Worked Structural Models For Oral Profiles

Oral Profiles Depend on Absorption as Well as Disposition

An oral concentration-time profile reflects both the rate at which drug reaches the systemic circulation and the rate at which it is removed. A model can therefore predict a different peak time and peak concentration even when its dose, bioavailability, volume, and elimination rate match another model. The examples below calculate profiles under three absorption structures: first-order absorption, first-order absorption with a lag, and transit absorption.

These calculations use hypothetical parameter values. They describe a single dose in a one-compartment model with linear, first-order elimination. They are deterministic structural predictions: no residual error or between-subject variability is added. The assumed values and model structure must be plausible for the application; arithmetic alone cannot establish that a structure is appropriate.

First-Order Absorption

With first-order absorption, the fraction of the dose remaining at the absorption site decreases exponentially. The systemic input rate is proportional to the amount still available for absorption. If absorption and elimination start at time zero, the predicted concentration for \(t\geq 0\) is:

This expression assumes a single well-mixed disposition compartment, complete adherence to the specified dose, constant \(F\), and linear first-order absorption and elimination. It also assumes there is no absorption delay or other input process. For \(k_a=k_e\), the continuous limiting form is \(C_{\mathrm{oral}}(t)=(FD/V)k_a t e^{-k_e t}\), rather than the apparent zero-over-zero form of the equation above.

Adding a Lag Time

A lag model assumes that systemic absorption does not begin until a specified time \(t_{\mathrm{lag}}\). After that delay, the same first-order absorption and elimination process applies. For a single oral dose, its profile is the first-order profile shifted along the time axis:

The zero-concentration prediction before the lag is an idealization: the model specifies no systemic input during that interval. It is suitable only when a fixed delay is a useful representation of the process. The lag shifts the profile in time; it does not, by itself, change its post-lag shape or its total systemic dose.

Transit Absorption

A transit model represents absorption as the result of passage through sequential first-order stages rather than as one direct first-order process or a fixed delay. Let \(m\) be the number of sequential stages from the dose to systemic absorption, each with rate constant \(k_{\mathrm{tr}}\). The resulting absorption-time density is a gamma density with integer shape \(m\):

This density integrates to one, so the model makes the full fraction \(F\) available for systemic absorption over time. Its mean absorption time is \(m/k_{\mathrm{tr}}\). In the convention used here, \(m=2\) means two sequential stages; the density is not a single exponential absorption rate.

For two stages (\(m=2\)), \(f_a(s)=k_{\mathrm{tr}}^2s e^{-k_{\mathrm{tr}}s}\). Substituting this density and integrating yields, when \(k_{\mathrm{tr}}\ne k_e\):

To see the integration step, set \(a=k_{\mathrm{tr}}-k_e\). The integral in the convolution becomes \(e^{-k_e t}k_{\mathrm{tr}}^2\int_0^t s e^{-as}\,ds\), and integration gives \(\int_0^t s e^{-as}\,ds=[1-e^{-at}(1+at)]/a^2\). The displayed profile follows directly. If \(k_{\mathrm{tr}}=k_e\), use the finite limit of the integral expression rather than dividing by zero.

Comparing the Structures and Checking Their Use

The equations make clear what distinguishes the three structures. First-order absorption begins immediately and has one absorption rate constant. A lagged model postpones the start of that same process by a fixed duration. A transit model spreads input over time through multiple stages, producing a gradual absorption-time distribution rather than a single fixed onset. The structures may predict similar concentrations over a limited observation window, so an apparent visual match at a few times is not enough to establish which one is best.

Useful checks begin with the model assumptions and the shape of the observed profile. Check that times and rate constants use compatible units, the dose and concentration units are consistent, and the selected equation matches the assumed number of transit stages. Compare predictions with observations around the onset and rising phase, as well as near and after the peak. A profile that rises more slowly than predicted, has a distinct onset delay, or shows systematic residual patterns may indicate a mismatch in absorption structure, but such patterns are not uniquely diagnostic. Sampling times, disposition, and other processes can also affect the apparent shape.

These models also have important limits. The equations assume fixed parameters over the profile, a single dose, linear elimination, and a constant systemic fraction \(F\). They do not represent dose-dependent bioavailability, saturable absorption, recirculation, or time-varying disposition. Those processes require different structures rather than a numerical adjustment to these formulas. In addition, a transit model's stage count and rate constant can be difficult to distinguish from other sources of delayed or gradual input using sparse observations alone.

Common Mistake

A frequent error is to evaluate a lagged profile at clock time \(t\) as though absorption had started at time zero. The post-lag equation instead uses elapsed time \(t-t_{\mathrm{lag}}\). For example, at \(t=2\) hours with a one-hour lag, the exponential terms use 1 hour, not 2 hours. Another error is to treat a transit mean time as a hard onset delay. A fixed lag predicts zero concentration before its specified onset; a transit distribution allows absorption to occur over a range of times. These are different assumptions and should not be interchanged merely because both can delay the peak.