Numerical prediction of peri-implant bone adaptation: Comparison of mechanical stimuli and sensitivity to modeling parameters

Article

Reference

Numerical prediction of peri-implant bone adaptation: Comparison of mechanical stimuli and sensitivity to modeling parameters

PICCININI, Marco, et al.

Abstract

Long term durability of osseointegrated implants depends on bone adaptation to stress and strain occurring in proximity of the prosthesis. Mechanical overloading, as well as disuse, may reduce the stability of implants by provoking bone resorption. However, an appropriate mechanical environment can improve integration. Several studies have focused on the definition of numerical methods to predict bone peri-implant adaptation to the mechanical environment. Existing adaptation models differ notably in the type of mechanical variable adopted as stimulus but also in the bounds and shape of the adaptation rate equation.

However, a general comparison of the different approaches on a common benchmark case is still missing and general guidelines to determine physically sound parameters still need to be developed. This current work addresses these themes in two steps. Firstly, the histograms of effective stress, strain and strain energy density are compared for rat tibiae in physiological (homeostatic) conditions. According to the Mechanostat, the ideal stimulus should present a clearly defined, position and tissue invariant lazy [...]

PICCININI, Marco, et al . Numerical prediction of peri-implant bone adaptation: Comparison of mechanical stimuli and sensitivity to modeling parameters. Medical Engineering & Physics , 2016, vol. 38, no. 11, p. 1348-1359

DOI : 10.1016/j.medengphy.2016.08.008 PMID : 27641659

Available at:

http://archive-ouverte.unige.ch/unige:89035

Disclaimer: layout of this document may differ from the published version.

1 / 1

Numerical prediction of peri-implant bone adaptation: Comparison of mechanical stimuli and sensitivity to modeling parameters

Marco Piccinini

^a

, Joel Cugnoni

^a,∗

, John Botsis

^a

, Patrick Ammann

^b

, Anselm Wiskott

^c

aLaboratory of Applied Mechanics and Reliability Analysis, École Polytechnique Fédérale de Lausanne, Station 9, CH-1015 Lausanne, Switzerland

bDivision of Bone Diseases, Department of Internal Medicine Specialities, Geneva University Hospitals and Faculty of Medicine, CH-1205 Geneva, Switzerland

cDivision of Fixed Prosthodontics and Biomaterials, School of Dental Medicine, University of Geneva, CH-1211 Geneva, Switzerland

a rt i c l e i nf o

Article history:

Received 29 January 2015 Revised 5 August 2016 Accepted 30 August 2016

Keywords:

Bone adaptation Implants integration Overloading Specimen-specific Finite element Gait

Mechanical stimulus

a b s t ra c t

Longtermdurabilityofosseointegratedimplantsdependsonboneadaptationtostressandstrainoccur- ringinproximityoftheprosthesis.Mechanicaloverloading,aswellasdisuse,mayreducethestability ofimplantsbyprovokingboneresorption.However,anappropriatemechanicalenvironmentcanimprove integration.Severalstudieshave focusedonthe definitionofnumericalmethodstopredict boneperi- implantadaptationtothemechanicalenvironment.Existingadaptationmodelsdiffernotablyinthetype ofmechanical variable adopted as stimulusbut alsointhe boundsand shape ofthe adaptation rate equation.However,ageneralcomparison ofthedifferentapproaches onacommonbenchmarkcaseis stillmissingandgeneralguidelinestodeterminephysicallysoundparametersstillneedtobedeveloped.

Thiscurrentworkaddressesthesethemesintwosteps.Firstly,thehistogramsofeffectivestress,strain andstrainenergydensityarecomparedforrattibiaeinphysiological(homeostatic)conditions.Accord- ingtotheMechanostat,theidealstimulusshouldpresentaclearlydefined,positionandtissueinvariant lazyzoneinhomeostaticconditions.Ourresultshighlightthatonlytheoctahedralshearstrainpresents thischaracteristic and canthus beconsidered theoptimal choiceforimplementation ofacontinuum levelboneadaptationmodel.Secondly,criticalmodelingparameterssuchaslazyzonebounds,typeof rateequationandboneoverloadingresponseareclassifieddependingontheirinfluenceonthenumeri- calpredictionsofboneadaptation.Guidelinesareproposedtoestablishthedominantmodelparameters basedonexperimentalandsimulateddata.

© 2016IPEM.PublishedbyElsevierLtd.Allrightsreserved.

1. Introduction

The process of bone adaptation to mechanical stimulationsis often modeled at the continuum scale through the Mechanostat [1].Thistheorypostulatesthataspecific mechanicalstimulusoc- curringinbone iskept within a physiologicalrange(i.e. thelazy zone,LZ)throughthevariationofbone mass[2,3]whichinturns affect the distribution of elastic modulus in the bone structure.

Thesephenomenologicaladaptation modelsrepresentatthe con- tinuumscalethe netresultof thelocal boneadaptation process:

thesensornetworkformedbyosteocytesandtheircomplexsignal- ingisdescribedby amechanicalmeasureofthestressandstrain inacontrol volume (the“stimulus”) whilethe activityoftheos- teoblastsandosteoclastsresulting indifferentrates oflocalbone appositionandresorptionarerepresentedbytherateofchangeof

∗ Corresponding author. Fax: + 41 21 693 73 40.

E-mail addresses: marco.piccinini@epfl.ch (M. Piccinini), joel.cugnoni@epfl.ch (J.

Cugnoni), john.botsis@epfl.ch (J. Botsis), Patrick.Ammann@hcuge.ch (P. Ammann), Anselm.Wiskott@unige.ch (A. Wiskott).

bonedensityatthecontinuumscale (the“bone appositionrate”).

Inextremeconditions,whenboneisoverloaded,continuousdam- ageaccumulationsupersedesthecapacityofbonetoadaptandre- pairitself,whichisrepresentedintheadaptationmodelsbyafast reductionofbone stiffnessand/or massabove acertainoverload- ing threshold.Severaladaptation modelshavebeen implemented in order to evaluate the integration of implants, for example in dentistry [4–6] by considering both bone apposition and resorp- tiondueto overloading.Theinterplaybetweenthesephenomena hasbeenseen toregulatethe peri-implantmarginalloss andde- terminethelongtermstabilityofdentalimplants[7,8].

Despite their versatility, these phenomenological approaches relyonmanyassumptionsthataredifficulttoverifybyexperimen- tationandrarelydiscussed[9],suchas:thechoiceofamechanical signal which drives the bone adaptation (‘stimulus’), the relation betweenthebone adaptation rateandthestimulus, thelimitsof boneadaptation andthesizeofthe zoneofstimulusdiffusionin non-localmodels.

