A model of tumor growth based on cell cycle kinetics

This work describes a mathematical model of growth based on the kinetics of the cell cycle. A traditional model of the cell cycle has been used, with the addition of a resting (G₀) state from which cells could reenter the reproductive cycle. The model assumes that a growth regulatory substance regulates the transition of cells to and from the resting state. Other transitions between the phases of the cycle were modeled as a first order process. Cell loss is an important feature of growth kinetics, and has been represented by a general but tractable mathematical form. The resulting model forms a system of ordinary nonlinear differential equations. Analytic methods are employed first in the study of this system. Simplifying assumptions regarding cell loss give rise to special cases for which equilibrium solutions can be found. One special case, which assumes first order loss from all cell cycle phases at equal rates, is presented here. For small time values, approximations corresponding to exponential growth were developed. The equations describing an intrinsic growth rate were derived. Simulation methods were used to further characterize the behavior of this model. Parameter values were chosen based on animal tumor cell cycle kinetic data, resulting in a set of 45 model simulations. Several tumor treatment protocols were simulated which illustrated the importance of the intrinsic growth rate and cell loss concepts. Although the qualitative behavior regarding absolute and relative growth is reasonable, this model awaits data for model fitting, parameter estimation, or revision of the equations.     

Cell cycle specificity of tumor necrosis factor and its receptor

Phase specificity in the TNF cytotoxic effect and the number of TNF binding receptors was investigated using L-M cells incubated synchronously from the S phase. TNF cytotoxicity was observed to occur at various levels during the cell cycle, with peak effect in the G2-M phase. Analysis with 125I-labeled TNF to determine the number of receptors binding TNF in the various cell phases shewed a phase specificity with the maximum number occurring in the G2-M phase, similar to the peak in cytotoxicity. The results suggest the existence of a cell cycle specificity in the cytotoxicity of TNF which is apparently related to changes in the number of receptors capable of binding TNF.     

Antitumor activity of L-asparaginase from Erwinia carotovora against different human and animal leukemic and solid tumor cell lines

The dose- and time-dependent antitumor and cytotoxic effects of L-asparaginases from Erwinia carotovora (ECAR LANS) and Escherichia coli (MEDAC) have been investigated using human leukemic cells and human and animal solid tumor cells. These included human T-cell acute lymphoblastic leukemia cell lines (Jurkat, Jurkat/A4, Molt-4), human chronic myeloid leukemia K562 cells, human promyelocytic leukemia HL-60, and also human solid tumor cells (prostate carcinoma LnCap, breast adenocarcinoma MCF7, ovarian adenocarcinoma SCOV-3 and carcinoma CaOV, hepatocarcinoma Hep G2, fibrosarcoma HT-1080) and animal solid tumor cells (rat Gasser??s ganglion neurinoma cells GGNC-1, mouse glioblastoma EPNT-5). We investigated sensitivity of tumor cells (seeded at different density) to L-asparaginases, as well the effect of L-asparaginases on cell growth rate, protein and DNA synthesis in the presence of various cytostatics. Cell cycle analysis by flow cytofluorimetry and detection of apoptotic cells before and after treatment with L-asparaginases indicate that ECAR LANS L-asparaginase suppressed growth of all tested solid tumor cells. Evaluation of leukemic cell number after treatment with L-asparaginases for 24, 48 and 72 h demonstrated that asparagine deficiency did not kill cells but stopped normal cell division. The cytofluorometric study of solid and leukemic cells revealed that except HL-60 cells the treatment with L-asparaginase for 72 h did not change cell cycle phase distribution and did not increase the number of apoptotic cells. Combined treatment of cells using a combination of L-asparaginase and doxorubicin significantly increased the number of apoptotic cells up to 60% (MCF-7 cells), 40% (Jurkat cells) and even 99% (HL-60). High levels of DNA and protein synthesis rates in asparaginase-treated tumor cells suggest lack of massive entry of tumor cells to apoptosis. This conclusion is based on the fact of sensitivity of multi-resistant Jurkat/A4 cells to L-asparaginases (it is nearly impossible to induce apoptosis in these cells). Since ECAR LANS did not influence growth of normal human fibroblasts it appears that the enzyme cytotoxicity is associated only asparagine deficiency.     

