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.     

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.     

Assessment and parameter identification of simplified models to describe the kinetics of semi-continuous biomethane production from anaerobic digestion of green and food waste

Biochemical reactions occurring during anaerobic digestion have been modelled using reaction kinetic equations such as first-order, Contois and Monod which are then combined to form mechanistic models. This work considers models which include between one and three biochemical reactions to investigate if the choice of the reaction rate equation, complexity of the model structure as well as the inclusion of inhibition plays a key role in the ability of the model to describe the methane production from the semi-continuous anaerobic digestion of green waste (GW) and food waste (FW). A parameter estimation method was used to investigate the most important phenomena influencing the biogas production process. Experimental data were used to numerically estimate the model parameters and the quality of fit was quantified. Results obtained reveal that the model structure (i.e. number of reactions, inhibition) has a much stronger influence on the quality of fit compared with the choice of kinetic rate equations. In the case of GW there was only a marginal improvement when moving from a one to two reaction model, and none with inclusion of inhibition or three reactions. However, the behaviour of FW digestion was more complex and required either a two or three reaction model with inhibition functions for both ammonia and volatile fatty acids. Parameter values for the best fitting models are given for use by other authors.     

Least-squares estimation of batch culture kinetic parameters

This article concerns the development of a simple and effective least-squares procedure for estimating the kinetic parameters in Monod expressions from batch culture data. The basic approach employed in this work was to translate the problem of parameter estimation to a mathematical model containing a single decision variable. The resulting model was then solved by an efficient one-dimensional search algorithm which can be adapted to any microcomputer or advanced programmable calculator. The procedure was tested on synthetic data (substrate concentrations) with different types and levels of error. The effect of endogeneous respiration on the estimated values of the kinetic parameters was also assessed. From the results of these analyses the least-squares procedure developed was concluded to be very effective.     

A decompositional approach to parameter estimation in pathway modeling: a case study of the Akt and MAPK pathways and their crosstalk

Parameter estimation is a critical problem in modeling biological pathways. It is difficult because of the large number of parameters to be estimated and the limited experimental data available. In this paper, we propose a decompositional approach to parameter estimation. It exploits the structure of a large pathway model to break it into smaller components, whose parameters can then be estimated independently. This leads to significant improvements in computational efficiency. We present our approach in the context of Hybrid Functional Petri Net modeling and evolutionary search for parameter value estimation. However, the approach can be easily extended to other modeling frameworks and is independent of the search method used. We have tested our approach on a detailed model of the Akt and MAPK pathways with two known and one hypothesized crosstalk mechanisms. The entire model contains 84 unknown parameters. Our simulation results exhibit good correlation with experimental data, and they yield positive evidence in support of the hypothesized crosstalk between the two pathways.     

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.     

Identifyability measures to select the parameters to be estimated in a solid‐state fermentation distributed parameter model

Process modeling can lead to of advantages such as helping in process control, reducing process costs and product quality improvement. This work proposes a solid‐state fermentation distributed parameter model composed by seven differential equations with seventeen parameters to represent the process. Also, parameters estimation with a parameters identifyability analysis (PIA) is performed to build an accurate model with optimum parameters. Statistical tests were made to verify the model accuracy with the estimated parameters considering different assumptions. The results have shown that the model assuming substrate inhibition better represents the process. It was also shown that eight from the seventeen original model parameters were nonidentifiable and better results were obtained with the removal of these parameters from the estimation procedure. Therefore, PIA can be useful to estimation procedure, since it may reduce the number of parameters that can be evaluated. Further, PIA improved the model results, showing to be an important procedure to be taken. © 2016 American Institute of Chemical Engineers Biotechnol. Prog., 32:905–917, 2016     

Parameter estimation of kinetic models from metabolic profiles: two-phase dynamic decoupling method

