Numerical prediction of peri-implant bone adaptation: comparison of

mechanical stimuli and sensitivity to modeling parameters.

Marco Piccinini1, Joel Cugnoni1*, John Botsis1, Patrick Ammann2 and Anselm

Wiskott3

1 Laboratory of applied mechanics and reliability analysis, École Polytechnique Fédérale de

Lausanne, Lausanne, Switzerland.

2 Division of Bone Diseases, Department of Internal Medicine Specialities, Geneva University

Hospitals and Faculty of Medicine, Geneva, Switzerland.

3 Division of fixed prosthodontics and biomaterials, School of dental medicine, University of

Geneva, Geneva, Switzerland.

* Corresponding author: Dr. Joël Cugnoni.

email: joel.cugnoni@epfl.ch

Address: Laboratory of Applied Mechanics and Reliability Analysis (LMAF)

Ecole Polytechnique Fédérale de Lausanne

Station 9, CH-1015 Lausanne

Switzerland

Phone: +41 21 693 59 73

Fax: +41 21 693 73 40

Authors’ email addresses:

Dr. Marco PiccininiProf. John BotsisProf. Patrick AmmannProf. Anselm Wiskott

marco.piccinini@epfl.chjohn.botsis@epfl.chPatrick.Ammann@hcuge.chAnselm.Wiskott@unige.ch

1

Numerical prediction of peri-implant bone adaptation: comparison of

mechanical stimuli and sensitivity to modeling parameters.

Abstract. Long term durability of osseointegrated implants depends on bone adaptation to

stresses and strains occurring in proximity of the prosthesis. Mechanical overloading, as well

as disuse, may reduce the stability of the implants by provoking bone resorption. On the

contraryHowever, an appropriate mechanical environment can improve the integration.

Several research studies have focused on the definition of numerical methods to predict bone

peri-implant adaptation to the mechanical environment. These adaptation models are based on

several different hypotheses, and differ in the type of mechanical variable adopted as

stimulus, in the relationship between bone adaptation rate and the stimulus, and also in

theconcerning the definition of the effective bone volume of bone considered to evaluate the

mechanotransduction stimuli. These different assumptions, combined with biovariability,

clearly affects and limits the accuracy of numerical predictions. This current work addresses

these themes in two steps. Firstly, the most suitable stimulus for the implementation of bone

adaptation theories at continuum scale is highlighted by comparing different mechanical

variables on the benchmark of whole rat tibiae being subjected to physiological deformation.

Secondly, critical modeling parameters are classified depending on their influence on the

numerical predictions of bone adaptation. The rResults highlight that the octahedral shear

strain appears to beas the most appropriate candidate for the implementation of Mechanostat-

inspired adaptation models at the continuum scale. Moreover, the sensitivity study allows the

establishment of ing a classification of parameters that which must be determined precisely

determined in order to obtain reliable numerical predictions.

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Keywords: bone, implants integration, overloading, specimen-specific, finite element, gait,

mechanical stimulus.

1. Introduction

The process of bone adaptation to mechanical stimulations is often modeled at the continuum

scale through the phenomenological approach of the Mechanostat [1]. This theory relies on

the assumption that a specific mechanical stimulus occurring in bone is kept within a

physiological range (i.e. the lazy zone, LZ) through the variation of the bone mass [2, 3].

Several adaptation models based on the Mechanostat have been implemented in order to

evaluate the integration of implants, for example in dentistry [4-6] in considering both bone

apposition and resorption because due to of overloading. The interplay between these

phenomena has been observed seen to regulate the peri-implant marginal loss and determine

the long term stability of dental implants [7, 8].

Despite their versatility, these phenomenological approaches rely on strong assumptions, and

the robustness of their predictions is rarely verified experimentally via experimentation [9].

Indeed, A there is an important open question concernings the mechanical variable chosen as

a triggering signal. Several investigations considering studying regulation signals based on

strain [1, 10], strain energy density [2] or stress [11, 12] lead to satisfying results in specific

applications, Hhowever, there is no clear agreement on which regulation signal provides the

best predictions in a general sense. Moreover, the signal selection is not frequently discussed,

and comparisons of between different sets of adaptation variables are rare [5, 13].

There are also Oother issues are related relating to the mathematical form of the adaptation

law relating which relates the level of mechanical stimulus to the bone apposition or

resorption rate. In order to preserve the natural structure of bone under physiological

conditions, continuum level isotropic bone adaptation models must at least exhibit a region of

homeostasis by defining a so-called Lazy Zone (LZ). The limits of the LZ controls the process

3

of bone adaptation by governing the transition between bone resorption, homeostasis and

apposition. However, because of their potential dependency on species, location and

biovariability, the bounds of the LZ are only rarely defined based on an experimentals basis

[14]., SecondFurthermore, the dependence of bone adaptation rate on the mechanical stimulus

has been formulated through linear [15], quadratic [16] or piecewise mathematic forms

including a rate saturation [17], but the sensitivity of the obtained predictions to these

different mathematical forms remains unclear.

Since bone reaction to stimulation is assumed to be driven by cells mechanotransduction [18],

several adaptation models involve a spatial averaging of the stimulus over a zone of influence

(ZOI) [3, 19]. This feature entails the assumption of a collective contribution to the local

stimulation, operated by sensitive cells interconnected by biological processes [20] through

the diffusion of chemical signals. Although the details of these cellular mechanisms are still

unclear, the size of the diffusion affected region represented by the ZOI is a key factor to in

obtaining accurate numerical predictions . However, the effective size and the dependence of

numerical predictions on the ZOI is scarcely investigated.

