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

A quantitative theory of solid tumor growth, metabolic rate and vascularization

The relationships between cellular, structural and dynamical properties of tumors have traditionally been studied separately. Here, we construct a quantitative, predictive theory of solid tumor growth, metabolic rate, vascularization and necrosis that integrates the relationships between these properties. To accomplish this, we develop a comprehensive theory that describes the interface and integration of the tumor vascular network and resource supply with the cardiovascular system of the host. Our theory enables a quantitative understanding of how cells, tissues, and vascular networks act together across multiple scales by building on recent theoretical advances in modeling both healthy vasculature and the detailed processes of angiogenesis and tumor growth. The theory explicitly relates tumor vascularization and growth to metabolic rate, and yields extensive predictions for tumor properties, including growth rates, metabolic rates, degree of necrosis, blood flow rates and vessel sizes. Besides these quantitative predictions, we explain how growth rates depend on capillary density and metabolic rate, and why similar tumors grow slower and occur less frequently in larger animals, shedding light on Peto's paradox. Various implications for potential therapeutic strategies and further research are discussed.

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.

Computational modeling of tumor response to vascular-targeting therapies--part I: validation

Mathematical modeling techniques have been widely employed to understand how cancer grows, and, more recently, such approaches have been used to understand how cancer can be controlled. In this manuscript, a previously validated hybrid cellular automaton model of tumor growth in a vascularized environment is used to study the antitumor activity of several vascular-targeting compounds of known efficacy. In particular, this model is used to test the antitumor activity of a clinically used angiogenesis inhibitor (both in isolation, and with a cytotoxic chemotherapeutic) and a vascular disrupting agent currently undergoing clinical trial testing. I demonstrate that the mathematical model can make predictions in agreement with preclinical/clinical data and can also be used to gain more insight into these treatment protocols. The results presented herein suggest that vascular-targeting agents, as currently administered, cannot lead to cancer eradication, although a highly efficacious agent may lead to long-term cancer control.

Coexistence and critical behavior in a lattice model of competing species

In the present paper we study a lattice model of two species competing for the same resources. Monte Carlo simulations for d = 1,2, and 3 show that when resources are easily available both species coexist. However, when the supply of resources is on an intermediate level, the species with slower metabolism becomes extinct. On the other hand, when resources are scarce it is the species with faster metabolism that becomes extinct. The range of coexistence of the two species increases with dimension. We suggest that our model might describe some aspects of the competition between normal and tumor cells. With such an interpretation, examples of tumor remission, recurrence, and different morphologies are presented. In the d = 1 and d = 2 models, we analyze the nature of phase transitions: they are either discontinuous or belong to the directed-percolation universality class, and in some cases they have an active subcritical phase. In the d = 2 case, one of the transitions seems to be characterized by critical exponents that differ from directed-percolation ones, but this transition could be also weakly discontinuous. In the d = 3 version, Monte Carlo simulations are in a good agreement with the solution of the mean-field approximation. This approximation predicts that oscillatory behavior occurs in the present model but only for d ≳ 2. For d ≥ 2, a steady state depends on the initial configuration in some cases.

Toward an Ising model of cancer and beyond

The holy grail of tumor modeling is to formulate theoretical and computational tools that can be utilized in the clinic to predict neoplastic progression and propose individualized optimal treatment strategies to control cancer growth. In order to develop such a predictive model, one must account for the numerous complex mechanisms involved in tumor growth. Here we review the research work that we have done toward the development of an 'Ising model' of cancer. The Ising model is an idealized statistical-mechanical model of ferromagnetism that is based on simple local-interaction rules, but nonetheless leads to basic insights and features of real magnets, such as phase transitions with a critical point. The review begins with a description of a minimalist four-dimensional (three dimensions in space and one in time) cellular automaton (CA) model of cancer in which cells transition between states (proliferative, hypoxic and necrotic) according to simple local rules and their present states, which can viewed as a stripped-down Ising model of cancer. This model is applied to study the growth of glioblastoma multiforme, the most malignant of brain cancers. This is followed by a discussion of the extension of the model to study the effect on the tumor dynamics and geometry of a mutated subpopulation. A discussion of how tumor growth is affected by chemotherapeutic treatment, including induced resistance, is then described. We then describe how to incorporate angiogenesis as well as the heterogeneous and confined environment in which a tumor grows in the CA model. The characterization of the level of organization of the invasive network around a solid tumor using spanning trees is subsequently discussed. Then, we describe open problems and future promising avenues for future research, including the need to develop better molecular-based models that incorporate the true heterogeneous environment over wide range of length and time scales (via imaging data), cell motility, oncogenes, tumor suppressor genes and cell-cell communication. A discussion about the need to bring to bear the powerful machinery of the theory of heterogeneous media to better understand the behavior of cancer in its microenvironment is presented. Finally, we propose the possibility of using optimization techniques, which have been used profitably to understand physical phenomena, in order to devise therapeutic (chemotherapy/radiation) strategies and to understand tumorigenesis itself.

Automated segmentation of cells with IHC membrane staining

This study presents a fully automated membrane segmentation technique for immunohistochemical tissue images with membrane staining, which is a critical task in computerized immunohistochemistry (IHC). Membrane segmentation is particularly tricky in immunohistochemical tissue images because the cellular membranes are visible only in the stained tracts of the cell, while the unstained tracts are not visible. Our automated method provides accurate segmentation of the cellular membranes in the stained tracts and reconstructs the approximate location of the unstained tracts using nuclear membranes as a spatial reference. Accurate cell-by-cell membrane segmentation allows per cell morphological analysis and quantification of the target membrane proteins that is fundamental in several medical applications such as cancer characterization and classification, personalized therapy design, and for any other applications requiring cell morphology characterization. Experimental results on real datasets from different anatomical locations demonstrate the wide applicability and high accuracy of our approach in the context of IHC analysis.

