Peak Misspecification Under Sparse Sampling

Why a Measured Peak Can Be Misleading

A concentration-time profile is observed only at its scheduled sampling times. If a short-lived peak falls between those times, the largest recorded concentration can be much lower than the profile’s underlying maximum, and the time assigned to the peak can be displaced. This is peak misspecification under sparse sampling: the data identify the maximum among sampled values, but that maximum may poorly represent the peak that occurred between samples.

The distinction matters whenever peak height or timing is used to describe an exposure profile. A low observed Cmax does not, by itself, show that the underlying profile had a low peak. Likewise, an observed Tmax is one of the sampling times, not necessarily the time at which the concentration was truly highest. The examples below use smooth, hypothetical profiles to isolate the effect of the sampling schedule.

Definition: For concentrations \(C(t_i)\) observed at scheduled times \(t_1,\ldots,t_n\), the observed Cmax is \(\max_i C(t_i)\). The observed Tmax is a sampling time \(t_i\) at which that maximum is attained. For an underlying continuous concentration profile \(C(t)\), its true maximum over the time interval is \(\max_t C(t)\), and its peak time is a time at which that maximum occurs. “True” here refers to the specified underlying profile, not a claim that a real study can observe an unknown continuous curve exactly.

These quantities coincide only if the schedule samples a time at which the underlying maximum occurs, subject to ties. In real data, measurement error and other complications can also affect the observed values; the worked calculations first set those aside so that the sampling effect is clear.

A Simple Profile and Its Sampling Consequences

Consider a hypothetical brief peak described by a Gaussian-shaped concentration curve:

C(t)=b+A\exp\left[-\frac{(t-t_p)^2}{2s^2}\right],

where \(b\) is the baseline concentration, \(A>0\) is the peak’s height above baseline, \(t_p\) is its peak time, and \(s>0\) controls its width. Concentration is measured in mg/L and time in hours in the examples. This curve is used as an illustrative profile, not as a general pharmacokinetic model. It assumes a smooth, single peak with no measurement error.

The sampled maximum depends on which scheduled times fall near \(t_p\). For this particular symmetric profile, the concentration declines as the absolute distance \(|t-t_p|\) increases. Thus, among sampled times, the largest value occurs at the time or times closest to \(t_p\). A narrow peak falls away quickly with distance, making a missed peak especially consequential. A broader peak may be less severely underestimated on the same schedule, although its observed Cmax and Tmax can still differ from their underlying values.

Worked Example: A Brief Peak Between Widely Spaced Samples

Suppose a hypothetical profile is \(C(t)=2+18\exp[-(t-4)^2/(2(0.5)^2)]\) mg/L. Its underlying peak is at \(t_p=4\) h, where \(C(4)=2+18=20\) mg/L. Measurements are taken at 0, 2, 5, and 8 h. Find the observed Cmax and Tmax.

At 0 h, the exponent is \(-(0-4)^2/[2(0.5)^2]=-32\), so \(C(0)=2+18e^{-32}\approx2.000\) mg/L. At 2 h, it is \(-(2-4)^2/0.5=-8\), giving \(C(2)=2+18e^{-8}\approx2.006\) mg/L. At 5 h, it is \(-(5-4)^2/0.5=-2\), giving \(C(5)=2+18e^{-2}\approx4.436\) mg/L. At 8 h, the exponent is again \(-32\), so \(C(8)\approx2.000\) mg/L.

The largest sampled value is therefore the 4.436 mg/L at 5 h: observed Cmax is 4.436 mg/L and observed Tmax is 5 h. Relative to the profile’s 20 mg/L maximum, the observed value is lower by \(20-4.436=15.564\) mg/L, or \(15.564/20\times100\approx77.82\%\). The sampled peak time is 1 h later than the underlying peak time. Adding a sample at 4 h would capture the specified maximum exactly; that does not imply that a real profile’s peak time is known in advance.

