3'5 Drug Dosage

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3.5 Drug Dosage
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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 ____ ?

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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

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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?

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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?
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(No Transcript)
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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!

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Finite Geometric Series
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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?

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Finite Geometric Series

• Googled on limit calculator, got
• http//www.numberempire.com/limitcalculator.php

• Lets check this against Vensim Dilantin model
  from lab .

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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

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Finite Geometric Series
53.86 mg 53,860 mg