Non-homogeneous Markov processes for biomedical data analysis

Some time ago, the Markov processes were introduced in biomedical sciences in order to study disease history events. Homogeneous and Non-homogeneous Markov processes are an important field of research into stochastic processes, especially when exact transition times are unknown and interval-censored observations are present in the analysis. Non-homogeneous Markov process should be used when the homogeneous assumption is too strong. However these sorts of models increase the complexity of the analysis and standard software is limited. In this paper, some methods for fitting non-homogeneous Markov models are reviewed and an algorithm is proposed for biomedical data analysis. The method has been applied to analyse breast cancer data. Specific software for this purpose has been implemented.     

Spatial pattern analysis in forest dynamics: deviation from power law and direction of regeneration waves

Grid-based models have been used to understand spatial heterogeneity of the vegetation height in forests and to analyze spatio-temporal dynamics of the forest regeneration process. In this report, we present two methods of identifying lattice models when spatio-temporal data are given. The first method detects directionality of regeneration waves based on the timing of local disturbance events. The second evaluates the forest pattern by recording the fraction of high and low vegetation areas at multiple spatial scales. We illustrate these methods by applying them to patterns generated using three simple stochastic lattice models: (1) two-state model, distinguishing sites with high and low vegetation, (2) three-state model, in which sites can be in an additional disturbed state, and (3) Shimagare model, which considers a continuous range of states. The combination of the two methods provides efficient means of distinguishing the models. The first method has a more direct ecological meaning, while the second is useful when forest data are limited in time.     

Minimizing model fitting objectives that contain spurious local minima by bootstrap restarting

Objective functions that arise when fitting nonlinear models often contain local minima that are of little significance except for their propensity to trap minimization algorithms. The standard methods for attempting to deal with this problem treat the objective function as fixed and employ stochastic minimization approaches in the hope of randomly jumping out of local minima. This article suggests a simple trick for performing such minimizations that can be employed in conjunction with most conventional nonstochastic fitting methods. The trick is to stochastically perturb the objective function by bootstrapping the data to be fit. Each bootstrap objective shares the large-scale structure of the original objective but has different small-scale structure. Minimizations of bootstrap objective functions are alternated with minimizations of the original objective function starting from the parameter values with which minimization of the previous bootstrap objective terminated. An example is presented, fitting a nonlinear population dynamic model to population dynamic data and including a comparison of the suggested method with simulated annealing. Convergence diagnostics are discussed.     

Parameter estimation and change-point detection from Dynamic Contrast Enhanced MRI data using stochastic differential equations

Dynamic Contrast Enhanced imaging (DCE-imaging) following a contrast agent bolus allows the extraction of information on tissue micro-vascularization. The dynamic signals obtained from DCE-imaging are modeled by pharmacokinetic compartmental models which integrate the Arterial Input Function. These models use ordinary differential equations (ODEs) to describe the exchanges between the arterial and capillary plasma and the extravascular-extracellular space. Their least squares fitting takes into account measurement noises but fails to deal with unpredictable fluctuations due to external/internal sources of variations (patients’ anxiety, time-varying parameters, measurement errors in the input function, etc.). Adding Brownian components to the ODEs leads to stochastic differential equations (SDEs). In DCE-imaging, SDEs are discretely observed with an additional measurement noise. We propose to estimate the parameters of these noisy SDEs by maximum likelihood, using the Kalman filter. In DCE-imaging, the contrast agent injected in vein arrives in plasma with an unknown time delay. The delay parameter induces a change-point in the drift of the SDE and ODE models, which is estimated also. Estimations based on the SDE and ODE pharmacokinetic models are compared to real DCE-MRI data. They show that the use of SDE provides robustness in the estimation results. A simulation study confirms these results.     

