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Abstract

The objective of the current study is to give three examples illustrating the good performance of an advanced mathematical modeling method based on the theory of dynamic systems in pharmacokinetics. The performance of the modeling method considered in the current study is illustrated in the following examples: 1) the first example illustrates the development of a mathematical model of the pharmacokinetic behavior of a drug administered orally; 2) the second example illustrates the determination of a physiologically realistic structure of the mean residence time of a drug administered orally; 3) the third example illustrates the development of a combined parent-metabolite mathematical model.

In all examples, the mathematical models developed successfully fitted to the experimental data.

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Keywords

Pharmacokinetics; Dynamic systems theory; Mathematical model

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Introduction

The basic idea behind the modeling method used in the current study may seem rather complicated. Over simplifying, this idea can be explained as follows: figure 1, (taken from the author’s web page http://www.uef.sav.sk/advanced.htm shows the working example used in the current study. The schematically illustrated drug administration is in the column headed “INPUTS”, and the schematically illustrated resulting plasma concentration-time profiles of the drug administered are in the column headed “OUTPUTS”. In the working example, the following key assumptions are applied: 1) a single bolus drug dose is administered intravenously to a hypothetical subject (see the first scheme in the column “INPUTS”); 2) a drug is administered by an infusion to a hypothetical subject (see the second scheme in the column “INPUTS”); 3) a drug is administeredby repeated multiple bolus doses to a hypothetical subject (see the third scheme in the column “INPUTS”); 4) the drug dose, the site of measurement of the plasma concentration-time profiles of the drug administered, and physiological properties of the body remind unchanged during the working experiment

By using traditional modeling methods such as compartment methods, significantly different models of the pharmacokinetic behavior of the drug administered are obtained. The reason for this is that compartment methods use only plasma concentration-time profiles of administered drugs. In the working example shown in [Figure1], the resulting plasma concentration-time profiles of the drug administered are different, therefore, the models developed by a compartment method are also different. On the contrary, modeling methods based on the theory of dynamic systems simultaneously uses both: the mathematically described inputs of drugs into the body and plasma (or blood) concentration-time profiles of drugsadministered. Models developed in this way are dependent only on physiological properties of the body, and on the ADME properties of the drugs administered. Therefore, the same models are obtained for the pharmacokinetic behavior of the drug administered, irrespective whether the drug is administered by a single-bolus dose, or by an infusion, or by multiple bolus doses, assumed in the working example shown in Figure 1. ADME is a well-known acronym commonly used in pharmacokinetics. It stands for absorption, distribution, metabolism, and excretion of a drug administered [1].

 It is well known that dynamic processes associated with the pharmacokinetic behavior of an administered drug are controlled by several mechanisms and are influenced not only by diverse interactions between the drug administered and a physiological environment but also by various factors [2,3]. Since the pharmacokinetic behavior of an administered drug is a complicated dynamic process, dependent upon a variety of independent factors, several studies described investigations of the pharmacokinetic behavior of administered drugs in the body using advanced modeling methods based on the theory of dynamic systems, see for example, the following studies [4-17] and the references therein. Three examples illustrating a successful use of an advanced modeling method based on the theory of dynamic systems in pharmacokinetics are also given in the current study.

The modeling method used in the current study was developed by Dedík [7,8]. It has been effectively used in several studies; see for example, the studies cited in the previous paragraph. Using the modeling method considered here, investigations of dynamic systems can be performed in a variety ways. In pharmacokinetics, the given modeling method can be advantageously used to investigate the pharmacokinetic behavior of an administered drug in the following way: In the first step of an investigation, an ADME related dynamic system (thereafter only dynamic system) is defined in the complexdomain, and is used to mathematically represent static and dynamic aspects of the pharmacokinetic behavior of an administered drug in the body, see, for example, the studies cited in the previous paragraph. In the second step of an investigation, a mathematical model of the dynamic system is developed and point estimates of model parameters are determined in the complex domain, using the noniterative modeling method published previously [18]. In the third step of an investigation, an optimal model of the dynamic system is selected, using the Akaike information criterion modified for the use in the complex domain [6,19].

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Examples

The first example illustrates the development of mathematical models of the pharmacokinetic behavior of pentacaine in healthy volunteers. Pentacaine is a long-acting carbanilate type local anaesthetic [20]. In the comparative study [4], the pharmacokinetic behavior of pentacaine in healthy volunteers was investigated using data collected during the clinical trial of pentacaine. A short description of the comparative study [4] is as follows: Pentacaine was orally administered to healthy volunteers. The modeling program CXT (based on the theory of dynamic systems) [7-10] and modeling program VisSim [21] were employed to develop mathematical models of the pharmacokinetic behavior of pentacaine in the healthy volunteers enrolled in the study [4]. The results obtained revealed that the best approximations of the plasma concentration-time profiles of pentacaine were obtained using the CXT in all healthy volunteers [4] (see figure 2, taken from the author’s previous study [4]).

