3.5 Drug Dosage

Single-Dose Model• Modeling concentration of drug in system for a single dose is straightforward,

using standard differential equation:

where Q is concentration of drug and k is rate of elimination from compartment (intestines, blood plasma).

• Refresher question: Q= ____ ?

Single-Dose Model• Modeling concentration of drug in system for a single dose is straightforward,

using standard differential equation:

where Q is concentration of drug and k is rate of elimination from compartment (intestines, blood plasma).

• Refresher question: Q= Q0ekt

Repeated-Dose Model• We saw in lab (Dilantin model) that repeated doses yield a concentration that tends

toward a fixed value:

• How does this work mathematically?

drug in system60,000

45,000

30,000

15,000

00 24 48 72 96 120 144 168

Time (Hour)

mcg

drug in system : Dilantin1compDS

Mathematics of Repeated Doses• Consider repeated dosage Q with fraction r retained at end of each dosage period.

• Want to compute concentration Qn in system at end of n dosage periods:

Is there a closed form (one-shot analytical formula) for this?

Finite Geometric Series

• The formula an-1 + an-2 + … + a0 (assuming a ≠ 1) is called a finite geometric series with base a.

• But this is still not in closed form!

Finite Geometric Series

Finite Geometric Series

• So closed form for n repeated dosages Q with fraction r retained at end of each dosage period is Q(1-rn) / (1-r)

• What happens as n approaches infinity?

Finite Geometric Series• Googled on limit calculator, got

http://www.numberempire.com/limitcalculator.php

• Let’s check this against Vensim Dilantin model from lab ….

Finite Geometric Series• Initial dosage Q0 = 100 mg• Absorption rate = 0.12• So effective dosage = 12 mg• Elimination rate = -ln(0.5)/22 = 0.0315• So after 8 hr, Q = 12e(-0.0315)(8) = 9.3264 mg• 9.3264 / 12 = 0.7772 = retention fraction

Finite Geometric Series

drug in system60,000

45,000

30,000

15,000

00 24 48 72 96 120 144 168

Time (Hour)

mcg

drug in system : Dilantin1compDS

53.86 mg = 53,860 g