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.     

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.     

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.     

Macrophage-Based Anti-Cancer Therapy: Modelling Different Modes of Tumour Targeting

Tumour hypoxia is associated with poor drug delivery and low rates of cell proliferation, factors that limit the efficacy of therapies that target proliferating cells. Since macrophages localise within hypoxic regions, a promising way to target hypoxic tumour cells involves engineering macrophages to express therapeutic genes under hypoxia. In this paper we develop mathematical models to compare the responses of avascular tumour spheroids to two modes of action: either the macrophages deliver an enzyme that activates an externally applied prodrug (bystander model), or they deliver cytotoxic factors directly (local model). The models we develop comprise partial differential equations for a multiphase mixture of tumour cells, macrophages and extracellular fluid, coupled to a moving boundary representing the spheroid surface. Chemical constituents, such as oxygen and drugs, diffuse within the multiphase mixture. Simulations of both models show the spheroid evolving to an equilibrium or to a travelling wave (multiple stable solutions are also possible). We uncover the parameter dependence of the wave speed and steady-state tumour size, and bifurcations between these solution forms. For some parameter sets, adding extra macrophages has a counterintuitive deleterious effect, triggering a bifurcation from bounded to unbounded tumour growth. While these features are common to the bystander and local models, the crucial difference is where cell death occurs. The bystander model is comparable to traditional chemotherapy, with poor targeting of hypoxic tumour cells; however, the local mode of action is more selective for hypoxic regions. We conclude that effective targeting of hypoxic tumour cells may require the use of drugs with limited mobility or whose action does not depend on cell proliferation.     

Cell Kinetic Characteristics In Different Parts of Multicellular Spheroids of Human Origin

The growth fraction, the cell cycle time, and the duration of the individual cell cycle phases were determined as a function of distance from the surface of multicellular spheroids of the human cell line NHIK 3025. the techniques employed were percentage of labelled mitoses and labelling index measurements after autoradiography and flow cytometric measurements of DNA histograms. to separate cell populations from the different parts of the spheroid, fractionated trypsinization was employed. The results were compared with corresponding values in NHIK 3025 cell populations grown as monolayer cultures. While practically all cells in exponentially growing monolayer populations are proliferating, the growth fraction was between 0.6 and 0.7 in the outer parts of the spheroid. the inner region was mainly occupied by a necrotic mass. the proliferating fraction of the recognizable cells in the inner region was slightly below 0.5. the mean cell cycle time of NHIK 3025 cells in monolayer culture is 18 hr. the mean cell cycle time of proliferating cells in the periphery of the spheroid was 30 hr, compared to 41 hr in the inner region (150 μm from the spheroid surface). All phases of the cell cycle were prolonged compared to populations of exponentially growing monolayer cells. Within each part of the spheroid the distribution of cell cycle times was considerably broadened compared with monolayer populations.     

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.     

Cell migration in multicell spheroids: Swimming against the tide

Multicell spheroids, small spherical clusters of cancer cells, have become an importantin vitro model for studying tumour development given the diffusion limited geometry associated with many solid tumour growths. Spheroids expand until they reach a dormant state where they exhibit a grossly static three-layered structure. However, at a cellular level, the spheroid is demonstrably dynamic with constituent cells migrating from the outer well-nourished region of the spheroid toward the necrotic central core. The mechanism that drives the migrating cells in the spheroid is not well understood. In this paper we demonstrate that recent experiments on internationalization can be adequately described by implicating pressure gradients caused by differential cell proliferation and cell death as the primary mechanism. Although chemotaxis plays a role in cell movement, we argue that it acts against the passive movement caused by pressure differences.     

Cell resensitization after delivery of a cycle-specific anticancer drug and effect of dose splitting: learning from tumour cords

After a single dose of an anticancer agent, changes due to cell death are expected to occur in the distribution of cells between proliferating and quiescent compartment as well as in the oxygenation and nutritional state of surviving cells. These changes are transient because tumour regrowth tends to restore the pretreatment status. The reoxygenation due to the decrease of oxygen consumption is expected to induce cell recruitment from quiescence into proliferation, and consequently to increase the sensitivity of the cell population to a successive treatment by a cycle-specific drug. In previous papers we proposed a model of the response of tumour cords (cylindrical arrangements of tumour cells growing around a blood vessel of the tumour) to single-dose treatments. The model included the motion of cells and oxygen diffusion and consumption. On the basis of that model suitably extended to better account for the action of anticancer drugs, we study the time course of the oxygenation and of the redistribution of cells between the proliferating and quiescent compartments. By means of simulations of the response to a dose delivered as two spaced equal fractions, we investigate the dependence of tumour response on the spacing between the fractions and on the main parameters of the system. A time window may be found in which the delivery of two fractions is more effective than the delivery of the undivided dose.     

