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\journal{Journal of Biomechanics}

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\hypersetup { pdftitle = {Valid $\upmu$Finite Element Models of
    Vertebral Trabecular Bone can be Obtained Using Tissue Properties
    Measured With Nanoindentation Under Wet Conditions}, pdfauthor =
  {Uwe Wolfram} }

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\begin{document}
\begin{frontmatter}

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  \title{Valid $\upmu$Finite Element Models of Vertebral Trabecular
    Bone can be Obtained Using Tissue Properties Measured With
    Nanoindentation Under Wet Conditions}

  \author[label1]{Uwe Wolfram\corref{cor1}} 
  \author[label1]{Hans-Joachim Wilke} 
  \author[label2]{Philippe K.\ Zysset}
  \address[label1]{Institute for Orthopaedic Research and
    Biomechanics, Ulm University, Ulm, Germany}
  \address[label2]{Institute for Lightweight Design and Structural
    Biomechanics, Vienna University of Technology, Vienna, Austria}

  \cortext[cor1]{Institute for Orthopaedic Research and Biomechanics,
    Ulm University, Helm\-hol\-tz\-stra{\ss}e 14, D -- 89081 Ulm,
    Germany. Phone: +49 731 55301, Fax: +49 731 55324, e-mail:
    \texttt{uwe.wolfram@uni-ulm.de}}

\begin{abstract}
% Text of abstract
  \indent Avoiding vertebral fractures is a major clinical
  challenge. Patient specific Finite Element (FE) simulations of whole
  vertebrae could help to decide which vertebra has the highest risk
  of fracture. To circumvent the huge effort necessary to obtain
  material properties on a length scale suitable for FE models of
  whole vertebrae, \mufe approaches were developed to determine the
  material properties numerically. These \mufe models depend in turn
  on properly determined micromechanical material properties of
  trabeculae. This study aimed to set up valid \mufe models by
  combining macroscopic mechanical testing (compression, tension and
  torsion) and microindentation under two different testing
  conditions.

  Lower trabecular tissue moduli were found for wet indentation
  conditions ($E^{wet}_{tiss} = 12.2\ \mathrm{GPa}$, $E_{tiss}^{dry} =
  15.4\ \mathrm{GPa}$) which lead to lower simulated apparent
  moduli. No significant difference could be found when comparing
  simulations with wet tissue properties and uniaxial experiments for
  tension, compression and torsion ($p < 0.05$). Significant
  regressions could be found with good concordance correlation
  coefficients for tension and compression ($r^{wet}_c \approx 0.97$,
  $p < 0.05$) and moderate concordance correlation coefficients for
  torsion ($r^{wet}_c \approx 0.81$, $p < 0.05$). The agreement
  between experiment and simulation was confirmed by Bland-Altman
  plots which showed mean differences close to zero and low lines of
  agreement.

  The results indicate that it is possible to directly set up valid
  \mufe models using tissue properties obtained under wet indentation
  conditions. Those models could form the basis for subsequent
  nonlinear analyses for the numerical determination of effective
  macroscopic material properties.
\end{abstract}
\begin{keyword}
% keywords here, in the form: keyword \sep keyword
  finite element \sep trabecular bone \sep nanoindentation \sep
  material properties \sep stiffness
% PACS codes here, in the form: \PACS code \sep code
  \begin{singlespacing}
    \begin{tabular}{cll}
      Word counter & Abstract:     & 230  \\ \hline
                   & Introduction: & 492  \\
                   & Methods:      & 981 \\
                   & Results:      & 336  \\
                   & Discussion:   & 1057 \\ \hline
                   & $\Sigma$      & 2867
    \end{tabular}
  \end{singlespacing}
\end{keyword}
\end{frontmatter}

