Model Of Tumor-immune Interactions Research Articles

A brief review: The application of mathematical modeling in cancer immunotherapy

In immunotherapy, scientists use drugs to enhance the host's own immune system and attenuate the suppression of the immune response by tumor cells. In this field, the research method of mathematical modeling can provide a way to quantitatively analyze and predict the effects of treatments for scientists. Starting with the model of tumor-immune interaction, this paper will then comprehensively review the application of mathematical models in predicting the effect of immunotherapy, analyze the optimal situation, and discuss feasible future research directions. Currently, mathematical models have been effective in classification and prediction. Scientists build the model to simulate the situation of humans biological systems and then validate it with real data, which makes it possible to predict the effect of various kinds of treatments. In the future, with the help of big data, models may play a role in the development of personalized treatment.

A Simple Model of Tumor-Immune Interaction: The Effect of Antigen Delay

We propose a simple model of tumor-immune interactions, which involves effector cells and tumor cells. In the model, the stimulation delay of tumor antigen in the immune system is incorporated. We investigate the dynamical behavior of the model via theoretic analysis and numerical simulations. The saddle-node bifurcation can occur in both cases with and without delay. In contrast to the case without delay, stimulation delay may result in some complex dynamical behaviors and biological phenomena. In the presence of delay, conditions on absolute/conditional stability of equilibria and the existence of Hopf bifurcations are obtained. We further discuss the effect of the tumor on the switch between absolute stability and conditional stability. Numerical simulations also show the existence of homoclinic bifurcation and the dependence of the asymptotic state of the tumor progression on initial conditions for different delay values. Effects of delay on the dynamics of the model and on the region of tumor extinction are illustrated by simulations with different sets of parameter values. Finally, the corresponding biological implications are demonstrated.

A study on fractional tumor-immune interaction model related to lung cancer via generalized Laguerre polynomials

BackgroundCancer, a complex and deadly health concern today, is characterized by forming potentially malignant tumors or cancer cells. The dynamic interaction between these cells and their environment is crucial to the disease. Mathematical models can enhance our understanding of these interactions, helping us predict disease progression and treatment strategies.MethodsIn this study, we develop a fractional tumor-immune interaction model specifically for lung cancer (FTIIM-LC). We present some definitions and significant results related to the Caputo operator. We employ the generalized Laguerre polynomials (GLPs) method to find the optimal solution for the FTIIM-LC model. We then conduct a numerical simulation and compare the results of our method with other techniques and real-world data.ResultsWe propose a FTIIM-LC model in this paper. The approximate solution for the proposed model is derived using a series of expansions in a new set of polynomials, the GLPs. To streamline the process, we integrate Lagrange multipliers, GLPs, and operational matrices of fractional and ordinary derivatives. We conduct a numerical simulation to study the effects of varying fractional orders and achieve the expected theoretical results.ConclusionThe findings of this study demonstrate that the optimization methods used can effectively predict and analyze complex phenomena. This innovative approach can also be applied to other nonlinear differential equations, such as the fractional Klein–Gordon equation, fractional diffusion-wave equation, breast cancer model, and fractional optimal control problems.

A tumor–immune interaction model with the effect of impulse therapy

A mathematical model describing the growth of tumor cells and the effect of impulsive therapy is studied in this paper. The governing equations and parameters related to the proposed model are based on the experimental and clinical outcome. Employing the quasi-steady-state approximation in delineating the dynamics of tumor–immune under different cytokines, a four-dimensional deterministic system is obtained. We explore the steady states, local and global stability criteria, transcritical bifurcation and boundary separation for the given model. The tumor cells are characterized by their growth rate and the tumor control region is determined in the parametric space. An impulsive differential equation is considered to determine the interval for perfect dosing of dendritic cell vaccination to boost the immune response. The effect of dendritic cell vaccination is determined by taking into account both perfect and imperfect therapeutics and the optimal number for therapeutics is computed. Our model demonstrates that the dendritic cell vaccination boosts the immune system and eradicates the tumor cell population.