Cell cycle dynamics inferred from the static properties of cells in balanced growth

The duration of a morphological phase of the cell cycle is reflected in the steady state distribution of the sizes of cells in that phase. Relationships presented here provide a method for estimating the timing and variability of any cell cycle phase. It is shown that the mean size of cells initiating and finishing any phase can be estimated from (1) the frequency of cells exhibiting the distinguishing morphological or autoradiographic features of the phase; (2) the mean size of cells in the phase; and (3) their coefficient of variation. The calculations are based on a submodel of the Koch-Schaechter Growth Controlled Model which assumes that (i) the distribution of division sizes is Gaussian; (ii) there is no correlation in division sizes between successive generations; and (iii) every cell division gives rise to two daughter cells of equal size. The calculations should be useful for a wider range of models, however, because the extrapolation factors are not sensitive to the chosen model. Criteria are proposed to allow the user to check the method's applicability for any experimental case. The method also provides a more efficient test of the dependence of growth on cell size than does the Collins-Richmond method. This is because the method uses the mean and coefficient of variation of the size of the total population, in conjunction with those of the cells in a final phase of the cell cycle, to test potential growth laws. For Escherichia coli populations studied by electron microscopy, an exponential growth model provided much better agreement than did a linear growth model. The computer simulations were used to generate rules for three types of cell phases: those that end at cell division, those that start at cell division, and those totally contained within a single cell cycle. For the last type, additional criteria are proposed to establish if the phase is well enough contained for the formulae and graphs to be used. The most useful rule emerging from these computer studies is that the fraction of the cell cycle time occupied by a phase is the product of the frequency of the phase and the ratio of the mean size of cells in that phase to the mean size of all cells in the population. A further advantage of the techniques presented here is that they use the 'extant' distributions that were actually measured, and not hypothesized distributions nor the special distributions needed for Collins-Richmond method that can only be calculated from the observed distributions of dividing or newborn cells on the basis of an assumed growth law.     

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.     

Kinetic analysis of cell size and DNA content distributions during tumor cell proliferation: Ehrlich ascites tumor study.

In order to study the growth dynamics of proliferating and non-proliferating cells utilizing discrete-time state equations, the cell cycle was divided into a finite number of age compartments. In analysing tumor growth, the kinetic parameters associated with a retardation in the growth rate of tumors were characterized by computer simulation in which the simulated results of the growth curve, the growth fraction, and the mean generation time were adjusted to fit the experimental data. The cell age distibution during the period of growth was obtained and by a linear transformation of the state transition matrices, was employed to specify the cell size and DNA content distributions. In an application of the model, the time-course behavior of cell cycle parameters of Ehrlich ascites tumor is illustrated, and the parameters important for the transition of cells in the proliferating compartment to the non-proliferating compartment are discussed, particularly in relation to the G1-G0 and G2-G0 transitions of non-cycling cells as revealed by the variation of cell size distribution.     

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.     

Cell cycle analysis of foreign gene (beta-galactosidase) expression in recombinant mouse cells under control of mouse mammary tumor virus promoter

The cell cycle dependency of foreign gene expression in recombinant mouse L cells was investigated. Two different recombinant mouse L cell lines having the glucocorticoid receptor-encoding gene and the lacZ reporter gene were used in this study. The lacZ gene expression was controlled by the glucocorticoid-inducible mouse mammary tumor virus (MMTV) promoter in both cell lines. In "M4" cells the gr gene was under the control of another MMTV promoter, but in "R2" cells it was under the control of the constitutive Rous sarcoma virus promoter. These normally attachment-grown cells were adapted to suspension culture, and a dual-laser flow cytometer was used to simultaneously determine the DNA and foreign protein (beta-galactosidase) content of single living cells. Expression of beta-galactosidase as a function of cell cycle phase was evaluated for cells in exponential growth without any addition of the glucocorticoid inducer, dexamethasone. Cell cycle positions in the S phase were estimated on the basis of DNA content per cell, and position in the G1 phase was estimated on the basis of cell size as measured by pulse-width time of flight. The results showed that beta-galactosidase synthesis occurred through all cell cycle phases, but the expression rate in the G1 phase was much lower than that in the S and G2/M phases in both cell lines. On the basis of cell size analysis, beta-galactosidase expression in M4 cells (with autoinducible promoter) was found to be higher than that in R2 cells (with inducible promoter) during the G1 phase. (c) 1996 John Wiley & Sons, Inc.     

The evolution of life cycles with haploid and diploid phases

Sexual eukaryotic organisms are characterized by an alternation between haploid and diploid phases. In vascular plants and animals, somatic growth and development occur primarily in the diploid phase, with the haploid phase reduced to the gametic cells. In many other eukaryotes, however, growth and development occur in both phases, with substantial variability among organisms in the length of each phase of the life cycle. A number of theoretical models and experimental studies have shed light on factors that may influence life cycle evolution, yet we remain far from a complete understanding of the diversity of life cycles observed in nature. In this paper we review the current state of knowledge in this field, and touch upon the many questions that remain unanswered. BioEssays 20 :453–462, 1998. © 1998 John Wiley & Sons, Inc.     

