Simulation of the influence of N2O on the chemical stage of water
radiolysis
JIŘÍ BARILLA, PAVEL SIMR, KVĚTUŠE SÝKOROVÁ
Faculty of Science, Department of Informatics,
J. E. Purkyne University in Usti nad Labem
Pasteurova 3544/1, 400 96 Usti nad Labem
CZECH REPUBLIC
Abstract: - The absorption of ionizing radiation causes the radiolysis of water to form aggressive
radicals. Water radiolysis plays an essential role in radiotherapy, radio sterilization, food irradiation,
and wastewater irradiation because living cells consist mainly of water. Radical clusters arise
immediately after irradiating water with ionizing radiation, and aggressive radicals damage living
cells. These damages are caused mainly by SSB and DSB formation on DNA molecules. The
mathematical simulation model, created with the help of Continuous Petri nets, is very suitable to
study the dynamics of the chemical stage of water radiolysis. This mathematical simulation model,
which includes the influence of oxygen on the chemical stage of radiobiological mechanism, was
created in our previous work. This paper is extended to include the influence of N2O. The presence of
N2O during irradiation of water plays a vital role because it increases OH radicals, which are mainly
responsible for DNA damage. The mathematical model enables us to simulate the dynamics of the
chemical reactions and the diffusion of radical clusters during chemical stage of water radiolysis.
Key-Words: - Continuous Petri nets, water radiolysis, the influence of N2O, radical clusters, simulation
model
Received: April 14, 2021. Revised: January 23, 2022. Accepted: February 24, 2022. Published: April 2, 2022.
1 Introduction
The radiolysis of water is caused by the absorption of
ionizing radiation in water medium to form
aggressive radicals
hydrated
electrons
These radicals diffuse
into surroundings and can damage the DNA molecule
if their concentration is sufficient when they meet the
DNA. Mainly OH radicals are responsible for the
formation of SSB (Single-Strand Breaks) and DSB
(Double-Strand Breaks) on DNA molecules. The
damages of DNA molecules caused by aggressive
radicals run mainly through indirect effects when
energy is absorbed outside the DNA molecule. The
indirect effect occurs when using low-LET radiation.
After the energy is transferred to the water, a radical
cluster is formed, and radicals inside the cluster
diffuse into surroundings and react mutually and with
other molecules present in water. The concentration
of radicals and, therefore, their radiobiological effect
is influenced by the presence of other substances
which can have radioprotective or radiosensitive
character. Radioprotective and radiosensitive
substances are used mainly in radiotherapy to protect
healthy cells against ionizing radiation or sensitize
cancer cells to obtain a greater radiobiological effect.
To study the dynamics of the chemical stage of water
radiolysis, the mathematical models created with the
help of Continuous Petri nets are very suitable. The
main reason is that Petri nets allow us to solve easily
complicated systems such as chemical reactions of
the radicals, ions, and other products of water
radiolysis together with diffusion of the radical
clusters which run simultaneously. Petri nets enable
us to use graphical tools to create mathematical
models, taking much less time than classical
programming. Also, the testing and the changes are
faster, too.
At the beginning of our research, we used the
programming language to create the mathematical
models describing the chemical stage of water
radiolysis [3][5]. Then, to quickly solve more
detailed and complicated mathematical models, we
further used the Continuous Petri nets in the research
[4][6][7][8][9]. We compared these simulated
mathematical models with the experimental data and
achieved good consent.
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The tested mathematical model has been used to
describe the influence of oxygen on the chemical
stage of the radiobiological mechanism in a paper [8].
The indirect effect of the radiobiological mechanism
is caused by aggressive radicals formed during the
radiolysis of water. The influence of oxygen on the
chemical stage of the radiobiological mechanism
plays an important role mainly in radiotherapy
because the oxygen concentration is higher in healthy
cells than in cancer cells. Therefore, radiation
sensitivity is different for healthy cells and cancer
cells. The dependence of the concentrations of
individual radicals on time under various oxygen
concentrations is showed to explain the oxygen
effect. The obtained results were compared with
experimental data, and a good agreement was
achieved.
