). In case of bio-logical tissue, electrostatic forces are often dominant.

Classical concepts, such as the transmembrane poten-tial of cells are directly associated with these elec-trostatic forces. Already years ago, Biot understoodthat his theories were closely associated with trans-membrane phenomena in living cells (). At least fourcomponents are involved in the swelling mechanics:a solid, a fluid, anions and cations. Lai et al. (Lai,Hou, and Mow 1991) developed a triphasic theory forsoft hydrated tissue and applied the theory to cartilagewhile neglecting geometric non-linearities. They ver-ified the theory for one-dimensional equilibrium re-sults. As soft tissues and cells are commonly subjectto large deformations, our group developed a finitedeformation theory of ionised media (Huyghe andJanssen 1997). In order to simplify the mathemeticsas much as possible a Lagrangian form of the entropyinequality has been derived which leads to equationsconsistent with Biot’s porous media theories in a morestraightforward way than the more familiar Eulerianapproach of Bowen (Bowen 1980). The incompress-ibility and electroneutrality conditions are introducedby means of two Lagrange multipliers; the latter isphysically interpreted as an electrical potential, theformer as a pressure.

6 Donnan OsmosisWhen an ionised medium is in contact with a mono-valent salt solution, diffusion of salt ions and flow offluid take place between the medium and the salt so-lution until equilibrium is reached:

����� ��� (1)

�� � �

�(2)

�� � �

�(3)

� � is the electrochemical potential of the cations, ��

is the electrochemical potential of the anions and ��

the chemical potential of the fluid in the medium. Thecorresponding overlined symbols refer to chemicalpotentials in the outer solution. Standard expressionsfor (electro)chemical potentials are found in the litera-ture (Richards 1980). If we assume incompressibilityfor each constituent, i.e. same partial molar volumesin either solution, we find:

����� ������ �

�� �������� � ������� (4)

�� � �

���� �

�� �������� � � ����� (5)

�� � �

�� ����� ��� �

����� �(6)

in which ���� are reference values,� � partial molar

volumes,� � activities, � the fluid pressure,

�abso-

5

lute temperature,

universal gas constant, � Fara-day’s constant and � the electrical potential. All ofthese (electro)chemical potentials are measured hereper unit volume constituent. Combination of equation(1) and (2) leads to:

� � � � � � � � � (7)

� � � � ��� � ���

� � � �� � �� (8)

where � � � is the Donnan potential between the in-ner and outer solution. If we define � ��� as the fixedcharge density per unit fluid volume of the inner solu-tion, taken positive for positive charges and negativefor negative charges, we can write the electroneutral-ity conditions as:

� � ��� � � � ��� (9)

� � � � � � � (10)

� � and � � are the cationic and anionic concentrationsper unit fluid volume in the inner solution, while thecorresponding overlined symbols pertain to the outersolution. From the previous equations we derive theDonnan equilibrium concentration of the ions:

� � � � � � ��� � � � � ��� �� �� ��� � � (11)

� � � ��� ��� � � � � ��� � � �� � � � � (12)

with

� � � � � � �� � � � (13)

and � � ����� ��� � � � � � the activity coefficient ofcomponent � . Eqs. (11-12) show that the cationicconcentration jumps to a higher and the anionic con-centration to a lower value when entering the porousmedium. These concentration jumps are responsiblefor the attraction of water into the porous mediumduring swelling and for the associated osmotic pres-sure � . Using eq. (6) one can derive Van ’t Hoff rela-tion from (3):

� � � � � � ������ � � � � � � � � � � � � ��� (14)

provided that the molar fractions of the ions are smallcompared to the molar fraction of the fluid.

