Mathematical Model of BCG Immunotherapy in Superficial Bladder Cancer

Mathematical model of MMC chemotherapy for non-invasive bladder cancer treatment

Mitomycin-C (MMC) chemotherapy is a well-established anti-cancer treatment for non-muscle-invasive bladder cancer (NMIBC). However, despite comprehensive biological research, the complete mechanism of action and an ideal regimen of MMC have not been elucidated. In this study, we present a theoretical investigation of NMIBC growth and its treatment by continuous administration of MMC chemotherapy. Using temporal ordinary differential equations (ODEs) to describe cell populations and drug molecules, we formulated the first mathematical model of tumor-immune interactions in the treatment of MMC for NMIBC, based on biological sources. Several hypothetical scenarios for NMIBC under the assumption that tumor size correlates with cell count are presented, depicting the evolution of tumors classified as small, medium, and large. These scenarios align qualitatively with clinical observations of lower recurrence rates for tumor size ≤ 30[mm] with MMC treatment, demonstrating that cure appears up to a theoretical x[mm] tumor size threshold, given specific parameters within a feasible biological range. The unique use of mole units allows to introduce a new method for theoretical pre-treatment assessments by determining MMC drug doses required for a cure. In this way, our approach provides initial steps toward personalized MMC chemotherapy for NMIBC patients, offering the possibility of new insights and potentially holding the key to unlocking some of its mysteries.

Revolutionizing cancer treatment: comprehensive insights into immunotherapeutic strategies

Cancer, characterized by the uncontrolled proliferation of aberrant cells, underscores the imperative for innovative therapeutic approaches. Immunotherapy has emerged as a pivotal constituent in cancer treatment, offering improved prognostic outcomes for a substantial patient cohort. Noteworthy for its precision, immunotherapy encompasses strategies such as adoptive cell therapy and checkpoint inhibitors, orchestrating the immune system to recognize and selectively target malignant cells. Exploiting the specificity of the immune response renders immunotherapy efficacious, as it selectively targets the body's immune milieu. Diverse mechanisms underlie cancer immunotherapies, leading to distinct toxicity profiles compared to conventional treatments. A remarkable clinical stride in the anticancer resources is immunotherapy. Remarkably, certain recalcitrant cancers like skin malignancies exhibit resistance to radiation or chemotherapy, yet respond favorably to immunotherapeutic interventions. Notably, combination therapies involving chemotherapy and immunotherapy have exhibited synergistic effects, enhancing overall therapeutic efficacy. Understanding the pivotal role of immunotherapy elucidates its complementary value, bolstering the therapeutic landscape. In this review, we elucidate the taxonomy of cancer immunotherapy, encompassing adoptive cell therapy and checkpoint inhibitors, while scrutinizing their distinct adverse event profiles. Furthermore, we expound on the unprecedented potential of immunogenic vaccines to bolster the anticancer immune response. This comprehensive analysis underscores the significance of immunotherapy in modern oncology, unveiling novel prospects for tailored therapeutic regimens.

Reparameterized multiobjective control of BCG immunotherapy

Bladder cancer is a cancerous disease that mainly affects elder men and women. The immunotherapy that uses Bacillus of Calmette and Guerin (BCG) effectively treats bladder cancer by stimulating the immune response of patients. The therapeutic performance of BCG relies on drug dosing, and the design of an optimal BCG regimen is an open question. In this study, we propose the reparameterized multiobjective control (RMC) approach for seeking an optimal drug dosing regimen and apply it to the design of BCG treatment. This approach utilizes constrained optimization based on a nonlinear bladder cancer model with impulsive drug instillation. We compare the performance of RMC with Koopman model predictive control (MPC) and validate the efficacy of optimal BCG dosing regimens through numerical simulations, demonstrating the efficient elimination of cancerous cells. The proposed control framework holds the potential for generalization to other model-based treatment designs.