In this paper, the influence of N2O on the chemical
stage of water radiolysis was analyzed with the help
of Continuous Petri nets. The time dependencies of
the concentrations of individual radicals under
various N2O concentrations were simulated to show
the impact of N2O molecules on radical
concentrations, mainly increasing concentration of
OH radicals, which results in more significant
damage to the DNA molecule. To create our
mathematical model, using Continuous Petri nets,
primary products of water radiolysis
(
), and associated products (,
, ,
,
, ) were taken into account.
Their initial yields and physicochemical constants
have been taken from the literature dealing with the
radiolysis of water, for example
[10][12][13][14][18][19][20][21][26][27]. As in
previous publications, the corresponding radical
clusters (at low-LET radiation) were described as
spherically symmetric systems [21].
2 Mathematical model of the influence
of N2O on the chemical stage of water
radiolysis
The basic mathematical model of the chemical stage
of water radiolysis was created in our previous papers
[6][7][8]. The mathematical model in this paper is
extended to include the influence of N2O on the
chemical stage of water radiolysis. Therefore, we
will not repeat the derivation of the mathematical
model, and only the short overview will be done.
The basic assumptions of the mathematical model
are:
Water radicals arise inside the diffusing
radical cluster, containing a nonhomogeneous
concentration of individual species [22].
Individual radicals diffuse differently,
according to their diffusion coefficients, must
also be included in the mathematical model.
Diffusion coefficients are taken from the
literature.
Immediately after cluster formation,
simultaneously with the diffusion of radicals
also the chemical reactions of radicals and
other species present in the cluster run. Their
rate constants are taken from the literature,
too.
In our mathematical model, we assume a
spherical symmetry of the diffusion clusters
[22], and initial conditions are given by the
volume of the cluster and by the
numbers of corresponding radicals.
All these conditions are described in detail in our
previous papers, where the general mathematical
model has been derived, too. In this paper, only the
final mathematical model, simulating the influence of
N2O, will be presented. The dynamics of the cluster
evolution can be described by a system of ordinary
differential equations, which solve the diffusion of
radicals and their chemical reactions.
To solve the system of ordinary differential
equations, the Continuous Petri nets have been used.
Petri nets enable us to easily create the mathematical
model with the help of graphical tools as well as to
perform the rapid simulations, which allows us to
optimize the parameters of the simulation model
[16][17][24][25]. The detailed description of the
solution of our mathematical model using Petri nets
can be found in [6][7][8], where Petri nets are also
sufficiently described.
The radicals, ions, and other substances arising inside
the cluster, and diffuse into the surroundings, are
presented in Table I, where we can also see their
diffusion coefficients and designation. All diffusion
coefficients are taken from the literature. We will
assume that at the beginning of the cluster diffusion
(at time t0), when the chemical reactions start, only
the following species will be present: , ,
,
and . In Table II, we can see chemical
reactions running after cluster formation, which is
considered in our mathematical model. The rate
constants of the chemical reactions presented in
Table II are also taken from the literature.
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Table 1. Diffusion coefficients [19]
Substance
Diffusion coefficient
(nm2.ns-1)
Species amount
Designation of diff. coefficients
1.
7.0
2.
2.2
3.
4.9
4.
9.5
5.
5.3
6.
5
7.
2.2
8.
1.8
9.
2.3
Table 2. Recombination reactions [14]
Reaction
Rate constants
(dm3.mole-1.s-1)
1.
10 × 1010
2.
2.5 × 1010
3.
6 × 109
4.
3 × 1010
5.
2.4 × 1010
6.
4 × 109
7.
2.3 × 1010
8.
1 × 1010
9.
1 × 1010
10.
2 × 106
11.
3 × 1010
12.
1 × 1011
13.
1 × 108
14.
1.2 × 1010
15.
5 × 107
16.
6 × 107
17.
1 × 106
18.
1.9 × 1010
19.
1 × 1010
20.
9.1× 109
Compared to our previous works, mainly [8], the
chemical reaction 20 (see Table II) has been added.
This chemical reaction expresses the influence of
N2O on the chemical stage of water radiolysis, such
that the number of OH radicals is increased, which
results in more significant damage to the DNA
molecule.
