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Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood
Open accessAbstract
MOTIVATION: Mathematical description of biological reaction networks by differential equations leads to large models whose parameters are calibrated in order to optimally explain experimental data. Often only parts of the model can be observed directly. Given a model that sufficiently describes the measured data, it is important to infer how well model parameters are determined by the amount and quality of experimental data. This knowledge is essential for further investigation of model predictions. For this reason a major topic in modeling is identifiability analysis. RESULTS: We suggest an approach that exploits the profile likelihood. It enables to detect structural non-identifiabilities, which manifest in functionally related model parameters. Furthermore, practical non-identifiabilities are detected, that might arise due to limited amount and quality of experimental data. Last but not least confidence intervals can be derived. The results are easy to interpret and can be used for experimental planning and for model reduction. AVAILABILITY: An implementation is freely available for MATLAB and the PottersWheel modeling toolbox at http://web.me.com/andreas.raue/profile/software.html. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.
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TL;DR
The paper presents a profile-likelihood method for assessing structural and practical identifiability in partially observed dynamical models and for deriving likelihood-based confidence intervals. Applied to a JAK-STAT signaling model, it identifies four structurally non-identifiable parameters and one practically non-identifiable parameter, and shows how additional measurements can resolve these ambiguities.
Key findings
- Profile likelihood distinguishes structural non-identifiability, seen as a flat profile, from practical non-identifiability, seen as a profile with a minimum that fails to cross the confidence threshold in one or both directions; identifiable parameters cross the threshold on both sides.
- In the JAK-STAT application, parameters p2, x1(0), s1, and s2 were structurally non-identifiable, with functional relations p2·x1(0)=const, s1·x1(0)=const, and s2·x1(0)=const; p3 was practically non-identifiable for increasing values.
- Table 1 reports infinite likelihood-based intervals for the structurally non-identifiable parameters, whereas Hessian-based asymptotic intervals were finite but misleadingly narrow, e.g. p2: −33 to +31, x1(0): −31 to +32, s1: −32 to +31, s2: −32 to +31; for p3, the profile-likelihood interval was −1.14 to +∞ versus −1.14 to +0.15 from the Hessian.
- Adding hypothetical measurements of x1(0)=200±20 nM and x3/(x2+x3)=0.90±0.05 at t=20 min made p2, x1(0), s1, and s2 structurally identifiable and p3 practically identifiable, while p2 remained practically non-identifiable. Profile-likelihood computation took 54±18 s per parameter on a 1 GHz CPU with 2 GB RAM.
Methods
- The approach uses an ODE model with weighted least-squares objective χ², computes the profile likelihood for each parameter by re-optimizing all other parameters at fixed values of the target parameter, and derives likelihood-based confidence intervals from χ² thresholds.
- The application is a JAK-STAT signaling model inspired by Swameye et al., calibrated to experimental data; observables are phosphorylated STAT in cytoplasm (y1) and total STAT in cytoplasm (y2) measured by quantitative western blotting, with cytoplasm and nucleus volumes of 1400 µm³ and 450 µm³.
- Parameters calibrated include p1–p4, s1, s2, and x1(0), with x2(0)–x4(0) set to zero; the model was implemented and fitted with PottersWheel, and experimental planning and model reduction were explored via profile-likelihood trajectories, Monte Carlo simulations, and likelihood-ratio thresholds.
Limitations
- The authors state that structural non-identifiability cannot be resolved by refining existing measurements or reducing noise; it requires a qualitatively new measurement that changes the observation mapping, such as observing additional species. Practical non-identifiability can be resolved by increasing the amount or quality of data and/or selecting better measurement time points.
- The authors note that recovering functional relations unambiguously from parameter changes along the profile likelihood is only possible when the corresponding manifold is one-dimensional; higher-dimensional manifolds require further analysis, such as the approach of Hengl et al.
- Evident from the text: the method is demonstrated on a single JAK-STAT model and dataset, with no broad empirical comparison across many model classes; the approach also assumes a sufficiently adequate model and normally distributed measurement noise.
