What is the importance of multiphysical phenomena in bone remodelling signals expression? A multiscale perspective

J O U R N A L O F T H E M E C H A N I C A L B E H AV I O R O F B I O M E D I C A L M A T E R I A L S 4 ( 2 0 1 1 ) 9 0 9 – 9 2 0

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Research paper

What is the importance of multiphysical phenomena in boneremodelling signals expression? A multiscale perspective

T. Lemairea,∗, E. Capiez-Lernoutb, J. Kaisera, S. Nailia, V. Sansalonea

aUniversité Paris Est, Laboratoire Modélisation et Simulation Multi Echelle, MSME-UMR-CNRS-8208, Bioméca - 61 Avenue du Général deGaulle, 94010 Créteil, FrancebUniversité Paris Est, Laboratoire Modélisation et Simulation Multi Echelle, MSME-UMR-CNRS-8208, Méca - 5 Boulevard Descartes, 77455Marne-La-Vallée, France

A R T I C L E I N F O

Article history:

Published online 9 March 2011

Keywords:

Biomechanics

Bone remodelling

Multiphysical phenomena

Multiscale analysis

Poromechanics

Numerical simulations

A B S T R A C T

Cortical bone, constituting the outer shell of long bones, is continuously renewed by bone

cells in response to daily stimuli. This process, known as bone remodelling, is essential

for proper bone functioning in both physiological and pathological conditions. Classical

bone remodelling models do not, or only implicitly do, take into account physico-chemical

phenomena, focussing on the mechanosensitivity property of the tissue. The aim of this

paper is to carry out an investigation of the multiphysical phenomena occuring in bone

life. Using a recent multiscale model combining piezoelectricity and electrokinetics to

poromechanics, the usual viewpoint of bone remodelling models is questioned and new

research avenues are proposed.c⃝ 2011 Elsevier Ltd. All rights reserved.

d

1. Introduction

The ability of bone tissue to adapt itself to its environmentas described by the Wolff Law (Wolff, 1892) is part of thegeneral trend in the biological community of the late 19thcentury. This trend can be crystallised by the statement ofSpencer: “Life is definable as the continuous adjustment ofinternal relations to external relations” (Spencer, 1867). Theliving materials, which are intrinsically active systems, aretypical adaptive systems that may change with the context.Their activity in response to external demands generateschanges in mass, composition and shape. In the case ofbone, this phenomenon is called adaptation. As such, bonetissue can sense, react and adapt itself to its environment.For instance, it is well known that the bone density decreasesduring space flight (Morey and Baylink, 1978; Cann and

∗ Corresponding author. Tel.: +33 145171572; fax: +33 145171433.E-mail address: [email protected] (T. Lemaire).

1751-6161/$ - see front matter c⃝ 2011 Elsevier Ltd. All rights reservedoi:10.1016/j.jmbbm.2011.03.007

Adachi, 1983). Thus, bone formation and resorption arethe result of a series of events transforming a physicalinformation into a biological response. This process includingall the phenomena characterising the bone cell’s ability tosense mechanical stimuli, and possibly to respond, is calledthe mechanotransduction of bone remodelling. As sketchedin Fig. 1, the cycle of the mechanotransduction of boneremodelling can be coarsely summarised as: a macroscopicexternal physical stimulus (i) is propagated within the bonetissue (ii), and then sensed at the microscale by sensitivecells (osteocytes, OCY) (iii), that induce signals emission (iv)to activate effector cells that will resorb (osteoclasts, OCL)old tissue and create (osteoblasts, OBL) new tissue (v), thusmodifying the macroscopic properties of the organ (vi).

In the commonly accepted remodelling theory, the effe-ctor cells are the osteoclasts and osteoblasts whereas the

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910 J O U R N A L O F T H E M E C H A N I C A L B E H AV I O R O F B I O M E D I C A L M A T E R I A L S 4 ( 2 0 1 1 ) 9 0 9 – 9 2 0

Fig. 1 – Simplified chain of the mechanotransduction of bone remodelling.

mechanosensitive cells are the osteocytes (Martin et al.,1998). The understanding of the nature of the triggeringsignals for the different cellular activities is much debated.Classical hypotheses stipulate that fluid shear effects (Burgerand Klein-Nulend, 1999; Westbroek et al., 2000), pericellularmatrix deformation (You et al., 2001; Han et al., 2004) and/ormicrocracks (Martin, 2002) do play a role in triggering theremodelling process.

The lack of possibilities for experimental investigationsof in situ bone poromechanical behaviour renders theoret-ical investigations valuable. In most of the representationsof bone remodelling, the mechanical stimuli acting on cells(pressure, shear stress, drag forces, etc.) are calculated fromthe poroelasticity theory (Biot, 1941), which adopts intrinsi-cally a macroscopic point of view. These mechanical inputsare then somehow downscaled and converted into biochem-ical microscopic signals regulating the remodelling activity(Adachi et al., 2010). In this manner, the nature of the incom-ing signals is thought to be purely mechanical. Moreover, themicroscopic phenomena are not directly involved since thefluid flow is quantified by a macroscopic textural parameter,the hydraulic permeability. In other words, even if involvingmicroscopic biochemical signals, these modelling strategiesremain purely macroscopic.

