# Multiscale modeling of viscoelastic behavior in demineralized human trabecular bone

Zhuang Xiong, Morad Bensidhoum, Daniel Ferry, Xingrong Huang, Esther Potier, Thierry Hoc

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Zhuang Xiong, Morad Bensidhoum, Daniel Ferry, Xingrong Huang, Esther Potier, et al. Multiscale modeling of viscoelastic behavior in demineralized human trabecular bone. International Journal of Mechanical Sciences, 2025, 307, pp.110887. 10.1016/j.ijmecsci.2025.110887. hal-05387059

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# Multiscale modeling of viscoelastic behavior in demineralized human trabecular bone

Zhuang Xiong<sup>a, b</sup>, Morad Bensidhoum<sup>a</sup>, Daniel Ferry<sup>b, c</sup>, Xingrong Huang<sup>c</sup>, Esther Potier<sup>a</sup>, Thierry Hoc<sup>a, d, *, e</sup>

## ARTICLE INFO

### Keywords:

Viscoelastic materials

Bone biomechanics

Multiscale modeling

Demineralized trabecular bone

Collagen matrix characterization

Raman spectroscopy analysis

## ABSTRACT

Bone is a hierarchical composite of inorganic and organic phases, with the former governing stiffness and strength, and the latter controlling toughness and post-yield behavior. The mechanical properties of the organic phase, particularly in demineralized trabecular bone, remain poorly understood, limiting insights into fracture mechanisms and scaffold design. In this study, we investigate the time-dependent viscoelastic behavior of human demineralized trabecular bone using an integrated approach combining stress-relaxation testing, micro-finite element simulations, and Raman spectroscopy at macro- and microscales. Multiscale equilibrium and instantaneous moduli were quantified and correlated with trabecular microarchitecture and collagen spectral markers. Our results reveal how collagen integrity and microarchitecture jointly govern viscoelastic behavior, providing a mechanistic link between tissue composition and macroscopic mechanical performance. These findings not only advance fundamental understanding of the organic matrix mechanics in trabecular bone but also establish a predictive framework for designing biomimetic scaffolds that replicate the native biomechanical and biochemical microenvironment.

## 1. Introduction

Bone tissue is a hierarchically structured composite material consisting of inorganic carbonated hydroxyapatite, organic matrix (90 % type I collagen and 10 % non-collagenous proteins), and water [1]. The inorganic phase is stiff and brittle, while the organic phase is softer and exhibits viscoelastic, time-dependent behavior [2,3]. Together, these two phases provide bone with exceptional strength and toughness for daily load-bearing [4].

Despite its robust structure, bone is susceptible to fractures caused by trauma, repetitive loading, aging, or disease [5–9]. For example, compact bone has a finite fatigue life of only 5–10 years [6,7], and patients with osteoporosis are prone to fragility fractures due to reduced bone quantity and quality [10]. These observations highlight the importance of understanding bone mechanics across multiple scales [11].

Bone's mechanical properties are governed by factors operating at two distinct scales: macroscopic architecture and microscopic tissue-

level composition [12,13]. At the macroscopic level, porosity strongly influences mechanical performance, with increased porosity correlating negatively with Young's modulus and fracture toughness in cortical bone [14,15]. For trabecular bone, porosity remains the primary determinant of apparent mechanical properties, although architecture also contributes [16–18]. At the microscopic level, both mineral content and collagen quality play crucial roles, where increased mineralization increases stiffness [13,19–21] and non-enzymatic collagen crosslinking increases brittleness [22,23].

While both organic and inorganic phases contribute to bone's mechanical behavior, the organic phase predominantly governs toughness and post-yield behavior [24]. Clinical studies demonstrate that collagen degradation through glycation reduces yield strain and ultimate stress in cortical bone [25], highlighting the importance of collagen integrity in maintaining mechanical competence. Through the denaturation of collagen, a reduced elastic modulus, failure stress, and failure strain can be observed in human cortical bone [26]. This suggests that investigating the organic matrix's properties could provide critical insights into

<footnote><sup>a</sup> Université Paris Cité, CNRS, INSERM, ENVA, B3OA, 75010, Paris, France</footnote>

<footnote><sup>b</sup> Aix Marseille Univ, CNRS, CINaM, 13288, Marseille</footnote>

<footnote><sup>c</sup> Sino-French Carbon Neutrality Research Center, Ecole Centrale de Pékin/School of General Engineering, Beihang University, 100191, Beijing, China</footnote>

<footnote><sup>d</sup> Mechanical Department, MSGMGC, Ecole Centrale de Lyon, 69134, Ecully, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: thierry.hoc@cnrs.fr (T. Hoc).</footnote>
fracture mechanisms and risk assessment [27].

Previous studies on demineralized cortical bone have revealed non-linear stress-strain behavior under monotonic [28–31] and cyclic loading [30,32]. Bracher et al. recently examined the macroscopic tensile properties of the demineralized collagen matrix from aging human cortical proximal femur [33]. In contrast, demineralized trabecular bone remains comparatively understudied. Existing research on demineralized trabecular bone has primarily focused on macroscopic properties [34,35], leaving tissue-level mechanics poorly understood. Chen et al.'s [34] work on demineralized trabecular bone and Xie et al.'s [35] constitutive modeling of bovine trabeculae under cyclic loading represent important advances, but neither study addressed human tissue nor explored the collagen quality-mechanics relationship at the tissue level.

In addition to elucidating the mechanical properties of native tissue in the context of bone fracture mechanisms, characterizing the mechanical behavior—particularly the viscoelasticity—of demineralized trabecular bone holds significant relevance for bone tissue engineering and bone-repair strategies. Demineralized trabecular bone has already been recognized as a promising biomaterial for tissue engineering applications due to its favorable properties, such as biocompatibility, high porosity, compatibility with composite materials, and biodegradability [36–38]. Recent studies have demonstrated that the viscoelastic properties of the tissue engineering scaffolds play a critical role in modulating cellular behavior, such as enhanced cell migration and reduced apoptosis [39–43]. In recent years, tuning the viscoelastic properties of synthetic scaffolds has gained attention as a promising strategy not only for tissue engineering but also for replicating the altered mechanical microenvironments observed in various pathological conditions [44–47]. However, the development of biomimetic bone tissue engineering scaffolds remains constrained by the limited understanding of the viscoelastic behavior of organic phase of bone tissue at the microscopic scale. To the best of our knowledge, comprehensive researches on the time-dependent mechanical properties of demineralized trabecular bone are lacking, hindering the rational design of scaffolds that accurately mimic native bone mechanics.