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35 Own interpretation · 14 Own result · 6 No full text of the sources · 6 No basis given · 1 Partly supported · 1 Sources not found · 1 Not checked yet · accuracy not yet measured
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“The largest discrepancy occurs for practically non-identifiable parameters, where asymptotic confidence intervals are significantly smaller than likelihood-based confidence intervals.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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“The variability of the predicted model observables y along the profile likelihood of p 3 remains consistent with the measurement errors, as shown in Figure 6.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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“The resulting plots of profile likelihood versus parameter reveal four structurally non-identifiable parameters p 2 , x 1 (0), s 1 , s 2 by their flat profile likelihood, see Figure 3, upper panel.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 5 APPLICATION
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“For identifiable parameters the discrepancies are small.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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“The functional relations h connecting these structurally non-identifiable parameters can be recovered from the changes of the remaining parameters, while calculating the profile likelihood of the structurally non-identifiable parameters, as shown in Figure 4.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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“The recalculated profile likelihood reveals, that parameters p 2 , x 1 (0), s 1 , s 2 become structurally identifiable (Fig. 3, lower panel), by measuring only one of them.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Experimental planning
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“The profile likelihood reveals that parameter p 3 is practically non-identifiable for increasing parameter values.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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“Nevertheless, a lower confidence bound can be derived.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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“In this case, asymptotic approximation leads to large but nevertheless finite confidence intervals for the structurally non-identifiable parameters.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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“All internal model states can only be identified up to a common factor.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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In the paper
“An approach to detect structural non-identifiability by the corresponding functional relations was introduced by Hengl et al. (2007).It is able to detect flatness of likelihood for arbitrary models, but it is not intended to detect practical non-identifiability.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 3 EXISTING METHODS
What the sources say
- Partly supportedcites Hengl et al., 2007 sourceExperimental — accuracy not yet measured
The Approach and Conclusion describe detecting structural non-identifiability through functionally related parameters in nonlinear models. But the Introduction explicitly lists practical identifiability analysis, accounting for data quality, as an aim of the method, contrary to the claim that it is not intended for that purpose. The passages do not specifically describe the method as detecting likelihood flatness. (LLM, conf=0.90)
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“Spots of largest variability suggest where and when a measurement of a species most efficiently determines theses parameters: x 1 at times t = 0 or t > 50; x 2 to x 4 at times 5 < t < 30.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Experimental planning
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“Calculating the profile likelihood takes 54 ± 18 s per parameter, using an implementation of the approach embedded in the Potters-Wheel fitting toolbox (1 GHz CPU, 2 GB RAM), which is described in the Supplementary Material.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 5 APPLICATION
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“Parameter p 3 becomes practically identifiable, only previously structurally non-identifiable parameter p 2 remains practically non-identifiable.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Experimental planning
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“The trajectories of the internal model states x along the profile likelihood of the practically non-identifiable parameter p 3 shown in Figure 6 comprises large antipodal variability of species x 2 and x 3 , revealing that the experimental setup is inappropriate to estimate this parameter with confidence.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Experimental planning
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“h 1 (θ sub ) = p 2 •x 1 (0)-const = 0 h 2 (θ sub ) = s 1 •x 1 (0)-const = 0 ( 1 1 ) h 3 (θ sub ) = s 2 •x 1 (0)-const = 0”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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In the paper
“The approach was applied to a model of the JAK-STAT signaling pathway.Non-identifiable parameters were detected, revealing limitations in the experimental setup.Additional measurements that efficiently improve parameter identification were suggested and validated.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 6 DISCUSSION
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“They are exact if the observables y depend linearly on θ.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 2.2.1 Asymptotic confidence intervals
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“Unfortunately, these methods become rapidly infeasible with increasing model size (Margaria et al., 2001;White et al., 2001).Practical non-identifiability cannot be detected, since experimental data are disregarded.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 3 EXISTING METHODS
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“Even identifiable parameters can only be determined within confidence intervals, which contain the true value of the parameter with a desired probability (Lehmann and Leo, 1983).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 1 INTRODUCTION
What the sources say
- Not matchedcites Lehmann and Leo, 1983
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In the paper
