Probabilistic methods for addressing uncertainty and variability in biological models: application to a toxicokinetic model

Population variability and uncertainty are important features of biological systems that must be considered when developing mathematical models for these systems. In this paper we present probability-based parameter estimation methods that account for such variability and uncertainty. Theoretical results that establish well-posedness and stability for these methods are discussed. A probabilistic parameter estimation technique is then applied to a toxicokinetic model for trichloroethylene using several types of simulated data. Comparison with results obtained using a standard, deterministic parameter estimation method suggests that the probabilistic methods are better able to capture population variability and uncertainty in model parameters.     

Importance of dose‐response model form in probabilistic risk assessment: A case study of health effects from methylmercury in fish

We compared the effect of uncertainty in dose‐response model form on health risk estimates to the effect of uncertainty and variability in exposure. We used three different dose‐response models to characterize neurological effects in children exposed in utero to methylmercury, and applied these models to calculate risks to a native population exposed to potentially contaminated fish from a reservoir in British Columbia. Uncertainty in model form was explicitly incorporated into the risk estimates. The selection of dose‐response model strongly influenced both mean risk estimates and distributions of risk, and had a much greater impact than altering exposure distributions. We conclude that incorporating uncertainty in dose‐response model form is at least as important as accounting for variability and uncertainty in exposure parameters in probabilistic risk assessment.     

Characterization and Simulation of Uncertain Frequency Distributions: Effects of Distribution Choice,Variability, Uncertainty,and Parameter Dependence

We use bootstrap simulation to characterize uncertainty in parametric distributions, including Normal, Lognormal, Gamma, Weibull, and Beta, commonly used to represent variability in probabilistic assessments. Bootstrap simulation enables one to estimate sampling distributions for sample statistics, such as distribution parameters, even when analytical solutions are not available. Using a two-dimensional framework for both uncertainty and variability, uncertainties in cumulative distribution functions were simulated. The mathematical properties of uncertain frequency distributions were evaluated in a series of case studies during which the parameters of each type of distribution were varied for sample sizes of 5, 10, and 20. For positively skewed distributions such as Lognormal, Weibull, and Gamma, the range of uncertainty is widest at the upper tail of the distribution. For symmetric unbounded distributions, such as Normal, the uncertainties are widest at both tails of the distribution. For bounded distributions, such as Beta, the uncertainties are typically widest in the central portions of the distribution. Bootstrap simulation enables complex dependencies between sampling distributions to be captured. The effects of uncertainty, variability, and parameter dependencies were studied for several generic functional forms of models, including models in which two-dimensional random variables are added, multiplied, and divided, to show the sensitivity of model results to different assumptions regarding model input distributions, ranges of variability, and ranges of uncertainty and to show the types of errors that may be obtained from mis-specification of parameter dependence. A total of 1,098 case studies were simulated. In some cases, counter-intuitive results were obtained. For example, the point value of the 95th percentile of uncertainty for the 95th percentile of variability of the product of four Gamma or Weibull distributions decreases as the coefficient of variation of each model input increases and, therefore, may not provide a conservative estimate. Failure to properly characterize parameter uncertainties and their dependencies can lead to orders-of-magnitude mis-estimates of both variability and uncertainty. In many cases, the numerical stability of two-dimensional simulation results was found to decrease as the coefficient of variation of the inputs increases. We discuss the strengths and limitations of bootstrap simulation as a method for quantifying uncertainty due to random sampling error.     

On parameter estimation for biaxial mechanical behavior of arteries

This article considers the parameter estimation of multi-fiber family models for biaxial mechanical behavior of passive arteries in the presence of the measurement errors. First, the uncertainty propagation due to the errors in variables has been carefully characterized using the constitutive model. Then, the parameter estimation of the artery model has been formulated into nonlinear least squares optimization with an appropriately chosen weight from the uncertainty model. The proposed technique is evaluated using multiple sets of synthesized data with fictitious measurement noises. The results of the estimation are compared with those of the conventional nonlinear least squares optimization without a proper weight factor. The proposed method significantly improves the quality of parameter estimation as the amplitude of the errors in variables becomes larger. We also investigate model selection criteria to decide the optimal number of fiber families in the multi-fiber family model with respect to the experimental data balancing between variance and bias errors.     

