Ordinary Differential Equation Models for Adoptive Immunotherapy

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.

Deep reinforcement learning-based control of chemo-drug dose in cancer treatment

BACKGROUND AND OBJECTIVE

Advancement in the treatment of cancer, as a leading cause of death worldwide, has promoted several research activities in various related fields. The development of effective treatment regimens with optimal drug dose administration using a mathematical modeling framework has received extensive research attention during the last decades. However, most of the control techniques presented for cancer chemotherapy are mainly model-based approaches. The available model-free techniques based on Reinforcement Learning (RL), commonly discretize the problem states and variables, which other than demanding expert supervision, cannot model the real-world conditions accurately. The more recent Deep Reinforcement Learning (DRL) methods, which enable modeling the problem in its original continuous space, are rarely applied in cancer chemotherapy.

METHODS

In this paper, we propose an effective and robust DRL-based, model-free method for the closed-loop control of cancer chemotherapy drug dosing. A nonlinear pharmacological cancer model is used for simulating the patient and capturing the cancer dynamics. In contrast to previous work, the state variables and control action are modeled in their original infinite spaces to avoid expert-guided discretization and provide a more realistic solution. The DRL network is trained to automatically adjust the drug dose based on the monitored states of the patient. The proposed method provides an adaptive control technique to respond to the special conditions and diagnosis measurements of different categories of patients.

RESULTS AND CONCLUSIONS

The performance of the proposed DRL-based controller is evaluated by numerical analysis of different diverse simulated patients. Comparison to the state-of-the-art RL-based method, which uses discretized state and action spaces, shows the superiority of the approach in the process and duration of cancer chemotherapy treatment. In the majority of the studied cases, the proposed model decreases the medication period and the total amount of administrated drug, while increasing the rate of reduction in tumor cells.

Merits and Demerits of ODE Modeling of Physicochemical Systems for Numerical Simulations

In comparison with the first-principles calculations mostly using partial differential equations (PDEs), numerical simulations with modeling by ordinary differential equations (ODEs) are sometimes superior in that they are computationally more economical and that important factors are more easily traced. However, a demerit of ODE modeling is the need of model validation through comparison with experimental data or results of the first-principles calculations. In the present review, examples of ODE modeling are reviewed such as sonochemical reactions inside a cavitation bubble, oriented attachment of nanocrystals, dynamic response of flexoelectric polarization, ultrasound-assisted sintering, and dynamics of a gas parcel in a thermoacoustic engine.

Dynamics of an Antitumour Model with Pulsed Radioimmunotherapy

In this paper, an antitumour model for characterising radiotherapy and immunotherapy processes at different fixed times is proposed. The global attractiveness of the positive periodic solution for each corresponding subsystem is proved with the integral inequality technique. Then, based on the differentiability of the solutions with respect to the initial values, the eigenvalues of the Jacobian matrix at a fixed point corresponding to the tumour-free periodic solution are determined, resulting in a sufficient condition for local stability. The solutions to the ordinary differential equations are compared, the threshold condition for the global attractiveness of the tumour-free periodic solution is provided in terms of an indicator R 0, and the permanence of a system with at least one tumour-present periodic solution is investigated. Furthermore, the effects of the death rate, effector cell injection dosage, therapeutic period, and effector cell activation rate on indicator R 0 are determined through numerical simulations, and the results indicate that radioimmunotherapy is more effective than either radiotherapy or immunotherapy alone.

Dose-dependent thresholds of dexamethasone destabilize CAR T-cell treatment efficacy

Chimeric antigen receptor (CAR) T-cell therapy is potentially an effective targeted immunotherapy for glioblastoma, yet there is presently little known about the efficacy of CAR T-cell treatment when combined with the widely used anti-inflammatory and immunosuppressant glucocorticoid, dexamethasone. Here we present a mathematical model-based analysis of three patient-derived glioblastoma cell lines treated in vitro with CAR T-cells and dexamethasone. Advanced in vitro experimental cell killing assay technologies allow for highly resolved temporal dynamics of tumor cells treated with CAR T-cells and dexamethasone, making this a valuable model system for studying the rich dynamics of nonlinear biological processes with translational applications. We model the system as a nonautonomous, two-species predator-prey interaction of tumor cells and CAR T-cells, with explicit time-dependence in the clearance rate of dexamethasone. Using time as a bifurcation parameter, we show that (1) dexamethasone destabilizes coexistence equilibria between CAR T-cells and tumor cells in a dose-dependent manner and (2) as dexamethasone is cleared from the system, a stable coexistence equilibrium returns in the form of a Hopf bifurcation. With the model fit to experimental data, we demonstrate that high concentrations of dexamethasone antagonizes CAR T-cell efficacy by exhausting, or reducing the activity of CAR T-cells, and by promoting tumor cell growth. Finally, we identify a critical threshold in the ratio of CAR T-cell death to CAR T-cell proliferation rates that predicts eventual treatment success or failure that may be used to guide the dose and timing of CAR T-cell therapy in the presence of dexamethasone in patients.

