Related papers: A Spatial Mutation Model with Increasing Mutation …
Motivated by models of cancer formation in which cells need to acquire $k$ mutations to become cancerous, we consider a spatial population model in which the population is represented by the $d$-dimensional torus of side length $L$.…
We study the spatial evolutionary dynamics of solid tumors as they obtain additional driver mutations. We start with a cancer clone that expands uniformly in three dimensions giving rise to a spherical shape. We assume that cell division…
Most cancers in humans are large, measuring centimeters in diameter, composed of many billions of cells. An equivalent mass of normal cells would be highly heterogeneous as a result of the mutations that occur during each cell division.…
The appearance of cancer in a tissue is thought to be the result of two or more successive mutations. We propose a stochastic model that allows for an exact computation of the distribution of the waiting time for a second mutation. This…
We consider a multistage cancer model in which cells are arranged in a $d$-dimensional integer lattice. Starting with all wild-type cells, we prove results about the distribution of the first time when two neutral mutations have accumulated…
Stochastic models of sequential mutation acquisition are widely used to quantify cancer and bacterial evolution. Across manifold scenarios, recurrent research questions are: how many cells are there with $n$ alterations, and how long will…
We consider a model of a population of fixed size N in which each individual gets replaced at rate one and each individual experiences a mutation at rate \mu. We calculate the asymptotic distribution of the time that it takes before there…
Cancer results from a sequence of genetic and epigenetic changes which lead to a variety of abnormal phenotypes including increased proliferation and survival of somatic cells, and thus, to a selective advantage of pre-cancerous cells. The…
Cancer progression is an evolutionary process that is driven by mutation and selection in a population of tumor cells. We discuss mathematical models of cancer progression, starting from traditional multistage theory. Each stage is…
Major efforts to sequence cancer genomes are now occurring throughout the world. Though the emerging data from these studies are illuminating, their reconciliation with epidemiologic and clinical observations poses a major challenge. In the…
We report on a simple model of spatial extend anti-tumor system with a fluctuation in growth rate, which can undergo a nonequilibrium phase transition. Three states as excited, sub-excited and non-excited states of a tumor are defined to…
Over time, a population acquires neutral genetic substitutions as a consequence of random drift. A famous result in population genetics asserts that the rate, $K$, at which these substitutions accumulate in the population coincides with the…
We consider a model of fixed size $N = 2^l$ in which there are $l$ generations of daughter cells and a stem cell. In each generation $i$ there are $2^{i-1}$ daughter cells. At each integral time unit the cells split so that the stem cell…
The problem of the onset and growth of solid tumour in homogeneous tissue is regarded using an approach based on local interaction between the tumoral and the sane tissue cells. The characteristic sizes and growth rates of spherical…
In this paper, a two-dimensional model for the growth of multi-layer tumors is presented. The model consists of a free boundary problem for the tumor cell membrane and the tumor is supposed to grow or shrink due to cell proliferation or…
Cancer results from genetic alterations that disturb the normal cooperative behavior of cells. Recent high-throughput genomic studies of cancer cells have shown that the mutational landscape of cancer is complex and that individual cancers…
In this work we explore the temporal dynamics of spatial heterogeneity during the process of tumorigenesis from healthy tissue. We utilize a spatial stochastic process model of mutation accumulation and clonal expansion in a structured…
We study a spatially inhomogeneous model of cancer growth based on Michaelis--Menten kinetics, subjected to additive Gaussian noise and multiplicative dichotomous noise. In presence of the latter, we can observe a transition between two…
Most human tumors result from the accumulation of multiple genetic and epigenetic alterations in a single cell. Mutations that confer a fitness advantage to the cell are known as driver mutations and are causally related to tumorigenesis.…
Spatial agent-based models are increasingly used to investigate the evolution of solid tumours subject to localised cell-cell interactions and microenvironmental heterogeneity. Here we present a non-technical step by step guide to…