Critical behavior of a tumor growth model: Directed percolation with a mean-field flavor

Abstract

We examine the critical behavior of a lattice model of tumor growth where supplied nutrients are correlated with the distribution of tumor cells. Our results support the previous report [Ferreira et al., Phys. Rev. E 85, 010901(R) (2012)], which suggested that the critical behavior of the model differs from the expected directed percolation (DP) universality class. Surprisingly, only some of the critical exponents (β, α, ν([perpendicular]), and z) take non-DP values while some others (β', ν(||), and spreading-dynamics exponents Θ, δ, z') remain very close to their DP counterparts. The obtained exponents satisfy the scaling relations β=αν(||), β'=δν(||), and the generalized hyperscaling relation Θ+α+δ=d/z, where the dynamical exponent z is, however, used instead of the spreading exponent z'. Both in d=1 and d=2 versions of our model, the exponent β most likely takes the mean-field value β=1, and we speculate that it might be due to the roulette-wheel selection, which is used to choose the site to supply a nutrient.

Two-dimensional percolation with multiple seeds

“…jamming percolation [14][15][16][17], and directed percolation [18][19][20][21][22][23][24][25] etc. Our model is certainly some kind of correlated percolation model.…”

Tumor proliferation and diffusion on percolation clusters

“…We used the Monte Carlo method and the percolation theory to produce a discrete twodimensional percolation cluster given by a square lattice of sides X × Y to represent the inhomogeneous tissue microenvironment [55,57]. Figure 1a is a typical tissue biopsy of tumor diffusion and invasion in a surrounding tissue and it was used to illustrate how to construct a percolation counterpart imitating tumor cells movement in surrounding tissue, as shown in Fig.…”

“…However, they did not compare the differences in the homogeneity of the tissues surrounding a tumor. Wendykier et al [54], Lipowski et al [55], Ferreira et al [56], and Moglia et al [57] used percolation to examine the critical behavior of avascular tumor growth, in which the survival and reproduction of tumor cells depended on the supplied nutrients. However, these studies did not systematically investigate the influence of the inhomogeneity of the surrounding tissue on tumor proliferation and diffusion.…”