Mathematical Model Creation for Cancer Chemo-Immunotherapy

Analysis of a combination of cancer treatments in efforts to overcome drug resistance

Tumor heterogeneity, the variability among cancer cells within a tumor, is a major contributor to drug resistance and poses challenges to effective treatment. To address this, we developed a mathematical model that captures tumor-immune interactions and the combined effects of chemotherapy and immunotherapy, focusing on their potential synergy. The model includes two tumor cell populations: chemosensitive (responsive to chemotherapy) and chemoresistant (unresponsive). We analyzed the stability of equilibrium states under three treatment scenarios: no treatment, chemotherapy alone, and combined therapy. To identify key parameters influencing the effectiveness of the combined treatment in reducing the overall tumor population, we performed a sensitivity analysis using Latin Hypercube Sampling (LHS) to calculate Partial Rank Correlation Coefficients (PRCCs) and their associated p-values. The analysis revealed that the killing rate of chemosensitive tumor cells by chemotherapy, the killing rate of chemosensitive tumor cells by immune cells, and the external doses of tumor-infiltrating lymphocyte (TIL) therapy were critical parameters. We formulated a quadratic optimal control problem to minimize tumor burden and treatment administration. Numerical simulations were conducted under two scenarios: with and without optimal control. Simulations without optimal control revealed that chemotherapy alone failed to eradicate tumors, while high-dose immunotherapy was more effective. However, combining the two therapies resulted in greater tumor reduction than either therapy alone. Under optimal control, our findings suggest that the most effective strategy involves administering a full chemotherapy dose along with gradually decreasing doses of TILs and interleukin-2 (IL-2).

Analysis of time-fractional cancer-tumor immunotherapy model using modified He-Laplace algorithm

Cancer encompasses various diseases characterized by the uncontrolled growth of abnormal cells, which can invade healthy tissues and spread throughout the body, making it the second leading cause of death worldwide. This study presents a fractional cancer treatment model with immunotherapy to enhance understanding of cancer's mathematical framework and behavior. The model comprises fractional differential equations analyzed using the Caputo-fractional derivative, aiming to control cancer growth while considering cell population metrics. A framework integrating various homotopies and Laplace transforms is developed to explore cancer's complexities. Simultaneous solution profiles for effector immune cells and tumor cells illustrate their mutual influence. The model examines parameters such as the death rate of immune cells, natural tumor growth rate, rate of immune cells killing fractional tumor cells and numerous others graphically for clarity. The fractional parameter β is visually represented through 2D, 3D, and contour plots. This comprehensive analysis validates the proposed approach, suggesting its applicability to other complex cancer treatment models for better decision-making in cancer treatment.

Advancing cancer drug development with mechanistic mathematical modeling: bridging the gap between theory and practice

Quantitative predictive modeling of cancer growth, progression, and individual response to therapy is a rapidly growing field. Researchers from mathematical modeling, systems biology, pharmaceutical industry, and regulatory bodies, are collaboratively working on predictive models that could be applied for drug development and, ultimately, the clinical management of cancer patients. A plethora of modeling paradigms and approaches have emerged, making it challenging to compile a comprehensive review across all subdisciplines. It is therefore critical to gauge fundamental design aspects against requirements, and weigh opportunities and limitations of the different model types. In this review, we discuss three fundamental types of cancer models: space-structured models, ecological models, and immune system focused models. For each type, it is our goal to illustrate which mechanisms contribute to variability and heterogeneity in cancer growth and response, so that the appropriate architecture and complexity of a new model becomes clearer. We present the main features addressed by each of the three exemplary modeling types through a subjective collection of literature and illustrative exercises to facilitate inspiration and exchange, with a focus on providing a didactic rather than exhaustive overview. We close by imagining a future multi-scale model design to impact critical decisions in oncology drug development.

The MOEO algorithm for multi-objective optimization of the cancer immuno-chemotherapy

In cancer treatment, chemotherapy has the disadvantage of killing both healthy and cancerous cells. Hence, the mixed-treatment of cancer such as chemo-immunotherapy is recommended. However, deriving the optimal dosage of each drug is a challenging issue. Although metaheuristic algorithms have received more attention in solving engineering problems due to their simplicity and flexibility, they have not consistently produced the best results for every problem. Thus, the need to introduce novel algorithms or extend the previous ones is felt for important optimization problems. Hence, in this paper, the multi-objective Equilibrium Optimizer algorithm, as an extension of the single-objective Equilibrium Optimizer algorithm, is recommended for cancer treatment problems. Then, the performance of the proposed algorithm is considered in both chemotherapy and mixed chemo-immunotherapy of cancer, considering the constraints of the tumor-immune dynamic system and the health level of the patients. For this purpose, two different patients with real clinical data are considered. The Pareto front obtained from the multi-objective optimization algorithm shows the points that can be selected for treatment under different criteria. Using the proposed multi-objective algorithm has reduced the total chemo-drug dose to 138.92 and 5.84 in the first patient, and 16.9 and 0.4384 in the second patient, for chemotherapy and chemo-immunotherapy, respectively. Comparing the results with previous studies demonstrates MOEO's superior performance in both chemotherapy and chemo-immunotherapy. However, it is shown that the proposed algorithm is more suitable for mixed-treatment approaches.

Dynamic analysis of a drug resistance evolution model with nonlinear immune response

Recent studies have utilized evolutionary mechanisms to impede the emergence of drug-resistant populations. In this paper, we develop a mathematical model that integrates hormonal treatment, immunotherapy, and the interactions among three cell types: drug-sensitive cancer cells, drug-resistant cancer cells and immune effector cells. Dynamical analysis is performed, examining the existence and stability of equilibria, thereby confirming the model's interpretability. Model parameters are calibrated using available prostate cancer data and literature. Through bifurcation analysis for drug sensitivity under different immune effector cells recruitment responses, we find that resistant cancer cells grow rapidly under weak recruitment response, maintain at a low level under strong recruitment response, and both may occur under moderate recruitment response. To quantify the competitiveness of sensitive and resistant cells, we introduce the comprehensive measures R1 and R2, respectively, which determine the outcome of competition. Additionally, we introduce the quantitative indicators CIE1 and CIE2 as comprehensive measures of the immune effects on sensitive and resistant cancer cells, respectively. These two indicators determine whether the corresponding cancer cells can maintain at a low level. Our work shows that the immune system is an important factor affecting the evolution of drug resistance and provides insights into how to enhance immune response to control resistance.

