Estimating Emptying Parameters from Physiological and PK Data

Why the Data Source Matters

A gastric-emptying model translates observed data into parameters that can be used in a PBPK model. Imaging can measure how much meal remains in the stomach over time. Pharmacokinetic (PK) marker data, by contrast, reflect the marker’s passage through gastric emptying and often also its absorption, distribution, metabolism, and excretion. The same emptying parameters may be estimated from either source, but the observations are not interchangeable: each source requires its own measurement model and assumptions.

This tutorial estimates a lag $L$ and a post-lag first-order emptying rate $k_g$. The focus is on estimating these parameters from physiological observations and appropriately interpreted PK-marker data, and on choosing sampling times that inform both parameters.

Definition: Let $R(t)$ be the fraction of the initial gastric meal or marker remaining in the stomach at time $t$, with $R(0)=1$. In the lagged first-order model, gastric emptying does not begin before time $L$. After $L$, the remaining fraction declines at rate $k_g$: $$ R(t)= \begin{cases} 1, & 0 \leq t \leq L,\\ \exp[-k_g(t-L)], & t>L, \end{cases} $$ where $L$ is measured in minutes and $k_g>0$ in inverse minutes. The corresponding emptied fraction is $F(t)=1-R(t)$.

The model assumes a well-defined time origin, a single lag, a constant post-lag fractional emptying rate, and no relevant loss of marker or meal mass except through gastric emptying. It also assumes that the measured quantity represents gastric retention, or has been validly converted to it. These assumptions are not guaranteed by collecting a time series.

Estimating Parameters from Imaging

Imaging provides direct estimates of gastric retention when the technique distinguishes stomach contents from material that has left the stomach. If measurements occur after the lag and are positive, taking logarithms gives a straight-line relationship:

\log R(t)=k_gL-k_gt,\qquad t>L.

Thus, a regression of $\log R(t)$ on time has slope $-k_g$ and intercept $k_gL$. With at least two distinct post-lag times, a line can be fit; with exactly two times, the slope is determined by the two observations. The lag estimate is the intercept divided by the rate estimate. In practice, fitting the original retention model by nonlinear least squares or likelihood may be preferable if measurement errors are better described on the original scale than on the log scale.

Worked Example: Estimating Lag and Rate from Imaging

Suppose a hypothetical imaging study measures gastric retention as $R(20)=0.7408$ and $R(50)=0.3012$. Assume both times are after the lag, the imaging measurements are unbiased estimates of retention, and the lagged first-order model is appropriate.

The post-lag log-retention slope gives the rate:

\widehat{k_g} =\frac{\log(0.7408)-\log(0.3012)}{50-20} =\frac{-0.300025-(-1.199981)}{30} =\frac{0.899956}{30} \approx 0.0299985\ \mathrm{min}^{-1}.

Using the first observation to estimate the lag, with the unrounded rate:

\widehat{L} =20+\frac{\log(0.7408)}{\widehat{k_g}} =20+\frac{-0.3000246}{0.02999854} \approx 9.99869\ \mathrm{min}.

The estimates are therefore $k_g\approx0.029999\ \mathrm{min}^{-1}$ and $L\approx9.999\ \mathrm{min}$, reported here to three decimal places. The estimated lag is below both observation times, consistent with the assumption that both observations lie on the post-lag part of the curve.

With more than two imaging observations, the fit can be checked rather than merely calculated. A plot of $\log R(t)$ against time should be approximately linear after the estimated lag. Systematic curvature, a changing slope, or a plateau after the putative lag suggests that the first-order post-lag model may be inadequate. Observations before the lag are predicted to remain at $R(t)=1$; measurement noise may produce small deviations, but a clear decline before the fitted lag is a warning sign.

Estimating from PK Marker Data

A PK marker does not usually measure gastric retention directly. For example, plasma concentration can rise after the marker leaves the stomach and is absorbed, then fall as the marker is distributed and eliminated. A concentration at one time is therefore not generally equal to the fraction emptied by that time. Marker-based estimation requires a measurement model that accounts for the processes between gastric exit and the recorded PK observation.

