Tumor-immune Dynamics Research Articles

Tumour-immune dynamics with an immune checkpoint inhibitor

The use of immune checkpoint inhibitors is becoming more commonplace in clinical trials across the nation. Two important factors in the tumour-immune response are the checkpoint protein programmed death-1 (PD-1) and its ligand PD-L1. We propose a mathematical tumour-immune model using a system of ordinary differential equations to study dynamics with and without the use of anti-PD-1. A sensitivity analysis is conducted, and series of simulations are performed to investigate the effects of intermittent and continuous treatments on the tumour-immune dynamics. We consider the system without the anti-PD-1 drug to conduct a mathematical analysis to determine the stability of the tumour-free and tumorous equilibria. Through simulations, we found that a normally functioning immune system may control tumour. We observe treatment with anti-PD-1 alone may not be sufficient to eradicate tumour cells. Therefore, it may be beneficial to combine single agent treatments with additional therapies to obtain a better antitumour response.

Tumour-Immune Dynamics with an Immune Checkpoint Inhibitor

Tumour-Immune Dynamics with an Immune Checkpoint Inhibitor

Modelling Tumour–Immune Dynamics, Disease Progression and Treatment

Modelling Tumour–Immune Dynamics, Disease Progression and Treatment

Abstract 4543: Local and systemic tumor-immune dynamics in metastatic cancer

Abstract Despite decades of effort, most disseminated cancers remain incurable, and progression from localized to metastatic disease is largely responsible for cancer-related mortality. In metastatic disease, an ongoing battle between tumor and host occurs at each site; tumor-associated antigens, stress proteins, and danger-associated molecular patterns can both initiate and continually stimulate an immune response against a tumor. Adding an additional layer of complexity, locally activated cytotoxic T cells traffic through the host circulatory system to also surveil metastasis elsewhere in the body. Thus, metastatic tumors are highly interdependent; changes in the nonlinear tumor-immune interactions in one tumor can perturb the systemic antitumor immune response, potentially facilitating spontaneous regression or aggressive outgrowth in distant sites. This can additionally influence the clinical outcome of therapeutic intervention, which depends on therapy-induced changes in tumor-immune dynamics both locally and systemically, as well as patient-specific initial conditions of the global disease. Using the tools of mathematical oncology we gain insights into this complex interconnectivity between metastatic sites. ODE models have been developed which incorporate both local tumor-immune interactions in each tumor site and the trafficking of activated T cells systemically, and can be parameterized by experimental and clinical investigations. Using four primary tumor sites (lung, liver, breast, kidney) we simulate the growth behavior of a primary site upon both seeding, growth, and treatment of a secondary (or several) metastatic site(s). In the presence of several metastatic sites, the dissemination of activated T cells is not necessarily intuitive and depends on many factors including the blood flow fraction to each organ and the tumor volume to organ size ratio. Certain sites may experience inhibition of growth, and other sites may be promoted upon seeding of an additional site. We computationally analyze all combinations of metastatic sites, and predict which sites are likely to benefit (tumor growth) and which to suffer (tumor shrinkage) in each combination based upon known properties of physiological blood flow and tumor volume at initial presentation. This allows the identification of which sites in each combination would be the optimal target for surgical therapy in order to induce the maximum reduction in overall tumor burden. Furthermore, we can simulate local radiation at each respective site, quantifying the difference in model-predicted decrease in overall tumor burden between individual targets. The results facilitate an improved understanding of general disease kinetics in the metastatic setting, emphasize that "local" therapy is highly likely to have systemic effects, and support the case for a paradigm shift in treatment target selection for metastatic disease. Citation Format: Rachel Walker, Jan Poleszczuk, Heiko Enderling. Local and systemic tumor-immune dynamics in metastatic cancer [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2017; 2017 Apr 1-5; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2017;77(13 Suppl):Abstract nr 4543. doi:10.1158/1538-7445.AM2017-4543

Abstract 1653: Modeling tumor immuno-dynamics to predict patient survival and immunotherapy efficacy

