Sequential Zero-Order and First-Order Absorption

Why Model Absorption in Two Stages?

Some absorption models represent a dose entering the systemic circulation at a constant rate for a finite period. A different assumption is that the dose first leaves its depot at a constant rate, then the remaining depot is absorbed more gradually according to first-order kinetics. This tutorial develops that sequential model and shows how each stage contributes to the concentration profile.

The key modeling distinction is that the two input mechanisms do not operate simultaneously. The zero-order stage ends at a specified time \(T\); only the depot amount left at that time supplies first-order input afterward. The model therefore needs to keep track of both the drug already absorbed and the drug still in the depot.

The model assumes one-compartment linear disposition, instantaneous mixing in the central compartment, and the same bioavailability fraction \(F\) for drug absorbed in either stage. There is no additional absorption or elimination from the depot beyond the stated processes. The condition \(R_0T\leq D\) prevents the constant-rate stage from absorbing more than the available dose.

Model Equations and Concentration Profile

During the first stage, the depot loses drug at rate \(R_0\). Once that stage ends, its remaining amount declines exponentially. The systemic input rate is \(F R_0\) before \(T\), and \(F k_a A_g(t)\) after \(T\).

Let \(A_c(t)\) denote the central-compartment amount and \(C(t)=A_c(t)/V\) its concentration. The central-compartment amount follows \(dA_c/dt=F(\text{depot depletion rate})-k_eA_c\), with \(A_c(0)=0\). Solving this equation gives the profile below. The post-switch time \(u=t-T\) is used to make the contributions from the two stages explicit.

After the switch, the first term is drug absorbed during the zero-order stage and subsequently eliminated. The second term is drug absorbed from the residual depot after the switch. This decomposition also helps check units: each bracketed term has units of amount, and multiplication by \(F/V\) gives concentration.

Derivation and Continuity at the Switch

During zero-order absorption, the central amount satisfies \(dA_c/dt=FR_0-k_eA_c\). Multiplying by the integrating factor \(e^{k_et}\) and integrating from 0 to \(t\) yields the first part of the concentration profile:

After the switch, put \(u=t-T\). The central amount at the switch is \(A_c(T)=\frac{FR_0}{k_e}(1-e^{-k_eT})\), and the subsequent input is \(Fk_aA_Te^{-k_au}\). Solving the central-compartment equation from this initial condition gives

For \(k_a\ne k_e\), evaluating the integral gives \(\frac{e^{-k_eu}-e^{-k_au}}{k_a-k_e}\). For \(k_a=k_e=k\), the integrand simplifies to \(e^{-ku}\) for every \(s\), so the integral is \(ue^{-ku}\). Dividing the resulting amount by \(V\) gives the stated profile, including the equal-rate limit.

The concentration is continuous at \(T\): setting \(u=0\) makes the residual-depot contribution zero and leaves exactly \(A_c(T)/V\), the concentration from the first-stage solution. The input rate itself need not be continuous; it changes from \(FR_0\) to \(Fk_aA_T\) at the specified switch. That abrupt change is an assumption of this model, not an algebraic error.

Assumptions, Diagnostics, and Alternatives

The formula is appropriate only when the change in absorption mechanism is a plausible structural description. Check the following before interpreting a fitted profile:

• Check the depot constraint. \(R_0T\) cannot exceed \(D\). A negative \(A_T\) is physically impossible under this model and signals incompatible parameter values or units.
• Check units and time origins. \(R_0\) must be dose per time; \(k_a\) and \(k_e\) must be inverse time; and \(T\) and each observation time must use the same time unit.
• Check the transition. The predicted concentration must be continuous at \(T\). A jump in concentration indicates a coding or implementation error; a change in input rate at \(T\) is allowed.
• Check limiting behavior. The profile should start at zero for an initially empty central compartment. With \(k_e>0\), it should eventually decline toward zero after the depot input becomes negligible.
• Check the model against the sampling times. Sparse observations near or after \(T\) may not distinguish the assumed switch from other absorption patterns. A plausible-looking curve alone does not establish that the sequence is supported by the data.

A common alternative is pure first-order absorption, in which the entire dose declines exponentially from the beginning, or pure zero-order absorption for a fixed duration, in which the input stops when that stage ends. Those models impose different input histories. The sequential model instead has a constant early depot-depletion rate and then a first-order decline of the remaining depot. It should not be interpreted as a constant-rate input that continues after \(T\), nor as two absorption mechanisms acting at once.

A practical pitfall is to use the post-switch expression with \(A_T=D\), forgetting that \(R_0T\) has already left the depot. That overcounts the dose and can inflate predicted concentrations and AUC. Another is to use the unequal-rate formula when \(k_a=k_e\); the denominator is then zero, although the underlying concentration has a well-defined limit. Use the equal-rate expression or a numerically stable limit near equality. Finally, this model assumes a sharp transition at \(T\). If data suggest a gradual change, a different structural model may be needed; the sequential formulation alone cannot describe every absorption profile.