� �and

� �are the osmotic coefficients:

� � ������ � �

������ � (15)

� � ������ � �

���� � � (16)

� � � ��� � � and� � � �!� � � are the molar fractions of the

fluid in the inner and outer solution. It may be clearfrom the above considerations that physical phenom-ena occuring in the porous medium are a combinationof mechanical, chemical and electrical effects. The in-terrelationship between these effects are well knownfor membrane processes (Staverman 1952). The pur-pose of this paper is to generalise these relationshipsfor porous media subjected to threedimensional finitedeformation. The four phases that we consider in themedium are: solid (superscript s), fluid (superscriptf), monovalent anions (superscript -) and monovalentcations (superscript +). Assuming all components in-trinsically incompressible and excluding mass trans-fer between phases, the mass balance of each phase iswritten as:

���"�$#

�&%')( ��� " %* " � �,+ � � � �

� � � � � � (17)

in which�-"

is the volume fraction and %* " the velocityof phase � . As we assume saturation

� � � � � � � � � � � � (18)

and as we neglect the volume fraction of the ions rel-ative to the other volume fractions, summation of theeqs. (17) yields the mass balance of the mixture:

%')( %* � � %'.($/ � � � %* � � %* � �10 �2+ (19)

It is useful to refer current descriptors of the mixturewith respect to an initial state of the porous solid. Wedefine the deformation gradient tensor 3 mapping aninfinitesimal material line segment in the initial stateof the solid onto the corresponding infinitesimal linesegment in the current state of the solid. The rela-tive volume change from the initial to the current stateis the determinant of the deformation gradient tensor4 �,5 6 # 3 . If we introduce volume fractions

7 " � 4 � " (20)

per unit initial volume, we can rewrite the mass bal-ance equation (17) as follows:8 � 7 "8 #

� 4:9;( � � " � %* " � %* � � � �2+ (21)

when using the identity:8 �8 #4 � 4 %'.( %* � (22)

Neglecting body forces and inertia, the momentumbalance takes the form:

%'<(>= " � %� " � 0 � � � �

� � � � � � (23)

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Figure 9: Electrohemical potential of anions andcations as a function of time during swelling of a onedimensional ionised medium. The solution from a 3Dfinite element code (van Loon et al. ) is compared tothe analytical solution (van Meerveld et al. ).

which after summation over the four phases, yields:

%' ( = � %' ( = � � %',( = � � %' ( = � � %',( = � � 0 (24)

if use is made of the balance condition:

%� � � %� � � %� � � %� � � 0 (25)= "is the partial stress tensor of constituent � , %� "

is the momentum interaction with constituents otherthan � . Balance of moment of momentum requiresthat the stress tensor

=be symmetric. If no moment of

momentum interaction between components occurs,the partial stresses

= "also are symmetric. We assume

all partial stresses to be symmetric. Under isothermaland incompressible conditions, the entropy inequal-ity for a unit volume of mixture reads (Huyghe andJanssen 1997): We introduce the strain energy func-tion as the Helmholtz free energy of a mixture volumewhich in the initial state of the solid equals unity.

� "is the Helmholz free energy of constituent � per unitmixture volume. The inequality for the entropy pro-duction per initial mixture volume reads:

�8 �8 #� � 4 =�� %' %* � � (26)

4 %'<(����� � � �

� � %* � � %* � � (>= � � � %* � � %* � � � � ��� +� Introducing a Lagrange multiplier � for the saturationcondition and � for the electroneutrality condition, theentropy inequality transforms into :

−1.5 −1 −0.5 0 0.5 1 1.50

0.2

0.4

0.6

time = 0 sec.

−1.5 −1 −0.5 0 0.5 1 1.50

0.2

0.4

0.6

time = 250 sec.

−1.5 −1 −0.5 0 0.5 1 1.50

0.2

0.4

0.6

time = 500 sec.

−1.5 −1 −0.5 0 0.5 1 1.50

0.2

0.4

0.6

time = 1700 sec.

−1.5 −1 −0.5 0 0.5 1 1.50

0.2

0.4

0.6

time = 40,000 sec.