Carbonic Anhydrase IX Suppression Shifts Partial Response to Checkpoint Inhibitors into Complete Tumor Eradication: Model-Based Investigation.

Immune checkpoint inhibitors (ICIs) have revolutionized the treatment of solid malignancies, including non-small-cell lung cancer. However, immunotherapy resistance constitutes a significant challenge. To investigate carbonic anhydrase IX (CAIX) as a driver of resistance, we built a differential equation model of tumor-immune interactions. The model considers treatment with the small molecule CAIX inhibitor SLC-0111 in combination with ICIs. Numerical simulations showed that, given an efficient immune response, CAIX KO tumors tended toward tumor elimination in contrast to their CAIX-expressing counterparts, which stabilized close to the positive equilibrium. Importantly, we demonstrated that short-term combination therapy with a CAIX inhibitor and immunotherapy could shift the asymptotic behavior of the original model from stable disease to tumor eradication. Finally, we calibrated the model with data from murine experiments on CAIX suppression and combination therapy with anti-PD-1 and anti-CTLA-4. Concluding, we have developed a model that reproduces experimental findings and enables the investigation of combination therapies. Our model suggests that transient CAIX inhibition may induce tumor regression, given a sufficient immune infiltrate in the tumor, which can be boosted with ICIs.

Dynamics and Stability of a Fractional-Order Tumor–Immune Interaction Model with B-D Functional Response and Immunotherapy

In this paper, we investigate a fractional-order tumor–immune interaction model with B-D function item and immunotherapy. First, the existence, uniqueness and nonnegativity of the solutions of the model are established. Second, the local and global asymptotic stability of some tumor-free equilibrium points and a unique positive equilibrium point are obtained. Finally, we use numerical simulation method to visualize and verify the theoretical conclusions. It is known that the fractional-order parameter β has a stabilization effect, and the tumor cells can be destroyed or controlled by using immunotherapy.

Stochastic tumor-immune interaction model with external treatments and time delays: An optimal control problem.

Herein, we discuss an optimal control problem (OC-P) of a stochastic delay differential model to describe the dynamics of tumor-immune interactions under stochastic white noises and external treatments. The required criteria for the existence of an ergodic stationary distribution and possible extinction of tumors are obtained through Lyapunov functional theory. A stochastic optimality system is developed to reduce tumor cells using some control variables. The study found that combining white noises and time delays greatly affected the dynamics of the tumor-immune interaction model. Based on numerical results, it can be shown which variables are optimal for controlling tumor growth and which controls are effective for reducing tumor growth. With some conditions, white noise reduces tumor cell growth in the optimality problem. Some numerical simulations are conducted to validate the main results.

Global Dynamics of the model of tumor-immune interaction

<p style='text-indent:20px;'>In this paper, we consider the model of the tumor and immune system, proposed by Han, He and Kuang (Discrete and Contin. Dyn. Syst. Ser. S 13:2347–2363, 2020). We give the necessary and sufficient conditions for the model to have no tumor equilibrium, one, two or three tumor equilibria respectively. Moreover, we prove that the model has no periodic solution, and give its global dynamics in the first quadrant. We find that the magnitude of the tumor reproduction number <inline-formula><tex-math id="M1">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula> relative to 1 can describe the final state of tumor and the level of the immune system. If <inline-formula><tex-math id="M2">\begin{document}$ R_0\leq1 $\end{document}</tex-math></inline-formula>, the tumor will be cleaned out by immune system in the end. However, if <inline-formula><tex-math id="M3">\begin{document}$ R_0>1 $\end{document}</tex-math></inline-formula>, the model appears monostable or bistable, i.e., the tumor coexists with the immune system and they are ultimately in a stable steady state.</p>

Evaluation of innate and adaptive immune system interactions in the tumor microenvironment via a 3D continuum model

