ABSTRACT

Cancer is a serious disease, that involves uncontrollable growth of abnormal cells that form a tumor (American cancer society). Cancer is induced by carcinogenesis, where cells suffer from damage to their DNA. Currently, research that is aimed at studying tumor growth is carried out according to some specific steps in the tumor formation. These steps correspond to avascular growing, angiogenesis and vascular growing (Alarcon, et al., 2005). In the first step, the tumor grows up to a maximum size, which depends on the blood supply as well as availability of oxygen and nutrients that are obtained through its surface. The second step is the angiogenesis, where the solid tumor releases diffusible substances named tumor angiogenic factors (TAF). This step is critical because the tumor develops its own blood supply and once vascularization is completed, the third step begins. During the latter, the vascular growing step, the tumor receives large amounts of nutrients and, as a result, it tumor grows much more than during the avascular step. The vascular growth increases the chance of developing metastases, due to the enhanced supply of resources (oxygen, nutrients, etc.) to the tumor, rendering it a dangerous step.
Carcinomas in the avascular growing step evolve in a similar way because they have a universal scheme of progression because they share similar characteristics regarding to the net of interactions between cells and extracellular matrix (Martins et al., 2007). This fact allows modeling and simulating the avascular growth of carcinomas. However, it is important to note that the modeling and simulation of cancer tumor growth is a complex task because it involves a great number of processes that are related at different scales, with cell-cell interactions and difficult biological phenomena. The present research presents a simple model of anthocyanins effect on carcinoma in avascular growing step. A modified Cellular Potts Model (CPM) and Metropolis Monte Carlo (MMC) method are used to model and simulate the tumor growth (Martins et al., 2007). Two cases were taken, tumor growth without and with anthocyanins effect.
CPM uses a lattice to describe N cells, which are composed by a number of pixels with a position (i,j). Each position is associated to a spin (?) to identify the space occupied for a cell. Therefore, a type of cell (? and ?â??) takes a set of discrete subcomponents that can be rearranged to produce the cellular movement (cellular change) which is made randomly by means of the dynamic of MMC method. The cellular interactions and changes are described in terms of energies according to equation (1).

(1) The first term describes the adhesion energy (J) between cells but when two cells are equal,

the delta Kronecker function (?) is used to ensure that only the energy between different cells is considered. The second term is the energy related to the area of the cell which is taken as a function of the elasticity parameter (?a) of the cell and the actual (a(?, ?)) and target (A(?, ?)T) areas are permitted. The third term (AF) represents the anthocyanin effect that is taken as a simple term according to the effect of anthocyanins reported in the literature. The first term is related to the attraction or repulsion between cells that can explore and reach the state of lower energy. The term that is related to area corresponds to the size of the cells because they cannot increase or decrease their dimensions without limits. The total energy (H) is calculated for one initial configuration (1) and for one new arrangement (2) and they are compared according to equation (2).

(2)

If the total energy is zero or negative then, the new configuration is accepted and the new configuration becomes the new starting point. If the total energy is positive then the change is accepted with a Boltzmann probability as is shown in equation (3).

(3) As a result, the tumor growing behavior was obtained for a different number of cells and

different Monte Carlo steps (MCS). For example, for a lattice of 150 x 150 representing only two types of cells, cancer cells and healthy cells are present in figure 1 in which the anthocyanins affect is not included yet in the model.
Figure 1. Tumor growing for a lattice of 150 x 150. Cancer cells (red) and healthy cells (blue) Finally, the effect of anthocyanins on tumor growth was analyzed for other cases with
different lattice and different MCS. It was possible to understand the effect of anthocyanins on tumor growth using this simple model.

References

Alarcon T., Byrne H. M., Maini P. K. A multiple scale model for tumor growth. Society for industrial and applied mathematics. Vol 3, No. 2, 2005. 440 â?? 475.
American Cancer Society. ¿Qué es el cáncer? 10 October 2010. Available in:
http://www.cancer.org (Accessed April 2014)
Martins M. L., Ferreira S. C., Vilela M. J. Multiscale models for the growth of avascular tumors. Physics of the life reviews. Vol 4 No. 2, 2007. 128 â?? 156.