Bioengineering and gene-editing technologies have paved the way for advance immunotherapies that can target patient-specific tumor cells. One of these therapies, chimeric antigen receptor (CAR) T-cell therapy has recently shown promise in treating glioblastoma, an aggressive brain cancer often with poor patient prognosis. Dexamethasone is a commonly prescribed anti-inflammatory medication due to the health complications of tumor associated swelling in the brain. However, the immunosuppressant effects of dexamethasone on the immunotherapeutic CAR T-cells are not well understood. To address this issue, we use mathematical modeling to study in vitro dynamics of dexamethasone and CAR T-cells in three patient-derived glioblastoma cell lines. We find that in each cell line studied there is a threshold of tolerable dexamethasone concentration. Below this threshold, CAR T-cells are successful at eliminating the cancer cells, while above this threshold, dexamethasone critically inhibits CAR T-cell efficacy. Our modeling suggests that in the presence of high dexamethasone reduced CAR T-cell efficacy, or increased exhaustion, can occur and result in CAR T-cell treatment failure.

Chimeric Antigen Receptor T Cell Therapies: A Review of Cellular Kinetic-Pharmacodynamic Modeling Approaches

Chimeric antigen receptor T cell (CAR-T cell) therapies have shown significant efficacy in CD19+ leukemias and lymphomas. There remain many challenges and questions for improving next-generation CAR-T cell therapies, and mathematical modeling of CAR-T cells may play a role in supporting further development. In this review, we introduce a mathematical modeling taxonomy for a set of relatively simple cellular kinetic-pharmacodynamic models that describe the in vivo dynamics of CAR-T cell and their interactions with cancer cells. We then discuss potential extensions of this model to include target binding, tumor distribution, cytokine-release syndrome, immunophenotype differentiation, and genotypic heterogeneity.

Alternative strategies for optimizing treatment of chronic lymphocytic leukemia with complex clonal architecture

In silico simulation of pre-clinical and clinical data may accelerate pre-clinical and clinical trial advances, leading to benefits for therapeutic outcomes, toxicity and cost savings. Combining this with clonal architecture data may permit truly personalized therapy. Chronic lymphocytic leukemia (CLL) exhibits clonal diversity, evolution and selection, spontaneously and under treatment pressure. We apply a dynamic simulation model to published CLL clonal architecture data to explore alternative therapeutic strategies, focusing on BTK inhibition. By deriving parameters of clonal growth and death behavior we model continuous vs time-limited ibrutinib therapy, and find that, despite persistence of disease, time to clinical progression may not differ. This is a testable hypothesis. We model IgVH-mutated CLL vs unmutated CLL by varying proliferation and find, based on the limited available data about clonal dynamics after such therapy, that there are differences predicted in response to anti-CD20 efficacy. These models can suggest potential clinical trials, and also indicate what additional data are needed to improve predictions. Ongoing work will expand modeling to agents such as venetoclax and to T cell therapies.

Mathematical Modelling Based on In Vivo Imaging Suggests CD137-Stimulated Cytotoxic T Lymphocytes Exert Superior Tumour Control Due to an Enhanced Antimitotic Effect on Tumour Cells

Simple Summary

Cytotoxic T lymphocytes (CTLs) play an important role in controlling tumours, and an improved understanding of how they accomplish this will benefit immunotherapeutic cancer treatment strategies. Stimulation of CTLs by targeting their CD137 receptor is a strategy currently under investigation for enhancing responses against tumours, yet so far only limited quantitative knowledge regarding the effects of such stimulation upon CTLs has been obtained. Here, we develop mathematical models to describe dynamic in vivo two-photon imaging of tumour infiltrating CTLs, to characterise differences in their function either in the presence or absence of a CD137 agonist antibody. We showed that an increased antiproliferative effect and a more sustained presence of CTLs within the tumour were the most important effects associated with anti-CD137 treatment.

Abstract

Several immunotherapeutic strategies for the treatment of cancer are under development. Two prominent strategies are adoptive cell transfer (ACT) of CTLs and modulation of CTL function with immune checkpoint inhibitors or with costimulatory antibodies. Despite some success with these approaches, there remains a lack of detailed and quantitative descriptions of the events following CTL transfer and the impact of immunomodulation. Here, we have applied ordinary differential equation models to two photon imaging data derived from a B16F10 murine melanoma. Models were parameterised with data from two different treatment conditions: either ACT-only, or ACT with intratumoural costimulation using a CD137 targeted antibody. Model dynamics and best fitting parameters were compared, in order to assess the mode of action of the CTLs and examine how the CD137 antibody influenced their activities. We found that the cytolytic activity of the transferred CTLs was minimal without CD137 costimulation, and that the CD137 targeted antibody did not enhance the per-capita killing ability of the transferred CTLs. Instead, the results of our modelling study suggest that an antiproliferative effect of CTLs exerted upon the tumour likely accounted for the majority of the reduction in tumour growth after CTL transfer. Moreover, we found that CD137 most likely improved tumour control via enhancement of this antiproliferative effect, as well as prolonging the period in which CTLs were inside the tumour, leading to a sustained duration of their antitumour effects following CD137 stimulation.