Figure 10: Meridional section of a cylinder subject toa stepwise decrease of the salt concentration of theexternal solution. From top to bottom sections of de-formed meshes are shown on time t = 0 s, t = 250 s,t = 500 s, t = 1700 s, t = 40000 s. The vertical dis-placement of the lower base of the cylinder is heldfixed. Along the upper and lower base of the cylindera no flow boundary condition is prescribed for wa-ter, cations and anions. Along the jacket of the cylin-der the chemical potential of the fluid and the electro-chemical potential of the anions and cations are pre-scribed , equal to their values in the external salt so-lution. Pressure, electrical potential and ion concen-trations are not prescribed along the jacket becausethese quantities exhibit deformation dependent jumpsacross the boundaries. Computation is done using a3D 27 node finite elements (van Loon et al. ). Thescale of the displacements and the positions are thesame. Although the overall response to a loweringof the external salt concentration is swelling, initialshrinking is observed at t = 250 s. This is explainedby the friction between ions and water and the initialoutflux of ions convecting water with it against thegradient in chemical potential of the water.

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�8 �8 #� � 4 =�� � � � %' %* �

� 4 ���� � � �

� = � � � � ��� � � �� � �� � � � � ��� �

� %' � %* � � %* � �� 4 �

��� � � �� %* � � %* � � ( � � %' � � � � ��� � � �� � � %

' � �� %'.( = � � � + (27)

in which � � is the valence of constituent � ,= � � �

isthe effective stress of the medium. We choose as inde-pendent variables the Green strain � , the Lagrangianform of the volume fractions of the fluid and the ions7 � , and of the relative velocities %* � � �.3 �� ( � %* � �%* � � , � � � � � � � . We apply the principle of equipres-ence asd the chain rule for time differentiation of

�:

� 4 = � � � � 3 ( � �� �( 3 � � � %' %* � �

���� � � �

� � �� %* � �

( 8 �8 # %* � �

� 4 � = � � � � � � � � � � ��� � � %' � %* � � %* � � (28)

� 4 � %* � � %* � � ( � � %' � � � � � %' � � � %')(>= � ��� � + in which � � are the electrochemical potentials of fluidand ions:

�� � � �

� 7� ���

����� � �� 7 �� �� � � � (29)

�� � � �� 7

� � �� � ���The Lagrange multiplier p can be interpreted as thefluid pressure and � as the electrical potential of themedium multiplied by the constant of Faraday. By astandard argument (Coleman and Noll 1963), we find:

= � � � � 4 3 ( � �� �( 3 � (30)

� �� %* � �

� 0 (31)

= � � � � � � � � � � ��� (32)

leaving as inequality:

���� � � �

4 � %* � � %* � � ( � � %' � � � � � %' � � � %' (�= � � � + (33)

Eq. (30) indicates that the effective stress of the mix-ture can be derived from a strain energy functionW which represents the free energy of the mixture.Eq. (31) shows that the strain energy function can-not depend on the relative velocities. Thus, the effec-tive stress of a quadriphasic medium can be derivedfrom a regular strain energy function, which phys-ically has the same meaning as in single phase orbiphasic media, but which can depend on both strainand ion concentrations in the medium. According toeq. (32) the partial stress of the fluid and the ions arescalars. Transforming the relative velocities to theirLagrangian equivalents, we find in stead of (33):

���� � � �

%* � � ( � � %' � � � � � � %' � � � � %' � ( = � ���2+� (34)

in which %' � � 3 � ( %' is the gradient operator withrespect to the initial configuration. If we assume thatthe system is not too far from equilibrium, we canexpress the dissipation (34) associated with relativeflow of fluid and ions as a quadratic function of therelative velocities:

� %' � � � � � � %' � � � � %' � (>= � � � � � � �

� ( %* �

� is a positive definite matrix of frictional tensors.Substituting eq. (32) into eq. (35) yields Lagrangianforms of the classical equations of irreversible ther-modynamics:

