Variational Tensor-Based Models for Image Diffusion in Non-Linear Domains

Numerical simulations in 3-dimensions of reaction–diffusion models for brain tumour growth

“…Table 1 shows the values chosen for the model parameters, that is values for the proliferation rate ρ and diffusion rates d w and d g for Low Grade Glioma (LGG) and High Grade Glioma (HGG). The variation of the parameter ρ influences the degree of the nonlinearity for the logistic source term, see (1), in the PDE model (12).…”

“…Thus, in the present work, we undertake the laborious task of doing full 3-dimensional simulations for two cases: The standard 3-dimensional Shepp-Logan phantom [42] and an MRI T1-weighted brain scan [40] from the Internet Brain Segmentation Repository (IBSR) (see [20]) describing a more complex geometry. We aim to not use any commercial solvers for partial differential equations but show instead how a rather straightforward finite difference scheme can be employed combined with recent imaging techniques [1] to render the simulations. We also show how existence and uniqueness of a solution to the reaction-diffusion model can be obtained using basic concepts of weak formulations.…”

“…The presented numerical simulations building on [1] with biological and mathematical background to the model are the main novelties of the present work. Since modelling of tumours involves both medical and mathematical insights it is believed that collecting such in one work with full simulations will be useful for interdisciplinary teams where participants typically have either a strong medical or mathematical background but not both.…”

Parallel Simulation of PDE-based Modelica Models using ParModelica

“…Those approximations can be derived using Taylor series, see for example (Eldén and Wittmeyer-Koch, 1996;Åström, 2015). As seen in ( 6), the discretisation (in one direction) will depend on the points on both sides.…”

Inverse Mathematical Models for Brain Tumour Growth

“…We did not use any commercial PDE solver but show 1.1. Outline of the thesis instead how a rather straightforward finite difference scheme can be employed to render the simulations, based on work of F.Åström [4]. In the simulations done on a realistic set up, that is the standard 3-dimensional Shepp-Logan phantom and an MRI T1-weighted brain scan from the Internet Brain Segmentation Repository (IBSR) describing a more complex geometry, we use different parameters including non-linearities for the forward model compared with the backward model as a way to simulate noise in the data.…”