Indeed, the first open question concerns the choice of the mechanical variable used as a triggering signal. Investigations http://dx.doi.org/10.1016/j.medengphy.2016.08.008

1350-4533/© 2016 IPEM. Published by Elsevier Ltd. All rights reserved.

studyingsignalsbasedonstrain[1,10],strainenergydensity[2]or stress [11,12] all lead to satisfying results in specific applications whenproperlycalibrated,however,thereisnoclearagreementon whichregulationsignal providesthebestconsistencyina general sense.Moreover, signalselectionandthedefinitionofboneappo- sition andresorptionthresholds are not frequentlydiscussed and comparisonsarerare[5,13].

Themathematicalformoftheadaptationlawwhichrelatesthe levelofmechanicalstimulustothebone appositionorresorption rate is also an open question. In order to preserve the natural structure ofboneunderphysiologicalconditions,continuum level isotropicboneadaptationmodelsmustatleastexhibit aregionof homeostasis by defininga so-calledlazy zone (LZ).The LZrepre- sent anequilibriumconditionatwhichnormalbone turnoveroc- curs, i.e. theresorption ratecontrolled by osteoclastsis equal to theappositionrateduetoosteoblastsactivity.ThelimitsoftheLZ andtheboneoverloadingthresholdarecriticalvariablesthat con- trol the adaptation process by governing the transition between bone resorption,homeostasis , apposition and damage. However, becauseoftheirpotentialdependencyonspecies,locationandbio- variability, those bounds are difficult to determine and are only rarelydefinedonarigorousexperimental basis[14].Furthermore, the dependenceofbone adaptation rateonthe mechanicalstim- ulus has been formulated through linear [15], quadratic [16] or piecewise functionsincluding a ratesaturation [17], butthesen- sitivityoftheobtainedpredictionstothesedifferentmathematical formsremainsunclear.

Sinceboneadaptationisassumedtobedrivenbycellmechan- otransduction[18],severaladaptationmodelsinvolveaspatialav- eragingofthestimulusoverazoneofinfluence(ZOI)[3,19,20]to representdiffusionprocesses.ThesizeoftheZOIaffectstheaccu- racy ofnumerical predictions butthisdependenceis scarcelyin- vestigated.

Furthermore,thepre-implantationbonestructureandgeometry differs significantly amongone group ofindividuals,which limits thevalidity ofpredictionsbasedon anaveragerepresentative ge- ometry.Biovariabilityisexpectedtoinduceasignificantscatterin resultsofboneadaptationandthispointisseldomdiscussed.

This workaims atestablishing guidelinesforthe definitionof thehypothesisneededtoobtainaccuratepredictionsofboneadap- tationaroundimplants.Themodelingparameters,adaptationthe- ories and mechanical stimuli are classified as critical, important ornegligiblewithrespectto theirinfluenceon resultsandmeth- ods are proposed tochoose the values ofthe dominant parame- ters. The ‘loaded implant’ animal model is adopted as a bench- mark [21]. This animal model allows investigating the effects of a controlled external stimulation of the bone tissue surrounding twotranscutaneousimplantsinsertedintheproximalpartofrats’

tibiae [22,23]. The ‘loaded implant’ model waschosen here asit allows aprecisecontrol ofthe loadinghistoryandimplantplace- ment but alsobecause it closelymimics thedifficulties found in clinicalimplantationsinwhichacomplexthreedimensionalstress state withlocalstress concentratorsarecommonlyobserved.Dif- ferentmechanicalstimuliare comparedonthe benchmarkoffull tibiaebeingsubjectedtophysiologicalloading conditions.Assum- ing that the Mechanostat hypothesis is valid, a clear lazy zone should beabletobe observedinthedistributionofproperadap- tationmechanicalstimuliundersuchconditions.Moreover,theLZ of the ideal mechanical stimulus should also satisfy the criteria of location independence, tissue independence and specimenin- dependence.The stimuluswhich bestsatisfiestheseconditionsis identifiedandusedincombinationwithaspecimen-specificadap- tationalgorithmtopredictbone peri-implantadaptation.Asensi- tivitystudysubsequentlyhighlightsthedependenceofboneadap- tationresultsontheLZ,ontheadaptationlaw,ontheZOI,onthe loadlevelandonbiovariability.

Fig. 1. Working principle of the ‘loaded implant’ model: two titanium implants are screwed mono and bi-cortically into the proximal part of the rat tibia (view cut). A controlled stimulation is provided daily by pulling the implants heads together.

2. Materialsandmethods 2.1. Animalmodel

Twotranscutaneous Ti implants were screwedmono- andbi- corticallyintotherighttibiaoffemaleSprague-Dawleyrats(Fig.1) followingtheproceduresdescribedin[22,23].After twoweeksof integration,fiveanimalswereeuthanized.This‘basal’grouprepre- sentedthepre-stimulationintegrationstateofthe‘loadedimplant’

modelandwasusedasabasisforboneadaptationsimulation.The remaining animals (‘stimulated group’) were subjected to a con- trolledexternal loadof 5N,applied on adaily basis to force the implant’sheads togetherandtostimulatethe bonetissuearound them.Theloadwasappliedwiththefollowingschedule:1Hzsi- nusoidalcyclefrom0Nto5N,900cycles/day,5days/week for4 weeks(totalof20daysofstimulation)withaprogressiveincrease ofloadamplitudeduringthefirstweek(+1N/day).Aftersacrifice, alltibiae weredissected,cleared oftheirsofttissue coverageand frozenat −21°C. The specimenswere then thawed out andana- lyzedusinga highresolutionCTimagingsystem(

μ

^CT-40,Scanco

MedicalAG,Brüttisellen,Switzerland,isotropicvoxelsize:20

μ

^m).

The technicalaspects of surgery,implant design,activation setup andCT imagingthat characterize the ‘loaded implant’ modelare describedindetailin[22–24].

2.2.Finiteelementmodels

Continuum-levelspecimen-specific FEmodels ofbare andim- plantedrattibiaeweregeneratedfromCTscansthroughaverified andvalidated procedure [24]. The CTimages were segmented to isolatethe continuum boneandimplantsdomains andprocessed withan open-source FE modelgenerator to quality second order tetrahedralmeshes.Thesemodelsrepresentthewholebonestruc- tureasa continuum, elastic, isotropic andinhomogeneous mate- rial. The local average BMD is calculated ineach element of the modelusingthemeanBMDoftheCTvoxelscontainedintheel- ements.Toavoidpotential checkerboardpatternsandoscillations, theBMD field is then averaged atthe nodesof the meshto ob- tainacontinuousdescription.Subsequently,theintegrationpoints were assigned their material properties by interpolation of the nodalBMDandbyusingthe density-elasticityrelationshipdevel- opedbyCoryetal.[25].Fivewholetibiaewere processedtogen- eratespecimen-specificFE models,which werethen subjected to a gait-based loading conditionthat hadbeen shown to generate asoundphysiologicalpatternofdeformation [26] (Fig.2a). These specimenswereadoptedasthebenchmarkforthecomparisonof mechanicalstimuli.