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.

Machine Learning-assisted immunophenotyping of peripheral blood identifies innate immune cells as best predictor of response to induction chemo-immunotherapy in head and neck squamous cell carcinoma - knowledge obtained from the CheckRad-CD8 trial

PURPOSE

Individual prediction of treatment response is crucial for personalized treatment in multimodal approaches against head-and-neck squamous cell carcinoma (HNSCC). So far, no reliable predictive parameters for treatment schemes containing immunotherapy have been identified. This study aims to predict treatment response to induction chemo-immunotherapy based on the peripheral blood immune status in patients with locally advanced HNSCC.

METHODS

The peripheral blood immune phenotype was assessed in whole blood samples in patients treated in the phase II CheckRad-CD8 trial as part of the pre-planned translational research program. Blood samples were analyzed by multicolor flow cytometry before (T1) and after (T2) induction chemo-immunotherapy with cisplatin/docetaxel/durvalumab/tremelimumab. Machine Learning techniques were used to predict pathological complete response (pCR) after induction therapy.

RESULTS

The tested classifier methods (LDA, SVM, LR, RF, DT, and XGBoost) allowed a distinct prediction of pCR. Highest accuracy was achieved with a low number of features represented as principal components. Immune parameters obtained from the absolute difference (lT2-T1l) allowed the best prediction of pCR. In general, less than 30 parameters and at most 10 principal components were needed for highly accurate predictions. Across several datasets, cells of the innate immune system such as polymorphonuclear cells, monocytes, and plasmacytoid dendritic cells are most prominent.

CONCLUSIONS

Our analyses imply that alterations of the innate immune cell distribution in the peripheral blood following induction chemo-immuno-therapy is highly predictive for pCR in HNSCC.

The impact of radio-chemotherapy on tumour cells interaction with optimal control and sensitivity analysis

Oncologists and applied mathematicians are interested in understanding the dynamics of cancer-immune interactions, mainly due to the unpredictable nature of tumour cell proliferation. In this regard, mathematical modelling offers a promising approach to comprehend this potentially harmful aspect of cancer biology. This paper presents a novel dynamical model that incorporates the interactions between tumour cells, healthy tissue cells, and immune-stimulated cells when subjected to simultaneous chemotherapy and radiotherapy for treatment. We analysed the equilibria and investigated their local stability behaviour. We also study transcritical, saddle-node, and Hopf bifurcations analytically and numerically. We derive the stability and direction conditions for periodic solutions. We identify conditions that lead to chaotic dynamics and rigorously demonstrate the existence of chaos. Furthermore, we formulated an optimal control problem that describes the dynamics of tumour-immune interactions, considering treatments such as radiotherapy and chemotherapy as control parameters. Our goal is to utilize optimal control theory to reduce the cost of radiotherapy and chemotherapy, minimize the harmful effects of medications on the body, and mitigate the burden of cancer cells by maintaining a sufficient population of healthy cells. Cost-effectiveness analysis is employed to identify the most economical strategy for reducing the disease burden. Additionally, we conduct a Latin hypercube sampling-based uncertainty analysis to observe the impact of parameter uncertainties on tumour growth, followed by a sensitivity analysis. Numerical simulations are presented to elucidate how dynamic behaviour of model is influenced by changes in system parameters. The numerical results validate the analytical findings and illustrate that a multi-therapeutic treatment plan can effectively reduce tumour burden within a given time frame of therapeutic intervention.

Understanding the Interplay of CAR-NK Cells and Triple-Negative Breast Cancer: Insights from Computational Modeling

Chimeric antigen receptor (CAR)-engineered natural killer (NK) cells have recently emerged as a promising and safe alternative to CAR-T cells for targeting solid tumors. In the case of triple-negative breast cancer (TNBC), traditional cancer treatments and common immunotherapies have shown limited effectiveness. However, CAR-NK cells have been successfully employed to target epidermal growth factor receptor (EGFR) on TNBC cells, thereby enhancing the efficacy of immunotherapy. The effectiveness of CAR-NK-based immunotherapy is influenced by various factors, including the vaccination dose, vaccination pattern, and tumor immunosuppressive factors in the microenvironment. To gain insights into the dynamics and effects of CAR-NK-based immunotherapy, we propose a computational model based on experimental data and immunological theories. This model integrates an individual-based model that describes the interplay between the tumor and the immune system, along with an ordinary differential equation model that captures the variation of inflammatory cytokines. Computational results obtained from the proposed model shed light on the conditions necessary for initiating an effective anti-tumor response. Furthermore, global sensitivity analysis highlights the issue of low persistence of CAR-NK cells in vivo, which poses a significant challenge for the successful clinical application of these cells. Leveraging the model, we identify the optimal vaccination time, vaccination dose, and time interval between injections for maximizing therapeutic outcomes.

A computational model for the cancer field effect

Introduction

The Cancer Field Effect describes an area of pre-cancerous cells that results from continued exposure to carcinogens. Cells in the cancer field can easily develop into cancer. Removal of the main tumor mass might leave the cancer field behind, increasing risk of recurrence.

Methods

The model we propose for the cancer field effect is a hybrid cellular automaton (CA), which includes a multi-layer perceptron (MLP) to compute the effects of the carcinogens on the gene expression of the genes related to cancer development. We use carcinogen interactions that are typically associated with smoking and alcohol consumption and their effect on cancer fields of the tongue.

Results

Using simulations we support the understanding that tobacco smoking is a potent carcinogen, which can be reinforced by alcohol consumption. The effect of alcohol alone is significantly less than the effect of tobacco. We further observe that pairing tumor excision with field removal delays recurrence compared to tumor excision alone. We track cell lineages and find that, in most cases, a polyclonal field develops, where the number of distinct cell lineages decreases over time as some lineages become dominant over others. Finally, we find tumor masses rarely form via monoclonal origin.

