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The goal of this project was to utilize mechanistic simulation to demonstrate a methodology that could determine drug combination dose schedules and dose intensities that would be most effective in eliminating multidrug-resistant cancer cells in early-stage colon cancer. An agent-based model of cell dynamics in human colon crypts was calibrated using measurements of human biopsy specimens. Mutant cancer cells were simulated as cells that were resistant to each of two drugs when the drugs were used separately. The drugs, 5-flurouracil and sulindac, have different mechanisms of action. An artificial neural network was used to generate nearly 200,000 two-drug dose schedules. A high-performance computer simulated each dose schedule as a in silico clinical trial and evaluated each dose schedule for its efficiency to cure (eliminate) multidrug-resistant cancer cells and its toxicity to the host, as indicated by continued crypt function. Among the dose schedules that were generated, 2,430 dose schedules were found to cure all multidrug-resistant mutants in each of the 50 simulated trials and retained colon crypt function. One dose schedule was optimal; it eliminated multidrug-resistant cancer cells with the minimum toxicity and had a time schedule that would be practical for implementation in the clinic. These results demonstrate a procedure to identify which combination drug dose schedules could be most effective in eliminating drug-resistant cancer cells. This was accomplished using a calibrated agent–based model of a human tissue, and a high-performance computer simulation of clinical trials. Significance: The results of computer-simulated clinical trials suggest a practical dose schedule for two drugs, 5-fluorouracil and sulindac, that could eliminate multidrug resistant early-stage colon cancer cells with minimum toxicity to the host.

The 2430 dose schedules that cured all doubly resistant mutants in 100% of 50 independent trials and allowed crypts to recover.

PURPOSE Adjuvant chemotherapy is used after surgery for stages II and III colorectal cancer to reduce recurrence. Nevertheless, recurrence may occur years later with the emergence of initially undetected minimal residual disease (MRD). Attempts to reduce recurrence by increasing the dose intensity and increasing the time of adjuvant therapy have been limited by the adverse effects of the recommended cytotoxic agents. The goals of this study were to suggest an alternative to the recommended cytotoxic agents and to determine optimal adjuvant therapy dose schedules that would reduce the percentage of recurrence at 5 years while retaining colon crypt function. METHODS A total of 84,400 dose schedules with different duration, interval between doses, and intensity of treatment were simulated with a high-performance computer. Simulated treatments used the drug sulindac, which had previously been used in primary prevention. With appropriate dose schedules, it can induce apoptosis at the crypt lumen surface while retaining crypt function. We used a computer model of cell dynamics in colon crypts that had been calibrated with measurements of human biopsy specimens. Proliferating mutant cells were assumed to emerge from MRD within crypts. Simulated outcomes included the recurrence percentage at 5 years and the retention of crypt function. RESULTS Optimal dose schedules were determined for adjuvant treatment of MRD that reduced the percentage of recurrence at 5 years of stages I, II, and III colon cancer to zero. CONCLUSION A new adjuvant therapy for colon cancer based upon optimum dose schedules of intermittent apoptotic treatment may prevent the recurrence of colon cancer from MRD and avoid the adverse effects of cytotoxic treatments.

We report on a statistical method for grouping anti-cancer drugs (GRAD) in single mouse trials (SMT). The method assigns candidate drugs into groups that inhibit or do not inhibit tumor growth in patient-derived xenografts (PDX). It determines the statistical significance of the group assignments without replicate trials of each drug. The GRAD method applies a longitudinal finite mixture model, implemented in the statistical package PROC TRAJ, to analyze a mixture of tumor growth curves for portions of the same tumor in different mice, each single mouse exposed to a different drug. Each drug is classified into an inhibitory or non-inhibitory group. There are several advantages to the GRAD method for SMT. It determines that probability that the grouping is correct, uses the entire longitudinal tumor growth curve data for each drug treatment, can fit different shape growth curves, accounts for missing growth curve data, and accommodates growth curves of different time periods. We analyzed data for 22 drugs for 18 human colorectal tumors provided by researchers in a previous publication. The GRAD method identified 18 drugs that were inhibitory against at least one tumor, and 10 tumors for which there was at least one inhibitory drug. Analysis of simulated data indicated that the GRAD method has a sensitivity of 84% and a specificity of 98%. A statistical method, GRAD, can group anti-cancer drugs into those that are inhibitory and those that are non-inhibitory in single mouse trials and provide probabilities that the grouping is correct.

