Structural Identifiability Analysis Research Articles

Identifiability investigation of within-host models of acute virus infection.

Uncertainty in parameter estimates from fitting within-host models to empirical data limits the model's ability to uncover mechanisms of infection, disease progression, and to guide pharmaceutical interventions. Understanding the effect of model structure and data availability on model predictions is important for informing model development and experimental design. To address sources of uncertainty in parameter estimation, we use four mathematical models of influenza A infection with increased degrees of biological realism. We test the ability of each model to reveal its parameters in the presence of unlimited data by performing structural identifiability analyses. We then refine the results by predicting practical identifiability of parameters under daily influenza A virus titers alone or together with daily adaptive immune cell data. Using these approaches, we present insight into the sources of uncertainty in parameter estimation and provide guidelines for the types of model assumptions, optimal experimental design, and biological information needed for improved predictions.

Structural and practical identifiability analysis in bioengineering: a beginner’s guide

Advancements in digital technology have brought modelling to the forefront in many disciplines from healthcare to architecture. Mathematical models, often represented using parametrised sets of ordinary differential equations, can be used to characterise different processes. To infer possible estimates for the unknown parameters, these models are usually calibrated using associated experimental data. Structural and practical identifiability analyses are a key component that should be assessed prior to parameter estimation. This is because identifiability analyses can provide insights as to whether or not a parameter can take on single, multiple, or even infinitely or countably many values which will ultimately have an impact on the reliability of the parameter estimates. Also, identifiability analyses can help to determine whether the data collected are sufficient or of good enough quality to truly estimate the parameters or if more data or even reparameterization of the model is necessary to proceed with the parameter estimation process. Thus, such analyses also provide an important role in terms of model design (structural identifiability analysis) and the collection of experimental data (practical identifiability analysis). Despite the popularity of using data to estimate the values of unknown parameters, structural and practical identifiability analyses of these models are often overlooked. Possible reasons for non-consideration of application of such analyses may be lack of awareness, accessibility, and usability issues, especially for more complicated models and methods of analysis. The aim of this study is to introduce and perform both structural and practical identifiability analyses in an accessible and informative manner via application to well established and commonly accepted bioengineering models. This will help to improve awareness of the importance of this stage of the modelling process and provide bioengineering researchers with an understanding of how to utilise the insights gained from such analyses in future model development.

Structural identifiability analysis of linear reaction–advection–diffusion processes in mathematical biology

Effective application of mathematical models to interpret biological data and make accurate predictions often requires that model parameters are identifiable. Approaches to assess the so-called structural identifiability of models are well established for ordinary differential equation models, yet there are no commonly adopted approaches that can be applied to assess the structural identifiability of the partial differential equation (PDE) models that are requisite to capture spatial features inherent to many phenomena. The differential algebra approach to structural identifiability has recently been demonstrated to be applicable to several specific PDE models. In this brief article, we present general methodology for performing structural identifiability analysis on partially observed reaction–advection–diffusion PDE models that are linear in the unobserved quantities. We show that the differential algebra approach can always, in theory, be applied to such models. Moreover, despite the perceived complexity introduced by the addition of advection and diffusion terms, consideration of spatial analogues of non-spatial models cannot exacerbate structural identifiability. We conclude by discussing future possibilities and the computational cost of performing structural identifiability analysis on more general PDE models.

Identifiability investigation of within-host models of acute virus infection

<p>Uncertainty in parameter estimates from fitting within-host models to empirical data limits the model's ability to uncover mechanisms of infection, disease progression, and to guide pharmaceutical interventions. Understanding the effect of model structure and data availability on model predictions is important for informing model development and experimental design. To address sources of uncertainty in parameter estimation, we used four mathematical models of influenza A infection with increased degrees of biological realism. We tested the ability of each model to reveal its parameters in the presence of unlimited data by performing structural identifiability analyses. We then refined the results by predicting practical identifiability of parameters under daily influenza A virus titers alone or together with daily adaptive immune cell data. Using these approaches, we presented insight into the sources of uncertainty in parameter estimation and provided guidelines for the types of model assumptions, optimal experimental design, and biological information needed for improved predictions.</p>

Quantitative Systems Pharmacology & Machine Learning - A match made in heaven or hell?

