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.     

Combining Gompertzian growth and cell population dynamics

A two-compartment model of cancer cells population dynamics proposed by Gyllenberg and Webb includes transition rates between proliferating and quiescent cells as non-specified functions of the total population, N. We define the net inter-compartmental transition rate function: Phi(N). We assume that the total cell population follows the Gompertz growth model, as it is most often empirically found and derive Phi(N). The Gyllenberg-Webb transition functions are shown to be characteristically related through Phi(N). Effectively, this leads to a hybrid model for which we find the explicit analytical solutions for proliferating and quiescent cell populations, and the relations among model parameters. Several classes of solutions are examined. Our model predicts that the number of proliferating cells may increase along with the total number of cells, but the proliferating fraction appears to be a continuously decreasing function. The net transition rate of cells is shown to retain direction from the proliferating into the quiescent compartment. The death rate parameter for quiescent cell population is shown to be a factor in determining the proliferation level for a particular Gompertz growth curve.     

Mathematical description and analysis of cell cycle kinetics and the application to Ehrlich ascites tumor

The linear and nonlinear aspects of the dynamics of the cell cycle kinetics of cell populations are studied. The dynamics are represented by difference equations. The characteristics of cell population systems are analyzed by applying the model to Ehrlich ascites tumor. The model applied for the simulations of the growth of Ehrlich ascites tumor cells incorporates processes of cell division, cell death, transition of cells to resting states and clearance of dead cells. Comparison of the results obtained with the model and the experimental data suggests that the duration of the mean generation time of the proliferating EAT cells increases with aging of the tumor. An attempt is made to relate the prolongation of cell mean generation time with processes of cell death and dead cell clearance. Studying the transition of cells to the resting states, it becomes apparent that in fact transition of proliferating cells to the resting states occurs somewhere close to the end of the cell cycle and with a rate that varies with the age of the tumor. Time course behavior of the cell age, cell size, and cell DNA distribution with aging of the tumor are obtained. Variations in average size and average DNA contents are determined.     

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.     

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.     

The Minimal, Phase-Transition Model for the Cell-Number Maintenance by the Hyperplasia-Extended Homeorhesis

Oncogenic hyperplasia is the first and inevitable stage of formation of a (solid) tumor. This stage is also the core of many other proliferative diseases. The present work proposes the first minimal model that combines homeorhesis with oncogenic hyperplasia where the latter is regarded as a genotoxically activated homeorhetic dysfunction. This dysfunction is specified as the transitions of the fluid of cells from a fluid, homeorhetic state to a solid, hyperplastic-tumor state, and back. The key part of the model is a nonlinear reaction-diffusion equation (RDE) where the biochemical-reaction rate is generalized to the one in the well-known Schlögl physical theory of the non-equilibrium phase transitions. A rigorous analysis of the stability and qualitative aspects of the model, where possible, are presented in detail. This is related to the spatially homogeneous case, i.e. when the above RDE is reduced to a nonlinear ordinary differential equation. The mentioned genotoxic activation is treated as a prevention of the quiescent G0-stage of the cell cycle implemented with the threshold mechanism that employs the critical concentration of the cellular fluid and the nonquiescent-cell-duplication time. The continuous tumor morphogeny is described by a time-space-dependent cellular-fluid concentration. There are no sharp boundaries (i.e. no concentration jumps exist) between the domains of the homeorhesis- and tumor-cell populations. No presumption on the shape of a tumor is used. To estimate a tumor in specific quantities, the model provides the time-dependent tumor locus, volume, and boundary that also points out the tumor shape and size. The above features are indispensable in the quantitative development of antiproliferative drugs or therapies and strategies to prevent oncogenic hyperplasia in cancer and other proliferative diseases. The work proposes an analytical-numerical method for solving the aforementioned RDE. A few topics for future research are suggested.     

Single-cell analysis demonstrates how nutrient deprivation creates apoptotic and quiescent cell populations in tumor cylindroids