Finally, it is worth noticing that the loading conditions and initial bone structure is also

affected by uncertainties and variability over time. Although average representative load

levels can indeed be estimated, implants involved in both orthopedics and dental application

are actually subjected to a wide range of time varyiedng load levels [21]. These stimulations

propagate through the bone tissue and may cause a bone adaptation that could differ from the

results of the average load case. Furthermore,inally the pre-implantation bone structure and

geometry differs significantly among onea group of individuals, which renders difficult

comparisons with experiments difficult and limits the validity of predictions based on an

average representative geometry. As bone adaptation is a strongly non-linear process, bone

4

adaptation results may therefore drastically vary between individuals, and biovariability is

expected to induce a significant scatter in the resulting bone structures.

This work aims to at clarifying the dependence of continuum level numerical predictions of

peri-implant bone adaptation on the mechanical variables adopted as stimuli, and on important

modeling parameters. The ‘loaded implant’ animal model is adopted as a benchmark [22].

This animal model allows investigating the effects of a controlled external stimulation of on

the bone tissue surrounding two transcutaneous implants inserted in the proximal part of rats’

tibiae [23, 24]. Different mechanical stimuli are compared on the benchmark of full tibiae

being subjected to physiological loading conditions. Assuming that the Mechanostat

hypothesis is valid, a clear Lazy Zone should be able to be observed in the distribution of

proper adaptation mechanical stimuli under those such conditions. Moreover, the LZ of the

ideal mechanical stimulus should also satisfy the criteria of location independence, tissue

independence and specimen independence. The stimulus that which best satisfies these

conditions is identified and used in combination with a specimen-specific adaptation

algorithm to predict bone peri-implant adaptation of a population of 5 specimens. A

sensitivity study subsequently highlights the dependence of bone adaptation results on the

bounds of the LZ, on the formulation of the adaptation law, on the parameters of the ZOI and

on the load level.

2. Materials and Methods

2.1 Animal model

Two SLA treated transcutaneous Ti implants were screwed mono- and bi-cortically into the

right tibia of female Sprague-Dawley rats (Figure 1). After two weeks of integration, a

controlled external load of 5 N was applied on a daily basis daily to force the implant’s heads

together and to stimulate the bone tissue around them implants. The load was applied with the

following schedule: 1Hz sinusoidal cycle, 900 cycles/day, 5 days/week, with a progressive

5

increase of loading during the first week (1 N/day). After the sacrifice, all tibiae were

dissected, cleared of their soft tissue coverage and frozen at to -21°C. The technical aspects of

surgery, implants design and activation setup characterizing the ‘loaded implant’ model are

described in details in [23, 24]. Prior to Computed Tomography (CT) scanning, the specimens

were maintained at 4°C for 24 hours in a 0.9% solution of NaCl and then gradually brought to

room temperature. The specimens were then analyzed using a high resolution CT imaging

system (μCT-40, Scanco Medical AG, Brüttisellen, Switzerland) with the following settings:

1022 slides × 360 degrees of rotation, isotropic voxel size: 20 μm, source potential: 70 kVp,

tube current: 114 μA, integration time: 320 ms, beam hardening correction: 200 mgHA/cm3.

2.2. Finite Element models

Continuum-level specimen-specific FE models of bare and implanted rat tibiae were

generated from mCT scans through a verified and validated procedure [25]. The CT images

were segmented and processed with an open-source FE model generator to quality second

order tetrahedral meshes. EventuallySubsequently, the nodes were assigned their material

properties depending on their original CT's BMD using the density-elasticity relationship

developed by Cory et al. [26]. These models represent the whole bone structure as a

continuum, isotropic and inhomogeneous material . The local elastic modulus is calculated

from using the mean BMD of the CT voxels contained in each element of the model. Five

whole tibiae were processed to generate specimen-specific FE models (element size ratio: 51

voxels/element, average model size: 1.2 M nodes), which. These models were then subjected

to a gait based loading condition, which had been shown to generate a sound physiological

pattern of deformation [27]. Loads were adjusted for each specimen with respect to the animal

body-weight, with. Tthe employed musculoskeletal loads are shown in Figure 2a. These

specimens were adopted as the benchmark for the comparison of mechanical stimuli.

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Five implanted specimens were processed in order to generate the FE models for iterative

computations on bone peri-implant adaptation (element size ratio: 300 voxels/element,

average model size: 100000 nodes), and mesh convergence was verified. These specimens

represented the pre-stimulation integration state of the ‘loaded implant’ model: they

underwent two weeks of post-surgery integration that which allowed the primary stability to

be established [23]. These FE models were then subjected to the boundary conditions shown

in Figure 2b, which correspond to the controlled loading provided during in-vivo stimulation.

The bone-implant contact was considered perfectly adherent as the SLA treatment of the

implants was verified to guaranteey proper bone ingrowths and good correct adhesion.

However, in order to prevent unrealistic transfer of loads to the tissue where the interface was

subjected to traction, a small elastic modulus was assigned to the tensile loaded regions of the

implants in contact with cortical bone in the distal and proximal directions, as discussed in

[28].

2.3. Mechanical stimuli

The three mechanical stimuli compared in this analysis were chosen to beas isotropic positive

scalar measures. Assuming a general interpretation of the Mechanostat at the continuum scale,

the LZ of the stimulus should be independent on the location within bone, and thus on the

tissue type, and ideally be as homogeneous as possible among a population. More

specificically , an adequate stimulus variable should respect these criteria under the following

physiological conditions:

Location invariance. The signal distribution (volume histogram of stimulus) is

location-independent meaning that different regions of interests of the same bone are

subjected to the same stimulus range of stimulus (i.e. similar bounds of the

distribution).

