Within-Patient Variability and Repeat Concentrations

Why Repeat Concentrations Need a Variability Model

A concentration measured in the same patient can differ from one occasion to another even when the prescribed dose and the patient's underlying exposure have not persistently changed. Variation can arise from biological fluctuations, imperfectly matched sampling times, dose-taking patterns, and residual analytical error. A difference between two results is therefore not, by itself, proof of a durable exposure change.

The task here is to describe the variation expected when conditions are stable, then compare an observed difference with that expectation. This requires comparable samples: for example, concentrations collected at the same point in the dosing interval under a stable regimen. The earlier material on The Minimum Metadata for Interpreting a Sample is relevant because dose, sampling time, and other context determine whether two concentrations can reasonably be compared.

Concentrations often vary multiplicatively: a fluctuation of a given percentage is more plausible than a fixed number of concentration units across a wide range of levels. A natural-log model represents this pattern and keeps fitted concentrations positive. Let \(C_{ij}\) be the concentration for patient \(i\) on occasion \(j\), and let \(Y_{ij}=\log(C_{ij})\), with concentration expressed in a fixed unit. For one patient observed in a stable period, a simple model is

Here, \(\mu\) is the patient's typical log concentration under the specified conditions, and \(\sigma_w\) is the within-patient standard deviation on the log scale. The model assumes that the occasions are comparable, errors have mean zero and a common variance, and errors are independent and approximately normal on the log scale. These are modeling assumptions, not guarantees supplied by having repeat results.

From Log-Scale Variation to Concentration Variation

The log-scale standard deviation is often easier to interpret after conversion to a geometric coefficient of variation (CV). Under the lognormal model, if \(Y\sim N(\mu,\sigma_w^2)\), then \(C=\exp(Y)\). Its mean is \(\exp(\mu+\sigma_w^2/2)\), and its variance is \(\exp(2\mu+\sigma_w^2)\{\exp(\sigma_w^2)-1\}\). Dividing the variance by the squared mean and taking the square root gives the CV.

This is the geometric CV: it describes multiplicative spread and does not depend on the typical concentration level. For modest \(\sigma_w\), the approximation \(100\sigma_w\%\) can be useful, but the exact expression is preferable when reporting a CV. In this tutorial, ratios and differences on the log scale are used to describe fluctuations; they are not interchangeable with additive differences in concentration units.

How Much Can Two Stable Measurements Differ?

If two independent measurements are made under the same stable conditions, their log-scale difference is \(\varepsilon_2-\varepsilon_1\). Independence gives variance \(\sigma_w^2+\sigma_w^2=2\sigma_w^2\), so the difference has standard deviation \(\sqrt{2}\sigma_w\). Exponentiating that difference gives the ratio of the two concentrations. Consequently, even with no persistent shift, two single measurements can differ substantially.

The range is a model-based prediction for the ratio of two single measurements under stable conditions, not a confidence interval for a persistent shift. It is centered at ratio 1 on the log scale. Its width grows with within-patient variability, so a given observed ratio is less surprising when ordinary fluctuation is large.

Replicates Help Separate a Shift from Fluctuation

A more informative comparison uses several comparable measurements in each of two periods. Let \(Y_{1j}\) and \(Y_{2j}\) be log concentrations in periods 1 and 2. Suppose each period has a stable typical log level, \(\mu_1\) or \(\mu_2\), and independent within-period errors with variance \(\sigma_w^2\). The quantity of interest is the shift \(\Delta=\mu_2-\mu_1\). Its exponentiation, \(\exp(\Delta)\), is the ratio of the periods' typical concentrations on the geometric scale.

With \(m_1\) and \(m_2\) independent observations in the periods, the difference of the sample log means estimates \(\Delta\). The variance of a sample mean of \(m\) independent observations is \(\sigma_w^2/m\). Since the two period means are independent, the variance of their difference is the sum of these variances:

If \(\sigma_w\) is known, a 95% interval for \(\Delta\) is \(\widehat{\Delta}\pm1.96\,\mathrm{SE}(\widehat{\Delta})\). Exponentiating both endpoints gives an interval for the ratio of typical concentrations. If the within-period variance is estimated from replicate data instead, its uncertainty must also be included; under the equal-variance normal model, a pooled within-period variance estimate leads to a \(t\) interval with the corresponding residual degrees of freedom. The known-\(\sigma_w\) calculation below is used to keep the example focused on the effect of replication.

Assumptions, Diagnostics, and Alternatives

Replicate measurements improve precision only when they provide information about the intended source of variation. In the two-period calculation, the log errors must be independent across observations and periods, share the specified variance, and be reasonably normal. The underlying typical level should be stable within each period, and sampling conditions should be comparable across periods. If measurements are serially correlated, the variance of a mean is not generally \(\sigma_w^2/m\), so treating every observation as independent can make an interval too narrow.

Inspect the design and the log concentrations rather than relying on a formula alone. Check whether dose and sample timing were consistent, whether the regimen or clinical state changed within a period, and whether one unusually high or low value dominates the mean. A log-scale plot can reveal skewness or outliers; a sequence plot can reveal a gradual drift or clusters that contradict independence. If sample timing or dosing differs, the apparent shift may reflect those differences rather than a persistent change in typical exposure. Analytical uncertainty is another component of observed variation; it should not be counted twice if the estimate of \(\sigma_w\) already includes it.

These calculations have limits. A statistically detectable shift can be small in practical terms, while a wide interval can leave substantial uncertainty even when the point estimate is large. A stable-variance model may be inappropriate if variability changes with concentration, or if dosing, sampling, adherence, or physiology changes systematically between periods. Where repeated observations are correlated or the conditions are not comparable, a simple independent-error calculation is not adequate. The model supports interpretation of a difference; it does not identify its cause.

Common Pitfall

Wrong reasoning: “The second concentration is 50% higher, so the patient's exposure must have risen by 50%.” This treats two noisy observations as exact measures of typical exposure and ignores their expected within-patient spread.

Correction: Compare like with like, quantify expected variation, and use replicate measurements when available. A single ratio should be interpreted relative to the stable ratio range; a replicate-based ratio should be accompanied by an interval that accounts for within-patient uncertainty. Neither calculation excuses mismatched timing or other changes in conditions.