Mathematical modeling of BCG-based bladder cancer treatment using socio-demographics

Cancer is one of the most widespread diseases around the world with millions of new patients each year. Bladder cancer is one of the most prevalent types of cancer affecting all individuals alike with no obvious "prototypical patient". The current standard treatment for BC follows a routine weekly Bacillus Calmette-Guérin (BCG) immunotherapy-based therapy protocol which is applied to all patients alike. The clinical outcomes associated with BCG treatment vary significantly among patients due to the biological and clinical complexity of the interaction between the immune system, treatments, and cancer cells. In this study, we take advantage of the patient's socio-demographics to offer a personalized mathematical model that describes the clinical dynamics associated with BCG-based treatment. To this end, we adopt a well-established BCG treatment model and integrate a machine learning component to temporally adjust and reconfigure key parameters within the model thus promoting its personalization. Using real clinical data, we show that our personalized model favorably compares with the original one in predicting the number of cancer cells at the end of the treatment, with [Formula: see text] improvement, on average.

Multiple colonies of cancer involved in mutual suppression with the immune system

We study the effects of the immune system on multiple cancer colonies. When cancer cells proliferate, cytotoxic T lymphocytes (CTLs) reactive to the cancer-specific antigens are activated, suppressing the growth of cancer colonies. The immune reaction activated by a large cancer colony may suppress and eliminate smaller colonies. However, cancer cells mitigate immune reactions by slowing down the activation of CTLs in dendritic cells with regulatory T cells and by inactivating CTLs attacking cancer cells with immune checkpoints. If cancer cells strongly suppress the immune reaction, the system may become bistable, where both the cancer-dominated and immunity-dominated states are locally stable. We study several models differing in the distance between colonies and the migration speeds of CTLs and regulatory T cells. We examine how the domains of attraction for multiple equilibria change with parameters. Nonlinear cancer-immunity dynamics may produce a sharp transition from a state with a small number of colonies and strong immunity to one with many colonies and weak immunity, resulting in the rapid emergence of many cancer colonies in the same organ or metastatic sites.

A Mathematical Study of the Role of tBregs in Breast Cancer

A model for the mathematical study of immune response to breast cancer is proposed and studied, both analytically and numerically. It is a simplification of a complex one, recently introduced by two of the present authors. It serves for a compact study of the dynamical role in cancer promotion of a relatively recently described subgroup of regulatory B cells, which are evoked by the tumour.

Cell-Level Spatio-Temporal Model for a Bacillus Calmette–Guérin-Based Immunotherapy Treatment Protocol of Superficial Bladder Cancer

Bladder cancer is one of the most widespread types of cancer. Multiple treatments for non-invasive, superficial bladder cancer have been proposed over the last several decades with a weekly Bacillus Calmette–Guérin immunotherapy-based therapy protocol, which is considered the gold standard today. Nonetheless, due to the complexity of the interactions between the immune system, healthy cells, and cancer cells in the bladder’s microenvironment, clinical outcomes vary significantly among patients. Mathematical models are shown to be effective in predicting the treatment outcome based on the patient’s clinical condition at the beginning of the treatment. Even so, these models still have large errors for long-term treatments and patients that they do not fit. In this work, we utilize modern mathematical tools and propose a novel cell-level spatio-temporal mathematical model that takes into consideration the cell–cell and cell–environment interactions occurring in a realistic bladder’s geometric configuration in order to reduce these errors. We implement the model using the agent-based simulation approach, showing the impacts of different cancer tumor sizes and locations at the beginning of the treatment on the clinical outcomes for today’s gold-standard treatment protocol. In addition, we propose a genetic-algorithm-based approach to finding a successful and time-optimal treatment protocol for a given patient’s initial condition. Our results show that the current standard treatment protocol can be modified to produce cancer-free equilibrium for deeper cancer cells in the urothelium if the cancer cells’ spatial distribution is known, resulting in a greater success rate.

Prediction of the Immune Phenotypes of Bladder Cancer Patients for Precision Oncology

Bladder cancer (BC) is the most common urinary malignancy; however accurate diagnosis and prediction of recurrence after therapies remain elusive. This study aimed to develop a biosignature of immunotherapy-based responses using gene expression data. Publicly available BC datasets were collected, and machine learning (ML) approaches were applied to identify a novel biosignature to differentiate patient subgroups. Immune phenotyping of BC in the IMvigor210 dataset included three subtypes: inflamed, excluded, and desert immune. Immune phenotypes were analyzed with gene expressions using traditional but powerful classification methods such as random forests, Deep Neural Networks (DNN), Support Vector Machines (SVM) together with boosting and feature selection methods. Specifically, DNN yielded the highest area under the curve (AUC) with precision and recall (PR) curves and receiver operating characteristic (ROC) curves for each phenotype ([Formula: see text] and [Formula: see text], respectively) resulting in the identification of gene expression features useful for immune phenotype classification. Our results suggest significant potential to further develop and utilize machine learning algorithms for analysis of BC and its precaution. In conclusion, the findings from this study present a novel gene expression assay that can accurately discriminate BC patients from controls. Upon further validation in independent cohorts, this gene signature could be developed into a predictive test that can support clinical evaluation and patient care.