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Tissue invariance. The Ssignal distribution is tissue-independent. Cortical and

trabecular bone undergo the same stimulus range of stimulus, i.e. the same histogram

of stimulus is observed in those two domains areas.

Specimen invariance. The distribution of the stimulus variable is homogeneous

within a population of specimens in under the same conditions. The standard deviation

of the histogram of stimulus within such a group of specimens in the same conditions

remains limited.

Three stimuli were compared by highlighting their compatibility with the proposed criteria, on

the benchmark of a whole tibiae subjected to physiological deformations [28].

The first stimulus considered was the elastic energy per unit of mass ψU (Eq. 1),. Tthis

variable was being adopted in remodeling algorithms for orthopedic [10] and dentistry studies

[16]. This energy-based stimulus combines the strain energy density U i that occurs during the

loading condition i, the local apparent bone apparent density ρ, and the number of loading

conditions N.

ψU= 1N ∑

i=1

N U i

ρ(1)

The second stimulus considered was the daily stress ψσ (Eq. 2) proposed initially by Carter,

Beaupré and co-workers et al [11, 12]. In this study, the trabecular tissue was modeled as a

continuum replicating the tissue macroscopic stiffness, without resolving the singular

trabeculae. Thus, the stress-based stimulus was formulated as

ψσ=( ρ c

ρ )2

∙(∑i=1

N

niσ im)

1 /m

(2)

where ρc is the apparent density of mineralized bone and m is an empirical constant adopted

to weight, the number of cycles and the stress depending on the physical activity. This

8

stimulus depends on multiple load cases N, on the number of loading cycles ni and on the

effective stress at the continuum level σ .

The third stimulus considered was the octahedral shear strain. Frost introduced the hypothesis

that the Mechanostat is driven by the peak daily strains that which occur in the bone tissue

[1]. The stain-based stimulus ψε can be expressed as shown in Equation 3.

ψε=max (ε1 , ε2 ,…, εN) (3)

where N is the number of loading conditions and ε i is the strain tensor generated during the

loading condition i at the continuum scale. To obtain a scalar measure, Eq. 3 was

reformulated as a function of the octahedral shear strain ε oct:

ψ ε=max (εoct ,1 , εoct ,2 …, εoct , N ) (4)

Among other scalar strain measures, the octahedral shear strain was chosen because several

investigations have highlighted the influence of shear on the tissue differentiation [28, 29].

The three signals are investigated through a single loading condition (N=i=1), based on the

peak musculoskeletal loads occurring during gait [27]. For the sake of comparison, the

number of loading cycles characterizing the daily stress was fixed to n = 1. As a consequence,

the considered energy-, stress- and strain-based stimuli are formulated as shown in Eqs. 5, 6

and 7, respectively.

ψU= Uρbmd

(5)

ψσ=( ρbmd ,c

ρbmd)

2

∙ σ(6)

ψε=ε oct (7)

where ρbmd and ρbmd , c are the local bone mineral density (BMD), and the fully mineralized

BMD (i.e. 1.2 gHA/cm3), respectively.

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The stimuli distribution was investigated in two fixed regions of interest fixed with respect to

the location of the implants, as shown in Figure 3a. The Inter-Implant region of interest ROIII

was obtained by a dilatation of +/- 0.3 mm of the plane where both implant axes lie. The

region of interest ROICY was defined as a cylinder surrounding the proximal implant location.

These two ROIs were chosen in order to capture two different loading states: ROIII

represented the region where the external stimulation was more effective, and ROICY provided

an estimation of the average signals all around the implant floating inside the trabecular bone.

The overall dimensions are showncan be seen in Figure 3b and c.

To analyze the tissue independence, the signals were also differentiatied between cortex,

trabecular tissue and marrow. A BMD threshold was fixed so to discriminate these tissues

and, modeled in the FE analysis through local, BMD-dependent material properties. In details,

stimuli were classified as cortical if BMD > 0.8 gHA/cm3, and trabecular if 0.3 < BMD < 0.8

gHA/cm3 [26]. These BMD thresholds approximately corresponded approximately to BV/TV

values of 0.25 and 0.6 respectively,. however, Tthe results belonging to “marrow” (BMD <

0.3 gHA/cm3) were neglected. These results belong to porous regions mostly composed

mostly by of marrow and blood vessels, in which the linear elastic model is not pertinent and

whose contribution to mechanotransduction is considered negligible. For each ROI, tissue

category and specimen, the volume histogram of the investigated stimulus variable were was

reconstructed from the FE simulation results. The mean value and SEM of the histograms of

the five specimens were finally calculated and compared in order to evaluate the

aforementioned invariance criteria.

2.4. Bone adaptation algorithm

The simulations of bone adaptation were based on the hypothesis that the bone adaptation

process leads to a normalization of its mechanical stimulus towards a constant range of

values, corresponding to the so called Lazy Zone [1] or Attractor States [11], by reducing or

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increasing the bone density. The density variation of each element was computed through the

formulation described in Eq. 8, inspired by the quadratic form proposed by Li et al. [16].

d ρbmd

dt={ 0 if ψ ≤ψa

K a (ψ−ψa )(1− ψ−ψa

ψd−ψa) if ψ>ψa

(8)

wWhere ψ is the chosen mechanical stimulus, K a and is the adaptation rate, and ψd and ψa are

respectively the damage and apposition attractor states respectively (Figure 4a). This

formulation implies that implant loading has no effects of the implants loading if the stimulus

is low (d ρbmd /dt=0 if ψ<ψa), a positive effect of the stimulation if the stimulus belongs to

the apposition zone (d ρbmd /dt >0 if ψa<ψ<ψd) and resorption due to overloading occurs if the

stimulus overcomes the damage limit (d ρbmd /dt <0 if ψ>ψd). The resorption caused by disuse

was neglected [16, 17] in the present study as it was verified that the normal daily activity of

the animals during the experiment was sufficient to maintain their bone structure and thus

prevent any disuse state of disuse.

