Simulated brain tumor growth dynamics using a three-dimensional cellular automaton

We have developed a novel and versatile three-dimensional cellular automaton model of brain tumor growth. We show that macroscopic tumor behavior can be realistically modeled using microscopic parameters. Using only four parameters, this model simulates Gompertzian growth for a tumor growing over nearly three orders of magnitude in radius. It also predicts the composition and dynamics of the tumor at selected time points in agreement with medical literature. We also demonstrate the flexibility of the model by showing the emergence, and eventual dominance, of a second tumor clone with a different genotype. The model incorporates several important and novel features, both in the rules governing the model and in the underlying structure of the model. Among these are a new definition of how to model proliferative and non-proliferative cells, an isotropic lattice, and an adaptive grid lattice.     

An evolutionary hybrid cellular automaton model of solid tumour growth

We propose a cellular automaton model of solid tumour growth, in which each cell is equipped with a micro-environment response network. This network is modelled using a feed-forward artificial neural network, that takes environmental variables as an input and from these determines the cellular behaviour as the output. The response of the network is determined by connection weights and thresholds in the network, which are subject to mutations when the cells divide. As both available space and nutrients are limited resources for the tumour, this gives rise to clonal evolution where only the fittest cells survive. Using this approach we have investigated the impact of the tissue oxygen concentration on the growth and evolutionary dynamics of the tumour. The results show that the oxygen concentration affects the selection pressure, cell population diversity and morphology of the tumour. A low oxygen concentration in the tissue gives rise to a tumour with a fingered morphology that contains aggressive phenotypes with a small apoptotic potential, while a high oxygen concentration in the tissue gives rise to a tumour with a round morphology containing less evolved phenotypes. The tissue oxygen concentration thus affects the tumour at both the morphological level and on the phenotype level.     

Mathematical modeling of biofilm structure with a hybrid differential-discrete cellular automaton approach

A hybrid differential-discrete mathematical model has been used to simulate biofilm structures (surface shape, roughness, porosity) as a result of microbial growth in different environmental conditions. In this study, quantitative two- and three-dimensional models were evaluated by introducing statistical measures to characterize the complete biofilm structure, both the surface structure and volume structure. The surface enlargement, coefficient of roughness, fractal dimension of surface, biofilm compactness, and solids hold-up were found to be good measures of biofilm structure complexity. Among many possible factors affecting the biofilm structure, the influence of biomass growth in relation to the diffusive substrate transport was investigated. Porous biofilms, with many channels and voids between the "finger-like" or "mushroom" outgrowth, were obtained in a substrate-transport-limited regime. Conversely, compact and dense biofilms occurred in systems limited by the biomass growth rate and not by the substrate transfer rate. The surface complexity measures (enlargement, roughness, fractal dimension) all increased with increased transport limitation, whereas the volume measures (compactness, solid hold-up) decreased, showing the change from a compact and dense to a highly porous and open biofilm.     

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.     

Cellular automaton of idealized brain tumor growth dynamics

A novel cellular automaton model of proliferative brain tumor growth has been developed. This model is able to simulate Gompertzian tumor growth over nearly three orders of magnitude in radius using only four microscopic parameters. The predicted composition and growth rates are in agreement with a test case pooled from the available medical literature. The model incorporates several new features, improving previous models, and also allows ready extension to study other important properties of tumor growth, such as clonal competition.     

A multiscale model for avascular tumor growth

The desire to understand tumor complexity has given rise to mathematical models to describe the tumor microenvironment. We present a new mathematical model for avascular tumor growth and development that spans three distinct scales. At the cellular level, a lattice Monte Carlo model describes cellular dynamics (proliferation, adhesion, and viability). At the subcellular level, a Boolean network regulates the expression of proteins that control the cell cycle. At the extracellular level, reaction-diffusion equations describe the chemical dynamics (nutrient, waste, growth promoter, and inhibitor concentrations). Data from experiments with multicellular spheroids were used to determine the parameters of the simulations. Starting with a single tumor cell, this model produces an avascular tumor that quantitatively mimics experimental measurements in multicellular spheroids. Based on the simulations, we predict: 1), the microenvironmental conditions required for tumor cell survival; and 2), growth promoters and inhibitors have diffusion coefficients in the range between 10(-6) and 10(-7) cm2/h, corresponding to molecules of size 80-90 kDa. Using the same parameters, the model also accurately predicts spheroid growth curves under different external nutrient supply conditions.     

