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.     

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 distribution 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.     

Sloppy size control of the cell division cycle

In an asynchronous, exponentially proliferating cell culture there is a great deal of variability among individual cells in size at birth, size at division and generation time (= age at division). To account for this variability we assume that individual cells grow according to some given growth law and that, after reaching a minimum size, they divide with a certain probability (per unit time) which increases with increasing cell size. This model is called sloppy size control because cell division is assumed to be a random process with size-dependent probability. We derive general equations for the distribution of cell size at division, the distribution of generation time, and the correlations between generation times of closely related cells. Our theoretical results are compared in detail with experimental results (obtained by Miyata and coworkers) for cell division in fission yeast, Schizosaccharomyces pombe. The agreement between theory and experiment is superior to that found for any other simple models of the coordination of cell growth and division.     

Cancer cell dynamics in presence of telomerase inhibitors: analysis of in vitro data

The inhibition of telomerase activity in actively dividing cells leads to suppression of cell growth after a time delay (inhibitory delay) required to reach a threshold telomeric DNA size. We developed a mathematical model of the dynamics of telomere size distribution and cell growth in the presence of telomere inhibitors that allowed quantification of the inhibitory delay. The model based on the solution of a system of differential equations described quantitatively recent experimental data on dynamics of cultured cells in presence of telomerase inhibitors. The analysis of the data by this model suggested the existence of at least two distinct subpopulations of cells with different proliferative activity. Size distribution of telomeres, fraction of proliferating cells, and tumor doubling times are of critical importance for the dynamics of cancer cells growth in presence of telomerase inhibitors. Rapidly growing cells with large telomeric DNA heterogeneity and small proliferating fractions as well as those with very short homogeneous telomeres would be the most sensitive to telomerase inhibitors.     

Tumor escape from immune elimination

A simple kinetic model is proposed for the interaction between a tumor and the immune system. Special attention is given to the phenomenon of “sneaking through” and the associated phenomena of tumor regression and recurrence. Sneaking through refers to the situations in which small antigenic tumors grow progressively, medium-sized tumors are rejected and large ones break through again. The combination of two factors is proposed as being essential for explaining this behavior: (a) The dependence of the immune response on antigen dose. (b) The negative intervention of immunosuppressors, or inhibitory factors.The immune response is described here as a repeated antigen-dependent stimulation of lymphoid cells to proliferate, with a parallel process of antigen-dependent differentiation to a terminal phase. For too small or too large antigen doses the growth in the number of cells is counter-balanced by loss from the proliferative pool through natural decay or by enhanced differentiation, respectively. When this is combined with the blocking of resting and proliferating precursor cells by factors originating from tumor cells, computer simulations demonstrate that the model is able to account for the observed patterns of tumor behavior. The model allows for a discussion of the significance of various biological parameters and is amenable for testing.     

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.     

Cancer treatment by telomerase inhibitors: predictions by a kinetic model

The inhibition of telomerase activity in actively dividing cells leads to shortening of their telomeres and suppression of cell growth when the telomere lengths become smaller than a certain threshold value (typically about 1-2 kb of DNA). We evaluated the time (efficacy delay) required to reach the threshold telomeric DNA size after initiation of treatment, which is of critical importance for the efficacy of telomerase inhibitors. A model based on the solution of a system of differential equations was developed to analyze the efficacy delay and dynamics of tumor growth. The efficacy delay was strongly dependent on the size distribution of telomere lengths at the treatment initiation. An increase in the heterogeneity of telomere size resulted in shortening of the delay. However, the long-term dynamics of tumors with homogeneous populations of telomeres were more significantly affected by telomerase inhibitors compared to tumors with heterogeneous size distribution of telomeres. Size distribution of telomeres and tumor doubling times are of critical importance for the dynamics of tumor growth in presence of telomerase inhibitors.     

Evolutionary dynamics of tumor progression with random fitness values

Most human tumors result from the accumulation of multiple genetic and epigenetic alterations in a single cell. Mutations that confer a fitness advantage to the cell are known as driver mutations and are causally related to tumorigenesis. Other mutations, however, do not change the phenotype of the cell or even decrease cellular fitness. While much experimental effort is being devoted to the identification of the functional effects of individual mutations, mathematical modeling of tumor progression generally considers constant fitness increments as mutations are accumulated. In this paper we study a mathematical model of tumor progression with random fitness increments. We analyze a multi-type branching process in which cells accumulate mutations whose fitness effects are chosen from a distribution. We determine the effect of the fitness distribution on the growth kinetics of the tumor. This work contributes to a quantitative understanding of the accumulation of mutations leading to cancer.     

