A tri-phasic mixture model of bone resorption: Theoretical investigations

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 ) 1 9 4 7 – 1 9 5 4

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Research paper

A tri-phasic mixture model of bone resorption: Theoreticalinvestigations

Gholamreza Rouhi∗

Department of Mechanical Engineering & School of Human Kinetics, University of Ottawa, ON, CanadaFaculty of Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran

A R T I C L E I N F O

Article history:

Received 5 July 2007

Received in revised form

17 June 2011

Accepted 18 June 2011

Published online 29 June 2011

Keywords:

Bone resorption

Mixture theory

Tri-phasic model

Chemical reactions

A B S T R A C T

In this paper, for the first time, a tri-phasic model of bone resorption using a mixture

with chemical reactions is proposed. Three constituents (matrix, fluid, and cells) are

considered. Conservation equations and entropy inequality are provided. The dependent

variables in the constitutive equations, such as the rate of resorption, are assumed to be a

function of temperature, deformation gradient, and the extent of the chemical reactions.

Using constitutive equations in the second law of thermodynamics, a criterion for the

thermodynamic equilibrium state is obtained which contains a bio–chemo–mechanical

affinity. Using the proposed model, one can find a theoretical explanation for some

clinically observed behavior of bone, for instance for the greater rate of bone resorption

in cortical than cancellous bone, using the conservation equations and/or consistency

requirements of continuum mixture theory. This work can be seen as a first step towards

establishing a new theoretical framework which could be developed in the future by

collaborative work, and with the hope of shedding some light on the multidisciplinary and

complex process of bone resorption.c⃝ 2011 Elsevier Ltd. All rights reserved.

v

d

1. Introduction

Bone is continuously remodeled through a coupled processof bone resorption and formation, the so-called bone remod-eling process. An early hypothesis about the dependence ofthe structure and form of bones, and the mechanical loadsthey carry, was proposed by Galileo in 1638 (Ascenzi, 1993).The nature of this dependence was first described in a semi-quantitative manner by Wolff (1892), who stated that “Ev-ery change in the form and function of a bone, or of theirfunction alone, is followed by certain definitive changes intheir internal architecture, and equally definite secondary

∗ Correspondence to: Department of Mechanical Engineering, UniCanada. Tel.: +1 613 562 5800x6268; fax: +1 613 562 5177.

E-mail address: [email protected].

1751-6161/$ - see front matter c⃝ 2011 Elsevier Ltd. All rights reservedoi:10.1016/j.jmbbm.2011.06.011

ersity of Ottawa, 161 Louis Pasteur, Room A017, Ontario, K1N 6N5,

alteration in their external conformation, in accordance withmathematical laws”. There are several theories about themechanisms of bone remodeling, but all of these considera unique governing equation for the whole remodeling pro-cess, and they are typically based on a single-phase contin-uum mechanics model (Cowin and Hegedus, 1976; Hegedusand Cowin, 1976; Martin, 1984; Huiskes et al., 1987; Beaupreet al., 1990; Prendergast and Taylor, 1994; Jacobs et al., 1997;Huiskes et al., 2000; Ruimerman et al., 2001; Doblare and Gar-cia, 2001, 2002; Ramtani and Zidi, 2001; Rouhi et al., 2004,2006; Ruimerman et al., 2005; McNamara and Prendergast,2007; Vahdati and Rouhi, 2009). It is well known that boneremodeling is comprised of bone resorption followed by

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1948 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 ) 1 9 4 7 – 1 9 5 4

Nomenclature

a•b Inner (or dot) product of the vector a and vectorb

N Number of different constituents in the mix-ture

B1, . . . ,BN N continuous bodies

Xa Position of a particle of the ath body in itsreference configuration

xa Position of a particle of the ath body in itsspatial configuration

x′a or Va Velocity of Xa at time t

aa Acceleration of the ath constituent at time t

GRAD Gradient with respect to material coordinates

grad Gradient with respect to spatial coordinates

ua Diffusion velocity of the ath constituent

ρa Density of the ath constituent

ρ Density of the mixture

ca Mass concentration of the ath constituent at(x, t)

ca Rate of mass supplied to the ath constituentper unit time from the other constituents thatoccupy x at time t, caused by chemical reactionsthat occur among them

