Related papers: Eden model for Pentagons
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…
Development combines three basic processes asymmetric --- cell division, signaling and gene regulation --- in a multitude of ways to create an overwhelming diversity of multicellular life-forms. Here, we attempt to chart this diversity…
This paper investigates the parallel complexity of several non-equilibrium growth models. Invasion percolation, Eden growth, ballistic deposition and solid-on-solid growth are all seemingly highly sequential processes that yield…
This paper is placed at the intersection-point between the study of theoretical computational models aimed at capturing the essence of genetic regulatory networks and the field of Artificial Embryology (or Computational Development). A…
We consider regular tessellations of the plane as infinite graphs in which $q$ edges and $q$ faces meet at each vertex, and in which $p$ edges and $p$ vertices surround each face. For $1/p + 1/q = 1/2$, these are tilings of the Euclidean…
Following Richardson and using results of Kesten on First-passage percolation, we obtain an upper bound on the expected perimeter in an Eden Growth Process. Using results of the author from a problem in Statistical Mechanics, we show that…
We describe a percolation-type approach to modeling of the processes of aging and certain other properties of tissues analyzed as systems consisting of interacting cells. Tissues are considered as structures made of regular healthy,…
We study the moduli space of euclidean structures with cone points on a surface, and describe a decomposition into cells each of which corresponds to a given combinatorial type of Delaunay tessellation. We use some of the ideas to study…
We consider a method of construction of self-similar dendrites on a plane and establish main topological and metric properties of resulting class of dendrites.
Vertex Models, as used to describe cellular tissue, have an energy controlled by deviations of each cell area and perimeter from target values. The constrained nonlinear relation between area and perimeter leads to new mechanical response.…
Phase-field models of tumour growth have proved useful as theoretical tools to investigate cancer invasion. A key implicit assumption underlying mathematical models of this type which have so far been proposed, though, is that cells in the…
Given recipe of qualitative, kinetic modelling by geometric methods of three-dimensional dendritic crystals. Characteristic features of the perturbations appearing on the surface of a spherical body, leading to different scenarios of the…
Epigenetic Tracking is a mathematical model of biological cells, originally conceived to study embryonic development. Computer simulations proved the capacity of the model to generate complex 3-dimensional cellular structures, and the…
In this survey article, a variety of systems modeling tumor growth are discussed. In accordance with the hallmarks of cancer, the described models incorporate the primary characteristics of cancer evolution. Specifically, we focus on…
We formulate and analyze interior penalty discontinuous Galerkin methods for coupled elliptic PDEs modeling excitable tissue, represented by intracellular and extracellular domains sharing a common interface. The PDEs are coupled through a…
A class of Cantor-type spaces and related geometric structures are discussed.
The invention of three-dimensional printers has led to major innovations in tissue engineering. They have enabled the printing of complex geometries such as those that occur in natural tissues, that were not possible with traditional…
We develop a phase-field model of eutectic growth that uses three phase fields, admits strictly binary interfaces as stable solutions, and has a smooth free energy functional. We use this model to simulate oscillatory limit cycles in…
We analyze the structure of two dimensional disordered cellular systems generated by extensive computer simulations. These cellular structures are studied as topological trees rooted on a central cell or as closed shells arranged…
The Waddington landscape serves as a metaphor illustrating the developmental process of cells, likening it to a small ball rolling down various trajectories into valleys. Constructing an epigenetic landscape of this nature aids in…