Immune cells in the tumor microenvironment (TME) are known to affect tumor growth, vascularization, and extracellular matrix (ECM) deposition. Marked interest in system-scale analysis of immune species interactions within the TME has encouraged progress in modeling tumor-immune interactions in silico. Due to the computational cost of simulating these intricate interactions, models have typically been constrained to representing a limited number of immune species. To expand the capability for system-scale analysis, this study develops a three-dimensional continuum mixture model of tumor-immune interactions to simulate multiple immune species in the TME. Building upon a recent distributed computing implementation that enables efficient solution of such mixture models, major immune species including monocytes, macrophages, natural killer cells, dendritic cells, neutrophils, myeloid-derived suppressor cells (MDSC), cytotoxic, helper, regulatory T-cells, and effector and regulatory B-cells and their interactions are represented in this novel implementation. Immune species extravasate from blood vasculature, undergo chemotaxis toward regions of high chemokine concentration, and influence the TME in proportion to locally defined levels of stimulation. The immune species contribute to the production of angiogenic and tumor growth factors, promotion of myofibroblast deposition of ECM, upregulation of angiogenesis, and elimination of living and dead tumor species. The results show that this modeling approach offers the capability for quantitative insight into the modulation of tumor growth by diverse immune-tumor interactions and immune-driven TME effects. In particular, MDSC-mediated effects on tumor-associated immune species’ activation levels, volume fraction, and influence on the TME are explored. Longer term, linking of the model parameters to particular patient tumor information could simulate cancer-specific immune responses and move toward a more comprehensive evaluation of immunotherapeutic strategies.

A TUMOR-IMMUNE MODEL WITH MIXED IMMUNOTHERAPY AND CHEMOTHERAPY: QUALITATIVE ANALYSIS AND OPTIMAL CONTROl

We develop a mathematical model of tumor-immune interactions, including six populations (tumor cells, CD8[Formula: see text]T cells, natural killer (NK) cells, dendritic cells, helper T cells, cytokine interleukin-12 (IL-12)) and three potential treatments (chemotherapy, Tumor-infiltrating lymphocyte (TIL) therapy and IL-12 therapy). We characterize the dynamics of our model without treatment through stability and sensitivity analysis, which provides a broad understanding of the long-term qualitative behavior. To find the best combination of the chemo-immunotherapy regimens to eliminate tumors, we formulate an optimal control problem with path constraints of total drug dose and solve it numerically with the optimal control software Pyomo. We also simulate the scenarios of traditional treatment protocols as a comparison and find that our optimal treatment strategies have a better therapeutic effect. In addition, numerical simulation results show that IL-12 therapy is a good adjunctive therapy and has a high potential for inhibiting a large tumor in combination with other therapy. In most cases, combination therapy is more effective than a single treatment.

Solving and Analysing Tumor–Immune Interaction Model by Generalized Differential Transformation Method

Solving and Analysing Tumor–Immune Interaction Model by Generalized Differential Transformation Method

Optimal Feedback Control of Cancer Chemotherapy Using Hamilton–Jacobi–Bellman Equation

Cancer chemotherapy has been the most common cancer treatment. However, it has side effects that kill both tumor cells and immune cells, which can ravage the patient’s immune system. Chemotherapy should be administered depending on the patient’s immunity as well as the level of cancer cells. Thus, we need to design an efficient treatment protocol. In this work, we study a feedback control problem of tumor‐immune system to design an optimal chemotherapy strategy. For this, we first propose a mathematical model of tumor‐immune interactions and conduct stability analysis of two equilibria. Next, the feedback control is found by solving the Hamilton–Jacobi–Bellman (HJB) equation. Here, we use an upwind finite‐difference method for a numerical approximate solution of the HJB equation. Numerical simulations show that the feedback control can help determine the treatment protocol of chemotherapy for tumor and immune cells depending on the side effects.