Studying the importance of regulatory T cells in chemoimmunotherapy mathematical modeling and proposing new approaches for developing a mathematical dynamic of cancer

Studying the mathematical dynamics of cancer has gained the attention of bioengineers in the past three decades. Different kinds of modelling considering various aspects of treatment have been proposed. In this paper, the key role of Regulatory T cells is discussed and a model in ordinary differential equation (ODE) form is proposed by adding this state to the system dynamics considering chemoimmunotherapy treatment. Regulatory T cells are considered as one of the main tumor cells' tactics to deceive the body's immune system. The improved model is verified mathematically and biologically and fits all criteria in both fields. The results show that entering Regulatory T cells state on cancer mathematical modelling for simulating body cells for chemoimmunotherapy provides a way to identify critical cases more carefully, which a simplified model is unable to accomplish. This point emphasizes the fact that this state must be present in cancer modelling to anticipate immune response more accurately. The advanced system fixed points are obtained by the Newton method and bifurcation diagrams are derived and discussed. New features and remarks are proposed during the journey of developing more accurate models that have the best fit with laboratory data. The sensitivity chart of the model is illustrated and novel aspects of discussions are made with the aim of personalizing a model for a patient and identifying critical conditions based on the chart before any treatment begins. This point enables physicians to determine whether critical conditions have occurred for a patient in a specific treatment or not.

Reinforcement learning strategies in cancer chemotherapy treatments: A review

BACKGROUND AND OBJECTIVE

Cancer is one of the major causes of death worldwide and chemotherapies are the most significant anti-cancer therapy, in spite of the emerging precision cancer medicines in the last 2 decades. The growing interest in developing the effective chemotherapy regimen with optimal drug dosing schedule to benefit the clinical cancer patients has spawned innovative solutions involving mathematical modeling since the chemotherapy regimens are administered cyclically until the futility or the occurrence of intolerable adverse events. Thus, in this present work, we reviewed the emerging trends involved in forming a computational solution from the aspect of reinforcement learning.

METHODS

Initially, this survey in-depth focused on the details of the dynamic treatment regimens from a broad perspective and then narrowed down to inspirations from reinforcement learning that were advantageous to chemotherapy dosing, including both offline reinforcement learning and supervised reinforcement learning.

RESULTS

The insights established in the chemotherapy-planning problem associated with the Reinforcement Learning (RL) has been discussed in this study. It showed that the researchers were able to widen their perspectives in comprehending the theoretical basis, dynamic treatment regimens (DTR), use of the adaptive control on DTR, and the associated RL techniques.

CONCLUSIONS

This study reviewed the recent researches relevant to the topic, and highlighted the challenges, open questions, possible solutions, and future steps in inventing a realistic solution for the aforementioned problem.

Optimal control design for drug delivery of immunotherapy in chemoimmunotherapy treatment

BACKGROUND AND OBJECTIVE

There are various approaches to control a mathematical dynamic of cancer, each of which is suitable for a special goal. Optimal control is considered as an applicable method to calculate the minimum necessary drug delivery in such systems.

METHODS

In this paper, a mathematical dynamic of cancer is proposed considering tumor cells, natural killer cells, CD8+T cells, circulating lymphocytes, IL-2 cytokine and Regulatory T cells as the system states, and chemotherapy, IL-2 and activated CD8+T cells injection rate as the control signals. After verifying the proposed mathematical model, the importance of the drug delivery timing and the effect of cancer cells initial condition are discussed. Afterwards, an optimal control is designed by defining a proper cost function with the goal of minimizing the number of tumor cells, and two immunotherapy drug amounts during treatment CONCLUSIONS: Results show that inappropriate injection of immunotherapy time schedule and the number of initial conditions of cancer cells might result in chemoimmunotherapy failure and auxiliary treatment must be prescribed to decrease tumor size before any treatment takes place. The obtained optimal control signals show that with lower amount of drug delivery and a suitable drug injection time schedule, tumor cells can be eliminated while a fixed immunotherapy time schedule protocol fails with larger amount of drug injection. This conclusion can be utilized with the aim of personalizing drug delivery and designing more accurate clinical trials based on the improved model simulations in order to save cost and time.

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.

Chemoimmunotherapy Administration Protocol Design for the Treatment of Leukemia through Mathematical Modeling and In Silico Experimentation

Cancer with all its more than 200 variants continues to be a major health problem around the world with nearly 10 million deaths recorded in 2020, and leukemia accounted for more than 300,000 cases according to the Global Cancer Observatory. Although new treatment strategies are currently being developed in several ongoing clinical trials, the high complexity of cancer evolution and its survival mechanisms remain as an open problem that needs to be addressed to further enhanced the application of therapies. In this work, we aim to explore cancer growth, particularly chronic lymphocytic leukemia, under the combined application of CAR-T cells and chlorambucil as a nonlinear dynamical system in the form of first-order Ordinary Differential Equations. Therefore, by means of nonlinear theories, sufficient conditions are established for the eradication of leukemia cells, as well as necessary conditions for the long-term persistence of both CAR-T and cancer cells. Persistence conditions are important in treatment protocol design as these provide a threshold below which the dose will not be enough to produce a cytotoxic effect in the tumour population. In silico experimentations allowed us to design therapy administration protocols to ensure the complete eradication of leukemia cells in the system under study when considering only the infusion of CAR-T cells and for the combined application of chemoimmunotherapy. All results are illustrated through numerical simulations. Further, equations to estimate cytotoxicity of chlorambucil and CAR-T cells to leukemia cancer cells were formulated and thoroughly discussed with a 95% confidence interval for the parameters involved in each formula.