F Deformation gradient

Fa Gradient of the deformation for Xa at time t•

F Rate of deformation gradient

La Velocity gradient for Ba at (x, t)

Pa Linear momentum supply caused by thepresence of other constituents

p Pressure

µ, λ and β Lame’s constants

Ta Partial stress tensor

E Strain tensor

D Strain rate tensor

εa Internal energy density of the ath constituent

qa Heat flux vector of the ath constituent

ra External heat supplied to the ath constituent

I Identity tensor

ψa Helmholtz free energy of the ath constituent

ψ Helmholtz free energy

ψMech. Mechanical free energy

ψChem. Chemical free energy

ηa Entropy density of the ath constituent

η Entropy of the mixture

θ Absolute temperature

εa Energy supplied to the ath constituent by otherconstituents in the mixture

T Stress tensor

q Heat flux vector

ζ Extent of a chemical reaction•

ζ Rate of the extent of a chemical reaction

ω Reaction vector

σ Chemical affinity

q Heat flux

Cs =ρ

Mρ and M are the density and the molar mass of

the matrix, respectively

Γ (x, t) A differentiable function of x and t

formation with a coupling between the two processes (Jee,2001). In 2000, Huiskes and co-workers (Huiskes et al., 2000)proposed a theory which is based on hypothetical osteocytestimulation of osteoblast bone formation, as an effect of ele-vated strain in the bone matrix, and a role for microcrack anddisuse in promoting osteoclast resorption. They assumed thata change in bone mass at a particular location is determinedby osteoblast bone formation minus osteoclast bone resorp-tion.Optimal remodeling is responsible for bone health andstrength throughout life. An imbalance in bone remodelingmay cause diseases, such as osteoporosis. When boneresorption outstrips bone formation for a long period of time,osteoporosis arises. Current methods of tackling osteoporosisplace the most attention on inhibiting or decreasingosteoclastic activity. Considering the time duration of thebone resorption process (1–3 weeks, Jee (2001)), and alsothe significant effect of resorption in osteoporotic cases,only the first phase of the remodeling process, i.e. boneresorption, is modeled here. Bone resorption is consideredas a chemical process that occurs between osteoclastsand the matrix. Osteoclasts dissolve bone mineral by acidsecretion that degrades the organic matrix (Margolis andMoreno, 1992; Russelle and Heymann, 2002; Teitelbaum,2000). There are different approaches to model multi-phasicmedia, e.g. mixture theory, the effective medium, andhomogenization approaches (Cowin, 1999). Recently, Rouhiet al. (2007) proposed a bi-phasic model for bone resorption.

In this work, we aim to develop a preliminary phenomeno-logical, tri-phasic, descriptive model of bone resorption withthe scope of deriving and introducing the governing equa-tions resulting from conservation laws (conservation of mass,momentum, and energy), the second law of thermodynamics,constitutive equations, and consistency requirements of mix-ture theory. It should be noted that, considering the fact thatboth mixture theory with chemical reactions and the boneresorption process from the mechanistic point of view are intheir infancy, the model proposed here at this stage of devel-opment is not able to offer verifiable predictions. The formerpoint can be seen as a drawback, but on the other hand canbe deemed as a positive point to encourage fellow researchersto get involved in this, almost virgin, field of research to shedmore light and make novel contributions.

2. Methods and assumptions

Bone is treated as a tri-phasic mixture of matrix (s), fluid(f) and cells (o). Bone resorption is considered as a chemicalreaction caused by the secretion of H+ and Cl− from

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 ) 1 9 4 7 – 1 9 5 4 1949

osteoclasts which creates an acidic environment in a sealedzone between the osteoclasts and the bone matrix. It isassumed that the solid phase obeys small deformation theoryand is isotropic and linearly elastic. The velocity of thematrix and cells is assumed to be zero. The fluid phase isassumed to be viscous, and inertial effects are neglectedbecause of the slow velocities that are at play. A non-rotational fluid is assumed for deriving the final form ofthe entropy inequality for the mixture as a whole. A non-polar mixture assumption is also made; thus the stresstensors and the inner part of the stress tensor are symmetric(Bowen, 1976). In the constitutive equations, it is assumedthat the free energy, enthalpy, specific entropy, heat flux,and stress tensor are functions of temperature, deformationgradient, and the extent of the chemical reactions. Boneresorption was considered as an isothermal and a quasi-static process. The last assumption is well justified becausethe characteristic time of bone resorption is much longerthan the characteristic time for inertial effects. Using theseassumptions, the governing equations for bone resorptionwere derived using the conservation laws (mass, momentum,and energy), as well as entropy inequality and the appropriateconstitutive equations.