Fig. 2. (a) Gait-based loading condition applied to FE models of whole tibiae for the comparison of stimuli [26] . Loads: medial and lateral condylar reactions, C mand C l, ankle joint reaction A j, bicep femoris B f, vastus medialis V mand lateralis V l, rectus femoris R f, tibialis anterior proximal T Apand distal T Ad. (b) Loading condition applied to FE models of implanted specimens for iterative computations. The dotted areas are characterized by a small elastic modulus. L is the inter-implant distance.

Fiveimplantedspecimensscannedatthe‘basal’statewerepro- cessed in order to generate the FE models for iterative compu- tationson bone peri-implant adaptation.These specimens repre- sentedthepre-stimulationintegrationstateofthe‘loadedimplant’

model: they underwent two weeks of post-surgery integration whichallowed theprimary stabilitytobeestablished[22],butno externalloading.TheseFEmodelsweresubjectedtotheboundary conditionsshowninFig.2bcorrespondingtothecontrolled load- ingprovidedduringin-vivostimulation.Thebone-implantcontact wasconsidered perfectly adherent. However, in order to prevent unrealistictransferofloadstothe tissuewheretheinterface was subjectedtotraction[26],asmallelasticmoduluswasassignedto anarrowstripofthetensileloadedregionsoftheimplantsincon- tactwithcorticalboneinthedistalandproximaldirections.

2.3.Mechanicalstimuli

Based on the hypothesis of the Mechanostat and Wolff’s law, anappropriatecontinuum‘stimulus’variableshouldexhibitaclear lazy zone in homeostatic conditions i.e. values of the stimulus variablesshouldbe strictlyboundedinhomeostaticconditions.To havean objective boneadaptation model, theLZ ofthe stimulus shouldalsobeindependenton thelocationwithin bone,ifpossi- blealsoindependent onthe tissue type andideallybe ashomo- geneousaspossibleamongapopulation.So,formally,anadequate stimulusvariableshouldrespectthefollowingcriteriaunderphys- iologicalhomeostaticconditions:

• Locationinvariance.Underhomeostaticconditions,differentre- gionsofthesameboneshouldshow thesamerangeofstimu- lus(i.e.thesamewidthofthehistogramofstimulusindiffer- entregions).

• Tissue invariance. Ideally, the ‘stimulus’ variable should be in- dependent on the bone structure. Thus cortical and trabecu- larboneshouldshowthesamestimulusrangeaswell(i.e.the samewidthofthehistogramofstimulusinbothtissues).

• Specimen invariance. The histogram ofthe stimulus should be homogeneouswithinapopulationofspecimens.

If theseconditions are not satisfied,the bone adaptation pro- cesswouldbecomesite-specific[10] meaningthat thebone cells shouldknowa prioritheir position inthe tissueandtheir target equilibriumcondition.

Three isotropic scalar stimuli were compared by highlighting theircompatibilitywiththesecriteria,onthebenchmarkofwhole tibiaesubjectedtophysiologicaldeformations[26].

The first stimulusconsidered was the elastic energy per unit ofmass

ψ

U (Eq.(1))[10,16].Thisenergy-basedstimuluscombines the continuum level strain energy densityU_i that occurs during theloadingconditioni,thelocalapparentbonedensity

ρ

^,andthe

numberofloadingconditionsN.

ψ

U= 1 N

N

i=1

Ui

ρ

⁽¹⁾

As described by Weinans et al. [10],

ψ

U represents the aver- age localstrain energy densityin the trabeculaeover a seriesof loadingcyclesandthusisaglobalmeasureoftheintensityofthe localmechanical environment perceived,directly orindirectly, by theosteocytes.

The second stimulus considered was daily stress

ψ

σ (Eq. (2)) [11,12],formulatedas

ψ

σ=

ρ

^c

ρ

2

·

_N

i=1

n_i

σ

i^m

1/m

(2)

ρ

^{c is the apparent densityof}mineralized bone andm is an em- piricalconstantadoptedtoweighthenumberofcyclesandstress dependingonthephysicalactivity.Thisstimulusdependsonmul- tipleloadcasesN,onthenumberofloadingcyclesn_i andonthe effectivestressatthecontinuumlevel

σ

^{.AsdescribedbyBeaupré}

etal.[11],

ψ

σ representsthedailyaveragestressatthetrabeculae level(hencethefactor(^ρ_ρ^c)^{2)andisindirectlyrelatedtothestrain} energymeasure

ψ

U throughtheelasticmodulusofbone.

Thethird stimulusconsideredwastheoctahedralshear strain.

Frost introduced the hypothesis that the Mechanostat is driven by the peakdaily strainsthat occur in thebone tissue [1].Since shear has been shown to influence tissue differentiation[27,28], thestrain-based stimulus

ψ

ε wasformulated as afunction ofthe octahedralshearstrain

ε

^oct:

ψ

ε=max

( ε

oct,1,

ε

oct,2...,

ε

oct,N

)

^with

ε

^oct = 2

3

( ε

1−

ε

2

)

²+

( ε

2−

ε

3

)

²+

( ε

3−

ε

1

)

^{2... (4)}

Fig. 3. ROIs for the comparison of mechanical stimuli. (a) Positioning of the inter-implant and cylindrical regions of interest with respect to the implants insertion coordinates (ROI IIand ROI CY, respectively). View cuts of the tibia: overall dimensions of (b) ROI IIand (c) ROI CY.

inwhich

ε

idenotestheithprincipalvalueofthecontinuumstrain tensor

ε

^{.In thecaseofa lattice ofstraight trabeculaein tension}

or compression, the stimulus

ψ

ε represents also the local strain experienced inthebone tissue.Thethree signalsareinvestigated through asingleloadingcondition(N=i=1),based onthepeak musculoskeletal loads occurring during gait [26]. The numberof loading cycles characterizing daily stress was fixed to n=1. As a consequence,theconsideredenergy,stress andstrain-based stim- uliareformulatedasshowninEqs.(5)–(7),respectively.

ψ

^U=

ρ

^Ubmd

(5)

ψ

σ=

ρ

bmd,c

ρ

bmd

2

·

σ

⁽⁶⁾

ψ

ε=

ε

^{oct (7)}

where

ρ

bmdand

ρ

bmd,carethelocalandthefullymineralizedBMD (i.e.1.2gHA/cm³).

The stimuli distribution wasinvestigated intwo fixed regions ofinterest(ROI) withrespect tothelocationoftheimplants(Fig.

3a). Theregion ROI_II wasobtainedby a dilatationof+/−0.3mm oftheplanewherebothimplantaxeslie.ROI_IIrepresentedthere- gionwheretheexternalstimulationwasmoreeffective.Theregion ROI_CYwasdefinedasacylindersurroundingtheproximalimplant location.Itprovidedanestimationofstimulusaroundthefloating implant.OveralldimensionscanbeseeninFig.3bandc.

