Efficient High-Order Space-Angle-Energy Polytopic Discontinuous Galerkin Finite Element Methods for Linear Boltzmann Transport.

We introduce an hp-version discontinuous Galerkin finite element method (DGFEM) for the linear Boltzmann transport problem. A key feature of this new method is that, while offering arbitrary order convergence rates, it may be implemented in an almost identical form to standard multigroup discrete ordinates methods, meaning that solutions can be computed efficiently with high accuracy and in parallel within existing software. This method provides a unified discretisation of the space, angle, and energy domains of the underlying integro-differential equation and naturally incorporates both local mesh and local polynomial degree variation within each of these computational domains. Moreover, general polytopic elements can be handled by the method, enabling efficient discretisations of problems posed on complicated spatial geometries. We study the stability and hp-version a priori error analysis of the proposed method, by deriving suitable hp-approximation estimates together with a novel inf-sup bound. Numerical experiments highlighting the performance of the method for both polyenergetic and monoenergetic problems are presented.

Identifying the spatial and temporal dynamics of molecularly-distinct glioblastoma sub-populations.

Glioblastomas (GBMs) are the most aggressive primary brain tumours and have no known cure. Each individual tumour comprises multiple sub-populations of genetically-distinct cells that may respond differently to targeted therapies and may contribute to disappointing clinical trial results. Image-localized biopsy techniques allow multiple biopsies to be taken during surgery and provide information that identifies regions where particular sub-populations occur within an individual GBM, thus providing insight into their regional genetic variability. These sub-populations may also interact with one another in a competitive or cooperative manner; it is important to ascertain the nature of these interactions, as they may have implications for responses to targeted therapies. We combine genetic information from biopsies with a mechanistic model of interacting GBM sub-populations to characterise the nature of interactions between two commonly occurring GBM sub-populations, those with EGFR and PDGFRA genes amplified. We study population levels found across image-localized biopsy data from a cohort of 25 patients and compare this to model outputs under competitive, cooperative and neutral interaction assumptions. We explore other factors affecting the observed simulated sub-populations, such as selection advantages and phylogenetic ordering of mutations, which may also contribute to the levels of EGFR and PDGFRA amplified populations observed in biopsy data.

Bayesian Calibration, Validation and Uncertainty Quantification for Predictive Modelling of Tumour Growth: A Tutorial.

In this work, we present a pedagogical tumour growth example, in which we apply calibration and validation techniques to an uncertain, Gompertzian model of tumour spheroid growth. The key contribution of this article is the discussion and application of these methods (that are not commonly employed in the field of cancer modelling) in the context of a simple model, whose deterministic analogue is widely known within the community. In the course of the example, we calibrate the model against experimental data that are subject to measurement errors, and then validate the resulting uncertain model predictions. We then analyse the sensitivity of the model predictions to the underlying measurement model. Finally, we propose an elementary learning approach for tuning a threshold parameter in the validation procedure in order to maximize predictive accuracy of our validated model.