Eur J Clin Pharmacol (2014) 70:1465–1470 DOI 10.1007/s00228-014-1759-x
PHARMACOKINETICS AND DISPOSITION
Adjustment of endogenous concentrations in pharmacokinetic modeling Alexander Bauer & Martin J. Wolfsegger
Received: 6 May 2014 / Accepted: 17 September 2014 / Published online: 3 October 2014 # Springer-Verlag Berlin Heidelberg 2014
Abstract Purpose Estimating pharmacokinetic parameters in the presence of an endogenous concentration is not straightforward as cross-reactivity in the analytical methodology prevents differentiation between endogenous and dose-related exogenous concentrations. This article proposes a novel intuitive modeling approach which adequately adjusts for the endogenous concentration. Methods Monte Carlo simulations were carried out based on a two-compartment population pharmacokinetic (PK) model fitted to real data following intravenous administration. A constant and a proportional error model were assumed. The performance of the novel model and the method of straightforward subtraction of the observed baseline concentration from post-dose concentrations were compared in terms of terminal half-life, area under the curve from 0 to infinity, and mean residence time. Results Mean bias in PK parameters was up to 4.5 times better with the novel model assuming a constant error model and up to 6.5 times better assuming a proportional error model. Conclusions The simulation study indicates that this novel modeling approach results in less biased and more accurate PK estimates than straightforward subtraction of the observed baseline concentration and overcomes the limitations of previously published approaches.
Keywords Endogenous concentration . Intrinsic concentration . Baseline adjustment . Pre-dose concentration . Pharmacokinetics A. Bauer (*) : M. J. Wolfsegger Baxter Innovations GmbH, Vienna, Austria e-mail: [email protected] M. J. Wolfsegger e-mail: [email protected]
Introduction In the development of biological drug products, pharmacokinetic (PK) studies are an essential part where circulating endogenous concentrations must be considered for PK analysis regardless of how small these concentrations are [1]. Previous simulation studies showed the importance of an adequate model to account for baseline data in pharmacodynamic (PD) responses [2–4]. However, awareness of this issue in PK analysis is often lacking. Endogenous concentrations, when assumed to show no circadian rhythm, are frequently handled by subtracting the baseline (pre-dose) concentration from post-dose concentrations (baseline subtracted or BS model). This procedure ignores analytic assay variability and results in post-dose differences which may have higher measurement errors than would occur in patients without an intrinsic concentration (depending on the covariance between baseline and post-dose concentrations). Gabrielsson and Weiner [5] proposed to model the endogenous concentration as a turnover rate using a system of differential equations which is a burden to many common software packages and may be difficult for non-experts to interpret. Schindel [6] suggested an approach based on a macro parameterized onecompartment model that includes an additional parameter for the endogenous concentration: C(t)=C0 +A·exp(−K·t), where C0 corresponds to the endogenous concentration, K to the terminal rate, A to the theoretical concentration at time point zero, and t to time after dose administration. However, this model does not take into account the observed concentration at baseline and is limited to the one-compartment model following intravenous bolus administration only. This paper presents a novel method to evaluate pharmacokinetics in the presence of temporarily constant endogenous concentrations that overcomes the limitations of the approaches mentioned above. A simulation study was carried out to show the superiority of this novel method over the BS
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Eur J Clin Pharmacol (2014) 70:1465–1470
method in a more complex scenario, based on real data, than presented by Schindel [6].
Model The introduced approach (baseline adjusted or BA model) models the concentration-time relationship by using a step function of the form: C0 if t < 0 C ðt Þ ¼ C 0 þ F ðθ; t Þ if t ≥ 0
( C ðt Þ ¼
C0
C 0 þ A1 ˙ exp K 1 ˙ t þ A2 ˙ exp K 2 ˙ t
if t < 0 if t ≥ 0
where F(θ, t) represents a general function based on a vector of parameters θ describing the concentration-time relationship. The model can therefore be applied independently of route of administration and number of compartments. The introduced step-function takes the observed concentration at baseline adequately into account and overcomes identifiability (i.e., model convergence) issues between endogenous and dose-related exogenous concentrations arising from more complex concentration-time relationships. For example, assuming a one-compartment model following intravenous bolus administration, F(θ, t)=A·exp(−K·t) where θ=(A, K).
