A multiscale moving boundary model arising in cancer invasion

Cancer invasion of tissue is a key aspect of the growth and spread of cancer and is crucial in the process of metastatic spread, i.e., the growth of secondary cancers. Invasion consists in cancer cells secreting various matrix degrading enzymes (MDEs) which destroy the surrounding tissue or extracellular matrix (ECM). Through a combination of proliferation and migration, the cancer cells then actively spread locally into the surrounding tissue. Thus processes occurring at the level of individual cells eventually give rise to processes occurring at the tissue level. In this paper we introduce a new type of multiscale model describing the process of cancer invasion of tissue. Our multiscale model is a two-scale model which focuses on the macroscopic dynamics of the distributions of cancer cells and of the surrounding extracellular matrix, and on the microscale dynamics of the MDEs, produced at the level of the individual cancer cells. These microscale dynamics take place at the interface of the cancer cells and the ECM and give rise to a moving boundary at the macroscale. On the computational side, in order to approximate the newly proposed model, we have developed a novel computational scheme based on a combination of finite elements at the microscale with a new finite difference technique at the macroscale, linking together in a moving boundary formulation of the problem. This two-scale numerical scheme is organized in such a way that it enables us to accurately model all the key processes of cancer invasion at both the macroscale and microscale