Evolutionary dynamics of cancer:From epigenetic regulation to cell population dynamics——mathematical model framework,applications,and open problems

Predictive modeling of the evolutionary dynamics of cancer is a challenging issue in computational cancer biology. In this paper, we propose a general mathematical model framework for the evolutionary dynamics of cancer, including plasticity and heterogeneity in cancer cells. Cancer is a group of diseases involving abnormal cell growth, during which abnormal regulation of stem cell regeneration is essential for the dynamics of cancer development. In general, the dynamics of stem cell regeneration can be simplified as a G0 phase cell cycle model, which leads to a delay differentiation equation. When cell heterogeneity and plasticity are considered, we establish a differential-integral equation based on the random transition of epigenetic states of stem cells during cell division. The proposed model highlights cell heterogeneity and plasticity;connects the heterogeneity with cell-to-cell variance in cellular behaviors(for example, proliferation, apoptosis, and differentiation/senescence);and can be extended to include gene mutation-induced tumor development. Hybrid computational models are developed based on the mathematical model framework and are applied to the processes of inflammationinduced tumorigenesis and tumor relapse after chimeric antigen receptor(CAR)-T cell therapy. Finally, we propose several mathematical problems related to the proposed differential-integral equation. Solutions to these problems are crucial for understanding the evolutionary dynamics of cancer.     

Circadian rhythm and cell population growth

Molecular circadian clocks, that are found in all nucleated cells of mammals, are known to dictate rhythms of approximately 24 h (circa diem) to many physiological processes. This includes metabolism (e.g., temperature, hormonal blood levels) and cell proliferation. It has been observed in tumor-bearing laboratory rodents that a severe disruption of these physiological rhythms results in accelerated tumor growth.The question of accurately representing the control exerted by circadian clocks on healthy and tumor tissue proliferation to explain this phenomenon has given rise to mathematical developments, which we review. The main goal of these previous works was to examine the influence of a periodic control on the cell division cycle in physiologically structured cell populations, comparing the effects of periodic control with no control, and of different periodic controls between them. We state here a general convexity result that may give a theoretical justification to the concept of cancer chronotherapeutics. Our result also leads us to hypothesize that the above mentioned effect of disruption of circadian rhythms on tumor growth enhancement is indirect, that is, this enhancement is likely to result from the weakening of healthy tissue that is at work fighting tumor growth.     

Modelling of glioblastoma growth by linking a molecular interaction network with an agent-based model

In this work, a mathematical model of malignant brain tumour growth is presented. In particular, the growth of glioblastoma is investigated on the intracellular and intercellular scale.

The Go or Grow principle of tumour cells states that tumour cells either migrate or proliferate. For glioblastoma, microRNA-451 has been shown to be an energy dependent key regulator of the LKB1 (liver kinase B1) and AMPK (AMP-activated protein kinase) pathway that influences the signalling for migration or cell division.

We introduce a mathematical model that reproduces these biological processes. The intracellular molecular interaction network is represented by a system of nine ordinary differential equations. This is put into a multiscale context by applying an agent-based approach: each cell is equipped with this interaction network and additional rules to determine its new phenotype as either migrating, proliferating or quiescent.

The evaluation of the proposed model by comparison of the results with in vitro experiments indicates its validity.     

Label structured cell proliferation models

We present a general class of cell population models that can be used to track the proliferation of cells which have been labeled with a fluorescent dye. The mathematical models employ fluorescence intensity as a structure variable to describe the evolution in time of the population density of proliferating cells. While cell division is a major component of changes in cellular fluorescence intensity, models developed here also address overall label degradation.     

A cellular automata model of chemotherapy effects on tumour growth: targeting cancer and immune cells

The effects of therapy on avascular cancer development based on a stochastic cellular automata model are considered. Making the model more compatible with the biology of cancer, the following features are implemented: intrinsic resistance of cancerous cells along with drug-induced resistance, drug-sensitive cells, immune system. Results are reported for no treatment, discontinued treatment after only one cycle of chemotherapy, and periodic drug administration therapy modes. Growth fraction, necrotic fraction, and tumour volume are used as output parameters beside a 2-D graphical growth presentation. Periodic drug administration is more effective to inhibit the growth of tumours. The model has been validated by the verification of the simulation results using in vivo literature data. Considering immune cells makes the model more compatible with the biological realities. Beside targeting cancer cells, the model can also simulate the activation of the immune system to fight against cancer.

Abbreviations CA: cellular automata; DSC: drug sensitive cell; DRC: drug resistant cell; GF: growth fraction; NF: necrotic fraction; ODE: ordinary differential equation; PDE: partial differential equation; SCAM: The proposed stochastic cellular automata model     

A Mathematical Model for Optimal Functional Disruption of Biochemical Networks

Biochemical networks are a particular kind of biological networks which describe the cell metabolism and regulate various biological functions, from biochemical pathways to cell growth. The relationship between structure, function and regulation in complex cellular networks is still a largely open question. This complexity calls for proper mathematical models and methods relating network structure and functional properties. In this paper we focus on the problem of drug targets’ identification by detecting network alteration strategies which lead to a cell functionality loss. We propose a mathematical model, based on a bi-level programming formulation, to obtain the minimum cost disruption policy through the identification of specific gene deletions. These deletions represent drug target identification of new drug treatments for hindering bacterial infections.     