Abstract State of the art models of cancer survival are increasingly utilizing molecular data that feature the number of non-synonymous mutations, or a network based stratification of mutations, as significant predictors. It is widely hypothesized that this association between mutations and survival is a result of neo-epitopes that induce a robust anti-tumor immune response. Naively this implies that patient neo-epitope burden should better predict survival than that of mutations alone, however this has generally been difficult to prove. Indeed, mutations and predicted neo-epitopes explain roughly the same survival variance, likely due to the intertwined difficulties of neo-epitope prioritization and the often immuno-suppressive tumor microenvironment. Reasoning that integrating leading order immune response with novel neo-epitope prioritization will lead to superior performance, we built a statistical model to quantify the influence of tumor immune-dynamics on patient survival. Neoepitopes (MHC-I/II) were predicted from mutations and filtered on a self-ligandome. The clonal structure of these mutations and the distribution of resulting epitopes were assessed as predictive features. Additionally, the existing response to epitopes was assayed by incorporating TCR and BCR sequence counts and entropy as a proxy for clonal tumor infiltrating lymphocyte expansion. Leveraging the cancer genome atlas (TCGA) for 9 major cancer types including bladder (BLCA), breast (BRCA), colorectal (COAD/READ), glioblastoma (GBM), liver (LIHC), lung (NSCLC), melanoma (SKCM), pancreatic (PAAD), and uterine (UCEC) cancer, we obtained omics data for a total of 2,866 patients. Random forest based recursive feature elimination was used to determine which features were most likely to be predictive while overfitting was controlled with k-fold cross validation. A cox proportional hazard model of survival as a function of mutation burden and functions of neoepitope and immune response predictors was constructed using the selected features. Our results confirm that neoepitope and immune response based predictors for survival often significantly outperform mutation burden alone and simultaneously suggest a quantitative classification of immunotherapy efficacy across cancer subtypes. Citation Format: Nicholas K. Akers, Eric E. Schadt, Bojan Losic. Modeling tumor immuno-dynamics to predict patient survival and immunotherapy efficacy [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2017; 2017 Apr 1-5; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2017;77(13 Suppl):Abstract nr 1653. doi:10.1158/1538-7445.AM2017-1653

Modeling the Dichotomy of the Immune Response to Cancer: Cytotoxic Effects and Tumor-Promoting Inflammation.

Although the immune response is often regarded as acting to suppress tumor growth, it is now clear that it can be both stimulatory and inhibitory. The interplay between these competing influences has complex implications for tumor development, cancer dormancy, and immunotherapies. In fact, early immunotherapy failures were partly due to a lack in understanding of the nonlinear growth dynamics these competing immune actions may cause. To study this biological phenomenon theoretically, we construct a minimally parameterized framework that incorporates all aspects of the immune response. We combine the effects of all immune cell types, general principles of self-limited logistic growth, and the physical process of inflammation into one quantitative setting. Simulations suggest that while there are pro-tumor or antitumor immunogenic responses characterized by larger or smaller final tumor volumes, respectively, each response involves an initial period where tumor growth is stimulated beyond that of growth without an immune response. The mathematical description is non-identifiable which allows an ensemble of parameter sets to capture inherent biological variability in tumor growth that can significantly alter tumor-immune dynamics and thus treatment success rates. The ability of this model to predict non-intuitive yet clinically observed patterns of immunomodulated tumor growth suggests that it may provide a means to help classify patient response dynamics to aid identification of appropriate treatments exploiting immune response to improve tumor suppression, including the potential attainment of an immune-induced dormant state.

A Mathematical Investigation on Tumor-Immune Dynamics: The Impact of Vaccines on the Immune Response

Mathematical models analyzing tumor-immune interactions provide a framework by which to address specific scenarios in regard to tumor-immune dynamics. Important aspects of tumor-immune surveillance to consider is the elimination of tumor cells from a host’s cell-mediated immunity as well as the implications of vaccines derived from synthetic antigen. In present studies, our mathematical model examined the role of synthetic antigen to the strength of the immune system. The constructed model takes into account accepted knowledge of immune function as well as prior work done by de Pillis et al. All equations describing tumor-immune growth, antigen presentation, immune response, and interaction rates were numerically simulated with MATLAB. Here, our work shows that a robust immune response can be generated if the immune system recognizes epitopes that are between 8 to 11 amino acids long. We show through mathematical modeling of how synthetic tumor vaccines can be utilized to mitigate a developing cancer.

