Free boundary problem for cancer cell protrusion formation. Mathematical model and numerical aspects for resolution.

Date(s) : 30/05/2017   iCal
11 h 00 min - 12 h 00 min

Metastatic cells are cancer cells that leave a primary tumor. They are able to invade sub-epithelial basement membranes and then to migrate through the healthy tissue towards the blood network or lymphatic system before invading the rest of the body and creating metastases. Numerous biological phenomena are involved inside the cell and they are strongly coupled to microenvironmental processes via the cell membrane. They lead to protrusion formation which are the early stages of invasion and directional migration, depending on though the protrusion is localized and proteolytic (invadopodia) or wider and non-proteolytic (pseudopodia).

For both invadopodia and pseudopodia, the protrusion formation at the cellular level can be mathematically described thanks to a free-boundary model based on PDEs. In our model, an interface accounts for the cell membrane, which is the main area of interest. This interface is between two harmonic phases. One phase stands for the outer phenomena: in the case of invadopodia, it accounts for the ligands which are created if some cellular MT1-MMP enzymes are embedded in the membrane and degrade the extracellular matrix. In the case of pseudopodia, the outer phase is a chemotactic signal which is diffused by a distant blood network or immune cells. In both cases, the outer phase is given with a Neumann boundary condition on the interface and provides the data of the Dirichlet boundary condition for the inner phase. The inner phase accounts for a cytoplasmic signal, which is triggered by the binding of outer ligands to receptors at the cell membrane. This signal is a simplification to describe the internal signalling pathways that lead to actin polymerization. The force exerted by the actin filaments on the membrane results in the protrusion formation. The interface velocity is then given as the gradient of the inner phase.

The positive feedback loop between inner and outer phases results in a strong mathematical coupling. From the numerical point of view, the coupling and successive derivations at each time step may result in nonconsistent solutions. The main focus of the presentation is to give the hints for building suitable numerical methods, which make it possible to overcome the issue thanks to the use of superconvergence properties. These methods are based on finite differences for solving the protrusion formation problem on Cartesian grid. A level set function is used to implicitly describe the interface in the usual Eulerian formalism. The core of the methods is the stabilization of the standard Ghost Fluid Method (Fedkiw et al.,1999), the use of a specific velocity extension, and linear, quadratic or cubic extrapolations of the ghost values. They result in different numerical schemes with different superconvergent behaviors. Hence, depending on the scheme, the solutions are either first order or second order accurate. Finally, the cubic method even leads to a second order accuracy of the interface curvature, which makes it possible to consider using interface regularization techniques to model the subsequent stages of cell migration. The presentation will be illustrated by convergence tests and simulation results showing the formation of membrane protrusions.


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