Research Article
Open Access
Mathematical Model of the Pharmacokinetic Behavior of
Orally Administered Erythromycin to Healthy Adult Male
Volunteers
Maria Durisova
Department of Pharmacology of Inflammation, Institute of Experimental Pharmacology and Toxicology, Slovak Academy of Sciences, Bratislava,
Dubravska cesta 4
*Corresponding author: Maria Durisova, Department of Pharmacology of Inflammation, Institute of Experimental Pharmacology and Toxicology,
Slovak Academy of Sciences, Bratislava, Dúbravska cesta 4, E-mail:
@
Received: September 06, 2015, Accepted: November 11, 2015, Published: December 16,2015
Citation: Durisova M (2015) Mathematical Model of the Pharmacokinetic Behavior of Orally Administered Erythromycin to
Healthy Adult Male Volunteers. SOJ Pharm Pharm Sci,2(2), 1-5.
Abstract
Objectives: The main objective of the current study was to
present a further example which showed a victorious use of an
superior mathematical modeling method based on the concept of a
dynamic system in mathematical modeling in pharmacokinetics. An
additional objective was to motivate researchers working in field of
pharmacokinetics to use of an alternative modeling method to those
modeling methods commonly used in pharmacokinetic studies. The
current study is a escort piece of a correlated (Yakatan et al. 1979)
study in volunteers published in the Journal of Pharmacokinetics and
Biopharmaceutics. In the study cited at this point, an investigation of
bioequivalence of erythromycin stearate tablets in man was described.
Methods: Data published in the study cited above and an advanced modeling method were used. (For the method used, please see e.g. the explanatory picture and the full-text studies at: http:// www.uef.sav.sk/advanced.htm)
Results: All mathematical models developed successfully fitted the measured data. Based on the mathematical models developed, main pharmacokinetic variables of erythromycin were determined. Conclusion: The mathematical modeling method used in the current study is universal, comprehensive, and flexible. Therefore, it can be used to develop mathematical models not only in pharmacokinetics but also in many other scientific fields.
Keywords: Erythromycin; Oral administration; Mathematical model
Methods: Data published in the study cited above and an advanced modeling method were used. (For the method used, please see e.g. the explanatory picture and the full-text studies at: http:// www.uef.sav.sk/advanced.htm)
Results: All mathematical models developed successfully fitted the measured data. Based on the mathematical models developed, main pharmacokinetic variables of erythromycin were determined. Conclusion: The mathematical modeling method used in the current study is universal, comprehensive, and flexible. Therefore, it can be used to develop mathematical models not only in pharmacokinetics but also in many other scientific fields.
Keywords: Erythromycin; Oral administration; Mathematical model
Introduction
Erythromycin is an antibiotic useful for the treatment of a
number of bacterial infections. This includes respiratory tract
infections, skin infections, chlamydia infections, and syphilis.
It may also be used during pregnancy to prevent Group B
streptococcal infection in newborns [1]. It is an effective inhibitor
of CYP3A4 that patently increases circulating levels of some other
HMG-CoA reductase inhibitors [20].
The main objective of the current study was to present a further example which showed a victorious use of an advanced mathematical modeling method based on the concept of a dynamic system in mathematical modeling in pharmacokinetics [3-14]. An additional objective was to motivate researchers in pharmacokinetics to use of an alternative modeling method to those modeling methods commonly used in pharmacokinetic studies. Previous examples presenting an advantageous use of the modeling method used in the current study can be found in full-text articles available completely free of cost on the authors’ web pages at: http://www.uef.sav.sk/durisova.htm and http:// www.uef.sav.sk/advanced.htm.
http://www.slovaklines.sk/fileadmin/user_upload/ cestovne_poriadky/pal/102426.pdf
The main objective of the current study was to present a further example which showed a victorious use of an advanced mathematical modeling method based on the concept of a dynamic system in mathematical modeling in pharmacokinetics [3-14]. An additional objective was to motivate researchers in pharmacokinetics to use of an alternative modeling method to those modeling methods commonly used in pharmacokinetic studies. Previous examples presenting an advantageous use of the modeling method used in the current study can be found in full-text articles available completely free of cost on the authors’ web pages at: http://www.uef.sav.sk/durisova.htm and http:// www.uef.sav.sk/advanced.htm.
