A novel COVID-19 epidemiological model with explicit susceptible and asymptomatic isolation compartments reveals unexpected consequences of timing social distancing.

Motivated by the current COVID-19 epidemic, this work introduces an epidemiological model in which separate compartments are used for susceptible and asymptomatic "socially distant" populations. Distancing directives are represented by rates of flow into these compartments, as well as by a reduction in contacts that lessens disease transmission. The dynamical behavior of this system is analyzed, under various different rate control strategies, and the sensitivity of the basic reproduction number to various parameters is studied. One of the striking features of this model is the existence of a critical implementation delay (CID) in issuing distancing mandates: while a delay of about two weeks does not have an appreciable effect on the peak number of infections, issuing mandates even slightly after this critical time results in a far greater incidence of infection. Thus, there is a nontrivial but tight "window of opportunity" for commencing social distancing in order to meet the capacity of healthcare resources. However, if one wants to also delay the timing of peak infections - so as to take advantage of potential new therapies and vaccines - action must be taken much faster than the CID. Different relaxation strategies are also simulated, with surprising results. Periodic relaxation policies suggest a schedule which may significantly inhibit peak infective load, but that this schedule is very sensitive to parameter values and the schedule's frequency. Furthermore, we considered the impact of steadily reducing social distancing measures over time. We find that a too-sudden reopening of society may negate the progress achieved under initial distancing guidelines, but the negative effects can be mitigated if the relaxation strategy is carefully designed.

Predicting the impact of placing an overdose prevention site in Philadelphia: a mathematical modeling approach.

BACKGROUND: Fatal overdoses from opioid use and substance disorders are increasing at an alarming rate. One proposed harm reduction strategy for reducing overdose fatalities is to place overdose prevention sites-commonly known as safe injection facilities-in proximity of locations with the highest rates of overdose. As urban centers in the USA are tackling legal hurdles and community skepticism around the introduction and location of these sites, it becomes increasingly important to assess the magnitude of the effect that these services might have on public health. METHODS: We developed a mathematical model to describe the movement of people who used opioids to an overdose prevention site in order to understand the impact that the facility would have on overdoses, fatalities, and user education and treatment/recovery. The discrete-time, stochastic model is able to describe a range of user behaviors, including the effects from how far they need to travel to the site. We calibrated the model to overdose data from Philadelphia and ran simulations to describe the effect of placing a site in the Kensington neighborhood. RESULTS: In Philadelphia, which has a non-uniform racial population distribution, choice of site placement can determine which demographic groups are most helped. In our simulations, placement of the site in the Kensington neighborhood resulted in White opioid users being more likely to benefit from the site's services. Overdoses that occur onsite can be reversed. Our results predict that for every 30 stations in the overdose prevention site, 6 per year of these would have resulted in fatalities if they had occurred outside of the overdose prevention site. Additionally, we estimate that fatalities will decrease further when referrals from the OPS to treatment are considered. CONCLUSIONS: Mathematical modeling was used to predict the impact of placing an overdose prevention site in the Kensington neighborhood of Philadelphia. To fully understand the impact of site placement, both direct and indirect effects must be included in the analysis. Introducing more than one site and distributing sites equally across neighborhoods with different racial and demographic characteristics would have the broadest public health impact. Cities and locales can use mathematical modeling to help quantify the predicted impact of placing an overdose prevention site in a particular location.

Evaluating optimal therapy robustness by virtual expansion of a sample population, with a case study in cancer immunotherapy.