% main text
\modulolinenumbers[1]
\linenumbers
\section{Introduction}
\label{sec:introduction}
Avoiding fractures of osteoporotic vertebrae is of major interest for
surgeons \citep{MeChCo92, EFFO}. Patient specific Finite Element
simulations could be a support to decide which vertebra has the
highest risk of fracture \citep{CrCaKe03}. The quality of the results
of these simulations depends on the used material properties. A huge
effort is necessary to determine material properties on a length scale
suitable for those models experimentally in vitro
\citep{RiZy09}. Therefore, micro Finite Element (\mufe) models were
developed to obtain the necessary material parameters
numerically. These, in turn, depend on the chosen micromechanical
tissue properties due to the hierarchical nature of bone
\citep{RhKuZi98}. \citet{RiOdKa96} used \mufe models to determine an
apparent stiffness tensor for trabecular bone. \citet{BaGuKw04,
  BaMoNi04} could show that it is possible to determine a yield
surface using \mufe models. These and other studies used calibrated
micro-mechanical properties for the trabecular tissue which were
obtained inversely from simulations of uniaxial macro-mechanical
experiments \citep{RiWeHu95, NiFeJo00, BaMoNi04,
  VeRiMu08}. \citet{BeEsFa09} stated that those calibrated effective
tissue stiffnesses can diverge for more than an order of magnitude and
that they show lower precision in individual measurements of the
tissue modulus. In contrast, \citet{ChPaAl07} showed that the tissue
properties for \mufe models of dried femoral trabecular bone can be
obtained using microindentation under dry conditions. So far this was
not shown for vertebral specimens under wet, i.\ e. more physiological
conditions. A direct measurement of tissue modulus could exclude
concerns connected to tissue heterogeneity or initial assumptions on
the absolute value.

Depth-sensing indentation techniques give the possibility to determine
the elastic properties of trabecular bone {\it ex vivo} with a high
spatial resolution \citep{OlPh92}. Different authors determined the
elastic moduli ($E$) of trabecular and cortical tissue
\citep{RhTsPh97, RhTsPh99, RoRhTs99, ZyGuHo99,
  LeNy08}. \citet{TuRhTa99} found a $\roundpar{20\%}$ reduction of $E$
when comparing acoustic microscopy and nanoindentation. The authors
argued that this was due to rehydration of the specimens for acoustic
microscopy. \citet{HeKuZy02} found reductions of $E$ when comparing
nanoindentation measurements in dry and wet condition. Furthermore,
\citet{HoGuZy05} found reductions of moduli of $\approx 20\%$ when
rehydrating the specimens.

Microindentation gives the possibility to directly obtain specimen
specific tissue properties of trabecular bone. The question remains
whether linear elastic \mufe models with these properties can be
validated against macroscopic experiments under moist conditions. The
aim of the study was therefore, to validate \mufe models of rehydrated
bone specimen with experimentally obtained tissue moduli against
apparent elastic properties obtained from uniaxial tension,
compression and torsion tests. Once validated those models could form
the basis to determine effective macroscopic mechanical properties of
vertebral trabecular bone.

\section{Materials and Methods}
\label{sec:materials_and_methods}
Thirty cylindrical samples of $8 \times 18$ to $25 \ \mathrm{mm}$
depending on the vertebral height were gained from 20 vertebrae of
donors with a median age of $56\ \mathrm{y}$ (range 37 -- $84\
\mathrm{y}$) and stored at \degr{-20} immediately after harvest.

Vertebrae were cleaned from surrounding marrow and the posterior parts
were moulded in Poly-Methyl Methacrylate (PMMA) (Technovit 3040,
Heraeus Kulzer GmbH, Wehrheim, Germany). Subsequently, the endplates
were removed with a high precision band saw (Exakt Apparatebau,
Norderstedt, Germany). The cylinders were cored from the remaining
slices using a custom made core drill. Marrow was removed from the top
and bottom ends of the cylinders using a pulsed water jet (Braun
Oral-B Professional Care 6500, Kronberg, Germany). The top and bottom
ends of the cylinders were moulded in Polyurethane (PU) (SG~2000,
ebalta Kunststoff GmbH, Rothenburg ob der Tauber, Germany) so that an
approximately $12\ \mathrm{mm}$ free gage length was left in-between
(Figure~\ref{fig:specimen})