In this study, we provide the first comprehensive investigation of the viscoelastic behavior of demineralized human trabecular bone by bridging macroscopic mechanics, tissue-scale simulations, and collagen quality. By integrating stress-relaxation testing, micro-finite element modeling, and Raman spectroscopy within a unified framework, we uncover how microarchitecture and collagen structure jointly govern mechanical performance across scales. Unlike previous work restricted to macroscopic characterization, our approach directly links Raman-derived biochemical markers to tissue viscoelasticity and, ultimately, to macroscopic mechanical behavior. These findings not only advance fundamental understanding of organic matrix mechanics in trabecular bone but also open new perspectives for fracture prediction and the rational design of biomimetic scaffolds with tunable viscoelastic properties.

## 2. Materials & methods

This study investigates the viscoelastic behavior of demineralized human bone allografts, where the mineral phase has been removed to specifically examine the collagenous matrix. A multi-scale experimental and numerical approach was employed to characterize structural and mechanical responses at both macroscopic and microscopic levels. The methodology included (i) sample preparation and controlled demineralization, (ii) micro-computed tomography analysis of trabecular architecture, (iii) compression-relaxation testing with viscoelastic modeling, (iv) micro-finite element simulations to assess scale-dependent mechanics, and (v) Raman spectroscopy to probe collagen structure. Statistical analyses were performed to assess the significance of the observed results. Detailed procedures are described below.

### 2.1. Sample preparation

Bone allograft samples were obtained from a bone bank (Biobank, France) that collects trabecular bone from femoral heads of human donors as illustrated in Fig. 1. The sampling direction and the donor information were not specified. Bone tissue samples were devitalized and defatted using the Supercrit® inactivation process and sterilized by gamma irradiation at 25 kGy, the bone marrow inside the samples were also removed during the treatment [48–50]. Then cylindrical allograft samples (6.9 mm diameter and 10 mm height) were retrieved from 30×20×10 mm³ trabecular bone blocks using a trephine in the water bath and then air-dried at room temperature (see Fig 1a and 1b). The study involved the utilization of fifteen samples from six distinct bone blocks.

### 2.2. X-ray computed tomography analysis

X-ray computed tomography (micro-CT) was used to obtain the 3D images of the fifteen samples. The micro-CT images were acquired using the Skyscan 1172 (Bruker, Belgium). The acquisition parameters were set to 40 kV, 100 µA, and an isotropic voxel size of 7.9 µm corresponding to protocol used in our previous study [51,52]. Then the images were reconstructed with NRecon software (v 1.7.1.0, Bruker, Belgium). The grey level threshold for bone tissue was calculated using the 3D auto-Otsu algorithm provided in the CTAn software (v 1.17.7.2, Bruker, Belgium). In the present study, the following bone microarchitectural parameters were determined [17,53]: bone volume fraction (BV/TV), bone surface-to-volume ratio (BS/BV), trabecular thickness (Tb.Th), trabecular number (Tb.N), trabecular separation (Tb.Sp), and degree of anisotropy (DA).

### 2.3. Bone sample demineralization

The mineralized samples were immersed in Osteosoft® solution (Merck, Germany) for a period of two weeks at room temperature with the objective of removing the inorganic phase in the tissue (see Fig. 1d). The Osteosoft solution was composed primarily of EDTA in order to ensure a steady and mild demineralization process without altering the structural integrity of each sample. The solution volume was ten times higher than that of the sample and the solution was replaced every 48 h.

Following the 2-week demineralization period, all samples were examined using the Skyscan 1172 micro-CT (Bruker, Belgium). This step confirmed the complete demineralization of our samples: the average BV/TV of the samples decreased from 25.54 ± 7.34 % before demineralization to 0.001 ± 0.001 % after demineralization (see Fig. 1c). Subsequent to the demineralization process, all samples were preserved in phosphate-buffered saline (PBS) solution at 4 °C.

### 2.4. Viscoelastic compression-relaxation tests

Following demineralization, all samples underwent compression-relaxation testing under ambient conditions (room temperature, open air) using an Instron 5566A universal testing machine (MecaBio platform, Aix Marseille Université, France) fitted with a 100 N load cell. The testing protocol consisted of three initial loading-unloading cycles (0–2 % strain) at a constant displacement rate of 1 mm/min. Subsequently, a uniaxial compression test was performed at the same displacement rate (1 mm/min) until a strain of 4 % was achieved. The displacement was then held constant for 600 s to allow stress relaxation until equilibrium was attained. Finally, unloading was conducted at 1 mm/min. Throughout the test, force and displacement data were recorded at a sampling frequency of 50 Hz.

### 2.5. Viscoelastic model

The viscoelastic behavior of the material was modeled using a
<figure><img src="image_4.png" /><figcaption>Fig. 1. Overview of the protocol combining material shaping, demineralization, and finite element simulation. (a) Human trabecular bone bloc (10 mm thick) obtained from a commercial biobank; (b) cylindrical sample (6.9 mm diameter, 10 mm height) extracted from the bloc; (c) bone volume fraction before and after demineralization, measured with an X-ray microscanner; (d) Demineralized sample image; (e) 3D mesh of the bone core generated with Avizo software; (f) distribution of Von Mises stresses within the bone core, calculated with the FEBio finite element code after a 4 % compression test.</figcaption></figure>

generalized 3-branch Maxwell model. At the macroscopic scale, the
relaxation modulus of each sample was described using a three-branch
Prony series:

$$
G^M(t) = G_0^M + \sum_{i=1}^{3} G_i^M e^{\left(\frac{t}{\tau_i}\right)} \quad (1)
$$

where $G_0^M$ and $G_i^M$ are the relaxation coefficients at macroscopic scale and $\tau_i$ are the viscoelastic retardation time for each Maxwell branch, derived from the force-time data obtained during compression-relaxation testing. $G^M$ represents the macroscopic shear stiffness and is related to the macroscopic Young's modulus $E^M$ through the standard relation $E^M = 2 G^M (1+\nu)$. To simplify the process, a Poisson's ratio $\nu$ of 0.49 was assumed for all samples, considering the material to be nearly incompressible.