微生物生活周期中的一个新的时期——长期稳定期

传统的对微生物生长时期的认识一般分为4个时期:迟缓期、对数期、稳定期、死亡期。实际上,这种划分不足以使我们认识到微生物生长过程的全貌。近年来许多研究表明,在死亡期之后存在一个完全不同意义的时期——长期稳定期。这个时期可能与微生物在环境中的生存状态更加相似。微生物细胞通过突变得以生存,并在选择中形成稳定期生长优势表型,深入研究微生物长期稳定期具有极其重要的意义。     

Selection and the Evolution of Genetic Life Cycles

The evolution of haploid and diploid phases of the life cycle is investigated theoretically, using a model where the relative length of haploid and diploid phases is under genetic control. The model assumes that selection occurs in both phases and that fitness in each phase is a function of the time spent in that phase. The equilibrium and stability conditions that allow for all-haploid, all-diploid, or polyphasic life cycles are considered for general survivorship functions. Types of stable life cycles possible depend on the form of the viability selection. If mortality rates are constant, either haploidy or diploidy is the only stable life cycle possible. Departures from constant mortality can give qualitatively different results. For example, when survivorship in each phase is a linear, decreasing function of the time spent in the phase, stable haploid, diploid or polyphasic life cycles are possible. The addition of genetic variation at a coevolving viability locus does not qualitatively affect the outcome with respect to the maintenance of polyphasic cycles but can lead to situations where more than one life cycle is concurrently stable. These results show that trade-offs between the advantages of being diploid and of being haploid may help explain the patterns of life cycles found in nature and that the type of selection may be critical to determining the results.     

Estimating the Total Rate of DNA Replication Using Branching Processes

Increasing the knowledge of various cell cycle kinetic parameters, such as the length of the cell cycle and its different phases, is of considerable importance for several purposes including tumor diagnostics and treatment in clinical health care and a deepened understanding of tumor growth mechanisms. Of particular interest as a prognostic factor in different cancer forms is the S phase, during which DNA is replicated. In the present paper, we estimate the DNA replication rate and the S phase length from bromodeoxyuridine-DNA flow cytometry data. The mathematical analysis is based on a branching process model, paired with an assumed gamma distribution for the S phase duration, with which the DNA distribution of S phase cells can be expressed in terms of the DNA replication rate. Flow cytometry data typically contains rather large measurement variations, however, and we employ nonparametric deconvolution to estimate the underlying DNA distribution of S phase cells; an estimate of the DNA replication rate is then provided by this distribution and the mathematical model.     

An intradermal assay for quantification and kinetics studies of tumor angiogenesis in mice.

A method is reported for the study of early phases of neovascularization in syngeneic murine tumors and human tumor xenografts in nude mice. Using this method, the effect of irradiation of tumor cells or tumor bed on tumor angiogenesis was studied. Tumor cells were injected intradermally in the abdominal skin flap, which was reopened at 2-day intervals to quantify newly formed blood vessels at the site of tumor cell injection. Both tumor cell injection and blood vessel counting were performed under a dissecting microscope. Using three syngeneic murine tumors and two clones of a human colonic adenocarcinoma, it was observed that new blood vessels started appearing within a few days after tumor cell injection and that this event preceded measurable tumor growth. The number of blood vessels increased exponentially for several days but then their further growth slowed. The extent of angiogenesis depended on the tumor type and the number of tumor cells injected. The exposure of the skin flap to ionizing radiation prior to tumor cell injection reduced neovascularization. We further observed that heavily irradiated tumor cells retained their ability to induce angiogenic response and that lymphoid cells (peritoneal exudate and spleen cells) could also elicit an angiogenic response, although it is weaker than the response elicited by tumor cells. Thus this method is suitable for quantification and kinetics of early phases of tumor angiogenesis in individual mice bearing transplants of syngeneic tumors or human tumor xenografts, and it can be useful for investigating various regulators of tumor angiogenesis.     

