Why Delayed Peaks Need an Absorption Model
After an oral dose, a concentration profile may rise slowly and reach its peak well after dosing. A delayed peak is not, by itself, evidence for a particular absorption mechanism. A fixed delay before absorption begins and a gradual sequence of transit steps can both shift the peak, but they imply different shapes during the early part of the profile. This tutorial compares those descriptions for a single oral dose.
The comparison assumes a one-compartment disposition model with linear, first-order elimination. The elimination rate constant is \(k_e=CL/V\), where \(CL\) is clearance and \(V\) is apparent volume of distribution. We use \(D\) for dose and \(F\) for the fraction of the dose that reaches the systemic circulation. Concentrations are assumed to be measured at known times after dosing. The residual error model used for fitting and simulation must also be specified; it should be appropriate to the assay and the observed variability.
Two Absorption Descriptions
For the lag-time model, the absorption input rate is zero before \(T_{\text{lag}}\), and is \(D k_a\exp[-k_a(t-T_{\text{lag}})]\) afterward. Under the stated linear one-compartment assumptions, the predicted concentration is
This expression assumes \(k_a\ne k_e\). If the two rate constants are equal, the limiting expression is used instead. The sharp onset at \(T_{\text{lag}}\) is a structural feature: absent residual error, the model predicts no concentration before that time.
For a transit model with \(m\) equal-rate transit compartments, let the initial dose amount \(D\) enter the first compartment. Each compartment transfers drug onward at rate \(k_{\text{tr}}\). The final compartment releases drug for absorption. With this definition, the systemic absorption input rate is
The predicted concentration is the accumulation of this input, reduced by elimination of drug that entered earlier:
Here \(m\) is a positive integer, usually chosen from a small set of plausible values, and \(k_{\text{tr}}\) is the transit rate constant. The input-rate curve has its maximum at \((m-1)/k_{\text{tr}}\) when \(m>1\), and its mean input time is \(m/k_{\text{tr}}\). These are properties of the absorption input, not generally the time of maximum plasma concentration: elimination acts while absorption is occurring.
Deriving the Lag-Model Peak Time
The peak time helps interpret a fitted lag model and check whether its estimated parameters produce a plausible profile. For \(t>T_{\text{lag}}\), write \(x=t-T_{\text{lag}}\). The concentration is a positive constant times \(e^{-k_e x}-e^{-k_a x}\). Differentiating with respect to \(x\), setting the derivative to zero, and solving gives
Thus \(t_{\max}=T_{\text{lag}}+\ln(k_a/k_e)/(k_a-k_e)\). This result assumes positive \(k_a\) and \(k_e\), and \(k_a\ne k_e\). It describes the maximum of the model’s concentration curve, not necessarily the time of the largest observed sample; sparse sampling can miss the true maximum.
Worked Example: Calculating a Lag-Model Peak
Suppose a hypothetical oral profile is described by \(k_a=1.20\ \text{h}^{-1}\), \(k_e=0.150\ \text{h}^{-1}\), and \(T_{\text{lag}}=1.00\ \text{h}\). The estimated \(FD/V\) is \(10.0\ \text{mg/L}\). Calculate the model peak time and concentration.
First calculate the time from the end of the lag to the peak:
Therefore \(t_{\max}=1.00+1.98=2.98\ \text{h}\) after dosing. The concentration coefficient is
At \(x=1.98\ \text{h}\), the exponential terms are approximately \(e^{-0.150(1.98)}=0.743\) and \(e^{-1.20(1.98)}=0.093\). Thus the peak prediction is \(11.43(0.743-0.093)=7.43\ \text{mg/L}\). These calculations check the implications of the specified parameters; they do not establish that a fixed delay is the true physical mechanism.
Worked Example: Interpreting a Transit Input
Consider a hypothetical transit model with \(m=3\) compartments and \(k_{\text{tr}}=0.750\ \text{h}^{-1}\). Find the mean input time and the time of maximum absorption input.