We proposed that electro-chemistry does also interferein this signalling expression (Lemaire et al., 2008). Thearguments developed in this paper are reinforced by therecent work of Ahn and Grodzinsky (2009) who proposedthat piezoelectricity due to the collagen-apatite matrix straincould engender a negatively charged environment for the

interstitial fluid flow. In this manner, a displacement ofthe solid phase of bone tissue may indirectly affect stressby increasing electro-osmosis. In order to investigate theorigin of the electric phenomena occurring in living corticalbone, we recently proposed a theoretical description (Lemaireet al., 2011) taking into account the electro-chemical effectsinherent to osmotic pressure, collagen piezo-mechanics,double-layers developing in the bone porous network andstreaming currents induced by the interstitial fluid flow. Thismodel, the first simultaneously treating piezo-mechanicsand electrokinetics in bone, is the outcome of severalearlier studies on multiphysical phenomena governing thebehaviour of saturated porous materials (Lemaire et al., 2002,2005, 2006, 2007, 2008, 2010a,b; Kaiser et al., 2009). Based on amultiscale description of the medium, this approach is welladapted to study bone tissue which is characterised by ahierarchical organization, from the Haversian system to thecanalicular pores. To capture the consequences of collagenpiezoelectricity or cellular electrochemistry, it is necessary toreach the submicrometric structures of bone.

The goal of this study thus is twofold. First, to demonstratethat classical treatments of bone mechanobiology neglectingelectro-chemical effects are quantitatively sufficient toevaluate the interstitial fluid variables at the bone tissue level.Second, to investigate the real implications of underlyingelectro-chemical couplings in the bone remodelling signalsexpression. Even if their consequences are not visible at themacroscale, these multiphysical effects could be significantat the cellular scale and thus should be taken into accountin new scenarios of bone adaptation. Our strategy consists

J O U R N A L O F T H E M E C H A N I C A L B E H AV I O R O F B I O M E D I C A L M A T E R I A L S 4 ( 2 0 1 1 ) 9 0 9 – 9 2 0 911

of discussing, at both the macroscale and the microscale,the importance of the multiphysical phenomena featuringin bone behaviour using physiologically-based simulations.That is why, in the first section, the ingredients of ourmodel are introduced. Then, some numerical examples arepresented to illustrate the interest of ourmultiscale approach.Through these simulations, we are able to distinguishthe phenomena that are purely microscopic from thosewhich can be macroscopically observed. Finally, from thesemultiscale considerations, we propose some avenues whichit is worth exploring when aiming at in silico representing thebone remodelling process.

2. Multiphysical description from the cellularvicinity to the macroscopic description of the bonetissue

In this section, a summary of the key elements of ourmultiphysical model of cortical bone behaviour is proposed.Readers interested in more details are invited to study theparent papers (Lemaire et al., 2006, 2008, 2010b, 2011). Thoseprevious studies present how a multiphysical description of asaturated porous material at the microscale (characterised bythe micro-coordinate x referring to a typical micro-length ℓ)is propagated at the upper scale (characterised by the macro-coordinate X referring to a typical macro-length L). Throughthe homogenization process, we are able to discriminatethe variables that vary only at the macroscale, called slowvariables, from the fast variables that may also vary at themicroscale. The ∇ spatial operator is used to represent thegradient of a quantity ∇⋆ or its divergence ∇ · ⋆. This operatoris split into two parts ∇ = η−1

∇x + ∇X, where ∇x and ∇Xcorrespond to differentiations at the micro- and macroscalerespectively, and where η = ℓ/L is the scaling ratio. Eventually,focussing on cortical bone analysis, the slow variables are:(i) the displacement vector field of the solid skeleton u –this result is quite classical in homogenization theories ofporoelasticity (Auriault and Sanchez-Palencia, 1977) –; (ii) thefluid pressure p; (iii) the salinity of the interstitial fluid n;(iv) the streaming potential ψ. In other words, ∇xu = 0 and∇xp = ∇xn = ∇xψ = 0.

These slow variables, in association with several otherfast ones, are linked through coupled equations formingthe multiphysical model. Before introducing all the differentphysical phenomena represented through our model, a briefpresentation of the hierarchy of cortical anatomy is given,to underline the necessity to invoke multiscale argumentsto understand the overall behaviour of bone, focusing onremodelling signals.

2.1. Cortical tissue: a multiscale strategy to understand amultiscale material

Cortical bone is seen as a porous medium with a solidpiezoelectric skeleton saturated by a dielectric fluid. Bonematerial is characterised by different hierarchical structuresas shown by Fig. 2.

