Biomedical subjects

M Kimmel

Publications and source records attributed to M Kimmel.

At least 37 records · Page 2Linked to original sources

PubMed1996

A discrete-time, multi-type generational inheritance branching process model of cell proliferation.

Mammalian cell populations, such as tumors, may contain subpopulations differing in parameters such as cell lifetimes, even if the populations are derived from single cells. The mode of inheritance of cell lifetimes has previously been the subject of experimental and mathematical investigation. To obtain data on cell lifetimes over more cell generations then previously available, Axelrod et al. [Cell Prolif. 26:235-249(1988)] measured the number of cells in primary colonies and secondary colonies derived form the primary colonies. The experimental results indicated large variance of cells per colony and highly significant correlations between the numbers of cells in primary and secondary colonies. To mathematically model these results we derive, for previously uninvestigated multi-type Galton-Watson branching process models, the covariance of the cell counts in the primary and secondary colonies. As a result, we are able to successfully model the data with two subpopulations having differing proliferation rates, in which the proliferation rate of a daughter cell is primarily determined by the proliferation rate of its mother. Interestingly, simulations display a trade-off between high values of variances and correlation coefficients. The values obtained from experiment are located on the boundary of the region attainable by simulation.

Animals↗

PubMed1996

Measures of variation at DNA repeat loci under a general stepwise mutation model.

Polymorphisms at tandem repeat loci are caused by mutations with allele sizes occasionally altered by more than one repeat unit in both forward and backward directions. Such mutational changes may occur with asymmetric probabilities. Therefore, a one-step symmetric stepwise mutation model may not be appropriate for studying the population dynamics at all repeat loci. In this work, we evaluated the expectation and variance of the within-population variance of the allele size distribution in a finite population, and the expected homozygosity at a locus by the coalescence approach under a general stepwise mutation model, where mutational transitions of allele sizes can be arbitrary, including being asymmetric. Under the special cases of symmetric one-step, two-step, and multi-step geometric distributions of mutations, our general results reduce to the corresponding results obtained by earlier investigators. The general results indicate that in a finite population, which has reached a steady state under the (general stepwise) mutation and drift balance, the within-population variance of allele sizes has a simple expectation (i.e., proportional to Nnu, the product of the mutation rate, nu, and effective population size, N). However, its stochastic variance is a quadratic function of this composite parameter, Nnu. Furthermore, this second-order variance does not decay with the number of alleles sampled from a population. Application of this theory to data on allele size distributions in unrelated Caucasians from the CEPH pedigree (obtained from the Genome Data Base) shows that the relationship of the variance and mean of within-population variance of allele sizes at tandem repeat loci, grouped by their chromosomal assignment, has a trend compatible with the theory. However, there is an indication that the second-order variance is generally underestimated. One reason for this departure might be that the CEPH sample may not represent a single homogeneous population that reached equilibrium at all tandem repeat loci.

DNA↗

PubMed1996

Dynamics of repeat polymorphisms under a forward-backward mutation model: within- and between-population variability at microsatellite loci.

Suggested molecular mechanisms for the generation of new tandem repeats of simple sequences indicate that the microsatellite loci evolve via some of forward-backward mutation. We provide a mathematical basis for suggesting a measure of genetic distance between populations based on microsatellite variation. Our results indicate that such a genetic distance measure can remain proportional to the divergence time of populations even when the forward-backward mutations produce variable and/or directionally biased alleles size changes. If the population size and the rate of mutation remain constant, then the measure will be proportional to the time of divergence of populations. This genetic distance is expressed in terms of a ratio of components of variance of allele sizes, based on expressions developed for studying population dynamics of quantitative traits. Application of this measure to data on 18 microsatellite loci in the nine human populations leads to evolutionary trees consistent with the known ethnohistory of the populations.

Alleles↗

PubMed1996

Error detection for genetic data, using likelihood methods.

As genetic maps become denser, the effect of laboratory typing errors becomes more serious. We review a general method for detecting errors in pedigree genotyping data that is a variant of the likelihood-ratio test statistic. It pinpoints individuals and loci with relatively unlikely genotypes. Power and significance studies using Monte Carlo methods are shown by using simulated data with pedigree structures similar to the CEPH pedigrees and a larger experimental pedigree used in the study of idiopathic dilated cardiomyopathy (DCM). The studies show the index detects errors for small values of theta with high power and an acceptable false positive rate. The method was also used to check for errors in DCM laboratory pedigree data and to estimate the error rate in CEPH-chromosome 6 data. The errors flagged by our method in the DCM pedigree were confirmed by the laboratory. The results are consistent with estimated false-positive and false-negative rates obtained using simulation.

Cardiomyopathy, Dilated↗

PubMed1996

Optimal control problems arising in cell-cycle-specific cancer chemotherapy.

