Modelling and mathematical problems related to tumor evolution and its interaction with the immune system

Modeling different strategies towards control of lung cancer: leveraging early detection and anti-cancer cell measures

The global population has encountered significant challenges throughout history due to infectious diseases. To comprehensively study these dynamics, a novel deterministic mathematical model, TCD I L 2 Z, is developed for the early detection and treatment of lung cancer. This model incorporates I L 2 cytokine and anti-PD-L1 inhibitors, enhancing the immune system's anticancer response within five epidemiological compartments. The TCD I L 2 Z model is analyzed qualitatively and quantitatively, emphasizing local stability given the limited data-a critical component of epidemic modeling. The model is systematically validated by examining essential elements such as equilibrium points, the reproduction number (R 0 ), stability, and sensitivity analysis. Next-generation techniques based on R 0 that track disease transmission rates across the sub-compartments are fed into the system. At the same time, sensitivity analysis helps model how a particular parameter affects the dynamics of the system. The stability on the global level of such therapy agents retrogrades individuals with immunosuppression or treated with I L 2 and anti-PD-L1 inhibitors admiring the Lyapunov functions' applications. NSFD scheme based on the implicit method is used to find the exact value and is compared with Euler's method and RK4, which guarantees accuracy. Thus, the simulations were conducted in the MATLAB environment. These simulations present the general symptomatic and asymptomatic consequences of lung cancer globally when detected in the middle and early stages, and measures of anticancer cells are implemented including boosting the immune system for low immune individuals. In addition, such a result provides knowledge about real-world control dynamics with I L 2 and anti-PD-L1 inhibitors. The studies will contribute to the understanding of disease spread patterns and will provide the basis for evidence-based intervention development that will be geared toward actual outcomes.

Fractional order cancer model infection in human with CD8+ T cells and anti-PD-L1 therapy: simulations and control strategy

In order to comprehend the dynamics of disease propagation within a society, mathematical formulations are essential. The purpose of this work is to investigate the diagnosis and treatment of lung cancer in persons with weakened immune systems by introducing cytokines ( I L 2 & I L 12 ) and anti-PD-L1 inhibitors. To find the stable position of a recently built system TCD I L 2 I L 12 Z, a qualitative and quantitative analysis are taken under sensitive parameters. Reliable bounded findings are ensured by examining the generated system's boundedness, positivity, uniqueness, and local stability analysis, which are the crucial characteristics of epidemic models. The positive solutions with linear growth are shown to be verified by the global derivative, and the rate of impact across every sub-compartment is determined using Lipschitz criteria. Using Lyapunov functions with first derivative, the system's global stability is examined in order to evaluate the combined effects of cytokines and anti-PD-L1 inhibitors on people with weakened immune systems. Reliability is achieved by employing the Mittag-Leffler kernel in conjunction with a fractal-fractional operator because FFO provide continuous monitoring of lung cancer in multidimensional way. The symptomatic and asymptomatic effects of lung cancer sickness are investigated using simulations in order to validate the relationship between anti-PD-L1 inhibitors, cytokines, and the immune system. Also, identify the actual state of lung cancer control with early diagnosis and therapy by introducing cytokines and anti-PD-L1 inhibitors, which aid in the patients' production of anti-cancer cells. Investigating the transmission of illness and creating control methods based on our validated results will both benefit from this kind of research.

Selected aspects of avascular tumor growth reproduced by a hybrid model of cell dynamics and chemical kinetics

We here propose a hybrid computational framework to reproduce and analyze aspects of the avascular progression of a generic solid tumor. Our method first employs an individual-based approach to represent the population of tumor cells, which are distinguished in viable and necrotic agents. The active part of the disease is in turn differentiated according to a set of metabolic states. We then describe the spatio-temporal evolution of the concentration of oxygen and of tumor-secreted proteolytic enzymes using partial differential equations (PDEs). A differential equation finally governs the local degradation of the extracellular matrix (ECM) by the malignant mass. Numerical realizations of the model are run to reproduce tumor growth and invasion in a number scenarios that differ for cell properties (adhesiveness, duplication potential, proteolytic activity) and/or environmental conditions (level of tissue oxygenation and matrix density pattern). In particular, our simulations suggest that tumor aggressiveness, in terms of invasive depth and extension of necrotic tissue, can be reduced by (i) stable cell-cell contact interactions, (ii) poor tendency of malignant agents to chemotactically move upon oxygen gradients, and (iii) presence of an overdense matrix, if coupled by a disrupted proteolytic activity of the disease.