Where will species go? Incorporating new advances in climate modelling into projections of species distributions

Bioclimatic models are the primary tools for simulating the impact of climate change on species distributions. Part of the uncertainty in the output of these models results from uncertainty in projections of future climates. To account for this, studies often simulate species responses to climates predicted by more than one climate model and/or emission scenario. One area of uncertainty, however, has remained unexplored: internal climate model variability. By running a single climate model multiple times, but each time perturbing the initial state of the model slightly, different but equally valid realizations of climate will be produced. In this paper, we identify how ongoing improvements in climate models can be used to provide guidance for impacts studies. In doing so we provide the first assessment of the extent to which this internal climate model variability generates uncertainty in projections of future species distributions, compared with variability between climate models. We obtained data on 13 realizations from three climate models (three from CSIRO Mark2 v3.0, four from GISS AOM, and six from MIROC v3.2) for two time periods: current (1985–1995) and future (2025–2035). Initially, we compared the simulated values for each climate variable (P, T_max, T_min, and T_mean) for the current period to observed climate data. This showed that climates simulated by realizations from the same climate model were more similar to each other than to realizations from other models. However, when projected into the future, these realizations followed different trajectories and the values of climate variables differed considerably within and among climate models. These had pronounced effects on the projected distributions of nine Australian butterfly species when modelled using the BIOCLIM component of DIVA-GIS. Our results show that internal climate model variability can lead to substantial differences in the extent to which the future distributions of species are projected to change. These can be greater than differences resulting from between-climate model variability. Further, different conclusions regarding the vulnerability of species to climate change can be reached due to internal model variability. Clearly, several climate models, each represented by multiple realizations, are required if we are to adequately capture the range of uncertainty associated with projecting species distributions in the future.     

A PDMP model of the epithelial cell turn-over in the intestinal crypt including microbiota-derived regulations

Mechanistic models are a powerful tool to gain insights into biological processes. The parameters of such models, e.g. kinetic rate constants, usually cannot be measured directly but need to be inferred from experimental data. In this article, we study dynamical models of the translation kinetics after mRNA transfection and analyze their parameter identifiability. That is, whether parameters can be uniquely determined from perfect or realistic data in theory and practice. Previous studies have considered ordinary differential equation (ODE) models of the process, and here we formulate a stochastic differential equation (SDE) model. For both model types, we consider structural identifiability based on the model equations and practical identifiability based on simulated as well as experimental data and find that the SDE model provides better parameter identifiability than the ODE model. Moreover, our analysis shows that even for those parameters of the ODE model that are considered to be identifiable, the obtained estimates are sometimes unreliable. Overall, our study clearly demonstrates the relevance of considering different modeling approaches and that stochastic models can provide more reliable and informative results.

     

Implications of Uncertainty and Variability in the Life Cycle Assessment of Pig Production Systems(7 pp)