Personalized Immunotherapy Treatment Strategies for a Dynamical System of Chronic Myelogenous Leukemia

Simple Summary

As computer performance continues to grow at more affordable costs, mathematical modelling and in silico experimentation begin to play a larger role in understanding cancer evolution. The aim of our work is to formulate a control strategy for the Adaptive Cellular Therapy (ACT) that can fully eradicates the Chronic Myelogenous Leukemia (CML) cells population in a mathematical model describing interactions between naive T cells, effector T cells and CML cancer cells in the circulatory blood system. Mathematical analysis and numerical simulations allow us to conclude that it is possible to design a personalized administration protocol for the ACT in the form of a pulse train with asymmetrical waves and a fixed amplitude to achieve complete CML cancer cells eradication. The amplitude of the impulse on which the treatment is applied is given by an arithmetical combination of the parameters of the system with at least a duty cycle of 45 min/day.

Abstract

This paper is devoted to exploring personalized applications of cellular immunotherapy as a control strategy for the treatment of chronic myelogenous leukemia described by a dynamical system of three first-order ordinary differential equations. The latter was achieved by applying both the Localization of Compact Invariant Sets and Lyapunov’s stability theory. Combination of these two approaches allows us to establish sufficient conditions on the immunotherapy treatment parameter to ensure the complete eradication of the leukemia cancer cells. These conditions are given in terms of the system parameters and by performing several in silico experimentations, we formulated a protocol for the therapy application that completely eradicates the leukemia cancer cells population for different initial tumour concentrations. The formulated protocol does not dangerously increase the effector T cells population. Further, complete eradication is considered when solutions go below a finite critical value below which cancer cells cannot longer persist; i.e., one cancer cell. Numerical simulations are consistent with our analytical results.

A Systematic Review of the Efforts and Hindrances of Modeling and Simulation of CAR T-cell Therapy

Chimeric antigen receptor (CAR) T-cell therapy is an immunotherapy that has recently become highly instrumental in the fight against life-threatening diseases. A variety of modeling and computational simulation efforts have addressed different aspects of CAR T-cell therapy, including T-cell activation, T- and malignant cell population dynamics, therapeutic cost-effectiveness strategies, and patient survival. In this article, we present a systematic review of those efforts, including mathematical, statistical, and stochastic models employing a wide range of algorithms, from differential equations to machine learning. To the best of our knowledge, this is the first review of all such models studying CAR T-cell therapy. In this review, we provide a detailed summary of the strengths, limitations, methodology, data used, and data gap in currently published models. This information may help in designing and building better models for enhanced prediction and assessment of the benefit-risk balance associated with novel CAR T-cell therapies, as well as with the data need for building such models.

Modeling CAR T-Cell Therapy with Patient Preconditioning

The Federal Drug Administration approved the first Chimeric Antigen Receptor T-cell (CAR T-cell) therapies for the treatment of several blood cancers in 2017, and efforts are underway to broaden CAR T technology to address other cancer types. Standard treatment protocols incorporate a preconditioning regimen of lymphodepleting chemotherapy prior to CAR T-cell infusion. However, the connection between preconditioning regimens and patient outcomes is still not fully understood. Optimizing patient preconditioning plans and reducing the CAR T-cell dose necessary for achieving remission could make therapy safer. In this paper, we test treatment regimens consisting of sequential administration of chemotherapy and CAR T-cell therapy on a system of differential equations that models the tumor-immune interaction. We use numerical simulations of treatment plans from within the scope of current medical practice to assess the effect of preconditioning plans on the success of CAR T-cell therapy. Model results affirm clinical observations that preconditioning can be crucial for most patients, not just to reduce side effects, but to even achieve remission at all. We demonstrate that preconditioning plans using the same CAR T-cell dose and the same total concentration of chemotherapy can lead to different patient outcomes due to different delivery schedules. Results from sensitivity analysis of the model parameters suggest that making small improvements in the effectiveness of CAR T-cells in attacking cancer cells will significantly reduce the minimum dose required for successful treatment. Our modeling framework represents a starting point for evaluating the efficacy of patient preconditioning in the context of CAR T-cell therapy.

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.