This phenomenological model was implemented in the finite element software FEBioStudio software (v 2.8.1), and in a custom-developed Matlab code (The MathWorks Inc., MA, USA) detailed in our previous published study [54]. This Matlab algorithm was used to determine the parameters $G_0^M$, $G_i^M$ and $\tau_i$ by fitting the macroscopic force-time relaxation data. Finally, the macroscopic instantaneous modulus $E_{ins}^M$ and macroscopic equilibrium modulus $E_{eq}^M$ were calculated as follows [55]:

$$
E_{\text{ins}}^{M} = 2(1 + \nu) \left( G_{0}^{M} + \sum_{i=1}^{3} G_{i}^{M} \right) \quad (2)
$$

$$
E_{eq}^{M} = 2(1 + \nu)G_{0}^{M} \qquad (3)
$$

where $\nu$ is the Poisson's ratio, $G_0^M$ and $G_i^M$ are the macroscopic shear
relaxation coefficients, and the summation accounts for contributions
from all three Maxwell branches. To validate the consistency between
the analytical and finite element models, compression-relaxation tests
were numerically simulated for all samples using FEBio software with
analytically derived parameters. An equivalent homogeneous cylindri-
cal core matching the sample dimensions was meshed and subjected to
uniaxial compression at a 4 % macroscopic strain level, applied at the
same strain rate as in the experimental tests, followed by a 600-second
relaxation step.

The force-time responses obtained from experimental measurements (Exp), analytical solutions (Ana), and finite element simulations (Fe) demonstrate excellent agreement at the macroscopic scale for three

samples with varying BV/TV levels, as shown in Supplementary Data
Fig. S1. This congruence was observed consistently across all BV/TV
variations, confirming the robustness of the modeling approach.

2.6. Micro-finite element simulation

The 3D micro-CT images were used to generate micro-finite element
meshes for each mineralized sample. The meshing process was per-
formed using Avizo software (v 2024.1, Germany) following the process
described in detail in our previous study [52]. In particular, a 3D median
filter of 3 voxels was firstly used to reduce the noise and increase the
image contrast for the 3D images. Then the unconnected components
were detected using the build-in Label analysis module. A
mesh-sensitivity analysis was conducted before the micro-finite element
simulation to maintain a good compromise between model accuracy and
computational cost. Each model has an average facet number of 1.5
million.

Tetrahedral meshes were generated and quasi-static uniaxial
compression simulations along the z-axis of the cylindrical core were
performed using FEBioStudio software (v 2.8.1). All samples were sub-
jected to a macroscopic compressive uniaxial strain level of 4 % as given
in the relaxation tests. Boundary conditions were defined such that all
nodes on the upper surface of the bone core were displaced by 4 % of the
sample height and those on the lower surface were fixed along the core
axis. The loading step lasted 24 s for each sample, and then the relax-
ation step last for 600 s.

For all the samples, a viscoelastic material property of bone with
three viscoelastic branches was assigned in FEBio. Assuming that macro-
to-micro mechanical behavior follows a linear rule, the relaxation
modulus of the bone material at the microscopic scale was described
using Prony series as follows:

$$
G^m(t) = \alpha \left( G_0^m + \sum_{i=1}^{3} G_i^m e^{\left(\frac{t}{\tau_i}\right)} \right) \quad (4)
$$

where $\alpha$ is the macro-to-micro relaxation modulus transformation ratio, and $G_0^M$, $G_i^M$ and $\tau_i$ corresponds to the macroscopic relaxation coefficients and retardation times obtained from Matlab-based identification. As a result, the microscopic instantaneous and equilibrium moduli of the material were calculated as follows:
$$E_{ins}^{M} = 2\alpha(1 + \nu) \left( G_{0}^{M} + \sum_{i=1}^{3} G_{i}^{M} \right) \qquad (5)$$

$$E_{eq}^{M} = 2\alpha(1 + \nu)G_{0}^{M} \qquad (6)$$

where α is the macro-to-micro relaxation modulus transformation ratio, ν is the Poisson's ratio, and $G_0^M$ and $G_i^M$ are the macroscopic relaxation coefficients from Matlab-based identification. For each specimen, the α ratio was manually adjusted in the micro-finite element simulation to reproduce the experimental macroscopic stress-relaxation curves. This optimization procedure yielded strong agreement between model predictions and empirical observations. The finalized mechanical parameters employed for each sample in the FEBioStudio simulations are provided in Supplementary Data Table S2.

## 2.7. Raman spectroscopy

Raman spectroscopy (InVia Raman Microscope, Renishaw, UK) was performed on all samples to analyze the amide I band. A 785 nm laser was used to excite the molecule in the material. The 20x objective and 0.4 numerical aperture produced a laser spot approximately 2 × 5 µm² in area on the sample. Acquisitions were made over the spectral range from 1550 cm⁻¹ to 1750 cm⁻¹ with an integration time of 30 s, three accumulations and a holographic grating (1200 lines.mm⁻¹) providing ~1 cm⁻¹ spectral resolution. WiRE™ software (Renishaw, UK) was used to despike the spectra. Background fluorescence was removed from all spectra using a third order polynomial fitting algorithm [56]. All spectra were then processed to minimize signal noise using Matlab software with a second order Savitzky-Golay filter and a frame length of 15 [57].

In the present study, the amide I band is considered to be composed of four sub-peaks that characterize the structure and properties of type I collagen as listed in Table 1 [57–60].

Identification of sub-peak ratios within Amide I band was performed using second derivative method. A typical curve of the spectrum obtained and the associated second derivative is shown in Fig. 2. Based on this method, the four sub-peaks were close to 1608, 1637, 1670, and 1691 cm⁻¹. All sub-peak ratios were calculated by directly taking the intensity value corresponding to the position of each sub-peak in the spectrum.

