research article three-scale multiphysics modeling of...

Research ArticleThree-Scale Multiphysics Modeling of TransportPhenomena within Cortical Bone

T. Lemaire,1 J. Kaiser,1,2 S. Naili,1 and V. Sansalone1

1Laboratoire Modelisation et Simulation Multi Echelle (MSME UMR 8208 CNRS), Universite Paris-Est,61 Avenue du General de Gaulle, 94010 Creteil, France2Comsol France, 5 Place Robert Schuman, 38000 Grenoble, France

Correspondence should be addressed to T. Lemaire; [email protected]

Received 1 July 2015; Accepted 8 September 2015

Academic Editor: Seungik Baek

Copyright © 2015 T. Lemaire et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Bone tissue can adapt its properties and geometry to its physical environment. This ability is a key point in the osteointegration ofbone implants since it controls the tissue remodeling in the vicinity of the treated site. Since interstitial fluid and ionic transporttaking place in the fluid compartments of bone plays a major role in the mechanotransduction of bone remodeling, this theoreticalstudy presents a three-scale model of the multiphysical transport phenomena taking place within the vasculature porosity and thelacunocanalicular network of cortical bone. These two porosity levels exchange mass and ions through the permeable outer wallof the Haversian-Volkmann canals. Thus, coupled equations of electrochemohydraulic transport are derived from the nanoscale ofthe canaliculi toward the cortical tissue, considering the intermediate scale of the intraosteonal tissue. In particular, the Onsagerreciprocity relations that govern the coupled transport are checked.

1. Introduction

Bone remodeling corresponds to a process by which bonetissue gradually alters its morphology and properties in orderto adapt to its external environment [1, 2].This ability is a keypoint of the success of different protocols in bone tissue engi-neering. It has thus been demonstrated that bone remodelingaround implants is essential to obtain a good osteointegrationof the implant [3]. Similarly, in bone tissue engineering, whengrafting a scaffold to fill a bone loss, there is increased boneremodelingwithin the treated site until the degradation of theentire scaffold [4].

That is why the study of the mechanisms governing boneremodeling process is a very important topic in bone biome-chanical engineering. Interstitial fluid and ionic transporttaking place in the fluid compartments of bone is thought toplay amajor role in the bone ability to transmit physical stim-uli toward bone cells and thus trigger the remodeling process[5, 6].

Cortical bone, which corresponds to the dense tissuelocated at the periphery of bones, is mostly made of cylindri-cal structures called osteons. These cylinders are crossed bya hierarchical porous network saturated by a fluid with bothmechanical and chemical functions. Axial Havers and radialVolkmann canals (HVC), few tens of microns in diameter,contain vasculature and nerves and provide a fast way totransmit chemomechanical information between differentosteons. Local messaging takes place inside the smaller lacu-nocanalicular porosity (LCP), a dense network of cavities(lacunae) and pseudocylindrical canals (canaliculi, few hun-dred nanometers in radius) crossing the osteonal matrix.TheLCP hosts the cellular network of the bone mechanosensingcells, the osteocytes, and is connected with the HVC.

Since the experimental description of the fluid flowwithin the cortical tissue is still a very challenging topic [7, 8],theoretical approaches are often carried out to understandthe in vivo phenomena that govern bone behavior. Mathe-matical descriptions of bone fluid flow often only consider

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 398970, 10 pageshttp://dx.doi.org/10.1155/2015/398970

2 Mathematical Problems in Engineering

(a) Cortical tissue (b) Osteons

(c) Lacunocanalicular network(d) Canalicular pore

Cortical boneTrabecular

bone

Haverscanal

Lacuna

Haversianporosity

Osteocyte

Collagen-apatite matrix

Pericellularmatrix

Osteocytedendrite

Figure 1: Multiscale structure of cortical bone.

the osteonal scale to describe the transport within the LCP[9–12] or sometimes involve a biporous treatment of thismedium [13–15], even if recent works tend to show that thenanopores inside the collagen-apatite matrix may also par-ticipate in the fluid movement inside bone volume [16, 17].

In geoscience, fluid and ionic transport in monoporousmedia is a classical application of upscalingmethods to deriveeffective transport coefficients [18–22]. Similarly, hydrochem-ical transport phenomena in the LCP were extensively stud-ied by our group [23, 24] highlighting their effects on bonephysiology [25, 26]. This paper stems from those results andextends them to the osteonal scale, focusing on the fluid andionic transport along the osteon. An asymptotic homogeniza-tion technique is used to upscale a coupled description of thetransport phenomena toward the tissue scale [23, 24]. Theoutcome is an explicit description of the velocity and the ionicflux along the osteon which turns out to be ruled by gener-alized forms of the Darcy law and macroscopic convection-diffusion equations.

To the best of our knowledge, this theoretical work is thefirst attempt to describe themultiphysical transport phenom-ena taking place within the HVC and the LCP at once. Thesetwo fluid compartments exchange mass and ions throughthe permeable outer wall of the HVC. The model developedin this paper accounts for these exchanges by identifyingthe normal fluxes of mass and ions through the outer wallsof the HVC with the longitudinal fluxes inside the canali-culi. Furthermore, a rewriting of the macroscopic equationswhen convection and diffusion mechanisms are comparableresulted in checking the Onsager reciprocal relations linking

the fluxes and driving forces that govern the coupled trans-port phenomena.