A Stochastic Optimal Control Model for BCG Immunotherapy in Superficial Bladder Cancer

Urologic studies have reported in several papers that the optimal dose of Bacillus Calmette-Guerin (BCG) immunotherapy in superficial bladder cancer, is still a subject of research. Our main goal from this paper, is to find treatment regimens that minimize the total number of tumors in the presence of a diffusion process. For this, we devise a stochastic model in the form of a nonlinear system of four stochastic differential equations (SDEs) that describe tumor-immune dynamics after BCG instillations. Therefore, we study the existence and the stability results. Then, we introduce a control function in the mathematical model, to represent the dose of BCG intravesical therapy, and we seek its optimal values through the application of a stochastic version of Pontryagin's maximum principle. Finally, we present some numerical simulations using iterative stochastic Runge-Kutta progressive-regressive schemes which we propose for solving the optimality system of the obtained stochastic two-point boundary value problem.

A model of the effects of cancer cell motility and cellular adhesion properties on tumour-immune dynamics.

We present a three-dimensional model simulating the dynamics of an anti-cancer T-cell response against a small, avascular, early-stage tumour. Interactions at the tumour site are accounted for using an agent-based model (ABM), while immune cell dynamics in the lymph node are modelled as a system of delay differential equations (DDEs). We combine these separate approaches into a two-compartment hybrid ABM-DDE system to capture the T-cell response against the tumour. In the ABM at the tumour site, movement of tumour cells is modelled using effective physical forces with a specific focus on cell-to-cell adhesion properties and varying levels of tumour cell motility, thus taking into account the ability of cancer cells to spread and form clusters. We consider the effectiveness of the immune response over a range of parameters pertaining to tumour cell motility, cell-to-cell adhesion strength and growth rate. We also investigate the dependence of outcomes on the distribution of tumour cells. Low tumour cell motility is generally a good indicator for successful tumour eradication before relapse, while high motility leads, almost invariably, to relapse and tumour escape. In general, the effect of cell-to-cell adhesion on prognosis is dependent on the level of tumour cell motility, with an often unpredictable cross influence between adhesion and motility, which can lead to counterintuitive effects. In terms of overall tumour shape and structure, the spatial distribution of cancer cells in clusters of various sizes has shown to be strongly related to the likelihood of extinction.

Numerical analysis of fractional-order tumor model

In this paper, the tumor-immune dynamics are simulated by solving a nonlinear system of differential equations. The fractional-order mathematical model incorporated with three Michaelis–Menten terms to indicate the saturated effect of immune response, the limited immune response to the tumor and to account the self-limiting production of cytokine interleukin-2. Two types of treatments were considered in the mathematical model to demonstrate the importance of immunotherapy. The limiting values of these treatments were considered, satisfying the stability criteria for fractional differential system. A graphical analysis is made to highlight the effects of antigenicity of the tumor and the fractional-order derivative on the tumor mass.

Cancer immunoediting: A process driven by metabolic competition as a predator–prey–shared resource type model

It is a well-established fact that tumors up-regulate glucose consumption to meet increasing demands for rapidly available energy by upregulating a purely glycolytic mode of glucose metabolism. What is often neglected is that activated cytotoxic cells of the immune system, integral players in the carcinogenesis process, also come to rely on glycolysis as a primary mode of glucose metabolism. Moreover, while cancer cells can revert back to aerobic metabolism, rapidly proliferating cytotoxic lymphocytes are incapable of performing their function when adequate resources are lacking. Consequently, it is likely that in the tumor microenvironment there may exist competition for shared resources between cancer cells and the cells of the immune system, which may underlie much of tumor–immune dynamics. Proposed here is a model of tumor–immune–glucose interactions, formulated as a predator–prey–common resource type system. The outcome of these interactions ranges from tumor elimination, to tumor dormancy, to unrestrained tumor growth. It is also predicted that the process of tumor escape can be preceded by periods of oscillatory tumor growth. A detailed bifurcation analysis of three subsystems of the model suggest that oscillatory regimes are a result of competition for shared resource (glucose) between the predator (immune cells) and the prey (cancer cells). Existence of competition for nutrients between cancer and immune cells may provide additional mechanistic insight as to why the efficacy of many immunotherapies may be limited.

