Application of global sensitivity analysis to determine goals for design of experiments: an example study on antibody-producing cell cultures

Global sensitivity analysis (GSA) can be used to quantify the importance of model parameters and their interactions with respect to model output. In this study, the Sobol' method for GSA is applied to a dynamic model of monoclonal antibody-producing mammalian cell cultures in order to identify the parameters that need to be accurately determined experimentally. Our results show that most parameters have low sensitivity indices and exhibit strong interactions with one another. These parameters can be set at their nominal values and unnecessary experimentation can therefore be avoided. In contrast, certain parameters are identified as sensitive, necessitating their estimation given sufficiently rich experimental data. Moreover, parameter sensitivity varies during culture time in a biologically meaningful manner. In conclusion, GSA can serve as an excellent precursor to optimal experiment design.     

Parameter estimation of a monod-type model. Part I: Theoretical identifiability and sensitivity analysis

Summary Parameter estimation of a Monod-type model based on the study of the theoretical identifiability of the model followed by the sensitivity analysis of the state variables with respect to parameters is presented. Theorerical identifiability allows to establish the unicity of the solution. On the other hand, sensitivity analysis throws light on the conditions that make parameters identifiable. Thus, the introduction of additional parameters, especially substrate maintenance and death constant, increases the estimation difficulty.     

Sensitivity analysis and parameter estimation in a model of anaerobic waste water treatment processes with substrate inhibition

Anaerobic waste water treatment processes are commonly presented by the fifth order Hill and Barth non-linear model, describing three main stages of anaerobic digestion. The model investigated in the present work is a modified version of the Hill and Barth model, which includes substrate inhibition of growth of methanogenic bacteria. Parameter estimation of this model is a difficult problem because of the high number of parameters to be estimated and the rather restricted information concerning the variables. The aim of the present work is to use sensitivity theory to find a method for selecting the most significant parameters. In particular, we develop a sensitivity model for anaerobic digestion using relative sensitivity functions. Simulation results from the sensitivity model show that the number of parameters to be estimated can be reduced from 12 to 4. We suggest a 2-step procedure for parameter estimation, which is also based on sensitivity analysis. This procedure gives results, which allow for off-line determination of parameter values if the experimental data for biogas production rate are known.     

Estimation of kinetic parameters in a structured yeast model using regularisation.

In this work, a procedure for estimating kinetic parameters in biochemically structured models was developed. The approach is applicable when the structure of a kinetic model has been set up and the kinetic parameters should be estimated. The procedure consists of five steps. First, initial values were found in or calculated from literature. Hereafter using sensitivity analysis the most sensitive parameters were identified. In the third step physiological knowledge was combined with the parameter sensitivities to manually tune the most sensitive parameters. In step four, a global optimisation routine was applied for simultaneous estimation of the most sensitive parameters identified during the sensitivity analysis. Regularisation was included in the simultaneous estimation to reduce the effect of insensitive parameters. Finally, confidence intervals for the estimated parameters were calculated. This parameter estimation approach was demonstrated on a biochemically structured yeast model containing 11 reactions and 37 kinetic constants as a case study.     

Parameter estimation and sensitivity analysis of a nonlinearly elastic static lung model

A model for the static pressure-volume behavior of the lung parenchyma based on a pseudo-elastic strain energy function was tested. Values of the model parameters and their variances were estimated by an optimal least-squares fit of the model-predicted pressures to the corresponding data from excised, saline-filled dog lungs. Although the model fit data from twelve lungs very well, the coefficients of variation for parameter values differed greatly. To analyze the sensitivity of the model output to its parameters, we examined an approximate Hessian, H, of the least-squares objective function. Based on the determinant and condition number of H, we were able to set formal criteria for choosing the most reliable estimates of parameter values and their variances. This in turn allowed us to specify a normal range of parameter values for these dog lungs. Thus the model not only describes static pressure-volume data, but also uses the data to estimate parameters from a fundamental constitutive equation. The optimal parameter estimation and sensitivity analysis developed here can be widely applied to other physiologic systems.     