Simulations A simulation study was carried out to compare the performance of the BA and BS models in terms of terminal half-life (t1/2), area under the curve (AUC) from 0 to infinity and mean residence time (MRT). Concentration-time profiles were generated based on a two-compartment population PK model fitted to real data following intravenous bolus administration with the following parameterization: C ðt Þ ¼ ðC 0 þ c0 Þ þ ðA1 þ a Þ exp −K t 1 1 ˙ ˙ þ ðA2 þ a2 Þ ˙ exp −K 2 ˙ t þ ε where fixed effects are expressed in upper case and random effects in lower case letters. Two different residual error models for ε were examined: normally distributed with (1)
constant error and (2) proportional error variance. The detailed parameterizations of these models are given in the Appendix. For each error model, 1E6 (=1,000,000) concentrationtime profiles were generated using the random number generator algorithm of Wichmann and Hill [7] as implemented in R version 3.0.2 [8]. Concentrations were drawn at baseline and at 0.083, 0.5, 2, 6, 12, 24, 28, 60, 72, 96, 108, and 120 h after study drug administration. Scenarios that resulted in negative values of C0 +c0 or A1 +a1 or A2 +a2 were excluded from analysis. Concentration-time profiles with a simulated negative baseline concentration (i.e., C0 +c0 +ε
PHARMACOKINETICS AND DISPOSITION
Adjustment of endogenous concentrations in pharmacokinetic modeling Alexander Bauer & Martin J. Wolfsegger
Received: 6 May 2014 / Accepted: 17 September 2014 / Published online: 3 October 2014 # Springer-Verlag Berlin Heidelberg 2014
Abstract Purpose Estimating pharmacokinetic parameters in the presence of an endogenous concentration is not straightforward as cross-reactivity in the analytical methodology prevents differentiation between endogenous and dose-related exogenous concentrations. This article proposes a novel intuitive modeling approach which adequately adjusts for the endogenous concentration. Methods Monte Carlo simulations were carried out based on a two-compartment population pharmacokinetic (PK) model fitted to real data following intravenous administration. A constant and a proportional error model were assumed. The performance of the novel model and the method of straightforward subtraction of the observed baseline concentration from post-dose concentrations were compared in terms of terminal half-life, area under the curve from 0 to infinity, and mean residence time. Results Mean bias in PK parameters was up to 4.5 times better with the novel model assuming a constant error model and up to 6.5 times better assuming a proportional error model. Conclusions The simulation study indicates that this novel modeling approach results in less biased and more accurate PK estimates than straightforward subtraction of the observed baseline concentration and overcomes the limitations of previously published approaches.
Keywords Endogenous concentration . Intrinsic concentration . Baseline adjustment . Pre-dose concentration . Pharmacokinetics A. Bauer (*) : M. J. Wolfsegger Baxter Innovations GmbH, Vienna, Austria e-mail: [email protected] M. J. Wolfsegger e-mail: [email protected]
Introduction In the development of biological drug products, pharmacokinetic (PK) studies are an essential part where circulating endogenous concentrations must be considered for PK analysis regardless of how small these concentrations are [1]. Previous simulation studies showed the importance of an adequate model to account for baseline data in pharmacodynamic (PD) responses [2–4]. However, awareness of this issue in PK analysis is often lacking. Endogenous concentrations, when assumed to show no circadian rhythm, are frequently handled by subtracting the baseline (pre-dose) concentration from post-dose concentrations (baseline subtracted or BS model). This procedure ignores analytic assay variability and results in post-dose differences which may have higher measurement errors than would occur in patients without an intrinsic concentration (depending on the covariance between baseline and post-dose concentrations). Gabrielsson and Weiner [5] proposed to model the endogenous concentration as a turnover rate using a system of differential equations which is a burden to many common software packages and may be difficult for non-experts to interpret. Schindel [6] suggested an approach based on a macro parameterized onecompartment model that includes an additional parameter for the endogenous concentration: C(t)=C0 +A·exp(−K·t), where C0 corresponds to the endogenous concentration, K to the terminal rate, A to the theoretical concentration at time point zero, and t to time after dose administration. However, this model does not take into account the observed concentration at baseline and is limited to the one-compartment model following intravenous bolus administration only. This paper presents a novel method to evaluate pharmacokinetics in the presence of temporarily constant endogenous concentrations that overcomes the limitations of the approaches mentioned above. A simulation study was carried out to show the superiority of this novel method over the BS
1466
Eur J Clin Pharmacol (2014) 70:1465–1470
method in a more complex scenario, based on real data, than presented by Schindel [6].
Model The introduced approach (baseline adjusted or BA model) models the concentration-time relationship by using a step function of the form: C0 if t < 0 C ðt Þ ¼ C 0 þ F ðθ; t Þ if t ≥ 0
( C ðt Þ ¼
C0
C 0 þ A1 ˙ exp K 1 ˙ t þ A2 ˙ exp K 2 ˙ t
if t < 0 if t ≥ 0
where F(θ, t) represents a general function based on a vector of parameters θ describing the concentration-time relationship. The model can therefore be applied independently of route of administration and number of compartments. The introduced step-function takes the observed concentration at baseline adequately into account and overcomes identifiability (i.e., model convergence) issues between endogenous and dose-related exogenous concentrations arising from more complex concentration-time relationships. For example, assuming a one-compartment model following intravenous bolus administration, F(θ, t)=A·exp(−K·t) where θ=(A, K).
Simulations A simulation study was carried out to compare the performance of the BA and BS models in terms of terminal half-life (t1/2), area under the curve (AUC) from 0 to infinity and mean residence time (MRT). Concentration-time profiles were generated based on a two-compartment population PK model fitted to real data following intravenous bolus administration with the following parameterization: C ðt Þ ¼ ðC 0 þ c0 Þ þ ðA1 þ a Þ exp −K t 1 1 ˙ ˙ þ ðA2 þ a2 Þ ˙ exp −K 2 ˙ t þ ε where fixed effects are expressed in upper case and random effects in lower case letters. Two different residual error models for ε were examined: normally distributed with (1)
constant error and (2) proportional error variance. The detailed parameterizations of these models are given in the Appendix. For each error model, 1E6 (=1,000,000) concentrationtime profiles were generated using the random number generator algorithm of Wichmann and Hill [7] as implemented in R version 3.0.2 [8]. Concentrations were drawn at baseline and at 0.083, 0.5, 2, 6, 12, 24, 28, 60, 72, 96, 108, and 120 h after study drug administration. Scenarios that resulted in negative values of C0 +c0 or A1 +a1 or A2 +a2 were excluded from analysis. Concentration-time profiles with a simulated negative baseline concentration (i.e., C0 +c0 +ε