The bone remodelling process originates in its complexmicrostructure. At the tissue scale (millimeter), cylindric

cavities are formed by osteoclasts before being filled byosteoblasts. The resulting structures are called secondaryosteons. Osteons are crossed by a hierarchical porous networksaturated by a fluid with both mechanical and chemicalfunctions. Axial Havers and radial Volkmann canals, a fewtens of micrometers in diameter, provide a fast way to passchemo-mechanical information between different osteons.Local messaging takes place inside the lacuno-canalicularporosity, a dense network of cavities (lacunae) and pseudo-cylindrical canals (canaliculi, few hundred nanometers inradius) crossing the osteonal matrix. The star-shapedosteocytes (OCY), the mechanosensing cells, and theirprocesses, are housed in lacunae and canaliculi, respectively.Osteocyte mechanosensation is thought to be dominated bythe fluid transport inside the canaliculi. Indeed, according toAdachi et al. (2009), the osteocyte processes developing insidethe canaliculi are much more sensitive to mechanical stimulithan the cellular body embedded in the lacuna. However,the electric potentials and ions effects are rarely studied.Note that the origin of all those phenomena is located atthe scale of the cell. Thus, when analysing the remodellingsignals, it is necessary to connect the macroscopic stimuliacting on bone with the canalicular scale. Paraphrasingthe Spencer quotation of the introduction, living tissueresults from the continuous interaction between internalmicroscopic phenomena andmacroscopic stimuli. That is whyour modelling strategy relies on a multiscale treatment of thebone behaviour.

2.2. Electricity: piezoelectricity, double-layer effects andstreaming potentials

Electricity in bone tissue has motivated studies since the late1950s with the emergence of the idea that bone adaptationcould be partly explained thanks to collagen piezoelectricity(Fukada and Yasuda, 1957; Yasuda, 1964; Bassett et al., 1964).Later on, the interest in piezoelectricity decreased whenmorecompelling mechanisms, such as streaming potential (Pollacket al., 1984) and fluid-generated shear stress (Weinbaumet al., 1994; Nguyen et al., 2010), began being studied.Nevertheless, recent advances in the understanding of bonephysiology point out the possible relevance of piezoelectriceffects in bone. For instance, putting forward experimentalstudies linking zeta potentials and collagen content (Otteret al., 1988), Ahn and Grodzinsky (2009) proposed thatpiezoelectricity could modify the zeta potential, so thatpiezoelectricity and streaming potentials could act in concert.

2.2.1. Electricity in the collagen-apatite matrixThe piezoelectric effect is a reversible process by whichsome materials are able to generate an electric potential inresponse to an appliedmechanical strain. The effect is closelyrelated to a change of polarization density within thematerialvolume. In dry bone, the piezoelectric properties are due tocollagen molecules (Fukada and Yasuda, 1957) which exhibitthe polar uniaxial orientation of molecular dipoles in theirstructure. As a result, they can be considered as a sort ofdielectric material exhibiting a quasi-permanent bulk chargeqs. Let the permittivity of the solid phase be quantified bythe second-order tensor ϵs and the piezoelectric coupling by


Fig. 2 – The different scales of bone.

the the piezoelectric third-order tensor 5. At the scale of thecollagen-apatite structure, the electric potential in the solidphase φs is thus governed by the Gauss–Maxwell law:

∇ · (5 : ∇u − ϵs · ∇φs) = qs. (1)

Here, the two operators · and : correspond to the singleand double contractions, respectively. Electricity in the solidphase involves both the displacement vector field u and theelectric potential in the solid φs. In fact, when homogenised,this equation remains purely microscopic (Lemaire et al.,2011) and, since the displacement u is a slow field, it onlyinvolves the electric potential:

− ∇x · (ϵs · ∇xφs) = qs. (2)

2.2.2. Double-layer and streaming potentials in the fluidAn important property inherent to most of the biologicalcharged porous media is the negative charge of their poresurface due to the presence of some negative sites suchas hydroxyl complexes. This negative charge is partiallycompensated by the adsorption of cations on the surfaceforming the inner compact layer commonly referred to as theimmobile Stern layer. Nevertheless, themajority of the excessof positively charged counter-ions is located in the electrolyteaqueous solution forming an outer diffuse layer composed ofmobile charges (see Fig. 3). Together with the fixed chargedgroups on the solid matrix, these ions form the so-calledelectric double-layer (Hunter, 1981).

The Boltzmann distributions of the cationic and anionicconcentrations n± are governed by the reduced double-layerpotential ϕ̄:

n±= nexp(∓ϕ̄). (3)

Note that the reduction of electric potentials ⋆ involves theFaraday constant F, the ideal gas constant R and the absolutetemperature T, so that ⋆̄ = F ⋆ /RT. The double-layer potentialϕ can be determined thanks to the purely microscopicPoisson–Boltzmann equation (Hunter, 1981; Lemaire et al.,2006):

∇x · ∇xϕ̄ =1

L2Dsinh ϕ̄. (4)

The Debye length LD =

ϵf RT/(2F2n) characterises the

thickness of the diffuse ionic layer. Here, ϵf is the dielectricpermittivity of the fluid phase (note that the permittivitytensor in the fluid is spherical, ϵf = ϵf I, I being the unitsecond-order tensor). When the pore size is large whencompared to the Debye length, the hyperbolic sine of Eq. (4)can be linearised (sinh ϕ̄ ≃ ϕ̄) to obtain the Debye–Hueckelapproximation.

When advected by the interstitial fluid, the mobile chargepopulation of the double-layer generates the macroscopicallyobserved streaming currents. In parallel, to conserve charge,the movement of the net charge generates an electricpotential, often referred to as streaming potential. In bone


Fig. 3 – Representation of the double-layer.

tissue, the flow of interstitial electrolyte may be driven by, forinstance, the deformation of the bone under external forces(Guzelsu andWalsh, 1990). Thus, the streaming potentials canbe attributed to strain-induced fluid flow (Pollack et al., 1984)and can be obtained from the hydraulic profiles within bone(Salzstein et al., 1987; Salzstein and Pollack, 1987; Lemaireet al., 2011).