To analyze tissue invariance, the signals were also differenti- ated betweencortex, trabeculartissue andmarrow throughBMD thresholds. In detail, stimuli were classified as cortical if BMD

>0.8gHA/cm^3, andtrabecular if 0.3 < BMD <0.8gHA/cm³ [25]. These BMDthresholds approximatelycorrespondedtoBV/TV val- ues of 0.25 and 0.6 respectively. Results belonging to “marrow”

(BMD <0.3gHA/cm³) were neglected. For each ROI, tissue cate- goryandspecimen,thevolumehistogramoftheinvestigatedstim- ulusvariablewasreconstructedfromtheFEsimulationresults.The meanvalueandSEMofthehistogramsoffivespecimenswerefi- nallycalculatedandcomparedinordertoevaluatetheaforemen- tionedinvariancecriteria.

2.4. Boneadaptationalgorithm

Thesimulationsofboneadaptationwerebasedonthehypoth- esisthat themechanicalstimulustendstowardsa constantrange

ofvalues,correspondingtotheLZ[1,11],byreducingorincreasing thebonedensity.Thedensityvariationwascomputedthroughthe formulationdescribedinEq.(8),inspiredbyLietal.[16]:

d

ρ

bmd

dt =

0 i f

ψ

≤

ψ

a

Ka

( ψ

−

ψ

a

)

¹⁻_ψ^ψ_d⁻₋^ψ_ψ^aa

i f

ψ

>

ψ

a (8)

where

ψ

^{isthechosen mechanicalstimulus, Ka istheadaptation}

rate,and

ψ

d and

ψ

a arethedamageandapposition LZlimits, re- spectively(Fig.4a). Thisformulaimpliesthat implantloadinghas noeffectifthestimulusislow(d

ρ

bmd/dt=0if

ψ

<

ψ

^a),apositive

effectofthestimulationifthestimulusisintheappositionrange (d

ρ

bmd/dt>0if

ψ

^a<

ψ

<

ψ

d) andresorptiondueto overloading occursifthestimulusovercomes thedamage limit (d

ρ

bmd/dt<0 if

ψ

>

ψ

d).Disusewasnotconsideredasphysiologicalloadswere sufficienttomaintainthebone structure[16,17].Thisformulation impliesaparabolic evolutionofthe apposition ratewithamaxi- mumatastimulus

ψ

=¹₂(

ψ

^a+

ψ

d)^followedbyaprogressivere- ductionrepresentingsubcriticaldamageaccumulationinthebone upto

ψ

=

ψ

d wherebone damageaccumulation cannotbe com- pensatedanymore.

Thecalculation oftheeffectivestimulusincludeda spatialav- eragingofthemechanicalvariablesoverasphericalzoneofinflu- ence(ZOI)[3]whichrepresentsthediffusiondistanceassociatedto mechanotransduction signaling.In detail,the signal ateach node wascalculatedthroughEq.(9)

ψ

i=

ψ

i+_Z

j=1f Dji

ψ

j

1+Z

j=1f D_ji

⁽⁹⁾

ZisthenumberofnodesincludedinthedefinedZOIandf(^Dji)^is afunction that weighsthesignal contributionofthe nodej with respecttoitsdistanceD_jifromthecurrentnodei.

Eq.(8)wassolvediterativelythroughforwardEulerintegration inorderto updatebone densityinrelationto tissuedeformation until convergence. A controlled BMD variation was calculated at eachiterationwithanadaptivetimesteptoguaranteethestability ofthesimulations[29].

2.5.Sensitivityanalyses

Fivespecimen-specificFEmodelsofimplantedtibiaewerepro- cessed withdifferent sets ofparameters to perform a sensitivity studyonthenumericalpredictionsofboneadaptation.Duetothe largenumberofparameters,thestudywasstructuredintoseveral

Fig. 4. Adaptation rate d ρ/d tversus mechanical stimulus ψ(a) quadratic form (b) linear form and (c) piecewise form with plateau.

Table 1

Details of sensitivity studies.

Parameter Nominal Levels # simulations

Adaptation thresholds (·10 ⁻²) ψa 1.25 1; 1.25; 1.5; 1.75 80 ψd 4.51 4.1; 4.51; 1.5; 5.5

ZOI Radius (mm) 0.3 0; 0.15; 0.3; 0.6; 0.9 65

Type Gaussian Linear, Gaussian, exponential

Law formulation Quadratic Linear, quadratic, plateau 15

External load (N) 5 3.3; 5; 6.7; 8.3; 10 25

separateparametric studies,whicharepresentedinthefollowing sectionsandsummarizedinTable1.

2.5.1. Nominalparameters

Based on the invariance analysis of the mechanical variables (Section3.1),theoctahedralstrainstimuluswaschosenastheref- erencesignal forthe sensitivityanalysis.Thecorrelation between mechanicalstimulusandboneappositionratewasprovidedbyEq.

(8), and the value of the apposition threshold,

ψ

^a=1.25×10⁻³, wasfixedinagreementwiththephysiologicaldeformationsoccur- ringinrattibiaeduringgait[26].Thevalueofthedamagethresh- old

ψ

d=4.51×10⁻³ wasfixed in agreementwith the longitudi- nalstrainlimitof4×10⁻³alreadyadoptedforastudyconcerning bonedamage[15].Asthesteadystatesolutionwasonlyofinterest inthepresentstudy,theadaptationrateK_awasgiventhevalueof 1(gHA/cm3)/(timeunit)similarlytoLietal.[16].ThenominalZOI radius waschosen corresponding to the limit of validity of con- tinuumdescription(‘LVC’)oftrabecular boneasdescribedin[30], corresponding to 3–5inter-trabecular lengthswhich is estimated at0.3mm in the present case. The decayof the signal with in- creasingdistancefromthecentralnodeoftheZOIwascomputed througha Gaussian function(Eq.(10)),which istypical formany diffusion-likenaturalprocesses[19].

f D_ji

=e⁻

Dji/r

2β² (10)

with

β

=0.4085.Thesesettingsareconsideredasthereferencefor thesensitivitystudy.

2.5.2. Sensitivitytothelazyzonethresholds

The ranges of octahedral shear strain thresholds

ψ

^{a and}

ψ

d

were discretized in four intervalsbetween 1 × 10⁻³ and 1.75× 10⁻³,and4.1×10⁻³and5.5×10⁻³,respectively.Theboneadap- tationsolutionswerecomputedforeachspecimenandeachpairof adaptationthresholds,whilekeepingallother parametersattheir nominalvaluesthereforeleadingto80simulations.

2.5.3. Sensitivitytotheadaptationlawformulation

Three mathematicalformulasoftheadaptationlawwerecom- pared:thequadraticforminEq.(8),alinearformulation[15]and apiecewiselawwithaplateau[17](Eq.(11) andFig.4b,Eq.(12) andFig.4c,respectively),foratotalof15simulations.

Table 2

Parameters of the adaptation laws.

