arXiv subjects

Kevin Leder

Publications and source records attributed to Kevin Leder.

At least 19 recordsLinked to original sources

arXiv2026

The Site Frequency Spectrum in an Exponentially-Growing Population with Selection

We consider a supercritical two-type continuous-time linear birth-death process with mutation and selection, in which wild-type individuals give rise to mutant offspring with a larger net growth rate. In this setting, we investigate the site frequency spectrum (SFS) of driver mutations, describing the number of driver mutations present at any given frequency in the population. First, we derive exact moments for the SFS and establish asymptotic power laws at large times and frequencies. Then, strong laws of large numbers for the driver SFS are proven by constructing suitable $L^2$-approximations. These results apply both to the case when all driver clones have the same selective advantage and when the selective advantage is random. Finally, we allow the frequency to vary with time to examine the number of "intermediate" and "large" driver clones, identifying a cutoff frequency at which there are order 1 number of mutant clones. Overall, our results provide quantitative insights into how selection shapes the site frequency spectrum both at small and large frequencies, which can in principle be leveraged to construct estimators of relevant evolutionary parameters, including the selective advantage of driver mutations.

math.PR↗

arXiv2025

Parameter Estimation in Recurrent Tumor Evolution with Finite Carrying Capacity

In this work, we investigate the population dynamics of tumor cells under therapeutic pressure. Although drug treatment initially induces a reduction in tumor burden, treatment failure frequently occurs over time due to the emergence of drug resistance, ultimately leading to cancer recurrence. To model this process, we employ a two-type branching process with state-dependent growth rates. The model assumes an initial tumor population composed predominantly of drug-sensitive cells, with a small subpopulation of resistant cells. Sensitive cells may acquire resistance through mutation, which is coupled to a change in cellular fitness. Furthermore, the growth rates of resistant cells are modulated by the overall tumor burden. Using stochastic differential equation techniques, we establish a functional law of large numbers for the scaled populations of sensitive cells, resistant cells, and the initial resistant clone. We then define the stochastic recurrence time as the first time the total tumor population regrows to its initial size following treatment. For this recurrence time, as well as for measures of clonal diversity and the size of the largest resistant clone at recurrence, we derive corresponding law of large number limits. These asymptotic results provide a theoretical foundation for constructing statistically consistent estimators for key biological parameters, including the cellular growth rates, the mutation rate, and the initial fraction of resistant cells.

math.PR↗

arXiv2024

A statistical framework for detecting therapy-induced resistance from drug screens

Resistance to therapy remains a significant challenge in cancer treatment, often due to the presence of a stem-like cell population that drives tumor recurrence post-treatment. Moreover, many anticancer therapies induce plasticity, converting initially drug-sensitive cells to a more resistant state, e.g. through epigenetic processes and de-differentiation programs. Understanding the balance between therapeutic anti-tumor effects and induced resistance is critical for identifying treatment strategies. In this study, we introduce a robust statistical framework, based on multi-type branching process models of the evolutionary dynamics of tumor cell populations, to detect and quantify therapy-induced resistance phenomena from high throughput drug screening data. Through comprehensive in silico experiments, we show the efficacy of our framework in estimating parameters governing population dynamics and drug responses in a heterogeneous tumor population where cell state transitions are influenced by the drug. Finally, using recent in vitro data from multiple sources, we demonstrate that our framework is effective for analyzing real-world data and generating meaningful predictions.

q-bio.PE↗

arXiv2024

Novel Optimization Techniques for Parameter Estimation

In this paper, we introduce a new optimization algorithm that is well suited for solving parameter estimation problems. We call our new method cubic regularized Newton with affine scaling (CRNAS). In contrast to so-called first-order methods which rely solely on the gradient of the objective function, our method utilizes the Hessian of the objective. As a result it is able to focus on points satisfying the second-order optimality conditions, as opposed to first-order methods that simply converge to critical points. This is an important feature in parameter estimation problems where the objective function is often non-convex and as a result there can be many critical points making it is near impossible to identify the global minimum. An important feature of parameter estimation in mathematical models of biological systems is that the parameters are constrained by either physical constraints or prior knowledge. We use an affine scaling approach to handle a wide class of constraints. We establish that CRNAS identifies a point satisfying $\epsilon$-approximate second-order optimality conditions within $O(\epsilon^{-3/2})$ iterations. Finally, we compare CRNAS with MATLAB's optimization solver fmincon on three different test problems. These test problems all feature mixtures of heterogeneous populations, a problem setting that CRNAS is particularly well-suited for. Our numerical simulations show CRNAS has favorable performance, performing comparable if not better than fmincon in accuracy and computational cost for most of our examples.

