# Intuiting Mathematical Objects using Kinetigrams - Kinetigrams: Absorption Models II

Author(s):
John Pais

Our final kinetigram definition addresses a situation of repetitive oral dosing of a drug. Here both the dissolution and absorption processes are modeled, but the excretion process is omitted in order to do something interesting while keeping the complexity to a minimum. The basic model described below was introduced in Spitznagel (1992) -- and further discussed in Yeargers et al. (1996) -- for a common decongestant assuming (essentially) a unit dose of 100 mg, a dosing interval of 6 hours, a uniform dissolution rate of 200 mg/hr, and consequent time for complete dissolution of each dose, one half hour. Furthermore, the rate parameters k1 and k2 are determined by the half-lives of this decongestant in the GI tract and in the blood, respectively. The setup for the example in Panel 1 of Kinetigram Definition 3 below is the same as for this decongestant.

The main idea here is that, since dissolution is turned on only for a part of each dosing interval, we need two functions on each interval to model drug in the gut, and two functions to model drug in the blood. Moreover, these functions need to be appropriately glued together, both inside each dosing interval and from interval to interval.

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 Panel 1 Panel 2

In Panel 2 we see that, in order to precisely specify the different pairs of functions on each dosing interval, their analytic description in terms of rate equations becomes still more involved compared to our previous definition. Once again, we immediately provide an example of how these functions might look visually, before considering their explicit analytic descriptions. The example displayed in Panel 1 has been constructed so that information about various features of these functions will be conveyed and further questions will be naturally evoked. One sees that steady state appears to be essentially achieved by dose eight, and that the (whole) drug in the blood function at steady state is periodic but not sinusoidal. Useful exercises that emerge while viewing Panel 1 include finding the minimum, maximum, and average quantity of drug in the blood for each dose and at steady state. An interesting question, whose answer is not clearly visible, is whether the minimum value of drug in the blood on each dosing interval is the same as the value at which it begins. Another question is how to find the steady state solution without going through a sequence of doses, or how to decrease the variation of drug in the blood on each dosing interval. All these exercises and questions now motivate and, in fact, require that we seriously consider the analytic descriptions of our functions and model.

 Here is a Java version of Kinetigram Definition 3, Panels 1 and 2:

John Pais, "Intuiting Mathematical Objects using Kinetigrams - Kinetigrams: Absorption Models II," Convergence (October 2004)

## JOMA

Journal of Online Mathematics and its Applications