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:
[email protected]
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 Piccinini
Prof. John Botsis
Prof. Patrick Ammann
Prof. Anselm Wiskott
[email protected]
[email protected]
[email protected]
[email protected]
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
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. 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 concerning the definition of effective bone volume
considered to evaluate the mechanotransduction stimuli. These different assumptions,
combined with biovariability, clearly affect and limit 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 results
highlight that the octahedral shear strain appears to be the most appropriate candidate for the
implementation of Mechanostat-inspired adaptation models at the continuum scale. Moreover,
the sensitivity study allows the establishment of a classification of parameters which must be
determined precisely in order to obtain reliable numerical predictions.
Keywords: bone, implants integration, overloading, specimen-specific, finite element, gait,
mechanical stimulus.
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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 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 due to overloading. The interplay between these phenomena has been 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 via experimentation [9]. Indeed, there is
an important open question concerning the mechanical variable chosen as a triggering signal.
Several investigations studying regulation signals based on strain [1, 10], strain energy
density [2] or stress [11, 12] lead to satisfying results in specific applications, however, there
is no clear agreement on which regulation signal provides the best predictions in a general
sense. Moreover, signal selection is not frequently discussed, and comparisons between
different sets of adaptation variables are rare [5, 13].
There are also other issues are relating to the mathematical form of the adaptation law 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 control the process 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
3
the LZ are only rarely defined on an experimental basis [14]. Furthermore, 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 cell mechanotransduction [18],
several adaptation models involve a spatial averaging of the stimulus over a zone of influence
(ZOI) [3, 19]. This 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 in obtaining
accurate numerical predictions . However, the effective size and 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 orthopedic and dental application
are actually subjected to a wide range of time varyied 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, the pre-implantation bone structure and
geometry differs significantly among one group of individuals, which renders 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 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 clarify the dependence of continuum level numerical predictions of periimplant bone adaptation on the mechanical variables adopted as stimuli, and on important
4
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 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 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 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 to force the implant’s heads
together and to stimulate the bone tissue around them. The load was applied with the
following schedule: 1Hz sinusoidal cycle, 900 cycles/day, 5 days/week, with a progressive
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 -21°C. The technical aspects of
surgery, implant design and activation setup characterizing the ‘loaded implant’ model are
described in detail 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
5
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. Subsequently, 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 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 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 the employed
musculoskeletal loads shown in Figure 2a. These specimens were adopted as the benchmark
for the comparison of mechanical stimuli.
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 which allowed the primary stability to be
established [23]. These FE models were then subjected to the boundary conditions shown in
6
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 guarantee proper bone ingrowth and 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 as 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
specifically, 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 interest of the same bone are
subjected to the same stimulus range (i.e. similar bounds of distribution).

Tissue invariance. Signal distribution is tissue-independent. Cortical and trabecular
bone undergo the same stimulus range , i.e. the same histogram of stimulus is
observed in those two areas.

Specimen invariance. The distribution of the stimulus variable is homogeneous
within a population of specimens under the same conditions. The standard deviation of
the histogram of stimulus within such a group 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].
7
The first stimulus considered was the elastic energy per unit of mass 𝜓𝑈 (Eq. 1), this variable
being adopted in remodeling algorithms for orthopedic [10] and dentistry studies [16]. This
energy-based stimulus combines the strain energy density 𝑈𝑖 that occurs during the loading
condition i, the local apparent bone density 𝜌, and the number of loading conditions N.
𝑁
1
𝑈𝑖
𝜓𝑈 = ∑
𝑁
𝜌
(1)
𝑖=1
The second stimulus considered was daily stress 𝜓𝜎 (Eq. 2) proposed initially by Carter,
Beaupré 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
𝑁
1/𝑚
𝜌𝑐 2
𝜓𝜎 = ( ) ∙ (∑ 𝑛𝑖 𝜎𝑖𝑚 )
𝜌
(2)
𝑖=1
𝜌𝑐 is the apparent density of mineralized bone and m is an empirical constant adopted to
weigh the number of cycles and stress depending on the physical activity. This 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 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)
N is the number of loading conditions and 𝜀𝑖 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 𝜀𝑜𝑐𝑡 :
𝜓𝜀 = max⁡(𝜀𝑜𝑐𝑡,1 , 𝜀𝑜𝑐𝑡,2 … , 𝜀𝑜𝑐𝑡,𝑁 )
8
(4)
Among other scalar strain measures, the octahedral shear strain was chosen because several
investigations have highlighted the influence of shear on 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 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 =
𝜓𝜎 = (
𝑈
𝜌𝑏𝑚𝑑
𝜌𝑏𝑚𝑑,𝑐 2
) ∙𝜎
𝜌𝑏𝑚𝑑
𝜓𝜀 = 𝜀𝑜𝑐𝑡
(5)
(6)
(7)
where 𝜌𝑏𝑚𝑑 and 𝜌𝑏𝑚𝑑,𝑐 are the local bone mineral density (BMD), and the fully mineralized
BMD (i.e. 1.2 gHA/cm3).
