|Cefamandole; Intravenous administration; Mathematical
|Cefamandole, also known as cephamandole is a second generation
broad-spectrum cephalosporin antibiotic. It is classified as a second
generation of cephalosporin, is very effective against certain Gramnegative
and Gram-positive microorganisms . It is well suited for
the use in situations which require rapid achievement of sufficient
drug concentrations in blood. In the study , pharmacokinetics of
cefamandole (administered intravenously over a period of 10 min to
six normal healthy adult male volunteers was investigated.
|The current study is a companion piece of a related Aziz study
published in June issue 1978 of Journal of pharmacokinetics and
Biopharmaceutics. The study cited here described an investigation
of the pharmacokinetic behavior of cephamandole, intravenously
administered, at a dose of 15 mg/kg over a period of 10 min to six
normal healthy adults, healthy adult male volunteers.
|The main goal of the current study was to present a further example
which illustrated a successful use of an advanced mathematical
modeling method based on the theory of dynamic systems in
mathematical modeling in pharmacokinetics [2-12]. An additional
goal was to motivate researchers in pharmacokinetics to use of an
alternative modeling method, namely a modeling method based on
the theory of dynamic systems to develop pharmacokinetic models.
Previous examples presenting an advantageous use of the modeling
method used in the current study can be found in the articles available
online, which could be downloaded free of cost from the following
Web pages of the author: https://www.uef.sav.sk/durisova.htm and
|The data published in the study  were used. For modeling purposes, an advanced mathematical modeling method based on
the theory of dynamic systems was employed; see e.g., the studies
cited above. The development of mathematical was performed in the
|In the first step of the method, a pharmacokinetic dynamic
system, here denoted as H, was defined for each volunteer by relating
the Laplace transform of the plasma concentration time profile of
cephamandole, here denoted as C(s), and the Laplace transform of the
intravenous input of cephamandole into the body, here denoted as I(s).
|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, were zero; b) the pharmacokinetic
processes occurring in the body after the intravenous cephamandole
administration; were both linear and time-invariant, c) concentrations
of cephamandole were the same throughout all subsystems of the
pharmacokinetic dynamic system H (where subsystems were integral
parts of the pharmacokinetic dynamic systems H); d) no barriers to the
distribution and/or elimination of cephamandole existed.
|In the second step of the method, the pharmacokinetic dynamic
systems H, were used to mathematically represent static and dynamic
aspects of the pharmacokinetic behavior of cephamandole in volunteers
|In the third step of the method, transfer functions, here denoted as
H(s), of pharmacokinetic dynamic systems H were derived by relating
the Laplace transforms of the mathematical representations of the
plasma concentration-time profiles of cephamandole, here denoted as
C(s), and the Laplace transforms of the mathematical representations
of the intravenous administration of cephamandole, here denoted as
I(s), (the lower case letter “S” denotes the complex Laplace variable), see
e.g., the following studies [3-12], references therein, and the following
|Thereafter, the pharmacokinetic dynamic system of each volunteer
was described with transfer function, here denoted as H(s) [3-12].
|For modeling purposes, the software named CTDB  and the
transfer function model HM(s), described by the following equation
|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 the gain of the dynamic system, a1…an,b1 …bm
are the additional model parameters, n is the highest degree of the
nominator polynomial, and m is the highest degree of the denominator
polynomial, where n<m [3-16].
|In the fourth step of the method, the transfer function H(s), was
converted to equivalent frequency response function, here denoted as
|In the fifth step of the method, the non-iterative method published
previously  was used to determine a mathematical model of the
frequency response function of each volunteer and point
estimates of parameters of the model frequency response function
in the complex domain. The model of the frequency response
function used in the current study is described by the following
|Analogously as in Eq. (2), n is the highest degree of the numerator
polynomial of the model frequency response function , m is the
highest degree of the denominator polynomial of the model frequency
response function , n≤m,i is the imaginary unit, and w is the
angular frequency in Equation 3. In the fifth step of the method, each
the model frequency response function , was refined using the
Monte-Carlo and the Gauss-Newton method in the time domain.
|In the sixth step of the method, the Akaike information criterion
 was used to select best models of frequency response functions
, of different complexity with the minimum value of the Akaike
information criterion. In the final step of the method, 95% confidence
intervals for parameters of the final models were determined.
|After the development of mathematical models of the
pharmacokinetic dynamic systems H, the following primary
pharmacokinetic variables were determined: The time of occurrence
of the maximum observed plasma concentration of cefamandole,
here denoted as tmax the maximum observed plasma concentration
of cefamandole, here denoted as Cmax, the elimination half-time
of cefamandole, here denoted as t1/2 the area under the plasma
concentration-time profile of cefamandole from time zero to infinity,
here denoted as, AUC0-∞, and total body clearance of cefamandole, here
denoted as CI
|The transfer function model, here denoted as HM(s) and the
frequency response function model have been implemented in
the computer program CTDB (8). A demo version of the computer
program CTDB is available at the following Web site: https://www.uef.
