Metamodeling tumor–immune system interaction, tumor evasion and immunotherapy

Bifurcations in coupled amyloid-β aggregation-inflammation systems

A complex interplay between various processes underlies the neuropathology of Alzheimer's disease (AD) and its progressive course. Several lines of evidence point to the coupling between Aβ aggregation and neuroinflammation and its role in maintaining brain homeostasis during the long prodromal phase of AD. Little is however known about how this protective mechanism fails and as a result, an irreversible and progressive transition to clinical AD occurs. Here, we introduce a minimal model of a coupled system of Aβ aggregation and inflammation, numerically simulate its dynamical behavior, and analyze its bifurcation properties. The introduced model represents the following events: generation of Aβ monomers, aggregation of Aβ monomers into oligomers and fibrils, induction of inflammation by Aβ aggregates, and clearance of various Aβ species. Crucially, the rates of Aβ generation and clearance are modulated by inflammation level following a Hill-type response function. Despite its relative simplicity, the model exhibits enormously rich dynamics ranging from overdamped kinetics to sustained oscillations. We then specify the region of inflammation- and coupling-related parameters space where a transition to oscillatory dynamics occurs and demonstrate how changes in Aβ aggregation parameters could shift this oscillatory region in parameter space. Our results reveal the propensity of coupled Aβ aggregation-inflammation systems to oscillatory dynamics and propose prolonged sustained oscillations and their consequent immune system exhaustion as a potential mechanism underlying the transition to a more progressive phase of amyloid pathology in AD. The implications of our results in regard to early diagnosis of AD and anti-AD drug development are discussed.

Intelligent solution predictive networks for non-linear tumor-immune delayed model

In this article, we analyze the dynamics of the non-linear tumor-immune delayed (TID) model illustrating the interaction among tumor cells and the immune system (cytotoxic T lymphocytes, T helper cells), where the delays portray the times required for molecule formation, cell growth, segregation, and transportation, among other factors by exploiting the knacks of soft computing paradigm utilizing neural networks with back propagation Levenberg Marquardt approach (NNLMA). The governing differential delayed system of non-linear TID, which comprised the densities of the tumor population, cytotoxic T lymphocytes and T helper cells, is represented by non-linear delay ordinary differential equations with three classes. The baseline data is formulated by exploiting the explicit Runge-Kutta method (RKM) by diverting the transmutation rate of Tc to Th of the Tc population, transmutation rate of Tc to Th of the Th population, eradication of tumor cells through Tc cells, eradication of tumor cells through Th cells, Tc cells' natural mortality rate, Th cells' natural mortality rate as well as time delay. The approximated solution of the non-linear TID model is determined by randomly subdividing the formulated data samples for training, testing, as well as validation sets in the network formulation and learning procedures. The strength, reliability, and efficacy of the designed NNLMA for solving non-linear TID model are endorsed by small/negligible absolute errors, error histogram studies, mean squared errors based convergence and close to optimal modeling index for regression measurements.

Dynamics aspects and bifurcations of a tumor-immune system interaction under stationary immunotherapy

We consider a three-dimensional mathematical model that describes the interaction between the effector cells, tumor cells, and the cytokine (IL-2) of a patient. This is called the Kirschner-Panetta model. Our objective is to explain the tumor oscillations in tumor sizes as well as long-term tumor relapse. We then explore the effects of adoptive cellular immunotherapy on the model and describe under what circumstances the tumor can be eliminated or can remain over time but in a controlled manner. Nonlinear dynamics of immunogenic tumors are given, for example: we prove that the trajectories of the associated system are bounded and defined for all positive time; there are some invariant subsets; there are open subsets of parameters, such that the system in the first octant has at most five equilibrium solutions, one of them is tumor-free and the others are of co-existence. We are able to prove the existence of transcritical and pitchfork bifurcations from the tumor-free equilibrium point. Fixing an equilibrium and introducing a small perturbation, we are able to show the existence of a Hopf periodic orbit, showing a cyclic behavior among the population, with a strong dominance of the parental anomalous growth cell population. The previous information reveals the effects of the parameters. In our study, we observe that our mathematical model exhibits a very rich dynamic behavior and the parameter μ̃ (death rate of the effector cells) and p̃1 (production rate of the effector cell stimulated by the cytokine IL-2) plays an important role. More precisely, in our approach the inequality μ̃2>p̃1 is very important, that is, the death rate of the effector cells is greater than the production rate of the effector cell stimulated by the cytokine IL-2. Finally, medical implications and a set of numerical simulations supporting the mathematical results are also presented.

