The Simple Model's Fit To Widmark's 1914 Alcohol Content Data And More

  In a 1914 paper by Widmark some more data is given for the concentration of alcohol in blood and urine versus time. The two are approximately the same as was shown in some earlier experiments. The data is from Table III. I tried fitting the simplified formula and wasn't impressed by the results.

So I tried a simple analogy that compares diffusion with electrical conduction.

For diffusion the concentration, c, of a substance acts like a potential which causes a quantity of mass m to move from one point to another. The rate of this flow is analogous to an electrical current. And there is an equivalent of Ohm's law I = ΔV/R which is dm/dt = Γ Δc. So we can draw an electrical circuit which will mimic the behavior of the diffusion.

The capacitors act like the compartments of the 3-compartment model representing the digestive tract, D, and the blood, B. One could add a extremely large capacitor to receive the eliminated alcohol but it would act like a short circuit since it would have a minute change in voltage with the addition of a small amount of charge q. The solution for the voltage across C_B is the sum of two exponentials.

The new model gave a better fit. The way the circuit works is the charges on the capacitors first tend to equalize with the voltage across C_D remaining slightly higher than C_B and then a small current from each drains through R_B. We would expect similar changes as the alcohol diffuses throughout the body.

Supplemental (Aug 29): The citation given for the article is,

Widmark, E. M. P. (1916), Über die Konzentration des genossenen Alkohols in Blut und Harn unter verschiedenen Umständen. Skandinavisches Archiv Für Physiologie, 33: 85–96. doi: 10.1111/j.1748-1716.1916.tb00107.x