Mathematical modeling and control of lung cancer with IL2 cytokine and anti-PD-L1 inhibitor effects for low immune individuals

Mathematical formulations are crucial in understanding the dynamics of disease spread within a community. The aim of this work is to examine that the Lung Cancer detection and treatment by introducing IL2 and anti-PD-L1 inhibitor for low immune individuals. Mathematical model is developed with the created hypothesis to increase immune system by antibody cell's and Fractal-Fractional operator (FFO) is used to turn the model into a fractional order model. A newly developed system TCDIL2Z is examined both qualitatively and quantitatively in order to determine its stable position. The boundedness, positivity and uniqueness of the developed system are examined to ensure reliable bounded findings, which are essential properties of epidemic models. The global derivative is demonstrated to verify the positivity with linear growth and Lipschitz conditions are employed to identify the rate of effects in each sub-compartment. The system is investigated for global stability using Lyapunov first derivative functions to assess the overall impact of IL2 and anti-PD-L1 inhibitor for low immune individuals. Fractal fractional operator is used to derive reliable solution using Mittag-Leffler kernel. In fractal-fractional operators, fractal represents the dimensions of the spread of the disease and fractional represents the fractional ordered derivative operator. We use combine operators to see real behavior of spread as well as control of lung cancer with different dimensions and continuous monitoring. Simulations are conducted to observe the symptomatic and asymptomatic effects of Lung Cancer disease to verify the relationship of IL2, anti-PD-L1 inhibitor and immune system. Also identify the real situation of the control for lung cancer disease after detection and treatment by introducing IL2 cytokine and anti-PD-L1 inhibitor which helps to generate anti-cancer cells of the patients. Such type of investigation will be useful to investigate the spread of disease as well as helpful in developing control strategies from our justified outcomes.

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 tumor growth using fractal calculus: Insights into tumor dynamics

Important concepts like fractal calculus and fractal analysis, the sum of squared residuals, and Aikaike's information criterion must be thoroughly understood in order to correctly fit cancer-related data using the proposed models. The fractal growth models employed in this work are classified in three main categories: Sigmoidal growth models (Logistic, Gompertz, and Richards models), Power Law growth model, and Exponential growth models (Exponential and Exponential-Lineal models)". We fitted the data, computed the sum of squared residuals, and determined Aikaike's information criteria using Matlab and the web tool WebPlotDigitizer. In addition, the research investigates "double-size cancer" in the fractal temporal dimension with respect to various mathematical models.

Tumor Evolution Models of Phase-Field Type with Nonlocal Effects and Angiogenesis

In this survey article, a variety of systems modeling tumor growth are discussed. In accordance with the hallmarks of cancer, the described models incorporate the primary characteristics of cancer evolution. Specifically, we focus on diffusive interface models and follow the phase-field approach that describes the tumor as a collection of cells. Such systems are based on a multiphase approach that employs constitutive laws and balance laws for individual constituents. In mathematical oncology, numerous biological phenomena are involved, including temporal and spatial nonlocal effects, complex nonlinearities, stochasticity, and mixed-dimensional couplings. Using the models, for instance, we can express angiogenesis and cell-to-matrix adhesion effects. Finally, we offer some methods for numerically approximating the models and show simulations of the tumor's evolution in response to various biological effects.

Computational systems biology approach for permanent tumor elimination and normal tissue protection using negative biasing: Experimental validation in malignant melanoma as case study

Complete spontaneous tumor regression (without treatment) is well documented to occur in animals and humans as epidemiological analysis show, whereby the malignancy is permanently eliminated. We have developed a novel computational systems biology model for this unique phenomenon to furnish insight into the possibility of therapeutically replicating such regression processes on tumors clinically, without toxic side effects. We have formulated oncological informatics approach using cell-kinetics coupled differential equations while protecting normal tissue. We investigated three main tumor-lysis components: (ⅰ) DNA blockade factors, (ⅱ) Interleukin-2 (IL-2), and (ⅲ) Cytotoxic T-cells (CD8⁺ T). We studied the temporal variations of these factors, utilizing preclinical experimental investigations on malignant tumors, using mammalian melanoma microarray and histiocytoma immunochemical assessment. We found that permanent tumor regression can occur by: 1) Negative-Bias shift in population trajectory of tumor cells, eradicating them under first-order asymptotic kinetics, and 2) Temporal alteration in the three antitumor components (DNA replication-blockade, Antitumor T-lymphocyte, IL-2), which are respectively characterized by the following patterns: (a) Unimodal Inverted-U function, (b) Bimodal M-function, (c) Stationary-step function. These provide a time-wise orchestrated tri-phasic cytotoxic profile. We have also elucidated gene-expression levels corresponding to the above three components: (ⅰ) DNA-damage G2/M checkpoint regulation [genes: CDC2-CHEK], (ⅱ) Chemokine signaling: IL-2/15 [genes: IL2RG-IKT3], (ⅲ) T-lymphocyte signaling (genes: TRGV5-CD28). All three components quantitatively followed the same activation profiles predicted by our computational model (Smirnov-Kolmogorov statistical test satisfied, α = 5%). We have shown that the genes CASP7-GZMB are signatures of Negative-bias dynamics, enabling eradication of the residual tumor. Using the negative-biasing principle, we have furnished the dose-time profile of equivalent therapeutic agents (DNA-alkylator, IL-2, T-cell input) so that melanoma tumor may therapeutically undergo permanent extinction by replicating the spontaneous tumor regression dynamics.

Can the Kuznetsov Model Replicate and Predict Cancer Growth in Humans?

