A mathematical modelling of rheumatoid arthritis

We developed a mathematical model to describe the inflammatory dynamics during arthritis development and resolution, focusing on the interaction between three key immune cell populations within the joint tissue, namely neutrophils (N), sublining synovial tissue-resident macrophages (SL-TRM, T), and circulating monocytes that are recruited into the inflamed joint and differentiate into monocyte-derived macrophages (MoMac, M). An inflammatory signal (S), representing the response to an external arthritogenic stimulus, drives the recruitment and activation of neutrophils and MoMac, while SL-TRM exert a regulatory and protective role. We propose a minimal ordinary differential equation model of acute inflammatory response into a joint that couples an inflammatory signal S to the three cell populations (N, T, M) and incorporates recruitment, stimulation, nonlinear self-limitation and mutual regulation. We show that the equilibrium structure depends on three adimensional parameters A, B, C and derive a cubic equation characterizing nontrivial steady states. We perform an analytical study of equilibria and their stability, and identify a bifurcation at A = 1 separating a disease-free regime from a sustained inflammatory state. Numerical simulations illustrate typical transient dynamics under various scenarios and show the ability of the model to capture the transition from acute to chronic inflammation, as well as the resolution of inflammation under certain conditions. In conclusion, our model provides a framework to explore mechanisms underlying acute versus sustained joint inflammation and offers a basis for parameter estimation from experimental data, as well as future extensions toward evaluation of therapeutic strategies.