� � � %' � � � � � � � � �

� ( %* � (35)

The momentum balance equation (24), the mass bal-ance eq. (21), the frictional eqs. (35), the constitutiverelationships for the electrochemical potentials(29)and of the effective stress (30) form a set of partialdifferential equations. The boundary conditions aregiven by a no-jump condition of the electrochemicalpotential of the ions and the fluid across the boundaryand the momentum balance of the boundary. In ourapplication we choose the mixing part of the energy

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function as� � 7 �

�7 � �

7 � � ���� 7 � � � �� 7 � � �

�� 7 �� ������ 7 ��� � � 7 ��� ����� � 7 � �� ��

7 ��� � � �� � 7 ��� � � � �� ��

7 ��� � � �� � 7 ��� � � � � (36)

Rearranging equation (29) yields,

�" � � � � "�� " � � �

� � � � � � ��" � 7 � (37)

in which,���� ������������� � ��� � � ����

���� � ! � " �#$$$$$%'&)(+*-, .0/12 /43 .057698 � .;:12 : 3 .<5=698�> ��?A@AB12 / .05 �C?A@AB12 : .<5��?A@AB12 / .05 ?A@12 / . / �

� ?A@AB12 : . 5 � ?A@12 : . :D EEEEEF � "(38)

is the inverse of the Hessian of the mixing energy.To obtain the weak formulation the equations aremultiplied by arbitrary, time independent weighingfunctions and integrated over the volume of the mix-ture ( G ). The momentum equation is multiplied by aweighing function %HJI . The saturation condition, massequation and equation for electroneutrality are multi-plied by the weighing functions HLK , H "M and HJN , re-spectively. After partial integration and applying thedivergence theorem, we find,OPPPPPPPPPPPPPPPPPPPPPQ PPPPPPPPPPPPPPPPPPPPPR

S T U %' %H'IAV � � = 5WG � S* %H'I ( � = ( %� � 5 � �S T H-K U %')( %* � V 5+G �ST H-K 4 �

�

8 � 7 �8 # 5WG �,+ �S T H "M 4 8 � 7 �8 # 5WG � S T X �"ZY " � %' �

"4[ ( %' H "M 5WG�

S* H "M X �

" Y " � %' �" [ ( %� 5 � �S T H'N X 4 �

�

� � ��� � 8 � 7 �8 #[ 5WG � +

(39)

in which�

is the outer surface of the mediumand Y " � � �-"�� � � " � � �� a generalised diffusion-permeability tensor. We choose to use an updated La-grange formulation. The total deformation tensor 3may be divided into,

3 �,3]\ ( 3_^ (40)

where 3_^ describes the deformation from the initialconfiguration G � to the reference configuration GJ^ ,and 3_\ denotes the deformation from the referenceconfiguration to the current state G . When transform-ing the balance equations to a known domain, the ref-erence configuration GJ^ is used. The gradient opera-tor is transformed according to,

%' � 3 � �\ ( %' ^ (41)

As the total deformation is divided, the volume ratiois divided in a similar way. From the definition of

4it

follows that,

4 � 4 \ 4 ^ (42)

Time discretization of the material time derivativesfor

4 \ and7 � yields,

4 \ U %'.( %* � V � `4 \ � 4 \ � a # (43)

8 � 7 �8 # �7 � � 7 �^a # (44)

For the mass balance a time discretization scheme isapplied,

%b �dc %b � # ^ � a # ��� � � c � %b ^ (45)

The time discretization scheme can be varied easilyfrom implicit Euler ( c��

) to explicit Euler ( c � + ).The Newton-Raphson iteration procedure is used todetermine a sequence of approximate solutions of thenon-linear equations. Quadratic interpolation func-tions ( e

˜) are used for the position field and weighing

function %HJf . Linear interpolation functions ( g˜

) aretaken for the discretization of the pressure, electro-chemical potentials, electric potential, volume frac-tions and their corresponding weighing functions. Thepredictor is a set of linearized equations,

