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.     

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.     

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.     

Practical identifiability of biokinetic parameters of a model describing two-step nitrification in biofilms

Parameter estimation and model calibration are key problems in the application of biofilm models in engineering practice, where a large number of model parameters need to be determined usually based on experimental data with only limited information content. In this article, identifiability of biokinetic parameters of a biofilm model describing two-step nitrification was evaluated based solely on bulk phase measurements of ammonium, nitrite, and nitrate. In addition to evaluating the impact of experimental conditions and available measurements, the influence of mass transport limitation within the biofilm and the initial parameter values on identifiability of biokinetic parameters was evaluated. Selection of parameters for identifiability analysis was based on global mean sensitivities while parameter identifiability was analyzed using local sensitivity functions. At most, four of the six most sensitive biokinetic parameters were identifiable from results of batch experiments at bulk phase dissolved oxygen concentrations of 0.8 or 5 mg O(2)/L. High linear dependences between the parameters of the subsets (KO2,AOB,muAOB) and (KO2,NOB,muNOB) resulted in reduced identifiability. Mass transport limitation within the biofilm did not influence the number of identifiable parameters but, in fact, decreased collinearity between parameters, especially for parameters that are otherwise correlated (e.g., muAOB) and KO2,AOB, or muNOB and KO2,NOB). The choice of the initial parameter values had a significant impact on the identifiability of two parameter subsets, both including the parameters muAOB and KO2,AOB. Parameter subsets that did not include the subsets muAOB and KO2,AOB or muNOB and KO2,NOB were clearly identifiable independently of the choice of the initial parameter values.     

The structural identifiability and parameter estimation of a multispecies model for the transmission of mastitis in dairy cows.

A structural identifiability analysis is performed on a mathematical model for the coupled transmission of two classes of pathogen. The pathogens, classified as major and minor, are aetiological agents of mastitis in dairy cows that interact directly and via the immunological reaction in their hosts. Parameter estimates are available from experimental data for all but four of the parameters in the model. Data from a longitudinal study of infection are used to estimate these unknown parameters. A novel approach and application of structural identifiability analysis is combined in this paper with the estimation of cross-protection parameters using epidemiological data.     

Review: To be or not to be an identifiable model. Is this a relevant question in animal science modelling?

What is a good (useful) mathematical model in animal science? For models constructed for prediction purposes, the question of model adequacy (usefulness) has been traditionally tackled by statistical analysis applied to observed experimental data relative to model-predicted variables. However, little attention has been paid to analytic tools that exploit the mathematical properties of the model equations. For example, in the context of model calibration, before attempting a numerical estimation of the model parameters, we might want to know if we have any chance of success in estimating a unique best value of the model parameters from available measurements. This question of uniqueness is referred to as structural identifiability; a mathematical property that is defined on the sole basis of the model structure within a hypothetical ideal experiment determined by a setting of model inputs (stimuli) and observable variables (measurements). Structural identifiability analysis applied to dynamic models described by ordinary differential equations (ODEs) is a common practice in control engineering and system identification. This analysis demands mathematical technicalities that are beyond the academic background of animal science, which might explain the lack of pervasiveness of identifiability analysis in animal science modelling. To fill this gap, in this paper we address the analysis of structural identifiability from a practitioner perspective by capitalizing on the use of dedicated software tools. Our objectives are (i) to provide a comprehensive explanation of the structural identifiability notion for the community of animal science modelling, (ii) to assess the relevance of identifiability analysis in animal science modelling and (iii) to motivate the community to use identifiability analysis in the modelling practice (when the identifiability question is relevant). We focus our study on ODE models. By using illustrative examples that include published mathematical models describing lactation in cattle, we show how structural identifiability analysis can contribute to advancing mathematical modelling in animal science towards the production of useful models and, moreover, highly informative experiments via optimal experiment design. Rather than attempting to impose a systematic identifiability analysis to the modelling community during model developments, we wish to open a window towards the discovery of a powerful tool for model construction and experiment design.     

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.     

