A Mathematical Model of TCR-T Cell Therapy for Cervical Cancer.

Engineered T cell receptor (TCR)-expressing T (TCR-T) cells are intended to drive strong anti-tumor responses upon recognition of the specific cancer antigen, resulting in rapid expansion in the number of TCR-T cells and enhanced cytotoxic functions, causing cancer cell death. However, although TCR-T cell therapy against cancers has shown promising results, it remains difficult to predict which patients will benefit from such therapy. We develop a mathematical model to identify mechanisms associated with an insufficient response in a mouse cancer model. We consider a dynamical system that follows the population of cancer cells, effector TCR-T cells, regulatory T cells (Tregs), and "non-cancer-killing" TCR-T cells. We demonstrate that the majority of TCR-T cells within the tumor are "non-cancer-killing" TCR-T cells, such as exhausted cells, which contribute little or no direct cytotoxicity in the tumor microenvironment (TME). We also establish two important factors influencing tumor regression: the reversal of the immunosuppressive TME following depletion of Tregs, and the increased number of effector TCR-T cells with antitumor activity. Using mathematical modeling, we show that certain parameters, such as increasing the cytotoxicity of effector TCR-T cells and modifying the number of TCR-T cells, play important roles in determining outcomes.

Designing experimental conditions to use the Lotka-Volterra model to infer tumor cell line interaction types.

The Lotka-Volterra model is widely used to model interactions between two species. Here, we generate synthetic data mimicking competitive, mutualistic and antagonistic interactions between two tumor cell lines, and then use the Lotka-Volterra model to infer the interaction type. Structural identifiability of the Lotka-Volterra model is confirmed, and practical identifiability is assessed for three experimental designs: (a) use of a single data set, with a mixture of both cell lines observed over time, (b) a sequential design where growth rates and carrying capacities are estimated using data from experiments in which each cell line is grown in isolation, and then interaction parameters are estimated from an experiment involving a mixture of both cell lines, and (c) a parallel experimental design where all model parameters are fitted to data from two mixtures (containing both cell lines but with different initial ratios) simultaneously. Each design is tested on data generated from the Lotka-Volterra model with noise added, to determine efficacy in an ideal sense. In addition to assessing each design for practical identifiability, we investigate how the predictive power of the model - i.e., its ability to fit data for initial ratios other than those to which it was calibrated - is affected by the choice of experimental design. The parallel calibration procedure is found to be optimal and is further tested on in silico data generated from a spatially-resolved cellular automaton model, which accounts for oxygen consumption and allows for variation in the intensity level of the interaction between the two cell lines. We use this study to highlight the care that must be taken when interpreting parameter estimates for the spatially-averaged Lotka-Volterra model when it is calibrated against data produced by the spatially-resolved cellular automaton model, since baseline competition for space and resources in the CA model may contribute to a discrepancy between the type of interaction used to generate the CA data and the type of interaction inferred by the LV model.

Study of dose-dependent combination immunotherapy using engineered T cells and IL-2 in cervical cancer.

Adoptive T cell based immunotherapy is gaining significant traction in cancer treatment. Despite its limited efficacy so far in treating solid tumors compared to hematologic cancers, recent advances in T cell engineering render this treatment increasingly more successful in solid tumors, demonstrating its broader therapeutic potential. In this paper we develop a mathematical model to study the efficacy of engineered T cell receptor (TCR) T cell therapy targeting the E7 antigen in cervical cancer cell lines. We consider a dynamical system that follows the population of cancer cells, TCR T cells, and IL-2 treatment concentration. We demonstrate that there exists a TCR T cell dosage window for a successful cancer elimination that can be expressed in terms of the initial tumor size. We obtain the TCR T cell dose for two cervical cancer cell lines: 4050 and CaSki. Finally, a combination therapy of TCR T cell and IL-2 treatment is studied. We show that certain treatment protocols can improve therapy responses in the 4050 cell line, but not in the CaSki cell line.

Modeling the Dynamics of Heterogeneity of Solid Tumors in Response to Chemotherapy.

In this paper, we extend the model of the dynamics of drug resistance in a solid tumor that was introduced by Lorz et al. (Bull Math Biol 77:1-22, 2015). Similarly to the original, radially symmetric model, the quantities we follow depend on a phenotype variable that corresponds to the level of drug resistance. The original model is modified in three ways: (i) We consider a more general growth term that takes into account the sensitivity of resistance level to high drug dosage. (ii) We add a diffusion term in space for the cancer cells and adjust all diffusion terms (for the nutrients and for the drugs) so that the permeability of the resource and drug is limited by the cell concentration. (iii) We add a mutation term with a mutation kernel that corresponds to mutations that occur regularly or rarely. We study the dynamics of the emerging resistance of the cancer cells under continuous infusion and on-off infusion of cytotoxic and cytostatic drugs. While the original Lorz model has an asymptotic profile in which the cancer cells are either fully resistant or fully sensitive, our model allows the emergence of partial resistance levels. We show that increased drug concentrations are correlated with delayed relapse. However, when the cancer relapses, more resistant traits are selected. We further show that an on-off drug infusion also selects for more resistant traits when compared with a continuous drug infusion of identical total drug concentrations. Under certain conditions, our model predicts the emergence of a heterogeneous tumor in which cancer cells of different resistance levels coexist in different areas in space.