Bringing consistency to simulation of population models--Poisson simulation as a bridge between micro and macro simulation

Population models concern collections of discrete entities such as atoms, cells, humans, animals, etc., where the focus is on the number of entities in a population. Because of the complexity of such models, simulation is usually needed to reproduce their complete dynamic and stochastic behaviour. Two main types of simulation models are used for different purposes, namely micro-simulation models, where each individual is described with its particular attributes and behaviour, and macro-simulation models based on stochastic differential equations, where the population is described in aggregated terms by the number of individuals in different states. Consistency between micro- and macro-models is a crucial but often neglected aspect. This paper demonstrates how the Poisson Simulation technique can be used to produce a population macro-model consistent with the corresponding micro-model. This is accomplished by defining Poisson Simulation in strictly mathematical terms as a series of Poisson processes that generate sequences of Poisson distributions with dynamically varying parameters. The method can be applied to any population model. It provides the unique stochastic and dynamic macro-model consistent with a correct micro-model. The paper also presents a general macro form for stochastic and dynamic population models. In an appendix Poisson Simulation is compared with Markov Simulation showing a number of advantages. Especially aggregation into state variables and aggregation of many events per time-step makes Poisson Simulation orders of magnitude faster than Markov Simulation. Furthermore, you can build and execute much larger and more complicated models with Poisson Simulation than is possible with the Markov approach.     

Integrating individual movement behaviour into dispersal functions

Dispersal functions are an important tool for integrating dispersal into complex models of population and metapopulation dynamics. Most approaches in the literature are very simple, with the dispersal functions containing only one or two parameters which summarise all the effects of movement behaviour as for example different movement patterns or different perceptual abilities. The summarising nature of these parameters makes assessing the effect of one particular behavioural aspect difficult. We present a way of integrating movement behavioural parameters into a particular dispersal function in a simple way. Using a spatial individual-based simulation model for simulating different movement behaviours, we derive fitting functions for the functional relationship between the parameters of the dispersal function and several details of movement behaviour. This is done for three different movement patterns (loops, Archimedean spirals, random walk). Additionally, we provide measures which characterise the shape of the dispersal function and are interpretable in terms of landscape connectivity. This allows an ecological interpretation of the relationships found.     

Time-varying Markov regression random-effect model with Bayesian estimation procedures: Application to dynamics of functional recovery in patients with stroke

The rates of functional recovery after stroke tend to decrease with time. Time-varying Markov processes (TVMP) may be more biologically plausible than time-invariant Markov process for modeling such data. However, analysis of such stochastic processes, particularly tackling reversible transitions and the incorporation of random effects into models, can be analytically intractable. We make use of ordinary differential equations to solve continuous-time TVMP with reversible transitions. The proportional hazard form was used to assess the effects of an individual’s covariates on multi-state transitions with the incorporation of random effects that capture the residual variation after being explained by measured covariates under the concept of generalized linear model. We further built up Bayesian directed acyclic graphic model to obtain full joint posterior distribution. Markov chain Monte Carlo (MCMC) with Gibbs sampling was applied to estimate parameters based on posterior marginal distributions with multiple integrands. The proposed method was illustrated with empirical data from a study on the functional recovery after stroke.     

Bayesian inference for stochastic epidemic models with time-inhomogeneous removal rates

Stochastic compartmental models of the SEIR type are often used to make inferences on epidemic processes from partially observed data in which only removal times are available. For many epidemics, the assumption of constant removal rates is not plausible. We develop methods for models in which these rates are a time-dependent step function. A reversible jump MCMC algorithm is described that permits Bayesian inferences to be made on model parameters, particularly those associated with the step function. The method is applied to two datasets on outbreaks of smallpox and a respiratory disease. The analyses highlight the importance of allowing for time dependence by contrasting the predictive distributions for the removal times and comparing them with the observed data.      

Stochastic Approximation Boosting for Incomplete Data Problems

Summary Boosting is a powerful approach to fitting regression models. This article describes a boosting algorithm for likelihood‐based estimation with incomplete data. The algorithm combines boosting with a variant of stochastic approximation that uses Markov chain Monte Carlo to deal with the missing data. Applications to fitting generalized linear and additive models with missing covariates are given. The method is applied to the Pima Indians Diabetes Data where over half of the cases contain missing values.     