MOTIVATION: Time-series measurements of metabolite concentration have become increasingly more common, providing data for building kinetic models of metabolic networks using ordinary differential equations (ODEs). In practice, however, such time-course data are usually incomplete and noisy, and the estimation of kinetic parameters from these data is challenging. Practical limitations due to data and computational aspects, such as solving stiff ODEs and finding global optimal solution to the estimation problem, give motivations to develop a new estimation procedure that can circumvent some of these constraints. RESULTS: In this work, an incremental and iterative parameter estimation method is proposed that combines and iterates between two estimation phases. One phase involves a decoupling method, in which a subset of model parameters that are associated with measured metabolites, are estimated using the minimization of slope errors. Another phase follows, in which the ODE model is solved one equation at a time and the remaining model parameters are obtained by minimizing concentration errors. The performance of this two-phase method was tested on a generic branched metabolic pathway and the glycolytic pathway of Lactococcus lactis. The results showed that the method is efficient in getting accurate parameter estimates, even when some information is missing.     

Application of Penalized Regression Techniques in Modelling Insulin Sensitivity by Correlated Metabolic Parameters

This paper aims to introduce penalized estimation techniques in clinical investigations of diabetes, as well as to assess their possible advantages and limitations. Data from a previous study was used to carry out the simulations to assess: a) which procedure results in the lowest prediction error of the final model in the setting of a large number of predictor variables with high multicollinearity (of importance if insulin sensitivity should be predicted) and b) which procedure achieves the most accurate estimate of regression coefficients in the setting of fewer predictors with small unidirectional effects and moderate correlation between explanatory variables (of importance if the specific relation between an independent variable and insulin sensitivity should be examined). Moreover a special focus is on the correct direction of estimated parameter effects, a non-negligible source of error and misinterpretation of study results. The simulations were performed for varying sample size to evaluate the performance of LASSO, Ridge as well as different algorithms for Elastic Net. These methods were also compared with automatic variable selection procedures (i.e. optimizing AIC or BIC).We were not able to identify one method achieving superior performance in all situations. However, the improved accuracy of estimated effects underlines the importance of using penalized regression techniques in our example (e.g. if a researcher aims to compare relations of several correlated parameters with insulin sensitivity). However, the decision which procedure should be used depends on the specific context of a study (accuracy versus complexity) and moreover should involve clinical prior knowledge.     

Parameter Estimation Methods for Single Neuron Models

With the advancement in computer technology, it has become possible to fit complex models to neuronal data. In this work, we test how two methods can estimate parameters of simple neuron models (passive soma) to more complex ones (neuron with one dendritic cylinder and two active conductances). The first method uses classical voltage traces resulting from current pulses injection (time domain), while the second uses measures of the neuron's response to sinusoidal stimuli (frequency domain). Both methods estimate correctly the parameters in all cases studied. However, the time-domain method is slower and more prone to estimation errors in the cable parameters than the frequency-domain method. Because with noisy data the goodness of fit does not distinguish between different solutions, we suggest that running the estimation procedure a large number of times might help find a good solution and can provide information about the interactions between parameters. Also, because the formulation used for the model's response in the frequency domain is analytical, one can derive a local sensitivity analysis for each parameter. This analysis indicates how well a parameter is likely to be estimated and helps choose an optimal stimulation protocol. Finally, the tests suggest a strategy for fitting single-cell models using the two methods examined.     

Sensitivity of a Hill-based muscle model to perturbations in model parameters

Musculoskeletal simulations of human movement commonly use Hill muscle models to predict muscle forces, but their sensitivity to model parameter values is not well understood. The purpose of this study was to evaluate muscle model sensitivity to perturbations in 14 Hill muscle model parameters in forward dynamic simulations of running and walking by varying each by +/-50%. Three evaluations of the muscle model were performed based on: (1) calculating the sensitivity of the muscle model only, (2) determining the continuous partial derivatives of the muscle equations with respect to each parameter, and (3) evaluating the effects on the running and walking simulations. Model evaluations were found to be very sensitive (percent change in outputs greater than parameter perturbation) to parameters defining the series elastic component (tendon), force-length curve of the contractile element and maximum isometric force. For some parameters, the range of literature values was larger than the model sensitivity. Model evaluations were insensitive to parameters defining the parallel elastic element, force-velocity curve of the contractile element and muscle activation time constants. The derivative method provided similar results, but also provided a generic, continuous equation that can easily be applied to other motions. The sensitivities of the running and walking simulations were reduced compared to the sensitivity of the muscle model alone. Results demonstrate the importance of evaluating sensitivity of a musculoskeletal simulation in a controlled manner and provide an indication of which parameters must be selected most carefully based on the sensitivity of a given movement.     

