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PubMed2002

Modeling epistasis of quantitative trait loci using Cockerham's model.

We use the orthogonal contrast scales proposed by Cockerham to construct a genetic model, called Cockerham's model, for studying epistasis between genes. The properties of Cockerham's model in modeling and mapping epistatic genes under linkage equilibrium and disequilibrium are investigated and discussed. Because of its orthogonal property, Cockerham's model has several advantages in partitioning genetic variance into components, interpreting and estimating gene effects, and application to quantitative trait loci (QTL) mapping when compared to other models, and thus it can facilitate the study of epistasis between genes and be readily used in QTL mapping. The issues of QTL mapping with epistasis are also addressed. Real and simulated examples are used to illustrate Cockerham's model, compare different models, and map for epistatic QTL. Finally, we extend Cockerham's model to multiple loci and discuss its applications to QTL mapping.

Analysis of Variance↗

PubMed2005

Bayesian calibration of process-based forest models: bridging the gap between models and data.

Process-based forest models generally have many parameters, multiple outputs of interest and a small underlying empirical database. These characteristics hamper parameterization. Bayesian calibration offers a solution to the calibration problem because it applies to models of any type or size. It provides parameter estimates, with measures of uncertainty and correlation among the parameters. The procedure begins by quantifying the uncertainty about parameter values in the form of a prior probability distribution. Then data on the output variables are used to update the parameter distribution by means of Bayes' Theorem. This yields a posterior calibrated distribution for the parameters, which can be summarized in the form of a mean vector and variance matrix. The predictive uncertainty of the model can be quantified by running it with different parameter settings, sampled from the posterior distribution. In a further step, one may evaluate the posterior probability of the model itself (rather than that of the parameters) and compare that against the probability of other models, to aid in model selection or improvement. Bayesian calibration of process-based models cannot be performed analytically, so the posterior parameter distribution must be approximated in the form of a representative sample of parameter values. This can be achieved by means of Markov Chain Monte Carlo simulation, which is suitable for process-based models because of its simplicity and because it does not require advance knowledge of the shape of the posterior distribution. Despite the suitability of Bayesian calibration, the technique has rarely been used in forestry research. We introduce the method, using the example of a typical forest model. Further, we show that reductions in parameter uncertainty, and thus in output uncertainty, can be effected by increasing the variety of data, increasing the accuracy of measurements and increasing the length of time series.

Algorithms↗

PubMed1981

Interactions among ventilation, the circulation, and the uptake and distribution of halothane--use of a hybrid computer multiple model: I. The basic model.

The authors describe an 18-compartment hybrid computer multiple model of the uptake and distribution of halothane. This model uses 88 equations and 124 parameter settings. Three submodels are incorporated into the basic model: 1) The mass transport of halothane is simulated on the digital portion of the hybrid computer. 2) A breath-by-breath pulmonary model with two compartments describes air pressure-flow relations in the airway system. 3) A beat-to-beat cardiovascular model with 15 compartments describes in detail blood pressure-flow relations. In addition, a baroreceptor-heart rate loop is included: an increase in arterial pressure causes a decrease in heart rate. The slope of the baroreceptor response is progressively decreased by halothane until at 2 per cent there is no response. The model of halothane uptake and distribution is separate from the blood and air pressure-flow models, but is, in effect, driven by them. Myocardial "contractility" (stroke volume) and certain regional vascular resistances can be affected by the concentration of halothane in one or any proportion of any combination of three compartments: arterial blood (arteriolar concentrations), cerebral gray matter, or myocardial. In turn, these factors significantly affect the uptake and distribution of halothane. The responses to three steady-state concentration, as well as to a step change in concentration from 0 to 2 per cent, were examined. Twenty-four outputs were recorded, including halothane concentrations in ten compartments; myocardial "contractility"; left and right ventricular and right atrial pressures; cardiac output; stroke volume, R-R interval; and blood flows in six regions. Two variables--alveolar concentration of halothane and arterial blood pressure--were recorded during a step change of 0 to 5 per cent. The model describes the appropriate steady-state and dynamic cardiovascular responses to halothane. It also demonstrates the complex interrelationships among caridac output, regional blood flow distribution, and the uptake and distribution of halothane. During step change in halothane concentration, most of the responses occur early, a phenomenon also seen in man and goats. Thus, the model is useful not only for representing organ and tissue halothane concentrations, but also for gaining new insights into cardiovascular alterations produced by rapidly changing concentrations of halothane and into the complex interactions between the circulation and the uptake and distribution of halothane.

