1. 3.5 Drug Dosage

2. 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= ____ ?

3. 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= Q 0 e kt

4. 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?

5. Mathematics of Repeated Doses Consider repeated dosage Q with fraction r retained at end of each dosage period. Want to compute concentration Q n in system at end of n dosage periods: Is there a closed form (one-shot analytical formula) for this?

7. Finite Geometric Series The formula a n-1 + a n-2 + … + a 0 (assuming a ≠ 1) is called a finite geometric series with base a. But this is still not in closed form!

8. Finite Geometric Series

9. So closed form for n repeated dosages Q with fraction r retained at end of each dosage period is Q(1-r n ) / (1-r) What happens as n approaches infinity?

10. Finite Geometric Series Googled on limit calculator, got http://www.numberempire.com/limitcalculator.php Let’s check this against Vensim Dilantin model from lab ….

11. Finite Geometric Series Initial dosage Q 0 = 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

12. Finite Geometric Series 53.86 mg = 53,860  g