Early prediction of clinical response to checkpoint inhibitor therapy in human solid tumors through mathematical modeling

Background:

Checkpoint inhibitor therapy of cancer has led to markedly improved survival of a subset of patients in multiple solid malignant tumor types, yet the factors driving these clinical responses or lack thereof are not known. We have developed a mechanistic mathematical model for better understanding these factors and their relations in order to predict treatment outcome and optimize personal treatment strategies.

Methods:

Here, we present a translational mathematical model dependent on three key parameters for describing efficacy of checkpoint inhibitors in human cancer: tumor growth rate (α), tumor-immune infiltration (Λ), and immunotherapy-mediated amplification of anti-tumor response (µ). The model was calibrated by fitting it to a compiled clinical tumor response dataset (n = 189 patients) obtained from published anti-PD-1 and anti-PD-L1 clinical trials, and then validated on an additional validation cohort (n = 64 patients) obtained from our in-house clinical trials.

Results:

The derived parameters Λ and µ were both significantly different between responding versus nonresponding patients. Of note, our model appropriately classified response in 81.4% of patients by using only tumor volume measurements and within 2 months of treatment initiation in a retrospective analysis. The model reliably predicted clinical response to the PD-1/PD-L1 class of checkpoint inhibitors across multiple solid malignant tumor types. Comparison of model parameters to immunohistochemical measurement of PD-L1 and CD8+ T cells confirmed robust relationships between model parameters and their underlying biology.

Conclusions:

These results have demonstrated reliable methods to inform model parameters directly from biopsy samples, which are conveniently obtainable as early as the start of treatment. Together, these suggest that the model parameters may serve as early and robust biomarkers of the efficacy of checkpoint inhibitor therapy on an individualized per-patient basis.

Funding:

We gratefully acknowledge support from the Andrew Sabin Family Fellowship, Center for Radiation Oncology Research, Sheikh Ahmed Center for Pancreatic Cancer Research, GE Healthcare, Philips Healthcare, and institutional funds from the University of Texas M.D. Anderson Cancer Center. We have also received Cancer Center Support Grants from the National Cancer Institute (P30CA016672 to the University of Texas M.D. Anderson Cancer Center and P30CA072720 the Rutgers Cancer Institute of New Jersey). This research has also been supported in part by grants from the National Science Foundation Grant DMS-1930583 (ZW, VC), the National Institutes of Health (NIH) 1R01CA253865 (ZW, VC), 1U01CA196403 (ZW, VC), 1U01CA213759 (ZW, VC), 1R01CA226537 (ZW, RP, WA, VC), 1R01CA222007 (ZW, VC), U54CA210181 (ZW, VC), and the University of Texas System STARS Award (VC). BC acknowledges support through the SER Cymru II Programme, funded by the European Commission through the Horizon 2020 Marie Skłodowska-Curie Actions (MSCA) COFUND scheme and the Welsh European Funding Office (WEFO) under the European Regional Development Fund (ERDF). EK has also received support from the Project Purple, NIH (U54CA210181, U01CA200468, and U01CA196403), and the Pancreatic Cancer Action Network (16-65-SING). MF was supported through NIH/NCI center grant U54CA210181, R01CA222959, DoD Breast Cancer Research Breakthrough Level IV Award W81XWH-17-1-0389, and the Ernest Cockrell Jr. Presidential Distinguished Chair at Houston Methodist Research Institute. RP and WA received serial research awards from AngelWorks, the Gillson-Longenbaugh Foundation, and the Marcus Foundation. This work was also supported in part by grants from the National Cancer Institute to SHC (R01CA109322, R01CA127483, R01CA208703, and U54CA210181 CITO pilot grant) and to PYP (R01CA140243, R01CA188610, and U54CA210181 CITO pilot grant). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Supermodeling in predictive diagnostics of cancer under treatment