We explore mathematical properties of models of cancer chemotherapy including cell-cycle dependence. Using the mathematical methods of control theory, we demonstrate two assertions of interest for the biomedical community: 1 Periodic chemotherapy protocols are close to the optimum for a wide class of models and have additional favourable properties. 2 Two possible approaches, (a) to minimize the final count of malignant cells and the cumulative effect of the drug on normal cells, or (b) to maximize the final count of normal cells and the cumulative effect of the drug on malignant cells, lead to similar principles of optimization. From the mathematical viewpoint, the paper provides a catalogue of simplest mathematical models of cell-cycle dependent chemotherapy. They can be classified based on the number of compartments and types of drug action modelled. In all these models the optimal controls are complicated by the singular and periodic trajectories and multiple solutions. However, efficient numerical methods have been developed. In simpler cases, it is also possible to provide an exhaustive classification of solutions. We also discuss developments in estimation of cell cycle parameters and cell-cycle dependent drug action.

Cell Compartmentation↗

PubMed1995

Mathematical modeling of the loss of telomere sequences.

hortening of telomeres is one of the supposed mechanisms of cellular aging and death. An important question related to this so-called "end-replication" hypothesis is whether it can explain in quantitative detail the dynamic of cell sensescence in vitro and in vivo. A natural way to answer this question is to use mathematical modeling. In this paper, the models were successfully fitted to data on cultured fibroblasts from two different sources assuming that after reaching the Hayflick checkpoint on a single chromosome cells cease to proliferate. The main conclusion is that the end-replication hypothesis provides an explanation for the cell aging process which is quantitatively consistent with the data. As a secondary outcome, estimates were obtained of the rate of shortening of telomeres and several interesting mathematical results for branching processes with infinite type spaces arise.

Animals↗

PubMed1994

Gene amplification by unequal sister chromatid exchange: probabilistic modeling and analysis of drug resistance data.

Unequal sister chromatid exchange has been proposed as one of several possible mechanisms for gene amplification resulting in tandemly repeated sequences on chromosomes. Two requirements for testing this hypothesis are analytical observations and a mathematical model. Recently observations were reported for the number of tandemly repeated sequences on chromosomes of cells growing in the presence of a toxic drug and the mechanism was proposed to be unequal sister chromatid exchange. We now develop a mathematical model of this process based on the following hypotheses, (i) the extent of slippage between paired sister chromatids is a random variable with geometric distribution, (ii) the number of crossover sites is a random variable with a Poisson distribution, and (iii) cells with less than a threshold number of copies of an essential gene are eliminated when grown in selective conditions. Iterating the model at successive cell divisions results in a Markov chain with a denumerable infinity of states. The resulting distributions of gene copy number per cell at a particular population size are compared to published data on the CAD gene in BHK cells growing in the presence of the drug PALA (Smith et al., 1990, Cell, 63, 1219). The mathematical model can reproduce the observed means and standard deviations of gene copy number per cell and allows construction of confidence region estimates of parameters describing the extent of slippage, density of crossover sites, and strength of selection. An important prediction of the model is that in non-selective conditions the cells with amplified sequences gradually disappear from the population even if they are not at a growth disadvantage, though rare cells with a very large number of amplified sequences might continue to exist. The success of modeling suggests that the proposed mechanism of gene amplification by unequal sister chromatid exchange is consistent with the number of tandemly repeated sequences on chromosomes observed in some circumstances.

Animals↗

PubMed1994

Fluctuation test for two-stage mutations: application to gene amplification.

The determination of mutation rates is an important experimental procedure for characterizing mutation processes. The accepted method of determining mutation rates, the fluctuation test, was introduced by Luria and Delbrück in 1943. Since then it has been applied to various microorganisms and cells. The Luria-Delbrück test is based on a restrictive hypothesis of mutations being due to single irreversible events. However, some inherited changes in phenotype, like gene amplification, may be due to two or more genetic changes, some of which may be reversible. The Luria-Delbrück model of mutation was compared to other models which included reversibility and more than one mutation stage. The Luria-Delbrück model has been confirmed to be consistent with the original bacteriophage resistance data. However, for gene amplification this model gives incompatible estimates of mutation rates by the P0 and r methods. Relaxing the hypotheses of the single-stage models did not improve the fit. In contrast, a two-stage reversible model provided a fit. Analysis of gene amplification data by the two-stage reversible model provides new information, including estimates of rates for each of the two forward stages and of the reverse step.

Animals↗

PubMed1994

Time-continuous branching walk models of unstable gene amplification.