2. Multiscale Structure of the Cortical Tissue

In Figure 1, the multiscale structure of cortical bone is pre-sented. From the organ to the cell, the hierarchical descriptionof cortical bone reads the following: (a) cortical tissue scale(10−2m): cortical bone is made of cylinders called osteonsthat are interconnected by macropores (axial Havers andradial Volkmann canals) containing the vasculature and thenerves; (b) osteon scale (10−3m): the osteon is made ofconcentric lamellae and contains a porous network of lacunaeinterconnected thanks to thin channels (canaliculi); (c) LCPscale (10−5m): located between the lamellae, each lacunahosts a mechanosensitive cell called osteocyte that developsdendritic protrusions inside the canaliculi to form a stellarnetwork; (d) canalicular pore scale (10−8m): the canalicularpore space, which is roughly annular between the collagen-apatite matrix and the cell dendrite surface, is partiallyoccupied by a pericellular matrix made of charged proteins(mainly glycosaminoglycans). Interstitial fluid transport atthis scale should take into account diffusive, convective,pericellular friction and electrochemical effects.

In this paper, we consider the interstitial fluid transportin two levels of these nested porous networks, namely,the Havers-Volkmann channels and the lacunocanalicularnetwork. Thus, in our three-scale homogenization process,the microscale corresponds to the canalicular pore scale(Figure 1(d); ℓ

𝜇

∼ 10−8m), the mesoscale corresponds to

Mathematical Problems in Engineering 3

the intraosteonal scale of the LCP and the pores correspondto the Haversian-Volkmann channels (Figures 1(b) and 1(c);ℓ𝑚

∼ 10−5m), and the macroscale corresponds to the cortical

tissue scale (Figure 1(a); ℓ𝑀

∼ 10−2m). As a consequence, the

micro-to-meso and meso-to-macro characteristic lengthsratios are comparable: ℓ

𝜇

/ℓ𝑚

∼ ℓ𝑚

/ℓ𝑀

∼ 𝜂 ≡ 10−3. Herein-

after, sub- and superscripts𝜇,𝑚, and𝑀will denote quantitiesat the micro-, meso-, and macroscale, respectively.

3. From the Microscale to the Mesoscale

At the microscale, the representative volume element (RVE)is a bone fraction surrounding the canaliculus (see Fig-ure 1(d)). Canaliculi are seen as two concentric cylinders.Theinterstitial fluid, which is assumed to be an incompressibleNewtonian monovalent electrolyte (typically Na-Cl), occu-pies the annular space between the outer canalicular walland the inner osteocyte process membrane. The cationic andanionic concentrations are noted 𝑛

+ and 𝑛−, respectively.

Under bulk conditions, electroneutrality reads 𝑛+ = 𝑛−

= 𝑛𝑏

,where 𝑛

𝑏

stands for the salinity.Accompanying the pressure induced hydraulic flow, other

electrochemical phenomena are generated by the electrolytesolution movement [27]. Indeed, due to the pore surfacecharge, double layers develop in the pore volume to main-tain electroneutrality. When advected by the fluid flow, themobile charge population of the double layer generates themacroscopically observed streaming potentials. The spatialvariability of these potentials generates electroosmotic flow[28].

Thus, the microscopic model combines an electrostaticsbalance to describe the double layers, Nernst-Planck equa-tions for the ionic species transport that include possibleelectromigration effects, and a Stokes equation consideringthe effects of the Coulombic body force. Moreover, the teth-ering pericellular fibers can sensibly reduce the permeabilityof the canalicular space. This friction effect is represented bya Brinkman-like term involving a pericellular permeability[29].

3.1. Statement of the Transport Model at the Microscale. Thissubsection briefly recalls the main results of the two-scaletreatment of multiphysical transport within the lacunocan-alicular system as presented in [24].

First, electrostatics is expressed thanks to the Poisson-Boltzmann equation:

∇ ⋅ ∇ (𝜓𝜇

+ 𝜑𝜇

) =1

(𝐿𝜇

𝐷

)2

sinh𝜑𝜇. (1)

In this equation, two reduced electric potentials appear:the reduced streaming potential 𝜓𝜇 and the reduced doublelayer potential 𝜑𝜇. Moreover the Debye length 𝐿

𝜇

𝐷

roughlycharacterizes the thickness of the electrical double layer.Electric flux continuity at the pore surfaces is the associatedboundary condition.

Then, the Coulombic force effects can be split to makeelectrochemical driving effects appear in addition to thehydraulic pressure effect. Fluid flowhas thus threefold nature:

the Poiseuille velocity k𝜇𝑃

in response to the gradient of thefluid pressure 𝑝

𝑏

, a chemoosmotic velocity k𝜇𝐶

in response tothe gradient of the salinity 𝑛

𝑏

, and an electroosmotic velocityk𝜇𝐸

in response to the gradient of the streaming potential𝜓𝜇. That is why, introducing the fluid dynamic viscosity 𝜇

𝑓

and the pericellular permeability 𝑘𝑓

, the three velocities aregoverned by the following equations:

𝜇𝑓

∇ ⋅ ∇k𝜇𝑃

−

𝜇𝑓

𝑘𝑓

k𝜇𝑃

= 𝑓𝜇

𝑃

∇𝑝𝑏

,

𝜇𝑓

∇ ⋅ ∇k𝜇𝐶

−

𝜇𝑓

𝑘𝑓

k𝜇𝐶

= 𝑓𝜇

𝐶

∇𝑛𝑏

,

𝜇𝑓

∇ ⋅ ∇k𝜇𝐸

−

𝜇𝑓

𝑘𝑓

k𝜇𝐸

= 𝑓𝜇

𝐸

∇𝜓𝜇

.

(2)

The coefficients 𝑓𝜇∙

(with ∙ = 𝑃, 𝐶, 𝐸) on the right-hand sideof these equations read [26]

𝑓𝜇

𝑃

= 1,

𝑓𝜇

𝐶

= 2𝑅𝑇 (cosh𝜑𝜇 − 1) ,

𝑓𝜇

𝐸

= −2𝑅𝑇𝑛𝑏

sinh𝜑𝜇,

(3)

where 𝑅 is the gas constant and 𝑇 the temperature. The totalfluid velocity vector at the micropore scale is thereby thesum of the three velocities k𝜇 = k𝜇

𝑃

+ k𝜇𝐶

+ k𝜇𝐸

. Moreover, atthe canalicular lateral surfaces, no-slip boundary conditionsare assumed for each elementary velocity. Note that thisassumption to consider no-slip condition for each coupledvelocity is necessary to perform the homogenization processas explained in [29].

