A fractional mathematical model approach on glioblastoma growth: tumor visibility timing and patient survival

Abstract

In this paper, we introduce a mathematical model given by \begin{equation} { }^c \mathfrak{D}_t^\alpha u = \nabla \cdot \mathrm{D} \nabla u + \rho f(u) \quad \text{in } \Omega, \end{equation} where $f(u)=\frac{1}{1-u/\mathrm{K}}, \, u/\mathrm{K} \neq 1, \, \mathrm{K} > 0$, to enhance established mathematical methodologies for better understanding glioblastoma dynamics at the macroscopic scale. The tumor growth model exhibits an innovative structure even within the conventional framework, including a proliferation term, $f(u)$, presented in a different form compared to existing macroscopic glioblastoma models. Moreover, it represents a further refined model by incorporating a calibration criterion based on the integration of a fractional derivative, $\alpha$, which differs from the existing models for glioblastoma. Throughout this study, we initially discuss the modeling dynamics of the tumor growth model. Given the frequent recurrence observed in glioblastoma cases, we then track tumor mass formation and provide predictions for tumor visibility timing on medical imaging to elucidate the recurrence periods. Furthermore, we investigate the correlation between tumor growth speed and survival duration to uncover the relationship between these two variables through an experimental approach. To conduct these patient-specific analyses, we employ glioblastoma patient data and present the results via numerical simulations. In conclusion, the findings on tumor visibility timing align with empirical observations, and the investigations into patient survival further corroborate the well-established inter-patient variability for glioblastoma cases.