Classical data assimilation (DA) techniques, synchronizing a computer model with observations, are highly demanding computationally, particularly, for complex over-parametrized cancer models. Consequently, current models are not sufficiently flexible to interactively explore various therapy strategies, and to become a key tool of predictive oncology. We show that, by using supermodeling, it is possible to develop a prediction/correction scheme that could attain the required time regimes and be directly used to support decision-making in anticancer therapies. A supermodel is an interconnected ensemble of individual models (sub-models); in this case, the variously parametrized baseline tumor models. The sub-model connection weights are trained from data, thereby incorporating the advantages of the individual models. Simultaneously, by optimizing the strengths of the connections, the sub-models tend to partially synchronize with one another. As a result, during the evolution of the supermodel, the systematic errors of the individual models partially cancel each other. We find that supermodeling allows for a radical increase in the accuracy and efficiency of data assimilation. We demonstrate that it can be considered as a meta-procedure for any classical parameter fitting algorithm, thus it represents the next - latent - level of abstraction of data assimilation. We conclude that supermodeling is a very promising paradigm that can considerably increase the quality of prognosis in predictive oncology.

The Role of Mathematical Models in Immuno-Oncology: Challenges and Future Perspectives

Immuno-oncology (IO) focuses on the ability of the immune system to detect and eliminate cancer cells. Since the approval of the first immune checkpoint inhibitor, immunotherapies have become a major player in oncology treatment and, in 2021, represented the highest number of approved drugs in the field. In spite of this, there is still a fraction of patients that do not respond to these therapies and develop resistance mechanisms. In this sense, mathematical models offer an opportunity to identify predictive biomarkers, optimal dosing schedules and rational combinations to maximize clinical response. This work aims to outline the main therapeutic targets in IO and to provide a description of the different mathematical approaches (top-down, middle-out, and bottom-up) integrating the cancer immunity cycle with immunotherapeutic agents in clinical scenarios. Among the different strategies, middle-out models, which combine both theoretical and evidence-based description of tumor growth and immunological cell-type dynamics, represent an optimal framework to evaluate new IO strategies.

Mathematical model of tumor immunotherapy for bladder carcinoma identifies the limitations of the innate immune response

Treatment for non-muscle invasive carcinoma of the bladder represents one of the few examples of successful tumor immunity. Six weekly intravesical instillations of Bacillus Calmette-Guerin (BCG), often followed by maintenance schedule, result in up to 50-70% clinical response. Current models suggest that the mechanism of action involves the non-specific activation of innate effector cells, which may be capable of acting in the absence of an antigen-specific response. For example, recent evidence suggests that BCG-activated neutrophils possess anti-tumor potential. Moreover, weekly BCG treatment results in a prime-boost pattern with massive influx of innate immune cells (107-108 PMN/ml urine). Calibrating in vivo data, we estimate that the number of neutrophil degranulations per instillation is approximately 106-107, more than sufficient to potentially eliminate ~106 residual tumor cells. Furthermore, neutrophils, as well as other innate effector cells are not selective in their targeting-thus surrounding cells may be influenced by degranulation and / or cytokine production. To establish if these observed conditions could account for clinically effective tumor immunity, we built a mathematical model reflecting the early events and tissue conditioning in patients undergoing BCG therapy. The model incorporates key features of tumor growth, BCG instillations and the observed prime / boost pattern of the innate immune response. Model calibration established that each innate effector cell must kill 90-95 bystander cells for achieving the expected 50-70% clinical response. This prediction was evaluated both empirically and experimentally and found to vastly exceed the capacity of the innate immune system. We therefore conclude that the innate immune system alone is unable to eliminate the tumor cells. We infer that other aspects of the immune response (e.g., antigen-specific lymphocytes) decisively contribute to the success of BCG immunotherapy.