A singular transport model describing cellular division

In this Note, we study a system of partial differential equations with a singular transport term describing blood cellular production. The population of cells considered is capable of simultaneous proliferation and maturation. We prove that uniqueness of solutions depends only on stem cells.     

Modelling hematopoiesis in health and disease

Hematopoiesis is the process responsible for maintaining the number of circulating blood cells that are undergoing continuous turnover. At the root of this process are the hematopoietic stem cells (HSC), that replicate slowly to self-renew and give rise to progeny cells that proceed along the path of differentiation. The process is complex, with the cells responding to a wide variety of cytokines and growth factors. We discuss the mathematics of hematopoiesis based on stochastic cell behavior. Multiple compartments are introduced to keep track of each cell division process and increasing differentiation. The same mathematical model that describes normal hematopoiesis across mammals as a stable steady state of a hierarchical stochastic process is also used to understand the detailed dynamics of various disorders both in humans and in animal models. The microecology of the multitude of cell lineages that constitute what we call troubled hematopoiesis evolves in time under mutation and selection, the paradigmatic components of Darwinian evolution. Thus, the present approach provides a novel perspective for looking at cancer progression and cure.     

Dynamics and numerical simulations in a production and development of red blood cells model with one delay

In this paper, we study an unstructured model of a cellular population in the spirit of Grabosch and Heijmans [Grabosch A, Heijmans HJAM. Production, development and maturation of red blood cells, A mathematical model. AM-R 8919, ISSN 0924-2953, 1989.] model. The cellular population is described by a system of differential equations with one delay. The basic assumption is that the cell population responsible for the production of blood cells consists of three compartments: the stem cells, the precursor cells, and the blood cells. We prove that the model has two possible steady states and their dynamics (depending on time delay) are studied in term of the local stability, we illustrate these results with numerical simulations for some different values of the time delay.     

Mathematical model of hematopoiesis dynamics with growth factor-dependent apoptosis and proliferation regulations

We consider a nonlinear mathematical model of hematopoietic stem cell dynamics, in which proliferation and apoptosis are controlled by growth factor concentrations. Cell proliferation is positively regulated, while apoptosis is negatively regulated. The resulting age-structured model is reduced to a system of three differential equations, with three independent delays, and existence of steady states is investigated. The stability of the trivial steady state, describing cells dying out with a saturation of growth factor concentrations is proven to be asymptotically stable when it is the only equilibrium. The stability analysis of the unique positive steady state allows the determination of a stability area, and shows that instability may occur through a Hopf bifurcation, mainly as a destabilization of the proliferative capacity control, when cell cycle durations are very short. Numerical simulations are carried out and result in a stability diagram that stresses the lead role of the introduction rate compared to the apoptosis rate in the system stability.     

Modelling forest fire spread using hexagonal cellular automata

In this paper a new mathematical model for predicting the spread of a fire front in homogeneous and inhomogeneous environments is presented. It is based on a bidimensional cellular automata model, whose cells stand for regular hexagonal areas of the forest. The results obtained are in agreement with the fire spreading in real forests.     

On analytical and numerical approaches to division and label structured population models

Even among cells in the same population, the concentration of a protein or cellular constituent can vary considerably. This heterogeneity can arise from several sources, including differences in kinetic rates between cells and distribution of cellular constituents through cell division. Compartmental models have been used to describe the distribution of the number of divisions undergone by cells in a population. More recently, such models have been coupled with the dynamics of intracellular labels and analytical solutions to the division and label structured population equations have been found. However, such approaches have thus far focused on simple models of intracellular dynamics such as the decay of an intracellular label. In this work, we demonstrate that analytical solutions are possible for more general forms of intracellular dynamics offering the promise to lend mathematical insight into population dynamics in more realistic biological settings.     

Characterization of stem cells using mathematical models of multistage cell lineages

Stem cells dynamics is an important field of research with promising clinical impacts. Due to the revolutionary new technologies of biological data collection, an enormous amount of information on specific factors and genes responsible for cell differentiation is available. However, the mechanisms controlling stem cell self-renewal, maintenance and differentiation are still poorly understood and there exists no general characterization of stem cells based on observable cell properties. We address these problems with the help of mathematical models. Stem cells are described as the cell type that is most responsive to certain environmental signals. This results in a dynamic characterization of stemness that depends on environmental conditions and is not necessarily linked to a unique cell population.     