We consider a stochastic mechanism of the loss of resistance of cancer cells to cytotoxic agents, in terms of unstable gene amplification. Two models being different versions of a time-continuous branching random walk are presented. Both models assume strong dependence in replication and segregation of the extrachromosomal elements. The mathematical part of the paper includes the expression for the expected number of cells with a given number of gene copies in terms of modified Bessel functions. This adds to the collection of rare explicit solutions to branching process models. Original asymptotic expansions are also demonstrated. Fitting the model to experimental data yields estimates of the probabilities of gene amplification and deamplification. The thesis of the paper is that purely stochastic mechanisms may explain the dynamics of reversible drug resistance of cancer cells. Various stochastic approaches and their limitations are discussed.

Animals↗

PubMed1994

Two simple models of almost the same population with very different dynamics.

We present our point of view of the controversy regarding complex dynamics of population models. We analyze two nonlinear models of population dynamics which may be used to model the same population, but which display very different dynamics. We prove that Model 1 stays globally asymptotically stable, while Model 2 is known to display instability, oscillations, and apparent period doubting leading to chaos. We also indicate situations in which both models are equivalent. We employ these results to argue that the "complex" behavior of some models may be due to apparently small differences in assumptions.

Animals↗

PubMed1993

Screening for lung cancer. The Mayo Lung Project revisited.

BACKGROUND: The Mayo Lung Project (MLP) reported lung cancer incidence and mortality in a population offered chest radiographs and sputum cytologic screening examinations every 4 months and a population offered only the Mayo Clinic advice to undergo annual examination. No mortality benefit attributable to screening was observed after 6 years of observation and at least 1 year of follow-up. METHODS: The authors describe a simulation study designed to estimate from Mayo data the parameters in a mathematical model of the natural history of lung cancer and to estimate the potential benefit associated with periodic screening of high-risk people starting at 45 years of age. RESULTS: It was found that the mean duration of Stage I non-small cell lung cancer is at least 4 years and that rates of Stage I detectability and curability are less than 25% and 35%, respectively. CONCLUSIONS: A trial of the magnitude, duration, and contamination of the MLP would have a less than 20% probability of showing significant benefit from screening; however, long-term annual screening might result in a modest decrease in lung cancer mortality, ranging from 0% to 13%. A greater benefit would accrue from improved detection and treatment.

Aged↗

PubMed1993

Thermodynamical model of mixed aggregation of intercalators with caffeine in aqueous solution.

Recently we presented evidence that some intercalating antitumor agents can form complexes with caffeine and that this process may be responsible for the modifying effect of caffeine on the pharmacological activity of these drugs (F. Traganos et al., Cancer Res. 51 (1991) 3682). Here we describe a statistical-thermodynamical model of mixed associations in which one component's self-association is limited to dimer formation while the second component has the ability of unlimited stacking. The system is controlled by three parameters which represent self-aggregation "neighborhood" association constants KCC and KAA and a mixed "neighborhood" association constant KAC. The model was tested using acridine orange and light absorption spectroscopy as an analytical method for detection of complex formation. The experiments performed at two NaCl concentrations (0.01 and 0.15 M) indicate interesting properties of the three-parameter system in which the first parameter (KCC) is practically independent of ionic strength, the second (KAA) is positively and the third parameter (KAC) is adversely affected by ionic strength.

Acridine Orange↗

PubMed1993

Hypocellular bladder wash specimens and their clinical significance.

Bladder wash flow cytometry specimens containing less than 1,000 cells were defined as hypocellular. Between June 1981 and December 1990, 5,676 bladder wash flow cytometry specimens were collected, of which 142 (2.5%) were deemed hypocellular. Of the 132 evaluable patients with hypocellular specimens 34 (30%) had documented evidence of carcinoma compared to 52% for the entire population. Among the 34 patients 30 had transitional cell carcinoma, of which 12 were carcinoma in situ, 9 stage Ta or T1 and 9 stage T2 or greater. Thus, there was no obvious relationship between stage or grade of carcinoma versus probability of a hypocellular specimen. Differences in irrigation techniques of different urologists in the operating room using general anesthesia versus catheter and cystoscopic irrigation in the outpatient clinic are discussed as possible explanations for the hypocellular specimens.

Biopsy↗

PubMed1992

A branching process model of gene amplification following chromosome breakage.

We have devised a mathematical model of gene amplification utilizing recent experimental observations concerning dihydrofolate reductase (DHFR) gene amplification in CHO cells. The mathematical model, based on a biological model which proposes that acentric elements are the initial intermediates in gene amplification, includes the following features: (1) initiation of amplification by chromosomal breakage to produce an acentric structure; (2) replication of acentric DNA, once per cell cycle; (3) dissociation of replicated acentric DNA; (4) unequal segregation of acentric DNA fragments to daughter cells at mitosis; (5) subsequent reintegration of acentric fragments into chromosomes. These processes are assumed to be independent for each element present in a cell at a given time. Thus, processes of unequal segregation and integration may occur in parallel, not necessarily in a unique sequence, and may be reiterated in one or multiple cell cycles. These events are described mathematically as a Galton-Watson branching process with denumerable infinity of object types. This mathematical model qualitatively and quantitatively reproduces the major elements of the dynamical behavior of DHFR genes observed experimentally. The agreement between the mathematical model and the experimental data lends credence to the biological model proposed by Windle et al. (1991), including the importance of chromosome breakage and subsequent gene deletion resulting from resection of the broken chromosome ends as initial events in gene amplification.

