Cell-based, continuum and hybrid models of tissue dynamics

Movement of amoeboid cells is involved in embryonic development, wound repair, the immune response to bacterial invasion, and tumor formation and metastasis. Individual cells detect extracellular chemical and mechanical signals via membrane receptors, and this initiates signal transduction cascades that produce intracellular signals. These signals control the motile machinery of the cell and thereby determine the spatial localization of contact sites with the substrate and the sites of force-generation needed to produce directed motion. The coordination and control of this complex process of direction sensing, amplification of spatial differences in the signal, assembly of the motile machinery, and control of the attachment to the substratum involves numerous molecules whose spatial distribution serves to distinguish the front from the rear of the cell, and whose temporal expression is tightly controlled. How chemical and mechanical signals are integrated, how spatial differences in signals are produced, and how propulsive and adhesive forces are controlled are issues that are amenable to mathematical modeling. An overview of some approaches to these complex problems is the subject of this chapter.