The endcapped cylinders were submerged in $0.9\%$ sodium chloride and
scanned in a $\upmu$CT ($\upmu$CT~40, SCANCO Medical AG,
Br{\"u}ttisellen, Switzerland) at a resolution of $\left(12\
  \mathrm{\upmu m} \right)^3$. Prior to mechanical testing, specimens
were rehydrated for one hour in $0.9\%$ sodium chloride. Subsequently,
aluminium rods were glued onto the endcaps using cyanoacrylate
(Loctite, Henkel Central Eastern Europe GmbH, Austria)
(Figure~\ref{fig:specimen}).
\begin{figure}[t]
  \centering
  \includegraphics[width=\textwidth]{fig_macroSpecimen.pdf}
  \caption{Workflow for an exemplary compression specimen. Top left,
    specimen with moulded endcaps. Top middle, midplane image from the
    CT data set of the gage length. Top right, glued metal rods for
    gripping. Bottom right, macroscopic test to determine the apparent
    elastic and shear moduli. Bottom left, axial trabecular
    cross-section with six microindentation imprints (black triangles)
    to obtain specimen specific trabecular tissue properties. Bottom
    middle, resulting FE model with deflection field $|u|$. The thick
    grey arrows illustrate the results which were input to the FE
    model and the subsequent validation.}
  \label{fig:specimen}
\end{figure}
An extensometer was attached to the PU endcaps after mounting the
specimen to the testing system (Mini-Bionix MTS system, Milwaukee, MN,
USA) (Figure~\ref{fig:specimen}). 

In torsion, angular data were recorded with the RVDT of the testing
system. The specimens were tested moist and preconditioned with three
cycles up to $\varepsilon = 0.004$ strain at a strain rate of
\strainrate{0.004}. Subsequently, the specimens were loaded up to
$\varepsilon = 0.004$ strain (in case of torsion to $\gamma = 0.008$)
at a strain rate of \strainrate{0.002} \citep{HaYePr04,
  BaMoNi04}. Tensile and compressive stress was calculated from the
recorded forces as $\sigma = \frac{F}{A_0}$ with $A_0$ the initial
cross-section of the sample. Torque-twist angle curves were denoised
with an average filter \citep{Jae05en}. Torsional stress was
calculated according to \citet{Na50} with $r$ the radius of the
specimen and $\theta$ the twist angle per unit length.
\begin{linenomath}
  \begin{equation}
    \label{eq:shearStress}
    \tau = \frac{1}{2 \pi r^3} \left( \theta \frac{dM_T}{d\theta} + 3M_T \right)
  \end{equation}
\end{linenomath}
The derivative $\frac{dM_T}{d\theta}$ was approximated from the
denoised torque-twist angle curve using a central difference operator.
Accounting for the influence of the fixation and the PU endcaps, the
shear strain $\gamma_B$ was calculated from:
\begin{linenomath}
  \begin{equation}
    \label{eq:shearStrain}
    \gamma_B = \left(\twistangle - \frac{M_T}{K_F} \right) \frac{r_S}{\ell_S} - \left(\frac{M_T \ell_{BP}}{J_{BP} G_{BP}}\right) \frac{r_{BP}}{\ell_{BP}}
  \end{equation}
\end{linenomath}
Here, $\twistangle$ denotes the twist angle, $M_T$ the torque moment,
$K_F$ the fixation stiffness and $\ell_{BP}$, $\ell_B$ and $\ell_s$, $r_s$
endcap length, free measurement length and sample length and radius
respectively. $J_{BP}$ and $G_{BP}$ denote the polar moment of inertia
and the shear modulus of the endcap. A rule of mixture was used to
calculate $G_{BP}$:
\begin{linenomath}
  \begin{equation}
    \label{eq:endcapShearModulus}
    G_{BP} = v_f G_P + \left( 1 - v_f \right) G_B
  \end{equation}
\end{linenomath}
Where, $G_P = E_P / 2 \roundpar{1 + \nu}$ is the shear modulus of PU
and $G_B$ of bone which was derived in a similar fashion. Young's
modulus of PU was determined to $2014\ \mathrm{MPa}$ in seven separate
uniaxial compression tests using the same protocol as for the bone
specimens. In case of bone, Young's modulus $E_B$ was directly
measured using microindentation. Young's and shear moduli were
obtained from the stress-strain curves using a moving regression
(Figure~\ref{fig:exemplaryDatasets}).
\begin{figure}[!ht]
  \centering
  \includegraphics[width=.9\textwidth]{fig_exemplaryDatasets.pdf}
  \caption{Loading part of exemplary datasets for compression, tension
    and torsion is shown. The black line indicates the data derived
    from the uniaxial measurements. The grey straight line is the
    result of a moving regression to determine Young's and Shear
    moduli at the stiffest part of the curve.}
  \label{fig:exemplaryDatasets}
\end{figure}