The calculation of the effective stimulus included a spatial averaging of the mechanical

variables over a spherical Zone Of Influence (ZOI) [3]. In details, the signal at each point was

calculated through Eq. 9

ψ i=ψ i+∑

j =1

Z

f ( D ji)ψ j

1+∑j=1

Z

f (D ji)(9)

Where Z is the number of nodes included in the defined ZOI and f (D ji ) is a function that

weights the signal contribution of the node j with respect to its distance D ji from the current

node i. With this formulation the conservation of the integral of ψ was ensured.

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Equation 8 was iteratively solved iteratively through forward Euler integration in order to

update the bone density in relation to the tissue deformation until equilibrium (negligible

change of densities over one iteration). A controlled maximum BMD variation was calculated

at each iteration with an adaptive time step to guaranteey the stability of the simulation

during the initial iterations, and when the larger variations of elastic modulus occurred., and

accelerate tThe convergence was accelerated when the increments of bone density becaome

small [30].

2.5. Sensitivity analyses

Five specimen-specific FE models of implanted tibiae were processed with different sets of

parameters in order to perform a detailed sensitivity study on the numerical predictions of

bone adaptation. The parameters changed in the sensitivity study are presented in the

following sections and summarized in Table 1.

2.5.1. Nominal parameters

The correlation between mechanical stimulus and bone apposition rate was provided by

Equation 8, and. Tthe value of the apposition threshold, ψa=1.25 × 10−3, was fixed in

agreement with the physiological deformations occurring in rat tibiae during gait (Section

3.1). The value of the damage threshold ψd=4.51 ×10−3 was fixed in agreement with the

longitudinal strain limit of 4 ×10−3 already adopted for a study concerning bone damage [15].

As the steady state solution was only the steady state solution was of interest in the present

study, the adaptation rate Ka was given the value of 1 (gHA/cm3)/(time unit) in agreement

with Li et al. [16]. To be compatible with the continuum modeling assumptions, the nominal

ZOI radius was chosen corresponding to the size of the representative volume element (RVE)

of trabecular bone, which is estimated at 0.3 mm in the present case (3 to 5 inter-trabecular

lengths [31]). The decay of the signal with increasing distance from the central node of the

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ZOI was computed through a gGaussian fFunction (Eq. 10), which is typical for many

diffusion-like natural processes [19].

f ( D ji )=e−D ji /r

2 β2 (10)

where β=0.4085. These settings are considered as the reference for the sensitivity study. Due

to the large number of parameters, the sensitivity study was structured into several separate

parametric studies, focusing on one mechanism at a time which are presented in the following

sections.

2.5.2. Effect of apposition and damage thresholds

Based on the invariance analysis of the mechanical variables, the octahedral strain stimulus

was chosen as the reference signal for the sensitivity analysis. The range of octahedral shear

strain thresholds ψa and ψd were as discretized in four values between 1×10−3 and 1.75×10−3,

and 4.1×10−3 and 5.5×10−3, respectively. The bone adaptation solutions were computed for

each specimen and each pair of adaptation thresholds (16 combinations), while keeping all

other parameters at their nominal values therefore, leading to a total of 80 adaptation

simulations.

2.5.3. Sensitivity to the adaptation law formulation

Three mathematical formulastions of the adaptation law were compared: the quadratic form

in Equation 8, a linear formulation [15] and a piecewise law with a plateau [17] (Equation 11

and 12 respectively), for a total of 15 iterative computations.

dρdt

={ 0 if ψ ≤ ψa

Ka (ψ−ψa ) if ψa<ψ ≤ψd

K d (ψ−ψd ) if ψ>ψd

(11)

13

dρdt

={0 if ψ≤ ψa

Ka (ψ−ψa ) if ψa<ψ ≤ψa 'Ka ( ψa '−ψa )+K a

' (ψa−ψ a ' )K a ( ψa '−ψa )+K a

' (ψa ' '−ψ a ' )K d (ψ−ψd )

ififif

ψa '<ψ ≤ψa ' 'ψa ' ' <ψ ≤ψd

ψd<ψ

(12)

The illustration of the adaptation rate versus the mechanical signal for Equations 11 and 12

are presented in Figure 4b and 4c, respectively. The adopted parameters were calculated to

preserve the nominal adaptation thresholds and are presented in Table 2.

2.5.4. Sensitivity to the Zone of Influence

A dedicated parametric study was performed by varying the ZOI radius r from 0 to 0.9 mm,

and considereding three types of weight functions: Gaussian, linear and exponential

(Equations 10, 13 and 14), therefore giving for a total of 65 iterative computations.

f ( D ij)=0.95(1−Dij

r )+0.05 (13)

f ( D ij)=e−2.99 Dij

r(14)

To achieve comparable effects, the constants used in the different forms of f ( D ij) were chosen

to achieve f ( D ij) = 1 for Dij = 0 and to f ( D ij) = 0.05 for Dij = r. The contribution of nodes

outside the ZOI (Dij > r) was neglected.

2.5.5. Sensitivity to load level

To analyze the dependency of results on the external load applied to the implants, the same

group of specimen-specific FE models were simulated, with the default adaptive modeling

parameters at five applied load levels, ranging from 3.3 to 10 N, therefore giving for a total of

25 iterative computations.

