Deriving D-Optimal Schedules Under Time Constraints

Why time limits change an optimal schedule

A locally D-optimal schedule chooses sampling times to maximize the determinant of the Fisher information matrix at specified nominal parameter values. The determinant summarizes information about all parameters jointly: a larger value corresponds to a smaller local confidence-ellipsoid volume when the regularity assumptions for that approximation hold. In practice, a mathematically attractive time can be unavailable. A sample may not be collected before a minimum processing time, or the study visit may end before the latest informative time. The design problem is therefore an optimization over a feasible time interval, not over all possible times.

The key distinction is between an unconstrained optimum that happens to be feasible and a constrained optimum that is forced to a boundary. The latter is not automatically a failure of the design. It can be the correct consequence of the allowable window. To see exactly how this happens, consider a simple individual IV bolus model and then state the boundary conditions that apply more generally.

Model, information, and assumptions

Use the one-compartment IV bolus model \(C(t)=D V^{-1}e^{-kt}\), where dose \(D\) is known, \(V\) is volume, and \(k\) is the elimination rate constant. For this derivation, assume the measured log concentration has independent, normally distributed errors with common known variance \(\sigma^2\). Thus the observation at time \(t_i\) is modeled as \(\log C(t_i)\) plus an error with mean zero and variance \(\sigma^2\). This constant-variance assumption on the log scale is part of this example; the resulting schedule need not remain optimal under a different error model.

Take the parameter vector to be \(\boldsymbol{\theta}=(\log V,\log k)\). The log-concentration mean is \(\log D-\log V-kt\). Its sensitivity row with respect to these parameters is \(\mathbf{s}(t)=(-1,-kt)\). Under the stated independent-error model, the Fisher information is the sum of sensitivity outer products divided by the error variance:

The sensitivities depend on \(k\), so this is a local design: evaluate the information at a nominal value of \(k\). The formula also assumes the model is appropriate over the selected times, the observations are independent, and the stated variance is applicable to each sample. If the information matrix is singular, its determinant is zero and it cannot support local estimation of both parameters.

Why the optimum uses the allowable endpoints

For this model the determinant simplifies to a direct measure of the spread of the sampling times. Expanding the two-by-two determinant gives

The second equality follows by expanding the pairwise sum: each \(t_i^2\) appears \(N-1\) times, and each cross-product \(t_it_j\) appears with coefficient \(-2\). The expression is therefore \(N\sum_i t_i^2-(\sum_i t_i)^2\). With \(k\) and \(\sigma\) fixed, maximizing the determinant is equivalent to maximizing the sum of squared pairwise time differences.

This objective is convex as a function of any one time when the other times are held fixed: it is a sum of squared linear differences. A convex function on a closed interval attains a maximum at an endpoint (or is constant, in which case an endpoint is also a maximizer). Apply this fact to one time at a time: starting from any feasible schedule, replace each time by an endpoint that does not decrease the objective. Thus at least one maximizer has every time at \(L\) or \(U\). If \(m\) samples are at \(L\), the only nonzero pairwise differences are between the two groups, giving

For integer \(m\) between 0 and \(N\), \(m(N-m)\) is largest when the two group sizes are as equal as possible. This follows from \(m(N-m)=N^2/4-(m-N/2)^2\). Hence the optimal schedule places approximately half the samples at each allowable endpoint. If \(N\) is odd, either of the two nearest integer allocations is optimal. At least two distinct times are needed for a nonsingular information matrix for these two parameters.

General boundary conditions for local designs

The endpoint result above is specific to the stated model and assumptions. For a more general PK model, information can vary nonlinearly with time, and the optimal schedule need not place every sample at an endpoint. A useful general check comes from the first-order conditions for constrained optimization. Write \(F(\mathbf{t})=\log\det I(\mathbf{t})\), provided \(I\) is positive definite. Maximizing \(F\) is equivalent to maximizing the determinant because the logarithm is strictly increasing.

For a small change \(dI\), the determinant identity \(\det(I+dI)=\det(I)\det(\mathrm{Id}+I^{-1}dI)\), where \(\mathrm{Id}\) is the identity matrix, implies \(d\log\det I=\operatorname{tr}(I^{-1}dI)\) to first order. Consequently,

For an interior time \(L<t_i<U\), a differentiable local optimum must have \(\partial F/\partial t_i=0\). At a lower boundary \(t_i=L\), only small increases are feasible, so a necessary first-order condition is \(\partial F/\partial t_i\leq 0\). At an upper boundary \(t_i=U\), only small decreases are feasible, so the necessary condition is \(\partial F/\partial t_i\geq 0\). These signs say that a feasible move into the interval cannot improve the objective to first order. They are necessary conditions, not proof of a global maximum for a general nonlinear design problem.

How to diagnose a constrained solution

A boundary-valued schedule should be checked, not rejected merely because some times equal \(L\) or \(U\). For the illustrative one-compartment model, the derivation establishes that endpoint times are optimal under its assumptions. For a more general model, verify the time bounds, calculate the information matrix at the proposed schedule, and check whether it is positive definite. If a claimed D-optimal design has a singular information matrix, it cannot jointly identify all modeled parameters locally, regardless of how attractive its time placement appears.

Numerical optimization is useful when the objective does not have the simple endpoint form derived here. Numerical Search Over Candidate Sampling Times can compare feasible schedules on a grid, but a grid is an approximation to continuous-time optimization. A coarse grid can miss a narrow optimum or make a genuinely active boundary appear slightly interior. Conversely, an optimizer may report a boundary because it has reached a constraint, not because it has established global optimality. Check the analytic derivative signs at active bounds, refine the time grid if using one, and compare feasible alternatives and starting values when the objective may have multiple local maxima.

The analytic derivation and numerical search serve different roles. The derivation explains why a boundary must be active in the stated model and gives the exact allocation. A numerical search can handle models for which no such closed-form argument is available, but it should be checked against feasibility and local behavior. Neither approach removes the dependence of a local design on its nominal parameter values or validates a misspecified PK or observation model.

Common mistake and limitations

• Mistake: “The unconstrained best time is outside the clinic window, so I will simply move that sample to the nearest boundary.” Correction: Sampling times contribute jointly to the information matrix. In a general model, moving one time can change the best placement of the others. Re-optimize the full feasible schedule or verify the relevant optimality conditions.
• Mistake: “A time at a boundary means the numerical optimization failed.” Correction: A boundary can be the true constrained optimum. Confirm the feasible-direction derivative sign, and distinguish a justified active constraint from a numerical result that has not converged.

The closed-form endpoint and balanced-allocation result depends on a two-parameter one-compartment IV bolus model, a fixed sample count, a common allowable interval, and independent equal-variance errors on the log-concentration scale. It is not a general rule for oral absorption, multi-compartment models, different residual error assumptions, or additional operational constraints. Those changes alter the sensitivities, information matrix, or feasible set and can change the optimal schedule.