Sparse Sampling During and After IV Infusion

Why Sampling Window Matters

A short infusion and a small number of concentration measurements can, in principle, provide information about both individual clearance and volume. But a sample’s value depends on when it is taken. While drug is entering the body, the concentration reflects both the ongoing input and elimination. After the infusion ends, the concentration declines, and the shape of that decline carries different information. A sparse schedule must use these changing sensitivities rather than assume that every measurable sample is equally informative.

The focus here is the one-compartment model with linear elimination and a constant-rate intravenous infusion. The design comparisons are local: they describe information at specified parameter values, with the stated observation-error model. They do not guarantee that one schedule is best for every patient or every parameter range.

Definition: Let $R_0$ be the constant infusion rate, $T$ the infusion duration, $CL$ individual clearance, and $V$ individual volume of distribution. Define $k=CL/V$. With time $t$ measured from infusion start, the predicted concentration is $$ C(t)= \begin{cases} \dfrac{R_0}{CL}\left(1-e^{-kt}\right), & 0\leq t\leq T,\\[6pt] \dfrac{R_0}{CL}\left(1-e^{-kT}\right)e^{-k(t-T)}, & t>T. \end{cases} $$

These expressions assume a known infusion rate and duration, a known dose delivered as $R_0T$, a well-mixed one-compartment disposition model, linear elimination, and no other drug input during the observation period. Concentrations are assumed to be observed without model misspecification. For the information calculations below, observations are conditionally independent, normally distributed about $C(t)$, and have the same known residual standard deviation $\sigma$ in concentration units. Different residual-error assumptions can change the preferred timing.

How the Two Windows Carry Information

Use log parameters $\log(CL)$ and $\log(V)$ so that the sensitivities describe changes per proportional change in each parameter. During infusion, set $u=kt$, and define $q(u)=u/(e^u-1)$. Differentiating the model gives

\mathbf{s}_{\mathrm{in}}(t) = \begin{pmatrix} \dfrac{\partial C(t)}{\partial \log(CL)}\\[4pt] \dfrac{\partial C(t)}{\partial \log(V)} \end{pmatrix} = C(t) \begin{pmatrix} -1+q(u)\\ -q(u) \end{pmatrix}.

For example, the factor $q(u)$ arises because $u$ changes proportionally with $CL$ and inversely with $V$. As $u$ approaches zero, $q(u)$ approaches one: the concentration is approximately $R_0t/V$, so its leading dependence is on volume rather than clearance. At larger $u$, $q(u)$ decreases toward zero, and the concentration during infusion becomes more sensitive to clearance and less sensitive to volume.

After infusion, let $a=kT$ and $b=k(t-T)$. The infusion-end factor depends on $a$, while the post-infusion decline depends on $b$. Differentiating both factors yields

\mathbf{s}_{\mathrm{post}}(t) = C(t) \begin{pmatrix} -1+q(a)-b\\ -q(a)+b \end{pmatrix}.

At the instant the infusion ends, $b=0$, so the post-infusion sensitivity equals the sensitivity at $t=T$ during infusion. A measurement at that boundary is one observation with the same model prediction from either expression, not two independent pieces of information. For later times, the term $b$ reflects the elapsed post-infusion decline. It can make the two parameter sensitivities point in different directions, which is useful for separating clearance from volume. However, the concentration itself eventually becomes small, weakening the information under additive error.

Interpretation: A sample is informative about a parameter when its predicted concentration changes appreciably with that parameter. A set of samples is useful for distinguishing clearance from volume when their sensitivity vectors are not nearly parallel. High concentrations alone do not ensure that the two parameters are separately well informed.

Worked Examples of Sparse Infusion Sampling

Worked Example: Comparing an Early, End-of-Infusion, and Post-Infusion Sample

Consider hypothetical values $R_0=100$ mg/h, $CL=10$ L/h, $V=20$ L, and $T=2$ h. Then $k=CL/V=0.5000$ h$^{-1}$, and $R_0/CL=10.00$ mg/L. Calculate sensitivities at $t=0.5$ h, $t=2$ h, and $t=4$ h.

At $t=0.5$ h, $u=0.25$. Thus $C=10(1-e^{-0.25})=2.212$ mg/L and $q(0.25)=0.8802$. The sensitivity vector is

\mathbf{s}(0.5) = 2.212 \begin{pmatrix} -1+0.8802\\ -0.8802 \end{pmatrix} \approx \begin{pmatrix} -0.2650\\ -1.947 \end{pmatrix} \text{ mg/L}.