2.5.6. Convergence criteria and selected output variables

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In all cases, the simulations were automatically stopped when 99.9 % of nodes showed null

adaptation errors (signal within Lazy Zone), and. Local BMD variations during bone

adaptation were assessed from the FE predictions in six Regions Of Interest (ROIs)

represented in Figure 5. The longitudinal stability of implants was monitored by estimating

the variation of inter-implant strain (calculated as d / L, where d and L are the relative inter-

implant heads displacement and inter-implant distance, respectively).

3. Results

3.1. Comparison of stimuli

The histogram distributions of the signals belonging to different ROIs and tissues are shown

in Figure 6. Each distribution is represented by the mean ± SEM calculated on the five

specimen- specific FE models of whole tibiae subjected to the gait-based loading condition.

Considering the energy-based stimulus ψU , a large amount of cortical bone in ROIII shows

levels of strain energy significantly larger to the ones measured in ROICY (40 % of the total,

black arrows in Figure 6a) and overall, the distributions of ψU in both ROIs are significantly

different. In oppositeYet conversely, for the trabecular bone, the distributions of ψU are

comparable similar between ROIs (i.e. the histograms are nearly superimposed, Figure 6b).

However, Tthe range of values of the strain energy stimulus ψUin trabecular bone (from 0 and

2 × 10−3 J/g ) and cortical bone (between 0 and 8 × 10−3 J/g) are very different, and. bBased

on these results, it is observed that the energy-based signal ψU does not satisfy the location

and tissue-invariance criteria.

The stress based stimulus ψσ shows location-invariant distributions, as shown by the

negligible differences between ROIs in Figure 6c and d, respectively. Nevertheless, the

distributions of stimulus ψσ in the cortical and trabecular bone are not comparable either in

terms of shape nor in terms of range. Indeed, a factor of more than two is found in the

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maximum the stress stimulus ψσ calculated in trabecular and cortical bone. As a

consequence, even though it satisfies the location invariance criterion, the stress-based signal

ψσ does not satisfy the tissue-invariance criterion.

Finally, the distributions of the strain based stimulus ψε are shown in Figure 6e and f, and.

Iinterestingly, the shape of the distributions is found to be similar in both ROIs and tissues,

and the the bounds of the strain based stimulus ψε remains homogeneous ( between 200 to

1500 με ¿. These results therefore demonstrate that the strain based stimulus ψ ε respects both

the tissue- and the location-invariance criteria in the present case, and. Ffor this reason, this

stimulus is chosen as reference for the sensitivity study of bone adaptation.

As a unique range of physiological strain stimulus can be identified, the obtained histograms

in physiological conditions can be used to set the apposition threshold of the strain- based

adaptation model in a systematic manner. Indeed, the reference value of the apposition

threshold for bone adaptation analyses,, ψa=1.25 × 10−3, is fixed by fitting the distribution of

deformation occurring in the rat tibia during gait. This value is quantified as the 95th

percentile of the octahedral shear strain distribution characterizing both cortical and trabecular

tissues between the implants and around the proximal implant (ROIII and ROICY) during

physiological activity.

The results of the sensitivity study are presented in the following paragraphs.

3.2. Effect of adaptation thresholds

The results of this parametric study are shown in Figure 7. The evolution of the inter-implant

strain (inversely proportional to stiffness) varies with both ψa and ψd (Figure 7a) within a

ranging of 0 % to −8 %, and. Tthe maximum and minimum and maximum increase of

stiffness coincides with the wider and narrower amplitude of the apposition zone (i.e. range of

overloading causing a positive BMD rate). Moreover, the effects of ψd reaches a plateau after

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5×10−3 , meaning that the damage controlled adaptation is no more longer activated above this

point. Overall, at the nominal load level, the value of the damage threshold ψd has a limited

influence on the inter implant stability and the inter-implant strain remains mostly controlled

by the apposition threshold ψa. The BMD variation in ROI1 and ROI3 (Figure 10b and c),

however, shows an even clearer trend: the average density variation in the different ROIs is

nearly independent of to ψd, while whereas it is definitively reduced with the increase of ψa.

3.3. Effect of adaptation law formulation

The results of this study in terms of inter-implant strain variation and BMD increment in ROIs

are reported in Figure 8a and b, respectively. Interestingly, the results of the quadratic and

linear formulations indeed show a clear agreement, with the latter slightly underestimating the

inter-implant strain and density increments with respect to the former one. However, the

mathematical form including a plateau involves a significant overestimation of both outputs.

This highlights the fact that even though all three adaptation laws have similar thresholds, the

shape of the adaptation law has an influence on the result, and thus different parameters

would be required to obtain consistent results. Considering that the adaptation threshold were

was set in a rigorous manner based on experimental data, and that both quadratic and linear

models provide consistent results with the experimental observations, it can therefore be

concluded that the multi-linear adapation law is not an appropriate choice in for the present

study.

3.4. Effect of the zone of influence

As shown in Figure 9a, if the ZOI the radius is lower than the RVE size estimated at 0.3 mm

(3 to 5 inter-trabecular lengths) the inter-implant strain unrealistically increases unrealistically

as the models predict a greater resorption because due to of overloading in regions close to the

implant. The extreme case of no ZOI (r = 0, purely local adaptation law) provokes a 17 %

17

increase in of inter-implant strain, which is exactly the opposite of what is found with the

nominal ZOI of 0.3mm (-5.5 %), and contradicts the experimental observations. This fact is

explained by the sharp stress concentration at nodes in contact with the implants. However, it

is remarkable that for radii above the RVE size there is nearly no variation of the inter-

implant strain, thus indicating that once the signal is averaged on a consistent volume of bone,

the solution remains stable. The BMD variation shown in Figure 9b shows a moderate

increase of in density with an increasing ZOI radius in ROI 1 and 2 both situated in the inter-

implant plane, and subjected to compression. Interestingly, the effects of the different decay

formulations (linear, exponential or Gaussian) on the adaptation results are found to be

negligible and the influence of the radius of the ZOI is the predominant parameter.

