Application of fast isogeometric L2 projection solver for tumor growth simulations

To employ computer models of tumor proliferation for planning cancer treatment, extremely fast numerical solvers are in high demand at least for two reasons: data adaptation and multiscale character of growth dynamics. The number of complex and non-linear phenomena influencing tumor dynamics involves hundreds of variable parameters. Its matching using incoming data requires reverse optimization or learning procedures, which are very demanding computationally. We present here the fast algorithm for isogeometric L2 projection, which we propose as a numerical engine for tumor modeling. First, we introduce the continuous-discrete model of melanoma described by the system of PDEs in a simplified computational layout of a fragment of skin. Apart from the continuous density fields such as tumor cell density, flux pressure, extracellular and degraded extracellular matrices, we also introduce a discrete model of the vasculature, which is the source of oxygen and nutrients. The discrete vasculature is coupled every given time interval with the continuous model influencing the vasculature remodeling. Our tests and the results of 2-D simulations of melanoma progression clearly show that isogeometric L2 projection utilizing the alternating directions solver is superior over classical approaches in terms of computational complexity, what makes it an excellent candidate for a numerical engine for continuous-discrete models of complex biological systems.