## 2.8. Statistical analysis

Statistical analyses were performed using GraphPad Prism (version 9.2, GraphPad Software, CA, USA, www.graphpad.com (accessed on November 10, 2023)). In all cases, p values < 0.05 were considered significant.

# 3. Results

This section presents a multi-scale investigation of the viscoelastic behavior of demineralized bone allografts. Macroscopic and microscopic mechanical properties are first analyzed in relation to bone volume fraction and other microstructural parameters. Micro-finite element

<caption>**Table 1**</caption>

Sub-peaks in Amide I band from Raman spectroscopy and their corresponding functionality.

<table><thead><tr><td>Raman spectra</td><td>Related functionality</td></tr></thead><tbody><tr><td>1690cm<sup>-1</sup>[57, 60]</td><td>Immature crosslinks and lack of hydrogen bonds, also relates to β-sheet or disordered secondary structure</td></tr><tr><td>1670 cm<sup>-1</sup>[58, 60]</td><td>Strongest amide I ν(C = O) component, β-sheet conformation</td></tr><tr><td>1640 cm<sup>-1</sup>[59, 60]</td><td>Ordered structure in α-helix conformation</td></tr><tr><td>1610 cm<sup>-1</sup>[60]</td><td>3<sub>10</sub> helix, stretching vibration of a carbon ring such as in tyrosine and phenylalanine</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Raman spectrum of a demineralized bone tissue in the 1600–1720 cm⁻¹ range, corresponding to the Amide I band of collagen. The raw spectrum (black line) and the smoothed spectrum obtained using a second-order Savitzky–Golay filter with a frame length of 15 (red line) are shown. The second derivative (blue line) reveals four sub-peaks at approximately 1609, 1637, 1670, and 1691 cm⁻¹.</figcaption></figure>

simulations are then employed to quantify the contributions of trabecular microarchitecture and local collagen composition, while Raman spectroscopy is used to link collagen features to microscale mechanics. Finally, the predictive capability of Raman-derived properties for macroscopic mechanical behavior is demonstrated.

## 3.1. BV/TV correlates with the macroscopic mechanical properties of demineralized samples

Micro-architectural parameters of the mineralized bone samples, derived from 3D micro-CT reconstructions, are summarized in Table 2. The mean values (± standard deviation) for the fifteen samples were as follows: bone volume fraction (BV/TV) = 25.54 ± 7.34 %, bone surface-to-volume ratio (BS/BV) = 17.72 ± 3.18 mm⁻¹, trabecular thickness (Tb. Th) = 0.20 ± 0.04 mm, trabecular number (Tb.N) = 1.30 ± 0.20, trabecular separation (Tb.Sp) = 0.64 ± 0.11 mm, and degree of anisotropy (DA) = 0.46 ± 0.09. Notably, BV/TV exhibited the highest normalized standard deviation (28 %), indicating greater variability compared to other parameters. The individual microarchitectural parameters for each sample are given in Supplementary Data Table S1.

Mechanical properties of demineralized samples revealed an average macroscopic instantaneous modulus of 0.24 ± 0.16 MPa and an average macroscopic equilibrium modulus of 0.13 ± 0.11 MPa (Table 2). A substantial range in mechanical properties was observed: the average macroscopic instantaneous modulus varied from 0.095 MPa to 0.738 MPa, while the average macroscopic equilibrium modulus ranged from 0.035 MPa to 0.477 MPa.

Pearson's linear regression analysis, assuming normal distributions,

<caption>**Table 2**</caption>

Average macroscopic instantaneous, equilibrium modulus and micro-architectural parameters of all the samples.

<table><thead><tr><td>Parameters</td><td>Average value</td><td>Standard deviation</td></tr></thead><tbody><tr><td>E<sup>M</sup><sub>ins</sub> (Mpa)</td><td>0.24</td><td>0.16</td></tr><tr><td>E<sup>M</sup><sub>eq</sub> (Mpa)</td><td>0.13</td><td>0.11</td></tr><tr><td>BV/TV (%)</td><td>25.54</td><td>7.34</td></tr><tr><td>BS/BV (mm<sup>-1</sup>)</td><td>17.72</td><td>3.18</td></tr><tr><td>Tb.Th (mm)</td><td>0.20</td><td>0.04</td></tr><tr><td>Tb.N</td><td>1.30</td><td>0.20</td></tr><tr><td>Tb.Sp (mm)</td><td>0.64</td><td>0.11</td></tr><tr><td>DA</td><td>0.46</td><td>0.09</td></tr></tbody></table>
was conducted to assess correlations between parameters (Fig. 3). BV/TV demonstrated strong positive correlations with both the average macroscopic instantaneous modulus ($E_{ins}^M$, $R = 0.87$) and average macroscopic equilibrium modulus ($E_{eq}^M$, $R = 0.85$). Additionally, BV/TV exhibited significant linear relationships with BS/BV, Tb.Th, Tb.N, and Tb.Sp. When BV/TV and DA are combined (Fig. 3c), the correlation coefficient is further improved, underscoring the importance of micro-architectural parameters in predicting macroscopic behavior.

## 3.2. Inverse method reveals the microscopic mechanical behavior of demineralized samples

Micro-finite element simulations of stress-relaxation tests were performed on demineralized samples, taking into account the micro-architecture of the trabecular bone. The macro-to-micro modulus transformation ratio ($\alpha$) was identified. For each sample, the obtained macro-to-micro modulus transformation ratio, equilibrium modulus, and instantaneous modulus at the microscopic scale are presented in Table 3.

The resulting macroscopic time-reaction force curves, denoted MicroFe, of three representative samples with low, medium, and high BV/TV levels are shown in Fig. 4b-d and compared with the experimental ones. Micro-finite element simulations, incorporating these parameters, achieved excellent agreement with experimental curves during both loading and relaxation steps. The equilibrium modulus at the microscopic scale, optimized for best fit, ranged from 1.1 MPa to 2.07 MPa (mean ± SD: 1.69 ± 0.31 MPa). The instantaneous modulus at the microscopic scale, optimized for best fit, ranged from 2.54 MPa to 4.05 MPa (mean ± SD: 3.43 ± 0.41 MPa). The variation in $E_{ins}^m$ (12 %) is smaller than that of $E_{eq}^m$ (19 %). The macro-to-micro modulus transformation ratio $\alpha$ varied widely (3.81–40.92), reflecting heterogeneity in macroscopic mechanical properties.