2.3. Ionic electro-diffusive transport

In the remodelling process, the paracrine communicationbetween the mechanosensors (osteocytes) and the effectorcells (osteoclasts and osteoblasts) requires to develop specifictransport processes. Neglecting the convective effects inthe narrow canaliculi but taking into account the ionicexchanges, the electrical phenomena induced by the double-layer and the streaming current require to modify theclassical diffusion equation. Thus, taking into account thepossible ionic exchanges between the cell and its fluidenvironment and the electromigration effects, a macroscopicelectro-diffusive Nernst-Planck equation can be obtained formonovalent ions (Lemaire et al., 2010a):

∂

∂t

n

ηf < exp(∓ϕ̄) >f +α±

= ∇X ·

D∗

±

∇Xn ± n∇Xψ̄

. (5)

Here, ⟨⋆⟩f represents the average value of the quantity overthe fluid domain. This equation exhibits three contributionsto the ionic transport: (i) the temporal term involvingthe influence of the porosity ηf , the averaged double-layereffects and the surface exchange term α±; (ii) a Browniandiffusion term in response to the salinity gradient; (iii) anelectromigration term in response to the gradient of thestreaming potential ψ. These two last terms are quantifiedusing effective diffusion tensors D∗

±involving, in addition to

the diffusion coefficients of the ions D±, the porosity ηf andthe electro-tortuosity tensors ϑ±:

D∗± = ηf D±ϑ−1

±. (6)

The explicit definition of ϑ± is obtained during the homog-enization process and can be found elsewhere (Kaiser et al.,2009; Lemaire et al., 2010a, 2011).

2.4. Interstitial fluid flow

Bone fluid flow, in addition to the hydraulic pressure gra-dient, may be governed by supplementary driving phenom-ena. Indeed, due to the variability of the streaming potential,an electrophoretic movement of the mobile charges is gen-erated. As a result, because of the viscous drag interac-tion, a concomitant electro-osmotic seepage flow is caused.Moreover, the chemical gradients engender osmotic fluidmovement too. In a previous study (Lemaire et al., 2008),a modified Darcy law taking into account electrokineticsand the fibrous pericellular matrix occupying the canalicularporosity (thanks to a pericellular matrix permeability Kf ) hasbeen derived. The strategy was to upscale the Stokes equa-tion including the Coulombic force to show that the fluid flowis caused by three driving effects, namely hydraulic transport(induced by a pressure gradient, indexed by P), osmosis (inresponse to a salinity gradient, indexed by C) and electro-osmosis (caused by a streaming potential gradient, indexedby E). Hence the macroscopic Darcy velocity V reads:

V = −KP∇Xp − KC∇Xn − KE∇Xψ̄. (7)

The macroscopic permeability tensors Kk = ⟨κk⟩ are obtainedthrough the homogenization process (Lemaire et al., 2008,2010b, 2011), ⟨κk⟩ being the average over the representativevolume of the local hydraulic conductivity parameters in re-sponse to each of the three driving effects (k = P,C,E).

2.5. Coupled poroelasticity

Bone fluid flow is generated by the strain of the solidmatrix. Classically, in bone biomechanics, the calculation ofthe hydraulic velocities caused by the mechanical loadingare based on the poroelasticity theory (Biot, 1941). Here,we consider the extension of this purely hydro-mechanicalmodelling to electrically charged and saturated porous mediarecently developed in Lemaire et al. (2011). Thus, transposingthe multiscale coupled representations of saturated clayeymaterials (Lemaire et al., 2002; Moyne and Murad, 2002)based on the seminal paper of Auriault and Sanchez-Palencia(1977), our strategy consists of deriving a Biot-like constitutive


equation from a microscale analysis. On the one hand,the solid phase of cortical bone obeys to the piezoelectricconstitutive equation:

Ss = C : ε(u)+ 5T· ∇φs, (8)

where Ss is the second-order stress tensor in the solid, ⋆T

stands for the transpose operator and ε(⋆) = (∇ ⋆ +∇ ⋆T)/2is the operator that gives the symmetric part of the gradientof the quantity ⋆. This operator is also used to express thesymmetric part of the interstitial fluid velocity v hereafter.

On the other hand, the constitutive equation for theNewtonian fluid phase involves the Donnan osmotic pressureπD and the second-order Maxwell tensor τM characterisingthe electrical effects:

Sf = −(p + πD)I + 2µf ε(v)+ τM, (9)

where Sf is the stress tensor in the fluid and µf is thedynamic viscosity of the fluid. The osmotic pressure typicallycorresponds to the swelling pressure, as observed in clayeymaterials (Low, 1994; Lemaire et al., 2004) and is expressedthanks to the double-layer potential:

πD = 2RTn(cosh ϕ̄− 1). (10)

Furthermore, the Maxwell tensor τM is defined thanks tothe electric field vector in the fluid Ef by:

τM =

ϵf

2(2Ef ⊗ Ef − (Ef · Ef )I), (11)

⊗ being the tensor product. The use of this constitutive equa-tion in the equilibrium equation of the fluid phase simplygives the Stokes equation used to describe the interstitialfluid movement. It can be noticed that the divergence of theMaxwell tensor τM simply represents the Coulombic force(Moyne and Murad, 2002).