Law Eqs. Adaptation thresholds (·10 ⁻²) Rate constant

ψa ψa ψa ψd K a K a’ K d

Quadratic ( 2 ) 1.25 – – 4.51 1 – –

Linear ( 5 ) 1.25 – – 4.51 1 – 1

Plateau ( 6 ) 1.25 1.5 1.75 4.51 0.05 1 1

d

ρ

dt =

⎧ ⎨

⎩

0 i f

ψ

^≤

ψ

^a

Ka

( ψ

⁻

ψ

^a

)

^{i f}

ψ

^a<

ψ

^≤

ψ

d

K_d

( ψ

⁻

ψ

d

)

^{i f}

ψ

^>

ψ

d

(11)

d

ρ

dt =

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎩

0 i f

ψ

≤

ψ

^a

Ka

( ψ

−

ψ

a

)

^{i f}

ψ

a<

ψ

≤

ψ

a

Ka

ψ

a−

ψ

^a

+Ka

ψ

^a−

ψ

a

i f

ψ

a <

ψ

^≤

ψ

a

Ka

ψ

a−

ψ

a

+Ka

ψ

a−

ψ

a

i f

ψ

a <

ψ

≤

ψ

d

K_d

( ψ

⁻

ψ

d

)

^{i f}

ψ

d<

ψ

(12) Theadoptedparameterswerecalculatedtopreserve thenomi- naladaptationthresholdsandarepresentedinTable2.Compared to the quadratic form in Eq. (8) which showsa peak apposition rateat

ψ

=2.88%,thelinearadaptationmodelshowsamaximum appositionrateatthebonedamagethreshold

ψ

d=4.51%followed byanabrupttransitiontodamagedrivenboneresorption.Incon- trast,thepiecewisemodelinEq.(12)presentsawiderangeofcon- stantappositionratewhichreachesitsmaximumrate(saturation) muchearlier(

ψ

a=1.75%)thantheothertwomodels.

2.5.4. Sensitivitytothezoneofinfluence

TheZOIradiusrwasvaried from0to0.9mm,andthreetypes of weight functions were considered: Gaussian, linear andexpo- nential(Eqs.(10),(13)and(14)),foratotalof65simulations.

f D_{i j}

=0.95

1−D_{i j} r

+0.05 (13)

f D_{i j}

=e^−2.99

Di j

r (14)

Fig. 5. Regions of interests (ROIs) where BMD is monitored.

Toachievecomparableeffects,theconstantsusedinthediffer- ent forms of f(^Di j) ^{were chosen to achieve f}(^Di j)=1 for D_ij=0 and f(^Di j)=0.05forD_ij=r.Thecontributionofnodesoutsidethe ZOI(D_ij>r)wasneglected.

2.5.5. Sensitivitytoloadlevel

The specimen-specificFE models were simulated withdefault parameters at five applied load levels ranging from 3.3 to 10N, thereforegiving25simulations.

2.5.6. Convergencecriteriaandselectedoutputvariables

Simulations were stopped when 99.9% of nodes showed null adaptation errors(signal within lazy zone).Local BMD variations wereassessedinsixROIs(Fig.5).Thelongitudinalstabilityofim- plantswasmonitoredbyestimatingthevariationofinter-implant straindefinedasd/L,wheredandLaretherelativeinter-implant headsdisplacementandinter-implantdistance,respectively.

Fig. 6. Distribution of signals (rows) with respect to the tissue type (columns) and the regions of interest ROI II and ROI CY(legend). In detail: energy-based signal ψU in cortical (a) and trabecular (b) tissue, stress-based signal ψσin cortical (c) and trabecular (d) tissue, and strain-based signal ψεin cortical (e) and trabecular (f ) tissue.

Fig. 7. Sensitivity of (a) inter-implant strain and BMD in (b) ROI1 and (c) ROI3 with respect to the perturbation of the apposition and damage thresholds ( ψaand ψd). The mean values (colored surface) and the SEM (upper and lower grids) of the group of five specimens adopted as benchmark are represented.

3. Results

3.1.Comparisonofstimuli

The histogram distributions of thesignals belongingto differ- entROIsandtissuesare showninFig.6.Eachdistributionisrep- resented by the mean ± SEM calculated on the five specimen- specificFEmodelsofwholetibiaesubjectedtothegait-basedload- ing condition [26]. Considering the energy-based stimulus

ψ

U, a large amountof cortical bone in ROI_II shows levels of strain en- ergysignificantly larger to theones measured inROI_CY (∼40% of thetotal,blackarrowsinFig.6a).Overall,thedistributionsinboth ROIsare significantly different. For the trabecular bone, the dis- tributions of

ψ

U are similar between ROIs (i.e. nearly superim- posedhistograms, Fig. 6b). However, the rangecovered by

ψ

Uin trabecular bone (from 0 and 2×10⁻³J/g) and cortical bone (be- tween 0 and 8×10⁻³J/g) are very different. Thus, the energy- based signal does not satisfy both location andtissue-invariance criteria.

The stress based stimulus

ψ

σ shows location-invariant distri- butions, as shownby the negligible differences between ROIs in

Fig. 6cand d,respectively. Nevertheless, the distributions incor- tical and trabecular bone are not comparable either in terms of shapeorintermsofrange.Asaconsequence,eventhoughitsat- isfies thelocation invariance criterion within atissue, the stress- basedsignaldoesnotsatisfythetissue-invariancecriterion.

Finally, the distributions of the strain based stimulus

ψ

ε are showninFig.6eandf.Theshapeofthedistributions isfoundto be similarin bothROIsandtissues, andtheboundsofthe strain basedstimulus

ψ

ε remainshomogeneous(between200and1500

με

)^{. These results therefore} demonstrate that the strain based stimulus

ψ

ε respects both the tissue and the location-invariance criteria under physiological conditions (i.e. during gait), and for thisreasonthisstimulusischosenasreferenceforthesensitivity studyofboneadaptation.

Asauniquerangeofphysiologicalstrain stimuluscanbeiden- tified,thehistogramscan beusedtoset theappositionthreshold of the adaptation model in physiological conditions. Indeed, the reference value of the apposition threshold,

ψ

^a=1.25×10⁻³, is quantifiedasthe95thpercentileoftheoctahedralshearstraindis- tributioncharacterizingbothtissuesandROIsduringphysiological activity.

Fig. 8. Sensitivity of (a) inter-implant strain and (b) BMD in ROIs with respect to the law formulation. Mean values and SEM are represented.

Fig. 9. Sensitivity of (a) inter-implant strain (mean values and SEM) and (b) BMD in ROIs with respect to the ZOI radius and weight function (i.e. L = linear, E = expo- nential, G = Gaussian).

3.2. Effectofadaptationthresholds

The results of this parametric study are shownin Fig. 7. The evolution of the inter-implant strain (inversely proportional to stiffness)variesbetween0and−8%withboth

ψ

^aand

ψ

d(Fig.7a).

