A Two Compartment, Two Drug Model with Linear Forcing
by Mike Martin

This page generates output plots for a two compartment, two drug delivery model with linear forcing

Consider a medication that has two drugs in it (say a decongestant and an antihistamine) on the time interval from [0, tfinal]. The drugs are administered orally and absorbed first in the gastrointestinal (GI) tract and that is the first compartment. Let x1(t) denote the concentration of the first drug at time t in the GI tract; further, let that first drug have a half-life of τx1 in the GI tract. Similarly, let x2(t) denote the concentration of the second drug at time t and τx2 be the half-life of the second drug in the GI tract. The drugs move from the GI tract to the bloodstream, the second compartment, and are eventually expelled or broken down. Let y1(t) denote the concentration of the first drug at time t and τy1 be the half-life of the first drug in the bloodstream. Let y2(t) denote the concentration of the second drug at time t and τy2 be the half-life of the second drug in the bloodstream. x1(0)=0, y1(0)=0, x2(0)=0, and y2(0)=0 are all initial conditions for the two two drugs in the respective compartments. We assume there are none of these drugs in the compartments to start with and, therefore, the initial conditions are all zero. The dosing function, f(t), is periodic on the interval from [0, Dinterval]. On this page we assume the dosing function is linear on a dosing interval of length Dinterval. The linear forcing function has slope m and y-intercept of b. In a the dosing interval, however, the dosing is "turned on" only on a subinterval, [0, Dadmin], and is otherwise zero (or "turned off").

Graphs are generated for the dynamics of this two compartment, two drug model. The first graph shows the dosing function on a dosing interval, [0, Dinterval]. The second graph shows the dosing function on the entire time interval, [0, tfinal]. The next graph shows the concentrations of the first drug in the GI tract (x1(t), plotted in blue) and in the bloodstream (y1(t), plotted in red). The final graph shows the concentrations of the second drug in the GI tract (x2(t), plotted in blue) and in the bloodstream (y2(t), plotted in red).

Enter in the values of the non-negative parameters tfinal, Dinterval, Dadmin, τx1, τy1, τx2, τy2, m, and b, then press the "Evaluate" button.

tfinal =  

Dinterval =  

Dadmin =  

τx1 =  

τy1 =  

τx2 =  

τy2 =  

m =  

b =  

References:

Spitznagel, E, Two Compartment Pharmacokinetic Models, CODEE Newsletter, Fall 1992, p. 2-4.

Macheras, P, A Iliadis Modeling in Biopharmaceutics, Pharmacokinetics, and Pharmacodynamics. Springer, New York, NY, 2006.