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Abstract:
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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 . |