The minimum andmaximum increase ofstiffnesscoincides with the widerandnarrower amplitudeoftheapposition zone.More- over, the effectof

ψ

d reachesa plateau after5 × 10⁻³,meaning

thatthedamagecontrolledadaptationisnolongeractivatedabove thispoint.At 5Nload,thevalue ofthedamagethreshold

ψ

d has alimitedinfluenceon inter-implantstrain,whichremains mostly controlled by the apposition threshold

ψ

^{a. The BMD variation in}

ROI1 andROI3(Fig.10b andc)shows an evenclearertrend: the averageBMD variation indifferent ROIsis nearly independent to

ψ

d,whereasitisreducedwiththeincreaseof

ψ

^a.

3.3.Effectofadaptationlawformulation

Theeffectofadaptationlawformulationoninter-implantstrain variationandBMD incrementinROIsare reportedinFig. 8aand b,respectively.Interestingly,theresultsofthequadraticandlinear formulationsshowclearagreement,withthelatterslightlyunder- estimatingtheinter-implantstrainanddensityincrementswithre- specttotheformer.Themathematicalformincludingaplateauin- volvesasignificantoverestimationofbothoutputs.Thishighlights the fact that even though all three adaptation laws havesimilar thresholds, the shape of the adaptation law has an influence on theresult.

3.4.Effectofthezoneofinfluence

AsshowninFig.9a,iftheZOIradiusislowerthanthelimitof validity of the continuum assumption (‘LVC’, 3–5 inter-trabecular lengths)theinter-implantstrainincreasesasthemodelspredicta greater resorptiondue tooverloading inregions close tothe im- plant.The extremecaseofnoZOI(r=0)provokesa 17%increase ininter-implantstrain,whichistheoppositeofwhatisfoundwith thenominalZOIof0.3mm(−5.5%).ForradiiabovetheLVC,there is nearly no variation of the inter-implant strain, thus indicating that once the signal is averaged on a consistent volume ofbone (i.e.compatible withthe continuum hypothesis),the solution re- mainsstable.TheBMDvariationshowninFig.9bshowsamoder- ateincreaseindensitywithanincreasingZOIradiusinROI 1and 2bothsituatedintheinter-implantplane,andsubjectedtocom- pression. Interestingly,the effectsof thedifferent decayformula- tionsarefoundtobenegligible.

3.5.Effectoftheloadlevel

InFig. 10, the inter-implantstrain variation after convergence is plotted against the applied load for all five specimens (S1 to S5).A3.3Nloadprovokesonlyaweakinter-implantstrainreduc- tion(−3%),whichisthennearlydoubledbyimposing5N(−5.5%).

Withhigherloads,thesystemstabilitydecreasessignificantlyand the effects of biovariability are strongly amplified. By applying

Fig. 10. Inter-implant strain sensitivity to the external load magnitude. Mean values and SEM are represented.

6.7N,theaverageleveloftheinter-implantstrainbecomespositive (+3.3%, i.e.inter-implantstiffnessdecreases)withalarger spread withrespecttolowerloadlevels.At8.6Ntheimbalancebetween resorption andapposition reachescritical levels for3 out ofthe 5 specimens, characterized by the failure of all the tissue sur- roundingthedistalimplant(Fig.11).Nevertheless,twospecimens didreachastableconvergedsolutionwithincreasedinter-implant strain(+20%).Finally,noneoftheFEmodels adaptedto10N:re- sorptionduetooverloadingdominatestheadaptation mechanism provokingthe instabilityofthe distalimplant.Inter-implantBMD fieldsestimatedwithallloadlevelsareshowninFig.11.

ThelocalBMDvariationsintheROIsshowasimilartrend(Fig.

12) andexplain the variationsinimplant stability.Indeed,atlow force,increasedBMDinthecompressiveregionsbetweenimplants iscorrelatedtothereductionofinter-implantstrain.However,the averageBMD in the ROIs continueto increase with higherloads (i.e.6.3 and8.7N) butregions ofsignificant bone resorption de- velop(Fig.11),leadingtoaprogressivereductionoftheimplant’s stability.Thus, itisworth noticingthatthe implantsstabilityand BMDvariationintheselectedROIsarenotcorrelatedathighloads.

4. Discussion

ThispaperfocusesonthereliabilityofMechanostat-basedpre- dictionsof bone adaptation estimated through specimen-specific, continuum FE models. The adopted modeling strategy relies on nominalparameters which are consistent withthe physiology of the ‘loaded implant’ model and with the problem length-scale.

The predictions obtained with these parameters were validated throughcomparisonwithin-vivoexperimentsandareusedasref- erence[31].Since thesephenomenological predictionsarethe re- sult of several assumptions, it is of great interest to clarify the methodswhichhavebeenused todefinethekey parametersand highlighttheirinfluenceontheresults.

Three mechanical variableswere compared on thebenchmark of full tibiae subjected to physiological loading conditions. The analysisofthestimulidistribution(Fig.6)withrespecttoposition, tissueandspecimeninvariancecriteriahighlightsthat,amongthe threestimulusconsidered,theoctahedral shearstrainisthemost appropriate stimulus for the implementation ofthe Mechanostat theory at the continuum scale. When physiological gait-based loadsareapplied, thissignalis theonlyone that showslocation, tissue andspecimen-independentdistributions asimplied by the Mechanostat and Wolff’s law. This result confirms the existence ofa unique range of selected stimulusvariable corresponding to physiological homeostatic conditions (i.e. the lazy zone), which drives bone macroscopic structural adaptation to mechanical

stimulations.However, not all mechanicalstimulusvariableshow thisproperty.

An exhaustive sensitivity study highlighted the robustness of the adopted strategy by clarifying the influence of biovariability, bone adaptation thresholds, adaptation law formulation,ZOI and external load on numerical predictions. The effects of all these variableswereevaluatedbyperformingmultiplespecimen-specific iterative computations that generated results dependent on the featuresofeachindividual.

The defaultboundsofthelazy zone (i.e.apposition anddam- agethresholds

ψ

^{a and}

ψ

d) weredefinedbyanalyzingphysiologi- caldeformationsoccurringduringrats’gaitandreferenceliterature data.Nevertheless,thedependencyofresultson theperturbation oftheseparametersisofgreatinterestconsidering thattheequi- libriumbetweenthe peri-implantbone apposition andtheapical resorptionfromoverloadingisakeyfactorinimplantstability.The results show that the perturbation of the adaptation thresholds withinconsistentrangesofstraindoesnotaffecttherobustnessof theinvestigated adaptationprocess (e.g.no worseningoftheim- plantlateral stabilityispredicted, Fig. 7). However, theseparam- eters clearlyaffectthe predictionof bothBMD andinter-implant strain variations.Thisthereforehighlights theneedforarigorous wayofdeterminingtheappositionthreshold

ψ

^ainparticular.The determinationofthisparameterthrough anhistogramanalysisof theoctahedralstrain stimulusinphysiologicalconditions,asused in this study, is highlyrecommended in order to obtain reliable predictions.Anotherpossibility,howevermorecomplexandinva- sive,is obviouslyin-situstrain measurements underphysiological conditions.