The stimuli distribution was investigated in two fixed regions of interest 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 can be seen in Figure 3b and c.
To analyze tissue independence, the signals were also differentiated 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 detail,
9
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 to BV/TV values of 0.25
and 0.6 respectively, however, the results belonging to “marrow” (BMD < 0.3 gHA/cm3)
were neglected. These results belong to porous regions mostly composed 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 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
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].
0 𝑖𝑓 𝜓 ≤ 𝜓𝑎
𝑑𝜌𝑏𝑚𝑑
𝜓 − 𝜓𝑎
={
𝐾𝑎 (𝜓 − 𝜓𝑎 ) (1 −
) 𝑖𝑓
𝑑𝑡
𝜓𝑑 − 𝜓𝑎
𝜓 > 𝜓𝑎
(8)
where 𝜓 is the chosen mechanical stimulus, 𝐾𝑎 is the adaptation rate, and 𝜓𝑑 and 𝜓𝑎 are the
damage and apposition attractor states respectively (Figure 4a). This formula implies that
implant loading has no effect if the stimulus is low (𝑑𝜌𝑏𝑚𝑑 ⁄𝑑𝑡 = 0 if 𝜓 < 𝜓𝑎 ), a positive
effect of the stimulation if the stimulus belongs to the apposition zone (𝑑𝜌𝑏𝑚𝑑 ⁄𝑑𝑡 > 0 if
𝜓𝑎 < 𝜓 < 𝜓𝑑 ) and resorption due to overloading occurs if the stimulus overcomes the
damage limit (𝑑𝜌𝑏𝑚𝑑 ⁄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
10
the experiment was sufficient to maintain their bone structure and thus prevent any 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 detail, the signal at each point was
calculated through Eq. 9
𝜓𝑖 + ∑𝑍𝑗=1 𝑓(𝐷𝑗𝑖 )𝜓𝑗
𝜓𝑖 =
1 + ∑𝑍𝑗=1 𝑓(𝐷𝑗𝑖 )
(9)
Z is the number of nodes included in the defined ZOI and 𝑓(𝐷𝑗𝑖 ) is a function that weighs the
signal contribution of the node j with respect to its distance 𝐷𝑗𝑖 from the current node i. With
this formula the conservation of the integral of 𝜓 was ensured.
Equation 8 was solved iteratively through forward Euler integration in order to update bone
density in relation to 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 guarantee the stability of the simulation during the initial iterations, and
when the larger variations of elastic modulus occurred. The convergence was accelerated
when the increments of bone density became 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 the value of the apposition threshold, 𝜓𝑎 = 1.25 × 10−3, was fixed in
11
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 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 ZOI was computed
through a Gaussian Function (Eq. 10), which is typical for many diffusion-like natural
processes [19].
𝑓(𝐷𝑗𝑖 ) = 𝑒
−
𝐷𝑗𝑖 /𝑟
2𝛽2
(10)
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 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.
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2.5.3. Sensitivity to the adaptation law formulation
Three mathematical formulas 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.
0
𝑖𝑓
𝑑𝜌
(𝜓
)
𝐾
−
𝜓
𝑖𝑓
={ 𝑎
𝑎
𝑑𝑡
𝐾𝑑 (𝜓 − 𝜓𝑑 ) 𝑖𝑓
𝜓 ≤ 𝜓𝑎
𝜓𝑎 < 𝜓 ≤ 𝜓𝑑
𝜓 > 𝜓𝑑
0
𝐾𝑎 (𝜓 − 𝜓𝑎 )
𝑑𝜌
= 𝐾𝑎 (𝜓𝑎 ′ − 𝜓𝑎 ) + 𝐾𝑎 ′ (𝜓𝑎 − 𝜓𝑎 ′)
𝑑𝑡
𝐾𝑎 (𝜓𝑎 ′ − 𝜓𝑎 ) + 𝐾𝑎 ′ (𝜓𝑎 ′′ − 𝜓𝑎 ′)
{
𝐾𝑑 (𝜓 − 𝜓𝑑 )
𝑖𝑓
𝑖𝑓
𝑖𝑓
𝑖𝑓
𝑖𝑓
𝜓 ≤ 𝜓𝑎
𝜓𝑎 < 𝜓 ≤ 𝜓𝑎 ′
𝜓𝑎 ′ < 𝜓 ≤ 𝜓𝑎 ′′
𝜓𝑎 ′′ < 𝜓 ≤ 𝜓𝑑
𝜓𝑑 < 𝜓
(11)
(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 considered three types of weight functions: Gaussian, linear and exponential (Equations
10, 13 and 14), therefore giving a total of 65 iterative computations.