Several mathematical models to predict tumor growth over time have been developed in the last decades. A central aspect of such models is the interaction of tumor cells with immune effector cells. The Kuznetsov model (Kuznetsov et al. in Bull Math Biol 56(2):295–321, 1994) is the most prominent of these models and has been used as a basis for many other related models and theoretical studies. However, none of these models have been validated with large-scale real-world data of human patients treated with cancer immunotherapy. In addition, parameter estimation of these models remains a major bottleneck on the way to model-based and data-driven medical treatment. In this study, we quantitatively fit Kuznetsov’s model to a large dataset of 1472 patients, of which 210 patients have more than six data points, by estimating the model parameters of each patient individually. We also conduct a global practical identifiability analysis for the estimated parameters. We thus demonstrate that several combinations of parameter values could lead to accurate data fitting. This opens the potential for global parameter estimation of the model, in which the values of all or some parameters are fixed for all patients. Furthermore, by omitting the last two or three data points, we show that the model can be extrapolated and predict future tumor dynamics. This paves the way for a more clinically relevant application of mathematical tumor modeling, in which the treatment strategy could be adjusted in advance according to the model’s future predictions.

Dose-dependent thresholds of dexamethasone destabilize CAR T-cell treatment efficacy

Chimeric antigen receptor (CAR) T-cell therapy is potentially an effective targeted immunotherapy for glioblastoma, yet there is presently little known about the efficacy of CAR T-cell treatment when combined with the widely used anti-inflammatory and immunosuppressant glucocorticoid, dexamethasone. Here we present a mathematical model-based analysis of three patient-derived glioblastoma cell lines treated in vitro with CAR T-cells and dexamethasone. Advanced in vitro experimental cell killing assay technologies allow for highly resolved temporal dynamics of tumor cells treated with CAR T-cells and dexamethasone, making this a valuable model system for studying the rich dynamics of nonlinear biological processes with translational applications. We model the system as a nonautonomous, two-species predator-prey interaction of tumor cells and CAR T-cells, with explicit time-dependence in the clearance rate of dexamethasone. Using time as a bifurcation parameter, we show that (1) dexamethasone destabilizes coexistence equilibria between CAR T-cells and tumor cells in a dose-dependent manner and (2) as dexamethasone is cleared from the system, a stable coexistence equilibrium returns in the form of a Hopf bifurcation. With the model fit to experimental data, we demonstrate that high concentrations of dexamethasone antagonizes CAR T-cell efficacy by exhausting, or reducing the activity of CAR T-cells, and by promoting tumor cell growth. Finally, we identify a critical threshold in the ratio of CAR T-cell death to CAR T-cell proliferation rates that predicts eventual treatment success or failure that may be used to guide the dose and timing of CAR T-cell therapy in the presence of dexamethasone in patients.

Bioengineering and gene-editing technologies have paved the way for advance immunotherapies that can target patient-specific tumor cells. One of these therapies, chimeric antigen receptor (CAR) T-cell therapy has recently shown promise in treating glioblastoma, an aggressive brain cancer often with poor patient prognosis. Dexamethasone is a commonly prescribed anti-inflammatory medication due to the health complications of tumor associated swelling in the brain. However, the immunosuppressant effects of dexamethasone on the immunotherapeutic CAR T-cells are not well understood. To address this issue, we use mathematical modeling to study in vitro dynamics of dexamethasone and CAR T-cells in three patient-derived glioblastoma cell lines. We find that in each cell line studied there is a threshold of tolerable dexamethasone concentration. Below this threshold, CAR T-cells are successful at eliminating the cancer cells, while above this threshold, dexamethasone critically inhibits CAR T-cell efficacy. Our modeling suggests that in the presence of high dexamethasone reduced CAR T-cell efficacy, or increased exhaustion, can occur and result in CAR T-cell treatment failure.

A phenotype-structured model to reproduce the avascular growth of a tumor and its interaction with the surrounding environment

We here propose a one-dimensional spatially explicit phenotype-structured model to analyze selected aspects of avascular tumor progression. In particular, our approach distinguishes viable and necrotic cell fractions. The metabolically active part of the disease is, in turn, differentiated according to a continuous trait, that identifies cell variants with different degrees of motility and proliferation potential. A parabolic partial differential equation (PDE) then governs the spatio-temporal evolution of the phenotypic distribution of active cells within the host tissue. In this respect, active tumor agents are allowed to duplicate, move upon haptotactic and pressure stimuli, and eventually undergo necrosis. The mutual influence between the emerging malignancy and its environment (in terms of molecular landscape) is implemented by coupling the evolution law of the viable tumor mass with a parabolic PDE for oxygen kinetics and a differential equation that accounts for local consumption of extracellular matrix (ECM) elements. The resulting numerical realizations reproduce tumor growth and invasion in a number scenarios that differ for cell properties (i.e., individual migratory behavior, duplication and mutation potential) and environmental conditions (i.e., level of tissue oxygenation and homogeneity in the initial matrix profile). In particular, our simulations show that, in all cases, more mobile cell variants occupy the front edge of the tumor, whereas more proliferative clones are selected at the more internal regions. A necrotic core constantly occupies the bulk of the mass due to nutrient deprivation. This work may eventually suggest some biomedical strategies to partially reduce tumor aggressiveness, i.e., to enhance necrosis of malignant tissue and to promote the presence of more proliferative cell phenotypes over more invasive ones.

