Ch 2.8 Case Study: Dull, Dizzy or Dead?

• Humans have been produced fermented beverages for consumptions for thousands of years.
• Early evidence dating back to at least 10,000 BC.
• Beer, wine and cider was often safer to drink than water.
  
• Consumed at every meal by people of all ages and classes.
  
• Excessive use of alcohol was not the accepted norm.
  

What do you call a belt made out of watches?

A waist of time.

• Alcohol requires no digestion and can be absorbed rapidly from stomach into bloodstream.
• This is particularly the case if stomach is empty.
• Alcohol is absorbed even more rapidly from intestines.
• Unless taken with food, little metabolism occurs in GI-tract.

• All of the alcohol is absorbed into the bloodstream.
• Alcohol is distributed freely to all body fluids.
• Concentration in brain rapidly approaches that in blood.
• 90-98% of alcohol is oxidized through liver and excreted.
• Remainder leaves through lungs, urine, saliva and sweat.
• Liver metabolizes alcohol at a constant rate.

• Blood alcohol level (BAL) is a measure of alcohol concentration in blood.
• Units for BAL are grams/100ml of blood.
• Sample Calculations
  • 50 g / 100 ml yields = 50 BAL
  • 100 mg / 100 ml = 0.1 g/100ml = 0.1 BAL
• Colorado legal driving limit:
  • 0.08 BAL = 0.08 g/100 ml = 80 mg/100 ml (DUI)
  • 0.05 BAL = 0.05 g/100 ml = 50 mg/100 ml (DWAI)

• Australian legal limit for driving (including boats and horse or camel drawn vehicles) is 0.05 BAL.
• A person with a BAL of 0.15 is 25 times more likely to have a fatal accident than one with no alcohol.

• Alcohol intake into GI-tract is controlled by drinker.
• Alcohol absorbed into bloodstream depends on:
  • Concentration of alcohol, liquids and food in GI-tract.
  • The weight and gender of individual.
• Rate of diffusion from GI tract and rate of absorption into bloodstream is proportional to amount of alcohol present.

• Alcohol is removed from bloodstream at constant rate by liver.
  • This is independent of body weight and gender of individual, as well as the concentration of alcohol in the bloodstream.
  • Assumes liver has not been damaged by large doses of alcohol.
• A small percentage leaves through sweat, saliva, breath, urine.
• This may mean BAL estimate slightly above true value.

• GI-tract
  • The GI-tract compartment has a single input and output.
• Bloodstream
  • The bloodstream compartment has a single input and output.

The balance law yields two word equations:

\[ \small{ \begin{aligned} \begin{Bmatrix} \mathrm{rate \, of \, change \, of } \\ \mathrm{alcohol \, in \, GI \, tract } \\ \end{Bmatrix} & = \begin{Bmatrix} \mathrm{rate \, of } \\ \mathrm{alcohol \, intake} \\ \end{Bmatrix} - \begin{Bmatrix} \mathrm{rate \, alcohol } \\ \mathrm{\, leaves \, GI \, tract} \\ \end{Bmatrix} \\ \\ \begin{Bmatrix} \mathrm{rate \, of \, change \, of } \\ \mathrm{alcohol \, in \, bloodstream } \\ \end{Bmatrix} &= \begin{Bmatrix} \mathrm{rate \, alcohol } \\ \mathrm{enters \, bloodstream} \\ \end{Bmatrix} - \begin{Bmatrix} \mathrm{rate \, alcohol } \\ \mathrm{leaves \, bloodstream} \\ \end{Bmatrix} \end{aligned} } \]

• \( x(t)= \) concentration of alcohol (BAL) in GI-tract at time \( t \).
  
• \( y(t)= \) concentration (BAL) in bloodstream at time \( t \).
• \( k_1 >0 \) is a proportionality constant between amount of alcohol in GI tract and diffusion rate (per hour).
• \( k_2 >0 \) is a proportionality constant between alcohol amount in bloodstream and absorption rate from GI tract (per hour).
  • For empty stomach, \( k_1 = k_2 \).
  • For full stomach, \( k_1 > k_2 \).
• \( k_3 >0 \) is a proportionality constant associated with alcohol removal by liver (BAL per hour).