Cell kinetics and radiation biology

The cell cycle, the growth fraction and cell loss influence the response of cells to radiation in many ways. The variation in radiosensitivity around the cell cycle, and the extent of radiation-induced delay in cell cycle progression have both been clearly demonstrated in vitro. This translates into a variable time of expression of radiation injury in different normal tissues, ranging from a few days in intestine to weeks, months or even years in slowly proliferating tissues like lung, kidney, bladder and spinal cord. The radiosensitivity of tumours, to single doses, is dominated by hypoxic cells which arise from the imbalance between tumour cell production and the proliferation and branching of the blood vessels needed to bring oxygen and other nutrients to each cell. The response to fractionated radiation schedules is also influenced by the cell kinetic parameters of the cells comprising each tissue or tumour. This is described in terms of repair, redistribution, reoxygenation and repopulation. Slowly cycling cells show much more curved underlying cell survival curves, leading to more dramatic changes with fractionation, dose rate or l.e.t. Rapidly cycling cells redistribute around the cell cycle when the cells in sensitive phases have been killed, and experience less mitotic delay than slowly proliferating cells. Reoxygenation seems more effective in tumours with rapidly cycling cells and high natural cell loss rates. Compensatory repopulation within a treatment schedule may spare skin and mucosa but does not spare slowly proliferating tissues. Furthermore, tumour cell proliferation during fractionated radiotherapy may be an important factor limiting the overall success of treatment.     

Cell kinetics in a tumour cord

In some tumours, the viable cells grow around blood vessels forming cylindrical structures called tumour cords, which are surrounded by regions of necrosis. In the present paper, we propose a mathematical model for the cell kinetics in a tumour cord at the stationary state. Both proliferating cells and quiescent cells are considered, and the proliferating cell population is structured by age. Cell migration towards cord periphery is accounted for from a continuum viewpoint. The age distribution of proliferating cells, the fraction of cells in S phase, the growth fraction and the velocity along the cord radius are computed. The predictions of the model are compared with literature data obtained from two experimental rat hepatomas. The model was used to compute the profile of the oxygen tension within the cord. Possible modifications and extensions are also presented.     

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.     

Implications of a simple mathematical model to cancer cell population dynamics

Recent research in cancer progression and treatment indicates that many forms of cancer arise from the development of a small subpopulation of abnormal cancer stem cells (CSCs) that promote cancer growth and spread. Many potential treatments preferentially interact with cells at certain stages of the cell cycle by either selective killing or halting the cell cycle, such as intense, nanosecond-duration pulsed electric fields (nsPEFs). Simple mathematical models of unfed cancer cell populations at the plateau of their growth characteristics may estimate the long-term consequences of these treatments on proliferating and quiescent cell populations. Applying such a model with no transition from the quiescent to proliferating state shows that it is possible for the proliferating cell population to fall below 1 if the quiescent cell population obtains a sufficient competitive advantage with respect to nutrient consumption and/or survival rate. Introducing small, realistic transition rates did not appreciably alter short-term or long-term population behaviour, indicating that the predicted small cell population behaviour (< 1 cell) is not an artefact of the simpler model. Experimental observations of nsPEF-induced effects on the cell cycle suggest that such a model may serve as a first step in assessing the viability of a given cancer treatment in vitro prior to clinical application.     

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.     

Cell cycle regulation by environmental pH

Purified populations of quiescent human tumour cells were isolated from plateau phase cultures of PMC-22 cells by centrifugal elutriation. Dilution into fresh medium resulted in these quiescent cells entering S phase exponentially with a t1/2 of 12 hr, after a 18-20-hr lag period during which cellular RNA content increased. Subsequent studies showed that recruitment of quiescent cells into the cell cycle could be regulated by extracellular pH. When exponentially growing PMC-22 cells were exposed to acidic extracellular pH levels, three growth patterns were observed: (1) Normal growth between pH 7.2 to pH 6.8; (2) A reduction in growth rate associated with accumulation of cells with a G₁ DNA content between pH 6.7 and 6.4 (this was also shown to occur in a number of other tumour cell lines); (3) Non-cell-cycle-phase-specific arrest of growth at pH levels less than 6.3. Further studies with purified quiescent cell populations showed the possible existence of a pH-dependent restriction point in the G₁ phase of these tumour cells. The implications of these observations to tumour biology are discussed.     

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.     

Cell migration in tumors

Invasion of cancer cells into surrounding tissue and the vasculature is an initial step in tumor metastasis. This requires chemotactic migration of cancer cells, steered by protrusive activity of the cell membrane and its attachment to the extracellular matrix. Recent advances in intravital imaging and the development of an in vivo invasion assay have provided new insights into how cancer cell migration is regulated by elements of the local microenvironment, including the extracellular matrix architecture and other cell types found in primary tumors. These results, combined with new findings from in vitro studies, have led to new insights into the molecular mechanisms of cell protrusive activity and chemotactic migration during invasion and metastasis.     

