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.     

Modeling of self-organized avascular tumor growth with a hybrid cellular automaton

Pattern formation in multicellular spheroids is addressed with a hybrid lattice-gas cellular automaton model. Multicellular spheroids serve as experimental model system for the study of avascular tumor growth. Typically, multicellular spheroids consist of a necrotic core surrounded by rings of quiescent and proliferating tumor cells, respectively. Furthermore, after an initial exponential growth phase further spheroid growth is significantly slowed down even if further nutrient is supplied. The cellular automaton model explicitly takes into account mitosis, apoptosis and necrosis as well as nutrient consumption and a diffusible signal that is emitted by cells becoming necrotic. All cells follow identical interaction rules. The necrotic signal induces a chemotactic migration of tumor cells towards maximal signal concentrations. Starting from a small number of tumor cells automaton simulations exhibit the self-organized formation of a layered structure consisting of a necrotic core, a ring of quiescent tumor cells and a thin outer ring of proliferating tumor cells.     

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.     

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.     

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.     

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.     

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.     

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.     

Population dynamics in tumor growth modeling

A new approach based on local interaction between cancer and tissue cells was applied to the problem of the onset and growth of solid tumors in homogeneous tissues and effects associated with dramatic changes in tumor growth after crossing the boundary between different tissues. The characteristic sizes and growth rates of spherical tumors, the points of the beginning and the end of spherical growth, and the further development of complex structures from the spherical ones (rough interface between the tumor and the host tissue, elongate outgrowths, dendritic structures, and metastases) were inferred assuming that the reproduction rate of a population of cancer cells is a nonmonotone function of their local concentration and thus of the local curvature of the tumor surface.     

The universal dynamics of tumor growth

Scaling techniques were used to analyze the fractal nature of colonies of 15 cell lines growing in vitro as well as of 16 types of tumor developing in vivo. All cell colonies were found to exhibit exactly the same growth dynamics-which correspond to the molecular beam epitaxy (MBE) universality class. MBE dynamics are characterized by 1), a linear growth rate, 2), the constraint of cell proliferation to the colony/tumor border, and 3), surface diffusion of cells at the growing edge. These characteristics were experimentally verified in the studied colonies. That these should show MBE dynamics is in strong contrast with the currently established concept of tumor growth: the kinetics of this type of proliferation rules out exponential or Gompertzian growth. Rather, a clear linear growth regime is followed. The importance of new cell movements-cell diffusion at the tumor border-lies in the fact that tumor growth must be conceived as a competition for space between the tumor and the host, and not for nutrients or other factors. Strong experimental evidence is presented for 16 types of tumor, the growth of which cell surface diffusion may be the main mechanism responsible in vivo. These results explain most of the clinical and biological features of colonies and tumors, offer new theoretical frameworks, and challenge the wisdom of some current clinical strategies.     

Population dynamics in modeling tumor growth

A new approach based on local interaction between cancer and tissue cells is applied to the problem of the onset and growth of solid tumors in homogeneous tissues and effects associated with dramatic changes in tumor growth after crossing the boundary between different tissues. The characteristic sizes and growth rates of spherical tumors, the points of the beginning and the end of spherical growth, and further development of complex structures from the spherical ones (rough interface between the tumor and the host tissue, elongate outgrowths, dendritic structures, and metastases) are inferred assuming that the reproduction rate of a population of cancer cells is a nonmonotonous function of their local concentration and thus of the local curvature of the tumor surface. The growth behavior changes dramatically when the tumor crosses a boundary between two tissues.     

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.     

Regulation of peptidase activity in a three-dimensional aggregate model of brain tumor vasculature

The abnormal vascular system of brain cancers inappropriately expresses membrane proteins, including proteolytic enzymes, ultimately resulting in blood extravasation. The production of inflammatory mediators, such as cytokines and nitric oxide, and tumor hypoxia have been implicated in these effects. We have previously shown that the activity of aminopeptidase A is increased in the abnormal vascular system of human and rat brain tumors. To study the mechanisms regulating the activities of peptidases in cerebral vasculature in brain tumors, we have developed a three-dimensional model of differentiated rat brain cells in aggregate cultures in which rat brain microvessels were incorporated. The secretion of interleukin-6 (IL-6) in the culture medium of aggregates was used as an indicator of inflammatory activation. Addition to these aggregates of C6 glioma cell medium (C6-CM) conditioned under hypoxic or normoxic conditions or serum mimicked tumor-dependent hypoxia or conditions of dysfunction of brain tumor vasculature. Hypoxic and normoxic C6-CM, but not serum, regulated peptidase activity in aggregates, and in particular it increased the activity of aminopeptidase A determined using histoenzymography. Serum, but not C6-CM, increased IL-6 production, but did not increase aminopeptidase A activity in aggregates. Thus soluble glioma-derived factors, but not serum-derived factors, induce dysfunctions of cerebral vasculature by directly regulating the activity of peptidases, not involving inflammatory activation. Tumor hypoxia is not necessary to modulate peptidase activity.     