An adaptive information-theoretic experimental design procedure for high-to-low fidelity calibration of prostate cancer models.

The use of mathematical models to make predictions about tumor growth and response to treatment has become increasingly prevalent in the clinical setting. The level of complexity within these models ranges broadly, and the calibration of more complex models requires detailed clinical data. This raises questions about the type and quantity of data that should be collected and when, in order to maximize the information gain about the model behavior while still minimizing the total amount of data used and the time until a model can be calibrated accurately. To address these questions, we propose a Bayesian information-theoretic procedure, using an adaptive score function to determine the optimal data collection times and measurement types. The novel score function introduced in this work eliminates the need for a penalization parameter used in a previous study, while yielding model predictions that are superior to those obtained using two potential pre-determined data collection protocols for two different prostate cancer model scenarios: one in which we fit a simple ODE system to synthetic data generated from a cellular automaton model using radiotherapy as the imposed treatment, and a second scenario in which a more complex ODE system is fit to clinical patient data for patients undergoing intermittent androgen suppression therapy. We also conduct a robust analysis of the calibration results, using both error and uncertainty metrics in combination to determine when additional data acquisition may be terminated.

Modeling interaction of Glioma cells and CAR T-cells considering multiple CAR T-cells bindings.

Chimeric antigen receptor (CAR) T-cell based immunotherapy has shown its potential in treating blood cancers, and its application to solid tumors is currently being extensively investigated. For glioma brain tumors, various CAR T-cell targets include IL13Rα2, EGFRvIII, HER2, EphA2, GD2, B7-H3, and chlorotoxin. In this work, we are interested in developing a mathematical model of IL13Rα2 targeting CAR T-cells for treating glioma. We focus on extending the work of Kuznetsov et al. (1994) by considering binding of multiple CAR T-cells to a single glioma cell, and the dynamics of these multi-cellular conjugates. Our model more accurately describes experimentally observed CAR T-cell killing assay data than the models which do not consider multi-cellular conjugates. Moreover, we derive conditions in the CAR T-cell expansion rate that determines treatment success or failure. Finally, we show that our model captures distinct CAR T-cell killing dynamics from low to high antigen receptor densities in patient-derived brain tumor cells.

Data driven model discovery and interpretation for CAR T-cell killing using sparse identification and latent variables.

In the development of cell-based cancer therapies, quantitative mathematical models of cellular interactions are instrumental in understanding treatment efficacy. Efforts to validate and interpret mathematical models of cancer cell growth and death hinge first on proposing a precise mathematical model, then analyzing experimental data in the context of the chosen model. In this work, we present the first application of the sparse identification of non-linear dynamics (SINDy) algorithm to a real biological system in order discover cell-cell interaction dynamics in in vitro experimental data, using chimeric antigen receptor (CAR) T-cells and patient-derived glioblastoma cells. By combining the techniques of latent variable analysis and SINDy, we infer key aspects of the interaction dynamics of CAR T-cell populations and cancer. Importantly, we show how the model terms can be interpreted biologically in relation to different CAR T-cell functional responses, single or double CAR T-cell-cancer cell binding models, and density-dependent growth dynamics in either of the CAR T-cell or cancer cell populations. We show how this data-driven model-discovery based approach provides unique insight into CAR T-cell dynamics when compared to an established model-first approach. These results demonstrate the potential for SINDy to improve the implementation and efficacy of CAR T-cell therapy in the clinic through an improved understanding of CAR T-cell dynamics.

Comparison of cell state models derived from single-cell RNA sequencing data: graph versus multi-dimensional space.

Single-cell sequencing technologies have revolutionized molecular and cellular biology and stimulated the development of computational tools to analyze the data generated from these technology platforms. However, despite the recent explosion of computational analysis tools, relatively few mathematical models have been developed to utilize these data. Here we compare and contrast two cell state geometries for building mathematical models of cell state-transitions with single-cell RNA-sequencing data with hematopoeisis as a model system; (i) by using partial differential equations on a graph representing intermediate cell states between known cell types, and (ii) by using the equations on a multi-dimensional continuous cell state-space. As an application of our approach, we demonstrate how the calibrated models may be used to mathematically perturb normal hematopoeisis to simulate, predict, and study the emergence of novel cell states during the pathogenesis of acute myeloid leukemia. We particularly focus on comparing the strength and weakness of the graph model and multi-dimensional model.

