A new mathematical model for avascular tumour growth

The early development of solid tumours has been extensively studied, both experimentally via the multicellular spheroid assay, and theoretically using mathematical modelling. The vast majority of previous models apply specifically to multicell spheroids, which have a characteristic structure of a proliferating rim and a necrotic core, separated by a band of quiescent cells. Many previous models represent these as discrete layers, separated by moving boundaries. Here, the authors develop a new model, formulated in terms of continuum densities of proliferating, quiescent and necrotic cells, together with a generic nutrient/growth factor. The model is oriented towards an in vivo rather than in vitro setting, and crucially allows for nutrient supply from underlying tissue, which will arise in the two-dimensional setting of a tumour growing within an epithelium. In addition, the model involves a new representation of cell movement, which reflects contact inhibition of migration. Model solutions are able to reproduce the classic three layer structure familiar from multicellular spheroids, but also show that new behaviour can occur as a result of the nutrient supply from underlying tissue. The authors analyse these different solution types by approximate solution of the travelling wave equations, enabling a detailed classification of wave front solutions.     

Modelling the formation of necrotic regions in avascular tumours

The mechanisms underlying the formation of necrotic regions within avascular tumours are not well understood. In this paper, we examine the relative roles of nutrient deprivation and of cell death, from both the proliferating phase of the cell cycle via apoptosis and from the quiescent phase via necrosis, in changing the structure within multicellular tumour spheroids and particularly the accumulation of dead cell material in the centre. A mathematical model is presented and studied that accounts for nutrient diffusion, changes in cell cycling rates, the two different routes to cell death as well as active motion of cells and passive motion of the dead cell material. In studying the accumulation of dead cell matter we do not distinguish between the route by which each was formed. The resulting mathematical model is examined for a number of scenarios. Results show that in many cases the size of the necrotic core is closely correlated with low levels in nutrient concentration. However, in certain cases, particularly where the rate of necrosis is large, the resulting necrotic core can lead to regions of non-negligible nutrient concentration-dependent upon the mode of cell death.     

The migration of cells in multicell tumor spheroids

A mathematical model is proposed to explain the observed internalization of microspheres and 3H-thymidine labelled cells in steady-state multicellular spheroids. The model uses the conventional ideas of nutrient diffusion and consumption by the cells. In addition, a very simple model of the progress of the cells through the cell cycle is considered. Cells are divided into two classes, those proliferating (being in G1, S, G2 or M phases) and those that are quiescent (being in G0). Furthermore, the two categories are presumed to have different chemotactic responses to the nutrient gradient. The model accounts for the spatial and temporal variations in the cell categories together with mitosis, conversion between categories and cell death. Numerical solutions demonstrate that the model predicts the behavior similar to existing models but has some novel effects. It allows for spheroids to approach a steady-state size in a non-monotonic manner, it predicts self-sorting of the cell classes to produce a thin layer of rapidly proliferating cells near the outer surface and significant numbers of cells within the spheroid stalled in a proliferating state. The model predicts that overall tumor growth is not only determined by proliferation rates but also by the ability of cells to convert readily between the classes. Moreover, the steady-state structure of the spheroid indicates that if the outer layers are removed then the tumor grows quickly by recruiting cells stalled in a proliferating state. Questions are raised about the chemotactic response of cells in differing phases and to the dependency of cell cycle rates to nutrient levels.     

Imaging- and Flow Cytometry-based Analysis of Cell Position and the Cell Cycle in 3D Melanoma Spheroids

Three-dimensional (3D) tumor spheroids are utilized in cancer research as a more accurate model of the in vivo tumor microenvironment, compared to traditional two-dimensional (2D) cell culture. The spheroid model is able to mimic the effects of cell-cell interaction, hypoxia and nutrient deprivation, and drug penetration. One characteristic of this model is the development of a necrotic core, surrounded by a ring of G1 arrested cells, with proliferating cells on the outer layers of the spheroid. Of interest in the cancer field is how different regions of the spheroid respond to drug therapies as well as genetic or environmental manipulation. We describe here the use of the fluorescence ubiquitination cell cycle indicator (FUCCI) system along with cytometry and image analysis using commercial software to characterize the cell cycle status of cells with respect to their position inside melanoma spheroids. These methods may be used to track changes in cell cycle status, gene/protein expression or cell viability in different sub-regions of tumor spheroids over time and under different conditions.     

