Minimum Spacing and Clinic-Window Constraints

Why operational constraints belong in the design

A sampling schedule can be statistically attractive and still be unusable. A predicted optimal time may fall outside clinic hours, two samples may be scheduled too close together for separate blood draws, or a visit may need to occur within a prespecified window. These are not minor implementation details: they change which schedules can actually be used, and therefore which schedule is optimal among the usable alternatives.

The design task is to choose ordered sample times that satisfy all operational constraints and then compare their information. The feasible schedule may differ from the unconstrained optimum, but its information loss is not necessarily large. It must be calculated for the particular model, nominal parameter values, and error assumptions rather than inferred from how far the times moved.

When clinic availability differs by visit, represent the allowed time for visit \(i\) by the intersection of its visit window and the clinic’s open hours: \([a_i,b_i]\cap[L,U]\). If the clinic is open in separate sessions, the allowed set can instead be a union of intervals. Times should use one consistent clock, such as hours after dosing; if actual clock time is used, all windows and gaps must be converted consistently.

Checking whether a schedule can exist

For a single interval per visit and fixed minimum gaps, a simple forward calculation checks feasibility. Begin at the earliest allowed time for the first visit. For every later visit, use the later of its own earliest time and the earliest time allowed by the preceding visit’s gap. If this earliest possible time exceeds the visit’s latest allowed time, the schedule is infeasible.

Here \(e_i\) is the earliest time at which visit \(i\) could occur while respecting the windows and all preceding gaps. Feasibility holds exactly when \(e_i\leq b_i\) for every visit. The forward rule is necessary because visit \(i\) cannot precede either its own window or the previous visit plus the required gap. It is sufficient because, if every \(e_i\) lies within its window, choosing each visit at \(e_i\) meets every gap and every window.

This check assumes that visits are ordered and each allowed set is a single closed interval. With disjoint clinic sessions, a feasible early session may lead to an infeasible later visit while a later session works. In that case, retain the possible interval alternatives and propagate feasibility across them, or enumerate the allowed combinations. Do not replace a disjoint set of clinic sessions with its overall minimum and maximum; that would incorrectly treat closed hours as available.

Information comparison under constraints

Operational constraints apply independently of the pharmacokinetic model. To illustrate how they can alter information, consider a one-compartment IV bolus concentration model \(C(t)=C_0e^{-kt}\), where \(C_0\) is the concentration at time zero and \(k\) is the elimination rate constant. Suppose two concentrations are measured with independent additive normal errors of common variance \(\sigma^2\). For this illustration, the nominal values are \(C_0=100\) concentration units, \(k=0.20\) per hour, and \(\sigma=5\) concentration units. These assumptions are specific to the calculation; different error assumptions or nominal values can change the comparison.

Use the parameter vector \((\log C_0,k)\). At time \(t_i\), the sensitivity vector is \((C_i,-t_iC_i)\), where \(C_i=C_0e^{-kt_i}\). The individual Fisher information matrix is the sum of the sensitivity outer products divided by \(\sigma^2\). For two samples, its determinant simplifies as follows.

To justify the determinant expression, put the two sensitivity vectors into the rows of a matrix \(G\). Then \(I=G^\mathsf{T}G/\sigma^2\), so \(\det I=(\det G)^2/\sigma^4\). The determinant of \(G\) is \(C_1C_2(t_1-t_2)\). Squaring it and substituting \(C_i=C_0e^{-kt_i}\) yields the stated formula. With two parameters, a larger determinant corresponds to a larger local D-criterion, provided the same parameterization, model, error assumptions, and nominal values are used for both schedules.

The expression shows why moving a sample later does not have a one-directional effect. Increasing the gap \((t_2-t_1)\) can improve separation of sensitivities, while moving both samples later reduces the concentrations and hence the information under this constant-variance error model. The operationally constrained design must be assessed by the full expression, not by the time gap alone.

Optimizing and diagnosing a constrained schedule

The proper comparison is between the unconstrained optimum and the best feasible schedule under the same information criterion. Do not take an unconstrained schedule and shift each visit independently until it appears operational: shifting one time can violate a minimum gap, leave a visit window, or create a new clinic conflict. Instead, define the feasible set first and optimize over that set. Candidate-time searches can discard infeasible schedules before ranking the remaining ones; continuous optimization must enforce the same constraints directly.

Useful checks include:

• Check every adjacent gap. Verify \(t_{i+1}-t_i\geq d_i\) using the final, rounded schedule. Rounding a time can turn a barely feasible gap into a violation.
• Check windows and clinic hours together. A time can satisfy its visit window but fall outside clinic hours, or vice versa.
• Inspect boundary solutions. If an optimum repeatedly lands at a window endpoint or exactly at a minimum gap, confirm that this is allowed by the protocol and not an artifact of a coarse candidate grid.
• Recalculate information at the implemented times. A schedule that is feasible before rounding or administrative adjustment may have different information after those changes.

A common mistake is to assume that the nearest feasible schedule to the unconstrained optimum must also be the best feasible design. That is not generally true: the determinant depends jointly on all sampling times and the sensitivity structure. The correction is to compare feasible schedules using the chosen information criterion—or optimize directly over the feasible set—rather than ranking them by distance from the unconstrained schedule.

A simple constrained optimization may be adequate when the feasible intervals are continuous and the model objective behaves smoothly. Enumerating candidate times is often convenient when clinic visits are available only at discrete slots, but the grid must be fine enough to avoid missing a better schedule between candidates. A penalty method can discourage constraint violations during numerical search, but a small penalty may leave violations and a large penalty may impair optimization. Whatever method is used, independently verify feasibility and recompute the objective at the final times.

The determinant comparison is local: it uses the specified nominal parameter values and the assumed error model. A schedule that is D-efficient at one nominal value may not retain the same ranking elsewhere. Also, a high determinant does not guarantee that every individual parameter is estimated precisely; it summarizes joint information about the parameter vector. Operational constraints can therefore motivate a feasible schedule, but the information calculation does not replace clinical judgment about visit burden, assay limitations, or the intended use of the estimates.