math.OC↗

arXiv2024

Prediction of cancer dynamics under treatment using Bayesian neural networks: A simulated study

Predicting cancer dynamics under treatment is challenging due to high inter-patient heterogeneity, lack of predictive biomarkers, and sparse and noisy longitudinal data. Mathematical models can summarize cancer dynamics by a few interpretable parameters per patient. Machine learning methods can then be trained to predict the model parameters from baseline covariates, but do not account for uncertainty in the parameter estimates. Instead, hierarchical Bayesian modeling can model the relationship between baseline covariates to longitudinal measurements via mechanistic parameters while accounting for uncertainty in every part of the model. The mapping from baseline covariates to model parameters can be modeled in several ways. A linear mapping simplifies inference but fails to capture nonlinear covariate effects and scale poorly for interaction modeling when the number of covariates is large. In contrast, Bayesian neural networks can potentially discover interactions between covariates automatically, but at a substantial cost in computational complexity. In this work, we develop a hierarchical Bayesian model of subpopulation dynamics that uses baseline covariate information to predict cancer dynamics under treatment, inspired by cancer dynamics in multiple myeloma (MM), where serum M protein is a well-known proxy of tumor burden. As a working example, we apply the model to a simulated dataset and compare its ability to predict M protein trajectories to a model with linear covariate effects. Our results show that the Bayesian neural network covariate effect model predicts cancer dynamics more accurately than a linear covariate effect model when covariate interactions are present. The framework can also be applied to other types of cancer or other time series prediction problems that can be described with a parametric model.

q-bio.QM↗

arXiv2024

Parameter Estimation from Single Patient, Single Time-Point Sequencing Data of Recurrent Tumors

In this study, we develop consistent estimators for key parameters that govern the dynamics of tumor cell populations when subjected to pharmacological treatments. While these treatments often lead to an initial reduction in the abundance of drug-sensitive cells, a population of drug-resistant cells frequently emerges over time, resulting in cancer recurrence. Samples from recurrent tumors present as an invaluable data source that can offer crucial insights into the ability of cancer cells to adapt and withstand treatment interventions. To effectively utilize the data obtained from recurrent tumors, we derive several large number limit theorems, specifically focusing on the metrics that quantify the clonal diversity of cancer cell populations at the time of cancer recurrence. These theorems then serve as the foundation for constructing our estimators. A distinguishing feature of our approach is that our estimators only require a single time-point sequencing data from a single tumor, thereby enhancing the practicality of our approach and enabling the understanding of cancer recurrence at the individual level.

stat.AP↗

arXiv2023

Limit theorems for the site frequency spectrum of neutral mutations in an exponentially growing population

The site frequency spectrum (SFS) is a widely used summary statistic of genomic data. Motivated by recent evidence for the role of neutral evolution in cancer, we investigate the SFS of neutral mutations in an exponentially growing population. Using branching process techniques, we establish (first-order) almost sure convergence results for the SFS of a Galton-Watson process, evaluated either at a fixed time or at the stochastic time at which the population first reaches a certain size. We finally use our results to construct consistent estimators for the extinction probability and the effective mutation rate of a birth-death process.

math.PR↗

arXiv2023

Statistical inference of the rates of cell proliferation and phenotypic switching in cancer

Recent evidence suggests that nongenetic (epigenetic) mechanisms play an important role at all stages of cancer evolution. In many cancers, these mechanisms have been observed to induce dynamic switching between two or more cell states, which commonly show differential responses to drug treatments. To understand how these cancers evolve over time, and how they respond to treatment, we need to understand the state-dependent rates of cell proliferation and phenotypic switching. In this work, we propose a rigorous statistical framework for estimating these parameters, using data from commonly performed cell line experiments, where phenotypes are sorted and expanded in culture. The framework explicitly models the stochastic dynamics of cell division, cell death and phenotypic switching, and it provides likelihood-based confidence intervals for the model parameters. The input data can be either the fraction of cells or the number of cells in each state at one or more time points. Through a combination of theoretical analysis and numerical simulations, we show that when cell fraction data is used, the rates of switching may be the only parameters that can be estimated accurately. On the other hand, using cell number data enables accurate estimation of the net division rate for each phenotype, and it can even enable estimation of the state-dependent rates of cell division and cell death. We conclude by applying our framework to a publicly available dataset.

