Adjoint method for a tumor invasion PDE-constrained optimization problem in 2D using adaptive finite element method

In this paper we present a method for estimating an unknown parameter that appears in a two dimensional non-linear reaction-diffusion model of cancer invasion. This model considers that tumor-induced alteration of micro-environmental pH provides a mechanism for cancer invasion. A coupled system reaction-diffusion describing this model is given by three partial differential equations for the 2D non-dimensional spatial distribution and temporal evolution of the density of normal tissue, the neoplastic tissue growth and the excess H+ ion concentration. Each of the model parameters has a corresponding biological interpretation, for instance, the growth rate of neoplastic tissue, the diffusion coefficient, the re-absorption rate and the destructive influence of H+ ions in the healthy tissue. The parameter is estimated by solving a minimization problem, in which the objective function is defined in order to compare both the real data and the numerical solution of the cancer invasion model. The real data can be obtained by, for example, fluorescence ratio imaging microscopy. We apply a splitting strategy joint with the adaptive finite element method to numerically solve the model. The minimization problem (the inverse problem) is solved by using a gradient-based optimization method, in which the functional derivative is provided through an adjoint approach.Fil: Quiroga, Andrés Agustin Ignacio. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; ArgentinaFil: Fernández, Damián Andrés. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; ArgentinaFil: Turner, Cristina Vilma. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; ArgentinaFil: Torres, German Ariel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin

Inverse problems in tumor modelling

In this project, we deepen the analysis of a tumour growth model, recently proposed by Garcke et al. in [1]. This model describes tumour and healthy cells evolution as well as tumour cells’ nutrients, mixture velocity and pressure in the domain. Furthermore, it takes into account chemotaxis and apoptosis death of tumour cells, through a system of parabolic nonlinear PDE, that is a Cahn-Hilliard Darcy model, together with an advection-diffusion-reaction equation describing the evolution of nutrients. We perform a dimensional analysis and we build a numerical solver by use of the finite element method in space, a Backward Euler in time and a Newton method to tackle the nonlinearity. We perform several numerical simulations in order to recover results obtained in the article and to catch a general growth of the tumour depending on parameters of interest. Finally, a PDE-constrained optimization problem is formulated and solved, aiming at determining the shape of the tumur after a fixed time from an initial guess of its location. From the numerical simulations we obtained for the nutrients, we notice that the concentration of nutrients in an observable zone around the tumor region could possibly bring enough information to achieve this goal. Therefore, a previous numerical simulation of nutrients will be taken as a target, in order to recover the controlled tumor function, previously simulated numerically. In this respect, preliminary numerical results show that, to some extents, it is possible to identify the general shape of the tumor, even if the exact result of the numerical simulation could not be recovered

Coupling schemes and inexact Newton for multi-physics and coupled optimization problems

This work targets mathematical solutions and software for complex numerical simulation and optimization problems. Characteristics are the combination of different models and software modules and the need for massively parallel execution on supercomputers. We consider two different types of multi-component problems in Part I and Part II of the thesis: (i) Surface coupled fluid- structure interactions and (ii) analysis of medical MR imaging data of brain tumor patients. In (i), we establish highly accurate simulations by combining different aspects such as fluid flow and arterial wall deformation in hemodynamics simulations or fluid flow, heat transfer and mechanical stresses in cooling systems. For (ii), we focus on (a) facilitating the transfer of information such as functional brain regions from a statistical healthy atlas brain to the individual patient brain (which is topologically different due to the tumor), and (b) to allow for patient specific tumor progression simulations based on the estimation of biophysical parameters via inverse tumor growth simulation (given a single snapshot in time, only). Applications and specific characteristics of both problems are very distinct, yet both are hallmarked by strong inter-component relations and result in formidable, very large, coupled systems of partial differential equations. Part I targets robust and efficient quasi-Newton methods for black-box surface-coupling of parti- tioned fluid-structure interaction simulations. The partitioned approach allows for great flexibility and exchangeable of sub-components. However, breaking up multi-physics into single components requires advanced coupling strategies to ensure correct inter-component relations and effectively tackle instabilities. Due to the black-box paradigm, solver internals are hidden and information exchange is reduced to input/output relations. We develop advanced quasi-Newton methods that effectively establish the equation coupling of two (or more) solvers based on solving a non-linear fixed-point equation at the interface. Established state of the art methods fall short by either requiring costly tuning of problem dependent parameters, or becoming infeasible for large scale problems. In developing parameter-free, linear-complexity alternatives, we lift the robustness and parallel scalability of quasi-Newton methods for partitioned surface-coupled multi-physics simulations to a new level. The developed methods are implemented in the parallel, general purpose coupling tool preCICE. Part II targets MR image analysis of glioblastoma multiforme pathologies and patient specific simulation of brain tumor progression. We apply a joint medical image registration and biophysical inversion strategy, targeting at facilitating diagnosis, aiding and supporting surgical planning, and improving the efficacy of brain tumor therapy. We propose two problem formulations and decompose the resulting large-scale, highly non-linear and non-convex PDE-constrained optimization problem into two tightly coupled problems: inverse tumor simulation and medical image registration. We deduce a novel, modular Picard iteration-type solution strategy. We are the first to successfully solve the inverse tumor-growth problem based on a single patient snapshot with a gradient-based approach. We present the joint inversion framework SIBIA, which scales to very high image resolutions and parallel execution on tens of thousands of cores. We apply our methodology to synthetic and actual clinical data sets and achieve excellent normal-to-abnormal registration quality and present a proof of concept for a very promising strategy to obtain clinically relevant biophysical information. Advanced inexact-Newton methods are an essential tool for both parts. We connect the two parts by pointing out commonalities and differences of variants used in the two communities in unified notation