Microindentation tests were performed on the surrounding cancellous
tissue of the cylinders obtained after coring. Two $5\ \mathrm{mm}$
thick slices were cut from the cancellous core using the high
precision band saw, one for wet and one for dry indentation. The
specimen preparation and the load protocol have been described
previously \citep{WoWiZy09}. The samples for wet indentation were
rehydrated in $99\%$ saturated Hank's Balanced Salt Solution (HBSS)
for $1.5\ \mathrm{h}$ prior to microindentation and submerged in HBSS
during testing. Indentation was performed using a depth-sensing
nanohardness tester equipped with a pyramidal Berkovich diamond tip
and in case of the wet indentation additionally with a wet cell (CSM
Instruments SA, Peseux, Switzerland). Three indentations were
performed on three axial trabecular cross-sections in the core and the
periphery of trabeculae giving 18 indentations per sample assuming a
Poisson's ratio of $\nu^{tiss} = 0.32$ \citep{RhTsPh97, HeEnPe03}.
Elastic moduli of the tissue $\left(E^{tiss}\right)$ were obtained
according to \citet{OlPh92} and averaged over all 18 indentations per
sample. This yielded a specific tissue modulus under wet and dry
conditions for each cylindrical specimen.

Micro FE models were setup for \emph{parFE}~\citep{ArLeMe08} from
CT-scans by converting image voxels to 8-noded volume
elements. Preliminarily, the greyscale datasets were scaled to
resolutions $\left(24\ \mathrm{\upmu m}\right)^3$, $\left(36\
  \mathrm{\upmu m}\right)^3$ and $\left(48\ \mathrm{\upmu m}
\right)^3$ and thresholded using IPLs (Image Processing Language,
SCANCO Medical AG, Br{\"u}ttisellen, Switzerland) single level
threshold. In-plane degree of anisotropy $DA_{ip} = 1.03 \pm 0.04$,
volume fraction $v_f = 0.11 \pm 0.03$ and deviation of trabecular main
direction from the cylindrical main axis $\alpha = 4.78^\circ \pm
2.36^\circ$ were determined using IPL. All nodes in the bottom
cross-section of the cylinders were constrained to $\left(u_x = u_y =
  u_z = 0\right)$. In case of uniaxial tension and compression, the
nodes of the top cross-section were loaded with a displacement in
$z$-direction and constraint in the other directions $\left(u_x = u_y
  = 0,\ u_z = 0.1\ \mathrm{mm}\right)$. In case of torsion, the top
plane was loaded with a rotation $\varphi$ around the middle axis of
the cylinder $\left( u_x = u_y \neq 0,\right.$ $\ u_z = 0,$
$\left.\varphi = 0.1^\circ \right)$. Forces and Moments were recorded
and apparent elastic and shear moduli $E_{app}$ and $G_{app}$ were
computed.

Statistical analyses were performed with R (R Foundation for
Statistical Computing) \citep{Cr05}. For nanoindentation, outliers
were defined as any measurement below the $5^{th}$ or above the
$95^{th}$ percentile leaving $580$ out of $648$
measurements. Simulations and measurements of apparent moduli were
first compared using Wilcoxon's Signed Rank Test to test for
differences in the distributions to a significance level of $p =
0.05$. The concordance correlation coefficient~$r_c$ \citep{Li89} in
combination with linear regression was used to evaluate the agreement
between experiment and simulation with respect to $E_{app}$ and
$G_{app}$. In addition, Bland-Altman plots \citep{BlAl86} were used to
compare experiment and simulation in a one by one fashion.