At local scale, the micro-finite element simulation results for three samples with different BV/TV are presented in Fig. 5. The von-Mises stress and the axial stress at two different time points, the end of loading step, and the end of relaxation step are presented. The local distribution of stress is heterogeneous among the samples and the stress level is higher for samples with a higher value of BV/TV. A distinct decrease in stress level can be observed from the end of the loading step to the end of the relaxation step.

Correlation analyses were performed between microarchitectural parameters and microscale mechanical properties, with the correlation matrix and associated plots presented in Supplementary Data Fig. S2. No statistically significant relationships were identified between micro-architectural parameters and either $E_{eq}^m$ or $E_{ins}^m$, suggesting that local mechanical behavior is predominantly governed by biochemical factors, such as collagen content and organization, rather than trabecular geometry. The correlation plot between the BV/TV and the transformation ratio $\alpha$ of each sample is presented in Fig. S2 as well and a negative correlation ($R^2=0.62$) was observed.

## 3.3. Homogeneous micro-finite element model reveals the contribution of the microarchitecture parameters on the macroscopic mechanical properties

To isolate the influence of microarchitecture on macroscopic mechanical properties, micro-finite element simulations were conducted using the same 3D models of each sample, with a microscopic equilibrium modulus identical for all samples and fixed to the mean value of $E_{eq}^m$ across all samples equal $\overline{E_{eq}^m} = 1.69$ MPa. Stress-relaxation curves were generated for each sample with this homogeneous equilibrium model, and the corresponding macroscopic instantaneous $E_{ins\_hom}^M$ and equilibrium modulus $E_{eq\_hom}^M$ were derived by using the analytical Matlab viscoelastic model.

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Correlations between microarchitectural parameters and macroscopic mechanical properties. (a) Heatmap showing correlations between various micro-architectural parameters and the instantaneous and equilibrium macroscopic moduli; (b) linear correlations between the macroscopic moduli (instantaneous and equilibrium) and bone volume fraction (BV/TV); (c) linear correlations between the macroscopic moduli (instantaneous and equilibrium) and a linear combination of bone volume fraction and degree of anisotropy, defined as $f(BV/TV, DA) = 0.019BV/TV + 0.24DA$.</figcaption></figure>
<table><caption>**Table 3**
Equilibrium modulus ($E_{eq}^m$), instantaneous modulus ($E_{ins}^m$) at the microscopic scale, and the macro-to-micro modulus transformation ratio ($\alpha$) for each sample.</caption><thead><tr><th>Sample n°</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th><th>12</th><th>13</th><th>14</th><th>15</th></tr></thead><tbody><tr><td>E<sub>eq</sub><sup>m</sup> (MPa)</td><td>1.60</td><td>1.45</td><td>2.00</td><td>1.25</td><td>2.07</td><td>1.53</td><td>2.13</td><td>1.40</td><td>1.90</td><td>1.70</td><td>1.94</td><td>1.10</td><td>2.04</td><td>1.40</td><td>1.82</td></tr><tr><td>E<sub>ins</sub><sup>m</sup> (MPa)</td><td>3.66</td><td>4.05</td><td>3.83</td><td>3.01</td><td>3.55</td><td>3.48</td><td>3.79</td><td>3.35</td><td>3.77</td><td>3.42</td><td>3.43</td><td>2.54</td><td>3.74</td><td>3.07</td><td>2.81</td></tr><tr><td>α</td><td>38.45</td><td>40.92</td><td>9.25</td><td>18.82</td><td>9.47</td><td>18.61</td><td>13.88</td><td>16.47</td><td>19.22</td><td>18.31</td><td>18.83</td><td>19.07</td><td>15.57</td><td>17.10</td><td>3.81</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Time-displacement and force-relaxation responses during stress-relaxation tests. (a) Time-displacement curve corresponding to the applied loading; (b–d) macroscopic force-relaxation curves for samples with different bone volume fractions (BV/TV): (b) sample No. 15, BV/TV = 46.2%; (c) sample No. 8, BV/TV = 32.9%; (d) sample No. 1, BV/TV = 17.9%. In each graph, the experimental response is the red line, and the micro-finite element simulation is the blue line. Across all BV/TV values, the experimental and numerical curves are nearly superimposed.</figcaption></figure>

The Pearson correlation analysis was performed between $E_{eq\_hom}^M$, $E_{ins\_hom}^M$, and BV/TV (Supplementary Data Fig. S3). A strong positive correlation ($R^2 = 0.78$) was observed, indicating that BV/TV accounts for 78% of the variation in macroscopic mechanical properties.

The remaining variation was hypothesized to arise from microscale factors, such as sample-specific chemical composition. To quantify the contribution of microscale factors, the variation parameter $\gamma$ was calculated as the relative difference between the measured macroscopic equilibrium modulus and the modulus predicted by a homogeneous microscale model, i.e., $\gamma = (E_{eq}^M - E_{eq\_hom}^M)/E_{eq}^M$.

Subsequent correlation analysis between $\gamma$ and $E_{eq}^m$ revealed a highly significant relationship ($R^2 = 0.93$), as shown in Supplementary Data Fig. S4. This robust correlation supports the use of $\gamma$ as a reliable indicator of microscale contributions to the mechanical behavior of demineralized bone samples. Furthermore, this strong linear relationship between $\gamma$ and $E_{eq}^m$ allows us to hypothesize that accounting for the characteristics of collagen at the microscopic scale, in combination with architectural parameters, should enable a better prediction of macroscopic viscoelastic behavior.

### 3.4. Multilinear regression results reveal the contribution of Raman spectrum intensity ratios for the microscopic mechanical properties prediction

Raman spectroscopy identified four characteristic peaks in the Amide I band (1610 cm$^{-1}$, 1640 cm$^{-1}$, 1670 cm$^{-1}$, and 1690 cm$^{-1}$), each associated with distinct collagen features (Table 1). The intensity ratio of individual sample is given in Supplementary Data Table S2. To evaluate the influence of collagen structure on mechanical properties, intensity ratios of these peaks were compared with both tissue equilibrium modulus and instantaneous modulus across samples and are presented in Supplementary Fig. S5 and S6. While no strong correlations emerged at the individual sample level, a consistent negative trend was observed between tissue elastic modulus and both 1690/1640 and 1670/1640 ratios, suggesting progressive mechanical weakening with increasing disordered collagen content.

