Cell-scale degradation of peritumoural extracellular matrix fibre network and its role within tissue-scale cancer invasion

Local cancer invasion of tissue is a complex, multiscale process which plays an essential role in tumour progression. Occurring over many different temporal and spatial scales, the first stage of invasion is the secretion of matrix degrading enzymes (MDEs) by the cancer cells that consequently degrade the surrounding extracellular matrix (ECM). This process is vital for creating space in which the cancer cells can progress and it is driven by the activities of specific matrix metalloproteinases (MMPs). In this paper, we consider the key role of two MMPs by developing further the novel two-part multiscale model introduced in [33] to better relate at micro-scale the two micro-scale activities that were considered there, namely, the micro-dynamics concerning the continuous rearrangement of the naturally oriented ECM fibres within the bulk of the tumour and MDEs proteolytic micro-dynamics that take place in an appropriate cell-scale neighbourhood of the tumour boundary. Focussing primarily on the activities of the membrane-tethered MT1-MMP and the soluble MMP-2 with the fibrous ECM phase, in this work we investigate the MT1-MMP/MMP-2 cascade and its overall effect on tumour progression. To that end, we will propose a new multiscale modelling framework by considering the degradation of the ECM fibres not only to take place at macro-scale in the bulk of the tumour but also explicitly in the micro-scale neighbourhood of the tumour interface as a consequence of the interactions with molecular fluxes of MDEs that exercise their spatial dynamics at the invasive edge of the tumour

Numerical interactions of random and directed motility during cancer invasion

The continuum modelling of cell migration during cancer invasion results in the coupling of parabolic and hyperbolic partial differential equations (PDEs) arising from the random motility of normal tissue and the directed movement up substrate gradients of malignant cells. The numerical solution of such systems of equations require different stability criteria being simultaneously satisfied. We show that in such a coupled system, the origins of numerical instability can be identified by analyzing the fastest growing mode in a numerically unstable solution. In general, stability can be achieved by choosing an appropriate grid size representing the more stringent of the conditions for hyperbolic and parabolic stability. However, this induces variable degrees of numerical diffusion because of a changing CFL (Courant, Friedrichs, and Lewy) number. Solving the hyperbolic and parabolic PDEs on separate grids results in a better convergence of the solution. Finally, we discuss the use of higher-order schemes for the solution of such problems. Cancer modelling brings together directed and random motility in a unique way thereby presenting interesting new numerical problems