Introduction. Allopurinol is used for the treatment of gout. The primary physiological response to allopurinol therapy is a reduction in serum urate concentrations. Attempts to model the pharmacokinetics and pharmacodynamics (PK & PD) of allopurinol using a turnover model have consistently produced unstable models with imprecise and biased estimates for some PD parameters (e.g. Kout) [1,2]. Given that a hysteresis could not be demonstrated in some individuals, we hypothesise that the allopurinol turnover model with a delay-driven by the kout parameter would not be identifiable.
Aim. To investigate the structural and deterministic identifiability characteristics of an allopurinol turnover model for urate-lowering response.
Methods. This research included an assessment of; 1) local structural identifiability (the uniqueness of a set of model parameters), 2) ‘external’ deterministic identifiability (EDI, related to the study design) and 3) ‘internal’ deterministic identifiability (IDI, related to an ‘internal’ aspects of an otherwise SI and EDI model and it’s parameter values) [3]. A simple example of IDI is a one compartment oral PK model with first-order input and output which may become non-EDI under some sets of parameter values (e.g. when the absorption rate constant approaches the value of the elimination rate constant creating a flip-flop scenario). The allopurinol PKPD model was deemed to be locally structurally identifiable if the -log(det(FIM)) value of was <0 and standard errors were provided using the ($DESIGN) function in NONMEM (v7.5.1) [4]. Successful external deterministic identifiability was concluded if relative standard errors (RSE) < 50%. Internal deterministic identifiability was explored using a stochastic simulation estimation (SSE) study. Three scenarios were considered for a hypothetical drug; 1) k > kout , 2) k ≈ kout and 3) k < kout, where k and kout are the elimination rate constants for the PK and PD biomarker respectively. Data for the PK and PD variables were simulated (100 datasets) under a geometric sampling design on days 1, 2 and 3, and day 30 and the data was fit to a sequential IPP model. The PD model included a turnover model linked to an inhibitory Emax model on the rate of the production. If RSE% for the PD parameter estimates was > 50% or approaching 0, the model was assumed to display non-IDI characteristics.
Results. The -log(det(FIM)) on value for the allopurinol turnover model was -65.5 and the covariance step was successful in $DESIGN suggesting a locally structurally identifiable model. RSE% values for the PD parameters were <50% so the study design was assumed to be sufficient to estimate the turnover model parameters and was therefore EDI. In the SSE study, all runs for the k > kout scenario produced RSE% values of < 50% (0.4-22%). When k ≈ kout, RSE% for the PD baseline parameter was either 0 or close to zero for most runs, and BSV for baseline and kout collapsed to 0 for 26% and 10% of the runs, respectively, suggesting IDI issues in some runs. When k < kout, all runs included RSE% values of either 0 or > 100% in one or more parameters suggesting that this scenario was not IDI.
Conclusion. The allopurinol PKPD model using a turnover model was locally structurally and externally deterministically identifiable. The analysis supports the hypothesis that the relative values of k (in the PK model) and kout (in the PD model), particularly when k < kout, rendered the model non-internally deterministically identifiable and contributed to poor parameter precision during model development. A simplified model structure was needed (e.g. direct effects model) for allopurinol PKPD.
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