3.5. Load level

In Figure 10, Tthe inter-implant strain variation after adaptation is plotted against the applied

load in Figure 10 for 5 different specimens, whereby. a A 3.3 N load provokes only a weak

inter-implant strain reduction (−3 %), which is then nearly doubled by imposing 5 N (−5.5

%). With higher loads, the system stability decreases significantly and the effects of

biovariability are strongly amplified. By applying 6.7 N, the mean level of the inter-implant

strain shows first signs of instability (increase of +3.3 %) with a spread that is becomesing

significantly larger with respect to lower load levels (range of inter-implant strain variation

between -5% and +10%). The density field variation reported in Figure 11 shows an

increasing apical resorption because due to of overloading. At 8.6 N the imbalance between

resorption and apposition reaches critical levels for 3 over out of the 5 specimens, which are

characterized by the failure of all the tissue surrounding the distal implant. Nevertheless, two

specimens did reach a stable converged solution with definitively increased inter-implant

strain (+20 %). Finally, none of the FE models adapted to 10 N: the resorption because due to

18

of overloading dominates the adaptation mechanism provoking the instability of the distal

implant.

The local BMD variations in the ROIs show a similar trend (Figure 12) and explain the

variations of in implant stability. Indeed, at low force, the increased of BMD in the

compressive regions between implants is correlated to the reduction of inter-implant strain.

However, the average BMD in the ROIs continue to increase with higher loads (i.e. 6.3 and

8.7 N) but regions of significant bone resorption develop, leading to a progressive reduction

of the implant’s stability. Thus, it is worth noticing that the implants stability and the BMD

variation in the selected ROIsS are not correlated at high loads.

4. Discussion

This paper focuses on the reliability of Mechanostat-based predictions of bone adaptation

estimated through specimen-specific, continuum FE models. The adopted modeling strategy

relies on nominal parameters that which are consistent with the physiology of the ‘loaded

implant’ model and with the problem length-scale. The predictions obtained with these

parameters where validated through comparison with in-vivo experiments and provide a

goodare therefore very accuractey [32]. Since these phenomenological predictions are the

result of several assumptions, it is of great interest to clarify the methods that which have been

used to define the key parameters and highlight their influence on the results.

Three mechanical variables were compared on the benchmark of full tibiae subjected to

physiological loading conditions. The analysis of the stimuli distribution (Figure 6) with

respect to position, tissue and specimen invariance criteria highlights that the octahedral shear

strain is the most appropriate stimulus for the implementation of the Mechanostat theory at

the continuum scale. When gait-based loads are applied, this signal is the only one that shows

location- , tissue- and specimen- independent distributions. This result confirms the existence

19

of a unique range of stimuli corresponding to physiological conditions (i.e. the Llazy zZone),

which drives bone macroscopic structural adaptation to mechanical stimulations.

An exhaustive sensitive study highlighted the robustness of the adopted strategy by clarifying

the influence of the biovariability, the bone adaptation thresholds, the adaptation law

formulation, the ZOI and the external load on numerical predictions of bone adaptation. The

effects of all these variables were evaluated by performing multiple specimen-specific

iterative computations that generated results dependent on the features of each individual.

The default bounds of the lazy zone (i.e. apposition and damage thresholds ψa and ψd) were

defined through the analysis ofanalysing physiological deformations occurring during rats’

gait and reliable literature data. Nevertheless, the dependency of results on the perturbation of

these parameters is of great interest considering that the equilibrium between the peri-implant

bone apposition and the apical resorption because offrom overloading is a key factor for thein

implant stability. The results show that the perturbation of the adaptation thresholds within

consistent ranges of strain does not affect the robustness of the investigated adaptation process

(e.g. no worsening of the implant lateral stability is predicted, Figure 7). However, these

parameters clearly affect the prediction of both BMD and inter-implant strain variations. This

therefore highlights the need of for a rigorous way of determining the apposition threshold ψa

in particular. The determination of this parameter through an histogram analysis of the

octahedral strain stimulus in physiological conditions, as used in this study, is highly

recommended in order to obtain reliable predictions.

The mathematical formulation of the density adaptation rate dependence on the stimulus also

has also a strong influence on the results. The quadratic form adopted as reference (Equation

8) allows describing both apposition and resorption because ofdue to overloading with few

parameters. Nevertheless, this phenomenological formulation may not be representative of the

actual correlation between bone mass variation and mechanical stimulus. In this work, the

20

results obtained with the quadratic adaptation law (Equation 8) are compared with the ones

obtained with a linear form and a piecewise formulation that which includes a plateau

(Equations 11 and 12, respectively). The Rresults shown in Figure 8 highlight that the

quadratic and linear formulations can be considered equivalent. On the contrary however, the

adoption of a formulation that which includes a plateau significantly affects the results, and

lead to unrealistic predictions when compared to experimental observations [33]. Thus, this

type of formulation should only be implemented only if the saturation of density rate is

supported by a sound experimental validation.

The stimulus average on a ZOI introduces the interesting concept of the transmission of a

local mechanical signal all around the stimulated area, through via biological processes

involving the tissue micro- and cellular-structures. Moreover, the obtained stimulus is

representative of the mechanical state that which characterizes a defined volume of bone.