Using the diffusion coefficients from Table I and the
chemical reactions from Table II, we can describe the
whole dynamics process of the chemical stage of
water radiolysis, where spherical symmetry of the
cluster evolution is assumed, by the following system
of ordinary differential equations:
(1)
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+
(2)
-
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
where , , , , , , ,
and represent numbers of
species , , , , , , ,
, ,
O2 and N2O, which are placed in corresponding
, , , , , , ,
and . k1, k2, …, k20 are rate constants of the
chemical reactions from Table II.
Immediately after cluster formation, radicals
diffuse into surroundings and volumes VH, ,
(12)
, ,, , ,
and are
increased according to the system of ordinary
differential equations:
(13)
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(14)
(15)
(16)
(17)
(18)
(19)
(20)
where , , , , , , ,
and are diffusion coefficients from Table I.
At the beginning of this chemical process, only
species , ,
, , and are present in
the radical cluster. Their initial yield is their initial
values for the radical correspondent cluster. The
initial volume is the same for all species and depends
on the energy transferred to the radical cluster.
3 Using Continuous Petri nets to solve
the mathematical model
Continuous Petri nets have been sufficiently
described in our previous papers [6][7][8], and here
only final results will be presented. The system
Visual object net ++ [23] has been used to solve the
given system of ordinary differential equations. This
visual system allows us to create the mathematical
model easily and quickly analyze various situations.
The graphical representation of the mathematical
simulation model is in Figure 1.
The circles in Figure 1 represent places, and
rectangles represent transitions. Each place can be
changed via transition connected to it by an arrow.
The connection can be performed only between a
place and a transition, but never between places or
between transitions. The places represent the system's
state, and each place is marked by an actual number
which determines the value of the monitored
parameter.
The places at the top of Figure 1 are not joined with
any transitions and represent constants from Table I
and Table II (diffusion coefficients and rate
coefficients of the chemical reactions).
The values of the cluster volumes are represented by
the places on the left of Figure 1 and are changed by
the transitions which are joined with them, and are
designated as VH, VOH, Ve, VH3O, VOHM,
VH2VH2O2, VOHM, and VHO2, representing the
volumes VH, , , ,, , ,
and . At the beginning of the chemical process,
all these places have the same value corresponds to
the initial volume of the radical cluster, and then the
individual volumes containing the corresponding
species are increased according to the relevant
diffusion coefficients.
The central part of the mathematical simulation
model is placed in the middle of Figure 1. The places
represent the number of individual species that
participate in chemical reactions and are designated
as H, OH, e, H3O, OHM, H2, H2O2, O2M, HO2, O2
and N2O, representing species , , , ,
, , ,
, , O2, and N2O. The value
of each place is changed via transitions connected
with them. The places and transitions are connected
to create a mathematical model that simulates the
dynamics of the chemical stage of water radiolysis
using the time evolution of the cluster.
The transitions in Figure 1 that cause changes of the
chemical species have transition functions in the
form:
(21)
(22)
(23)
(24)
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(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
The volumes on the left in Figure 1 are changed
according to the transition functions:
(41)
(42)
(43)
(44)
(45)
(46)
(47)
(48)
(49)
To use the mathematical simulation model presented
in Figure 1 for concrete radical cluster, it is necessary
to enter the initial values of the individual places,
which represent constants from Table I and Table II,
the initial number of chemical species, and the initial
value of cluster volume.
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Figure 1. Processes running in radical clusters, represented with the help of Petri nets.
4 Application of the simulation model
on the influence of N2O on the time
evolution of cluster radicals
The mathematical simulation model, created
with the help of Continuous Petri nets, enables us
to simultaneously solve the time evolution of the
cluster diffusion with the chemical reactions
running inside the cluster. As has been
mentioned, the main significance of our
mathematical simulation model is the possibility
to study not only the influence of various
substances on the chemical stage of water
radiolysis, but also to study the influence of
radioprotective and radiosensitive substances on
DNA damage (mainly SSB and DSB formation),
which can be very useful not only in
radiotherapy, but also in the other fields where
living cells are irradiated by ionizing radiation.
The initial size of the radical cluster depends on
the amount of transferred energy. Based on our
earlier analysis [3][5] of the experimental data
presented by Blok and Loman [11], using the
optimization procedure, it has been derived that
the average initial size of the radical clusters
efficient in DNA damage (SSB and DSB
formation) corresponds to energy 300 eV and its
volume diameter is circa 27 nm.
