Adjoint method for a tumor growth PDE-constrained optimization problem

Abstract

In this paper we present a method for estimating unknown parameters that appear on an avascular, spheric tumour growth model. The model for the tumour is based on nutrient driven growth of a continuum of live cells, whose birth and death generate volume changes described by a velocity field.The model consists on a coupled system of partial differential equations whose spatial domain is the tumour, that changes in size over time. Thus, the situation can be formulated as a free boundary problem.After solving the forward problem properly, we use the model for the estimation of parameters by fitting the numerical solution with real data, obtained via in vitro experiments and medical imaging. We define an appropriate functional to compare both the real data and the numerical solution. We use the adjoint method for the minimization of this functional, getting a better performance than the obtained with the pattern search method.

A mathematical method for parameter estimation in a tumor growth model

“…We have defined the objective functions in terms of variables that can be experimentally measured as explained in Knopoff et al (2013). For example, the density of living cells could be measured via biomedical imaging like PET technique for a tumor in vivo, or via immunofluorescence and electronic scan microscopy technique for in vitro cases (Taylor et al 1986;Martin et al 1994).…”

An inverse problem for an immobilized enzyme model

“…For this purpose, we use the adjoint method (see Hinze et al), which is known to be more efficient than the finite difference method. Indeed, the adjoint method does not depend on the number of parameters to be recovered, which differs from the finite difference method or even from derivative‐free methods (see Knopoff et al).…”

“…First, we formulate the inverse problem of interest into the framework described in subsection . We follow the ideas introduced in Knopoff . Let .…”

“…Using the well-posedness result for the linear system proven in Theorem 2.1 and some results about abstract semilinear Cauchy problems, we will establish the existence and uniqueness of solutions for system (12).…”

“…where (S, S b ) is the solution of the direct problem (12) for each choice of p. Let us consider the function spaces…”