## Tuesday, August 4, 2015

### Modeling Blood Alcohol Content Over Time

Suppose you were drinking and you wondered how your Blood Alcohol Content changed over time. We can use a 3-cell model that tracks the amount of alcohol in the digestive tract, in the blood and the amount that has been eliminated over time. The model and the equations expressing the changes over time is shown below.

We can solve the rate of change equations for different situations such as chugging a drink at t = 0 or drinking at a steady rate over a time interval from an initial time t = 0 to a final time t = tf. The general solution for blood alcohol for drinking rate R0, the amount in the digestive tract, D0, and the amount in the blood, B0, at t = 0 is a function involving simple exponential functions,

The constants λ and μ depend on the individuals tolerance for alcohol.

What is the solution if one drinks over time tf? Initially there is no alcohol in the digestive tract and blood so D0 = 0 and B0 = 0 so the constant coefficients, F, G and H, in the solution only depend on R0. At time tf the consumption rate drops to zero and the amounts in the digestive tract and blood are Df and Bf. The solution consists of two segments that are equal at t = tf

If one chugs a drink at t = 0 then R0 and B0 equal zero and D0 equals the amount of alcohol drunk. So the solution is,

One can plot the two solutions and compare the results. The two solutions are similar in shape but slightly shifted in time.

The values for λ and μ chosen are similar to those of someone who rarely drinks.