“These methods are appropriate if functional relations h between the parameters emerging from structural non-identifiability are linear.This is often not the case for reaction networks modeled by ODE, because observables depend non-linearly on the parameters.Practical non-identifiability cannot be detected, because a quadratic approximation is not able to explain increasing but limited behavior of”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 3 EXISTING METHODS
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In the paper
“However, even for the simplest reaction network the observables y depend nonlinearly on θ and the amount and quality of experimental data is often insufficient.Therefore, asymptotic confidence intervals might not be appropriate (Joshi et al., 2006).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 2.2.1 Asymptotic confidence intervals
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- Not checked — no full text of the sourcecites Joshi et al., 2006 source
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“Likelihood-based confidence intervals are considered superior to asymptotic confidence intervals for finite samples (Neale and Miller, 1997).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 2.2.2 Finite sample confidence intervals
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- Not checked — no full text of the sourcecites Neale and Miller, 1997 source
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In the paper
“Along a practical non-identifiability, the observable y change only negligibly remaining compliant with the given measurement accuracy.Nevertheless, model behavior in terms of internal states x might vary strongly.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > )]:
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“This arises frequently if amount and quality of experimental data is insufficient and manifests in a confidence interval that is infinite.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 2.3.2 Practical non-identifiability
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In the paper
“The points of passover represent likelihoodbased confidence intervals as defined in Equation ( 8) (Royston, 2007).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 4 APPROACH
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- Not checked — no full text of the sourcecites Royston, 2007 source
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In the paper
“The results are easy to interpret”
- Also stated in: § 6 DISCUSSION · plain
“The approach results in easily interpretable plots of profile likelihood versus parameter.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Abstract
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- Also stated in: § 6 DISCUSSION · plain
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In the paper
“The profile likelihood of a practically non-identifiable parameter has a minimum, but is not excessing a threshold α for increasing and/or decreasing values of θ i (see Definition 1).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 4 APPROACH
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“This gives insight into a complex and high-dimensional parameter space.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 6 DISCUSSION
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“It can be automated, but an explicit advantage is that the output might be evaluated visually.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 6 DISCUSSION
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In the paper
“The approach can be applied to any parameter estimation problem, where a likelihood or a similar objective criterion is available, e.g.partial differential equations (PDE) or stochastic differential equations (SDE).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 6 DISCUSSION
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In the paper
“The profile likelihood can be calculated for each parameter separately.Thereby it is possible to restrict the analysis to the parameters relevant for the biological question.Moreover, this allows to perfectly parallelize the approach, which is a major benefit for its scalability.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 6 DISCUSSION
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In the paper
“Working in logarithmic parameter space linearizes the functional relations given in Equation (11).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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In the paper
“The impact of new measurements can be evaluated by Monte Carlo simulations.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Experimental planning
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In the paper
“Thus, identifiability analysis ensures that the model complexity is tailored to the information content given by the experimental data.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 6 DISCUSSION
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The authors state this as an assumption.
The measurement noise is normally distributed.
“The measurement noise (t) is assumed to be normally distributed.”
The authors state this as an assumption.
An appropriate model M that sufficiently describes the available experimental data is assumed.
“Furthermore, an appropriate model M that sufficiently describes the available experimental data is assumed.”
The paper relies on this without support.
No approach for explicit testing of practical non-identifiability is available.
“Similar to a clear formal definition, an approach for explicit testing of practical non-identifiability is not available to our knowledge.”
The authors state this as an assumption.
The initial values of species x2 to x4 are equal to zero.
“The initial values of species x 2 to x 4 are assumed to be equal to zero.”
The authors state this as an assumption.
For the model-reduction argument, a parameter θi is practically non-identifiable for decreasing parameter value.
“Assume that a parameter θ i is practically non-indentifiable for decreasing parameter value.”
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“Such parameter estimations usually yield statistical information related to parameter precision (importantly, not accuracy) such as standard errors or correlation matrices, and sometimes information derived from more sophisticated approaches such as profile likelihood.39”
From Best Practices to Maximize the Use and Reuse of Quantitative and Systems Pharmacology Models: Recommendations From the United Kingdom Quantitative and Systems Pharmacology Network (2019) · cites this paper as [39]
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In the citing paper
“Raue et al. (2009) referred to likelihood ratio C.I.s as finite sample C.I.s, to underscore the superiority of likelihood ratio C.I.s over Wald-type C.I.s.”