Structural identifiability of systems biology models: a critical comparison of methods

Analysing the properties of a biological system through in silico experimentation requires a satisfactory mathematical representation of the system including accurate values of the model parameters. Fortunately, modern experimental techniques allow obtaining time-series data of appropriate quality which may then be used to estimate unknown parameters. However, in many cases, a subset of those parameters may not be uniquely estimated, independently of the experimental data available or the numerical techniques used for estimation. This lack of identifiability is related to the structure of the model, i.e. the system dynamics plus the observation function. Despite the interest in knowing a priori whether there is any chance of uniquely estimating all model unknown parameters, the structural identifiability analysis for general non-linear dynamic models is still an open question. There is no method amenable to every model, thus at some point we have to face the selection of one of the possibilities. This work presents a critical comparison of the currently available techniques. To this end, we perform the structural identifiability analysis of a collection of biological models. The results reveal that the generating series approach, in combination with identifiability tableaus, offers the most advantageous compromise among range of applicability, computational complexity and information provided.     

Exploration of the intercellular heterogeneity of topotecan uptake into human breast cancer cells through compartmental modelling

A mathematical multi-cell model for the in vitro kinetics of the anti-cancer agent topotecan (TPT) following administration into a culture medium containing a population of human breast cancer cells (MCF-7 cell line) is described. This non-linear compartmental model is an extension of an earlier single-cell type model and has been validated using experimental data obtained using two-photon laser scanning microscopy (TPLSM). A structural identifiability analysis is performed prior to parameter estimation to test whether the unknown parameters within the model are uniquely determined by the model outputs. The full model has 43 compartments, with 107 unknown parameters, and it was found that the structural identifiability result could not be established even when using the latest version of the symbolic computation software Mathematica. However, by assuming that a priori knowledge is available for certain parameters, it was possible to reduce the number of parameters to 81, and it was found that this (Stage Two) model was globally (uniquely) structurally identifiable. The identifiability analysis demonstrated how valuable symbolic computation is in this context, as the analysis is far too lengthy and difficult to be performed by hand.     

Structural identifiability of a model for the acetic acid fermentation process

Modelling has proved an essential tool for addressing research into biotechnological processes, particularly with a view to their optimization and control. Parameter estimation via optimization approaches is among the major steps in the development of biotechnology models. In fact, one of the first tasks in the development process is to determine whether the parameters concerned can be unambiguously determined and provide meaningful physical conclusions as a result. The analysis process is known as 'identifiability' and presents two different aspects: structural or theoretical identifiability and practical identifiability. While structural identifiability is concerned with model structure alone, practical identifiability takes into account both the quantity and quality of experimental data. In this work, we discuss the theoretical identifiability of a new model for the acetic acid fermentation process and review existing methods for this purpose.     

Identifiability and online estimation of diagnostic parameters with in the glucose insulin homeostasis

Today, diagnostic decisions about pre-diabetes or diabetes are made using static threshold rules for the measured plasma glucose. In order to develop an alternative diagnostic approach, dynamic models as the Minimal Model may be deployed. We present a novel method to analyze the identifiability of model parameters based on the interpretation of the empirical observability Gramian. This allows a unifying view of both, the observability of the system's states (with dynamics) and the identifiability of the system's parameters (without dynamics). We give an iterative algorithm, in order to find an optimized set of states and parameters to be estimated. For this set, estimation results using an Unscented Kalman Filter (UKF) are presented. Two parameters are of special interest for diagnostic purposes: the glucose effectiveness S(G) characterizes the ability of plasma glucose clearance, and the insulin sensitivity S(I) quantifies the impact from the plasma insulin to the interstitial insulin subsystem. Applying the identifiability analysis to the trajectories of the insulin glucose system during an intravenous glucose tolerance test (IVGTT) shows the following result: (1) if only plasma glucose G(t) is measured, plasma insulin I(t) and S(G) can be estimated, but not S(I). (2) If plasma insulin I(t) is captured additionally, identifiability is improved significantly such that up to four model parameters can be estimated including S(I). (3) The situation of the first case can be improved, if a controlled external dosage of insulin is applied. Then, parameters of the insulin subsystem can be identified approximately from measurement of plasma glucose G(t) only.     

