Choosing D-, A-, and c-Optimal PK Times

One Sample Budget, Different Design Objectives

A fixed number of PK samples can be placed to answer different questions. A design intended to estimate all model parameters jointly need not be the best design for reducing their average uncertainty, and neither is necessarily best for estimating one specified parameter combination. D-, A-, and c-optimality formalize these distinct objectives. Their rankings can differ even when every candidate schedule has the same number of samples.

The criteria are applied to a local approximation of an individual PK model. They do not identify a universally best schedule: they identify a best schedule relative to a model, nominal parameter values, residual-error assumptions, feasible sampling times, and a stated objective.

Local Information for an Individual PK Schedule

Suppose an individual has observations at times \(t_1,\ldots,t_m\), and the model predicts concentration \(f(t,\theta)\), where \(\theta\) is a vector of \(p\) parameters. At specified nominal values \(\theta_0\), the sensitivity of the predicted concentration to the parameters at time \(t_i\) is the row vector \(g_i=\partial f(t_i,\theta)/\partial\theta\), evaluated at \(\theta_0\). Let the observation errors be independent, mean-zero Gaussian errors with known variances \(\sigma_i^2\), treated as fixed for this local calculation. The schedule's local Fisher information matrix is

Equivalently, if \(J\) is the matrix with rows \(g_i\) and \(R\) is the diagonal matrix of observation variances, then \(F=J^{\mathsf T}R^{-1}J\). Thus samples are informative when their predictions are sensitive to parameter changes, with less weight assigned to observations having greater residual variance. The calculations below use whitened sensitivities, meaning that each sensitivity has already been divided by its observation-error standard deviation.

The inverse-information approximation is local: it describes uncertainty near \(\theta_0\), not every possible property of a nonlinear model or estimator. It is most defensible when the model is correctly specified, the nominal values are relevant, the sensitivities are informative, and the estimator is operating in a region where a first-order approximation is reasonable.

What the Three Criteria Optimize

Write \(C=F^{-1}\) for the local parameter covariance approximation. D-optimality maximizes \(\det(F)\), equivalently minimizing \(\det(C)\). This targets the overall volume of the joint uncertainty region. A-optimality minimizes \(\operatorname{tr}(C)\), the sum of the parameter variances in the chosen parameterization. c-optimality minimizes \(c^{\mathsf T}Cc\), the approximate variance of the specified linear parameter contrast \(c^{\mathsf T}\theta\), for a fixed vector \(c\).

For D-optimality, a joint confidence ellipsoid based on covariance \(C\) has volume proportional to \(\sqrt{\det(C)}\), for a fixed confidence level and number of parameters. Maximizing \(\det(F)\) therefore shrinks that volume. This is a joint-uncertainty objective, not a promise that every individual standard error is smallest.

A-optimality sums the diagonal entries of \(C\), so it minimizes the sum of the parameter variances. The sum is meaningful only with attention to parameter scales and units. If, for example, one parameter is expressed in milligrams and another in grams, the numerical trace can change under a unit conversion. Standardizing parameters or explicitly weighting their variances may be appropriate when a raw sum does not represent the scientific objective.

For c-optimality, the vector \(c\) specifies the target. If \(c\) selects one coordinate, the criterion targets that parameter's variance; if it combines coordinates, it targets the uncertainty in their linear combination. Its value incorporates covariance between parameters, so it cannot generally be calculated by adding marginal variances alone.

Why These Criteria Follow from the Local Approximation

The distinctions can be justified by linearizing the model around \(\theta_0\). Let \(\varepsilon\) denote the vector of observation errors and \(\delta=\hat{\theta}-\theta_0\). Locally, the model is \(f(\theta)\approx f(\theta_0)+J(\theta-\theta_0)\). The weighted least-squares estimation error in this linear approximation is

Because the errors have covariance \(R\), direct multiplication gives

This establishes the covariance approximation under the local linear model and fixed Gaussian error covariance. The determinant, trace, and contrast variance then summarize different aspects of this same covariance: joint volume, summed marginal variances, and uncertainty in a specified linear function, respectively.

Contrasts, Covariance, and Parameterization

The parameterization must match the stated objective. Under a nonsingular linear re-expression of parameters, the numerical determinant of information changes by a factor that is constant across schedules, so the D-optimal ranking is unchanged when the same transformation is applied to every candidate. The A criterion can change its ranking because the unweighted sum of variances depends on parameter scale. For c-optimality, the contrast must be transformed consistently so it continues to describe the same scientific quantity.

Choosing and Checking a Criterion

A practical comparison starts by stating what uncertainty matters. Use D-optimality when the objective is joint precision across the parameter vector and a common parameter scaling is defensible. Use A-optimality when the summed variances in a chosen parameterization represent the desired overall precision. Use c-optimality when a particular linear parameter combination is the target. If the scientific target is nonlinear, a linear contrast may be an approximation near \(\theta_0\); its local gradient can be used as the contrast vector, with the approximation stated explicitly.

Before interpreting any score, check the design and its assumptions:

• Check rank. If \(F\) is singular or nearly singular, inverse-based criteria are undefined or highly unstable. A large determinant alone should not be used to conceal a practically weak parameter direction.
• Check weights and units. Verify that sensitivities use the intended parameterization and that observation variances have been included consistently. An unweighted calculation can favor times with large raw sensitivities even when those observations are imprecise.
• Check the target. State the entries and meaning of \(c\), and calculate the full quadratic form \(c^{\mathsf T}F^{-1}c\), including covariance terms.
• Check feasibility. Compare only schedules that satisfy the same sample budget and relevant time, assay, and operational constraints. The criterion does not make an infeasible time acceptable.
• Check local assumptions. The calculation is conditional on the nominal parameter values, model form, and residual-error assumptions. A locally optimal schedule may perform differently if these inputs are inaccurate.

These criteria are alternatives rather than interchangeable labels. A D-optimal design is not automatically A-optimal or c-optimal. Other practical approaches include choosing times from clinical considerations, comparing a small set of prespecified schedules, or using a weighted objective that combines several scientific priorities. Such choices can be sensible, but the objective and any weights should be stated rather than presented as a consequence of one of the three criteria.

Common Pitfall: Treating “Optimal” as an Absolute Ranking

A common mistake is to calculate a determinant and conclude that the schedule with the largest value is best for every PK purpose. That reasoning is incorrect: the determinant summarizes joint uncertainty volume, not the variance of every parameter or contrast. The correction is to define the target first, compute the matching criterion, and report its interpretation in context. A second error is to use \(\operatorname{tr}(F^{-1})\) without considering the parameter units; the correction is to choose and report a meaningful parameter scaling or variance weighting before comparing A scores.