Insights into the behaviour of systems biology models from dynamic sensitivity and identifiability analysis: a case study of an NF-kappaB signalling pathway

Mathematical modelling offers a variety of useful techniques to help in understanding the intrinsic behaviour of complex signal transduction networks. From the system engineering point of view, the dynamics of metabolic and signal transduction models can always be described by nonlinear ordinary differential equations (ODEs) following mass balance principles. Based on the state-space formulation, many methods from the area of automatic control can conveniently be applied to the modelling, analysis and design of cell networks. In the present study, dynamic sensitivity analysis is performed on a model of the IkappaB-NF-kappaB signal pathway system. Univariate analysis of the Euclidean-form overall sensitivities shows that only 8 out of the 64 parameters in the model have major influence on the nuclear NF-kappaB oscillations. The sensitivity matrix is then used to address correlation analysis, identifiability assessment and measurement set selection within the framework of least squares estimation and multivariate analysis. It is shown that certain pairs of parameters are exactly or highly correlated to each other in terms of their effects on the measured variables. The experimental design strategy provides guidance on which proteins should best be considered for measurement such that the unknown parameters can be estimated with the best statistical precision. The whole analysis scheme we describe provides efficient parameter estimation techniques for complex cell networks.     

Identification in an anaerobic batch system: global sensitivity analysis, multi-start strategy and optimization criterion selection

Several mathematical models have been developed in anaerobic digestion systems and a variety of methods have been used for parameter estimation and model validation. However, structural and parametric identifiability questions are relatively seldom addressed in the reported AD modeling studies. This paper presents a 3-step procedure for the reliable estimation of a set of kinetic and stoichiometric parameters in a simplified model of the anaerobic digestion process. This procedure includes the application of global sensitivity analysis, which allows to evaluate the interaction among the identified parameters, multi-start strategy that gives a picture of the possible local minima and the selection of optimization criteria or cost functions. This procedure is applied to the experimental data collected from a lab-scale sequencing batch reactor. Two kinetic parameters and two stoichiometric coefficients are estimated and their accuracy was also determined. The classical least-squares cost function appears to be the best choice in this case study.     

基于小波分析的大豆叶绿素a含量高光谱反演模型

 2003和2004年分别在长春市良种场和中国科学院海伦黑土生态实验站实测了大田耕作与水肥耦合作用下大豆（Glycine max）冠层高光谱反射率 与叶绿素a含量数据,对光谱反射率、微分光谱与叶绿素a含量进行了相关分析;采用归一化植被指数(Normalized diffe rence vegetation index, NDVI)、土壤调和植被指数(Soil-adjusted vegetation index, SAVI)、再归一植被指数(Renormalized difference vegetation index, RDVI)、第二修正比值植被指数(Modified second ratio index, MSRI)等建立了大豆叶绿素a反演模型;应用小波分析对采集的光谱反 射率数据进行了能量系数提取,并以小波能量系数作为自变量进行了单变量与多变量回归分析,对大豆叶绿素a进行了估算。研究结果表明,大 豆叶绿素a 与可见光光谱反射率相关性较好,并在红光波段取得最大值(R²>0.70),但在红边处,微分光谱与大豆叶绿素a的相关性较反射率好 得多,在其它波段则相反;由NDVI、SAVI、RDVI、MSRI等植被指数建立的估算模型可以提高大豆叶绿素a的估算精度（R²>0.75）;小波能量系 数回归模型可以进一步提高大豆叶绿素a含量的估算水平,以一个特定小波能量系数作为自变量的回归模型,大豆叶绿素a回归决定系数R²高达 0.78;多变量回归分析结果表明,大豆叶绿素a实测值与预测值的线性回归决定系数R²均高达0.85。以上结果表明, 小波分析可以对高光谱进 行特征变量提取,并可在一定程度上提高大豆生理参数反演精度。     

Parameter estimation of a monod-type model. Part II: Use of a neural network with associative memories

Summary On the basis of a previous study related with parameter identifiability and sensitivity analysis of a Monod-type model, a parameter estimation method based on Artificial Neural Networks (ANN) with Associative Memories (AMs) is presented. A combination of an iterative procedure and a convergence index given by AMs allows to confirm the nature of relations existing between state variables and parameters which were found in the first part of the study. The convergence criterion is particularly well adapted to showing various influences of state variables on parameter estimation of such a model.     