3. Kinematics

Consider N continuous bodies B1, . . . ,BN, each of which isvisualized by the region it occupies in three-dimensionalphysical space. As in theories of single materials, each bodycan be assigned a fixed but otherwise arbitrary referenceconfiguration and a motion:

x = xa(Xa, t), (1)

where Xa is the position of a particle of the ath constituentin its reference configuration, and x is the position occupiedat time t by the particle labeled Xa. The function xa in Eq.(1) is called deformation function for the ath constituent.The gradient of the deformation for Xa at time t is a lineartransformation defined by

Fa = GRAD xa(Xa, t). (2)

The symbol GRAD in Eq. (2) denotes the gradient with respectto material coordinates. The velocity gradient for Ba at (x, t) isdefined by

La = grad x′a(x, t) = F′

aF−1a . (3)

For a mixture, the N bodies B1, . . . ,BN are allowed to occupythe common portions of physical space. Thus, each spatialposition x in the mixture is occupied by the N particles, onefrom each constituent. Each constituent is assigned a density,and for the ath constituent the density is denoted by ρa. Themass concentrations of the different phases for this tri-phasicmodel (ca, a = s, f , and o) can be written as

cs =ρs

ρs + ρo + ρf; co =

ρo

ρs + ρo + ρf;

cf =

ρf

ρs + ρo + ρf.

(4)

It is assumed that velocity of the solid phase and bone-resorbing cells are nearly zero, thus

Vs&Vo ∼= 0. (5)

As a result, the diffusion velocities (ua, a = s, f , and o) of thedifferent phases can be derived as

us = uo = −

ρf

ρs + ρo + ρfVf ; uf =

ρs + ρc

ρs + ρo + ρfVf . (6)

Method of derivation of some of the equations appeared inthis article can be found in the Appendices.

4. Conservation laws

The axiom of mass balance for the bone matrix, bone fluid,bone-resorbing cells, and the whole mixture are, respectively,

∂ρs

∂t(ρf + ρo)− ρs

∂ρf

∂t+ grad ρf .Vf +

∂ρo

∂t

= (ρf + ρo + ρs)

div

ρfρsVf

ρf + ρo + ρs

+ cs

(7)

∂ρf

∂t+ grad ρf .Vf

(ρs + ρo)− ρf

∂ρs

∂t+∂ρo

∂t

= −(ρf + ρo + ρs)

div

ρf (ρs + ρo)Vf

ρf + ρo + ρs

+ cf

(8)

∂ρo

∂t(ρs + ρf )− ρo

∂ρs

∂t+

∂ρf

∂t+ grad ρf .Vf

= −(ρf + ρo + ρs)

div

ρoρfVf

ρf + ρo + ρs

+ co

(9)

cs + cf + co = 0, (10)

where ca (a = s, o, and f ) are the rates of mass transferred tothe ath constituent caused by chemical reactions with otherconstituents.

It is assumed that inertia effects and body forces arenegligible compared to the external loads applied on thebone. Thus, the linear momentum equations for the solidphase, fluid phase, cells, and the entire mixture become,respectively,

ps + div(2µE + λ trE) = 0 (11)

∂(ρfVf )

∂t+ div(ρfVf ⊗ Vf ) = div(−pI + 2ηD + 2µE + λ trE) (12)

po + divTo = 0 (13)

div(2µE + λ trE + To − pI + 2ηD) = 0, (14)

where µ, η, and λ are Lame’s constants; E and D are strainand strain rate tensors; To is the stress tensor of the resorbingcells; the product Vf ⊗ Vf is a linear transformation ((Vf ⊗

Vf )u = Vf (Vf • u) for all vectors u); and trE, pa, and p are thetrace of the strain tensor, the momentum supplied to the athconstituent, and the hydrostatic pressure of the fluid phase,respectively.