One useful representation is a calibrated estimate of cumulative gastric exit, $F_{\mathrm{PK}}(t)$, derived from marker data using known or independently characterized absorption and disposition. If the marker is conserved through the measurement process or its recovery is correctly accounted for, then $R_{\mathrm{PK}}(t)=1-F_{\mathrm{PK}}(t)$ can be used in the lagged first-order model. This conversion is valid only under the calibration assumptions; raw plasma concentrations cannot simply be substituted for $R(t)$.

Worked Example: Estimating Emptying from a Calibrated PK Marker

In a hypothetical marker study, calibration of the marker’s PK behavior yields estimated gastric retention fractions of $0.8187$ at 15 minutes and $0.3679$ at 45 minutes. Assume the calibration correctly accounts for absorption and disposition, both observations are post-lag, and the lagged first-order model applies.

Estimate the rate from the difference in log retention:

\widehat{k_g} =\frac{\log(0.8187)-\log(0.3679)}{45-15} =\frac{-0.2000376-(-0.9999441)}{30} =\frac{0.7999065}{30} \approx 0.02666355\ \mathrm{min}^{-1}.

Then use the 15-minute observation to estimate the lag:

\widehat{L} =15+\frac{\log(0.8187)}{0.02666355} =15+\frac{-0.2000376}{0.02666355} \approx 7.49772\ \mathrm{min}.

The estimated parameters are $k_g\approx0.026664\ \mathrm{min}^{-1}$ and $L\approx7.498\ \mathrm{min}$. These are estimates conditional on the calibration and model assumptions, not direct readings from plasma concentration. In particular, an error in the assumed marker disposition could be mistaken for a change in gastric emptying.

For a marker with a known disposition model, a more direct approach is to fit the marker observations and the gastric input model together. For example, if marker input to the systemic circulation is proportional to the gastric emptying rate, predicted concentration is obtained by combining that input with the marker’s absorption and elimination model. The parameters are then estimated by finding values that make predicted concentrations agree with the observed concentration-time data. This avoids treating concentration as retention, but makes the result depend on the correctness of the full PK measurement model.

Key distinction: Imaging estimates retention through a physiological measurement of stomach contents. A PK marker estimates emptying through a chain of processes that must be represented or calibrated. Agreement between the estimates is informative only if their measurement models and assumptions are also credible.

Sampling Schedules and Parameter Precision

A schedule should include informative observations both near the onset of emptying and across the post-lag decline. Early measurements help locate the transition from the lag period; later post-lag measurements help determine the rate. If all measurements occur well after the lag, the rate may be estimable but the lag is inferred by extrapolating the fitted line back to $R=1$. If all measurements occur before emptying begins, they provide little information about $k_g$.

The effect of time spacing can be illustrated with two post-lag measurements. If errors in log retention are independent with standard deviation $\sigma_{\log R}$, the standard error of the two-point slope is $\sigma_{\log R}/\sqrt{\sum_i(t_i-\bar{t})^2}$. Wider spacing increases the denominator and, under this error model, can improve precision of the rate estimate. But very late measurements may have retention close to zero, where measurement error, detection limits, or background signal can dominate.

Worked Example: Comparing Two Post-Lag Sampling Spacings

Consider a hypothetical design comparison under a model with $L=10$ minutes and $k_g=0.03\ \mathrm{min}^{-1}$. Suppose the standard deviation of each log-retention measurement is $0.05$, independently, and compare post-lag times of 20 and 25 minutes with times of 20 and 50 minutes. Both schedules observe after the assumed lag.

For two times, $\sum_i(t_i-\bar{t})^2=(t_2-t_1)^2/2$. With the 20- and 25-minute schedule, this sum is $5^2/2=12.5\ \mathrm{min}^2$, so:

SE(\widehat{k_g})=\frac{0.05}{\sqrt{12.5}} \approx 0.01414\ \mathrm{min}^{-1}.

