A General Model-Based Design of Experiments Approach to Achieve Practical Identifiability

A General Model-Based Design of Experiments Approach to Achieve Practical Identifiability

A general model-based design of experiments approach to achieve practical identifiability of pharmacokinetic and pharmacodynamic models

F. Galvanin, C.C. Ballan, M. Barolo, F. Bezzo

Online Supplemental Material 1

Structural identifiability analysis

The procedure proposed for structural identifiability analysis, coupling SLI and SGI tests is illustrated in Figure S1. The procedure starts given a model structure M(in the form given by Eq. 1 in the main text) and prior knowledge on the parametric set in terms of initial guess (θ0) and preliminary variance-covariance matrix of model parameters (Σ0), defining the expected variability domain for the parametric set.

Figure S1 Procedure for structural identifiability analysis.

After the SLI test is performed, a MBDoE session will provide a preliminary set of experimentalsettings (φ0) ensuring the local identifiability of the model with structure M at the currently available information on the parametric set (θ0). The approach is based on the assumption that MBDoE techniques are used to design the experiments. If the determinant of the Fisher information matrix is null, the model structure has to be modified until a suitable design vector can be determined adopting MBDoE procedure. Once a suitable design vector is determined, the global identifiability of the model is tested by performing a SGI test on the entire variability domain of model parameters (Θ), identified by θ0 and by the preliminary variance-covariance matrix of model parameters (Σ0).

The SLI test is based on the definition of local identifiability proposed by Shaw [1]:

Definition (local identifiability, SLI test): the model M with output trajectory y is locally identifiable if, in an open neighborhood of ,for the set of system inputs u and the initial conditions y0, the NyNsp × Nθ estimability matrix PE

(S1)

has full rank. In (S1) Sy(ti) is the Ny× Nθ sensitivity matrix evaluated at the sampling time ti.

Two alternative and equivalent formulations for SLI test can be defined

  1. a SLI test based on the analysis of PE columns: if the correlation between PE columns is different from 1 the model is locally identifiable [2-3].
  2. a SLI test based on the analysis of the Fisher information matrix: if the Fisher information matrix is non singular, the model is deemed locally identifiable [4].

The SLI test performed in this paper follows the second formulation, which is directly related to the classical mathematical formulation for MBDoE. In fact, under the hyphotesis of constant measurement errors

(S2)

anda D-optimal MBDoE can be carried out where the experimental conditions aiming at maximising the determinant of are the ones producing the lowest correlation for PE. Conversely, experimental conditions providing high correlation among PE columns leads to singularity of H matrix with the information related to specific subsets of model parameters becoming close to zero [5]. The SLI test is used to test the applicability of a D-optimal MBDoE acting on (S2) where the maximisation of the expected information is realised by adopting a high number of sampling times in order to approximate the continuous profiles of dynamic sensitivities in the evaluation of PE. Correlation analysis of PE columns can be used to analyse the relationship between identifiability and parameter sensitivity for each single model parameter. For over-parameterised models, PE columns are highly correlated and the matrix becomes rank-deficient: one parameter can be expressed as a linear combination of the others, suggesting possible alternative parameterisations of the model to be used for ensuring local identifiability.

Following the seminal definition by Ljung and Glad [6] themodel with structure M is said to be structurally globally identifiable (SGI) if

M() = M(*)   = *,* (S3) where Θ.

In order to test the structural global identifiability of the model (described by Eq. (1) in the main text) at the experimental settings φ0satisfying the SLI test, it must be verified that different parametric sets do not provide the same model response in the entire variability domain of model parameters. The SGI test adopted in this paper follows the optimisation-based approach to test global identifiability suggested by Asprey and Macchietto [7]:

Definition (global identifiability, SGI test): the model with structure M and output trajectory y is globally identifiable if, for any two parametric sets , and a time horizon of interest , for the set of system inputs u0 and the same initial conditions y0 for the measured outputs, the distancebetween two parameter vectors θ and θ* providing the same model output is such that

= (S4)

subject to

(S5)

(S6) whereand are two proper weighting matrices and εθ and εy are arbitrarily small numbers.

The test (S4-S6) imply the direct numerical evaluation of (S3) over the entire variability domain of model parameters Θfor the single set of manipulated inputs () determined from SLI test, stating that the model can be deemed structurally globally identifiable (SGI) if the distance between two parameters vectors providing the same model response is arbitrarily small. The optimisation is carried out, usually defined by θ0 and Σ0. As the theoretical validity of SGI test can be influenced by the choice of εθ, Walter and coworkers [8] suggest a generalisation of the approach through the definition of δ-identifiability (identifiability where εθ is a function of a proper δ which is specific for the given parameter).

References

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