From rSalvador: An R Package for the Fluctuation Experiment (2017) · cites this paper as Raue et al. 2009
In the citing paper
“When two models are nested (Mð hPL Þ and Mð hopt Þ) their likelihood ratio is approximately distributed according to a v 2 p distribution.Here p refers to the number of degrees of freedom which is defined as the difference in the number of parameters (which in this case is 1).”
From Parameter uncertainty in biochemical models described by ordinary differential equations (2013) · cites this paper as [36]
In the citing paper
“In such cases, more reliable confidence intervals can be obtained using the Profile Likelihood method [36].”
From Parameter uncertainty in biochemical models described by ordinary differential equations (2013) · cites this paper as [36]
In the citing paper
“Techniques for this exist in both a frequentist [13,36,91] as well as a Bayesian setting [87,90].”
From Parameter uncertainty in biochemical models described by ordinary differential equations (2013) · cites this paper as [36]
In the citing paper
“Fortunately, recent literature is correcting this (e.g. [129–140]).”
From Reverse engineering and identification in systems biology: strategies, perspectives and challenges (2014) · cites this paper as [133]
In the citing paper
“Given the data, not all parameters are identifiable (Raue et al., 2009).”
From A bayesian approach to targeted experiment design (2012) · cites this paper as Raue et al., 2009
In the citing paper
“Methods for an approximate sampling of the parameter space, e.g. the Markov Chain Monte Carlo (MCMC) methods [11,12], and bootstrap based approaches [4,13] are numerically demanding and provide only rough approximations for ODE models. Therefore, it is difficult to control the coverage of the prediction confidence intervals for such approaches.”
From Likelihood based observability analysis and confidence intervals for predictions of dynamic models (2012) · cites this paper as [4]
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Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 5 APPLICATION
“Species concentrations are modeled in nanometer.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
“time / min x 1 / nM”
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Generalised likelihood profiles for models with intractable likelihoods
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“Therefore, we conclude our method can be employed for the purposes of identifiability analysis for intractable likelihoods in the same way standard likelihood profiles are used when the likelihood is available (Raue et al., 2009).”
InfluentialKramer, A., & Radde, N. E. (2010) · Procedia Computer Science · doi:10.1016/j.procs.2010.04.184 · Cited by 8
Dufresne, E., Harrington, H. A., & Raman, D. V. (2018) · Journal of Algebraic Statistics · doi:10.18409/jas.v9i1.64 · Cited by 8
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MethodResultInfluentialDifferences in respiratory mechanics estimation with respect to manoeuvres and mathematical models
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MethodBayesian Calibration of Multistate Stochastic Simulators
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A Dynamic Mathematical Model of Bile Acid Clearance in HepaRG Cells
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“Practical nonidentifiability and parameter confidence bounds were analyzed using the profile likelihood method (Kreutz et al., 2013; Raue et al., 2009).”
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Digital Seed Train Twins and Statistical Methods
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Experimental Design in Dynamical System Identification: A Bandit-Based Active Learning Approach
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BackgroundStructural Identifiability Analysis via Extended Observability and Decomposition
Villaverde, A. F., Barreiro, A., & Papachristodoulou, A. (2016) · IFAC-PapersOnLine · doi:10.1016/j.ifacol.2016.12.121 · Cited by 8
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BackgroundMathematical Tools in Cancer Signalling Systems Biology
Vera, J., & Wolkenhauer, O. (2011) · Springer eBooks · doi:10.1007/978-94-007-1567-7_7 · Cited by 7
“Until now, our reduced ability to produce enough quantitative data together with some intrinsic mathematical problems of this approach such as identifiability ( Raue et al. 2009 ), or sloppiness (Gutenkunst et al. 2007), make it fully operative only for small-size mathematical models of cell signalling systems.”
