 *Corresponding Author:
 J. Chrenova
Institute of Automation, Measurement and Applied Informatics, Faculty of Mechanical Engineering, Slovak University of Technology, Namestie Slobody 17, 812 31 Bratislava, Slovakia
Email: jana.chrenova@stuba.sk
Date of Submission  06 June 2012 
Date of Revision  15 February 2013 
Date of Acceptance  20 February 2013 
Indian J Pharm Sci 2013;75(2):171177 
Abstract
This study aimed to identify the cause of atypical shape of measured concentrationtime profile in the peak area by one compartment open model with a lag time (Bateman function with a lag) after single dose oral administration of drug published in "Pharmacokinetic and Pharmacodynamic Data Analysis: Concepts and Application" by Gabrielsson and Weiner (1997) and two concentration profiles after frequent sampling oral glucose tolerance test. Following the oral administration of 100 μg of substance A to human volunteer, frequent sampling was carried out and concentrationtime profiles were obtained. Our hemodynamic circulatory structural model capable of parameters estimation of circulation and gastrointestinal subsystem to explain the plateau within the interval 40100 min (substance A) and 1530 min (glucose) of the measured concentrationtime profile was developed. The mean residence time, the rate constants of absorption and elimination parameters of our model were calculated. Comparing to the Bateman function, our results demonstrate better approximation of the substance A and glucose concentrationtime profile and estimation of absorption rate constant by our structural model. Obtained model results indicate that the atypical shape of measured concentrationtime profile of single dose oral administration of drug was probably caused by the gastrointestinal and circulation system with deep compartment. This applies to the substances with high coefficient of absorption.
Keywords
Absorption rate constant, elimination rate constant, gastric emptying, hemodynamic circulatory model, plateau
The main inspiration for creation of this work was from the book by Gabrielsson and Weiner [1]. Authors of chapter “PK 2  One compartment oral data” (pages 333340) aimed at the modeling of oral administered substance A using one compartment model presented first order input model known as Bateman function with a lag time [26]. Parameter estimation by WinNonlin version 1.1 (Scientific Consulting Inc., Apex, NC) were performed.
Bateman function in fig. 1 presents a concave function on the interval 0100 min, but it does not satisfactorily describe the measured points on the interval 40100 min (dashed line in fig. 1). Our working hypothesis for the explanation of the pseudolinear phenomena  plateau was that the atypical shape of the measured profile on the interval 40100 min is caused by the gastrointestinal (GI) and the circulation system. Glucose is used as second example with the atypical shape of the measured concentration profile on the interval 1530 min. There are also other examples of the atypical shape of the concentrationtime profile of oral administered drug, such as Larginine [7], vitamin C [8], ibuprofen [9], fenobam [10], paracetamol [11] and anthocyanins [12].
Materials and Methods
A human volunteer was given a single oral dose comprising 100 μg of substance A. Consequently, the frequent sampling to obtain the concentrationtime data was done at 10, 15, 20, 30, 40, 60, 90, 120, 180, 210, 240, 300, and 360 min after substance A administration [1]. Two other human volunteers (one male and one female) were given 75 g of anhydrous glucose [13] in 250 ml water solution within 12 min at time zero. Approval of the study protocol was obtained from the Ethic Committee of the Institute of Experimental Endocrinology of Slovak Academy of Sciences [13]. Frequent sampling oral glucose tolerance test was performed by the criteria of Expert Committee on Diagnosis and Classification of Diabetes Mellitus [14]. The blood samples were collected at 15 min before the glucose was administered and at 8, 15, 22, 30, 45, 60, 90, 120, 140, 160, and 180 min after the glucose was administered. Determination of glucose plasma concentrations was carried out by using the glucose oxidize method (Boehringer Manheim, Germany) [13].