\section{Results}
No difference could be found between the degree of anisotropy and the
bone volume fraction of the samples for compression, tension and
torsion $\left(p < 0.05\right)$. Different resolutions of the \mufe
models did not lead to differences in simulated $E_{app}$ or $G_{app}$
with $r_c \geq 0.98$ except for torsion at $48\ \mathrm{\upmu m}$
where a drop of $r_c$ to $0.95$ was observable
(Figure~\ref{fig:influenceRes}). Therefore, the focus is laid on the
results for $\roundpar{36\ \upmu m}^3$.
\begin{figure}[!ht]
  \centering
  \includegraphics[width=.9\textwidth]{fig_influenceRes.pdf}
  \caption{Influence of the resolution of the \mufe models on the
    apparent modulus for the three load cases. Concordance correlation
    coefficients $\left(r_c\right)$ were computed for the
    comparison. The results at $24\ \upmu \mathrm{m}$ resolution were
    defined to serve as basis against which the comparison was carried
    out. Resolutions are colour encoded in the figure from $24\ \upmu
    \mathrm{m}$ (black-square), $36\ \upmu \mathrm{m}$ (grey circle)
    to $48\ \upmu \mathrm{m}$ (light grey triangles). The dashed black
    line is the concordance line, i.\ e.\ $24\ \upmu \mathrm{m}$ results}
\label{fig:influenceRes}
\end{figure}

Significantly lower $E^{wet}_{tiss} = 12.2\ \mathrm{GPa}$ was found
for wet indentation conditions compared to dry indentation conditions
$E_{tiss}^{dry} = 15.4\ \mathrm{GPa}$ $\left(p < 0.05\right)$. No
significant difference was found between $E_{app}$ in tension and
compression $\left( p > 0.05 \right)$. Lower apparent moduli were
found when simulating the experiments with tissue properties obtained
under wet $\left( E_{app, UC}^{wet} = 413\ \mathrm{MPa} \right.$,
$E_{app, UT}^{wet} = 436\ \mathrm{MPa}$, $\left.G_{app}^{wet} = 76\
  \mathrm{MPa}\right)$ and dry indentation conditions $\left( E_{app,
    UC}^{dry}\right.$ $\left.= 513\ \mathrm{MPa}\right.$, $E_{app,
  UT}^{dry}$ $= 536\ \mathrm{MPa}$, $\left.G_{app}^{dry} = 100\
  \mathrm{MPa}\right)$ ($p < 0.05$,
Figure~\ref{fig:anovaBarplots36}). No significant difference could be
found for $E_{app}$ or $G_{app}$ between experiment and simulations
with tissue properties from wet indentation $\roundpar{p < 0.05}$ but
between experiment and simulations with dry tissue properties
$\roundpar{p > 0.05}$.
\begin{figure}[!ht]
  \centering
  \includegraphics[width=.9\textwidth]{fig_anovaBarplots36.pdf}
  \caption{Mean apparent elastic moduli. Top left, tissue moduli for
    two different microindentation testing conditions. Top right,
    comparison compressive apparent moduli against experiment. Bottom
    left, comparison of tensile apparent moduli against
    experiment. Bottom right, comparison of apparent shear moduli
    against experiment. Whiskers denote $\pm 1$ standard
    deviation. The asterisk denotes significance to a significance
    level of $p = 0.05$.}
  \label{fig:anovaBarplots36}
\end{figure}

Significant linear regressions between experiment and simulation were
found for compression, tension and torsion for both wet and dry tissue
properties $\left(p < 0.05 \right)$. Good concordance between
experiment and simulation with wet tissue properties was found for
compression $\left(r_c^{wet} = 0.97\right)$ and tension
$\left(r_c^{wet} = 0.97\right)$. Moderate concordance between
experiment and simulation with wet tissue properties was found for
torsion $\left(r_c^{wet} = 0.85\right)$. When comparing experiment and
simulation with dry tissue properties $r_c$ was distinctly lower in all
load cases (Figure~\ref{fig:regression36}).
\begin{figure}[!ht]
  \centering
  \includegraphics[width=\textwidth]{fig_concordance36.pdf}
  \caption{Concordance correlation coefficient of experimental
    apparent elastic moduli and computed apparent elastic moduli at a
    resolution of $\left(36\ \mathrm{\upmu m}\right)^3$. Squared
    points encode compression, circular points encode tension,
    triangular points encode torsion and diamonds encode
    outliers. Different tissue properties obtained from
    microindentation under wet and dry conditions are colour encoded
    with black and grey respectively. The dashed line indicates the
    concordance line $y = x$.}
  \label{fig:regression36}
\end{figure}

