On structural and practical identifiability

We discuss issues of structural and practical identifiability of partially observed differential equations which are often applied in systems biology. The development of mathematical methods to investigate structural non-identifiability has a long tradition. Computationally efficient methods to detect and cure it have been developed recently. Practical non-identifiability on the other hand has not been investigated at the same conceptually clear level. We argue that practical identifiability is more challenging than structural identifiability when it comes to modelling experimental data. We discuss that the classical approach based on the Fisher information matrix has severe shortcomings. As an alternative, we propose using the profile likelihood, which is a powerful approach to detect and resolve practical non-identifiability.

be fitted. Often, this leads to an over-parameterised model that over-fits the data. The parameters of such a model and in turn its predictions are not well-determined and it thus remains a bad model.
The path from such a bad model towards a good model is laborious: additional data needs to be measured and integrated, the model complexity needs to be reduced and balanced to the available data, or a combination of both.
This process needs to be iterated until a good model is found, which has well determined parameters and predictions.
However, such a good model also needs to deliver biological insights in order to be useful. Only this third property turns a good model into a useful model. In this sense, the final goal of mathematical modelling in systems biology is not the model itself but to use the model to understand biology. One example of how a model can be used to gain biological insight, which would be unattainable by merely assessing the data by itself, was given by Becker et al. [2].

Parameter identifiability
The concept of identifiability is strongly linked to the transition from bad models to good models. Identifiability analysis is necessary to create good models that can describe the data and have well-determined parameters and predictions. It is especially important when modelling biological systems because the limited amount and quality of the experimental data with large measurement noise in only partially observed systems often leads to bad models during the modelling process. Concerning identifiability, one distinguishes between structural identifiability dealing with inherently indeterminable parameters due to the model structure itself, and practical identifiability, dealing with insufficiently informative measurements to determine the parameters with adequate precision.

Partially observed dynamical systems
A biological system is translated into ordinary differential equations (ODEs) comprising n model states x(t), unknown parameters p to be estimated from time-resolved experimental data, and external stimuli u(t). Since data is often recorded on a relative scale, scaling and offset parameters for background corrections need to be estimated in parallel. Furthermore, in typical applications not all components of a cell-biological system can be measured, e.g. because of the limited availability or restricted capability of antibodies to discriminate between un-phosphorylated, i.e. inactive, and phosphorylated, i.e. active, proteins. Thus, an observation function g(·) is required that maps the internal states x to the observations: Typically, the dimension m of y is smaller than the dimension n of x. We are therefore dealing with parameter estimation in partially observed systems. Moreover, in systems biology, these ODE models are typically stiff, nonlinear, sparse and non-autonomous, and the discrete time observations are noisy.
Parameter estimation is usually performed based on the weighted residual sum of squares, the negative loglikelihood assuming Gaussian errors to determine the agreement of experimental data with the model trajectories, where y D kl and σ D kl represent d k data points and measurement errors at time points t l for each observable. A common point estimate for the best parameter vector is the maximum likelihood estimator

Structural identifiability
Definition of structural identifiability and connection to observability Partially observed dynamical systems often exhibit structural non-identifiability. Structural identifiability is the ability to uniquely estimate parameters from any given model output. A parameter p i is globally structurally identifiable [3], if for all parameter vectors p, it holds An individual parameter p i is structurally non-identifiable, if changing the parameter does not alter the model trajectory y, because the changes can be fully compensated by altering other parameters. Local structural identifiability of a parameter is defined by reducing the definition to a neighbourhood v(p) instead of the entire parameter space. A model is structurally identifiable, if all of its parameters are structurally identifiable. Multiple related definitions for structural identifiability exist, for a comprehensive discussion see a recent overview [4].
A structurally non-identifiable parameter implies the existence of a manifold in parameter space upon which the trajectory y is unchanged. However, on this manifold the dynamic variables x of the model can change, e.g. by a scaling factor, and are thus not uniquely determinable. This is denoted as non-observability, a concept closely related to parameter non-identifiability [5,6,7,8,9].

A priori analysis of structural identifiability
Two basic approaches exist to assess structural identifiability of non-linear dynamic models. A priori methods only use the model definition, while a posteriori methods use the available data to find non-identifiable parameters.
Many a priori algorithms have been developed based on a variety of approaches. Powerful methods use Lie group theory, since non-identifiabilities are closely related to symmetries in the system [10,11,12,13]. Furthermore, a variety of notable methods exist, which are based on power series expansion [14], generating series [15,16], seminumerical approaches [17,18], differential algebra [19,20,21,22,23,24,25,26,27], differential geometry [28], and numerical algebraic geometry [29]. For reviews of some of these approaches, see [28,30,31]. Many of these approaches, especially the early developed methods, can only be applied to rather low-dimensional systems because of their computational complexity. Thus, recent developments have mainly focused on improving the computational efficiency of the algorithms, e.g. by local sensitivity calculations.
As a promising example, Joubert et al. [32] proposed a comprehensive and computationally fast pipeline to cure structural non-identifiabilities by re-parameterisation of the model in a five-step procedure: (i) a numerical identifiability analysis based on sensitivities, (ii) symbolic identifiability calculations for the low-dimensional candidates from (i), this renders the procedure fast, (iii) computation of new model parameters, this step is not unique, but requires decisions of the modeler, (iv) simplify the original model leading to a lower dimensional parameter vector, and finally (v) check the identifiability of the re-parameterised model. In an application to a model with 21 states and 75 parameters, two groups of non-identifiable parameters were detected and the model was re-parameterised within minutes.