A performance evaluation framework for association mining in spatial data

The evaluation of the process of mining associations is an important and challenging problem in database systems and especially those that store critical data and are used for making critical decisions. Within the context of spatial databases we present an evaluation framework in which we use probability distributions to model spatial regions, and Bayesian networks to model the joint probability distribution and the structural relationships among spatial and non-spatial predicates. We demonstrate the applicability of the proposed framework by evaluating representatives from two well-known approaches that are used for learning associations, i.e., dependency analysis (using statistical tests of independence) and Bayesian methods. By controlling the parameters of the framework we provide extensive comparative results of the performance of the two approaches. We obtain measures of recovery of known associations as a function of the number of samples used, the strength, number and type of associations in the model, the number of spatial predicates associated with a particular non-spatial predicate, the prior probabilities of spatial predicates, the conditional probabilities of the non-spatial predicates, the image registration error, and the parameters that control the sensitivity of the methods. In addition to performance we investigate the processing efficiency of the two approaches.

A sub-cellular viscoelastic model for cell population mechanics

Understanding the biomechanical properties and the effect of biomechanical force on epithelial cells is key to understanding how epithelial cells form uniquely shaped structures in two or three-dimensional space. Nevertheless, with the limitations and challenges posed by biological experiments at this scale, it becomes advantageous to use mathematical and 'in silico' (computational) models as an alternate solution. This paper introduces a single-cell-based model representing the cross section of a typical tissue. Each cell in this model is an individual unit containing several sub-cellular elements, such as the elastic plasma membrane, enclosed viscoelastic elements that play the role of cytoskeleton, and the viscoelastic elements of the cell nucleus. The cell membrane is divided into segments where each segment (or point) incorporates the cell's interaction and communication with other cells and its environment. The model is capable of simulating how cells cooperate and contribute to the overall structure and function of a particular tissue; it mimics many aspects of cellular behavior such as cell growth, division, apoptosis and polarization. The model allows for investigation of the biomechanical properties of cells, cell-cell interactions, effect of environment on cellular clusters, and how individual cells work together and contribute to the structure and function of a particular tissue. To evaluate the current approach in modeling different topologies of growing tissues in distinct biochemical conditions of the surrounding media, we model several key cellular phenomena, namely monolayer cell culture, effects of adhesion intensity, growth of epithelial cell through interaction with extra-cellular matrix (ECM), effects of a gap in the ECM, tensegrity and tissue morphogenesis and formation of hollow epithelial acini. The proposed computational model enables one to isolate the effects of biomechanical properties of individual cells and the communication between cells and their microenvironment while simultaneously allowing for the formation of clusters or sheets of cells that act together as one complex tissue.

In silico models of cancer

Cancer is a complex disease that involves multiple types of biological interactions across diverse physical, temporal, and biological scales. This complexity presents substantial challenges for the characterization of cancer biology, and motivates the study of cancer in the context of molecular, cellular, and physiological systems. Computational models of cancer are being developed to aid both biological discovery and clinical medicine. The development of these in silico models is facilitated by rapidly advancing experimental and analytical tools that generate information-rich, high-throughput biological data. Statistical models of cancer at the genomic, transcriptomic, and pathway levels have proven effective in developing diagnostic and prognostic molecular signatures, as well as in identifying perturbed pathways. Statistically inferred network models can prove useful in settings where data overfitting can be avoided, and provide an important means for biological discovery. Mechanistically based signaling and metabolic models that apply a priori knowledge of biochemical processes derived from experiments can also be reconstructed where data are available, and can provide insight and predictive ability regarding the behavior of these systems. At longer length scales, continuum and agent-based models of the tumor microenvironment and other tissue-level interactions enable modeling of cancer cell populations and tumor progression. Even though cancer has been among the most-studied human diseases using systems approaches, significant challenges remain before the enormous potential of in silico cancer biology can be fully realized.

Front instabilities and invasiveness of simulated 3D avascular tumors

We use the Glazier-Graner-Hogeweg model to simulate three-dimensional (3D), single-phenotype, avascular tumors growing in an homogeneous tissue matrix (TM) supplying a single limiting nutrient. We study the effects of two parameters on tumor morphology: a diffusion-limitation parameter defined as the ratio of the tumor-substrate consumption rate to the substrate-transport rate, and the tumor-TM surface tension. This initial model omits necrosis and oxidative/hypoxic metabolism effects, which can further influence tumor morphology, but our simplified model still shows significant parameter dependencies. The diffusion-limitation parameter determines whether the growing solid tumor develops a smooth (noninvasive) or fingered (invasive) interface, as in our earlier two-dimensional (2D) simulations. The sensitivity of 3D tumor morphology to tumor-TM surface tension increases with the size of the diffusion-limitation parameter, as in 2D. The 3D results are unexpectedly close to those in 2D. Our results therefore may justify using simpler 2D simulations of tumor growth, instead of more realistic but more computationally expensive 3D simulations. While geometrical artifacts mean that 2D sections of connected 3D tumors may be disconnected, the morphologies of 3D simulated tumors nevertheless correlate with the morphologies of their 2D sections, especially for low-surface-tension tumors, allowing the use of 2D sections to partially reconstruct medically-important 3D-tumor structures.

Assessment of Anti-angiogenic Drug in Cancer Therapy

It has been documented in the literature that a solid tumor survives by the generation of micro-vessels around it. This phenomenon is known as angiogenesis. Angiogenesis is governed by two factors, namely Tumor Angiogenic Factor (TAF) secreted by the tumor cells and tissue Fibronectin (FNT) concentration in the extra-cellular space. These two factors help in mobilization of endothelial cells from nearby blood vessels. At the initial phase of angiogenesis, neighboring blood vessels affect in formation of capillary sprouts. In this work, to the authors develop a clinically relevant analytical model that could act as an effective tracing system of tumor growth. The author has performed a quantitative assessment of tumor angiogenesis. This analytical method is a correlation between tumor system and vasculature system through an analytical assessment at peripheral blood circulatory of tumor milieu.

Safety margin in irradiation of colorectal liver metastases: assessment of the control dose of micrometastases

Backround

Micrometastases of colorectal liver metastases are present in up to 50% of lesions. In this study we sought to determine the threshold dose for local control of occult micrometastases in patients undergoing CT (computed tomography)-guided brachytherapy of colorectal liver metastases.

