Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth

In this paper we examine spatio-temporal pattern formation in reaction-diff usion systems on the surface of the unit sphere in 3D. We first generalise the usual linear stability analysis for a two-chemical system to this geome trical context. Noting the limitations of this approach (in terms of rigoro us prediction of spatially heterogeneous steady-states) leads us to develop , as an alternative, a novel numerical method which can be applied to syste ms of any dimension with any reaction kinetics. This numerical method is ba sed on the method of lines with spherical harmonics and uses fast Fourier t ransforms to expedite the computation of the reaction kinetics. Numerical e xperiments show that this method efficiently computes the evolution of spat ial patterns and yields numerical results which coincide with those predict ed by linear stability analysis when the latter is known. Using these tools , we then investigate the role that pre-pattern (Turing) theory may play in the growth and development of solid rumours. The theoretical steady-stale distributions of two chemicals (one a growth activating factor, the other a growth inhibitory factor) are compared with the experimentally and clinica lly observed spatial heterogeneity of cancer cells in small. solid spherica l tumours such as multicell spheroids and carcinomas. Moreover, we suggest a number of chemicals which are known to be produced by rumour cells (autoc rine growth factors). and are also known to interact with one another. as p ossible growth promoting and growth inhibiting factors respectively. In ord er to connect more concretely the numerical method to this application, we compute spatially heterogeneous patterns on the surface of a growing spheri cal tumour, modelled as a moving-boundary problem. The numerical results st rongly support the theoretical expectations in this case. Finally in an app endix we give a brief analysis of the numerical method.