|Clindamycin; Oral administration; Mathematical model
|Lincomycin and clindamycin are antibiotics often used in clinical
practice [1-6]. Both antibiotics are bacteriostatic and inhibit protein
synthesis in sensitive bacteria . Clindamycin bioavailability from
Clindamycin-2-Palmitate and Cindamimycin-2 Hexadecylcarbonate
in man was determined in the study by Forist et al. . The current
study is a companion piece of the related Forist et al. study published
in the February 1973 issue of Journal of Pharmacokinetics and
Biopharmaceutics. The goal of the current study was to provide a
further example of a successful use of a non-traditional method in
the development of mathematical models in pharmacokinetics [7-17]. Previous examples showing an advantageous use of the modeling
method used in the current study can be found in the articles available
online. The articles considered here can be downloaded free of cost
from the following Web pages of the author: http://www.uef.sav.sk/
durisova.htm and http://www.uef.sav.sk/advanced.htm.
|As stated above, an advanced mathematical modeling method
based on the theory of dynamic systems was employed to develop a
mathematical model of the pharmacokinetic behavior of clindamycin
in healthy male volunteers enrolled in the study by Forist et al. , and
in the current study. The development of the mathematical model of the
pharmacokinetic behavior of clindamycin after the oral administration
of 150 mg of clindaminycin to each volunteer was performed in the
|In the first step of the model development process, a pharmacokinetic
dynamic system, here denoted as H, was defined for each volunteer
using the Laplace transform of the mathematically described plasma
concentration time profile of clindamycin of the volunteer, here
denoted as C(s), and the Laplace transform of the mathematically
described oral administration of 150 mg of clindamycin (here called:
input) to the volunteer.
|The following simplifying assumptions were made prior the model
development process: a) initial conditions of all pharmacokinetic
dynamic systems H were zero; b) pharmacokinetic processes occurring in the body after the oral administration of clindamycin were
asymptotically time-invariant linear; c) concentrations of clindamycin
were the same throughout all subsystems of the pharmacokinetic
dynamic system H (where each subsystem was an integral part of a
pharmacokinetic dynamic system H); d) no barriers to the distribution
and/or elimination of clindamycin existed.
|In the second step of the model development process, the
pharmacokinetic dynamic systems H, were used to mathematically
represent static and dynamic aspects [18-20] of the pharmacokinetic
behavior of clindamycin in the volunteers enrolled in the study  and
in the current study.
|In the third step of the model development process, the transfer
function, here denoted as H(s) of each pharmacokinetic dynamic
system H was derived using the Laplace transform of the mathematically
described the plasma concentration-time profile of clindamycin, here
denoted as C(s), and the Laplace transform of the mathematically
described the oral input of clindamycin to the body, here denoted as
I(s), see Eq. (1), (the lower case letter “S” denotes the complex Laplace
variable), see e.g., the following studies [8-17] references therein.
|The pharmacokinetic dynamic systems of the volunteers were
described with the transfer functions H(s). For modeling purposes, the
software package CTDB  and the transfer function model HM(s)
described by the following equation were used:
|On the right-hand-side of Eq. (2) is the Padé approximant
 of the model transfer function HM(s), G is an estimator of the
model parameter called a gain of a dynamic system, a1…an, b1 …bm are additional model parameters, and n is the highest degree
of the nominator polynomial, and m is the highest degree of the
denominator polynomial, where n<m [8-18].
|In the fourth step of the model development process, each transfer
function H(s) was converted into an equivalent frequency response
function here denoted as F(iωj) .
|In the fifth step of the model development process, the non-iterative
method published previously  was used to determine mathematical models of frequency response functions F(iωj)of the volunteers
and point estimates of parameters of the model frequency response
functions F(iωj) in the complex domain. The model of the frequency
response function F(iωj) used in the current study is described by the
|Analogously as in Eq. (2), n is the highest degree of the numerator
polynomial of the model frequency response function F(iωj), m is the
highest degree of the denominator polynomial of the model frequency
response function F(iωj), n≤m, i is the imaginary unit, and ω is the
angular frequency in Eq. (3).
