Why Transit Time Needs More Than One Summary
A statement such as “gastric emptying takes three hours” sounds precise, but it leaves several questions unanswered. Is three hours a mean or a median? Does it describe one person, a group, or a fitted distribution? Was the measurement made after a meal or during fasting, and what exactly counted as “emptying”? These distinctions matter because gastrointestinal timing varies between individuals and changes with physiological conditions and the measurement protocol.
For PBPK applications, a timing value should therefore be treated as a summary of a defined physiological measurement, not as a universal clock for every person or dose. Gastric emptying, small-intestinal transit, and colonic transit are separate processes measured over different anatomical spans. Their reported times should not be combined or compared unless their definitions and conditions are sufficiently alike.
What the Timing Measures Describe
Gastric emptying measurements depend on what is being followed and how emptying is defined. A reported half-emptying time, for example, is the time at which half of the measured meal or marker has left the stomach under the stated measurement procedure. It is not the time at which the entire stomach is empty. A single half-emptying value also does not describe the full emptying curve: meals can have an initial lag, and emptying can proceed at different rates over time.
Small-intestinal transit time is commonly reported as the time for a marker to pass through the small intestine, using a specified endpoint such as arrival at the ileocecal region. Colonic transit time concerns passage through the colon to a defined endpoint. These measures are not interchangeable. In particular, a reported marker transit time is not automatically a direct estimate of the residence time of every drug molecule or of the contents of each anatomical region.
The earlier material on regional gastrointestinal compartments establishes the anatomical spans being discussed. For any reported time, the measurement record should also make clear the marker or meal, the start and end points, the study condition, and how the endpoint was detected. Without those details, even a correctly calculated average can be difficult to interpret.
Conditions and Variability
Gastric emptying is sensitive to whether a person is fasting or has consumed a meal. Meal composition, energy content, physical form, and volume can also affect the measured pattern. Solids and liquids need not empty in the same way, and a liquid containing nutrients is not necessarily equivalent to a non-nutritive liquid. “Fed” and “fasted” are therefore not minor labels: they identify different physiological conditions under which a timing value was obtained.
Small-intestinal and colonic transit also vary across people and across measurement occasions. Differences in protocol, marker, endpoint definition, and observation schedule can contribute to apparent differences between reported estimates. A published value should be read together with its qualifiers and measurement context, rather than copied as though it were a context-free constant.
A group summary describes the observations collected under the study’s conditions. It does not imply that every individual has that time, nor does it show how much one person’s time varies from occasion to occasion. When the number of observations is small, the observed minimum and maximum can be especially sensitive to which people happened to be measured.
Worked Example: Mean, Median, and Range of Gastric Emptying Times
Suppose, purely hypothetically, that seven participants complete the same specified meal protocol, and their measured gastric half-emptying times are 1.8, 2.1, 2.4, 2.7, 3.0, 3.6, and 5.4 hours. These numbers illustrate summaries of one defined measurement; they are not reference values for all meals or people.
Mean. Add the seven observations and divide by seven:
Median. The observations are already ordered. With seven values, the fourth is the middle value, so the median is 2.7 hours.
Range. The observed minimum is 1.8 hours and the observed maximum is 5.4 hours. The range is therefore 1.8–5.4 hours, with a width of \(5.4-1.8=3.6\) hours.
The mean exceeds the median because the largest observation pulls the arithmetic average upward. Neither summary tells us the complete shape of the distribution, and this small sample’s range should not be mistaken for the full range possible in the population.
Distributional Estimates and a Lognormal Illustration
Transit times are positive, and observations may be right-skewed: many people have times near the center while a smaller number have longer times. A lognormal distribution is one possible model for positive, right-skewed data. It is an assumption to assess, not a physiological law. If \(T\) is a transit time, the lognormal assumption is that \(\ln T\) follows a normal distribution with mean \(\mu_{\ln T}\) and variance \(\sigma_{\ln T}^{2}\).
These formulas follow by transforming the normal variable. Since \(T=\exp(\ln T)\) and the exponential function is increasing, the \(q\)-quantile on the original scale is the exponential of the \(q\)-quantile on the log scale. The median corresponds to \(z_{0.50}=0\), giving \(\exp(\mu_{\ln T})\). For the mean, write \(Y=\ln T\), whose density is normal. In the expectation integral for \(\exp(Y)\), completing the square gives
The remaining normal-shaped term integrates to the same normalizing constant as a normal density, leaving \(E[T]=\exp(\mu_{\ln T}+\sigma_{\ln T}^{2}/2)\). Because this factor exceeds \(\exp(\mu_{\ln T})\) when \(\sigma_{\ln T}>0\), the lognormal mean is greater than its median.