Optimal dosage protocols for mathematical models of synergy of chemo- and immunotherapy

The release of tumor antigens during traditional cancer treatments such as radio- or chemotherapy leads to a stimulation of the immune response which provides synergistic effects these treatments have when combined with immunotherapies. A low-dimensional mathematical model is formulated which, depending on the values of its parameters, encompasses the 3 E's (elimination, equilibrium, escape) of tumor immune system interactions. For the escape situation, optimal control problems are formulated which aim to revert the process to the equilibrium scenario. Some numerical results are included.

Modeling interaction of Glioma cells and CAR T-cells considering multiple CAR T-cells bindings

Chimeric antigen receptor (CAR) T-cell based immunotherapy has shown its potential in treating blood cancers, and its application to solid tumors is currently being extensively investigated. For glioma brain tumors, various CAR T-cell targets include IL13Rα2, EGFRvIII, HER2, EphA2, GD2, B7-H3, and chlorotoxin. In this work, we are interested in developing a mathematical model of IL13Rα2 targeting CAR T-cells for treating glioma. We focus on extending the work of Kuznetsov et al. (1994) by considering binding of multiple CAR T-cells to a single glioma cell, and the dynamics of these multi-cellular conjugates. Our model more accurately describes experimentally observed CAR T-cell killing assay data than the models which do not consider multi-cellular conjugates. Moreover, we derive conditions in the CAR T-cell expansion rate that determines treatment success or failure. Finally, we show that our model captures distinct CAR T-cell killing dynamics from low to high antigen receptor densities in patient-derived brain tumor cells.

The Dynamics of a Fractional-Order Mathematical Model of Cancer Tumor Disease

This article explores the application of the reduced differential transform method (RDTM) for the computational solutions of two fractional-order cancer tumor models in the Caputo sense: the model based on cancer chemotherapeutic effects which explain the relation between chemotherapeutic drugs, tumor cells, normal cells, and immune cells using a fractional partial differential equations, and the model that describes the different cases of killing rate K of cancer cells (the killing percentage of cancer cells K (I) is dependent on the number of cells, (II) is a function of time only, and (III) is a function of space only). The solutions are presented using Mathematica software as a convergent power series with elegantly computed terms using the suggested technique. The proposed method gives new series form results for various values of gamma. To clarify the complexity of the models, we plot the two- and three-dimensional and contour graphics of the obtained solutions at varied values of fractional-order gamma and the selected system parameters. The solutions are analyzed with fractional and reduced differential transform methods to obtain an idea of invariance regarding the computed solution of the designed mathematical model. The obtained results demonstrate the efficiency and preciseness of the proposed method to achieve a better understanding of chemotherapy effects. It is observed that chemotherapy drugs boost immunity against the specific cancer by decreasing the number of tumor cells, and the killing rate K of cancerous cells depend on the cells concentration.

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.

Integrative lymph node-mimicking models created with biomaterials and computational tools to study the immune system

The lymph node (LN) is a vital organ of the lymphatic and immune system that enables timely detection, response, and clearance of harmful substances from the body. Each LN comprises of distinct substructures, which host a plethora of immune cell types working in tandem to coordinate complex innate and adaptive immune responses. An improved understanding of LN biology could facilitate treatment in LN-associated pathologies and immunotherapeutic interventions, yet at present, animal models, which often have poor physiological relevance, are the most popular experimental platforms. Emerging biomaterial engineering offers powerful alternatives, with the potential to circumvent limitations of animal models, for in-depth characterization and engineering of the lymphatic and adaptive immune system. In addition, mathematical and computational approaches, particularly in the current age of big data research, are reliable tools to verify and complement biomaterial works. In this review, we first discuss the importance of lymph node in immunity protection followed by recent advances using biomaterials to create in vitro/vivo LN-mimicking models to recreate the lymphoid tissue microstructure and microenvironment, as well as to describe the related immuno-functionality for biological investigation. We also explore the great potential of mathematical and computational models to serve as in silico supports. Furthermore, we suggest how both in vitro/vivo and in silico approaches can be integrated to strengthen basic patho-biological research, translational drug screening and clinical personalized therapies. We hope that this review will promote synergistic collaborations to accelerate progress of LN-mimicking systems to enhance understanding of immuno-complexity.