On the growth of filamentary structures in planar media

We analyse a mathematical model for the growth of thin filaments into a two dimensional medium. More exactly, we focus on a certain reaction/diffusion system, describing the interaction between three chemicals (an activator, an inhibitor and a growth factor), and including a fourth cell variable characterising irreversible incorporation to a filament. Such a model has been shown numerically to generate structures shaped like nets. We perform an asymptotical analysis of the behaviour of solutions, in the case when the system has parameters very large and very small, thereby allowing the onset of different time and space scales. In particular, we describe the motion of the tip of a filament, and the changes in the relevant chemical species nearby. Copyright © 2004 John Wiley & Sons, Ltd.     

The development of a cellular automaton-finite volume model for dendritic growth

A cellular automaton to track the solid–liquid interface movement is linked to finite volume computations of solute diffusion to simulate the behavior of dendritic structures in binary alloys during solidification. A significant problem encountered in the CA formulation has been the presence of artificial anisotropy in growth kinetics introduced by a Cartesian CA grid. A new technique to track the interface movement is proposed to model dendritic growth in different crystallographic orientations while reducing the anisotropy due to grid orientation. The model stability with respect to the numerical parameters (cell size and time step) for various operating conditions is examined. A method for generating an operating window in Δt and Δx has been identified, in which the model gives a grid-independent set of results for calculated dendrite tip radius and tip undercooling. Finally, the model is compared to published experimental and analytical results for both directional and equiaxed growth conditions.     

Parameter estimation in metabolic flux balance models for batch fermentation—Formulation & Solution using Differential Variational Inequalities (DVIs)

Recent years have witnessed a surge in research in cellular biology. There has been particular interest in the interaction between cellular metabolism and its environment. In this work we present a framework for fitting fermentation models that include this interaction. Differential equations describe the evolution of extracellular metabolites, while a Linear Program (LP) models cell metabolism, and piecewise smooth functions model the links between cell metabolism and its environment. We show that the fermentation dynamics can be described using Differential Variational Inequalities (DVIs). Discretization of the system and reformulation of the VIs using optimality conditions converts the DVI to a Mathematical Program with Complementarity Constraints (MPCC). We briefly describe an interior point algorithm for solving MPCCs. Encouraging numerical results are presented in estimating model parameters to fit model prediction and data obtained from fermentation, using cultures of Saccharomyces cerevisiae reported in the literature.     

A stochastic model for wound healing

Three separate activities of wound healing have been identified: migration, proliferation and differentiation. In this paper we present a mathematical model for the activities of migration and proliferation in an invitro system. The motion of a cell is modelled by a two-dimensional Brownian motion in the “unwounded” media. To reflect the proliferative activity in the wound area, we shall impose growth dynamics on the cells which are position dependent. From the resulting motile-growth stochastic model, we are able to estimate the expected number of cells in the wound at time t. From this, the expected time of wound closure can be predicted.     

A Planning Algorithm for Correction Therapies After Allogeneic Stem Cell Transplantation

In this paper we continue the investigation of a basic mathematical model describing the dynamics of three cell lines after allogeneic stem cell transplantation: normal host cells, leukemic host cells and donor cells, whose evolution ultimately lead either to the normal hematopoietic state achieved by the expansion of the donor cells and the elimination of the host cells, or to the leukemic hematopoietic state characterized by the proliferation of the cancer line and the suppression of the other cell lines. A theoretical basis for the control of post-transplant evolution is provided. We describe several scenarios of change of system parameters by which a bad post-transplant evolution can be corrected and turned into a good one and we propose therapy planning algorithms for guiding the correction treatment.     

Simulating brain tumor heterogeneity with a multiscale agent-based model: Linking molecular signatures,phenotypes and expansion rate

We have extended our previously developed 3D multi-scale agent-based brain tumor model to simulate cancer heterogeneity and to analyze its impact across the scales of interest. While our algorithm continues to employ an epidermal growth factor receptor (EGFR) gene–protein interaction network to determine the cells’ phenotype, it now adds an implicit treatment of tumor cell adhesion related to the model’s biochemical microenvironment. We simulate a simplified tumor progression pathway that leads to the emergence of five distinct glioma cell clones with different EGFR density and cell ‘search precisions’. The in silico results show that microscopic tumor heterogeneity can impact the tumor system’s multicellular growth patterns. Our findings further confirm that EGFR density results in the more aggressive clonal populations switching earlier from proliferation-dominated to a more migratory phenotype. Moreover, analyzing the dynamic molecular profile that triggers the phenotypic switch between proliferation and migration, our in silico oncogenomics data display spatial and temporal diversity in documenting the regional impact of tumorigenesis, and thus support the added value of multi-site and repeated assessments in vitro and in vivo. Potential implications from this in silico work for experimental and computational studies are discussed.     

Experimental versus numerical data for breast cancer progression

This paper deals with a mouse model of breast cancer based on two mammary adenocarcinoma cell lines derived from a spontaneous tumor of the mammary gland in a female BALB/c mouse. We investigate both animal and mathematical models of tumor progression, and demonstrate a correspondence between the experimental and predicted data. The mathematical model is solved numerically and the laboratory data are utilized in order to find unknown parameters for the model equations. The results of the numerical experiments illustrate that the mathematical model has a potential to describe the growth of cancer cells in vivo.