Developing and studying the dynamical behavior of a nonlinear mathematical model for cancers with tumor by considering immune system role

We are constrained by widespread cancerous diseases to improve treatment methods which save patients and provide better living conditions during and after the treatment period. Because of the complexity of the treatment process, mathematical models need to be used in order to have a better understanding of the process. However, deriving an adequate complex model that can capture the disease pattern which could be confirmed by simulations and experiments has its own barriers. In this paper, a new mathematical model is developed concerning immune system effect on cancer. The model is introduced using nonlinear ordinary differential equations. Also, the qualitative behavior of the proposed system is studied in order to examine the extent of the model with respect to the nature of tumor evolution. Thus, number and status of equilibria points in line with the existence of limit cycles are obtained for sub-systems and the whole system. Meanwhile, possible bifurcations are mentioned, and the consequent evolutions are described. It is shown that the model conforms well to natural possibilities, cancer growth or remission. Thus, the model would be fit for further studies for prediction and contemplating treatment method, especially for immune stimulating drugs and immunotherapy.

A NUMERICAL PROOF THAT CERTAIN CELLS FOLLOW THE TRAILS OF THE DIFFUSIONS OF SOME CHEMICALS IN THE EXTRACELLULAR MATRIX

In this work, we obtain the numerical solutions of a 2D mathematical model of tumor angiogenesis originally presented in [Pamuk S, ÇAY İ, Sazci A, A 2D mathematical model for tumor angiogenesis: The roles of certain cells in the extra cellular matrix, Math Biosci 306:32–48, 2018] to numerically prove that the certain cells, the endothelials (EC), pericytes (PC) and macrophages (MC) follow the trails of the diffusions of some chemicals in the extracellular matrix (ECM) which is, in fact, inhomogeneous. This leads to branching, the sprouting of a new neovessel from an existing vessel. Therefore, anastomosis occurs between these sprouts. In our figures we do see these branching and anastomosis, which show the fact that the cells diffuse according to the structure of the ECM. As a result, one sees that our results are in good agreement with the biological facts about the movements of certain cells in the Matrix.

Stability and Hopf bifurcations in a competitive tumour-immune system with intrinsic recruitment delay and chemotherapy

In this paper, a three-dimensional nonlinear delay differential system including Tumour cells, cytotoxic-T lymphocytes, T-helper cells is constructed to investigate the effects of intrinsic recruitment delay and chemotherapy. It is found that the introduction of chemotherapy and time delay can generate richer dynamics in tumor-immune system. In particular, there exists bistable phenomenon and the tumour cells would be cleared if the effect of chemotherapy on depletion of the tumour cells is strong enough or the side effect of chemotherapy on the hunting predator cells is under a threshold. It is also shown that a branch of stable periodic solutions bifurcates from the coexistence equilibrium when the intrinsic recruitment delay of tumor crosses the threshold which is new mechanism, which can help understand the short-term oscillations in tumour sizes as well as long-term tumour relapse. Numerical simulations are presented to illustrate that larger intrinsic recruitment delay of tumor leads to larger amplitude and longer period of the bifurcated periodic solution, which indicates that there exists longer relapse time and then contributes to the control of tumour growth.

Computational modelling of modern cancer immunotherapy

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

Data Driven Mathematical Model of Colon Cancer Progression

Every colon cancer has its own unique characteristics, and therefore may respond differently to identical treatments. Here, we develop a data driven mathematical model for the interaction network of key components of immune microenvironment in colon cancer. We estimate the relative abundance of each immune cell from gene expression profiles of tumors, and group patients based on their immune patterns. Then we compare the tumor sensitivity and progression in each of these groups of patients, and observe differences in the patterns of tumor growth between the groups. For instance, in tumors with a smaller density of naive macrophages than activated macrophages, a higher activation rate of macrophages leads to an increase in cancer cell density, demonstrating a negative effect of macrophages. Other tumors however, exhibit an opposite trend, showing a positive effect of macrophages in controlling tumor size. Although the results indicate that for all patients the size of the tumor is sensitive to the parameters related to macrophages, such as their activation and death rate, this research demonstrates that no single biomarker could predict the dynamics of tumors.

Towards Personalized Diagnosis of Glioblastoma in Fluid-Attenuated Inversion Recovery (FLAIR) by Topological Interpretable Machine Learning

Glioblastoma multiforme (GBM) is a fast-growing and highly invasive brain tumor, which tends to occur in adults between the ages of 45 and 70 and it accounts for 52 percent of all primary brain tumors. Usually, GBMs are detected by magnetic resonance images (MRI). Among MRI, a fluid-attenuated inversion recovery (FLAIR) sequence produces high quality digital tumor representation. Fast computer-aided detection and segmentation techniques are needed for overcoming subjective medical doctors (MDs) judgment. This study has three main novelties for demonstrating the role of topological features as new set of radiomics features which can be used as pillars of a personalized diagnostic systems of GBM analysis from FLAIR. For the first time topological data analysis is used for analyzing GBM from three complementary perspectives—tumor growth at cell level, temporal evolution of GBM in follow-up period and eventually GBM detection. The second novelty is represented by the definition of a new Shannon-like topological entropy, the so-called Generator Entropy. The third novelty is the combination of topological and textural features for training automatic interpretable machine learning. These novelties are demonstrated by three numerical experiments. Topological Data Analysis of a simplified 2D tumor growth mathematical model had allowed to understand the bio-chemical conditions that facilitate tumor growth—the higher the concentration of chemical nutrients the more virulent the process. Topological data analysis was used for evaluating GBM temporal progression on FLAIR recorded within 90 days following treatment completion and at progression. The experiment had confirmed that persistent entropy is a viable statistics for monitoring GBM evolution during the follow-up period. In the third experiment we developed a novel methodology based on topological and textural features and automatic interpretable machine learning for automatic GBM classification on FLAIR. The algorithm reached a classification accuracy up to 97%.