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.

Metaheuristics and Pontryagin's minimum principle for optimal therapeutic protocols in cancer immunotherapy: a case study and methods comparison

In this paper, the performance appropriateness of population-based metaheuristics for immunotherapy protocols is investigated on a comparative basis while the goal is to stimulate the immune system to defend against cancer. For this purpose, genetic algorithm and particle swarm optimization are employed and compared with modern method of Pontryagin's minimum principle (PMP). To this end, a well-known mathematical model of cell-based cancer immunotherapy is described and examined to formulate the optimal control problem in which the objective is the annihilation of tumour cells by using the minimum amount of cultured immune cells. In this regard, the main aims are: (i) to introduce a single-objective optimization problem and to design the considered metaheuristics in order to appropriately deal with it; (ii) to use the PMP in order to obtain the necessary conditions for optimality, i.e. the governing boundary value problem; (iii) to measure the results obtained by using the proposed metaheuristics against those results obtained by using an indirect approach called forward-backward sweep method; and finally (iv) to produce a set of optimal treatment strategies by formulating the problem in a bi-objective form and demonstrating its advantages over single-objective optimization problem. A set of obtained results conforms the performance capabilities of the considered metaheuristics.

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

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

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

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Bistability could be harnessed to devise effective combination immunotherapy approaches.

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

Mathematical Modeling Reveals That the Administration of EGF Can Promote the Elimination of Lymph Node Metastases by PD-1/PD-L1 Blockade

In the advanced stages of cancers like melanoma, some of the malignant cells leave the primary tumor and infiltrate the neighboring lymph nodes (LNs). The interaction between secondary cancer and the immune response in the lymph node represents a complex process that needs to be fully understood in order to develop more effective immunotherapeutic strategies. In this process, antigen-presenting cells (APCs) approach the tumor and initiate the adaptive immune response for the corresponding antigen. They stimulate the naive CD4+ and CD8+ T lymphocytes which subsequently generate a population of helper and effector cells. On one hand, immune cells can eliminate tumor cells using cell-cell contact and by secreting apoptosis inducing cytokines. They are also able to induce their dormancy. On the other hand, the tumor cells are able to escape the immune surveillance using their immunosuppressive abilities. To study the interplay between tumor progression and the immune response, we develop two new models describing the interaction between cancer and immune cells in the lymph node. The first model consists of partial differential equations (PDEs) describing the populations of the different types of cells. The second one is a hybrid discrete-continuous model integrating the mechanical and biochemical mechanisms that define the tumor-immune interplay in the lymph node. We use the continuous model to determine the conditions of the regimes of tumor-immune interaction in the lymph node. While we use the hybrid model to elucidate the mechanisms that contribute to the development of each regime at the cellular and tissue levels. We study the dynamics of tumor growth in the absence of immune cells. Then, we consider the immune response and we quantify the effects of immunosuppression and local EGF concentration on the fate of the tumor. Numerical simulations of the two models show the existence of three possible outcomes of the tumor-immune interactions in the lymph node that coincide with the main phases of the immunoediting process: tumor elimination, equilibrium, and tumor evasion. Both models predict that the administration of EGF can promote the elimination of the secondary tumor by PD-1/PD-L1 blockade.

Integrated cancer tissue engineering models for precision medicine

Tumors are not merely cancerous cells that undergo mindless proliferation. Rather, they are highly organized and interconnected organ systems. Tumor cells reside in complex microenvironments in which they are subjected to a variety of physical and chemical stimuli that influence cell behavior and ultimately the progression and maintenance of the tumor. As cancer bioengineers, it is our responsibility to create physiologic models that enable accurate understanding of the multi-dimensional structure, organization, and complex relationships in diverse tumor microenvironments. Such models can greatly expedite clinical discovery and translation by closely replicating the physiological conditions while maintaining high tunability and control of extrinsic factors. In this review, we discuss the current models that target key aspects of the tumor microenvironment and their role in cancer progression. In order to address sources of experimental variation and model limitations, we also make recommendations for methods to improve overall physiologic reproducibility, experimental repeatability, and rigor within the field. Improvements can be made through an enhanced emphasis on mathematical modeling, standardized in vitro model characterization, transparent reporting of methodologies, and designing experiments with physiological metrics. Taken together these considerations will enhance the relevance of in vitro tumor models, biological understanding, and accelerate treatment exploration ultimately leading to improved clinical outcomes. Moreover, the development of robust, user-friendly models that integrate important stimuli will allow for the in-depth study of tumors as they undergo progression from non-transformed primary cells to metastatic disease and facilitate translation to a wide variety of biological and clinical studies.

