Don’t Drink and Derive By Rob Golden & Wendy Pitchko

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?

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

Assumptions Amount of fluid in body is proportional to body mass and constant depends on gender (m=0.68k, f=0.65k) 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

Assumptions Cont.. An elimination rate of 12 g/L common to all people Consumption rateElimination rate Alcohol Level (g) Body Weight (lbs) Body Weight (kg) Body Fluid (L) Alcohol Concentration

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

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

Outline of the Mathematical Methods Used to Solve the Model Continuous Time:

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.

dY/dt=Rate 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. Euler’s Method was used to solve for the three different drinking styles

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/dt=rate out, which is -12 grams per hour

Style 2: Sipping The sipping drinkers drink consistently throughout the drinking interval. Now the rate of change becomes a little more complex. dY/dt=Rate In – Rate Out So now there is a constant amount being added and removed in each time interval

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

The Piece Wise Function If ((t * 10) Mod 5 = 0) Then RHS2 = Else RHS2 = -flowrateOut End If If (t > 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

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 t+1 based on time t

Equations and Variables A t+1 = A t + C t – E t A t = amount of alcohol in body at start of minute t (g) C t = amount of alcohol consumed in minute t (g) E t = amount of alcohol eliminated in minute t (g) G t = total amount of alcohol in body at time t

Chuggers Created a formula where only C 0 had a positive value Used loop function: While (t < timetotal) g = g + c - e a = a + (c - e) / fluid If g >= 0.2 Then e = 0.2 Else e = g End If t = t + 1 c = 0

Shooters Posed a unique problem of piecewise function, but was solved with a Mod loop Code section: While (t < drinktime) g = g + c - e a = a + (c - e) / fluid If g >= 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

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.

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

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)

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?

Mr. Fermat Final alcohol concentration of 0.95g/L Unable to drive home Maximum concentration of 2.3g/L

Ms. Taylor Final Concentration of 1.42g/L Unable to drive home Maximum concentration of 3.18g/L

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

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

Mr. Pythagoras Final Alcohol Concentration of 1.17g/L Max concentration of 1.6g/L Unable to Drive Home

Mr. Euler Final Concentration of 1.4g/L Unable to drive home Max concentration of 2.02g/L

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

Weight Comparison

Fitness Comparison

Alcohol Type Comparison

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

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

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

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

Continuous Time Discrete Time