Anaerobic digestion model no. 1-based distributed parameter model of an anaerobic reactor: I. Model development

This work presents a distributed parameter model of the anaerobic digestion process. The model is based on the Anaerobic digestion model no. 1 (ADM1) and was developed to simulate anaerobic digestion process in high-rate reactors with significant axial dispersion, such as in upflow anaerobic sludge bed (UASB) reactors. The model, which was named ADM1d, combines ADM1's kinetics of biomass growth and substrate transformation with axial dispersion material balances. ADM1d uses a hyperbolic tangent function to describe biomass distribution within a one compartment model. A comparison of this approach with a two-compartment, sludge bed - liquid above the bed, model showed similar simulation results while the one-compartment model had less equations. A comparison of orthogonal collocation and finite difference algorithms for numerical solution of ADM1d showed better stability of the finite difference algorithm.     

Optimal experiment selection for parameter estimation in biological differential equation models

ABSTRACT: BACKGROUND: Parameter estimation in biological models is a common yet challenging problem. In this work we explore the problem for gene regulatory networks modeled by differential equations with unknown parameters, such as decay rates, reaction rates, Michaelis-Menten constants, and Hill coefficients. We explore the question to what extent parameters can be efficiently estimated by appropriate experimental selection. RESULTS: A minimization formulation is used to find the parameter values that best fit the experiment data. When the data is insufficient, the minimization problem often has many local minima that fit the data reasonably well. We show that selecting a new experiment based on the local Fisher Information of one local minimum generates additional data that allows one to successfully discriminate among the many local minima. The parameters can be estimated to high accuracy by iteratively performing minimization and experiment selection. We show that the experiment choices are roughly independent of which local minima is used to calculate the local Fisher Information. CONCLUSIONS: We show that by an appropriate choice of experiments, one can, in principle, efficiently and accurately estimate all the parameters of gene regulatory network. In addition, we demonstrate that appropriate experiment selection can also allow one to restrict model predictions without constraining the parameters using many fewer experiments. We suggest that predicting model behaviors and inferring parameters represent two different approaches to model calibration with different requirements on data and experimental cost.     

Model‐based identifiable parameter determination applied to a simultaneous saccharification and fermentation process model for bio‐ethanol production

In this work, a methodology for the model‐based identifiable parameter determination (MBIPD) is presented. This systematic approach is proposed to be used for structure and parameter identification of nonlinear models of biological reaction networks. Usually, this kind of problems are over‐parameterized with large correlations between parameters. Hence, the related inverse problems for parameter determination and analysis are mathematically ill‐posed and numerically difficult to solve. The proposed MBIPD methodology comprises several tasks: (i) model selection, (ii) tracking of an adequate initial guess, and (iii) an iterative parameter estimation step which includes an identifiable parameter subset selection (SsS) algorithm and accuracy analysis of the estimated parameters. The SsS algorithm is based on the analysis of the sensitivity matrix by rank revealing factorization methods. Using this, a reduction of the parameter search space to a reasonable subset, which can be reliably and efficiently estimated from available measurements, is achieved. The simultaneous saccharification and fermentation (SSF) process for bio‐ethanol production from cellulosic material is used as case study for testing the methodology. The successful application of MBIPD to the SSF process demonstrates a relatively large reduction in the identified parameter space. It is shown by a cross‐validation that using the identified parameters (even though the reduction of the search space), the model is still able to predict the experimental data properly. Moreover, it is shown that the model is easily and efficiently adapted to new process conditions by solving reduced and well conditioned problems. © 2013 American Institute of Chemical Engineers Biotechnol. Prog., 29:1064–1082, 2013     

Production of fine and speciality chemicals: procedure for the estimation of LCIs

