Management of invasive cells in soft biological tissues through modulated nonlinear excitations: Long-range effects

Abstract

In the present paper, we investigate the contribution of long-range diffusion and diffusive stresses on cellular waves generated during an invasion in soft biological tissues. To that end, we consider a set of three coupled nonlinear partial differential equations describing spatiotemporal evolutions of cell density, extracellular matrix (ECM), and spatial displacements of a material point taken in the ECM. In addition to long-range diffusion, diffusive stresses, and proliferation, the model also accounts for ECM production-degradation. The linear stability analysis reveals that in presence of temporal perturbations, there is no stability-instability transition in the system. In presence of spatiotemporal perturbations, instability dominates but the stability-instability transition is controlled by proliferation rate and long-range diffusion of cells. Analytical solutions are constructed as well as their existence conditions provided. Amongst the structures recovered, breathers are associated with the transport of a high number of cells. Long-range diffusive stresses, saturation, and ECM depletion rate significantly modify the wave structure as well as the number of cells carried. On the other hand, Proliferation, ECM production rate, long-range haptotaxis, diffusion, and traction affect both spatial rearrangement and the number of cells carried by the wave. Through our analysis, numerical solutions remain positive, do not diverge from their analytical counterpart, meaning that the solutions proposed are stable and may be used to probe the inherent biological realities of the model considered.