Analysis of a multi-species Cahn-Hilliard-Keller-Segel tumor growth model with chemotaxis and angiogenesis

We introduce a multi-species diffuse interface model for tumor growth, characterized by its incorporation of essential features related to chemotaxis, angiogenesis and proliferation mechanisms. We establish the weak well-posedness of the system within an appropriate variational framework, accommodating various choices for the nonlinear potentials. One of the primary novelties of the work lies in the rigorous establishment of the existence of a weak solution through the introduction of delicate approximation schemes. To our knowledge, this represents a novel advancement for both the intricate Cahn-Hilliard-Keller-Segel system and the Keller-Segel subsystem with source terms. Moreover, when specific conditions are met, such as having more regular initial data, a smallness condition on the chemotactic constant with respect to the magnitude of initial conditions and potentially focusing solely on the two-dimensional case, we provide regularity results for the weak solutions. Finally, we derive a continuous dependence estimate, which, in turn, leads to the uniqueness of the smoothed solution as a natural consequence

Large deformations in terms of stretch and rotation and local solution to the non-stationary problem

In this paper we consider and generalize a model, recently proposed and analytically investigated in its quasi-stationary approximation by the authors, for visco-elasticity with large deformations and conditional compatibility, where the independent variables are the stretch and the rotation tensors. The model takes the form of a system of integro-differential coupled equations. Here, its derivation is generalized to consider mixed boundary conditions, which may represent a wider range of physical applications then the case with Dirichlet boundary conditions considered in our previous contribution. This also introduces nontrivial technical difficulties in the theoretical framework, related to the definition and the regularity of the solutions of elliptic operators with mixed boundary conditions. As a novel contribution, we develop the analysis of the fully non-stationary version of the system where we consider inertia. In this context, we prove the existence of a local in time weak solution in three space dimensions, employing techniques from PDEs and convex analysis.Comment: arXiv admin note: text overlap with arXiv:2307.0299

Large deformations in terms of stretch and rotation and global solution to the quasi-stationary problem

In this paper we derive a new model for visco-elasticity with large deformations where the independent variables are the stretch and the rotation tensors which intervene with second gradients terms accounting for physical properties in the principle of virtual power. Another basic feature of our model is that there is conditional compatibility, entering the model as kinematic constraint and depending on the magnitude of an internal force associated to dislocations. Moreover, due to the kinematic constraint, the virtual velocities depend on the solutions of the problem. As a consequence, the variational formulation of the problem and the related mathematical analysis are neither standard nor straightforward. We adopt the strategy to invert the kinematic constraints through Green propagators, obtaining a system of integro-differential coupled equations. As a first mathematical step, we develop the analysis of the model in a simplified setting, i.e. considering the quasi-stationary version of the full system where we neglect inertia. In this context, we prove the existence of a global in time strong solution in three space dimensions for the system, employing techniques from PDEs and convex analysis, thus obtaining a novel contribution in the field of three dimensional finite visco-elasticity described in terms of the stretch and rotation variables. We also study a limit problem, letting the magnitude of the internal force associated to dislocations tend to zero, in which case the deformation becomes incompatible and the equations takes the form of a coupled system of PDEs. For the limit problem we obtain global existence, uniqueness and continuous dependence from data in three space dimensions

Strict separation and numerical approximation for a non-local Cahn-Hilliard equation with single-well potential

In this paper we study a non-local Cahn-Hilliard equation with singular single-well potential and degenerate mobility. This results as a particular case of a more general model derived for a binary, saturated, closed and incompressible mixture, composed by a tumor phase and a healthy phase, evolving in a bounded domain. The general system couples a Darcy-type evolution for the average velocity field with a convective reaction-diffusion type evolution for the nutrient concentration and a non-local convective Cahn-Hilliard equation for the tumor phase. The main mathematical difficulties are related to the proof of the separation property for the tumor phase in the Cahn-Hilliard equation: up to our knowledge, such problem is indeed open in the literature. For this reason, in the present contribution we restrict the analytical study to the Cahn-Hilliard equation only. For the non-local Cahn- Hilliard equation with singular single-well potential and degenerate mobility, we study the existence and uniqueness of weak solutions for spatial dimensions $d\leq 3$. After showing existence, we prove the strict separation property in three spatial dimensions, implying the same property also for lower spatial dimensions, which opens the way to the proof of uniqueness of solutions. Finally, we propose a well posed and gradient stable continuous finite element approximation of the model for $d\leq 3$, which preserves the physical properties of the continuos solution and which is computationally efficient, and we show simulation results in two spatial dimensions which prove the consistency of the proposed scheme and which describe the phase ordering dynamics associated to the system