Predicting nucleic acid binding interfaces from structural models of proteins

The function of DNA‐ and RNA‐binding proteins can be inferred from the characterization and accurate prediction of their binding interfaces. However, the main pitfall of various structure‐based methods for predicting nucleic acid binding function is that they are all limited to a relatively small number of proteins for which high‐resolution three‐dimensional structures are available. In this study, we developed a pipeline for extracting functional electrostatic patches from surfaces of protein structural models, obtained using the I‐TASSER protein structure predictor. The largest positive patches are extracted from the protein surface using the patchfinder algorithm. We show that functional electrostatic patches extracted from an ensemble of structural models highly overlap the patches extracted from high‐resolution structures. Furthermore, by testing our pipeline on a set of 55 known nucleic acid binding proteins for which I‐TASSER produces high‐quality models, we show that the method accurately identifies the nucleic acids binding interface on structural models of proteins. Employing a combined patch approach we show that patches extracted from an ensemble of models better predicts the real nucleic acid binding interfaces compared with patches extracted from independent models. Overall, these results suggest that combining information from a collection of low‐resolution structural models could be a valuable approach for functional annotation. We suggest that our method will be further applicable for predicting other functional surfaces of proteins with unknown structure. Proteins 2012. © 2011 Wiley Periodicals, Inc.     

Analytical methods for predicting the behaviour of population models with general spatial interactions

Many biologists use population models that are spatial, stochastic and individual based. Analytical methods that describe the behaviour of these models approximately are attracting increasing interest as an alternative to expensive computer simulation. The methods can be employed for both prediction and fitting models to data. Recent work has extended existing (mean field) methods with the aim of accounting for the development of spatial correlations. A common feature is the use of closure approximations for truncating the set of evolution equations for summary statistics. We investigate an analytical approach for spatial and stochastic models where individuals interact according to a generic function of their distance; this extends previous methods for lattice models with interactions between close neighbours, such as the pair approximation. Our study also complements work by Bolker and Pacala (BP) [Theor. Pop. Biol. 52 (1997) 179; Am. Naturalist 153 (1999) 575]: it treats individuals as being spatially discrete (defined on a lattice) rather than as a continuous mass distribution; it tests the accuracy of different closure approximations over parameter space, including the additive moment closure (MC) used by BP and the Kirkwood approximation. The study is done in the context of an susceptible-infected-susceptible epidemic model with primary infection and with secondary infection represented by power-law interactions. MC is numerically unstable or inaccurate in parameter regions with low primary infection (or density-independent birth rates). A modified Kirkwood approximation gives stable and generally accurate transient and long-term solutions; we argue it can be applied to lattice and to continuous-space models as a substitute for MC. We derive a generalisation of the basic reproduction ratio, R(0), for spatial models.     

Stochastic models for cell motion and taxis

Certain biological experiments investigating cell motion result in time lapse video microscopy data which may be modeled using stochastic differential equations. These models suggest statistics for quantifying experimental results and testing relevant hypotheses, and carry implications for the qualitative behavior of cells and for underlying biophysical mechanisms. Directional cell motion in response to a stimulus, termed taxis, has previously been modeled at a phenomenological level using the Keller-Segel diffusion equation. The Keller-Segel model cannot distinguish certain modes of taxis, and this motivates the introduction of a richer class of models which is nevertheless still amenable to statistical analysis. A state space model formulation is used to link models proposed for cell velocity to observed data. Sequential Monte Carlo methods enable parameter estimation via maximum likelihood for a range of applicable models. One particular experimental situation, involving the effect of an electric field on cell behavior, is considered in detail. In this case, an Ornstein- Uhlenbeck model for cell velocity is found to compare favorably with a nonlinear diffusion model.     

A Stochastic Compartmental Model with Long Lasting Infusion

Most of the compartmental models in current use to model pharmacokinetic systems are deterministic. Stochastic formulations of pharmacokinetic compartmental models introduce stochasticity through either a probabilistic transfer mechanism or the randomization of the transfer rate constants. In this paper we consider a linear stochastic differential equation (LSDE) which represents a stochastic version of a one‐compartment linear model when input function undergoes random fluctuations. The solution of the LSDE, its mean value and covariance structure are derived. An explicit likelihood function is obtained either when the process is observed continuously over a period of time or when sampled data are available, as it is generally feasible. We discuss some asymptotic properties of the maximum likelihood estimators for the model parameters. Furthermore we develop expressions for two random variables of interest in pharmacokinetics: the area under the time‐concentration curve, M₀(T), and the plateau concentration, x_ss. Finally the estimation procedure is illustrated by an application to real data.     