Materials and methods

Nineteen patients demonstrated 34 local tumor recurrences originating from micrometastases after CT-guided brachytherapy of 27 colorectal liver metastases. We considered a local tumor recurrence as originating from a micrometastasis if tumor regrowth occurred adjacent to a formerly irradiated lesion and the distance of the 3D isocenter of the new lesion was ≤ 23.5 mm from the previous tumor margin. Follow-up MRI was fused with the planning-CT and dosimetry data. Two reviewers independently indicated the dose exposure at the isocenter of the micrometastases. Statistical analysis included an analysis of variance (ANOVA) using backward selection. 95% tolerance intervals with coverage of 87.5 and 75% of the data of the normal distribution were calculated.

Results

The median distance of the micrometastases to the margin of the originating colorectal metastases was 8.75 mm (1-21 mm). Dose exposure at the isocenter was 12.25 Gy (7-19.8) in median. We stratified according to the distance from the isocenter to the initial tumor margin: ≤ 9 mm, > 9-15 mm and > 15 mm. The median dose in the according isocenters was 13.18, 11.6 and 11.85 Gy. The threshold dose failing to prevent micrometastasis growth was sigificantly higher in a subgroup of lesions with ≤ 9 mm distance as compared to > 15 mm (13.18 vs 11.85 Gy). Adjuvant chemotherapy correlated with greater distance of micrometastasis growth to the tumor but not with the threshold dose.

Conclusion

To prevent loss of local tumor control by continuous growth of micrometastases a threshold dose of 15,4 Gy (single fraction) should be delivered at a distance of 21 mm to the gross tumor margin.

Promotion of variant human mammary epithelial cell outgrowth by ionizing radiation: an agent-based model supported by in vitro studies

Introduction

Most human mammary epithelial cells (HMEC) cultured from histologically normal breast tissues enter a senescent state termed stasis after 5 to 20 population doublings. These senescent cells display increased size, contain senescence associated β-galactosidase activity, and express cyclin-dependent kinase inhibitor, p16^INK4A (CDKN2A; p16). However, HMEC grown in a serum-free medium, spontaneously yield, at low frequency, variant (v) HMEC that are capable of long-term growth and are susceptible to genomic instability. We investigated whether ionizing radiation, which increases breast cancer risk in women, affects the rate of vHMEC outgrowth.

Methods

Pre-stasis HMEC cultures were exposed to 5 to 200 cGy of sparsely (X- or γ-rays) or densely (1 GeV/amu ⁵⁶Fe) ionizing radiation. Proliferation (bromodeoxyuridine incorporation), senescence (senescence-associated β-galactosidase activity), and p16 expression were assayed in subcultured irradiated or unirradiated populations four to six weeks following radiation exposure, when patches of vHMEC became apparent. Long-term growth potential and p16 promoter methylation in subsequent passages were also monitored. Agent-based modeling, incorporating a simple set of rules and underlying assumptions, was used to simulate vHMEC outgrowth and evaluate mechanistic hypotheses.

Results

Cultures derived from irradiated cells contained significantly more vHMEC, lacking senescence associated β-galactosidase or p16 expression, than cultures derived from unirradiated cells. As expected, post-stasis vHMEC cultures derived from both unirradiated and irradiated cells exhibited more extensive methylation of the p16 gene than pre-stasis HMEC cultures. However, the extent of methylation of individual CpG sites in vHMEC samples did not correlate with passage number or treatment. Exposure to sparsely or densely ionizing radiation elicited similar increases in the numbers of vHMEC compared to unirradiated controls. Agent-based modeling indicated that radiation-induced premature senescence of normal HMEC most likely accelerated vHMEC outgrowth through alleviation of spatial constraints. Subsequent experiments using defined co-cultures of vHMEC and senescent cells supported this mechanism.

Conclusions

Our studies indicate that ionizing radiation can promote the outgrowth of epigenetically altered cells with pre-malignant potential.

Image guided personalization of reaction-diffusion type tumor growth models using modified anisotropic eikonal equations

Reaction-diffusion based tumor growth models have been widely used in the literature for modeling the growth of brain gliomas. Lately, recent models have started integrating medical images in their formulation. Including different tissue types, geometry of the brain and the directions of white matter fiber tracts improved the spatial accuracy of reaction-diffusion models. The adaptation of the general model to the specific patient cases on the other hand has not been studied thoroughly yet. In this paper, we address this adaptation. We propose a parameter estimation method for reaction-diffusion tumor growth models using time series of medical images. This method estimates the patient specific parameters of the model using the images of the patient taken at successive time instances. The proposed method formulates the evolution of the tumor delineation visible in the images based on the reaction-diffusion dynamics; therefore, it remains consistent with the information available. We perform thorough analysis of the method using synthetic tumors and show important couplings between parameters of the reaction-diffusion model. We show that several parameters can be uniquely identified in the case of fixing one parameter, namely the proliferation rate of tumor cells. Moreover, regardless of the value the proliferation rate is fixed to, the speed of growth of the tumor can be estimated in terms of the model parameters with accuracy. We also show that using the model-based speed, we can simulate the evolution of the tumor for the specific patient case. Finally, we apply our method to two real cases and show promising preliminary results.

In silico models of cancer

Cancer is a complex disease that involves multiple types of biological interactions across diverse physical, temporal, and biological scales. This complexity presents substantial challenges for the characterization of cancer biology, and motivates the study of cancer in the context of molecular, cellular, and physiological systems. Computational models of cancer are being developed to aid both biological discovery and clinical medicine. The development of these in silico models is facilitated by rapidly advancing experimental and analytical tools that generate information-rich, high-throughput biological data. Statistical models of cancer at the genomic, transcriptomic, and pathway levels have proven effective in developing diagnostic and prognostic molecular signatures, as well as in identifying perturbed pathways. Statistically inferred network models can prove useful in settings where data overfitting can be avoided, and provide an important means for biological discovery. Mechanistically based signaling and metabolic models that apply a priori knowledge of biochemical processes derived from experiments can also be reconstructed where data are available, and can provide insight and predictive ability regarding the behavior of these systems. At longer length scales, continuum and agent-based models of the tumor microenvironment and other tissue-level interactions enable modeling of cancer cell populations and tumor progression. Even though cancer has been among the most-studied human diseases using systems approaches, significant challenges remain before the enormous potential of in silico cancer biology can be fully realized.

