The objectives of the proposed project are to develop physically sound and mathematically rig- orous diffuse-interface models for tumor growth, to analyze the well-posedness of problems based on these models, to design efficient time-stepping schemes and finite-element discretization algorithms, to build the computer software implementing these algorithms, and to develop solution verification methods based on a posteriori error estimation. The focus of the research work will be on the devel- opment and analysis of mathematical models that describe at the continuum scale avascular growth of tumors, i.e. in the absence of nearby blood vessels, and aim at predicting the evolution of large tu- morous regions while ignoring the behavior of individual cells. Continuum models of tumor growth can be derived from first principles through the continuum theory of mixtures. Mixture theory pro- vides an elegant and general framework for modeling multicomponent media such, as living tissue, composed of several species of interacting constituents. A remarkable property of phenomenological models based on mixture theory is that, when considering the concentration gradients of various constituents into the Helmholtz free energy functionals, one obtains diffuse-interface models that introduce smooth transitional boundaries between the various constituents. The resulting equa- tions are systems of the Cahn-Hilliard type, namely complex systems of nonlinear time-dependent fourth-order partial-differential equations. Such diffuse-interface tumor-growth models have been proposed only recently in the literature and the mathematical analysis and development of efficient discretizations for such systems are only in the initial stage. The main objectives in this research project are thus to address several important open issues related to the development of spatio- temporal diffuse-interface phase field models for computer predictions of tumor growth, including: (1) development of formulations that satisfy thermodynamical properties of the system; (2) devel- opment of a rigorous mathematical framework for the analysis of tumor-growth models based on gradient ow theory; (3) development of new stable and high-order accurate time-stepping schemes by using semi-implicit splitting approaches; and (4) development of efficient goal-oriented error estimation algorithms for the control of spatial and temporal discretization errors for the highly nonlinear time-dependent coupled problem embodied by the proposed tumor-growth models.
Cancer is a disease of the genome, characterized by uncontrolled cellular growth and invasion, that afflicts every year millions of Americans from all age categories. The primary motivation of the research project is thus concerned with one of the grand challenges of our times, that is, to understand the mechanisms of cancer so that reliable treatments, or better, preventative measures, can be determined to relieve the impact this disease has on so many people. It has turned out to be a difficult endeavor, due to many reasons, but the most important ones could be that there are more than one hundred different types of cancer and the causes and effects of each type occur on a wide range of scales-specific mutations happen at the molecular scale while tumors may invade a significant portion of the human body. It is not too uncommon to believe that biologists or medical physicians are the main players in cancer research; however, more and more scientists from other disciplines, such as mathematics or physics, are getting involved in the investigation of possible causes of tumor development and behavior. It is in fact our hope that mathematical and computational tools could provide new insights that could help guide more fundamental research issues to be addressed by biologists and medical physicians. We believe that computer simulations represent powerful means for furthering discovery and acquiring scientific knowledge. These simulations will help in the future explore detailed mechanisms of tumor growth in natural environments that cannot be directly studied in patients.