Cellular and extracellular markers of hemangioma

Several cellular and extracellular markers that distinguish the phases of the hemangioma life cycle have been described previously. However, details of the phenotypic changes of; the various cellular elements during hemangioma development have not been fully reported, and the extracellular matrix composition, especially in the vicinity of the proliferating endothelial cells, is poorly described. This study examined the expression of cellular and extracellular molecules and cytokines in the proliferative, involuting, and involuted phases of hemangioma. Paraffin-embedded hemangioma specimens, four from each phase, were examined histochemically and immunohistochemically. Throughout the three phases, vascular endothelial cells stained positive for CD31 and von Willebrand factor, although in the involuted phase, not all vessels in the tissue expressed these endothelial markers. Proliferating cell nuclear antigen was expressed by the majority of endothelial cells and pericytes in the proliferative and early involuting phases, but its expression was negligible in the involuted phase. In addition to finding that the total number of mast cells was highest in the involuting phase, the authors observed that the proportion of chymase-positive mast cells decreased with the progression of hemangioma and that virtually all mast cells expressed the biogenic amine phenotype throughout the hemangioma life cycle. The localization of vascular endothelial growth factor predominantly to the pericytes and endothelial cells during the proliferative phase and of basic fibroblast growth factor to the endothelial cells in both the proliferative and early involuting phases is consistent with previous reports, although the latter growth factorwas also observed in mast cells. Type IV collagen and the beta chain of laminin and perlecan were detected in the basement membranes in all phases. Interestingly, collagen types I, III, and V were present in basal membranes throughout the phases and with increasing density in the stromal areas with involution, although type I collagen was less prominent during the proliferative phase. Short-chain collagen type VIII was localized extracellularly throughout the development of hemangioma but, during the early proliferative phase, it was also detected within mast cells. The expression of specific cytokines and cellular and extracellular markers may help distinguish the different clinical phases of the hemangioima life cycle. These results provide further insight into the biology of hemangioma.     

Expression of mast cell proteases correlates with mast cell maturation and angiogenesis during tumor progression

Tumor cells are surrounded by infiltrating inflammatory cells, such as lymphocytes, neutrophils, macrophages, and mast cells. A body of evidence indicates that mast cells are associated with various types of tumors. Although role of mast cells can be directly related to their granule content, their function in angiogenesis and tumor progression remains obscure. This study aims to understand the role of mast cells in these processes. Tumors were chemically induced in BALB/c mice and tumor progression was divided into Phases I, II and III. Phase I tumors exhibited a large number of mast cells, which increased in phase II and remained unchanged in phase III. The expression of mouse mast cell protease (mMCP)-4, mMCP-5, mMCP-6, mMCP-7, and carboxypeptidase A were analyzed at the 3 stages. Our results show that with the exception of mMCP-4 expression of these mast cell chymase (mMCP-5), tryptases (mMCP-6 and 7), and carboxypeptidase A (mMC-CPA) increased during tumor progression. Chymase and tryptase activity increased at all stages of tumor progression whereas the number of mast cells remained constant from phase II to III. The number of new blood vessels increased significantly in phase I, while in phases II and III an enlargement of existing blood vessels occurred. In vitro, mMCP-6 and 7 are able to induce vessel formation. The present study suggests that mast cells are involved in induction of angiogenesis in the early stages of tumor development and in modulating blood vessel growth in the later stages of tumor progression.     

Cell population dynamics during the differentiative phase of tissue development

The dynamics of a cell population whose numbers are growing exponentially have been described well by a mathematical model based on the theory of age-dependent branching processes. Such a model, however, does not cover the period following exponential growth when cell differentiation curtails population size. This paper offers an extension to the branching process model to remedy this deficiency. The extended model is ideal for describing embryonic growth; its use is illustrated with data from embryonic retina. The model offers a better computational framework for the interpretation of a variety of data (growth curves of cell numbers, DNA histograms, thymidine labelling indices, FLM curves, BUdR-labelled mitoses curves) because age-distributions can be calculated at any stage of development, not just during exponential growth. Proportions of cells in the various phases of the cell cycle can be computed as growth slows. Such calculations show the gradual transition from a population dominated by cells which are young with respect to cell cycle age to one dominated by those which are old, and the effects such biases have on the proportions of cells in each phase.     