Although the biological mechanism driving these phenomena is still unclear, it is of great

interest to investigate the sensitivity of the numerical predictions to different ZOI dimensions

and stimulus decay functions (Equations 10, 13 and 14), and. Tthe results of this parametric

study highlights that the definition of a ZOI is essential to predict consistent results (Figure 9).

The key factor is the radius of the ZOI, which should at least correspond to the RVE size (3-5

trabeculae), while the weight function only plays only a secondary role. Although these

results are limited to the treatment of a strain-based stimulus derived from a continuum, and a

linear and elastic framework, a similar trend is expected for stress- and energy-based

variables.

The magnitude of the external load regulates the mechanical stimulation transferred from the

implant to the surrounding tissue, and. Iin the ‘loaded implant model’ the applied load is

controlled (5 N nominal value). However, the a study of the results’ sensitivity to different

ranges of loads underlines the stimulation limits, above which harmful effects are dominant.

21

Despite the limits of the proposed analysis, which does not account for eventual fatigue cracks

or inflammatory reactions taking place in case due to of critical overloading, several

interesting conclusions can be drawn. The results are very sensitive to the load magnitude

(Figure 10) and evolve in a complex non-linear manner. The range of external forces

generating a positive effect on integration (i.e. augmentation of peri-implant density and

improved implant stability) is found to be relatively narrow for the present experiment.

Indeed, with 3.3 N the bone reaction is quite limited while at 5 N the optimum is already

reached. Higher loads provoke larger increases of BMD associated with dangerous peri-

implant bone loss because of due to overloading (Figure 10), and . Iinterestingly, the results

spread increases with load because of due to the difference between specimens. As a

consequence, it can be postulated that the maximum forces to which each individual can adapt

depends on specimen-specific features., and therefore Tthe results of this sensitivity study

highlight the need to follow a specimen specific approach in order to study critical

overloading, and shows the potential of the proposed methodology to perform such analysis.

Moreover, the performed parametric studies indeed allow establishing a ranking of the

modeling parameters based on their effects on the inter-implant strain and maximum BMD

variation (Figure 13). Subsequently, Tthree categories are identified and can be adopted to

optimize similar approaches:

Critical Parameters. This category includes applied load overestimation and the

absence of ZOI, which provokes more than 100% variation of both inter-implant strain

and BMD in ROIs. If these parameters are not considered or wrongly implemented,

the numerical predictions can become totally inconsistent. The ZOI should be set at

least equal to the RVE size. If critical loads are of interest, specimen specific

simulation should be carried out.

22

Important Parameters. This category includes the perturbations which provokeing

variations between 10 % and 100 % and 10 % . The adaptation thresholds, the

piecewise law formulation with a plateau, load underestimation and the ZOI radius

overestimation all fall in this category. and Aan ambiguous implementation of these

parameters does not lead to unstable solutions, but however the results are indeed

inaccurate. At the very least, the bone adaptation threshold should be identified on the

basis of experimental data for the damage limit ψd , and from an histogram analysis in

physiological conditions for the apposition limit ψa.

Negligible Parameters. This category involves the type of formulation (quadratic or

linear law) and the type of decay function in the ZOI. Compared to the nominal

settings, these perturbations scarcely affect both the inter-implant strain and the BMD

predictions, and thus can be chosen arbitrarily.

In conclusion, the results of the presented numerical analyses highlights that a quadratic

adaptation model based on octahedral shear strain stimulus, with a ZOI equal to the trabecular

bone RVE size and apposition threshold determined by histogram analysis of physiological

conditions, appears to be the most appropriate formulation to perform prediction of predict

peri-implant bone adaptation at the continuum level.

5. Acknowledgments

Mrs Severine Clement and Mrs. Isabelle Badoud (University of Geneva) are gratefully

acknowledged for their work on surgery, animal care, and data acquisition. This work was

partially funded by Swiss National Science Fundation (SNF) project 315230-127612.

6. References

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23

[3] Mullender MG, Huiskes R. Proposal for the regulatory mechanism of Wolff's law. Journal of Orthopaedic Research. 1995;13:503--12.[4] Chou HY, Jagodnik JJ, Müftü S. Predictions of bone remodeling around dental implant systems. Journal of Biomechanics. 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. Clinical Oral Implants Research. 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. Journal of Biomechanics. 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. 1990. p. 391-4.[8] Qian J, Wennerberg A, Albrektsson T. Reasons for Marginal Bone Loss around Oral Implants. Clinical Implant Dentistry and Related Research. 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 activity, mechanotransduction, and tissue properties. Biomechanics and Modeling in Mechanobiology. 2011;10:701-12.[10] Weinans H, Huiskes R, Grootenboer HJ. The behavior of adaptive bone-remodeling simulation models. Journal of Biomechanics. 1992;25:1425-41.[11] Beaupré GS, Orr TE, Carter DR. An approach for time-dependent bone modeling and remodeling - theoretical development. Journal of Orthopaedic Research. 1990;8:651--61.[12] Carter DR, Orr TE, Fyhrie DP. Relationships between loading history and femoral cancellous bone architecture. Journal of Biomechanics. 1989;22:231-44.[13] Terrier A, Rakotomanana RL, Ramaniraka AN, Leyvraz PF. Adaptation models of anisotropic bone. Computer methods in biomechanics and biomedical engineering. 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. Clinical Biomechanics. 2005;20:998-1006.[15] McNamara LM, Prendergast PJ. Bone remodelling algorithms incorporating both strain and microdamage stimuli. Journal of Biomechanics. 2007;40:1381 - 91.[16] Li J, Li H, Shi L, Fok ASL, Ucer C, Devlin H, et al. A mathematical model for simulating the bone remodeling process under mechanical stimulus. Dental Materials. 2007;23:1073-8.[17] Crupi V, Guglielmino E, La Rosa G, Vander Sloten J, Van Oosterwyck H. Numerical analysis of bone adaptation around an oral implant due to overload stress. Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine. 2004;218:407-15.[18] Turner CH, Forwood MR, Otter MW. Mechanotransduction in bone: Do bone cells act as sensors of fluid flow? FASEB Journal. 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. Journal of Mechanics of Materials and Structures. 2011;6:303-19.[21] O'Connor CF, Franciscus RG, Holton NE. Bite force production capability and efficiency in neandertals and modern humans. American Journal of Physical Anthropology. 2005;127:129-51.[22] Wiskott HWA, Cugnoni J, Scherrer SS, Ammann P, Botsis J, Belser UC. Bone reactions to controlled loading of endosseous implants: A pilot study. Clinical Oral Implants Research. 2008;19:1093-102.[23] Wiskott HWA, Bonhote P, Cugnoni J, Durual S, Zacchetti G, Botsis J, et al. Implementation of the "loaded implant" model in the rat using a miniaturized setup - description of the method and first results. Clinical Oral Implants Research. 2011;0.

