A Further Example Showing Efficiency of a Modeling Method Based on the Theory of Dynamic Systems in Pharmacokinetics

The data available in the study by Plusquellec et al. published in the October Issue of the Journal Medical Engineering & Physics were used to exemplify the method considered here. For modeling purpose an advanced mathematical modeling method was employed. Modeling was performed using the computer program named CTDB described in the study by Dedík et al. published in September 2007 issue of the Journal Diabetes Research and Clinical Practice.


INTRODUCTION
Ranitidine is a histamine H2-receptor antagonist with a potent and long-acting antisecretory effect in humans that signi icantly improves the quality of gastric ulcer healing and histological scores of gastric mucosa in patients with gastric ulcers. In addition, ranitidine was successfully utilized in the treatment of active duodenal ulcers and gastric hypersecretory, where the inhibitory effect of ranitidine on the gastric secretion was much longer than that of cimetidine. Ranitidine is a widely used drug and is known to be well tolerated by patients. Ranitidine is commonly used in treatment of peptic ulcer disease, gastroesophageal re lux disease, and Zollinger-Ellison syndrome it is possibly more effective than cimetidine [1][2][3][4][5][6][7][8][9][10].
The goals of the current study were twofold: 1) to present a further example showing ef iciency of a modeling method based on the theory of dynamic systems in pharmacokinetics; 2) to exemplify the modeling method considered here, using the data available in the study by Plusquellec et al. published
(3) The static and dynamic properties of the pharmacokinetic behavior of orally administered ranitidine [25][26][27] were described with the ADME-related dynamic pharmacokinetic system; (4) The transfer function, denoted by H(s), of the ADME-related dynamic pharmacokinetic system was derived, using the pro iles C(s) and I(s), see Eq. (1).
The ADME-related dynamic pharmacokinetic system was described with the transfer function H(s) in the complex domain.
In the following text, the ADME-related dynamic pharmacokinetic system was simply called the dynamic system (6) The mathematical model of the dynamic system was developed using the computer program named CTDB [15] and the transfer function model H M (s) described by the following equation: On the right-hand-side of Eq. (2) is the Padé approximant [28,29] The transfer function H(s) was converted into equivalent frequency response function, denoted by F(i ω j ) [29]. Table 1: Parameters of the fourth-order model of the dynamic system describing the dynamic pharmacokinetic behavior of orally administered ranitidine [1].   (8) The non-iterative met hod described in the study published previously [29] was used to develop a mathematical model of the frequency response function F M ( iω j ) and to determine point estimates of parameters of the model of the frequency response function F M (iω j ) in the complex domain. The model of the frequency response function F M (iω j ) used in the current study is described by the following equation:

Model parameters Estimates of model parameters (95% CI)
Analogously as in Eq. (2), n is the highest degree of the numerator polynomial of the model of the frequency response function F M (iω j ), m is the highest degree of the denominator polynomial of the mathematical model of the frequency response function F M (iω j ), n ≤ m, i is the imaginary unit, and ω is the angular frequency in Eq. (3).
The Akaike information criterion, modi ied for the use in the complex domain [9,30] was employed to select the best mathematical model of the frequency response function F M (iω j ) and to determine point estimates of the parameters of the best mathematical model of the frequency response function F M (i j ).
Finally, the Monte-Carlo and the Gauss-Newton method [31,32] were used to re ine the mathematical model of the frequency response function F M (iω j ) and to determine 95 % con idence intervals of the parameters of the best mathematical model of the frequency response function F M (iω j ) in the time domain.
After the development of the best mathematical model F M ( i ω j ) of the dynamic system investigated, the following primary pharmacokinetic variables of ranitidine were determined: the elimination half-time of ranitidine, denoted by t ½ , the area under the serum concentration-time pro ile of ranitidine from time zero to in inity, denoted by, AUC ο -∞ , and total body clearance of ranitidine, denoted by CI.
The mathematical model of the transfer function H M (s) and the mathematical model of the frequency response function F M (iω j ) are implemented in the computer program CTDB [15]. A demo version of the computer program CTDB is available at the following web site of the author: http://www.uef.sav.sk/advanced.htm