Internal Deterministic Identifiability of PK and PKPD Models

Introduction: Identifiability analysis is an important aspect of the PKPD modelling framework and is divided into structural and deterministic identifiability. Structural identifiability (SI) relates to the uniqueness of outputs to parameters and inputs. Deterministic identifiability (DI) relates to the precision of parameter estimates. Here we introduce two subcategories of DI: (1) external deterministic identifiability (EDI) and (2) internal deterministic identifiability (IDI). EDI is a function of factors that are external to the model, such as study design. For example, a model will not be identifiable, even if it is theoretically structurally identifiable, when the number of unique observations (unique(n)) is less than the number of fixed effects parameters (p). IDI is a function of factors that are internal to the model. This might manifest as a theoretically structurally identifiable model (i.e. where unique(n) >> p) with parameter sets that yield unacceptably poor parameter precision.

Methods:  In this study, 3 models were used to explore IDI: (1) a locally structurally identifiable model characterised by a 1-compartment PK model with first-order input and output (FOIO), (2) a globally structurally identifiable parent-metabolite model with bolus input and complete first-order metabolism of parent to metabolite and elimination of metabolite (PM-model), (3) a globally structurally identifiable 1- compartment bolus input first-order output PK model linked to an Emax turnover model (IVBTO-model). The design, for all models, consisted of 96 geometrically spaced samples for each response variable. Each study included 100 simulates and the dose administered was 1000 mg. For the FOIO and PM-models the initial set of parameter values were arbitrary and for the IVBTO-model the parameters were adapted from published data [1]. Between subject log normal variance for all parameters for all models was set to 0.1 and random unexplained variability was 10% for proportional error and a variance of 1 (mg/mL)2 for the additive error. Random parameter vector variates were generated that covered a plausible profile of the response, within pre-defined boundaries, and the relative asymptotic standard error (RSE) values computed for each set of parameters. The sets of parameter values with high RSE values (a relative standard error > 100%) were additionally evaluated at the D-optimal design using POPT. Any model with a set of parameter values with RSE values greater than 100% under the optimal design was designated to be not IDI.

Results: There was clear evidence that the FOIO model was not-IDI.  It is seen in this trivial example that as the first-order rate constant of input (ka) approaches the value for output (k; CL/V), the RSE values of ka, V and their between-subject variance tend to infinity. The PM-model was found to be IDI. In the case of the IVBTO model, we found several sets of parameter values that yielded high RSE values for the fixed and between-subject variance for IC50 under an optimal design.

Conclusion: From this work, it is seen that sets of parameter values exist that can render models not deterministically identifiable.  This occurs in models that are otherwise structurally identifiable as well as externally deterministically identifiable, e.g. under an appropriately optimal design for some sets of parameter values.  We have designated deterministic identifiability issues that are not amenable to the design as internal deterministic identifiability. In the case of the FOIO-model, the presence of an IDI issue was explicable on the basis of the model structure.  However for the IVBTO-model this was not as obvious.  There is a need to consider that IDI issues may be present during model development even in circumstances where the design is rich or has been optimally tailored to the problem at hand.

References:

1.      Sharma A, Jusko WJ (1998) Characteristics of indirect pharmacodynamic models and applications to clinical drug responses. Br J Clin Pharmacol 45 (3):229-239.

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