This chapter develops a robust (experimental) model system which will give epigenetic studies at the cell and population levels the convenience and quantitation that phage and bacterial systems provided for molecular genetics. Proliferating populations of cells have been studied using many experimental techniques, perhaps most extensively, time-lapse photography and flow cytometry. Each method of studying cell proliferation views a different aspect of cell populations and has specific advantages and disadvantages. For instance, time-lapse photography has the advantage that precise cell life times can be determined for individual cells and their relatives. Compared to time-lapse photography, flow cytometry has the disadvantage that information cannot be obtained about related cells within a pedigree, and that some information about subpopulations may be lost. The regression coefficient of primary colony size on the average of secondary colony sizes is an indication of the persistence of colony sizes over several cell generations.

The effectiveness of cancer chemotherapy is limited by intra-tumor heterogeneity, the emergence of spontaneous and induced drug-resistant mutant subclones, and the maximum dose to which normal tissues can be exposed without adverse side effects. The goal of this project was to determine if intermittent schedules of the maximum dose that allows colon crypt maintenance could overcome these limitations, specifically by eliminating mixtures of drug-resistant mutants from heterogeneous early colon adenomas while maintaining colon crypt function. A computer model of cell dynamics in human colon crypts was calibrated with measurements of human biopsy specimens. The model allowed simulation of continuous and intermittent dose schedules of a cytotoxic chemotherapeutic drug, as well as the drug’s effect on the elimination of mutant cells and the maintenance of crypt function. Colon crypts can tolerate a tenfold greater intermittent dose than constant dose. This allows elimination of a mixture of relatively drug-sensitive and drug-resistant mutant subclones from heterogeneous colon crypts. Mutants can be eliminated whether they arise spontaneously or are induced by the cytotoxic drug. An intermittent dose, at the maximum that allows colon crypt maintenance, can be effective in eliminating a heterogeneous mixture of mutant subclones before they fill the crypt and form an adenoma.

Cancer chemotherapy dose schedules are conventionally applied intermittently, with dose duration of the order of hours, intervals between doses of days or weeks, and cycles repeated for weeks. The large number of possible combinations of values of duration, interval, and lethality has been an impediment to empirically determine the optimal set of treatment conditions. The purpose of this project was to determine the set of parameters for duration, interval, and lethality that would be most effective for treating early colon cancer. An agent-based computer model that simulated cell proliferation kinetics in normal human colon crypts was calibrated with measurements of human biopsy specimens. Mutant cells were simulated as proliferating and forming an adenoma, or dying if treated with cytotoxic chemotherapy. Using a high performance computer, a total of 28,800 different parameter sets of duration, interval, and lethality were simulated. The effect of each parameter set on the stability of colon crypts, the time to cure a crypt of mutant cells, and the accumulated dose was determined. Of the 28,800 parameter sets, 434 parameter sets were effective in curing the crypts of mutant cells before they could form an adenoma and allowed the crypt normal cell dynamics to recover to pretreatment levels. A group of 14 similar parameter sets produced a minimal time to cure mutant cells. A different group of 9 similar parameter sets produced the least accumulated dose. These parameter sets may be considered as candidate dose schedules to guide clinical trials for early colon cancer.

The question of stem cell control is at the center of our understanding of tissue functioning, both in healthy and cancerous conditions. It is well accepted that cellular fate decisions (such as divisions, dif- ferentiation, apoptosis) are orchestrated by a network of regulatory signals emitted by different cell pop- ulations in the lineage and the surrounding tissue. The exact regulatory network that governs stem cell lineages in a given tissue is usually unknown. Here we propose an algorithm to identify a set of candi- date control networks that are compatible with (a) measured means and variances of cell populations in different compartments, (b) qualitative information on cell population dynamics, such as the existence of local controls and oscillatory reaction of the system to population size perturbations, and (c) statistics of correlations between cell numbers in different compartments. Using the example of human colon crypts, where lineages are comprised of stem cells, transit amplifying cells, and differentiated cells, we start with a theoretically known set of 32 smallest control networks compatible with tissue stability. Utilizing near-equilibrium stochastic calculus of stem cells developed earlier, we apply a series of tests, where we compare the networks��� expected behavior with the observations. This allows us to exclude most of the networks, until only three, very similar, candidate networks remain, which are most compatible with the measurements. This work demonstrates how theoretical analysis of control networks combined with only static biological data can shed light onto the inner workings of stem cell lineages, in the absence of direct experimental assessment of regulatory signaling mechanisms. The resulting candidate networks are dom- inated by negative control loops and possess the following properties: (1) stem cell division decisions are negatively controlled by the stem cell population, (2) stem cell differentiation decisions are negatively controlled by the transit amplifying cell population.

Information about tumors is usually obtained from a single assessment of a tumor sample, performed at some point in the course of the development and progression of the tumor, with patient characteristics being surrogates for natural history context. Differences between cells within individual tumors (intratumor heterogeneity) and between tumors of different patients (intertumor heterogeneity) may mean that a small sample is not representative of the tumor as a whole, particularly for solid tumors which are the focus of this paper. This issue is of increasing importance as high-throughput technologies generate large multi-feature data sets in the areas of genomics, proteomics, and image analysis. Three potential pitfalls in statistical analysis are discussed (sampling, cut-points, and validation) and suggestions are made about how to avoid these pitfalls.

