Zero-Order Absorption as a Constant-Rate Input

Constant-Rate Input Over a Finite Duration

Some formulations or delivery processes are represented as supplying drug to the body at a constant rate for a limited time. This is called zero-order absorption: while the process is active, the amount leaving the absorption depot per unit time is constant, rather than being proportional to the amount still in the depot. The input ends when the specified duration is reached. This tutorial defines that input and tracks the depot amount; it does not derive the resulting concentration profile.

Let \(D\) be the dose placed in the depot at the start of absorption, in units such as mg. Let \(T\) be the absorption duration, in hours, and let \(\tau\) be the absorption start time. The depot is assumed to contain \(D\) immediately before time \(\tau\). If there is no lag, \(\tau=0\). Let \(F\) be the fraction of the depot dose that reaches the systemic circulation, with \(0\leq F\leq1\). Separating depot depletion from systemic availability keeps the physical amount remaining in the depot distinct from the amount that ultimately enters systemic circulation.

The rate units follow directly from the dose and duration: mg divided by hours gives mg/h. The depot amount is linear in elapsed absorption time, not exponential. Define elapsed absorption time as \(u=t-\tau\) after absorption begins. The piecewise form matters: before the start the depot has not begun to decline, and after the end it cannot become negative.

Why the Dose-to-Rate Relationship Holds

During the active interval, a constant rate \(R_{\mathrm{dep}}\) removes \(R_{\mathrm{dep}}\Delta t\) amount over any interval of length \(\Delta t\). If the entire depot dose \(D\) is depleted over duration \(T\), conservation of amount requires that the total removed amount equal \(D\).

For \(0<u<T\), the amount removed by elapsed time \(u\) is \(R_{\mathrm{dep}}u\). Subtracting that amount from the initial depot amount gives \(A_{\mathrm{dep}}=D-R_{\mathrm{dep}}u\). Substitution of \(R_{\mathrm{dep}}=D/T\) yields the linear depletion equation in the definition. At \(u=T\), it gives \(A_{\mathrm{dep}}=0\), as required. If only a fraction \(F\) of the depot outflow reaches systemic circulation, multiplying the depot-depletion rate by \(F\) gives the systemic input rate; it does not change the physical depot-depletion rate under this formulation.

This formulation assumes the rate is constant from the start through the end of absorption, the full depot amount is depleted by time \(\tau+T\), and there is no interruption or additional input to that depot. It also treats \(F\) as a fixed fraction of the outflow. Those are modeling assumptions, not conclusions guaranteed by the dose and duration alone.

Using the Piecewise Definition at Boundaries

A finite-duration input is active only during a specified interval. If its start time is \(\tau\), the active interval is from \(\tau\) to \(\tau+T\). The depot amount stays at \(D\) before the start, declines linearly during the interval, and is zero from the endpoint onward. This prevents two common arithmetic errors: subtracting absorption before it starts and extending a constant rate beyond the time needed to deplete the dose.

At exactly \(t=\tau+T\), the during-absorption expression also gives zero. The endpoint can therefore be included in either adjacent piece without changing the depot amount. For the input rate, it is convenient to say the rate applies during the active interval and is zero after completion; the single instant at the boundary does not change the total amount delivered.

Duration, Rate, and Dose Must Agree

Any two of dose, rate, and duration determine the third, provided they refer to the same amount and the same process. A proposed rate and duration imply a total amount \(R_{\mathrm{dep}}T\). If that amount differs from the specified depot dose, the proposed values do not describe complete absorption of that dose under a constant-rate model.

Assumptions, Diagnostics, and Alternatives

A zero-order input is useful when the scientific model supports a nearly constant delivery rate over a finite interval. It is a compact idealization: the model does not explain why the rate is constant, nor does it accommodate a rate that gradually accelerates or slows unless the formulation is extended. Before using it, check that the dose, rate, and duration are compatible; that the start time is defined; and that the assumed depot amount cannot become negative over the times being modeled.

Potential warning signs include a rate-duration product inconsistent with the dose, evidence that input begins gradually, or systematic disagreement with observations near the presumed start or end of absorption. A flat input rate is an assumption about drug entry, not a claim that measured concentrations must rise linearly: disposition processes also affect concentration. The concentration consequences require a separate model.

The main alternative in the earlier material on first-order absorption is an input rate proportional to the amount remaining in the depot. Under first-order absorption, the depot declines exponentially and the input rate changes over time; under zero-order absorption, depot depletion is linear and the rate remains fixed until the finite endpoint. A lag-time model shifts when input begins but does not, by itself, specify whether the subsequent input is constant-rate or first-order. These features can be combined only by specifying both the start time and the post-start input mechanism.

A practical pitfall is to equate a constant systemic input rate with a constant concentration increase. The former describes drug entering the systemic circulation per unit time; it does not account for drug distribution or elimination. Another is to confuse the dose \(D\) with the systemic amount \(FD\). State which amount is being used whenever rate units or remaining amounts are reported.