Why Width and Grid Alignment Matter

The size of the miss is not determined by the gap between samples alone. It also depends on how rapidly the profile changes around its peak and on where the peak lies relative to the sampling grid. Two schedules with the same largest interval can produce different observed peaks if one places a sample near the peak and the other does not.

Worked Example: A Broader Peak on the Same Schedule

Now consider \(C(t)=1+12\exp[-(t-4)^2/(2(1.5)^2)]\) mg/L, which has an underlying maximum of 13 mg/L at 4 h. Use the same schedule: 0, 2, 5, and 8 h.

At 0 and 8 h, the exponent is \(-16/4.5\approx-3.5556\), so each concentration is \(1+12e^{-3.5556}\approx1.343\) mg/L. At 2 h, the exponent is \(-4/4.5\approx-0.8889\), giving \(C(2)=1+12e^{-0.8889}\approx5.933\) mg/L. At 5 h, the exponent is \(-1/4.5\approx-0.2222\), giving \(C(5)=1+12e^{-0.2222}\approx10.609\) mg/L.

The observed Cmax is 10.609 mg/L at the observed Tmax of 5 h. It is \(13-10.609=2.391\) mg/L below the underlying maximum, a deficit of about \(2.391/13\times100=18.39\%\). This is a smaller proportional miss than in the brief-peak example, despite using the same schedule. A sample at 4 h would record 13 mg/L in this specified profile.

A different alignment can also change the apparent timing. A schedule may place its nearest sample after the peak rather than before it, making observed Tmax late. On an asymmetric profile, the closest time by clock hours need not give the largest concentration; the local shape on each side of the peak matters. The Gaussian examples illustrate a mechanism, not a rule that observed Tmax must always be the nearest sample in time.

Worked Example: A Shifted Brief Peak and a Late Observed Tmax

Let \(C(t)=0.5+9\exp[-(t-4.25)^2/(2(0.4)^2)]\) mg/L, with an underlying maximum of 9.5 mg/L at 4.25 h. Suppose samples are taken at 0, 3, 5, and 7 h.

At 3 h, the exponent is \(-(3-4.25)^2/0.32=-4.8828\), so \(C(3)=0.5+9e^{-4.8828}\approx0.568\) mg/L. At 5 h, the exponent is \(-(5-4.25)^2/0.32=-1.7578\), so \(C(5)=0.5+9e^{-1.7578}\approx2.052\) mg/L. At 0 and 7 h, the exponents are approximately \(-56.4453\) and \(-23.6328\), respectively, so both concentrations are approximately 0.500 mg/L at the displayed precision.

The observed Cmax is approximately 2.052 mg/L at 5 h. It is \(9.5-2.052=7.448\) mg/L below the underlying peak, and the observed Tmax is 0.75 h later than 4.25 h. A sample at 4.25 h would capture the specified maximum. The contrast shows why the actual placement of samples around the peak matters, not merely the number of samples in a profile.

Justifying the Miss and Its Limits

For the illustrative Gaussian curve, the derivative shows where its maximum occurs:

\frac{dC(t)}{dt} =-\frac{t-t_p}{s^2}\bigl(C(t)-b\bigr).

Because \(C(t)-b>0\) and \(s^2>0\), the derivative is positive when \(t<t_p\), zero at \(t=t_p\), and negative when \(t>t_p\). Therefore \(C(t)\) increases up to \(t_p\) and decreases afterward, so its unique maximum is \(C(t_p)=b+A\). If the sampled times are a subset of the time interval being considered, every sampled concentration is no greater than this continuous maximum. Hence the observed Cmax cannot exceed the profile maximum in this error-free example. It equals that maximum if a scheduled sample falls at \(t_p\); otherwise it is lower.

That inequality does not say how large the miss will be. The profile shape and the sample locations are needed to quantify it. Nor does it guarantee that observed Cmax will be below the unknown peak in actual data: random measurement error can push an observed concentration upward or downward. With noisy measurements, a high observed value may partly reflect error, while a peak between samples may still go unseen. The simple curve separates these issues; it does not resolve them for real profiles.