q-bio.QM↗

arXiv2022

Dynamics of advantageous mutant spread in spatial death-birth and birth-death Moran models

The spread of an advantageous mutation through a population is of fundamental interest in population genetics. While the classical Moran model is formulated for a well-mixed population, it has long been recognized that in real-world applications, the population usually has an explicit spatial structure which can significantly influence the dynamics. In the context of cancer initiation in epithelial tissue, several recent works have analyzed the dynamics of advantageous mutant spread on integer lattices, using the biased voter model from particle systems theory. In this spatial version of the Moran model, individuals first reproduce according to their fitness and then replace a neighboring individual. From a biological standpoint, the opposite dynamics, where individuals first die and are then replaced by a neighboring individual according to its fitness, are equally relevant. Here, we investigate this death-birth analogue of the biased voter model. We construct the process mathematically, derive the associated dual process, establish bounds on the survival probability of a single mutant, and prove that the process has an asymptotic shape. We also briefly discuss alternative birth-death and death-birth dynamics, depending on how the mutant fitness advantage affects the dynamics. We show that birth-death and death-birth formulations of the biased voter model are equivalent when fitness affects the former event of each update of the model, whereas the birth-death model is fundamentally different from the death-birth model when fitness affects the latter event.

q-bio.PE↗

arXiv2021

Clonal Diversity at Cancer Recurrence

Despite initial success, cancer therapies often fail due to the emergence of drug-resistant cells. In this study, we use a mathematical model to investigate how cancer evolves over time, specifically focusing on the state of the tumor when it recurs after treatment. We use a two-type branching process to capture the dynamics of both drug-sensitive and drug-resistant cells. We analyze the clonal diversity of drug-resistant cells at the time of cancer recurrence, which is defined as the first time the population size of drug-resistant cells exceeds a specified proportion of the initial population size of drug-sensitive cells. We examine two clonal diversity indices: the number of clones and the Simpson's Index. We calculate the expected values of these indices and utilize them to develop statistical methods for estimating model parameters. Additionally, we examine these two indices conditioned on early recurrence in the special case of a deterministically decaying sensitive population, with the aim of addressing the question of whether early recurrence is driven by a single mutation that generates an unusually large family of drug-resistant cells (corresponding to a low clonal diversity), or if it is due to the presence of an unusually large number of mutations causing drug resistance (corresponding to a high clonal diversity). Our findings, based on both indices, support the latter possibility. Furthermore, we demonstrate that the time of cancer recurrence can serve as a valuable indicator of clonal diversity, offering new insights for the treatment of recurrent cancers.

math.PR↗

arXiv2021

Exact site frequency spectra of neutrally evolving tumors: a transition between power laws reveals a signature of cell viability

The site frequency spectrum (SFS) is a popular summary statistic of genomic data. While the SFS of a constant-sized population undergoing neutral mutations has been extensively studied in population genetics, the rapidly growing amount of cancer genomic data has attracted interest in the spectrum of an exponentially growing population. Recent theoretical results have generally dealt with special or limiting cases, such as considering only cells with an infinite line of descent, assuming deterministic tumor growth, or taking large-time or large-population limits. In this work, we derive exact expressions for the expected SFS of a cell population that evolves according to a stochastic branching process, first for cells with an infinite line of descent and then for the total population, evaluated either at a fixed time (fixed-time spectrum) or at the stochastic time at which the population reaches a certain size (fixed-size spectrum). We find that while the rate of mutation scales the SFS of the total population linearly, the rates of cell birth and cell death change the shape of the spectrum at the small-frequency end, inducing a transition between a $1/j^2$ power-law spectrum and a $1/j$ spectrum as cell viability decreases. We show that this insight can in principle be used to estimate the ratio between the rate of cell death and cell birth, as well as the mutation rate, using the site frequency spectrum alone. Although the discussion is framed in terms of tumor dynamics, our results apply to any exponentially growing population of individuals undergoing neutral mutations.