To further investigate the influence of Raman spectral features on microscale mechanical properties, multilinear regression analysis was performed between the six intensity ratios of the Amide I band and the variation parameter ($\gamma$). The resulting regression model:
<figure><img src="image_8.png" /><figcaption>Fig. 5. Local stress distribution obtained by micro-finite element simulations during stress-relaxation tests. (a-f) Von Mises stress distributions for samples with different bone volume fractions (BV/TV) at the end of the elastic loading (t = 24 s) and at the end of relaxation (t = 624 s): sample No. 15, BV/TV = 46.2 %, at (a) t = 24 s and (b) t = 624 s; sample No. 8, BV/TV = 32.9 % at (c) t = 24 s and (d) t = 624 s; sample No. 1, BV/TV = 17.9 % at (e) t = 24 s and (f) t = 624 s. (g-l) Axial stress distributions for the same samples at the end of the elastic loading (t = 24 s) and at the end of relaxation (t = 624 s): sample No. 15, BV/TV = 46.2 %, at (g) t = 24 s and (h) t = 624 s; sample No. 8, BV/TV = 32.9 % at (i) t = 24 s and (j) t = 624 s; sample No. 1, BV/TV = 17.9 % at (k) t = 24 s and (l) t = 624 s.</figcaption></figure>

$$ \gamma_{\text{ram}} = \beta_0 + \sum_{i=1}^{6} \beta_i r_i \qquad (7) $$

(where $r_1 = I_{1690}/I_{1670}$, $r_2 = I_{1690}/I_{1610}$, $r_3 = I_{1670}/I_{1640}$, $r_4 = I_{1670}/I_{1640}$, $r_5 = I_{1670}/I_{1610}$, $r_6 = I_{1640}/I_{1610}$) captured the influence of collagen structure on microscale behavior. The regression coefficients determined from the multilinear analysis were $\beta_0 = -74.16$, $\beta_1 = 83.61$, $\beta_2 = -50.56$, $\beta_3 = 1.127$, $\beta_4 = 41.78$, $\beta_5 = -3.376$, $\beta_6 = 5.517$.

Fig. 6 shows a significant correlation between $E_{\text{eq}}^m$ and $\gamma_{\text{ram}}$ supporting

the utility of Raman spectral data in predicting microscale mechanical behavior.

Based on this linear relationship, the microscopic equilibrium modulus $E_{\text{eq}}^m$ was directly predicted from the Raman-derived variation parameter $\gamma_{\text{ram}}$ as follows:

$$ E_{\text{eq-ram}}^{m} = 1.206 * \gamma_{\text{ram}} + 1.785 \qquad (8) $$

### 3.5. Predicted microscopic mechanical properties accurately represent macroscopic behavior

Using the predicted microscopic equilibrium modulus $E_{\text{eq-ram}}^m$ for each sample, micro-finite element simulations were conducted to generate macroscopic stress-relaxation curves. Macroscopic equilibrium ($E_{\text{eq-ram}}^M$) and instantaneous ($E_{\text{ins-ram}}^M$) moduli were derived using the analytical three-branch viscoelastic model. The correlation plots between the macroscopic equilibrium modulus $E_{\text{eq-ram}}^M$ derived from micro-finite element using the microscopic Raman modulus (respectively $E_{\text{ins-ram}}^M$) and the macroscopic equilibrium modulus $E_{\text{eq}}^M$ derived from experiments ($E_{\text{ins}}^M$ respectively) are shown in Fig. 7. The results demonstrate exceptional agreement, with Pearson correlation coefficients of $R^2 = 0.99$ for the equilibrium modulus and $R^2 = 0.97$ for the instantaneous modulus, respectively. This strong correlation confirms that the micro-scale mechanical properties predicted from Raman spectral data effectively capture the macroscopic mechanical behavior of the samples. These results demonstrate that Raman spectroscopy provides a non-destructive means to predict the microscale mechanical properties of demineralized bone, which in turn accurately reflect macroscopic mechanical behavior. This highlights the potential of Raman-derived collagen metrics as a reliable tool for assessing tissue mechanics without physically altering the sample.

<figure><img src="image_9.png" /><figcaption>Fig. 6. Relationship between the microscopic equilibrium Young's modulus and the Raman intensity ratio $\gamma_{\text{ram}}$. The linear equation $E_{\text{eq}}^m = 1.206 \gamma_{\text{ram}} + 1.785$ allows prediction of the microscopic Young's modulus $E_{\text{eq}}^m$, with a coefficient of determination ($R^2$) of 62 %.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 7. Correlation plots between measured and Raman-based predicted macroscopic moduli. (a) Macroscopic instantaneous modulus of each sample ($E_{ins}^M$) versus the predicted macroscopic instantaneous modulus ($E_{ins,ram}^M$) computed using a mechanical behavior law based on the combination of Raman intensity ratios ($\gamma_{ram}$), with a coefficient of determination of 97%; (b) Macroscopic equilibrium modulus of each sample ($E_{eq}^M$) versus the predicted macroscopic equilibrium modulus ($E_{eq,ram}^M$) computed using the same Raman-based mechanical behavior law, with a coefficient of determination of 99%.</figcaption></figure>

## 4. Discussion

Understanding the time-dependent mechanical behavior of human demineralized trabecular bone and its relationship with microarchitecture and biochemical composition is essential for gaining insights into bone fracture mechanisms [11,22,24,27]. It is also critical for the development of biomimetic scaffolds in bone tissue engineering, which require detailed knowledge of the structural and mechanical properties of native bone, particularly viscoelasticity, as it plays a crucial role in cellular responses [40–43]. In this study, we systematically characterized the micro-architectural, mechanical, and biochemical features of demineralized human trabecular bone allografts at both macroscopic and tissue levels. Stress-relaxation tests combined with micro-CT scanning revealed that bone volume fraction (BV/TV) is a key determinant of mechanical behavior at the macroscopic scale. At the tissue level, the viscoelastic properties of demineralized trabecular bone were comprehensively assessed through stress-relaxation testing and micro-finite element modeling. Integration with Raman spectroscopy further elucidated structure–function relationships at the molecular scale, showing that collagen helical integrity contributes to tissue elasticity. Finally, multilinear regression analysis provided robust correlations between chemical composition and mechanical behavior across scales. Together, these findings not only inform the rational design of scaffolds that replicate the biomechanical and biochemical microenvironment of native bone, but also enhance our understanding of the mechanical role of the organic matrix in bone tissue.

