Mathematical models for optimizing tumour radiotherapy. I: A simple two compartment cell-kinetic model for the unperturbed growth of transplantable tumours

Tumour growth kinetics has been analysed on the basis of interactions between two compartments comprising the proliferating and non-proliferating cells. Starting from the differential equations of growth of the cell-populations in the two compartments and assuming the process of intercompartmental cell transfers to be linear, an analytic expression on the variation of growth-fraction with the age of the tumour is obtained. The restricted conditions on the cell-cycle time and cell-loss-rate, under which these differential equations lead to a Gompertzian growth of the tumour, are critically analysed. The formalism permits the estimation of some important cell-kinetic parameters, like growth-fraction or cell-loss-factor, from a knowledge of the tumour-growth curve, cell-cycle-time and a single measurement of the cell-loss-rate of the matured tumour, provided the tumour follows a Gompertzian growth. The validity of the model has been verified with the experimental data on 4 different transplantable murine tumour systems. Usefulness of the model has been demonstrated by making some interesting predictions regarding the radiation response of the tumours from the cell-kinetic parameters.     

A stochastic formulation of the gompertzian growth model for in vitro bactericidal kinetics: parameter estimation and extinction probability

Time-kill curves have frequently been employed to study the antimicrobial effects of antibiotics. The relevance of pharmacodynamic modeling to these investigations has been emphasized in many studies of bactericidal kinetics. Stochastic models are needed that take into account the randomness of the mechanisms of both bacterial growth and bacteria-drug interactions. However, most of the models currently used to describe antibiotic activity against microorganisms are deterministic. In this paper we examine a stochastic differential equation representing a stochastic version of a pharmacodynamic model of bacterial growth undergoing random fluctuations, and derive its solution, mean value and covariance structure. An explicit likelihood function is obtained both when the process is observed continuously over a period of time and when data is sampled at time points, as is the custom in these experimental conditions. Some asymptotic properties of the maximum likelihood estimators for the model parameters are discussed. The model is applied to analyze in vitro time-kill data and to estimate model parameters; the probability of the bacterial population size dropping below some critical threshold is also evaluated. Finally, the relationship between bacterial extinction probability and the pharmacodynamic parameters estimated is discussed.     

Longitudinal Data Analysis with Event Time as a Covariate

We consider the estimation of a nonparametric smooth function of some event time in a semiparametric mixed effects model from repeatedly measured data when the event time is subject to right censoring. The within-subject correlation is captured by both cross-sectional and time-dependent random effects, where the latter is modeled by a nonhomogeneous Ornstein–Uhlenbeck stochastic process. When the censoring probability depends on other variables in the model, which often happens in practice, the event time data are not missing completely at random. Hence, the complete case analysis by eliminating all the censored observations may yield biased estimates of the regression parameters including the smooth function of the event time, and is less efficient. To remedy, we derive the likelihood function for the observed data by modeling the event time distribution given other covariates. We propose a two-stage pseudo-likelihood approach for the estimation of model parameters by first plugging an estimator of the conditional event time distribution into the likelihood and then maximizing the resulting pseudo-likelihood function. Empirical evaluation shows that the proposed method yields negligible biases while significantly reduces the estimation variability. This research is motivated by the project of hormone profile estimation around age at the final menstrual period for the cohort of women in the Michigan Bone Health and Metabolism Study.     

A random walk model of oligodendrocyte generation in vitro and associated estimation problems.

A branching stochastic process proposed earlier to model oligodendrocyte generation by O-2A progenitor cells under in vitro conditions does not allow invoking the maximum likelihood techniques for estimation purposes. To overcome this difficulty, we propose a partial likelihood function based on an embedded random walk model of clonal growth and differentiation of O-2A progenitor cells. Under certain conditions, the partial likelihood function yields consistent estimates of model parameters. The usefulness of this approach is illustrated with computer simulations and data analyses.     