Anesthesia, Inhalation↗

PubMed1995

Testing the conservativeness of a screening model in a model validation exercise.

Although models for calculating derived emission limits have been around for many years, the opportunity to test them against independent data sets did not arise until the Biospheric Model Validation Study (BIOMOVS) was organized in 1985. Within BIOMOVS, two scenarios tested predictions of the movement of 131I and 137Cs from air to soil, pasture, milk, and beef. One of these scenarios was a model intercomparison of a chronic release over 30 y. The second scenario, using data gathered world-wide after the Chernobyl accident, allowed predictions to be compared directly with observations. The Canadian Standards Association's Guidelines for Calculating Derived Release Limits for Radioactive Material in Airborne and Liquid Effluents for Normal Operation of Nuclear Facilities was tested in both these scenarios to see whether its predictions were suitably conservative, as they should be for a screening model. A comparison was made between results of the Canadian Standards Association's model and those of other screening models on the one hand and results of models attempting to predict best estimates on the other hand. This analysis shows that often screening models are not conservative, and thus there should be much more effort to test the models against observations.

Air Pollutants, Radioactive↗

PubMed2000

More realistic models of sexually transmitted disease transmission dynamics: sexual partnership networks, pair models, and moment closure.

BACKGROUND: Mathematical models of sexually transmitted disease transmission have proven powerful tools for interpreting observed epidemiologic pattern. However, the most commonly used formulation of such models largely fail to capture the effect of partnership concurrency and contact network structure on transmission. GOAL: The development of a compartmental model of partnership formation and dissolution that includes approximations for the influence of the sexual-partner network. STUDY DESIGN: Theoretical analysis of ordinary differential equation models for sexually transmitted disease transmission within sex-partner networks. RESULTS: The approach developed advances earlier pair models, allows for the influence of concurrent sexual partnerships, and illustrates the importance of concurrency to the persistence of diseases with relatively short durations of infectiousness. The authors also illustrate that heterogeneity in risk is possible even in model populations in which all individuals follow the same behavioral rules. CONCLUSION: Deterministic extended pair models offer a powerful approach to modelling sexually transmitted disease transmission that usefully complement computationally intensive microsimulation models.

Female↗

PubMed2005

Linking population-level models with growing networks: a class of epidemic models.

We introduce a class of growing network models that are directly applicable to epidemiology. We show how to construct a growing network model (individual-level model) that generates the same epidemic-level outcomes as a population-level ordinary differential equation (ODE) model. For concreteness, we analyze the susceptible-infected (SI) ODE model of disease invasion. First, we give an illustrative example of a growing network whose population-level variables are compatible with those of this ODE model. Second, we demonstrate that a growing network model can be found that is equivalent to the Crump-Mode-Jagers (CMJ) continuous-time branching process of the SI ODE model of disease invasion. We discuss the computational advantages that our growing network model has over the CMJ branching process.

Animals↗

PubMed2003

Improving macromolecular atomic models at moderate resolution by automated iterative model building, statistical density modification and refinement.

An iterative process for improving the completeness and quality of atomic models automatically built at moderate resolution (up to about 2.8 A) is described. The process consists of cycles of model building interspersed with cycles of refinement and combining phase information from the model with experimental phase information (if any) using statistical density modification. The process can lead to substantial improvements in both the accuracy and completeness of the model compared with a single cycle of model building. For eight test cases solved by MAD or SAD at resolutions ranging from 2.0 to 2.8 A, the fraction of models built and assigned to sequence was 46-91% (mean of 65%) after the first cycle of building and refinement, and 78-95% (mean of 87%) after 20 cycles. In an additional test case, an incorrect model of gene 5 protein (PDB code 2gn5; r.m.s.d. of main-chain atoms from the more recent refined structure 1vqb at 1.56 A) was rebuilt using only structure-factor amplitude information at varying resolutions from 2.0 to 3.0 A. Rebuilding was effective at resolutions up to about 2.5 A. The resulting models had 60-80% of the residues built and an r.m.s.d. of main-chain atoms from the refined structure of 0.20 to 0.62 A. The algorithm is useful for building preliminary models of macromolecules suitable for an experienced crystallographer to extend, correct and fully refine.

Algorithms↗

PubMed1998

Effect of conductivity uncertainties and modeling errors on EEG source localization using a 2-D model.