Coupling solid and fluid stresses with brain tumour growth and white matter tract deformations in a neuroimaging-informed model

Brain tumours are among the deadliest types of cancer, since they display a strong ability to invade the surrounding tissues and an extensive resistance to common therapeutic treatments. It is therefore important to reproduce the heterogeneity of brain microstructure through mathematical and computational models, that can provide powerful instruments to investigate cancer progression. However, only a few models include a proper mechanical and constitutive description of brain tissue, which instead may be relevant to predict the progression of the pathology and to analyse the reorganization of healthy tissues occurring during tumour growth and, possibly, after surgical resection. Motivated by the need to enrich the description of brain cancer growth through mechanics, in this paper we present a mathematical multiphase model that explicitly includes brain hyperelasticity. We find that our mechanical description allows to evaluate the impact of the growing tumour mass on the surrounding healthy tissue, quantifying the displacements, deformations, and stresses induced by its proliferation. At the same time, the knowledge of the mechanical variables may be used to model the stress-induced inhibition of growth, as well as to properly modify the preferential directions of white matter tracts as a consequence of deformations caused by the tumour. Finally, the simulations of our model are implemented in a personalized framework, which allows to incorporate the realistic brain geometry, the patient-specific diffusion and permeability tensors reconstructed from imaging data and to modify them as a consequence of the mechanical deformation due to cancer growth

A Cahn-Hilliard phase field model coupled to an Allen-Cahn model of viscoelasticity at large strains

We propose a new Cahn-Hilliard phase field model coupled to incompressible viscoelasticity at large strains, obtained from a diffuse interface mixture model and formulated in the Eulerian configuration. A new kind of diffusive regularization, of Allen-Cahn type, is introduced in the transport equation for the deformation gradient, together with a regularizing interface term depending on the gradient of the deformation gradient in the free energy of the system. We study the global existence of a weak solution for the model. While standard diffusive regularizations of the transport equation for the deformation gradient presented in literature allows the existence study only for simplified cases, i.e. in two space dimensions and for convex elastic free energy densities of Neo-Hookean type which are independent from the phase field variable, the present regularization allows to study more general cases. In particular, we obtain the global existence of a weak solution in three space dimensions and for generic nonlinear elastic energy densities with polynomial growth. Our analysis considers elastic free energy densities which depend on the phase field variable and which can possibly degenerate for some values of the phase field variable. By means of an iterative argument based on elliptic regularity bootstrap steps, we find the maximum allowed polynomial growths of the Cahn-Hilliard potential and the elastic energy density which guarantee the existence of a solution in three space dimensions. We propose two unconditionally energy stable finite element approximations of the model, based on convex splitting ideas and on the use of a scalar auxiliary variable, proving the existence and stability of discrete solutions. We finally report numerical results for different test cases with shape memory alloy type free energy with pure phases characterized by different elastic properties

Numerical simulation of geochemical compaction with discontinuous reactions

The present work deals with the numerical simulation of porous media subject to the coupled effects of mechanical compaction and reactive flows that can significantly alter the porosity due to dissolution, precipitation or transformation of the solid matrix. These chemical processes can be effectively modelled as ODEs with discontinuous right hand side, where the discontinuity depends on time and on the solution itself. Filippov theory can be applied to prove existence and to determine the solution behaviour at the discontinuities. From the numerical point of view, tailored numerical schemes are needed to guarantee positivity, mass conservation and accuracy. In particular, we rely on an event-driven approach such that, if the trajectory crosses a discontinuity, the transition point is localized exactly and integration is restarted accordingly

Deep Anatomical Federated Network (Dafne): an open client/server framework for the continuous collaborative improvement of deep-learning-based medical image segmentation

Semantic segmentation is a crucial step to extract quantitative information from medical (and, specifically, radiological) images to aid the diagnostic process, clinical follow-up. and to generate biomarkers for clinical research. In recent years, machine learning algorithms have become the primary tool for this task. However, its real-world performance is heavily reliant on the comprehensiveness of training data. Dafne is the first decentralized, collaborative solution that implements continuously evolving deep learning models exploiting the collective knowledge of the users of the system. In the Dafne workflow, the result of each automated segmentation is refined by the user through an integrated interface, so that the new information is used to continuously expand the training pool via federated incremental learning. The models deployed through Dafne are able to improve their performance over time and to generalize to data types not seen in the training sets, thus becoming a viable and practical solution for real-life medical segmentation tasks.Comment: 10 pages (main body), 5 figures. Work partially presented at the 2021 RSNA conference and at the 2023 ISMRM conference In this new version: added author and change in the acknowledgmen