Understanding how quiescent and apoptotic populations form in tumors is necessary because these cell types can considerably diminish therapeutic efficacy. Most cancer therapeutics are ineffective against quiescent cells because they target rapidly proliferating cells. Distinguishing apoptosis is important because apoptotic cells are committed to death and do not require treatment. Regrowth of quiescent cell can lead to tumor re-occurrence and metastasis, which are the leading causes of cancer mortality. We hypothesized that cylindroid cultures and acridine orange staining could be used to determine how nutrient diffusion creates apoptotic and quiescent regions in tumors. To test this hypothesis we developed a microscopy technique to measure cellular DNA and RNA content in single cells using thin cylindroids and acridine orange staining. Cell classification was compared to flow cytometry of cells grown in defined monolayer cultures. The presence of apoptosis was confirmed by morphological nuclear analysis. The effect of diffusion was determined by varying incubation time, cylindroid size, and exposing cylindroids to nutrient-deficient media. Four overlapping regions were identified as a function of cylindroid radius: an outer viable/quiescent region; a second quiescent/apoptotic region; a third late-stage apoptotic region; and an inner dead region. In monolayer cultures the absence of glutamine and growth factors induced apoptosis and hypoxia induced quiescence. Treating with nutrient-deficient media suggested that cells became quiescent near the periphery because of glucose and oxygen limitations, and became apoptotic and died further from the edge because of glutamine and growth factor limitations. These results show that cellular microenvironments can be identified in cylindroids using simple acridine orange staining and that single cell fluorescence can be measured in three-dimensional culture. The developed techniques will be useful for developing cancer therapies and determining how cell death and apoptosis are induced in three-dimensional tumor tissue.     

From growth arrest to growth suppression.

Since the introduction of the cell cycle concept two approaches to study growth regulation of cells have been proposed. One claims that cells are naturally quiescent, requiring a stimulatory encouter with growth factors for induction of cell division. The other considers cellular multiplication as the natural steady-state; cessation of multiplication is thus a restriction imposed on the system. In the latter case emphasis is mainly on the signals involved in arrest of multiplication. This Prospect focuses on specific events occurring in mammalian cells at growth arrest, senescence, and terminal differentiation, specifically emphasizing the growth inhibitory factors, tumor suppressor genes, and other signals for growth suppression.     

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.     

Cell-cycle-dependent drug-resistant quiescent cancer cells induce tumor angiogenesis after chemotherapy as visualized by real-time FUCCI imaging

We previously demonstrated that quiescent cancer cells in a tumor are resistant to conventional chemotherapy as visualized with a fluorescence ubiquitination cell cycle indicator (FUCCI). We also showed that proliferating cancer cells exist in a tumor only near nascent vessels or on the tumor surface as visualized with FUCCI and green fluorescent protein (GFP)-expressing tumor vessels. In the present study, we show the relationship between cell-cycle phase and chemotherapy-induced tumor angiogenesis using in vivo FUCCI real-time imaging of the cell cycle and nestin-driven GFP to detect nascent blood vessels. We observed that chemotherapy-treated tumors, consisting of mostly of quiescent cancer cells after treatment, had much more and deeper tumor vessels than untreated tumors. These newly-vascularized cancer cells regrew rapidly after chemotherapy. In contrast, formerly quiescent cancer cells decoyed to S/G₂ phase by a telomerase-dependent adenovirus did not induce tumor angiogenesis. The present results further demonstrate the importance of the cancer-cell position in the cell cycle in order that chemotherapy be effective and not have the opposite effect of stimulating tumor angiogenesis and progression.     

Salmonella typhimurium specifically chemotax and proliferate in heterogeneous tumor tissue in vitro

Multi-drug resistance greatly limits the efficacy of conventional blood-born chemotherapeutics, which have limited ability to penetrate tumor tissue and are ineffective at killing quiescent cells far from tumor vasculature. Nonpathogenic, motile bacteria can overcome both of theses limitations. We hypothesize that the accumulation of S. typhimurium in tumors is controlled by two mechanisms: (1) chemotaxis towards compounds produced by quiescent cancer cells and (2) preferential growth within tumor tissue. We tested this hypothesis by quantifying the relative contributions of these mechanisms using the tumor cylindroid model, which mimics the microenvironments of in vivo tumors. Time-lapse fluorescence microscopy was used to measure the accumulation of GFP-labeled S. typhimurium into cylindroids of different size. Cylindroids larger than 500 microm in diameter contain quiescent cells, whereas cylindroids smaller than 500 microm do not. Spatio-temporal profiles of bacterial concentration were fit to a mathematical model to calculate two parameters that describe bacterial interaction with tumors: intratumoral bacterial growth, M, and intratumoral bacterial chemoattraction, K. It was observed that S. typhimurium is attracted to cylindroids and accumulate at long time points in the central region of large cylindroids. Both intratumoral bacterial growth and chemotaxis were significantly greater in large cylindroids, suggesting that quiescent cells secrete bacterial chemoattractants and the presence of necrotic and quiescent cells enable S. typhimurium to replicate in tumor tissue. In this study, several mechanisms of S. typhimurium accumulation in solid tumors have been quantified, which we believe is an important step in the development of bacterial-based therapeutics to target tumor quiescence.     