Computational modelling of modern cancer immunotherapy

Modern cancer immunotherapy has revolutionised oncology and carries the potential to radically change the approach to cancer treatment. However, numerous questions remain to be answered to understand immunotherapy response better and further improve the benefit for future cancer patients. Computational models are promising tools that can contribute to accelerated immunotherapy research by providing new clues and hypotheses that could be tested in future trials, based on preceding simulations in addition to the empirical rationale. In this topical review, we briefly summarise the history of cancer immunotherapy, including computational modelling of traditional cancer immunotherapy, and comprehensively review computational models of modern cancer immunotherapy, such as immune checkpoint inhibitors (as monotherapy and combination treatment), co-stimulatory agonistic antibodies, bispecific antibodies, and chimeric antigen receptor T cells. The modelling approaches are classified into one of the following categories: data-driven top-down vs mechanistic bottom-up, simplistic vs detailed, continuous vs discrete, and hybrid. Several common modelling approaches are summarised, such as pharmacokinetic/pharmacodynamic models, Lotka-Volterra models, evolutionary game theory models, quantitative systems pharmacology models, spatio-temporal models, agent-based models, and logic-based models. Pros and cons of each modelling approach are critically discussed, particularly with the focus on the potential for successful translation into immuno-oncology research and routine clinical practice. Specific attention is paid to calibration and validation of each model, which is a necessary prerequisite for any successful model, and at the same time, one of the main obstacles. Lastly, we provide guidelines and suggestions for the future development of the field.

Stability Analysis of Delayed Tumor-Antigen-ActivatedImmune Response in Combined BCG and IL-2Immunotherapy of Bladder Cancer

We use a system biology approach to translate the interaction of Bacillus Calmette-Gurin (BCG) + interleukin 2 (IL-2) for the treatment of bladder cancer into a mathematical model. The main goal of this research is to predict the outcome of BCG + IL-2 treatment combinations. We examined whether the delay effect caused by the proliferation of tumor antigen-specific effector cells after the immune system destroys BCG-infected urothelium cells after BCG and IL-2 immunotherapy influences success in bladder cancer treatment. To do this, we introduce a system of differential equations where the variables are the main participants in the immune response after BCG installations to fight cancer: the number of tumor cells, BCG cells, immune cells, and cytokines involved in the tumor-immune response. The relevant parameters describing the dynamics of the system are taken from a variety of biological, clinical literature and estimated using the mathematical models. We examine the local stability analysis of non-negative equilibrium states of the model. In theory, treatment could improve system stability, and we analyze the stability of all equilibria using the method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). Our results prove that the period for the proliferation of tumor antigen-specific effector cells does not influence to the success of the non-responsive patients after an intensified combined BCG + IL-2 treatment.

Quantitative Systems Pharmacology Approaches for Immuno-Oncology: Adding Virtual Patients to the Development Paradigm

Drug development in oncology commonly exploits the tools of molecular biology to gain therapeutic benefit through reprograming of cellular responses. In immuno‐oncology (IO) the aim is to direct the patient’s own immune system to fight cancer. After remarkable successes of antibodies targeting PD1/PD‐L1 and CTLA4 receptors in targeted patient populations, the focus of further development has shifted toward combination therapies. However, the current drug‐development approach of exploiting a vast number of possible combination targets and dosing regimens has proven to be challenging and is arguably inefficient. In particular, the unprecedented number of clinical trials testing different combinations may no longer be sustainable by the population of available patients. Further development in IO requires a step change in selection and validation of candidate therapies to decrease development attrition rate and limit the number of clinical trials. Quantitative systems pharmacology (QSP) proposes to tackle this challenge through mechanistic modeling and simulation. Compounds’ pharmacokinetics, target binding, and mechanisms of action as well as existing knowledge on the underlying tumor and immune system biology are described by quantitative, dynamic models aiming to predict clinical results for novel combinations. Here, we review the current QSP approaches, the legacy of mathematical models available to quantitative clinical pharmacologists describing interaction between tumor and immune system, and the recent development of IO QSP platform models. We argue that QSP and virtual patients can be integrated as a new tool in existing IO drug development approaches to increase the efficiency and effectiveness of the search for novel combination therapies.

Quantitative Mechanistic Modeling in Support of Pharmacological Therapeutics Development in Immuno-Oncology

Following the approval, in recent years, of the first immune checkpoint inhibitor, there has been an explosion in the development of immuno-modulating pharmacological modalities for the treatment of various cancers. From the discovery phase to late-stage clinical testing and regulatory approval, challenges in the development of immuno-oncology (IO) drugs are multi-fold and complex. In the preclinical setting, the multiplicity of potential drug targets around immune checkpoints, the growing list of immuno-modulatory molecular and cellular forces in the tumor microenvironment-with additional opportunities for IO drug targets, the emergence of exploratory biomarkers, and the unleashed potential of modality combinations all have necessitated the development of quantitative, mechanistically-oriented systems models which incorporate key biology and patho-physiology aspects of immuno-oncology and the pharmacokinetics of IO-modulating agents. In the clinical setting, the qualification of surrogate biomarkers predictive of IO treatment efficacy or outcome, and the corresponding optimization of IO trial design have become major challenges. This mini-review focuses on the evolution and state-of-the-art of quantitative systems models describing the tumor vs. immune system interplay, and their merging with quantitative pharmacology models of IO-modulating agents, as companion tools to support the addressing of these challenges.

