An image-informed Cahn–Hilliard Keller–Segel multiphase field model for tumor growth with angiogenesis

In this paper we develop a new four-phase tumor growth model with angiogenesis, derived from a diffuse-interface mixture model composed by a viable tumor component, a necrotic component, a liquid component and an angiogenetic component, coupled with two massless chemicals representing a perfectly diluted nutrient and an angiogenetic factor. This model is derived from variational principles complying with the second law of thermodynamics in isothermal situations, starting from biological constitutive assumptions on the tumor cells adhesion properties and on the infiltrative mechanics of tumor-induced vasculature in the tumor tissues, and takes the form of a coupled degenerate Cahn–Hilliard Keller–Segel system for the mixture components with reaction diffusion equations for the chemicals. The model is informed by neuroimaging data, which give informations about the patient-specific brain geometry and tissues microstructure, the distribution of the different tumor components, the white matter fiber orientations and the vasculature density. We describe specific and robust preprocessing steps to extract quantitative informations from the neuroimaging data and to construct a computational platform to solve the model on a patient-specific basis. We further introduce a finite element approximation of the model equations which preserve the qualitative properties of the continuous solutions. Finally, we show simulation results for the patient-specific tumor evolution of a patient affected by GlioBlastoma Multiforme, considering two different test cases before surgery, corresponding to situations with high or low nutrient supply inside the tumor, and a test case after surgery. We further perform a sensitivity based patient-specific parameters estimation based on longitudinal neuroimaging data for a test case with acquired data at two time points before surgery. We show that our model correctly predicts the overall extension of the tumor distribution and the intensity of the angiogenetic process, paving the way for assisting the clinicians in properly assessing the therapy outcomes and in designing optimal patient-specific therapeutic schedules.