Goal, Scope and Background Calculating LCA outcomes implies the use of parameters, models, choices and scenarios which introduce uncertainty, as they imperfectly account for the variability of both human and environmental systems. The analysis of the uncertainty of LCA results, and its reduction by an improved estimation of key parameters and through the improvement of the models used to convert emissions into regional impacts, such as eutrophication, are major issues for LCA. Methods In a case study of pig production systems, we propose a simple quantification of the uncertainty of LCA results (intra-system variability) and we explore the inter-system variability to produce more robust LCA outcomes. The quantification of the intra-system uncertainty takes into account the variability of the technical performance (crop yield, feed efficiency) and of emission factors (for NH3, N2O and NO3) and the influence of the functional unit (FU) (kg of pig versus hectare used). For farming systems, the inter-system variability is investigated through differentiating the production mode (conventional, quality label, organic (OA)), and the farmer practices (Good Agricultural Practice (GAP) versus Over Fertilised (OF)), while for natural systems, variability due to physical and climatic characteristics of catchments expected to modify nitrate fate is explored. Results and Conclusion For the eutrophication and climate change impact categories, the uncertainty associated with field emissions contributes more to the overall uncertainty than the uncertainty associated with emissions from livestock buildings, with crop yield and with feed efficiency. For acidification, the uncertainty of emissions from livestock buildings is the single most important contributor to the overall uncertainty. The influence of the FU on eutrophication results is very important when comparing systems with different degrees of intensification such as GAP and OA. Concerning the inter-system variability, differences in farmer practices have a larger effect on eutrophication than differences between production modes. Finally, the physical characteristics of the catchment and the climate strongly affect the results for eutrophication. In conclusion, in this case study, the main sources of uncertainty are in the estimation of emission factors, due both to the variability of environmental conditions and to lack of knowledge (emissions of N2O at the field level), but also in the model used for assessing regional impacts such as eutrophication. Recommendation and Perspective Suitable deterministic simulation models integrating the main controlling variables (environmental conditions, farmer practices, technology used) should be used to predict the emissions of a given system as well as their probabilistic distribution allowing the use of stochastic modelling. Finally, our simulations on eutrophication illustrate the necessity of integrating the fate of pollutants in models of impact assessment and highlight the important margin of improvement existing for the eutrophication impact assessment model.     

Stability,sensitivity and uncertainty rates in the flow equations of ecological models

In previous work by the authors about dynamic system modeling, basic ecosystems concepts and their application to ecological modeling theory were formalized. Measuring how a variable effects certain processes leads to improvements in dynamic systems modelling and facilitates the author’s study of system diversity in which model sensitivity is a key theme. Initially, some variables and their numeric data are used for modeling. Predictions from the constructed models depend on these data. Generic study of sensitivity aims to show to what degree model behavior is altered by modification of some specific data. If small variations cause important modifications in the model’s global behavior then the model is very sensitive in relation to the variables used. If uncertain systems are considered, then it is important to submit the system to extreme situations and analyze its behavior. In this article, some indexes of uncertainty will be defined in order to determine the variables' influence in the case of extreme changes. This will permit analysis of the system’s sensitivity in relation to several simulations.     

Dynamic activation model for a glutamatergic neurovascular unit

This article considers a dynamic spatially lumped model for brain energy metabolism and proposes to use the results of a Markov chain Monte Carlo (MCMC) based flux balance analysis to estimate the kinetic model parameters. By treating steady state reaction fluxes and transport rates as random variables we are able to propagate the uncertainty in the steady state configurations to the predictions of the dynamic model, whose responses are no longer individual but ensembles of time courses. The kinetic model consists of five compartments and is governed by kinetic mass balance equations with Michaelis-Menten type expressions for reaction rates and transports between the compartments. The neuronal activation is implemented in terms of the effect of neuronal activity on parameters controlling the blood flow and neurotransmitter transport, and a feedback mechanism coupling the glutamate concentration in the synaptic cleft and the ATP hydrolysis, thus accounting for the energetic cost of the membrane potential restoration in the postsynaptic neurons. The changes in capillary volume follow the balloon model developed for BOLD MRI. The model follows the time course of the saturation levels of the blood hemoglobin, which link metabolism and BOLD FMRI signal. Analysis of the model predictions suggest that stoichiometry alone is not enough to determine glucose partitioning between neuron and astrocyte. Lactate exchange between neuron and astrocyte is supported by the model predictions, but the uncertainty on the direction and rate is rather elevated. By and large, the model suggests that astrocyte produces and effluxes lactate, while neuron may switch from using to producing lactate. The level of ATP hydrolysis in astrocyte is substantially higher than strictly required for neurotransmitter cycling, in agreement with the literature.     

