Mathematical Investigations of Current Health Issues Theresa Laurent St. 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 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

Graph of Patty’s BAC

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:

Graph of Dave’s BAC

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 calculators o alculator.htm alculator.htm o rist/drunkdriving/calculator.htm rist/drunkdriving/calculator.htm o

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 A 0 is the amount of drug in the blood at time t = 0 and k = ln(2)/half-life.

The Mathematical Model After a single dose, the number of milligrams of drug in the bloodstream after t hours is given by

Repeating Doses If a drug is taken every 48 hours, the amount of drug in the bloodstream t hours after the n th dose is given by:

Visualizing the Problem

Solving Problem 2 The amount of drug in the bloodstream immediately after the 7 th dose is

The number of milligrams of drug in the bloodstream t hours after the last dose is given by

If one nanogram can be detected in a 10 milliliter blood sample, how many milligrams could be in 4 liters x =

How long will it take for 286 mg in the bloodstream to reduce to mg in the bloodstream? t = 539 hours or 22 days

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: k a = the rate at which the estrogen is absorbed into the bloodstream. k = ln(2)/ elimination half-life.

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 ( k a = 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:

Estrogen Graph for the Patch

Oral Contraceptives The amount of estrogen in the body t hours after a dose is taken is given by:

Constants F = the fraction of the dose that will absorb into the bloodstream. D = the amount of estrogen in the dose. k a = 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 ( k a =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:

Repeating Doses If a dose is given every 24 hours, the number of micrograms of estrogen in the bloodstream t hours after the n th dose is given by:

Graph of Estrogen – Oral Tablets

Comparing Graphs

Comparing Models to Real Data

Contact Information Theresa Laurent