A kinetic model of T cell autoreactivity in autoimmune diseases

We construct a mathematical model of kinetic type in order to describe the immune system interactions in the context of autoimmune disease. The interacting populations are self-antigen presenting cells, self-reactive T cells and the set of immunosuppressive cells consisting of regulatory T cells and Natural Killer cells. The main aim of our work is to develop a qualitative analysis of the model equations and investigate the existence of biologically realistic solutions. Having this goal in mind we describe the interactions between cells during an autoimmune reaction based on biological considerations that are given in the literature and we show that the corresponding system of integro-differential equations has finite positive solutions. The asymptotic behaviour of the solution of the system is also studied. We complement our mathematical analysis with numerical simulations that study the sensitivity of the model to parameters related to proliferation of immunosuppressive cells, destruction of self-antigen presenting cells and self-reactive T cells and tolerance of SRTCs to self-antigens.

Asymptotic analysis of a mathematical model of the atherosclerosis development

A mathematical model, which takes into account new experimental results about diverse roles of macrophages in the atherosclerosis development, is proposed. Using technic of upper and lower solutions, the existence and uniqueness of its positive solution are justified. After the nondimensionalization, small parameters are found and the multiscale analysis of the corresponding perturbed problem is performed when those parameters tend to zero. In particular, the limit two-dimensional problem, which is a coupled system of reaction–diffusion equations and ordinary differential equations, is derived; the asymptotic approximation is constructed; the uniform pointwise estimate for the difference between the solution of the original problem and the solution of the limit problem as well as the respective [Formula: see text]-estimates for the fluxes are proved.

Analysis of tumour-immune evasion with chemo-immuno therapeutic treatment with quadratic optimal control

A simple mathematical model for the growth of tumour with discrete time delay in the immune system is considered. The dynamical behaviour of our system by analysing the existence and stability of our system at various equilibria is discussed elaborately. We set up an optimal control problem relative to the model so as to minimize the number of tumour cells and the chemo-immunotherapeutic drug administration. Sensitivity analysis of tumour model reveals that parameter value has a major impact on the model dynamics. We numerically illustrate how does these delay can change the stability region of the immune-control equilibrium and display the different impacts to the control of tumour. Finally, epidemiological implications of our analytical findings are addressed critically.

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.

Dynamic interplay between tumour, stroma and immune system can drive or prevent tumour progression

In the tumour microenvironment, cancer cells directly interact with both the immune system and the stroma. It is firmly established that the immune system, historically believed to be a major part of the body's defence against tumour progression, can be reprogrammed by tumour cells to be ineffective, inactivated, or even acquire tumour promoting phenotypes. Likewise, stromal cells and extracellular matrix can also have pro-and anti-tumour properties. However, there is strong evidence that the stroma and immune system also directly interact, therefore creating a tripartite interaction that exists between cancer cells, immune cells and tumour stroma. This interaction contributes to the maintenance of a chronically inflamed tumour microenvironment with pro-tumorigenic immune phenotypes and facilitated metastatic dissemination. A comprehensive understanding of cancer in the context of dynamical interactions of the immune system and the tumour stroma is therefore required to truly understand the progression toward and past malignancy.

From short-range repulsion to Hele-Shaw problem in a model of tumor growth

We investigate the large time behavior of an agent based model describing tumor growth. The microscopic model combines short-range repulsion and cell division. As the number of cells increases exponentially in time, the microscopic model is challenging in terms of computational time. To overcome this problem, we aim at deriving the associated macroscopic dynamics leading here to a porous media type equation. As we are interested in the long time behavior of the dynamics, the macroscopic equation obtained through usual derivation method fails at providing the correct qualitative behavior (e.g. stationary states differ from the microscopic dynamics). We propose a modified version of the macroscopic equation introducing a density threshold for the repulsion. We numerically validate the new formulation by comparing the solutions of the micro- and macro- dynamics. Moreover, we study the asymptotic behavior of the dynamics as the repulsion between cells becomes singular (leading to non-overlapping constraints in the microscopic model). We manage to show formally that such asymptotic limit leads to a Hele-Shaw type problem for the macroscopic dynamics. We compare the micro- and macro- dynamics in this asymptotic limit using explicit solutions of the Hele-Shaw problem (e.g. radially symmetric configuration). The numerical simulations reveal an excellent agreement between the two descriptions, validating the formal derivation of the macroscopic model. The macroscopic model derived in this paper therefore enables to overcome the problem of large computational time raised by the microscopic model, but stays closely linked to the microscopic dynamics.