A mathematical model of tumor growth and its response to single irradiation

Background

Mathematical modeling of biological processes is widely used to enhance quantitative understanding of bio-medical phenomena. This quantitative knowledge can be applied in both clinical and experimental settings. Recently, many investigators began studying mathematical models of tumor response to radiation therapy. We developed a simple mathematical model to simulate the growth of tumor volume and its response to a single fraction of high dose irradiation. The modelling study may provide clinicians important insights on radiation therapy strategies through identification of biological factors significantly influencing the treatment effectiveness.

Methods

We made several key assumptions of the model. Tumor volume is composed of proliferating (or dividing) cancer cells and non-dividing (or dead) cells. Tumor growth rate (or tumor volume doubling time) is proportional to the ratio of the volumes of tumor vasculature and the tumor. The vascular volume grows slower than the tumor by introducing the vascular growth retardation factor, θ. Upon irradiation, the proliferating cells gradually die over a fixed time period after irradiation. Dead cells are cleared away with cell clearance time. The model was applied to simulate pre-treatment growth and post-treatment radiation response of rat rhabdomyosarcoma tumors and metastatic brain tumors of five patients who were treated with Gamma Knife stereotactic radiosurgery (GKSRS).

Results

By selecting appropriate model parameters, we showed the temporal variation of the tumors for both the rat experiment and the clinical GKSRS cases could be easily replicated by the simple model. Additionally, the application of our model to the GKSRS cases showed that the α-value, which is an indicator of radiation sensitivity in the LQ model, and the value of θ could be predictors of the post-treatment volume change.

Conclusions

The proposed model was successful in representing both the animal experimental data and the clinically observed tumor volume changes. We showed that the model can be used to find the potential biological parameters, which may be able to predict the treatment outcome. However, there is a large statistical uncertainty of the result due to the small sample size. Therefore, a future clinical study with a larger number of patients is needed to confirm the finding.

     

Apoptosis in skin cancer development and regression

Non-melanoma skin cancers (NMSC) are the most common malignant tumors in white population and their incidence has been increasing worldwide. Molecular events regulating cell survival, apoptosis, growth arrest as well as cell differentiation, are important contributors to the overall kinetics of benign and malignant cell growth and play a role in their development, progression and regression. Failure of these pathways can result in the loss of control over proliferation and lead to tumor development through the inactivation of tumor suppressor genes or the activation of oncogenes. Also, immunological mechanisms have been implicated in a phenomenon of tumor progression as well as spontaneous tumor regression. We have tried to summarize the main events in etiopatogenesis, development, progression and in some cases skin cancer regression. Further studies are needed to elucidate completely the details of apoptotic control in normal skin and determine factors resulting in apoptotic disbalance and disease.     

An explicit representation of the Luria–Delbrück distribution

The probability distribution of the number of mutant cells in a growing single-cell population is presented in explicit form. We use a discrete model for mutation and population growth which in the limit of large cell numbers and small mutation rates reduces to certain classical models of the Luria-Delbrück distribution. Our results hold for arbitrarily large values of the mutation rate and for cell populations of arbitrary size. We discuss the influence of cell death on fluctuation experiments and investigate a version of our model that accounts for the possibility that both daughter cells of a non-mutant cell might be mutants. An algorithm is presented for the quick calculation of the distribution. Then, we focus on the derivation of two essentially different limit laws, the first of which applies if the population size tends to infinity while the mutation rate tends to zero such that the product of mutation rate times population size converges. The second limit law emerges after a suitable rescaling of the distribution of non-mutant cells in the population and applies if the product of mutation rate times population size tends to infinity. We discuss the distribution of mutation events for arbitrary values of the mutation rate and cell populations of arbitrary size, and, finally, consider limit laws for this distribution with respect to the behavior of the product of mutation rate times population size. Thus, the present paper substantially extends results due to Lea and Coulson (1949), Bartlett (1955), Stewart et al. (1990), and others.     