Structural model construction
Inclusion of the GI system to the hemodynamic circulatory model (fig. 2) gives the option to analyze the plateau phenomena on the measured concentrationtime profile shown in fig. 1. The scheme of the proposed structural model, with the oral substance dose D as input and the substance blood concentration in the sampling subsystem C_{S} as output, includes the cardiopulmonary subsystem CP, the portal subsystem P, the liver subsystem L, the GI subsystem, the other subsystems O and the sampling subsystem S. Assuming that all of the significant subsystems shown in fig. 2 within the range of measured concentrations formalized behave as linear dynamic systems, then i^{th} subsystem can be described by the transfer function H_{i} presented the general mathematical model of the subsystem as wellstirred model with time delay:
where s is Laplace operator, T is time constant of the subsystem and τ is time delay of the subsystem. The constant gi represented the attenuation of the subsystem and quantified the uptake of substance A in specific subsystem is expressed by the form:
The definition and the transfer function of the GI subsystem comprising absorption part A and gastric empting (GE) (fig. 3) and respecting the mass balance is defined as
where D(t) = Dose ·(t), δ(t) is Dirac function, M_{A} is absorbed amount per unit of time in GI subsystem in the firstpass metabolism of the drug, T_{A} is mean residence time of the absorption subsystem, F is fraction and τ is time delay of the subsystem.
For substance blood concentration in the right atrium C_{RA} is valid
where Q_{CP} is plasmatic blood flow through CP system expressed as follows:
and M_{L}, M_{O}, M_{S} are mass quantities that flow from the liver, the other organs and the sampling subsystems into right atrium RA per unit of time. Q_{L}, Q_{O}, Q_{S} are plasmatic blood flow via the liver, the other organs and the sampling subsystems.
The definitions and transfer functions of the elementary subsystems, respecting the mass balance, are considered as follows equations:
Regarding the CP subsystem is valid
where C_{LV} is the substance concentration in the left ventricle, M_{RA} is the substance amount per unit of time in the right atrium and Q_{CP} is plasmatic blood flow via CP subsystem.
Regarding the portal subsystem P is valid
where M_{P}, g_{P}, Q_{P}, G_{P} and T_{P} are the substance amount per unit of time, the attenuation, the plasmatic blood flow, the gain and the time constant, respectively, related to the portal subsystem.
Regarding the liver subsystem L is valid
where M_{Hv} is the substance amount per unit of time in the hepatic vein. M_{L}, g_{L} and T_{L} are the substance amount per unit of time, the attenuation and the time constant, respectively, related to the liver subsystem, where g_{L} is dimensionless value.
Regarding the other organ subsystem O is valid
where M_{O}, g_{O}, Q_{O}, G_{O}, τ_{O} and T_{O} are the substance amount per unit of time, the attenuation, the plasmatic blood flow, the gain, the time delay and the time constant, respectively, related to the other organ subsystem.
The model of peripheral sampling subsystem S was considered as ideal subsystem for which is valid
For the mean residence time of the drug in the whole body (MRT_{W}) after the oral administration is valid following equation:
(1)
and for the mean residence time of the GI subsystem (MRT_{GI}) is valid
where T_{A} is mean residence time of the absorption subsystem, F is absorbed fraction and τ is a time delay of the subsystem.
Mean residence time of the substance of the circulation system (MRT_{CIRC}) is calculated according to:
Numerical calculation of mean residence time of the whole system MRT_{W} from zero to infinity is expressed as
The absorption rate constant k_{a} is expressed as
where T_{A} is mean residence time of the absorption subsystem.
The elimination rate constant k_{el} is expressed as
(2)
where C_{S} is the concentration of the substance in the sampling subsystem S.
All model calculation and parameter estimation using the Clinical Trials DataBase software [15] were performed. Employing the parameters of the developed structural mathematical model (figs. 2 and 3), the vector λ of estimated parameters was determined as follows:
The point estimate the model parameters by the Monte Carlo method [16] implemented in Computer Controlled Sequential Simulation method [17,18].
Results and Discussion
The final outcome of the processed data obtained from the human volunteer is presented in fig. 4 (substance A), 56 (glucose) and Tables 13.