It should be noted that the stress tensor for our tri-phasicmixture is defined by

T = Ts + Tf + To − (ρsus ⊗ us + ρfuf ⊗ uf+ρouo ⊗ uo). (15)

But, since the products of the diffusion velocities, i.e. ui ⊗ ui,are too small, the stress tensor for the mixture will take thefollowing form:

T = Ts + Tf + To. (16)

For the sake of simplicity, here, it is assumed that the mixtureis non-polar. Thus, the axiom of angular momentum can besimplified to the following form, which means that the stresstensor of each constituent in the mixture is symmetric:

Ta = TTa . (17)


The axiom of angular momentum for our mixture reduces to

div(x × (2µE + λ trE + To − pI + 2ηD)) = 0. (18)

Assuming that bone resorption is an isothermal process, andby employing the first law of thermodynamic for this tri-phasic system, one can find

(ρs + ρo + ρf )∂

∂t

ρsεs + ρoεo + ρf εf

ρs + ρo + ρf

+

ρfVf

ρs + ρo + ρfgrad(ρsεs + ρoεo + ρf εf )

=

ρf

ρs + ρo + ρftr[(Ts + To + Tf )gradVf ]

+ (ρsrs + ρoro + ρf rf )

+

ρfVf

ρs(bf − bs)+ ρo(bf − bo)

ρs + ρo + ρf

−div

(qs + qf + qo)+

Vf

ρs + ρo + ρf

×

[ρf (T

To + TTs )− (ρs + ρo)TTf

− ρf

ρsεs + ρoεo + (ρs + ρo)εf

], (19)

where the εa (a = s, f , and o) are the internal energy densities,the qa (a = s, f , and o) are the heat flux vectors, and thera (a = s, f , and o) are the external heat supplies to the athconstituent.

If one assumes that the body forces, i.e. bs, bf , and bo,are negligible, then the third term on the right-hand sideof Eq. (19) will disappear. Moreover, if we assume that boneresorption is an adiabatic process, i.e. qs = qf = qo = 0,and also that there is no heat supply from external sources(rs = rf = ro = 0), then the first law of thermodynamics willtake the following form:

(ρs + ρo + ρf )∂

∂t


ρs + ρo + ρf

+

ρfVf


= −div

Vf

ρs + ρo + ρf

ρf (T


− ρf


+

ρf

ρs + ρo + ρftr[(Ts + To + Tf )gradVf ]. (20)

Assuming that bone resorption is an isothermal process,i.e. θs = θf = θo = θ, the entropy inequality for this tri-phasicsystem can be written as

− cs(ηs − ηf )θ+ (ρsrs + ρf rf + ρoro)+ div(qs + qf + qo)

+ θ

ρo∂ηo

∂t+ ρs

∂ηs

∂t+ ρf

∂ηf

∂t+ Vf • grad ηf

≥ 0, (21)

where ηa, qa, and ra are the specific entropy, heat fluxvector, and the external heat supply for the ath constituent,respectively.

Moreover, assuming that the bone resorption process isadiabatic, and also postulating that there is no externalheat supply to any of the mixture components during theresorption process, i.e. rs, rf and ro = 0, from the second lawof thermodynamics, the minimal value for the rate of boneresorption, cf MIN, will be

cf MIN =

ρs∂ηs∂t + ρf

∂ηf∂t + Vf • grad ηf

+ ρo

∂ηo∂t

ηf − ηs. (22)

Considering that the apparent density of osteoclasts is muchless than that of the other two phases, i.e. bone matrix andthe bone fluid phase, from Eq. (22) one can conclude that

cf ≥

ρs∂ηs∂t + ρf

∂ηf∂t + Vf • grad ηf

ηf − ηs

. (23)

From inequality (23), the minimum rate of bone resorptioncan be found as a function of the specific entropies ofbone matrix and fluid, the apparent density of bone matrixand bone fluid, and also the fluid phase velocity. From theinequality above one can also conclude that an increase in thedifference between the specific entropy of bone matrix andfluid will result in a reduction in the rate of bone resorption.