𝑓(𝐷𝑖𝑗 ) = 0.95 (1 −
𝐷𝑖𝑗
) + 0.05
𝑟
𝐷𝑖𝑗
𝑟
𝑓(𝐷𝑖𝑗 ) = 𝑒 −2.99
(13)
(14)
To achieve comparable effects, the constants used in the different forms of 𝑓(𝐷𝑖𝑗 ) were
chosen to achieve 𝑓(𝐷𝑖𝑗 ) = 1 for Dij = 0 and to 𝑓(𝐷𝑖𝑗 ) = 0.05 for Dij = r. The contribution of
nodes outside the ZOI (Dij > r) was neglected.
2.5.5. Sensitivity to load level
13
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 a total of 25
iterative computations.
2.5.6. Convergence criteria and selected output variables
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 interimplant 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. Yet conversely, for the trabecular bone, the distributions of 𝜓𝑈 are similar between
ROIs (i.e. the histograms are nearly superimposed, Figure 6b). However, the range of values
of the strain energy stimulus 𝜓𝑈 in 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 based on these results, it is
14
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 or in terms of range. Indeed, a factor of more than two is found in the
maximum 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
interestingly, the shape of the distributions is found to be similar in both ROIs and tissues, and
the bounds of the strain based stimulus 𝜓𝜀 remains homogeneous (between 200 to 1500 𝜇𝜀).
These results therefore demonstrate that the strain based stimulus 𝜓𝜀 respects both the tissue
and the location-invariance criteria in the present case, and for 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.
15
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 the 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 5×10−3 , meaning that
the damage controlled adaptation is no longer activated above this point. Overall, at the
nominal load level, the value of the damage threshold 𝜓𝑑 has a limited influence on inter
implant stability and 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 to
𝜓𝑑 , 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 clear agreement, with the latter slightly underestimating the
inter-implant strain and density increments with respect to the former. 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 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 for the present
study.
16
3.4. Effect of the zone of influence
As shown in Figure 9a, if the ZOI radius is lower than the RVE size estimated at 0.3 mm (3 to
5 inter-trabecular lengths) the inter-implant strain unrealistically increases as the models
predict a greater resorption due to overloading in regions close to the implant. The extreme
case of no ZOI (r = 0, purely local adaptation law) provokes a 17 % increase in 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 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, the inter-implant strain variation after adaptation is plotted against the applied
load for 5 different specimens, whereby 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 becomes 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 due to
overloading. At 8.6 N the imbalance between resorption and apposition reaches critical levels
17
for 3 out of the 5 specimens,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:
resorption due to 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 in implant stability. Indeed, at low force, increased 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 BMD variation in
the selected ROIs 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 which are consistent with the physiology of the ‘loaded implant’
model and with the problem length-scale. The predictions obtained with these parameters
were validated through comparison with in-vivo experiments and are therefore very accuracte
[32]. Since these phenomenological predictions are the result of several assumptions, it is of
great interest to clarify the methods 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
18
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 of
a unique range of stimuli corresponding to physiological conditions (i.e. the Lazy Zone),
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 biovariability, bone adaptation thresholds, adaptation law formulation, ZOI
and 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 analysing 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 from overloading is a key factor in 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
for a rigorous way of determining the apposition threshold 𝜓𝑎 in particular. The determination
of this parameter through a 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 a strong influence on the results. The quadratic form adopted as reference (Equation 8)
allows describing both apposition and resorption due to overloading with few parameters.
19
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 adaptation law (Equation 8) are compared with the ones obtained
with a linear form and a piecewise formulation which includes a plateau (Equations 11 and
12, respectively). The results shown in Figure 8 highlight that the quadratic and linear
formulations can be considered equivalent. On the contrary however, the adoption of a
formulation 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 if the saturation of density rate is supported by a sound
experimental validation.
The stimulus average on a ZOI introduces the interesting concept of transmission of a local
mechanical signal all around the stimulated area, via biological processes involving the tissue
micro and cellular-structures. Moreover, the obtained stimulus is representative of the
mechanical state 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 the 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 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 mechanical stimulation transferred from the
implant to the surrounding tissue, and in the ‘loaded implant model’ the applied load is
controlled (5 N nominal value). However, a study of the results’ sensitivity to different ranges
20
of loads underlines the stimulation limits, above which harmful effects are dominant. Despite
the limits of the proposed analysis, which does not account for eventual fatigue cracks or
inflammatory reactions taking place due to 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 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 due to overloading
(Figure 10), and interestingly, the results spread increases with load 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 the 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, three 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.
21

Important Parameters. This category includes perturbations which provoke
variations between 10 % and 100 % . The adaptation thresholds, the piecewise law
formulation with a plateau, load underestimation and the ZOI radius overestimation all
fall in this category and an ambiguous implementation of these parameters does not
lead to unstable solutions, 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 a 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 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.
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