Collective dynamics of cancer cells confined in a confluent monolayer of normal cells

Tumorigenesis often involves specific changes in cell motility and intercellular adhesion. Understanding the collective cancer cell behavior associated with these specific changes could facilitate the detection of malignant characteristics during tumor growth and invasion. In this study, a cellular vertex model is developed to investigate the collective dynamics of a disk-like aggregate of cancer cells confined in a confluent monolayer of normal cells. The effects of intercellular adhesion and cell motility on tumor progression are examined. It is found that the stresses in both the cancer cells and the normal cells increase with tumor growth, resulting in a crowded environment and enhanced cell apoptosis. The intercellular adhesion between cancer cells and normal cells is revealed to promote tumor growth and invasion. The tumor invasion dynamics hinges on the motility of cancer cells. The cancer cells could orchestrate into different collective migration modes, e.g., directional migration and rotational oscillations, dictated by the competition between cell persistence and local coordination. Phase diagrams are established to reveal the competitive mechanisms. This work highlights the role of mechanics in regulating tumor growth and invasion.     

A stochastic model of a cell population with quiescence

A cell population in which cells are allowed to enter a quiescent (nonproliferating) phase is analyzed using a stochastic approach. A general branching process is used to model the population which, under very mild conditions, exhibits balanced exponential growth. A formula is given for the asymptotic fraction of quiescent cells, and a numerical example illustrates how convergence toward the asymptotic fraction exhibits a typical oscillatory pattern. The model is compared with deterministic models based on semigroup analysis of systems of differential equations.     

Kinetic study of a heterogeneous tumor cell population using a mathematical model

A mathematical model of a heterogenous tumor as a system of interrelating cell populations is described, including a pool of quiescent cells, cell-to-cell variability in maturation rates, and cell migration from growth area to necrotic one. Computer simulation results are given, model labeled mitoses and labeled index curves for the Lewis carcinoma are compared with experimental data.     

Histological Characterization of the Tumorigenic “Peri-Necrotic Niche” Harboring Quiescent Stem-Like Tumor Cells in Glioblastoma

Background

Characterization of the niches for stem-like tumor cells is important to understand and control the behavior of glioblastomas. Cell-cycle quiescence might be a common mechanism underlying the long-term maintenance of stem-cell function in normal and neoplastic stem cells, and our previous study demonstrated that quiescence induced by hypoxia-inducible factor (HIF)-1α is associated with a high long-term repopulation capacity of hematopoietic stem cells. Based on this, we examined human astrocytoma tissues for HIF-1α-regulated quiescent stem-like tumor cells as a candidate for long-term tumorigenic cells and characterized their niche histologically.

Methods

Multi-color immunohistochemistry was used to visualize HIF-1α-expressing (HIF-1α⁺) quiescent stem-like tumor cells and their niche in astrocytoma (WHO grade II–IV) tissues. This niche was modeled using spheroids of cultured glioblastoma cells and its contribution to tumorigenicity was evaluated by sphere formation assay.

Results

A small subpopulation of HIF-1α⁺ quiescent stem-like tumor cells was found in glioblastomas but not in lower-grade astrocytomas. These cells were concentrated in the zone between large ischemic necroses and blood vessels and were closer to the necrotic tissues than to the blood vessels, which suggested that a moderately hypoxic microenvironment is their niche. We successfully modeled this niche containing cells of HIF-1α⁺ quiescent stem-like phenotype by incubating glioblastoma cell spheroids under an appropriately hypoxic condition, and the emergence of HIF-1α⁺ quiescent stem-like cells was shown to be associated with an enhanced sphere-forming activity.

Conclusions

These data suggest that the “peri-necrotic niche” harboring HIF-1α⁺ quiescent stem-like cells confers a higher tumorigenic potential on glioblastoma cells and therefore may be a therapeutic target to control the behavior of glioblastomas.     

A Microenvironment Based Model of Antimitotic Therapy of Gompertzian Tumor Growth

A model of tumor growth, based on two-compartment cell population dynamics, and an overall Gompertzian growth has been previously developed. The main feature of the model is an inter-compartmental transfer function that describes the net exchange between proliferating (P) and quiescent (Q) cells and yields Gompertzian growth for tumor cell population N = P + Q. Model parameters provide for cell reproduction and cell death. This model is further developed here and modified to simulate antimitotic therapy. Therapy decreases the reproduction-rate constant and increases the death-rate constant of proliferating cells with no direct effect on quiescent cells. The model results in a system of two ODE equations (in N and P/N) that has an analytical solution. Net tumor growth depends on support from the microenvironment. Indirectly, this is manifested in the transfer function, which depends on the proliferation ratio, P/N. Antimitotic therapy will change P/N, and the tumor responds by slowing the transfer rate from P to Q. While the cellular effects of therapy are modeled as dependent only on antimitotic activity of the drug, the tumor response also depends on the tumor age and any previous therapies—after therapy, it is not the same tumor. The strength of therapy is simulated by the parameter λ, which is the ratio of therapy induced net proliferation rate constant versus the original. A pharmacodynamic factor inversely proportional to tumor size is implemented. Various chemotherapy regimens are simulated and the outcomes of therapy administered at different time points in the life history of the tumor are explored. Our analysis shows: (1) for a constant total dose administered, a decreasing dose schedule is marginally superior to an increasing or constant scheme, with more pronounced benefit for faster growing tumors, (2) the minimum dose to stop tumor growth is age dependent, and (3) a dose-dense schedule is favored. Faster growing tumors respond better to dose density.     