We propose that there is an opportunity to devise new cancer therapies based on the recognition that tumors have properties of ecological systems. Traditionally, localized treatment has targeted the cancer cells directly by removing them (surgery) or killing them (chemotherapy and radiation). These modes of therapy have not always been effective because many tumors recur after these therapies, either because not all of the cells are killed (local recurrence) or because the cancer cells had already escaped the primary tumor environment (distant recurrence). There has been an increasing recognition that the tumor microenvironment contains host noncancer cells in addition to cancer cells, interacting in a dynamic fashion over time. The cancer cells compete and/or cooperate with nontumor cells, and the cancer cells may compete and/or cooperate with each other. It has been demonstrated that these interactions can alter the genotype and phenotype of the host cells as well as the cancer cells. The interaction of these cancer and host cells to remodel the normal host organ microenvironment may best be conceptualized as an evolving ecosystem. In classic terms, an ecosystem describes the physical and biological components of an environment in relation to each other as a unit. Here, we review some properties of tumor microenvironments and ecological systems and indicate similarities between them. We propose that describing tumors as ecological systems defines new opportunities for novel cancer therapies and use the development of prostate cancer metastases as an example. We refer to this as "ecological therapy" for cancer.

The Galton–Watson Process is the oldest, simplest and best known branching process. It can be described as follows.

One of the important questions in population dynamics and particularly in population genetics is how to gain information about a population’s past, given its present status. Sources, historical in nature such as written records, archeological such as cemeteries, paleontological such as fossils, or even biological such as ancient DNA, are often of assistance. However in many cases, all that is available is a sample from a contemporary population, with information about its demography or genetic make-up. Sometimes, a mathematical model of population growth may be assumed or statistically inferred from paleo-ecology or by other means. Human populations are of major interest, as are populations of endangered species. Other categories of biological genealogies are gaining prominence. Among them are genealogies of cells in cancerous tumors.

The Bellman–Harris branching process is more general than the processes considered in the preceding chapters. Lifetimes of particles are nonnegative random variables with arbitrary distributions. It is described as follows. A single ancestor particle is born at t = 0. It lives for time τ which is a random variable with cumulative distribution function \(G(\tau)\). At the moment of death, the particle produces a random number of progeny according to a probability distribution with pgf f(s). Each of the first generation progeny behaves, independently of each other and the ancestor, as the ancestor particle did, i.e., it lives for a random time distributed according to \(G(\tau)\) and produces a random number of progeny according to f(s). If we denote Z(t) the particle count at time t, we obtain a stochastic process \(\{Z(t),\ t\geq 0\}\). This so-called age-dependent process is generally non-Markov, but two of its special cases are Markov: the Galton–Watson process and the age-dependent branching process with exponential lifetimes. The Bellman–Harris process is more difficult to analyze, but it has many properties similar to these two processes.

This chapter is devoted to the use of time-continuous branching process with exponential life-time distributions. This process also has the Markov property and is closely related to the Galton–Watson process. The exponential distribution to model lifetimes of particles is not well motivated by any biological assumptions. Indeed, the exponential distribution admits lifetimes which are arbitrarily close to 0, while it is known that life cycles of organisms and cells have lower bounds of durations, which are greater than 0. The advantage of using the exponential distribution is that it leads, in many cases, to computable expressions. These expressions allow one to deduce properties which can then be conjectured for more general models.

The evolution of cooperation has a well established theoretical framework based on game theory. This approach has made valuable contributions to a wide variety of disciplines, including political science, economics, and evolutionary biology. Existing cancer theory suggests that individual clones of cancer cells evolve independently from one another, acquiring all of the genetic traits or hallmarks necessary to form a malignant tumor. It is also now recognized that tumors are heterotypic, with cancer cells interacting with normal stromal cells within the tissue microenvironment, including endothelial, stromal, and nerve cells. This tumor cell–stromal cell interaction in itself is a form of commensalism, because it has been demonstrated that these nonmalignant cells support and even enable tumor growth. Here, we add to this theory by regarding tumor cells as game players whose interactions help to determine their Darwinian fitness. We marshal evidence that tumor cells overcome certain host defenses by means of diffusible products. Our original contribution is to raise the possibility that two nearby cells can protect each other from a set of host defenses that neither could survive alone. Cooperation can evolve as by-product mutualism among genetically diverse tumor cells. Our hypothesis supplements, but does not supplant, the traditional view of carcinogenesis in which one clonal population of cells develops all of the necessary genetic traits independently to form a tumor. Cooperation through the sharing of diffusible products raises new questions about tumorigenesis and has implications for understanding observed phenomena, designing new experiments, and developing new therapeutic approaches.