Identifiability: the first step in parameter estimation

The observations in an experiment define a set of observational parameters that are functions of the basic kinetic parameters of the model of the system. The problem of identifiability is concerned with whether the observational parameters uniquely specify the basic kinetic parameters. As such, it depends only on the functional relation between the two levels of parameters and not on errors of observation and the estimation procedure. It should be checked before doing the experiment. Given initial estimates of the basic kinetic parameters, identifiability can be checked, in a local sense, from data generated by simulating the experiment on the model.     

Structural identifiability of polynomial and rational systems

Since analysis and simulation of biological phenomena require the availability of their fully specified models, one needs to be able to estimate unknown parameter values of the models. In this paper we deal with identifiability of parametrizations which is the property of one-to-one correspondence of parameter values and the corresponding outputs of the models. Verification of identifiability of a parametrization precedes estimation of numerical values of parameters, and thus determination of a fully specified model of a considered phenomenon. We derive necessary and sufficient conditions for the parametrizations of polynomial and rational systems to be structurally or globally identifiable. The results are applied to investigate the identifiability properties of the system modeling a chain of two enzyme-catalyzed irreversible reactions. The other examples deal with the phenomena modeled by using Michaelis–Menten kinetics and the model of a peptide chain elongation.     

Cellular signaling identifiability analysis: A case study

Two primary purposes for mathematical modeling in cell biology are (1) simulation for making predictions of experimental outcomes and (2) parameter estimation for drawing inferences from experimental data about unobserved aspects of biological systems. While the former purpose has become common in the biological sciences, the latter is less common, particularly when studying cellular and subcellular phenomena such as signaling—the focus of the current study. Data are difficult to obtain at this level. Therefore, even models of only modest complexity can contain parameters for which the available data are insufficient for estimation. In the present study, we use a set of published cellular signaling models to address issues related to global parameter identifiability. That is, we address the following question: assuming known time courses for some model variables, which parameters is it theoretically impossible to estimate, even with continuous, noise-free data? Following an introduction to this problem and its relevance, we perform a full identifiability analysis on a set of cellular signaling models using DAISY (Differential Algebra for the Identifiability of SYstems). We use our analysis to bring to light important issues related to parameter identifiability in ordinary differential equation (ODE) models. We contend that this is, as of yet, an under-appreciated issue in biological modeling and, more particularly, cell biology.     

On the identifiability of metabolic network models

A major problem for the identification of metabolic network models is parameter identifiability, that is, the possibility to unambiguously infer the parameter values from the data. Identifiability problems may be due to the structure of the model, in particular implicit dependencies between the parameters, or to limitations in the quantity and quality of the available data. We address the detection and resolution of identifiability problems for a class of pseudo-linear models of metabolism, so-called linlog models. Linlog models have the advantage that parameter estimation reduces to linear or orthogonal regression, which facilitates the analysis of identifiability. We develop precise definitions of structural and practical identifiability, and clarify the fundamental relations between these concepts. In addition, we use singular value decomposition to detect identifiability problems and reduce the model to an identifiable approximation by a principal component analysis approach. The criterion is adapted to real data, which are frequently scarce, incomplete, and noisy. The test of the criterion on a model with simulated data shows that it is capable of correctly identifying the principal components of the data vector. The application to a state-of-the-art dataset on central carbon metabolism in Escherichia coli yields the surprising result that only $4$ out of $31$ reactions, and $37$ out of $100$ parameters, are identifiable. This underlines the practical importance of identifiability analysis and model reduction in the modeling of large-scale metabolic networks. Although our approach has been developed in the context of linlog models, it carries over to other pseudo-linear models, such as generalized mass-action (power-law) models. Moreover, it provides useful hints for the identifiability analysis of more general classes of nonlinear models of metabolism.     