The overestimation of $E_{app}$ and $G_{app}$ when using dry tissue
properties in the simulation (Figures~\ref{fig:regression36}) can as
well be observed in the Bland-Altman plots
(Figures~\ref{fig:BlandAltman36}). Negative mean difference between
experimental and simulated moduli were found with $-89.2\
\mathrm{MPa}$ in compression, $-97.3\ \mathrm{MPa}$ in tension and
$-14.9\ \mathrm{MPa}$ in torsion. In contrast the wet tissue
properties deliver mean differences of $10.6\ \mathrm{MPa}$ in
compression, $2.3\ \mathrm{MPa}$ in tension and $9.2\ \mathrm{MPa}$ in
torsion. Compression and tension show low lines of agreement ($\pm
1.96$ standard deviation) which contain all data points in case of wet
tissue properties. In case of torsion these are more severe due to the
outliers (Figures \ref{fig:BlandAltman36}).
\begin{figure}[!ht]
  \centering
  \includegraphics[width=\textwidth]{fig_BlandAltman36.pdf}
  \caption{Bland-Altman plots of measured apparent elastic moduli and
    simulated apparent elastic moduli at a resolution of the FE models
    of $\left(36\ \mathrm{\upmu m}\right)^3$. Squared points encode
    compression and circular points encode tension. Wet and dry
    microindentation conditions are colour encoded with black and
    light grey respectively. The solid line denotes respective means
    of the differences and the dashed lines the lines of agreement
    ($\pm 1.96$ standard deviation from the mean difference).}
  \label{fig:BlandAltman36}
\end{figure}

\section{Discussion}
The goal of this study was to set up valid specimen specific \mufe
models of vertebral trabecular bone using experimentally obtained
specimen specific effective tissue properties. 

Judging from the Wilcoxon tests, no difference could be found between
experiment and simulation under moist testing conditions
(Figure~\ref{fig:anovaBarplots36}). The apparent Young's moduli were
slightly higher than those found by \citet{KePiCr97}. However,
$E_{app}$ lay within the ranges reported by \citet{KoKe98} and
\citet{MoBaKe03}. The differences could be due to different endcap
techniques and slightly different testing setups and preparation
protocols. For the apparent shear moduli no differences between
experiment and simulation under moist conditions could be found as
well. The shear moduli obtained in the present study were lower than
those presented by \citet{BrDuRu99} for femoral and by \citet{RiZy09}
for different sites. This difference could be due to the higher volume
fraction in these studies and as well to different testing setups and
preparation protocols.

The linear regressions for tension and compression indicate that
simulation and experiment could be related. The small deviation of
$r_c$ from one indicated small errors in the comparison of experiment
and simulation in a one by one fashion. Only small location and scale
shifts from the concordance line were observable in the comparison of
experiment and simulation with wet tissue properties
(Figure~\ref{fig:regression36}). This was as well observable in the
scatter in the Bland-Altman plots
(Figure~\ref{fig:BlandAltman36}). These small errors were presumably
due to the fact that the endcaps were not modelled explicitly. Instead
just the free gage length with idealised fixed-end boundary conditions
was modelled. It was assumed that this procedure causes just minor
errors \citep{BeEsFa09}. In case of torsion, $r_c$ was worse than in
case of tension and compression with worse location and scale shifts
(Figure~\ref{fig:regression36}). The lines of agreement in the
Bland-Altman plots were relatively higher as in case of uniaxial
tension and compression relative to the value range. This was
presumably due to the applied boundary conditions. \citet{BeEsFa09}
states that in case of torsion those moulded endcaps could lead to
slight pooling/wicking of the PU into the gage length. This would
cause boundary conditions different from those used for the \mufe
models. Furthermore, the rather unphysiological load case could have
led to early damage which in turn could have softened the material. In
addition, torsion could have lead to more bending and geometrical
nonlinearities. These effects were not included in the linear \mufe
models.
\begin{figure}[!ht]
  \centering
  \includegraphics[width=.9\textwidth]{fig_torsionSED.pdf}
  \caption{Strain energy density (SED) pattern in two extreme torsion
    samples. Left, a sample below and right, a sample above the
    concordance line in Figure~\ref{fig:regression36}. Rising SED is
    color encoded from blue to red. The grey circles point to areas
    with high SED where damage damage could have occured.}
  \label{fig:SED}
\end{figure}