Analysis of structural identifiability using experimental data
In contrast to the aforementioned methods, a posteriori methods use the available data to perform identifiability analysis. They infer structural non-identifiability based on model fits to experimental data. Similar to some sensitivitybased a priori approaches, these approaches only assess local structural identifiability.
One approach by Hengl et al. [33] suggested to perform numerous fits and investigate non-parametrically whether the final parameter estimates form a low-dimensional manifold in parameter space. This approach also allows to disentangle different sets of coupled non-identifiable parameters.
An informative and successful method is based on the profile likelihood [34]. The idea of the profile likelihood is to vary one parameter p i after the other around the maximum likelihood estimate (Equation (4)) and re-optimise the remaining ones For the two-parameter examples in Figure 1, the blue dashed lines show the path in the parameter space determined by Equation (6). Figure 1A shows the profile likelihood of an identifiable parameter. For a structurally non-identifiable parameter the profile likelihood yields a flat line as shown in Figure 1B. Plotting the remaining parameters along the profiled parameter reveals which parameters are coupled to the non-identifiable one [35]. The profile likelihood was recently extended to include two-dimensional profiles to allow for the identification of parameter interdependence [36].
Profile likelihood calculation can be computationally demanding for larger systems due to the numerical reoptimisation. Addressing this issue, a fast a posteriori method to test identifiability without the need to calculate complete profiles using radial penalisation was recently developed [37]. Structural non-identifiability can also be investigated a posteriori by a Bayesian Markov chain Monte Carlo (MCMC) sampling approach. However, for non-identifiable systems efficient mixing and thus convergence of the Markov chains is difficult [38]. This problem can be cured by informative priors but these would mask the problem and should only be implemented if they are based on actual biological insights and prior information. One recent application in the field identified a minimal subset of reactions in a signalling network with a combination of parallel tempering and LASSO regression methods [39].

Re-parameterising structurally non-identifiable models
Given the recent advances in the computational efficiency of methods, we essentially consider determining structural identifiability no longer a bottleneck in the modelling of non-linear dynamic systems with ODEs. When the structurally non-identifiable parameters are determined, the problem is usually fixed by a re-parameterisation of the model. In the simplest case this is accomplished by fixing some of the involved parameters to a certain value. The price to be paid is typically that the information about the scale of some components is lost. Nevertheless, biolog-ically meaningful re-parameterisation of the models after finding non-identifiabilites remains a challenging task (G. Massonis et al. , arXiv:2012.09826v2).

Practical identifiability
From structural to practical identifiability Structural identifiability implies practical identifiability only for an infinite amount of data with zero noise. Practical identifiability is important for obtaining precise parameter estimates. Moreover, it is especially crucial to ensure that model predictions are well-determined. It is analyzed increasingly often to judge a model's predictivity [40,41,42,43,44]. The notion of practical identifiability has been rather vague in the literature, mainly referring to large confidence intervals [45,46,47]. Some approaches exist that define practical identifiability as a combination of model structure and experimental protocol without actual data [48,49]. In contrast, we consider a combination of model and data as practically identifiable if the confidence intervals of all estimated parameters are of finite size [35].

Parameter confidence intervals and identifiability
The profile likelihood (Equation (6)) provides a proper assessment of confidence intervals of estimated parameters in ODE models (Figure 1) by where ∆ α denotes the α quantile of the χ 2 distribution with df = 1 degrees of freedom for point-wise confidence intervals [34].
The traditional method for determining confidence intervals based on the Fisher information matrix (FIM) leads to accurate confidence intervals for linear regression models. Since the solutions of all nontrivial ODE models are non-linear in their parameters, using this method for analysing identifiability of such models is questionable [50]. Furthermore, in contrast to FIM-based confidence intervals, profile likelihood-based confidence intervals can be asymmetric and are invariant under re-parameterisations of the model, e.g. the often applied logarithmic transformation of the parameters. Figure 2 shows five parameters with FIM-and profile likelihood-based confidence intervals, mainly taken from applications in synthetic biology [51,52].
Identifiability is obtained if all estimated parameters are structurally and practically identifiable, i.e. have finite confidence intervals. A non-identifiable parameter is called practically non-identifiable if its confidence interval can be narrowed by adding additional measurements for the existing observables and experimental conditions ( Figure 1C).

Bayesian methods for identifiability analysis
Bayesian sampling approaches, e.g. MCMC, can be used to assess practical identifiability [53,54,55]. This, however is only feasible if the model is structurally identifiable, since structural non-identifiabilities will lead to bad mixing of the sampling algorithms. Given a structurally identifiable model, MCMC sampling yields similar results as the profile likelihood analysis [38]. However, a recent application in a spatio-temporal reaction-diffusion model showed, that it is one order of magnitude slower than the profile likelihood [56].