Mathematical modeling and optimal control problems in brain tumor targeted drug delivery strategies

In this paper, we present a mathematical model that describes tumor-normal cells interaction dynamics focusing on role of drugs in treatment of brain tumors. The goal is to predict distribution and necessary quantity of drugs delivered in drug-therapy by using optimal control framework. The model describes interactions of tumor and normal cells using a system of reactions–diffusion equations involving the drug concentration, tumor cells and normal tissues. The control estimates simultaneously blood perfusion rate, reabsorption rate of drug and drug dosage administered, which affect the effects of brain tumor chemotherapy. First, we develop mathematical framework which models the competition between tumor and normal cells under chemotherapy constraints. Then, existence, uniqueness and regularity of solution of state equations are proved as well as stability results. Afterwards, optimal control problems are formulated in order to minimize the drug delivery and tumor cell burden in different situations. We show existence and uniqueness of optimal solution, and we derive necessary conditions for optimality. Finally, to solve numerically optimal control and optimization problems, we propose and investigate an adjoint multiple-relaxation-time lattice Boltzmann method for a general nonlinear coupled anisotropic convection–diffusion system (which includes the developed model for brain tumor targeted drug delivery system).

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.

In silico tumor control induced via alternating immunostimulating and immunosuppressive phases

Despite recent advances in the field of Oncoimmunology, the success potential of immunomodulatory therapies against cancer remains to be elucidated. One of the reasons is the lack of understanding on the complex interplay between tumor growth dynamics and the associated immune system responses. Toward this goal, we consider a mathematical model of vascularized tumor growth and the corresponding effector cell recruitment dynamics. Bifurcation analysis allows for the exploration of model's dynamic behavior and the determination of these parameter regimes that result in immune-mediated tumor control. In this work, we focus on a particular tumor evasion regime that involves tumor and effector cell concentration oscillations of slowly increasing and decreasing amplitude, respectively. Considering a temporal multiscale analysis, we derive an analytically tractable mapping of model solutions onto a weakly negatively damped harmonic oscillator. Based on our analysis, we propose a theory-driven intervention strategy involving immunostimulating and immunosuppressive phases to induce long-term tumor control.

Modelling circulating tumour cells for personalised survival prediction in metastatic breast cancer

Ductal carcinoma is one of the most common cancers among women, and the main cause of death is the formation of metastases. The development of metastases is caused by cancer cells that migrate from the primary tumour site (the mammary duct) through the blood vessels and extravasating they initiate metastasis. Here, we propose a multi-compartment model which mimics the dynamics of tumoural cells in the mammary duct, in the circulatory system and in the bone. Through a branching process model, we describe the relation between the survival times and the four markers mainly involved in metastatic breast cancer (EPCAM, CD47, CD44 and MET). In particular, the model takes into account the gene expression profile of circulating tumour cells to predict personalised survival probability. We also include the administration of drugs as bisphosphonates, which reduce the formation of circulating tumour cells and their survival in the blood vessels, in order to analyse the dynamic changes induced by the therapy. We analyse the effects of circulating tumour cells on the progression of the disease providing a quantitative measure of the cell driver mutations needed for invading the bone tissue. Our model allows to design intervention scenarios that alter the patient-specific survival probability by modifying the populations of circulating tumour cells and it could be extended to other cancer metastasis dynamics.

Numerical resolution of a model of tumour growth

We consider and solve numerically a mathematical model of tumour growth based on cancer stem cells (CSC) hypothesis with the aim of gaining some insight into the relation of different processes leading to exponential growth in solid tumours and into the evolution of different subpopulations of cells. The model consists of four hyperbolic equations of first order to describe the evolution of four subpopulations of cells. A fifth equation is introduced to model the evolution of the moving boundary. The coefficients of the model represent the rates at which reactions occur. In order to integrate numerically the four hyperbolic equations, a formulation in terms of the total derivatives is posed. A finite element discretization is applied to integrate the model equations in space. Our numerical results suggest the existence of a pseudo-equilibrium state reached at the early stage of the tumour, for which the fraction of CSC remains small. We include the study of the behaviour of the solutions for longer times and we obtain that the solutions to the system of partial differential equations stabilize to homogeneous steady states whose values depend only on the values of the parameters. We show that CSC may comprise different proportions of the tumour, becoming, in some cases, the predominant type of cells within the tumour. We also obtain that possible effective measure to detain tumour progression should combine the targeting of CSC with the targeting of progenitor cells.

Mathematical modeling for novel cancer drug discovery and development

INTRODUCTION

Mathematical modeling enables: the in silico classification of cancers, the prediction of disease outcomes, optimization of therapy, identification of promising drug targets and prediction of resistance to anticancer drugs. In silico pre-screened drug targets can be validated by a small number of carefully selected experiments.

AREAS COVERED

This review discusses the basics of mathematical modeling in cancer drug discovery and development. The topics include in silico discovery of novel molecular drug targets, optimization of immunotherapies, personalized medicine and guiding preclinical and clinical trials. Breast cancer has been used to demonstrate the applications of mathematical modeling in cancer diagnostics, the identification of high-risk population, cancer screening strategies, prediction of tumor growth and guiding cancer treatment.

EXPERT OPINION

Mathematical models are the key components of the toolkit used in the fight against cancer. The combinatorial complexity of new drugs discovery is enormous, making systematic drug discovery, by experimentation, alone difficult if not impossible. The biggest challenges include seamless integration of growing data, information and knowledge, and making them available for a multiplicity of analyses. Mathematical models are essential for bringing cancer drug discovery into the era of Omics, Big Data and personalized medicine.