H广义线性模型中的参数估计的比较

本文研究H广义线性模型中未知参数的两种估计方法,一种是边际似然函数法,另一种是Lee和Nelder提出来的L-N法．对于一类具有两个随机效应的典型的Poisson-Gamma类模型,在一些正则性条件之下,我们已经证明了其中固定效应卢的L-N估计的强相合性及渐近正态性,并得到了其收敛于真值的速度．针对这类模型,本文进一步给出了其边际似然函数的解析表达式,并且通过Monte Carlo模拟,对模型中固定效应β的边际似然估计和L—N估计进行了比较,模拟表明L—N估计比边际似然估计在拟Poisson-Gamma模型中有着更加优良的表现,具有更高的精度。     

Gompertzian growth pattern correlated with phenotypic organization of colon carcinoma,malignant glioma and non-small cell lung carcinoma cell lines

Abstract. In the current study we present a Gompertzian model for cell growth as a function of cell phenotype using six human tumour cell lines (A-549, NCI-H596, NCI-H520, HT-29, SW-620 and U-251). Monolayer cells in exponential growth at various densities were quantified over a week by sulforhodamine B staining assay to produce cell-growth curves. A Gompertz equation was fitted to experimental data to obtain, for each cell line, three empirical growth parameters (initial cell density, cell-growth rate and carrying capacity – the maximal cell density). A cell-shape parameter named deformation coefficient D (a morphological relationship among spreading and confluent cells) was established and compared by regression analysis with the relative growth rate parameter K described by the Gompertz equation. We have found that coefficient D is directly proportional to the growth parameter K . The fit curve significantly matches the empirical data ( P  < 0.05), with a correlation coefficient of 0.9152. Therefore, a transformed Gompertzian growth function was obtained accordingly to D . The degree of correlation between the Gompertzian growth parameter and the coefficient D allows a new interpretation of the growth parameter K on the basis of morphological measurements of a set of tumour cell types, supporting the idea that cell-growth kinetics can be modulated by phenotypic organization of attached cells.     

A model for the analysis of growth data from designed experiments

A model for growth data from designed experiments is presented which extends the stochastic differential equation of Sandland and McGilchrist (1979, Biometrics 35, 255-272). Residual maximum likelihood (REML) is used to estimate the parameters of the model. The model is easily extended to incomplete data and is shown to overcome some of the practical difficulties encountered with the profile model. The procedure is applied to data from experiments on pigs and sheep.     

Estimating the number of species in a stochastic abundance model

Consider a stochastic abundance model in which the species arrive in the sample according to independent Poisson processes, where the abundance parameters of the processes follow a gamma distribution. We propose a new estimator of the number of species for this model. The estimator takes the form of the number of duplicated species (i.e., species represented by two or more individuals) divided by an estimated duplication fraction. The duplication fraction is estimated from all frequencies including singleton information. The new estimator is closely related to the sample coverage estimator presented by Chao and Lee (1992, Journal of the American Statistical Association 87, 210-217). We illustrate the procedure using the Malayan butterfly data discussed by Fisher, Corbet, and Williams (1943, Journal of Animal Ecology 12, 42-58) and a 1989 Christmas Bird Count dataset collected in Florida, U.S.A. Simulation studies show that this estimator compares well with maximum likelihood estimators (i.e., empirical Bayes estimators from the Bayesian viewpoint) for which an iterative numerical procedure is needed and may be infeasible.     

A diffusion process to model generalized von Bertalanffy growth patterns: Fitting to real data

The von Bertalanffy growth curve has been commonly used for modeling animal growth (particularly fish). Both deterministic and stochastic models exist in association with this curve, the latter allowing for the inclusion of fluctuations or disturbances that might exist in the system under consideration which are not always quantifiable or may even be unknown. This curve is mainly used for modeling the length variable whereas a generalized version, including a new parameter b≥1, allows for modeling both length and weight for some animal species in both isometric (b=3) and allometric (b≠3) situations.In this paper a stochastic model related to the generalized von Bertalanffy growth curve is proposed. This model allows to investigate the time evolution of growth variables associated both with individual behaviors and mean population behavior. Also, with the purpose of fitting the above-mentioned model to real data and so be able to forecast and analyze particular characteristics, we study the maximum likelihood estimation of the parameters of the model. In addition, and regarding the numerical problems posed by solving the likelihood equations, a strategy is developed for obtaining initial solutions for the usual numerical procedures. Such strategy is validated by means of simulated examples. Finally, an application to real data of mean weight of swordfish is presented.     

Using maximum likelihood to estimate population size from temporal changes in allele frequencies.

