Linking Cellular and Mechanical Processes in Articular Cartilage Lesion Formation: A Mathematical Model

Abstract

Post-traumatic osteoarthritis affects almost 20% of the adult US population. An injurious impact applies a significant amount of physical stress on articular cartilage and can initiate a cascade of biochemical reactions that can lead to the development of osteoarthritis. In our effort to understand the underlying biochemical mechanisms of this debilitating disease, we have constructed a multiscale mathematical model of the process with three components: cellular, chemical, and mechanical. The cellular component describes the different chondrocyte states according to the chemicals these cells release. The chemical component models the change in concentrations of those chemicals. The mechanical component contains a simulation of a blunt impact applied onto a cartilage explant and the resulting strains that initiate the biochemical processes. The scales are modeled through a system of partial-differential equations and solved numerically. The results of the model qualitatively capture the results of laboratory experiments of drop-tower impacts on cartilage explants. The model creates a framework for incorporating explicit mechanics, simulated by finite element analysis, into a theoretical biology framework. The effort is a step toward a complete virtual platform for modeling the development of post-traumatic osteoarthritis, which will be used to inform biomedical researchers on possible non-invasive strategies for mitigating the disease.

Maximum shear strain-based algorithm can predict proteoglycan loss in damaged articular cartilage

“…The algorithms investigated in this study were only concentrated on the FCD loss due to a mechanical stimulus and excluded the effect of any biological processes such as inflammation or altered PG synthesis due to injurious loading of the chondrocytes. The presence of biochemical stimuli is evident in PTOA, and the damage models by Kapitanov et al (2016) and Gardiner et al (2016) included them. The former model combined cellular (healthy, diseased and dead cells), chemical (pro-and anti-inflammatory cytokines) and mechanical (tissue internal strains which induce the biochemical processes) components.…”

Mathematics as a conduit for translational research in post‐traumatic osteoarthritis

“…Once we have the steady‐state strain computations from Abaqus, we solve the age‐ and space‐structured equations using our own software and methods. The methods have a rigorous mathematical foundation, and have been used in a range of applications such as Proteus mirabilis swarm colony formation, avascular tumor invasion, biofilm growth and senescence, dormancy in bacteria, and articular cartilage lesion formation related to the work in this article …”

“…Biomechanical models and computational simulations have a long, rich history in helping to understand the mechanics of bone and cartilage . Incorporation of cell biology and biochemistry into these rigorous systems requires what we refer to as “biomathematical” models and simulations that are just beginning to be developed for PTOA . Here, we propose that the chain of events leading from acute joint injury to PTOA, including mechanical stress and specific cellular pathways, can ultimately be described in sets of equations that represent known biomechanics and cellular responses to mechanical stress.…”

“…Thus, as an example of this conduit of translation we have built a mathematical model for cartilage response to injury from overloading. Here, as in our group's prior work, cartilage cell populations are separated into three model compartments, based on cells being “healthy,” “sick,” or “dead.” Within these compartments, cells are further subdivided based on signaling state or being necrotic or apoptotic. This approach originally relied upon delay differential equations, and later a mathematical construct called “age structure,” to represent the delays in the various signaling processes over time.…”

“…This approach originally relied upon delay differential equations, and later a mathematical construct called “age structure,” to represent the delays in the various signaling processes over time. Going forward, instead of representing the role of mechanical stress implicitly through certain functional forms in our biomathematical model, we are using finite element analysis (FEA) to incorporate biomechanical criteria explicitly . The project presented here was designed to qualitatively predict the cartilage tissue's reaction to systemic overload, in the context of inflammation and oxidative stress, with our simulations covering a period of 14 days post‐injury.…”