Immediately at the end of the physicochemical
stage (after radical cluster formation), only
species , ,
, and are present
inside the cluster. Their initial numbers , ,
VH2
H
PI
PI
DH
DH
DOH
DOH
De
De
K1
K1
K2
K2
K3
K3
VOH
OH
Ve
e
K4
K4
K5
K5
K6
K6
VH3O
DH3O
DH3O
K7
K7
H3O
K8
K8
K9
K9
K10
K10
K11
K11
K12
K12
K13
K13
K14
K14
DHO2
DHO2
DO2M
DO2M
DOHM
DOHM
VHO2
VO2M
VOHM
HO2
O2M
OHM
O2
VO2
VO2
K15
K15
K16
K16
K17
K17
K18
K18
K19
K19
DH2
DH2
DH2O2
DH2O2
VH2O2
H2
H2O2
VH
N2O
VN2O
VN2O
K20
K20
T(H+H)
TVH2
TVOH
T(OH+OH)
TVe
T(e+e)
T(H+OH)
T(H+e)
T(OH+e)
TVH3O
T(e+H3O)
TVHO2
TVO2M
TVOHM
T(H+HO2)
T(H+O2)
T(OH+HO2)
T(e+O2)
T(HO2+HO2)
T(H3O+O2M)
T(H3O+OHM)
TVH2O2
TVH T(H+H2O2)
T(OH+H2O2)
T(OH+H2)
T(e+H2O2)
T(HO2)
T(H+H)
T(e+e)
T(OH+OH)
T(HO2+HO2)
T(e+N2O)
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, , and can be characterized by
radical yield values under anoxic conditions
[14][18][19][20][22]. For transferred energy 300
eV [8] it will be put at the time t = 0
, , =14.34,
, ,
while the initial numbers of other species will be
equal to zero.
The values of diffusion coefficients , , ,
, , , ,
, and of rate
constants are presented in Table I and
Table II and are taken from the literature.
A good agreement with experimental data has
been achieved to verify our mathematical
simulation model under the initial conditions
introduced above (cluster diameter 27 nm and
energy 300 eV and others) (see Table III).
Table 3. Comparison of the calculated value with experimental results
Substance
Initial yield
(G0)
Experimental
yield (G)
Petri nets
(G)
1.
0.42
0.62
0.620
2.
5.5
2.8
2.804
3.
4.78
2.8
2.808
4.
4.78
2.8
2.802
5.
0.15
0.47
0.473
6.
0
0.73
0.733
To study the influence of N2O on the chemical stage
of water radiolysis, and mainly the influence of N2O
on the radiobiological mechanism, the analysis of the
time evolution of the radical cluster is instrumental
because the prominent role is played by the
concentration of radicals at the moment of DNA
collision with the radical cluster. The concentration
of radicals depends on the distance of DNA molecule
from the cluster when radicals begin diffuse into
surroundings, and their concentration decreases. The
near is DNA to radical cluster, the higher probability
of DNA damage is. The presence of N2O during
irradiation increases the concentration of OH
radicals. Therefore, the analyses of the dependences
of OH radical concentration on the concentration of
N2O are fundamental. As we will see in the following
analyses, a higher concentration of N2O results in a
higher concentration of OH radicals and results in a
more significant damage of DNA molecules.
In the following part, we will show the time
dependencies of individual radical concentrations
under various N2O concentrations, which is very
important to analyze the effect of ionizing radiation
on living cells. All analyses will be done for cluster
diameter 27 nm and energy 300 eV.
In Figure 2, we will start with the time-dependent
radical concentrations in saturated solution
without the presence of N2O.
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Figure 2. Concentrations depending on the time in saturated solution without the presence of N2O.
As shown in Figure 2, the concentration of OH
radicals is higher than that of other radicals and
decreases slower. As a result, OH radicals have the
most excellent effect on DNA damage, according to
generally accepted opinion. The concentration of H
radicals is deficient and can hardly cause DNA
damage. At time t = 0, the concentration of hydrated
electrons
is nearly as high as the concentration of
OH radicals but immediately drops sharply with time
and also can hardly cause damage to DNA molecules.