As pharmaceutical development moves from early stage in vitro experimentation to later in vivo and subsequent clinical trials, data and knowledge are acquired across multiple time and length scales, from the subcellular to whole patient cohort scale. Realising the potential of this data for informing decision making in pharmaceutical development requires the individual and combined application of machine learning (ML) and mechanistic multiscale mathematical modelling approaches. Here we outline how these two approaches, both individually and in tandem, can be applied at different stages of the drug discovery and development pipeline to inform decision making compound development. The importance of discerning between knowledge and data is highlighted in informing the initial use of ML or mechanistic Quantitative Systems Pharmacology (QSP) models. We discuss the application of sensitivity and structural identifiability analyses of QSP models in informing future experimental studies, to which ML may be applied, as well as how ML approaches can be used to inform mechanistic model development. Relevant literature studies are highlighted and we close by discussing caveats regarding the application of each approach in an age of constant data acquisition. Significance Statement We consider when best to apply Machine Learning (ML) and mechanistic Quantitative Systems Pharmacology (QSP) approaches in the context of the drug discovery and development pipeline. We discuss the importance of prior knowledge and data available for the system of interest and how this informs the individual and combined application of ML and QSP approaches at each stage of the pipeline.

Structural identifiability of biomolecular controller motifs with and without flow measurements as model output.

Controller motifs are simple biomolecular reaction networks with negative feedback. They can explain how regulatory function is achieved and are often used as building blocks in mathematical models of biological systems. In this paper we perform an extensive investigation into structural identifiability of controller motifs, specifically the so-called basic and antithetic controller motifs. Structural identifiability analysis is a useful tool in the creation and evaluation of mathematical models: it can be used to ensure that model parameters can be determined uniquely and to examine which measurements are necessary for this purpose. This is especially useful for biological models where parameter estimation can be difficult due to limited availability of measureable outputs. Our aim with this work is to investigate how structural identifiability is affected by controller motif complexity and choice of measurements. To increase the number of potential outputs we propose two methods for including flow measurements and show how this affects structural identifiability in combination with, or in the absence of, concentration measurements. In our investigation, we analyze 128 different controller motif structures using a combination of flow and/or concentration measurements, giving a total of 3648 instances. Among all instances, 34% of the measurement combinations provided structural identifiability. Our main findings for the controller motifs include: i) a single measurement is insufficient for structural identifiability, ii) measurements related to different chemical species are necessary for structural identifiability. Applying these findings result in a reduced subset of 1568 instances, where 80% are structurally identifiable, and more complex/interconnected motifs appear easier to structurally identify. The model structures we have investigated are commonly used in models of biological systems, and our results demonstrate how different model structures and measurement combinations affect structural identifiability of controller motifs.

Classical structural identifiability methodology applied to low-dimensional dynamic systems in receptor theory.

Mathematical modelling has become a key tool in pharmacological analysis, towards understanding dynamics of cell signalling and quantifying ligand-receptor interactions. Ordinary differential equation (ODE) models in receptor theory may be used to parameterise such interactions using timecourse data, but attention needs to be paid to the theoretical identifiability of the parameters of interest. Identifiability analysis is an often overlooked step in many bio-modelling works. In this paper we introduce structural identifiability analysis (SIA) to the field of receptor theory by applying three classical SIA methods (transfer function, Taylor Series and similarity transformation) to ligand-receptor binding models of biological importance (single ligand and Motulsky-Mahan competition binding at monomers, and a recently presented model of a single ligand binding at receptor dimers). New results are obtained which indicate the identifiable parameters for a single timecourse for Motulsky-Mahan binding and dimerised receptor binding. Importantly, we further consider combinations of experiments which may be performed to overcome issues of non-identifiability, to ensure the practical applicability of the work. The three SIA methods are demonstrated through a tutorial-style approach, using detailed calculations, which show the methods to be tractable for the low-dimensional ODE models.

A target-cell limited model can reproduce influenza infection dynamics in hosts with differing immune responses

We consider a hierarchy of ordinary differential equation models that describe the within-host viral kinetics of influenza infections: the IR model explicitly accounts for an immune response to the virus, while the simpler, target-cell limited TEIV and TV models do not. We show that when the IR model is fitted to pooled experimental murine data of the viral load, fraction of dead cells, and immune response levels, its parameters values can be determined. However, if, as is common, only viral load data are available, we can estimate parameters of the TEIV and TV models but not the IR model. These results are substantiated by a structural and practical identifiability analysis. We then use the IR model to generate synthetic data representing infections in hosts whose immune responses differ. We fit the TV model to these synthetic datasets and show that it can reproduce the characteristic exponential increase and decay of viral load generated by the IR model. Furthermore, the values of the fitted parameters of the TV model can be mapped from the immune response parameters in the IR model. We conclude that, if only viral load data are available, a simple target-cell limited model can reproduce influenza infection dynamics and distinguish between hosts with differing immune responses.

