Wednesday, September 2, 2015

Adding Compartments to the Blood Diffusion Model Can Raise the Peak BAC

  I used an electrical RC circuit to model the flow of alcohol from an initial compartment to two others with no removal of the alcohol from the system to simplify the problem. Alcohol diffusing from the digestive tract into the blood and from there into body tissues can show a slightly higher peak than the 3-compartment model. The circuit used, component values and some of its initial conditions are as follows.

One can compute successive values for the charges on the capacitors which are analogous to the amount of alcohol in each compartment using the simultaneous equations or attempt to solve the system of equations analytically. The analytical equations are easier to compute but can be rather difficult to find. The solutions for the charges on the capacitors are shown in the figure below. I included the analytical equation for the charge, qB, corresponding to the blood alcohol content. Notice that the digestive tract and blood can approach equilibrium before the equilibrium is reached for the tissues accounting for the slightly raised peak.

Since qB and qT are initially zero the simultaneous equations tell us that the initial rate of change of qT is also zero and its curve is slow to change at first. The characteristic equation for the circuit is cubic but fortunately the constant term is missing so one only has to solve a quadratic equation to get the values of α1 and α2.

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