BackgroundWhat if the Fit is Unfit? Criteria for Biological Systems Estimation beyond Residual Errors
Voit, E. O. (2011) · doi:10.1002/9783527638079.ch9 · Cited by 8
Computational Systems Biology in Disease Modeling and Control, Review and PerspectivesPreprint
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Modeling resistance to the broadly neutralizing antibody PGT121 in people living with HIV-1
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“Simply, the parameter θ j , i is practically identifiable if the confidence interval is finite [49, 50, 53].”
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Mathematical Modeling of Influenza Dynamics: Integrating Seasonality and Gradual Waning Immunity
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“Since in this case we only vary the value of the parameter β 0 we are performing a practical identifiability by using the profile likelihood (Kreutz et al. 2012; Raue et al. 2009, 2013).”
Dynamic causal modelling in probabilistic programming languages
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Optimal Experimental Design Based on Two-Dimensional Likelihood Profiles
Litwin, T., Timmer, J., & Kreutz, C. (2022) · Frontiers in Molecular Biosciences · doi:10.3389/fmolb.2022.800856 · Cited by 8
“Confidence intervals generated by the profile likelihood approach have more desirable properties in the finite sample case (Meeker and Escobar, 1995) and allow for the conceptional and operational definition of practical identifiability (Raue et al., 2009).”
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Longitudinal single-cell data informs deterministic modelling of inflammatory bowel disease
Kilian, C., Ulrich, H., Zouboulis, V. A., Sprezyna, P., Schreiber, J., Landsberger, T., Büttner, M., Biton, M., Villablanca, E. J., Huber, S., et al. (2024) · npj Systems Biology and Applications · doi:10.1038/s41540-024-00395-9 · Cited by 8
Achieving Occam’s razor: Deep learning for optimal model reduction
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Dynamic modelling of signalling pathways when ordinary differential equations are not feasible
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“Thus, for the estimation of confidence intervals (CIs), we apply the profile likelihood approach (Raue et al. 2009), which has been proven to account for non-linearity and is one of the most frequently applied approaches for pathway models (Kreutz et al. 2013).”
Self-citationMathematical modeling and analysis of leptospirosis–COVID-19 co-infection with real data
Said, M., Roh, Y., & Jung, I. H. (2024) · The European Physical Journal Plus · doi:10.1140/epjp/s13360-024-05846-0 · Cited by 7
Mechanistic models of humoral kinetics following COVID-19 vaccination
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Identifiability of population models via a measure theoretical approach
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“While there is a broad literature on identifiability of single cell models (Farina et al., 2006; Raue et al., 2009), we are not aware of any results concerning identifiability of the population models discussed above.”
BackgroundModeling and Model Simplification to Facilitate Biological Insights and Predictions
Eriksson, O., & Tegner, J. N. (2015) · Studies in mechanobiology, tissue engineering and biomaterials · doi:10.1007/978-3-319-21296-8_12 · Cited by 7
Towards precise reconstruction of gene regulatory networks by data integration
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PARAMETER IDENTIFIABILITY THROUGH INFORMATION THEORY
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“matrix [10, 5] or confidence intervals [11].”
Dirac mixture distributions for the approximation of mixed effects models
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“using parametric bootstraps or profile likelihoods (Raue et al., 2009), is planned.”
MethodMechanistic inference of the metabolic rates underlying $$^{13}$$C breath test curves
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“nificance a is fhi : logðLðĥÞÞ logðLðĥj61⁄4i; hiÞ\Dag, where Da is v(2)ða; dfÞ=2 where v(2)ða; dfÞ is the chi-squared distribution with a number of degrees of freedom equal to the number of parameters and a is the level of significance [25].”
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Predictive power of non-identifiable models
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“The prior parameter distributions are LogNormal(μ = 1, σ = 3) clipped to [0, 104], which gives δi,prior ≈ exp(3) ≈ 20 (for all i), implying that “on average” all prior parameter values can vary 20-fold from the median.”