Tumour-Natural Killer and CD8+ T Cells Interaction Model with Delay

The literature suggests that effective defence against tumour cells requires contributions from both Natural Killer (NK) cells and CD8+ T cells. NK cells are spontaneously active against infected target cells, whereas CD8+ T cells take some times to activate cell called as cell-specific targeting, to kill the virus. The interaction between NK cells and tumour cells has produced the other CD8+ T cell, called tumour-specific CD8+ T cells. We illustrate the tumour–immune interaction through mathematical modelling by considering the cell cycle. The interaction of the cells is described by a system of delay differential equations, and the delay, τ represent time taken for tumour cell reside interphase. The stability analysis and the bifurcation behaviour of the system are analysed. We established the stability of the model by analysing the characteristic equation to produce a stability region. The stability region is split into two regions, tumour decay and tumour growth. By applying the Routh–Hurwitz Criteria, the analysis of the trivial and interior equilibrium point of the model provides conditions for stability and is illustrated in the stability map. Numerical simulation is carried out to show oscillations through Hopf Bifurcation, and stability switching is found for the delay system. The result also showed that the interaction of NK cells with tumour cells could suppress tumour cells since it can increase the population of CD8+ T cells. This concluded that the inclusion of delay and immune responses (NK-CD8+ T cells) into consideration gives us a deep insight into the tumour growth and helps us understand how their interactions contribute to kill tumour cells.

Algorithmic asymptotic analysis: Extending the arsenal of cancer immunology modeling

The recent advances in cancer immunotherapy boosted the development of tumor-immune system models, with the aim to indicate more efficient treatments. Physical understanding is however difficult to be acquired, due to the complexity and the multi-scale dynamics of these models. In this work, the dynamics of a fundamental model formulating the interactions of tumor cells with natural killer cells, CD8⁺ T cells and circulating lymphocytes is examined. It is first shown that the long-term evolution of the system towards high-tumor or tumor-free equilibria is determined by the dynamics of an initial explosive stage of tumor progression. Focusing on this stage, the algorithmic Computational Singular Perturbation methodology is employed to identify the underlying mechanisms confining the system's evolution and the governing slow dynamics along them. These insights are preserved along different tumor-immune system and patient-dependent realizations. On top of these identifications, a novel reduced model is algorithmically constructed, which accurately predicts the dynamics of the system during the explosive stage and includes half of the parameters of the detailed model. The present analysis demonstrates the potential of algorithmic asymptotic analysis for acquiring physical understanding and for simplifying the complexity of cancer immunology models. Along with the current techniques on the field, this analysis can provide guidelines for more effective treatment development.

Mathematical modeling of tumor-immune system interactions: the effect of rituximab on breast cancer immune response

tBregs are a newly discovered subcategory of B regulatory cells, which are generated by breast cancer, resulting in the increase of Tregs and therefore in the death of NK cells. In this study, we use a mathematical and computational approach to investigate the complex interactions between the aforementioned cells as well as CD8⁺ T cells, CD4⁺ T cells and B cells. Furthermore, we use data fitting to prove that the functional response regarding the lysis of breast cancer cells by NK cells has a ratio-dependent form. Additionally, we include in our model the concentration of rituximab - a monoclonal antibody that has been suggested as a potential breast cancer therapy - and test its effect, when the standard, as well as experimental dosages, are administered.

Modeling codelivery of CD73 inhibitor and dendritic cell-based vaccines in cancer immunotherapy

Dendritic cells (DCs) are the dominant class of antigen-presenting cells in humans; therefore, a range of DC-based approaches have been established to promote an immune response against cancer cells. The efficacy of DC-based immunotherapeutic approaches is markedly affected by the immunosuppressive factors related to the tumor microenvironment, such as adenosine. In this paper, based on immunological theories and experimental data, a hybrid model is designed that offers some insights into the effects of DC-based immunotherapy combined with adenosine inhibition. The model combines an individual-based model for describing tumor-immune system interactions with a set of ordinary differential equations for adenosine modeling. Computational simulations of the proposed model clarify the conditions for the onset of a successful immune response against cancer cells. Global and local sensitivity analysis of the model highlights the importance of adenosine blockage for strengthening effector cells. The model is used to determine the most effective suppressive mechanism caused by adenosine, proper vaccination time, and the appropriate time interval between injections.

A NEW MATHEMATICAL MODELING AND SUB-OPTIMAL CHEMOTHERAPY OF CANCER

The development of more accurate cancer mathematical models leads to present more realistic treatment protocols, especially in model-based treatment protocol design. Hence, a cancer mathematical model is presented by considering tumor cells, immune cells, interleukins, macrophages polarization, and chemotherapy based on biological concepts. Both local and global sensitivity analyses are done to examine the effect of changing the parameters on the final tumor population. Then, the tumor-free equilibrium points of the system are derived, and their stabilities are studied. The main target of chemotherapy is to eliminate the tumor while limiting drug toxicity. The SDRE method is used to construct a sub-optimal control strategy by using the developed nonlinear cancer model. For simulation, three patients are considered: a young patient, an old patient, and a pregnant patient. These cases have different immune system strengths. Also, three initial tumor sizes are regarded for each case. So, different treatment strategies are suggested. Eradication of tumor cells in a finite duration with a desired amount of chemo-drug is shown in the simulation results. The results confirmed that the immune system’s ability plays an important role in treatment success. It is shown that there are different treatment protocols for different patients, and the SDRE method is more flexible and effective than the others.

Investigating Optimal Chemotherapy Options for Osteosarcoma Patients through a Mathematical Model

Simple Summary

Osteosarcoma is a rare type of cancer with poor prognoses. However, to the best of our knowledge, there are no mathematical models that study the impact of chemotherapy treatments on the osteosarcoma microenvironment. In this study, we developed a data driven mathematical model to analyze the dynamics of the important players in three groups of osteosarcoma tumors with distinct immune patterns in the presence of the most common chemotherapy drugs. The results indicate that the treatments’ start times and optimal dosages depend on the unique growth rate of the tumor, which implies the necessity of personalized medicine. Furthermore, the developed model can be extended by others to build models that can recommend individual-specific optimal dosages.