A data-driven clustering method for time course gene expression data

Gene expression over time is, biologically, a continuous process and can thus be represented by a continuous function, i.e. a curve. Individual genes often share similar expression patterns (functional forms). However, the shape of each function, the number of such functions, and the genes that share similar functional forms are typically unknown. Here we introduce an approach that allows direct discovery of related patterns of gene expression and their underlying functions (curves) from data without a priori specification of either cluster number or functional form. Smoothing spline clustering (SSC) models natural properties of gene expression over time, taking into account natural differences in gene expression within a cluster of similarly expressed genes, the effects of experimental measurement error, and missing data. Furthermore, SSC provides a visual summary of each cluster's gene expression function and goodness-of-fit by way of a 'mean curve' construct and its associated confidence bands. We apply this method to gene expression data over the life-cycle of Drosophila melanogaster and Caenorhabditis elegans to discover 17 and 16 unique patterns of gene expression in each species, respectively. New and previously described expression patterns in both species are discovered, the majority of which are biologically meaningful and exhibit statistically significant gene function enrichment. Software and source code implementing the algorithm, SSClust, is freely available (http://genemerge.bioteam.net/SSClust.html).     

Identifying Predator–Prey Processes from Time-Series

The functional response is a key element in predator–prey models as well as in food chains and food webs. Classical models consider it as a function of prey abundance only. However, many mechanisms can lead to predator dependence, and there is increasing evidence for the importance of this dependence. Identification of the mathematical form of the functional response from real data is therefore a challenging task. In this paper we apply model-fitting to test if typical ecological predator–prey time series data, which contain both observation error and process error, can give some information about the form of the functional response. Working with artificial data (for which the functional response is known) we will show that with moderate noise levels, identification of the model that generated the data is possible. However, the noise levels prevailing in real ecological time-series can give rise to wrong identifications. We will also discuss the quality of parameter estimation by fitting differential equations to such time-series.     

Functional bioinformatics for Arabidopsis thaliana

MOTIVATION: The genome of Arabidopsis thaliana, which has the best understood plant genome, still has approximately one-third of its genes with no functional annotation at all from either MIPS or TAIR. We have applied our Data Mining Prediction (DMP) method to the problem of predicting the functional classes of these protein sequences. This method is based on using a hybrid machine-learning/data-mining method to identify patterns in the bioinformatic data about sequences that are predictive of function. We use data about sequence, predicted secondary structure, predicted structural domain, InterPro patterns, sequence similarity profile and expressions data. RESULTS: We predicted the functional class of a high percentage of the Arabidopsis genes with currently unknown function. These predictions are interpretable and have good test accuracies. We describe in detail seven of the rules produced.     

Smoothing spline ANOVA frailty model for recurrent event data

Gap time hazard estimation is of particular interest in recurrent event data. This article proposes a fully nonparametric approach for estimating the gap time hazard. Smoothing spline analysis of variance (ANOVA) decompositions are used to model the log gap time hazard as a joint function of gap time and covariates, and general frailty is introduced to account for between-subject heterogeneity and within-subject correlation. We estimate the nonparametric gap time hazard function and parameters in the frailty distribution using a combination of the Newton-Raphson procedure, the stochastic approximation algorithm (SAA), and the Markov chain Monte Carlo (MCMC) method. The convergence of the algorithm is guaranteed by decreasing the step size of parameter update and/or increasing the MCMC sample size along iterations. Model selection procedure is also developed to identify negligible components in a functional ANOVA decomposition of the log gap time hazard. We evaluate the proposed methods with simulation studies and illustrate its use through the analysis of bladder tumor data.     

Varying coefficient model with unknown within-subject covariance for analysis of tumor growth curves

SUMMARY: In this article we develop a nonparametric estimation procedure for the varying coefficient model when the within-subject covariance is unknown. Extending the idea of iterative reweighted least squares to the functional setting, we iterate between estimating the coefficients conditional on the covariance and estimating the functional covariance conditional on the coefficients. Smoothing splines for correlated errors are used to estimate the functional coefficients with smoothing parameters selected via the generalized maximum likelihood. The covariance is nonparametrically estimated using a penalized estimator with smoothing parameters chosen via a Kullback-Leibler criterion. Empirical properties of the proposed method are demonstrated in simulations and the method is applied to the data collected from an ovarian tumor study in mice to analyze the effects of different chemotherapy treatments on the volumes of two classes of tumors.     

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.