Modeling growth of a heterogeneous tumor

It has long been recognized that the growth of tumor population depends on the initial age distribution of the cells in the tumor and the age-dependent cellular birth rate. Deterministic dual-cell models have been available for sometime; these models take into account the effects of the resultant cell heterogeneity. Nevertheless, these models ignore various variables significantly affecting the growth, such as those characterizing the cells' inherent properties and environmental factors. Uncertainties, or fluctuations, arise when the growth is simulated with the models. Stochastic analysis of these fluctuations is the focus of the current work.Two types of cells are visualized to proliferate separately and to transform mutually during the process. The master equations of the system have been formulated through probabilistic population balance around a particular state by considering all mutually exclusive events. The governing equations for the means, variances, and covariance of the random variables have been derived through the system-size expansion of these nonlinear master equations. The stochastic pathways of the two different types of cells have been numerically simulated by the algorithm derived from the master equation for two different physical situations, one without and, the other, with the chemotherapeutic treatment. The results of the current study illuminate the significance of stochastically modeling the responses of the tumor to a variety of medicinal treatments: The coefficient of variation of the malignant cells' population magnifies with time under chemotherapeutic regimens. Consequently, the impact of the uncertainties in the exact number of malignant cells as expressed by this coefficient of variation is highly unpredictable. For example, it becomes increasingly uncertain if or how fast these cells will reactivate to become a full-blown carcinogenic tumor after treatment.     

A cellular automaton model for tumour growth in inhomogeneous environment

Most of the existing mathematical models for tumour growth and tumour-induced angiogenesis neglect blood flow. This is an important factor on which both nutrient and metabolite supply depend. In this paper we aim to address this shortcoming by developing a mathematical model which shows how blood flow and red blood cell heterogeneity influence the growth of systems of normal and cancerous cells. The model is developed in two stages. First we determine the distribution of oxygen in a native vascular network, incorporating into our model features of blood flow and vascular dynamics such as structural adaptation, complex rheology and red blood cell circulation. Once we have calculated the oxygen distribution, we then study the dynamics of a colony of normal and cancerous cells, placed in such a heterogeneous environment. During this second stage, we assume that the vascular network does not evolve and is independent of the dynamics of the surrounding tissue. The cells are considered as elements of a cellular automaton, whose evolution rules are inspired by the different behaviour of normal and cancer cells. Our aim is to show that blood flow and red blood cell heterogeneity play major roles in the development of such colonies, even when the red blood cells are flowing through the vasculature of normal, healthy tissue.     

Modeling tumor growth with random onset

The longitudinal assessment of tumor volume is commonly used as an endpoint in small animal studies in cancer research. Groups of genetically identical mice are injected with mutant cells from clones developed with different mutations. The interest is on comparing tumor onset (i.e., the time of tumor detection) and tumor growth after onset, between mutation groups. This article proposes a class of linear and nonlinear growth models for jointly modeling tumor onset and growth in this situation. Our approach allows for interval-censored time of onset and missing-at-random dropout due to early sacrifice, which are common situations in animal research. We show that our approach has good small-sample properties for testing and is robust to some key unverifiable modeling assumptions. We illustrate this methodology with an application examining the effect of different mutations on tumorigenesis.     

Formulation and numerical simulations of a continuum model of avascular tumor growth

In this paper we present a continuum mathematical model for a multicellular spheroid that mimics the micro-environment within avascular tumor growth. The model consists of a coupled system of non-linear convection-diffusion-reaction equations. This system is solved using a previously developed conservative Galerkin characteristics method. In the model considered, there are three cell types: the proliferative cells, the quiescent non-dividing cells which stay in the G₀ phase of the cell cycle and the necrotic cells. The model includes viable cell diffusion, diffusion of cellular material and the removal of necrotic cells. We assume that the nutrients diffuse passively and are consumed by the proliferative and quiescent tumor cells depending on the availability of resources (oxygen, glucose, etc.). The numerical simulations are performed using different sets of parameters, including biologically realistic ones, to explore the effects of each of these model parameters on reaching the steady state. The present results, taken together with those reported earlier, indicate that the removal of necrotic cells and the diffusion of cellular material have significant effects on the steady state, reflecting growth saturation, the number of viable cells, and the spheroid size.     

Modeling tumor growth.

The meaning and limitations of certain mathematical models of tumor growth are discussed, and some new derivations of the existing models are given. A theoretical justification for Gompertz's law of growth for tumors is presented. An age-dependent Von Bertalanffy's equation and diffusion models are introduced, and existence and uniqueness problems are addressed.     

Hybrid cellular automaton modeling of nutrient modulated cell growth in tissue engineering constructs

Mathematic models help interpret experimental results and accelerate tissue engineering developments. We develop in this paper a hybrid cellular automata model that combines the differential nutrient transport equation to investigate the nutrient limited cell construct development for cartilage tissue engineering. Individual cell behaviors of migration, contact inhibition and cell collision, coupled with the cell proliferation regulated by oxygen concentration were carefully studied. Simplified two-dimensional simulations were performed. Using this model, we investigated the influence of cell migration speed on the overall cell growth within in vitro cell scaffolds. It was found that intense cell motility can enhance initial cell growth rates. However, since cell growth is also significantly modulated by the nutrient contents, intense cell motility with conventional uniform cell seeding method may lead to declined cell growth in the final time because concentrated cell population has been growing around the scaffold periphery to block the nutrient transport from outside culture media. Therefore, homogeneous cell seeding may not be a good way of gaining large and uniform cell densities for the final results. We then compared cell growth in scaffolds with various seeding modes, and proposed a seeding mode with cells initially residing in the middle area of the scaffold that may efficiently reduce the nutrient blockage and result in a better cell amount and uniform cell distribution for tissue engineering construct developments.     