A mathematical model of tumour growth with Beddington-DeAngelis functional response: a case of cancer without disease

A previously published mathematical model, governing tumour growth with mixed immunotherapy and chemotherapy treatments, is modified and studied. The search time, which is assumed to be neglectable in the previously published model, is incorporated into the functional response for tumour-cell lysis by effector cells. The model exhibits bistability where a tumour-cell population threshold exists. A tumour with an initial cell population below the threshold can be controlled by the immune system and remains microscopic and asymptomatic called cancer without disease while that above the threshold grows to lethal size. Bifurcation analysis shows that (a) the chemotherapy-induced damage may cause a microscopic tumour, which would never grow to become lethal if untreated, to grow to lethal size, (b) applying chemotherapy alone requires a large dosage to be successful,

Stability Analysis of Delayed Immune Response BCG Infection in Bladder Cancer Treatment Model by Stochastic Perturbations

We present a revised mathematical model of the immune response to Bacillus Calmette-Guérin (BCG) treatment of bladder cancer, optimized according to biological and clinical data accumulated during the last decade. The improved model accounts for cytotoxic T lymphocyte differentiation as an integral element of the delayed immune response, as well as the logistic growth terms for cancer cell proliferation. Three equilibria are demonstrated for the proposed model, which is assumed to be influenced by white noise stochastic perturbations that are directly proportional to the system state deviation from an equilibrium. Stability conditions for all equilibria are analyzed using the Kolmanovskii-Shaikhet general method of Lyapunov functionals construction.

Method of directly defining the inverse mapping applied to prostate cancer immunotherapy — Mathematical model

In this paper, we apply the method of directly defining the inverse mapping introduced by Liao and Zhao [On the method of directly defining inverse mapping for nonlinear differential equations, Numer. Algorithms 72(4) (2016) 989–1020] to the problem of prostate cancer immunotherapy. We extend this method in two directions: first, we apply the method to a system of nonlinear ordinary differential equation, and second, we propose a new technique for finding the base functions in the considered algorithm.

Addressing current challenges in cancer immunotherapy with mathematical and computational modelling

The goal of cancer immunotherapy is to boost a patient's immune response to a tumour. Yet, the design of an effective immunotherapy is complicated by various factors, including a potentially immunosuppressive tumour microenvironment, immune-modulating effects of conventional treatments and therapy-related toxicities. These complexities can be incorporated into mathematical and computational models of cancer immunotherapy that can then be used to aid in rational therapy design. In this review, we survey modelling approaches under the umbrella of the major challenges facing immunotherapy development, which encompass tumour classification, optimal treatment scheduling and combination therapy design. Although overlapping, each challenge has presented unique opportunities for modellers to make contributions using analytical and numerical analysis of model outcomes, as well as optimization algorithms. We discuss several examples of models that have grown in complexity as more biological information has become available, showcasing how model development is a dynamic process interlinked with the rapid advances in tumour-immune biology. We conclude the review with recommendations for modellers both with respect to methodology and biological direction that might help keep modellers at the forefront of cancer immunotherapy development.

Dynamics of Immune Checkpoints, Immune System, and BCG in the Treatment of Superficial Bladder Cancer

This paper aims to study the dynamics of immune suppressors/checkpoints, immune system, and BCG in the treatment of superficial bladder cancer. Programmed cell death protein-1 (PD-1), cytotoxic T-lymphocyte-associated antigen 4 (CTLA4), and transforming growth factor-beta (TGF-β) are some of the examples of immune suppressors/checkpoints. They are responsible for deactivating the immune system and enhancing immunological tolerance. Moreover, they categorically downregulate and suppress the immune system by preventing and blocking the activation of T-cells, which in turn decreases autoimmunity and enhances self-tolerance. In cancer immunotherapy, the immune checkpoints/suppressors prevent and block the immune cells from attacking, spreading, and killing the cancer cells, which leads to cancer growth and development. We formulate a mathematical model that studies three possible dynamics of the treatment and establish the effects of the immune checkpoints on the immune system and the treatment at large. Although the effect cannot be seen explicitly in the analysis of the model, we show it by numerical simulations.