To better understand the progression of heterogeneous breast cancers, four models of progression pathways have been evaluated. The models describe the progression through the grades of ductal carcinoma in situ (DCIS) 1, 2, and 3, and through the grades of invasive ductal carcinoma (IDC) 1, 2, and 3. The first three pathways, termed linear, nonlinear, and branched, describe DCIS as aprogenitor of IDC, and grades of DCIS progressing into grades of IDC. The fourth pathway, termed parallel, describes DCIS and IDC as diverging from a common progenitor and progressing through grades in parallel. The best transition rates for the linear, nonlinear, and branched pathways were sought using a random search in combination with a directed search based on the Nelder���Mead simplex method. Parameter values for the parallel pathway were determined with heuristic graphs. Results of computer simulation were compared with clinically observed frequencies of grades of DCIS and grades of IDC that were reported to occur together in heterogeneous tumors. Each of the four pathways could simulate frequencies that resembled, to varying degrees, the clinical observations. The parallel pathway produced the best correspondence with clinical observations. These results quantify the traditional descriptions in which grades of DCIS are the progenitors of grades of IDC. The results also raise the alternative possibility that, in some tumors with both components, DCIS and IDC may have diverged from a common progenitor.

Intermittent treatment schedules have been proposed to improve the tolerance of drugs for cancer chemoprevention. However, determining a maximum tolerated dose, and the extent of the improvement, has been challenging experimentally and clinically. In order to determine the quantitative advantage of intermittent pulse treatment schedules for the chemoprevention of colon cancer we have used a computer model of human colon crypts calibrated with measurements of human biopsy specimens. In simulations, crypts were treated with an agent that increases the probability that cells, both normal and mutant, would be removed at the top of the crypt. Sulindac, which increases apoptosis at the lumen surface, is such an agent. The effect of intermittent pulse drug treatment schedules were compared with constant drug treatment schedules. Crypts treated with intermittent pulse schedules have three times the maximum tolerated dose than crypts treated with constant schedules, and have a 10 year delay in the appearance of adenomas. Intermittent treatment schedules have previously been proposed for chemoprevention. Here computer simulations have quantified the effect on human colon crypts of intermittent treatment schedules and constant treatment schedules of a chemotherapeutic drug. Intermittent pulses have an advantage, they allow an increased maximum tolerated dose, and result in an increased chemoprevention by delay.

Two possible pathways of breast tumor progression were investigated by searching for values of transition rates that could reproduce the clinically observed co-occurrence frequencies of grades of ductal carcinoma in situ and grades of invasive ductal carcinoma in heterogeneous tumors. Two different pathways were analysed, a linear pathway with seven parameters, and a nonlinear pathway with three parameters. In each pathway ductal carcinoma in situ (DCIS) is a progenitor of invasive carcinoma (IDC). In the linear pathway breast tumor progression is along increasing grades: DCIS 1-DCIS 2-DCIS 3-IDC 1-IDC 2-IDC 3. In the nonlinear pathway progression of DCIS and progression of IDC can proceed in parallel steps, and in addition, with transitions from each grade of DCIS to a corresponding grade of IDC. The biological pathways were interpreted mathematically as compartment models with transition rates between stages in an explicit series of coupled differential equations. Two methods were used to search for transition rates that could reproduce the observed co-occurrence frequencies, a limited empirical search and an extensive genetic algorithmic search. Neither search method, with either pathway, could find a combination of transition rates that would reproduce the set of observed co-occurrence frequencies. We conclude that neither the linear pathway, nor the nonlinear pathway considered here, is an adequate description of progression in heterogeneous breast tumors. This quantitative investigation lends support to previous evidence from histopathology and molecular biology that the grades of DCIS and IDC seen together in heterogeneous breast tumors may not be obligate steps in tumor progression.

This is a report from the Kananaskis working group on quantitative methods in tumour heterogeneity. Tumour progression is currently believed to result from genetic instability and consequent acquisition of new genetic properties in some of the tumour cells. Cross-sectional assessment of genetic markers for human tumours requires quantifiable measures of intratumour heterogeneity for each parameter or characteristic observed; the relevance of heterogeneity to tumour progression can best be ascertained by repeated assessment along a tumour progressional time line. This paper outlines experimental and analytic considerations that, with repeated use, should lead to a better understanding of tumour heterogeneity, and hence, to improvements in patient diagnosis and therapy. Four general principles were agreed upon at the Symposium: (1) the concept of heterogeneity requires a quantifiable definition so that it can be assessed repeatably; (2) the quantification of heterogeneity is necessary so that testable hypotheses may be formulated and checked to determine the degree of support from observed data; (3) it is necessary to consider (a) what is being measured, (b) what is currently measurable, and (c) what should be measured; and (4) the proposal of working models is a useful step that will assist our understanding of the origins and significance of heterogeneity in tumours. The properties of these models should then be studied so that hypotheses may be refined and validated. Cytometry 31:67–73, 1998. © 1998 Wiley-Liss, Inc.