Worked Example: Percentiles From a Hypothetical Transit-Time Distribution
Suppose a hypothetical fitted lognormal model for a defined transit measurement has median 36 hours and log-scale standard deviation 0.4700. Thus \(\mu_{\ln T}=\ln(36)\), and the model describes variation in transit time across the population represented by the fit. It does not describe uncertainty in the estimated median.
Mean. The lognormal mean is:
Central 90% of modeled times. The standard normal quantiles for the 5th and 95th percentiles are approximately \(-1.645\) and \(1.645\). Therefore:
Under this assumed model, the 5th and 95th percentiles are approximately 16.62 and 78.00 hours. The interval from 16.62 to 78.00 hours is a central 90% interval for individual times described by the model, not a 90% confidence interval for the median. The substantial distance between the mean and median also shows why reporting only one central value can conceal the distribution’s right tail.
Comparing Groups Without Losing the Context
A difference in summaries is interpretable only if the compared measurements are sufficiently comparable. Consider a hypothetical study that reports a median gastric half-emptying time of 2.4 hours after a specified test meal in one condition and 3.1 hours after a different meal in another condition. The numerical difference is 0.7 hours, but it cannot be attributed solely to a condition effect if meal composition, measurement method, or endpoint also differs. The comparison is descriptive unless the study design and analysis justify a stronger interpretation.
Means are useful when the arithmetic average is relevant and the distribution is not dominated by extreme values. Medians are often more representative of a typical observation in a skewed distribution. Ranges show the extremes observed in a sample but are sensitive to sample size. Quantiles describe selected parts of a distribution, while a fitted distribution can summarize a broader pattern under its assumptions. These summaries answer different questions; none is a universal replacement for the others.
| Summary | What it describes | Main caution |
|---|---|---|
| Mean | Arithmetic average of observed times | Can be pulled upward by long times in a right-skewed sample |
| Median | Middle observed time | Does not describe the spread by itself |
| Range | Observed minimum and maximum | Depends strongly on which and how many people were measured |
| Distributional estimate | Modeled shape and selected quantiles | Depends on the model, data, and population to which it is applied |
Diagnostics, Failure Modes, and Limitations
Before using a reported value, check that the anatomical span, marker, endpoint, meal or fasting condition, units, and summary type are stated. If a paper reports “transit time” without these details, the value may not be comparable with another measurement bearing the same label. A mean from one study and a median from another are not directly interchangeable, especially when their distributions are skewed.
For a distributional model, inspect whether the assumed shape is plausible against the observations. A lognormal model may be unsuitable if the data show multiple clusters, substantial censoring, or a shape inconsistent with a single right-skewed distribution. Sparse or widely spaced measurements can also make an apparent endpoint uncertain: the recorded time may be bounded by the observation schedule rather than known exactly. Such limitations should be reported rather than hidden behind a precise-looking estimate.
A frequent mistake is to treat the range as a confidence interval or as the limits of all physiologically possible times. The observed range only spans the values in the measured sample. Another mistake is to interpret a population percentile as uncertainty about the population median. A 5th-to-95th percentile interval describes the modeled spread of individual times; uncertainty in a fitted parameter requires a separate inferential analysis.
Check Your Understanding
Use the definitions and calculations in this tutorial. The solutions follow the questions.
- For the ordered times 2, 3, 3, 4, and 8 hours, calculate the mean, median, and range.
- Why is a reported gastric half-emptying time not equivalent to the time at which the stomach is completely empty?
- A lognormal transit-time model has median 24 hours and log-scale standard deviation 0.30. Calculate its mean using the lognormal formula.
- Does a central 90% interval for modeled individual transit times give a 90% confidence interval for the population median? Explain.
- Name two details needed before comparing transit times from two studies.
Solutions. (1) The mean is \((2+3+3+4+8)/5=20/5=4\) hours. The median is the third ordered value, 3 hours. The range is 2–8 hours, with width 6 hours. (2) Half-emptying refers to the time when half the measured meal or marker has left under the stated method; it does not specify when the remainder leaves. (3) The mean is \(24\exp(0.30^2/2)=24\exp(0.045)\approx25.10\) hours. (4) No. It describes the modeled spread of individual times, whereas a confidence interval concerns uncertainty in an estimated parameter. (5) Suitable details include the anatomical span, marker and endpoint, meal or fasting condition, measurement protocol, and whether the reported value is a mean, median, range, or modeled quantile.