Dynamics of cancer recurrence

Mutation-induced drug resistance in cancer often causes the failure of therapies and cancer recurrence, despite an initial tumor reduction. The timing of such cancer recurrence is governed by a balance between several factors such as initial tumor size, mutation rates and growth kinetics of drug-sensitive and resistance cells. To study this phenomenon we characterize the dynamics of escape from extinction of a subcritical branching process, where the establishment of a clone of escape mutants can lead to total population growth after the initial decline. We derive uniform in-time approximations for the paths of the escape process and its components, in the limit as the initial population size tends to infinity and the mutation rate tends to zero. In addition, two stochastic times important in cancer recurrence will be characterized: (i) the time at which the total population size first begins to rebound (i.e., become supercritical) during treatment, and (ii) the first time at which the resistant cell population begins to dominate the tumor.Comment: Published in at http://dx.doi.org/10.1214/12-AAP876 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Analysis of a Splitting Estimator for Rare Event Probabilities in Jackson Networks

We consider a standard splitting algorithm for the rare-event simulation of overflow probabilities in any subset of stations in a Jackson network at level n, starting at a fixed initial position. It was shown in DeanDup09 that a subsolution to the Isaacs equation guarantees that a subexponential number of function evaluations (in n) suffice to estimate such overflow probabilities within a given relative accuracy. Our analysis here shows that in fact O(n^{2{\beta}+1}) function evaluations suffice to achieve a given relative precision, where {\beta} is the number of bottleneck stations in the network. This is the first rigorous analysis that allows to favorably compare splitting against directly computing the overflow probability of interest, which can be evaluated by solving a linear system of equations with O(n^{d}) variables.Comment: 23 page

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 $i$

Minimizing Metastatic Risk in Radiotherapy Fractionation Schedules

Metastasis is the process by which cells from a primary tumor disperse and form new tumors at distant anatomical locations. The treatment and prevention of metastatic cancer remains an extremely challenging problem. This work introduces a novel biologically motivated objective function to the radiation optimization community that takes into account metastatic risk instead of the status of the primary tumor. In this work, we consider the problem of developing fractionated irradiation schedules that minimize production of metastatic cancer cells while keeping normal tissue damage below an acceptable level. A dynamic programming framework is utilized to determine the optimal fractionation scheme. We evaluated our approach on a breast cancer case using the heart and the lung as organs-at-risk (OAR). For small tumor $\alpha/\beta$ values, hypo-fractionated schedules were optimal, which is consistent with standard models. However, for relatively larger $\alpha/\beta$ values, we found the type of schedule depended on various parameters such as the time when metastatic risk was evaluated, the $\alpha/\beta$ values of the OARs, and the normal tissue sparing factors. Interestingly, in contrast to standard models, hypo-fractionated and semi-hypo-fractionated schedules (large initial doses with doses tapering off with time) were suggested even with large tumor $\alpha$/$\beta$ values. Numerical results indicate potential for significant reduction in metastatic risk.Comment: 12 pages, 3 figures, 2 table

Optimized Treatment Schedules for Chronic Myeloid Leukemia

Over the past decade, several targeted therapies (e.g. imatinib, dasatinib, nilotinib) have been developed to treat Chronic Myeloid Leukemia (CML). Despite an initial response to therapy, drug resistance remains a problem for some CML patients. Recent studies have shown that resistance mutations that preexist treatment can be detected in a substan- tial number of patients, and that this may be associated with eventual treatment failure. One proposed method to extend treatment efficacy is to use a combination of multiple targeted therapies. However, the design of such combination therapies (timing, sequence, etc.) remains an open challenge. In this work we mathematically model the dynamics of CML response to combination therapy and analyze the impact of combination treatment schedules on treatment efficacy in patients with preexisting resistance. We then propose an optimization problem to find the best schedule of multiple therapies based on the evolution of CML according to our ordinary differential equation model. This resulting optimiza- tion problem is nontrivial due to the presence of ordinary different equation constraints and integer variables. Our model also incorporates realistic drug toxicity constraints by tracking the dynamics of patient neutrophil counts in response to therapy. Using realis- tic parameter estimates, we determine optimal combination strategies that maximize time until treatment failure.Comment: 26 pages, 7 figure

Evolutionary dynamics of tumor progression with random fitness values

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. Other mutations, however, do not change the phenotype of the cell or even decrease cellular fitness. While much experimental effort is being devoted to the identification of the different functional effects of individual mutations, mathematical modeling of tumor progression generally considers constant fitness increments as mutations are accumulated. In this paper we study a mathematical model of tumor progression with random fitness increments. We analyze a multi-type branching process in which cells accumulate mutations whose fitness effects are chosen from a distribution. We determine the effect of the fitness distribution on the growth kinetics of the tumor. This work contributes to a quantitative understanding of the accumulation of mutations leading to cancer phenotypes.Comment: 33 pages, 2 Figure