Animals↗

PubMed1992

The effect of surgical treatment on survival from early lung cancer. Implications for screening.

We assessed the effect of surgery on survival from stage I non-small-cell lung cancer based on data collected in these screening programs. The majority of patients diagnosed in each program were treated by surgical resection, but 5 percent of the Sloan-Kettering group, 21 percent of the Hopkins group and 11 percent of the Mayo group failed to receive surgical treatment. Approximately 70 percent of the stage I patients in each program who were treated surgically survived more than five years, but there were only two five-year survivors among those who did not have surgery. We conclude that patients with lung cancers detected in stage I by chest x-ray film and treated surgically have a good chance of remaining free of disease for many years. Those stage I lung cancers which are not resected progress and lead to death within five years. Therefore, every effort should be made to detect and treat lung cancer early in high-risk populations.

Age Factors↗

PubMed1991

Unequal cell division, growth regulation and colony size of mammalian cells: a mathematical model and analysis of experimental data.

This work describes mathematically the dynamics of expansion of cell populations from the initial division of single cells to colonies of several hundred cells. This stage of population growth is strongly influenced by stochastic (random) elements including, among others, cell death and quiescence. This results in a wide distribution of colony sizes. Experimental observations of the NIH3T3 cell line as well as for the NIH3T3 cell line transformed with the ras oncogene were obtained for this study. They include the number of cells in 4-day-old colonies initiated from single cells and measurements of sizes of sister cells after division, recorded in the 4-day-old colonies. The sister cell sizes were recorded in a way which enabled investigation of their interdependence. We developed a mathematical model which includes cell growth and unequal cell division, with three possible outcomes of each cell division: continued cell growth and division, quiescence, and cell death. The model is successful in reproducing experimental observations. It provides good fits to colony size distributions for both NIH3T3 mouse fibroblast cells and the same cells transformed with the rasEJ human cancer gene. The difference in colony size distributions could be fitted by assuming similar cell lifetimes (12-13 hr) and similar probabilities of cell death (q = 0.15), but using different probabilities of quiescence, r = 0 for the ras oncogene transformed cells and r = 0.1 for the non-transformed cells. The model also reproduces the evolution of distributions of sizes of cells in colonies, from a single founder cell of any specified size to the stable limit distribution after eight to ten cell divisions. Application of the model explains in what way both random events and deterministic control mechanisms strongly influence cell proliferation at early stages in the expansion of colonies.

Animals↗

PubMed1991

DNA/RNA content and proliferative fractions of colorectal carcinomas: a five-year prospective study relating flow cytometry to survival.

In our prospective study, flow cytometric analysis of cellular DNA and RNA content was performed on unfixed fresh specimens of colorectal adenocarcinoma taken from 176 patients. Of the 176 tumors, 113 (64%) were aneuploid. There was no correlation between aneuploidy and tumor stage, grade, location, or size. After a median follow-up of 5.6 years, no correlation between DNA or RNA content and patient survival was found. DNA content alone was not an independent prognostic factor when the colorectal carcinomas were segregated by curable and incurable stages. However, normal mucosa, diploid tumors, and aneuploid tumors showed progressively higher proliferation and higher RNA and DNA indices. Proliferative fraction--defined as the percentage of cells in S + G2 and M phases of the cell cycle--was significantly related to ploidy and to Dukes' stage. Despite these correlations, we did not detect a significant influence of proliferative fraction on survival when patients were segregated above or below the mean proliferative fraction for all tumors. More accurate methods of identifying the proliferative fraction of tumor cells are currently being pursued. While the role of flow cytometry in the evaluation and management of patients with colorectal carcinoma is still undefined for a number of other cellular parameters, it seems unlikely that DNA index, RNA index, or the proliferative fractions calculated from the DNA histogram, will, of themselves, represent independent prognostic factors.

Adenocarcinoma↗

PubMed1991

Asymptotic behavior of nonlinear semigroup describing a model of selective cell growth regulation.

A new scheme of regulation of cell population growth is considered, called the selective growth regulation. The principle is that cells are withdrawn from proliferation depending on their contents of certain biochemical species. The dynamics of the cell population structured by the contents of this species is described by the functional integral equation model, previously introduced by the authors. The solutions of the model equations generate a semigroup of nonlinear positive operators. The main problem solved in this paper concerns stability of the equilibria of the model. This requires stating and proving of an original abstract result on the spectral radius of a perturbation of a semigroup of positive linear operators. Biological applications are discussed.

Cell Division↗