Regrowth kinetics of cells from different regions of multicellular spheroids of four cell lines

A basic understanding of the recruitment of quiescent tumor cells into the cell cycle would be an important contribution to tumor biology and therapy. As a first step in pursuing this goal, we have investigated the regrowth kinetics of cells from different regions in multicellular spheroids of rodent and human origin. Cells were isolated from four different depths within the spheroids using a selective dissociation technique. The outer cells were proliferating and resumed growth after replating with a 0-8-hour lag period, similar to cells from exponentially growing monolayers. With increasing depth of origin, the lag periods prior to regrowth increased to 2-3 times the monolayer doubling time; cells from plateau-phase monolayers showed a lag period of 1-1.5 times the doubling period. After resuming growth, all cells of a given cell line grew with the same doubling time and achieved the same confluency level. The inner spheroid cells and cells from plateau-phase monolayers had reduced clonogenic efficiencies. The inner cells were initially 1.5-3 times smaller than the outer cells, but began to increase in volume within 4 hours of replating. The fractions of S-phase cells were greatly decreased with increasing depth of origin in the spheroids; there were long delays prior to S-phase recovery after plating, to a maximum of 1-1.5 times the normal doubling time. These results suggest that those quiescent cells from spheroids and monolayers which are able to reenter the cell cycle are predominantly in the G1-phase. However, quiescent cells from the innermost spheroid region require approximately twice as long to enter normal cell cycle traverse as cells from plateau-phase monolayers. The selective dissociation method can isolate very pure populations of proliferating and quiescent cells in a rapid and nonperturbing manner; this system will be valuable in further characterizing quiescent cells from spheroids.     

Necrotic core in EMT6/Ro tumour spheroids: Is it caused by an ATP deficit?

Although commonly related to nutrient deprivation, the cause of the formation of the necrotic core in the multicellular tumour spheroids is still a controversial issue. We propose a simple model for the cell ATP production that assumes glucose and lactate as the only fuel substrates, and describes the main reactions occurring in the glycolytic and the oxidative pathways. Under the key assumption that cell death occurs when ATP production falls to a critical level, we formulate a multiscale model that integrates the energy metabolism at the cellular level with the diffusive transport of the metabolites in the spheroid mass. The model has been tested by predicting the measurements of the necrotic radius obtained by Freyer and Sutherland (1986a) in EMT6/Ro spheroids under different concentrations of glucose and oxygen in the culture medium. The results appear to be in agreement with the hypothesis that necrosis is caused by ATP deficit.     

A reduction in the in situ rates of oxygen and glucose consumption of cells in EMT6/Ro spheroids during growth

The rates of consumption of oxygen and glucose by EMT6/Ro cells in multicellular spheroids were measured at various times during normal growth. In situ spheroid cellular consumption rates were similar to those of exponentially growing single cells up to a spheroid diameter of 150 micron. Further growth resulted in decreases in the rates of both oxygen and glucose consumption which were correlated with the increase in spheroid diameter and cell number. At a diameter of 1300 micron, both rates of cellular consumption had decreased by a factor of 2.5. The rates of consumption per unit of nonnecrotic spheroid volume decreased in a similar manner. Measurements with single cells demonstrated that the rate of oxygen consumption was coupled with glucose concentration, and vice versa. The rates of consumption for cells dissociated from small spheroids indicated that there was some effect of the spheroid environment. As the spheroids grew, however, association in the spheroid structure accounted for a smaller proportion of the total observed reduction in the rates of nutrient consumption. The presence of central necrosis also appeared to have no effect on the rates of consumption of these nutrients. Spheroid-derived cells showed a decrease in cell volume with growth as the cells accumulated in a quiescent state. Measurements with single cells demonstrated that oxygen and glucose consumption were correlated with cell volume and with the development of nonproliferating cells. We conclude that the observed decrease in oxygen and glucose consumption with growth in spheroids is largely due to the progressive accumulation of cells in a quiescent state characterized by an inherently lower cellular rate of nutrient utilization.     

