Experiment design for nonparametric models based on minimizing Bayes Risk: application to voriconazole¹

Abstract

An experimental design approach is presented for individualized therapy in the special case where the prior information is specified by a nonparametric (NP) population model. Here, a NP model refers to a discrete probability model characterized by a finite set of support points and their associated weights. An important question arises as to how to best design experiments for this type of model. Many experimental design methods are based on Fisher information or other approaches originally developed for parametric models. While such approaches have been used with some success across various applications, it is interesting to note that they largely fail to address the fundamentally discrete nature of the NP model. Specifically, the problem of identifying an individual from a NP prior is more naturally treated as a problem of classification, i.e., to find a support point that best matches the patient's behavior. This paper studies the discrete nature of the NP experiment design problem from a classification point of view. Several new insights are provided including the use of Bayes Risk as an information measure, and new alternative methods for experiment design. One particular method, denoted as MMopt (multiple-model optimal), will be examined in detail and shown to require minimal computation while having distinct advantages compared to existing approaches. Several simulated examples, including a case study involving oral voriconazole in children, are given to demonstrate the usefulness of MMopt in pharmacokinetics applications.

Keywords: Bayes Risk; Experiment design; Nonparametric; Population model; Voriconazole.

Figure 1

Responses for the two-support-point example. The Dopt design is indicated for each support point taken separately.

Figure 2

The ED measure (solid) is equal to one half the Dopt measure for the slow a₂ support point |M (a₂)| (dot), and one half the Dopt measure for the fast a₁ support point |M (a₁)| (dash). Since the slow support point measure dominates the fast support point, the ED sample at t = 3.9991 is nearly identical to that of the slow a₂ model’s Dopt sample at t = 4, but has been drawn slightly lower by the weighted influence of the fast a₁ model’s Dopt sample at t = 0.6667.

Figure 3

Depiction of response separation r(t) between responses μ(t, a₁) and μ(t, a₂).

Figure 4

Bayes Risk is depicted as the grey area between curves, corresponding to the sum of Type I and Type II errors. Bayes Risk decreases as response separation r(t) increases, which “pulls apart” the two Gaussian distributions and shrinks the grey area between them.

Figure 5

Responses for two-support-point example with closely-spaced parameters.

Figure 6

Responses for four-support-point example.

Figure 7

Ten support point responses to dose input d(t).

Figure 8

Bayes Risk computed using 1 × 10⁶ Monte Carlo runs, showing the 1-sample Bayesian design at Bopt=4.25 hr with an optimal value of prob=0.5474.

Figure 9

Oral dosing responses of time versus concentration, simulated from a voriconazole nonparametric model with 125 support points. Patients are given loading doses at 0 and 12 hours, and a maintenance dose at 24 hours. MMopt samples times are determined on a grid constrained to 0.5 hour intervals. The two optimal sample times (MMopt2) are shown in dashed lines at 25 and 36 hours. The one optimal sample time (MMopt1) is indicated by the dash-dot line at 35 hours.