Cancer is a highly heterogeneous disease, exhibiting spatial and temporal variations that pose challenges for designing robust therapies. Here, we propose the VEPART (Virtual Expansion of Populations for Analyzing Robustness of Therapies) technique as a platform that integrates experimental data, mathematical modeling, and statistical analyses for identifying robust optimal treatment protocols. VEPART begins with time course experimental data for a sample population, and a mathematical model fit to aggregate data from that sample population. Using nonparametric statistics, the sample population is amplified and used to create a large number of virtual populations. At the final step of VEPART, robustness is assessed by identifying and analyzing the optimal therapy (perhaps restricted to a set of clinically realizable protocols) across each virtual population. As proof of concept, we have applied the VEPART method to study the robustness of treatment response in a mouse model of melanoma subject to treatment with immunostimulatory oncolytic viruses and dendritic cell vaccines. Our analysis (i) showed that every scheduling variant of the experimentally used treatment protocol is fragile (nonrobust) and (ii) discovered an alternative region of dosing space (lower oncolytic virus dose, higher dendritic cell dose) for which a robust optimal protocol exists.

Finding causative genes from high-dimensional data: an appraisal of statistical and machine learning approaches.

Modern biological experiments often involve high-dimensional data with thousands or more variables. A challenging problem is to identify the key variables that are related to a specific disease. Confounding this task is the vast number of statistical methods available for variable selection. For this reason, we set out to develop a framework to investigate the variable selection capability of statistical methods that are commonly applied to analyze high-dimensional biological datasets. Specifically, we designed six simulated cancers (based on benchmark colon and prostate cancer data) where we know precisely which genes cause a dataset to be classified as cancerous or normal - we call these causative genes. We found that not one statistical method tested could identify all the causative genes for all of the simulated cancers, even though increasing the sample size does improve the variable selection capabilities in most cases. Furthermore, certain statistical tools can classify our simulated data with a low error rate, yet the variables being used for classification are not necessarily the causative genes.

Microenvironment-Mediated Modeling of Tumor Response to Vascular-Targeting Drugs.

The tumor-associated microvasculature is one of the key elements of the microenvironment that helps shape, and is shaped by, tumor progression. Given the important role of the vasculature in tumor progression, and the fact that tumor and normal vasculature are physiologically and molecularly distinct, much effort has gone into the development of vascular-targeting drugs that in theory should target tumors without significant risk to normal tissue. In this chapter, a multiscale hybrid mathematical model of tumor-vascular interactions is presented to provide a theoretical basis for assessing tumor response to vascular-targeting drugs. Model performance is calibrated to quantitative clinical data on tumor response to angiogenesis inhibitors (AIs), preclinical data on response to a cytotoxic chemotherapy, and qualitative preclinical data on response to vascular disrupting agents (VDAs). The calibrated model is then used to explore two questions of clinical interest. First, the hypothesis that AIs and VDAs are complementary treatments, rather than redundant, is explored. The model predicts a minimal increase in antitumor activity as a result of adding a VDA to an AI treatment regimen, and in fact at times the combination can exert less antitumor activity than stand-alone AI treatment. Second, the question of identifying an optimal dosing strategy for treating with an AI and a cytotoxic agent is addressed. Using a stochastic optimization scheme, an intermittent schedule for both chemotherapy and AI administration is identified that can eradicate the simulated tumors. We propose that this schedule may have increased clinical antitumor activity compared to currently used treatment protocols.

Microenvironmental Niches and Sanctuaries: A Route to Acquired Resistance.

A tumor vasculature that is functionally abnormal results in irregular gradients of metabolites and drugs within the tumor tissue. Recently, significant efforts have been committed to experimentally examine how cellular response to anti-cancer treatments varies based on the environment in which the cells are grown. In vitro studies point to specific conditions in which tumor cells can remain dormant and survive the treatment. In vivo results suggest that cells can escape the effects of drug therapy in tissue regions that are poorly penetrated by the drugs. Better understanding how the tumor microenvironments influence the emergence of drug resistance in both primary and metastatic tumors may improve drug development and the design of more effective therapeutic protocols. This chapter presents a hybrid agent-based model of the growth of tumor micrometastases and explores how microenvironmental factors can contribute to the development of acquired resistance in response to a DNA damaging drug. The specific microenvironments of interest in this work are tumor hypoxic niches and tumor normoxic sanctuaries with poor drug penetration. We aim to quantify how spatial constraints of limited drug transport and quiescent cell survival contribute to the development of drug resistant tumors.