Modelling the Cell Cycle and Cell Movement in Multicellular Tumour Spheroids

This paper analyses a recent mathematical model of avascular tumour spheroid growth which accounts for both cell cycle dynamics and chemotactic driven cell movement. The model considers cells to exist in one of two compartments: proliferating and quiescent, as well as accounting for necrosis and apoptosis. One particular focus of this paper is the behaviour created when proliferating and quiescent cells have different chemotactic responses to an extracellular nutrient supply. Two very different steady-state behaviours are identified corresponding to those cases where proliferating cells move either more quickly or more slowly than quiescent cells in response to a gradient in the extracellular nutrient supply. The case where proliferating cells move more rapidly leads to the commonly accepted spheroid structure of a thin layer of proliferating cells surrounding an inner quiescent core. In the case where proliferating cells move more slowly than quiescent cells the model predicts an interesting structure of a thin layer of quiescent cells surrounding an inner core of proliferating and quiescent cells. The sensitivity of this tumour structure to the cell cycle model parameters is also discussed. In particular variations in the steady-state size of the tumour and the types of transient behaviour are explored. The model reveals interesting transient behaviour with sharply delineated regions of proliferating and quiescent cells.     

Autophagy Protects from Trastuzumab-Induced Cytotoxicity in HER2 Overexpressing Breast Tumor Spheroids

Multicellular tumor spheroids represent a 3D in vitro model that mimics solid tumor essential properties including assembly and development of extracellular matrix and nutrient, oxygen and proliferation gradients. In the present study, we analyze the impact of 3D spatial organization of HER2-overexpressing breast cancer cells on the response to Trastuzumab. We cultured human mammary adenocarcinoma cell lines as spheroids with the hanging drop method and we observed a gradient of proliferating, quiescent, hypoxic, apoptotic and autophagic cells towards the inner core. This 3D organization decreased Trastuzumab sensitivity of HER2 over-expressing cells compared to monolayer cell cultures. We did not observe apoptosis induced by Trastuzumab but found cell arrest in G0/G1 phase. Moreover, the treatment downregulated the basal apoptosis only found in tumor spheroids, by eliciting protective autophagy. We were able to increase sensitivity to Trastuzumab by autophagy inhibition, thus exposing the interaction between apoptosis and autophagy. We confirmed this result by developing a resistant cell line that was more sensitive to autophagy inhibition than the parental BT474 cells. In summary, the development of Trastuzumab resistance relies on the balance between death and survival mechanisms, characteristic of 3D cell organization. We propose the use of spheroids to further improve the understanding of Trastuzumab antitumor activity and overcome resistance.     

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

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

Development of intracytoplasmic lumens in a colon cancer cell line cultured on a non-adhesive surface

Cell-matrix interactions have important effects on phenotypic features, such as morphology, differentiation and cell growth. Several papers have suggested that when cell-matrix interactions are interrupted, cells grow as multicellular spheroids and eventually undergo apoptosis. We found that when ET(-), a laminin non-adherent colon cancer cell line, was cultured on poly-2-hydroxyethyl methacrylate (HEMA) coated plastic, the cells floated as cellular aggregates of spheroids or as single cells. Some of the single cells contained a very large intracytoplasmic lumen (ICL) and appeared similar to signet ring cells. These ICL were lined by a layer of short microvilli. The number of the cell did not increased cells when cultured on poly-HEMA. Another type of single cells, usually without ICL, demonstrated the characteristics of apoptotic cells by histologic examination. Acridine orange staining, flow cytometry and electron microscopy confirmed the apoptotic nature of those cells. In immunohistochemical staining for proliferating cell nuclear antigen, spheroids of cells and single cells with ICL were immunoreactive, while most of the single cells without ICL were negative. These results suggest that multicellular aggregation and formation of ICL were induced by the adaptation of ET(-) colon cancer cells in a harmful environment caused by reduced adhesiveness, and these changes might be related to cell survival.     

Numerical simulation and graphical analysis of in vitro benign tumor growth: application of single-particle state bosonic matter equation with length scaling

We describe the application of a non-linear single-particle state bosonic condensate equation to simulate multicellular tumor growth by treating it as a coupling of two classical wave equations with real components. With one component representing the amplitude of the cells in their volume growth phase and the other representing the amplitude of the cells in their proliferation or mitosis phase, the two components of the coupled equation feed each other during the time evolution and are coupled together through diffusion and other linear and non-linear terms. The features of quiescent and necrotic cells, which result from poor nutrient diffusion into a tumor, have been found to correspond quite well to experimental data when they are modeled as depending on higher cell density. Classical hallmarks of benign tumor growth, such as the initial rapid growth, followed by a dramatic collapse in the proliferating cell count and a strong re-growth thereafter appear quite encouragingly in the theoretical results. A tool for graphical analysis of the tumor simulation results has been developed to provide morphological information about tumors at various growth stages. The model and the graphical analysis can be extended further to create an effective tool to predict/monitor tumor growth. 1 Screen shot from the graphical analysis tool showing simulation results after ten days: clustering of cells of the tumor (up); cell density profile (down) Dedicated to Professor Dr. Paul von Ragué Schleyer on the occasion of his 75th birthday     