Nonlinear modelling of cancer: bridging the gap between cells and tumours

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

Growing heterogeneous tumors in silico

An in silico tool that can be utilized in the clinic to predict neoplastic progression and propose individualized treatment strategies is the holy grail of computational tumor modeling. Building such a tool requires the development and successful integration of a number of biophysical and mathematical models. In this paper, we work toward this long-term goal by formulating a cellular automaton model of tumor growth that accounts for several different inter-tumor processes and host-tumor interactions. In particular, the algorithm couples the remodeling of the microvasculature with the evolution of the tumor mass and considers the impact that organ-imposed physical confinement and environmental heterogeneity have on tumor size and shape. Furthermore, the algorithm is able to account for cell-level heterogeneity, allowing us to explore the likelihood that different advantageous and deleterious mutations survive in the tumor cell population. This computational tool we have built has a number of applications in its current form in both predicting tumor growth and predicting response to treatment. Moreover, the latent power of our algorithm is that it also suggests other tumor-related processes that need to be accounted for and calls for the conduction of new experiments to validate the model's predictions.

Introduction of hypermatrix and operator notation into a discrete mathematics simulation model of malignant tumour response to therapeutic schemes in vivo. Some operator properties

The tremendous rate of accumulation of experimental and clinical knowledge pertaining to cancer dictates the development of a theoretical framework for the meaningful integration of such knowledge at all levels of biocomplexity. In this context our research group has developed and partly validated a number of spatiotemporal simulation models of in vivo tumour growth and in particular tumour response to several therapeutic schemes. Most of the modeling modules have been based on discrete mathematics and therefore have been formulated in terms of rather complex algorithms (e.g. in pseudocode and actual computer code). However, such lengthy algorithmic descriptions, although sufficient from the mathematical point of view, may render it difficult for an interested reader to readily identify the sequence of the very basic simulation operations that lie at the heart of the entire model. In order to both alleviate this problem and at the same time provide a bridge to symbolic mathematics, we propose the introduction of the notion of hypermatrix in conjunction with that of a discrete operator into the already developed models. Using a radiotherapy response simulation example we demonstrate how the entire model can be considered as the sequential application of a number of discrete operators to a hypermatrix corresponding to the dynamics of the anatomic area of interest. Subsequently, we investigate the operators’ commutativity and outline the “summarize and jump” strategy aiming at efficiently and realistically address multilevel biological problems such as cancer. In order to clarify the actual effect of the composite discrete operator we present further simulation results which are in agreement with the outcome of the clinical study RTOG 83–02, thus strengthening the reliability of the model developed.

At the biological modeling and simulation frontier

We provide a rationale for and describe examples of synthetic modeling and simulation (M&S) of biological systems. We explain how synthetic methods are distinct from familiar inductive methods. Synthetic M&S is a means to better understand the mechanisms that generate normal and disease-related phenomena observed in research, and how compounds of interest interact with them to alter phenomena. An objective is to build better, working hypotheses of plausible mechanisms. A synthetic model is an extant hypothesis: execution produces an observable mechanism and phenomena. Mobile objects representing compounds carry information enabling components to distinguish between them and react accordingly when different compounds are studied simultaneously. We argue that the familiar inductive approaches contribute to the general inefficiencies being experienced by pharmaceutical R&D, and that use of synthetic approaches accelerates and improves R&D decision-making and thus the drug development process. A reason is that synthetic models encourage and facilitate abductive scientific reasoning, a primary means of knowledge creation and creative cognition. When synthetic models are executed, we observe different aspects of knowledge in action from different perspectives. These models can be tuned to reflect differences in experimental conditions and individuals, making translational research more concrete while moving us closer to personalized medicine.

Rule-Based Simulation of Multi-Cellular Biological Systems-A Review of Modeling Techniques

Emergent behaviors of multi-cellular biological systems (MCBS) result from the behaviors of each individual cells and their interactions with other cells and with the environment. Modeling MCBS requires incorporating these complex interactions among the individual cells and the environment. Modeling approaches for MCBS can be grouped into two categories: continuum models and cell-based models. Continuum models usually take the form of partial differential equations, and the model equations provide insight into the relationship among the components in the system. Cell-based models simulate each individual cell behavior and interactions among them enabling the observation of the emergent system behavior. This review focuses on the cell-based models of MCBS, and especially, the technical aspect of the rule-based simulation method for MCBS is reviewed. How to implement the cell behaviors and the interactions with other cells and with the environment into the computational domain is discussed. The cell behaviors reviewed in this paper are division, migration, apoptosis/necrosis, and differentiation. The environmental factors such as extracellular matrix, chemicals, microvasculature, and forces are also discussed. Application examples of these cell behaviors and interactions are presented.

Rule-Based Simulation of Multi-Cellular Biological Systems-A Review of Modeling Techniques

Emergent behaviors of multi-cellular biological systems (MCBS) result from the behaviors of each individual cells and their interactions with other cells and with the environment. Modeling MCBS requires incorporating these complex interactions among the individual cells and the environment. Modeling approaches for MCBS can be grouped into two categories: continuum models and cell-based models. Continuum models usually take the form of partial differential equations, and the model equations provide insight into the relationship among the components in the system. Cell-based models simulate each individual cell behavior and interactions among them enabling the observation of the emergent system behavior. This review focuses on the cell-based models of MCBS, and especially, the technical aspect of the rule-based simulation method for MCBS is reviewed. How to implement the cell behaviors and the interactions with other cells and with the environment into the computational domain is discussed. The cell behaviors reviewed in this paper are division, migration, apoptosis/necrosis, and differentiation. The environmental factors such as extracellular matrix, chemicals, microvasculature, and forces are also discussed. Application examples of these cell behaviors and interactions are presented.