Inhibition of subcutaneous growth of Ehrlich ascites carcinoma (EAC) tumor by post-immunization with EAC-cell gangliosides and its anti-idiotype antibody in relation to tumor angiogenesis, apoptosis, cell cycle and infiltration of CD4+, CD8+ lymphocytes, NK cells, suppressor cells and APC-cells in tumor

Both EAC-tumor associated gangliosides and its anti-idiotype antibody inhibited growth of this tumor significantly. Immuno-histological studies with von Willebrand Factor (vWF) antibody indicated that tumor angiogenesis as determined by expression of vWF decreased in tumors of mice, post-immunized with EAC-cell gangliosides as well as its anti-idiotype antibody. Infiltration of various immune cells of the host in the tumor correlated to some extent with tumor-growth inhibition. Apoptosis study using AnnexinV-FITC and propidium iodide indicated that tumor growth inhibition in mice post-immunized with EAC-gangliosides and its anti-idiotype antibody were due to enhanced apoptosis and cell death. Cell cycle analysis by FACS indicated that EAC-cell associated gangliosides and its anti-idiotype antibody were acting both at the M2 i.e. S and M3 i.e. G2/M phases of the cell cycle to arrest tumor growth.     

Immunoediting: evidence of the multifaceted role of the immune system in self-metastatic tumor growth

ABSTRACT: BACKGROUND: The role of the immune system in tumor progression has been subject to discussion for many decades. Numerous studies suggest that a low immune response might be beneficial, if not necessary, for tumor growth, and only a strong immune response can counter tumor growth and thus inhibit progression. METHODS: We implement a cellular automaton model previously described that captures the dynamical interactions between the cancer stem and non-stem cell populations of a tumor through a process of self-metastasis. By overlaying on this model the diffusion of immune reactants into the tumor from a peripheral source to target cells, we simulate the process of immune-system-induced cell kill on tumor progression. RESULTS: A low cytotoxic immune reaction continuously kills cancer cells and, although at a low rate, thereby causes the liberation of space-constrained cancer stem cells to drive self-metastatic progression and continued tumor growth. With increasing immune system strength, however, tumor growth peaks, and then eventually falls below the intrinsic tumor sizes observed without an immune response. With this increasing immune response the number and proportion of cancer stem cells monotonically increases, implicating an additional unexpected consequence, that of cancer stem cell selection, to the immune response. CONCLUSIONS: Cancer stem cells and immune cytotoxicity alone are sufficient to explain the three-step "immunoediting" concept - the modulation of tumor growth through inhibition, selection and promotion.     

Division Cycle of Myxococcus xanthus III. Kinetics of Cell Growth and Protein Synthesis

The kinetics of cell growth and protein synthesis during the division cycle of Myxococcus xanthus was determined. The distribution of cell size for both septated and nonseptated bacteria was obtained by direct measurement of the lengths of 8,000 cells. The Collins-Richmond equation was modified to consider bacterial growth in two phases: growth and division. From the derived equation, the growth rate of individual cells was computed as a function of size. Nondividing cells (growth phase) comprised 91% of the population and took up 87% of the time of the division cycle. The absolute and specific growth rates of nondividing cells were observed to increase continually throughout the growth phase; the growth rate of dividing cells could not be determined accurately by this technique because of changes in the geometry of cells between the time of septation and physical separation. The rate of protein synthesis during the division cycle was measured by pulselabeling an exponential-phase culture with radio-active valine or arginine and then preparing the cells for quantitative autoradiography. By measuring the size of individual cells as well as the number of grains, the rate of protein synthesis as a function of cell size was obtained. Nondividing cells showed an increase in both the absolute and specific rates of protein synthesis throughout the growth phase; the specific rate of protein synthesis for dividing cells was low when compared to growthphase cells. Cell growth and protein synthesis are compared to the previously reported kinetics of deoxyribonucleic acid and ribonucleic acid synthesis during the division cycle.     

Rates of accumulation of glycosidase activities during growth and differentiation of Dictyostelium discoideum.

1. The rates of accumulation (enzyme units/h per 10(8) cells) of a number of glycosidase activities were studied in Dictyostelium discoideum cells during the growth and differentiation phases of this organism's life cycle. 2. The rates of accumulation of the enzymes beta-N-acetylglucosaminidase, alpha-glucosidase and beta-galactosidase remain unchanged during the growth and early differentiation phases. 3. The considerable changes in specific activity of the enzymes which occur in the early differentiation phase are due to the massive loss of total cellular protein which occurs at this time. 4. Significant alterations can occur in the rates of accumulation of alpha-mannosidase during both the growth and differentiation phases, and since, on the onset of differentiation, beta-glucosidase activity is excreted and degraded, the rate of accumulation of this enzyme differs in the growth and differentiation phases. 5. The characteristic rates of accumulation of all these glycosidases change markedly with changes in the growth conditions of the myxamoebae, and thus these rates of synthesis must be regulated independently; however, addition of cyclic AMP to the growth medium has no effect on them.