After upscaling, the piezoelectric effects fade away at themacroscale (Lemaire et al., 2011) and the overall momentumbalance equation, involving the total stress tensor, Stot =

⟨Ss⟩ + ⟨Sf ⟩, expressed from the averaged stress tensors in thesolid and fluid phases, reads:

∇X · Stot = 0. (12)

The total stress tensor obeys a coupled Biot-like equationexplicitly derived elsewhere (Lemaire et al., 2011):

Stot = C∗: εX(u)− α∗p + τ∗. (13)

In this equation, εX corresponds to the macroscopic part ofthe operator ε, that is to say built from ∇X. Furthermore,the homogenised fourth-order elasticity tensor C∗ and thehomogenised Biot second-order tensor α∗ are obtainedfollowing the classical treatment of poroelasticity as proposedby Auriault and Sanchez-Palencia (1977). Moreover, themacroscopic electro-chemical tensor τ∗ representing themacroscopic effects of the fluid electro-chemical phenomenais similar to the one previously obtained for the multiphysicaldescription of clayey materials (Lemaire et al., 2002; Moyneand Murad, 2002). This tensor accounts for: (i) the sphericalDonnan pressure effect; (ii) the action of the Maxwell tensor;(iii) the electro-chemical effects occurring at the solid–fluidinterface (Lemaire et al., 2011).

2.6. Interface conditions

Let us now consider the interface conditions. At the solid–fluid interface characterised by its unit normal vector n, anelectrical jump σfs of the electric potential gradient may existand the potential continuity is considered:

[[−ϵk · ∇xφk · n]] = σfs, (14)

[[φk]] = 0, (15)

where k = s, f and considering that φf coincides with ϕ.In parallel, at the cell-fluid interface, the electric flux isdetermined thanks to a negative surface charge σOCY:

−ϵf · ∇φf

· n = σOCY. (16)

Moreover, at the interface, no-slip conditions are considered:

∂u∂t

= v, (17)

and ionic exchanges may occur between the solid and fluidphases (Kaiser et al., 2009). Finally, the normal stress continu-ity is considered:

[[−Sk · n]] = 0, (18)

where k = s, f .

3. Investigation of the remodelling signals

The multiphysical model briefly introduced in the previoussection is now used to investigate the role of physico-chemical phenomena in the expression of bone remodellingsignals.

3.1. Micro-electricity

The first investigation deals with the micro-electric phenom-ena, that is to say the piezoelectricity of the collagen-apatitematrix and the double-layer phenomena occurring in thecanaliculi.

We consider a two-dimensional canalicular network char-acterised by a canalicular density dc of around 2.8 canaliculiper square micrometer (Can µm−2) (Marotti et al., 1995) andby an annular geometry of the canalicular pores specified bythe canaliculus radius Rc of 100 nm and the osteocyte pro-cess radius Rm of 50 nm (You et al., 2004). Noting Cc thecanalicular center-to-center distance, a square periodic cell of3Cc = 1.8 µmcontaining nine equidistant canaliculi is built (seeFig. 4). In the solid (a) and fluid (b) phases of bone, the elec-tric field is governed by Eq. (2) and by the linearised form ofEq. (4). Electrical phenomena within the cell process (c) arenon studied and the cell membrane is assumed to presenta negative surface charge σOCY of −0.1 C m−2. This surfacecharge is used to compute the charge density in the solidqs (see Table 1) and to assign the Neumann boundary condi-tions on the double-layer potential at the cell-fluid interfacethrough Eq. (16). At the edge of the periodic cell, symmetricboundary conditions are taken whereas, at the solid–fluid in-terface, the electric transmission conditions of Eqs. (14) and(15) are considered.


Fig. 4 – 2D periodic representation of the canalicularnetwork.

Using the characteristics given in Table 1, this problemhas been implemented in the finite element softwareCOMSOL MULTIPHYSICS (COMSOL Multiphysics, 2005). In thissimulation, we assumed a spherical form for the permittivitytensor in the solid ϵs = ϵsI. The mesh contains 2,800,000degrees of freedom. This is large enough to obtain areasonable precision, especially in the high gradient zones.The simulation results are shown in Fig. 5. On the leftside (Fig. 5(a)), the surface plot of the electric potential ispresented. In the pores, it decreases from zero to −0.1 Vnear the walls. This high gradient corresponds to the double-layers location. In the collagen apatite matrix, the potentialdecreases from the pore surface to reach a minimum around−0.2 V. On the right side (Fig. 5(b)), the electric potential

profile along the diagonal of the cell is presented (cf. thediagonal line represented in Fig. 5(a)). Note that the greyboxes correspond to the pore space. Following the diagonalpath from the left to the right, the potential first graduallyincreases from its minimal negative value in the solid phase.Then, when crossing the pore interface, it sharply increasesto reach zero. This high gradient area corresponds to thedouble-layer developing in the fluid near the bone matrix.Then, in the central zone of the pore, the electric potentialremains zero. When approaching the cellular negativelycharged membrane, the potential sharply decreases in thecell side double-layer to reach the cell membrane potential.Then, by periodicity, this scenario is replicated.