Also, using the first and second laws of thermodynamics,one can find an expression for the rate of mass supplied tothe bone fluid phase, i.e. the rate of bone resorption (see Eq.(24), given in Box I). The ηa (a = s, o, and f ) are the specificentropies of the constituents.

From inequality (24), it can be concluded that, byincreasing the difference between the specific entropies ofthe solid and fluid phases of bone, the rate of bone resorptionwill decrease, i.e. similar to the conclusion made based oninequality (23). Moreover, from equations (22)–(24), it canbe concluded that when the specific entropy of the bonematrix approaches that of the bone fluid, based on thefirst and second laws of thermodynamics, excessive andabnormal bone resorption occurs. This pathological statemaycorrespond to osteoporosis, so-called bone silent disease,which in most cases lead to bone fracture. Further researchneeds to be done, mostly on experimental grounds, in orderto find a physical interpretation of the expressions resultingfrom the first and second laws of thermodynamics for thistri-phasic bone resorption process.

From the consistency requirement for energy balancefor the constituents and the whole mixture, the followingrelation can be concluded for the rate of bone resorption:

Cf =

PfVf (ρf−(ρs+ρo))

ρf+ρs+ρo− (εs + εf + εo)

V2f2 + (εf − εs)

, (25)

where the εa (a = s, f , and o) are the energies supplied to theath constituent, the quantity εa is the local interaction forenergy, just as ca and pa which are the local interactions formass, and linear momentum, respectively. It should be notedthat, in deriving Eq. (25), it is assumed that both the rate ofmass transferred to the resorbing cells caused by chemicalreactions and the momentum supplied to the osteoclasts arenegligible.


4)
cf ≥
ρs∂ψs∂t + ρf

∂ψf∂t + Vf • gradψf

+ ρo

∂ψo∂t + tr(TTf gradVf )− (εs + εf + εo)

ηf − ηs(2

Box I.

Moreover, if one assumes that the apparent density of os-teoclasts is much less than that of the other two constituentspresent in the mixture, i.e. the bone matrix and bone fluid(ρo ≪ ρs& ρf ), then Eq. (25) will take the following form:

Cf =

PfVf (ρf − ρs)− (ρf + ρs)(εs + εf + εo)

(ρf + ρs)V2f2 + (ρf + ρs)(εf − εs)

. (26)

Eq. (26) shows that the rate of bone resorption is a functionof different parameters as follows: (a) the apparent densityof bone matrix and bone fluid (ρa, a = s and f ); (b) the fluidvelocity (Vf ); (c) momentum supplied to the fluid or solid

phase (Pf or Ps); (d) the internal energy densities of bonematrix and bone fluid (εa, a = s and f ); and the energy suppliedto the ath constituent by other constituents (εa,a = s, f , and o).

Assuming that the momentum supplied to the osteoclastsis negligible, one can find the following relation between themomentum supplied to the fluid and solid phases:

Pf = CsVf − Ps. (27)

As Eq. (27) shows, when there is no resorption, themomentum supplied to the bone matrix and bone fluidwould have the same magnitude. Also, when the momentumsupplied to the matrix and the fluid is the same and has anon-zero value (Ps = Pf = P = 0), the rate of mass supplied tothe fluid phase, i.e. the rate of bone resorption, can be foundusing the following equation:

Cf =−2PVf

, (28)

where the fluid phase velocity (Vf ) is assumed to have a non-zero value.

Furthermore, Eq. (28) shows that the rate of boneresorption is inversely proportional to the fluid velocity. Also,in high-porosity spongy bone, by increasing the porosity (orequivalently, increasing ρf ), the rate of bone resorption willdecrease, and vice versa. The former result indicates thatthere might be a negative feedback in the bone resorptioncontrol system. Roughly speaking, this can be seen as atheoretical verification of the experimental results of Martin(1984) relating a specific surface to bone porosity.

