Journal of Theoretical Biology 292 (2012) 86–92

Contents lists available at SciVerse ScienceDirect

Journal of Theoretical Biology

0022-51

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/yjtbi

Piezoelectricity could predict sites of formation/resorption in boneremodelling and modelling

J.R. Fernandez a, J.M. Garcıa-Aznar b,n, R. Martınez c

a Departamento de Matematica Aplicada I, Universidade de Vigo ETSI Telecomunicacion, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spainb Multiscale in Mechanical and Biological Engineering (M2BE), Aragon Institute of Engineering Research (I3A), Universidad de Zaragoza, Zaragoza, Spainc Departamento de Matematica Aplicada, Universidade de Santiago de Compostela, Santiago de Compostela, Spain

a r t i c l e i n f o

Article history:

Received 7 June 2011

Received in revised form

27 September 2011

Accepted 30 September 2011Available online 8 October 2011

Keywords:

Bone modelling

Piezoelectricity

Finite element modelling

Mechanosensor mechanism

Mechano-electrical modelling

93/$ - see front matter & 2011 Elsevier Ltd. A

016/j.jtbi.2011.09.032

esponding author. Tel.: þ34 976 762796.

ail address: jmgaraz@unizar.es (J.M. Garcıa-A

a b s t r a c t

We have developed a mathematical approach for modelling the piezoelectric behaviour of bone tissue

in order to evaluate the electrical surface charges in bone under different mechanical conditions. This

model is able to explain how bones change their curvature, where osteoblasts or osteoclasts could

detect in the periosteal/endosteal surfaces the different electrical charges promoting bone formation or

resorption. This mechanism also allows to understand the BMU progression in function of the electro-

mechanical bone behaviour.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Although it is widely accepted that mechanical loading canregulate bone adaptation and that osteocytes are the mechan-osensor cells, it is unclear how actuator cells, osteoclasts andosteoblasts, are able to control bone resorption and formation infunction of mechanical conditions. Bone remodelling (adaptation ofbone without a change of the overall shape, where only changesassociated to internal mechanical properties of bone material areconsidered) and modelling (adaptation of bone in relation to boneshape) occur in the available bone surfaces due to the action ofosteoclasts and osteoblasts. In the case of bone remodelling acoupled mechanism occurs between osteoclasts and osteoblaststhat regulate the process of bone formation and resorption throughBasic Multicellular Units (BMUs) (Frost, 1969; Garcıa-Aznar et al.,2005; Hernandez et al., 1999). However, in bone modelling thiscoupled mechanism does not occur, and the exact spatial relation-ship between mechanical conditions and consequent bone model-ling is not well understood. In fact, many different theoreticalassumptions have been hypothesized in order to understand bothregulatory mechanisms in a unified theory.

One problem that has been widely analysed to understand boneadaptation capacity is the self-straightening of a fractured bone,when it has healed in an angulated position (a clear example ofbone modelling). This fact motivated that several authors (Basset

ll rights reserved.

znar).

and Becker, 1962; Basset et al., 1964) proposed that tensile stresson the convex surface causes (Martin et al., 1998) bone resorption,while compressive stress on the concave surface produces boneformation. However, Frost (1964) noted that this explanation isappropriate for the periosteal surface and not the endosteal. Hence,Frost (1964) proposed an intuitive theory which is known as theflexural neutralization theory (Frost, 1964). He proposed that boneresponse in the surface depends on the relative curvature of thesurface, where increased surface convexity produces bone resorp-tion, whereas a decreased surface convexity causes bone forma-tion. The ma\in disadvantage of this theory is to understand howcells are able to detect surface curvature. An alternative theory hasbeen proposed based on the strong correspondence betweencircumferential gradients of longitudinal normal strain with thespecific sites of periosteal bone formation (Gross et al., 1997; Judexet al., 1997). Given the association between strain gradients andfluid flow, another theory has been based on the fluid flow along apressure gradient from more compressed regions to more tensileregions (Mi et al., 2005a,b; Qin et al., 2003).

