In this work, we develop a computational tool to predict the patient-specific evolution of a highly malignant brain tumour, the glioblastoma multiforme (GBM), and its response to therapy. A diffuse-interface mathematical model based on mixture theory is fed by clinical neuroimaging data that provide the anatomical and microstructural characteristics of the patient brain. The model is numerically solved using the finite element method, on the basis of suitable numerical techniques to deal with the resulting Cahn-Hilliard type equation with degenerate mobility and single-well potential.The results of simulations performed on the real geometry of a patient brain quantitatively show how the tumour expansion dependens on the local tissue structure. We also report the results of a sensitivity analysis concerning the effects of the different therapeutic strategies employed in the clinical Stupp protocol. The simulated results are in qualitative agreement with the observed evolution of GBM during growth, recurrence and response to treatment. Taken as a proof-of-concept, these results open the way to a novel personalized approach of mathematical tools in clinical oncology. K E Y W O R D S diffuse-interface model, finite element, glioblastoma multiforme, mixture theory, personalized medicine M S C ( 2 0 1 0 ) 35K57, 35K65, 35Q92, 65M60, 92C50
We consider a Cahn-Hilliard-type equation with degenerate mobility and single-well potential of Lennard-Jones type. This equation models the evolution and growth of biological cells such as solid tumors. The degeneracy set of the mobility and the singularity set of the cellular potential do not coincide, and the absence of cells is an unstable equilibrium configuration of the potential. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. We give existence results for different classes of weak solutions. Moreover, we formulate a continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality. We prove the existence and uniqueness of the discrete solution for any spatial dimension together with the convergence to the weak solution for spatial dimension d = 1. We present simulation results in 1 and 2 space dimensions. We also study the dynamics of the spinodal decomposition and the growth and scaling laws of phase ordering dynamics. In this case, we find similar results to the ones obtained in standard phase ordering dynamics and we highlight the fact that the asymptotic behavior of the solution is dominated by the mechanism of growth by bulk diffusion. KEYWORDScontinuous Galerkin finite element approximation, degenerate Cahn-Hilliard equation, single well potential, tumor growth models 7598
In this paper we deal with the numerical analysis of an upscaled model of a reactive flow in a porous medium, which describes the transport of solutes undergoing precipitation and dissolution, leading to the formation/degradation of crystals inside the porous matrix. The model is defined at the Darcy scale, and it is coupled to a Darcy flow characterized by a permeability field that changes in space and time according to the precipitated crystal concentration. The model involves a non-linear multi-valued reaction term, which is treated exactly by solving an inclusion problem for the solutes and the crystals dynamics. We consider a weak formulation for the coupled system of equations expressed in a dual mixed form for the Darcy field and in a primal form for the solutes and the precipitate, and show its well posedness without resorting to regularization of the reaction term. Convergence to the weak solution is proved for its finite element approximation. We perform numerical experiments to study the behavior of the system and to assess the effectiveness of the proposed discretization strategy. In particular we show that a method that captures the discontinuity yields sharper dissolution fronts with respect to methods that regularize the discontinuous term
Glioblastoma extensively infiltrates the brain; despite surgery and aggressive therapies, the prognosis is poor. A multidisciplinary approach combining mathematical, clinical and radiological data has the potential to foster our understanding of glioblastoma evolution in every single patient, with the aim of tailoring therapeutic weapons. In particular, the ultimate goal of biomathematics for cancer is the identification of the most suitable theoretical models and simulation tools, both to describe the biological complexity of carcinogenesis and to predict tumor evolution. In this report, we describe the results of a critical review about different mathematical models in neuro-oncology with their clinical implications. A comprehensive literature search and review for English-language articles concerning mathematical modelling in glioblastoma has been conducted. The review explored the different proposed models, classifying them and indicating the significative advances of each one. Furthermore, we present a specific case of a glioblastoma patient in which our recently proposed innovative mechanical model has been applied. The results of the mathematical models have the potential to provide a relevant benefit for clinicians and, more importantly, they might drive progress towards improving tumor control and patient’s prognosis. Further prospective comparative trials, however, are still necessary to prove the impact of mathematical neuro-oncology in clinical practice.
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