5. Constitutive equations

For chemically reacting mixtures, similarly to the assump-tions made by Bowen (1968), we assume that the Helmholtzfree energy (ψ), specific entropy (η), stress tensor (T), heat fluxvector (q), and reaction vector (ω) are functions of temperature(θ), deformation gradient (F), and the extent of the chemical

reaction vector (ζ):

ψ = ψ(θ, F, ζ)

η = η(θ, F, ζ)

T = T(θ, F, ζ)

q = q(θ, F, ζ)

ω = ω(θ, F, ζ).

(29)

The material time derivative of Eq. (29) is

•

ψ =∂ψ

∂θ(θ, F, ζ)

•

θ+tr

∂ψ

∂F

(θ, F, ζ)T

•

F+∂ψ

∂ζ(θ, F, ζ)

•

ζ, (30)

where•

K (K = ψ, θ, F and ζ) is the material time derivative of K.Transferring Eq. (30) into the Clausius–Duhem inequality

(Bowen, 1976) yields

−•

θ

qi(θ, F, ζ)

θ+ ρ

∂ψ(θ, F, ζ)∂θ

+ ρη(θ, F, ζ)

+ trT(θ, F, ζ)− ρ

∂ψ(θ, F, ζ)∂F

•

F

L − ρ

∂ψ(θ, F, ζ)∂ζ

•

ζ ≥ 0, (31)

where L =•

F F−1 is the velocity gradient.Coleman and Curtin (1967) stated that there is at least

one thermodynamic process that satisfies the constitutive

equations, i.e. Eq. (31), in which the values of•

θ and L can bespecified independently of any other term in the inequality.Thus, one can conclude that

η(θ, F, ζ) = −

∂ψ(θ, F, ζ)

∂θ+

qiρθ

(32)

T = ρF∂ψ(θ, F, ζ)

∂F(33)

−∂ψ(θ, F, ζ)

∂ζ

•

ζ ≥ 0. (34)

The first term in Eq. (34) is called the chemical affinity, σ:

σ = −∂ψ(θ, F, ζ)

∂ζ. (35)

Eq. (34) says that, in accordance with the second law ofthermodynamics, the chemical affinity times the rate of theextent of the chemical reactions should always be greaterthan or equal to zero. In other words, Eq. (34) shows that, ifthe chemical affinity is negative, the rate of the extent of thechemical reactions should be positive in order to have boneresorption, and vice versa. This axiom can help us to find anew driving force for the bone resorption process.

For a solid–fluid mixture, in a non-equilibrium state, achemical process will occur at the interface of the solidand fluid phases, resulting in a flow of mass through theboundary, and a change in boundary position. Assumingthat dissolution occurs only at the solid–fluid interface, onecan express the Helmholtz free energy of the mixture (ψ)as a summation of mechanical (ψMech.) and chemical energy(ψChem.):

ψ = ψMech.+ ψChem.. (36)


Using a dissipation law, the following expression for thedriving force of bone resorption, σ, can be obtained (Rouhiet al., 2007):

σ = ψMech.+ P + Cs(µs − µext.), (37)

where P, µs, and µext. are the hydrostatic pressure of thebone fluid, the chemical potential of the solid phase in theunstressed condition, and the chemical potential generatedby the resorbing cells, respectively. Also, Cs is defined as(Rouhi et al., 2007)

Cs =ρ

M, (38)

where ρ and M are the density and the molar mass of thematrix, respectively.

Using the driving force of bone resorption, i.e. Eq. (37),instead of Gibb’s free energy in the rate of dissolution ofthe mineral phase, and also using conservation of massequations, one can find a theoretical explanation for thecorrespondence between damage and rate of resorption. Byintroducing microdamage in bone, because of the stressconcentration in the vicinity of the cracks, the driving forceof bone resorption (Eq. (37)) would increase, and thus therate of resorptionwill increase. Experimental studies in whichdamage is produced by cyclic loading demonstrated thatresorption is primarily associated with microcracks (Burr andMartin, 1993; Mori and Burr, 1993).