Finally, noting 𝐷±

the ionic diffusion parameters, theionic fluxes at the microscale J𝜇

±

= −𝐷±

exp(∓𝜑𝜇)(∇𝑛𝑏

±

𝑛𝑏

∇𝜓𝜇

) are due to diffusion and electromigration effects.Then, the Nernst-Planck convection-electrodiffusion equa-tions read then

𝜕

𝜕𝑡(𝑛𝑏

exp (∓𝜑𝜇)) + ∇ ⋅ (𝑛𝑏

exp (∓𝜑𝜇) k𝜇) + ∇ ⋅ J𝜇±

= 0. (4)

Ionic exchanges are possible at the pore surface and quan-tified thanks to exchanges coefficients 𝛼𝜇

±

:

J𝜇±

⋅ n = 𝛼𝜇

±

𝜕𝑛𝑏

𝜕𝑡. (5)

3.2. Description of the Transport Phenomena at the Mesoscaleof the Lamellar Tissue. Thefirst upscaling procedure to derivethe transport phenomena at the mesoscale from the micro-scopic description is based on the asymptotic periodic homo-genization process as proposed in [23, 30, 31]. The two-levelhomogenization procedure for the same Poisson-Boltzmannequation (1) and Brinkman equations (2) can be found in [23,29], whereas the upscaling of the Nernst-Planck equations isgiven in [24].

In particular, it was shown in these references that thefluid pressure 𝑝

𝑏

, the salinity 𝑛𝑏

, and the streaming potential


𝜓𝜇

= 𝜓𝑏

do not vary at the microscale whereas the doublelayer potential 𝜑𝜇 is purely microscopic.

At the mesoscale, the average fluid velocity along thecanalicular network V𝑚 (the superscript 𝑚 referring to themesoscale) can be described by a modified Darcy law:

V𝑚 = ⟨k𝜇⟩𝑚

= ⟨k𝜇𝑃

⟩𝑚

+ ⟨k𝜇𝐶

⟩𝑚

+ ⟨k𝜇𝐸

⟩𝑚

. (6)

Thus, the elementary velocities at the mesoscale are

⟨k𝜇𝑃

⟩𝑚

= −K𝑚𝑃

∇𝑚

𝑝𝑏

,

⟨k𝜇𝐶

⟩𝑚

= −K𝑚𝐶

∇𝑚

𝑛𝑏

,

⟨k𝜇𝐸

⟩𝑚

= −K𝑚𝐸

∇𝑚

𝜓𝑏

,

(7)

where K𝑚𝑃

, K𝑚𝐶

, and K𝑚𝐸

represent the effective permeabilitytensors quantifying the Poiseuille, osmotic, and electroos-motic effects at the mesoscale. The expressions of these ten-sors are detailed in our previous works (see, e.g., [23]). More-over,∇

𝑚

represents the space derivative operator with respectto the coordinates of the mesoscopic level whereas ⟨⋆⟩

𝑚

represents the averaging operator on the microscopic RVE.Finally, to derive the effective ionic fluxes at themesoscale

⟨J𝜇±

⟩𝑚

, we have to consider two cases. On the one hand, ifthe diffusive and convective effects are of the same order ofmagnitude, the canalicular Peclet number 𝑃𝑒𝜇 ≡ 𝑂(1) andwe obtain

⟨J𝜇±

⟩𝑚

= −D𝑚±

(∇𝑚

𝑛𝑏

± 𝑛𝑏

∇𝑚

𝜓𝑏

)

+ 𝑛𝑏

⟨exp (∓𝜑𝜇) k𝜇⟩𝑚

.

(8)

The complete expression of the effective diffusion tensor atthe mesoscaleD𝑚

±

is detailed in [24]. This quantity combinestextural and electrical effects. These ionic fluxes along thecanalicular system can then be separated into convective andelectrodiffusive parts J𝑐

±

and J𝑑±

:

⟨J𝜇±

⟩𝑚

= J𝑐±

+ J𝑑±

, (9)

where J𝑐±

= 𝑛𝑏

⟨exp(∓𝜑𝜇)k𝜇⟩𝑚

and J𝑑±

= −D𝑚±

(∇𝑚

𝑛𝑏

±𝑛𝑏

∇𝑚

𝜓𝑏

).On the other hand, if diffusion is the main transport

process at the canalicular scale, the Peclet number is smalland the convective term J𝑐

±

vanishes so that

⟨J𝜇±

⟩𝑚

= J𝑑±

. (10)

4. From the Lamellar Tissue tothe Cortical Tissue

The purpose of this section is now to describe the flow phe-nomena and chemical transport at the macroscale of the cor-tical tissue. At this scale, the porous network corresponds tothe Havers-Volkmann canals (named Haversian pores here-after), whereas the effective solid phase of the porousmediumcorresponds to the collagen-apatite tissue and the micro-porosity of the lacunocanalicular system.Hydroelectrochem-ical phenomena occurring across the canalicular networkare nevertheless integrated in the description of transportphenomena across Haversian porosity.

4.1. Statement of the Transport Model at the Haversian PoreScale. Since capillaries and nerves are present in the centralpart of Haversian pores [32], the interstitial fluid flows inthe pseudo-annular space between the outer wall of theHaversian canal and the walls of the capillaries and nerves.Again, the pore geometry can be roughly represented by twoconcentric cylinders representing the Haversian wall (outercylinder) and capillary and nerves (inner cylinder).

The walls of the Haversian pores are permeable [33],allowing exchanges of matter and ions with the blood vessels(inner wall, indexed vas) and with the lacunocanalicularnetwork (outer wall, indexed LCP). To use again periodichomogenization, we assume that a 3D periodic representativecell can be defined. Note that the possibility to obtain areliable elementary volume at the scale of cortical tissue isa tricky point in bone biomechanics due to the high spatialheterogeneity of this tissue.