Information dynamics in carcinogenesis and tumor growth

The storage and transmission of information is vital to the function of normal and transformed cells. We use methods from information theory and Monte Carlo theory to analyze the role of information in carcinogenesis. Our analysis demonstrates that, during somatic evolution of the malignant phenotype, the accumulation of genomic mutations degrades intracellular information. However, the degradation is constrained by the Darwinian somatic ecology in which mutant clones proliferate only when the mutation confers a selective growth advantage. In that environment, genes that normally decrease cellular proliferation, such as tumor suppressor or differentiation genes, suffer maximum information degradation. Conversely, those that increase proliferation, such as oncogenes, are conserved or exhibit only gain of function mutations. These constraints shield most cellular populations from catastrophic mutator-induced loss of the transmembrane entropy gradient and, therefore, cell death. The dynamics of constrained information degradation during carcinogenesis cause the tumor genome to asymptotically approach a minimum information state that is manifested clinically as dedifferentiation and unconstrained proliferation. Extreme physical information (EPI) theory demonstrates that altered information flow from cancer cells to their environment will manifest in-vivo as power law tumor growth with an exponent of size 1.62. This prediction is based only on the assumption that tumor cells are at an absolute information minimum and are capable of "free field" growth that is, they are unconstrained by external biological parameters. The prediction agrees remarkably well with several studies demonstrating power law growth in small human breast cancers with an exponent of 1.72+/-0.24. This successful derivation of an analytic expression for cancer growth from EPI alone supports the conceptual model that carcinogenesis is a process of constrained information degradation and that malignant cells are minimum information systems. EPI theory also predicts that the estimated age of a clinically observed tumor is subject to a root-mean square error of about 30%. This is due to information loss and tissue disorganization and probably manifests as a randomly variable lag phase in the growth pattern that has been observed experimentally. This difference between tumor size and age may impose a fundamental limit on the efficacy of screening based on early detection of small tumors. Independent of the EPI analysis, Monte Carlo methods are applied to predict statistical tumor growth due to perturbed information flow from the environment into transformed cells. A "simplest" Monte Carlo model is suggested by the findings in the EPI approach that tumor growth arises out of a minimally complex mechanism. The outputs of large numbers of simulations show that (a) about 40% of the populations do not survive the first two-generations due to mutations in critical gene segments; but (b) those that do survive will experience power law growth identical to the predicted rate obtained from the independent EPI approach. The agreement between these two very different approaches to the problem strongly supports the idea that tumor cells regress to a state of minimum information during carcinogenesis, and that information dynamics are integrally related to tumor development and growth.     

Collective bacterial dynamics revealed using a three-dimensional population-scale defocused particle tracking technique

An ability to monitor bacterial locomotion and collective dynamics is crucial to our understanding of a number of well-characterized phenotypes including biofilm formation, chemotaxis, and virulence. Here, we report the tracking of multiple swimming Escherichia coli cells in three spatial dimensions and at single-cell resolution using a novel three-dimensional (3D) defocused particle tracking (DPT) method. The 3D trajectories were generated for wild-type Escherichia coli strain RP437 as well as for isogenic derivatives that display smooth swimming due to a cheA deletion (strain RP9535) or incessant tumbling behavior due to a cheZ deletion (strain RP1616). The 3D DPT method successfully differentiated these three modes of locomotion and allowed direct calculation of the diffusion coefficient for each strain. As expected, we found that the smooth swimmer diffused more readily than the wild type, and both the smooth swimmer and the wild-type cells exhibited diffusion coefficients that were at least two orders of magnitude larger than that of the tumbler. Finally, we found that the diffusion coefficient increased with increasing cell density, a phenomenon that can be attributed to the hydrodynamic disturbances caused by neighboring bacteria.     

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.     

Agnostic particle tracking for three-dimensional motion of cellular granules and membrane-tethered bead dynamics

The ability to detect biological events at the single-molecule level provides unique biophysical insights. Back-focal-plane laser interferometry is a promising technique for nanoscale three-dimensional position measurements at rates far beyond the capability of standard video. We report an in situ calibration technique for back-focal-plane, low-power (nontrapping) laser interferometry. The technique does not rely on any a priori model or calibration knowledge, hence the name “agnostic”. We apply the technique to track long-range (up to 100 μm) motion of a variety of particles, including magnetic beads, in three-dimensions with high spatiotemporal resolution (∼2 nm, 100 μs). Our tracking of individual unlabeled vesicles revealed a previously unreported grouping of mean-squared displacement curves at short timescales (<10 ms). Also, tracking functionalized magnetic beads attached to a live cell membrane revealed an anchorage-dependent nonlinear response of the membrane. The software-based technique involves injecting small perturbations into the probe position by driving a precalibrated specimen-mounting stage while recording the quadrant photodetector signals. The perturbations and corresponding quadrant photodetector signals are analyzed to extract the calibration parameters. The technique is sufficiently fast and noninvasive that the calibration can be performed on-the-fly without interrupting or compromising high-bandwidth, long-range tracking of a particle.     

Banded vegetation: development under uniform rainfall from a simple cellular automaton model

Banded vegetation communities are known from semi-arid and arid landscapes in many parts of the world, in grasslands, shrublands, and woodlands. The origin of the distinctive patterning has been the subject of speculation, a common view being that banding evolves through the decline of more complete vegetation cover because of climatic deterioration or through grazing disturbance. A simple model based on cellular automata is employed to test the hypothesis that plausible mechanisms of water partitioning in spatially unstructured plant communities can bring about the development of banding. It is shown that without any climatic change or external disturbance, strongly developed banding can emerge from an initially random distribution of plants. Physical processes underlying the water partitioning, some of which remain unresearched, are discussed, and management implications noted.