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.

Network science in clinical trials: A patient-centered approach

There has been a paradigm shift in translational oncology with the advent of novel molecular diagnostic tools in the clinic. However, several challenges are associated with the integration of these sophisticated tools into clinical oncology and daily practice. High-throughput profiling at the DNA, RNA and protein levels (omics) generate a massive amount of data. The analysis and interpretation of these is non-trivial but will allow a more thorough understanding of cancer. Linear modelling of the data as it is often used today is likely to limit our understanding of cancer as a complex disease, and at times under-performs to capture a phenotype of interest. Network science and systems biology-based approaches, using machine learning and network science principles, that integrate multiple data sources, can uncover complex changes in a biological system. This approach will integrate a large number of potential biomarkers in preclinical studies to better inform therapeutic decisions and ultimately make substantial progress towards precision medicine. It will however require development of a new generation of clinical trials. Beyond discussing the challenges of high-throughput technologies, this review will develop a framework on how to implement a network science approach in new clinical trial designs in order to advance cancer care.

Dual role of delay effects in a tumour-immune system

In this paper, a previous tumour-immune interaction model is simplified by neglecting a relatively weak direct immune activation by the tumour cells, which can still keep the essential dynamics properties of the original model. As the immune activation process is not instantaneous, we now incorporate one delay for the activation of the effector cells (ECs) by helper T cells (HTCs) into the model. Furthermore, we investigate the stability and instability regions of the tumour-presence equilibrium state of the delay-induced system with respect to two parameters, the activation rate of ECs by HTCs and the HTCs stimulation rate by the presence of identified tumour antigens. We show the dual role of this delay that can induce stability switches exhibiting destabilization as well as stabilization of the tumour-presence equilibrium. Besides, our results reveal that an appropriate immune activation time delay plays a significant role in control of tumour growth.

The mathematics of cancer: integrating quantitative models

Mathematical modelling approaches have become increasingly abundant in cancer research. The complexity of cancer is well suited to quantitative approaches as it provides challenges and opportunities for new developments. In turn, mathematical modelling contributes to cancer research by helping to elucidate mechanisms and by providing quantitative predictions that can be validated. The recent expansion of quantitative models addresses many questions regarding tumour initiation, progression and metastases as well as intra-tumour heterogeneity, treatment responses and resistance. Mathematical models can complement experimental and clinical studies, but also challenge current paradigms, redefine our understanding of mechanisms driving tumorigenesis and shape future research in cancer biology.

A study of tumour growth based on stoichiometric principles: a continuous model and its discrete analogue

In this paper, we consider a continuous mathematically tractable model and its discrete analogue for the tumour growth. The model formulation is based on stoichiometric principles considering tumour-immune cell interactions in potassium (K (+))-limited environment. Our both continuous and discrete models illustrate 'cancer immunoediting' as a dynamic process having all three phases namely elimination, equilibrium and escape. The stoichiometric principles introduced into the model allow us to study its dynamics with the variation in the total potassium in the surrounding of the tumour region. It is found that an increase in the total potassium may help the patient fight the disease for a longer period of time. This result seems to be in line with the protective role of the potassium against the risk of pancreatic cancer as has been reported by Bravi et al. [Dietary intake of selected micronutrients and risk of pancreatic cancer: An Italian case-control study, Ann. Oncol. 22 (2011), pp. 202-206].

Evasion of tumours from the control of the immune system: consequences of brief encounters

BACKGROUND

In this work a mathematical model describing the growth of a solid tumour in the presence of an immune system response is presented. Specifically, attention is focused on the interactions between cytotoxic T-lymphocytes (CTLs) and tumour cells in a small, avascular multicellular tumour. At this stage of the disease the CTLs and the tumour cells are considered to be in a state of dynamic equilibrium or cancer dormancy. The precise biochemical and cellular mechanisms by which CTLs can control a cancer and keep it in a dormant state are still not completely understood from a biological and immunological point of view. The mathematical model focuses on the spatio-temporal dynamics of tumour cells, immune cells, chemokines and "chemorepellents" in an immunogenic tumour. The CTLs and tumour cells are assumed to migrate and interact with each other in such a way that lymphocyte-tumour cell complexes are formed. These complexes result in either the death of the tumour cells (the normal situation) or the inactivation of the lymphocytes and consequently the survival of the tumour cells. In the latter case, we assume that each tumour cell that survives its "brief encounter" with the CTLs undergoes certain beneficial phenotypic changes.

