Mean Squared Error as Criterion for Sampling Schedule Optimization for Individual Dose Targeting in IV Busulfan
David H. Salinger (1), Jeannine S. McCune (2), David K. Blough (2), Paolo Vicini (1)
(1) Department of Bioengineering and (2) Department of Pharmacy; University of Washington, Seattle, WA, USA
Objectives: Pharmacokinetics-based dose targeting of daily IV busulfan is routinely performed to achieve a narrow plasma exposure of busulfan to lower toxicity while maximizing efficacy in hematopoietic cell transplant (HCT) recipients. Dose targeting is performed using maximum a posteriori (MAP) estimation informed on pharmacokinetic sampling conducted following a standard initial dose and subject to population pharmacokinetic prior. Current sampling schedules require inpatient admission, which is expensive. Our objectives were (1) to employ a simulation approach to determine the optimal outpatient sampling schedule for dose adjustment; (2) to employ simulation to compare this new sampling schedule to the current standard of care (inpatient sampling at 3, 3.25, 4.5, 6, and 8 hr post infusion initiation); and (3) to evaluate the mean squared error (MSE)  (a measure of estimate bias and precision) as compared to D-optimality (a measure of precision only) as a criterion for sampling schedule optimization for MAP estimation. To accomplish this, we examined an aggressively reduced sampling schedule (a single time point, instead of 6), since the difference between criteria is expected to be minimal for rich data. As it is well known, MAP estimation for determining an individual's PK causes "shrinkage" of the individual's (true but unknown) PK parameters towards the population mean. We propose to treat this shrinkage as if it were individual estimator bias, hence the use of MSE as optimization criterion.
Methods: To create a daily IV busulfan sampling schedule feasible in the outpatient clinic, we sought to determine six optimal blood sampling times between 0.25 hr and 6 hr post infusion initiation. Three of the six sampling times were fixed at 3 hr (end of infusion), 3.25 hr (as a backup), and 6 hr (latest feasible time), because these sampling times were deemed critical based on previous clinical experience of daily IV busulfan targeting. Sampling times were restricted to 0.25 hr increments. No repetition of sampling times was allowed.
To employ MSE as optimization criterion, it is necessary to use a simulation approach (to capture the shrinkage as estimation bias). We simulated 1000 subjects from our population model of IV busulfan kinetics. Each subject was then simulated with 15 instantiations of the (additive and proportional) error. All subject's parameters were then re-estimated for each error instantiation on every potential time grid (of 6 sampling times). The MSE for a model parameter is computed as the mean prediction error squared plus the prediction error variance. Scaled MSE (MSE divided by the true parameter value squared) was computed for each parameter and averaged for each individual on each potential time grid. The figure of merit was the mean (over all 1000 subjects) MSE evaluated on each potential sampling time grid. A similar approach was undertaken to compare the new outpatient sampling schedule to the inpatient standard of care sampling schedule.
Results: For rich sampling (6 data points, times 3, 3.25 and 6 fixed) and for a simple model the difference between the MSE and D-optimal schedule was, as expected, not large. The best schedule by MSE was sampling at 2, 2.75, 3, 3.25, 5.75, and 6 hrs. The best schedule by D-optimality was sampling at 1.75, 2, 3, 3.25, 5.75, and 6 hrs. Of the 1330 potential schedules, both the D-optimal and MSE-optimal schedules were amongst the top 20 in terms of opposing criteria. This was not unexpected, due to the rich sampling schedules.
In Objective 2, the MSE-optimal schedule (outpatient sampling at 2, 2.75, 3, 3.25, 5.75, and 6 hrs) with parameter prior resulted in an RMSE of 6.35% in a simulation of 1000 synthetic subjects, each with 15 replicates (error instantiations). By comparison, the standard of care schedule (inpatient sampling at 3, 3.25, 4.5, 6, and 8 hrs) without parameter prior resulted in an RMSE of 8.63%. Thus, we conclude that outpatient sampling is feasible.
In Objective 3, we employed single time point MAP estimation to highlight differences between the optimality criteria. The D-optimal schedule had its optimal sample at T=1.9 hr and the MSE-optimal schedule at T=1.6 hr. Bias and precision (as prediction error variance) are two components of MSE. The precision-optimal schedule coincided with the D-optimal schedule (as expected). The bias-optimal schedule had its optimal sample at T=1.3 hr.
Conclusions: For richly sampled data, the difference between the MSE and D-optimal schedule should not be expected to be large. For sparsely sampled data, the MSE criterion may provide a useful approach to achieve optimal scheduling for individual MAP estimation for the purpose of individual dose adjustment.
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