Repeated-Dose Accumulation With Delayed and Extended Absorption

Why Repeated Doses Can Overlap in More Than One Way

A new dose does not necessarily arrive only after the previous dose has finished entering the body. A lag time can postpone absorption, a zero-order input can continue at a fixed rate for hours, and transit compartments can spread absorption over time. With repeated dosing, these inputs may overlap. The concentration at a sampling time must then reflect both drug already in the body and drug still entering from one or more earlier doses.

The calculations below use linear one-compartment disposition: drug entering the systemic circulation is distributed into a fixed apparent volume \(V\), and is eliminated with first-order rate constant \(k_e\). The central practical result is to calculate the contribution from each dose at the time of interest, then add those contributions. This addition is valid only under the stated linearity assumptions.

Conditions for Adding Dose Contributions

Assume each dose has the same bioavailability \(F\), apparent volume \(V\), and absorption and elimination parameters, unless a worked example says otherwise. For a dose \(D\), let \(R_{\mathrm{in}}(s)\) denote its systemic input rate at time \(s\), measured in amount per time. The concentration contribution at time \(t\) is

The exponential factor is the fraction of drug entering at time \(s\) that remains in the body at time \(t\), under first-order elimination. If the total input is the sum of inputs from doses at times \(t_j\), then the integral of that sum equals the sum of the integrals. Therefore,

This superposition requires linear disposition: \(V\) and \(k_e\) do not change with concentration, and one dose does not alter another dose’s absorption or disposition. It also assumes that the specified input profiles correctly represent each dose. Saturable elimination, concentration-dependent bioavailability, or dose-dependent absorption can invalidate simple addition.

Lagged First-Order Absorption

For a dose administered at time \(t_j\), suppose absorption begins after lag time \(T_{\mathrm{lag}}\) and is then first order with rate constant \(k_a\). Its absorption start time is \(\tau_j=t_j+T_{\mathrm{lag}}\). For \(t\geq\tau_j\), the contribution is

Before \(\tau_j\), this dose contributes zero concentration in this model. After absorption begins, the expression accounts for both continuing absorption and elimination. It is not appropriate to treat a lagged dose as if the full dose entered the body instantaneously at \(\tau_j\).

Extended Zero-Order Absorption

Suppose each dose’s bioavailable amount \(FD\) enters at a constant rate over duration \(T\). Its input rate is \(FD/T\) from dose time \(t_j\) through \(t_j+T\), and zero outside that interval. Let \(a=t-t_j\) be elapsed time since that dose. While input is ongoing, \(0\leq a\leq T\), integration gives

If input has ended, \(a>T\), the concentration at the end of input is reduced by elimination over the remaining time:

These expressions assume the full bioavailable amount is delivered uniformly across \(T\). If the next dose is given before \(T\) has elapsed, the input rates from the two doses overlap, even though their concentration contributions are still computed separately and added.

Transit Absorption and Repeated Doses

Transit absorption distributes the time to systemic input rather than assigning one fixed delay. For an Erlang transit-time distribution with \(N\) transit compartments and transit rate \(k_{\mathrm{tr}}\), the absorption-time density for \(u\geq0\) is

This density integrates to one. A dose’s input rate is \(FD f(u)\) at elapsed time \(u\), and its concentration contribution follows by integrating that input against the elimination kernel. The earlier material on transit compartments describes this distributed delay; here the important repeated-dose step is to evaluate each dose’s contribution at its own elapsed time and sum.

For \(N=2\), \(f(u)=k_{\mathrm{tr}}^2u e^{-k_{\mathrm{tr}}u}\). With \(a=t-t_j\geq0\), the contribution is

When \(k_{\mathrm{tr}}\ne k_e\), integrating \(u e^{-xu}\) gives \(\{1-e^{-xa}(1+xa)\}/x^2\), with \(x=k_{\mathrm{tr}}-k_e\). This yields a directly evaluable expression for this two-compartment transit input.

Derivation, Diagnostics, and Common Pitfalls

The justification for superposition follows directly from the concentration integral. Write the total input as \(R_{\mathrm{in}}(s)=\sum_j R_j(s)\). Then

The equality is valid because integration is linear and the same elimination kernel applies to every dose. It does not imply that absorption profiles are identical at a given clock time: each is shifted by its dose time and has its own elapsed absorption time.

When checking a calculation, verify that each dose has been included only after its administration, and use elapsed time relative to that dose’s lag or input start. For finite-duration zero-order input, use the ongoing-input expression only when \(0\leq a\leq T\); after \(T\), use the post-input expression. For transit input, the contribution must be zero before the dose and nonnegative afterward. If the model parameters are positive, a computed negative concentration signals an algebraic, timing, or numerical error.

A frequent mistake is to assume a new dose resets or replaces absorption from the earlier dose. It does not under the models above: each input continues according to its own profile. Another mistake is to multiply a single-dose concentration by a dose count as a shortcut. That is justified only in special circumstances, such as identical contributions at the time of interest; generally, doses have different elapsed times and different elimination histories. At periodic steady state, repeated profiles may be represented by a sum over prior doses, but the same dose-specific timing logic still applies.

The alternatives differ in their representation of input, not in the superposition principle. Lagged first-order absorption uses a waiting period followed by an exponentially declining input rate. Zero-order absorption uses a constant rate over a defined duration. Transit absorption spreads input according to a distribution of transit times. The choice should follow the intended absorption mechanism and data support; these formulas do not determine which mechanism is appropriate. They also do not cover nonlinear disposition or changing bioavailability.