These are standard multiple-dosing pharmacokinetics equations (as covered in Biopharmaceutics and Pharmacokinetics texts, e.g. Brahmankar & Jaiswal). My library search did not surface dedicated equation sections, so this answer is from standard pharmacokinetic principles.
i. Problem of a Missed Dose
If a dose is missed during a multiple-dosing regimen, the plasma concentration at any time t after the missed dose (but before the next scheduled dose) is calculated by ignoring the missed dose in the superposition equation. The plasma concentration is simply predicted as if that dose was never administered:
Cn = C∞ × [(1 - e^(-nKt·τ)) / (1 - e^(-Kt·τ))] × e^(-Kt·t')
More practically, using the superposition principle, the concentration at any time is obtained by summing the contributions of all previously given doses only (the missed dose contributes zero):
Cp = Σ [ (D/Vd) × e^(-Kt·t_i) ] (for i = each dose actually administered, excluding the missed one)
where:
- D = dose, Vd = volume of distribution
- Kt (or K) = elimination rate constant
- t_i = time elapsed since each administered dose
- τ = dosing interval
The key operational rule: recompute the expected Cp trajectory by dropping the missed dose term from the summation, then resume the regular superposition calculation from the next dose actually taken.
ii. Repetitive I.V. Injection (Multiple IV Bolus Doses)
For a drug given as repeated IV bolus injections at fixed interval τ, plasma concentration after the nth dose, at time t after that dose, is given by the superposition principle:
Cn = (D/Vd) × [(1 - e^(-nKτ)) / (1 - e^(-Kτ))] × e^(-Kt)
where:
- D = IV bolus dose
- Vd = apparent volume of distribution
- K = first-order elimination rate constant
- τ = dosing interval
- n = number of doses administered
- t = time elapsed after the nth dose
As n → ∞ (steady state), the term (1 - e^(-nKτ)) → 1, giving the steady-state equation:
C∞ = (D/Vd) × [1 / (1 - e^(-Kτ))] × e^(-Kt)
Maximum and minimum steady-state concentrations:
C∞max = (D/Vd) × [1 / (1 - e^(-Kτ))]
C∞min = (D/Vd) × [e^(-Kτ) / (1 - e^(-Kτ))]
iii. Early or Late Dose Administration During Multiple Dosing
When a dose in a multiple-dose regimen is administered earlier or later than the scheduled time (i.e., the actual interval τ' differs from the planned τ), the plasma concentration is calculated by substituting the actual elapsed time for that interval in place of the fixed τ, for that specific dosing interval only, while keeping all other (correctly timed) intervals unchanged in the superposition sum.
General form for the concentration after such an irregular dose:
Cp = (D/Vd) × Σ e^(-K·t_i)
where t_i is recalculated as the actual time elapsed since each dose (using the true administration times rather than the assumed regular τ spacing) for the dose given early/late, and the standard τ-based spacing for all doses given on schedule.
Practically:
- Early dose: t' (time since previous dose) is shorter than τ, so the residual concentration from the previous dose is higher than expected at the time of the new dose → the new peak (Cmax) will be higher than the normal steady-state Cmax.
- Late dose: t' is longer than τ, so more drug has already been eliminated before the new dose is given → the concentration just before the delayed dose (Cmin) will be lower than expected, and there is a trough period of subtherapeutic levels.
The overall plasma level curve is then reconstructed by re-applying the one-compartment decay equation C = C0 × e^(-Kt) stepwise, using the actual (not nominal) time intervals between the irregular dose and its neighboring doses, and resuming the normal superposition equation from the next correctly-timed dose onward.