q-bio.PE↗

arXiv2020

Large Deviations of Cancer Recurrence Timing

We study large deviation events in the timing of disease recurrence. In particular, we are interested in modeling cancer treatment failure due to mutation-induced drug resistance. We first present a two-type branching process model of this phenomenon, where an initial population of cells that are sensitive to therapy can produce mutants that are resistant to the therapy. In this model, we investigate two random times, the recurrence time and the crossover time. Recurrence time is defined as the first time that the population size of mutant cells exceeds a given proportion of the initial population size of drug-sensitive cells. Crossover time is defined as the first time that the resistant cell population dominates the total population. We establish convergence in probability results for both recurrence and crossover time. We then develop expressions for the large deviations rate of early recurrence and early crossover events. We characterize how the large deviation rates and rate functions depend on the initial size of the mutant cell population. We finally look at the large deviations rate of early recurrence conditioned on the number of mutant clones present at recurrence in the special case of a deterministically decaying sensitive population. We find that if recurrence occurs before the predicted law of large numbers limit then there will likely be an increase in the number of clones present at recurrence time.

math.PR↗

arXiv2020

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.

math.PR↗

arXiv2020

Understanding the role of phenotypic switching in cancer drug resistance

The emergence of acquired drug resistance in cancer represents a major barrier to treatment success. While research has traditionally focused on genetic sources of resistance, recent findings suggest that cancer cells can acquire transient resistant phenotypes via epigenetic modifications and other non-genetic mechanisms. Although these resistant phenotypes are eventually relinquished by individual cells, they can temporarily 'save' the tumor from extinction and enable the emergence of more permanent resistance mechanisms. These observations have generated interest in the potential of epigenetic therapies for long-term tumor control or eradication. In this work, we develop a mathematical model to study how phenotypic switching at the single-cell level affects resistance evolution in cancer. We highlight unique features of non-genetic resistance, probe the evolutionary consequences of epigenetic drugs and explore potential therapeutic strategies. We find that even short-term epigenetic modifications and stochastic fluctuations in gene expression can drive long-term drug resistance in the absence of any bona fide resistance mechanisms. We also find that an epigenetic drug that slightly perturbs the average retention of the resistant phenotype can turn guaranteed treatment failure into guaranteed success. Lastly, we find that combining an epigenetic drug with an anti-cancer agent can significantly outperform monotherapy, and that treatment outcome is heavily affected by drug sequencing.

q-bio.PE↗

arXiv2020

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 $i$.

math.PR↗

arXiv2019

Splitting Algorithms for Rare Events of Semimartingale Reflecting Brownian motions

We study rare event simulations of semimartingale reflecting Brownian motions (SRBMs) in an orthant. The rare event of interest is that a $d$-dimensional positive recurrent SRBM enters the set $B_n = \{z\in\mathbb{R}^d: \sum_{k=1}^d z_k = n\}$ before reaching a small neighborhood of the origin as $n\to\infty$. We show that under a proper scaling and some regularity conditions, the probability of interest satisfies a large deviation principle. We then construct a subsolution to the variational problem for our rare event, and based on this subsolution construct particle based simulation algorithms to estimate the probability of the rare event. It is shown that the proposed algorithm is stable and theoretically superior to standard Monte Carlo for a broad class of positive recurrent SRBMs.

math.PR↗

arXiv2016

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.

q-bio.TO↗

arXiv2016

Optimizing chemoradiotherapy to target multi-site metastatic disease and tumor growth

The majority of cancer-related fatalities are due to metastatic disease. In chemoradiotherapy, chemotherapeutic agents are administered along with radiation to increase damage to the primary tumor and control systemic disease such as metastasis. This work introduces a mathematical model to obtain optimal drug and radiation protocols in a chemoradiotherapy scheduling problem with the objective of minimizing metastatic cancer cell populations at multiple potential sites while maintaining a minimum level of damage to the primary tumor site. We derive closed-form expressions for an optimal chemotherapy fractionation regimen. A dynamic programming framework is used to determine the optimal radiotherapy fractionation regimen. Results show that chemotherapeutic agents do not change the optimal radiation fractionation regimens, and vice-versa. Interestingly, we observe that regardless of radio-sensitivity parameters, hypo-fractionated schedules are optimal solutions for the radiotherapy fractionation problem. Furthermore, it is optimal to immediately start radiotherapy. However, for chemotherapy, we find that the structure of the optimal schedule depends on model parameters such as chemotherapy-induced cell-kill at primary and metastatic sites, as well as the ability of primary tumor cells to initiate successful metastasis at different body sites. We quantify the trade-off between the new and traditional objectives of minimizing the metastatic population size and maximizing the tumor control probability, respectively, for a cervical cancer case. The trade-off information indicates the potential for significant reduction in the metastatic population with minimal loss in primary tumor control.

math.OC↗