Osteoarthritis (OA) is the most common cause of disability in ageing societies, with no effective therapies available to date. Two preclinical models are widely used to validate novel OA interventions (MCL-MM and DMM). Our aim is to discern disease dynamics in these models to provide a clear timeline in which various pathological changes occur. OA was surgically induced in mice by destabilisation of the medial meniscus. Analysis of OA progression revealed that the intensity and duration of chondrocyte loss and cartilage lesion formation were significantly different in MCL-MM vs DMM. Firstly, apoptosis was seen prior to week two and was narrowly restricted to the weight bearing area. Four weeks post injury the magnitude of apoptosis led to a 40–60% reduction of chondrocytes in the non-calcified zone. Secondly, the progression of cell loss preceded the structural changes of the cartilage spatio-temporally. Lastly, while proteoglycan loss was similar in both models, collagen type II degradation only occurred more prominently in MCL-MM. Dynamics of chondrocyte loss and lesion formation in preclinical models has important implications for validating new therapeutic strategies. Our work could be helpful in assessing the feasibility and expected response of the DMM- and the MCL-MM models to chondrocyte mediated therapies.
The reaction counts chemical master equation (CME) is a high-dimensional variant of the classical population counts CME. In the reaction counts CME setting, we count the reactions which have fired over time rather than monitoring the population state over time. Since a reaction either fires or not, the reaction counts CME transitions are only forward stepping. Typically there are more reactions in a system than species, this results in the reaction counts CME being higher in dimension, but simpler in dynamics. In this work, we revisit the reaction counts CME framework and its key theoretical results. Then we will extend the theory by exploiting the reactions counts’ forward stepping feature, by decomposing the state space into independent continuous-time Markov chains (CTMC). We extend the reaction counts CME theory to derive analytical forms and estimates for the CTMC decomposition of the CME. This new theory gives new insights into solving hitting times-, rare events-, and a priori domain construction problems.
We present a numerical method to model dynamical systems from data. We use the recently introduced method Scalable Probabilistic Approximation (SPA) to project points from a Euclidean space to convex polytopes and represent these projected states of a system in new, lower-dimensional coordinates denoting their position in the polytope. We then introduce a specific nonlinear transformation to construct a model of the dynamics in the polytope and to transform back into the original state space. To overcome the potential loss of information from the projection to a lower-dimensional polytope, we use memory in the sense of the delay-embedding theorem of Takens. By construction, our method produces stable models. We illustrate the capacity of the method to reproduce even chaotic dynamics and attractors with multiple connected components on various examples.