Cellular and geometric control of tissue growth and mitotic instability

Agent-based modeling for the tumor microenvironment (TME)

Cancer is a disease that arises from the uncontrolled growth of abnormal (tumor) cells in an organ and their subsequent spread into other parts of the body. If tumor cells spread to surrounding tissues or other organs, then the disease is life-threatening due to limited treatment options. This work applies an agent-based model to investigate the effect of intra-tumoral communication on tumor progression, plasticity, and invasion, with results suggesting that cell-cell and cell-extracellular matrix (ECM) interactions affect tumor cell behavior. Additionally, the model suggests that low initial healthy cell densities and ECM protein densities promote tumor progression, cell motility, and invasion. Furthermore, high ECM breakdown probabilities of tumor cells promote tumor invasion. Understanding the intra-tumoral communication under cellular stress can potentially lead to the design of successful treatment strategies for cancer.

Coherent light scattering from cellular dynamics in living tissues

This review examines the biological physics of intracellular transport probed by the coherent optics of dynamic light scattering from optically thick living tissues. Cells and their constituents are in constant motion, composed of a broad range of speeds spanning many orders of magnitude that reflect the wide array of functions and mechanisms that maintain cellular health. From the organelle scale of tens of nanometers and upward in size, the motion inside living tissue is actively driven rather than thermal, propelled by the hydrolysis of bioenergetic molecules and the forces of molecular motors. Active transport can mimic the random walks of thermal Brownian motion, but mean-squared displacements are far from thermal equilibrium and can display anomalous diffusion through Lévy or fractional Brownian walks. Despite the average isotropic three-dimensional environment of cells and tissues, active cellular or intracellular transport of single light-scattering objects is often pseudo-one-dimensional, for instance as organelle displacement persists along cytoskeletal tracks or as membranes displace along the normal to cell surfaces, albeit isotropically oriented in three dimensions. Coherent light scattering is a natural tool to characterize such tissue dynamics because persistent directed transport induces Doppler shifts in the scattered light. The many frequency-shifted partial waves from the complex and dynamic media interfere to produce dynamic speckle that reveals tissue-scale processes through speckle contrast imaging and fluctuation spectroscopy. Low-coherence interferometry, dynamic optical coherence tomography, diffusing-wave spectroscopy, diffuse-correlation spectroscopy, differential dynamic microscopy and digital holography offer coherent detection methods that shed light on intracellular processes. In health-care applications, altered states of cellular health and disease display altered cellular motions that imprint on the statistical fluctuations of the scattered light. For instance, the efficacy of medical therapeutics can be monitored by measuring the changes they induce in the Doppler spectra of livingex vivocancer biopsies.

Mathematical Models of Cancer Cell Plasticity

Quantitative modelling is increasingly important in cancer research, helping to integrate myriad diverse experimental data into coherent pictures of the disease and able to discriminate between competing hypotheses or suggest specific experimental lines of enquiry and new approaches to therapy. Here, we review a diverse set of mathematical models of cancer cell plasticity (a process in which, through genetic and epigenetic changes, cancer cells survive in hostile environments and migrate to more favourable environments, respectively), tumour growth, and invasion. Quantitative models can help to elucidate the complex biological mechanisms of cancer cell plasticity. In this review, we discuss models of plasticity, tumour progression, and metastasis under three broadly conceived mathematical modelling techniques: discrete, continuum, and hybrid, each with advantages and disadvantages. An emerging theme from the predictions of many of these models is that cell escape from the tumour microenvironment (TME) is encouraged by a combination of physiological stress locally (e.g., hypoxia), external stresses (e.g., the presence of immune cells), and interactions with the extracellular matrix. We also discuss the value of mathematical modelling for understanding cancer more generally.

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.

Effects of anatomical constraints on tumor growth

Competition for available nutrients and the presence of anatomical barriers are major determinants of tumor growth in vivo. We extend a model recently proposed to simulate the growth of neoplasms in real tissues to include geometrical constraints mimicking pressure effects on the tumor surface induced by the presence of rigid or semirigid structures. Different tissues have different diffusivities for nutrients and cells. Despite the simplicity of the approach, based on a few inherently local mechanisms, the numerical results agree qualitatively with clinical data (computed tomography scans of neoplasms) for the larynx and the oral cavity.