Traveling wavefronts of a nonlinear reaction–diffusion model of tumor growth under the acid environment

In this paper, a reaction–diffusion model describing temporal development of tumor tissue, normal tissue and excess H+ ion concentration is considered. Based on a combination of perturbation methods, the Fredholm theory and Banach fixed point theorem, we theoretically justify the existence of the traveling wave solution for this model.

Modeling of human tumor xenografts and dose rationale in oncology

Xenograft models are commonly used in oncology drug development. Although there are discussions about their ability to generate meaningful data for the translation from animal to humans, it appears that better data quality and better design of the preclinical experiments, together with appropriate data analysis approaches could make these data more informative for clinical development. An approach based on mathematical modeling is necessary to derive experiment-independent parameters which can be linked with clinically relevant endpoints. Moreover, the inclusion of biomarkers as predictors of efficacy is a key step towards a more general mechanism-based strategy.

Evolution and morphology of microenvironment-enhanced malignancy of three-dimensional invasive solid tumors

The emergence of invasive and metastatic behavior in malignant tumors can often lead to fatal outcomes for patients. The collective malignant tumor behavior resulting from the complex tumor-host interactions and the interactions between the tumor cells is currently poorly understood. In this paper, we employ a cellular automaton (CA) model to investigate microenvironment-enhanced malignant behaviors and morphologies of in vitro avascular invasive solid tumors in three dimensions. Our CA model incorporates a variety of microscopic-scale tumor-host interactions, including the degradation of the extracellular matrix by the malignant cells, nutrient-driven cell migration, pressure buildup due to the deformation of the microenvironment by the growing tumor, and its effect on the local tumor-host interface stability. Moreover, the effects of cell-cell adhesion on tumor growth are explicitly taken into account. Specifically, we find that while strong cell-cell adhesion can suppress the invasive behavior of the tumors growing in soft microenvironments, cancer malignancy can be significantly enhanced by harsh microenvironmental conditions, such as exposure to high pressure levels. We infer from the simulation results a qualitative phase diagram that characterizes the expected malignant behavior of invasive solid tumors in terms of two competing malignancy effects: the rigidity of the microenvironment and cell-cell adhesion. This diagram exhibits phase transitions between noninvasive and invasive behaviors. We also discuss the implications of our results for the diagnosis, prognosis, and treatment of malignant tumors.

Inter-cellular signaling network reveals a mechanistic transition in tumor microenvironment

We conducted inter-cellular cytokine correlation and network analysis based upon a stochastic population dynamics model that comprises five cell types and fifteen signaling molecules inter-connected through a large number of cell-cell communication pathways. We observed that the signaling molecules are tightly correlated even at very early stages (e.g. the first month) of human glioma, but such correlation rapidly diminishes when tumor grows to a size that can be clinically detected. Further analysis suggests that paracrine is shown to be the dominant force during tumor initiation and priming, while autocrine supersedes it and supports a robust tumor expansion. In correspondence, the cytokine correlation network evolves through an increasing to decreasing complexity. This study indicates a possible mechanistic transition from the microenvironment-controlled, paracrine-based regulatory mechanism to self-sustained rapid progression to fetal malignancy. It also reveals key nodes that are responsible for such transition and can be potentially harnessed for the design of new anti-cancer therapies.

Understanding cancer mechanisms through network dynamics

Cancer is a complex, multifaceted disease. Cellular systems are perturbed both during the onset and development of cancer, and the behavioural change of tumour cells usually involves a broad range of dynamic variations. To an extent, the difficulty of monitoring the systemic change has been alleviated by recent developments in the high-throughput technologies. At both the genomic as well as proteomic levels, the technological advances in microarray and mass spectrometry, in conjunction with computational simulations and the construction of human interactome maps have facilitated the progress of identifying disease-associated genes. On a systems level, computational approaches developed for network analysis are becoming especially useful for providing insights into the mechanism behind tumour development and metastasis. This review emphasizes network approaches that have been developed to study cancer and provides an overview of our current knowledge of protein-protein interaction networks, and how their systemic perturbation can be analysed by two popular network simulation methods: Boolean network and ordinary differential equations.

In silico modelling of tumour margin diffusion and infiltration: review of current status

As a result of advanced treatment techniques, requiring precise target definitions, a need for more accurate delineation of the Clinical Target Volume (CTV) has arisen. Mathematical modelling is found to be a powerful tool to provide fairly accurate predictions for the Microscopic Extension (ME) of a tumour to be incorporated in a CTV. In general terms, biomathematical models based on a sequence of observations or development of a hypothesis assume some links between biological mechanisms involved in cancer development and progression to provide quantitative or qualitative measures of tumour behaviour as well as tumour response to treatment. Generally, two approaches are taken: deterministic and stochastic modelling. In this paper, recent mathematical models, including deterministic and stochastic methods, are reviewed and critically compared. It is concluded that stochastic models are more promising to provide a realistic description of cancer tumour behaviour due to being intrinsically probabilistic as well as discrete, which enables incorporation of patient-specific biomedical data such as tumour heterogeneity and anatomical boundaries.

