# Replicating bone behavior using 3D-printed structure based on Triply Periodic Minimal Surfaces

Joris Claude, Rémy Gauthier, Christian Olagnon, Hélène Follet

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Joris Claude, Rémy Gauthier, Christian Olagnon, Hélène Follet. Replicating bone behavior using 3D-printed structure based on Triply Periodic Minimal Surfaces. Multidisciplinary Biomechanics Journal, 2025, 50th congress of the.., 10.46298/mbj.16151. hal-05191703v2

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# Replicating bone behavior using 3D-printed structure based on Triply Periodic Minimal Surfaces

Joris Claude<sup>a, b</sup>, Rémy Gauthier<sup>a</sup>, Christian Olagnon<sup>a</sup>, Hélène Follet<sup>b*</sup>

<sup>a</sup> INSA LYON, Univ Claude Bernard Lyon 1, CNRS, MatéIS UMR 5510, Lyon, France

<sup>b</sup> Univ Claude Bernard Lyon 1, INSERM, LYOS U1033, Lyon, France

* Corresponding author: helene.follet@inserm.fr

Received date: 04/04/2025

Accepted date: 27/06/2025

Publication date: 27/10/2025

**Keywords:** osteoporosis, metastasis, 3D printing, bone model, trabecular bone

© 2025 The Authors

Licence CC-BY 4.0

Published by Société de Biomécanique

## 1. Introduction

The utilization of 3D-printed bone models has the potential to address the ethical and practical challenges associated with the collection of human bone samples. These models could serve as a valuable tool in predicting the risk of bone fractures associated with specific pathologies such as osteoporosis or bone metastasis. They can also contribute to the development of novel medical devices and biomaterials (Nizam et al. 2023). To develop such models, it is proposed to use a single 3D-printed material with heterogeneous architecture, with the aim of reproducing mechanical properties of bone.

Triply Periodic Minimal Surfaces (TPMS) represent a state-of-the-art approach to create structures with tunable parameters such as relative porosity, pore size, and pore shape, all of which influence the mechanical properties of the structure. We mimic bone-like behavior by creating a structure with graded TPMS mechanical properties. The purpose of this study is to reproduce the elastic modulus of bone in order to have a similar approach to the numerical models used in finite element simulation for predicting bone fracture risk.

## 2. Methods

### 2.1 Specimen preparation

Four cylindrical cancellous bone samples ($h = 20$ mm, $d = 18$ mm) were extracted from bovine humeri epiphyses and CT-scanned at 40 µm voxel size. A calibration

phantom is used to obtain a bone mineral density value for each pixel of the images.

### 2.1 Construction of mechanical twins

Firstly, bone mineral density values from the CT-scans of the specimens are averaged into $2 \times 2 \times 2$ mm³ elements. These values are converted to elastic modulus values according to the conversion equation from Kopperdhal et al. (2002). Secondly, the elastic modulus of each element is mapped to a TPMS (Gyroid) relative density value according to a Gibson-Ashby law that has been previously determined through experimental studies. Finally, mechanical twins are generated from the relative density mapping of gyroid elements measuring $2 \times 2 \times 2$ mm³. The boundaries between elements are recursively generated using the method based on Sigmoid Functions developed by Yoo (2013).

### 2.3 3D printing of the Mechanical Twins

For each bone specimen, two identical Mechanical Twins (MT) are 3D-printed using a low-cost LCD 3D-printer (Elegoo Saturn S) and a commercial light-cured resin with 30% ceramic loading (Jamg He CR-30; Hardness: 90 Shore D, Flexural Modulus: 2.3 GPa, Flexural Strength: 65 MPa).

A thermal initiator, azobisisobutyronitrile (AIBN), is incorporated into the resin prior to printing using a 1% by mass concentration. After printing, MTs undergo cleaning in an isopropanol bath, followed by drying
and a 24-hour thermal treatment at 80°C. This process releases additional free radicals, which facilitates the completion of the polymerization process in a uniform manner across the structure (Uzcategui et al. 2018).

## 2.4 Mechanical characterisation

All bone samples and Mechanical Twins were tested in axial compression until failure. Elastic moduli are defined as the slope at the inflection point of the elastic part of the stress-strain curve.

# 3. Results and discussion

The results derived from the uniaxial compression of bone samples and mechanical twins are illustrated in (Fig. 1.) and (Tab. 1.).