The community effect and ectoderm-mesoderm interaction in Xenopus muscle differentiation

Community effects are believed to play an important role in the patterning of many tissues during development. They involve an interaction between neighbouring equivalent cells that is necessary for them to proceed to their fully differentiated state. However, the mechanisms underlying these effects remain unclear. In this paper, diffusion-based mathematical models are constructed and analysed in order to study possible mechanisms for the community effect in Xenopus muscle differentiation. These models differ from each other in the assumptions that are made about the nature of an inhibitory effect that ectodermal tissue has been observed to have on muscle differentiation. It is possible to construct consistent models based on all the forms of inhibition considered. However, each model requires the diffusible factors on which it is based to have different properties. The current data from tissues reaggregate experiments are insufficient to determine the mechanisms underlying the community effect; the work presented here suggests that quantitative analysis of a further series of reaggregate experiments will make it possible to distinguish between the proposed mechanisms.

Application of competition theory to tumour growth: implications for tumour biology and treatment

To assess critical parameters controlling tumour growth and response to therapy, competition theory models the tumour-host interface as a network of interacting normal and malignant cell populations using coupled, non-linear differential equations. When the equations are analysed under conditions which simulate tumour development, three phases of tumour growth, each with different critical parameters, can be predicted. Transitions between these phases correspond to the initiation, promotion and invasion stages demonstrated in experimental models of carcinogenesis. Critical cellular properties for each transition are predicted including phenomena already demonstrated experimentally such as the linkage of invasive tumour growth with acquisition of angiogenesis. The model also predicts the previously unknown phenomenon of "functional equivalence" in which disparate tumour traits can play identical roles in tumour growth and invasion. This approach allows the diverse but inconsistent properties of transformed cells to be understood according to their specific contribution to tumorigenesis. The models have significant implications for treatment strategies.

Nonlinear diffusion of a growth inhibitory factor in multicell spheroids

A mathematical model is presented for the production of a growth inhibitory factor (GIF) within a multicell spheroid. The model is based on the assumption that the GIF diffuses within the spheroid in a nonlinear spatially dependent manner. This is in contrast with previous models, in which the nonlinearity was assumed in the production term. The results of the new model are compared with those of previous models and with experimental data.

On the concentration profile of a growth inhibitory factor in multicell spheroids

A mathematical model is presented for the production of a growth inhibitory factor (GIF) within a multicell spheroid. The main assumption of the model is that the GIF is produced by cells within the spheroid in some prescribed nonlinear, spatially dependent manner. Given that the diffusion of the GIF is known to take place over a much shorter time scale than that of spheroid growth, the steady-state profile of the GIF in various spheroids of differing radii is examined and theoretical results are compared with actual experimental data.

Modelling the growth of solid tumours and incorporating a method for their classification using nonlinear elasticity theory

Medically, tumours are classified into two important classes--benign and malignant. Generally speaking, the two classes display different behaviour with regard to their rate and manner of growth and subsequent possible spread. In this paper, we formulate a new approach to tumour growth using results and techniques from nonlinear elasticity theory. A mathematical model is given for the growth of a solid tumour using membrane and thick-shell theory. A central feature of the model is the characterisation of the material composition of the model through the use of a strain-energy function, thus permitting a mathematical description of the degree of differentiation of the tumour explicitly in the model. Conditions are given in terms of the strain-energy function for the processes of invasion and metastasis occurring in a tumour, being interpreted as the bifurcation modes of the spherical shell which the tumour is essentially modelled as. Our results are compared with actual experimental results and with the general behaviour shown by benign and malignant tumours. Finally, we use these results in conjunction with aspects of surface morphogenesis of tumours (in particular, the Gaussian and mean curvatures of the surface of a solid tumour) in an attempt to produce a mathematical formulation and description of the important medical processes of staging and grading cancers. We hope that this approach may form the basis of a practical application.

A qualitative analysis of some models of tissue growth

Using maximum principles for parabolic and elliptic operators, we examine, in a general way, some models of tissue growth. These typically consist of a model mechanism for the diffusion of a mitotic inhibitor (growth inhibitory factor, GIF) throughout the tissue. Central to the modeling is the inclusion of a source function that models the production of GIF throughout the tissue. We examine the effect this term has on the resulting distribution of GIF in the tissue and comment on the appropriateness of different source functions, in particular a uniform production rate or a nonuniform production rate of inhibitor. Given that it is more appropriate to infer from the patterns of mitosis that are observed experimentally in various tissues the GIF concentration profile rather than the source function profile, it may be more appropriate to use these types of models to determine the qualitative form of the source term rather than proposing this function a priori.

