From PK Data to Modeling Questions
Exploratory data analysis (EDA) is most useful when several displays are considered together. A concentration-time profile can show how concentrations change within a participant; a distribution plot shows how much participants differ at a selected time; and dose, visit, and covariate displays help reveal whether those differences align with study features. None of these plots establishes a pharmacokinetic mechanism. Their purpose is to describe the observed data, detect problems, and identify questions worth evaluating in a model.
This worked case uses a small, entirely hypothetical dataset. Six adults received oral doses once daily: three received 50 mg and three received 100 mg. Concentrations were recorded at 1, 4, and 8 hours after a dose on Day 1 and Day 14. All values below are hypothetical and in mg/L. The displayed values are assumed to be valid, quantifiable samples collected at the stated times; a real analysis would verify that assumption using the actual timestamps and data records. Weight is in kg and estimated glomerular filtration rate (eGFR) in mL/min/1.73 m².
| Participant | Dose (mg) | Weight (kg) | eGFR | Day 1: 1 h, 4 h, 8 h (mg/L) | Day 14: 1 h, 4 h, 8 h (mg/L) |
|---|---|---|---|---|---|
| A | 50 | 55 | 110 | 4.0, 2.0, 0.80 | 7.2, 3.6, 1.44 |
| B | 50 | 62 | 95 | 5.0, 2.5, 1.00 | 8.0, 4.0, 1.60 |
| C | 50 | 74 | 82 | 3.2, 1.6, 0.64 | 5.8, 2.9, 1.16 |
| D | 100 | 100 | 90 | 8.0, 4.0, 1.60 | 14.4, 7.2, 2.88 |
| E | 100 | 78 | 45 | 12.0, 6.0, 2.40 | 18.0, 9.0, 3.60 |
| F | 100 | 88 | 50 | 7.0, 3.5, 1.40 | 11.0, 5.5, 2.20 |
The table is a compact representation, not a substitute for checking the underlying records. Before plotting, confirm participant identifiers, dose and visit labels, units, time origin, and the link between each concentration and its sample. In a real dataset, actual elapsed time since dose matters: nominal labels such as “1 h” can conceal meaningful differences in collection time. Also check for duplicate records, missing samples, implausible values, and concentrations below the assay’s quantification limit.
Begin With Individual Profiles and Distributions
The first view should preserve the participant-level trajectories. Plot each participant’s concentration against time, keeping the participants identifiable and separating visits when needed. A summary curve can help orient the eye, but it should not obscure whether the pattern is shared or driven by one or two individuals. The earlier material on readable subject overlays and within- and between-subject variation provides the plot principles; here the task is to use the displays together to build a case-level interpretation.
Worked Example: Reading Profiles Alongside a Distribution
For participant C on Day 1, the displayed concentrations are 3.2 mg/L at 1 hour, 1.6 mg/L at 4 hours, and 0.64 mg/L at 8 hours. The values decrease at the sampled times. This describes the observed profile, but it does not by itself establish when the peak occurred, estimate an elimination rate, or identify a structural PK model. The samples could miss an earlier or later peak.
Now compare all six participants at each Day 1 time. At 1 hour, the sorted concentrations are 3.2, 4.0, 5.0, 7.0, 8.0, and 12.0 mg/L. With an even number of observations, the median is the average of the two middle values:
At 4 hours, the sorted values are 1.6, 2.0, 2.5, 3.5, 4.0, and 6.0 mg/L, so the median is \((2.5+3.5)/2=3.0\) mg/L. At 8 hours, the sorted values are 0.64, 0.80, 1.00, 1.40, 1.60, and 2.40 mg/L, giving a median of \((1.00+1.40)/2=1.20\) mg/L. These medians summarize a downward pattern across the three sampled times. The ranges—3.2 to 12.0, 1.6 to 6.0, and 0.64 to 2.40 mg/L, respectively—also show substantial separation among participants at each time.
The distribution at each time adds context to the median: a median alone cannot show the spread, skew, or a potentially influential observation. With only six participants, a plot of individual points is more informative than a smooth density curve or elaborate summary. A modeling question raised by these displays is whether the between-participant differences are compatible with variation in dose, covariates, sampling, or other sources that a PK model should represent.
Compare Dose Groups With and Without Normalization
Dose-stratified concentration-time displays retain the concentrations in their measured units and show how profiles differ across dose groups. Dose-normalized displays divide each concentration by the administered dose. These are complementary views, not competing versions of the same analysis. Raw concentrations answer what was observed at each dose; normalized concentrations can help assess whether concentrations appear to scale approximately with dose under comparable conditions.