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Clinical Validation of the Quick Dynamic Insulin Sensitivity Test
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“However, the concurrent identification of SI and glucose sensitivity SG terms in the minimal model becomes practically nonidentifiable [13], [14] in some clinically relevant cohorts [13], [15], [16], despite the”
BackgroundMathematical Modeling of Biochemical Systems with PottersWheel
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Self-citationBatch Cultivation Model for Biopolymer Production
Torres-Cerna, C. E. (2017) · Chemical and Biochemical Engineering Quarterly · doi:10.15255/cabeq.2016.952 · Cited by 7
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Relative sensitivity analysis of the predictive properties of sloppy models
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Covid-19 in Deutschland – Erklärung, Prognose und Einfluss gesundheitspolitischer Maßnahmen
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Self-citationA quantitative map of nuclear pore assembly reveals two distinct mechanismsPreprint
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Quantitative Metabolic Flux Analysis Based on Isotope Labeling
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Structural and practical identifiability of contrast transport models for DCE-MRI
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“Contrary to structural non-identifiability due to the inherent model parameterization, the above issues typically occur when dealing with real data, hence the name of practical non-identifiability [34].”
InfluentialBultereys, V., Matsunami, K., Descamps, L., et al. (2025) · Pharmaceutics · doi:10.3390/pharmaceutics17091197 · Cited by 6
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A Step‐by‐Step Workflow for Performing In Silico Clinical Trials With Nonlinear Mixed Effects Models
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Identifiability analysis for models of the translation kinetics after mRNA transfection
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Optimized Bayesian system identification in quantum devices
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A Comprehensive Guideline for Hybrid Modeling of Engineered Microbial Processes
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“with δα being the α quantile of the χ2 distribution with df = 1 (pointwise) or df = nθ (simultaneous) degrees of freedom [9].”
BackgroundMethodInfluentialIdentifiability Analysis of a Pressure-Depending Alveolar Recruitment Model
Schranz, C., Docherty, P. D., Chiew, Y. S., et al. (2012) · IFAC Proceedings Volumes · doi:10.3182/20120829-3-hu-2029.00015 · Cited by 6
“In these regions, changes in SSE are negligible, despite significant variation in model parameter values (Raue et al., 2009).”
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“(35) or generalized sensitivity functions of Thomaseth and Cobelli (36) [extended to multiple output models by Kappel and Munir (37)].”
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Modeling and control in physiology
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Genetic algorithm applied to simultaneous parameter estimation in bacterial growth
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The Impact of Exogenous Insulin Input on Calculating Hepatic Clearance Parameters
McHugh, A. D., Chase, J. G., Knopp, J. L., Ormsbee, J. J., Kulawiec, D. G., Merry, T. L., Murphy, R., Shepherd, P. R., Burden, H. J., & Docherty, P. D. (2021) · Journal of Diabetes Science and Technology · doi:10.1177/1932296820986878 · Cited by 6
Regularization and concave loss functions for estimation of chemical kinetic models
Opara, K. R., & Oh, P. P. (2021) · Applied Soft Computing · doi:10.1016/j.asoc.2021.108286 · Cited by 6
Determining robust reaction kinetics from limited data
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Combining multiple data sources with different biases in state‐space models for population dynamics
Polansky, L., Mitchell, L., & Newman, K. B. (2023) · Ecology and Evolution · doi:10.1002/ece3.10154 · Cited by 5
“SSMs seem to have a high potential for being nearly redundant, with model parameters esti-mable in theory that are practically non-identifiable with unbounded confidence intervals (Raue et al., 2009).”
Phenotypic evolution of SARS-CoV-2: a statistical inference approach
Benhamou, W., Lion, S., Choquet, R., & Gandon, S. (2023) · Evolution · doi:10.1093/evolut/qpad133 · Cited by 5
Accurately summarizing an outbreak using epidemiological models takes time
Case, B., Young, J.-G., & Hébert‐Dufresne, L. (2023) · Royal Society Open Science · doi:10.1098/rsos.230634 · Cited by 7
Studying Cellular Signal Transduction with OMIC Technologies
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Model-Based Computation of Total Stressed Blood Volume from a Preload Reduction Experiment
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“Using such data, the model is practically non-identifiable [25].”