The overall good $r_c$ in combination with the mean difference close
to zero and the low limits of agreement in the Bland-Altman plots
between experiment and simulation with wet tissue properties for
compression and tension indicates validity of these
models. Nevertheless, minor inter-specimen variations were obvious and
the one by one comparison between experiment and simulation was not
perfect. In case of torsion, the moderate agreement should be
attributed to the boundary conditions and not to the models and their
parameters. The worse concordance and the bigger limits of agreement
of the models with dry tissue properties were relatable to the dry
tissue moduli which were too stiff. This presumably caused as well the
greater scatter between experiment and simulation. Using a tissue
modulus of around $15\ \mathrm{GPa}$ leads to a clear overestimation
of the apparent stiffness. Hence, tissue moduli gained under similar
testing conditions between macroscopic and microscopic experiments
(dry, wet or alcohol dehydrated) should be used.

The experimentally obtained tissue moduli under wet conditions were
approximately twice as high than the numerically calibrated found by
\citet{RiWeHu95} and \citet{VeRiMu08}. Both tissue moduli obtained
under wet and dry indentation conditions were lower than those found
by \citet{NiFeJo00} and \citet{BaMoNi04} for femoral trabecular
bone. Interestingly, \citet{BeEsFa09} used similar methods as the two
last studies on vertebral trabecular bone and found just a slightly
lower mean effective tissue modulus than the here presented, obtained
under wet indentation conditions. The authors investigated the
influence of the boundary conditions of the FE model on the numerical
determination of an effective tissue modulus. This could indicate that
their effective tissue moduli are in a sound range and that the here
presented models were properly constrained. Tissue moduli found in the
present study were lower than those used in \citet{RiHuEc03}. This was
probably due to the fact that the tissue moduli used in that study
were obtained from alcohol dehydrated bone specimens
\citep{TuRhTa99}. It is known that alcohol dehydration increases the
stiffness of bone tissue \citep{BuFeBo04}. However, these moduli lay
in the range of the tissue moduli found under dry conditions in the
present study. Wet tissue moduli were approximately $25\%$ higher than
those found by \citet{HoMoKo00} who measured on transverse trabecular
cross-sections under wet conditions.  This is almost the ratio between
elastic moduli measured on axial trabecular cross-sections and elastic
moduli measured on transverse trabecular cross-sections as presented
by \citet{RhTsPh99}. This indicates that the tissue moduli found in
this study are in good agreement with the literature.

The chosen resolution of $\left(36\ \mathrm{\upmu m} \right)^3$ was
approximately one fifth of a trabecular diameter in
vertebrae. \citet{VeRiMu08} found dependencies of the results on the
resolution of \mufe models. The authors started at $\left(40\
  \mathrm{\upmu m} \right)^3$ and coarsened to $\left(60\
  \mathrm{\upmu m}\right)^3$. \citet{BeKe09} found that nonlinear
\mufe models at resolutions of $\left(80\ \mathrm{\upmu m}\right)^3$
and less are proper tools to examine micro-architecture related
problems. Hence, the here chosen resolution seems to be above standard
values in the literature. Since the here investigated linear models
are considered to form the basis for nonlinear computations the chosen
resolution of $\left(36\ \mathrm{\upmu m} \right)^3$ was regarded as a
good compromise between accuracy and computation time.

This study suggests that it is possible to directly set up valid \mufe
models of vertebral trabecular bone in the linear range. This was
possible by using experimentally obtained specimen specific effective
tissue moduli under wet, i.e.\ more physiologic conditions. Except
isotropy, no initial guess on the effective tissue properties was
made. Furthermore, no numerical calibration process to obtain them was
necessary. The presented models can be considered valid in the linear
range and thus eligible to form the basis for nonlinear computations.

\section*{Conflict of interest statement}
The authors have no potential conflict of interest regarding this
article.

\section*{Acknowledgements}
This study was supported by the Deutsche Forschungsgemeinschaft
(German Research Foundation), grand number WI 1352/13 -- 1. We want to
thank the team of the Institute of Lightweight Design and Structural
Biomechanics. Special thanks goes to Enrico Dall'Ara, Dieter Pahr and
Peter Varga for fruitful discussions and advice.

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