The mean input time is \(m/k_{\text{tr}}=3/0.750=4.00\ \text{h}\). Since \(m>1\), the input-rate maximum occurs at \((m-1)/k_{\text{tr}}=2/0.750=2.67\ \text{h}\). The different values are consistent: the input rate rises to a maximum and then has a declining tail, which pulls its mean later than its mode. Neither value should be reported as the predicted plasma \(t_{\max}\) without also accounting for elimination through the concentration integral.
Comparing Fits With Residual Patterns
Fit the two models to the same observations using the same disposition assumptions and the same residual error model. A residual at time \(t_i\) can be defined as \(e_i=y_i-\widehat{C}(t_i)\), where \(y_i\) is the observed concentration and \(\widehat{C}(t_i)\) is the model prediction. Residuals should be examined against time and against predictions. If variability increases with concentration, raw residuals can appear larger near the peak even when the relative fit is reasonable; use diagnostics consistent with the fitted error model, such as appropriately standardized residuals.
A lag model can miss a gradual early rise if its fitted \(T_{\text{lag}}\) places the onset too late. A transit model can miss a sharp onset if its input is too spread out. Repeated residuals of the same sign over consecutive early times are more informative than one isolated discrepancy: they indicate that the model’s time course may be systematically wrong. Also inspect the peak and terminal decline, since an absorption model should not be selected solely because it fits the rising limb.
Worked Example: Comparing Residuals and RMSE
The following hypothetical observations are in mg/L. The two prediction columns are outputs from fits using the same observations and residual-error specification. Calculate residuals and compare the root mean squared error (RMSE), defined here as the square root of the mean squared raw residual.
| Time (h) | Observed | Lag prediction | Transit prediction |
|---|---|---|---|
| 0.5 | 0.20 | 0.10 | 0.25 |
| 1 | 0.80 | 0.40 | 0.75 |
| 2 | 2.70 | 2.00 | 2.50 |
| 3 | 4.10 | 4.00 | 4.00 |
| 4 | 4.50 | 4.60 | 4.40 |
| 6 | 3.80 | 3.90 | 3.90 |
| 8 | 2.60 | 2.50 | 2.70 |
For the lag fit, the observed-minus-predicted residuals are \(0.10, 0.40, 0.70, 0.10, -0.10, -0.10, 0.10\ \text{mg/L}\). Therefore
For the transit fit, the residuals are \(-0.05, 0.05, 0.20, 0.10, 0.10, -0.10, -0.10\ \text{mg/L}\). Thus
In this example, the lag fit underpredicts the first three concentrations, producing a consecutive positive-residual pattern. The transit fit has smaller raw residuals and no equally clear early pattern. That supports the transit description for these data, but RMSE alone does not account for the error model, parameter count, or uncertainty. The residual plot and predictive checks still matter.
Predictive Checks and Model Choice
A predictive check asks whether a fitted model can generate concentration profiles resembling the observed profile at the actual sampling times. Simulate replicate datasets from the fitted model, including its estimated parameter uncertainty when feasible and its residual variability. Compare features chosen in advance, such as early concentrations, peak time, peak concentration, or the number of observations below a quantification limit. For example, if observed early concentrations repeatedly exceed the lag model’s simulated range but fall within the transit model’s range, the lag model may be inadequate for the early rise.
Prediction bands are not confidence intervals for the mean curve: they describe variation in new observations under the simulated model and error assumptions. Nor does one observation falling outside a band automatically invalidate a model. Check whether discrepancies recur and whether the model reproduces the features relevant to the analysis.
Worked Example: Checking Early-Time Predictions
For the observations in the preceding table, suppose predictive simulations at 0.5 h and 1 h give the following hypothetical 90% prediction intervals. The remaining five observations fall within both models’ prediction intervals.
| Time | Observed (mg/L) | Lag 90% prediction interval (mg/L) | Transit 90% prediction interval (mg/L) |
|---|---|---|---|
| 0.5 h | 0.20 | 0.00 to 0.15 | 0.05 to 0.40 |
| 1 h | 0.80 | 0.25 to 0.70 | 0.50 to 1.00 |
At 0.5 h, the observed \(0.20\ \text{mg/L}\) exceeds the lag-model upper limit of \(0.15\ \text{mg/L}\), but lies within the transit interval. At 1 h, \(0.80\ \text{mg/L}\) likewise exceeds the lag-model upper limit of \(0.70\ \text{mg/L}\), while lying within the transit interval. Across all seven times, the lag model covers five observations and the transit model covers seven, given the stated results for the other five times. This small illustrative check favors the transit model for the early rise; it does not establish that future observations must behave similarly.
