Mathematical Models and Adaptive Algorithms for Tumor Growth

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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 onthese models, to design efficient time-stepping schemes and finite-element discretization algorithms,to build the computer software implementing these algorithms, and to develop solution verificationmethods 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 growthof 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 growthcan 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 phenomenologicalmodels based on mixture theory is that, when considering the concentration gradients of variousconstituents into the Helmholtz free energy functionals, one obtains diffuse-interface models thatintroduce smooth transitional boundaries between the various constituents. The resulting equa-tions are systems of the Cahn-Hilliard type, namely complex systems of nonlinear time-dependentfourth-order partial-differential equations. Such diffuse-interface tumor-growth models have beenproposed only recently in the literature and the mathematical analysis and development of efficientdiscretizations for such systems are only in the initial stage. The main objectives in this researchproject 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 ongradient ow theory; (3) development of new stable and high-order accurate time-stepping schemesby using semi-implicit splitting approaches; and (4) development of efficient goal-oriented errorestimation algorithms for the control of spatial and temporal discretization errors for the highlynonlinear 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 motivationof the research project is thus concerned with one of the grand challenges of our times, thatis, to understand the mechanisms of cancer so that reliable treatments, or better, preventativemeasures, can be determined to relieve the impact this disease has on so many people. It hasturned out to be a difficult endeavor, due to many reasons, but the most important ones couldbe that there are more than one hundred different types of cancer and the causes and effects ofeach type occur on a wide range of scales-specific mutations happen at the molecular scale whiletumors may invade a significant portion of the human body. It is not too uncommon to believethat biologists or medical physicians are the main players in cancer research; however, more andmore scientists from other disciplines, such as mathematics or physics, are getting involved inthe investigation of possible causes of tumor development and behavior. It is in fact our hopethat mathematical and computational tools could provide new insights that could help guide morefundamental research issues to be addressed by biologists and medical physicians. We believe thatcomputer simulations represent powerful means for furthering discovery and acquiring scientificknowledge. These simulations will help in the future explore detailed mechanisms of tumor growthin natural environments that cannot be directly studied in patients.