9

� ��� � � ���˜

� & ˜��� @ ���

���� �

�� ���

��������

� K˜ ˜ ��� �

�˜ �

�� ��� ���

�� � �

�� �

������

� M˜

��˜ � � �

� � ( ˜ " � ( ˜ � �� �����������

�������

���� � �� �

�������

� N˜

��� � �����˜

�� �

����� @�� � � � � � � � ��� � �T

� ������

˜ "! ˜ @ �T

��� � �

� � �! ˜

#%$&#%' ��� ! ˜ @ �

T��� �)( �

� � � � @*+� � � � * \-, �T

& ˜ � ��� � @+. ˜ �T

˜ ����! ˜ � # ' � # ' � �

T( ˜ " �/( � �0� @* �

��� * \0, � T M

˜

�( ˜ � � � � ( � � � # ' � @* � � � @ � * \-, �

T M˜

�$

�˜ � �

���! ˜

#%$1# ' ! ˜ @ ��� ˜ � � � ˜ � $ � �T (46)

The matrices8 ,

8 2and

8 # result from the lineari-sation of 3_\ ,

= � � �and

4 \ respectively. The matrices3and

3 M contain the derivatives of the quadratic andlinear interpolation functions respectively. The col-umn

3˜f also contains of derivatives of the quadratic

interpolation functions. For calculation a 27-nodebrick element is chosen with 3 displacements inevery node. In each corner of the brick one pressure,3 chemical potentials and an electric potential iscalculated, resulting in a total of 121 degrees offreedom per element. The code is verified using ana-lytical solutions of the linearised equations for a 1Dmedium subject to stepwise change in external saltconcentration (van Meerveld et al. ). The comparisonis shown in fig. 8 for the chemical potential of thefluid and in fig. 9 for the electrochemical potentialsof the cations and anions. The analytical solutionis obtained by reducing the linearised equations to3 diffusion equations. Both the numerical solutionas the analytical solution c solution learly show twotime constants, one for the diffusion of the ions andthe other for the pressure diffusion. As an example ofapplication we computed the response of a cylindricalmesh of ... elements subject to a stepwise changein external salt concentration (fig. 10. Diffusioncoeffients of the ions are 4 + ���� � � � �� . Hydraulicpermeability is

+ ��65 � �87 � �� � �� . The formationfactor is 0.4. Neo-Hookean material behaviour is as-sumed with a shear modulus of 0.06 MPa and a bulkmodulus of 0.08 MPa. Initial fixed charge density is-0.2 Meq and an initial fluid volume fraction of 0.7.A two way response is calculated; an initial shrink isfollowed by swelling to an new equilibrium.

The initial shrinking is associated with negativeosmosis, because the outflux of ions towards the low-ered external salt concentration convects water awayfrom the sample, against the gradient of chemical po-tential of the water. Non-uniform electrical potentialchanges within the medium are computed. Presentresearch has focussed on the experimental verifi-cation of the predicted electrical potential changesduring swelling and consolidation. In order to havereproducible and homogenous samples samples, wedo a good deal of the research on synthetic hydrogelswhich exhibit properties similar to biological tissues.

7 CONCLUSIONS

The application of porous media mechanics in biol-ogy is growing field of interest, that presents chal-lenges which in many aspects are similar to thoseencountered in geomechanics. At small pore scales,electrical effects become very significant. However,the biological materials have a structure that is farmore sophisticated. Hence, cooperation with manymore disciplines is mandatory in order to cover thewhole range expertise needed to address all issues. Itis vital that the community of porous media expertscontribute their share to understanding of our ownbodies and those of other species.

8 ACKNOWLEDGEMENT

The authors acknowledge financial support fromthe Technology Foundation STW, the technologicalbranch of the Netherlands Organisation of Scientific

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Research NWO and the ministry of Economic Affairs(grant MGN 3759)

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