Modeling mechanical inhomogeneities in small populations of proliferating monolayers and spheroids

Understanding the mechanical behavior of multicellular monolayers and spheroids is fundamental to tissue culture, organism development, and the early stages of tumor growth. Proliferating cells in monolayers and spheroids experience mechanical forces as they grow and divide and local inhomogeneities in the mechanical microenvironment can cause individual cells within the multicellular system to grow and divide at different rates. This differential growth, combined with cell division and reorganization, leads to residual stress. Multiple different modeling approaches have been taken to understand and predict the residual stresses that arise in growing multicellular systems, particularly tumor spheroids. Here, we show that by using a mechanically robust agent-based model constructed with the peridynamic framework, we gain a better understanding of residual stresses in multicellular systems as they grow from a single cell. In particular, we focus on small populations of cells (1–100 s) where population behavior is highly stochastic and prior investigation has been limited. We compare the average strain energy density of cells in monolayers and spheroids using different growth and division rules and find that, on average, cells in spheroids have a higher strain energy density than cells in monolayers. We also find that cells in the interior of a growing spheroid are, on average, in compression. Finally, we demonstrate the importance of accounting for stochastic fluctuations in the mechanical environment, particularly when the cellular response to mechanical cues is nonlinear. The results presented here serve as a starting point for both further investigation with agent-based models, and for the incorporation of major findings from agent-based models into continuum scale models when explicit representation of individual cells is not computationally feasible.     

Variations in tumor cell growth rates and metabolism with oxygen concentration, glucose concentration, and extracellular pH.

Tumors and multicellular tumor spheroids can develop gradients in oxygen concentration, glucose concentration, and extracellular pH as they grow. In order to calculate these gradients and assess their impact on tumor growth, it is necessary to quantify the effect of these variables on tumor cell metabolism and growth. In this work, the oxygen consumption rates, glucose consumption rates, and growth rates of EMT6/Ro mouse mammary tumor cells were measured at a variety of oxygen concentrations, glucose concentrations, and extracellular pH levels. At an extracellular pH of 7.25, the oxygen consumption rate of EMT6/Ro cells increased by nearly a factor of 2 as the glucose concentration was decreased from 5.5 mM to 0.4 mM. This effect of glucose concentration on oxygen consumption rate, however, was slight at an extracellular pH of 6.95 and disappeared completely at an extracellular pH of 6.60. The glucose consumption rate of EMT6/Ro cells increased by roughly 40% when the oxygen concentration was reduced from 0.21 mM to 0.023 mM and decreased by roughly 60% when the extracellular pH was decreased from 7.25 to 6.95. The growth rate of EMT6/Ro cells decreased with decreasing oxygen concentration and extracellular pH; however, severe conditions were required to stop cell growth (0.0082 mM oxygen and an extracellular pH of 6.60). Empirical correlations were developed from these data to express EMT6/Ro cell growth rates, oxygen consumption rates, and glucose consumption rates, as functions of oxygen concentration, glucose concentration, and extracellular pH. These empirical correlations make it possible to mathematically model the gradients in oxygen concentration, glucose concentration, and extracellular pH in EMT6/Ro multicellular spheroids by solution of the diffusion/reaction equations. Computations such as these, along with oxygen and pH microelectrode measurements in EMT6/Ro multicellular spheroids, indicated that nutrient concentration and pH levels in the inner regions of spheroids were low enough to cause significant changes in nutrient consumption rates and cell growth rates. However, pH and oxygen concentrations measured or calculated in EMT6/Ro spheroids where quiescent cells have been observed were not low enough to cause the cessation of cell growth, indicating that the observed quiescence must have been due to factors other than acidic pH, oxygen depletion, or glucose depletion.     

Parameter estimation and determinability analysis applied to Drosophila gap gene circuits

Background

Our objective was to discover in silico axioms that are plausible representations of the operating principles realized during characteristic growth of EMT6/Ro mouse mammary tumor spheroids in culture. To reach that objective we engineered and iteratively falsified an agent-based analogue of EMT6 spheroid growth. EMT6 spheroids display consistent and predictable growth characteristics, implying that individual cell behaviors are tightly controlled and regulated. An approach to understanding how individual cell behaviors contribute to system behaviors is to discover a set of principles that enable abstract agents to exhibit closely analogous behaviors using only information available in an agent's immediate environment. We listed key attributes of EMT6 spheroid growth, which became our behavioral targets. Included were the development of a necrotic core surrounded by quiescent and proliferating cells, and growth data at two distinct levels of nutrient.