Abstract

Since all tumors are unique, they may respond differently to the same treatments. Therefore, it is necessary to study their characteristics individually to find their best treatment options. We built a mathematical model for the interactions between the most common chemotherapy drugs and the osteosarcoma microenvironments of three clusters of tumors with unique immune profiles. We then investigated the effects of chemotherapy with different treatment regimens and various treatment start times on the behaviors of immune and cancer cells in each cluster. Saliently, we suggest the optimal drug dosages for the tumors in each cluster. The results show that abundances of dendritic cells and HMGB1 increase when drugs are given and decrease when drugs are absent. Populations of helper T cells, cytotoxic cells, and IFN-γ grow, and populations of cancer cells and other immune cells shrink during treatment. According to the model, the MAP regimen does a good job at killing cancer, and is more effective than doxorubicin and cisplatin combined or methotrexate alone. The results also indicate that it is important to consider the tumor’s unique growth rate when deciding the treatment details, as fast growing tumors need early treatment start times and high dosages.

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.

Data-Driven Mathematical Model of Osteosarcoma

As the immune system has a significant role in tumor progression, in this paper, we develop a data-driven mathematical model to study the interactions between immune cells and the osteosarcoma microenvironment. Osteosarcoma tumors are divided into three clusters based on their relative abundance of immune cells as estimated from their gene expression profiles. We then analyze the tumor progression and effects of the immune system on cancer growth in each cluster. Cluster 3, which had approximately the same number of naive and M2 macrophages, had the slowest tumor growth, and cluster 2, with the highest population of naive macrophages, had the highest cancer population at the steady states. We also found that the fastest growth of cancer occurred when the anti-tumor immune cells and cytokines, including dendritic cells, helper T cells, cytotoxic cells, and IFN-γ, switched from increasing to decreasing, while the dynamics of regulatory T cells switched from decreasing to increasing. Importantly, the most impactful immune parameters on the number of cancer and total cells were the activation and decay rates of the macrophages and regulatory T cells for all clusters. This work presents the first osteosarcoma progression model, which can be later extended to investigate the effectiveness of various osteosarcoma treatments.

Building mean field ODE models using the generalized linear chain trick & Markov chain theory

The well known linear chain trick (LCT) allows modellers to derive mean field ODEs that assume gamma (Erlang) distributed passage times, by transitioning individuals sequentially through a chain of sub-states. The time spent in these sub-states is the sum of k exponentially distributed random variables, and is thus gamma distributed. The generalized linear chain trick (GLCT) extends this technique to the broader phase-type family of distributions, which includes exponential, Erlang, hypoexponential, and Coxian distributions. Phase-type distributions are the family of matrix exponential distributions on [0,∞) that represent the absorption time distributions for finite-state, continuous time Markov chains (CTMCs). Here we review CTMCs and phase-type distributions, then illustrate how to use the GLCT to efficiently build ODE models from underlying stochastic model assumptions. We introduce two novel model families by using the GLCT to generalize the Rosenzweig-MacArthur predator-prey model, and the SEIR model. We illustrate the kinds of complexity that can be captured by such models through multiple examples. We also show the benefits of using a GLCT-based model formulation to speed up the computation of numerical solutions to such models. These results highlight the intuitive nature, and utility, of using the GLCT to derive ODE models from first principles.

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.

α-Catenin levels determine direction of YAP/TAZ response to autophagy perturbation

The factors regulating cellular identity are critical for understanding the transition from health to disease and responses to therapies. Recent literature suggests that autophagy compromise may cause opposite effects in different contexts by either activating or inhibiting YAP/TAZ co-transcriptional regulators of the Hippo pathway via unrelated mechanisms. Here, we confirm that autophagy perturbation in different cell types can cause opposite responses in growth-promoting oncogenic YAP/TAZ transcriptional signalling. These apparently contradictory responses can be resolved by a feedback loop where autophagy negatively regulates the levels of α-catenins, LC3-interacting proteins that inhibit YAP/TAZ, which, in turn, positively regulate autophagy. High basal levels of α-catenins enable autophagy induction to positively regulate YAP/TAZ, while low α-catenins cause YAP/TAZ activation upon autophagy inhibition. These data reveal how feedback loops enable post-transcriptional determination of cell identity and how levels of a single intermediary protein can dictate the direction of response to external or internal perturbations.

Artificial immune system features added to breast cancer clinical data for machine learning (ML) applications

We here propose a new method of combining a mathematical model that describes a chemotherapy treatment for breast cancer with a machine-learning (ML) algorithm to increase performance in predicting tumor size using a five-step procedure. The first step involves modeling the chemotherapy treatment protocol using an analytical function. In the second step, the ML algorithm is trained to predict the tumor size based on clinico-pathological data and data obtained from magnetic resonance imaging results at different time points of treatment. In the third step, the model is solved according to adjustments made at the individual patient level based on the initial tumor size. In the fourth step, the important variables are extracted from the mathematical model solutions and inserted as added features. In the final step, we applied various ML algorithms on the merged data. Performance comparison among algorithms showed that the root mean square error of the linear regression decreased with the addition of the mathematical results, and the accuracy of prediction as well as the F1-scores increased with the addition of the mathematical model to the neural network. We established these results for four different cohorts of women at different ages with breast cancer who received chemotherapy treatment.

Translational approaches to treating dynamical diseases through in silico clinical trials

The primary goal of drug developers is to establish efficient and effective therapeutic protocols. Multifactorial pathologies, including dynamical diseases and complex disorders, can be difficult to treat, given the high degree of inter- and intra-patient variability and nonlinear physiological relationships. Quantitative approaches combining mechanistic disease modeling and computational strategies are increasingly leveraged to rationalize pre-clinical and clinical studies and to establish effective treatment strategies. The development of clinical trials has led to new computational methods that allow for large clinical data sets to be combined with pharmacokinetic and pharmacodynamic models of diseases. Here, we discuss recent progress using in silico clinical trials to explore treatments for a variety of complex diseases, ultimately demonstrating the immense utility of quantitative methods in drug development and medicine.

