Geometrical approach to tumor growth

Abstract

Tumor growth has a number of features in common with a physical process known as molecular beam epitaxy. Both growth processes are characterized by the constraint of growth development to the body border, and surface diffusion of cells/particles at the growing edge. However, tumor growth implies an approximate spherical symmetry that makes necessary a geometrical treatment of the growth equations. The basic model was introduced in a former article [C. Escudero, Phys.Rev. E 73, 020902(R) (2006)], and in the present work we extend our analysis and try to shed light on the possible geometrical principles that drive tumor growth. We present two-dimensional models that reproduce the experimental observations, and analyse the unexplored three-dimensional case, for which new conclusions on tumor growth are derived.

Statistics of interfacial fluctuations of radially growing clusters

“…In order to construct radial growth equations one may invoke the reparametrization invariance principle [21,22], as has already been done a number of times [10,11,[13][14][15][16]. In the case of white and Gaussian fluctuations, the d-dimensional spherical noise is given by…”

“…A series of works constitutes an exception to this rule [10][11][12][13][14][15][16], as they proposed a partial differential equation with stochastic terms as a benchmark for analyzing the dynamics of radial interfaces. Because studying this sort of equations is complicated by the nonlinearities implied by reparametrization invariance, a simplified version in which only the substrate growth was considered was introduced in [17].…”

Dynamic Scaling of Non-Euclidean Interfaces

“…expression that will be evaluated with the help of a small noise expansion [21]: r(θ, t) = R(t) + √ ǫρ(θ, t), where the noise amplitude ǫ is assumed to be small enough. We get R(t) = F t, for t > t 0 , and…”

“…However, the long time behavior shows a marked weakening of the fluctuations, and, as we will show, the logarithmic dependence is the footprint of critical dimensionality. The two-dimensional random deposition equation is [21] …”

Dynamics of curved interfaces

“…Subsequent works were devoted to the radial (1 + 1d) KPZ equation [16,17,18], the radial (1 + 1d) and spherical (2 + 1d) Mullins-Herring (MH) equa-tion [19,20], and the general reparametrization invariant formulation of stochastic growth equations [21]. An analytical approach to these equations [18] showed that for short spatial scales and time intervals the dynamics of radial interfaces was equivalent to that of the planar case; however, long time intervals yielded a different output.…”