Mathematical analysis and classification of tumor immune dynamics in T cell transfer treatment

Immunotherapy for cancer is a forthcoming therapeutic option in cancer treatment. A variety of approaches, including adoptive cell transfer, adjuvant therapy, and monoclonal antibody therapy, are currently used as complementary or alternative cancer treatments. Since immunity against tumors is established as a result of complex interactions among tumors and immune cells, a theoretical framework to elucidate the dynamics of tumor killing by immune cells initiated by clinical intervention needs to be developed. We construct a simple mathematical model that describes the killing of tumors by T cells. We focus on a mathematical characterization of the effect of adoptive cell transfer therapy, the injection of a patient's own ex vivo activated T cells to initiate tumor immunity. On the basis of rigorous mathematical analyses and numerical computations, we provide biological interpretations for the possible outcomes of adoptive cell transfer under three different scenarios in terms of proliferation. In our modeling framework, adoptive cell transfer therapy can be understood as a driving force that potentially operates to shift the state of an immune response from inactive to active. On the other hand, our modeling framework suggests the possibility that a tumor can exploit a self-augmenting proliferation of T cells mediated by autocrine/paracrine signaling as an escape strategy from an immune response.

A dynamical model of tumour immunotherapy

A coupled ordinary differential equation model of tumour-immune dynamics is presented and analysed. The model accounts for biological and clinical factors which regulate the interaction rates of cytotoxic T lymphocytes on the surface of the tumour mass. A phase plane analysis demonstrates that competition between tumour cells and lymphocytes can result in tumour eradication, perpetual oscillations, or unbounded solutions. To investigate the dependence of the dynamic behaviour on model parameters, the equations are solved analytically and conditions for unbounded versus bounded solutions are discussed. An analytic characterisation of the basin of attraction for oscillatory orbits is given. It is also shown that the tumour shape, characterised by a surface area to volume scaling factor, influences the size of the basin, with significant consequences for therapy design. The findings reveal that the tumour volume must surpass a threshold size that depends on lymphocyte parameters for the cancer to be completely eliminated. A semi-analytic procedure to calculate oscillation periods and determine their sensitivity to model parameters is also presented. Numerical results show that the period of oscillations exhibits notable nonlinear dependence on biologically relevant conditions.

Synthesis of optimal control in a mathematical model of tumour–immune dynamics

SummaryAn optimal control problem for a mathematical model of tumour–immune dynamics under the influence of chemotherapy is considered. The toxicity effect of the chemotherapeutic agent on both tumour and immunocompetent cells is taken into account. A standard linear pharmacokinetic equation for the chemotherapeutic agent is added to the system. The aim is to find an optimal strategy of treatment to minimize the tumour volume while keeping the immune response not lower than a fixed permissible level as far as possible. Sufficient conditions for the existence of not more than one switching and not more than two switchings without singular regimes are obtained. The surfaces in the extended phase space, on which the last switching appears, are constructed analytically.Copyright © 2013 John Wiley & Sons, Ltd.

