Dont Drink and Derive

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Dont Drink and Derive

• By
• Rob Golden
• Wendy Pitchko

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Problem

• At what rate does alcohol accumulate in the body?
• What effect does drinking style have on alcohol
  concentration?
• How do different drinks measure up?
• How do body type and gender affect concentration?

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Alcohol Digestion
20 digested by alcohol dehydrogenase
20 absorbed by brain
50 into blood
Liver breaks down and eliminates ethanol at about
½ oz. per hour
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Assumptions

• Amount of fluid in body is proportional to body
  mass and constant depends on gender (m0.68k,
  f0.65k)
• Alcohol does not add to net fluid
• Drinking itself does not take time
• 0.8g/L is legal limit in Alberta
• 4g/L puts you in a comma
• 4.5g/L results in death

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Assumptions Cont..

• An elimination rate of 12 g/L common to all
  people

Alcohol Level (g)
Consumption rate
Elimination rate
Alcohol Concentration
Body Weight (lbs)
Body Fluid (L)
Body Weight (kg)
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Model Systems

• 1) Chuggers Drink quantity of alcohol at start
  of time
• 2) Shooters Drink a specific amount at constant
  time intervals
• 3) Sippers Drink steadily over time period

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Model Translation

• Two possible choices discrete time or continuous
  time
• Using model assumptions examine both options for
  benefits and limitations

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Outline of the Mathematical Methods Used to Solve
the Model

• Continuous Time

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The Mathematical Model

• Differential equations were used to keep track of
  the amount of alcohol in the body.
• The rate at which the amount of alcohol was
  changing was the amount coming in minus the
  amount leaving the body.
• Y(t) was the amount of alcohol in the body in
  grams.

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• dY/dtRate In Rate Out
• In our case, this was equal to the amount of
  drinks times the amount of alcohol in that drink
  minus the amount of alcohol being metabolized by
  the body.
• Eulers Method was used to solve for the three
  different drinking styles

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Style 1 Chugging

• This model was really easy because they consume
  all of their alcohol at the beginning and no
  other alcohol is entering the body after that.
  The model simply becomes
• dY/dtrate out, which is -12 grams per hour

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Style 2 Sipping

• The sipping drinkers drink consistently
  throughout the drinking interval.
• Now the rate of change becomes a little more
  complex.
• dY/dtRate In Rate Out
• So now there is a constant amount being added and
  removed in each time interval

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Style 3 Shooters

• In this case we have drinkers having drinks or
  shots in a non-continuous fashion.
• A piecewise function had to be developed to
  account for the periods of non consumption
• This makes the model much more complex than the
  2 previous cases

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The Piece Wise Function

• If ((t 10) Mod 5 0) Then
• RHS2 134 - 12
• Else
• RHS2 -flowrateOut
• End If
• If (t gt 5.6) Then
• RHS2 -12
• End If
• The mod function in the equation allows us to
  only consume alcohol in periods that are
  divisible by 5 with no remainder
• Now when t is equal to 0 and 0.5, which is every
  half hour, the person takes a shot of alcohol

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Discrete Time Model

• Using time steps of one minute, translate rates
  from h to min
• Create spreadsheet to obtain necessary user
  information (stats, drinking style, type)
• Create Macro to compute alcohol concentration at
  time t1 based on time t

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Equations and Variables

• At1 At Ct Et
• At amount of alcohol in body at start of minute
  t (g)
• Ct amount of alcohol consumed in minute t (g)
• Et amount of alcohol eliminated in minute t (g)
• Gt total amount of alcohol in body at time t

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Chuggers

• Created a formula where only C0 had a positive
  value
• Used loop function

While (t lt timetotal) g g c - e
a a (c - e) / fluid If g gt 0.2
Then e 0.2 Else
e g End If t
t 1 c 0
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Shooters
While (t lt drinktime) g g c - e
a a (c - e) / fluid If g gt
0.2 Then e 0.2
Else e g End If
t t 1 If t Mod Frequency 0
Then c totalhit Else
c 0 End If Wend

• Posed a unique problem of piecewise function, but
  was solved with a Mod loop
• Code section

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Sippers

• We had a continuous input of alcohol by dividing
  the grams per hour based on user input by 60 and
  using a piecewise function to continue time after
  drinking was complete.

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Party Scenario

• -Six People
• -Different drinking styles and drink choice
• -Challenge to see who can drive home at end of
  evening

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Guests

• Mr. Fermat Chug Beers (poor shape, 240lbs, 0.65k
  L )
• Mr. Pythagoras Sip Highballs (180 lbs , 0.68k L)
• Mr. Euler shoot Vodka (150 lbs, 0.68k L)
• Ms. Isosceles Sip Wine (130 lbs, 0.65k L)
• Ms. Reimann Shoot Margaritas (100 lbs, 0.65k L)
• Ms. Taylor Chug coolers (poor Shape, 180 lbs,
  0.62k L)

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Facts and Queries

• After 6 hours of drinking, 12 drinks a piece,
  there are two hours before someone must be able
  to drive home sober.
• NB out of shape people have less fluid in their
  bodies
• Who will drive home?
• Who will not make it?

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Mr. Fermat

• Final alcohol concentration of 0.95g/L
• Unable to drive home
• Maximum concentration of 2.3g/L

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Ms. Taylor

• Final Concentration of 1.42g/L
• Unable to drive home
• Maximum concentration of 3.18g/L

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Ms. Reimann

• Final Alcohol Concentration 4.08g/L
• Maximum concentration of 5.09g/L
• She went into a coma at 271 min and pronounced
  dead at 301 min

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Ms. Isosceles

• Final Alcohol concentration of 0.63g/L
• Maximum concentration of 1.28g/L
• The only responsible person to be able to drive
  home
• Ms. Isosceles will never host a party again
  because she hates driving

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Mr. Pythagoras

• Final Alcohol Concentration of 1.17g/L
• Max concentration of 1.6g/L
• Unable to Drive Home

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Mr. Euler

• Final Concentration of 1.4g/L
• Unable to drive home
• Max concentration of 2.02g/L

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Party Conclusion

• At the end of the rowdy math party, only 5 guests
  remained to be taken home.
• Ms. Reimann consumed too much alcohol and died
• Ms. Isosceles was the only person able to drive
  the others home

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Weight Comparison
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Fitness Comparison
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Alcohol Type Comparison
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Model Critique

• Obviously the model is a simplified version of a
  complex system, so some errors are going to
  result from this
• Not all alcohol is eliminated from the body, some
  is absorbed by the brain and other organs
• The first major concern is the rate of alcohol
  removal from the human system
• The rate of 12g/hour is not very accurate for all
  the different groups of people

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• The alcohol removal rate depends on the activity
  level at the time, eating habits, and the type of
  alcohol being metabolized
• Another problem with this model is that it is
  assumed that when people are drinking that they
  are drinking at a constant rate over that period
  of time. This is unreasonable because no single
  person drinks at a perfectly steady rate

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• In our model, we manipulated the fitness level of
  the party guests without fully understanding the
  impact of fitness level on the fluid level of an
  individual
• The model had a very high entertainment level and
  is very user friendly

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Discrete vs. Continuous

• The continuous model allowed for alcohol to be
  added and removed in a more realistic fashion
• Discrete is more user friendly for manipulation
  of variables
• Discrete has a problem with discrepancy of
  occurrences during a time interval versus the
  beginning or end

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Continuous Time
Discrete Time