Minimally sufficient experimental design using identifiability analysis.

Mathematical models are increasingly being developed and calibrated in tandem with data collection, empowering scientists to intervene in real time based on quantitative model predictions. Well-designed experiments can help augment the predictive power of a mathematical model but the question of when to collect data to maximize its utility for a model is non-trivial. Here we define data as model-informative if it results in a unique parametrization, assessed through the lens of practical identifiability. The framework we propose identifies an optimal experimental design (how much data to collect and when to collect it) that ensures parameter identifiability (permitting confidence in model predictions), while minimizing experimental time and costs. We demonstrate the power of the method by applying it to a modified version of a classic site-of-action pharmacokinetic/pharmacodynamic model that describes distribution of a drug into the tumor microenvironment (TME), where its efficacy is dependent on the level of target occupancy in the TME. In this context, we identify a minimal set of time points when data needs to be collected that robustly ensures practical identifiability of model parameters. The proposed methodology can be applied broadly to any mathematical model, allowing for the identification of a minimally sufficient experimental design that collects the most informative data.

Mitigating non-genetic resistance to checkpoint inhibition based on multiple states of immune exhaustion.

Despite the revolutionary impact of immune checkpoint inhibition on cancer therapy, the lack of response in a subset of patients, as well as the emergence of resistance, remain significant challenges. Here we explore the theoretical consequences of the existence of multiple states of immune cell exhaustion on response to checkpoint inhibition therapy. In particular, we consider the emerging understanding that T cells can exist in various states: fully functioning cytotoxic cells, reversibly exhausted cells with minimal cytotoxicity, and terminally exhausted cells. We hypothesize that inflammation augmented by drug activity triggers transitions between these phenotypes, which can lead to non-genetic resistance to checkpoint inhibitors. We introduce a conceptual mathematical model, coupled with a standard 2-compartment pharmacometric (PK) model, that incorporates these mechanisms. Simulations of the model reveal that, within this framework, the emergence of resistance to checkpoint inhibitors can be mitigated through altering the dose and the frequency of administration. Our analysis also reveals that standard PK metrics do not correlate with treatment outcome. However, we do find that levels of inflammation that we assume trigger the transition from the reversibly to terminally exhausted states play a critical role in therapeutic outcome. A simulation of a population that has different values of this transition threshold reveals that while the standard high-dose, low-frequency dosing strategy can be an effective therapeutic design for some, it is likely to fail a significant fraction of the population. Conversely, a metronomic-like strategy that distributes a fixed amount of drug over many doses given close together is predicted to be effective across the entire simulated population, even at a relatively low cumulative drug dose. We also demonstrate that these predictions hold if the transitions between different states of immune cell exhaustion are triggered by prolonged antigen exposure, an alternative mechanism that has been implicated in this process. Our theoretical analyses demonstrate the potential of mitigating resistance to checkpoint inhibitors via dose modulation.

Guiding model-driven combination dose selection using multi-objective synergy optimization.

Despite the growing appreciation that the future of cancer treatment lies in combination therapies, finding the right drugs to combine and the optimal way to combine them remains a nontrivial task. Herein, we introduce the Multi-Objective Optimization of Combination Synergy - Dose Selection (MOOCS-DS) method for using drug synergy as a tool for guiding dose selection for a combination of preselected compounds. This method decouples synergy of potency (SoP) and synergy of efficacy (SoE) and identifies Pareto optimal solutions in a multi-objective synergy space. Using a toy combination therapy model, we explore properties of the MOOCS-DS algorithm, including how optimal dose selection can be influenced by the metric used to define SoP and SoE. We also demonstrate the potential of our approach to guide dose and schedule selection using a model fit to preclinical data of the combination of the PD-1 checkpoint inhibitor pembrolizumab and the anti-angiogenic drug bevacizumab on two lung cancer cell lines. The identification of optimally synergistic combination doses has the potential to inform preclinical experimental design and improve the success rates of combination therapies. Jel classificationDose Finding in Oncology.