Model predictions
To test the predictive power of a model, confidence intervals for the predictions can be computed. For this purpose, forward evaluations of the model are utilized, e.g. bootstrap approaches [57] or sensitivity analysis [58]. They typically require large numerical efforts in the context of non-linear biological models with a high-dimensional parameter space.
A more powerful approach is the prediction profile likelihood which is obtained by minimising χ 2 res (p) (Equation (3)) under the constraint that the model response g pred (p) is equal to the prediction z. The prediction profile likelihood propagates the uncertainty from the experimental data to the prediction by exploring the prediction space instead of the parameter space [59].
If the model predictions are not of sufficient precision, one has two principal options to tailor the model complexity to the information content of the data: (i) measure additional data, corresponding to an increase of the dimension of the observation function g in Equation (2), or (ii) reduce the model complexity according to the available data, corresponding to a decrease of the dimension of the parameter space and/or of the ODE system f in Equation (1).
Both options increase the practical identifiability of the model.

Achieving practical identifiability by new measurements with optimal experimental design
Practical identifiability can be achieved by adding new data [44,60]. The process of determining the most informative targets and time points for the new measurements is known as optimal experimental design and is frequently applied in different modelling fields, e.g. metabolic models [61], animal science [62], linear perturbation networks [63] or synthetic biology [64]. The task is related to the search of an additional measurement that contains the maximal information about the system or parts of it. For improving the identifiability of a specific parameter, the model trajectories along the corresponding parameter profile can be investigated [65,66]. Thereby, measurement points with maximal information content for the parameter of interest can be determined, which corresponds to trajectories with high spread. Similarly, the prediction profile likelihood (Equation (8)) determines the prediction uncertainty of the model at a potential new measurement time point [59] thus promoting the identifiability of the whole model. Measurement points with high prediction uncertainty are effective to constrain the model further, whereas measurements with a low prediction uncertainty are better suited for model selection purposes.

Achieving practical identifiability by reducing model complexity
If measuring additional data is not feasible, the complexity of the model has to be reduced. One way is to fix parameter values or ratios of parameters by means of prior knowledge [67], sensitivity analysis [68,69] or profile likelihood [70]. However, fixing parameters can decrease the interpretative relevance of the model's predictions.
Taking this into account, a systematic model reduction strategy that tailors model complexity to the available data was suggested by Maiwald et al. [71]. Based on likelihood profiles, they discuss four basic scenarios that are discriminated based on the profile likelihood by the combinations of: either (i) the profile flattens out for a logarithmised parameter going to infinity or (ii) to minus infinity, and either other parameters are (a) coupled to the investigated one or (b) not. For all four possible combinations, there is a cure. For case (i/a), one differential equation is replaced by an algebraic equation, for (i/b), states can be lumped, for (ii/a), a variable is fixed, leading to a structural nonidentifiability that can be cured by the methods discussed above, and for (ii/b), a reaction can be removed from the model. This model reduction strategy has been applied e.g. in [51,52,72]. Independent of the applied method, model reduction steps, and in particular the conclusions thereof, should always be documented together with the model according to good scientific practice to facilitate reproducibility.

Conclusions
Given the multitude of recently developed methods [13,16,27,32], we consider the file of identifying structurally non-identifiable parameters as closed. Future research in this field could focus on identifying biologically plausible re-parameterisations of the model, for which no comprehensive method yet exists. Furthermore, the extension of the concept of identifiability to different model types, e.g. mixed effects models [73,74], is of interest.
Achieving practical identifiability for model and data is more laborious in practice. Practical non-identifiabilities can be detected reliably, e.g. by the profile likelihood method [31]. In order to achieve identifiability, the model complexity has to be reduced or additional data must be added. Profile likelihood-based model reduction [71] and optimal experimental design [66] provide valuable methods for these purposes. A flowchart locating structural and practical identifiability analysis as discussed in this review within the entire modelling process is given in Figure 3.
Although the availability of advanced methods for the detection and cure of structural and practical non-identifiabilities is promising, two related challenges remain. In many applications identifiability analysis is not performed with stateof-the-art methods. Particularly, identifiability analysis based on the Fisher information matrix can be misleading in typical applications in systems biology. We propose a more consequent use of the discussed methods for structural identifiability and especially profile likelihood for practical identifiability analysis in order to check the limitations and predictive power of mathematical models. In summary, we believe the focal point of research in systems biology should always remain on the biological insights that can be gained from mathematical models which are structurally and practically identifiable. The topics discussed in this review related to structural identifiability (blue) and practical identifiability (red) are highlighted with colours in the flowchart. The remaining tiles in gray represent aspects that are beyond the scope of this review. The intricacy of the flowchart shows, that the path to biological insights requires multiple iterations of different methods. Identifiability analysis is an integral part of this workflow and should be performed to gain insights from predictive models with well-determined parameters. Furthermore, methods dealing with structural and practical identifiability should always be focused on ultimately progressing along the path towards biological insights.

Conflict of interest statement
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.