Tumor–Immune Dynamics Regulated in the Microenvironment Inform the Transient Nature of Immune-Induced Tumor Dormancy

Cancer in a host induces responses that increase the ability of the microenvironment to sustain the growing mass, for example, angiogenesis, but cancer cells can have varying sensitivities to these sustainability signals. Here, we show that these sensitivities are significant determinants of ultimate tumor fate, especially in response to treatments and immune interactions. We present a mathematical model of cancer-immune interactions that modifies generalized logistic growth with both immune-predation and immune-recruitment. The role of a growing environmental carrying capacity is discussed as a possible regulatory mechanism for tumor growth, and this regulation is shown to modify cancer-immune interactions and the possibility of achieving immune-induced tumor dormancy. This mathematical model qualitatively matches experimental observations of immune-induced tumor dormancy as it predicts dormancy as a transient period of growth that necessarily ends in either tumor elimination or tumor escape. As dormant tumors may exist asymptomatically and may be easier to treat with conventional therapy, an understanding of the mechanisms behind tumor dormancy may lead to new treatments aimed at prolonging the dormant state or converting an aggressive cancer to the dormant state.

Delay differential model for tumour–immune dynamics with HIV infection of CD4+ T-cells

In this paper, we introduce ordinary and delay differential equations to describe the interactions between a malignant tumour and the immune system in vivo in the presence of human immunodeficiency virus (HIV) infection of CD4+ T-cells. In the delay model, we take into account the time lags required by the healthy effector cell components to recognize the pathogens and tumour cells. The models consist of four populations: tumour cells, healthy effector cells (CD4+ T-cells), effector cells infected by HIVs and free viral particles. The presence of delay term in the model leads to a notable increase in the complexity of the observed behaviour. We investigate the qualitative behaviour of the models and find the conditions that guarantee the asymptotic stability of the steady states. Numerical simulations are provided to illustrate and extend the theoretical results. The obtained results are consistent with the real phenomena and give a better understanding of cancer immunity and viral oncogenesis.

Numerical Modeling of Fractional-Order Biological Systems

We provide a class of fractional-order differential models of biological systems with memory, such as dynamics of tumor-immune system and dynamics of HIV infection of CD4<sup >+</sup> T cells. Stability and nonstability conditions for disease-free equilibrium and positive equilibria are obtained in terms of a threshold parameter <svg style="vertical-align:-3.25793pt;width:22.5px;" id="M1" height="16.025" version="1.1" viewBox="0 0 22.5 16.025" width="22.5" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.737)"><path id="x211B" d="M852 669l7 -19q-33 -13 -66 -38q83 -53 83 -139q0 -65 -49 -111q-49 -47 -131 -50v-2q54 -40 54 -84q0 -36 -38 -97t-38 -77q0 -34 32 -34q52 0 145 101l17 -16q-100 -115 -176 -115q-38 0 -65.5 23.5t-27.5 58.5q-1 28 20.5 61t43.5 63t20 51q0 48 -33 48q-30 0 -47 11&#xA;q-81 -166 -177 -242.5t-219 -76.5q-77 0 -125 36t-48 87q0 31 17 51t42 20q22 0 36.5 -13.5t14.5 -34.5t-11.5 -34.5t-31.5 -13.5q-23 0 -30 12h-2q0 -28 35 -54q34 -25 85 -25q94 0 152 52q60 53 145 203q75 133 131 214q54 75 121 124q-76 48 -192 48q-150 0 -250 -75&#xA;q-101 -75 -101 -174q0 -41 30 -68.5t73 -27.5q158 0 205 245h20q-8 -107 -69 -191q-62 -84 -158 -84q-63 0 -102.5 35.5t-39.5 94.5q0 107 111 191t282 84q124 0 221 -59q38 24 84 41zM814 486q0 64 -48 102q-7 -6 -14 -14t-14.5 -18l-10.5 -14q-4 -5 -35 -55l-6 -11&#xA;q-6 -11 -10 -20t-10 -20l-6 -11l-17 -36q-4 -7 -7.5 -14.5l-6 -13t-3.5 -7.5q34 0 60 -17q57 6 92 50q36 43 36 99z" /></g> <g transform="matrix(.012,-0,0,-.012,16.112,15.825)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105&#xA;q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g> </svg> (minimum infection parameter) for each model. We provide unconditionally stable method, using the Caputo fractional derivative of order <svg style="vertical-align:-0.1254pt;width:9.7749996px;" id="M2" height="7.9499998" version="1.1" viewBox="0 0 9.7749996 7.9499998" width="9.7749996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D6FC" d="M545 106q-67 -118 -134 -118q-24 0 -40 37.5t-30 129.5h-2q-47 -72 -103 -119.5t-108 -47.5q-47 0 -76 45.5t-29 119.5q0 113 85 204t174 91q47 0 70 -33.5t43 -119.5h3q32 47 80 140l55 13l10 -9q-47 -80 -138 -201q17 -99 27.5 -136t22.5 -37q23 0 69 61zM333 204&#xA;q-14 98 -31 149.5t-50 51.5q-49 0 -94 -70t-45 -164q0 -55 15.5 -86t40.5 -31q70 0 164 150z" /></g> </svg> and implicit Euler&#x2019;s approximation, to find a numerical solution of the resulting systems. The numerical simulations confirm the advantages of the numerical technique and using fractional-order differential models in biological systems over the differential equations with integer order. The results may give insight to infectious disease specialists.