With the 20- and 50-minute schedule, the sum is $30^2/2=450\ \mathrm{min}^2$, giving:

SE(\widehat{k_g})=\frac{0.05}{\sqrt{450}} \approx 0.002357\ \mathrm{min}^{-1}.

Under the stated log-error assumptions, the wider spacing is more informative about the slope. However, the model predicts $R(50)=\exp[-0.03(50-10)]\approx0.3012$, while at 80 minutes it predicts only $\exp[-0.03(80-10)]\approx0.1225$. If still later measurements approach the assay’s detection limit, they may not retain the assumed $0.05$ log-scale error. A practical schedule balances time separation against reliable measurement and includes observations around the expected onset.

Model Checks, Alternatives, and Limitations

Before interpreting estimates, check whether the data support the model’s basic conditions. Confirm that the time origin is defined consistently, that retention is positive for log-scale fitting, and that the selected post-lag measurements are actually after the lag. Plot observed and fitted retention on the original scale and, for post-lag observations, on the log scale. Look for systematic residual patterns, early decline inconsistent with a fixed lag, or a tail that declines more slowly or rapidly than a straight line on the log scale. For PK markers, also check whether the calibration or disposition model is supported independently and whether marker recovery is plausibly complete or correctly adjusted.

The lagged first-order model is compact and often useful, but it imposes a sharp onset and a single exponential decline. If emptying is gradual from the start, varies across meal components, or has a broad distribution of transit times, the fitted lag and rate may summarize the observations without representing a literal physiological delay and constant rate. Alternatives include fitting a distributed transit-time model, using a more flexible time-varying emptying rate, or estimating emptying from imaging without forcing a first-order curve. These approaches can better represent complex time courses but require additional assumptions and data.

A common mistake is to fit a straight line to log plasma concentrations and call its negative slope the gastric-emptying rate. Plasma concentration generally reflects both input and elimination; its slope is not generally $-k_g$. The correction is to use a validated conversion from PK data to gastric retention, or fit the gastric input together with the marker’s absorption and disposition model. Likewise, rounding logarithms or rate estimates before calculating the lag can shift the reported result; retain unrounded intermediate values and round only the final estimates.

Specify the observation model.
State whether the data are imaging retention, calibrated PK-derived retention, or raw PK measurements requiring a joint input-disposition model.

Check assumptions and timing.
Verify a consistent time origin, positive retention for log fitting, and a defensible set of post-lag observations.

Fit and assess.
Estimate $k_g$ and $L$, inspect fitted curves and residual patterns, and report units and the data source.

Key takeaway: Emptying estimates are conditional on both the gastric model and the observation model. Imaging can measure retention more directly; PK markers require calibration or an explicit account of absorption and disposition. Sampling should locate the onset while retaining enough reliable post-lag measurements to estimate the decline.

Check Your Understanding

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

Why can a raw plasma concentration of a gastric marker not generally be treated as the gastric retention fraction?
For post-lag retention measurements, what do the slope and intercept of a regression of $\log R(t)$ on time represent?
A schedule measures retention only at times before the estimated lag. Which parameter is poorly informed, and why?
Why might very late imaging observations be less useful than their wide spacing suggests?

Solutions. (1) Plasma concentration reflects gastric input plus absorption and disposition, so it is not generally equal to the fraction remaining in the stomach. A calibrated conversion or joint PK model is needed. (2) The slope is $-k_g$, and the intercept is $k_gL$; therefore $k_g$ is the negative slope and $L$ is the intercept divided by $k_g$. (3) The post-lag rate $k_g$ is poorly informed because the model predicts no decline before $L$; measurements that remain at $R(t)=1$ do not reveal the post-lag slope. (4) Very late retention may approach zero, where detection limits, background signal, and larger relative measurement error can undermine the assumed error model.