6. Discussion

This paper is a contribution toward the understanding ofthe bone resorption process as a chemical reaction occurringin multi-phasic media. Knowing that bone has differentconstituents, and that resorption is caused by chemicalreactions between the cells and the bone matrix, a tri-phasic model is used in this research. By using mixturetheory with chemical reactions, first, the contributions ofthe different phases present in the mixture can be observed.Second, using a consistency requirement for energy balance,it is found that rate of bone resorption is a function ofdifferent factors, including the apparent density of the bonematrix and the bone fluid, the fluid velocity, the energysupplied to the bone matrix, bone fluid and osteoclasts, themomentum supplied to the fluid or solid phase, and theinternal energy densities of the different constituents. Third,using a relation between the momentum supplied to thesolid and fluid phases, one can conclude that the rate ofbone resorption is inversely proportional to the bone fluidvelocity. Also, in a high-porosity spongy bone, by increasingthe porosity, the rate of resorption will decrease, and viceversa. This result encourages one to look at normal boneresorption as a control system with a negative feedback. Itis speculated that bone resorption in cortical and cancellousbones should be affected by a control system which resultsfrom a relation between the specific surface of bone and itsapparent density. Another result of this mixture approach isentering the extent of chemical reactions as an independentvariable in the constitutive equations (Eq. (29)), which, inturn can introduce a new driving force for bone resorption

(Eq. (37)). The resulting driving force contains not onlymechanical factors, but also the effects of chemical andbiological factors. Recent studies of the central regulation ofbone mass (Haberland et al., 2001; Bajayo et al., 2005; Takeda,2005) encourage one to put more thought on the drivingforce of bone resorption and remodeling as a vital subjectrelated to bone resorption, as well as remodeling processes.Furthermore, introducing inequality (34) as a criterion for thethermodynamic equilibrium state might be seen as anotherinteresting feature of this work. For the sake of simplicity,the presence of osteocytes in the bone matrix was discarded,despite the fact that fluid flow in the bone matrix (e.g. inthe lacuno-canalicular network) has a definite effect on theosteocytes, and, most likely, on the osteoclasts, and thuson the rate of bone resorption. It seems, at least at themoment, that adding osteocytes to the model would makeit more complex and decreases its feasibility. There are somelimitations in the proposed model in this study, such as a lackof numerical analysis, which can likely shed more light onthe process, and lack of mechanosensor cells, i.e. osteocytes,in the model. Considering that bone resorption is causedby chemical reactions between osteoclasts and bone matrixas an exchange of mass between the bone matrix and thefluid phase, and knowing that the rate of the reactions isdependent not only on the concentrations of the differentions present in the system but also on mechanical stimuli, itseems that one of the best approaches to model this processis mixture theory, incorporating chemical reactions and alsoa more general affinity than either pure mechanical or purechemical. The model presented here is a naïve contributionwith the hope of shedding some light on the bone resorptionprocess, and is a starting point to encourage other researchersto contribute to the field of bone resorption, and to the boneremodeling process.

Acknowledgment

Special thanks go to the University of Ottawa, Canada, andto Amirkabir University of Technology, Iran, for their support.The author would also like to express his sincere appreciationto Prof. M. Epstein from the University of Calgary, and Dr. E.Silva.

Appendix A. Conservation of mass equation forthe solid phase

ρ•ca = −div(ρaca)+ ca

ca =ρa

ρ =

N∑1ρa

cs =ρs

ρs + ρf + ρo

cs + cf + co = 0

Assumptions: co ∼= 0, Vs&Vo ∼= 0

(ρs + ρf + ρo)

∂ρs∂t (ρf + ρo)− ρs

∂ρf∂t + Vf • grad ρf +

∂ρo∂t

(ρs + ρf + ρo)2

= div

ρsρfVf

ρs + ρf + ρo

+ cs. (7)


Appendix B. Consistency requirement for the lin-ear momentum

ua(x, t) = Va(x, t)−•x(x, t)

•x(x, t) =

1ρ

N−1ρaVa

us = uo = −

ρfVf

ρs + ρf + ρo

uf = −ρs + ρo

ρs + ρf + ρoVf

N−1

pa + caua = 0

Assumption: po = 0

ps + pf = csVf .

Appendix C. Conservation of linear momentumfor the fluid phase

∂(ρaVa)

∂t+ div(ρaVa ⊗ Va) = divTa + ρaba + pa + caVa

(Va ⊗ Va)u = Va(Va • u), where u is an arbitrary vector.

