November 6, 2003 15:0 WSPC/Trim Size: 9in x 6in for Proceedings encapsulation SYMMETRY REDUCTION OF A MODEL IN SPHERICAL SYMMETRY FOR BENIGN TUMOR G. GAMBINO, A. M. GRECO, M. C. LOMBARDO Dip. Matematica, Universit` a di Palermo Via Archirafi 34, 90123 Palermo, Italy E-mail: {gaetana},{greco},{lombardo}@math.unipa.it A PDEs system, describing the expansive growth of a benign tumor and the phe- nomenon of encapsulation, is studied via a group analysis approach. A weak equiv- alence classification is obtained and the original PDEs system is reduced to an ODEs system. Numerical simulations are performed both for ODEs and PDEs, which turn out to be in perfect agreement between each other, showing a realistic enough description of the biological process. 1. Introduction In this work exact solutions of a nonlinear system of PDEs, describing the formation of a capsule around a benign tumor, are searched in the frame- work of group analysis. By assuming the hypothesis of spherical symmetry in the process of encapsulation for the tumor cell density u(r, t) and the ex- tracellular matrix density c(r, t), the following 3-D evolution system arises as an extension of a previously proposed 1-D model 1 : - ∂u ∂t + f (u)+ 1 r 2 ∂ ∂r  r 2 h(c) ∂u ∂r  =0 (1a) - ∂c ∂t + k r 2 ∂ ∂r  r 2 cϑ(c)h(c) ∂u ∂r  =0. (1b) Here f (u),h(c) and θ(c) are three arbitrary nonnegative continuously differ- entiable functions, representing respectively the source term, the reduction of the cell motility at high matrix densities and the effect of saturation on the rate of matrix movement and convection per cell at high matrix densities. Moreover, they are supposed to satisfy the following constraints 1 : f (0) = f (1) = 0; ∀u ∈ (0, 1); f ′ (0) > 0,f (u) <f ′ (0) u, ∀u ∈ (0, 1]; h ′ (c) ≤ 0, ∀ c ≥ 0,h(1) > 0; θ(1) = 1,θ ′ (c) ≤ 0, ∀ c ≥ 0; . (2) 1