Similarly to the multiphysical approach proposed at thecanalicular scale, the description of the transport phenomenaat the scale of theHaversian porosity should take into accountboth hydraulic and electrochemical effects.

4.1.1. Electrostatics. At themesoscale, the electrical potentialsare governed by a similar equation as obtained at the micro-scale:

∇ ⋅ ∇ (𝜓𝑚

+ 𝜑𝑚

) =1

(𝐿𝑚

𝐷

)2

sinh𝜑𝑚. (11)

The electric flux continuity at the walls completes theelectrostatics problem.

4.1.2. Fluid Movement with Coulombic Force. Since the fric-tion effect of the pericellular matrix is no more present at themesoscale of the Haversian porosity, the coupled fluid flow isnow represented by the following Stokes equations:

𝜇𝑓

∇ ⋅ ∇k𝑚𝑃

= 𝑓𝑚

𝑃

∇𝑝𝑏

,

𝜇𝑓

∇ ⋅ ∇k𝑚𝐶

= 𝑓𝑚

𝐶

∇𝑛𝑏

,

𝜇𝑓

∇ ⋅ ∇k𝑚𝐸

= 𝑓𝑚

𝐸

∇𝜓𝑚

,

(12)

where the coefficients 𝑓𝑚∙

are the same as before when trans-posed at the mesoscale:

𝑓𝑚

𝑃

= 1,

𝑓𝑚

𝐶

= 2𝑅𝑇 (cosh𝜑𝑚 − 1) ,

𝑓𝑚

𝐸

= −2𝑅𝑇𝑛𝑏

sinh𝜑𝑚.

(13)

Note that these equations correspond to (2) consideringan infinite fiber permeability (𝑘

𝑓

→ ∞). Again, the velocityin theHaversian porosity k𝑚 is the sumof three contributions:

k𝑚 = k𝑚𝑃

+ k𝑚𝐶

+ k𝑚𝐸

. (14)

Noting n and t the outer normal and longitudinal tangentunit vectors, the boundary conditions at the walls of theHaversian porosity are given hereafter.


(i) Tangent no-slip condition:

k𝑚∙

⋅ t = 0 (15)

(ii) Exchanges with the vasculature or the LCP:

k𝑚∙

⋅ n = V⋆∙

⋅ n, (16)

where ∙ = 𝑃, 𝐶, 𝐸 and ⋆ = vas, LCP depending on thelocation of the wall. The filtration velocities across the bloodvessels membranes Vvas

∙

have to be given. The filtrationvelocities at the outer wall of the pore VLCP

∙

are equal tothe velocities ⟨k𝜇

∙

⟩𝑚

determined from the previous upscalingprocedure in (7).

Note finally that pressure continuity between the Haver-sian and canalicular pores should be considered [34] and thatthe incompressibility of the fluid should also be assumed foreach elementary velocity:

∇ ⋅ k𝑚∙

= 0. (17)

4.1.3. Ionic Transport. As at the microscale, the ionic fluxvector at the mesoscale of the vascular porosity J𝑚

±

=

−𝐷±

exp(∓𝜑𝑚)(∇𝑛𝑏

±𝑛𝑏

∇𝜓𝑚

) follows theNernst-Planck equa-tions:

𝜕

𝜕𝑡(𝑛𝑏

exp (∓𝜑𝑚)) + ∇ ⋅ (𝑛𝑏

exp (∓𝜑𝑚) k𝑚) + ∇ ⋅ J𝑚±

= 0.

(18)

At the inner porewalls, the normal fluxes J𝑚±

⋅n are equal tothe ionic fluxes through the blood vessels walls Jvas

±

⋅n (whichhas to be provided):

J𝑚±

⋅ n = Jvas±

⋅ n. (19)

At the outer pore walls, the normal fluxes J𝑚±

⋅ n areequal to the fluxes derived from the canalicular networkanalysis ⟨J𝜇

±

⟩𝑚

⋅ n (see (9) or (10) according to the value ofthe canalicular Peclet number):

J𝑚±

⋅ n = ⟨J𝜇±

⟩𝑚

⋅ n. (20)

4.2. Homogenization to Reach the Macroscale. The homoge-nization procedure consists in the nondimensional writing ofthe problems, the asymptotic expansion, the collection of theslow variables, and the proposition of the closure problems(see, e.g., [23]). Here, this meso- to macroscale upscalingstage is often similar to the one performed to get to themesoscopic description from the microscale. Thus, we willbriefly present the similarities and be more specific about thedifferences.

4.2.1. Electrostatics. The homogenization process of thePoisson-Boltzmann equation (11) is the same as the oneproposed to perform the passage between the micro- andmesoscale.

4.2.2. Treatment of the Fluid Flow. Again, the homogeniza-tion procedure of the fluid transport equation and the massconservation are similar as previously done considering𝑘𝑓

→ ∞. Only the boundary conditions should be treated ina different way. Indeed, the effects of the hydraulic exchangesbetween canalicular and Haversian pores may vary depend-ing on the scale ratios ℓ

𝜇

/ℓ𝑚

and ℓ𝑚

/ℓ𝑀

[34, 35]. Here, ifthe no-slip condition of (15) remains the same, accordingto the arguments of [36], the canalicular permeability hasto be scaled by 𝜂

2, giving the following scaling rules of thenondimensional exchange conditions of (16):

k𝑚 ⋅ n = 𝜂2V⋆ ⋅ n, (21)

with ⋆ = vas, LCP depending on the considered pore surface.