With the huge development of computers, numerical modellingis one of the techniques more used in biomechanics to test andvalidate these theories, and the research on bone modelling is not anexception. First numerical models were developed with the aim ofsimulating coupled bone remodelling and modelling (Huiskes et al.,1987; Beaupre et al., 1990). In all these cases, bone response wasregulated by means of the strain energy density using the ‘‘lazyzone’’ or ‘‘dead zone’’ concept (Huiskes et al., 1987; Beaupre et al.,1990). However, other authors have followed the computer-aidedoptimization (CAO) hypothesis (Garcıa et al., 2001; Mattheck, 1998),

J.R. Fernandez et al. / Journal of Theoretical Biology 292 (2012) 86–92 87

which follows a homogeneous stress distribution at its surface toregulate the bone adaptative mechanism. Another differentapproach was proposed by Martinez et al. (2006) who havenumerically demonstrated that the level of damage and the strainenergy near the periosteum and endosteum may also control theexternal bone remodelling response in the cortex of long bones. Infact, they successfully simulated the experiments developed byRubin and Lanyon (1982), in which they evaluated bone modellingin the sheep’s radius when ulna’s osteotomy was performed. Morerecently, periosteal surface pressures have been assumed to inhibitbone formation and induce bone resorption, while tensile strainsperpendicular to the periosteal surface have been hypothesized toinhibit bone resorption and induce new bone deposition (Carpenterand Carter, 2008). In all these models an appropriate periostealsurface remodelling is predicted, but predictions fail to estimateendosteal remodelling. In addition, these theories have been eval-uated to study bone modelling in the external bone surfaces, andthey have not used to simulate BMUs progression through existingbone tissue during the process of bone renewal (bone remodelling).

However, more recently, a coupled transport and mechanicalapproach is solved based on biphasic porous media by Burger et al.(2003), who analysed the role of the fluid flow pattern in a BMUprogression during remodelling. They showed that volumetricexpansion leads to influx of canalicular fluid at the tip of the cone,whereas at the basis of the cone, high volumetric compressionproduces high efflux of canalicular fluid. These results allowedpostulating the hypothesis that BMU progression during remodel-ling occurs as a result of different canalicular flow patterns aroundcutting cone and reversal zone during loading. But, this model isnot able to predict the bone alignment of a long bone with amalaligned fracture, given that the model cannot predict the fluidflow at the periosteal surface because it is impermeable.

Therefore, as far as authors know, currently there are nomodels able to explain in one unified theory the phenomena ofbone modelling and remodelling under loading conditions. Onepossible unified theory that could justify osteoclasts/osteoblaststend to work in some bone surfaces instead of others, could be thedifferent electric change on each surface. In fact, the piezoelectricbehaviour of bone could help to elucidate how cells are able tosense this mechano-regulation process. Actually in recent worksFerreira et al. (2009) and Noris-Suarez et al. (2007) have arguedthat bone healing and growth are controlled by the rate ofdeposition of hydroxyapatite, and this process has been so farattributed to the work of osteoblasts, which are attracted by theelectrical dipoles produced either by piezoelectricity or due todeformation of the bone (specially the collagen in it). However,currently there are not many models that justify bone remodel-ling and modelling in a unified theory based on bone piezo-electricity. In fact, only one theoretical mathematical model hasbeen recently developed and computationally analysed (Qin andYang, 2008), but this model uses a large number of parametersthat are really difficult to measure experimentally.

Nevertheless, recently, a renovated interest has appeared toshow the importance of bone piezoelectricity in bone responsive-ness to mechanical environment (Ahn and Grodzinsky, 2009;Ramtani, 2008; Ferreira et al., 2009; Noris-Suarez et al., 2007).However, this interest already appeared in the 1960s, when bonepiezoelectricity was invoked as a potential mechanism to explainmechanical bone adaptation (Basset and Becker, 1962; Basset et al.,1964; Anderson and Eriksson, 1970). Two different mechanisms areresponsible for bone piezoelectricity: extracellular matrix piezo-electricity mainly due to the molecular asymmetry of collagen andstreaming potentials generated by the flow of a liquid acrosscharged surfaces. Despite the relevance of piezoelectricity, it hasnot been normally used to understand bone modelling and remo-delling phenomena.

