Adjoint method for a tumor invasion PDE-constrained optimization problem

ANDRÉS QUIROGA; DAMIÁN FERNÁNDEZ; GERMÁN TORRES; CRISTINA TURNER

Carlos Paz

Workshop; XIII Latin American Workshop on Nonlinear Phenomena; 2013

We present [1] a method for estimating unknown parameter that appear on a non-linear reaction-diusion model of cancer invasion [2]. This model describe for the non dimensional spatial distribution and temporal evolution of the density of normal tissue (u1 ), the neoplastic tissue growth (u2 ) and the excess concentration of $H^+$ ions ($u_3$ ). A coupled system reaction- diusion describing this model is given by three partial dierential equations: egin{eqnarray*} rac{ partial u_1 }{ partial t } &=& u_1 left( 1 - u_1 ight) - delta_1 u_1 u_3, rac{ partial u_2 }{ partial t } &=& ho_2u_2 left( 1 - u_2 ight)+ abla cdot left( D left( 1 - u_1 ight) abla u_2 ight), rac{ partial u_3}{ partial t } &=& delta_3(u_2 - u_3 ) + DDelta u_3. end{eqnarray*} Each of the model parameters has a corresponding biological interpretation, for instance, the growth rate of neoplastic tissue ($ ho_2$ ), the diusion coecient($D_2$ ), the reabsorption rate ($delta_3$ ) and the destructive inuence of $H^+$ ions in the healthy tissue ($delta_1$ ). The last one cant be estimated in the direct form. After solving the forward problem properly, we use the model for the estimation of parameters by tting the numerical solution with real data, obtained via in vitro experiments and uorescence ratio imaging microscopy [3]. We dene an appropriate functional to compare both the real data and the numerical solution using the adjoint method [4] for the minimization of this functional. We apply Finite Element Method (FEM) to solve both the direct and inverse problem.