Multifocality and recurrence risk: a quantitative model of field cancerization

Primary tumors often emerge within genetically altered fields of premalignant cells that appear histologically normal but have a high chance of progression to malignancy. Clinical observations have suggested that these premalignant fields pose high risks for emergence of secondary recurrent tumors if left behind after surgical removal of the primary tumor. In this work, we develop a spatio-temporal stochastic model of epithelial carcinogenesis, combining cellular reproduction and death dynamics with a general framework for multi-stage genetic progression to cancer. Using this model, we investigate how macroscopic features (e.g. size and geometry of premalignant fields) depend on microscopic cellular properties of the tissue (e.g.\ tissue renewal rate, mutation rate, selection advantages conferred by genetic events leading to cancer, etc). We develop methods to characterize how clinically relevant quantities such as waiting time until emergence of second field tumors and recurrence risk after tumor resection. We also study the clonal relatedness of recurrent tumors to primary tumors, and analyze how these phenomena depend upon specific characteristics of the tissue and cancer type. This study contributes to a growing literature seeking to obtain a quantitative understanding of the spatial dynamics in cancer initiation.Comment: 36 pages, 11 figure

Recurrence Plots in Nonlinear Time Series Analysis: Free Software

Recurrence plots are graphical devices specially suited to detect hidden dynamical patterns and nonlinearities in data. However, there are few programs available to apply such a mehodology. This paper reviews one of the best free programs to apply nonlinear time series analysis: Visual Recurrence Analysis (VRA). This program is targeted to recurrence analysis and the so-called Recurrence Quantitative Analysis (RQA, the quantitative counterpart of recurrence plots), although it includes many procedures in a friendly visual environment. Comparisons with alternative programs are performed.

Quantitative recurrence in two-dimensional extended processes

Under some mild condition, a random walk in the plane is recurrent. In particular each trajectory is dense, and a natural question is how much time one needs to approach a given small neighborhood of the origin. We address this question in the case of some extended dynamical systems similar to planar random walks, including \ZZ^2-extension of hyperbolic dynamics. We define a pointwise recurrence rate and relate it to the dimension of the process, and establish a convergence in distribution of the rescaled return times near the origin

Quantitative recurrence properties in conformal iterated function systems

Let $\Lambda$ be a countable index set and $S=\{\phi_i: i\in \Lambda\}$ be a conformal iterated function system on $[0,1]^d$ satisfying the open set condition. Denote by $J$ the attractor of $S$. With each sequence $(w_1,w_2,...)\in \Lambda^{\mathbb{N}}$ is associated a unique point $x\in [0,1]^d$. Let $J^\ast$ denote the set of points of $J$ with unique coding, and define the mapping $T:J^\ast \to J^\ast$ by $Tx= T (w_1,w_2, w_3...) = (w_2,w_3,...)$. In this paper, we consider the quantitative recurrence properties related to the dynamical system $(J^\ast, T)$. More precisely, let $f:[0,1]^d\to \mathbb{R}^+$ be a positive function and $$R(f):=\{x\in J^\ast: |T^nx-x|<e^{-S_n f(x)}, \ {\text{for infinitely many}}\ n\in \mathbb{N}\},$$ where $S_n f(x)$ is the $n$th Birkhoff sum associated with the potential $f$. In other words, $R(f)$ contains the points $x$ whose orbits return close to $x$ infinitely often, with a rate varying along time. Under some conditions, we prove that the Hausdorff dimension of $R(f)$ is given by $\inf\{s\ge 0: P(T, -s(f+\log |T'|))\le 0\}$, where $P$ is the pressure function and $T'$ is the derivative of $T$. We present some applications of the main theorem to Diophantine approximation.Comment: 25 page

Approximation of symmetrizations by Markov processes

Under continuity and recurrence assumptions, we prove that the iteration of successive partial symmetrizations that form a time-homogeneous Markov process, converges to a symmetrization. We cover several settings, including the approximation of the spherical nonincreasing rearrangement by Steiner symmetrizations, polarizations and cap symmetrizations. A key tool in our analysis is a quantitative measure of the asymmetry

The recurrence time for ergodic systems of infinite measures

We investigate quantitative recurrence in systems having an infinite measure. We extend the Ornstein-Weiss theorem for a general class of infinite systems estimating return time in decreasing sequences of cylinders. Then we restrict to a class of one dimensional maps with indifferent fixed points and calculate quantitative recurrence in sequences of balls, obtaining that this is related to the behavior of the map near the fixed points