Therefore, although there are many theories that try to eluci-date the mechanism that regulate bone cell function in differentbone surfaces, most of them have focused on bone remodelling ormodelling. In fact, as far as authors know, there is no one unifiedtheory able to justify both phenomena. Hence, this manuscriptproposes a hypothesis in which we show, through a computationalapproach, that bone matrix piezoelectricity is able to explain howbone is selectively deposited or removed at different endosteal andperiosteal surfaces in bone modelling under loading conditions.Moreover, the same hypothesis can be used to estimate thelocation of osteoclasts and osteoblasts in the advance of oneBMU in bone remodelling under loading conditions.

2. Material and methods

2.1. Piezoelectric bone model

In order to characterize the mechanical properties of the bone,we employ the laws used by Weinans et al. (1992). In this model thebone is considered as an isotropic elastic solid assuming thatPoisson’s ratio n is independent of the apparent density r whereasYoung’s modulus is given by EðrÞ ¼Mrg, where M and g are positiveconstants which characterize the bone behaviour. To obtain thenumerical results that we present in the next section, we considerthe following parameters used by Weinans et al. (1992):

M¼ 3790, g¼ 3, n¼ 0:3:

In order to introduce the piezoelectricity in the model, weextend the classical electro-mechanical dependence regulatingthe coupling between the mechanical and electric fields by thefunction ðr=rnÞ

g (rn being a reference value of 1.0 g/cm3). Thisfunction guarantees that the electric field increases with thedensity of the bone. Hence, the constitutive law for the stresstensor r (N/mm2) and the electric displacement D (C/mm2) is thefollowing:

r¼ 2 mðrÞeðuÞþlðrÞDivðuÞI� rrn

� �gEnEðjÞ, ð1Þ

D¼DeþDE ¼rrn

� �gEeðuÞþ r

rn

� �gbEðjÞ, ð2Þ

where u is the displacement field, e is the strain tensor, EðjÞ is theelectric field, En denotes the transpose of the third-order piezo-electric tensor E, b is the electric permittivity tensor, I denotesthe identity operator, Div represents the divergence operator andmðrÞ and lðrÞ are Lame’s coefficients of the material, assumed todepend on the apparent density of the bone. These constitutiveequations define the bone piezoelectric effect, so when bone issubject to a mechanical load, it generates an electric charge.Conversely, when bone is electrically charged by a voltage,strains/stresses can appear.

In this work, we assume as other authors (see Fotiadis et al.,1999; Qin and Ye, 2004) that bone behaves like a crystal withhexagonal symmetry. Therefore, the piezoelectric tensor E isdefined with four values and the permittivity tensor b is adiagonal matrix given by two constants. These tensors take thefollowing matrix expression:

E ¼0 0 0 e14 e15 0

0 0 0 e15 �e14 0

e31 e31 e33 0 0 0

0B@

1CA and b¼

b11 0 0

0 b11 0

0 0 b33

0B@

1CA,

where the third direction coincides with the longitudinal direc-tion of a long bone. According to Fotiadis et al. (1999) we consider

J.R. Fernandez et al. / Journal of Theoretical Biology 292 (2012) 86–9288

the following piezoelectric and permittivity coefficients:

e31 ¼ 1:50765� 10�9 C=mm2, e33 ¼ 1:87209� 10�9 C=mm2,

e15 ¼ 3:57643� 10�9 C=mm2, e14 ¼ 17:88215� 10�9 C=mm2,

b11 ¼ 88:54� 10�12 C2=N mm2, b33 ¼ 106:248� 10�12 C2=N mm2:

The numerical results presented in the next section are obtainedby using the classical finite element method to approximate thespatial variable. In this case, continuous piecewise affine functions areemployed. Therefore, since the initial bone remodelling is known, wemust use the coupled equations (1) and (2) to obtain both displace-ment and electric fields at this initial time. The resulting fully linearsystem is nonsymmetric and it is then solved by applying the well-known LU method, implemented by using MATLAB.

2.2. Examples of application: compressive loads

Our aim here is to numerically show that bone formation andresorption may be related to electrical charges in the bone surfaces,due to contributions produced by mechanical loading De. Hence, weconsider a diaphysis of a long bone and an osteon under compressiveloads, in order to understand its mechano-electric behaviour.

