Why Sampling Phase Matters
A concentration measurement is informative only to the extent that the model prediction changes when a parameter changes. In a one-compartment IV bolus model, that change depends on when the sample is taken. Early concentrations are strongly influenced by the initial concentration, while later concentrations reflect the elimination rate as well as the declining amount of drug in the body. The practical question is not simply whether a sample is “early” or “late,” but which parameter changes could plausibly explain its observed concentration.
The earlier material on time-specific sensitivities defines a sensitivity as the change in a model prediction with respect to a parameter. Here we use those profiles to interpret sampling phases for an individual IV bolus model. The conclusions depend on the model, the parameter scale, and the residual-error assumptions; phase labels alone do not guarantee that a parameter is well estimated.
Model, Parameters, and Sensitivity Profiles
Assume a known IV bolus dose \(D\), a one-compartment model with instantaneous mixing, constant volume \(V\), and first-order elimination with clearance \(CL\). The predicted concentration at time \(t\) after dosing is
For the sensitivity calculations below, use log parameters \(\log V\) and \(\log CL\). This expresses a parameter perturbation in relative rather than absolute units: for example, a small change in \(\log V\) corresponds approximately to the same fractional change in \(V\). Assume independent concentration errors with constant variance \(\sigma^2\), so sensitivity magnitudes can be compared directly across times. This is a local, model-based assessment; it does not account for model misspecification or uncertainty in the dose.
Since \(k=CL/V\), changing \(V\) at fixed \(CL\) changes both the initial concentration and the elimination rate. Applying the chain rule gives
The factor \(C(t)\) matters. A large fractional effect on a very small terminal concentration can still yield a small absolute change in measured concentration. At \(t=0\), the volume sensitivity is \(-C(0)\) and clearance sensitivity is zero. As time increases, clearance sensitivity grows in magnitude initially, while volume sensitivity changes sign at \(kt=1\). Far into the terminal phase, both sensitivities eventually shrink in absolute concentration units as \(C(t)\) approaches zero.
Worked Example: Reading Sensitivities Across the Curve
Suppose a hypothetical patient receives \(D=100\) mg, with \(V=20\) L and \(CL=4\) L/h. Then \(k=CL/V=0.2000\) h\(^{-1}\), and the initial concentration is \(D/V=5.000\) mg/L. Calculate the concentrations and log-parameter sensitivities at 0.5, 5, and 15 h.
At \(t=0.5\) h, \(kt=0.1000\) and \(C=5e^{-0.1}=4.524\) mg/L. Thus \(s_V=4.524(-1+0.1)=-4.072\) mg/L and \(s_{CL}=-4.524(0.1)=-0.4524\) mg/L. The early observation is much more sensitive to volume than to clearance.
At \(t=5\) h, \(kt=1.000\) and \(C=5e^{-1}=1.839\) mg/L. The sensitivities are \(s_V=1.839(-1+1)=0\) mg/L and \(s_{CL}=-1.839\) mg/L. At this particular time, a small log-volume change has no first-order effect on concentration when \(CL\) is held fixed; clearance has a nonzero effect.
At \(t=15\) h, \(kt=3.000\) and \(C=5e^{-3}=0.2489\) mg/L. Then \(s_V=0.2489(2)=0.4979\) mg/L and \(s_{CL}=-0.2489(3)=-0.7468\) mg/L. The clearance sensitivity is larger in absolute value here, but both sensitivities are small compared with the early volume sensitivity because little concentration remains.
These calculations show why “terminal samples inform clearance” needs qualification. A terminal slope can be useful for identifying the elimination rate, but a late concentration may have little absolute information under constant-variance additive error. The useful balance depends on the error model and assay range as well as on the model sensitivities.
From Sensitivities to Information and Correlation
For independent Gaussian observations with known constant variance \(\sigma^2\), the local Fisher information for the two log parameters is the sum of outer products of their sensitivity vectors, divided by \(\sigma^2\). If \(s(t_i)=(s_V(t_i),s_{CL}(t_i))^\mathsf{T}\), then
The direction of each sensitivity vector matters as well as its length. If the vectors from all sampling times point in nearly the same direction, the data can identify only a similar combination of parameters, even if individual sensitivities are substantial. The local covariance approximation is \(F^{-1}\), when \(F\) is nonsingular. For a two-parameter information matrix with entries \(F_{11}, F_{12}, F_{22}\), the corresponding local parameter correlation is
This is the correlation in the inverse information matrix, not the correlation between the two sensitivity columns. The latter is \(F_{12}/\sqrt{F_{11}F_{22}}\), so for two parameters the signs are opposite. A correlation near \(1\) or \(-1\) indicates strong local trade-off and uncertainty in separating the parameters; it does not by itself say that the model is structurally unidentifiable.
Worked Example: A Three-Time Schedule and Its Local Correlation
Use the three hypothetical sampling times and parameter values in the preceding example. First form the unweighted sensitivity information matrix \(G=\sum_i s(t_i)s(t_i)^\mathsf{T}\). A common variance factor would multiply every entry by \(1/\sigma^2\) and does not change the resulting correlation.
Using the rounded sensitivity vectors \((-4.072,-0.4524)\), \((0,-1.839)\), and \((0.4979,-0.7468)\), the entries are \(G_{11}=(-4.072)^2+(0)^2+(0.4979)^2\approx16.83\), \(G_{22}=(-0.4524)^2+(-1.839)^2+(-0.7468)^2\approx4.144\), and \(G_{12}=(-4.072)(-0.4524)+(0)(-1.839)+(0.4979)(-0.7468)\approx1.470\).