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 effect of immunotherapies on human castration-resistant prostate cancer

In this paper we analyze the potential effect of immunotherapies on castration-resistant form of human Prostate Cancer (PCa). In particular, we examine the potential effect of the dendritic vaccine sipuleucel-T, the only currently available immunotherapy option for advanced PCa, and of ipilimumab, a drug targeting the Cytotoxic T-Lymphocyte Antigen 4 (CTLA4), exposed on the CTLs membrane, currently under Phase II clinical trial. The model, building on the one by Rutter and Kuang, includes different types of immune cells and interactions and is parameterized on available data. Our results show that the vaccine has only a very limited effect on PCa, while repeated treatments with ipilimumab appear potentially capable of controlling and even eradicating an androgen-independent prostate cancer. From a mathematical analysis of a simplified model, it seems likely that, under continuous administration of ipilimumab, the system lies in a bistable situation where both the no-tumor equilibrium and the high-tumor equilibrium are attractive. The schedule of periodic treatments could then determine the outcome, and mathematical models could help determine an optimal schedule.

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.

A QSP model of prostate cancer immunotherapy to identify effective combination therapies

Immunotherapy, by enhancing the endogenous anti-tumor immune responses, is showing promising results for the treatment of numerous cancers refractory to conventional therapies. However, its effectiveness for advanced castration-resistant prostate cancer remains unsatisfactory and new therapeutic strategies need to be developed. To this end, systems pharmacology modeling provides a quantitative framework to test in silico the efficacy of new treatments and combination therapies. In this paper we present a new Quantitative Systems Pharmacology (QSP) model of prostate cancer immunotherapy, calibrated using data from pre-clinical experiments in prostate cancer mouse models. We developed the model by using Ordinary Differential Equations (ODEs) describing the tumor, key components of the immune system, and seven treatments. Numerous combination therapies were evaluated considering both the degree of tumor inhibition and the predicted synergistic effects, integrated into a decision tree. Our simulations predicted cancer vaccine combined with immune checkpoint blockade as the most effective dual-drug combination immunotherapy for subjects treated with androgen-deprivation therapy that developed resistance. Overall, the model presented here serves as a computational framework to support drug development, by generating hypotheses that can be tested experimentally in pre-clinical models.

Mathematical Modelling for the Role of CD4+T Cells in Tumor-Immune Interactions

Mathematical modelling has been used to study tumor-immune cell interaction. Some models were proposed to examine the effect of circulating lymphocytes, natural killer cells, and CD8+T cells, but they neglected the role of CD4+T cells. Other models were constructed to study the role of CD4+T cells but did not consider the role of other immune cells. In this study, we propose a mathematical model, in the form of a system of nonlinear ordinary differential equations, that predicts the interaction between tumor cells and natural killer cells, CD4+T cells, CD8+T cells, and circulating lymphocytes with or without immunotherapy and/or chemotherapy. This system is stiff, and the Runge-Kutta method failed to solve it. Consequently, the "Adams predictor-corrector" method is used. The results reveal that the patient's immune system can overcome small tumors; however, if the tumor is large, adoptive therapy with CD4+T cells can be an alternative to both CD8+T cell therapy and cytokines in some cases. Moreover, CD4+T cell therapy could replace chemotherapy depending upon tumor size. Even if a combination of chemotherapy and immunotherapy is necessary, using CD4+T cell therapy can better reduce the dose of the associated chemotherapy compared to using combined CD8+T cells and cytokine therapy. Stability analysis is performed for the studied patients. It has been found that all equilibrium points are unstable, and a condition for preventing tumor recurrence after treatment has been deduced. Finally, a bifurcation analysis is performed to study the effect of varying system parameters on the stability, and bifurcation points are specified. New equilibrium points are created or demolished at some bifurcation points, and stability is changed at some others. Hence, for systems turning to be stable, tumors can be eradicated without the possibility of recurrence. The proposed mathematical model provides a valuable tool for designing patients' treatment intervention strategies.

Review on biofabrication and applications of heterogeneous tumor models

Resolving the origin and development of tumor heterogeneity has proven to be a crucial challenge in cancer research. In vitro tumor models have been widely used for both scientific and clinical research. Currently, tumor models based on 2D cell culture, animal models, and 3D cell-laden constructs are widely used. Heterogeneous tumor models, which consist of more than one cell type and mimic cell-cell as well as cell-matrix interactions, are attracting increasing attention. Heterogeneous tumor models can serve as pathological models to study the microenvironment and tumor development such as tumorigenesis, invasiveness, and malignancy. They also provide disease models for drug screening and personalized therapy. In this review, the current techniques, models, and oncological applications regarding 3D heterogeneous tumor models are summarized and discussed.

A stochastic individual-based model to explore the role of spatial interactions and antigen recognition in the immune response against solid tumours

Spatial interactions between cancer and immune cells, as well as the recognition of tumour antigens by cells of the immune system, play a key role in the immune response against solid tumours. The existing mathematical models generally focus only on one of these key aspects. We present here a spatial stochastic individual-based model that explicitly captures antigen expression and recognition. In our model, each cancer cell is characterised by an antigen profile which can change over time due to either epimutations or mutations. The immune response against the cancer cells is initiated by the dendritic cells that recognise the tumour antigens and present them to the cytotoxic T cells. Consequently, T cells become activated against the tumour cells expressing such antigens. Moreover, the differences in movement between inactive and active immune cells are explicitly taken into account by the model. Computational simulations of our model clarify the conditions for the emergence of tumour clearance, dormancy or escape, and allow us to assess the impact of antigenic heterogeneity of cancer cells on the efficacy of immune action. Ultimately, our results highlight the complex interplay between spatial interactions and adaptive mechanisms that underpins the immune response against solid tumours, and suggest how this may be exploited to further develop cancer immunotherapies.