2.2.1. Diaphysis of a long bone with a malaligned fracture

In the first example, we consider a diaphysis of a fractured longbone which is healed in an angulated position. It is known that, in thissituation, the bone tends to become straighter and bone formationoccurs on concave surfaces while bone resorption takes place onconvex surfaces. We assume that the length of the diaphysis is150 mm and the diameter 27 mm (see Croker et al., 2009). Moreover,

Fig. 1. Diaphysis of a long bone where we assume that the bone is healed in an

angulated position, forming 101 with the vertical direction. We have fixed the

potential and displacement fields on the lower boundary and we have applied a

compressive load on the upper boundary.

it has an internal part formed by cancellous bone, with a density of0.8 g/cm3 and two layers of 7 mm thickness of cortical bone with ahigher density, 1.6 g/cm3. We apply a compression load on the upperboundary, with a maximum value of 2.5 MPa. We have chosen thisload in order to obtain a maximum deformation of 2000 me in thecortical bone (see Martin et al., 1998). Finally we have fixed thepotential and displacement fields on the lower boundary (see Fig. 1)to a value of zero. To use the piezoelectric tensor we consider thelongitudinal direction of the bone in the three directions of the tensor.

2.2.2. Analysis of one osteon in progression

In the second example, we analyse the bone remodelling in anosteon. In Fig. 2 we can see the three-dimensional finite elementmesh of this osteon. The cylindrical tunnel and the sphericalcutting cone have a diameter of 200 mm, which is representativefor a resorption cavity in human cortical bone. The outer diameterof the piece of bone in the model is 700 mm. The bone osteon isassumed to be loaded in compression (20 MPa) along its long-itudinal direction and under torsion (0.0084 N mm) at maximumload during a walking cycle. Moreover, we fix the electric potentialfield on the outer boundary. In order to simulate a piece of corticalbone, we assume that the density is 1.6 g/cm3 (Baca et al., 2007).

3. Results

3.1. Diaphysis of a long bone with a malaligned fracture

In order to show the behaviour of the electric displacement wehave analysed in Fig. 3 the normal electric displacement due to

Fig. 2. Finite element model of an osteon under compressive and torsional loads.

Moreover, the potential field is fixed to zero on the outer boundary.

Fig. 3. The normal electric displacement due to mechanical effects (De) takes

positive values where the bone is forming and negative values where the

resorption takes place. As we can see, the bone is forming on the concave surfaces

of the layer whereas the resorption takes place on the convex surfaces.

8.923e-0045.516e-0042.110e-004

-1.297e-004-4.703e-004

1.716e-0035.087e-004

-6.987e-004-1.906e-003-3.113e-003

4.201e-004

J.R. Fernandez et al. / Journal of Theoretical Biology 292 (2012) 86–92 89

mechanical conditions on the boundaries of the cortical layer (De).We have plotted it on a horizontal section of the bone corre-sponding to the fracture line. We can observe that this value ispositive on the concave surfaces, whereas on the convex surfacesis negative. This result is obtained due to the expression of thepiezoelectric tensor E and the distribution of the strains in thecortex. Since the normal vector to the surfaces is perpendicular tothe z-direction, then the normal strains have no effect on theelectrical displacement which is regulated by the shear strains onthe bone surfaces (see Fig. 4) that occur as a consequence of theinclination of the load with respect to the bone axis.

1.844e-004-5.131e-005-2.870e-004-5.227e-004

Fig. 4. Strain distribution on the bone: (a) x-component, (b) z-component and

(c) xz-component.

z

x

Fig. 5. Normal electric displacement due to mechanical effects under compression

and torsion. We can observe the positive values where the osteoclasts are

generating new bone and the negative values into the cutting cone because of

the activity of the osteoclasts.