3.2. Poromechanics

If the piezoelectricity does not appear at the macroscale,the electro-mechanical coupling emerges through the macro-scopic electro-chemical tensor τ∗. According to the experi-mental studies of Low (1994) and the theoretical works ofDerjaguin et al. (1987), the most important contribution to thetensor τ∗ is the Donnan osmotic pressure −πDI. An estimationπD of the pressure πD is proposed for a platelet geometry ofthe pores by the Langmuir formula (Israelachvili, 1991) whichonly depends on the half inter-platelets distance d:

πD =

ϵf (πRT)2

2(Fd)2. (19)

This approximation is acceptable for a pore size d largerthan a few nanometers (what is true in cortical bone) andfor high values of the surface charge density of the interface.Since our goal is to check whether the electro-chemicaleffects can be neglected at the macroscale in bone tissue, weconsider now a larger negative surface charge σ = −0.2 Cm−2.Moreover, the lower the salinity, the thicker the double-layerand so the stronger the electro-chemical effects are going tobe. As a result, we consider a low salinity n = 10−4 mol l−1 toplot in Fig. 6 the evolution of the Donnan pressure πD versus

× 10e-6 m

× 10e-6 m

0

-0.02

0

-0.06

-0.04

-0.08

-0.1

-0.12

-0.14

-0.16

-0.18

Electric Potential (V)1.5

1

0.5

1.5

1

0.50

0

-0.05

-0.1

-0.15

V

0

-0.05

-0.1

-0.15

-0.2

Ele

ctric

pot

entia

l (V

)

0 1 1.5 2 2.50.5

Position along the diagonal (m)

a

b

Fig. 5 – Electric potential distribution at the microscale of the canalicular network: (a) surface plotting on the representativecell (left); (b) plotting along the diagonal of the representative cell; the grey boxes represent the pore space.


Table 1 – Parameters used for the micro-electric simulations. The morphological data are adapted from Marotti et al.(1995) and You et al. (2004) and the physico-chemical data are adapted from Lemaire et al. (2006, 2011).

Geometry

Canaliculus radius Rc = 100 nmWall to wall distance between two canaliculi Lc = 400 nmCenter-to-center distance between two canaliculi Cc = Lc + 2Rc

Osteocyte process radius Rm = 50 nmCanalicular density dc = 2.8 Can µm−2

Physico-chemical parameters

Ionic concentration n = 0.1 mol l−1

Dielectric permittivity in the fluid ϵf = 7 × 10−10 A.s.V−1 m−1

Dielectric permittivity in the solid ϵs = 1 × 10−7 A.s.V−1 m−1

Surface charge of the osteocyte membrane σOCY = −0.1 C m−2

Volume charge density in the solid qs ≡ σOCY2πRcC−2c

Electric charge jump σfs ≪ σOCYTemperature 310 K

× 10-8

109

108

107

106

105

104

Don

nan

osm

otic

pre

ssur

e (P

a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Pore half size (m)

Simulation ofLangmuir approximation

Fig. 6 – Donnan pressure versus the pore size.

the pore half size d. The geometry of the pore consists of face-to-face platelets presenting the negative surface charge σ andseparated by a distance 2d. The calculation of the Donnanpressure results from Eq. (10), the double-layer potentialbeing calculated by a custom code developed to study clayeymaterials (Lemaire et al., 2004, 2007).

The Langmuir approximation becomes more pertinentfor large pores. Nevertheless, this approximation tends toslightly overestimate the Donnan pressure. Moreover, fortypical pore sizes of the bone canaliculi (10−8–10−7 m), theDonnan pressure is one order of magnitude lower than theatmospheric pressure. Since the typical stresses within bonelay around 0.1–1 MPa (Piekarski and Munro, 1977; Zhanget al., 1998), the electro-chemical effects included in themodified Biot equation (13) can be neglected and classicalporoelasticity is appropriate to describe the averaged hydro-mechanical phenomena of cortical tissue.

3.3. Effective diffusion

The effective diffusion tensor defined by Eq. (6) dependson the canalicular porosity ηf ≡ 5% and the electro-tortuosity ϑ whose twofold nature is both electrical and

textural (Kaiser et al., 2009). The electric part of this tortuosityis due to the difference between the cationic and anionicdistributions (see Fig. 3 and Eq. (3)). Indeed, the double-layers are characterised by higher concentrations of thecompensating ions. The second part of this tortuosity is theclassical textural property depicting the twisted geometry ofthe pores. This morphological concept corresponds to thetortuous description of the osteocytic processes reported byMarotti et al. (1992). On the one hand, the effects of thismorphological tortuosity may affect the diffusion coefficientsby a factor that is less than an order of magnitude, perhapsthree. On the other hand, the effects of the double-layerhighly depend on the salinity and may vary by a factor ofone order of magnitude (Lemaire et al., 2010a). This electricalinfluence on the ionic transport decreases when the pore sizeis large enough (around 100 nm) and when the fluid salinityis not too low (more than 0.05 mol l−1).