The rationale follows directly from proportional scaling. If, at a fixed time and under comparable conditions, concentration can be represented as \(C=kD\) for a common proportionality factor \(k\), then dividing by dose gives \(C/D=k\). If dose-normalized values differ systematically, possible explanations include nonproportional PK, group differences in participants or sampling, and random variation. The display alone cannot distinguish those explanations.
Worked Example: Comparing Day 1 Concentrations by Dose
At 1 hour on Day 1, the 50 mg group has concentrations 4.0, 5.0, and 3.2 mg/L. In sorted order these are 3.2, 4.0, and 5.0 mg/L, so the group median is 4.0 mg/L. The 100 mg group has concentrations 8.0, 12.0, and 7.0 mg/L; the sorted values are 7.0, 8.0, and 12.0 mg/L, so its median is 8.0 mg/L. The raw median is twice as high in the 100 mg group, a descriptive comparison in this small dataset.
For the 50 mg group, the dose-normalized 1-hour values are \(4.0/50=0.080\), \(5.0/50=0.100\), and \(3.2/50=0.064\) (mg/L)/mg. Their median is 0.080 (mg/L)/mg. For the 100 mg group, the values are \(8.0/100=0.080\), \(12.0/100=0.120\), and \(7.0/100=0.070\) (mg/L)/mg; their median is also 0.080 (mg/L)/mg. The similar medians are consistent with approximate dose scaling at this time in these observed data. They do not establish dose proportionality: each group has only three participants, and individual normalized values vary.
A useful next display is the dose-stratified profile across all three times, paired with the corresponding normalized profile. If the apparent agreement occurs only at 1 hour but not at 4 or 8 hours, that time dependence is informative. Conversely, apparent similarity at all displayed times still does not prove that the entire concentration-time curves scale proportionally.
Compare Visits Without Over-Labeling Them
A visit display can show whether concentrations differ between occasions, but its interpretation depends on the dosing history and sampling design. Here participants received daily doses, and Day 14 is a later visit. The data do not establish that steady state was reached, so the visit should not be labeled “steady state” solely because it occurred on Day 14. The timing of the sample relative to the most recent dose must also be comparable across visits before concentrations are interpreted as a visit-related difference.
Worked Example: Summarizing the Day 1 to Day 14 Pattern
Participant A’s 1-hour concentration changes from 4.0 mg/L on Day 1 to 7.2 mg/L on Day 14. The within-participant ratio is:
Thus, A’s observed Day 14 concentration at that sampling time is 1.80 times the Day 1 value, or 80% higher. The analogous 1-hour ratios for B through F are \(8.0/5.0=1.60\), \(5.8/3.2=1.81\), \(14.4/8.0=1.80\), \(18.0/12.0=1.50\), and \(11.0/7.0=1.57\). All six observed ratios exceed 1. This is a consistent descriptive visit pattern in this dataset, but it does not prove accumulation: differences in dosing, adherence, timing, or other conditions would need to be considered, and Day 14 need not represent steady state.
Plotting connected participant-specific points makes clear that this is a within-participant comparison, while a visit-level distribution shows the group spread. Treating the two visits as unrelated groups would discard the pairing. A modeling question is whether the visit pattern is expected from repeated dosing under the proposed model and dosing history, rather than whether a higher later-visit concentration automatically requires a visit effect.
Use Covariate Plots to Generate Questions
Plot concentrations or dose-normalized concentrations against continuous covariates, retaining participant identity and indicating dose group where helpful. In this example, eGFR and weight are available. A visible association may motivate consideration of a covariate relationship, but it does not show that the covariate causes a change in exposure. Dose group, participant characteristics, and other study features can be related, and a small dataset gives limited coverage of the covariate range.
Worked Example: Inspecting eGFR Against Normalized Concentration
At 1 hour on Day 1, participants E and F have eGFR values of 45 and 50 mL/min/1.73 m² and dose-normalized concentrations of \(12.0/100=0.120\) and \(7.0/100=0.070\) (mg/L)/mg. Participant A has eGFR 110 mL/min/1.73 m² and a normalized concentration of \(4.0/50=0.080\) (mg/L)/mg. These points do not form a simple monotonic pattern: E has the largest displayed normalized value, while F, with similarly low eGFR, does not. The six participants therefore provide a reason to inspect renal function further, not evidence of a reliable eGFR effect.