一个良性肿瘤细胞生长的计算机仿真模型

本文建立了一个良性肿瘤在正常细胞组织中生长的计算机仿真模型。初始细胞模型采用二维Voronoi结构,使用繁特卡罗法将其离散成x^*y个小细胞,对肿瘤与正常细胞分别施加蒙特卡罗工关过程使细胞生长。根据细胞组织内营养水平可以决定肿瘤细胞（分别为休眠细胞和有生长繁殖能力的细胞）的生长状态。     

Mathematical modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumour sites

Poor drug delivery and low rates of cell proliferation are two factors associated with hypoxia that diminish the efficacy of many chemotherapeutic drugs. Since macrophages are known to migrate specifically towards, and localize within, hypoxic tumour regions, a promising resolution to these problems involves genetically engineering macrophages to perform such anti-tumour functions as inducing cell lysis and inhibiting angiogenesis. In this paper we outline a modelling approach to characterize macrophage infiltration into early avascular solid tumours, and extensions to study the interaction of these cells with macrophages already present within the tumour. We investigate the role of chemotaxis and chemokine production, and the efficacy of macrophages as vehicles for drug delivery to hypoxic tumour sites. The model is based upon a growing avascular tumour spheroid, in which volume is filled by tumour cells, macrophages and extracellular material, and tumour cell proliferation and death is regulated by nutrient diffusion. Crucially, macrophages occupy volume, and hence contribute to the volume balance and hence the size of the tumour. We also include oxygen-dependent production of macrophage chemokines, which can lead to accumulations in the hypoxic region of the tumour. We find that the macrophage chemotactic sensitivity is a key determinant of macrophage infiltration and tumour size. Although increased infiltration should be beneficial from the point of view of macrophage-based therapies, such infiltration in fact leads to increased tumour sizes. Finally, we include terms representing the induced death of tumour cells by hypoxic engineered macrophages. We demonstrate that reductions in tumour size can be achieved, but predict that a combination of therapies would be required for complete eradication. We also highlight some counter-intuitive predictions-for example, absolute and relative measures of tumour burden lead to different conclusions about prognosis. In summary, this paper illustrates how mathematical models may be used to investigate promising macrophage-based therapies.     

Enhanced serine palmitoyltransferase expression in proliferating fibroblasts, transformed cell lines, and human tumors.

Metastatic processes, including cell invasion, extracellular matrix degradation, and tissue remodeling, require cellular reorganization and proliferation. The cell signaling molecules required and the proteins involved in cell restructuring have not been completely elucidated. We have been studying the role of sphingolipids in normal cell activity and in several pathophysiological states. In this study we used immunohistochemistry to observe the presence of the two known subunits of serine palmitoyltransferase (SPT) in proliferating cells, in an in vitro model of wound repair, and in human malignant tissue. We report increased expression of the two subunits, SPT1 and SPT2, in the proliferating cells in these models. We also demonstrate a change in subcellular localization of the SPT subunits from predominantly cytosolic in quiescent cells to nuclear in proliferating cells. In addition, we observed SPT1 and SPT2 immunoreactivity in reactive stromal fibroblasts surrounding the carcinoma cells of some of the tumors. This enhanced SPT expression was absent in the stromal fibroblasts surrounding normal epithelial cells. Our results suggest a potential role for overexpression of SPT in the processes of cell metastasis.     

Murine Mammary Tumour Cells In Vitro. Ii. Recruitment of Quiescent Cells

Abstract The development of a pure quiescent (Q) tumour cell population can be induced in three mouse mammary tumour lines (66, 67 and 68H) by nutrient deprivation. When these Q cells were removed from nutrient-deprived cultures and replated in fresh medium at a lower cell concentration within 72 hr of entering quiescence virtually all of the Q cells could re-enter the proliferating (P) state. This recruitment was characterized by an increase in cell volume, an increase in total cellular RNA, and a resumption of cell division. the length of the Q to P transition varied among the three cell lines and the depth of the quiescent state depended on the amount of time the cells had been quiescent. Once re-entry into the P compartment was completed, cell-cycle times, as estimated by the culture doubling time, were the same as the cells that had not entered the Q state. however, after 72 hr in quiescence, not all of the 66 cells could reattach after trypsinization and of those that could reattach 50% were incapable of either increasing their RNA levels to that of proliferating G₁ cells or entering S. Clonogenicity of the nutrient-deprived Q cells in these lines decreases exponentially from time the cells enter quiescence with approximate half-times of 32, 34, and 96 hr for the 66, 68H and 67 cells, respectively. Slnce clonogenicity was already declining at a time when all the Q cells could re-enter the P compartment, the ability of a Q cell to form a colony is not determined solely by its capacity to re-enter the proliferating compartment.