3.4. Bone fluid flow and shear stress effects

In a previous study (Lemaire et al., 2006), it has been shownthat, for physiological conditions, the osmotic and electro-osmotic parts of the macroscopic Darcian velocity introducedby Eq. (7) represent less than 7% of the whole interstitialmacroscopic flow. Thus, a classical purely hydraulic Darcy lawis sufficient to roughly describe the bone fluid macroscopicmovement. Moreover, following the arguments of Salzsteinand Pollack (1987), the macroscopically observed streamingpotentials can be deduced from the macroscopic fluid flow(Lemaire et al., 2011).

However, when focussing on the phenomena occurringwithin the canalicular space partially occupied by thepericellular matrix, we showed that electro-chemical over-velocities develop in the double-layers (Lemaire et al., 2008),generating coupled shear effects acting on the osteocytemembrane of the same order of magnitude of the hydraulicshear stress (Lemaire et al., 2010b). For instance, typicalcoupled velocity profiles are presented in Fig. 7. These profilesare calculated using the values of Table 2. These profilesconfirm that, quantitatively, the main part of the averagefluid velocity is due to the hydraulic driving force. In fact,the electro-chemical velocities are almost zero across the


Table 2 – Parameters of the fluid flow model. When notspecified here, the parameters are those presented inTable 1; the data are adapted from Zhou et al. (2009) andLemaire et al. (2010b).

Properties of the saturated pores

Canaliculus length ℓc = 36 µmIonic concentration n = 0.01 mol l−1

Dynamic viscosity of the saturating fluid µf = 0.65 × 10−3 Pl

Surface potential of the pores −20 mVPericellular permeability Kf = 5 × 10−17 m2

Macroscopic driving gradients

Hydraulic effect 5000 Pa/ℓcElectric effect 10 mV/ℓcChemical effect 0.01 M/ℓc

Fig. 7 – Coupled velocity profiles in the radial direction ofthe canalicular pore.

Table 3 – Estimated coupled shear stress effects.

Hydraulic shearstress (Pa)

Osmotic shearstress (Pa)

Electro-osmotic shear

stress (Pa)

1.44 0.47 0.85

pore, except on the left, near the cell membrane, and onthe right, near the canalicular wall, where electro-chemicalover-velocities develop in the double-layers. These coupledflows cause higher shear stress effects as shown by the slopeof the different profiles at the pore interfaces. In Table 3,the different contributions to the shear stress acting on thecell membrane are given. They are comparable and typicallylay around 1 Pa. The consequences of this remark in theviewpoint of bone remodelling have now to be discussed.

4. Discussion: new remodelling transductionscenarios

Due to the difficulty in carrying out in vivo experimentsto understand the bone remodelling process, modellingand computational approaches become increasingly commontools for testing hypothesis for the regulation of boneremodelling (van Oers et al., 2008; Adachi et al., 2010). In a

multiscale perspective, two difficulties arise when aiming atin silico representation of bone remodelling. The first one ishow to localise the consequences of the macroscopic stimuliin the vicinity of the bone cells. The second one is thequantification of the cellular activity at the microscale fromthe limited information that can be identified as an input intoa remodelling model at the organ scale. This study is onlyfocussed on the first point.

Different hypotheses have been proposed to explain thenature of the signals felt by osteocytes: shear stress effects(Weinbaum et al., 1994; Burger and Klein-Nulend, 1999;Westbroek et al., 2000), pericellular matrix deformation (Youet al., 2001; Han et al., 2004), microcracks (O’Brien et al.,2000; Hazenberg et al., 2006; Nguyen et al., 2011). Whateverthe adopted viewpoint, it is generally agreed that propertieslocalised at the cellular scale can be correlated with themacroscopic demands of bone tissue. Thus, since the hydro-mechanical coupling is the principal factor to connect thebody activity and the cellular environment, poromechanics iswidely used to mimic the in vivo implications of bone activity.The skeleton strain induces interstitial bone fluid flow andmicrocracks that are somehow felt by the mechanosensitivecells (Fritton and Weinbaum, 2009; Martin, 2002). The mainflaw in this classical treatment is that such macroscopicmodels provide homogenised fields associated with the fluidphase (pressure, velocity, etc.) that have somehow to bedownscaled to translate them to the cellular environment andconverted into mechanical remodelling signals (microscopicshear stress, pericellular drag force, etc.). Nevertheless, even ifliving tissue is known to be highly electro-chemically reactive,the majority of the biomechanical models of bone simplyneglects possible electro–chemo–mechanical interactions. Wepropose that it is worth detailing the consequences of thosecouplings at both scales, the tissue level and the cellularvicinity.

This study, adopting the multiscale approach extensivelypresented in Lemaire et al. (2011), shows that a classicaltreatment of the bone behaviour only involving the hydro-mechanical effects is more or less sufficient to explain thebone hydro-mechanical macroscopic behaviour. In fact, dueto the electro-chemical phenomena being originated by theelectric charge of the solid–fluid interface, the canalicularpores are large enough to limit their consequences atthe tissue level. The double-layer and piezoelectric effectsare thus purely microscopic. They only slightly modifythe macroscopic ionic transport, whereas the behaviour ofbone can be fairly well described by classical poroelasticitycombined with the usual Darcy law. Consequently, thisstudy confirms that, quantitatively, classical bone modellingefforts involving poroelasticity adequately quantify the fieldsassociated with the fluid phase (pressure field, Darcianvelocity, etc.) at the tissue level.