3.2. Osteon in progression

It is known that bone remodelling occurs in local groups ofosteoblasts and osteoclasts called Bone Multicellular Units(BMUs), where each unit is organized into a ‘‘cutting cone’’ ofosteoclasts reabsorbing bone followed by osteoblasts refilling thebone defect left by osteoclasts (Frost, 1969). In order to under-stand this behaviour in function of the electro-mechanical beha-viour of the bone, we have analysed the normal electricdisplacement due to mechanical effects (De) on the cylindricaltunnel and the spherical cutting cone. In Fig. 5 we can observethese values on a vertical section of the osteon. When we applythe compressive and torsional loads we can observe that thisvalue is positive in the cylindrical part, where normally osteo-blasts deposit bone matrix and it is negative at the tip of thecutting cone. Moreover, the normal vector to the surface coincidesthere with the direction 3 (z-direction), and therefore, the normalstrains contribute to the electrical displacement, mainly the strainin the direction 3 (see Fig. 6). However, in the cylindrical surfaceof the osteon, the normal to the surface is perpendicular to thedirection 3 and therefore, only the shear strains influence on theelectrical displacement. In the transition zone between the cut-ting zone and the cylindrical zone the normal is changing andthere is a coupled contribution of the normal and shear strains.

Then, we have to remark the importance of the torsional load,because if we only consider the compressive load, we can see inFig. 7 how only negative electrical displacements occur in thecutting cone. Meanwhile, in the cylindrical tunnel of the osteonthese electrical displacements are zero. It is due to the fact that,under compression loads, shear strains are zero in this zone.However, with torsional loads, shear strains are different fromzero producing an electrical displacement perpendicular to thebone surface that induces bone formation in the cylindrical partrefilling the osteon.

4. Discussion

In this work, we have analysed if bone matrix piezoelectricityis able to explain bone modelling in an angulated theoreticalstraight bone under bending conditions and bone remodelling inthe advance of one osteon under physiological loads. Usingpiezoelectric parameters of the literature (Fotiadis et al., 1999;

3.181e-004

4.962e-004

6.742e-004

8.523e-004

1.030e-003

-7.805e-005

2.094e-004

4.969e-004

7.843e-004

1.072e-003

-3.600e-003

-2.894e-003

-2.187e-003

-1.481e-003

-7.750e-004

-1.748e-003

-8.784e-004

-8.800e-006

8.609e-004

1.731e-003

-8.951e-004

-4.309e-004

3.324e-005

4.974e-004

9.615e-004

-2.839e-003

-1.450e-003

-6.035e-005

1.329e-003

2.718e-003

Fig. 6. Strain distribution around the osteon: (a) X component, (b) Y component, (c) Z component, (d) XY component, (e) XZ component and (f) YZ component.

z

x

Fig. 7. Normal electric displacement due to mechanical effects under compres-

sion. We only have plotted the values with a modulus bigger than 5�10�13.

J.R. Fernandez et al. / Journal of Theoretical Biology 292 (2012) 86–9290

Qin and Ye, 2004) we have been able to find a justification tounderstand how the mechanical loads can generate an electricpolarization of bone, obtaining negative charges where bone willbe removed and positive charges where bone will be formed.Therefore, this assumption could help to understand a possiblemechanism by which bone cells are able to sense differentsurfaces, showing the key role that shear strains present on thismechanism. Nevertheless, we have to keep in mind that this

conclusion has been obtained with the piezoelectric tensordefined by several authors (Fotiadis et al., 1999; Qin and Ye,2004). However, there are strong discrepancies in the piezo-electric tensor depending on different works. In fact, otherauthors consider that the piezoelectric tensor is defined by non-zero constants in the normal components (see Ahn andGrodzinsky, 2009; Anderson and Eriksson, 1970), where however,shear components were not considered. To understand theinfluence of the piezoelectric tensor on the bone response, wehave developed a sensitive analysis, finding that this model isvery sensitive to the piezoelectric tensor values, obtaining invalidresults with those parameters that are very far from thoseproposed by Fotiadis et al. (1999) and Qin and Ye (2004).

Therefore and despite of having found some parameter valuesin the literature, bone piezoelectricity is a research field whichhas hardly been considered relevant in bone adaptation andregeneration mechanisms. This fact has motivated that thenumber of biomechanical works in this field has been reduceddramatically since 1960 until now, being really difficult to findexperiments to validate models of the electro-mechanical beha-viour of the bone. Therefore, a strong effort should be done in thisdirection in order to unravel the multiphysics character of bonephysiology.