Front instabilities and invasiveness of simulated avascular tumors

We study the interface morphology of a 2D simulation of an avascular tumor composed of identical cells growing in an homogeneous healthy tissue matrix (TM), in order to understand the origin of the morphological changes often observed during real tumor growth. We use the Glazier-Graner-Hogeweg model, which treats tumor cells as extended, deformable objects, to study the effects of two parameters: a dimensionless diffusion-limitation parameter defined as the ratio of the tumor consumption rate to the substrate transport rate, and the tumor-TM surface tension. We model TM as a nondiffusing field, neglecting the TM pressure and haptotactic repulsion acting on a real growing tumor; thus, our model is appropriate for studying tumors with highly motile cells, e.g., gliomas. We show that the diffusion-limitation parameter determines whether the growing tumor develops a smooth (noninvasive) or fingered (invasive) interface, and that the sensitivity of tumor morphology to tumor-TM surface tension increases with the size of the dimensionless diffusion-limitation parameter. For large diffusion-limitation parameters, we find a transition (missed in previous work) between dendritic structures, produced when tumor-TM surface tension is high, and seaweed-like structures, produced when tumor-TM surface tension is low. This observation leads to a direct analogy between the mathematics and dynamics of tumors and those observed in nonbiological directional solidification. Our results are also consistent with the biological observation that hypoxia promotes invasive growth of tumor cells by inducing higher levels of receptors for scatter factors that weaken cell-cell adhesion and increase cell motility. These findings suggest that tumor morphology may have value in predicting the efficiency of antiangiogenic therapy in individual patients.

Microenvironment driven invasion: a multiscale multimodel investigation

Cancer is a complex, multiscale process, in which genetic mutations occurring at a subcellular level manifest themselves as functional and morphological changes at the cellular and tissue scale. The importance of interactions between tumour cells and their microenvironment is currently of great interest in experimental as well as computational modelling. Both the immediate microenvironment (e.g. cell-cell signalling or cell-matrix interactions) and the extended microenvironment (e.g. nutrient supply or a host tissue structure) are thought to play crucial roles in both tumour progression and suppression. In this paper we focus on tumour invasion, as defined by the emergence of a fingering morphology, which has previously been shown to be dependent upon harsh microenvironmental conditions. Using three different modelling approaches at two different spatial scales we examine the impact of nutrient availability as a driving force for invasion. Specifically we investigate how cell metabolism (the intrinsic rate of nutrient consumption and cell resistance to starvation) influences the growing tumour. We also discuss how dynamical changes in genetic makeup and morphological characteristics, of the tumour population, are driven by extreme changes in nutrient supply during tumour development. The simulation results indicate that aggressive phenotypes produce tumour fingering in poor nutrient, but not rich, microenvironments. The implication of these results is that an invasive outcome appears to be co-dependent upon the evolutionary dynamics of the tumour population driven by the microenvironment.

Multi-scale, multi-resolution brain cancer modeling

In advancing discrete-based computational cancer models towards clinical applications, one faces the dilemma of how to deal with an ever growing amount of biomedical data that ought to be incorporated eventually in one form or another. Model scalability becomes of paramount interest. In an effort to start addressing this critical issue, here, we present a novel multi-scale and multi-resolution agent-based in silico glioma model. While 'multi-scale' refers to employing an epidermal growth factor receptor (EGFR)-driven molecular network to process cellular phenotypic decisions within the micro-macroscopic environment, 'multi-resolution' is achieved through algorithms that classify cells to either active or inactive spatial clusters, which determine the resolution they are simulated at. The aim is to assign computational resources where and when they matter most for maintaining or improving the predictive power of the algorithm, onto specific tumor areas and at particular times. Using a previously described 2D brain tumor model, we have developed four different computational methods for achieving the multi-resolution scheme, three of which are designed to dynamically train on the high-resolution simulation that serves as control. To quantify the algorithms' performance, we rank them by weighing the distinct computational time savings of the simulation runs versus the methods' ability to accurately reproduce the high-resolution results of the control. Finally, to demonstrate the flexibility of the underlying concept, we show the added value of combining the two highest-ranked methods. The main finding of this work is that by pursuing a multi-resolution approach, one can reduce the computation time of a discrete-based model substantially while still maintaining a comparably high predictive power. This hints at even more computational savings in the more realistic 3D setting over time, and thus appears to outline a possible path to achieve scalability for the all-important clinical translation.

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.

On the basic reproduction number and the topological properties of the contact network: An epidemiological study in mainly locally connected cellular automata

We study the spreading of contagious diseases in a population of constant size using susceptible-infective-recovered (SIR) models described in terms of ordinary differential equations (ODEs) and probabilistic cellular automata (PCA). In the PCA model, each individual (represented by a cell in the lattice) is mainly locally connected to others. We investigate how the topological properties of the random network representing contacts among individuals influence the transient behavior and the permanent regime of the epidemiological system described by ODE and PCA. Our main conclusions are: (1) the basic reproduction number (commonly called R0) related to a disease propagation in a population cannot be uniquely determined from some features of transient behavior of the infective group; (2) R0 cannot be associated to a unique combination of clustering coefficient and average shortest path length characterizing the contact network. We discuss how these results can embarrass the specification of control strategies for combating disease propagations.

Designing convergent cellular automata

Cellular automata (CA) have been used by biologists to study dynamic non-linear systems where the interaction between cell behaviour and end-pattern is investigated. It is difficult to achieve convergence of a CA towards a specific static pattern and a common solution is to use genetic algorithms and evolve a ruleset that describes cell behaviour. This paper presents an alternative means of designing CA to converge to specific static patterns. A matrix model is introduced and analysed then a design algorithm is demonstrated. The algorithm is significantly less computationally intensive than equivalent evolutionary algorithms, and not limited in scale, complexity or number of dimensions.

In silico cancer modeling: is it ready for prime time?

At the dawn of the era of personalized, systems-driven medicine, computational or in silico modeling and the simulation of disease processes is becoming increasingly important for hypothesis generation and data integration in both experiments and clinics alike. Arguably, the use of these techniques is nowhere more visible than in oncology. To illustrate the field's vast potential, as well as its current limitations, we briefly review selected studies on modeling malignant brain tumors. Implications for clinical practice, and for clinical trial design and outcome prediction, are also discussed.

