Cellular transport through nonlinear mechanical waves in fibrous and absorbing biological tissues

The present paper investigates the spatiotemporal evolution of a population of mesenchymal cells during invasion. In that end, a mathematical model accounting for haptotaxis, chemotaxis, zero-proliferation, viscous, and traction forces is considered. Injecting plane wave ansatz in the model reveals viscoelastic properties of the medium, as well as the existence of parameters regions where expansive and compressive waves may be found. The model reduction leads to a cubic complex Ginzburg-Landau equation. Stability line versus envelope wave vector is plotted to prove that instability domains may be controlled by increasing traction or by reducing substrate attachment. Analytical solutions constructed reveal that the number of particles carried increases with increasing values of substrate attachment or by reduction of traction forces. Our solutions keep their positivity, and do not deviate from their numerical counterparts. Our investigations summarize the fact that mechanical behaviors coupled with biochemical processes of cells are at great interplay during the invasion in biological tissues.