The expansion of DNA triplet repeats has been shown to be responsible for about a dozen hereditary diseases. In this paper we are concerned with a computer model of such expansion, applied to the fragile X syndrome and Huntington's disease, for which enough quantitative data have been collected. The nucleotide sequence associated with the fragile X consists of CGG repeats and is located inside theFMR1 gene. In normal individuals there is a variable number of triplet repeats less than 60; in asymptomatic carriers the number of repeats is 60–200 (premutation). From the premutation range, the number of triplet repeats can increase within one generation to more than 200 producing affected individuals. In Huntington's disease the CAG repeats are located inside theHDgene. In normal individuals the number of repeats varies from around 11, up to 34. In the intermediate range (34–37 repeats), the mutability is increased, frequently leading to alleles of more than 37 repeats, and the disease phenotype. The rapid increase of the number of triplet repeats in affected individuals has been proposed to be due to the formation of folded DNA structures (hairpins) and their repair or misrepair. In order to determine if this proposed mechanism is adequate to account for the rapid increase of repeats and the large number of repeats in affected individuals we developed a mathematical model that includes the known mechanisms of hairpin formation, and strand synthesis and repair. Simulations based on the model using realistic probabilities of hairpin formation produced results that corresponded with the observed range of repeats and transition probabilities from normal to affected individuals. Similar modelling has been published for the Huntington's disease data. However, in this paper we demonstrate that a uniform approach works for fragile X and Huntington's disease, although the detailed assumptions of the model have to be different. These difference provide insight into the mechanisms of expansion in both cases. Among these insights is that an apparent threshold in the number of repeats for rapid expansion, and the preference for expansion over contraction, may be accounted for by relative probabilities of hairpin formation, replication, slippage and repair.

We utilize an extension of the variance-components models for cell lineage data in Huggins and Staudte [1] (R. M. Huggins and R. G. Staudte, Variance components models for dependent cell populations. J. Am. Stat. Assoc. 89:19–29 (1994)) to analyze NIH3T3 cells grown in two different media. This modeling approach has the advantage of a simple built-in correlation structure between familial members and allows for estimating experimental effects, rather than treating them as random effects. In addition, this methodology gives robust estimates of model parameters together with standard errors required for statistical inference. The importance of clonal heterogeneity and interexperiment variability in modeling eukaryotic cell cycles was previously pointed out by Kuczek and Axelrod [2] (T. Kuczek and D. E. Axelrod, The importance of clonal heterogeneity and interexperimental variability in modeling the eukaryotic cell cycle. Math. Biosci. 79:87–96 (1986)). This analysis confirms significantly positive sister-sister correlation when cells are grown in rich or poor medium and negative mother-daughter correlation when cells are grown in poor medium. However, for cells grown in rich medium, Kuczek and Axelrod's analysis gives negative mother-daughter correlations, whereas this analysis gives significant positive mother-daughter correlations.

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.

p53 protein detected immunohistochemically has not been accepted as a biomarker for breast cancer patients because of disparate reports of the relationship between the amount of p53 protein detected and patient survival. The purpose of this study was to determine experimental conditions and methods of data analysis for which p53 stain intensity could be prognostic for survival of young breast cancer patients. A tissue microarray of specimens from 93 patients was stained with anti-p53 antibody, and stain intensity measured with a computer-aided image analysis system. A cut-point at one standard deviation below the mean of the distribution of p53 stain intensity separated patients into two groups with significantly different survival. These results were confirmed by Quantitative Nuclear Grade determined by DNA-specific Feulgen staining. P53 provided information beyond ER and PR status. Therefore, under the conditions reported here, p53 protein can be an effective prognostic factor for young breast cancer patients.

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.

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.

The persistence of cell lifetimes during about 10 successive cell generations was investigated by comparing the number of cells in primary colonies and in secondary colonies derived from primary colonies. Primary colonies were grown from single cells for 3 or 4 days (a time equivalent to an average of five cell generations) and the number of cells in each primary colony determined. Cells in each primary colony were dispersed to initiate secondary colonies, grown for the same time, and the number of cells in secondary colonies determined. Several criteria were used to compare primary and related secondary colonies, the most informative was found to be regression and correlation coefficients between number of cells in primary colonies and mean numbers of cells in related secondary colonies. For two non-transformed mouse fibroblast cell lines, NIH 3T3 and BALB 3T3, the regression and correlation coefficients of cell number in primary and secondary colonies were positive. This suggests inheritance of cell lifetimes over many cell generations. After the addition of an activated ras oncogene (human cellular Harvey ras, or viral Kirsten ras) some regression and correlation coefficients changed in magnitude but all remained positive. Comparison of experimental data and the results of computer simulations suggest that several models of inheritance of cell lifetimes are not adequate to explain the results, including a model of independence between lifetimes of mother and daughter cells and the common model that describes daughter cells as inheriting the lifetime of their mother with deviation. Simulations do suggest that cell lifetimes are inherited within clones as deviation from the lifetime of the initial cell, and that the ras oncogene does not destroy persistence within clones but does increase heterogeneity of cell lifetimes.

