Nonlinear optimization for a tumor invasion PDE model
- PDF / 811,277 Bytes
- 15 Pages / 439.37 x 666.142 pts Page_size
- 27 Downloads / 248 Views
Nonlinear optimization for a tumor invasion PDE model A. A. I. Quiroga1 · G. A. Torres2 · D. Fernández3 · C. V. Turner3
Received: 22 December 2015 / Revised: 4 May 2016 / Accepted: 23 May 2016 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2016
Abstract In this work, we introduce a methodology to approximate unknown parameters that appear on a non-linear reaction–diffusion model of tumor invasion. These equations consider that tumor-induced alteration of micro-environmental pH furnishes a mechanism for cancer invasion. A coupled system reaction–diffusion explaining this model is given by three partial differential equations for the non-dimensional spatial distribution and temporal evolution of the density of normal tissue, the neoplastic tissue growth and the excess concentration of H+ ions. The tumor model parameters have a corresponding biological meaning: the reabsorption rate, the destructive influence of H+ ions in the healthy tissue, the growth rate of tumor tissue and the diffusion coefficient. We propose to solve the direct problem using the Finite Element Method (FEM) and minimize an appropriate functional including both the
Communicated by Luz de Teresa. The work of the authors was partially supported by Grants from CONICET 2012-2015, SECYT-UNC 2015 and AGENCIA PICT-2014.
B
C. V. Turner [email protected] A. A. I. Quiroga [email protected] G. A. Torres [email protected] D. Fernández [email protected]
1
Centro Atómico Bariloche, CNEA, Bustillo km. 9.5, CRUB-UNCo, Quintral 1250, 8400 San Carlos de Bariloche, Rio Negro, Argentina
2
Facultad de Ciencias Exactas y Naturales y Agrimensura, IMIT-CONICET, Universidad Nacional del Nordeste, Av. Libertad 5470, 3400 Corrientes, Argentina
3
Facultad de Matemática, Astronomía y Física, CIEM-CONICET, Universidad Nacional de Córdoba, Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba, Argentina
123
A. A. I. Quiroga et al.
real data (obtained via in-vitro experiments and fluorescence ratio imaging microscopy) and the numerical solution. The gradient of the functional is computed by the adjoint method. Keywords Reaction–diffusion equation · Tumor invasion · PDE-constrained optimization · Adjoint method · Finite element method Mathematics Subject Classification 35Q92 · 35R30 · 65M60
1 Introduction Cancer is one of the greatest killers in the world although medical activity has been successful, despite great difficulties, at least for some pathologies. A great effort of human and economical resources is devoted, with successful outputs, to cancer modeling (Cristini and Lowengrub 2010; Adam and Bellomo 1997; Bellomo et al. 2008a, b; Byrne 2010; Preziosi 2003). Some comments on the importance of mathematical modeling in cancer can be found in the literature. In the work of Bellomo et al. (2008b) it is mentioned that “Cancer modelling has, over the years, grown immensely as one of the challenging topics involving applied mathematicians working with researchers active in the biological sciences. The motiva
Data Loading...
