Coupled Analysis of Active Biological Processes for Meniscus Tissue Regeneration
During the last decades, mathematical modeling and numerical simulation have become valuable tools for investigating complex biomedical systems. They significantly contribute to the understanding of a biological process, often allowing to extend the study results to related problem classes. At the University of Kaiserslautern-Landau (Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau RPTU), we started in 2022 with a project that tackles a prominent example of a biomedical application: the meniscus regeneration and involved cell and tissue-level phenomena.
Clinical studies indicate that partial and total meniscectomies lead to prevalence of premature osteoarthritis in knee joints. Therefore, substantial efforts are being made towards finding adequate regenerative tissue for meniscus replacement. Most regenerative approaches are clinically motivated and focus rather on the practical application than on the micro- and macroscopic cellular mechanisms and the interactions with the scaffold material. The latter viewpoint is promising in the sense that it aims to understand the basic control mechanisms in cell-scaffold interactions under different environmental parameters. In this way, the in-silico model can help to improve the in-vitro studies and eventually can also pave the way for a promising in-vivo usage.
Challenges and Co Simulation Strategies
With respect to the in-silico modeling and simulation, a major challenge lies in the well-posed and numerically efficient coupling of the processes at the cell level with the macroscopic behavior and the mechanical properties of the tissue. The active processes at the cell level, such as cell differentiation and matrix synthesis, have a strong impact on the resulting tissue structure and quality, while macroscopic effects in turn are important stimuli for the processes at the microscopic level. Moreover, the time scales of the different processes differ vastly and call for appropriate co simulation strategies.
A key feature of the experimental study in this project, which is carried out by Andreas Seitz at the University of Ulm, is the use of a nonwoven scaffold in a novel 3D printed bioreactor that allows in-vitro investigations of scaffolds in interaction with chondrocytes and adipose tissue-derived stem cells. In this framework, crucial stimuli to trigger relevant proliferation, migration and differentiation can be identified by state-of-the-art experimental measurements. While meaningful clinical data is very difficult to obtain from the interior meniscus tissue, this off-the-wall approach provides comprehensive underpinnings for the mathematical modeling and numerical simulation.
A second computational model has been developed for the dynamics of mesenchymal stem cells and chondrocytes inside the scaffold, which leads to a system of diffusion-reaction-advection equations. Here we built on experience with similar models for glioma growth and used a first order Non-symmetric Interior Penalty discontinuous Galerkin (NIP dG) scheme in space to account for local mass conservation. The temporal discretization was based on the simple and robust implicit Euler scheme. Again, FreeFEM++ provided the framework for the programming. We started here with a simplified model where the motility terms are independent of the fibre orientation in the scaffold. Moreover, the influence of fluid-induced mechanical stress on the cell dynamics was analyzed. For this purpose, a one-way coupling was established were first the porous medium simulation above is performed to evaluate the stress field in the scaffold, which is then used in combination with a periodic forcing function for stimulating the cell dynamics.
The figure above shows the result of the in-silico experiment in terms of densities of chondrocytes, hyaluronic acid, which is important for chemotaxis, and the produced cartilage structure. At the moment, we are working on refining the model by taking the fibre orientation distribution into account, by establishing a collaboration with the group of Claudia Redenbach, also at RPTU Kaiserslautern-Landau, who performs a directional analysis of CT scans of real scaffold samples. From a computational viewpoint, the fibre orientation leads to a cell diffusion tensor where an elliptic integral has to be evaluated to account for the second moments of the fibre orientation distribution.
Authors: Bernd Simeon and Christina Surulescu, University of Kaiserslautern-Landau (Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau RPTU), Department of Mathematics, Differential-Algebraic Systems Group
More information about the DFG-Priority Programme 2311: www.spp2311.de