The role of extracellular matrix in glioma invasion: a cellular Potts model approach

In this work, a cellular Potts model based on the differential adhesion hypothesis is employed to analyze the relative importance of select cell-cell and cell-extracellular matrix (ECM) contacts in glioma invasion. To perform these simulations, three types of cells and two ECM components are included. The inclusion of explicit ECM with an inhomogeneous fibrous component and a homogeneously dispersed afibrous component allows exploration of the importance of relative energies of cell-cell and cell-ECM contacts in a variety of environments relevant to in vitro and in vivo experimental investigations of glioma invasion. Simulations performed here focus chiefly on reproducing findings of in vitro experiments on glioma spheroids embedded in collagen I gels. For a given range and set ordering of energies associated with key cell-cell and cell-ECM interactions, our model qualitatively reproduces the dispersed glioma invasion patterns found for most glioma cell lines embedded as spheroids in collagen I gels of moderate concentration. In our model, we find that invasion is maximized at intermediate collagen concentrations, as occurs experimentally. This effect is seen more strongly in model gels composed of short collagen fibers than in those composed of long fibers, which retain significant connectivity even at low density. Additional simulations in aligned model matrices further elucidate how matrix structure dictates invasive patterns. Finally, simulations that allow invading cells to both dissolve and deposit ECM components demonstrate how Q-Potts models may be elaborated to allow active cell alteration of their surroundings. The model employed here provides a quantitative framework with which to bound the relative values of cell-cell and cell-ECM interactions and investigate how varying the magnitude and type of these interactions, as well as ECM structure, could potentially curtail glioma invasion.

Simulating tumor growth in confined heterogeneous environments

The holy grail of computational tumor modeling is to develop a simulation tool that can be utilized in the clinic to predict neoplastic progression and propose individualized optimal treatment strategies. In order to develop such a predictive model, one must account for many of the complex processes involved in tumor growth. One interaction that has not been incorporated into computational models of neoplastic progression is the impact that organ-imposed physical confinement and heterogeneity have on tumor growth. For this reason, we have taken a cellular automaton algorithm that was originally designed to simulate spherically symmetric tumor growth and generalized the algorithm to incorporate the effects of tissue shape and structure. We show that models that do not account for organ/tissue geometry and topology lead to false conclusions about tumor spread, shape and size. The impact that confinement has on tumor growth is more pronounced when a neoplasm is growing close to, versus far from, the confining boundary. Thus, any clinical simulation tool of cancer progression must not only consider the shape and structure of the organ in which a tumor is growing, but must also consider the location of the tumor within the organ if it is to accurately predict neoplastic growth dynamics.

Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching

We develop a thermodynamically consistent mixture model for avascular solid tumor growth which takes into account the effects of cell-to-cell adhesion, and taxis inducing chemical and molecular species. The mixture model is well-posed and the governing equations are of Cahn-Hilliard type. When there are only two phases, our asymptotic analysis shows that earlier single-phase models may be recovered as limiting cases of a two-phase model. To solve the governing equations, we develop a numerical algorithm based on an adaptive Cartesian block-structured mesh refinement scheme. A centered-difference approximation is used for the space discretization so that the scheme is second order accurate in space. An implicit discretization in time is used which results in nonlinear equations at implicit time levels. We further employ a gradient stable discretization scheme so that the nonlinear equations are solvable for very large time steps. To solve those equations we use a nonlinear multilevel/multigrid method which is of an optimal order O(N) where N is the number of grid points. Spherically symmetric and fully two dimensional nonlinear numerical simulations are performed. We investigate tumor evolution in nutrient-rich and nutrient-poor tissues. A number of important results have been uncovered. For example, we demonstrate that the tumor may suffer from taxis-driven fingering instabilities which are most dramatic when cell proliferation is low, as predicted by linear stability theory. This is also observed in experiments. This work shows that taxis may play a role in tumor invasion and that when nutrient plays the role of a chemoattractant, the diffusional instability is exacerbated by nutrient gradients. Accordingly, we believe this model is capable of describing complex invasive patterns observed in experiments.

Efficient Monte Carlo modelling of individual tumour cell propagation for hypoxic head and neck cancer

A Monte Carlo tumour model has been developed to simulate tumour cell propagation for head and neck squamous cell carcinoma. The model aims to eventually provide a radiobiological tool for radiation oncology clinicians to plan patient treatment schedules based on properties of the individual tumour. The inclusion of an oxygen distribution amongst the tumour cells enables the model to incorporate hypoxia and other associated parameters, which affect tumour growth. The object oriented program FORTRAN 95 has been used to create the model algorithm, with Monte Carlo methods being employed to randomly assign many of the cell parameters from probability distributions. Hypoxia has been implemented through random assignment of partial oxygen pressure values to individual cells during tumour growth, based on in vivo Eppendorf probe experimental data. The accumulation of up to 10 million virtual tumour cells in 15 min of computer running time has been achieved. The stem cell percentage and the degree of hypoxia are the parameters which most influence the final tumour growth rate. For a tumour with a doubling time of 40 days, the final stem cell percentage is approximately 1% of the total cell population. The effect of hypoxia on the tumour growth rate is significant. Using a hypoxia induced cell quiescence limit which affects 50% of cells with and oxygen levels less than 1 mm Hg, the tumour doubling time increases to over 200 days and the time of tumour growth for a clinically detectable tumour (10(9) cells) increases from 3 to 8 years. A biologically plausible Monte Carlo model of hypoxic head and neck squamous cell carcinoma tumour growth has been developed for real time assessment of the effects of multiple biological parameters which impact upon the response of the individual patient to fractionated radiotherapy.

An image-driven parameter estimation problem for a reaction-diffusion glioma growth model with mass effects

We present a framework for modeling gliomas growth and their mechanical impact on the surrounding brain tissue (the so-called, mass-effect). We employ an Eulerian continuum approach that results in a strongly coupled system of nonlinear Partial Differential Equations (PDEs): a reaction-diffusion model for the tumor growth and a piecewise linearly elastic material for the background tissue. To estimate unknown model parameters and enable patient-specific simulations we formulate and solve a PDE-constrained optimization problem. Our two main goals are the following: (1) to improve the deformable registration from images of brain tumor patients to a common stereotactic space, thereby assisting in the construction of statistical anatomical atlases; and (2) to develop predictive capabilities for glioma growth, after the model parameters are estimated for a given patient. To our knowledge, this is the first attempt in the literature to introduce an adjoint-based, PDE-constrained optimization formulation in the context of image-driven modeling spatio-temporal tumor evolution. In this paper, we present the formulation, and the solution method and we conduct 1D numerical experiments for preliminary evaluation of the overall formulation/methodology.