The mathematical formulation of the density adaptation rate dependence on the stimulus also has a strong influence on the results. The quadratic formadopted as reference(Eq.(8)) allows describingbothappositionandresorptionduetooverloadingwith fewparameters. Thismodelalsopresentsasmooth transitionbe- tweenappositionanddamage-drivenresorption.Nevertheless,this phenomenological formulation may not be representative of the actual correlation between bone mass variation and mechanical stimulus. In this work, the results obtained with the quadratic lawarecomparedwiththeonesobtainedwitha linearformand a piecewise formulation which includes a plateau (Eqs. (11) and (12), respectively). The resultsshown in Fig.8 highlight that the quadratic and linear formulations can be considered equivalent.

On thecontrary, the adoption of a formulationwhich includes a largeplateausignificantlyaffectstheresults,andleadtounrealis- ticpredictions when compared to experimental observations [31, 32].Thus,thistypeofformulationshouldonlybeimplementedif thesaturationofdensityrateissupportedby asoundexperimen- talvalidation.

Fig. 11. BMD field variation with respect to the external load magnitude. Implants are hidden.

The stimulusaverageon a ZOIintroduces the interesting con- cept of transmission of a local mechanical signal all around the stimulatedarea,viabiologicalprocessesinvolvingthetissuemicro and cellular-structures.Moreover, the filtered stimulusis a mea- sureofthemechanicalstateinastatisticallyrepresentativevolume ofbone.Itisthusofgreatinteresttoinvestigatethesensitivityof thenumericalpredictionstodifferentZOIdimensionsandstimulus decayfunctions(Eqs.(10),(13)and(14)).Theresultsofthisstudy highlightsthatthedefinitionofaZOIisessentialtopredictconsis- tentresults(Fig.9). ThekeyfactoristheradiusoftheZOI,which should atleastcorrespondtotheLVC size(3–5 trabeculae)while theweightfunctiononlyplaysasecondaryrole.

Themagnitudeoftheexternalloadregulates mechanicalstim- ulationtransferredfromtheimplanttothesurroundingtissue,and a study of the results’ sensitivity to different load levels under- linesthestimulationlimitsabove whichharmfuleffectsaredom- inant.Indeed,theresultsareverysensitivetotheloadmagnitude (Fig. 10) and evolve in a complex non-linear manner. The range offorcesgenerating a positiveeffect onintegration (i.e.augmen- tationof peri-implant density and improvedimplant stability) is foundtobe relativelynarrow forthe presentexperiment. Indeed, with3.3Nbonereactionisquitelimitedwhileat5Ntheoptimum isalreadyreached. Higher loads provoke largerincreasesof BMD associatedwithdangerousperi-implantbonelossduetooverload- ing(Fig.11).Interestingly,theresultsspreadincreasessignificantly withload dueto the difference between specimens.As a conse- quence, it can be postulated that the maximum forces to which each individual can adapt dependson specimen-specific features suchastheirinitialbonestructure,andthereforetheresultsofthis sensitivitystudyhighlight the need to adopta specimenspecific approachtostudycriticaloverloading.

Moreover,theperformedparametric studiesallowestablishing arankingofthemodelingparametersbasedontheireffectsonthe inter-implantstrain andmaximumBMDvariation (Fig. 13).Three categories are identified and can be adopted to optimizesimilar approaches:

• Critical parameters: the load overestimation and the absence of ZOI.They provoke morethan 100% variation ofboth inter- implant strain andBMD in ROIs.If these parameters are not controlled or not implemented,the numerical predictionscan becometotallyinconsistent.TheZOIshouldbesetatleastequal to 3–5inter-trabecularspacingtoavoidspurious meshdepen- denteffects.Ifcriticalloadsareofinterest,resultsextrapolated frompopulations oranaveragespecimencan beimprecise.In thiscaseaspecimenspecificapproachisrecommended.

• Important parameters: theadaptationthresholds,thepiecewise law formulationwithaplateau,the loadunderestimation and the ZOI radius overestimation. An ambiguous implementation of theseparametersdoesnot leadto unstablesolutions, how- ever the results are indeed inaccurate (between 10 and 100%

error).Atleast,theboneadaptationthresholdshouldbeidenti- fiedfromphysiologicalconditionsandlinearorquadraticadap- tationlawbeimplementedasafirstoption.

• Negligibleparameters:thetypeofformulation(quadraticorlin- ear adaptationlaw)andthetypeofdecayfunctionintheZOI.

Theseperturbations scarcelyaffectthe predictions,andcanbe chosenarbitrarily.

Inconclusion,amongthedifferentmodelinghypothesisinvesti- gatedheretopredict peri-implantboneadaptation atthecontin- uum level,a modelformulation basedon octahedral shearstrain stimulus, a linear or quadratic evolution law anda ZOI equal to 3–5inter-trabecularspacingishighlyrecommendedbasedonthe theoreticalconsistency considerationsdescribedinthiswork.The useofanhistogramanalysisofthestimuliinphysiologicalloading conditionsisalsoshowntoprovideasoundbasisforthedetermi- nationoftheLZboundsandfortheselectionofrobustadaptation stimulusvariable.

However,it isimportanttoremindthat thisworkisbased on aphenomenologicaldescriptionoftheboneadaptationatthecon- tinuumlengthscalewhichisatleasttwotothreehierarchicallev- elsabove thecellular levels.The aboverecommendationsconclu- sionsshouldthus notbe consideredashintsonhowbone physi- ologyandmechano-transductionactually worksatasmallerscale wheretheactualbiophysicalprocessestakeplace.

Finally,it should be notedthat theproposed bone adaptation method has been successfully applied and compared to experi- mentaldata,asdescribedintheworksbyPiccininietal.[31]and

Fig. 12. Density in ROIs sensitivity to the external load magnitude. Mean values and SEM are represented.

Fig. 13. Ranking of the perturbations analyzed by sensitivity studies. The variations are computed with respect to the results obtained with nominal settings ( Table 1 ).

Piccinini[32]. The proposed model hasshown consistent predic- tion with respect to the experimental observations in terms of BMD evolution and inter-implant strain evolution and was able tocaptureatleastpartiallythespecimen-specificresponsewithin a group. However, the physiological relevance of the above rec- ommendationsstillneed tobeestablished forother experimental conditions.

FundingandEthicalApprovement

ThisworkwaspartiallyfundedbySwissNationalScienceFoun- dations (SNF) project 315230-127612 and internal funding from EPFL and UNIGE. All animal experiments were approved by the University ofGenevaanimal rights committeeand supervisedby thelocalveterinaryboardinaccordancetoSwissregulations.

Conflictofinterest Noconflictofinterest.

Acknowledgments

Mrs. Severine ClementandMrs. IsabelleBadoudare gratefully acknowledgedfortheirwork onexperiments.Thisworkwaspar- tiallyfunded bySwissNationalScienceFoundations(SNF) project 315230-127612.