|In the fifth step of the model development process, each model
frequency response function F(iωj) was refined using the Monte-Carlo
and the Gauss-Newton method in the time domain.
|In the sixth step of the model development process, the Akaike
information criterion  was used to select the best models of the
frequency response functions F(iωj) with minimum values of the Akaike
information criterion. In the final step of the model development
process, 95% confidence intervals for parameters of the best models
F(iωj) were determined.
|After the development of mathematical models of all
pharmacokinetic dynamic systems H, the following primary
pharmacokinetic variables were determined: the elimination halftime
of clindamycin, here denoted as t1/2 the area under the plasma
concentration-time profile of clindamycin from time zero to infinity,
here denoted as, AUC0-∞, and total body clearance of clindamycin, here
denoted as Cl.
|The transfer function model HM(s) and the frequency response
function model F(iωj) have been implemented in the computer
program CTDB . A demo version of the computer program CTDB
is available at the following Web site: http://www.uef.sav.sk/advanced.
Results and Discussion
|The best-fit third-order models of F(iωj) were selected using the
Akaike information criterion . The general form of the third order
model of F(iωj) is described by the following equation:
|The model described by Eq. (1) was suitable also for the
development of models of the frequency response functions derived
using the clindamycin concentration data of all volunteers enrolled in
the study by Forist et al. , and in the current study. Estimates of the
model parameters a0, a1, b1, b2, b3 are in Table 1. Model-based estimates
of primary pharmacokinetic variables of clindamycin are in Table 2.
|In order to show results obtained, volunteer L.N. was arbitrarily
chosen from the volunteers enrolled in the study by Forist et al. 
and in the current study. Figure 1 illustrated the observed plasma
concentration time profile of clindamycin of volunteer L.N. and the
description of the observed profile with the developed model of the
pharmacokinetic dynamic system, defined for subject L.N. Analogous
results hold for all subjects investigated in the study by Smith and the
|The pharmacokinetic dynamic systems used in the current study
were mathematical objects, without any physiological relevance. They
were used to model static and dynamic aspects of the pharmacokinetic behavior of clindamycin [18-20] in the healthy male subjects enrolled
in the study by Smith and in the current study. The method used in the
current study has been described in detail in the previous studies [8-17], authored or co-authored by the author of the current study.
|As in previous studies [8-17], the development of mathematical
models of pharmacokinetic dynamic systems was based on the
known inputs and outputs of pharmacokinetic dynamic systems, in
the current study. In general, if a dynamic system is modeled using
a transfer function model, as it was done in the current study (see
Equation 2), then the accuracy of the model depends on the degrees of
the polynomials of the transfer function model used to fit the data, see
e.g., the following studies [8-17].
|The parameter gain is also called gain coefficient, or gain factor.
In general, a parameter gain is defined as a relationship between
magnitudes of an output of a dynamic system to a magnitude of an
input into a dynamic system in steady state. Or in other words, a
parameter gain of a dynamic system is a proportional value that shows
a relationship between a magnitude of an output to a magnitude of an
input of a dynamic system in the steady state.
|The pharmacokinetic meaning of a parameter gain depends on the
nature of the dynamic system investigated; see e.g., studies available at:
|The non-iterative method published in the study  and used in
the current study enables quick identification of optimal structures
of model frequency responses. It is a great advantage of this method,
because this significantly speeds up the development of frequency
|The reason for conversion of HM(s) to F(iωj) can be explained as
follows: the variable:“S” in the transfer function model HM(s) described
by Eq. (2) is a complex Laplace variable, while the angular frequency
„ω” in the model F(iωj) described by Eq. (4) is a real variable. Therefore,
the model F(iωj) can be determined in time domain.
|The linear mathematical models developed in the current study
sufficiently approximated static and dynamic aspects [18-20] of the pharmacokinetic behavior of clindamycin in all subjects enrolled in the
study by Smith and in the current study.
|The current study showed again that mathematical and
computational tools from the theory of dynamic systems can be
successfully used in mathematical modeling in pharmacokinetics.