http://www.slovaklines.sk/fileadmin/user_upload/ cestovne_poriadky/pal/102426.pdf
Methods
The data published in the study [1] were employed. For
modeling purposes, an advanced mathematical modeling method
modeling method based on the concept of a dynamic system
was used; see e.g. the studies cited above. The development of
a mathematical model of each dynamic system H was based on
the following simplifying assumptions: a) initial conditions of
each dynamic system H be zero; b) pharmacokinetic processes
occurring in the body once oral erythromycin administration;
were linear and time-invariant, c) concentrations of erythromycin
were the same throughout all subsystems of the dynamic
systems H (where subsystems were integral parts of whole
dynamic systems H); d) no barriers to the distribution and/or
elimination of erythromycin existed. The modeling process used
in the present study can be described as follows:
In the first step of the method, a dynamic system H, was defined for each volunteer by relating the Laplace transform of the serum concentration time profile of erythromycin, denoted C(s), and the Laplace transform of the erythromycin oral input into the body, denoted I(s).
In the second step of the method, the dynamic systems H, were used to mathematically represent dynamic relations between erythromycin inputs into the body and erythromycin behavior in the body [14-16].
In the third step of the method, the transfer function, denoted H(s), (see Eq. 1) was derived for each active system H by relating Laplace transform of the mathematical illustration of the serum concentration-versus-time profile of erythromycin denoted C(s), and the Laplace transform of the mathematical illustration of the oral administration of erythromycin, denoted I(s), (the lower case letter "s" denotes the complex Laplace variable), see e.g. the studies cited above and the following equation:
$$H(s)=\frac{C(s)}{I(s)}.$$
In the first step of the method, a dynamic system H, was defined for each volunteer by relating the Laplace transform of the serum concentration time profile of erythromycin, denoted C(s), and the Laplace transform of the erythromycin oral input into the body, denoted I(s).
In the second step of the method, the dynamic systems H, were used to mathematically represent dynamic relations between erythromycin inputs into the body and erythromycin behavior in the body [14-16].
In the third step of the method, the transfer function, denoted H(s), (see Eq. 1) was derived for each active system H by relating Laplace transform of the mathematical illustration of the serum concentration-versus-time profile of erythromycin denoted C(s), and the Laplace transform of the mathematical illustration of the oral administration of erythromycin, denoted I(s), (the lower case letter "s" denotes the complex Laplace variable), see e.g. the studies cited above and the following equation:
Thereafter, the dynamic system H of each volunteer was
described with transfer function denoted H(s), see, e.g. the
following studies [3-14] and references therein.
For modeling purpose, the software named CTDB [8] and transmit the function model H_{M}(s ), described in the following equation were used:
$${H}_{M}(s)=G\frac{{a}_{0}+{a}_{1}s+\mathrm{...}+{a}_{n}{s}^{n}}{1+{b}_{1}s+\mathrm{...}+{b}_{m}{s}^{m}}.$$
For modeling purpose, the software named CTDB [8] and transmit the function model H_{M}(s ), described in the following equation were used:
On the right-hand-side of Eq. [2] is filling a approximant of
H_{M}( s)[19], G is an estimator of the model parameter known as
the gain of the dynamic system H, a_{1},… a_{n}, b_{1},… b_{m} are the additional
model parameters, n is the maximum degree of the nominator
polynomial, and m is the maximum degree of the denominator
polynomial, where n < m, see e.g. the following studies [3-14].
In the fourth step of the method, the transfer function H(s) was converted into equivalent frequency response function, denoted F(iω_{j})[14].
In the fifth step of the method, the non-iterative method published earlier [3-14] was used to determine a mathematical model of the frequency response function F_{M}(iω_{j}) and point estimates the parameters of the model frequency response function F_{M}(iω_{j}) in the complex domain for each volunteer. The model of the frequency response function F_{M}(iω_{j}) used in the recent study is described by the following equation:
$${F}_{M}(i{\omega}_{j})=G\frac{{a}_{0}+{a}_{1}i{\omega}_{j}+\mathrm{...}+{a}_{n}{(i\omega )}^{n}}{1+{b}_{1}i{\omega}_{j}+\mathrm{...}+{b}_{m}{(i{\omega}_{j})}^{m}}.$$
In the fourth step of the method, the transfer function H(s) was converted into equivalent frequency response function, denoted F(iω_{j})[14].
In the fifth step of the method, the non-iterative method published earlier [3-14] was used to determine a mathematical model of the frequency response function F_{M}(iω_{j}) and point estimates the parameters of the model frequency response function F_{M}(iω_{j}) in the complex domain for each volunteer. The model of the frequency response function F_{M}(iω_{j}) used in the recent study is described by the following equation:
Analogously as in Eq. [2]: n is the maximum degree of
the numerator polynomial of the model frequency response
function F_{M}(iω_{j}) m is the maximum degree of the denominator
polynomial of the model frequency response function F_{M}(iω_{j})
n ≤m, . i is the invented unit, and ω is the angular frequency in
Eq.(3). In the fifth step of the method, each the model frequency
response function F_{M}(iω_{j}) was advanced, using the Monte-Carlo
and the Gauss-Newton method in the time domain.