Treatments of Uncertainty and Variability in Ecological Risk Assessment of Single-Species Populations

The selection of the most appropriate model for an ecological risk assessment depends on the application, the data and resources available, the knowledge base of the assessor, the relevant endpoints, and the extent to which the model deals with uncertainty. Since ecological systems are highly variable and our knowledge of model input parameters is uncertain, it is important that models include treatments of uncertainty and variability, and that results are reported in this light. In this paper we discuss treatments of variation and uncertainty in a variety of population models. In ecological risk assessments, the risk relates to the probability of an adverse event in the context of environmental variation. Uncertainty relates to ignorance about parameter values, e.g., measurement error and systematic error. An assessment of the full distribution of risks, under variability and parameter uncertainty, will give the most comprehensive and flexible endpoint. In this paper we present the rationale behind probabilistic risk assessment, identify the sources of uncertainty relevant for risk assessment and provide an overview of a range of population models. While all of the models reviewed have some utility in ecology, some have more comprehensive treatments of uncertainty than others. We identify the models that allow probabilistic assessments and sensitivity analyses, and we offer recommendations for further developments that aim towards more comprehensive and reliable ecological risk assessments for populations.     

The dynamical consequences of developmental variability and demographic stochasticity for host-parasitoid interactions

Few age-structured models of species dynamics incorporate variability and uncertainty in population processes. Motivated by laboratory data for an insect and its parasitoid, we investigate whether such assumptions are appropriate when considering the population dynamics of a single species and its interaction with a natural enemy. Specifically, we examine the effects of developmental variability and demographic stochasticity on different types of cyclic dynamics predicted by traditional models. We show that predictions based on the deterministic fixed-development approach are differentially sensitive to variability and noise in key life stages. In particular, we find that the demonstration of half-generation cycles in the single-species model and the multigeneration cycles in the host-parasitoid model are sensitive to the introduction of developmental variability and noise, whereas generation cycles are robust to the intrinsic variability and uncertainty that may be found in nature.     

The human hepatocyte cell lines IHH and HepaRG: models to study glucose, lipid and lipoprotein metabolism

Metabolic diseases reach epidemic proportions. A better knowledge of the associated alterations in the metabolic pathways in the liver is necessary. These studies need in vitro human cell models. Several human hepatoma models are used, but the response of many metabolic pathways to physiological stimuli is often lost. Here, we characterize two human hepatocyte cell lines, IHH and HepaRG, by analysing the expression and regulation of genes involved in glucose and lipid metabolism. Our results show that the glycolysis pathway is activated by glucose and insulin in both lines. Gluconeogenesis gene expression is induced by forskolin in IHH cells and inhibited by insulin in both cell lines. The lipogenic pathway is regulated by insulin in IHH cells. Finally, both cell lines secrete apolipoprotein B-containing lipoproteins, an effect promoted by increasing glucose concentrations. These two human cell lines are thus interesting models to study the regulation of glucose and lipid metabolism.     

Are Subject-Specific Musculoskeletal Models Robust to the Uncertainties in Parameter Identification?