Frequency response functions are complex functions, therefore
modeling is performed in the complex domain. The modeling
methods used to develop model frequency response functions are
computationally intensive, and an accurate modeling requires at least a
partial knowledge of the theory of dynamic system, and an abstract way
of thinking about dynamic systems investigated.
|The principal difference between traditional pharmacokinetic
modeling methods and modeling methods that use of mathematical
and computational tools from the theory of dynamic systems can be
explained as follows: the former methods are based on mathematical
modeling plasma (or blood) concentration-time profiles of drugs
administered, however the latter methods are based on mathematical
modeling dynamic relationships between a mathematically represented
drug inputs to the body and mathematically represented resulting
plasma (or blood) concentration-time profiles of drugs administered.
See e.g., the articles and an explanatory example available at the author’s
Web site http://www.uef.sav.sk/advanced.htm.
|The computational and modeling methods that use computational
and modeling tools from the theory of dynamic systems can be used
for example for adjustment of a drug (or a substance) dosing aimed
at achieving and then maintaining required drug (or a substance)
concentration–time profile in patients as exemplified in the following
study . Moreover, the methods considered here can be used for
safe and cost-effective individualization of dosing of of drugs, or
substances, for example using computer-controlled infusion pumps.
This is very important e.g., for an administration of a clotting factor to a
hemophilia patient, as exemplified in the simulation study cited above.
| The advantages of the model and modeling method used in the
current study are evident here: The models developed overcome one
of the well-known limitations of compartmental models: For the
development and use of the models considered here, an assumption of
well-mixed spaces in the body (in principle unrealistic) is not necessary.
The basic structure of the models is broadly applicable. Therefore, this structure can be used in the development of mathematical models not
only in the field of pharmacokinetics but also in several other scientific
and practical fields. From a point of view of the pharmacokinetic
community, an advantage of the models developed using computational
tools from the theory of dynamic systems is that the models considered
here emphasize dynamical aspects of the pharmacokinetic behavior of
administered drugs in a human or an animal body. Transfer functions
of dynamic systems are not unknown in pharmacokinetics; see e.g., the
following studies [24-26]. In pharmacokinetics, transfer functions are
usually called disposition functions [27,28].
|The models developed and used in the current study successfully
described the pharmacokinetic behavior of clindamycin in the body
after its oral administration to healthy male adult subjects, enrolled in
the study by Forist et al. , and in the current study. The modeling
method used in the current study can be used for mathematical
modeling dynamic systems not only in the field of pharmacokinetics
and in many other scientific or practical fields. The current study again
showed that mathematical and computational tools from the theory
of dynamic systems can be advantageously used in pharmacokinetic
modeling. To see the previous examples illustrating a successful use
of the modeling method employed in the current study please visits
the author’s Web site (an English version): http://www.uef.sav.sk/
advanced.htm. The current study showed that an integration of key
concepts from pharmacokinetic and bioengineering is a good and
efficient way to study dynamic processes in pharmacokinetics, because
such integration combines mathematical rigor with biological insight.
|The author gratefully acknowledges the financial support obtained from the
Slovak Academy of Sciences in Bratislava, Slovak Republic.
|This study is dedicated to the memory of the late Professor Luc Balant
who passed away unexpectedly in December 2013. Internationally, Professor
Luc Balant was widely known for his work in the COST Domain Committee for
Biomedicine and Molecular Biosciences and in the COST Action B15: “Modeling
During Drug Development”.
|The motto of this study is: “The undergoing physical laws necessary for the
mathematical theory of a large part of physics and of the whole chemistry are thus
completely known, and difficulty is only that the exact application of these laws
lead to equations much more complicated to be soluble”. (One of the outstanding
theoretical physicists P. A. M. Dirac (1902-1984)).
Tables at a glance
Figures at a glance
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