Results

We then created an analogue made up of quasi-autonomous software agents and an abstract environment in which they could operate. The system was designed so that upon execution it could mimic EMT6 cells forming spheroids in culture. Each agent used an identical set of axiomatic operating principles. In sequence, we used the list of targeted attributes to falsify and revise these axioms, until the analogue exhibited behaviors and attributes that were within prespecified ranges of those targeted, thereby achieving a level of validation.

Conclusion

The finalized analogue required nine axioms. We posit that the validated analogue's operating principles are reasonable representations of those utilized by EMT6/Ro cells during tumor spheroid development.     

An on-lattice agent-based Monte Carlo model simulating the growth kinetics of multicellular tumor spheroids

PurposeTo develop an on-lattice agent-based model describing the growth of multicellular tumor spheroids using simple Monte Carlo tools.MethodsCells are situated on the vertices of a cubic grid. Different cell states (proliferative, hypoxic or dead) and cell evolution rules, driven by 10 parameters, and the effects of the culture medium are included. About twenty spheroids of MCF-7 human breast cancer were cultivated and the experimental data were used for tuning the model parameters.ResultsSimulated spheroids showed adequate sizes of the necrotic nuclei and of the hypoxic and proliferative cell phases as a function of the growth time, mimicking the overall characteristics of the experimental spheroids. The relation between the radii of the necrotic nucleus and the whole spheroid obtained in the simulations was similar to the experimental one and the number of cells, as a function of the spheroid volume, was well reproduced. The statistical variability of the Monte Carlo model described the whole volume range observed for the experimental spheroids. Assuming that the model parameters vary within Gaussian distributions it was obtained a sample of spheroids that reproduced much better the experimental findings.ConclusionsThe model developed allows describing the growth of in vitro multicellular spheroids and the experimental variability can be well reproduced. Its flexibility permits to vary both the agents involved and the rules that govern the spheroid growth. More general situations, such as, e. g., tumor vascularization, radiotherapy effects on solid tumors, or the validity of the tumor growth mathematical models can be studied.     

A multiscale model for avascular tumor growth

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

Multicell spheroids as a model for cell kinetic studies

Cells growing in tissue culture as three-dimensional, multicellular aggregates called 'spheroids' typically show a decreasing growth fraction and development of quiescent subpopulations as the spheroids enlarge. Kinetic studies in a number of spheroid systems have indicated that the primary reason for the tumour-like growth is a progressive decrease in growth fraction, with only a modest elongation of cell cycle time in larger spheroids. In this paper, the cellular growth kinetics for spheroids of V79 Chinese hamster lung cells are reviewed, and the regrowth kinetics of cells resuming growth after recovery from quiescent regions of the spheroids are described. Further, the role of regrowth/repopulation in determining the spheroid response to anti-tumour cytotoxics is explored, with particular emphasis on treatment with cisplatin and etoposide. By separating the effects of cytotoxicity and regrowth in the overall spheroid response to anti-neoplastic drugs, it is suggested that 'drug resistance' in tumours can be a kinetic as well as a genetic problem.     

Invited review Multicell spheroids as a model for cell kinetic studies

Abstract. Cells growing in tissue culture as three-dimensional, multicellular aggregates called 'spheroids' typically show a decreasing growth fraction and development of quiescent subpopulations as the spheroids enlarge. Kinetic studies in a number of spheroid systems have indicated that the primary reason for the tumour-like growth is a progressive decrease in growth fraction, with only a modest elongation of cell cycle time in larger spheroids. In this paper, the cellular growth kinetics for spheroids of V79 Chinese hamster lung cells are reviewed, and the regrowth kinetics of cells resuming growth after recovery from quiescent regions of the spheroids are described. Further, the role of regrowth/repopulation in determining the spheroid response to anti-tumour cytotoxics is explored, with particular emphasis on treatment with cisplatin and etoposide. By separating the effects of cytotoxicity and regrowth in the overall spheroid response to anti-neoplastic drugs, it is suggested that 'drug resistance' in tumours can be a kinetic as well as a genetic problem.