4.2.3. Treatment of the Ionic Transport. First, we consider thatthe convective and diffusive driving effects can be compared;that is to say that 𝑃𝑒𝑚 ≡ 𝑂(1). Analogously to the results ofthe previous paragraph, if the upscaling of the Nernst-Planckequations (18) is very close to the one performed previously(see (8)), only the specificity of the nondimensional writingof the boundary conditions (19) and (20) should be herediscussed.

On the one hand, remembering the decomposition of⟨J𝜇±

⟩𝑚

in (9), the nondimensional writing of the boundaryequations (20) at the outer wall of the Haversian pores reads

J𝑚±

⋅ n = (𝐽𝑐

±

𝑛

J𝑐±

+ 𝐽𝑑

±

𝑛

J𝑑±

) ⋅ n, (22)

where the nondimensional convective flux numbers 𝐽𝑐±

𝑛

anddiffusive flux numbers 𝐽𝑑

±

𝑛

resulting from the LCP transportare introduced. To propose scaling rules of these parameters,we have to estimate their order of magnitude. A referencevalue of the convective flux could be obtained from the lacu-nocanalicular porosity 𝜙

𝜇, the reference fluid velocity in theLCP V𝑚ref , the reference length of the mesoscale 𝐿𝑚ref , and thereference ionic diffusion coefficients𝐷ref as follows:

𝐽𝑐

±

𝑛

≡V𝑚ref𝜙𝜇

𝐿𝑚

ref𝐷ref

. (23)

Since we have V𝑚ref ≡ 10−7m⋅s−1 [7], 𝜙𝜇 ≡ 10

−3 [37], 𝐿𝑚ref ≡

10−5 (see Figure 1(c)), and 𝐷ref ≡ 10

−9m2⋅s−1 (sodium bulkdiffusivity in water), we obtain 𝐽

𝑐

±

𝑛

≡ 𝜂2. Furthermore, a

reference value of the diffusive flux reads immediately

𝐽𝑑

±

𝑛

≡𝜙𝜇

𝐷ref𝐷ref

≡ 𝜙𝜇

≡ 𝜂. (24)

Then, at the outer wall of the Haversian pores, the exchangesconditions of (22) read

J𝑚±

⋅ n = (𝜂2J𝑐±

+ 𝜂J𝑑±

) ⋅ n. (25)

On the other hand, at the vasculature wall, if we considerthat the ionic exchange term with vasculature is mainly dueto convective terms, we have no dimensional conditionsinvolving 𝜂2:

J𝑚±

⋅ n = 𝜂2Jvas±

⋅ n. (26)


4.2.4. Slow Variables. Through the homogenization process,we recover that the fluid pressure 𝑝

𝑏

, the salinity 𝑛𝑏

, and thestreaming potential 𝜓𝑚 = 𝜓

𝑏

are macroscopic fields.

4.3. Macroscopic Model at the Scale of Cortical Tissue. In thissubsection, themacroscopic equations of themodel are given.We note ⟨⋆⟩

𝑀

the volume average on the representative cellused to pass from the meso- to the macroscale and ⟨⋆⟩

𝑆

𝑀

thesolid-fluid interface average (integral over the surface per unitof volume) of a given flux.

At the macroscale, the Poisson-Boltzmann problem is nomore visible. However, it still has to be solved inside therepresentative volume since the double layer potential conse-quences do exist in the transport phenomena. Moreover, theDarcy fluid velocity k𝑀 is expressed through a coupled law asobtained before:

k𝑀 = ⟨k𝑚⟩𝑀

= −K𝑀𝑃

∇𝑀

𝑝𝑏

− K𝑀𝐶

∇𝑀

𝑛𝑏

− K𝑀𝐸

∇𝑀

𝜓𝑏

, (27)

where ∇𝑀

is the macroscopic space derivation operator andK𝑀∙

are the effective coupled permeability tensors (with ∙ =

𝑃, 𝐶, 𝐸). Their expressions can be derived from cell problemssimilar to those presented in [23].

Finally, the ionic transport at the macroscopic scaleinvolves submacroscopic exchange terms Γ

±

= ⟨J𝑑±

⋅ n⟩𝑆𝑀

:

𝜕

𝜕𝑡[𝑛𝑏

⟨exp (∓𝜑𝑚)⟩𝑀

] + Γ±

+ ∇𝑀

⋅ [𝑛𝑏

⟨exp (∓𝜑𝑚) k∗⟩𝑀

]

= ∇𝑀

⋅ [DM±

(∇𝑀

𝑛𝑏

± 𝑛𝑏

∇𝑀

𝜓𝑏

)] ,

(28)

where the effective velocity k∗ = k𝑚 − 𝑉𝜇, which is

the difference between the fluid velocity in the mesoporesk𝑚 and a chemical accumulation velocity term 𝑉

𝜇. Thischemical accumulation velocity represents the effects dueto the downer scale. Furthermore, the complete expressionsof the effective diffusion tensors at the macroscale DM

±

aresimilar to those proposed in [24].

5. Discussion on the OnsagerReciprocal Relations

TheOnsager reciprocal relations allow highlighting the inter-actions between transport phenomena occurring simultane-ously in a thermodynamic system. In this section, the ques-tion of the validity of the Onsager reciprocal relationships isexamined for the above problems. This analysis is based on astudy of transport phenomena in coupled clayey media [38].

5.1. Discussion on the Fluid Flow in the Haversian Porosity.The coupled Darcy law (27) is the average expression of ageneralizedHagen-Poiseuille law expressing the fluid velocityat the Haversian pore scale k𝑚:

k𝑚 = −𝜅𝑚

𝑃

∇𝑀

𝑝𝑏

− 𝜅𝑚

𝐶

∇𝑀

𝑛𝑏

− 𝜅𝑚

𝐸

∇𝑀

𝜓𝑏

, (29)

where 𝜅𝑚∙

are conductivity parameters quantifying the cou-pled contributions in the Haversian porosity.These transportparameters at the Haversian pore scale are obtained from thecellular problem appearing in the meso-to-macro homoge-nization process. Indeed, the classical closure statements forcoupled flow are (see [23])

k𝑚𝑃

= −𝜅𝑚

𝑃

∇𝑀

𝑝𝑏

,

k𝑚𝐶

= −𝜅𝑚

𝐶

∇𝑀

𝑛𝑏

,

k𝑚𝐸

= −𝜅𝑚

𝐸

∇𝑀

𝜓𝑏

.

