Cancer stem cells (CSCs) possess capacity to both self-renew and generate all cells within a tumor, and are thought to drive tumor recurrence. Targeting the stem cell niche to eradicate CSCs represents an important area of therapeutic development. The complex nature of many interacting elements of the stem cell niche, including both intracellular signals and microenvironmental growth factors and cytokines, creates a challenge in choosing which elements to target, alone or in combination. Stochastic stimulation techniques allow for the careful study of complex systems in biology and medicine and are ideal for the investigation of strategies aimed at CSC eradication. We present a mathematical model of the breast cancer stem cell (BCSC) niche to predict population dynamics during carcinogenesis and in response to treatment. Using data from cell line and mouse xenograft experiments, we estimate rates of interconversion between mesenchymal and epithelial states in BCSCs and find that EMT/MET transitions occur frequently. We examine bulk tumor growth dynamics in response to alterations in the rate of symmetric self-renewal of BCSCs and find that small changes in BCSC behavior can give rise to the Gompertzian growth pattern observed in breast tumors. Finally, we examine stochastic reaction kinetic simulations in which elements of the breast cancer stem cell niche are inhibited individually and in combination. We find that slowing self-renewal and disrupting the positive feedback loop between IL-6, Stat3 activation, and NF-κB signaling by simultaneous inhibition of IL-6 and HER2 is the most effective combination to eliminate both mesenchymal and epithelial populations of BCSCs. Predictions from our model and simulations show excellent agreement with experimental data showing the efficacy of combined HER2 and Il-6 blockade in reducing BCSC populations. Our findings will be directly examined in a planned clinical trial of combined HER2 and IL-6 targeted therapy in HER2-positive breast cancer.
The SARS-CoV-2 pandemic led to the closure of nearly all K-12 schools in the United States of America in March 2020. Although reopening K-12 schools for in-person schooling is desirable for many reasons, officials also understand that risk reduction strategies and detection of cases must be in place to allow children to safely return to school. Furthermore, the consequences of reclosing recently reopened schools are substantial and impact teachers, parents, and ultimately the educational experience in children. Using a stratified Susceptible-Exposed-Infected-Removed model, we explore the influences of reduced class density, transmission mitigation (such as the use of masks, desk shields, frequent surface cleaning, or outdoor instruction), and viral detection on cumulative prevalence. Our model predicts that a combination of all three approaches will substantially reduce SARS-CoV-2 prevalence. The model also shows that reduction of class density and the implementation of rapid viral testing, even with imperfect detection, have greater impact than moderate measures for transmission mitigation.
The SARS-CoV-2 pandemic led to closure of nearly all K-12 schools in the United States of America in March 2020. Although reopening K-12 schools for in-person schooling is desirable for many reasons, officials understand that risk reduction strategies and detection of cases are imperative in creating a safe return to school. Furthermore, consequences of reclosing recently opened schools are substantial and impact teachers, parents, and ultimately educational experiences in children. To address competing interests in meeting educational needs with public safety, we compare the impact of physical separation through school cohorts on SARS-CoV-2 infections against policies acting at the level of individual contacts within classrooms. Using an age-stratified Susceptible-Exposed-Infected-Removed model, we explore influences of reduced class density, transmission mitigation, and viral detection on cumulative prevalence. We consider several scenarios over a 6-month period including (1) multiple rotating cohorts in which students cycle through in-person instruction on a weekly basis, (2) parallel cohorts with in-person and remote learning tracks, (3) the impact of a hypothetical testing program with ideal and imperfect detection, and (4) varying levels of aggregate transmission reduction. Our mathematical model predicts that reducing the number of contacts through cohorts produces a larger effect than diminishing transmission rates per contact. Specifically, the latter approach requires dramatic reduction in transmission rates in order to achieve a comparable effect in minimizing infections over time. Further, our model indicates that surveillance programs using less sensitive tests may be adequate in monitoring infections within a school community by both keeping infections low and allowing for a longer period of instruction. Lastly, we underscore the importance of factoring infection prevalence in deciding when a local outbreak of infection is serious enough to require reverting to remote learning.
Background and Objectives: Biological systems with intertwined feedback loops pose a challenge to mathematical modeling efforts. Moreover, rare events, such as mutation and extinction, complicate system dynamics. Stochastic simulation algorithms are useful in generating time-evolution trajectories for these systems because they can adequately capture the influence of random fluctuations and quantify rare events. We present a simple and flexible package, BioSimulator.jl, for implementing the Gillespie algorithm, τ-leaping, and related stochastic simulation algorithms. The objective of this work is to provide scientists across domains with fast, user-friendly simulation tools. Methods: We used the high-performance programming language Julia because of its emphasis on scientific computing. Our software package implements a suite of stochastic simulation algorithms based on Markov chain theory. We provide the ability to (a) diagram Petri Nets describing interactions, (b) plot average trajectories and attached standard deviations of each participating species over time, and (c) generate frequency distributions of each species at a specified time. Results: BioSimulator.jl’s as interface allows users to build models programmatically within Julia. A model is then passed to the simulate routine to generate simulation data. The built-in tools allow one to visualize results and compute summary statistics. Our examples highlight the broad applicability of our software to systems of varying complexity from ecology, systems biology, chemistry, and genetics. Conclusion: The user-friendly nature of BioSimulator.jl encourages the use of stochastic simulation, minimizes tedious programming efforts, and reduces errors during model specification.
Differential sensitivity analysis is indispensable in fitting parameters, understanding uncertainty, and forecasting the results of both thought and lab experiments. Although there are many methods currently available for performing differential sensitivity analysis of biological models, it can be difficult to determine which method is best suited for a particular model. In this paper, we explain a variety of differential sensitivity methods and assess their value in some typical biological models. First, we explain the mathematical basis for three numerical methods: adjoint sensitivity analysis, complex perturbation sensitivity analysis, and forward mode sensitivity analysis. We then carry out four instructive case studies. (a) The CARRGO model for tumor-immune interaction highlights the additional information that differential sensitivity analysis provides beyond traditional naive sensitivity methods, (b) the deterministic SIR model demonstrates the value of using second-order sensitivity in refining model predictions, (c) the stochastic SIR model shows how differential sensitivity can be attacked in stochastic modeling, and (d) a discrete birth-death-migration model illustrates how the complex perturbation method of differential sensitivity can be generalized to a broader range of biological models. Finally, we compare the speed, accuracy, and ease of use of these methods. We find that forward mode automatic differentiation has the quickest computational time, while the complex perturbation method is the simplest to implement and the most generalizable.
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