Identifiability and retrievability of unique parameters describing intrinsic Andrews kinetics

A key factor contributing to the variability in the microbial kinetic parameters reported from batch assays is parameter identifiability, i.e., the ability of the mathematical routine used for parameter estimation to provide unique estimates of the individual parameter values. This work encompassed a three-part evaluation of the parameter identifiability of intrinsic kinetic parameters describing the Andrews growth model that are obtained from batch assays. First, a parameter identifiability analysis was conducted by visually inspecting the sensitivity equations for the Andrews growth model. Second, the practical retrievability of the parameters in the presence of experimental error was evaluated for the parameter estimation routine used. Third, the results of these analyses were tested using an example data set from the literature for a self-inhibitory substrate. The general trends from these analyses were consistent and indicated that it is very difficult, if not impossible, to simultaneously obtain a unique set of estimates of intrinsic kinetic parameters for the Andrews growth model using data from a single batch experiment.     

Comparative study of parameter sensitivity analyses of the TCR-activated Erk-MAPK signalling pathway

Parameter estimation is a major challenge for mathematical modelling of biological systems. Given the uncertainties associated with model parameters, it is important to understand how sensitive the model output is to variations in parameter values. A local sensitivity analysis determines the model sensitivity to parameter variations over a localised region around the nominal parameter values, whereas a global sensitivity analysis (GSA) investigates the sensitivity over the entire parameter space. Using a T-cell receptor-activated Erk-MAPK signalling pathway model as an example, the authors present a comparative study of a variety of different sensitivity analysis techniques. These techniques include: local sensitivity analysis, existing GSA methods of partial rank correlation coefficient, Sobol's, extended Fourier amplitude sensitivity test, as well as a weighted average of local sensitivities and a new GSA method to extract global parameter sensitivities from a parameter identification routine. Results of this study revealed critical reactions in the signalling pathway and their impact on the signalling dynamics and provided insights into embedded regulatory mechanisms such as feedback loops in the pathway. From this study, a recommendation emerges for a general sensitivity analysis strategy to efficiently and reliably infer quantitative, dynamic as well as topological properties from systems biology models.     

Unscented Kalman filter with parameter identifiability analysis for the estimation of multiple parameters in kinetic models

In systems biology, experimentally measured parameters are not always available, necessitating the use of computationally based parameter estimation. In order to rely on estimated parameters, it is critical to first determine which parameters can be estimated for a given model and measurement set. This is done with parameter identifiability analysis. A kinetic model of the sucrose accumulation in the sugar cane culm tissue developed by Rohwer et al. was taken as a test case model. What differentiates this approach is the integration of an orthogonal-based local identifiability method into the unscented Kalman filter (UKF), rather than using the more common observability-based method which has inherent limitations. It also introduces a variable step size based on the system uncertainty of the UKF during the sensitivity calculation. This method identified 10 out of 12 parameters as identifiable. These ten parameters were estimated using the UKF, which was run 97 times. Throughout the repetitions the UKF proved to be more consistent than the estimation algorithms used for comparison.     

On construction of stochastic genetic networks based on gene expression sequences

Reconstruction of genetic regulatory networks from time series data of gene expression patterns is an important research topic in bioinformatics. Probabilistic Boolean Networks (PBNs) have been proposed as an effective model for gene regulatory networks. PBNs are able to cope with uncertainty, corporate rule-based dependencies between genes and discover the sensitivity of genes in their interactions with other genes. However, PBNs are unlikely to use directly in practice because of huge amount of computational cost for obtaining predictors and their corresponding probabilities. In this paper, we propose a multivariate Markov model for approximating PBNs and describing the dynamics of a genetic network for gene expression sequences. The main contribution of the new model is to preserve the strength of PBNs and reduce the complexity of the networks. The number of parameters of our proposed model is O(n2) where n is the number of genes involved. We also develop efficient estimation methods for solving the model parameters. Numerical examples on synthetic data sets and practical yeast data sequences are given to demonstrate the effectiveness of the proposed model.     