ASAS-NANP symposium: Mathematical Modeling in Animal Nutrition: The power of identifiability analysis for dynamic modeling in animal science:a practitioner approach.

Constructing dynamic mathematical models of biological systems requires estimating unknown parameters from available experimental data, usually using a statistical fitting procedure. This procedure is usually called parameter identification, parameter estimation, model fitting, or model calibration. In animal science, parameter identification is often performed without analytic considerations on the possibility of determining unique values of the model parameters. These analytical studies are related to the mathematical property of structural identifiability, which refers to the theoretical ability to recover unique values of the model parameters from the measures defined in an experimental setup and use the model structure as the sole basis. The structural identifiability analysis is a powerful tool for model construction because it informs whether the parameter identification problem is well-posed (i.e., the problem has a unique solution). Structural identifiability analysis is helpful to determine which actions (e.g., model reparameterization, choice of new data measurements, and change of the model structure) are needed to render the model parameters identifiable (when possible). The mathematical technicalities associated with structural identifiability analysis are very sophisticated. However, the development of dedicated, freely available software tools enables the application of identifiability analysis without needing to be an expert in mathematics and computer programming. We refer to such a non-expert user as a practitioner for hands-on purposes. However, a practitioner should be familiar with the model construction and software implementation process. In this paper, we propose to adopt a practitioner approach that takes advantage of available software tools to integrate identifiability analysis in the modeling practice in the animal science field. The application of structural identifiability implies switching our regard of the parameter identification problem as a downstream process (after data collection) to an upstream process (before data collection) where experiment design is applied to guarantee identifiability. This upstream approach will substantially improve the workflow of model construction toward robust and valuable models in animal science. Illustrative examples with different levels of complexity support our work. The source codes of the examples were provided for learning purposes and to promote open science practices.

Modeling the interplay between albumin-globulin metabolism and HIV infection.

Human immunodeficiency virus (HIV) infection is a major public health concern with 1.2 million people living with HIV in the United States. The role of nutrition in general, and albumin/globulin in particular in HIV progression has long been recognized. However, no mathematical models exist to describe the interplay between HIV and albumin/globulin. In this paper, we present a family of models of HIV and the two protein components albumin and globulin. We use albumin, globulin, viral load and target cell data from simian immunodeficiency virus (SIV)-infected monkeys to perform model selection on the family of models. We discover that the simplest model accurately and uniquely describes the data. The selection of the simplest model leads to the observation that albumin and globulin do not impact the infection rate of target cells by the virus and the clearance of the infected target cells by the immune system. Moreover, the recruitment of target cells and immune cells are modeled independently of globulin in the selected model. Mathematical analysis of the selected model reveals that the model has an infection-free equilibrium and a unique infected equilibrium when the immunological reproduction number is above one. The infection-free equilibrium is locally stable when the immunological reproduction number is below one, and unstable when the immunological reproduction number is greater than one. The infection equilibrium is locally stable whenever it exists. To determine the parameters of the best fitted model we perform structural and practical identifiability analysis. The structural identifiability analysis reveals that the model is identifiable when the immune cell infection rate is fixed at a value obtained from the literature. Practical identifiability reveals that only seven of the sixteen parameters are practically identifiable with the given data. Practical identifiability of parameters performed with synthetic data sampled a lot more frequently reveals that only two parameters are practically unidentifiable. We conclude that experiments that will improve the quality of the data can help improve the parameter estimates and lead to better understanding of the interplay of HIV and albumin-globulin metabolism.

Nonlinear Electrochemical Impedance Spectroscopy for Lithium-Ion Battery Model Parameterization

In this work we analyze the local nonlinear electrochemical impedance spectroscopy (NLEIS) response of a lithium-ion battery and estimate model parameters from measured NLEIS data. The analysis assumes a single-particle model including nonlinear diffusion of lithium within the electrode particles and asymmetric charge transfer kinetics at their surface. Based on this model and assuming a moderately-small excitation amplitude, we systematically derive analytical formulae for the impedances up to the second harmonic response, allowing the meaningful interpretation of each contribution in terms of physical processes and nonlinearities in the model. The implications of this for parameterization are explored, including structural identifiability analysis and parameter estimation using maximum likelihood, with both synthetic and experimentally measured impedance data. Accurate fits to impedance data are possible, however inconsistencies in the fitted diffusion timescales suggest that a nonlinear diffusion model may not be appropriate for the cells considered. Model validation is also demonstrated by predicting time-domain voltage response using the parameterized model and this is shown to have excellent agreement with measured voltage time-series data (11.1 mV RMSE).