A mathematical model for the growth and classification of a solid tumor: a new approach via nonlinear elasticity theory using strain-energy functions

Medically, tumors are classified into two important classes--benign and malignant. Generally speaking, the two classes display different behaviour with regard to their rate and manner of growth and subsequent possible spread. In this paper, we formulate a new approach to tumor growth using results and techniques from nonlinear elasticity theory. A mathematical model is given for the growth of a solid tumor using membrane and thick-shell theory. A central feature of the model is the characterization of the material composition of the tumor through the use of a strain energy function, thus permitting a mathematical description of the degree of differentiation of the tumor explicitly in the model. Conditions are given in terms of the strain energy function for the processes of invasion and metastasis occurring in a tumor, being interpreted as the bifurcation modes of the spherical shell, which the tumor is essentially modeled as. Our results are compared with actual medical experimental results and with the general behavior shown by benign and malignant tumors. Finally, we use these results in conjunction with aspects of surface morphogenesis of tumors (in particular, the Gaussian and mean curvatures of the surface of a solid tumor) in an attempt to produce a mathematical formulation and description of the important medical processes of staging and grading cancers. We hope that this approach may form the basis of a practical application.

The diffusion of an inhibitor in a spherical tumor

The purpose of this paper is to show the correct mathematical behavior of the solution to a diffusion equation that has previously been used in a modeling situation involving a spherical tumor.

Mathematical modelling of microenvironment and growth in EMT6/Ro multicellular tumour spheroids

In order to determine the role of micromilieu in tumour spheroid growth, a mathematical model was developed to predict EMT6/Ro spheroid growth and microenvironment based upon numerical solution of the diffusion/reaction equation for oxygen, glucose, lactate ion, carbon dioxide, bicarbonate ion, chlorine ion and hydrogen ion along with the equation of electroneutrality. This model takes into account the effects of oxygen concentration, glucose concentration and extracellular pH on cell growth and metabolism. Since independent measurements of EMT6/Ro single cell growth and metabolic rates, spheroid diffusion constants, and spinner flask mass transfer coefficients are available, model predictions using these parameters were compared with published data on EMT6/Ro spheroid growth and micro-environment. The model predictions of reduced spheroid growth due to reduced cell growth rates and cell shedding fit experimental spheroid growth data below 700 microns, but overestimated the spheroid growth rate at larger diameters. Predicted viable rim thicknesses based on predicted near zero glucose concentrations fit published viable rim thickness data for 1000 microns spheroids grown at medium glucose concentrations of 5.5 mM or less. However, the model did not accurately predict the onset of necrosis. Moreover, the model could not predict the observed decreases in oxygen and glucose metabolism seen in spheroids with time, nor could it predict the observed growth plateau. This suggests that other unknown factors, such as inhibitors or cell-cell contact effects, must also be important in affecting spheroid growth and cellular metabolism.

Diffusion regulated growth characteristics of a spherical prevascular carcinoma

Recently a mathematical model of the prevascular phases of tumor growth by diffusion has been investigated (S.A. Maggelakis and J.A. Adam, Math. Comput. Modeling, in press). In this paper we examine in detail the results and implications of that mathematical model, particularly in the light of recent experimental work carried out on multicellular spheroids. The overall growth characteristics are determined in the present model by four parameters: Q, gamma, b, and delta, which depend on information about inhibitor production rates, oxygen consumption rates, volume loss and cell proliferation rates, and measures of the degree of non-uniformity of the various diffusion processes that take place. The integro-differential growth equation is solved for the outer spheroid radius R0(t) and three related inner radii subject to the solution of the governing time-independent diffusion equations (under conditions of diffusive equilibrium) and the appropriate boundary conditions. Hopefully, future experimental work will enable reasonable bounds to be placed on parameter values referred to in this model: meanwhile, specific experimentally-provided initial data can be used to predict subsequent growth characteristics of in vitro multicellular spheroids. This will be one objective of future studies.

Mathematical models of tumor growth. IV. Effects of a necrotic core

Two mathematical models for the control of the growth of a tumor by diffusion of mitotic inhibitor are presented. The inhibitor production rate is taken to be uniform in a necrotic core for the first model and in the nonnecrotic region for the second model. Regions of stable and unstable growth are determined, and conclusions are drawn about the limiting peripheral widths of stable tissue growth for both models. Comparisons of the results from the two models indicate that the models are sensitive to the source distributions of inhibitor production.