This paper presents a sensitivity study of electroencephalography-based source localization due to errors in the head-tissue conductivities and to errors in modeling the conductivity variation inside the brain and scalp. The study is conducted using a two-dimensional (2-D) finite element model obtained from a magnetic resonance imaging (MRI) scan of a head cross section. The effect of uncertainty in the following tissues is studied: white matter, gray matter, cerebrospinal fluid (CSF), skull, and fat. The distribution of source location errors, assuming a single-dipole source model, is examined in detail for different dipole locations over the entire brain region. We also present a detailed analysis of the effect of conductivity on source localization for a four-layer cylinder model and a four-layer sphere model. These two simple models provide insight into how the effect of conductivity on boundary potential translates into source location errors, and also how errors in a 2-D model compare to errors in a three-dimensional model. Results presented in this paper clearly point to the following conclusion: unless the conductivities of the head tissues and the distribution of these tissues throughout the head are modeled accurately, the goal of achieving localization accuracy to within a few millimeters is unattainable.

Brain↗

PubMedDate not supplied

The JigCell model builder: a spreadsheet interface for creating biochemical reaction network models.

Converting a biochemical reaction network to a set of kinetic rate equations is tedious and error prone. We describe known interface paradigms for inputing models of intracellular regulatory networks: graphical layout (diagrams), wizards, scripting languages, and direct entry of chemical equations. We present the JigCell Model Builder, which allows users to define models as a set of reaction equations using a spreadsheet (an example of direct entry of equations) and outputs model definitions in the Systems Biology Markup Language, Level 2. We present the results of two usability studies. The spreadsheet paradigm demonstrated its effectiveness in reducing the number of errors made by modelers when compared to hand conversion of a wiring diagram to differential equations. A comparison of representatives of the four interface paradigms for a simple model of the cell cycle was conducted which measured time, mouse clicks, and keystrokes to enter the model, and the number of screens needed to view the contents of the model. All four paradigms had similar data entry times. The spreadsheet and scripting language approaches require significantly fewer screens to view the models than do the wizard or graphical layout approaches.

Algorithms↗

PubMed2005

Interval fuzzy modeling applied to Wiener models with uncertainties.

This correspondence addresses the problem of interval fuzzy model identification and its use in the case of the robust Wiener model. The method combines a fuzzy identification methodology with some ideas from linear programming theory. On a finite set of measured data, an optimality criterion which minimizes the maximum estimation error between the data and the proposed fuzzy model output is used. The min-max optimization problem can then be seen as a linear programming problem that is solved to estimate the parameters of the fuzzy model in each fuzzy domain. This results in lower and upper fuzzy models that define the confidence interval of the observed data. The model is called the interval fuzzy model and is used to approximate the static nonlinearity in the case of the Wiener model with uncertainties. The resulting model has the potential to be used in the areas of robust control and fault detection.

Algorithms↗

PubMed1999

Relating the classical covariance adjustment techniques of multivariate growth curve models to modern univariate mixed effects models.

The relationship between the modern univariate mixed model for analyzing longitudinal data, popularized by Laird and Ware (1982, Biometrics 38, 963-974), and its predecessor, the classical multivariate growth curve model, summarized by Grizzle and Allen (1969, Biometrics 25, 357-381), has never been clearly established. Here, the link between the two methodologies is derived, and balanced polynomial and cosinor examples cited in the literature are analyzed with both approaches. Relating the two models demonstrates that classical covariance adjustment for higher-order terms is analogous to including them as random effects in the mixed model. The polynomial example clearly illustrates the relationship between the methodologies and shows their equivalence when all matrices are properly defined. The cosinor example demonstrates how results from each method may differ when the total variance-covariance matrix is positive definite, but that the between-subjects component of that matrix is not so constrained by the growth curve approach. Additionally, advocates of each approach tend to consider different covariance structures. Modern mixed model analysts consider only those terms in a model's expectation (or linear combinations), and preferably the most parsimonious subset, as candidates for random effects. Classical growth curve analysts automatically consider all terms in a model's expectation as random effects and then investigate whether "covariance adjusting" for higher-order terms improves the model. We apply mixed model techniques to cosinor analyses of a large, unbalanced data set to demonstrate the relevance of classical covariance structures that were previously conceived for use only with completely balanced data.

Biometry↗

PubMed1999

Fitting a regression model for genotype-by-environment data on heading dates in grasses by methods for nonlinear mixed models.