At the beginning of the chemical stage, the
concentration of HO2 radicals increases according to
reaction 19 (see Table II), and then their
concentration decreases by the chemical reactions
with other species and their diffusion into
surroundings. Nevertheless, their concentration is
much lower than the concentration of OH radicals,
and therefore HO2 radicals have a much smaller
radiobiological effect than OH radicals.
Figure 3. Time dependence of cluster diameters for radicals H, OH, and hydrated electrons
.
0
20
40
60
80
100
120
140
010 20 30 40 50 60 70 80 90 100
diameter [nm]
time [ns]
H
OH
e
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In Figure 3, one can see the time changes of cluster
diameters for radicals H, OH, and hydrated electrons
. The time dependence of the HO2 radicals cluster
diameter has been omitted since that OH, and HO2
radicals have similar time dependence because they
have nearly the same diffusion coefficients ( =
2.2 and = 2.3). The diameter of the OH radicals
cluster increases slower than the cluster diameter of
other radicals, and its result is such that a decrease of
concentration levels of OH radicals is also slowed
down.
Comparing Figures 2 and 3, it is possible to find out
radical concentration and corresponding cluster size
at any time. Only radicals that have sufficient
concentration can cause damage to the DNA
molecule. We can estimate the distance from the
cluster center to the DNA molecule when the
concentration of radicals is still sufficient to cause
damage to the DNA molecule. It is a fact that the
smaller the cluster, the higher the concentration of
radicals and vice versa.
Figure 2 shows the time-dependent radical
concentrations in saturated O2 solution without the
presence of N2O. In the following figures, the time-
dependent radical concentrations at different N2O
concentrations will be presented. Comparing Figures
2, 4, – 7, one can see that N2O influences the OH
concentration the most. The higher the N2O
concentration, the higher OH concentration (see
Figures 2, 4 – 7, and 9). In a saturated N2O solution,
the concentration of the OH radicals is nearly twice
as high as in the solution without N2O (see Figures 2
and 7). As can be predicted from the previous
statements, a higher concentration of OH radicals
should cause a larger number of SSB and DSB on a
DNA molecule.
To validate this fact, we can use experimental data
presented by Blok and Loman [11], and used in our
previous paper [3]. For example, from experimental
data in Figure 1 and Table III [3], one can find out
that without N2O, the number of DSB is 0.041 per
DNA molecule, and at saturated N2O solution, the
number of DSB is 0.06 per DNA molecule which is
nearly twice. Thus, the experimental data are in good
agreement with the results obtained using our
simulation model.
With increasing N2O concentration, the concentration
of hydrated electrons
slightly decreases (see
Figures 2, 4 – 7, and 10), which means that hydrated
electrons
can hardly cause DNA damage.
The same applies to radicals H, the concentration is
minimal from the beginning (see Figures 2, 4 – 7, and
8), and they can hardly cause DNA damage.
The concentration of HO2 radicals also decreases
with the increasing concentration of N2O (see Figures
2, 4 – 7, and 11) and is much lower than the
concentration of OH radicals.
Figure 4. Concentrations depending on the time in saturated solution at N2O concentration
1.660 mmol.dm-3.
0
0,5
1
1,5
2
2,5
3
010 20 30 40 50
concentration [mmol/dm3]
time [ns]
H
OH
e
HO2
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Figure 5. Concentrations depending on the time in saturated solution at N2O concentration
8.303 mmol.dm-3.
Figure 6. Concentrations depending on the time in saturated solution at N2O concentration
16.605 mmol.dm-3.
0
0,5
1
1,5
2
2,5
3
3,5
010 20 30 40 50
concentration [mmol/dm3]
time [ns]
H
OH
e
HO2
0
0,5
1
1,5
2
2,5
3
3,5
4
010 20 30 40 50
concentration [mmol/dm3]
time [ns]
H
OH
e
HO2
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Figure 7. Concentrations depending on the time in saturated solution at N2O concentration
24.161 mmol.dm-3.
Figure 8. Concentrations depending on the time of H radicals in saturated solution at various N2O
concentrations [mmol.dm-3].
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
010 20 30 40 50
concentration [mmol/dm3]
time [ns]
H
OH
e
HO2
0
0,05
0,1
0,15
0,2
0,25
010 20 30 40 50
concentration [mmol/dm3]
time [ns]
1.660
8.303
16.605
24.161
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Figure 9. Concentrations depending on the time of OH radicals in saturated solution at various N2O
concentrations [mmol.dm-3].