Tumor Clearance Analysis on a Cancer Chemo-Immunotherapy Mathematical Model

Mathematical models may allow us to improve our knowledge on tumor evolution and to better comprehend the dynamics between cancer, the immune system and the application of treatments such as chemotherapy and immunotherapy in both short and long term. In this paper, we solve the tumor clearance problem for a six-dimensional mathematical model that describes tumor evolution under immune response and chemo-immunotherapy treatments. First, by means of the localization of compact invariant sets method, we determine lower and upper bounds for all cells populations considered by the model and we use these results to establish sufficient conditions for the existence of a bounded positively invariant domain in the nonnegative orthant by applying LaSalle's invariance principle. Then, by exploiting a candidate Lyapunov function we determine sufficient conditions on the chemotherapy treatment to ensure tumor clearance. Further, we investigate the local stability of the tumor-free equilibrium point and compute conditions for asymptotic stability and tumor persistence. All conditions are given by inequalities in terms of the system parameters, and we perform numerical simulations with different values on the chemotherapy treatment to illustrate our results. Finally, we discuss the biological implications of our work.

Mathematical Analysis of a Mathematical Model of Chemovirotherapy: Effect of Drug Infusion Method

A mathematical model for the treatment of cancer using chemovirotherapy is developed with the aim of determining the efficacy of three drug infusion methods: constant, single bolus, and periodic treatments. The model is in the form of ODEs and is further extended into DDEs to account for delays as a result of the infection of tumor cells by the virus and chemotherapeutic drug responses. Analysis of the model is carried out for each of the three drug infusion methods. Analytic solutions are determined where possible and stability analysis of both steady state solutions for the ODEs and DDEs is presented. The results indicate that constant and periodic drug infusion methods are more efficient compared to a single bolus injection. Numerical simulations show that with a large virus burst size, irrespective of the drug infusion method, chemovirotherapy is highly effective compared to either treatments. The simulations further show that both delays increase the period within which a tumor can be cleared from body tissue.

Mathematical modeling of tumor-immune cell interactions

The anti-tumor activity of the immune system is increasingly recognized as critical for the mounting of a prolonged and effective response to cancer growth and invasion, and for preventing recurrence following resection or treatment. As the knowledge of tumor-immune cell interactions has advanced, experimental investigation has been complemented by mathematical modeling with the goal to quantify and predict these interactions. This succinct review offers an overview of recent tumor-immune continuum modeling approaches, highlighting spatial models. The focus is on work published in the past decade, incorporating one or more immune cell types and evaluating immune cell effects on tumor progression. Due to their relevance to cancer, the following immune cells and their combinations are described: macrophages, Cytotoxic T Lymphocytes, Natural Killer cells, dendritic cells, T regulatory cells, and CD4+ T helper cells. Although important insight has been gained from a mathematical modeling perspective, the development of models incorporating patient-specific data remains an important goal yet to be realized for potential clinical benefit.

A structural methodology for modeling immune-tumor interactions including pro- and anti-tumor factors for clinical applications

The immune system turns out to have both stimulatory and inhibitory factors influencing on tumor growth. In recent years, the pro-tumor role of immunity factors such as regulatory T cells and TGF-β cytokines has specially been considered in mathematical modeling of tumor-immune interactions. This paper presents a novel structural methodology for reviewing these models and classifies them into five subgroups on the basis of immune factors included. By using our experimental data due to immunotherapy experimentation in mice, these five modeling groups are evaluated and scored. The results show that a model with a small number of variables and coefficients performs efficiently in predicting the tumor-immune system interactions. Though immunology theorems suggest to employ a larger number of variables and coefficients, more complicated models are here shown to be inefficient due to redundant parallel pathways. So, these models are trapped in local minima and restricted in prediction capability. This paper investigates the mathematical models that were previously developed and proposes variables and pathways that are essential for modeling tumor-immune. Using these variables and pathways, a minimal structure for modeling tumor-immune interactions is proposed for future studies.

MATHEMATICAL MODELING OF IN-VIVO TUMOR-IMMUNE INTERACTIONS FOR THE CANCER IMMUNOTHERAPY USING MATURED DENDRITIC CELLS

To develop an anticancer drug, the mathematical models are nowadays indispensable because of complex immunological mechanisms defying with high experimentation costs as well as a large number of parameters. Based on immunological theories and vision of experimentation data, a simple and sufficient compartment model is designed that can accurately interpret and predict the effects of dendritic cell (DC)-based immunotherapy in accordance with experimentation data. The model includes effector cells, regulatory T cells, helper T cells, and DCs. A new key feature is the inclusion of immunotherapy with DCs matured with different materials. All the parameters of the model have been optimally obtained by fitting the experimental data using genetic algorithm. The proposed model has been used to predict a near-optimal pattern that minimizes tumor size after vaccination. This pattern has been validated by carrying out the associated in-vivo experimentation. The model recommends maturation materials and doses that activate a small amount of Treg in the early days and a large Th1/Treg ratio in the next days. The performance of the model compared with the previous study was shown to be superior, both qualitatively and quantitatively.

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.

Identifying the optimal anticancer targets from the landscape of a cancer-immunity interaction network

Cancer immunotherapy, an approach of targeting immune cells to attack tumor cells, has been suggested to be a promising way for cancer treatment recently. However, the successful application of this approach warrants a deeper understanding of the intricate interplay between cancer cells and the immune system. Especially, the mechanisms of immunotherapy remain elusive. In this work, we constructed a cancer-immunity interplay network by incorporating interactions among cancer cells and some representative immune cells, and uncovered the potential landscape of the cancer-immunity network. Three attractors emerge on the landscape, representing the cancer state, the immune state, and the hybrid state, which can correspond to escape, elimination, and equilibrium phases in the immunoediting theory, respectively. We quantified the transition processes between the cancer state and the immune state by calculating transition actions and identifying the corresponding minimum action paths (MAPs) between these two attractors. The transition actions, directly calculated from the high dimensional system, are correlated with the barrier heights from the landscape, but provide a more precise description of the dynamics of a system. By optimizing the transition actions from the cancer state to the immune state, we identified some optimal combinations of anticancer targets. Our combined approach of the landscape and optimization of transition actions offers a framework to study the stochastic dynamics and identify the optimal combination of targets for the cancer-immunity interplay, and can be applied to other cell communication networks or gene regulatory networks.