Notwithstanding this efficiency of classical approachesto recover the macroscopic fluid variables, they ignore thephenomena occurring at the cellular scale. In particular,the mechanotransduction of the remodelling signals, whichis microscopic by nature, is only represented through adhoc macroscopic laws (Adachi et al., 2010; van Oers et al.,in press). As a consequence, the electro-chemical effectslocated at the cellular level are simply left aside. Even if


invisible at the tissue scale, the microscopic implications ofelectro-chemistry in the cellular mechanosensation cannotbe neglected. The example of Table 3 has been especiallychosen to illustrate this point. Osteocytes are known torespond to fluid flow stimulation (Burger and Klein-Nulend,1999; Westbroek et al., 2000), and in particular to fluid shearstress (Williams et al., 1994). More generally, bone cells in vitrorespond to fluid shear effects of 0.2–6 Pa over their surface(Cowin, 2002). The results reported in Table 3 indicate thatthe three contributions to the fluid shear felt by the cells stayin this range. However, for some specific cases, the hydraulicand the electro-chemical parts of the shear may act inopposition, possibly causing an annihilation of the total shearstimulation. This situation can be caused by the clogging ofsome canaliculi (such as a “closed tube” configuration), or byopposite chemical gradients due to the cellular activity. If thethree contributions to the fluid movement act in concert, thetotal shear stress felt by the osteocyte is around 2.8 Pa. On thecontrary, if the Poiseuille effect is compensated by the electro-chemical ones, the total shear collapses below the limit valueof 0.2 Pa.

In this numerical example, we focussed on the shearstress as the key mechanotransduction signal. Since, theelectro-chemical contributions to the local fluid flow aremore important near the cell membrane where the double-layers develop, there surely exists other implications on thetransmission of the drag effect exerted on the pericellularmatrix to the cell membrane and its underlying central actinfilament. Anyway, we illustrate how models that adopt apurely macroscopic point of view sometimes neglect somesignificant microscopic multiphysical effects. As a result, apurely macroscopic description of bone remodelling providesa somewhat skewed representation of the in vivo remodellingsignals.

From this study, different avenues worth exploring canbe drawn. To go into more depth in our understanding ofbone remodelling, additional work is required in order toimprove the description of the cells in their environment.New experiments or innovative models (from the moleculeto the organ) should be proposed to improve our knowledgeof the phenomena at the cellular scale. The multiscalestrategy derived in this paper is a step toward this direction.However, the cellular activity in its environment requiresto explore many nanoscopic possibilities. For instance, theionic and molecular exchanges between the cells and thesurrounding fluid, the role of the pericellular matrix andthe cell membrane, the cytoskeleton behaviour, etc., areimportant research issues that cannot be apprehendedthrough classical continuum mechanics and that requirethe use of other approaches such as molecular dynamicsor statistical mechanics. Such a task requires to merge theskills and to superpose the viewpoints of different researchgroups involving biologists, biomechanicians, biochemists,physico-chemists, surgeons, medical engineers, etc. Througha transversal approach, the description of the macroscopicadaptation of the tissue structure may be able to includemicroscopic information such as cell-to-cell paracrinalcommunication, cellular morphology, and more generally themicroscopic implications of different parameters such as age,gender, way of life, etc.

Finally, another important point to explore is the perti-nence of transposing in vitro observations or in silico consid-erations to describe the in vivo phenomena. Indeed, existingin vitro studies do not reproduce the osteocyte–flow interac-tions in the in vivo cellular environment. Thus, further in-vestigations directly linking osteocyte stimulation and in vivomechanotransduction mechanisms have to be carried out.

5. Conclusion

The aim of this study was to offer new perspectives for boneremodelling models. Based on a multiphysical hierarchicaltreatment of the phenomena governing the cortical tissuebehaviour, this work quantitatively proved the accuracy ofcommon macroscopic bone tissue models. Nevertheless,qualitatively, it also puts into relief the weakness of purelyhydro-mechanical approaches when studying remodellingsignals at the cellular scale. Indeed, even if washed out at thetissue level, themicroscopic electro-chemical phenomena arevisible in the neighbourhood of the cell.

The consequences of the microscopic effects in theclassical remodelling transduction scenarios are considerablesince they imply that classical approaches adopting a purelymacroscopic point of view and neglecting the coupledphenomena have to be revised. Since the key phenomena arelocated at the scale of the cell, it would be vain to carry onworking only at the organ scale.

Revisiting the common point of view of bone remodellingmodels, this paper offers more questions than answers. Ifhuge steps have been made since the early works of JuliusWolff (Wolff, 1892), there are still a lot of ways to improveour understanding of bone remodelling. In particular, thefocus should be made on the phenomena occurring at thecellular scale. Concomitantly to the advances in modellingat the micro- and nanoscale (homogenization, moleculardynamics, etc.) and in microscopic experiments (imagingprocess, microsensors, etc.), the bone remodelling descriptionhas to be revisited.

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