Conference Agenda

Our bones are impressive structures that constantly adapt to habitual mechanical load through a dynamic process known as bone remodelling. This continuous renewal allows bones to maintain their mechanical stability. Nevertheless, excessive overload can disrupt the delicate balance of bone remodelling leading to the formation of microcracks. However, as long as these microdamages remain within a certain threshold, they even trigger an enhanced remodelling response in these areas, allowing the bones to heal these microcracks in a targeted manner [1].

Recent research [2] has shown that the so-called flexoelectric effect—the ability of bone material to generate an electric potential under inhomogeneous deformation—plays a crucial role in coordinating the cellular mechanisms essential for the healing of microcracks. A deeper understanding of these regenerative processes, combined with their modelling and simulation, is therefore essential for developing more effective medical treatments and therapeutic strategies.

We propose a novel methodology that phenomenologically accounts for the trabecular microstructure of bone and size-dependent material effects using a micromorphic framework [3]. This framework has been extended to include nonlinear electro-elastic and flexoelectric energy contributions within the constitutive equations. To evaluate the approach, we use a cracked cantilever beam as a representative model, approximating a bone sample with a microcrack—such as those that may occur in the femoral neck region—and commonly employed in experimental studies [2]. In the presence of a bone crack, the flexoelectric effect causes an electric field to be generated around the crack when asymmetric deformation (e.g., bending) is applied. The stress gradient is highest at the tip of the crack, resulting in the greatest electric field. The effect is therefore most pronounced at the crack tip and decreases with further distance. The generated electric field acts as a biological trigger, inducing osteocyte apoptosis, which is followed by the bone-building activities of the osteoblasts [1,2]. The activation of these cellular mechanisms is captured in our model by an increase in nominal bone density in the vicinity of the crack tip. The presented analysis investigates the influence of the key model parameters on the magnitude of the electric potential and, consequently, on the nominal bone density. Additionally, we present an enhanced approach that specifically analyses the critical range of electric fields between 1 and 10 kV/m, which induce programmed osteocyte cell death (apoptosis), and examines their impact on the growth of nominal bone density in these regions.

[1] Heino T, Kurata K, Higaki H, Väänänen K. 2009. Evidence for the role of osteocytes in the initiation of targeted remodeling. Technology and Health Care. 17:49–56
[2] Núñez Toldrà R, Vasquez-Sancho F, Barroca N, Catalan G. 2020. Investigation of the cellular response to bone fractures: Evidence for flexoelectricity. Scientific Reports. 10:254
[3] Titlbach A, Papastavrou A, McBride A, Steinmann P. 2023. A novel micromorphic approach captures non-locality in continuum bone remodelling. Computer Methods in Biomechanics and Biomedical Engineering. 27(8):1042–1055

Recent advancements in manufacturing technologies have enabled the development of microstructurally architected materials, with functionally graded lattices emerging as promising candidates for biomedical applications. Their ability to replicate the complex structure of native bone makes them suitable for orthopedic and dental implants. The performance of these materials depends on both biomechanical compatibility and biological integration. Accordingly, recent studies have explored the mechanical behavior of TPMS structures within manufacturability constraints [1], and employed density-based topology optimization on homogenized lattices to design patient-specific hip implants [2]. This study aims to define a more accurate mechanical optimization approach to mitigate the stress shielding effect that may occur in the prosthesis-bone system due to differences in compliance between the bone and implant materials. The stress shielding effect can progressively compromise the implant's load-bearing capacity, as the surrounding bone progressively thins over time, thereby negatively affecting the implant's durability and functionality. The developed methodology is based on a dataset obtained from micro-Computed Tomography (μCT) scans of a series of human hip bone samples. From this dataset, specific morphological and mechanical characteristics have been extracted. In particular, the volume fraction is calculated as the ratio between the dimensions of the bone tissue islands and the total sample volume, while the trabecular thickness is estimated through direct interpolation by superimposing ellipsoidal domains onto the segmented bone tissue. The anisotropy is evaluated using the Mean Intercepts Length (MIL) method. The statistical behavior of these geometrical features is incorporated into a computational study to produce a continuum equivalent of the resected bone prior to implantation. Given the high computational costs related to stochastic computational homogenization, it is demonstrated that the complex bone material with random microstructure can be substituted with a simplified ergodic model. Several numerical homogenization techniques are employed to represent composites with random structures: from stochastic collocation methods to polynomial chaos, as well as Monte Carlo methods and Stochastic Finite Elements (SFEM). The latter are the most commonly used for such composites [3,4]. The homogenized continuum equivalent allows for the evaluation of the minimal achievable stress discontinuity at the prosthesis-bone interface, with accuracy governed by the level of geometric detail incorporated in the model. The stress jump is quantified through finite element analysis of the implant under various loading conditions. Future developments of this work will involve the formulation of an inverse problem, where the homogenized continuum serves as the mechanical target for a density-based topology optimization of a graded, periodic lattice structure—such as TPMS.

