Checking Local Design Dependence on Nominal Values

Why Recheck a Locally Optimized Schedule?

An individual PK schedule is often optimized using a specified set of nominal parameter values. The resulting times are locally optimal: they perform well for the model sensitivities evaluated at those particular values. If the true fixed parameters differ, the concentration-time curve and its sensitivities can change, and another schedule may be more informative. A design that looks excellent at one nominal point is not automatically reliable across a plausible range.

The practical question is not whether one can find a single schedule that is best for every possible parameter value. It is whether the proposed schedule remains sufficiently informative over the parameter values considered plausible for the design problem. Answering that requires recalculating sensitivities and design diagnostics at each value, rather than assuming that the nominal result carries over unchanged.

The calculations below use a one-compartment bolus model as a clear setting for checking local dependence. The same logic applies to other individual PK models: specify the model and error assumptions, evaluate the sensitivities at each fixed parameter vector, and compare candidate schedules on that common basis. This is a conditional, local assessment; it is not a claim that the unknown parameters have been estimated or averaged over a population.

Model, Sensitivities, and Information

Suppose a hypothetical individual receives an intravenous bolus dose \(D\), with concentration at time \(t\) described by \(C(t)=A e^{-kt}\), where \(A=D/V\), \(V\) is volume of distribution, and \(k\) is the elimination rate constant. Use the parameter vector \(\theta=(\log A,k)\). Assume observations at fixed times have independent, normally distributed additive errors with known variance \(\sigma^2\), the model is correct, and \(A\), \(k\), and \(\sigma\) are fixed when evaluating a local design.

The sensitivity vector at time \(t_j\) is the gradient of the predicted concentration with respect to the parameters. For this parameterization, its components are concentration per unit change in \(\log A\), and concentration times hours per unit change in \(k\), respectively. With independent errors of common variance, the local Fisher information matrix is the sum of the weighted outer products of these vectors.

For a two-parameter design with exactly two distinct sampling times \(t_1\) and \(t_2\), the determinant has a particularly useful form. A zero determinant means the two sensitivity vectors do not provide independent information about both parameters. A larger determinant indicates greater joint information for this parameterization at the fixed parameter values and under the stated error model.

Why the Determinant Changes

Write \(C_i=C(t_i;\theta)\). The two sensitivity vectors form the columns of a matrix whose rows are \((C_1,-t_1C_1)\) and \((C_2,-t_2C_2)\). The information matrix is the product of the transpose and the matrix, divided by \(\sigma^2\). Its determinant is therefore the squared determinant of that sensitivity matrix, divided by \(\sigma^4\):

This expression shows two sources of local design dependence. The time separation \(t_2-t_1\) helps distinguish the amplitude and elimination rate, while the concentrations \(C_1\) and \(C_2\) depend on the fixed values of \(A\) and \(k\). If elimination is faster, later concentrations can be much lower, changing the information contributed by late samples. Merely keeping the same sample times does not keep the sensitivity profiles or the information matrix fixed.

The derivation relies on the stated assumptions: the predicted concentration is differentiable in the parameters, observation errors are independent with common known variance, and the design contains exactly two observations for the displayed two-time determinant. With more observations, the information is still the sum of weighted sensitivity outer products, but the two-time closed form does not generally apply. If the residual variance model changes with concentration, the weights must also be recalculated according to that model.

Check Whether Design Rankings Persist

The determinant at one parameter value is not enough to establish that a schedule is robust. A useful check evaluates all candidate schedules at each fixed parameter vector using the same model, parameterization, and error assumptions. Then compare their information and determine whether their relative ranking changes. For two parameters, D-efficiency of a candidate design relative to a reference design at a given parameter vector is the square root of their determinant ratio. It is a relative measure of joint information, not a probability or a guarantee about estimation error in a particular dataset.

The comparison is meaningful only when both determinants use the same parameterization and the same fixed parameter values. A parameterization change can change the numerical scale of a determinant. Likewise, comparing determinants computed under different error variances would mix parameter-value sensitivity with a change in the assumed observation precision.

A Practical Reassessment Workflow

A useful assessment is systematic rather than based on a single alternative value. Specify a plausible set of fixed parameter vectors, recalculate the relevant sensitivities and information matrices at each vector, and report how the candidate schedules compare. The parameter values are held fixed for each calculation; the procedure does not assign probabilities to those values unless such probabilities are supplied as part of a separate design criterion.

A parameter grid can include combinations at the low and high ends of plausible ranges, as well as intermediate values. Checking only one parameter at a time may miss important interactions: for example, a change in amplitude may affect concentration magnitudes while a change in elimination rate changes the relative contribution of early and late samples. For a small grid, reporting each result directly is transparent. For a larger grid, summarize the minimum relative efficiency or the range of each diagnostic, while retaining enough detail to identify where the worst performance occurs.

Diagnostics, Alternatives, and Limitations

Several diagnostics help identify a schedule that is fragile across nominal values. First, look for a large drop in determinant or target-specific information at plausible parameter values. Second, inspect whether candidate rankings reverse. Third, check for nearly linearly dependent sensitivity vectors or an information matrix with a very small eigenvalue; these indicate that some parameter combination is poorly distinguished locally. Parameter correlations from candidate designs can also flag near-confounding, but a correlation alone does not show how much information is available overall. These diagnostics should be interpreted together.

If the determinant remains positive but becomes very small, the design may technically identify both parameters in the local calculation while still providing weak information. A zero or nearly zero determinant can arise when sample times coincide, when sensitivities vanish, or when the sensitivity directions become indistinguishable. In the exponential example, very late samples at rapid elimination can have concentrations close to zero, making both sensitivity components small. Computational checks should also confirm that times respect the design constraints and that the model and units have been implemented consistently.

The main alternative to selecting the schedule optimized at one nominal vector is to adopt an explicit robustness criterion. One may choose a schedule that performs acceptably at every point on a prespecified grid, for example by maximizing the minimum relative efficiency over that grid, or compare a weighted average if justified weights over parameter values are available. These are different decision rules, not consequences of local D-optimality. A grid-based worst-case check is only as comprehensive as its grid and plausible parameter ranges.

A further limitation is that this assessment treats the fixed parameter values as known separately at each evaluation and uses local, model-based information. It does not account for between-subject variability, uncertainty in the residual model, model misspecification, or parameter distributions unless those features are explicitly included in the design calculation. Nor does a stable determinant guarantee that every parameter estimate will be accurate in a realized dataset. The method is a diagnostic of local information, not a substitute for checking model assumptions or practical feasibility.

Common Mistake

Another frequent error is to change parameter values and then compare only concentration predictions. A design comparison concerns derivatives and the information they imply, not concentration fit alone. Conversely, if the model, variance, or parameterization changes between evaluations, an information difference cannot be attributed solely to nominal-value dependence.