Why Sparse Sampling Blurs Variability Sources

Why the Same Concentrations Can Tell Different Stories

A concentration measured at a particular time records the patient’s concentration at that time; it does not reveal how the concentration varied between measurements. This matters when sampling is sparse. A patient who is consistently above the typical concentration may have a stable individual difference, while a patient whose concentration moves up and down may happen to be above typical at the sampled times. With only a few measurements, these explanations can be difficult—or impossible—to distinguish.

The deterministic pharmacokinetic model provides a useful reference curve. We will compare it with two simple ways to represent departures from that curve: a stable patient-specific multiplier and a multiplier that changes over time. These are illustrations of competing explanations, not a claim that every changing multiplier follows a particular stochastic differential equation.

A Reference Curve for a Single Oral Dose

Consider a one-compartment model with first-order oral absorption and elimination. Let the dose be 100 mg, bioavailability \(F=1\), apparent volume \(V=25\) L, absorption rate constant \(k_a=1.2\) h\(^{-1}\), and elimination rate constant \(k_e=0.15\) h\(^{-1}\). For time \(t\) in hours after the dose, the deterministic concentration is

C_0(t)=\frac{F\,\mathrm{Dose}\,k_a}{V(k_a-k_e)} \left(e^{-k_e t}-e^{-k_a t}\right) =4.5714286\left(e^{-0.15t}-e^{-1.2t}\right)\ \text{mg/L}.

The coefficient follows from \(100(1.2)/[25(1.2-0.15)]=120/26.25=4.5714286\). This curve is the reference prediction, not a claim that every patient follows it exactly.

Worked Example: Computing Sparse Reference Concentrations

Suppose samples are collected at 2 and 8 hours. At 2 hours,

C_0(2)=4.5714286(e^{-0.30}-e^{-2.40}) \approx4.5714286(0.7408182-0.09071795) \approx2.9719\ \text{mg/L}.

At 8 hours, \(e^{-1.20}\approx0.3011942\) and \(e^{-9.60}\approx0.0000677\), so

C_0(8)=4.5714286(0.3011942-0.0000677) \approx1.37658\ \text{mg/L} \approx1.377\ \text{mg/L}.

These two reference values describe the curve at the sample times. By themselves, they provide no direct record of what happened between 2 and 8 hours.

Stable and Time-Varying Explanations

Definition: A stable patient effect in this illustration is a patient-specific multiplier \(A_i\) that remains fixed over the observation period, so that the patient’s concentration is \(C_i(t)=A_iC_0(t)\). A time-varying multiplier is a positive function \(G_i(t)\) that may change during the period, giving \(C_i(t)=G_i(t)C_0(t)\). The subscript \(i\) identifies the patient.

A stable multiplier can represent a persistent difference from the reference curve. A time-varying multiplier can represent departures that change over time. The labels describe how the multiplier behaves; they do not, on their own, identify a biological mechanism.

Worked Example: A Stable Effect at Two Sample Times

Take a patient with stable multiplier \(A_i=1.30\). The predicted concentrations at the two sample times are

C_i(2)=1.30(2.9719)=3.8635\ \text{mg/L}, \qquad C_i(8)=1.30(1.37658)=1.78955\ \text{mg/L} \approx1.790\ \text{mg/L}.

At both observations, the patient is 30% above the reference curve. This is consistent with a stable difference. But two observations do not establish that the multiplier stayed fixed between them.

The limitation can be stated exactly. Let the sample times be \(t_1,\ldots,t_m\). If a time-varying multiplier takes the same values as a stable multiplier at those times, then the two models give the same sampled concentrations, even if their predictions differ between samples.

Result: Suppose \(C_0(t_j)\) is defined at each of the finite sample times \(t_j\), and \(G_i(t_j)=A_i\) for every \(j=1,\ldots,m\). Then the stable model \(C_i(t)=A_iC_0(t)\) and the time-varying model \(C_i(t)=G_i(t)C_0(t)\) have identical concentrations at every sample time. No analysis of those concentrations alone can distinguish the two trajectories without additional assumptions or observations.

Proof. At any sample time \(t_j\), the time-varying model gives \(G_i(t_j)C_0(t_j)\). By the stated condition, this equals \(A_iC_0(t_j)\), which is exactly the stable model’s value. This holds for each \(j\), so the full vector of sampled concentrations is identical under the two models. The result concerns the sampled values; it does not say that the trajectories agree at unsampled times. \(\square\)

Worked Example: Different Paths, Identical Samples

Keep the stable multiplier \(A_i=1.30\), and consider samples at 2 and 8 hours. On the interval from 2 to 8 hours, define an alternative multiplier by

G_i(t)=1.30\exp\left\{\ln(1.5)\sin\left(\frac{\pi(t-2)}{6}\right)\right\}.

At \(t=2\) and \(t=8\), the sine is zero, so \(G_i=1.30\). The sampled concentrations therefore match the stable model: approximately 3.8635 mg/L at 2 hours and 1.790 mg/L at 8 hours. At 5 hours, however, the sine is one, giving \(G_i(5)=1.30(1.5)=1.95\). The reference concentration there is

C_0(5)=4.5714286(e^{-0.75}-e^{-6}) \approx4.5714286(0.4723666-0.0024788) \approx2.14806\ \text{mg/L}.

The stable model predicts \(1.30(2.14806)\approx2.79248\) mg/L at 5 hours. The time-varying model predicts \(1.95(2.14806)\approx4.18871\) mg/L. Thus, the models agree at both observed times but differ substantially at an unobserved time. The construction is an explicit illustration of what sparse samples leave unknown; it is not evidence that this particular path is pharmacologically plausible.