The analysis of agricultural crop variety trials is usually complicated by the presence of genotype-by-environment interaction. A number of methods and models have been proposed to tackle this problem. One of the most common methods is the regression approach due to Yates and Cochran (1938, Journal of Agricultural Science 28, 556-580), in which performances of genotypes in the environments are regressed onto environmental means. The underlying regression model contains a multiplicative term with two unknown parameters (one for genotypes and one for environments). In the present paper, the model is modified by exchanging the role of genotypes and environments. Various diagnostic plots show that this modified model is adequate for a data set on heading dates in the grass species Dactylis glomerata. If environments are considered as a random factor while genotypes are taken as fixed, the model falls into the class of nonlinear mixed models. Recently, a number of procedures have been suggested for this class of models, which are based on first-order Taylor series expansion. Alternatively, the model can be estimated by maximum likelihood. This paper discusses the application of these methods for estimating parameters of the model.

Agriculture↗

PubMed2006

Physiologically-based pharmacokinetic modeling of genistein in rats, Part I: Model development.

Genistein is a phytoestrogen-a plant-derived compound that binds to and activates the estrogen receptor-occurring at high levels in soy beans and food products, leading to widespread human exposure. The numerous scientific publications available describing genistein's dosimetry, mechanisms of action, and identified or putative health effects in both experimental animals and humans make it ideal for examination as an example of endocrine-active compound (EAC). We developed a physiologically-based pharmacokinetic (PBPK) model to quantify the internal, target-tissue dosimetry of genistein in adult rats. Complexities of the model include enterohepatic circulation, binding of both genistein and its conjugates to plasma proteins, and the multiple compartments used to describe transport through the bile duct and gastrointestinal tract. Other aspects of the model are simple perfusion-limited transport to the tissue groups and first-order rates of metabolism, uptake, and excretion. We describe here the model structure and initial calibration of the model by fitting to a large data set for Wistar rats. The model structure can be readily extrapolated to describe genistein dosimetry in humans or modified to describe the dosimetry of other phytoestrogens and phenolic EACs. The model does a fair job of capturing the pharmacokinetics. Although it does not describe the interindividual variability and we have not identified a single set of parameters that provide a good fit to the data for both oral and intravenous exposures, we believe it provides a good initial attempt at PBPK modeling for genistein, which can serve as a template for other phytoestrogens and in the design of future experiments and research that can be used to fill data gaps and better estimate model parameters.

Animals↗

PubMed1998

A model for binaural response properties of inferior colliculus neurons. I. A model with interaural time difference-sensitive excitatory and inhibitory inputs.

A model was developed that simulates the binaural response properties of low-frequency inferior colliculus (IC) neurons in response to several types of stimuli. The model incorporates existing models for auditory-nerve fibers, bushy cells in the cochlear nucleus, and cells in medial superior olive (MSO). The IC model neuron receives two inputs, one excitatory from an ipsilateral MSO model cell and one inhibitory from a contralateral MSO model cell. The membrane potential of the IC model neuron (and the other model neurons) is described by Hodgkin-Huxley type equations. Responses of IC neurons are simulated for pure-tone stimuli, binaural beat stimuli, interaural phase-modulated tones, single binaural clicks, and pairs of binaural clicks. The simulation results show most of the observed properties of IC discharge patterns, including the bimodal and unimodal interaural time difference (ITD) functions, sensitivities to direction and rate of change of ITD, ITD-dependent echo suppression, and early and late inhibitions in response to clicks. This study demonstrates that these response properties can be generated by a simple model incorporating ITD-dependent excitation and inhibition from binaural neurons.

Acoustic Stimulation↗

PubMed1998

A model for binaural response properties of inferior colliculus neurons. II. A model with interaural time difference-sensitive excitatory and inhibitory inputs and an adaptation mechanism.

The inferior colliculus (IC) model of Cai et al. [J. Acoust. Soc. Am. 103, 475-493 (1998)] simulated the binaural response properties of low-frequency IC neurons in response to various acoustic stimuli. This model, however, failed to simulate the sensitivities of IC neurons to dynamically changing temporal features, such as the sharpened dynamic interaural phase difference (IPD) functions. In this paper, the Cai et al. (1998) model is modified such that an adaptation mechanism, viz., an additional channel simulating a calcium-activated, voltage-independent potassium channel which is responsible for afterhyperpolarization, is incorporated in the IC membrane model. Simulations were repeated with this modified model, including the responses to pure tones, binaural beat stimuli, interaural phase-modulated stimuli, binaural clicks, and pairs of binaural clicks. The discharge patterns of the model in response to current injection were also studied and compared with physiological data. It was demonstrated that this model showed all the properties that were simulated by the Cai et al. (1998) model. In addition, it showed some properties that were not simulated by that model, such as the sharpened dynamic IPD functions and adapting discharge patterns in response to current injection.