Improving dynamic predictions with ensembles of observable models.

Dynamic mechanistic modelling in systems biology has been hampered by the complexity and variability associated with the underlying interactions, and by uncertain and sparse experimental measurements. Ensemble modelling, a concept initially developed in statistical mechanics, has been introduced in biological applications with the aim of mitigating those issues. Ensemble modelling uses a collection of different models compatible with the observed data to describe the phenomena of interest. However, since systems biology models often suffer from a lack of identifiability and observability, ensembles of models are particularly unreliable when predicting non-observable states. We present a strategy to assess and improve the reliability of a class of model ensembles. In particular, we consider kinetic models described using ordinary differential equations with a fixed structure. Our approach builds an ensemble with a selection of the parameter vectors found when performing parameter estimation with a global optimization metaheuristic. This technique enforces diversity during the sampling of parameter space and it can quantify the uncertainty in the predictions of state trajectories. We couple this strategy with structural identifiability and observability analysis, and when these tests detect possible prediction issues we obtain model reparameterizations that surmount them. The end result is an ensemble of models with the ability to predict the internal dynamics of a biological process. We demonstrate our approach with models of glucose regulation, cell division, circadian oscillations and the JAK-STAT signalling pathway. The code that implements the methodology and reproduces the results is available at https://doi.org/10.5281/zenodo.6782638. Supplementary data are available at Bioinformatics online.

Parameter Estimation of Dynamic Beer Fermentation Models.

In this study, two dynamic models of beer fermentation are proposed, and their parameters are estimated using experimental data collected during several batch experiments initiated with different sugar concentrations. Biomass, sugar, ethanol, and vicinal diketone concentrations are measured off-line with an analytical system while two on-line immersed probes deliver temperature, ethanol concentration, and carbon dioxide exhaust rate measurements. Before proceeding to the estimation of the unknown model parameters, a structural identifiability analysis is carried out to investigate the measurement configuration and the kinetic model structure. The model predictive capability is investigated in cross-validation, in view of opening up new perspectives for monitoring and control purposes. For instance, the dynamic model could be used as a predictor in receding-horizon observers and controllers.

Structural identifiability of compartmental models for infectious disease transmission is influenced by data type

If model identifiability is not confirmed, inferences from infectious disease transmission models may not be reliable, so they might result in misleading recommendations. Structural identifiability analysis characterises whether it is possible to obtain unique solutions for all unknown model parameters, given the model structure. In this work, we studied the structural identifiability of some typical deterministic compartmental models for infectious disease transmission, focusing on the influence of the data type considered as model output on the identifiability of unknown model parameters, including initial conditions. We defined 26 model versions, each having a unique combination of underlying compartmental structure and data type(s) considered as model output(s). Four compartmental model structures and three common data types in disease surveillance (incidence, prevalence and detected vector counts) were studied. The structural identifiability of some parameters varied depending on the type of model output. In general, models with multiple data types as outputs had more structurally identifiable parameters, than did models with a single data type as output. This study highlights the importance of a careful consideration of data types as an integral part of the inference process with compartmental infectious disease transmission models.

75 The Power of Theoretical and Practical Identifiability Analysis for Modeling (Micro-) Biological Processes

Abstract When modeling biological systems, one major step of the modeling exercise is the confrontation between the theory (the model) and the reality (the data). Such a confrontation passes through the resolution of the parameter identification (model calibration) problem, which aims at finding a set of parameters that best fits the variables predicted by the model to the data. The parameter identification step is often addressed like a downstream process. Using this approach, once the data are collected, the modeler has a minimal degree of freedom to improve the accuracy of her model. Instead, the modeler can identify opportunities for improving model accuracy if the parameter identification problem is addressed like an upstream process. For instance, one of these opportunities is to determine for a given experimental setup (before the actual data collection), whether the optimization problem is well-posed (that it is, if the problem has unique solution). Knowing this information beforehand is useful to set up an experimental protocol that guarantees the uniqueness of the parameter estimates. This question of uniqueness is related to the notion of structural (theoretical) identifiability. The modeler might also be interested in knowing whether the available data that will result from the experimental protocol will be informative enough to identify the model parameters accurately. The question of parameter accuracy based on the quality of available data is related to the notion of practical identifiability. This work aims to present the theoretical basis of structural and practical identifiability analysis. By capitalizing on the use of dedicated software tools, we will illustrate further the power of structural and practical identifiability analysis in the modeling exercise using relevant examples for the modeling of microbiological processes such as the rumen microbial fermentation.