In the sixth step of the method, the Akaike information was used to differentiate among models of frequency response functions F_{M}(iω_{j}) of dissimilar complication and to select the best model of the frequency response function F_{M}(iω_{j}) with the minimum value of the Akaike information criterion [15]. In the final step of the method, 95 % confidence intervals for parameters of the final models F_{M}(iω_{j}) were determined.
After the improvement of mathematical models of the dynamic systems H, the following primary pharmacokinetic variables were determined: The time occurrence of the maximum observed plasma concentration of erythromycin, denoted t_{max}, the maximum observed plasma concentration of erythromycin, denoted C_{max}, the elimination half-time of erythromycin, denoted t_{1/2}, area under the plasma concentration versus time profile of erythromycin from time zero to infinity, denoted, $AU{C}_{o\text{-}\infty},$ and total body clearance of erythromycin, denoted by CI.
The transfer function model H_{M}(s)and the frequency response function model F_{M}(iω_{j}) are implemented in the computer program CTDB [8]. A sample version of the computer program CTDB is available at: http://www.uef.sav.sk/advanced. htm. Transfer functions and frequency response functions are not unknown in pharmacokinetics, see e.g. the following studies [20,21].
In the sixth step of the method, the Akaike information was used to differentiate among models of frequency response functions F_{M}(iω_{j}) of dissimilar complication and to select the best model of the frequency response function F_{M}(iω_{j}) with the minimum value of the Akaike information criterion [15]. In the final step of the method, 95 % confidence intervals for parameters of the final models F_{M}(iω_{j}) were determined.
After the improvement of mathematical models of the dynamic systems H, the following primary pharmacokinetic variables were determined: The time occurrence of the maximum observed plasma concentration of erythromycin, denoted t_{max}, the maximum observed plasma concentration of erythromycin, denoted C_{max}, the elimination half-time of erythromycin, denoted t_{1/2}, area under the plasma concentration versus time profile of erythromycin from time zero to infinity, denoted, $AU{C}_{o\text{-}\infty},$ and total body clearance of erythromycin, denoted by CI.
The transfer function model H_{M}(s)and the frequency response function model F_{M}(iω_{j}) are implemented in the computer program CTDB [8]. A sample version of the computer program CTDB is available at: http://www.uef.sav.sk/advanced. htm. Transfer functions and frequency response functions are not unknown in pharmacokinetics, see e.g. the following studies [20,21].
Results
The best-fit third-order model of F_{M}(iω_{j}) selected with the
Akaike information criterion was described by Eq. (4):
$${F}_{M}(i{\omega}_{j})=G\frac{{a}_{0}+{a}_{1}i{\omega}_{j}}{1+{b}_{1}i{\omega}_{j}+{b}_{2}i{\omega}_{2}+{b}_{3}i{\omega}_{3}}.$$
This model provided an adequate fit to the erythromycin
concentration data in all volunteers investigated in the previous
[1] and the current study. Estimates of the model parameters a_{0},
a_{1}, a_{2}, a_{3} are listed in Table 1. Model-based estimates of primary
pharmacokinetic variables were listed in Table 2.
Volunteer No.1 was arbitrarily chosen from fourteen volunteers investigated in the previous study [1] and in the current study, to illustrate the results obtained. Figure 1 showed the experimental plasma concentration versus time profile of erythromycin and the report of the observed profile with the developed model of the dynamic system H. Analogous results also hold for all volunteers participating in the previous [1] and the current study.
Volunteer No.1 was arbitrarily chosen from fourteen volunteers investigated in the previous study [1] and in the current study, to illustrate the results obtained. Figure 1 showed the experimental plasma concentration versus time profile of erythromycin and the report of the observed profile with the developed model of the dynamic system H. Analogous results also hold for all volunteers participating in the previous [1] and the current study.
Discussion
The dynamic systems used in the current study were
mathematical objects, without any physiological application.
They were used to model dynamic relationships between
Table 1: Parameters of the third-order model of the dynamic system
describing the pharmacokinetic behavior of orally administered
erythromycin to volunteer No.1.
Table 2: Pharmacokinetic variables of orally administered erythromycin
to volunteer No.1.
