Download 3.3 Drug Dosage

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:
drug in system
60,000
mcg
45,000
30,000
15,000
0
0
24
48
72
96
Time (Hour)
120
144
168
drug in system : Dilantin1compDS
• How does this work mathematically?
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
53.86 mg = 53,860 mg
drug in system
60,000
mcg
45,000
30,000
15,000
0
0
24
48
drug in system : Dilantin1compDS
72
96
Time (Hour)
120
144
168