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[24] Zacchetti G, Wiskott A, Cugnoni J, Botsis J, Ammann P. External mechanical microstimuli modulate the osseointegration of titanium implants in rat tibiae. BioMed Research International. 2013;2013.[25] Piccinini M, Cugnoni J, Botsis J, Zacchetti G, Ammann P, Wiskott A. Factors affecting subject-specific finite element models of implant-fitted rat bone specimens: Critical analysis of a technical protocol. Computer methods in biomechanics and biomedical engineering. 2014;17:1403-17.[26] Cory E, Nazarian A, Entezari V, Vartanians V, Müller R, Snyder BD. Compressive axial mechanical properties of rat bone as functions of bone volume fraction, apparent density and micro-ct based mineral density. Journal of Biomechanics. 2010;43:953-60.[27] Piccinini M, Cugnoni J, Botsis J, Ammann P, Wiskott A. Influence of gait loads on implant integration in rat tibiae: Experimental and numerical analysis. Journal of Biomechanics. 2014;47:3255-63.[28] Lacroix D, Prendergast PJ. A mechano-regulation model for tissue differentiation during fracture healing: analysis of gap size and loading. Journal of Biomechanics. 2002;35:1163 - 71.[29] Prendergast PJ, Huiskes R, Søballe K. Biophysical stimuli on cells during tissue differentiation at implant interfaces. Journal of Biomechanics. 1997;30:539 - 48.[30] 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. Journal of Biomechanics. 1993;26:369-82.[31] Bouxsein ML, Boyd SK, Christiansen BA, Guldberg RE, Jepsen KJ, Müller R. Guidelines for assessment of bone microstructure in rodents using micro-computed tomography. Journal of Bone and Mineral Research. 2010;25:1468-86.[32] Piccinini M. Prediction of peri-implant bone adaptation to mechanical environments - theoretical developments and experimental validation: EPFL; 2014.

25

Table 1. Details of sensitivity studies.

Parameter Default Levels#

models

Adaptation thresholds (

∙ 10−3 ¿

❑❑ 1.25 1; 1.25; 1.5; 1.7580

❑❑ 4.51 4.1; 4.51; 1.5; 5.5

ZOI Radius (mm) 0.3 0; 0.15; 0.3; 0.6; 0.965

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

Table 2. Parameters of the adaptation laws.

26

Adaptation thresholds (∙ 10−3 ¿ Rate constant

Law Eqn. ❑❑ ❑❑ ❑❑ ❑❑ Ka Ka’ Kd

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

Figure 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.

27

Figure 2. (a) gait-based loading condition applied to FE models of whole tibiae for the

comparison of stimuli [27]. Loads: medial and lateral condylar reactions, Cm and Cl, ankle

joint reaction Aj, bicep femoris Bf, vastus medialis Vm and lateralis Vl, rectus femoris Rf,

tibialis anterior proximal TAp and distal TAd. (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.

28

Figure 3. ROIs for the comparison of mechanical stimuli. (a) Ppositioning of the Inter-Implant

and Cylindrical regions of interest with respect to the implants insertion coordinates (ROI II

and ROICY, respectively). View cuts of the tibia: overall dimensions of (b) ROI II and (c)

ROICY.

29

Figure 4. Adaptation rate dρ /dt versus mechanical stimulus ψ. (a) quadratic form. (b) linear

form and (c) piecewise form with plateau.

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

30

Figure 6. Distribution of signals (rows) with respect to the tissue type (columns) and the

regions of interest ROIII and ROICY (legend). In details: energy-based signal ψU in cortical (a)

and trabecular (b) tissue, stress-based signal ψσ in cortical (c) and trabecular (d) tissue, and

strain-based signal ψ ε in cortical (e) and trabecular (f ) tissue.

31

Figure 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 (ψa and ψd). The mean

values (colored surface) and the SEM (upper and lower grids) of the group of five specimens

32

adopted as benchmark are represented.

Figure 8. Sensitivity of (a) inter-implant strain and (b) BMD in ROIs with respect to the law

formulation. Mean values and SEM are represented.

33

Figure 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 = exponential, G =

gaussian).

34

Figure 10. Inter-implant strain sensitivity to the external load magnitude. Mean values and

SEM are represented.

35

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

hidden.

36

Figure 12. Density in ROIs sensitivity to the external load magnitude. Mean values and SEM

are represented.

Figure 13. Ranking of the perturbations analyzed by sensitivity studies. The variations are

computed with respect to the results obtained with nominal settings (Table 1).

37

38