Which of the following expresses the accumulation factor R for first-order kinetics with dosing interval τ?

• A. R = 1 / (1 − e^(−kτ))
• B. R = e^(−kτ)
• C. R = 1 − e^(−kτ)
• D. R = 1 + e^(−kτ)

Answer: (A)

In first‑order kinetics with repeated dosing at interval τ, each dose adds to the residual amount from all prior doses, and between doses that residual amount decays by a factor e^(−kτ). So the total amount after multiple doses is the sum D[1 + e^(−kτ) + e^(−2kτ) + …], an infinite geometric series. The sum of that series is D / (1 − e^(−kτ)). The accumulation factor R is the ratio of the steady‑state amount after dosing to the amount from a single dose, which yields R = 1 / (1 − e^(−kτ)). This expression captures how much the drug accumulates: if τ is short or elimination is slow (k small), e^(−kτ) is close to 1 and R becomes large; if τ is long or elimination is rapid (k large), e^(−kτ) is small and R approaches 1. The other forms don’t represent the infinite sum of accumulating contributions from all previous doses: e^(−kτ) is the fraction remaining between doses, 1 − e^(−kτ) is the fraction eliminated in a dosing interval, and 1 + e^(−kτ) does not reflect the accumulation across multiple doses.