Substance  F_{1}  F_{2}  T_{a}  Τ_{1}  Τ_{2}  MRT_{GI} 

(%)  (%)  (min)  (min)  (min)  (min)  
Substance A  34.348  65.652  44.717  11.503  27.227  68.354 
Glucose 1  33  66  2.852  1.731  13.398  12.271 
Glucose2  100  0  1.085  7.751  0  8.836 
F_{1,2}=Absorbed fractions of the substance dose, τ_{1,2}=Time delays of the subsystem, T_{a}=Mean residence time of the absorption subsystem, MRT_{GI}=Mean residence time of the gastrointestinal subsystem
Table 1: Model Estimated Parameters of the Gastrointestinal Subsystem
Substance  Q_{CP} (l/min)  g_{l} (min/l)  G_{P} (min/l)  G_{O} (min/l)  τ_{O}(min)  T_{L} (min)  T_{P} (min)  T_{O} (min)  MRT_{CIRC} (min) 

Substance A  1.6  0.969  1.023  0.323  43.575  4.239  1.623  6.733  62.292 
Glucose 1  2  0.229  0.453  1.032  9.704  10.636  3.877  1.498  26.485 
Glucose 2  2.48  1.022  0.445  0.872  13.568  15.072  4.689  1.274  34.231 
Q_{CP}=Plasmatic blood flow via cardiopulmonary subsystem, g is attenuation of the subsystem, G=Gain of the subsystem, τ=Time delay of the subsystem, T=Mean residence time of the subsystem, L=Liver subsystem, P=Portal subsystem, O=Other organ subsystem, MRT_{CIRC}=Mean residence time of the circulation subsystem
Table 2: Model Estimated Parameters of the Circulation Subsystem
k_{a} (1/min) 
kel (1/min) 
k_{a}* (1/min) 
k_{el}* (1/min) 
MRT_{W} (min) 


Substance A  0.022  0.043  0.008  0.009  130.646 
Glucose 1  0.35  0.047  0.041  0.045  38.756 
Glucose 2  0.922  0.365  0.039  0.048  43.067 
k_{a}=Absorption rate constant, k_{el}=Elimination rate constant, MRT_{W}=Mean residence time of the whole system, *Derived or estimated parameters by using Bateman function
Table 3: Model Derived Parameters and Compared Parameters by Hemodynamic Circulatory Model and Estimated Using Bateman Function
Figs. 4 and 5 show the measured and modeled concentrationtime profile C of substance A and glucose 1 and the influence of the first absorbed fraction F_{1} and the second absorbed fraction F_{2}. The absorbed fractions F_{1} and F_{2} expressed by partial concentrationtime profiles C_{1} (dashed line) and C_{2} (dotted line), respectively, are responsible for the main peak 1 of the final concentrationtime profile C and presents the result of the effect of gastric emptying. The final shape of the concentrationtime profile C for t≤60 min is expressed as:
where C_{1} and C_{2} are the partial concentrationtime profiles developed by fraction F_{1} and F_{2}, respectively. Consequently, the secondary peak 2 is the time transformed peak 1 probably influenced by the deep compartment and the circulation system. Fig. 6 shows the measured and modeled concentrationtime profile of glucose 2. In comparison with concentrationtime profile of glucose 1 and substance A, concentrationtime profile glucose 2 comprises only one absorbed fraction.
The model estimated parameters of the GI and the circulation subsystems are listed in Tables 1 and 2, respectively. The model parameters of the GI subsystem include absorbed fractions F_{1} and F_{2}, time delays τ_{1} and τ_{2}, mean residence time of the absorption subsystem T_{A} and MRT_{GI}. Obtained results of the model approximation show approximately twice more value of the second fraction contrary to the first fraction (Table 1).
The model parameters of the circulation subsystem include plasmatic blood flow in the CP subsystem Q_{CP}, gain of the liver, portal, and the other organ subsystems as G_{L}, G_{P}, and G_{O}, respectively, time delay of the other subsystem τ_{O}, mean residence time of the liver, portal, and the other organ subsystems as T_{L}, T_{P}, and T_{O}, respectively, and MRTCIRC are listed in Table 2.