Dynamical properties and tumor clearance conditions for a nine-dimensional model of bladder cancer immunotherapy

Understanding the global interaction dynamics between tumor and the immune system plays a key role in the advancement of cancer therapy. Bunimovich-Mendrazitsky et al. (2015) developed a mathematical model for the study of the immune system response to combined therapy for bladder cancer with Bacillus Calmette-Guérin (BCG) and interleukin-2 (IL-2) . We utilized a mathematical approach for bladder cancer treatment model for derivation of ultimate upper and lower bounds and proving dissipativity property in the sense of Levinson. Furthermore, tumor clearance conditions for BCG treatment of bladder cancer are presented. Our method is based on localization of compact invariant sets and may be exploited for a prediction of the cells populations dynamics involved into the model.

Treatment of non-muscle invasive bladder cancer with Bacillus Calmette-Guerin (BCG): Biological markers and simulation studies

Intravesical Bacillus Calmette-Guerin (BCG) vaccine is the preferred first line treatment for non-muscle invasive bladder carcinoma (NMIBC) in order to prevent recurrence and progression of cancer. There is ongoing need for the rational selection of i) BCG dose, ii) frequency of BCG administration along with iii) synergistic adjuvant therapy and iv) a reliable set of biochemical markers relevant to tumor response. In this review we evaluate cellular and molecular markers pertinent to the immunological response triggered by the BCG instillation and respective mathematical models of the treatment. Specific examples of markers include diverse immune cells, genetic polymorphisms, miRNAs, epigenetics, immunohistochemistry and molecular biology 'beacons' as exemplified by cell surface proteins, cytokines, signaling proteins and enzymes. We identified tumor associated macrophages (TAMs), human leukocyte antigen (HLA) class I, a combination of Ki-67/CK20, IL-2, IL-8 and IL-6/IL-10 ratio as the most promising markers for both pre-BCG and post-BCG treatment suitable for the simulation studies. The intricate and patient-specific nature of these data warrants the use of powerful multi-parametral mathematical methods in combination with molecular/cellular biology insight and clinical input.

Improving Bacillus Calmette-Guérin (BCG) immunotherapy for bladder cancer by adding interleukin 2 (IL-2): a mathematical model

One of the treatments offered to non-invasive bladder cancer patients is BCG instillations, using a well-established, time-honoured protocol. Some of the patients, however, do not respond to this protocol. To examine possible changes in the protocol, we provide a platform for in silico testing of alternative protocols for BCG instillations and combinations with IL-2, to be used by urologists in planning new treatment strategies for subpopulations of bladder cancer patients who may benefit from a personalized protocol. We use a systems biology approach to describe the BCG-tumour-immune interplay and translate it into a set of mathematical differential equations. The variables of the equation set are the number of tumour cells, bacteria cells, immune cells, and cytokines participating in the tumour-immune response. Relevant parameters that describe the system's dynamics are taken from a variety of independent literature, unrelated to the clinical trial results assessed by the model predictions. Model simulations use a clinically relevant range of initial tumour sizes (tumour volume) and tumour growth rates (tumour grade), representative of a virtual population of fifty patients. Our model successfully retrieved previous clinical results for BCG induction treatment and BCG maintenance therapy with a complete response (CR) rate of 82%. Furthermore, we designed alternative maintenance protocols, using IL-2 combinations with BCG, which improved success rates up to 86% and 100% of the patients, albeit without considering possible side effects. We have shown our simulation platform to be reliable by demonstrating its ability to retrieve published clinical trial results. We used this platform to predict the outcome of treatment combinations. Our results suggest that the subpopulation of non-responsive patients may benefit from an intensified combined BCG IL-2 maintenance treatment.