At the macroscopic scale, the instantaneous ($E_{ins}^M$) and equilibrium ($E_{eq}^M$) moduli of demineralized samples were derived from stress-relaxation curves. A significant decrease in elastic moduli was observed following demineralization, consistent with the loss of mineral-phase reinforcement [33]. Notably, demineralized bone specimens exhibited substantially greater variability in mechanical properties compared to their mineralized counterparts, likely due to the disruption of the mineral network that normally contributes to stiffness. In the study by Xie et al., demineralized bovine tibial trabecular bone exhibited an instantaneous compliance of 0.33–0.66 MPa⁻¹ under tensile loading [35]. This order-of-magnitude difference is likely due to inter-species variations, such as thicker collagen fibrils and higher crosslink density in bovine tissue, as well as differences in testing modality (tension versus compression) [32].

Regression analysis revealed a strong positive correlation between bone volume fraction (BV/TV) and both $E_{ins}^M$ ($R = 0.87$) and $E_{eq}^M$ ($R = 0.85$). This observation aligns with previous findings in mineralized trabecular bone, where BV/TV is a dominant predictor of macroscopic mechanical behavior [61–63]. These results indicate that tissue volume

fraction remains a key determinant of macroscopic mechanical properties even in the absence of mineral content. It should also be noted that accounting for microarchitectural anisotropy, as suggested by Maquer et al. [18], improves the prediction of macroscopic behavior. Consequently, the fabric tensor is another important microarchitectural parameter to consider when predicting macroscopic mechanical properties for both mineralized and demineralized tissue. While other microarchitectural parameters, such as trabecular thickness (Tb.Th) and trabecular spacing (Tb.Sp), showed significant associations with mechanical behavior, their collinearity with BV/TV limits the interpretation of their independent effects and emphasizes the dominant influence of tissue volume fraction. These findings underscore that even in the absence of mineral content, microarchitecture—including both volume fraction and anisotropy—remains a key driver of apparent mechanical behavior.

At the tissue scale, this study provides the first reported quantification of tissue-level viscoelastic moduli for human demineralized trabecular bone, with values ranging from 1.1 to 2.07 MPa. Interestingly, the weak correlation between BV/TV and tissue modulus ($R = 0.39$) suggests that tissue mechanics are primarily determined by molecular-level structure rather than by bulk architecture. The inverse method by micro-finite element simulation results achieved exceptional agreement with stress-relaxation mechanical test results, validating this methodology previously limited to mineralized tissue [64]. On the other hand, the multiscale approach used in this study shows that the macro-to-micro modulus transformation ratio ($\alpha$) depends on micro-architectural heterogeneity. Indeed, this heterogeneity is reflected in a wide range of macroscopic-to-tissue modulus ratios (2.47–16.79), with lower BV/TV further amplifying the ratio (Supplementary Fig. S2c). This scaling effect underscores the critical role of pore network topology in governing mechanical behavior across different scales [17].

In terms of bone tissue viscoelasticity [3,65,66], the demineralized bone matrix is primarily composed of type I collagen, a triple-helical protein that self-assembles into fibrils via quarter-staggered molecular packing [11]. These viscoelastic fibrils [67–69] subsequently organize into higher-order fibers and networks, stabilized by both enzymatic and non-enzymatic cross-links [70,71]. Stress-relaxation tests revealed the time-dependent behavior in demineralized trabecular bone, which was quantitatively described using a three-branch generalized Maxwell model. The relaxation response exhibited three characteristic time regimes: (1) rapid short-term relaxation ($\tau_1 = 4.3 \pm 1.5$ s); (2) intermediate relaxation ($\tau_2 = 25.0 \pm 9.0$ s); and (3) prolonged long-term relaxation ($\tau_3 = 385.0 \pm 104.7$ s). These distinct relaxation timescales correspond to the hierarchical deformation mechanisms operating at
different structural levels of the collagenous matrix such as fibril sliding, interstitial fluid flow, and proteoglycan-mediated interactions [72,73].

While the hierarchical viscoelastic behavior of demineralized trabecular bone reflects the structural organization of collagen at multiple scales, we further investigated how specific collagen features—particularly crosslinking patterns and triple-helical integrity—affect tissue mechanical properties, using Raman spectroscopic analysis to link molecular structure with microscopic viscoelasticity. Collagen crosslinking has been shown to significantly influence the mechanical properties of the organic phase in bone tissue [59,74]. In particular, several previous studies [5,26,57,75] have reported that changes in bone toughness—defined as the energy absorbed per unit volume prior to fracture [70,76,77]—are associated with altered collagen fibril packing and increased intermolecular crosslinking. Consequently, modifications in collagen crosslinking within the organic matrix can directly impact bone tissue mechanical behavior, including toughness, fracture resistance, and viscoelasticity. In the present study, Raman spectroscopic analysis revealed significant associations between collagen structural indices and tissue elastic modulus in demineralized trabecular bone. Although no statistically significant correlation was observed between individual intensity ratios and the microscopic equilibrium modulus, two notable negative trends emerged: the 1690 cm⁻¹/1670 cm⁻¹ ratio (R² = 0.17) and the 1690 cm⁻¹/1640 cm⁻¹ ratio (R² = 0.20). The 1690 cm⁻¹/1670 cm⁻¹ ratio reflects the relative abundance of reducible to non-reducible crosslinks, serving as an indicator of collagen aging, while the 1690 cm⁻¹/1640 cm⁻¹ ratio correlates with disruptions in collagen's triple-helical integrity [78,79]. Overall, these results confirm that alterations in collagen structure, whether in crosslinking patterns or triple-helical integrity, directly compromise the mechanical properties of demineralized bone tissue.