Observer-based techniques for the identification and analysis of avascular tumor growth

Cancer represents one of the most challenging issues for the biomedical research, due its large impact on the public health state. For this reason, many mathematical methods have been proposed to forecast the time evolution of cancer size and invasion. In this paper, we study how to apply the Gompertz’s model to describe the growth of an avascular tumor in a realistic setting. To this aim, we introduce mathematical techniques to discretize the model, an important requirement when discrete-time measurements are available. Additionally, we describe observed-based techniques, borrowed from the field of automation theory, as a tool to estimate the model unknown parameters. This identification approach is a promising alternative to traditional statistical methods, and it can be easily extended to other models of cancer growth as well as to the evaluation of not measurable variables, on the basis of the available measurements. We show an application of this method to the analysis of solid tumor growth and parameters estimation in presence of a chemotherapy agent.     

Simulating complex tumor dynamics from avascular to vascular growth using a general level-set method

A comprehensive continuum model of solid tumor evolution and development is investigated in detail numerically, both under the assumption of spherical symmetry and for arbitrary two-dimensional growth. The level set approach is used to obtain solutions for a recently developed multi-cell transport model formulated as a moving boundary problem for the evolution of the tumor. The model represents both the avascular and the vascular phase of growth, and is able to simulate when the transition occurs; progressive formation of a necrotic core and a rim structure in the tumor during the avascular phase are also captured. In terms of transport processes, the interaction of the tumor with the surrounding tissue is realistically incorporated. The two-dimensional simulation results are presented for different initial configurations. The computational framework, based on a Cartesian mesh/narrow band level-set method, can be applied to similar models that require the solution of coupled advection-diffusion equations with a moving boundary inside a fixed domain. The solution algorithm is designed so that extension to three-dimensional simulations is straightforward.     

A quantitative cellular automaton model of in vitro multicellular spheroid tumour growth

We report numerical results from a 2D cellular automaton (CA) model describing the dynamics of the in vitro cultivated multicellular spheroid obtained from EMT6/Ro (mammary carcinoma) cell line. Significantly, the CA model relaxes the often assumed one-to-one correspondence between cells and CA sites so as to correctly model the peripheral mitotic boundary region, and to enable the study of necrosis in large avascular tumours. By full calibration and scaling to available experimental data, the model produces with good accuracy experimentally comparable data on a range of bulk tumour kinetics and necrosis measures. Our main finding is that the metabolic production of H⁺ ions is not sufficient to cause central necrosis prior to the sub-viable nutrient-deficient stage of tumour development being reached. Thus, the model suggests that an additional process is required to explain the experimentally observable onset of necrosis prior to the non-viable nutrient-deficient point being reached.     

A hybrid cellular automaton model of clonal evolution in cancer: the emergence of the glycolytic phenotype

We present a cellular automaton model of clonal evolution in cancer aimed at investigating the emergence of the glycolytic phenotype. In the model each cell is equipped with a micro-environment response network that determines the behaviour or phenotype of the cell based on the local environment. The response network is modelled using a feed-forward neural network, which is subject to mutations when the cells divide. This implies that cells might react differently to the environment and when space and nutrients are limited only the fittest cells will survive. With this model we have investigated the impact of the environment on the growth dynamics of the tumour. In particular, we have analysed the influence of the tissue oxygen concentration and extra-cellular matrix density on the dynamics of the model. We found that the environment influences both the growth and the evolutionary dynamics of the tumour. For low oxygen concentration we observe tumours with a fingered morphology, while increasing the matrix density gives rise to more compact tumours with wider fingers. The distribution of phenotypes in the tumour is also affected, and we observe that the glycolytic phenotype is most likely to emerge in a poorly oxygenated tissue with a high matrix density. Our results suggest that it is the combined effect of the oxygen concentration and matrix density that creates an environment where the glycolytic phenotype has a growth advantage and consequently is most likely to appear.     

Coexistence of sexual and asexual conspecifics: a cellular automaton model

Most previous models of populations mixed for reproductive mode have omitted important local interactions between sexual and asexual individuals. We propose a cellular automaton model where local rules focus on fertilization and colonization. This model produces rich sets of data which are then studied by means of spatial statistics. Results point to the fixation of one of the two reproductive modes in the landscape. However, some examples of coexistence of sexual and asexual conspecifics over long periods of time are also found. This model is an example of a CA that diverges from its mean field approximation. The formation of sexual and asexual clusters reduces effective colonization rate in the CA and may account for this behaviour.     

Parameter estimation with a novel gradient-based optimization method for biological lattice-gas cellular automaton models

Lattice-gas cellular automata (LGCAs) can serve as stochastic mathematical models for collective behavior (e.g. pattern formation) emerging in populations of interacting cells. In this paper, a two-phase optimization algorithm for global parameter estimation in LGCA models is presented. In the first phase, local minima are identified through gradient-based optimization. Algorithmic differentiation is adopted to calculate the necessary gradient information. In the second phase, for global optimization of the parameter set, a multi-level single-linkage method is used. As an example, the parameter estimation algorithm is applied to a LGCA model for early in vitro angiogenic pattern formation.