Diagnostics, Alternatives, and Practical Limits

A useful diagnostic begins with the sampling schedule, not just the reported maximum. Plot concentrations against actual sampling times and inspect the interval around the observed high value. A steep increase before that value or a steep decrease afterward indicates that the local peak may be poorly resolved. A long gap spanning the apparent rise and fall is a particular concern. Also note whether the likely peak region is represented by only one sample, whether an important scheduled sample is missing, and whether the observed high value lies near the edge of the sampled interval. These are warning signs, not proof that the true peak was missed.

The main practical alternatives address different parts of the problem:

Approach	What it can do	Key limitation
Add samples around the expected peak	Can reduce the time gaps and make an observed maximum more representative.	Requires selecting feasible sample times; a peak outside the anticipated window may still be missed.
Interpolate between observed points	Can summarize a value between samples under a chosen interpolation rule.	Cannot establish an unobserved sharp peak without assumptions about the curve; sparse points may support very different shapes.
Fit a concentration-time model	Can estimate a continuous peak and its time if the model adequately represents the profile.	The result depends on model form, parameter estimation, and available information; a smooth fitted curve is not direct observation.

Denser sampling is generally the most direct way to improve temporal resolution around a peak, but it cannot guarantee exact capture of an unknown peak. Interpolation and model-based estimation can provide estimates between observations; they do not create data and should not be presented as if the peak were measured. In particular, simply joining sparse points does not rule out a narrow excursion between them. A model-based peak estimate is only as credible as its assumptions and fit.

Sparse-sampling concerns should therefore qualify interpretation of observed Cmax and Tmax. They do not, on their own, show that another exposure measure is unaffected or affected by the same amount; that requires a separate analysis. This tutorial’s calculations address peak height and timing only.

Common Mistake

Common pitfall: “The observed Cmax is 4.436 mg/L, so the profile’s peak was about 4.436 mg/L.” This treats the maximum among sampled values as if it were necessarily the continuous maximum. The correction is to call 4.436 mg/L the observed Cmax, report the sampling schedule, and state that the underlying peak may have occurred between samples. A continuous peak estimate requires additional observations or explicit assumptions and analysis.

Key takeaway: Observed Cmax and Tmax are determined jointly by the concentration-time shape and the sampling times. A brief peak can be substantially underestimated and its time displaced when no sample falls near it; the observed maximum alone cannot reveal the size or exact timing of an unobserved peak.

Check Your Understanding

Answer each question using the distinction between sampled values and the underlying profile.

A smooth hypothetical profile has a unique maximum of 15 mg/L at 6 h. Samples are collected at 4 and 8 h, and both values are below 15 mg/L. Can either sampled value be reported as the measured observed Cmax? What can be said about the relation between the observed Cmax and 15 mg/L if measurements are error-free?
For \(C(t)=3+7\exp[-(t-5)^2/(2(1)^2)]\) mg/L, what is the underlying peak concentration and peak time? What observed Cmax and Tmax result if the only samples are at 4 and 6 h?
Two schedules have the same largest gap, but one has a sample close to the peak and the other does not. Must their observed Cmax values be equal? Explain.
Why does straight-line interpolation between two sparse samples not prove that no brief peak occurred between them?

Solutions. (1) Yes: the larger of the two measured concentrations is the observed Cmax, and its sampling time is the observed Tmax. Under the stated error-free condition, that observed Cmax is below 15 mg/L because neither sample is at the unique maximum. (2) The underlying maximum is \(3+7=10\) mg/L at 5 h. At 4 and 6 h, the exponent is \(-1/2\), so each concentration is \(3+7e^{-1/2}\approx7.246\) mg/L. The observed Cmax is approximately 7.246 mg/L, tied at both times; either 4 h or 6 h is an observed Tmax under a convention that reports one tied time, and both times should be noted if ties are retained. (3) No. The location and shape of the peak relative to each schedule can differ even when the largest gap is the same. (4) Straight-line interpolation imposes a chosen shape between the observed endpoints. The endpoints alone do not exclude a sharper rise and fall between them, so interpolation cannot establish the unobserved profile.