Figure 1: Observed serum concentration versus time profile of erythromycin
and the description of the observed profile with the model of the
volunteer’s No.1 dynamic system which mathematically represented
the dynamic relation between erythromycin input to the body and behavior
of erythromycin in volunteer No. 1.
Table 3: Observed and model predicted blood concentrations of
erythromycin in volunteer No.1.
Time (h) |
Observed concentration of erythromycin (ng/ml) |
Model predicted concentration of erythromycin (ng/ml) |
0 |
0 |
0 |
0,5 |
47.4 |
48 |
1.0 |
185 |
185 |
2 |
293 |
293 |
4 |
254 |
254 |
6 |
158 |
158 |
9 |
76.9 |
76.9 |
12 |
130 |
130 |
erythromycin inputs into the body [15-17] and behavior of
erythromycin in the body. The method used in the recent study
has been described in detail in the previous studies [3-13].
As in previous studies, authored or co-authored by the author of the recent study, the development of mathematical models of the dynamic systems H, was based on the known inputs and outputs of the dynamic systems H. In general, a dynamic system is modeled using the transfer function models H_{M}(s). as it was the case in the recent study, then the accuracy of the model depends in large part of the degrees of polynomials of the transfer function models H_{M}(s) used to fit the data, see e.g. the following studies [3-14].
The parameter gain is also known as gain coefficient, or gain factor. Generally, the parameter gain is defined as relationship between the magnitudes of an output of the dynamic system to a magnitude of an input to the dynamic system in steady state. Or in other words, the parameter gain of a dynamic system is a proportional value that shows the relationship between the magnitudes of an output to a magnitude of an input of a dynamic system in steady state.
The pharmacokinetic meaning of the parameter gain depends on the nature of the dynamic system; see e.g. studies available at: http://www.uef.sav.sk/advanced.htm.
The non-iterative method published in the study [14] and used in the recent study is capable of providing quick identification of an optimal structure of a model frequency response. This is a great advantage of this method, because this significantly speeds up the development of frequency response models. The reason for conversion of H_{M}(s) to ( ) M j F iω was as explained in the following text. The variable: "s" in H_{M}(s) is a complex Laplace variable (see Eq. [2]), while the angular frequency ω(see Eq. [4]) is a real variable, what is suitable for modeling purposes.
The mathematical models developed in the recent study sufficiently approximated the dynamic relationships between erythromycin input to the body and behavior of erythromycin in the body in the volunteers investigated in the previous [1] and the current study.
The current study again showed that mathematical and computational tools from system engineering can be successfully used in mathematical modeling in pharmacokinetics. Frequency response functions are complex functions, therefore modeling is performed in the complex domain. In addition, the modeling methods used to develop model frequency response functions are computationally intensive, and modeling require as a minimum partial knowledge of the theory of dynamic system, and an abstract way of thinking about the dynamic system under study.
The principle difference between pharmacokinetic modeling methods traditionally used in pharmacokinetic studies and modeling methods that use mathematical and computational tools from the theory of dynamic systems, is as follows: the former methods are based on modeling plasma (or blood) concentrationtime profiles of drugs, however the latter methods are based on modeling dynamic relationships between a mathematically represented drug inputs into the body and mathematically represent resulting plasma (or blood) concentration-time profiles of administered drugs. See e.g. the previous studies authored and/or coauthored by the author of the current study and the explanatory example, available free of cost at the author’s Web page: http://www.uef.sav.sk/advanced.htm.
The computational and modeling methods that use computational and modeling tools from the theory of dynamic systems that can be example for adjustment of drug administration aimed at achieving and then maintaining required drug concentration–time profiles in patients see e.g. the following study [6]. The methods considered here can be used for safe and cost-effective individualization of drug dosing e.g. through computer-controlled infusion pumps. This is important example for administration of clotting factors to hemophilia patients, as exemplified in the study [6].
The advantages of the model and modeling method used in the recent study are evident here: The models developed and used overcome one of the typical limitations of compartmental models: For the development and use of the models considered here, an assumption of well-mixed spaces in the body is not necessary. The basic structures of the models developed are used largely applicable to mathematical modeling different dynamic systems in the field of pharmacokinetics and in many other scientific as well as practical fields. From a point of view pharmacokinetic community, is an advantage of the models developed using computational tools from the theory of dynamic systems is that the models considered here emphasize dynamic relationships between drug inputs into the body and behavior of a drug in a human and/or an animal body. The method used in the current study can be easily generalized. Therefore, there is no problem to use the method considered here in several scientific and practical fields. Transfer functions of dynamic systems are not unknown in pharmacokinetics; see e.g. the following studies [20-22].