The comparison between derived or estimated parameters by our developed hemodynamic circulatory model and parameters estimated by using the Bateman function is listed in Table 3.
Our work as a reanalysis of the study of Gabrielsson and Weiner [1], was focused on the identification of the system defined by the oral administered substance A and frequent sampling oral glucose tolerance test data as the input and the measured concentrationtime profile as the output by hemodynamic circulatory structural model included GI subsystem. This is for the substances with high coefficient of absorption.
While one compartment open model (Bateman function) was not capable of fitting the measured data within the time interval 40100 min (fig. 1) in case of substance A and the time interval 1530 min in case of glucose, our model approximation presents a good fitting of the measured values within this interval (figs. 46). The shape of final concentrationtime profile C as the result of the parameters estimation by our structural model is characterized by the individual peaks 1 and 2. Regarding obtained results of our modeling, the peak 1 is expressed by the sum of the partial concentrationtime profiles C_{1} and C_{2} related to the individual absorbed fractions F_{1} and F_{2}, respectively, which suggest the effect of the GI system. The second peak 2 presents the influence mainly of the circulation system included the deep compartment to the final concentrationtime profile of substance A in the observed human subject. In Table 1, the first fraction F_{1} (34.425%  substance A) and (33%  glucose 1) was absorbed in the small intestine with the time delay of 11.579 min (substance A) and 1.731 min (glucose 1) compared to the second absorbed fraction F_{2} (65.575%  substance A) and (66%  glucose 1) with the time delay of 27.262 min (substance A) and 13.398 min (glucose 1). Glucose 2 had only one fraction with time delay of 7.751 min.
The mean residence time of the substance A in the GI system MRT_{GI} (68.354 min) is almost similar to the MRT_{CIRC} (62.292 min). As for the glucose 1 and glucose 2, MRT_{GI} is (12.271 min) and (8.836 min) and MRTCIRC (26.485 min) and (34.231 min), respectively (Table 2).
The attenuation of the liver subsystem g_{L} expressed in the steady state SS as where C_{in} is the input substance concentration to the system and C_{out} is the output substance concentration from the system, characterizes the substance uptake in the liver subsystem. In the case of C_{out} < C_{in} is g_{L}<1 else g_{L}=1. Observed g_{L}=0.969 (Table 2) then indicates the uptake of the substance A in the liver. The uptake of the glucose in the liver is 0.229 (glucose 1) and 1.022 (glucose 2).
Table 3 shows to the comparison between the individual absorption k_{a} and elimination kel rate constants. Absorption rate constant k_{a}^{*} according to study [1] of 0.043 l/min appears 2 times higher values in comparison with our estimated value k_{a} of 0.022 l/min. The elimination rate constant k_{el}* (0.009 l/min) according to study [1] presents similar values compared to calculations (Eq. 2) related to our developed structural model (0.008 l/min and 0.295 l/min, respectively). The value of mean residence time MRT_{W} of the whole body calculated by our structural model was 130.646 min (substance A), 38.756 min (glucose 1) and 43.067 (glucose 2). Model estimated value of k_{a} for glucose 1 is 0.35, which is similar to the value estimated by Bateman function (0.47) and for glucose 2 is 0.922 is 3 times higher as compared to the value estimated by using Bateman function (0.365). Model estimated values k_{e} for glucose 1 and 2 are 0.41 and 0.39, respectively and they are very similar to the values estimated by using Bateman function (Table 2).
In summary, obtained model results show a good approximation of the final concentrationtime profile of substance A and glucose by our hemodynamic circulatory structural model compared to the Bateman function. Our work presents the validation of the hypothesis that the atypical shape of measured concentrationtime profile of oral administered substance A and glucose single dose was due to the effect of the GI subsystem and the circulation system included the deep compartment.