Sensitivity analysis for an improved estimation of respiratory mechanics parameters

In this paper a respiratory mechanics model is considered, which is characterized by a biquadratic input impedance, and a sensitivity analysis has been carried out to determine the influence of experimental conditions on parameter estimation. This analysis was effected with data obtained experimentally, in three different patients under intermittent positive pressure ventilation. In all three cases, the model's input impedance demonstrated a maximum sensitivity in relation to the various parameters included in the field of frequencies from 0 to 10 Hz. This seems to suggest therefore, that the use of a low-pass filter with a cut-off frequency equal to 10 Hz could improve the signal/noise ratio and, consequently, the accuracy of the estimation of the parameters. Furthermore, the use of a system input with a bandwidth of 0–10 Hz provides the experimental conditions, under which good estimates of the parameters can be obtained. This conclusion has also been confirmed by simulation studies which have been conducted with different types of input signals.     

Temperature tolerance and energetics: a dynamic energy budget-based comparison of North Atlantic marine species

Temperature tolerance and sensitivity were examined for some North Atlantic marine species and linked to their energetics in terms of species-specific parameters described by dynamic energy budget (DEB) theory. There was a general lack of basic information on temperature tolerance and sensitivity for many species. Available data indicated that the ranges in tolerable temperatures were positively related to optimal growth temperatures. However, no clear relationships with temperature sensitivity were established and no clear differences between pelagic and demersal species were observed. The analysis was complicated by the fact that for pelagic species, experimental data were completely absent and even for well-studied species, information was incomplete and sometimes contradictory. Nevertheless, differences in life-history strategies were clearly reflected in parameter differences between related species. Two approaches were used in the estimation of DEB parameters: one based on the assumption that reserve hardly contributes to physical volume; the other does not make this assumption, but relies on body-size scaling relationships, using parameter values of a generalized animal as pseudo-data. Temperature tolerance and sensitivity seemed to be linked with the energetics of a species. In terms of growth, relatively high temperature optima, sensitivity and/or tolerance were related to lower relative assimilation rates as well as lower maintenance costs. Making the step from limited observations to underlying mechanisms is complicated and extrapolations should be carefully interpreted. Special attention should be devoted to the estimation of parameters using body-size scaling relationships predicted by the DEB theory.     

Influence of body segment parameters and modeling assumptions on the estimate of center of mass trajectory

This study sought to determine the effect of inaccuracies in body segment parameters and modeling assumptions on the estimate of antero-posterior center of mass (COM) trajectory. Four different methods, one based on segmental kinematics, and three methods based on kinetic recordings were compared via simulation. Kinematic patterns (quiet stance, ankle-related sway, hip-ankle-related sway, sit-up and sit-up-sit-down) were tested with a 2D four-link model of the body and the ground reaction force vector was obtained by inverse dynamics. Errors in the estimation of body segment parameters were simulated by applying a +/-10% variation to one or more parameters at a time. These errors propagated differently to the COM estimated location between methods, between parameters within the same method, and between tasks. The kinematics-based method was the most sensitive to body segment parameters, with special regards to segment lengths and head-arms-trunk parameters. Root mean square error between estimated and simulated COM location reached 19mm in balance-related tasks and 38.3mm in sit-up-sit-down. The kinetics-based methods were largely less sensitive to inaccuracies in body segment parameters. In particular, the technique proposed by Zatsiorsky and King (J. Biomech. 31 (1998) 161), was completely insensitive to segment parameters. On the other hand the kinetics-based methods showed an intrinsic estimation error, due to the underlying model assumptions. The methods based on the double integration of horizontal force had better outcomes with tasks challenging such assumptions, with a maximal error in COM location of 15mm in the sit-up-sit-down. The method proposed by Shimba (J. Biomech. 17 (1984) 53) showed the best trade-off between sensitivity to body segment parameters and estimation performances given the ideal test conditions.     

Development of an in vivo method for determining material properties of passive myocardium

Calculation of mechanical stresses and strains in the left ventricular (LV) myocardium by the finite element (FE) method relies on adequate knowledge of the material properties of myocardial tissue. In this paper, we present a model-based estimation procedure to characterize the stress-strain relationship in passive LV myocardium. A 3D FE model of the LV myocardium was used, which included morphological fiber and sheet structure and a nonlinear orthotropic constitutive law with different stiffness in the fiber, sheet, and sheet-normal directions. The estimation method was based on measured wall strains. We analyzed the method's ability to estimate the material parameters by generating a set of synthetic strain data by simulating the LV inflation phase with known material parameters. In this way we were able to verify the correctness of the solution and to analyze the effects of measurement and model error on the solution accuracy and stability. A sensitivity analysis was performed to investigate the observability of the material parameters and to determine which parameters to estimate. The results showed a high degree of coupling between the parameters governing the stiffness in each direction. Thus, only one parameter in each of the three directions was estimated. For the tested magnitudes of added noise and introduced model errors, the resulting estimated stress-strain characteristics in the fiber and sheet directions converged with good accuracy to the known relationship. The sheet-normal stress-strain relationship had a higher degree of uncertainty as more noise was added and model error was introduced.     

