The focus of this study is the development of a statistical modelling procedure for characterising intra-tumour heterogeneity, motivated by recent clinical literature indicating that a variety of tumours exhibit a considerable degree of genetic spatial variability. A formal spatial statistical model has been developed and used to characterise the structural heterogeneity of a number of supratentorial primitive neuroectodermal tumours (PNETs), based on diffusion-weighted magnetic resonance imaging. Particular attention is paid to the spatial dependence of diffusion close to the tumour boundary, in order to determine whether the data provide statistical evidence to support the proposition that water diffusivity in the boundary region of some tumours exhibits a deterministic dependence on distance from the boundary, in excess of an underlying random 2D spatial heterogeneity in diffusion. Tumour spatial heterogeneity measures were derived from the diffusion parameter estimates obtained using a Bayesian spatial random effects model. The analyses were implemented using Markov chain Monte Carlo (MCMC) simulation. Posterior predictive simulation was used to assess the adequacy of the statistical model. The main observations are that the previously reported relationship between diffusion and boundary proximity remains observable and achieves statistical significance after adjusting for an underlying random 2D spatial heterogeneity in the diffusion model parameters. A comparison of the magnitude of the boundary-distance effect with the underlying random 2D boundary heterogeneity suggests that both are important sources of variation in the vicinity of the boundary. No consistent pattern emerges from a comparison of the boundary and core spatial heterogeneity, with no indication of a consistently greater level of heterogeneity in one region compared with the other. The results raise the possibility that DWI might provide a surrogate marker of intra-tumour genetic regional heterogeneity, which would provide a powerful tool with applications in both patient management and in cancer research.
Stochastic differential equations and comparison of financial models with levy process using Markov chain Monte Carlo (MCMC) simulation
