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

Mutation timing in a spatial model of evolution

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 $L$. Initially, no sites have mutations, but sites with $i-1$ mutations acquire an $i$th mutation at rate $\mu_i$ per unit area. Mutations spread to neighboring sites at rate $\alpha$, so that $t$ time units after a mutation, the region of individuals that have acquired the mutation will be a ball of radius $\alpha t$. We calculate, for some ranges of the parameter values, the asymptotic distribution of the time required for some individual to acquire $k$ mutations. Our results, which build on previous work of Durrett, Foo, and Leder, are essentially complete when $k = 2$ and when $\mu_i = \mu$ for all $i$

In Sickness and in Health, Until Death Do Us Part : An Examination of FMLA Rights for Same-Sex Spouses and a Case Note on Obergefell v. Hodges

This note discusses the history of the lesbian, gay, bisexual, and transgender (LGBT) struggle for equal rights alongside the Supreme Court\u27s recent ruling in Obergefell v. Hodges and uses this to examine the potential effect on the rights granted to same-sex spouses by the Family Medical Leave Act (FMLA). Part II records the jurisprudence that has slowly evolved over the past forty to fifty years to make the present a more hospitable era for same-sex marriage to take root today. Part III gives a general overview of the FMLA\u27s history and current form. Part IV reviews the facts prompting the Court\u27s decision in Obergefell. Part V analyzes the majority opinion alongside the dissenting opinions by comparing and contrasting the dissenting judges\u27 separate and underlying interests. Part VI theorizes Obergefell\u27s potential legal and social impact on FMLA requirements and the definition of spouse, and it concludes by examining how these changes might work to open the way for further advancements in same-sex rights in employment and healthcare law

A Comparison of Mutation and Amplification-Driven Resistance Mechanisms and Their Impacts on Tumor Recurrence

Tumor recurrence, driven by the evolution of drug resistance is a major barrier to therapeutic success in cancer. Resistance is often caused by genetic alterations such as point mutation, which refers to the modification of a single genomic base pair, or gene amplification, which refers to the duplication of a region of DNA that contains a gene. Here we investigate the dependence of tumor recurrence dynamics on these mechanisms of resistance, using stochastic multi-type branching process models. We derive tumor extinction probabilities and deterministic estimates for the tumor recurrence time, defined as the time when an initially drug sensitive tumor surpasses its original size after developing resistance. For models of amplification-driven and mutation-driven resistance, we prove law of large numbers results regarding the convergence of the stochastic recurrence times to their mean. Additionally, we prove sufficient and necessary conditions for a tumor to escape extinction under the gene amplification model, discuss behavior under biologically relevant parameters, and compare the recurrence time and tumor composition in the mutation and amplification models both analytically and using simulations. In comparing these mechanisms, we find that the ratio between recurrence times driven by amplification vs. mutation depends linearly on the number of amplification events required to acquire the same degree of resistance as a mutation event, and we find that the relative frequency of amplification and mutation events plays a key role in determining the mechanism under which recurrence is more rapid. In the amplification-driven resistance model, we also observe that increasing drug concentration leads to a stronger initial reduction in tumor burden, but that the eventual recurrent tumor population is less heterogeneous, more aggressive, and harbors higher levels of drug-resistance.Comment: 52 Pages, 5 figure

Spread of premalignant mutant clones and cancer initiation in multilayered tissue

Over 80% of human cancers originate from the epithelium, which covers the outer and inner surfaces of organs and blood vessels. In stratified epithelium, the bottom layers are occupied by stem and stem-like cells that continually divide and replenish the upper layers. In this work, we study the spread of premalignant mutant clones and cancer initiation in stratified epithelium using the biased voter model on stacked two-dimensional lattices. Our main result is an estimate of the propagation speed of a premalignant mutant clone, which is asymptotically precise in the cancer-relevant weak-selection limit. We use our main result to study cancer initiation under a two-step mutational model of cancer, which includes computing the distributions of the time of cancer initiation and the size of the premalignant clone giving rise to cancer. Our work quantifies the effect of epithelial tissue thickness on the process of carcinogenesis, thereby contributing to an emerging understanding of the spatial evolutionary dynamics of cancer.Comment: 44 pages, 11 figure