PurposeNuclear grade of breast DCIS is considered during patient management decision-making although it may have only a modest prognostic association with therapeutic outcome. We hypothesized that visual inspection may miss substantive differences in nuclei classified as having the same nuclear grade. To test this hypothesis, we measured subvisual nuclear features by quantitative image cytometry for nuclei with the same grade, and tested for statistical differences in these features.Experimental design and statistical analysisThirty-nine nuclear digital image features of about 100 nuclei were measured in digital images of HResultsStatistically significant differences were detected in nuclear features between ducts with the same nuclear grade, both in different ducts of the same patient, and between ducts in different patients with DCIS of more than one grade.ConclusionNuclei in ducts visually described as having the same nuclear grade had significantly different subvisual digital image features. These subvisual differences may be considered additional manifestations of heterogeneity over and above differences that can be observed microscopically. This heterogeneity may explain the inconsistency of nuclear grading as a prognostic factor.

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.

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.

Traits such as birth size and lifetime can vary widely even among non-mutated progeny of the same cell proliferating in the same environment. On the other hand, population parameters of these traits may remain stable over many generations, and there may be a distinct inheritance of these traits from mother to daughters. We have reconsidered the implication of mother-daughter correlations in light of linear regression analysis. It is proposed that a non-mutant cell whose phenotype deviates from the population mean produces progeny whose rate of regression toward the mean is proportional to 1-r, where r is the mother-daughter correlation coefficient of the trait under study. Theoretical support for this proposition is derived from linear regression analysis. Empirical support is found in pedigree analysis of cell growth constants among NIH3T3 mouse fibroblast cells, where the presence of an activated human ras oncogene is associated with a decreased r and an increased rate at which the growth constants of progeny regress toward the population mean.

Background Nuclear grade has been associated with breast DCIS recurrence and progression to invasive carcinoma; however, our previous study of a cohort of patients with breast DCIS did not find such an association with outcome. Fifty percent of patients had heterogeneous DCIS with more than one nuclear grade. The aim of the current study was to investigate the effect of quantitative nuclear features assessed with digital image analysis on ipsilateral DCIS recurrence. Methods Hematoxylin and eosin stained slides for a cohort of 80 patients with primary breast DCIS were reviewed and two fields with representative grade (or grades) were identified by a Pathologist and simultaneously used for acquisition of digital images for each field. Van Nuys worst nuclear grade was assigned, as was predominant grade, and heterogeneous grading when present. Patients were grouped by heterogeneity of their nuclear grade: Group A: nuclear grade 1 only, nuclear grades 1 and 2, or nuclear grade 2 only (32 patients), Group B: nuclear grades 1, 2 and 3, or nuclear grades 2 and 3 (31 patients), Group 3: nuclear grade 3 only (17 patients). Nuclear fine structure was assessed by software which captured thirty-nine nuclear feature values describing nuclear morphometry, densitometry, and texture. Step-wise forward Cox regressions were performed with previous clinical and pathologic factors, and the new image analysis features. Results Duplicate measurements were similar for 89.7% to 97.4% of assessed image features. The rate of correct classification of nuclear grading with digital image analysis features was similar in the two fields, and pooled assessment across both fields. In the pooled assessment, a discriminant function with one nuclear morphometric and one texture feature was significantly (p = 0.001) associated with nuclear grading, and provided correct jackknifed classification of a patient's nuclear grade for Group A (78.1%), Group B (48.4%), and Group C (70.6%). The factors significantly associated with DCIS recurrence were those previously found, type of initial presentation (p = 0.03) and amount of parenchymal involvement (p = 0.05), along with the morphometry image feature of ellipticity (p = 0.04). Conclusion Analysis of nuclear features measured by image cytometry may contribute to the classification and prognosis of breast DCIS patients with more than one nuclear grade.