We develop a maximum-likelihood framework for using temporal changes in allele frequencies to estimate the number of breeding individuals in a population. We use simulations to compare the performance of this estimator to an F-statistic estimator of variance effective population size. The maximum-likelihood estimator had a lower variance and smaller bias. Taking advantage of the likelihood framework, we extend the model to include exponential growth and show that temporal allele frequency data from three or more sampling events can be used to test for population growth.     

Cellular automaton of idealized brain tumor growth dynamics

A novel cellular automaton model of proliferative brain tumor growth has been developed. This model is able to simulate Gompertzian tumor growth over nearly three orders of magnitude in radius using only four microscopic parameters. The predicted composition and growth rates are in agreement with a test case pooled from the available medical literature. The model incorporates several new features, improving previous models, and also allows ready extension to study other important properties of tumor growth, such as clonal competition.     

Maximum likelihood estimation in the additive hazards model

The additive hazards model specifies the effect of covariates on the hazard in an additive way, in contrast to the popular Cox model, in which it is multiplicative. As the non-parametric model, additive hazards offer a very flexible way of modeling time-varying covariate effects. It is most commonly estimated by ordinary least squares. In this paper, we consider the case where covariates are bounded, and derive the maximum likelihood estimator under the constraint that the hazard is non-negative for all covariate values in their domain. We show that the maximum likelihood estimator may be obtained by separately maximizing the log-likelihood contribution of each event time point, and we show that the maximizing problem is equivalent to fitting a series of Poisson regression models with an identity link under non-negativity constraints. We derive an analytic solution to the maximum likelihood estimator. We contrast the maximum likelihood estimator with the ordinary least-squares estimator in a simulation study and show that the maximum likelihood estimator has smaller mean squared error than the ordinary least-squares estimator. An illustration with data on patients with carcinoma of the oropharynx is provided.     

Semiparametric regression for periodic longitudinal hormone data from multiple menstrual cycles

We consider semiparametric regression for periodic longitudinal data. Parametric fixed effects are used to model the covariate effects and a periodic nonparametric smooth function is used to model the time effect. The within-subject correlation is modeled using subject-specific random effects and a random stochastic process with a periodic variance function. We use maximum penalized likelihood to estimate the regression coefficients and the periodic nonparametric time function, whose estimator is shown to be a periodic cubic smoothing spline. We use restricted maximum likelihood to simultaneously estimate the smoothing parameter and the variance components. We show that all model parameters can be easily obtained by fitting a linear mixed model. A common problem in the analysis of longitudinal data is to compare the time profiles of two groups, e.g., between treatment and placebo. We develop a scaled chi-squared test for the equality of two nonparametric time functions. The proposed model and the test are illustrated by analyzing hormone data collected during two consecutive menstrual cycles and their performance is evaluated through simulations.     

Cox Regression in Nested Case–Control Studies with Auxiliary Covariates

Summary Nested case–control (NCC) design is a popular sampling method in large epidemiological studies for its cost effectiveness to investigate the temporal relationship of diseases with environmental exposures or biological precursors. Thomas' maximum partial likelihood estimator is commonly used to estimate the regression parameters in Cox's model for NCC data. In this article, we consider a situation in which failure/censoring information and some crude covariates are available for the entire cohort in addition to NCC data and propose an improved estimator that is asymptotically more efficient than Thomas' estimator. We adopt a projection approach that, heretofore, has only been employed in situations of random validation sampling and show that it can be well adapted to NCC designs where the sampling scheme is a dynamic process and is not independent for controls. Under certain conditions, consistency and asymptotic normality of the proposed estimator are established and a consistent variance estimator is also developed. Furthermore, a simplified approximate estimator is proposed when the disease is rare. Extensive simulations are conducted to evaluate the finite sample performance of our proposed estimators and to compare the efficiency with Thomas' estimator and other competing estimators. Moreover, sensitivity analyses are conducted to demonstrate the behavior of the proposed estimator when model assumptions are violated, and we find that the biases are reasonably small in realistic situations. We further demonstrate the proposed method with data from studies on Wilms' tumor.     

Closed-Form Approximations to the REML Estimator of a Variance Ratio (or Heritability) in a Mixed Linear Model

In this article, we estimate heritability or intraclass correlation in a mixed linear model having two sources of variation. In most applications, the commonly used restricted maximum likelihood (REML) estimator can only be obtained via an iterative approach. In some cases, the algorithm used to compute REML estimates may be slow or may even fail to converge. We develop a set of closed-form approximations to the REML estimator, and the performance of these estimators is compared with that of the REML estimator. We provide guidelines regarding how to choose the estimator that best approximates the REML estimator. Examples presented in the article suggest that the closed-form estimators compete with and, in some cases, outperform the REML estimator.     

