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. / text
| Identifer | oai:union.ndltd.org:UTEXAS/oai:repositories.lib.utexas.edu:2152/ETD-UT-2011-05-3395 |
| Date | 16 June 2011 |
| Creators | Hawkins-Daarud, Andrea Jeanine |
| Source Sets | University of Texas |
| Language | English |
| Detected Language | English |
| Type | thesis |
| Format | application/pdf |
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