The second example illustrates a section from the author’s previous study [15]. The section shown corresponds to the determination of the physiologically realistic structure of a mean residence time MRTpo of a drug administered orally in an immediate release dosage form to a hypothetical subject. The physiologically realistic structure of a meanresidence time MRTiv of an intravenously administered drug to a human subject was determined in the previous study by the author [14]. It is described by the following equation:

       

Five interconnected structural components related to the pharmacokinetic behavior of an intravenously administered drug to a human subject occur on the right-hand side of Eq. (1). The structural component Fcp describes the contribution to MRTiv by the dynamic process associated with the drug dynamic transit [22]in the cardiopulmonary subsystem Hcp [23]. (A subsystem of a dynamic system is a part of a dynamic system which itself has characteristics of a dynamic system). The structural component Fp describes the contribution to MRTiv by the dynamic process associated with the drug dynamic transit in the portal-venous subsystem Hp [24]. The structural component Fh describes the contribution to iv MRT by the process associated with the drug dynamic transit in the hepatic-portal subsystem Hp [22,24]. The structural component o F describes the contribution to iv MRT by the process associated with the drug dynamic transit in the subsystem . o H The subsystem o H mathematically represents drug fate and disposition in noneliminating tissues. If the drug is subject to the enterohepatic circulation (EHC) [25], the structural component r F describes the contribution to iv MRT by the dynamic process associated with the drug dynamic transit in the subsystem . r H The subsystem r H mathematically represents the dynamic process associated with EHC, and the structural components p F and 0 F are as follows: 

           

        

   

In Eqs. (2)-(4), p Q is blood flow in the portal vein, o Q is blood flow in non-eliminating tissues, h Cl is the hepatic clearance of the drug administered, p MT is the mean time of the drug dynamic transit in the portal-venous subsystem, h MT is the mean time of the drug dynamic transport in the hepatic-portal subsystem, and o MT is the mean time of the drug dynamic transit in the subsystem , o H where  

   

 and i MT is the mean time of the drug dynamic transit in noneliminating tissues, the i subscript specifies a tissue [24].

In order to mathematically represent dynamic processes associated with drug fate and disposition after an oral administration, the dynamic system po H was defined, and a circulatory model of the dynamic system po H was developed. The developed circulatory model of the dynamic system po H is diagrammed in Figure 3, which was taken from the previous study by the author [15], preformed under equal condition as the current study. The study cited here can be briefly described as follows: The determination of the physiologically realistic structure of po MRT was performed in the following steps: In the first step, a working example was prepared and used to schematically illustrate the drug dynamic transit [22] in the body after an oral administration to a hypothetical subject. In the second step, the following simplifying assumptions were made: a) the drug was uniformly distributed in the body; b) the drug was eliminated mainly by hepatic excretion; c) the drug was not bound to plasma proteins or tissues; d) the drug was completely dissolved in the stomach, and emptied into the duodenum; e) no barriers to the distribution (or elimination) of the drug existed. In the third step, a circulatory model of the dynamic system po H was developed. In the fourth step, the following equation was derived for the description of po MRT in the Laplace (s) domain:

          

In Eq. (6), H_po(s) is the transfer function [4,17] of the dynamic system . po H In the last step, the physiologically realistic structure of po MRT was determined, using Eq. (6), the circulatory model of the dynamic system po H developed, the previously published method [5,8], and all assumptions concerning the pharmacokinetic behavior of the drug administered. The determined physiologically realistic structure of po MRT is described by the following equation: 

      

The right-hand side of Eq. (7) comprises four interconnected structural components involved in the pharmacokinetic behavior of the orally administered drug. s MT is the mean time of the drug dynamic transit [22] in the subsystem , s H i.e. in the subsystem mathematically representing the following dynamic processes: disintegration of a drug formulation, liberation of a drug from a drug formulation, drug dissolution, hepatic and intestinal first-pass effects (if present), and gastric emptying; p MT is the mean time of the drug dynamic transit in the portal subsystem , p H h MT is the mean time of the drug dynamic transit in the hepatic-portal subsystem . h H iv MRT in Eq. (7) is the same as that in Eq. (1). The sum of the mean times s p h MT + MT + MT in Eq. (7) is total mean time of the drug dynamic transit from the gastrointestinal tract to the blood circulation, see Figure 3. Assuming that EHC [25] was not present, the following equation was developed for the description of the structural component r F :

       

The sum of the mean transit times s p h MT + MT + MT in Eq. (8) is total mean time of the drug dynamic transit from the gastrointestinal tract to the blood circulation, see Figure 3.

The third example illustrates the development of a combined mathematical model of the dynamic process of the formation of 7-hydroxymethotrexate (7OH-MTX) from methotrexate (MTX) in patients undergoing treatment for psoriasis with MTX [26]. The chemotherapeutic agent methotrexate (MTX) is widely used in tumor therapy for different forms of leukemia, as well as for the therapy of patients with arthritis and/or psoriasis [26-29]. In the study [26], MTX was administered to psoriatic patients in a single oral dose of 15 mg once per week, and the investigation of pharmacokinetics and pharmacodynamics of MTX was performed in the early phase (3 months) after the start of therapy of psoriatic patients with MTX. In the current study, combined models for MTX and 7OH-MTX aredeveloped [figure 4a,figure 4b,figure 4c] using data kindly provided by the authors of the study [26]. The development of the models started with the definition of the dynamic systems H in the complex domain. Thereafter, the transfer functions H(s) (s is the Laplace variable) of the dynamic systems H were derived by relating the Laplace transforms of the blood concentration-time profiles of 7OH-MTX ( 7 - ( ) OH MTX C s ) to the Laplace transforms of the blood concentration-time profiles of MTX ( ( ) MTX C s) :

        

Employing the models developed, the following quantities were determined: metabolic ratios, mean times of the formation dynamic process of 7OH-MTX from MTX, and rates of the formation dynamic process of 7OH-MTX from MTX. The results obtained revealed that the metabolic ratio were approximately constant (0.67, 0.58, 0.59) during the first three months of the treatment of the patients with psoriasis with MTX. However, the mean times of the dynamic process of the formation of 7OH-MTX increased, from the value of 9.35 h (after the first dose of MTX) to the value of 15.59 h (after thirteenth dose of MTX). The formation rates of 7OH-MTX from MTX decreased from a maximal value of about 0.06 (1/h) after the first MTX dose to a maximal value of about 0.031 (1/h) after the thirteenth MTX dose, see [Table 1].