Acknowledgements
This work was supported by Competence Centre for SMART Technologies for Electronics and Informatics Systems and Services, ITMS 26240220072, funded by the Research and Development Operational Programme from the ERDF.
References
 Gabrielsson J, Weiner D. Pharmacokinetic and Pharmacodynamic Data Analysis: Concepts and Application. 2nd ed. Stockholm: Swedish Pharmaceutical Press; 1997.
 Bateman H. The solution of a system of differential equations occurring in the theory of radioactive transformations. Proc Cambridge Phil Soc 1910;15:42327.
 Edler L, Wahrendorf J, Frank N. Computational methods for the screening of pharmacokinetic parameters in metabolism experiments. Int J Biomed Comput 1983;14:36980.
 Garrett ER. The Bateman function revisited: A critical reevaluation of the quantitative expressions to characterize concentrations in the one compartment body model as a function of time with firstorder invasion and firstorder elimination. J Pharmacokinet Biopharm 1994;22:10328.
 Volosov A, Bialer M. Use of mean residence time to determine the magnitude of difference between rate constants and to calculate tmax in the Bateman equation. Biopharm Drug Dispos 1999;20:39.
 RoanesLozano E, GonzalezBermejo A, RoanesMacias E, Cabezas J. An application of computer algebra to pharmacokinetics: The Bateman equation. Soc Ind Appl Math Rev 2006;48:13346.
 Tangphao O, Grossmann M, Chalon S, Hoffman BB, Blaschke TF. Pharmacokinetics of intravenous and oral Larginine in normal volunteers. Br J Clin Pharmacol 1999;47:2616.
 Viscovich M, Lykkesfeldt J, Poulsen HE. Vitamin C pharmacokinetics of plain and slow release formulations in smokers. Clin Nutr 2004;23:104350.
 Dewland PM, Reader S, Berry P. Bioavailability of ibuprofen following oral administration of standard ibuprofen, sodium ibuprofen or ibuprofen acid incorporating poloxamer in healthy volunteers. BMC Clin Pharmacol 2009;9:19.
 BerryKravis E, Hessl D, Coffey S, Hervey C, Schneider A, Yuhas J, et al. A pilot open label, single dose trial of fenobam in adults with fragile X syndrome. J Med Genet 2009;46:26671.
 Tanner T, Aspley S, Munn A, Thomas T. The pharmacokinetic profile of a novel fixed dose combination tablet of ibuprofen and paracetamol. BMC Clin Pharmacol 2010;10:10.
 Cao G, Muccitelli HU, SanchezMoreno C, Prior RL. Anthocyanins are absorbed in glycated forms in elderly women: A pharmacokinetic study. Am J Clin Nutr 2001;73:9206.
 Dedík L, Durisová M, Penesová A, Miklovicová D, Tvrdonová M. Estimation of influence of gastric emptying on shape of glucose concentrationtime profile measured in oral glucose tolerance test. Diabetes Res Clin Pract 2007;77:37784.
 Expert committee on the diagnosis and classification of diabetes mellitus. Report of the expert committee on the diagnosis and classification of diabetes mellitus. Diabetes Care 2002;25:S520.
 Dedík L, Ďurišová M. System approach in technical, environmental, and biomedical studies. Bratislava: Slovak University of Technology; 1999.
 Manno I. Introduction to the MonteCarlo Method. Budapest: Akademiai Kiado; 1999.
 Dedík L, Tvrdonová M, Durisová M, Penesová A, Miklovicová D, Kozlovský M. Computer controlled sequential simulation method: Reconsidering evaluation of measurements from frequently sampled intravenous glucose tolerance test. Comput Methods Programs Biomed 2009;95:19.
 Tvrdonova M, Dedik L, Mircioiu C, Miklovicova D, Durisova M. Physiologically motivated timedelay model to account for mechanisms underlying enterohepatic circulation of piroxicam in human beings. Basic Clin Pharmacol Toxicol 2009;104:3542.