From Fitting the Average to Fitting the Individual: A Cautionary Tale for Mathematical Modelers.

An outstanding challenge in the clinical care of cancer is moving from a one-size-fits-all approach that relies on population-level statistics towards personalized therapeutic design. Mathematical modeling is a powerful tool in treatment personalization, as it allows for the incorporation of patient-specific data so that treatment can be tailor-designed to the individual. Herein, we work with a mathematical model of murine cancer immunotherapy that has been previously-validated against the average of an experimental dataset. We ask the question: what happens if we try to use this same model to perform personalized fits, and therefore make individualized treatment recommendations? Typically, this would be done by choosing a single fitting methodology, and a single cost function, identifying the individualized best-fit parameters, and extrapolating from there to make personalized treatment recommendations. Our analyses show the potentially problematic nature of this approach, as predicted personalized treatment response proved to be sensitive to the fitting methodology utilized. We also demonstrate how a small amount of the right additional experimental measurements could go a long way to improve consistency in personalized fits. Finally, we show how quantifying the robustness of the average response could also help improve confidence in personalized treatment recommendations.

Model-informed experimental design recommendations for distinguishing intrinsic and acquired targeted therapeutic resistance in head and neck cancer.

The promise of precision medicine has been limited by the pervasive resistance to many targeted therapies for cancer. Inferring the timing (i.e., pre-existing or acquired) and mechanism (i.e., drug-induced) of such resistance is crucial for designing effective new therapeutics. This paper studies cetuximab resistance in head and neck squamous cell carcinoma (HNSCC) using tumor volume data obtained from patient-derived tumor xenografts. We ask if resistance mechanisms can be determined from this data alone, and if not, what data would be needed to deduce the underlying mode(s) of resistance. To answer these questions, we propose a family of mathematical models, with each member of the family assuming a different timing and mechanism of resistance. We present a method for fitting these models to individual volumetric data, and utilize model selection and parameter sensitivity analyses to ask: which member(s) of the family of models best describes HNSCC response to cetuximab, and what does that tell us about the timing and mechanisms driving resistance? We find that along with time-course volumetric data to a single dose of cetuximab, the initial resistance fraction and, in some instances, dose escalation volumetric data are required to distinguish among the family of models and thereby infer the mechanisms of resistance. These findings can inform future experimental design so that we can best leverage the synergy of wet laboratory experimentation and mathematical modeling in the study of novel targeted cancer therapeutics.

A novel three-phase model of brain tissue microstructure.

We propose a novel biologically constrained three-phase model of the brain microstructure. Designing a realistic model is tantamount to a packing problem, and for this reason, a number of techniques from the theory of random heterogeneous materials can be brought to bear on this problem. Our analysis strongly suggests that previously developed two-phase models in which cells are packed in the extracellular space are insufficient representations of the brain microstructure. These models either do not preserve realistic geometric and topological features of brain tissue or preserve these properties while overestimating the brain's effective diffusivity, an average measure of the underlying microstructure. In light of the highly connected nature of three-dimensional space, which limits the minimum diffusivity of biologically constrained two-phase models, we explore the previously proposed hypothesis that the extracellular matrix is an important factor that contributes to the diffusivity of brain tissue. Using accurate first-passage-time techniques, we support this hypothesis by showing that the incorporation of the extracellular matrix as the third phase of a biologically constrained model gives the reduction in the diffusion coefficient necessary for the three-phase model to be a valid representation of the brain microstructure.

Mathematical Approach to Differentiate Spontaneous and Induced Evolution to Drug Resistance During Cancer Treatment.