References:

1. T. Poltue, et al, IJES2 211,106762, 2021

2. P. Muller et al, Nature/Scientific Reports 14:5719, 2024

3. M. Ostoja-Starzewski, Int. J. Solid Struct., 5, 19, 2429-2455, 1998.

4. P. Steinmann et al, Comput. Methods Appl. Mech. Engrg. 357, 112563, 2019.

Mechanics-based in silico models of the human brain are gaining importance in understanding the role of mechanics in physiological and pathological processes such as brain folding, brain injury, ageing, and age-related diseases. Those models rely on experimental data for multiple purposes. The development or choice of appropriate models requires an understanding of underlying mechanisms that relies on observations made in experiments. Once a model is chosen, data are required for parameter identification and validation. One major challenge for further improvements of existing models is the limited access to data. Due to the vulnerability of the brain, in vivo testing is restricted to small strains. Human tissue from body donations is not always available and the quick degradation of brain tissue leads to a short time frame for post mortem data acquisition. As a response to the limited data availability, recent studies have attempted to gain insights into the mechanics of brain tissue through purely computational methods. Examples are micromechanical material models of white matter that are based on the assumption that aligned axons cause transversely isotropic material behavior. Here, we present results from ex vivo mechanical testing and histological analysis of human brain white matter and brain stem. Histological analyses of mechanically tested samples suggest alternative explanations for the direction-dependent behavior of tissue from the corpus callosum: in this brain region, not only axons and glial cells, but also blood vessels, which are thicker and stiffer than axons, are highly aligned. We further show that, while axons in the brain stem may appear unidirectional in diffusion tensor imaging data of lower resolution, the brain stem does not show transversely isotropic material behavior. This aligns with the complex anatomy of the brain stem, which not only contains axons of varying directions but also gray matter. With respect to brain modeling, our results suggest that continuum-mechanics based models combined with in vivo data and detailed anatomical knowledge may be a more robust, flexible, and cost-effective approach than those that explicitly model microstructural tissue components such as axons and cells. More generally, our results show that it is necessary to gain a more thorough understanding of brain tissue mechanics before focusing on a single modeling approach. A close collaboration with anatomists, clinicians, and biologists is beneficial to ensure that imaging data are interpreted correctly. In conclusion, while in vivo data are the best choice to be fed into ready-for-application, patient-specific models, a combination of in vivo and ex vivo data is needed to first elucidate underlying mechanisms and choose appropriate modeling approaches.

The brain is recognized as one of the most intricate organs in the human body, and modeling its mechanical behavior has posed significant challenges for researchers over many years. Understanding the response of this multiphase tissue to mechanical loading and the reliable calibration of material parameters remains limited. Previous viscoelastic models utilizing two relaxation times have effectively captured aspects of brain tissue response; however, the Theory of Porous Media offers a continuum mechanical perspective to investigate the fundamental physical mechanisms, particularly the interactions between the solid matrix and the freely flowing interstitial fluid. Leveraging our established experimental testing protocol, this study conducts finite element simulations of cyclic compression-tension loading and relaxation experiments on human visual cortex (grey matter) and corona radiata (white matter) specimens. The findings indicate that solid volumetric stress is a critical element influencing the biphasic behavior of the tissue, as it significantly interacts with the porous effects governed by the permeability. Through inverse parameter identification, it is revealed that poroelasticity alone fails to adequately capture the time-dependent material behavior, particularly the significant hysteresis observed during cyclic loading. Incorporating viscous effects to introduce a second time-dependent mechanism greatly enhances model accuracy, facilitating satisfactory representations for both loading modes simultaneously. Our analysis illustrates that the first Lamé parameter is instrumental in influencing fluid flow and porous dissipation: lower values facilitate volumetric deformation and augment permeability effects, whereas higher values inhibit fluid flow and reduce poroelastic behavior. Additionally, the first Lamé parameter governs the temporal progression of deformation within the biphasic material in conjunction with the permeability. The insights provided shed light on the distinct contributions of viscous and porous effects. Nonetheless, the strong interdependencies among porous, viscous, and volumetric effects underscore the necessity for further experimental investigations to reliably ascertain all material parameters. Thereby, our findings stress the importance of calibrating parameter models in accordance with expected deformation and fluid flow regimes.