To further explore the relationship between collagen structure and mechanical behavior, we next assessed whether a combination of multiple spectral indices could more reliably predict mechanical properties across different scales. Multilinear regression analysis revealed that a linear combination of six spectral ratios could effectively predict both microscopic (equilibrium modulus $E_{eq}^m$: R² = 0.62) and macroscopic mechanical properties (equilibrium modulus $E_{eq}^M$: R² = 0.99; instantaneous modulus $E_{ins}^M$: R² = 0.97) of the demineralized samples. The difference of correlation between the micro- and macro-scopic scales shows that the mechanical behavior at tissue level is only partially influenced by the molecular-level structural changes and contribute not as much as the BV/TV to the mechanical behavior at the apparent level. Taken together, these results demonstrate that collagen's hierarchical organization governs bone mechanical properties across scales, highlighting that both microarchitecture and molecular composition are critical for tissue performance. Importantly, this hierarchical influence has direct implications for tissue engineering, where viscoelastic cues derived from collagen structure play a central role in guiding cell behavior and informing the design of biomimetic scaffolds.

Viscoelasticity is a fundamental mechanical property present in nearly all living tissues. Unlike pure elasticity, this time-dependent behavior enables cells to dynamically interact with their environment, providing essential mechanical cues that influence cellular behavior [39]. Consequently, understanding viscoelasticity is critical for the design of soft biomaterials and the development of physiologically relevant *in vitro* tissue models. The present study has investigated the time-dependent mechanical behavior of demineralized human trabecular bone at both macro- and micro-scales, providing important insights for the design of biomimetic collagen-based scaffolds in bone tissue engineering. Robin et al. [80] demonstrated that collagen's microstructure is equally as critical as its biochemical composition in driving bone regeneration. Their findings highlight the necessity of biomimetic hierarchical organization over mere mineral content when engineering bone substitutes. Our results align with their conclusions, reinforcing that 3D microarchitecture plays a dominant role in determining the

macroscale mechanical properties of demineralized bone, whereas chemical composition exerts a more localized influence. These findings underscore the importance of incorporating structural hierarchy—not just biochemical signals—into future scaffold designs to better replicate native tissue mechanics and function for biomimetic applications [81–87].

Several limitations should be considered when interpreting the results of the present study. First, the donor metadata (age, medical history, and anatomical orientation) were unavailable for the bone samples, potentially introducing uncontrolled biological variability. While gamma irradiation and supercritical CO₂ processing have been shown to preserve macroscopic mechanical properties, their effects on tissue-level mechanics remain unclear. Second, in this study, the behavior of bone trabeculae was considered to be homogeneous and isotropic. However, trabeculae are composed of collagen bundles whose density and orientation vary within the bone architecture. Quantifying collagen bundle orientation and density using 3D imaging could provide valuable microstructural insights [88–90]. Recent work by Alberini et al. [91] offers an advanced methodological framework to address this challenge, employing a 3D Fourier-based analysis to quantify collagen fiber orientation in biological tissues. Third, although EDTA demineralization is generally regarded as minimally disruptive, its long-term effects on collagen fibril mechanics remain largely unexplored. In light of previous findings [92], it should be emphasized that future investigations on the effects of demineralization and other treatments (e. g., CO₂, EDTA) across the full Raman spectrum are warranted. Such systematic analyses would allow the identification of sub-bands least sensitive to treatment, thereby facilitating a more reliable transposition of the results to fresh, untreated bone samples. Taken together, these limitations underscore the importance of conducting future studies with donor-matched controls and comprehensive multiscale characterization, in order to strengthen the translational relevance of the present findings.

## 5. Conclusion

This study provides the first comprehensive multiscale characterization of human demineralized trabecular bone, bridging microarchitecture, tissue-scale mechanics, and molecular composition. By integrating stress-relaxation testing, micro-finite element modeling, and Raman spectroscopy, we established a novel methodological framework linking collagen triple-helical integrity and crosslink density to tissue-scale viscoelasticity, while confirming bone volume fraction (BV/TV) as the dominant determinant of macroscopic mechanical behavior.

Importantly, our findings reveal a decoupling of macro- and micro-scale mechanics: macroscopic competence can be predicted from architecture, whereas tissue mechanics remain governed by molecular organization. This highlights the necessity of capturing hierarchical structure when designing biomimetic scaffolds for bone regeneration. These insights advance current understanding beyond mineralized tissue models and highlight the organic matrix as a critical regulator of bone toughness.

Beyond fundamental contributions, our results set the stage for the rational design of collagen-based biomimetic scaffolds that replicate both the hierarchical structure and the time-dependent behavior of native bone. Future studies incorporating donor-matched controls and anisotropic modeling explicitly accounting for collagen bundle orientation and density will refine predictive accuracy and facilitate the translation of these insights into clinically relevant tissue engineering strategies. Overall, this work provides a multiscale, predictive framework for developing collagen-based scaffolds that faithfully replicate the structural, mechanical, and biochemical environment of native bone, offering a significant advance beyond previous approaches focused solely on mineral content or bulk mechanics.
## The use and declaration of AI and AI-assisted technologies

During the preparation of this work the author(s) used DeepSeek V3 in order to improve readability and language. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication.

### CRediT authorship contribution statement

**Zhuang Xiong:** Writing – review & editing, Writing – original draft, Software, Resources, Investigation, Formal analysis, Data curation. **Morad Bensidhoum:** Writing – review & editing, Validation, Supervision, Funding acquisition, Conceptualization. **Daniel Ferry:** Writing – review & editing, Resources, Data curation. **Xingrong Huang:** Writing – review & editing, Validation. **Esther Potier:** Writing – review & editing, Validation, Supervision. **Thierry Hoc:** Writing – review & editing, Validation, Supervision, Project administration, Funding acquisition, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgement

The authors would like to thank the Smart Bone Organoid project (ANR-21-CE19–0013–01), the Chinese Scholarship Council (CSC), and Dr. C. Camy from the MecaBio platform, Aix Marseille Université for their support.

### Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ijmecsci.2025.110887.

### Data availability

Data will be made available on request.

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