mathematical investigations of current health issues theresa laurent st. louis college of pharmacy...
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Mathematical Investigations of Current Health Issues
Theresa LaurentSt. Louis College of Pharmacy
2007 AMATYC Conference
Investigating Blood Alcohol Content (BAC)
The Questions
Consider two different drinking scenarios. What are they drinking and how frequently are they drinking it?
How high will the BACs become for each of our drinkers?
When will our drinkers’ BACs reach legal levels in order to drive home?
Factors Affecting BAC
BAC = amount of alcohol consumed amount of water in the body
What additional information do you think we will need to collect in order to determine when our drinkers will be at a legal BAC to drive home?
General Information
For women, about 49% of their body weight is water. For men, it’s about 58%.
A legal BAC is less than 0.08%g/mL.
The body eliminates alcohol at a rate of about 0.018 to 0.02%g/mL every hour.
Conversion Formulas
1 oz = 28 g
2.2 lbs = 1 kg
1 L of water = 1 kg
Problem Number 1
Dave and Patty arrive at a bar at 8 pm. Dave downs four 12-ounce Shiner Bock beers, one every half-hour until 9:30 pm. Patty drinks three 1-ounce shots of tequila as soon as she arrives and then stops drinking. Are either Dave or Patty at a legal blood alcohol level by 11 pm when they decide to leave?
Calculating Patty’s BAC
Tequila has 50% alcohol per volume.
Patty weighs 120 lbs.
For women, about 49% of their body weight is water.
Patty’s BAC
mLg
mL
g
/%157.0
)1000)(5.54(49.0
)84%(50BAC
Graph of Patty’s BAC
Patty's BAC
0.157
0.139
0.121
0.103
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
8:00 8:30 9:00 9:30 10:00 10:30 11:00
time
BA
C
Calculating Dave’s BAC
Shiner Bock has 4.4% alcohol per volume.
Dave weighs 200 lbs.
For men, about 58% of their body weight is water.
Dave’s BAC
After one drink:
mLg
mL
g
/%028.0
)1000)(9.90(58.0
)28)(12%(4.4BAC
Graph of Dave’s BAC
Dave's BAC
0.028
0.019
0.047
0.038
0.066
0.057
0.085
0.076
0.067
0.058
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
8:00 8:30 9:00 9:30 10:00 10:30 11:00
time
BA
C
Sample of Student Work
Other Projects using BAC Evaluate the consistency of the
mathematical model with BAC tables
Evaluate the consistency of the mathematical model with BAC calculatorso http://www.globalrph.com/blood_alco
hol_calculator.htmo http://www.dot.wisconsin.gov/safety/
motorist/drunkdriving/calculator.htmo http://www.ou.edu/oupd/bac.htm
Steroids
Problem Number 2:
An athlete has taken 200 mg of Trenbolone (an anabolic steroid) by intravenous injection once every two days for the last two weeks. If he discontinues taking the drug, how long will he have to wait until he could pass a drug test?
Additional Information
The half-life of Trenbolone is about 28 hours.
In a 10 milliliter blood sample, drug testing instruments can detect as little as one nanogram of Trenbolone.
Adult humans have approximately 4 liters of blood in their bodies.
The Mathematical Model After a single dose, the amount of drug
in the bloodstream after t hours is given by
where A0 is the amount of drug in the blood at time t = 0 and k = ln(2)/half-life.
kteAtA 0)(
The Mathematical Model After a single dose, the number of
milligrams of drug in the bloodstream after t hours is given by
tetA 025.0200)(
Repeating Doses
If a drug is taken every 48 hours, the amount of drug in the bloodstream t hours after the nth dose is given by:
))1(48(...)96()48()()( ntAtAtAtAtAn
Visualizing the ProblemAmount of Drug in the Bloodstream
200 mg given every 48 hours for two weeks
0
50
100
150
200
250
300
350
0 48 96 144 192 240 288 336
hours
mg
Solving Problem 2 The amount of drug in the
bloodstream immediately after the 7th dose is
mg
A
286
5.06.15.51.182.60200)0(7
The number of milligrams of drug in the bloodstream t hours after the last dose is given by
tetA 025.0286)(
If one nanogram can be detected in a 10 milliliter blood sample, how many milligrams could be in 4 liters
x = 0.0004
L
mgx
mL
ng
410
1
How long will it take for 286 mg in the bloodstream to reduce to 0.0004 mg in the bloodstream?
t = 539 hours or 22 days
te 025.02860004.0
Birth Control
Birth Control Patch Users Warned
FDA: It Puts Women At Higher Risk For Clots Than Pill Does
11/10/2005
Problem Number 3
Compare the amount of estrogen in a patient who takes oral contraceptives to the amount in a patient who uses the patch.
The Patch The amount of estrogen in the body, t
hours after the patch is applied is given by:
ka = the rate at which the estrogen is
absorbed into the bloodstream. k = ln(2)/elimination half-life.
kta ek
ktA 1)(
Drug Information
The half-life of estrogen is about 10 hours (k = 0.069)
The Ortho Evra patch releases 20 micrograms of estrogen every 24 hours (ka = 0.83 mcg/hr)
Estrogen and the Patch The number of micrograms of
estrogen in the body, t hours after the patch is applied is given by:
tetA 69.0112)(
Estrogen Graph for the Patch
Estrogen in the Body - Ortho Evra Patch
0
2
4
6
8
10
12
14
0 168 336 504 672
hours
mcg
of
estr
og
en
Oral Contraceptives
The amount of estrogen in the body t hours after a dose is taken is given by:
tkkt
a
a aeekk
FDktA
)(
Constants
F = the fraction of the dose that will absorb into the bloodstream.
D = the amount of estrogen in the dose.
ka = ln(2)/absorption half-life.
k = ln(2)/elimination half-life.
Drug Information Each Ortho Tri-Cyclen tablet
contains 35 mcg of estrogen (D = 35).
50% of the dose will absorb into the bloodstream (F = 0.5).
The absorption half-life is 30 minutes (ka=1.4).
The elimination half-life is 10 hours (k = 0.069).
Estrogen and the Pill The number of micrograms of
estrogen in the body t hours after a single dose is taken is given by:
tt eetA 4.1069.04.18)(
Repeating Doses
If a dose is given every 24 hours, the number of micrograms of estrogen in the bloodstream t hours after the nth dose is given by:
))1(24(...)48()24()()( ntAtAtAtAtAn
Graph of Estrogen – Oral Tablets
Estrogen in the Body - Ortho Tri-Cyclen Tablets
0
5
10
15
20
0 168 336 504 672
hours
mcg
of
estr
og
en
Comparing Graphs
Comparison of Estrogen in the Body
0
5
10
15
20
0 168 336 504 672
hours
mcg
of
estr
og
en
Patch
Pill
Comparing Models to Real Data