Modeling Macrophage Polarization and Its Effect on Cancer Treatment Success

Positive feedback loops drive immune cell polarization toward a pro-tumor phenotype that accentuates immunosuppression and tumor angiogenesis. This phenotypic switch leads to the escape of cancer cells from immune destruction. These positive feedback loops are generated by cytokines such as TGF-β, Interleukin-10 and Interleukin-4, which are responsible for the polarization of monocytes and M1 macrophages into pro-tumor M2 macrophages, and the polarization of naive helper T cells intopro-tumor Th2 cells. In this article, we present a deterministic ordinary differential equation (ODE) model that includes key cellular interactions and cytokine signaling pathways that lead to immune cell polarization in the tumor microenvironment. The model was used to simulate various cancer treatments in silico. We identified combination therapies that consist of M1 macrophages or Th1 helper cells, coupled with an anti-angiogenic treatment, that are robust with respect to immune response strength, initial tumor size and treatment resistance. We also identified IL-4 and IL-10 as the targets that should be neutralized in order to make these combination treatments robust with respect to immune cell polarization. The model simulations confirmed a hypothesis based on published experimental evidence that a polarization into the M1 and Th1 phenotypes to increase the M1-to-M2 and Th1-to-Th2 ratios plays a significant role in treatment success. Our results highlight the importance of immune cell reprogramming as a viable strategy to eradicate a highly vascularized tumor when the strength of the immune response is characteristically weak and cell polarization to the pro-tumor phenotype has occurred.

Maximization of viability time in a mathematical model of cancer therapy

In this paper, we study a dynamic optimization problem for a general nonlinear mathematical model for therapy of a lethal form of cancer. The model describes how the populations of cancer and normal cells evolve under the influence of the concentrations of nutrients (oxygen, glucose, etc.) and the applied therapeutic agent (drug). Regulated intensity of the therapy is interpreted as a time-dependent control strategy. The therapy (control) goal is to maximize the viability time, i. e., the duration of staying in a so-called safety region (which specifies safe living conditions of a patient in terms of constraints on the amounts of cancer and normal cells), subject to limited resources of the therapeutic agent. In a specific benchmark case, a novel optimality principle for admissible therapy strategies is established. It states that the optimal trajectories should finally reach a certain corner of the safety region or at least the upper constraint on the quantity of cancer cells. The results of numerical simulations appear to be in good agreement with the proposed principle. Typical qualitative structures of optimal treatment strategies are also obtained. Furthermore, important characteristics of the model are the competition coefficient (describing the negative influence of cancer cells on normal cells), the upper bound in the drug integral constraint, and the ratio between the therapy and damage coefficients (i. e., the ratio between the positive primary and negative side effects of the therapy).

An extended mathematical model of tumor growth and its interaction with the immune system, to be used for developing an optimized immunotherapy treatment protocol

BACKGROUND

Chemotherapy is usually known as the main modality for cancer treatment. Nevertheless, most of chronic cancers could not be treated with chemotherapy alone. Immunotherapy is a new modality for cancer treatment that is effective for early stages of cancer and it has fewer side effects compared to chemotherapy, specifically for those types of cancer that are resistant to it.

METHOD

This work presents an extended mathematical model to depict interactions between cancerous and adaptive immune system in mouse. We called the model an extended model, because we embedded all those compartments that have important roles in response to tumor in one model. The model includes tumor cells, natural killers, naïve and mature cytotoxic T cells, naïve and mature helper T cells, regulatory T cells, dendritic cells and interleukin 2 cytokine. Whole cycle of cell division program of immune cells is also considered in the model. We also optimized protocol of immunotherapy with DC vaccine based on the proposed mathematical model.

RESULT

Simulation results of the proposed model are in conformity with the experimental data recorded from mouse in immunology department of Tehran University of Medical Science as well as what has been explained in the literature. Our results explain dynamics of the immune cells from the first day of cancer growth and progression. Simulation result shows that reducing intervals between immunotherapy injections, efficacy of the treatment will be increased because CD8+ cells are boosted more rapidly. Optimized protocol for immunotherapy suggests that if the effect of DC vaccines on increasing number of anti-tumor immune cells be just before the maximum number of CD8+ cells, the effect of treatment will be maximized.

Mathematical model of tumor-immune surveillance

We present a novel mathematical model involving various immune cell populations and tumor cell populations. The model describes how tumor cells evolve and survive the brief encounter with the immune system mediated by natural killer (NK) cells and the activated CD8(+) cytotoxic T lymphocytes (CTLs). The model is composed of ordinary differential equations describing the interactions between these important immune lymphocytes and various tumor cell populations. Based on up-to-date knowledge of immune evasion and rational considerations, the model is designed to illustrate how tumors evade both arms of host immunity (i.e. innate and adaptive immunity). The model predicts that (a) an influx of an external source of NK cells might play a crucial role in enhancing NK-cell immune surveillance; (b) the host immune system alone is not fully effective against progression of tumor cells; (c) the development of immunoresistance by tumor cells is inevitable in tumor immune surveillance. Our model also supports the importance of infiltrating NK cells in tumor immune surveillance, which can be enhanced by NK cell-based immunotherapeutic approaches.

Mathematical Modelling and Analysis of the Tumor Treatment Regimens with Pulsed Immunotherapy and Chemotherapy

To begin with, in this paper, single immunotherapy, single chemotherapy, and mixed treatment are discussed, and sufficient conditions under which tumor cells will be eliminated ultimately are obtained. We analyze the impacts of the least effective concentration and the half-life of the drug on therapeutic results and then find that increasing the least effective concentration or extending the half-life of the drug can achieve better therapeutic effects. In addition, since most types of tumors are resistant to common chemotherapy drugs, we consider the impact of drug resistance on therapeutic results and propose a new mathematical model to explain the cause of the chemotherapeutic failure using single drug. Based on this, in the end, we explore the therapeutic effects of two-drug combination chemotherapy, as well as mixed immunotherapy with combination chemotherapy. Numerical simulations indicate that combination chemotherapy is very effective in controlling tumor growth. In comparison, mixed immunotherapy with combination chemotherapy can achieve a better treatment effect.