<table><caption>**Table 1.** Experimental elastic moduli of bone specimens (E_Bone) and their respective Mechanical Twins (E_MT_1 and E_MT_2).</caption><thead><tr><th></th><th>B1</th><th>B2</th><th>B3</th><th>B4</th></tr></thead><tbody><tr><td>E_Bone (MPa)</td><td>570</td><td>300</td><td>370</td><td>860</td></tr><tr><td>E_MT_1 (MPa)</td><td>560</td><td>380</td><td>430</td><td>660</td></tr><tr><td>E_MT_2 (MPa)</td><td>580</td><td>420</td><td>520</td><td>550</td></tr></tbody></table>

The elastic moduli of the mechanical twins deviate by up to 30% from the corresponding bone samples, with an average relative difference of approximately 23%. By considering a deformation failure criterion of 1.9% total strain of the specimen, traditionally used in finite element simulation (Saillard 2024), the experimental method introduced in the current study allows to estimate the risk of failure of a bone sample (Fig. 1.) in a similar way compared to numerical modelling.

Despite the similarity observed in the elastic moduli of the Mechanical Twins and the bone samples, the remaining disparities can be attributed to the use of a relationship experimentally established using human bone. A relationship based on bovine bone samples would be more appropriate to use.

The present study will be extended to new bone samples with the aim of investigating the mechanical variability of Mechanical Twins and efforts will be made to reproduce the ultimate load, ultimate stress and ultimate deformation of a bone specimen in complement of its elastic modulus.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Stress-Strain curve of bovine bone specimen in black B1(left) and B2 (right) and their respective 3D-printed mechanical twins (orange)</figcaption></figure>

This TPMS-based architecture can also be modified to consider bone micro-architectural parameters, such as degree of anisotropy or trabecular thickness, thereby achieving a greater degree of similarity between the mechanical model and the original bone. The use of a ceramic loaded resin allows the parameterization of the size and density of the ceramic particles to better monitor the mechanical properties of the Mechanical Twin. The failure mode of the Mechanical Twins can also be selected according to the type of TPMS cell used.

# 4. Conclusions

The development of this architecture represents an exciting step towards the design of 3D printed mechanical bone models, offering a compelling alternative to the use of human cadaveric specimens. The use of 3D printed bone models has the significant advantage of allowing mechanical tests to be performed with different loading conditions on the same specimen, a limitation inherent in the use of bone specimens.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Contributor Roles

JC: Conceptualization, Data curation, Methodology, Writing original draft. RG: Conceptualization, Methodology, Supervision, Validation, Writing-review & editing, Funding acquisition. CO: Methodology,
Supervision, Validation, Writing-review & editing.
HF: Conceptualization, Methodology, Supervision,
Validation, Writing-review & editing, Funding
acquisition.

Data, software, code availability

Data will be made available on request. The python
code used to generate the mechanical twins will be
available on request after publication.

References

Nizam, M., Purohit, R., & Taufik, M. (2025). Role of 3D printing in healthcare: A comprehensive review on treatment and training. *Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine*, 09544119251321585. https://doi.org/10.1177/09544119251321585

Kopperdahl, D. L., Morgan, E. F., & Keaveny, T. M. (2002). Quantitative computed tomography estimates of the mechanical properties of human vertebral trabecular bone. *Journal of Orthopaedic Research*, 20(4), 801–805. https://doi.org/10.1016/S0736-0266(01)00185-1

Yoo, D.-J. (2013). Heterogeneous porous scaffold design using the continuous transformations of triply periodic minimal surface models. *International Journal of Precision Engineering and Manufacturing*, 14(10), 1743–1753. https://doi.org/10.1007/s12541-013-0234-4

Uzcategui, A. C., Muralidharan, A., Ferguson, V. L., Bryant, S. J., & McLeod, R. R. (2018). Understanding and improving mechanical properties in 3D printed parts using a dual-cure acrylate-based resin for stereolithography. *Advanced Engineering Materials*, 20(12), 1800876. https://doi.org/10.1002/adem.201800876

Saillard, E., Gardegaront, M., Levillain, A., Bermond, F., Mitton, D., Pialat, J.-B., ... Follet, H. (2024). Finite element models with automatic computed tomography bone segmentation for failure load computation. *Scientific Reports*, 14(1), 16576. https://doi.org/10.1038/s41598-024-66934-w