Theoretical and numerical analysis of fractal fractional model of tumor-immune interaction with two different kernels

Fractal fractional operators in Caputo and Caputo-Fabrizio sense are being used in this manuscript to explore the interaction between the immune system and cancer cells. The tumour-immune model has been investigated numerically and theoretically by the singular and non-singular fractal fractional operators. Via fixed point theorems, the existence and uniqueness of the model under the Caputo fractal fractional operator have been demonstrated. Using the fixed point theory, the existence of a unique solution has been derived under the Caputo-Fabrizio case. Through nonlinear analysis, the Ulam-Hyres stability of the model has been derived. For the singular and nonsingular fractal fractional operators, numerical results have been developed by Lagrangian-piece wise interpolation. We simulate the numerical results for the various sets of fractional and fractal orders to describe the relationship between immune and cancer cells under the novel operators with two different kernels. We compared the dynamics of the tumor-immune model using a power law and an exponential-decay kernel to explore that the nonsingular fractal fractional operator provides better dynamics for the considered model.

Analysis of fractal-fractional model of tumor-immune interaction

Recently, Atangana proposed new operators by combining the fractional and fractal calculus. These recently proposed operators, referred to as fractal-fractional operators, have been widely used to study the complex dynamics of a problem. Cancer is a prevalent disease today and is difficult to cure. The immune system tends to fight it as cancer sets up in the body. In this manuscript, the novel operators have been used to analyze the relationship between the immune system and cancer cells. The tumor-immune model has been studied qualitatively and quantitatively via Atangana-Baleanu fractal-fractional operator. The existence and uniqueness results of the model under Atangana-Baleanu fractal-fractional operator have proved through fixed point theorems. The Ulam-Hyres stability for the model has derived through non-linear analysis. Numerical results have developed through Lagrangian-piece wise interpolation for the different fractal-fractional operators. To visualize the relationship between immune cells and cancers cells under novel operators in a various sense, we simulate the numerical results for the different sets of fractional and fractal orders.

Optimal control strategy for cancer remission using combinatorial therapy: A mathematical model-based approach

Abstract In this article, we develop and analyze a non-linear mathematical model of tumor-immune interactions with combined therapeutic drug and treatment controls. To understand under what circumstances the cancerous cells can be destroyed, an optimal control problem is formulated with treatments as control parameters. By designing a quadratic control based functional, we establish the optimal treatment strategies that maximize the number of immune-effector cells, minimize the number of cancer cells, and detrimental effects caused by the amount of drugs. The necessary and sufficient conditions for optimal control are established. We prove the existence and uniqueness of an optimal control problem. To recognize significant system’s parameters, sensitivity analysis are performed for the drug administration and cost functional respectively. We also carry out a cost-effectiveness analysis to determine the most cost-effective therapeutic strategy. The numerical results validate analytical findings and also elucidates that the combinatorial drug therapy can alleviate the cancerous cells under different scenarios.

Analysis of a fractional tumor-immune interaction model with exponential kernel

In this paper, a tumor-immune interaction model has been analyzed via Caputo-Fabrizio fractional derivative operator with exponential kernel. Existence of solution of the model has been established with a fixed-point method and then it demonstrated the uniqueness of solution also. The stability of the model has been analyzed with the help of Hyers-Ulam stability approach and then numerical solution by using the Adam-Basford method. The results are further examined in detail with simulations for different fractional derivative values.

Exploring dynamical complexity in a time-delayed tumor-immune model.

The analysis of dynamical complexity in nonlinear phenomena is an effective tool to quantify the level of their structural disorder. In particular, a mathematical model of tumor-immune interactions can provide insight into cancer biology. Here, we present and explore the aspects of dynamical complexity, exhibited by a time-delayed tumor-immune model that describes the proliferation and survival of tumor cells under immune surveillance, governed by activated immune-effector cells, host cells, and concentrated interleukin-2. We show that by employing bifurcation analyses in different parametric regimes and the 0-1 test for chaoticity, the onset of chaos in the system can be predicted and also manifested by the emergence of multi-periodicity. This is further verified by studying one- and two-parameter bifurcation diagrams for different dynamical regimes of the system. Furthermore, we quantify the asymptotic behavior of the system by means of weighted recurrence entropy. This helps us to identify a resemblance between its dynamics and emergence of complexity. We find that the complexity in the model might indicate the phenomena of long-term cancer relapse, which provides evidence that incorporating time-delay in the effect of interleukin in the tumor model enhances remarkably the dynamical complexity of the tumor-immune interplay.