Integrative mathematical oncology

Cancer research attracts broad resources and scientists from many disciplines, and has produced some impressive advances in the treatment and understanding of this disease. However, a comprehensive mechanistic view of the cancer process remains elusive. To achieve this it seems clear that one must assemble a physically integrated team of interdisciplinary scientists that includes mathematicians, to develop mathematical models of tumorigenesis as a complex process. Examining these models and validating their findings by experimental and clinical observations seems to be one way to reconcile molecular reductionist with quantitative holistic approaches and produce an integrative mathematical oncology view of cancer progression.

Biocomputing: numerical simulation of glioblastoma growth using diffusion tensor imaging

Glioblastoma multiforma (GBM) is one of the most aggressive tumors of the central nervous system. It can be represented by two components: a proliferative component with a mass effect on brain structures and an invasive component. GBM has a distinct pattern of spread showing a preferential growth in the white fiber direction for the invasive component. By using the architecture of white matter fibers, we propose a new model to simulate the growth of GBM. This architecture is estimated by diffusion tensor imaging in order to determine the preferred direction for the diffusion component. It is then coupled with a mechanical component. To set up our growth model, we make a brain atlas including brain structures with a distinct response to tumor aggressiveness, white fiber diffusion tensor information and elasticity. In this atlas, we introduce a virtual GBM with a mechanical component coupled with a diffusion component. These two components are complementary, and can be tuned independently. Then, we tune the parameter set of our model with an MRI patient. We have compared simulated growth (initialized with the MRI patient) with observed growth six months later. The average and the odd ratio of image difference between observed and simulated images are computed. Displacements of reference points are compared to those simulated by the model. The results of our simulation have shown a good correlation with tumor growth, as observed on an MRI patient. Different tumor aggressiveness can also be simulated by tuning additional parameters. This work has demonstrated that modeling the complex behavior of brain tumors is feasible and will account for further validation of this new conceptual approach.

Reconstruction of DNA sequences using genetic algorithms and cellular automata: towards mutation prediction?

Change of DNA sequence that fuels evolution is, to a certain extent, a deterministic process because mutagenesis does not occur in an absolutely random manner. So far, it has not been possible to decipher the rules that govern DNA sequence evolution due to the extreme complexity of the entire process. In our attempt to approach this issue we focus solely on the mechanisms of mutagenesis and deliberately disregard the role of natural selection. Hence, in this analysis, evolution refers to the accumulation of genetic alterations that originate from mutations and are transmitted through generations without being subjected to natural selection. We have developed a software tool that allows modelling of a DNA sequence as a one-dimensional cellular automaton (CA) with four states per cell which correspond to the four DNA bases, i.e. A, C, T and G. The four states are represented by numbers of the quaternary number system. Moreover, we have developed genetic algorithms (GAs) in order to determine the rules of CA evolution that simulate the DNA evolution process. Linear evolution rules were considered and square matrices were used to represent them. If DNA sequences of different evolution steps are available, our approach allows the determination of the underlying evolution rule(s). Conversely, once the evolution rules are deciphered, our tool may reconstruct the DNA sequence in any previous evolution step for which the exact sequence information was unknown. The developed tool may be used to test various parameters that could influence evolution. We describe a paradigm relying on the assumption that mutagenesis is governed by a near-neighbour-dependent mechanism. Based on the satisfactory performance of our system in the deliberately simplified example, we propose that our approach could offer a starting point for future attempts to understand the mechanisms that govern evolution. The developed software is open-source and has a user-friendly graphical input interface.

Classifying the expansion kinetics and critical surface dynamics of growing cell populations

We systematically study the growth kinetics and the critical surface dynamics of cell monolayers by a class of computationally efficient cellular automaton models avoiding lattice artifacts. Our numerically derived front velocity relationship indicates the limitations of the Fisher-Kolmogorov-Petrovskii-Piskounov equation for tumor growth simulations. The critical surface dynamics corresponds to the Kardar-Parisi-Zhang universality class, which disagrees with the interpretation by Bru et al. of their experimental observations as generic molecular-beam-epitaxy-like growth, questioning their conjecture that a successful therapy should lead away from molecular beam epitaxy.

A cellular automaton to model the development of primary shoot meristems of Arabidopsis thaliana

Development of organisms is a very complex process in which a lot of gene networks of different cell types are integrated. Development of a cellular automaton (Ermentrout and Edelshtein-Keshet, J Theor Biol 160:97-133, 1993) that models the morphodynamics of different cell types is the first step in understanding and analysis of the regulatory mechanisms underlying the functioning of developmental gene networks. A model of a cellular automaton has been developed, which simulates the embryonic development of shoot meristem in Arabidopsis thaliana. The model adequately describes the basic stages in development of this organ in wild and mutant types.

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.

Ab initio phenomenological simulation of the growth of large tumor cell populations

In a previous paper we have introduced a phenomenological model of cell metabolism and of the cell cycle to simulate the behavior of large tumor cell populations (Chignola and Milotti 2005 Phys. Biol. 2 8). Here we describe a refined and extended version of the model that includes some of the complex interactions between cells and their surrounding environment. The present version takes into consideration several additional energy-consuming biochemical pathways such as protein and DNA synthesis, the tuning of extracellular pH and of the cell membrane potential. The control of the cell cycle, which was previously modeled by means of ad hoc thresholds, has been directly addressed here by considering checkpoints from proteins that act as targets for phosphorylation on multiple sites. As simulated cells grow, they can now modify the chemical composition of the surrounding environment which in turn acts as a feedback mechanism to tune cell metabolism and hence cell proliferation: in this way we obtain growth curves that match quite well those observed in vitro with human leukemia cell lines. The model is strongly constrained and returns results that can be directly compared with actual experiments, because it uses parameter values in narrow ranges estimated from experimental data, and in perspective we hope to utilize it to develop in silico studies of the growth of very large tumor cell populations (10(6) cells or more) and to support experimental research. In particular, the program is used here to make predictions on the behavior of cells grown in a glucose-poor medium: these predictions are confirmed by experimental observation.