Differential responses of proliferating versus quiescent cells to adriamycin.

The relative sensitivity of proliferating and quiescent cells to DNA-damaging agents is a key factor for cancer chemotherapy. Here we undertook a reevaluation of the way that proliferating and quiescent cells differ in their responses and fate to adriamycin-induced damage. Distinct types of assays that measure membrane integrity, metabolic activity, cell size, DNA content, and the ability to proliferate were used to compare growing and quiescent Swiss3T3 fibroblasts after adriamycin treatment. We found that immediately after adriamycin treatment of growing cells, p53 and p21(Cip1/Waf1) were induced but the cells remained viable. In contrast, less p53 and p21(Cip1/Waf1) were induced in quiescent cells after adriamycin treatment, but the cells were more prone to immediate cell death, possibly involving apoptosis. Adriamycin induced a G2/M cell cycle arrest in growing cells and a concomitant increase in cell size. In contrast, adriamycin induced an increase in sub-G1 DNA content in quiescent cells and a decrease in cell size. In contrast to the short-term responses, adriamycin-treated quiescent cells have a better long-term survival and proliferation potential than adriamycin-treated growing cells in colony formation assays. These data suggest that proliferating and resting cells are remarkably different in their short-term and long-term responses to adriamycin.     

Curcumin induces cell death without oligonucleosomal DNA fragmentation in quiescent and proliferating human CD8+ cells

Cytotoxic CD8+ cells play an important role in determining host response to tumor, thus chemotherapy is potentially dangerous as it may lead to T cells depletion. The purpose of this study was to elucidate the propensity of quiescent and proliferating human CD8+ cells to undergo cell death upon treatment with curcumin, a natural dye in Phase I of clinical trials as a prospective chemopreventive agent. Methods: We treated human quiescent or proliferating CD8+ cells with 50 microM curcumin or irradiated them with UVC. Cell death symptoms such as decreased cell viability, chromatin condensation, activation of caspase-3 and specific DFF40/CAD endonuclease and oligonucleosomal DNA fragmentation were analyzed using MTT test, microscopic observation, Western blotting and flow cytometry. Results: Curcumin decreased cell viability, activated caspase-3 and decreased the level of DFF45/ICAD, the inhibitor of the DFF40/CAD endonuclease. However, this did not lead to oligonucleosomal DNA degradation. In contrast, UVC-irradiated proliferating, but not quiescent CD8+ cells revealed molecular and morphological changes characteristic for apoptosis, including oligonucleosomal DNA fragmentation. Curcumin can induce cell death in normal human lymphocytes both quiescent and proliferating, without oligonucleosomal DNA degradation which is considered as a main hallmark of apoptotic cell death. Taking into account the role of CD8+ cells in tumor response, their depletion during chemotherapy could be particularly undesirable.     

Cell-cell contact and growth regulation of pinocytosis in 3T3 cells

In sub-confluent cultures of Balb/c-3T3 cells, pinocytosis rates were increased after exposure to specific growth factors (serum; platelet-derived growth factor, PDGF; epidermal growth factor, EGF). Conversely, as cells became growth-inhibited with increasing culture density, there was a corresponding decline in pinocytosis rate per cell. In order to test whether density-inhibition of pinocytosis was influenced either by the growth cycle or by cell contact independently of growth, cells were induced into a quiescent state at a range of subconfluent and confluent densities. Under such conditions, cell density did not significantly inhibit pinocytosis rate. When confluent quiescent cultures in 2.5% serum were exposed to 10% serum, the resulting round of DNA synthesis was accompanied by enhanced pinocytosis per cell, even though the cells were incontact with one another. Furthermore, in a SV40-viral transformed 3T3 cell line, both the growth fraction and the pinocytosis rate per cell remained unchanged over a wide range of culture densities. These studies indicate that density-dependent inhibition of pinocytosis in 3T3 cells appears to be secondary to growth-inhibition rather than to any direct physical effects of cell–cell contact.