c-Optimal Sampling for Clearance or Half-Life

Why the Target Changes the Sampling Schedule

A sampling schedule is not “optimal” in the abstract. It is optimal relative to an objective. A schedule designed to estimate several model parameters jointly can differ from one designed to estimate a single clinically relevant quantity, such as clearance or terminal half-life. The distinction matters when the number of samples is limited: allocating observations to improve one target can leave other parameters less precisely estimated.

This tutorial derives that distinction for a one-compartment IV bolus model. The derivation is local: it uses sensitivities and an approximate covariance matrix evaluated at nominal parameter values. It assumes that the observed log concentrations follow a linear terminal profile with independent, constant-variance errors. These assumptions make the target-specific allocation explicit; more complex models require their own sensitivity calculations.

Model and Derivation of the Endpoint Allocation

Suppose a dose \(D\) is administered as an IV bolus to a one-compartment model. The concentration at time \(t\) is \(C(t)=(D/V)e^{-kt}\), where \(V\) is volume and \(k=CL/V\) is the elimination rate constant. Write the log-concentration model as \(Y(t)=\log C(t)=a+bt+\varepsilon(t)\), with \(a=\log(D/V)\), \(b=-k\), and \(\varepsilon(t)\sim N(0,\sigma^2)\). Assume errors are independent across sampling times, \(\sigma^2\) is constant, the dose is known, and \(b<0\). The model is appropriate only when the one-compartment exponential adequately describes the observed profile over the design window.

Consider \(n\) observations at times 0 and \(T\), with proportion \(p\) at time 0 and \(q=1-p\) at time \(T\). For the linear parameter vector \((a,b)\), the approximate covariance of the estimates is determined by the information matrix. Equivalently, the fitted endpoint means are \(a\) and \(a+bT\), with variances \(\sigma^2/(np)\) and \(\sigma^2/(nq)\), respectively. For any differentiable target \(g(a,b)\), let \(g_a\) and \(g_b\) denote its derivatives at the nominal values. Since \(b\) is the difference of the endpoint means divided by \(T\), the delta-method variance is

The terms in brackets show how the target combines information from the two endpoints. Define \(A=g_a-g_b/T\) and \(B=g_b/T\). For nonzero \(A\) and \(B\), minimizing \(A^2/p+B^2/(1-p)\) gives \(p=|A|/(|A|+|B|)\). One justification is the Cauchy–Schwarz inequality:

with equality when \(p:q=|A|:|B|\). This is a target-specific allocation, not a universal rule. If one coefficient is zero, the formal minimum puts all observations at the endpoint associated with the nonzero coefficient; that boundary design may not estimate both model parameters. In practice, a positive minimum allocation or another constraint may be needed to retain parameter identifiability.

Clearance and Half-Life Have Different Targets

In the \((a,b)\) parameterization, clearance is \(CL=Dk/V\), so \(\log CL=\log D-a+\log(-b)\). Terminal half-life is \(t_{1/2}=\log(2)/k\), so \(\log t_{1/2}=\log\{\log(2)\}-\log(-b)\). The delta-method gradients are therefore different:

For log clearance, the endpoint coefficients are \(A=-1+1/(kT)\) and \(B=-1/(kT)\). Thus the locally optimal proportion at time 0 is \(p^*=\lvert-1+1/(kT)\rvert/\{\lvert-1+1/(kT)\rvert+\lvert1/(kT)\rvert\}\), with the remainder at \(T\). For log half-life, \(A=-1/(kT)\) and \(B=1/(kT)\), so the optimum is balanced: half the observations at each endpoint. These formulas assume both endpoints are feasible and that their errors have the same variance.

For comparison, the determinant of the information matrix for estimating \((a,b)\) is proportional to \(pqT^2\). At fixed \(n\) and fixed endpoints, this determinant is largest when \(p=q=1/2\). D-optimality therefore balances the endpoints for this model, regardless of whether the scientific target is clearance or half-life. The half-life schedule happens to agree with that D-optimal allocation under these assumptions; the clearance schedule need not.

Checking Candidate-Design Calculations

A target variance is obtained from the target gradient and the inverse information matrix, not from the determinant alone. The following arithmetic illustrates how a target-specific comparison should be checked. Suppose a hypothetical local calculation reports \(c^\mathsf{T}I^{-1}c=12/48\) for a three-to-one allocation and \(20/64\) for a balanced allocation, with log-scale residual SD \(\sigma=0.10\). If these quantities are the design factors before multiplication by \(\sigma^2\), the variances are:

and

The three-to-one allocation has the smaller variance for this specified target, even if the balanced design has the larger information determinant. The comparison is valid only if both design factors were computed for the same target, parameterization, nominal values, and residual-error assumptions.

Diagnostics, Limitations, and Alternatives

Before using an endpoint formula, check that the design and model support its assumptions. Plot or inspect the predicted and observed log concentrations over time: systematic curvature can indicate that the assumed exponential profile is inadequate. Check whether residual spread changes with concentration or time; unequal variances invalidate the equal-variance formulas and can change the optimal allocation. Verify that the nominal \(k\) is plausible, since the clearance allocation depends on \(kT\). A schedule highly sensitive to that nominal value may not perform well for the actual individual.

The formula also presumes that both endpoints are available and measurable. A nominal sample at time zero may be physically impossible after an IV bolus, and a late concentration may fall below quantification limits. If the earliest and latest feasible times are instead \(L\) and \(U\), use the model sensitivities at those times and recompute the target variance; do not simply assume the 0-to-\(T\) formula still applies. Minimum spacing, clinic windows, assay precision, and a requirement to estimate nuisance parameters can also alter the design.

In a two-compartment model, the terminal profile and its relation to clearance depend on more than one exponential component. The one-compartment derivation should not be transferred directly. For such a model, calculate the sensitivity of the chosen target to all relevant parameters and evaluate \(c^\mathsf{T}I^{-1}c\) for candidate schedules, respecting the residual-error model and feasibility constraints.

D-optimality is a natural alternative when joint estimation of the model parameters is the priority: it maximizes the information determinant, a measure of overall parameter information. c-optimality instead minimizes uncertainty for the stated scalar target. Neither criterion is universally superior. A D-optimal design may sacrifice precision for a particular target; a c-optimal design may leave some parameters poorly estimated or even fail to identify all parameters if it concentrates too heavily. A constrained c-optimal design can impose minimum sampling proportions or require a nonsingular information matrix when broader parameter estimation is also needed.

• Common mistake: “D-optimal means best for clearance.” A large determinant summarizes joint information; it does not guarantee minimum variance for clearance. Calculate the target-specific variance.
• Common mistake: using the allocation formula without naming the target. The coefficients \(A\) and \(B\) come from that target’s gradient. Derive them for log clearance or log half-life rather than reusing a schedule automatically.
• Common mistake: treating a local result as universal. The schedule depends on nominal parameters, feasible times, and the assumed error model. Check sensitivity to those choices and compare candidate designs under realistic constraints.