RESULTS

We explore the dynamics of the model under these assumptions and show that the process of immuno-evasion can arise as a consequence of these encounters. We show that the proposed mechanism not only shape the dynamics of the total number of tumor cells and of CTLs, but also the dynamics of their spatial distribution. We also briefly discuss the evolutionary features of our model, by framing them in the recent quasi-Lamarckian theories.

CONCLUSIONS

Our findings might have some interesting implication of interest for clinical practice. Indeed, immuno-editing process can be seen as an "involuntary" antagonistic process acting against immunotherapies, which aim at maintaining a tumor in a dormant state, or at suppressing it.

Separable transition density in the hybrid model for tumor-immune system competition

A hybrid model, on the competition tumor cells immune system, is studied under suitable hypotheses. The explicit form for the equations is obtained in the case where the density function of transition is expressed as the product of separable functions. A concrete application is given starting from a modified Lotka-Volterra system of equations.

A STUDY OF DIFFERENTIAL EQUATION MODELING MALIGNANT TUMOR CELLS IN COMPETITION WITH IMMUNE SYSTEM

In this paper, we present a competition model of malignant tumor growth that includes the immune system response. The model considers two populations: immune system (effector cells) and population of tumor (tumor cells). Ordinary differential equations are used to model the system to take into account the delay of the immune response. The existence of positive solutions of the model (with/without delay) is showed. We analyze the stability of the possible steady states with respect to time delay and the existence of positive solutions of the model (with and without delay). We show theoretically and through numerical simulations that periodic oscillations may arise through Hopf bifurcation. An algorithm for determining the stability of bifurcating periodic solutions is proved.

Simple biophysical model of tumor evasion from immune system control

The competitive nonlinear interplay between a tumor and the host's immune system is not only very complex but is also time-changing. A fundamental aspect of this issue is the ability of the tumor to slowly carry out processes that gradually allow it to become less harmed and less susceptible to recognition by the immune system effectors. Here we propose a simple epigenetic escape mechanism that adaptively depends on the interactions per time unit between cells of the two systems. From a biological point of view, our model is based on the concept that a tumor cell that has survived an encounter with a cytotoxic T-lymphocyte (CTL) has an information gain that it transmits to the other cells of the neoplasm. The consequence of this information increase is a decrease in both the probabilities of being killed and of being recognized by a CTL. We show that the mathematical model of this mechanism is formally equal to an evolutionary imitation game dynamics. Numerical simulations of transitory phases complement the theoretical analysis. Implications of the interplay between the above mechanisms and the delivery of immunotherapies are also illustrated.

Numerical solutions for a model of tissue invasion and migration of tumour cells

The goal of this paper is to construct a new algorithm for the numerical simulations of the evolution of tumour invasion and metastasis. By means of mathematical model equations and their numerical solutions we investigate how cancer cells can produce and secrete matrix degradative enzymes, degrade extracellular matrix, and invade due to diffusion and haptotactic migration. For the numerical simulations of the interactions between the tumour cells and the surrounding tissue, we apply numerical approximations, which are spectrally accurate and based on small amounts of grid-points. Our numerical experiments illustrate the metastatic ability of tumour cells.

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

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

Bounded-noise-induced transitions in a tumor-immune system interplay

By studying a recent biophysical model of tumor growth in the presence of the immune system, here we propose that the phenomenon of evasion of tumors from immune control at a temporal mesoscale might, in some cases, be due to random fluctuations in the levels of the immune system. Bounded noises are considered, but the Gaussian approach is also used for analytical reference. After showing that in the case of bounded noises there may be multiple attractors in the space of probability densities, we numerically show that the velocity of convergence toward asymptotic density is very slow and that a transitory analysis is needed. Then, by simulations using the sine-Wiener and the Tsallis noises, we show that if the level of the noise is sufficiently large then there may be the onset of noise-induced transitions in the transitory density evaluated at realistic times. Namely, the transitions are from unimodal density centered at low values of tumor burden to bimodal densities that have a second maximum centered at higher values. However, those transitions depend on the distribution of the noise.