Predictive oncology: a review of multidisciplinary, multiscale in silico modeling linking phenotype, morphology and growth

Empirical evidence and theoretical studies suggest that the phenotype, i.e., cellular- and molecular-scale dynamics, including proliferation rate and adhesiveness due to microenvironmental factors and gene expression that govern tumor growth and invasiveness, also determine gross tumor-scale morphology. It has been difficult to quantify the relative effect of these links on disease progression and prognosis using conventional clinical and experimental methods and observables. As a result, successful individualized treatment of highly malignant and invasive cancers, such as glioblastoma, via surgical resection and chemotherapy cannot be offered and outcomes are generally poor. What is needed is a deterministic, quantifiable method to enable understanding of the connections between phenotype and tumor morphology. Here, we critically assess advantages and disadvantages of recent computational modeling efforts (e.g., continuum, discrete, and cellular automata models) that have pursued this understanding. Based on this assessment, we review a multiscale, i.e., from the molecular to the gross tumor scale, mathematical and computational "first-principle" approach based on mass conservation and other physical laws, such as employed in reaction-diffusion systems. Model variables describe known characteristics of tumor behavior, and parameters and functional relationships across scales are informed from in vitro, in vivo and ex vivo biology. We review the feasibility of this methodology that, once coupled to tumor imaging and tumor biopsy or cell culture data, should enable prediction of tumor growth and therapy outcome through quantification of the relation between the underlying dynamics and morphological characteristics. In particular, morphologic stability analysis of this mathematical model reveals that tumor cell patterning at the tumor-host interface is regulated by cell proliferation, adhesion and other phenotypic characteristics: histopathology information of tumor boundary can be inputted to the mathematical model and used as a phenotype-diagnostic tool to predict collective and individual tumor cell invasion of surrounding tissue. This approach further provides a means to deterministically test effects of novel and hypothetical therapy strategies on tumor behavior.

An agent-based computational approach for representing aspects of in vitro multi-cellular tumor spheroid growth

There have been many efforts to explain and simulate tumor growth with mathematical and computational models. However, none have systematically examined the behaviors of tumor spheroids during growth. The interactions among tumor cells during growth are also not well understood. We have implemented an agent-based computational approach to study the macro- and micro- behaviors of avascular tumor spheroids during growth. Our simulations of tumor spheroid growth begin with a single tumor cell in optimal environmental conditions. We observe an initial phase of rapid growth, during which the shape of the collective approximates a spheroid. Subsequently a characteristic layered structure develops, consisting of an outermost proliferating cell layer, an intermediate quiescent cell layer, and a central necrotic core. These behaviors of our in silico spheroids map well to experimental in vitro observations.

A pattern-theoretic characterization of biological growth

Mathematical and statistical modeling of biological growth is an important problem in medical diagnostics. Here, we seek tools to analyze changes in anatomical parts using images collected over time. We introduce a structured model, called Growth by Random Iterated Diffeomorphisms (GRID), that treats a cumulative growth deformation as a composition of several elementary deformations. Each elementary deformation applies to a small region by capturing deformation local to that region and is characterized by a seed and a radial deformation pattern around that seed. These GRID variables--seed locations and radial deformation patterns---are estimated from observed images in two steps: 1) estimate a cumulative deformation over an observation interval; 2) estimate GRID variables using maximum-likelihood criterion from this estimated cumulative deformation. We demonstrate this framework using an MRI image data of a rat's brain growth. For future statistical analysis, we propose a time-varying Poisson process for the seed placements and a random drawing from a predetermined catalog of deformations for the radial deformation patterns.

Cellular automaton simulation examining progenitor hierarchy structure effects on mammary ductal carcinoma in situ

A computer simulation is used to model ductal carcinoma in situ, a form of non-invasive breast cancer. The simulation uses known histological morphology, cell types, and stochastic cell proliferation to evolve tumorous growth within a duct. The ductal simulation is based on a hybrid cellular automaton design using genetic rules to determine each cell's behavior. The genetic rules are a mutable abstraction that demonstrate genetic heterogeneity in a population. Our goal was to examine the role (if any) that recently discovered mammary stem cell hierarchies play in genetic heterogeneity, DCIS initiation and aggressiveness. Results show that simpler progenitor hierarchies result in greater genetic heterogeneity and evolve DCIS significantly faster. However, the more complex progenitor hierarchy structure was able to sustain the rapid reproduction of a cancer cell population for longer periods of time.

Modeling the effects of vasculature evolution on early brain tumor growth

Mathematical modeling of both tumor growth and angiogenesis have been active areas of research for the past several decades. Such models can be classified into one of two categories: those that analyze the remodeling of the vasculature while ignoring changes in the tumor mass, and those that predict tumor expansion in the presence of a non-evolving vasculature. However, it is well accepted that vasculature remodeling and tumor growth strongly depend on one another. For this reason, we have developed a two-dimensional hybrid cellular automaton model of early brain tumor growth that couples the remodeling of the microvasculature with the evolution of the tumor mass. A system of reaction-diffusion equations has been developed to track the concentration of vascular endothelial growth factor (VEGF), Ang-1, Ang-2, their receptors and their complexes in space and time. The properties of the vasculature and hence of each cell are determined by the relative concentrations of these key angiogenic factors. The model exhibits an angiogenic switch consistent with experimental observations on the upregulation of angiogenesis. Particularly, we show that if the pathways that produce and respond to VEGF and the angiopoietins are properly functioning, angiogenesis is initiated and a tumor can grow to a macroscopic size. However, if the VEGF pathway is inhibited, angiogenesis does not occur and tumor growth is thwarted beyond 1-2mm in size. Furthermore, we show that tumor expansion can occur in well-vascularized environments even when angiogenesis is inhibited, suggesting that anti-angiogenic therapies may not be sufficient to eliminate a population of actively dividing malignant cells.