The early measurement is substantially more sensitive to log volume than to log clearance. At $t=T=2$ h, $u=1$, $C=10(1-e^{-1})=6.321$ mg/L, and $q(1)=0.5820$. Therefore

\mathbf{s}(2) = 6.321 \begin{pmatrix} -0.4180\\ -0.5820 \end{pmatrix} \approx \begin{pmatrix} -2.642\\ -3.679 \end{pmatrix} \text{ mg/L}.

At $t=4$ h, the infusion has been over for $\tau=2$ h. Here $a=1$, $b=1$, and $C=10(1-e^{-1})e^{-1}=2.325$ mg/L. The post-infusion sensitivities are

\mathbf{s}(4) = 2.325 \begin{pmatrix} -1+0.5820-1\\ -0.5820+1 \end{pmatrix} \approx \begin{pmatrix} -3.298\\ 0.9721 \end{pmatrix} \text{ mg/L}.

The three sensitivity vectors differ in direction. In particular, the post-infusion measurement has a positive volume sensitivity, unlike the early infusion measurement. This contrast can help distinguish the parameters when measurements are combined, provided the observation model and parameter values are reasonable.

Worked Example: Checking a Later Post-Infusion Concentration and Its Sensitivities

For another hypothetical regimen, take $CL=5$ L/h, $V=50$ L, $R_0=50$ mg/h, and $T=8$ h. Then $k=0.1000$ h$^{-1}$, and $R_0/CL=10.00$ mg/L. At $t=16$ h, the infusion ended eight hours earlier, so $a=b=0.8000$. Since $q(0.8)=0.6528$,

C(16) = 10.00(1-e^{-0.8000})e^{-0.8000} = 10.00(0.5507)(0.4493) \approx 2.474\text{ mg/L}.

The sensitivity coefficients are $-1+0.6528-0.8000=-1.1472$ for log clearance and $-0.6528+0.8000=0.1472$ for log volume. Therefore

\mathbf{s}(16) = 2.474 \begin{pmatrix} -1.1472\\ 0.1472 \end{pmatrix} \approx \begin{pmatrix} -2.839\\ 0.3643 \end{pmatrix} \text{ mg/L}.

The calculation shows why it is important to check both the predicted concentration and the sensitivity multiplier: an arithmetic error in the concentration propagates directly into every component of the sensitivity vector. Here, clearance sensitivity is negative and larger in magnitude, while the volume sensitivity is positive and smaller.

Worked Example: Comparing Two Two-Sample Schedules

Return to the regimen in the first example and suppose $\sigma=0.2000$ mg/L. Compare a schedule with samples at $0.5$ and $2$ h to one with samples at $0.5$ and $4$ h. Under independent normal errors with this common standard deviation, the local Fisher information matrix for the two log parameters is $\mathbf{F}=\sigma^{-2}\sum_i \mathbf{s}(t_i)\mathbf{s}(t_i)^{\mathsf T}$. For a two-sample, two-parameter design, its determinant is $\det(\mathbf{F})=\{\det(\mathbf{S})\}^2/\sigma^4$, where the rows of $\mathbf{S}$ are the two sensitivity vectors.

Using the rounded sensitivities above, the determinant of the sensitivity matrix for $0.5$ and $2$ h is $\det(\mathbf{S})=(-0.2650)(-3.679)-(-1.947)(-2.642)\approx-4.17$. Thus $\det(\mathbf{F})\approx 4.17^2/(0.2000)^4\approx 1.09\times10^4$. For $0.5$ and $4$ h, $\det(\mathbf{S})=(-0.2650)(0.9721)-(-1.947)(-3.298)\approx-6.68$, giving $\det(\mathbf{F})\approx 6.68^2/(0.2000)^4\approx 2.79\times10^4$.

For these assumed parameter and error values, the schedule that replaces the end-of-infusion sample with the $4$-h sample has the larger local determinant, about 2.57 times as large. This is a comparison at one parameter point, not a claim that post-infusion sampling is always preferable. The end-of-infusion sample may be useful for other parameter values, error models, or practical constraints.

Why Very Late Samples Can Fail

After infusion, increasing $b$ changes the relative sensitivity coefficients, but it also reduces $C(t)$ exponentially. With a common additive error standard deviation, a single sample contributes $\mathbf{s}(t)\mathbf{s}(t)^{\mathsf T}/\sigma^2$ to the information matrix. Thus a large relative sensitivity cannot compensate indefinitely for a concentration approaching zero.

Worked Example: The Cost of Sampling Too Late

Use $R_0/CL=10.00$ mg/L, $a=1$, and $k=0.5000$ h$^{-1}$ from the first example. Compare post-infusion samples at $t=4$, $8$, and $12$ h, corresponding to $b=1$, $3$, and $5$. Their concentrations are $2.325$, $0.3147$, and $0.04259$ mg/L, respectively. Applying the post-infusion sensitivity formula gives approximately $(-3.298,\ 0.9721)^{\mathsf T}$, $(-1.075,\ 0.7609)^{\mathsf T}$, and $(-0.2308,\ 0.1882)^{\mathsf T}$ mg/L.

