Individual Sampling at Multiple-Dose Steady State

Why Sampling at Steady State Requires Its Own Design

At multiple-dose steady state, a concentration measured after a dose reflects both that dose and drug remaining from earlier doses. A sample time that seems informative in a single-dose profile may therefore have a different role within a repeated-dose interval. Designing sparse sampling means choosing times whose concentrations respond differently to the individual’s unknown parameters—not simply choosing times with high concentrations or evenly spacing collections.

This tutorial considers a single individual with a known dose and dosing interval, described by a one-compartment model with first-order absorption and elimination. The design question is how to choose a small set of times within one steady-state interval to estimate clearance, volume, and absorption rate. The earlier material on oral PK sampling established why different portions of a concentration profile can convey different parameter information; here, repeated-dose accumulation is included explicitly.

The Steady-State Profile and Its Sensitivities

For \(k_a\ne k\), define

Summing the contributions from the current and all preceding doses gives the steady-state concentration

The denominator terms account for accumulation across dosing intervals. Omitting them gives a single-dose profile, not the steady-state profile. A concentration just before the next dose is the limit as \(t\) approaches \(\tau\) from below; the sample belongs to the interval ending at that dose.

For design, use log parameters \(\theta=(\log CL,\log V,\log k_a)\). The sensitivity of a concentration to a parameter is its partial derivative with respect to that parameter, evaluated at the nominal individual values. Log parameterization expresses changes on a relative scale: for example, a small change in \(\log CL\) corresponds approximately to the same fractional change in \(CL\).

Differentiating the steady-state expression gives the following useful formulas. Here \(A_k\) denotes the derivative of \(A(t)\) with respect to \(k\), and \(B_{k_a}\) denotes the derivative of \(B(t)\) with respect to \(k_a\):

The derivatives of the accumulation terms are

The volume identity follows because changing \(\log V\) changes the factor \(1/V\) directly and changes \(k=CL/V\) in the opposite direction: \(\partial\log k/\partial\log V=-1\). Thus the volume derivative is the negative concentration minus the clearance derivative. The absorption derivative includes both the derivative of the prefactor \(k_a/(k_a-k)\) and the derivative of the accumulation term \(B\); dropping either contribution can materially distort the sensitivity.

From Sensitivities to Schedule Information

Suppose a schedule samples at \(m\) times \(t_1,\ldots,t_m\). Stack the sensitivity rows to form the \(m\times3\) matrix \(S\). Under independent assay errors with known variances \(\sigma_i^2\), the local Fisher information matrix for the log parameters is

The information matrix summarizes how strongly and distinctly the observations respond to parameter changes. For a fixed number of observations and the same parameterization, a local D-optimal comparison selects the schedule with the largest \(\det(I)\), equivalently the largest \(\log\det(I)\). This criterion rewards information spread across parameter directions; it does not mean that each parameter is individually estimated with equal precision.

With three parameters, fewer than three observations cannot give a full-rank information matrix. Three observations can provide full rank if their sensitivity rows are linearly independent. Additional observations may improve precision, but repeated rows or nearly collinear rows can add little information about directions already represented in the schedule.

Choosing Times and Checking Whether the Design Is Credible

For a sparse design, begin with feasible candidate times inside \([0,\tau)\), calculate each sensitivity row at the nominal individual parameters, and compare feasible schedules using their weighted information matrices. The practical search may be exact over a small candidate set or approximate over a grid; either way, the selected schedule should be recalculated using the actual feasible collection times and the intended residual error model.

A useful diagnostic is the rank and conditioning of \(S\) or \(I\). If sensitivity columns are nearly proportional across the chosen times, the corresponding parameters are difficult to distinguish, even if concentrations are measured precisely. Very large or unstable numerical derivatives can also signal a calculation problem, especially near \(k_a=k\), where the displayed parameterization has a removable limiting form but direct evaluation can suffer cancellation. In that region, use a numerically stable limit or an equivalent implementation and verify it against nearby parameter values.

This local design also depends on the nominal parameters. If the individual’s true absorption or elimination rate differs substantially from the assumed value, the sensitivity pattern—and hence the preferred times—can change. A design should therefore be checked over plausible nominal values when those values are uncertain. This is a sensitivity check, not a guarantee that a locally optimal schedule remains optimal for every individual.

Alternatives, Limitations, and a Common Pitfall

A D-criterion is a joint-parameter criterion: it compares the overall local volume of uncertainty for the selected parameter vector. Other design objectives can prioritize one parameter or a particular combination; the appropriate choice depends on the estimation goal. Simple peak, trough, or mid-interval sampling rules can be easier to communicate and may be operationally convenient, but they do not by themselves establish that the three sensitivity directions are well separated. Conversely, a numerically high D-information schedule may be impractical because of clinic timing, assay limits, or patient burden.

The calculation assumes a correct structural model, known dose history and interval, steady state, known error variances, and locally valid nominal parameters. If bioavailability is also unknown, it can be confounded with scale parameters such as volume unless the data or external information separate them. Correlated assay errors require a non-diagonal error covariance in the information calculation. These limitations should be considered before interpreting a numerical schedule ranking as a clinical recommendation.

• Pitfall: using a single-dose curve. The reasoning “the current dose determines the concentration at time \(t\), so the single-dose expression is enough” ignores residual drug from previous doses. Use the steady-state accumulation denominators for both elimination and absorption.
• Pitfall: treating a high concentration as high information. A large concentration alone does not show that sensitivities distinguish clearance, volume, and absorption. Compare the full sensitivity rows, weighted for assay precision.
• Pitfall: mixing parameter scales. A design based on derivatives with respect to raw parameters cannot be compared directly with one based on log parameters without transforming the sensitivities consistently.