Subject-specific musculoskeletal modeling can be applied to study musculoskeletal disorders, allowing inclusion of personalized anatomy and properties. Independent of the tools used for model creation, there are unavoidable uncertainties associated with parameter identification, whose effect on model predictions is still not fully understood. The aim of the present study was to analyze the sensitivity of subject-specific model predictions (i.e., joint angles, joint moments, muscle and joint contact forces) during walking to the uncertainties in the identification of body landmark positions, maximum muscle tension and musculotendon geometry. To this aim, we created an MRI-based musculoskeletal model of the lower limbs, defined as a 7-segment, 10-degree-of-freedom articulated linkage, actuated by 84 musculotendon units. We then performed a Monte-Carlo probabilistic analysis perturbing model parameters according to their uncertainty, and solving a typical inverse dynamics and static optimization problem using 500 models that included the different sets of perturbed variable values. Model creation and gait simulations were performed by using freely available software that we developed to standardize the process of model creation, integrate with OpenSim and create probabilistic simulations of movement. The uncertainties in input variables had a moderate effect on model predictions, as muscle and joint contact forces showed maximum standard deviation of 0.3 times body-weight and maximum range of 2.1 times body-weight. In addition, the output variables significantly correlated with few input variables (up to 7 out of 312) across the gait cycle, including the geometry definition of larger muscles and the maximum muscle tension in limited gait portions. Although we found subject-specific models not markedly sensitive to parameter identification, researchers should be aware of the model precision in relation to the intended application. In fact, force predictions could be affected by an uncertainty in the same order of magnitude of its value, although this condition has low probability to occur.     

Flux-balance analysis of mitochondrial energy metabolism: consequences of systemic stoichiometric constraints

Mitochondrial metabolism is a critical component in the functioning and maintenance of cellular organs. The stoichiometry of biochemical reaction networks imposes constraints on mitochondrial function. A modeling framework, flux-balance analysis (FBA), was used to characterize the optimal flux distributions for maximal ATP production in the mitochondrion. The model predicted the expected ATP yields for glucose, lactate, and palmitate. Genetic defects that affect mitochondrial functions have been implicated in several human diseases. FBA can characterize the metabolic behavior due to genetic deletions at the metabolic level, and the effect of mutations in the tricarboxylic acid (TCA) cycle on mitochondrial ATP production was simulated. The mitochondrial ATP production is severely affected by TCA-cycle mutations. In addition, the model predicts the secretion of TCA-cycle intermediates, which is observed in clinical studies of mitochondriopathies such as those associated with fumarase deficiency. The model provides a systemic perspective to characterize the effect of stoichiometric constraints and specific metabolic fluxes on mitochondrial function.     

Bayesian Dynamical Systems Modelling in the Social Sciences

Data arising from social systems is often highly complex, involving non-linear relationships between the macro-level variables that characterize these systems. We present a method for analyzing this type of longitudinal or panel data using differential equations. We identify the best non-linear functions that capture interactions between variables, employing Bayes factor to decide how many interaction terms should be included in the model. This method punishes overly complicated models and identifies models with the most explanatory power. We illustrate our approach on the classic example of relating democracy and economic growth, identifying non-linear relationships between these two variables. We show how multiple variables and variable lags can be accounted for and provide a toolbox in R to implement our approach.     

Evaluation of regression models in metabolic physiology: predicting fluxes from isotopic data without knowledge of the pathway

This study explores the ability of regression models, with no knowledge of the underlying physiology, to estimate physiological parameters relevant for metabolism and endocrinology. Four regression models were compared: multiple linear regression (MLR), principal component regression (PCR), partial least-squares regression (PLS) and regression using artificial neural networks (ANN). The pathway of mammalian gluconeogenesis was analyzed using [U−¹³C]glucose as tracer. A set of data was simulated by randomly selecting physiologically appropriate metabolic fluxes for the 9 steps of this pathway as independent variables. The isotope labeling patterns of key intermediates in the pathway were then calculated for each set of fluxes, yielding 29 dependent variables. Two thousand sets were created, allowing independent training and test data. Regression models were asked to predict the nine fluxes, given only the 29 isotopomers. For large training sets (>50) the artificial neural network model was superior, capturing 95% of the variability in the gluconeogenic flux, whereas the three linear models captured only 75%. This reflects the ability of neural networks to capture the inherent non-linearities of the metabolic system. The effect of error in the variables and the addition of random variables to the data set was considered. Model sensitivities were used to find the isotopomers that most influenced the predicted flux values. These studies provide the first test of multivariate regression models for the analysis of isotopomer flux data. They provide insight for metabolomics and the future of isotopic tracers in metabolic research where the underlying physiology is complex or unknown.We acknowledge the support of NIH Grant DK58533 and the DuPont-MIT Alliance.     