The comparison also spans two dose groups, and eGFR values are unevenly distributed across the small sample. A useful display would show all six points, annotate participants, and distinguish dose groups. The next question is whether the available data cover enough of the covariate range within dose groups to separate a covariate pattern from group composition. This is a design and data-coverage question as much as a plotting question.
Diagnostics, Alternatives, and Limits
A coherent EDA workflow moves between individual and pooled views rather than relying on one plot. A practical sequence is to inspect the data records and sampling times, overlay participant profiles, display concentration distributions at comparable times, stratify by dose and visit, and then examine covariates with participant-level points. Each view should preserve enough context to reveal whether a pattern is shared, sparse, or driven by a small number of observations.
- Check timing and dose history. Actual sample times and the time since the last dose can differ from nominal labels. A mismatch can create an apparent profile or visit difference.
- Check visibility and scale. Linear and log concentration scales emphasize different features. Zero or below-quantification values require explicit handling before log-scale displays; they must not be treated as ordinary positive measurements.
- Check group coverage. A dose or covariate comparison is weak when groups have little overlap in participant characteristics or only a few observations.
- Check what summaries conceal. Medians and quantiles are useful, but with sparse samples they can hide individual trajectories and cannot repair unbalanced sampling.
The main alternative to a pooled distribution plot is a set of subject-level displays; the former summarizes the group, while the latter preserves individual structure. Raw concentration plots retain the measured scale, while dose-normalized plots are easier to compare under a dose-proportionality hypothesis. Neither representation is universally preferable. Likewise, a scatterplot of concentration against a covariate is a descriptive alternative to stratifying by covariate categories; categorizing a continuous covariate can discard information, while a scatterplot can be difficult to interpret when there are very few participants.
EDA has important limits. Sparse sampling cannot reveal an entire concentration-time curve, and a handful of observations cannot reliably distinguish competing explanations for between-participant variation. Apparent associations may reflect confounding or chance. These displays do not estimate population PK parameters, quantify uncertainty in a formal model, or establish a causal covariate effect. They provide a disciplined set of observations and questions to take into model development.
Turning Patterns Into Questions
A concise EDA summary should separate what is seen from what needs evaluation. For this hypothetical dataset, the sampled Day 1 medians decrease from 6.0 mg/L at 1 hour to 3.0 mg/L at 4 hours and 1.20 mg/L at 8 hours, with visible between-participant spread. At 1 hour, the raw dose-group medians differ by a factor of two and the dose-normalized medians are both 0.080 (mg/L)/mg. Each participant’s 1-hour Day 14 value exceeds their Day 1 value, but the data alone do not establish steady state or explain the visit pattern. The eGFR display is too small and uneven to support a firm covariate conclusion.
Those observations raise focused questions: Are sampling times and dose histories comparable? Does a PK model reproduce the profile shapes and participant-level differences? Is the apparent dose scaling consistent at other times? Can the later-visit pattern be explained by the dosing history? Is there sufficient covariate coverage to evaluate a relationship? The value of EDA is not that it answers every question in advance, but that it makes the questions specific and makes potential data limitations visible.
Check Your Understanding
Use the hypothetical dataset above. Answers follow the questions.
- Calculate the median Day 1 concentration at 8 hours across all six participants.
- At 1 hour on Day 1, compare the raw median concentration for the 50 mg group with the dose-normalized median. Include units.
- What does the fact that all six 1-hour Day 14 to Day 1 ratios exceed 1 support, and what does it not establish?
- Why is the eGFR pattern insufficient to claim that lower eGFR causes higher concentration?
Solutions. (1) The sorted 8-hour values are 0.64, 0.80, 1.00, 1.40, 1.60, and 2.40 mg/L, so the median is \((1.00+1.40)/2=1.20\) mg/L. (2) The 50 mg group’s 1-hour median is 4.0 mg/L; dividing its individual values by 50 mg gives 0.080, 0.100, and 0.064 (mg/L)/mg, whose median is 0.080 (mg/L)/mg. (3) It supports a consistent descriptive increase at that sampled time between these visits; it does not prove accumulation, steady state, or the cause of the increase. (4) There are only six participants, low-eGFR observations do not share the same normalized concentration, and dose and other participant differences may affect the pattern. The plot is hypothesis-generating, not causal evidence.