Currently, recent works have appeared to show the impor-tance of bone piezoelectricity in bone responsiveness to mechan-ical environment (Ahn and Grodzinsky, 2009; Isaacson andBloebaum, 2010), specially focused on the role of collagen

J.R. Fernandez et al. / Journal of Theoretical Biology 292 (2012) 86–92 91

piezoelectricity in bone physiology. In fact, in this recent work(Ahn and Grodzinsky, 2009) authors discuss that only fluid shearstresses are not able to give an explanation to the wet bone’scomplex response to loading. For instance, how bone cells candifferentiate among different types of loads: bending vs. twistingvs. shear. In fact, authors discuss that it is necessary to incorpo-rate bone matrix piezoelectricity.

However, until now, more works and therefore more impor-tance have received the role of streaming potentials due to theinterstitial fluid flow than bone matrix piezoelectricity in therecent understanding of bone physiology (Riddle and Donahue,2009). In fact, more than 25 years ago, Pienkowski and Pollack(1983) concluded that the deformation of skeletal tissue, underfour-point bending, generates electrical potentials, resulting fromstreaming potential associated with the movement of interstitialfluid through the porosities of bone. More recently, Burger et al.(2003) analysed the role of the fluid flow pattern in a BMUprogression during remodelling. Nevertheless, these models arenot able to simulate the bone alignment of a long bone with amalaligned fracture. Therefore, as far as authors know, currentlythere are no models able to explain in one unified theory thephenomena of bone modelling and remodelling under loadingconditions. In this work, despite the uncertainty of the piezo-electric tensor and the additional simplifications assumed in themodel here analysed, we show the relevant role of extracellularmatrix piezoelectricity in bone modelling and remodelling underphysiological loading conditions, predicting the initial sites ofbone formation and resorption. Nevertheless, it is important toanalyse these simplifications and their possible impact on theobtained results. Firstly, we have assumed bone as isotropic whena clear anisotropy defines its mechanical behaviour (Martınez-Reina et al., 2011). Certainly, this isotropic assumption allows tobetter show the effect of piezoelectricity in the stress results,which could be altered due to anisotropic mechanical behaviour.Secondly, the proposed mathematical approach is fully mechan-istic and neglects other relevant aspects, such as electrochemicalaction. Indeed, recent experimental works have shown the role ofdeformation-induced piezoelectricity in bone adaptation (Ferreiraet al., 2009; Noris-Suarez et al., 2007). Moreover, this modelpredicts the initial sites of bone formation and resorption indifferent adaptive bone processes under loading conditions, butnot the progression of these sites over time. Nevertheless, thesesimplificative hypotheses do not affect to our qualitative conclu-sion about the key role of bone matrix piezoelectricity asmechanosensor mechanism.

Therefore, this work shows the relevance of bone matrixpiezoelectricity as a mediator for regulation of bone modellingand remodelling under loading conditions in one unified modelthat had not been previously explored. Nevertheless, this model isnot able to simulate other phenomena, such as bone resorptioninduced by non-loading conditions and the effect of loadingfrequency on bone response. These phenomena have been nor-mally associated to the interstitial fluid movement, effect which isout of the scope of this work. However, this fact and the multi-physics character of bone require the use of more complexmodels that allow the modelling of multiple phenomena inconjunction: matrix piezoelectricity, electrochemical action,streaming potentials and strain generated fluid flow. Additionally,the modelling of this fluid movement with charge densitiesoccurs at different levels of fluid containment: matrix porosity,lacuna/canaliculi and Haversian/Volkmann canals (Cowin et al.,2009). Hence, the computer modelling of bone requires the use ofmultiphysics and multiscale approaches, including matrix piezo-electricity as an additional relevant factor.

To efficiently implement and validate these sophisticatedmodels, the physiological role of piezoelectricity has to be fully

evaluated and unravelled. Therefore, novel experimental techni-ques have to be developed for studying bone piezoelectricityunder physiological conditions of temperature and moisture.

Acknowledgements

The authors gratefully acknowledge the research support ofthe Spanish Ministry of Science and Technology through ResearchProjects DPI2009-14115-C03-01 and MTM2006-13981, and theFPI graduate research fellowship program.

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