Problems of determinacy in compartmental modeling with application to Bilirubin Kinetics

A new concept, called determinacy, is introduced. The concept is applied to compartmental modeling of the metabolism of bilirubin in the human biological system. Necessarily gross simplifications must be made in deriving the model. This paper treats the consequent problems of determinacy: (a) selection of model structure from a finite set of alternatives; (b) selection of parameter values from a finite set of solutions (resolution of ambiguities); (c) selection of parameter values from a continuum of solutions.     

Stochastic compartmental model with branching particles

A multi-compartmental model with particles producing offspring according to the Markov branching process has been studied. Explicit results are given for the two-compartmental system and for irreversible general multicompartmental systems. The known models in stochastic compartmental analysis are shown to be particular cases of this model and applications are cited.     

Stochastic modeling of cellular colonies with quiescence: An application to drug resistance in cancer

Several cancers are thought to be driven by cells with stem cell like properties. An important characteristic of stem cells, which also applies to primitive tumor cells, is the ability to undergo quiescence, where cells can temporarily stop the cell cycle. Cellular quiescence can affect the kinetics of tumor growth, and the susceptibility of the cells to therapy. To study how quiescence affects treatment, we formulate a stochastic birth–death process with quiescence, on a combinatorial cellular mutation network, and consider the pre-treatment (growth) and treatment (decay) regimes. We find that, in the absence of mutations, treatment (if sufficiently strong) will proceed as a biphasic decline with the first (faster) phase driven by the elimination of the cycling cells and the second (slower) phase limited by the process of cell awakening. Other regimes are possible for weaker treatments. We also describe how the process of mutant generation is influenced by quiescence. Interestingly, for single-drug treatments, the probability to have resistance at start of treatment is independent of quiescence. For two or more drugs, the probability to have generated resistant mutants before treatment grows with quiescence. Finally, we study the influence of quiescence on the treatment phase. Starting from a given composition of mutants, the chances of treatment success are not influenced by the presence of quiescence.     

Stochastic modeling of cellular colonies with quiescence: an application to drug resistance in cancer

Several cancers are thought to be driven by cells with stem cell like properties. An important characteristic of stem cells, which also applies to primitive tumor cells, is the ability to undergo quiescence, where cells can temporarily stop the cell cycle. Cellular quiescence can affect the kinetics of tumor growth, and the susceptibility of the cells to therapy. To study how quiescence affects treatment, we formulate a stochastic birth-death process with quiescence, on a combinatorial cellular mutation network, and consider the pre-treatment (growth) and treatment (decay) regimes. We find that, in the absence of mutations, treatment (if sufficiently strong) will proceed as a biphasic decline with the first (faster) phase driven by the elimination of the cycling cells and the second (slower) phase limited by the process of cell awakening. Other regimes are possible for weaker treatments. We also describe how the process of mutant generation is influenced by quiescence. Interestingly, for single-drug treatments, the probability to have resistance at start of treatment is independent of quiescence. For two or more drugs, the probability to have generated resistant mutants before treatment grows with quiescence. Finally, we study the influence of quiescence on the treatment phase. Starting from a given composition of mutants, the chances of treatment success are not influenced by the presence of quiescence.     

Stochastic compartmental models with banching and immigrant particles

A multicompartmental model in which particles enter the system from the environment and reproduce according to a Markov branching process has been considered. Explicit expressions have been obtained for the mean vector and the correlation structure for the numbers of particles in different compartments in different time points of the system. Growth rates of the mean vector and some special cases of the system are also discussed.     

A regularized estimation approach for case-cohort periodic follow-up studies with an application to HIV vaccine trials

This paper discusses regression analysis of the failure time data arising from case-cohort periodic follow-up studies, and one feature of such data, which makes their analysis much more difficult, is that they are usually interval-censored rather than right-censored. Although some methods have been developed for general failure time data, there does not seem to exist an established procedure for the situation considered here. To address the problem, we present a semiparametric regularized procedure and develop a simple algorithm for the implementation of the proposed method. In addition, unlike some existing procedures for similar situations, the proposed procedure is shown to have the oracle property, and an extensive simulation is conducted and it suggests that the presented approach seems to work well for practical situations. The method is applied to an HIV vaccine trial that motivated this study.     