Integrative Approaches to Cancer Immunotherapy

Cancer immunotherapy aims to arm patients with cancer-fighting immunity. Many new cancer-specific immunotherapeutic drugs have gained approval in the past several years, demonstrating immunotherapy's efficacy and promise as an anticancer modality. Despite these successes, several outstanding questions remain for cancer immunotherapy, including how to make immunotherapy more efficacious in a broader range of cancer types and patients, and how to predict which patients will respond or not respond to therapy. We present a case for integrative systems approaches that will answer these questions. This involves applying mechanistic and statistical modeling, establishing consistent and widely adopted experimental tools to generate systems-level data, and creating sustained mechanisms of support. If implemented, these approaches will lead to major advances in cancer treatment.

Cancer-induced immunosuppression can enable effectiveness of immunotherapy through bistability generation: A mathematical and computational examination

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The presence of an immunological barrier in cancer- immune system interaction (CISI) is consistent with the bistability patterns in that system.

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In CISI models, bistability patterns are consistent with immunosuppressive effects dominating immunoproliferative effects.

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Bistability could be harnessed to devise effective combination immunotherapy approaches.

Cancer immunotherapies rely on how interactions between cancer and immune system cells are constituted. The more essential to the emergence of the dynamical behavior of cancer growth these interactions are, the more effectively they may be used as mechanisms for interventions. Mathematical modeling can help unearth such connections, and help explain how they shape the dynamics of cancer growth. Here, we explored whether there exist simple, consistent properties of cancer-immune system interaction (CISI) models that might be harnessed to devise effective immunotherapy approaches. We did this for a family of three related models of increasing complexity. To this end, we developed a base model of CISI, which captures some essential features of the more complex models built on it. We find that the base model and its derivates can plausibly reproduce biological behavior that is consistent with the notion of an immunological barrier. This behavior is also in accord with situations in which the suppressive effects exerted by cancer cells on immune cells dominate their proliferative effects. Under these circumstances, the model family may display a pattern of bistability, where two distinct, stable states (a cancer-free, and a full-grown cancer state) are possible. Increasing the effectiveness of immune-caused cancer cell killing may remove the basis for bistability, and abruptly tip the dynamics of the system into a cancer-free state. Additionally, in combination with the administration of immune effector cells, modifications in cancer cell killing may be harnessed for immunotherapy without the need for resolving the bistability. We use these ideas to test immunotherapeutic interventions in silico in a stochastic version of the base model. This bistability-reliant approach to cancer interventions might offer advantages over those that comprise gradual declines in cancer cell numbers.

Spatio-temporal aspects of the interplay of cancer and the immune system

The conventional mean-field kinetic models describing the interplay of cancer and the immune system are temporal and predict exponential growth or elimination of the population of tumour cells provided their number is small and their effect on the immune system is negligible. More complex kinetics are associated with non-linear features of the response of the immune system. The generic model presented in this communication takes into account that the rates of the birth and death of tumour cells inside a tumour spheroid can significantly depend on the radial coordinate due to diffusion limitations in the supply of nutrients and/or transport of the species (cells and proteins) belonging to the immune system. In this case, non-trivial kinetic regimes are shown to be possible even without appreciable perturbation of the immune system.

Bridging blood cancers and inflammation: The reduced Cancitis model

A novel mechanism-based model - the Cancitis model - describing the interaction of blood cancer and the inflammatory system is proposed, analyzed and validated. The immune response is divided into two components, one where the elimination rate of malignant stem cells is independent of the level of the blood cancer and one where the elimination rate depends on the level of the blood cancer. A dimensional analysis shows that the full 6-dimensional system of nonlinear ordinary differential equations may be reduced to a 2-dimensional system - the reduced Cancitis model - using Fenichel theory. The original 18 parameters appear in the reduced model in 8 groups of parameters. The reduced model is analyzed. Especially the steady states and their dependence on the exogenous inflammatory stimuli are analyzed. A semi-analytic investigation reveals the stability properties of the steady states. Finally, positivity of the system and the existence of an attracting trapping region in the positive octahedron guaranteeing global existence and uniqueness of solutions are proved. The possible topologies of the dynamical system are completely determined as having a Janus structure, where two qualitatively different topologies appear for different sets of parameters. To classify this Janus structure we propose a novel concept in blood cancer - a reproduction ratio R. It determines the topological structure depending on whether it is larger or smaller than a threshold value. Furthermore, it follows that inflammation, affected by the exogenous inflammatory stimulation, may determine the onset and development of blood cancers. The body may manage initial blood cancer as long as the self-renewal rate is not too high, but fails to manage it if an inflammation appears. Thus, inflammation may trigger and drive blood cancers. Finally, the mathematical analysis suggests novel treatment strategies and it is used to discuss the in silico effect of existing treatment, e.g. interferon-α or T-cell therapy, and the impact of malignant cells becoming resistant.