Our system of IVPs is

\[ \begin{aligned} \frac{dx}{dt} & = I - k_1x, \,\,\, x(0)= x_0 \\ \frac{dy}{dt} & = k_2x - \frac{k_3 y}{y+M}, \,\,\, y(0)= 0 \end{aligned} \]

Compartment diagram

• Alcohol rapidly absorbed in first hour (\( k_1 = k_2 = 6 \)).
• Number of initial drinks \( n \) consumed rapidly.
• No further drinking after initial \( n \) amount, so \( I = 0 \).
• Initial amount of alcohol in GI-tract is \( x_0 = nx_s \)
  • \( x_s = \) effective BAL produced by single drink.

• Recall IVP for bloodstream:

\[ \small{ \frac{dy}{dt} = k_2x - \frac{k_3 y}{y+M} } \]

• Liver removes alcohol from blood at 8 g/hr; corresponding BAL reduction depends on total body fluids. For 68 kg man:

\[ \small{ \begin{aligned} k_3 & = \frac{8 g/hr}{(0.82 L/kg)(68 kg)} = \frac{8 g/hr}{(0.82)(68)(10)100 ml} \cong 0.0143 \, \mathrm{BAL/hr} \end{aligned} } \]

8/(0.82*68*10)

[1] 0.0143472

DrinksModel1 <- function(W,n,P,kfactor) {
# W = Weight (lbs), n = Number drinks at start
# P = percent of blood fluids in body.  
# P = 0.87 for males, P = 0.67 for females

Wkg <- W/2.205 #Weight in kg from pounds
xs <- 14/(P*Wkg*10)  #Effective BAL for one drink
k3 <- 8/(P*Wkg*10)   #k3 value for bloodstream
k2 <- kfactor*k1  #k2 value for bloodstream 
I  <- 0  #Zero drinks after initial amount

#System of ODEs
f1 <- function(x) {I - k1*x}   
f2 <- function(x,y) {k2*x - k3*y/(y + M)}   

DrinksModel2 <- function(W,n,P,kfactor) {
# W = Weight (lbs), n = Ave # drinks/hour
# P = percent of blood fluids in body  
# P = 0.87 for males, P = 0.67 for females

Wkg <- W/2.205 #Weight in kg from pounds
xs <- 14/(P*Wkg*10)  #Effective BAL for one drink
k3 <- 8/(P*Wkg*10)   #k3 value for bloodstream
k2 <- kfactor*k1  #k2 value for bloodstream 
I  <- 0.025*n  #BAL input continuously drinking

#System of ODEs
f1 <- function(x) {I - k1*x}   
f2 <- function(x,y) {k2*x - k3*y/(y + M)}   

• Rate \( k_1 \) at which alcohol leaves GI-tract on full stomach is greater than rate \( k_2 \) at which alcohol enters bloodstream.
  
• After substantial meal, rate of absorption into bloodstream is approximately halved: \( k_1 > k_2 = 0.5k_1 \).
• Otherwise our previous model stays same.

• Men have a higher percentage of blood fluid levels than women, and generally have a higher body weight as well.
• Men will tend to have a lower BAL than women for the same number of drinks, or even for one or two additional drinks.
• In figure below, 170 lb male (left) with three drinks and 120 lb female (right) with two drinks, both on an empty stomach.

• Continuous drinking quickly leads to high BAL levels.
• A full stomach greatly reduces the BAL levels, both in terms of max BAL level and length of time for an elevated BAL level.
• In graphs, peak level reduced by about half, as is elevated BAL time length (left = empty stomach, right = full stomach)

Mathematical Modeling with Case Studies, Barnes and Fulford, CRC Press, 2015.

History of alcoholic drinks, https://en.wikipedia.org/wiki/History_of_alcoholic_drinks, retrieved on 9/16/2022.