An increased number of copies of specific genes may offer an advantage to cells when they grow in restrictive conditions such as in the presence of toxic drugs, or in a tumor. Three mathematical models of gene amplification and deamplification are proposed to describe the kinetics of unstable phenotypes of cells with amplified genes. The models differ in details but all assume probabilistic mechanisms of increase and decrease in gene copy number per cell (gene amplification/deamplification). Analysis of the models indicates that a stable distribution of numbers of copies of genes per cell, observed experimentally, exists only if the probability of deamplification exceeds the probability of amplification. The models are fitted to published data on the loss of methotrexate resistance in cultured cell lines, due to the loss of amplified dihydrofolate reductase gene. For two mouse cell lines unstably resistant to methotrexate the probabilities of amplification and deamplification of the dihydrofolate reductase gene on double minute chromosomes are estimated to be approximately 2% and 10%, respectively. These probabilities are much higher than widely presumed. The models explain the gradual disappearance of the resistant phenotype when selective pressure is withdrawn, by postulating that the rate of deamplification exceeds the rate of amplification. Thus it is not necessary to invoke a growth advantage of nonresistant cells which has been the standard explanation. For another analogous process, the loss of double minute chromosomes containing the myc oncogene from SEWA tumor cells, the growth advantage model does seem to be superior to the amplification and deamplification model. In a more theoretical section of the paper, it is demonstrated that gene amplification/deamplification can result in reduction to homozygosity, such as is observed in some tumors. Other applications are discussed.

Cancer is a physical alteration of the relation of cells and their tissues resulting in aberrant social organization. These alterations are detected as masses (tumors) or, in the case of leukemia, as an abnormal number of white blood cells. However, many inflammatory lesions also form masses, and not all neoplasms are malignant. Therefore, histological criteria of malignancy are used for the diagnosis of cancer. Histopathology is the sine qua non of cancer diagnosis. Any other putative marker of cancer must be validated by histopathologic examination. Grading and staging of cancers are used to predict the clinical course and outcome of individual cancers. Grading of cancer is based on the histological criteria of the neoplasm including the degree of deviation from normal of tissue architecture and the differentiation and proliferation of individual cells. Extensive studies have correlated these microscopic characteristics with clinical outcome. Grading of most neoplasms, whether invasive or preinvasive, is useful in prognosis and consequently in therapy decisions. Staging, in most neoplasms, is based on the local extent of tissue involvement and/or the detection of the neoplastic cells in distant sites with microscopic confirmation. The molecular revolution has provided great insight into the genetic alterations leading to cancer and has given hope, in some quarters, that molecular techniques will supplement, or even supplant, microscopic examination. Each biomolecule can now claim its own view of organismic disease states with the suffix ‘‘-omics’’ (genomics, metabolomics, proteomics, glycomics, lipomics, etc.), and each promises to perfect biomarkers as the knowledge base evolves. The discovery and characterization of prostate-specific antigen (PSA) has been one of the great triumphs of this approach. However, some studies indicate that PSA lacks the specificity and sensitivity needed for a biomarker (1–3). The advocates of ‘‘systems biology’’ are diligently using their tools to find the next biomarker. However, their technologies will be validated using the gold standard of cancer diagnosis, microscopic histopathology. Early detection, without doubt, has had a major impact on the successful treatment of cancer. The improved cure rates of diseases such as cervical and breast cancers are indications of the impact of early detection on treatment. Early detection of most solid cancer involves the recognition and understanding of precancers known as carcinoma-in-situ or intraepithelial neoplasia. These foci of atypical cells are considered the precursors to malignancy. Students of specific cancers have identified apparent morphological continua that suggest a sequential acquisition of characteristics leading from normal to malignancy (cancer) (4–14). Although histopathology has successfully detected and characterized these early lesions, these studies also illustrate the limitations of histopathology as a predictive biomarker. Our current concepts of neoplastic progression are largely based on the ‘‘multiple genetic hit’’ hypothesis and the ‘‘linear sequential acquisition’’ models of neoplastic progression. Although somewhat successfully applied to colon cancer, the successive acquisition model has been less predictive in cervical, prostate, and breast cancer, where the relation between ‘‘low-grade’’ intraepithelial neoplasia and invasive cancer has been questioned (15–17). Cervical intraepithelial neoplasia is an excellent example where a specific viral infection may be required to develop a high-grade, progressive in-situ lesion (17). Thus, alternative models of neoplastic evolution have been proposed in which there are several lineages that progress in parallel. These are found to fit observed histological (18) and molecular observations (19) better than a simple linear acquisition model. Mouse models of tumor biology are informative. The insertion or manipulation of genes that are associated with human malignancy has resulted in murine tumors that mimic the 1 Published in a supplement to The Journal of Nutrition. Presented as part of the conference ‘‘The Use and Misuse of Biomarkers as Indicators of Cancer Risk Reduction Following Dietary Manipulation’’ held July 12–13, 2005 in Bethesda, MD. This conference was sponsored by the Center for Food Safety and Applied Nutrition (CFSAN), Food and Drug Administration (FDA), Department of Health and Human Services (DHHS); the Office of Dietary Supplements (ODS), National Institutes of Health, DHHS; and the Division of Cancer Prevention (DCP), National Cancer Institute, National Institutes of Health, DHHS. Guest Editors for the supplement publication were Harold E. Seifried, National Cancer Institute, NIH; and Claudine Kavanaugh, CFSAN, FDA. Guest editor disclosure: H.E. Seifried, no relationships to disclose; C. Kavanaugh, no relationships to disclose. 2 Supported by grants RO1CA089140, U01 CA105490-01 from the National Cancer Institute and U42 RR14905 from the NCRR and 1076-CCR-S0 from the New Jersey Commission on Cancer Research. 3 Author disclosure: no relationships to disclose. * To whom correspondence should be addressed. E-mail: rdcardiff@ucdavis. edu. 7 Abbreviations used: DCIS, ductal carcinoma in situ; GEM, genetically engineered mouse; HPO, hyperplastic outgrowth; MIN, mammary intraepithelial neoplasia; MIN-O, mammary intraepithelial neoplasia outgrowths; PSA, prostatespecific antigen.