Optimal response to chemotherapy for a mathematical model of tumor–immune dynamics

An optimal control problem for cancer chemotherapy is considered that includes immunological activity. In the objective a weighted average of several quantities that describe the effectiveness of treatment is minimized. These terms include (i) the number of cancer cells at the terminal time, (ii) a measure for the immunocompetent cell densities at the terminal point (included as a negative term), (iii) the overall amount of cytotoxic agents given as a measure for the side effects of treatment and (iv) a small penalty on the terminal time that limits the overall therapy horizon which is assumed to be free. This last term is essential in obtaining a well-posed problem formulation. Employing a Gompertzian growth model for the cancer cells, for various scenarios optimal controls and corresponding responses of the system are calculated. Solutions initially follow a full dose treatment, but then at one point switch to a singular regimen that only applies partial dosages. This structure is consistent with protocols that apply an initial burst to reduce the tumor volume and then maintain a small volume through lower dosages. Optimal controls end with either a prolonged period of no dose treatment or, in a small number of scenarios, this no dose interval is still followed by one more short burst of full dose treatment.

A Validated Mathematical Model of Cell-Mediated Immune Response to Tumor Growth

Mathematical models of tumor-immune interactions provide an analytic framework in which to address specific questions about tumor-immune dynamics. We present a new mathematical model that describes tumor-immune interactions, focusing on the role of natural killer (NK) and CD8+ T cells in tumor surveillance, with the goal of understanding the dynamics of immune-mediated tumor rejection. The model describes tumor-immune cell interactions using a system of differential equations. The functions describing tumor-immune growth, response, and interaction rates, as well as associated variables, are developed using a least-squares method combined with a numerical differential equations solver. Parameter estimates and model validations use data from published mouse and human studies. Specifically, CD8+ T-tumor and NK-tumor lysis data from chromium release assays as well as in vivo tumor growth data are used. A variable sensitivity analysis is done on the model. The new functional forms developed show that there is a clear distinction between the dynamics of NK and CD8+ T cells. Simulations of tumor growth using different levels of immune stimulating ligands, effector cells, and tumor challenge are able to reproduce data from the published studies. A sensitivity analysis reveals that the variable to which the model is most sensitive is patient specific, and can be measured with a chromium release assay. The variable sensitivity analysis suggests that the model can predict which patients may positively respond to treatment. Computer simulations highlight the importance of CD8+ T-cell activation in cancer therapy.

A model of tumor and lymphocyte interactions

The interactions between a solid tumor and the immune systemare described both prior to and after neovascularizationby a predator-prey model, and predictions about tumor behaviorin a host are made. Trajectory analysis of phase-planeportraits as well as standard perturbation analysis showthat most system steady states are unstable but thatstability is theoretically possible. Reasonable parametervalue estimation enables meaningful analysis of systembehavior, and Mathematica is used to simulate modeldynamics. The model accounts for many observed tumorbehaviors, and regions of uncontrolled tumor growth,tumor extinction in finite time, and irreversible lymphocytedecline are found either analytically or numerically. Abetter understanding of tumor-immune dynamics is obtained,allowing for improved research on treatment specifically inthe area of immunotherapy.