For $\sigma=0.2000$ mg/L, the trace of each sample’s information contribution is $\|\mathbf{s}(t)\|^2/\sigma^2$. The values are approximately $295.5$, $43.41$, and $2.22$, respectively. Under this additive-error assumption, the latest sample contributes far less information, even though its sensitivity coefficients include larger elapsed-time terms. It may also be below the assay’s quantification limit, in which case the ordinary concentration observation model would not apply.

Assumptions, Diagnostics, and Design Alternatives

Before using these calculations, verify that the planned infusion and observation times correspond to the actual administration record. A delayed start, interruption, or inaccurate duration changes the input history and hence the predicted concentration and sensitivities. Check that the one-compartment assumption is plausible over the proposed windows; a systematic early distribution phase or a changing elimination process can produce residual patterns that this model cannot represent. Also check whether the assay error is approximately constant in concentration units. If variability grows with concentration or includes a proportional component, the information contributions need to reflect that error structure rather than use a common $\sigma$.

A useful diagnostic is to inspect the sensitivity vectors, not only the concentrations. If the vectors from all chosen times are nearly parallel, the information matrix may be poorly conditioned and clearance and volume may be difficult to separate. If a proposed late sample is predicted near the assay limit, its nominal sensitivity may overstate its practical usefulness. A design should also be checked over plausible parameter values: a schedule selected at a single nominal $CL$ and $V$ can perform poorly when an individual’s elimination rate differs substantially.

The main alternatives are not interchangeable. Sampling only during infusion can capture the rise toward a plateau; very early points are especially volume-sensitive, while later points can become more clearance-sensitive. Sampling only after infusion uses the decay and may provide useful parameter contrast, but measurements taken too late may be weak or unquantifiable. A schedule spanning the end of infusion and a suitable post-infusion window can combine these features, though a sample exactly at the end does not create a second observation merely because it is described as belonging to both phases. The best balance depends on the parameter target, residual error, feasible clinic times, and assay limits.

These results are local and model-dependent. They do not account automatically for uncertainty in the nominal parameter values, missed samples, infusion-record error, or model misspecification. If the errors are correlated, heteroscedastic, censored, or otherwise different from the assumptions stated above, the information calculation must be modified. Nor does a larger determinant by itself guarantee precise estimation of each parameter under every practical analysis; it summarizes joint information under the specified local model.

Common Pitfalls

Calling the end-of-infusion concentration a post-infusion decline sample. At the boundary, elapsed post-infusion time is zero; the prediction and sensitivities coincide with those at the end of the infusion.
Assuming the latest sample is automatically best for clearance. The elapsed-time term can increase relative sensitivity, but concentration decay reduces absolute sensitivity under additive error.
Comparing concentrations instead of sensitivities. A high predicted concentration may carry little information about one parameter, and two high-concentration samples may have nearly parallel sensitivity vectors.
Ignoring the residual-error model. Information depends on sensitivity relative to observation variability. A schedule comparison made under constant additive error may not hold under proportional or combined error.
Treating a local comparison as universal. The calculations depend on the assumed clearance, volume, infusion, and error values; evaluate the design over the plausible range relevant to its use.

Key takeaway: For a one-compartment IV infusion, early infusion samples tend to emphasize volume, while later infusion and post-infusion samples change the balance of clearance and volume sensitivities. Sparse sampling should combine useful sensitivity directions while keeping concentrations measurable, and every comparison must be interpreted under its model and error assumptions.

Check Your Understanding

Answer each question before reading the brief solutions.

During infusion, what does the concentration approach when $kt$ is very small, and which parameter does this approximation emphasize?
Why does a sample taken exactly at infusion end not provide one during-infusion and one post-infusion observation?
For a post-infusion sample, explain what happens to the concentration as $b=k(t-T)$ becomes large under the stated model.
What diagnostic feature of sensitivity vectors suggests that clearance and volume may be difficult to distinguish?

Solutions. 1. Since $1-e^{-kt}\approx kt$, $C(t)\approx(R_0/CL)kt=R_0t/V$, so the leading dependence is on volume. 2. The model is continuous at $T$, and the post-infusion elapsed time is zero there; both expressions give the same prediction and sensitivity for a single measurement. 3. The factor $e^{-b}$ drives concentration toward zero, so under additive error the absolute sensitivity and information contribution eventually diminish. 4. Nearly parallel sensitivity vectors provide little contrast between the parameter effects and can make the information matrix poorly conditioned.