Parameter estimation in longitudinal studies with outcome-dependent follow-up

In many observational studies, individuals are measured repeatedly over time, although not necessarily at a set of prespecified occasions. Instead, individuals may be measured at irregular intervals, with those having a history of poorer health outcomes being measured with somewhat greater frequency and regularity; i.e., those individuals with poorer health outcomes may have more frequent follow-up measurements and the intervals between their repeated measurements may be shorter. In this article, we consider estimation of regression parameters in models for longitudinal data where the follow-up times are not fixed by design but can depend on previous outcomes. In particular, we focus on general linear models for longitudinal data where the repeated measures are assumed to have a multivariate Gaussian distribution. We consider assumptions regarding the follow-up time process that result in the likelihood function separating into two components: one for the follow-up time process, the other for the outcome process. The practical implication of this separation is that the former process can be ignored when making likelihood-based inferences about the latter; i.e., maximum likelihood (ML) estimation of the regression parameters relating the mean of the longitudinal outcomes to covariates does not require that a model for the distribution of follow-up times be specified. As a result, standard statistical software, e.g., SAS PROC MIXED (Littell et al., 1996, SAS System for Mixed Models), can be used to analyze the data. However, we also demonstrate that misspecification of the model for the covariance among the repeated measures will, in general, result in regression parameter estimates that are biased. Furthermore, results of a simulation study indicate that the potential bias due to misspecification of the covariance can be quite considerable in this setting. Finally, we illustrate these results using data from a longitudinal observational study (Lipshultz et al., 1995, New England Journal of Medicine 332, 1738-1743) that explored the cardiotoxic effects of doxorubicin chemotherapy for the treatment of acute lymphoblastic leukemia in children.     

Mathematical modeling and parameter estimation of axonal cargo transport

This paper is about how cortical recurrent interactions in primary visual cortex (V1) together with feedback from extrastriate cortex can account for spectral peaks in the V1 local field potential (LFP). Recent studies showed that visual stimulation enhances the γ-band (25–90 Hz) of the LFP power spectrum in macaque V1. The height and location of the γ-band peak in the LFP spectrum were correlated with visual stimulus size. Extensive spatial summation, possibly mediated by feedback connections from extrastriate cortex and long-range horizontal connections in V1, must play a crucial role in the size dependence of the LFP. To analyze stimulus-effects on the LFP of V1 cortex, we propose a network model for the visual cortex that includes two populations of V1 neurons, excitatory and inhibitory, and also includes feedback to V1 from extrastriate cortex. The neural network model for V1 was a resonant system. The model’s resonance frequency (ResF) was in the γ-band and varied up or down in frequency depending on cortical feedback. The model’s ResF shifted downward with stimulus size, as in the real cortex, because increased size recruited more activity in extrastriate cortex and V1 thereby causing stronger feedback. The model needed to have strong local recurrent inhibition within V1 to obtain ResFs that agree with cortical data. Network resonance as a consequence of recurrent excitation and inhibition appears to be a likely explanation for γ-band peaks in the LFP power spectrum of the primary visual cortex.     

Polygenic modeling of genome-wide association studies: an application to prostate and breast cancer

Genome-wide association studies (GWAS) have successfully detected and replicated associations with numerous diseases, including cancers of the prostate and breast. These findings are helping clarify the genomic basis of such diseases, but appear to explain little of disease heritability. This limitation might reflect the focus of conventional GWAS on a small set of the most statistically significant associations with disease. More information might be obtained by analyzing GWAS using a polygenic model, which allows for the possibility that thousands of genetic variants could impact disease. Furthermore, there may exist common polygenic effects between potentially related phenotypes (e.g., prostate and breast cancer). Here we present and apply a polygenic model to GWAS of prostate and breast cancer. Our results indicate that the polygenic model can explain an increasing--albeit low--amount of heritability for both of these cancers, even when excluding the most statistically significant associations. In addition, nonaggressive prostate cancer and breast cancer appear to share a common polygenic model, potentially reflecting a similar underlying biology. This supports the further development and application of polygenic models to genomic data.     