MethodReconstruction of promoter activity statistics from reporter protein population snapshot data
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Doeschl‐Wilson, A., Wilson, A., Nielsen, J. P., Nauwynck, H., Archibald, A. L., & Aït-Ali, T. (2016) · BMC Systems Biology · doi:10.1186/s12918-016-0345-5 · Cited by 7
“By definition, mechanistic models representing processes that are difficult to measure are often over-parameterized resulting in poorly identifiable or non-identifiable model parameters [29, 30].”
BackgroundAyabina, D. V., Hendon-Dunn, C. L., Bacon, J., & Colijn, C. (2016) · Journal of The Royal Society Interface · doi:10.1098/rsif.2016.0745 · Cited by 6
Extending existing structural identifiability analysis methods to mixed-effects models
Janzén, D. L. I., Jirstrand, M., Chappell, M. J., & Evans, N. D. (2017) · Mathematical Biosciences · doi:10.1016/j.mbs.2017.10.009 · Cited by 6
“bility analysis including the Taylor series approach [3], the Laplace transformation approach [4], the similarity transformation approach [5], the Exact Arithmetic Rank (EAR) approach [6], differential algebra based approaches [7], input-output approaches [8], and the profile likelihood approach [9].”
MethodReconstructing Complex Fluid Properties from the Behavior of Fluctuating Immersed Particles
Hohenegger, C., & McKinley, S. A. (2018) · SIAM Journal on Applied Mathematics · doi:10.1137/17m1131660 · Cited by 6
Zazueta, C. L., Stavdahl, Ø., & Fougner, A. L. (2021) · IEEE Transactions on Biomedical Engineering · doi:10.1109/tbme.2021.3125839 · Cited by 6
“Indeed, in typical Systems Biology applications likelihood is equivalent to the least-squares criterion [41, 42].”
BackgroundMethodInfluentialAlgebra, Geometry and Topology of ERK Kinetics
Marsh, L., Dufresne, E., Byrne, H. M., & Harrington, H. A. (2022) · Bulletin of Mathematical Biology · doi:10.1007/s11538-022-01088-2 · Cited by 6
“Real data is often noisy; testing whether parameter recovery is possible with imperfect data is the problem of practical identifiability [68, 21].”
BackgroundMethodInfluentialGanegoda, N. C., Wijaya, K. P., Chávez, J. P., Aldila, D., Erandi, K. K. W. H., & Amadi, M. (2021) · Nonlinear Dynamics · doi:10.1007/s11071-021-07111-w · Cited by 6
“As the parameter dimension is much smaller than the data size, the standard asymptotic confidence interval [73] has been suggested to delineate the parameter uncertainty [74,75].”
BackgroundA variational approach to parameter estimation in ordinary differential equations
Kaschek, D., & Timmer, J. (2012) · BMC Systems Biology · doi:10.1186/1752-0509-6-99 · Cited by 7
“Commonly, if measurements for the inputs are available, non-parametric estimates like smoothing splines are employed to describe the input data [2,3].”
MethodSelf-citationQi, F., & Wright, M. M. (2016) · AIMS energy · doi:10.3934/energy.2016.1.52 · Cited by 6
An Algorithm and R Program for Fitting and Simulation of Pharmacokinetic and Pharmacodynamic Data
Li, J., Yan, K., Hou, L., et al. (2016) · European Journal of Drug Metabolism and Pharmacokinetics · doi:10.1007/s13318-016-0358-x · Cited by 6
“Moreover, if the modeling is performed with sparse data, the STS method may run into practically unidentifiable parameters or problems resulting from the model selection [34, 35].”
MethodStructural and practical identifiability analysis of S‐system
Zhan, C., Li, B. Y. S., & Yeung, L. F. (2015) · IET Systems Biology · doi:10.1049/iet-syb.2015.0014 · Cited by 6
Structural Identifiability Analysis of Fractional Order Models with Applications in Battery SystemsPreprint
Alavi, S. M. M., Mahdi, A., Jacob, P. E., Payne, S. J., & Howey, D. A. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1511.01402 · Cited by 6
“Practical identifiability, on the other hand, takes into account the practical aspects of the problem that come with real data including noise, bias and signal quality such as the shape of excitation, its magnitude, length, frequency ranges, etc, [8].”