(30)

Introducing these equations in the Stokes equations (12),we obtain the cell problems:

−𝜇𝑓

∇ ⋅ ∇𝜅𝑚

∙

= 𝑓𝑚

∙

, (31)

where the 𝑓𝑚∙

are those defined by (13). These cell problemsare completed by introducing the closure statements in theprevious boundary conditions (21).The coupled permeabilityparameters at the macroscale are then determined by anaveraging procedure: ⟨𝜅𝑚

∙

⟩𝑀

= K𝑀∙

.Equation (29) can thus be rewritten remembering that

the definition of the reduced streaming potential involves theFaraday constant 𝐹, the gas constant 𝑅, and the temperature𝑇, so that 𝜓

𝑏

= 𝐹𝜓𝑏

/𝑅𝑇, and noticing that ∇𝑛𝑏

=

(𝑛𝑏

/𝑅𝑇)𝑅𝑇∇ ln 𝑛𝑏

:

k𝑚 = −𝜅𝑚

𝑃

∇𝑀

𝑝𝑏

− 𝜅𝑚

𝐶

𝑛𝑏

𝑅𝑇𝑅𝑇∇𝑀

ln 𝑛𝑏

− 𝜅𝑚

𝐸

𝐹

𝑅𝑇∇𝑀

𝜓𝑏

. (32)

The averaged version of this equation provides thus analternative expression of the Darcy law (27):

k𝑀 = ⟨k𝑚⟩𝑀

= − ⟨𝜅𝑚

𝑃

⟩𝑀

∇𝑀

𝑝𝑏

− ⟨𝜅𝑚

𝐶

⟩𝑀

𝑛𝑏


ln 𝑛𝑏

− ⟨𝜅𝑚

𝐸

⟩𝑀

𝐹

𝑅𝑇∇𝑀

𝜓𝑏

.

(33)

5.2. Macroscopic Ionic Flux and Electric Current. After hav-ing reformulated the fluid flow at the macroscale, we thenintend to derive the macroscopic chemical flux and electriccurrent. First, the sum of the cationic and anionic transport(28) reads

𝜕

𝜕𝑡[2𝑛𝑏

⟨cosh (𝜑𝑚)⟩𝑀

] + (Γ+

+ Γ−

) + ∇𝑀

⋅ [2𝑛𝑏

⟨cosh (𝜑𝑚) k∗⟩𝑀

] − ∇𝑀

⋅ [(DM+

+DM−

) ∇𝑀

𝑛𝑏

+ (DM+

−DM−

) 𝑛𝑏

∇𝑀

𝜓𝑏

]

= 0.

(34)


This equation can be reformulated to obtain the conservationof the chemical flux:

𝜕

𝜕𝑡[2𝑛𝑏

⟨cosh (𝜑𝑚)⟩𝑀

] + (Γ+

+ Γ−

) + ∇𝑀

⋅ [2𝑛𝑏

⟨k∗⟩𝑀

] + ∇𝑀

⋅ JM = 0,

(35)

where the chemical flux JM is defined by

JM = 𝑛𝑏

⟨2 (cosh (𝜑𝑚) − 1) k∗⟩𝑀

−𝑛𝑏

𝑅𝑇(DM+

+DM−

) 𝑅𝑇∇𝑀

ln 𝑛𝑏

−𝐹

𝑅𝑇(DM+

−DM−

) 𝑛𝑏

∇𝑀

𝜓𝑏

.

(36)

Similarly, the difference of the cationic and anionic trans-port equations (28) gives the conservation of the electric cur-rent:

𝜕

𝜕𝑡[𝐹𝑛𝑏

⟨−2 sinh (𝜑𝑚)⟩𝑀

] + 𝐹 (Γ+

− Γ−

) + ∇𝑀

⋅ IM

= 0,

(37)

where the electric flux IM is

IM = 𝐹𝑛𝑏

− ⟨2 sinh (𝜑𝑚) k∗⟩𝑀

−𝐹𝑛𝑏

𝑅𝑇(DM+

−DM−

) 𝑅𝑇∇𝑀

ln 𝑛𝑏

−𝐹2

𝑅𝑇(DM+

+DM−

) 𝑛𝑏

∇𝑀

𝜓𝑏

.

(38)

5.3. Matrix Form of theMultiphysics Transport within CorticalTissue. If we assume that the chemical accumulation effectscan be neglected, the effective velocity appearing in (28) isequal to the local fluid velocity in theHaversian porosity k∗ =k𝑚 and the use of (32) in (36) and (38) gives

JM = 𝑛𝑏

⟨2 (cosh (𝜑𝑚) − 1) (−𝜅𝑚

𝑃

∇𝑀

𝑝𝑏

− 𝜅𝑚

𝐶

𝑛𝑏

𝑅𝑇

⋅ 𝑅𝑇∇𝑀

ln 𝑛𝑏

− 𝜅𝑚

𝐸

𝐹

𝑅𝑇∇𝑀

𝜓𝑏

)⟩𝑀

−𝑛𝑏

𝑅𝑇(DM+

+DM−

) 𝑅𝑇∇𝑀

ln 𝑛𝑏

−𝐹

𝑅𝑇(DM+

−DM−

) 𝑛𝑏

∇𝑀

𝜓𝑏

,

(39)

IM = 𝐹𝑛𝑏

⟨−2 sinh (𝜑𝑚) (−𝜅𝑚𝑃

∇𝑀

𝑝𝑏

− 𝜅𝑚

𝐶

𝑛𝑏


ln 𝑛𝑏

− 𝜅𝑚

𝐸

𝐹

𝑅𝑇∇𝑀

𝜓𝑏

)⟩𝑀

−𝐹𝑛𝑏

𝑅𝑇(DM+

−DM−

) 𝑅𝑇∇𝑀

ln 𝑛𝑏

−𝐹2

𝑅𝑇(DM+

+DM−

)

⋅ 𝑛𝑏

∇𝑀

𝜓𝑏

.