Figure 10. Concentrations depending on the time of hydrated electrons
in saturated solution at
various N2O concentrations [mmol.dm-3].
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
010 20 30 40 50
concentration [mmol/dm3]
time [ns]
1.660
8.303
16.605
24.161
-0,5
0
0,5
1
1,5
2
2,5
010 20 30 40 50
concentration [mmol/dm3]
time [ns]
1.660
8.303
16.605
24.161
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Figure 11. Concentrations depending on the time of HO2 radicals in saturated solution at various N2O
concentrations [mmol.dm-3].
The analysis performed based on our simulation
model supports our previous results, and it is
following the known general opinion that OH
radicals determine the decisive influence on DNA
damage. All these characteristics can be explained
based on of reactions considered and summarized in
Table II. The chemical reaction of hydrated electrons
with N2O results in the form of OH radicals
which significantly contributes to DNA damage. The
severe DNA molecule damage is represented by a
DSB that is always formed by a pair of radicals from
one cluster that meets the corresponding DNA
molecule. DSB being formed only when two SSB are
formed in different strands.
As the clusters and DNA molecules are distributed
randomly in the corresponding space, they meet at
different times due to thermal motion and cluster
diffusion. The probability of DSB formation can be
then given as
(50)
where pS can be expressed approximately as
Parameters αj represent the efficiency of individual
radicals, that take part in forming individual SSB in
dependence on the applied dose, and
corresponds to the concentration of radicals j; DSB is
formed only when two SSB are formed in different
strands. In the given formula, it has been assumed
(for simplicity) that DNA molecules and diffusing
clusters have come into mutual contact with the same
frequency in different time instants.
The mathematical model of the chemical stage of
water radiolysis based on using Continuous Petri nets
brings new possibilities in studying regularities in
radiobiological processes in dependence on other
species present in the corresponding medium.
5 Conclusion
As shown in this work, the presence of N2O during
irradiation of water by ionizing radiation significantly
influences radiobiological effect on living cells,
mainly if low-LET radiation is used [1][2].
Furthermore, the presence of N2O molecules in water
during irradiation causes the increase of OH radical
concentration resulting in more significant damage on
DNA molecules because OH radicals have decisive
influence on SSB and DSB formation. This fact can
be used in radiotherapy, radiosterilization, food
irradiation, and wastewater irradiation.
An indirect effect of ionizing radiation is assumed in
our work, where aggressive radicals cause SSB and
DSB on DNA molecules. The DNA is damaged
during the chemical stage of water radiolysis when
energy is transferred to the radical cluster, and the
concentration of radicals decreases by their chemical
reactions and their diffusion into surroundings. To
analyze the effect of ionizing radiation on DNA
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0,5
010 20 30 40 50
concentration [mmol/dm3]
time [ns]
1.660
8.303
16.605
24.161
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2022.19.7
Jiří Barilla, Pavel Simr, Květuše Sýkorová
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damage, the mathematical simulation model is very
suitable. However, this mathematical model is very
complicated because it includes the diffusion of
radicals and their chemical reactions simultaneously.
Furthermore, spherical symmetry is assumed for the
time evolution of the radical cluster.
Continuous Petri nets were used to solve this
complicated mathematical model, because they
provide us with the graphical tools to create the
mathematical model easily. Creating the
mathematical simulation model with the help of
Continuous Petri nets is much faster than by classical
programming. We are able to perform a great deal of
analysis of the studied system and quickly adapt the
model to analyze another similar system. Especially
research into the influence of radioprotective and
radiosensitive substances on the irradiation of living
cells with ionizing radiation using mathematical
simulation models is very vital.
In our further research, the damage of living cells by
ionizing radiation will be simulated with the help of
Colored Petri nets because they provide us with
better tools for creating a simulation model and its
analysis. We will aim to simulate the physical,
chemical, and biological stages of the radiobiological
mechanism together with reparation processes in a
cell.
Acknowledgment
This work was supported by the Faculty of science, J.
E. Purkinje University in Usti nad Labem, Czech
Republic. English language correction was performed
by Saliha Afzaal.
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