An application of a mixed-effects location scale model for analysis of Ecological Momentary Assessment (EMA) data

Summary .   For longitudinal data, mixed models include random subject effects to indicate how subjects influence their responses over repeated assessments. The error variance and the variance of the random effects are usually considered to be homogeneous. These variance terms characterize the within-subjects (i.e., error variance) and between-subjects (i.e., random-effects variance) variation in the data. In studies using ecological momentary assessment (EMA), up to 30 or 40 observations are often obtained for each subject, and interest frequently centers around changes in the variances, both within and between subjects. In this article, we focus on an adolescent smoking study using EMA where interest is on characterizing changes in mood variation. We describe how covariates can influence the mood variances, and also extend the standard mixed model by adding a subject-level random effect to the within-subject variance specification. This permits subjects to have influence on the mean, or location, and variability, or (square of the) scale, of their mood responses. Additionally, we allow the location and scale random effects to be correlated. These mixed-effects location scale models have useful applications in many research areas where interest centers on the joint modeling of the mean and variance structure.     

Noise-induced transition in excitable neuron models

 We studied the influence of noisy stimulation on the Hodgkin-Huxley neuron model. Rather than examining the noise-related variability of the discharge times of the model – as has been done previously – our study focused on the effect of noise on the stationary distributions of the membrane potential and gating variables of the model. We observed that a gradual increase in the noise intensity did not result in a gradual change of the distributions. Instead, we could identify a critical intermediate noise range in which the shapes of the distributions underwent a drastic qualitative change. Namely, they moved from narrow unimodal Gaussian-like shapes associated with low noise intensities to ones that spread widely at large noise intensities. In particular, for the membrane potential and the sodium activation variable, the distributions changed from unimodal to bimodal. Thus, our investigation revealed a noise-induced transition in the Hodgkin-Huxley model. In order to further characterize this phenomenon, we considered a reduced one-dimensional model of an excitable system, namely the active rotator. For this model, our analysis indicated that the noise-induced transition is associated with a deterministic bifurcation of approximate equations governing the dynamics of the mean and variance of the state variable. Finally, we shed light on the possible functional importance of this noise-induced transition in neuronal coding by determining its effect on the spike timing precision in models of neuronal ensembles. Received: 19 September 2000 / Accepted in revised form: 4 March 2001     

Model-Based Quantification of the Systemic Interplay between Glucose and Fatty Acids in the Postprandial State

In metabolic diseases such as Type 2 Diabetes and Non-Alcoholic Fatty Liver Disease, the systemic regulation of postprandial metabolite concentrations is disturbed. To understand this dysregulation, a quantitative and temporal understanding of systemic postprandial metabolite handling is needed. Of particular interest is the intertwined regulation of glucose and non-esterified fatty acids (NEFA), due to the association between disturbed NEFA metabolism and insulin resistance. However, postprandial glucose metabolism is characterized by a dynamic interplay of simultaneously responding regulatory mechanisms, which have proven difficult to measure directly. Therefore, we propose a mathematical modelling approach to untangle the systemic interplay between glucose and NEFA in the postprandial period. The developed model integrates data of both the perturbation of glucose metabolism by NEFA as measured under clamp conditions, and postprandial time-series of glucose, insulin, and NEFA. The model can describe independent data not used for fitting, and perturbations of NEFA metabolism result in an increased insulin, but not glucose, response, demonstrating that glucose homeostasis is maintained. Finally, the model is used to show that NEFA may mediate up to 30–45% of the postprandial increase in insulin-dependent glucose uptake at two hours after a glucose meal. In conclusion, the presented model can quantify the systemic interactions of glucose and NEFA in the postprandial state, and may therefore provide a new method to evaluate the disturbance of this interplay in metabolic disease.