What One Observation Can Establish

A related ambiguity arises when each patient contributes only one concentration measurement. At that single time, the distribution of patient-to-patient differences can look the same whether each patient’s deviation is stable or whether it is a snapshot of a changing deviation. This can be shown without assuming a complicated concentration model.

Result: At a fixed sampling time \(t^\ast\), suppose the observed concentration is \(C_0(t^\ast)e^Z\), where \(Z\sim N(0,0.16)\). If \(Z\) is interpreted as a stable patient effect, or instead as the value at \(t^\ast\) of a time-varying effect whose distribution at that time is \(N(0,0.16)\), the single-time concentration has the same distribution under both interpretations.

Proof. In either interpretation, the value at the observed time is \(C_0(t^\ast)e^Z\), with \(Z\) having the same normal distribution. Applying the same function \(z\mapsto C_0(t^\ast)e^z\) to identically distributed variables gives identically distributed concentrations. The interpretation of \(Z\) as stable or time-varying does not alter this one-time distribution. Since only one time is observed, no within-patient change is recorded, and this sample cannot establish whether \(Z\) would remain fixed at other times. \(\square\)

Worked Example: Interpreting a One-Time Deviation

Suppose the log multiplier \(Z\) at a sample time has distribution \(N(0,0.16)\), so its standard deviation is \(0.4\). A multiplier \(e^Z\) differs from 1 by more than a factor \(e^{0.5}\) in either direction when \(|Z|>0.5\). Standardizing gives

P(|Z|>0.5) =2\left[1-\Phi\left(\frac{0.5}{0.4}\right)\right] =2[1-\Phi(1.25)] \approx2(1-0.894350) =0.2113.

Thus, about 21.13% of such one-time log deviations lie outside \([-0.5,0.5]\), corresponding to multipliers below \(e^{-0.5}\approx0.6065\) or above \(e^{0.5}\approx1.6487\). This calculation describes the size of deviations at the sampled time. It does not tell us whether an individual’s deviation is persistent or changes later.

What Sparse Sampling Can and Cannot Tell You

Sparse observations are not automatically useless. They can show that concentrations depart from a deterministic reference curve, and observations from several patients may reveal a distribution of values at particular times. The difficulty is attributing those departures to a stable or evolving source without enough information to support that distinction.

Approach	What it represents	What sparse samples leave uncertain
Deterministic ODE reference	A single predicted concentration-time curve under specified parameters	Whether departures reflect persistent patient differences or changes over time
Stable-effects model	Patient-specific departures that remain fixed over the observation period	Whether the apparent persistence is real or merely unobserved between samples
Time-varying model	Departures that can evolve during the observation period	The unobserved trajectory, especially across long gaps between samples

A mixed-effects model is a natural way to represent stable patient-to-patient differences. A stochastic differential equation can represent a state that evolves over time. Choosing between them is not settled merely by observing that patients have different concentrations: both kinds of model can produce such differences. Model assumptions matter, and sparse data may not contain enough information to test them sharply.

More observations within the same patient can reveal changes that two distant samples miss. Repeated measurements at carefully chosen times may therefore make an evolving pattern more visible. But additional sampling does not guarantee identification: if competing models make nearly identical predictions at the selected times, or if the data are highly variable, the distinction can remain uncertain. A good design targets times at which the models predict meaningfully different concentrations, rather than simply adding measurements without regard to the competing predictions.

Diagnostics and Common Pitfalls

Do not infer a stable effect from a consistent-looking pair of samples. The multiplier may have changed between them and returned to the same value by the second sample.
Do not infer time evolution from between-patient scatter alone. Stable patient differences can also produce scatter across individuals.
Check what is actually observed. A study with one measurement per patient does not directly show within-patient change, regardless of how many patients it includes.
Inspect the sampling times and gaps. If important changes could occur inside a long unsampled interval, the observations cannot directly rule them out.
State the model assumptions. A restrictive model may distinguish explanations because it rules out some trajectories. That conclusion then depends on the restriction, not on sparse observations alone.

A common mistake is to reason, “The patient’s two concentrations are both above the reference curve, so the patient must have a stable effect.” The correction is that both sampled multipliers could be equal while the multiplier varies between those times, as the worked construction demonstrates. The observations support an above-reference pattern at the sample times; they do not establish the intervening path.

The same caution applies in the opposite direction. A single concentration that is far from the reference curve does not demonstrate that the patient’s concentration was changing. It could reflect a stable patient effect. Sparse-data interpretation should separate what the samples show from what a model assumes about the unobserved trajectory.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

For a stable multiplier of 1.30, what concentration does the reference model predict at 8 hours, using \(C_0(8)\approx1.37658\) mg/L?
Two models agree at every sample time but differ between samples. Does that contradict the result about identical sampled concentrations? Explain.
If each patient has one sample and the deviations have the same distribution under a stable-effect model and a time-varying model, what can the one-time data establish about whether deviations change over time?
Give one reason why collecting more samples does not automatically guarantee that stable and evolving variation can be distinguished.

Solutions. (1) \(1.30(1.37658)=1.789554\) mg/L, which rounds to 1.790 mg/L. (2) No. The result concerns equality at the finite observed times; it places no requirement on the paths between those times. (3) They can describe the distribution of deviations at the sampled time, but cannot establish from one observation per patient whether a deviation persists or changes later. (4) The added samples may fall at times when both models predict nearly the same values, or the observations may be too variable to resolve the difference.