Cellular automaton in cancerous growth with metastasis C. C. Martins 1 , K. C. Iarosz 1 , A. M. Batista 1 , R. L. Viana 2 , S. R. Lopes 2 , I. L. Caldas 3 and T. J. P. Penna 4 1 Departamento de Matem´atica e Estat´ ıstica, Universidade Estadual de Ponta Grossa, 84030-900, Ponta Grossa, PR, Brazil 2 Departamento de F´ ısica, Universidade Federal do Paran´a, 81531-990, Curitiba, PR, Brazil 3 Instituto de F´ ısica, Universidade de S˜ao Paulo, Caixa Postal 66316, 05315-970, S˜ao Paulo, SP, Brazil and 4 Instituto de F´ ısica, Universidade Federal Fluminense, 24210-340, Niter´oi, RJ, Brazil (Dated: April 9, 2009) In this work we considered cellular automaton model to study cancerous growth. We analyzed the effect of the tumor suppression and proliferation rate through the Hamming distance. According to this distance the spatio-temporal patterns were identified in terms of classes. Metastasis is investigated from a primary to a secondary cellular automaton. The average time to start the metastasis increases exponential with the increasing size of the primary tumor, moreover there is a dependence related to the tumor proliferation and suppression rate. PACS numbers: 05.50+q,87.18.Hf Cancerous growth is one of the most intriguing mod- ern science [1]. The interest for the problem, since the past two decades, has led to the formulation of numer- ous growth models [2]. The models have been proposed in order to analyze one or several basic features, such as the metastasis [3], the lack of nutrients [4], the competi- tion for resources and the cytotoxic activity made by the immune response [5]. Continuous and discrete models are types of approaches used to describe the cancerous growth. Cellular automaton is a kind of discrete model used to simulate the proliferation of cancerous cell [6]. In this study it was considered cellular automaton aim- ing to analyze cancerous growth. Let the cancerous (abnormal) cells, the dead cancerous cells, the effector (cytotoxic) cells (macrophages, etc) and the complexes produced by the cytotoxic process be respectively repre- sented by C, D, E 0 , and E. The cell-mediated immune which responses to cancer can be depicted by the follow- ing reactions: C k ′ 1 -→ 2C, (1) C + E 0 k2 -→ E k3 -→ E 0 + D, (2) D k4 -→ normal. (3) Reaction (1) describes the proliferation of cancerous cells, with k ′ 1 = k 1  1 - N c φ  , (4) where k 1 is the proliferation rate of cancerous cells, N c is the total number of cancerous cells and φ is a constant, thus N c reaches the maximum φ. The first reaction in (2) denotes the cytotoxic process, in that reaction (2) a single effector binds to one abnormal cell at a time. The sec- ond reaction of (2) depicts the dissolution of complexes. Equation (3) describes the dissolution of dead cells [7]. C 0 1 2 C C C 3 4 (b) (a) 0 1 2 C C C 3 4 C FIG. 1: The number 0 indicates the center of the square lat- tice, where 1,2,3 and 4 denote the four quadrants. When ρ ≤ ρc the second daughter cell will occupy one of the two shadowed sites with equal probability, as it is showed in (a), and (b) for ρ>ρc. The values adopted for the parameters k 1 , k 2 , k 3 and k 4 are listed in reference [7]. To overcome the lack of the experimental data for k 4 , we considered the value in the wide range (0.1-0.4). The density of cancerous cells ρ describes the effect of the mechanical pressure on cancer development, ρ = N ′ R 2 , (5) where N ′ = N c + N E + N D , R =( ∑ R ij )/N ′ and R ij depicts the distance from the site (i, j ) occupied by a cancerous cell to the origin. Considering a critical value of ρ, ρ = ρ c , we have that if ρ ≤ ρ c , the second daugh- ter cell resulting from the proliferation can only occupy one of the inside nearest neighboring sites occupied by normal cell (Fig. 1a), if ρ>ρ c , the second daughter cell may invade outside nearest neighboring sites with nor- mal cell (Fig. 1b). This way we used equal probability of the second daughter to occupy one of the two possible neighboring sites. The shape of the tumor is shown by Fig. 2 for k 1 =0.4, k 2 =0.1, k 3 = k 4 =0.35, ρ c =3.7, Φ c = 10 3 and t = 200. We considered at t = 0, 5 cancerous cells in the central part of the square lattice 101 × 101 and all of the remaining sites are occupied by normal cells. The black, red and green colors denote C, E and D, respectively.