Goal, Scope, Background  To improve the environmental performance of chemical products or services, especially via comparisons of chemical products, LCA is a suitable evaluation method. However, no procedure to obtain comprehensive LCI-data on the production of fine and speciality chemicals is available to date, and information on such production processes is scarce. Thus, a procedure was developed for the estimation of LCIs of chemical production process-steps, which relies on only a small amount of input data. Methods  A generic input-output scheme of chemical production process-steps was set up, and equations to calculate inputs and outputs were established. For most parameters in the resulting estimation procedure, default values were derived from on-site data on chemical production processes and from heuristics. Uncertainties in the estimated default values were reflected as best-case and worst-case scenarios. The procedure was applied to a case study comparing the production of two active ingredients used for crop protection. Verification and a sensitivity analysis were carried out. Results and Discussion  It was found that the impacts from the mass and energy flows estimated by the procedure represent a significant share of the impacts assessed in the case study. In a verification, LCI-data from existing processes yielded results within the range of the estimated best-case and worst-case scenarios. Note that verification data could not be obtained for all process steps. From the verification results, it was inferred that mass and energy flows of existing processes for the production of fine and speciality chemicals correspond more frequently to the estimated best-case than to the worst-case scenario. In the sensitivity analysis, solvent demand was found to be the most crucial parameter in the environmental performance of the chemical production processes assessed. Conclusion  Mass and energy flows in LCIs of production processes for fine and speciality chemicals should not be neglected, even if only little information on a process is available. The estimation procedure described here helps to overcome lacking information in a transparent, consistent way. Recommendations and Outlook  Additional verifications and a more detailed estimation of the default parameters are desirable to learn more about the accuracy of the estimation procedure. The procedure should also be applied to case studies to gain insight into the usefulness of the estimation results in different decision-making contexts.     

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.     

Dynamical model development and parameter identification for an anaerobic wastewater treatment process.

This paper deals with the development and the parameter identification of an anaerobic digestion process model. A two-step (acidogenesis-methanization) mass-balance model has been considered. The model incorporates electrochemical equilibria in order to include the alkalinity, which has to play a central role in the related monitoring and control strategy of a treatment plant. The identification is based on a set of dynamical experiments designed to cover a wide spectrum of operating conditions that are likely to take place in the practical operation of the plant. A step by step identification procedure to estimate the model parameters is presented. The results of 70 days of experiments in a 1-m(3) fermenter are then used to validate the model.     

Analysis and application of ADM1 for anaerobic methane production

An anaerobic model for the serum bottle test was developed and analyzed with sensitivities of stoichiometric and kinetic parameters to the components in order to establish a basis for appropriate application of the model. Anaerobic glucose degradation in a serum bottle was selected as an example. The anaerobic model was developed based on the anaerobic digestion model no. 1 (ADM1), which had five processes with 17 kinetic and stoichiometric parameters. Sensitivity analysis showed that the yield of product on the substrate (f) has high sensitivities to model components, and that the methane concentration was the most sensitive component. Important parameters including yield of product on the substrate (f), yield of biomass on the substrate (Y), and half-saturation values (K) were estimated using genetic algorithms, which optimized the parameters with experimental results. The Monod maximum specific uptake rate (k) was, however, so strongly associated with the concentration of biomass, that values could not be estimated individually. Simulation with estimated parameters showed good agreement with experimental results in the case of methane production. However, there were some differences in acetate and propionate concentrations.     

A simple work flow for biologically inspired model reduction - application to early JAK-STAT signaling

Background  

Modeling of biological pathways is a key issue in systems biology. When constructing a model, it is tempting to incorporate all known interactions of pathway species, which results in models with a large number of unknown parameters. Fortunately, unknown parameters need not necessarily be measured directly, but some parameter values can be estimated indirectly by fitting the model to experimental data. However, parameter fitting, or, more precisely, maximum likelihood parameter estimation, only provides valid results, if the complexity of the model is in balance with the amount and quality of the experimental data. If this is the case the model is said to be identifiable for the given data. If a model turns out to be unidentifiable, two steps can be taken. Either additional experiments need to be conducted, or the model has to be simplified.     

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.