Toward a predictive model of tumor growth
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In this work, an attempt is made to lay out a framework in which models of tumor growth can be built, calibrated, validated, and differentiated in their level of goodness in such a manner that all the uncertainties associated with each step of the modeling process can be accounted for in the final model prediction. The study can be divided into four basic parts. The first involves the development of a general family of mathematical models of interacting species representing the various constituents of living tissue, which generalizes those previously available in the literature. In this theory, surface effects are introduced by incorporating in the Helmholtz free ` gradients of the volume fractions of the interacting species, thus providing a generalization of the Cahn-Hilliard theory of phase change in binary media and leading to fourth-order, coupled systems of nonlinear evolution equations. A subset of these governing equations is selected as the primary class of models of tumor growth considered in this work. The second component of this study focuses on the emerging and fundamentally important issue of predictive modeling, the study of model calibration, validation, and quantification of uncertainty in predictions of target outputs of models. The Bayesian framework suggested by Babuska, Nobile, and Tempone is employed to embed the calibration and validation processes within the framework of statistical inverse theory. Extensions of the theory are developed which are regarded as necessary for certain scenarios in these methods to models of tumor growth. The third part of the study focuses on the numerical approximation of the diffuse-interface models of tumor growth and on the numerical implementations of the statistical inverse methods at the core of the validation process. A class of mixed finite element models is developed for the considered mass-conservation models of tumor growth. A family of time marching schemes is developed and applied to representative problems of tumor evolution. Finally, in the fourth component of this investigation, a collection of synthetic examples, mostly in two-dimensions, is considered to provide a proof-of-concept of the theory and methods developed in this work.