|The best-fit third-order model of selected with the Akaike
information criterion is described by Eq. (4):
|This model provided an adequate fit to the cefamandole
concentration data in all volunteers investigated in the previous (1) and
the current study. Estimates of the model parameters a0,a1,b1,b2,b3 are
listed in Table 1. Model-based estimates of primary pharmacokinetic
variables of cefamandole are given in Tables 2 and 3.
|Volunteer No. 1 was arbitrarily chosen from six volunteers
investigated in the previous study  and in the current study, to
illustrate the results obtained. Figure 1 illustrates the observed plasma
concentration time profile of cefamandole and the description of the
observed profile with the developed model of the pharmacokinetic
dynamic system, defined for volunteer No. 1. Analogous results also
hold for all volunteers investigated in the previous  and the current
|The first peak of the plasma concentration of cefamandole was
achieved immediately after the end of cefamandole infusion and
the second peak of the plasma cefamandole concentration was
approximately achieved at 45 min after the start of the cefamandole
infusion (Figure 1).
|The pharmacokinetic dynamic systems used in the current study are
mathematical objects, without any physiological relevance. They were
used to model some static and dynamic aspects of the pharmacokinetic
behavior of cefamandole [13-15] in the healthy male volunteers
investigated in the previous  and current study. The method used in
the current study has been described in detail in the previous studies
[3-13], authored or co-authored by the author of the current study.
|As in previous studies, the development of mathematical models of
the pharmacokinetic dynamic systems was based on known inputs and
outputs of the pharmacokinetic dynamic systems, in the current study.
In general, if a dynamic system is modeled using a transfer function
model; as it was the case in the current study (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
|The parameter gain is also called gain coefficient, or gain factor.
Generally, parameters gains are defined as relationships between
magnitudes of outputs of dynamic systems to magnitudes of inputs
to the dynamic systems at steady state. Or in other words, the
parameter gains of dynamic systems are proportional values that show
relationships between magnitudes of outputs to magnitudes of inputs
of dynamic systems at the steady state. The pharmacokinetic meaning
of parameters gains depend on the nature of investigated dynamic systems;
see e.g., studies available at: https://www.uef.sav.sk/advanced.htm.
|The non-iterative method published in the study  and used in
the current study provides quick identification of optimal structures
of models 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 can be explained
as follows: the variable:“s” in the transfer function model HM(s) is a
complex Laplace variable (Equation 2), while the angular frequency
w in (Equation 4), is a real variable, this proved to be suitable for
|The linear mathematical models developed in the current
study sufficiently approximated static and dynamic aspects of
the pharmacokinetic behavior of cefamandole in the volunteers
investigated in the previous  and the current study.
|The current study showed again 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. The modeling methods used to develop model
frequency response functions are computationally intensive, for
accurate modeling they require at least a partial knowledge of the theory
of dynamic system, and an abstract way of thinking about 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, is as
follows: the former methods are based on mathematical modeling of
plasma (or blood) concentration-time profiles of administered drugs,
however the latter methods are based on mathematical modeling of
dynamic relationships between a mathematically represented drug
administration and a mathematically represented resulting plasma (or
blood) concentration-time profile of the drug administered. See e.g.,
the studies and an explanatory example available at https://www.uef.sav.
|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 see e.g., the following study .
Moreover, the methods considered here can be used for safe and costeffective
individualization of drug (or a substance) dosing of a drug
and/or a substance, for example using computer-controlled infusion
pumps. This is very important for administration of a clotting factor to
a hemophilia patient, as exemplified in the simulation study .
|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 developed using
computational and modeling tools from the theory of dynamic systems
are broadly applicable to develop mathematical models not only in the
field of pharmacokinetics but also in several other scientific as well as
practical fields. From a point of view of 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 a drug in a human and/or an animal body. Transfer functions of
dynamic systems are not unknown in pharmacokinetics; see e.g., the
following studies [19-21]. In pharmacokinetics, transfer functions are
usually called disposition functions [22,23].
|The models developed and used in the current study successfully
described the pharmacokinetic behavior of cefamandole after its
intravenous administration over a period of 10 minutes to healthy
male adult volunteers. The modeling method used in the current study
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 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 the successful use of the
modeling method employed in the current study please visit the
author’s Web page, English version: https://www.uef.sav.sk/advanced.
htm. The current study showed that an integration of pharmacokinetic
and bioengineering approaches is a good and efficient way to study
processes in pharmacokinetics, because such integration combines
mathematical rigor with biological insight.
Tables at a glance
Figures at a glance
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