Diversity of dynamics and morphologies of invasive solid tumors

Complex tumor-host interactions can significantly affect the growth dynamics and morphologies of progressing neoplasms. The growth of a confined solid tumor induces mechanical pressure and deformation of the surrounding microenvironment, which in turn influences tumor growth. In this paper, we generalize a recently developed cellular automaton model for invasive tumor growth in heterogeneous microenvironments [Y. Jiao and S. Torquato, PLoS Comput. Biol.7, e1002314 (2011)] by incorporating the effects of pressure. Specifically, we explicitly model the pressure exerted on the growing tumor due to the deformation of the microenvironment and its effect on the local tumor-host interface instability. Both noninvasive-proliferative growth and invasive growth with individual cells that detach themselves from the primary tumor and migrate into the surrounding microenvironment are investigated. We find that while noninvasive tumors growing in "soft" homogeneous microenvironments develop almost isotropic shapes, both high pressure and host heterogeneity can strongly enhance malignant behavior, leading to finger-like protrusions of the tumor surface. Moreover, we show that individual invasive cells of an invasive tumor degrade the local extracellular matrix at the tumor-host interface, which diminishes the fingering growth of the primary tumor. The implications of our results for cancer diagnosis, prognosis and therapy are discussed.

Emergent behaviors from a cellular automaton model for invasive tumor growth in heterogeneous microenvironments

Understanding tumor invasion and metastasis is of crucial importance for both fundamental cancer research and clinical practice. In vitro experiments have established that the invasive growth of malignant tumors is characterized by the dendritic invasive branches composed of chains of tumor cells emanating from the primary tumor mass. The preponderance of previous tumor simulations focused on non-invasive (or proliferative) growth. The formation of the invasive cell chains and their interactions with the primary tumor mass and host microenvironment are not well understood. Here, we present a novel cellular automaton (CA) model that enables one to efficiently simulate invasive tumor growth in a heterogeneous host microenvironment. By taking into account a variety of microscopic-scale tumor-host interactions, including the short-range mechanical interactions between tumor cells and tumor stroma, degradation of the extracellular matrix by the invasive cells and oxygen/nutrient gradient driven cell motions, our CA model predicts a rich spectrum of growth dynamics and emergent behaviors of invasive tumors. Besides robustly reproducing the salient features of dendritic invasive growth, such as least-resistance paths of cells and intrabranch homotype attraction, we also predict nontrivial coupling between the growth dynamics of the primary tumor mass and the invasive cells. In addition, we show that the properties of the host microenvironment can significantly affect tumor morphology and growth dynamics, emphasizing the importance of understanding the tumor-host interaction. The capability of our CA model suggests that sophisticated in silico tools could eventually be utilized in clinical situations to predict neoplastic progression and propose individualized optimal treatment strategies.

The goal of the present work is to develop an efficient single-cell based cellular automaton (CA) model that enables one to investigate the growth dynamics and morphology of invasive solid tumors. Recent experiments have shown that highly malignant tumors develop dendritic branches composed of tumor cells that follow each other, which massively invade into the host microenvironment and ultimately lead to cancer metastasis. Previous theoretical/computational cancer modeling neither addressed the question of how such chain-like invasive branches form nor how they interact with the host microenvironment and the primary tumor. Our CA model, which incorporates a variety of microscopic-scale tumor-host interactions (e.g., the mechanical interactions between tumor cells and tumor stroma, degradation of the extracellular matrix by the tumor cells and oxygen/nutrient gradient driven cell motions), can robustly reproduce experimentally observed invasive tumor evolution and predict a wide spectrum of invasive tumor growth dynamics and emergent behaviors in various different heterogeneous environments. Further refinement of our CA model could eventually lead to the development of a powerful simulation tool for clinical purposes capable of predicting neoplastic progression and suggesting individualized optimal treatment strategies.

NUMERICAL VERSUS EXPERIMENTAL DATA FOR PROSTATE TUMOUR GROWTH

The goal of this paper is to solve mathematical model equations on solid tumour growth and compute their parameter values by applying growth rates of prostate cancer cell lines in vivo. For these computations, we investigate previously developed C3(1)/Tag transgenic models of prostate cancer. To make the computations fast, we have constructed an algorithm, which is based on small amounts of spatial grid-points and obtained a correspondence between the in vivo growth of tumours and the solutions of the model equations.

Model for in vivo progression of tumors based on co-evolving cell population and vasculature

With countless biological details emerging from cancer experiments, there is a growing need for minimal mathematical models which simultaneously advance our understanding of single tumors and metastasis, provide patient-personalized predictions, whilst avoiding excessive hard-to-measure input parameters which complicate simulation, analysis and interpretation. Here we present a model built around a co-evolving resource network and cell population, yielding good agreement with primary tumors in a murine mammary cell line EMT6-HER2 model in BALB/c mice and with clinical metastasis data. Seeding data about the tumor and its vasculature from in vivo images, our model predicts corridors of future tumor growth behavior and intervention response. A scaling relation enables the estimation of a tumor's most likely evolution and pinpoints specific target sites to control growth. Our findings suggest that the clinically separate phenomena of individual tumor growth and metastasis can be viewed as mathematical copies of each other differentiated only by network structure.

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.