Mathematical modeling of therapeutic neural stem cell migration in mouse brain with and without brain tumors.

Neural stem cells (NSCs) offer a potential solution to treating brain tumors. This is because NSCs can circumvent the blood-brain barrier and migrate to areas of damage in the central nervous system, including tumors, stroke, and wound injuries. However, for successful clinical application of NSC treatment, a sufficient number of viable cells must reach the diseased or damaged area(s) in the brain, and evidence suggests that it may be affected by the paths the NSCs take through the brain, as well as the locations of tumors. To study the NSC migration in brain, we develop a mathematical model of therapeutic NSC migration towards brain tumor, that provides a low cost platform to investigate NSC treatment efficacy. Our model is an extension of the model developed in Rockne et al. (PLoS ONE 13, e0199967, 2018) that considers NSC migration in non-tumor bearing naive mouse brain. Here we modify the model in Rockne et al. in three ways: (i) we consider three-dimensional mouse brain geometry, (ii) we add chemotaxis to model the tumor-tropic nature of NSCs into tumor sites, and (iii) we model stochasticity of migration speed and chemosensitivity. The proposed model is used to study migration patterns of NSCs to sites of tumors for different injection strategies, in particular, intranasal and intracerebral delivery. We observe that intracerebral injection results in more NSCs arriving at the tumor site(s), but the relative fraction of NSCs depends on the location of injection relative to the target site(s). On the other hand, intranasal injection results in fewer NSCs at the tumor site, but yields a more even distribution of NSCs within and around the target tumor site(s).

Bayesian Information-Theoretic Calibration of Radiotherapy Sensitivity Parameters for Informing Effective Scanning Protocols in Cancer.

With new advancements in technology, it is now possible to collect data for a variety of different metrics describing tumor growth, including tumor volume, composition, and vascularity, among others. For any proposed model of tumor growth and treatment, we observe large variability among individual patients' parameter values, particularly those relating to treatment response; thus, exploiting the use of these various metrics for model calibration can be helpful to infer such patient-specific parameters both accurately and early, so that treatment protocols can be adjusted mid-course for maximum efficacy. However, taking measurements can be costly and invasive, limiting clinicians to a sparse collection schedule. As such, the determination of optimal times and metrics for which to collect data in order to best inform proper treatment protocols could be of great assistance to clinicians. In this investigation, we employ a Bayesian information-theoretic calibration protocol for experimental design in order to identify the optimal times at which to collect data for informing treatment parameters. Within this procedure, data collection times are chosen sequentially to maximize the reduction in parameter uncertainty with each added measurement, ensuring that a budget of n high-fidelity experimental measurements results in maximum information gain about the low-fidelity model parameter values. In addition to investigating the optimal temporal pattern for data collection, we also develop a framework for deciding which metrics should be utilized at each data collection point. We illustrate this framework with a variety of toy examples, each utilizing a radiotherapy treatment regimen. For each scenario, we analyze the dependence of the predictive power of the low-fidelity model upon the measurement budget.

Mathematical deconvolution of CAR T-cell proliferation and exhaustion from real-time killing assay data.

Chimeric antigen receptor (CAR) T-cell therapy has shown promise in the treatment of haematological cancers and is currently being investigated for solid tumours, including high-grade glioma brain tumours. There is a desperate need to quantitatively study the factors that contribute to the efficacy of CAR T-cell therapy in solid tumours. In this work, we use a mathematical model of predator-prey dynamics to explore the kinetics of CAR T-cell killing in glioma: the Chimeric Antigen Receptor T-cell treatment Response in GliOma (CARRGO) model. The model includes rates of cancer cell proliferation, CAR T-cell killing, proliferation, exhaustion, and persistence. We use patient-derived and engineered cancer cell lines with an in vitro real-time cell analyser to parametrize the CARRGO model. We observe that CAR T-cell dose correlates inversely with the killing rate and correlates directly with the net rate of proliferation and exhaustion. This suggests that at a lower dose of CAR T-cells, individual T-cells kill more cancer cells but become more exhausted when compared with higher doses. Furthermore, the exhaustion rate was observed to increase significantly with tumour growth rate and was dependent on level of antigen expression. The CARRGO model highlights nonlinear dynamics involved in CAR T-cell therapy and provides novel insights into the kinetics of CAR T-cell killing. The model suggests that CAR T-cell treatment may be tailored to individual tumour characteristics including tumour growth rate and antigen level to maximize therapeutic benefit.