1. Case Study: Drug Dosage CS 170: Computing for the Sciences and Mathematics

2. Administrivia Last time (in P265)  Constrained Growth Today  Case Study: Drug Dosage  HW3 Due  HW4 Assigned

3. Pharmacokinetics Scenario: Your company is developing a drug to help with something. You’ve been tasked with determining what the dosage instructions are going to be. What you know:  A little biology  Minimum effective (and Minimum toxic) drug concentrations  Half-life of the drug

4. One-Compartment Model of Single Dose Concentration of drug in system = amount of drug/volume of blood MEC = minimum effective concentration MTC = maximum therapeutic concentration or minimum toxic concentration

5. Example: Aspirin Blood in an adult's body ≈ 5 liters Amount of plasma ≈ 3 liters Two 325 mg tablets: 2(325)1000 µg Plasma half-life (t 1/2 ) of dose ~ 3.1 to 3.2 hr Q = aspirin_in_plasma dQ/dt = -KQ with K = -ln(0.5)/t 1/2 Therapeutic range  150-300 µg/ml Consider only a single dose

6. One-Compartment Model of Single Dose

7. Example: Dilantin Amount of plasma ≈ 3 liters One 100 mg tablet: 100,000 µg Plasma half-life (t 1/2 ) of dose ~ 22 hrs Q = dilantin_in_plasma dQ/dt = -KQ with K = -ln(0.5)/t 1/2 Therapeutic range  10-20 µg/ml  Toxicity only occurs at > 20,000 µg/ml Consider repeated doses. What is a good range?

8. One-Compartment Model of Repeated Doses ingested = absorption_fraction * (pulse of dosage over each interval)

9. Mathematics of Repeated Doses Absorption level ≈ 0.12 Elimination rate of –ln(0.5)/22 ≈ 0.0315 Amount of drug in the system after 8 hr is Q = Q 0 e - 0.0315(8) ≈ (12)(0.7772) = 9.3264 mg

10. Mathematics of Repeated Doses Q n - amount of drug in system immediately after dose n Q 1 = 12 mg Q 2 = 12(0.7772) + 12 Q 3 = Q 2 (0.7772) + 12 = 12(0.7772) 2 + 12(0.7772) + 12 Q n = Q n - 1 (0.7772) + 12 = 12(0.7772) n - 1 + … + 12(0.7772) 2 + 12(0.7772) + 12

11. Finite geometric series a n - 1 + … + a 2 + a 1 + a 0 = (1 - a n )/(1 - a) for a ≠ 1 Q n = 12(0.7772) n - 1 + … + 12(0.7772) + 12 = 12(1 - 0.7772 n )/(1 - 0.7772)  12(1)/(1 - 0.7772) = 53.8599 mg

12. HOMEWORK! READ Module 3.5 in the textbook Homework 4  READ Module 7.4 in the textbook  COMPLETE Projects 1 and 2 (page 276)  Due next Monday, October 4 th