## Classical Mathematical Modeling for Description and Prediction of Experimental Tumor Growth: the case of melanoma

Jueves 30 de marzo, 17:00, auditorio N. Bralic

Patricio Cumsille ∗ Andrés Rodríguez †

∗Group of Investigation in Tumor Angiogenesis (GIANT), Department of Basic Sciences, Faculty of Sciences, University of Bío-Bío, Campus Fernando May, Avda. Andrés Bello 720, Casilla 447, Chillán, Chile, and Centre for Biotechnology and Bioengineering, University of Chile, Beaucheff 851, Santiago, Chile, e-mail: Esta dirección de correo electrónico está protegida contra spambots. Usted necesita tener Javascript activado para poder verla. †Group of Investigation in Tumor Angiogenesis (GIANT), Department of Basic Sciences, Faculty of Sciences, University of Bío-Bío, Campus Fernando May, Avda. Andrés Bello 720, Casilla 447, Chillán, Chile, e-mail: Esta dirección de correo electrónico está protegida contra spambots. Usted necesita tener Javascript activado para poder verla.

Abstract

In this work we study classical mathematical models of tumor growth (see [2, 5]) in order to describe and predict melanoma development (melanoma is the most serious type of skin cancer). To do so, firstly an experimental methodology will be carry out: human melanoma cells will be grown in a culture medium RPMI-1640 with a pH of 7.2, supplemented with serum at 5% in order to stimulate division. Once confluent, cells will be suspended in a sterile medium. Next, 200,000 of these melanoma cells will be injected in the skin of two group of mice, from which daily growth will be observed starting from the 7th day post injection, by using a Caliper, measuring the length and the width of melanoma tumor for each group, control and experimental, data which will be used in order to make parameters estimation of mentioned models, thereby describing and forecasting tumor growth. Control group is composed by wild- type mice, whereas experimental group consists of mice to which Adenosine receptor of type A2 has been knocked out, with the aim to investigate the influence of Adenosine in tumor growth, since this proteine has a role as an angiogenesis modulator by enhancing vascular endothelial growth factor (VEGF)1 activity (see [1]).

The numerical methodology required in order to make parameters estimation of the classical mathematical models, based upon ordinary differential equations (ODE), consists in employing Trust-Region-Reflective Algorithm (see [3]), a non linear optimiza- tion method specially designed to solve parameters estimation problems by non linear least squares criterion. This algorithm is implemented in Matlab through lsqnonlin function (see [7]). In order to use it, we will give as input data of this function: the experimental data (the length and the width measured at certain times, in days), the ODE solution, if an explicit formula is available (in the case where this is not available, we use the ode45 solver of Matlab in order to numerically solve the ODE for each model), an initial guess of the optimal value of the parameter vector (which in gen- eral is unknown, but exploring the relative square error, in evaluating it as a function of parameters space, we achieve an initial guess to start the optimization procedure), the Jacobian matrix of the residuals (when an explicit solution of the ODE model is available), and lower and upper bounds for the parameters (obtained also by the ex- ploratory method mentioned before). Next, we will apply statistical methods based upon non linear least squares regression tools (see [4, 6]) in order to assess goodness of fit for parameters estimated, as well as, performance of prediction to each model studied.

The expected results of this work are: 1) establish the descriptive power of each model, using several goodness-of-fit metrics and a study of parametric identifiability, 2) assess the models ability to forecast tumor growth, this for both groups of individu- als, under experimental conditions described before, and 3) conclude that both groups come from different populations, that is to say, the fact of knocking out the type A2 Adenosine receptor influences tumor growth, which will be achieved through validation of mathematical models with experimental data.

References [1] F. Azuaje, F. Le ́onard, M. Rolland-Turner, Y. Devaux, and D. R. Wagner, Proof-of-principle investigation of an algorithmic model of adenosine-mediated angio- genesis, Theoretical Biology and Medical Modelling, 8 (2011), p. 7. [2] S. Benzekry, C. Lamont, A. Beheshti, A. Tracz, J. Ebos, L. Hlatky, and P. Hahnfeldt, Classical mathematical models for description and prediction of exper- imental tumor growth, PLoS Comput. Biol., 10 (2014), p. e1003800. [3] M. A. Branch, T. F. Coleman, and Y. Li, A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems, SIAM Journal on Scientific Computing, 21 (1999), pp. 1–23. [4] K. P. Burnham and D. R. Anderson, Model selection and multimodel inference : a practical information-theoretic approach, Springer, New York, 2002. [5] P. Cumsille, A. Coronel, C. Conca, C. Quin ̃inao, and C. Escudero, Proposal of a hybrid approach for tumor progression and tumor-induced angiogenesis, Theoretical Biology and Medical Modelling, 12 (2015), p. 13. [6] C. J. W. George A. F. Seber, Nonlinear Regression, Wiley Series in Probability and Statistics, Wiley, 2003. [7] The Mathworks Inc., Nonlinear least squares (curve fitting). https://www.mathworks.com/help/optim/nonlinear-least-squares-curve-fitting.html, 2016. [Online; accessed 2016].