Background(2025) · doi:10.1093/bioinformatics/btaf581 · Cited by 6
“In general, parameters are considered structurally identifiable when a parameter set maps to a unique model output (Raue et al. 2009, Komorowski et al. 2011).”
Parameter Estimation in Hybrid Machine Learning and Mechanistic Models of Infectious Diseases
Jhutty, S. S., & Hernandez-Vargas, E. A. (2022) · IFAC-PapersOnLine · doi:10.1016/j.ifacol.2022.09.020 · Cited by 5
Gao, M., Barile, M., Chabra, S., Haltalli, M. L. R., Calderbank, E. F., Chao, Y., Zheng, W., Wilson, N. K., Laurenti, E., Göttgens, B., et al. (2025) · Nature Communications · doi:10.1038/s41467-025-63150-6 · Cited by 16
On linear models and parameter identifiability in experimental biological systems
Lamberton, T. O., Condon, N. D., Stow, J. L., & Hamilton, N. (2014) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2014.05.028 · Cited by 5
“…is generally approached by developing confidence intervals for estimated parameters and considering the parameter space of an objective function near solutions, either using quadratic approximations or numerical methods (Cobelli and DiStefano, 1980; Raue et al., 2009, Berthoumieux et al., 2013).”
MethodUsing rheological monitoring to determine the gelation kinetics of chitosan-based systems
Duarte, B. P. M., & Moura, M. J. (2022) · Mathematical Biosciences & Engineering · doi:10.3934/mbe.2023054 · Cited by 5
“The computation of the variance-covariance and correlation matrices of the parameters is based on the procedures in [43, 44].”
MethodHaus, E. S., Drengstig, T., & Thorsen, K. (2023) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1011398 · Cited by 5
Anderson, H. G., Takacs, G. P., Harrison, J. K., Rong, L., & Stepien, T. L. (2024) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2024.111951 · Cited by 5
A new statistical model with simulation study and reliability analysis of an audio and podcast tool
Yuan, H. (2025) · Alexandria Engineering Journal · doi:10.1016/j.aej.2024.12.044 · Cited by 5
Metadata
- Authors
- Raue, A., Kreutz, C., Maiwald, T., Bachmann, J., Schilling, M., Klingmüller, U., & Timmer, J.
- Affiliations
- University of Freiburg (DE)
- Corresponding author
- Raue, A.
- Venue
- Bioinformatics
- Publisher
- Oxford University Press (OUP)
- Published
- 2009-06-08
- Volume
- 25
- Issue
- 15
- Pages
- 1923-1929
- ISSN
- 1367-4803 (ISSN-L)
- 1367-4811
- OpenAlex
- W2168648059
- Cited by
- 1,442
- References
- 29
- Citations by year
- 2026: 209 · 2025: 107 · 2024: 103 · 2023: 82 · 2022: 98 · 2021: 118 · 2020: 97 · 2019: 116 · 2018: 94 · 2017: 78 · 2016: 87 · 2015: 64 · 2014: 52 · 2013: 54 · 2012: 39
- FWCI
- 10.99
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Bioinformatics (publishedVersion)
- PubMed (publishedVersion)
- OPUS (Augsburg University) (submittedVersion)
- Funders
- Deutsches Krebsforschungszentrum
- European Commission (223188, EU-FP7, FP7 HEALTH)
- Bundesministerium für Bildung und Forschung (FRISYS 0313921, 0313074D, 0313921)
- FP7 Health
- Topics
- Life Sciences › Biochemistry, Genetics and Molecular Biology › Molecular Biology › Gene Regulatory Network Analysis
- Physical Sciences › Engineering › Control and Systems Engineering › Control Systems and Identification
- Life Sciences › Biochemistry, Genetics and Molecular Biology › Molecular Biology › Microbial Metabolic Engineering and Bioproduction
- Keywords
- identifiability analysis, profile likelihood, parameter estimation, confidence intervals, experimental design, model reduction
- MeSH terms
- Algorithms, Models, Biological, Probability, Computational Biology
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-10-11, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-11 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Cited sources are checked with a language model’s help (gpt-6-sol).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
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