A semiparametric mixture cure survival model for left‐truncated and right‐censored data

In follow‐up studies, the disease event time can be subject to left truncation and right censoring. Furthermore, medical advancements have made it possible for patients to be cured of certain types of diseases. In this article, we consider a semiparametric mixture cure model for the regression analysis of left‐truncated and right‐censored data. The model combines a logistic regression for the probability of event occurrence with the class of transformation models for the time of occurrence. We investigate two techniques for estimating model parameters. The first approach is based on martingale estimating equations (EEs). The second approach is based on the conditional likelihood function given truncation variables. The asymptotic properties of both proposed estimators are established. Simulation studies indicate that the conditional maximum‐likelihood estimator (cMLE) performs well while the estimator based on EEs is very unstable even though it is shown to be consistent. This is a special and intriguing phenomenon for the EE approach under cure model. We provide insights into this issue and find that the EE approach can be improved significantly by assigning appropriate weights to the censored observations in the EEs. This finding is useful in overcoming the instability of the EE approach in some more complicated situations, where the likelihood approach is not feasible. We illustrate the proposed estimation procedures by analyzing the age at onset of the occiput‐wall distance event for patients with ankylosing spondylitis.     

A Likelihood Approach to Populations Samples of Microsatellite Alleles

This paper presents a likelihood approach to population samples of microsatellite alleles. A Markov chain recursion method previously published by GRIFFITHS and TAVARE is applied to estimate the likelihood function under different models of microsatellite evolution. The method presented can be applied to estimate a fundamental population genetics parameter θ as well as parameters of the mutational model. The new likelihood estimator provides a better estimator of θ in terms of the mean square error than previous approaches. Furthermore, it is demonstrated how the method may easily be applied to test models of microsatellite evolution. In particular it is shown how to compare a one-step model of microsatellite evolution to a multi-step model by a likelihood ratio test.     

Analysis of free growth and post-treatment regrowth of C3H mammary carcinoma in individual mice

Free growth and post-Doxorubicin treatment regrowth of the C3H mammary carcinoma were analysed in individual mice. In both cases, the Gompertzian function provided a better fit than the exponential function, and the difference was statistically significant (P less than 0.001, chi 2 test). No comprehensive Gompertzian function was found, and each individual tumour growth or regrowth was described by a specific curve. Nevertheless, although both individually measured alpha 0 and beta, Gompertzian parameters varied from one animal to another, in both free-growing and post-treatment regrowing tumours a strong linear correlation between alpha 0 and beta was found. A parallelism test was performed to verify if there exists any treatment-induced alteration. The two regression lines appeared to be identical, however.     

Estimation of Parameters for Macroparasite Population Evolution Using Approximate Bayesian Computation

Summary We estimate the parameters of a stochastic process model for a macroparasite population within a host using approximate Bayesian computation (ABC). The immunity of the host is an unobserved model variable and only mature macroparasites at sacrifice of the host are counted. With very limited data, process rates are inferred reasonably precisely. Modeling involves a three variable Markov process for which the observed data likelihood is computationally intractable. ABC methods are particularly useful when the likelihood is analytically or computationally intractable. The ABC algorithm we present is based on sequential Monte Carlo, is adaptive in nature, and overcomes some drawbacks of previous approaches to ABC. The algorithm is validated on a test example involving simulated data from an autologistic model before being used to infer parameters of the Markov process model for experimental data. The fitted model explains the observed extra‐binomial variation in terms of a zero‐one immunity variable, which has a short‐lived presence in the host.     

Estimation of regression parameters and the hazard function in transformed linear survival models

An estimator of the regression parameters in a semiparametric transformed linear survival model is examined. This estimator consists of a single Newton-like update of the solution to a rank-based estimating equation from an initial consistent estimator. An automated penalized likelihood algorithm is proposed for estimating the optimal weight function for the estimating equations and the error hazard function that is needed in the variance estimator. In simulations, the estimated optimal weights are found to give reasonably efficient estimators of the regression parameters, and the variance estimators are found to perform well. The methodology is applied to an analysis of prognostic factors in non-Hodgkin's lymphoma.