Growth and cellular characteristics of multicell spheroids

The data reviewed here demonstrate that there are many similarities in growth and cellular characteristics for different types of tumor cells grown as multicell spheroids. Furthermore, where comparisons have been made many of the features of spheroids also occur in tumors in vivo. However, as for tumors, there are also many characteristics of individual types of spheroids which are relatively specific and cannot be generalized as properties of all spheroid model systems. The results also demonstrate the marked influence which cellular microenvironments regulated by a supply of oxygen and nutrients may have on the development of cellular heterogeneity. Furthermore, using spheroids it was shown that dynamic cellular and metabolic interactions exist in regulating the development of cellular subpopulations and microenvironments. Spheroids are more sensitive to alterations in culture environment than are monolayer or single-cell suspension cultures. Consequently, researchers who use this model system must characterize, optimize, and standardize the growth conditions for the spheroid cell type being investigated. This information then provides a base from which to undertake detailed studies, which are not possible in experimental tumors, of controlled manipulation of microenvironments in spheroids. The ranges of cellular microenvironments and cellular heterogeneity which exist at different stages of spheroid growth provide a model, at least in part, for coexisting size ranges of microregions in many solid tumors. Thus, spheroids provide a model, which at different stages of growth is readily manipulated and controlled experimentally, to facilitate studies of contributions of individual environmental factors, or concomitant changes in these, on cellular phenotypic expression. It is probable that the cellular changes which can be demonstrated to occur during spheroid growth, also occur in vivo. Modulation of cellular characteristics revealed by research with spheroids requires much more study to determine the mechanisms and effects on tumor cell behavior, as well as response to therapeutic agents and their relevance to tumors in vivo.

Biochemical modulation of angiogenesis in the chorioallantoic membrane of the chick embryo

A variety of substances potentially having effects on angiogenesis in the skin were assayed on the chorioallantoic membrane of the chicken embryo (CAM). Millipore filter discs alone and saturated with saline 0.9% (controls), keratinocyte-conditioned medium, lactic acid 10(-1) M, adenosine 10(-4) M, sodium fluoride 10(-4) M, dinitrophenol 10(-4) M, histamine 10(-4) M, 5-hydroxytryptamine 10(-4) M, acetylcholine 10(-4) M, prostaglandin E2 3 X 10(-4) M, prostaglandin F2 alpha 3 X 10(-4) M, arachidonic acid 10(-4) M, epidermal growth factor 5 X 10(-5) g/ml, human plasma fibronectin 10(-4) g/ml, acetylsalicylic acid 10(-3) M, and arachis oil were applied to the CAM and the vascular responses quantitated 4 days later. None of the agents with the exception of keratinocyte-conditioned medium stimulated new vessel growth as compared to the controls. However, arachis oil (p less than 0.001) and ADP (p less than 0.01) were associated with significantly decreased vascular responses relative to controls. The specimens incubated with saline, fibronectin, ADP, and arachis oil were examined histologically; with the exception of arachis oil all displayed ectodermal epithelial and mesenchymal hyperplasia of the membrane in association with increased vascularity. Almost no perceptible change was noted histologically with arachis oil.

Oscillating spatial structures of a tissue

We present a simple mathematical model for the self-controlled growth of a tissue giving rise to an oscillating tissue size under certain conditions. The control is brought about by two substances (two inhibitors or one inhibitor and one nutrient) which influence the cell kinetics locally. The inhibitors are produced by the tissue itself (whereas the nutrient comes from outside but is consumed by the tissue which produces the same effect). Both diffuse freely throughout the tissue, and thus realize a communication between different parts of the tissue. In any case the tissue approaches a self-maintaining space-time structure with properties depending on the parameters of proliferation, death and inhibiting control. We discuss the conditions for this structure not to be time-independent but oscillating.

The glucose distribution in 9L rat brain multicell tumor spheroids and its effect on cell necrosis

The glucose consumption rate of 9L rat brain tumor cells was determined as a function of concentration. Glucose uptake followed Michaelis-Menten kinetics with a Km of 0.58 mM and a Vmax of 1.6 pg/cell-min. The glucose diffusion coefficient in spheroids of 9L tumor cells was determined to be 1.5 x 10(-6) cm2/s at 37 degrees C. Using these parametric values, the glucose distribution in 9L multicell spheroids was calculated and related to the viable and necrotic zones.

A model for the growth of multicellular spheroids

Based on biological observations and the basic physical properties of tri-dimensional structures, a mathematical expression is derived to relate the growth rate of multicellular spheroids to some easily measurable parameters. This model involves properties both of the individual cells and of the spheroid structure, such as the cell doubling time in monolayer, the rate of cell shedding from the spheroid and the depth of the external rim of cycling cells. The derived growth equation predicts a linear expansion of the spheroid diameter with time. The calculated growth rate for a number of spheroid cell types is in good agreement with experimental data. The model provides a simple and practical view of growth control in spheroids, and is further adapted to include parameters presumably responsible for the growth saturation in large spheroids.

The effects of cell density and metabolite flux on cellular dynamics

Density-dependent regulation of cell growth in tissue culture is a well-known phenomenon but the mechanism of regulation remains obscure. Here we explore the effects of cell density and metabolite flux on the collective dynamics of a cell population. The intracellular dynamics are modelled by positive feedback kinetic mechanisms of the kind known to apply to yeast cells. Several experimental observations related to glycolytic oscillations are predicted and it is suggested that the general conclusions may be applicable in a broader context.