Assumptions: bs,bf &bo are negligible

∂(ρfVf )

∂t+ div(ρfVf ⊗ Vf ) = div tf + pf + cfVf

ps + pf = csVf

∂(ρfVf )

∂t+ div(ρfVf ⊗ Vf ) = div(−pI + 2ηD + 2µE + λtrE).

Appendix D. Energy consistency requirement

N−1

[εa + ua • pa + ca

εa +

12u2a

]= 0

Cf =

PfVf (ρf−(ρs+ρo))

ρf+ρs+ρo− (εs + εf + εo)

V2f2 + (εf − εs)

. (25)

Appendix E. Conservation of energy for the wholemixture

ρ•εI = tr(TIL)− divqI + ρr +

N−1ρaua • ba

εI =1ρ

N−1ρaεa

•

Γ (x, t) =∂Γ

∂t(x, t)+ [gradΓ (x, t)]

•x(x, t),

where Γ is a differentiable function of x and t.

TI =

N−1

Ta

L = grad•x(x, t) =

ρf

ρs + ρf + ρogradVf

ρr =

N−1ρara = ρsrs + ρf rf + ρoro

qI =

N−1(qa − TTaua + ρaεaua)

=

Vf

ρs + ρf + ρo

ρf (T

To + TTs )− (ρf + ρo)TTf

− ρf [ρoεo + ρsεs + (ρs + ρo)εf ]

+ qs + qf + qo

N−1ρaua • ba = (ρsbs + ρobo)

−ρfVf

ρs + ρf + ρo

+ ρfbfVf

ρs + ρo

ρs + ρf + ρo

.

So, conservation of energy for the whole mixture will take thefollowing form:

(ρs + ρo + ρf )∂

∂t


ρs + ρo + ρf

+

ρfVf


=

ρf


+ (ρsrs + ρoro + ρf rf )

+

ρfVf

ρs(bf − bs)+ ρo(bf − bo)

ρs + ρo + ρf

−div

(qs + qf + qo)+

Vf

ρs + ρo + ρf

×

ρf (T


− ρf


. (19)

But, if one assumes that bs, bf , and bo are negligible comparedto the other forces, the first law of thermodynamics will besimplified to the following form for our tri-phasic mixture:

(ρs + ρo + ρf )∂

∂t


ρs + ρo + ρf

+

ρfVf


= −div

(qs + qf + qo)+

Vf

ρs + ρo + ρf

×

ρf (T


− ρf


+

ρf


+ (ρsrs + ρoro + ρf rf ).


Appendix F. Second law of thermodynamics (orentropy inequality)

N−1

[ρaη

′a + div

haθa

− ρaηaua

−ρaraθa

+ caηa

]≥ 0

ha = qa + ρaθaηaua

Γ ′a(x, t) =

∂Γ

∂t(x, t)+ [gradΓ (x, t)]x′

a(x, t)

=∂Γ

∂t(x, t)+ [gradΓ (x, t)]Va(x, t),

where Γ is a differentiable function of x and t.Assumptions: θs = θf = θo = θ& co = 0

cs(ηs − ηf )θ+ (ρsrs + ρf rf + ρoro)+ div(qs + qf + qo)

+ θ

ρo∂ηo

∂t+ ρs

∂ηs

∂t+ ρf

∂ηf

∂t+ Vf • grad ηf

≥ 0.

Appendix G. First and second laws of thermody-namics

N−1

1θa

[−ρa

ψ′a + ηaθ

′a+ tr(TTaLa)−

qa • gaθa

+ ea − ua • pa − ca

ψa +

12u2a

]≥ 0

ψa = εa − ηaθa

ea = εa + uapa + ca

εa +

12u2a

ga = grad θa

Assumptions: θs = θf = θo = θ, so gs = gf = go = 0Since Vs = Vo = 0, Ls = gradVs = 0 and Lo = gradVo = 0,

−ρs∂ψs

∂t− ρf

∂ψf

∂t+ Vf • gradψf

− ρo

∂ψo

∂t

+ tr(TTf gradVf )+ (εs + εf + εo)

+ cf (ψs − ψf + εf − εs) ≥ 0.

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A tri-phasic mixture model of bone resorption: Theoretical investigations