PURPOSE: Drug resistance is a major impediment to the success of cancer treatment. Resistance is typically thought to arise from random genetic mutations, after which mutated cells expand via Darwinian selection. However, recent experimental evidence suggests that progression to drug resistance need not occur randomly, but instead may be induced by the treatment itself via either genetic changes or epigenetic alterations. This relatively novel notion of resistance complicates the already challenging task of designing effective treatment protocols. MATERIALS AND METHODS: To better understand resistance, we have developed a mathematical modeling framework that incorporates both spontaneous and drug-induced resistance. RESULTS: Our model demonstrates that the ability of a drug to induce resistance can result in qualitatively different responses to the same drug dose and delivery schedule. We have also proven that the induction parameter in our model is theoretically identifiable and propose an in vitro protocol that could be used to determine a treatment's propensity to induce resistance.

Simulating tumor growth in confined heterogeneous environments.

The holy grail of computational tumor modeling is to develop a simulation tool that can be utilized in the clinic to predict neoplastic progression and propose individualized optimal treatment strategies. In order to develop such a predictive model, one must account for many of the complex processes involved in tumor growth. One interaction that has not been incorporated into computational models of neoplastic progression is the impact that organ-imposed physical confinement and heterogeneity have on tumor growth. For this reason, we have taken a cellular automaton algorithm that was originally designed to simulate spherically symmetric tumor growth and generalized the algorithm to incorporate the effects of tissue shape and structure. We show that models that do not account for organ/tissue geometry and topology lead to false conclusions about tumor spread, shape and size. The impact that confinement has on tumor growth is more pronounced when a neoplasm is growing close to, versus far from, the confining boundary. Thus, any clinical simulation tool of cancer progression must not only consider the shape and structure of the organ in which a tumor is growing, but must also consider the location of the tumor within the organ if it is to accurately predict neoplastic growth dynamics.

Developing a Minimally Structured Mathematical Model of Cancer Treatment with Oncolytic Viruses and Dendritic Cell Injections.

Mathematical models of biological systems must strike a balance between being sufficiently complex to capture important biological features, while being simple enough that they remain tractable through analysis or simulation. In this work, we rigorously explore how to balance these competing interests when modeling murine melanoma treatment with oncolytic viruses and dendritic cell injections. Previously, we developed a system of six ordinary differential equations containing fourteen parameters that well describes experimental data on the efficacy of these treatments. Here, we explore whether this previously developed model is the minimal model needed to accurately describe the data. Using a variety of techniques, including sensitivity analyses and a parameter sloppiness analysis, we find that our model can be reduced by one variable and three parameters and still give excellent fits to the data. We also argue that our model is not too simple to capture the dynamics of the data, and that the original and minimal models make similar predictions about the efficacy and robustness of protocols not considered in experiments. Reducing the model to its minimal form allows us to increase the tractability of the system in the face of parametric uncertainty.

Limiting the development of anti-cancer drug resistance in a spatial model of micrometastases.

While chemoresistance in primary tumors is well-studied, much less is known about the influence of systemic chemotherapy on the development of drug resistance at metastatic sites. In this work, we use a hybrid spatial model of tumor response to a DNA damaging drug to study how the development of chemoresistance in micrometastases depends on the drug dosing schedule. We separately consider cell populations that harbor pre-existing resistance to the drug, and those that acquire resistance during the course of treatment. For each of these independent scenarios, we consider one hypothetical cell line that is responsive to metronomic chemotherapy, and another that with high probability cannot be eradicated by a metronomic protocol. Motivated by experimental work on ovarian cancer xenografts, we consider all possible combinations of a one week treatment protocol, repeated for three weeks, and constrained by the total weekly drug dose. Simulations reveal a small number of fractionated-dose protocols that are at least as effective as metronomic therapy in eradicating micrometastases with acquired resistance (weak or strong), while also being at least as effective on those that harbor weakly pre-existing resistant cells. Given the responsiveness of very different theoretical cell lines to these few fractionated-dose protocols, these may represent more effective ways to schedule chemotherapy with the goal of limiting metastatic tumor progression.