Uncertainty analysis of forest carbon sink forecast with varying measurement errors: a data assimilation approach

Aims Accurate forecast of ecosystem states is critical for improving natural resource management and climate change mitigation. Assimilating observed data into models is an effective way to reduce uncertainties in ecological forecasting. However, influences of measurement errors on parameter estimation and forecasted state changes have not been carefully examined. This study analyzed the parameter identifiability of a process-based ecosystem carbon cycle model, the sensitivity of parameter estimates and model forecasts to the magnitudes of measurement errors and the information contributions of the assimilated data to model forecasts with a data assimilation approach.Methods We applied a Markov Chain Monte Carlo method to assimilate eight biometric data sets into the Terrestrial ECOsystem model. The data were the observations of foliage biomass, wood biomass, fine root biomass, microbial biomass, litter fall, litter, soil carbon and soil respiration, collected at the Duke Forest free-air CO₂ enrichment facilities from 1996 to 2005. Three levels of measurement errors were assigned to these data sets by halving and doubling their original standard deviations.Important findings Results showed that only less than half of the 30 parameters could be constrained, though the observations were extensive and the model was relatively simple. Higher measurement errors led to higher uncertainties in parameters estimates and forecasted carbon (C) pool sizes. The long-term predictions of the slow turnover pools were affected less by the measurement errors than those of fast turnover pools. Assimilated data contributed less information for the pools with long residence times in long-term forecasts. These results indicate the residence times of C pools played a key role in regulating propagation of errors from measurements to model forecasts in a data assimilation system. Improving the estimation of parameters of slow turnover C pools is the key to better forecast long-term ecosystem C dynamics.     

Regression modeling of semicompeting risks data

Semicompeting risks data are often encountered in clinical trials with intermediate endpoints subject to dependent censoring from informative dropout. Unlike with competing risks data, dropout may not be dependently censored by the intermediate event. There has recently been increased attention to these data, in particular inferences about the marginal distribution of the intermediate event without covariates. In this article, we incorporate covariates and formulate their effects on the survival function of the intermediate event via a functional regression model. To accommodate informative censoring, a time-dependent copula model is proposed in the observable region of the data which is more flexible than standard parametric copula models for the dependence between the events. The model permits estimation of the marginal distribution under weaker assumptions than in previous work on competing risks data. New nonparametric estimators for the marginal and dependence models are derived from nonlinear estimating equations and are shown to be uniformly consistent and to converge weakly to Gaussian processes. Graphical model checking techniques are presented for the assumed models. Nonparametric tests are developed accordingly, as are inferences for parametric submodels for the time-varying covariate effects and copula parameters. A novel time-varying sensitivity analysis is developed using the estimation procedures. Simulations and an AIDS data analysis demonstrate the practical utility of the methodology.     

Mathematical and kinetic modeling of biofilm reactor based on ant colony optimization

Inverse estimation of model parameters via mathematical modeling route, known as inverse modeling (IM), is an attractive alternative approach to the experimental methods. This approach makes use of efficient optimization techniques in the course of solution of an inverse problem with the aid of measured data. In this study, a novel optimization method based on ant colony optimization (ACO), denoted by ACO-IM, is presented for inverse estimation of kinetic and film thickness parameters of biofilm models that describe an experimental fixed bed anaerobic reactor. The proposed optimization method for parameter estimation emulates the fact that ants are capable of finding the shortest path from a food source to their nest by depositing a trial of pheromone during their walk. The efficacy of the ACO-IM for numerical estimation of bio-kinetic parameters is demonstrated through its application for the anaerobic treatment of industry wastewater in a fixed bed biofilm process. The results explain the rigorousness of mathematical models, the form of kinetic and film thickness models and the type of packing to be used with the biofilm process for accurate determination of kinetic and film thickness parameters so as to ensure reliable predictive performance of the biofilm reactor models.