The structural identifiability and parameter estimation of a multispecies model for the transmission of mastitis in dairy cows with postmilking teat disinfection

A mathematical model for the transmission of two interacting classes of mastitis causing bacterial pathogens in a herd of dairy cows is presented and applied to a specific data set. The data were derived from a field trial of a specific measure used in the control of these pathogens, where half the individuals were subjected to the control and in the others the treatment was discontinued. The resultant mathematical model (eight non-linear simultaneous ordinary differential equations) therefore incorporates heterogeneity in the host as well as the infectious agent and consequently the effects of control are intrinsic in the model structure. A structural identifiability analysis of the model is presented demonstrating that the scope of the novel method used allows application to high order non-linear systems. The results of a simultaneous estimation of six unknown system parameters are presented. Previous work has only estimated a subset of these either simultaneously or individually. Therefore not only are new estimates provided for the parameters relating to the transmission and control of the classes of pathogens under study, but also information about the relationships between them. We exploit the close link between mathematical modelling, structural identifiability analysis, and parameter estimation to obtain biological insights into the system modelled.     

Conceptual, methodological and computational issues concerning the compartmental modeling of a complex biological system: Postprandial inter-organ metabolism of dietary nitrogen in humans

A multi-compartmental model has been developed to describe dietary nitrogen (N) postprandial distribution and metabolism in humans. This paper details the entire process of model development, including the successive steps of its construction, parameter estimation and validation. The model was built using experimental data on dietary N kinetics in certain accessible pools of the intestine, blood and urine in healthy adults fed a [15N]-labeled protein meal. A 13-compartment, 21-parameter model was selected from candidate models of increasing order as being the minimum structure able to properly fit experimental data for all sampled compartments. Problems of theoretical identifiability and numerical identification of the model both constituted mathematical challenges that were difficult to solve because of the large number of unknown parameters and the few experimental data available. For this reason, new robust and reliable methods were applied, which enabled (i) a check that all model parameters could theoretically uniquely be determined and (ii) an estimation of their numerical values with satisfactory precision from the experimental data. Finally, model validation was completed by first verifying its a posteriori identifiability and then carrying out external validation.     

Identifiability of causal effects for binary variables with baseline data missing due to death

We discuss identifiability and estimation of causal effects of a treatment in subgroups defined by a covariate that is sometimes missing due to death, which is different from a problem with outcomes censored by death. Frangakis et al. (2007, Biometrics 63, 641-662) proposed an approach for estimating the causal effects under a strong monotonicity (SM) assumption. In this article, we focus on identifiability of the joint distribution of the covariate, treatment and potential outcomes, show sufficient conditions for identifiability, and relax the SM assumption to monotonicity (M) and no-interaction (NI) assumptions. We derive expectation-maximization algorithms for finding the maximum likelihood estimates of parameters of the joint distribution under different assumptions. Further we remove the M and NI assumptions, and prove that signs of the causal effects of a treatment in the subgroups are identifiable, which means that their bounds do not cover zero. We perform simulations and a sensitivity analysis to evaluate our approaches. Finally, we apply the approaches to the National Study on the Costs and Outcomes of Trauma Centers data, which are also analyzed by Frangakis et al. (2007) and Xie and Murphy (2007, Biometrics 63, 655-658).     

Construction of a large scale integrated map of macrophage pathogen recognition and effector systems

Background

Mathematical models provide abstract representations of the information gained from experimental observations on the structure and function of a particular biological system. Conferring a predictive character on a given mathematical formulation often relies on determining a number of non-measurable parameters that largely condition the model's response. These parameters can be identified by fitting the model to experimental data. However, this fit can only be accomplished when identifiability can be guaranteed.

Results

We propose a novel iterative identification procedure for detecting and dealing with the lack of identifiability. The procedure involves the following steps: 1) performing a structural identifiability analysis to detect identifiable parameters; 2) globally ranking the parameters to assist in the selection of the most relevant parameters; 3) calibrating the model using global optimization methods; 4) conducting a practical identifiability analysis consisting of two (a priori and a posteriori) phases aimed at evaluating the quality of given experimental designs and of the parameter estimates, respectively and 5) optimal experimental design so as to compute the scheme of experiments that maximizes the quality and quantity of information for fitting the model.

Conclusions

The presented procedure was used to iteratively identify a mathematical model that describes the NF-κB regulatory module involving several unknown parameters. We demonstrated the lack of identifiability of the model under typical experimental conditions and computed optimal dynamic experiments that largely improved identifiability properties.