(40)

Finally, from the macroscopic Darcy law (33) and theexpressions of the chemical flux (39) and of the electric cur-rent (40) we can derive the matrix form of the multiphysicstransport phenomena at the cortical scale:

(

k𝑀

JM

IM)

= (

−L𝑃𝑃 −L𝑃𝐶 −L𝑃𝐸

−L𝐶𝑃 −L𝐶𝐶 −L𝐶𝐸

−L𝐸𝑃 −L𝐸𝐶 −L𝐸𝐸)(

∇𝑀

𝑝𝑏

𝑅𝑇∇𝑀

ln 𝑛𝑏

∇𝑀

𝜓𝑏

).

(41)

The diagonal terms of the matrix represent the directparameters whereas the off-diagonal terms quantify the cou-pling phenomena. Their expressions are then

L𝑃𝑃 = ⟨𝜅𝑚

𝑃

⟩𝑀

,

L𝑃𝐶 = 𝑛𝑏

𝑅𝑇⟨𝜅𝑚

𝐶

⟩𝑀

,

L𝑃𝐸 = 𝐹

𝑅𝑇⟨𝜅𝑚

𝐸

⟩𝑀

,

L𝑃𝐶 = 𝑛𝑏

⟨2 (cosh (𝜑𝑚) − 1) 𝜅𝑚

𝑃

⟩𝑀

,

L𝐶𝐶 =𝑛2

𝑏

𝑅𝑇⟨2 (cosh (𝜑𝑚) − 1) 𝜅

𝑚

𝐶

⟩𝑀

+𝑛𝑏

𝑅𝑇(DM+

+DM−

) ,

L𝐶𝐸 = 𝐹𝑛𝑏


𝑚

𝐸

⟩𝑀

+𝐹𝑛𝑏

𝑅𝑇(DM+

−DM−

) ,

L𝐸𝑃 = −𝐹 ⟨2𝑛𝑏

sinh (𝜑𝑚) 𝜅𝑚𝑃

⟩𝑀

,

L𝐸𝐶 = −𝐹𝑛𝑏

𝑅𝑇⟨2𝑛𝑏

sinh (𝜑𝑚) 𝜅𝑚𝐶

⟩𝑀

+𝐹𝑛𝑏

𝑅𝑇(DM+

−DM−

) ,

L𝐸𝐸 = −𝐹2


sinh (𝜑𝑚) 𝜅𝑚𝐸

⟩𝑀

+𝐹2

𝑛𝑏

𝑅𝑇(DM+

−DM−

) .

(42)

The Onsager reciprocal relations would require thismatrix to be symmetric. This point is checked in Appendixby showing that L𝑃𝐶 = L𝐶𝑃, L𝑃𝐸 = L𝐸𝑃, and L𝐶𝐸 = L𝐸𝐶.


6. Concluding Remarks

The use of modeling approaches to mimic the in vivo behav-ior of bone tissue is a common alternative to the difficultyto perform convenient in vivo experiments. When adoptinga theoretical approach, it is necessary to find a way tovalidate the model results. In bone poromechanics, thisvalidation is often performed through a method correlatingthe macroscopic streaming potentials induced by the stress-generated fluid flow within the lacunocanalicular pores. Thisidea was first proposed by [39] and was later used by severalother groups [10, 23, 40]. However, all these studies arebased on strong assumptions. First, they involve a two-scaleapproach connecting phenomena occurring in the canali-culi to phenomena at the intermediate scale of osteons orosteonal tissue whereas experimental in vivo measurementsare performed at the organ scale (see, e.g., [41]). Further-more, these studies consider homeostasis to neglect chemicalgradient effects and put forward the Onsager reciprocityrelations to obtain the streaming potential distribution fromthe upper scale fluid velocity. When steady state is reached,there is no net charge transfer, implying that the electriccurrent is zero. This is equivalent to setting the Ohmic andconvective currents equal and opposite. Thus, the streamingpotential can be linearly connected to the macroscopicpressure field.

The present study makes it possible to reach the macro-scopic scale of bone tissue and shows the validity of theOnsager reciprocity relations.

A multiscale model of the transport phenomena occur-ring at different scales of cortical tissue was developed adopt-ing two successive upscaling procedures. It takes into accountthe existing hydroelectrochemical phenomena occurring atthe different scales of pores (Haversian porosity and canali-culi).

Furthermore, a rewriting of the macroscopic equationswhen convection and diffusion mechanisms are compa-rable resulted in checking Onsager reciprocal relationslinking the coupled quantities that govern the coupledtransport phenomena. This approach thus reinforces theseminal idea of [39] to use macroscopic streaming poten-tials measurements to validate poromechanical models ofbone that are extensively used in bone biomechanical engi-neering.