Cytokines as effectors and predictors of responses in the treatment of bladder cancer by bacillus Calmette-Guérin

The most effective intravesical treatment of non-muscle-invasive bladder cancer is instillation of live Mycobacterium bovis bacillus Calmette-Guérin (BCG). BCG stimulates the release of cytokines, contributing directly or indirectly to its effectiveness. However, the function of specific cytokines is not well understood. We have undertaken a nonsystematic review of primary evidence regarding cytokine detection, activation and response in BCG patients. Cytokines IL-2, IL-8 and TNF-α appear to be essential for effective BCG therapy and nonrecurrence, while IL-10 may have an inhibitory effect on BCG responses. IL-2, IL-8, TRAIL and TNF-α are potentially predictive of response to BCG. Alterations in genes encoding cytokines may also affect responses. There are significant data showing the association of certain cytokines with successful BCG treatment, and which may be useful predictive markers. Isolating those cytokines mediating efficacy may hold the key to ameliorating BCG's side effects and improving efficacy and patient compliance.

BCG-mediated bladder cancer immunotherapy: identifying determinants of treatment response using a calibrated mathematical model

Intravesical Bacillus Calmette Guérin (BCG) immunotherapy is considered the standard of care for treatment of non-muscle invasive bladder cancer; however the treatment parameters were established empirically. In order to evaluate potential optimization of clinical parameters of BCG induction therapy, we constructed and queried a new mathematical model. Specifically, we assessed the impact of (1) duration between resection and the first instillation; (2) BCG dose; (3) indwelling time; and (4) treatment interval of induction therapy – using cure rate as the primary endpoint. Based on available clinical and in vitro experimental data, we constructed and parameterized a stochastic mathematical model describing the interactions between BCG, the immune system, the bladder mucosa and tumor cells. The primary endpoint of the model was the probability of tumor extinction following BCG induction therapy in patients with high risk for tumor recurrence. We theoretically demonstrate that extending the duration between the resection and the first BCG instillation negatively influences treatment outcome. Simulations of higher BCG doses and longer indwelling times both improved the probability of tumor extinction. A remarkable finding was that an inter-instillation interval two times longer than the seven-day interval used in the current standard of care would substantially improve treatment outcome. We provide insight into relevant clinical questions using a novel mathematical model of BCG immunotherapy. Our model predicts an altered regimen that may decrease side effects of treatment while improving response to therapy.

Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of BCG treatment of bladder cancer

Understanding the dynamics of human hosts and tumors is of critical importance. A mathematical model was developed by Bunimovich-Mendrazitsky et al., who explored the immune response in bladder cancer as an effect of BCG treatment. This treatment exploits the host's own immune system to boost a response that will enable the host to rid itself of the tumor. Although this model was extensively studied using numerical simulation, no analytical results on global tumor dynamics were originally presented. In this work, we analyze stability in a mathematical model for BCG treatment of bladder cancer based on the use of quasi-normal form and stability theory. These tools are employed in the critical cases, especially when analysis of the linearized system is insufficient. Our goal is to gain a deeper insight into the BCG treatment of bladder cancer, which is based on a mathematical model and biological considerations, and thereby to bring us one step closer to the design of a relevant clinical protocol.

Theoretical modeling techniques and their impact on tumor immunology

Currently, cancer is one of the leading causes of death in industrial nations. While conventional cancer treatment usually results in the patient suffering from severe side effects, immunotherapy is a promising alternative. Nevertheless, some questions remain unanswered with regard to using immunotherapy to treat cancer hindering it from being widely established. To help rectify this deficit in knowledge, experimental data, accumulated from a huge number of different studies, can be integrated into theoretical models of the tumor-immune system interaction. Many complex mechanisms in immunology and oncology cannot be measured in experiments, but can be analyzed by mathematical simulations. Using theoretical modeling techniques, general principles of tumor-immune system interactions can be explored and clinical treatment schedules optimized to lower both tumor burden and side effects. In this paper, we aim to explain the main mathematical and computational modeling techniques used in tumor immunology to experimental researchers and clinicians. In addition, we review relevant published work and provide an overview of its impact to the field.

Interactions between the immune system and cancer: a brief review of non-spatial mathematical models

We briefly review spatially homogeneous mechanistic mathematical models describing the interactions between a malignant tumor and the immune system. We begin with the simplest (single equation) models for tumor growth and proceed to consider greater immunological detail (and correspondingly more equations) in steps. This approach allows us to clarify the necessity for expanding the complexity of models in order to capture the biological mechanisms we wish to understand. We conclude by discussing some unsolved problems in the mathematical modeling of cancer-immune system interactions.