Evaluation of the delta-method to efficiently compute probability of target attainment of antibiotics
Sebastian G. Wicha (1), Alexander Solms (2), Wilhelm Huisinga (2) and Charlotte Kloft (1)
(1) Dept. of Clinical Pharmacy and Biochemistry, Institute of Pharmacy, Freie Universitaet Berlin, Germany, (2) Institute of Mathematics, University of Potsdam, Germany
Objectives: To assess therapeutic success or failure of antibiotic treatments pharmacokinetic (PK)/pharmacodynamic (PD) breakpoints are frequently used in probability of target attainment (PTA) analyses. For this purpose, commonly time-consuming Monte-Carlo simulations (MCS) considering the interindividual variability in PK are performed. PTA is then calculated as the fraction of scenarios for which the PK/PD breakpoint is attained.
For an empiric probabilistic dosing module in the recently developed web-based dosing support software TDMx (www.tdmx.eu) , MCS was found too slow for convenient usage. Instead, interindividual variability bands around the typical PK profile and resulting PTAs were to be approximated by use of the delta-method (DM) and results were compared to conventional MCS.
Methods: A published population PK model of the beta-lactam antibiotic meropenem (MER)  was used for evaluation of MCS- and DM-based PTAs. PK covariates were set to their typical values , serum creatinine to 0.7 mg/dL, minimal inhibitory concentration to 4 mg/L and the PK/PD breakpoint for MER to fT>MIC of 40% . Short (1 h TID), prolonged (4 h TID) and continuous infusion (24 h) dosing regimens were assessed. Interindividual variability of the PK parameters was varied from 20% to 70% CV.
MCS-based PTAs were calculated based upon 1000 simulations each. For DM-based PTAs, the ‘apparent’ variance of the PK profile var(f(θ,t)) was computed at each time point using the delta method with the Jacobian of the PK model J and the variance-covariance matrix Ω (var(f(θ,t))=J*Ω*J^T). With var(f(θ,t)), prediction intervals up to the 95th (in 1.25 steps) were derived for PTA calculation. Both methods were compared with respect to correlation and required CPU time.
Results: For MCS, the variability of PTA was 0.014 (SD) at n=1000 replicates. Differences between MCS-based and DM-based PTAs ranged from -0.05 and 0.03 (mean: -0.004) and were independent of the set interindividual PK variability. Both methods correlated well (R²=0.995, DM=MCS×1.04-0.031). CPU time was ca. 1.3 sec. for DM and ca. 48 sec. for MCS for computation of a single dosing scenario.
Conclusion: DM-based computation of PTAs was in well agreement with the conventionally used MCS-based approach thereby reducing the required CPU time by > factor 35. The DM-based algorithm for PTA calculation was hence integrated in TDMx facilitating rapid empiric dosing decisions prior to initialising antibiotic treatment.
 S.G. Wicha, M.G. Kees, A. Solms, I.K. Minichmayr, A. Kratzer, C. Kloft. TDMx – a novel web-based open-access support tool for optimising antimicrobial dosing regimens in clinical routine. Int J Antimicrob Agents. Epub ahead of print (2015).
 C. Li, J.L. Kuti, C.H. Nightingale, et al. Population pharmacokinetic analysis and dosing regimen optimization of meropenem in adult patients. J Clin Pharmacol (2006) 46(10): 1171–8.