Internal Deterministic Identifiability

Aims: Identifiability is an important component of model development. An identifiability analysis can provide the basis for understanding the limits of model structure and parameterisation. There are two types of identifiability analysis, structural and deterministic. A structural identifiability analysis refers to the formal identifiability of model structure (see (1) for a description and expansion to a population PK model). Deterministic identifiability (DI) is concerned with the influence of study design on the precision of the parameter estimates in a structurally (at least locally) identifiable model given imperfect input-output data. We further classify DI into two sub-types external deterministic identifiability (EDI) and internal deterministic identifiability (IDI). EDI relates to the DI of models conditioned on the study design controlled by the investigator. In a simple case, a design where the number of unique observations (n) is less than the number of parameters (p) in the model will be deterministically unidentifiable despite being structurally identifiable. By contrast, IDI describes the situation in which a specific set of parameters yield unreasonably imprecise parameter estimates, despite the model being structurally identifiable and the design fulfilling the needs of EDI. Here EDI can be considered to be any unconstrained optimal design where unique (n) > p. In this setting a non-IDI situation is defined as one where ∃θ ⃗:SE(θ,ξ^D )>ψ , Where, θ ⃗ is a vector of parameter values, ξ^D is the vector of design variables from D-optimum design and ψ represents the level of imprecision that is important to the investigator. For the purposes of this research, we define a relative standard error (RSE) 100 % to be upper limit of acceptable imprecision. The aim of this study is to investigate whether IDI can be identified in a common set of PK and PKPD models.

Methods: Three models were selected for the IDI analysis: 1) a first-order input and output PK model (FOIO), 2) a parent-metabolite (PM) model with iv-bolus input to the parent compartment (P), first-order metabolism of P to a metabolite (M) and first-order elimination of M. It is assumed that P is completely metabolised to M and that sampling occurs from both the P and M compartment. Finally, 3) a turnover model with iv-bolus input (IVBTO). The drug was assumed to reduce the rate of production of a hypothetical biomeasure of interest. An intensive, 78 sample, design was chosen for each response variable. For the FOIO and PM models the initial set of parameter values were arbitrary and for IVBTO the parameters were adapted from (2). Using boundaries on these selected initial sets of initial parameters, random variates were generated that cover a plausible profile of response. The sets of parameter values with high RSE values (aboveψ; 100%) were evaluated at their D-optimal design using POPT. This served the purpose of ruling out EDI. Any set of parameter values that retained RSE greater than 100% under the optimal design indicates that the respective model is 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) that the RSE values ka, V and their between-subject variance tends to infinity. Whereas for the PM-model, there is no such observation and the model was IDI. In case of the IVBTO model there existed several sets of parameter values that provided high RSE for the fixed and between-subject variance for IC50 for the optimal design.

Conclusion: From this work it is evident that sets of parameter values exist that can render our example models not identifiable. This occurs in models that are otherwise structurally identifiable as well as externally deterministically identifiable (i.e. under the optimal design). We have termed this as internal deterministic identifiability. This has been explored with three models. In the case of the FOIO, the presence of an IDI issue was explicable on the basis of the model structure. However for IVBTO this was not as obvious. There is a need to consider that IDI issues may be present during model development.

1. Shivva V. An Approach for Identifiability of population pharmacokinetic–pharmacodynamic models. CPT: Pharmacometrics Syst Pharmacol. 2013; 2(e49):1-9.
2. Jusko W J. Characteristics of indirect pharmacokinetic models and applications to clinical drug responses. Br J Clin Pharmacol. 1998; 45:229-239.