Structural identifiability of impedance spectroscopy fractional-order equivalent circuit models with two constant phase elements

Structural identifiability analysis of fractional-order equivalent circuit models (FO-ECMs) obtained through electrochemical impedance spectroscopy (EIS) remains as a challenging problem. Aside from practical challenges such as measurement noise and the selection of excitation signals, no widely accepted analytical or numerical proof addressing the structural identifiability of impedance spectroscopy FO-ECMs exists. Through the use of coefficient mapping technique, this paper proposes novel computationally-efficient algebraic equations for numerical structural identifiability analysis of a widely used FO-ECM with Grünwald–Letnikov fractional derivative approximation and two constant phase elements (CPEs) including the Warburg term. The effects of the length of the data and sampling time on the proposed method for the structural identifiability analysis of the two-CPE FO-ECMs are discussed along with two examples and a discussion of the results.

Modeling the uncertainty in epidemiological models through interval analysis considering actual data from two municipalities in Colombia affected by dengue

Epidemiological models have become powerful tools for studying and understanding the characteristics and impact of transmitted diseases in a population. However, these models usually require specifying several values of input parameters obtained from experimental data, characterized by high uncertainty levels due to biological variation. This situation is evident for models that simulate the transmission of vector-borne diseases such as dengue, our case study. Therefore, treating and modeling this uncertainty is essential to ensure the robustness of designed models. For this, we propose to model the uncertainty through interval analysis by representing the input parameters and initial conditions by real closed intervals in the forward problem. This approach has the advantage of making a minimal number of assumptions concerning uncertainties, unlike the traditional methods (probabilistic and fuzzy). To illustrate the performance of this methodology, we consider a coupled ODE system of seven state variables and nine parameters, representing the transmission of Dengue between host-vector populations. Additionally, to enhance the use of the numerical method utilized for solving the system, the uncertain quantities (parameters and initial conditions) are determined based on the results of (i) the sensitivity analysis of R0, (ii) the structural identifiability analysis of the model, (iii) the characteristics of the available information about mosquito population, and (iv) dengue incidence data in two municipalities in Colombia, Itagüí and Neiva, during the outbreaks in 2016. We believe that the methodology proposed here to select and incorporate uncertainty in epidemiological models through interval analysis is widely applicable to other phenomena and models in science and engineering.

Practical identifiability of parametrised models: A review of benefits and limitations of various approaches

This systematic review of practical identifiability (PI) explores the challenging issue of how parameter identification of models is affected by both experimental considerations and model structure. Structural identifiability (SI) analyses that yield binary assessment of parameter uniqueness have been historically dominant in the field. However, recent developments in the less explored PI domain have facilitated more nuanced estimates of identified model parameter trade-off and variance. As PI acknowledges variation in parameter estimates due to real-world limitations in data quality and quantity, it can both explore how parameters may trade-off, and guide more informative experimental design.In this review, PI analysis methodologies used across various fields of study are compared, and their role in aiding experimental design is discussed. The methods presented show that the choice of PI approach requires careful consideration based on the modelling context and desired research outcomes. Illustrative examples are included for common methodologies, and some ongoing research is briefly reviewed. Overall, the concept of PI brings value to model-based analyses across a broad range of disciplines.

Parametrization of physics-based battery models from input–output data: A review of methodology and current research

Physics-based battery models are important tools in battery research, development, and control. To obtain useful information from the models, accurate parametrization is essential. A complex model structure and many unknown and hard-to-measure parameters make parametrization challenging. Furthermore, numerous applications require non-invasive parametrization relying on parameter estimation from measurements of current and voltage. Parametrization of physics-based battery models from input–output data is a growing research area with many recent publications. This paper aims to bridge the gap between researchers from different fields that work with battery model parametrization, since successful parametrization requires both knowledge of the underlying physical system as well as understanding of theory and concepts behind parameter estimation. The review encompasses sensitivity analyses, methods for parameter optimization, structural and practical identifiability analyses, design of experiments and methods for validation as well as the use of machine learning in parametrization. We highlight that not all model parameters can accurately be identified nor are all relevant for model performance. Nonetheless, no consensus on parameter importance could be shown. Local methods are commonly chosen because of their computational advantages. However, we find that the implications of local methods for analysis of non-linear models are often not sufficiently considered in reviewed literature.

Structural identifiability analysis of nonlinear time delayed systems with generalized frequency response functions

Structural identifiability analysis of nonlinear time delayed systems with generalized frequency response functions