Application of the hybrid discrete-continuum technique

Following on from the previous chapter, here we present a number of specific biological problems involving cell migration that can be modeled using the hybrid discrete-continuum (HDC) technique. The three problems considered are (i) angiogenesis (blood vessel formation) in response to a solid tumor, (ii) tissue invasion by cancer cells, and (iii) aggregation in the slime mold Dictyostelium discoideum. Tumor-induced angiogenesis is the formation of blood vessels (vasculature) from a preexisting vasculature in response to chemical stimuli (tumor angiogenic factors, TAF) produced and secreted by the solid tumor. The formation of the new vasculature facilitates further tumor growth and invasion. The mathematical model describes the formation of the blood-vessel network in response to chemical stimuli (TAF) supplied by a solid tumor. The model also takes into account essential cell-matrix interactions via the inclusion of the matrix macromolecule fibronectin. A discretized form of the partial differential equation model is then used to develop a biased random walk model that enables us to track individual endothelial cells at the blood vessel tips and incorporate anastomosis, mitosis, and branching explicitly into the model. A second important aspect of solid tumor growth is the invasion of the local tissue by the cancer cells. A crucial part of the this process is the ability of the cancer cells to degrade the surrounding tissue or extracellular matrix (ECM) through the secretion of matrix degrading enzymes (MDEs). As the ECM is degraded, the cancer cells actively migrate into the space created, thereby spreading locally into the healthy host tissue. If there are blood or lymph vessels in the vicinity, these can then provide a route for the cancer cells to spread widely throughout the host's body. An initial system of PDEs is formulated considering the evolution of cancer cell density, MDEs, and ECM density. An HDC model is then developed that enables us to track the migration and spread of individual cancer cells into the host tissue. The results have implications for the surgical removal of solid tumors. The social amoeba Dictyostelium discoideum is a widely studied paradigm for biological pattern formation. Mathematical models can show how a collective pattern of cell communication and motion driven by the chemoattractant cyclic adenosine 3'5'-monophosphate (cAMP) arises from excitable local cAMP kinetics and cAMP diffusion. The master continuum PDE model describes the evolution of cAMP concentration, the fraction of active receptors on Dictyostelium cells, and cell density. The HDC technique is then used to develop a biased random walk model that enables us to track individual Dictyostelium amoeba as they move through the aggregation territory in response to the local cAMP gradients. The theoretical aggregation patterns generated by computer simulations of the discrete model are compared with the morphology of such patterns observed in in vivo experiments.