The determinant is approximately \(16.83(4.144)-(1.470)^2\approx67.58\), which is positive, so this two-parameter matrix is nonsingular. The local parameter correlation is \(-1.470/\sqrt{16.83(4.144)}\approx-0.1760\). Thus, in this local approximation, the parameter estimates have only modest negative correlation. The result reflects the combined information from the selected times; it is not a universal property of a three-sample schedule.
As an arithmetic check on interpreting a two-parameter information matrix, suppose another hypothetical local calculation gives \(F=\begin{pmatrix}300&410\\410&1001\end{pmatrix}\), in scaled parameter coordinates. Its determinant is \(300(1001)-410^2=132200\), so it is nonsingular. The local parameter correlation is \(-410/\sqrt{300(1001)}=-0.748180\ldots\), or approximately \(-0.7482\). The fairly strong negative correlation signals a pronounced local trade-off, despite the positive off-diagonal information entry.
What Each Sampling Region Can and Cannot Tell You
The sensitivity profiles support a useful interpretation, provided it remains tied to the model and error assumptions.
- Early samples: when \(kt\) is small, \(s_V\) is approximately \(-C(t)\) and \(s_{CL}\) is approximately \(-C(t)kt\). With known dose, this makes early observations particularly useful for the concentration scale and volume. If all samples are extremely early, however, there is little direct separation of clearance from volume through the elimination slope.
- Intermediate samples: as the curve declines, the clearance sensitivity becomes material. Around \(kt=1\), the log-volume sensitivity is zero in this parameterization while the clearance sensitivity is nonzero. Sampling on both sides of such a point can help distinguish changes in scale from changes in decline rate.
- Terminal samples: the declining curve’s slope contains information about \(k=CL/V\). But under constant additive error, a very late sample can be close to the assay’s lower range and have weak absolute sensitivity to both parameters. Several nearly redundant terminal observations may improve precision in the terminal trend without resolving a volume-clearance trade-off.
These are not rules that assign one parameter exclusively to one phase. In this model, \(V\) affects both \(D/V\) and \(k\), while \(CL\) affects the rate of decline through \(k\). If dose is unknown, early concentrations cannot identify \(V\) independently of dose. If the assay error is proportional to concentration rather than constant in concentration units, the effective weighting of late samples changes. The sensitivity derivatives remain derivatives of the mean model, but the information calculation must reflect the specified error model.
Compute \(kt\), using the individual’s parameter values or a plausible nominal value. This gives a more transferable description than clock time alone.
Evaluate \(s_V(t)\) and \(s_{CL}(t)\), with the assumed observation-error scale in mind. Large absolute sensitivities indicate stronger local response of the predicted concentration.
Inspect the information matrix and its inverse. Similar sensitivity directions imply parameter correlation and potentially unstable separate estimates.
Compare predicted concentrations with assay limits and verify that the assumed error model is reasonable. A mathematically sensitive time may be infeasible or imprecise in practice.
Diagnostics, Alternatives, and Limitations
A useful diagnostic is to inspect the sensitivity profiles across the entire feasible sampling window, not just at the chosen times. A profile that is nearly zero across that window signals weak local information for that parameter. A design whose sensitivity vectors are nearly collinear may yield a poorly conditioned information matrix and large standard errors or strong parameter correlation. If the fitted model is outside the region where the local approximation is credible, sensitivities evaluated at a single nominal parameter vector may not describe the information well.
Sensitivity interpretation complements, rather than replaces, design criteria. A D-optimal design summarizes joint information through the determinant of the information matrix; it can favor a schedule that balances information across parameters, but the determinant alone does not explain which phase contributes to which parameter or reveal all practical concerns. Direct standard-error or correlation summaries make those concerns more visible. Profile likelihood or simulation-based assessment can be useful when the local, approximately quadratic likelihood assumption is doubtful, although these approaches require more computation.
Important failure modes include treating “terminal” as automatically informative, ignoring the assumed error model, and interpreting a small parameter correlation as proof that both parameters are precise. Precision also depends on the absolute information levels. Finally, the analysis is conditional on a one-compartment IV bolus model. If the observed curve has a distribution phase, delayed mixing, or other systematic structure, these phase interpretations may be misleading; residual and model diagnostics must support the model before its design implications are trusted.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- For the stated model, what are \(s_V(t)\) and \(s_{CL}(t)\) at \(t=0\)?
- At what dimensionless time does the log-volume sensitivity equal zero, and what does that mean locally?
- Why may a very late sample have limited information under constant-variance additive error even if it lies in the terminal phase?
- If \(F_{12}>0\) for a nonsingular two-parameter information matrix, what is the sign of the local parameter correlation?
- Why can two samples with large sensitivity magnitudes still provide weak separation of parameters?
Solutions. At \(t=0\), \(s_V=-D/V\) and \(s_{CL}=0\). Since \(s_V=C(t)(-1+kt)\), it is zero at \(kt=1\); locally, a small log-volume change at fixed clearance has no first-order effect on concentration at that time. A very late concentration is small, so both absolute sensitivities can be small relative to constant assay noise. For two parameters, the correlation is \(-F_{12}/\sqrt{F_{11}F_{22}}\), so it is negative when \(F_{12}>0\). Finally, large sensitivity vectors that point in nearly the same direction are collinear or nearly collinear; they may support a combination of parameters well while separating the individual parameters poorly.