Hormone-dependent changes of blood vessels in DMBA-induced rat mammary carcinoma and its regression studied by 3H-thymidine autoradiography

Using DMBA-induced rat breast cancer, the changes in the histology and proliferative activity underlying the phenomenon of tumor regression by hormone therapy were studied by 3H-thymidine autoradiography. The control tumor was found to essentially consist of two histologically different areas, medullary (A area) and tubular or cystic (B area). The cancer cells in the A area were homogeneously proliferating with a cell cycle time of 51h, and among those in the B area, 65% were proliferating with a cell cycle time of 81h while 35% were non-proliferating. Among the various-kinds of hormone therapies, ovariectomy plus male sex hormone administration was most effective in inducing tumor regression. In the regressed tumor, the A area was greatly diminished due to central necrosis and replaced by cystic B area. In the remaining A area, the cell cycle time was lengthened to 97h, and that for the proliferating cells in the B area was as long as 118h. The most striking histological change after ovariectomy plus male sex hormone administration was the diffuse necrosis of the capillary endothelial cell within 24h, followed by hemorrhage, central necrosis in the A area (1W), and final stage of fibrosis (2W). The tumor administered with female sex hormone after ovariectomy showed a rebound growth from the regression, due to the initial reactivation of the endothelial cell proliferation and following stimulation of cancer cell mitotic activity. From these observations, it is concluded that the capillary endothelial cells in DMBA-induced rat breast cancer are estrogen dependent, and that the tumor regression induced by decreased estrogen-level is attributable to the massive necrosis from capillary insufficiency and anoxia.     

Pathology of spontaneous and immunotherapy‐induced tumor regression in a murine model of melanoma

We evaluated the spontaneous and immunotherapy‐induced histological changes in the tumor microenvironment of a mouse melanoma regression model consisting of immunocompetent C57BL/6J mice implanted with syngeneic YUMMER1.7 melanoma cells. We focused on tumor regression phenotypes and spatial relationships of melanoma cells with B cells and neutrophils since this was not previously described. We found common themes to the host response to cancer irrespective of the mode of tumor regression. In nonregression tumors, melanoma cells were epithelioid shaped and tightly packed. In regression tumors, melanoma cells were spindle shaped and discohesive. B cells including plasmablasts and plasma cells were numerous and were increased with immunotherapy. Neutrophils were in direct contact with dead or dying melanoma cells. Immunotherapy increased neutrophil counts and induced neutrophil extracellular traps (NETs)‐like formations and geographic necrosis. Beyond tumor regression, the increase in the B cell and neutrophil response could play a role in immunotherapy‐induced adverse reactions.     

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.     

Ribosome accumulation and the regulation of epidermal hyperplastic growth.

Ribosomes accumulate in large numbers during epidermal hyperplastic growth, and their number returns to normal levels during the period of regression when the hyperplastic epidermis returns to its normal size. There is a rough correlation between the degree of epidermal hyperplastic growth and the amount of ribosome accumulation in each epidermal cell. Elevated ribosome levels are also seen in chronic epidermal hyperplasia and in epidermal papillomas. The suggestion is put forward that the marked accumulation of ribosomes in epidermal cells, as well as other as yet unspecified linked molecular changes, produce an imbalance which keeps epidermal cells proliferating resulting in hyperplastic growth. In epidermal tumors this imbalance is locked in place resulting in a permanent epidermal hyperplastic growth condition.     

Mice bearing late-stage tumors have normal functional systemic T cell responses in vitro and in vivo

Immune suppression in tumor-bearing hosts is considered to be one factor causally associated with the growth of antigenic tumors. Support for this hypothesis has come from reports that spleen T cells in tumor-bearing mice are deficient in either priming or effector phase functions. We have reexamined this hypothesis in detail using multiple murine tumor models, including transplantable adenocarcinoma, melanoma, sarcoma, and thymoma, and also a transgenic model of spontaneous breast carcinoma. In both in vitro and in vivo assays of T cell function (proliferation, cytokine production, induction of CD8+ alloreactive CTL, and development of anti-keyhole limpet hemocyanin CD4+ T cells, rejection of allogeneic or syngeneic regressor tumors, respectively) we show that mice bearing sizable tumor burdens are not systemically suppressed and do not have diminished T cell functions. Therefore, if immune suppression is a causal function in the growth of antigenic tumor, the basis for escape from immune destruction is likely to be dependent upon tumor-induced T cell dysfunction at the site of tumor growth.