Individual based modeling and parameter estimation for a Lotka-Volterra system

Stochastic component, inevitable in biological systems, makes problematic the estimation of the model parameters from a single sequence of measurements, despite the complete knowledge of the system. We studied the problem of parameter estimation using individual-based computer simulations of a 'Lotka-Volterra world'. Two kinds (species) of particles--X (preys) and Y (predators)--moved on a sphere according to deterministic rules and at the collision (interaction) of X and Y the particle X was changed to a new particle Y. Birth of preys and death of predators were simulated by addition of X and removal of Y, respectively, according to exponential probability distributions. With this arrangement of the system, the numbers of particles of each kind might be described by the Lotka-Volterra equations. The simulations of the system with low (200-400 particles on average) number of individuals showed unstable oscillations of the population size. In some simulation runs one of the species became extinct. Nevertheless, the oscillations had some generic properties (e.g. mean, in one simulation run, oscillation period, mean ratio of the amplitudes of the consecutive maxima of X and Y numbers, etc.) characteristic for the solutions of the Lotka-Volterra equations. This observation made it possible to estimate the four parameters of the Lotka-Volterra model with high accuracy and good precision. The estimation was performed using the integral form of the Lotka-Volterra equations and two parameter linear regression for each oscillation cycle separately. We conclude that in spite of the irregular time course of the number of individuals in each population due to stochastic intraspecies component, the generic features of the simulated system evolution can provide enough information for quantitative estimation of the system parameters.     

Implications of empirical data quality to metapopulation model parameter estimation and application

Parameter estimation is a critical step in the use of any metapopulation model for predictive purposes. Typically metapopulation studies assume that empirical data are of good quality and any errors are so insignificant that they can be ignored. However, three types of errors occur commonly in metapopulation data sets. First, patch areas can be mis-estimated. Second, unknown habitat patches may be located within or around the study area. Third, there may be false zeros in the data set, that is, some patches were observed to be empty while there truly was a population in the patch. This study investigates biases induced into metapopulation model parameter estimates by these three types of errors. It was found that mis-estimated areas influence the scaling of extinction risk with patch area; extinction probabilities for large patches become overestimated. Missing patches cause overestimation of migration distances and colonization ability of the species. False zeros can affect very strongly all model components, the extinction risk in large patches, intrinsic extinction rates in general, migration distances and colonization ability may become all overestimated. Biases in parameter estimates translate into corresponding biases in model predictions, which are serious particularly if metapopulation persistence becomes overestimated. This happens for example when the migration capability of the species is overestimated. Awareness of these biases helps in understanding seemingly anomalous parameter estimation results. There are also implications for field work: it may be reasonable to allocate effort so that serious errors in data are minimized. It is particularly important to avoid observing false zeros for large and/or isolated patches.     

Stochastic modeling of drug resistance in cancer

One of the main causes of failure in the treatment of cancer is the development of drug resistance by the cancer cells. Employing multi-drug therapeutic strategies is a promising way to prevent resistance and improve the chances of treatment success. We formulate and analyse a stochastic model for multi-drug resistance and investigate the dependence of treatment outcomes on the initial tumor load, mutation rates and the turnover rate of cancerous cells. We elucidate the general principles of the emergence and evolution of resistant cells inside the tumor, before and after the start of treatment. We discover that for non-mutagenic drugs, pre-existence contributes more to resistance generation than the treatment phase; this result holds for the case where all drugs are applied simultaneously, and is not applicable for sequential therapy models. The application of mathematical modelling to aspects of adjuvant chemotherapy scheduling. J. Math. Biol. 48(4), 375-422]. Also, we find that treatment success is independent on the turnover rate for one drug, and it depends strongly on it for multi-drug therapies. For low-turnover rates, increasing the number of drugs will increase the probability of successful therapy. For very high-turnover rates, increasing the number of drugs used does not significantly increase the chances of treatment success.     

Frequency response of the chest: modeling and parameter estimation.

The frequency response of the respiratory system was studied in the range from 3 to 70 Hz in 15 normal subjects by applying sinusoidal pressure variations around the chest and measuring gas flow at the mouth. The observed input-output relationships were systematically compared to those predicted on the basis of linear differential equations of increasing order. From 3 to 20 Hz the behavior of the system was best described by a 3rd-order equation, and from 3 to 50 Hz by a 4th-order one. A mechanistic model of the 4th order, featuring tissue compliance (Ct), resistance (Rt) and inertance (It), alveolar gas compressibility (Cg) and airway resistance (Raw), and inertance (Iaw) was developed. Using that model, the following mean values were found: Ct = 2.08-10(-2)1-hPa-1 (1 hPa congruent to 1 cm of water); Rt = 1.10-hPa-1(-1)-s; It = 0.21-10(-2)hPa-1(-1)-s2; Raw = 1.35-hPa-1(-1)-s; Iaw = 2.55-10(-2)hPa-1(-1)-s2. Additional experiments devised to validate the model were reasonably successful, suggesting that the physical meaning attributed to the coefficients was correct. The validity of the assumptions and the physiological meaning of the coefficients are discussed.     