When both descriptions fit adequately, compare their residual patterns, predictive performance, and complexity. If likelihood-based fitting uses the same observations and the same error model, an information criterion such as \(\mathrm{AIC}=-2\log L+2k\) can summarize fit and parameter-count trade-offs; \(L\) is the maximized likelihood and \(k\) is the number of estimated parameters. Its value is comparative, not a probability that a model is true. The transit model’s integer \(m\) is a structural choice, so state which values were considered and how they were compared. A lower criterion does not replace examination of predictions and residuals.
A lag model is a compact description when the profile is consistent with a distinct period of negligible absorption followed by a relatively sharp onset. A transit model is useful when the early rise is gradual and a sequence of release steps gives a plausible smooth input. Both are approximations: observed concentration data alone may not identify a unique absorption mechanism. Sparse early sampling, uncertain dose timing, assay limitations, or misspecified residual variability can make the comparison inconclusive. A plausible model should reproduce the observed profile without systematic residual structure and should make acceptable predictions at the sampling times of interest.
Assumptions, Diagnostics, and Common Pitfalls
The formulas and comparison above rely on linear disposition with a single elimination rate, a known dose time, and a correctly specified concentration observation and residual-error model. They also assume that the absorption description is applied consistently across the observations being compared. If concentration-dependent clearance, changing bioavailability, or other processes are important, an apparent absorption discrepancy may instead reflect misspecification elsewhere. The lag and transit models do not by themselves identify a formulation mechanism.
- Inspect the early profile. A lag model’s abrupt onset and a transit model’s gradual input make early observations especially informative. With no samples near onset, the two models may be difficult to distinguish.
- Inspect residuals across the full profile. A good early fit does not compensate for a systematic mismatch near the peak or during decline.
- Check parameter plausibility and stability. Estimates at bounds, large changes with small data perturbations, or strong parameter trade-offs signal weak identification. Do not interpret a precise-looking estimate as reliable without considering those diagnostics.
- Use the same comparison basis. Changing the residual model or observations between candidate fits confounds the absorption-model comparison.
A common mistake is to say that a transit model is correct simply because it has the smaller RMSE or because its predicted peak occurs later. The correction is to treat these as evidence about fit, not proof of mechanism. Check the residual pattern, simulate predictions under the fitted assumptions, account for model complexity where relevant, and report when the data do not distinguish the alternatives.
Check Your Understanding
Answer each question, then compare with the solutions.
- For \(k_a=0.90\ \text{h}^{-1}\), \(k_e=0.10\ \text{h}^{-1}\), and \(T_{\text{lag}}=0.50\ \text{h}\), what is the lag-model peak time?
- For \(m=4\) and \(k_{\text{tr}}=0.80\ \text{h}^{-1}\), calculate the mean transit input time and the input-rate peak time.
- What residual pattern would suggest that a fitted lag time postpones absorption too long?
- Why does an observation outside a 90% predictive interval not, by itself, prove that the model is wrong?
- Name two conditions that should be held constant when comparing lag and transit fits.
Solutions. (1) \(t_{\max}=0.50+\ln(0.90/0.10)/(0.90-0.10)=0.50+\ln(9)/0.80=3.25\ \text{h}\) after dosing. (2) The mean input time is \(4/0.80=5.00\ \text{h}\); the input-rate peak is \((4-1)/0.80=3.75\ \text{h}\). These are input features, not the plasma peak time. (3) Consecutive positive observed-minus-predicted residuals early in the profile indicate underprediction and may show that modeled absorption begins too late. (4) A prediction interval allows some observations to fall outside it, and a single miss can occur by chance under the model; repeated or patterned misses are more informative. (5) Examples include using the same observations, the same disposition assumptions, and the same residual-error model.