The aim of this study was to assess components of variation in nocturia and to determine any putative geomagnetic influence. A 54-year old man with benign prostatic hyperplasia had recorded for about 4 years the number of times he awoke each night to urinate. The data have been reanalyzed for chronomics, the mapping of time structures (chronomes), involving the computation of least squares spectra of the urinary record and of environmental variables recorded during the same 4-year span. In addition to the previously reported monthly variation, other periodicities have been documented, including two separate components with periods of one week and of a near-week. The precise 7-day period may be a mainly exogenous resonance with external influences such as a weekly social schedule, whereas the near-week may be a partial resonance with natural changes in geomagnetics, reflecting in part changes in other non-photic natural environmental factors.

Objective:: The goal of this study is to re-examine the oligonucleotide microarray dataset of Shipp et al. (www.genome.wi.mit.du/MPR/lymphoma), which contains the intensity levels of 6817 genes of 58 patients with diffuse large B-cell lymphoma (DLBCL) and 19 with follicular lymphoma (FL), by means of the combinatorics, optimisation, and logic-based methodology of logical analysis of data (LAD). The motivations for this new analysis included the previously demonstrated capabilities of LAD and its expected potential (1) to identify different informative genes than those discovered by conventional statistical methods, (2) to identify combinations of gene expression levels capable of characterizing different types of lymphoma, and (3) to assemble collections of such combinations that if considered jointly are capable of accurately distinguishing different types of lymphoma. Methods and materials:: The central concept of LAD is a pattern or combinatorial biomarker, a concept that resembles a rule as used in decision tree methods. LAD is able to exhaustively generate the collection of all those patterns which satisfy certain quality constraints, through a systematic combinatorial process guided by clear optimization criteria. Then, based on a set covering approach, LAD aggregates the collection of patterns into classification models. In addition, LAD is able to use the information provided by large collections of patterns in order to extract subsets of variables, which collectively are able to distinguish between different types of disease. Results:: For the differential diagnosis of DLBCL versus FL, a model based on eight significant genes is constructed and shown to have a sensitivity of 94.7% and a specificity of 100% on the test set. For the prognosis of good versus poor outcome among the DLBCL patients, a model is constructed on another set consisting also of eight significant genes, and shown to have a sensitivity of 87.5% and a specificity of 90% on the test set. The genes selected by LAD also work well as a basis for other kinds of statistical analysis, indicating their robustness. Conclusion:: These two models exhibit accuracies that compare favorably to those in the original study. In addition, the current study also provides a ranking by importance of the genes in the selected significant subsets as well as a library of dozens of combinatorial biomarkers (i.e. pairs or triplets of genes) that can serve as a source of mathematically generated, statistically significant research hypotheses in need of biological explanation.

In this chapter, we consider a number of examples of branching processes with infinite type spaces. No systematic theory can be presented. However, in Sect. 7.1 we review various approaches generalizing the denumerable case. Also, general processes (Sect. C.1) include the denumerable type space as a special case. We will base considerations on an analogy with the finite mutitype case whenever possible. However, the stress is on interesting and diverse properties, which are different from the finite multitype setup and on biologically motivated examples.

The branching process is a system of particles (individuals, cells , molecules, etc.) which live for a random time and, at some point during lifetime or at the moment of death, produce a random number of progeny. Processes allowing production of new individuals during a parent individual’s lifetime are called the general or Jagers-Crump-Mode processes . They are suitable for description of populations of higher organisms, like vertebrates and plants. Processes that assume production of progeny at the terminal point of parent entity’s lifetime are called the classical processes. They are usually sufficient for modeling populations of biological cells, genes or biomolecules . In some processes, like the time-continuous Markov process , the distinction is immaterial since one of the progeny of a particle may be considered an extension of the parent.