Extension of compartmental parameters to blocks of compartments with application to lipoprotein kinetics

Suppose that the compartments of a compartmental model are separated into blocks (sets of compartments). In general, the blocks can not be regarded as compartments but it may be possible to construct a “condensation model,” the compartments of which correspond to the blocks, in such fashion so as to retain certain salient properties of the blocks. Condensation is a way of formally summarizing a large model by presenting a smaller one to emphasize certain characteristics of the larger model. Suppose that the parameters to be retained are the mean residence times through the blocks; this paper deals with the construction of the condensation model, the properties of the condensation model, and the possible applications of condensation to data analysis, particularly in regard to lipoprotein kinetics.     

Optimal sampling time selection for parameter estimation in dynamic pathway modeling

Systems Biology is an emerging research area, which considers mathematical representations of inter- and intra-cellular dynamics. Among the many research problems that have been addressed, dynamic modeling of signal transduction pathways has received increasing attention. The usual approach to represent intra-cellular dynamics are nonlinear, usually ordinary, differential equations. The purpose of the models is to test and generate hypothesis of specific pathways and it is therefore required to estimate model parameters from experimental data. The experiments to generate data are complex and expensive, as a consequence of which the time series available are usually rather short, with few if any replicates. Almost certainly, not all variables one would like to include in a model can be measured. Parameter estimation is therefore an important research problem in Systems Biology and the focus of this paper. In particular, we are interested in optimizing the sampling time selection in order to minimize the variance of the parameter estimation error. With few sampling time points feasible, their selection is of practical importance in experimental design. Finally, the theoretical results are supported with an application.     

A parameter estimation technique for stochastic self-assembly systems and its application to human papillomavirus self-assembly

Virus capsid assembly has been a key model system for studies of complex self-assembly but it does pose some significant challenges for modeling studies. One important limitation is the difficulty of determining accurate rate parameters. The large size and rapid assembly of typical viruses make it infeasible to directly measure coat protein binding rates or deduce them from the relatively indirect experimental measures available. In this work, we develop a computational strategy to deduce coat-coat binding rate parameters for viral capsid assembly systems by fitting stochastic simulation trajectories to experimental measures of assembly progress. Our method combines quadratic response surface and quasi-gradient descent approximations to deal with the high computational cost of simulations, stochastic noise in simulation trajectories and limitations of the available experimental data. The approach is demonstrated on a light scattering trajectory for a human papillomavirus (HPV) in vitro assembly system, showing that the method can provide rate parameters that produce accurate curve fits and are in good concordance with prior analysis of the data. These fits provide an insight into potential assembly mechanisms of the in vitro system and give a basis for exploring how these mechanisms might vary between in vitro and in vivo assembly conditions.     

Natural and orthogonal interaction framework for modeling gene-environment interactions with application to lung cancer

Objectives: We aimed at extending the Natural and Orthogonal Interaction (NOIA) framework, developed for modeling gene-gene interactions in the analysis of quantitative traits, to allow for reduced genetic models, dichotomous traits, and gene-environment interactions. We evaluate the performance of the NOIA statistical models using simulated data and lung cancer data. Methods: The NOIA statistical models are developed for additive, dominant, and recessive genetic models as well as for a binary environmental exposure. Using the Kronecker product rule, a NOIA statistical model is built to model gene-environment interactions. By treating the genotypic values as the logarithm of odds, the NOIA statistical models are extended to the analysis of case-control data. Results: Our simulations showed that power for testing associations while allowing for interaction using the NOIA statistical model is much higher than using functional models for most of the scenarios we simulated. When applied to lung cancer data, much smaller p values were obtained using the NOIA statistical model for either the main effects or the SNP-smoking interactions for some of the SNPs tested. Conclusion: The NOIA statistical models are usually more powerful than the functional models in detecting main effects and interaction effects for both quantitative traits and binary traits.