Appendix

Recovery of the Onsager Reciprocity Property

If u is a quantity, the divergence theorem gives a link betweenthe volume and surface average values. Moreover, due to theno-slip boundary condition which infers that 𝜅𝑚

∙

= 0 on thepore surface, a strong simplification is obtained:

⟨∇ ⋅ (𝜅𝑚

∙

u)⟩𝑀

= ⟨𝜅𝑚

∙

u ⋅ n⟩𝑆𝑀

= 0. (A.1)

The use of this result and (31) for ∙ = 𝑃, 𝐶 gives L𝐶𝑃 = L𝑃𝐶:

L𝐶𝑃 = 𝑛𝑏

⟨2 (cosh (𝜑𝑚) − 1) 𝜅𝑚

𝑃

⟩𝑀

=𝑛𝑏

𝑅𝑇⟨2𝑅𝑇 (cosh (𝜑𝑚) − 1) 𝜅

𝑚

𝑃

⟩𝑀

=𝑛𝑏

𝑅𝑇⟨𝑓𝑚

𝐶

𝜅𝑚

𝑃

⟩𝑀

=𝑛𝑏

𝑅𝑇⟨−𝜇𝑓

(∇ ⋅ ∇𝜅𝑚

𝐶

) 𝜅𝑚

𝑃

⟩𝑀

=𝑛𝑏


𝜅𝑚

𝑃

∇ ⋅ ∇𝜅𝑚

𝐶

⟩𝑀

=𝑛𝑏

𝑅𝑇(−𝜇𝑓

⟨𝜅𝑚

𝑃

∇ ⋅ ∇𝜅𝑚

𝐶

⟩𝑀

)

=𝑛𝑏


⟨∇ ⋅ (𝜅𝑚

𝑃

∇𝜅𝑚

𝐶

) + ∇𝜅𝑚

𝑃

⋅ ∇𝜅𝑚

𝐶

⟩𝑀

)

=𝑛𝑏


⟨∇ ⋅ (𝜅𝑚

𝑃

∇𝜅𝑚

𝐶

)⟩𝑀

+ ⟨∇𝜅𝑚

𝑃

⋅ ∇𝜅𝑚

𝐶

⟩𝑀

)

=𝑛𝑏


⟨∇𝜅𝑚

𝑃

⋅ ∇𝜅𝑚

𝐶

⟩𝑀

)

=𝑛𝑏


⟨∇ ⋅ (𝜅𝑚

𝐶

∇𝜅𝑚

𝑃

)⟩𝑀

+ ⟨∇𝜅𝑚

𝑃

⋅ ∇𝜅𝑚

𝐶

⟩𝑀

)

=𝑛𝑏


⟨𝜅𝑚

𝐶

∇ ⋅ ∇𝜅𝑚

𝑃

⟩𝑀

)

=𝑛𝑏


(∇ ⋅ ∇𝜅𝑚

𝑃

) 𝜅𝑚

𝐶

⟩𝑀

=𝑛𝑏

𝑅𝑇⟨𝑓𝑚

𝑃

𝜅𝑚

𝐶

⟩𝑀

=𝑛𝑏

𝑅𝑇⟨𝜅𝑚

𝐶

⟩𝑀

= L𝑃𝐶.

(A.2)

Similarly, from (31) for ∙ = 𝑃, 𝐸, we obtain L𝐸𝑃 = L𝑃𝐸:

L𝐸𝑃 = −𝐹 ⟨2𝑛𝑏


⟩𝑀

=𝐹

𝑅𝑇⟨−2𝑅𝑇𝑛

𝑏


⟩𝑀

=𝐹

𝑅𝑇⟨𝑓𝑚

𝐸

𝜅𝑚

𝑃

⟩𝑀

=𝐹

𝑅𝑇⟨(−𝜇𝑓

∇ ⋅ ∇𝜅𝑚

𝐸

) 𝜅𝑚

𝑃

⟩𝑀

= ⋅ ⋅ ⋅ =𝐹

𝑅𝑇⟨(−𝜇𝑓

∇ ⋅ ∇𝜅𝑚

𝑃

) 𝜅𝑚

𝐸

⟩𝑀

=𝐹

𝑅𝑇⟨𝑓𝑚

𝑃

𝜅𝑚

𝐸

⟩𝑀

=𝐹

𝑅𝑇⟨𝜅𝑚

𝐸

⟩𝑀

= L𝑃𝐸.

(A.3)

The last equality L𝐶𝐸 = L𝐸𝐶 is obtained from (31) for ∙ =

𝐶, 𝐸:

L𝐶𝐸 = 𝐹𝑛𝑏


𝑚

𝐸

⟩𝑀

+𝐹𝑛𝑏

𝑅𝑇(DM+

−DM−

)


=𝐹𝑛𝑏

𝑅2𝑇2⟨2𝑅𝑇 (cosh (𝜑𝑚) − 1) 𝜅

𝑚

𝐸

⟩𝑀

+𝐹𝑛𝑏

𝑅𝑇(DM+

−DM−

)

=𝐹𝑛𝑏

𝑅2𝑇2⟨𝑓𝑚

𝐶

𝜅𝑚

𝐸

⟩𝑀

+𝐹𝑛𝑏

𝑅𝑇(DM+

−DM−

)

=𝐹𝑛𝑏

𝑅2𝑇2⟨(−𝜇𝑓

∇ ⋅ ∇𝜅𝑚

𝐶

) 𝜅𝑚

𝐸

⟩𝑀

+𝐹𝑛𝑏

𝑅𝑇(DM+

−DM−

) = ⋅ ⋅ ⋅

=𝐹𝑛𝑏

𝑅2𝑇2⟨(−𝜇𝑓

∇ ⋅ ∇𝜅𝑚

𝐸

) 𝜅𝑚

𝐶

⟩𝑀

+𝐹𝑛𝑏

𝑅𝑇(DM+

−DM−

)

=𝐹𝑛𝑏

𝑅2𝑇2⟨𝑓𝑚

𝐸

𝜅𝑚

𝐶

⟩𝑀

+𝐹𝑛𝑏

𝑅𝑇(DM+

−DM−

)

=𝐹𝑛𝑏

𝑅2𝑇2⟨−2𝑅𝑇𝑛

𝑏


⟩𝑀

+𝐹𝑛𝑏

𝑅𝑇(DM+

−DM−

)

= −𝐹𝑛𝑏



⟩𝑀

+𝐹𝑛𝑏

𝑅𝑇(DM+

−DM−

) = L𝐸𝐶.

(A.4)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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