Acoustic Stimulation↗

PubMed2005

Modeling error and stability of endothelial cytoskeletal membrane parameters based on modeling transendothelial impedance as resistor and capacitor in series.

Transendothelial impedance across an endothelial monolayer grown on a microelectrode has previously been modeled as a repeating pattern of disks in which the electrical circuit consists of a resistor and capacitor in series. Although this numerical model breaks down barrier function into measurements of cell-cell adhesion, cell-matrix adhesion, and membrane capacitance, such solution parameters can be inaccurate without understanding model stability and error. In this study, we have evaluated modeling stability and error by using a chi(2) evaluation and Levenberg-Marquardt nonlinear least-squares (LM-NLS) method of the real and/or imaginary data in which the experimental measurement is compared with the calculated measurement derived by the model. Modeling stability and error were dependent on current frequency and the type of experimental data modeled. Solution parameters of cell-matrix adhesion were most susceptible to modeling instability. Furthermore, the LM-NLS method displayed frequency-dependent instability of the solution parameters, regardless of whether the real or imaginary data were analyzed. However, the LM-NLS method identified stable and reproducible solution parameters between all types of experimental data when a defined frequency spectrum of the entire data set was selected on the basis of a criterion of minimizing error. The frequency bandwidth that produced stable solution parameters varied greatly among different data types. Thus a numerical model based on characterizing transendothelial impedance as a resistor and capacitor in series and as a repeating pattern of disks is not sufficient to characterize the entire frequency spectrum of experimental transendothelial impedance.

Cells, Cultured↗

PubMed2004

Predictive pharmacokinetic-pharmacodynamic modeling of tumor growth kinetics in xenograft models after administration of anticancer agents.

The available mathematical models describing tumor growth and the effect of anticancer treatments on tumors in animals are of limited use within the drug industry. A simple and effective model would allow applying quantitative thinking to the preclinical development of oncology drugs. In this article, a minimal pharmacokinetic-pharmacodynamic model is presented, based on a system of ordinary differential equations that link the dosing regimen of a compound to the tumor growth in animal models. The growth of tumors in nontreated animals is described by an exponential growth followed by a linear growth. In treated animals, the tumor growth rate is decreased by a factor proportional to both drug concentration and number of proliferating tumor cells. A transit compartmental system is used to model the process of cell death, which occurs at later times. The parameters of the pharmacodynamic model are related to the growth characteristics of the tumor, to the drug potency, and to the kinetics of the tumor cell death. Therefore, such parameters can be used for ranking compounds based on their potency and for evaluating potential differences in the tumor cell death process. The model was extensively tested on discovery candidates and known anticancer drugs. It fitted well the experimental data, providing reliable parameter estimates. On the basis of the parameters estimated in a first experiment, the model successfully predicted the response of tumors exposed to drugs given at different dose levels and/or schedules. It is, thus, possible to use the model prospectively, optimizing the design of new experiments.

Antineoplastic Agents↗

PubMedDate not supplied

Comparison of emission models with computational fluid dynamic simulation and a proposed improved model.

Understanding source behavior is important in controlling exposure to airborne contaminants. Industrial hygienists are often asked to infer emission information from room concentration data. This is not easily done, but models that make simplifying assumptions regarding contaminant transport are frequently used. The errors resulting from these assumptions are not yet well understood. This study compares emission estimates from the single-zone completely mixed (CM-1), two-zone completely mixed (CM-2), and uniform diffusivity (UD) models with the emissions set as boundary conditions in computational fluid dynamic (CFD) simulations of a workplace. The room airflow and concentration fields were computed using Fluent 4. These numerical experiments were factorial combinations of three source locations, five receptor locations, three dilution airflow rates, and two generation rate profiles, constant and time-varying. The aim was to compute plausible concentration fields, not to simulate exactly the processes in a real workroom. Thus, error is defined here as the difference between model and CFD predictions. For the steady-state case the UD model had the lowest error. When the source near-field contained the breathing zone receptor, the CM-2 model was applied. Then, in decreasing agreement with CFD were UD, CM-2, and CM-1. Averaging over all source and receptor locations (CM-2 applied for only one), in decreasing order of agreement with CFD were UD, CM-1, and CM-2. Source and receptor location had large effects on emission estimates using the CM-1 model and some effect using the UD model. A location-specific mixing factor (location factor) derived from steady-state concentration gradients was used to build a more accurate time-dependent emission model, CM-L. Total mass emitted from a time-varying source was modeled most accurately by CM-L, followed by CM-1 and CM-2.

Air Movements↗