Analysis of tumor-immune dynamics in a delayed dendritic cell therapy model.

We formulate a tumor-immune interaction model with a constant delay to capture the behavior following application of a dendritic cell therapy. The model is validated using experimental data from melanoma-induced mice. Through theoretical and numerical analyses, the model is shown to produce rich dynamics, such as a Hopf bifurcation and bistability. We provide thresholds for tumor existence and, in a special case, tumor elimination. Our work indicates a sensitivity in model outcomes to the immune response time. We provide a stability analysis for the high tumor equilibrium. For small delays in response, the tumor and immune system coexist at a low level. Large delays give rise to fatally high tumor levels. Our computational and theoretical work reveals that there exists an intermediate region of delay that generates complex oscillatory, even chaotic, behavior. The model then reflects uncertainty in treatment outcomes for varying initial tumor burdens, as well as tumor dormancy followed by uncontrolled growth to a lethal size, a phenomenon seen in vivo. Theoretical and computational analyses suggest efficacious treatments to use in conjunction with the dendritic cell vaccine. Additional analysis of a highly aggressive tumor additionally confirms the importance of representation with a time delay, as periodic solutions are strictly able to be generated when a delay is present.

A Tumor-Immune Interaction Model for Synergistic Combinations of Anti PD-L1 and Ionizing Irradiation Treatment.

Combination therapy with immune checkpoint blockade and ionizing irradiation therapy (IR) generates a synergistic effect to inhibit tumor growth better than either therapy does alone. We modeled the tumor-immune interactions occurring during combined IT and IR based on the published data from Deng et al. The mathematical model considered programmed cell death protein 1 and programmed death ligand 1, to quantify data fitting and global sensitivity of critical parameters. Fitting of data from control, IR and IT samples was conducted to verify the synergistic effect of a combination therapy consisting of IR and IT. Our approach using the model showed that an increase in the expression level of PD-1 and PD-L1 was proportional to tumor growth before therapy, but not after initiating therapy. The high expression level of PD-L1 in T cells may inhibit IT efficacy. After combination therapy begins, the tumor size was also influenced by the ratio of PD-1 to PD-L1. These results highlight that the ratio of PD-1 to PD-L1 in T cells could be considered in combination therapy.

Mathematical Modelling of the Inhibitory Role of Regulatory T Cells in Tumor Immune Response

The immune system against tumors acts through a complex dynamical process showing a dual role. On the one hand, the immune system can activate some immune cells to kill tumor cells (TCs), such as cytotoxic T lymphocytes (CTLs) and natural killer cells (NKs), but on the other hand, more evidence shows that some immune cells can help tumor escape, such as regulatory T cells (Tregs). In this paper, we propose a tumor immune interaction model based on Tregs-mediated tumor immune escape mechanism. When helper T cells’ (HTCs) stimulation rate by the presence of identified tumor antigens is below critical value, the coexistence (tumor and immune) equilibrium is always stable in its existence region. When HTCs stimulation rate is higher than the critical value, the inhibition rate of effector cells (ECs) by Tregs can destabilize the coexistence equilibrium and cause Hopf bifurcations and produce a limit cycle. This model shows that Tregs might play a crucial role in triggering the tumor immune escape. Furthermore, we introduce the adoptive cellular immunotherapy (ACI) and monoclonal antibody immunotherapy (MAI) as the treatment to boost the immune system to fight against tumors. The numerical results show that ACI can control TCs more, while MAI can delay the inhibitory effect of Tregs on ECs. The result also shows that the combination of both immunotherapies can control TCs and reduce the inhibitory effect of Tregs better than a single immunotherapy can control.