As in previous studies, authored or co-authored by the author of the recent study, the development of mathematical models of the dynamic systems H, was based on the known inputs and outputs of the dynamic systems H. In general, a dynamic system is modeled using the transfer function models H_{M}(s). as it was the case in the recent study, then the accuracy of the model depends in large part of the degrees of polynomials of the transfer function models H_{M}(s) used to fit the data, see e.g. the following studies [3-14].
The parameter gain is also known as gain coefficient, or gain factor. Generally, the parameter gain is defined as relationship between the magnitudes of an output of the dynamic system to a magnitude of an input to the dynamic system in steady state. Or in other words, the parameter gain of a dynamic system is a proportional value that shows the relationship between the magnitudes of an output to a magnitude of an input of a dynamic system in steady state.
The pharmacokinetic meaning of the parameter gain depends on the nature of the dynamic system; see e.g. studies available at: http://www.uef.sav.sk/advanced.htm.
The non-iterative method published in the study [14] and used in the recent study is capable of providing quick identification of an optimal structure of a model frequency response. This is a great advantage of this method, because this significantly speeds up the development of frequency response models. The reason for conversion of H_{M}(s) to ( ) M j F iω was as explained in the following text. The variable: "s" in H_{M}(s) is a complex Laplace variable (see Eq. [2]), while the angular frequency ω(see Eq. [4]) is a real variable, what is suitable for modeling purposes.
The mathematical models developed in the recent study sufficiently approximated the dynamic relationships between erythromycin input to the body and behavior of erythromycin in the body in the volunteers investigated in the previous [1] and the current study.
The current study again showed that mathematical and computational tools from system engineering can be successfully used in mathematical modeling in pharmacokinetics. Frequency response functions are complex functions, therefore modeling is performed in the complex domain. In addition, the modeling methods used to develop model frequency response functions are computationally intensive, and modeling require as a minimum partial knowledge of the theory of dynamic system, and an abstract way of thinking about the dynamic system under study.
The principle difference between pharmacokinetic modeling methods traditionally used in pharmacokinetic studies and modeling methods that use mathematical and computational tools from the theory of dynamic systems, is as follows: the former methods are based on modeling plasma (or blood) concentrationtime profiles of drugs, however the latter methods are based on modeling dynamic relationships between a mathematically represented drug inputs into the body and mathematically represent resulting plasma (or blood) concentration-time profiles of administered drugs. See e.g. the previous studies authored and/or coauthored by the author of the current study and the explanatory example, available free of cost at the author’s Web page: http://www.uef.sav.sk/advanced.htm.
The computational and modeling methods that use computational and modeling tools from the theory of dynamic systems that can be example for adjustment of drug administration aimed at achieving and then maintaining required drug concentration–time profiles in patients see e.g. the following study [6]. The methods considered here can be used for safe and cost-effective individualization of drug dosing e.g. through computer-controlled infusion pumps. This is important example for administration of clotting factors to hemophilia patients, as exemplified in the study [6].
The advantages of the model and modeling method used in the recent study are evident here: The models developed and used overcome one of the typical limitations of compartmental models: For the development and use of the models considered here, an assumption of well-mixed spaces in the body is not necessary. The basic structures of the models developed are used largely applicable to mathematical modeling different dynamic systems in the field of pharmacokinetics and in many other scientific as well as practical fields. From a point of view pharmacokinetic community, is an advantage of the models developed using computational tools from the theory of dynamic systems is that the models considered here emphasize dynamic relationships between drug inputs into the body and behavior of a drug in a human and/or an animal body. The method used in the current study can be easily generalized. Therefore, there is no problem to use the method considered here in several scientific and practical fields. Transfer functions of dynamic systems are not unknown in pharmacokinetics; see e.g. the following studies [20-22].
Conclusion
The models developed and used in the current study are
successfully described the pharmacokinetic behavior of oral
administration to healthy male adult volunteers. The modeling
method used is universal, comprehensive and flexible and thus
it can be applied to a broad range of dynamic systems in the
field of pharmacokinetics and in many other fields. The current
study repeatedly presented an attempt to visualize the successful
use of mathematical and computational tools from the theory
of dynamic systems in pharmacokinetic modeling. For the
previous attempts with the use of the modeling method used in
the current study please visit http://www.uef.sav.sk/advanced.
htm. The current study repeatedly showed that an integration
of pharmacokinetic and bioengineering approaches is a good and
efficient way to study processes in pharmacokinetics, because for
this is that such integration combines mathematical rigor with
biological insight.
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