On the efficient numerical simulation of heterogenous anisotropic diffusion models of tumor invasion using GPUs

Mathematical models for tumor invasion are often used to predict the behaviour of cancer evolution and can produce strikingly nontrivial patterns. Therefore, their numerical solution often requires high spatial resolution to capture detailed biophysical phenomena. As a consequence, long computational times are required when using a serial implementation of numerical schemes. Parallel computing can improve dramatically the time efficiency of some numerical methods such as finite differences algo- rithms, which are relatively simple to implement and apply to tumor invasion models. For clinical operators and applied scientists involved in setting up realistic experiments, the possibility of running fast comparative simulations using simple algorithms implemented into affordable processors is of primary interest, and that is where Graphical Processing Units (GPUs) excel. In this work we focus on a mathematical model of anisotropic and heterogeneous diffusion of tu- mor cells, a set of time-evolution parabolic equations. The model includes chemotaxis and haptotaxis effects, that appear in other biophysical phenomena [1]. We compute numerical solutions consider- ing spatial discretization by centered finite differences and time integration through an explicit Euler method. The choice of time-explicit algorithms is due to their greater ease of implementation, and performance, on GPU devices, despite the limitations related to their reduced stability properties. The codes used in this work are designed using CUDA. The CUDA platform (Compute Unified De- vice Architecture NVIDIA 2007), was designed to support GPU execution of programs and focuses on data parallelism. With CUDA, graphics cards can be programmed with a medium-level language, that can be seen as an extension to C/C++/Fortran, without requiring a great deal of hardware exper- tise. We refer to [6] and [5] for a comprehensive introduction to GPU-based parallel computing