References

[1] Frost HM . Bone ‘mass’ and the ‘mechanostat’: a proposal. Anat Rec 1987;219:1–9 .

[2] Huiskes R , Weinans H , Grootenboer HJ , Dalstra M , Fudala B , Slooff TJ . Adap- tive bone-remodeling theory applied to prosthetic-design analysis. J Biomech 1987;20:1135–50 .

[3] Mullender MG , Huiskes R . Proposal for the regulatory mechanism of Wolff’s law. J Orthop Res 1995;13:503–12 .

[4] Chou HY , Jagodnik JJ , Müftü S . Predictions of bone remodeling around dental implant systems. J Biomech 2008;41:1365–73 .

[5] Mellal A , Wiskott HWA , Botsis J , Scherrer SS , Belser UC . Stimulating effect of implant loading on surrounding bone. Comparison of three numerical models and validation by in vivo data. Clin Oral Implant Res 2004;15:239–48 . [6] Reina JM , García-Aznar JM , Domínguez J , Doblaré M . Numerical estimation

of bone density and elastic constants distribution in a human mandible. J Biomech 2007;40:828–36 .

[7] Hoshaw SJ , Brunski JB , Cochran GVB , Higuchi KW . Theories of Bone Modeling and Remodeling in Response to Mechanical Usage: Experimental Investigation of an In Vivo Bone-Implant Interface. In: Goldstein SA, editor. 1990 Advances in Bioengineering. ASME, NY; 1990. p. 391–4 .

[8] Qian J , Wennerberg A , Albrektsson T . Reasons for marginal bone loss around oral implants. Clin Implant Dent Relat Res 2012;14:792–807 .

[9] Cox LGE , Van Rietbergen B , Van Donkelaar CC , Ito K . Analysis of bone architecture sensitivity for changes in mechanical loading, cellular activ- ity, mechanotransduction, and tissue properties. Biomech Model Mechanobiol 2011;10:701–12 .

[10] Weinans H , Huiskes R , Grootenboer HJ . The behavior of adaptive bone-remod- eling simulation models. J Biomech 1992;25:1425–41 .

[11] Beaupré GS , Orr TE , Carter DR . An approach for time-dependent bone model- ing and remodeling - theoretical development. J Orthop Res 1990;8:651–61 . [12] Carter DR , Orr TE , Fyhrie DP . Relationships between loading history and

femoral cancellous bone architecture. J Biomech 1989;22:231–44 .

[13] Terrier A , Rakotomanana RL , Ramaniraka AN , Leyvraz PF . Adaptation models of anisotropic bone. Comput Methods Biomech Biomed Eng. 1997;1:47–59 . [14] Terrier A , Miyagaki J , Fujie H , Hayashi K , Rakotomanana L . Delay of intracortical

bone remodelling following a stress change: a theoretical and experimental study. Clin Biomech 20 05;20:998–10 06 .

[15] McNamara LM , Prendergast PJ . Bone remodelling algorithms incorporating both strain and microdamage stimuli. J Biomech 2007;40:1381–91 .

[16] Li J , Li H , Shi L , Fok ASL , Ucer C , Devlin H , et al. A mathematical model for sim- ulating the bone remodeling process under mechanical stimulus. Dent Mater 2007;23:1073–8 .

[17] Crupi V , Guglielmino E , La Rosa G , Vander Sloten J , Van Oosterwyck H . Numer- ical analysis of bone adaptation around an oral implant due to overload stress.

Proc Inst Mech Eng Part H J Eng Med 2004;218:407–15 .

[18] Turner CH , Forwood MR , Otter MW . Mechanotransduction in bone: do bone cells act as sensors of fluid flow? FASEB J 1994;8:875–8 .

[19] Schulte FA , Zwahlen A , Lambers FM , Kuhn G , Ruffoni D , Betts D , et al. Strain-adaptive in silico modeling of bone adaptation - A computer simulation validated by in vivo micro-computed tomography data. Bone 2013;52:485–92 .

[20] Kumar NC , Jasiuk I , Dantzig J . Dissipation energy as a stimulus for cortical bone adaptation. J Mech Mater Struct 2011;6:303–19 .

[21] Wiskott HWA , Cugnoni J , Scherrer SS , Ammann P , Botsis J , Belser UC . Bone reactions to controlled loading of endosseous implants: a pilot study. Clin Oral Implant Res 2008;19:1093–102 .

[22] Wiskott HWA , Bonhote P , Cugnoni J , Durual S , Zacchetti G , Botsis J , et al. Im- plementation of the “loaded implant” model in the rat using a miniaturized setup - description of the method and first results. Clin Oral Implant Res 2011;23(12):1352–9 .

[23] Zacchetti G , Wiskott A , Cugnoni J , Botsis J , Ammann P . External mechanical microstimuli modulate the osseointegration of titanium implants in rat tibiae.

BioMed Res Int 2013:1–9 Article id: 234093 .

[24] Piccinini M , Cugnoni J , Botsis J , Zacchetti G , Ammann P , Wiskott A. Factors af- fecting subject-specific finite element models of implant-fitted rat bone spec- imens: Critical analysis of a technical protocol. Comput Methods Biomech Biomed Eng 2012;17(13):1403–17 .

[25] Cory E , Nazarian A , Entezari V , Vartanians V , Müller R , Snyder BD . Com- pressive axial mechanical properties of rat bone as functions of bone vol- ume fraction, apparent density and micro-ct based mineral density. J Biomech 2010;43:953–60 .

[26] Piccinini M , Cugnoni J , Botsis J , Ammann P , Wiskott A . Influence of gait loads on implant integration in rat tibiae: experimental and numerical analysis. J Biomech 2014;47:3255–63 .

[27] Lacroix D , Prendergast PJ . A mechano-regulation model for tissue differen- tiation during fracture healing: analysis of gap size and loading. J Biomech 2002;35:1163–71 .

[28] Prendergast PJ , Huiskes R , Søballe K . Biophysical stimuli on cells during tissue differentiation at implant interfaces. J Biomech 1997;30:539–48 .

[29] Van Rietbergen B , Huiskes R , Weinans H , Sumner DR , Turner TM , Galante JO . The mechanism of bone remodeling and resorption around press-fitted tha stems. J Biomech 1993;26:369–82 .

[30] Harrigan TPJM , Mann RW , Harris WH . Limitations of the continuum assump- tion in cancellous bone. J Biomech 1988;21:269–75 .

[31] Piccinini M, Cugnoni J, Botsis J, Ammann P, Wiskott A. Peri-implant bone adap- tations to overloading in rat tibiae: experimental investigations and numerical predictions. Clin Oral Implant Res 2016. doi: 10.1111/clr.12760 .

[32] Piccinini M. Prediction of peri-implant bone adaptation to mechanical environ- ments - theoretical developments and experimental validation, Ph.D. thesis, DOI: 10.5075/epfl-thesis-6273, URL: http://infoscience.epfl.ch/record/200508:

EPFL; 2014.