Choosing Sampling Times From Parameter Sensitivities

Why Sampling Time Changes What a Concentration Can Tell You

A concentration measured soon after an intravenous bolus and one measured later do not provide the same information about pharmacokinetic parameters. In a one-compartment model, volume sets the initial concentration scale, while clearance controls how quickly concentration declines. A useful sampling schedule therefore needs times at which these effects are distinguishable, not merely several measurements taken close together.

Here, “distribution-scale” refers narrowly to the concentration scale associated with volume in this one-compartment model. The model has no separate distribution phase or peripheral compartment. Sensitivity calculations can help screen candidate times, but they are local to specified parameter values and do not replace evaluation of the full design under its error model.

Define the elimination rate constant \(k=CL/V\), with units of inverse time, and the dimensionless time \(x=kt\). Then \(C(t)=(D/V)\exp(-x)\). The initial concentration scale is \(D/V\); the exponential decline depends on \(CL/V\). Because both parameters enter the exponent as well as the overall expression, their effects should be compared using derivatives rather than assigned to completely separate parts of the curve.

Deriving and Reading the Sensitivities

A raw sensitivity is the derivative of the predicted concentration with respect to a parameter. Its units depend on the parameter’s units, which can make comparisons awkward. For positive parameters, a more interpretable local measure is the derivative with respect to the log parameter. For example, \(\partial C/\partial\log(CL)=CL\,\partial C/\partial CL\); it describes the concentration change associated with a small relative change in clearance.

Differentiating this expression gives the log-concentration sensitivities. These are also relative sensitivities of concentration, because \(d\log C=dC/C\) for a small change.

At \(t=0\), the clearance sensitivity is zero and the volume sensitivity is \(-1\): a 1% increase in volume gives approximately a 1% decrease in the immediate concentration. As time increases, clearance sensitivity becomes more negative. Volume sensitivity moves from \(-1\), passes through zero when \(x=1\), and becomes positive after that point. This does not mean volume has ceased to affect concentration at \(x=1\); it means the local effects of volume on the scale and on the rate constant cancel for log concentration at that particular time.

Why Separated Times Help Distinguish Parameters

A pair of measurements is useful for separating two parameters when their sensitivity vectors are not parallel. For two times \(t_1\) and \(t_2\), write \(x_i=kt_i\). The matrix whose rows are the two sensitivity vectors has determinant \((-x_1)(x_2-1)-(-x_2)(x_1-1)=x_1-x_2\). If the times differ, this determinant is nonzero in the idealized two-observation calculation; if the times coincide, the rows are identical and the two local effects cannot be separated from those observations alone.

A nonzero determinant is not a guarantee of precise estimation. If the times are very close, the sensitivity vectors are nearly parallel, and the distinction can be overwhelmed by measurement error. Nor does this algebra account for other parameters, random effects, or constraints on when samples can be collected. It is a simple diagnostic for whether candidate times span different local parameter effects.

Precision, Feasibility, and Choosing a Candidate Time

Sensitivity magnitude must be interpreted alongside the observation error. If log concentrations have independent errors with common variance \(\sigma_{\log}^{2}\), the local information contribution from one observation is proportional to the outer product of its sensitivity vector, divided by \(\sigma_{\log}^{2}\). This is a two-parameter illustration of the local Fisher information used in design calculations. It assumes the model is correct at the nominal parameters and the specified error model is appropriate.

For example, take the first pair above and suppose the log-scale standard deviation is \(0.10\), so \(\sigma_{\log}^{2}=0.01\). The sum of sensitivity outer products is \(\begin{pmatrix}1.04&0.16\\0.16&0.64\end{pmatrix}\). Dividing by \(0.01\) gives the information matrix \(\begin{pmatrix}104&16\\16&64\end{pmatrix}\), whose determinant is \(104(64)-16^2=6400\). For the pair with \(x=0.2,0.4\), the corresponding matrix is \(\begin{pmatrix}20&40\\40&100\end{pmatrix}\), with determinant \(20(100)-40^2=400\). Under these particular assumptions, the more separated pair has a larger determinant and better joint information for the two log parameters. This comparison holds the error variance fixed and should not be carried over unchanged to another error model.

With additive concentration error of constant standard deviation \(\sigma\), the relevant derivatives are instead \(\partial C/\partial\log(CL)=-Cx\) and \(\partial C/\partial\log(V)=C(x-1)\). Their magnitudes shrink as concentration declines. Thus a very late observation may have a useful sensitivity direction but contribute little information if the concentration is small relative to assay error. Under lognormal residual error, a constant log-scale variance gives a different weighting. Choosing times from sensitivities therefore requires stating whether errors are additive, proportional, or otherwise modeled.

Diagnostics, Alternatives, and Limitations

Several warning signs indicate that a sensitivity-based choice may be misleading:

• Candidate times are too close. Similar \(x\) values yield nearly parallel sensitivity vectors, so clearance and volume effects are difficult to distinguish.
• A late concentration is too small. Under additive error, low concentrations can have little information even when their relative sensitivities look large. Check predicted concentrations against assay performance.
• The nominal parameters are unrepresentative. Since \(x=CLt/V\), the same clock time can be early for one individual and late for another. Evaluate a plausible range of parameter values rather than relying on one profile when variability is material.
• The structural model is inadequate. If the actual concentration-time course has a distribution phase, delayed input, or other features absent from the one-compartment bolus model, its sensitivity pattern may not describe the data-generating process.

Sensitivity inspection is a fast, interpretable alternative to choosing times by intuition alone, such as taking equally spaced samples. Equal spacing does not ensure that the observations have distinct or useful parameter sensitivities. Conversely, a formal design criterion based on the full Fisher information can incorporate multiple parameters and the residual error model, but it requires more modeling assumptions and computation. Sensitivity plots or tables are best treated as a diagnostic and screening tool before evaluating a complete candidate schedule.

This calculation has deliberate limits. It describes local sensitivity at nominal fixed effects in a one-compartment model; it does not by itself account for between-subject variability, uncertainty in nuisance parameters, or an entire population design objective. It also does not imply that the time with the largest derivative is automatically best. A strong derivative in one parameter can coincide with poor separation from another parameter, poor assay precision, or impractical collection. Use sensitivity direction, magnitude, error weighting, and feasibility together.