Part IV: From Cellular to Multicellular Models
Cell migration is an essential feature of physiologic and pathologic phenomena in biology, such as embryonic development, wound healing, and tumor invasion. According to the local micro-environment and the function of the migrating cell, the characteristics of migration may vary considerably. In connective tissue, cells (cancer cells, ﬁbroblasts, etc.) interact both with other cells and with the surrounding tissue (ECM, extracellular matrix), which provides them with a natural complex scaﬀold to which to adhere in order to migrate. Recently much attention has been devoted to the description of the mechanics of cell motion as a result of their interactions with the ECM. Experiments have evidenced two types of motion, amoeboid and mesenchymal, that relate to diﬀerent migration strategies. The amoeboid motion corresponds to a “path ﬁnding” strategy involving morphological adaptation of cells, while the mesenchymal motion corresponds to a “path generating” strategy involving proteolytic activity of cells to degrade the ﬁbers of the ECM. This chapter takes a closer look at the individual interaction mechanisms to develop a model for amoeboid cell migration that includes both a preferential movement of cells along the collagen ﬁbers of the ECM-a phenomenon called “contact guidance”—and a randomly oriented migration due to interactions among cells in denser areas. A modeling framework is derived at the mesoscopic (kinetic) scale, and a continuous (macroscopic) model is deduced through a diﬀusive limit of the kinetic one. The response of the cells to external stimuli (taxis), capable of inﬂuencing and biasing the motion, is also included. Finally, numerical simulations are presented to illustrate the ability of the model to account for the inﬂuence of (1) the heterogeneity and/or the anisotropy of the ECM medium and (2) various sorts of taxes (chemotaxis, haptotaxis, repellent behavior).