Treatment strategies for combining immunostimulatory oncolytic virus therapeutics with dendritic cell injections.

Oncolytic viruses (OVs) are used to treat cancer, as they selectively replicate inside of and lyse tumor cells. The efficacy of this process is limited and new OVs are being designed to mediate tumor cell release of cytokines and co-stimulatory molecules, which attract cytotoxic T cells to target tumor cells, thus increasing the tumor-killing effects of OVs. To further promote treatment efficacy, OVs can be combined with other treatments, such as was done by Huang et al., who showed that combining OV injections with dendritic cell (DC) injections was a more effective treatment than either treatment alone. To further investigate this combination, we built a mathematical model consisting of a system of ordinary differential equations and fit the model to the hierarchical data provided from Huang et al. We used the model to determine the effect of varying doses of OV and DC injections and to test alternative treatment strategies. We found that the DC dose given in Huang et al. was near a bifurcation point and that a slightly larger dose could cause complete eradication of the tumor. Further, the model results suggest that it is more effective to treat a tumor with immunostimulatory oncolytic viruses first and then follow-up with a sequence of DCs than to alternate OV and DC injections. This protocol, which was not considered in the experiments of Huang et al., allows the infection to initially thrive before the immune response is enhanced. Taken together, our work shows how the ordering, temporal spacing, and dosage of OV and DC can be chosen to maximize efficacy and to potentially eliminate tumors altogether.

Computational modeling of tumor response to vascular-targeting therapies--part I: validation.

Mathematical modeling techniques have been widely employed to understand how cancer grows, and, more recently, such approaches have been used to understand how cancer can be controlled. In this manuscript, a previously validated hybrid cellular automaton model of tumor growth in a vascularized environment is used to study the antitumor activity of several vascular-targeting compounds of known efficacy. In particular, this model is used to test the antitumor activity of a clinically used angiogenesis inhibitor (both in isolation, and with a cytotoxic chemotherapeutic) and a vascular disrupting agent currently undergoing clinical trial testing. I demonstrate that the mathematical model can make predictions in agreement with preclinical/clinical data and can also be used to gain more insight into these treatment protocols. The results presented herein suggest that vascular-targeting agents, as currently administered, cannot lead to cancer eradication, although a highly efficacious agent may lead to long-term cancer control.

Modeling the effects of vasculature evolution on early brain tumor growth.

Mathematical modeling of both tumor growth and angiogenesis have been active areas of research for the past several decades. Such models can be classified into one of two categories: those that analyze the remodeling of the vasculature while ignoring changes in the tumor mass, and those that predict tumor expansion in the presence of a non-evolving vasculature. However, it is well accepted that vasculature remodeling and tumor growth strongly depend on one another. For this reason, we have developed a two-dimensional hybrid cellular automaton model of early brain tumor growth that couples the remodeling of the microvasculature with the evolution of the tumor mass. A system of reaction-diffusion equations has been developed to track the concentration of vascular endothelial growth factor (VEGF), Ang-1, Ang-2, their receptors and their complexes in space and time. The properties of the vasculature and hence of each cell are determined by the relative concentrations of these key angiogenic factors. The model exhibits an angiogenic switch consistent with experimental observations on the upregulation of angiogenesis. Particularly, we show that if the pathways that produce and respond to VEGF and the angiopoietins are properly functioning, angiogenesis is initiated and a tumor can grow to a macroscopic size. However, if the VEGF pathway is inhibited, angiogenesis does not occur and tumor growth is thwarted beyond 1-2mm in size. Furthermore, we show that tumor expansion can occur in well-vascularized environments even when angiogenesis is inhibited, suggesting that anti-angiogenic therapies may not be sufficient to eliminate a population of actively dividing malignant cells.