CC BY
28
Известия Саратовского университета. Новая серия. Серия: Математика. Механика. Информатика. 2021. Т. 21, вып. 2. С. 259-266
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 2, pp. 259-266
Article
https://doi.org/10.18500/1816-9791-2021-21-2-259-266
Mathematical and computer simulation of the electrophysical properties of a multicellular structure exposed to nanosecond
electrical pulses
R. P. Kim10, S. A. Korchagin2
1 Yuri Gagarin State Technical University of Saratov, 77 Polytechnicheskaya St., Saratov 410054, Russia 2Financial University under the Government of the Russian Federation, 49 Leningradsky Prospekt, Moscow 125993, Russia
Roman P. Kim, kimrp1988@gmail.com, https://orcid.org/0000-0002-7986-5810 Sergey A. Korchagin, korchaginser@gmail.com, https://orcid.org/0000-0001-8042-4089
Abstract. The article presents mathematical and computer models which allow to study the electrophysical properties (permittivity, impedance) of a multicellular structure exposed to nanosecond electrical pulses. The paper proposes a simulation approach that includes complex use of the classical theory of describing the electrodynamic properties of dispersed systems and the effective medium theory. We describe cell geometry using Gielis equations, which allow us to take account of the irregular shapes of cell membranes. We carry out a computational experiment with cell models to study the frequency dependences of permittivity and impedance exposed to nanosecond electrical pulses. The article considers the influence of membrane porosity on cell conductivity and permittivity as well. We carry out computer simulation of the plasma membrane electroporation mechanism. The obtained results will help to understand better the fundamental processes in the cell membrane exposed to electrical pulses and can be used in various practical applications, such as targeted drug delivery, incorporation of DNA and RNA genes into bacterial and mammalian cells, as well as the selective destruction of cancer cells.
Keywords: mathematical simulation, cellular membrane, electroporation, impedance, permittivity
For citation: Kim R. P., Korchagin S. A. Mathematical and computer simulation of the electro-physical properties of a multicellular structure exposed to nanosecond electrical pulses. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 2, pp. 259-266 (in English). https://doi.org/10.18500/1816-9791-2021-21-2-259-266
This is an open access article distributed under the terms of Creative Commons Attribution License (CC-BY 4.0)
Научная статья УДК 517.98
https://doi.org/10.18500/1816-9791-2021-21-2-259-266
Математическое и компьютерное моделирование электрофизических свойств многоклеточной структуры при воздействии наносекундных электрических импульсов
Р. П Ким10, С. А. Корчагин2
Саратовский государственный технический университет имени Гагарина Ю. А., Россия, 410054, г. Саратов, ул. Политехническая, д. 77
2Финансовый университет при Правительстве Российской Федерации, Россия, 125993, г. Москва, Ленинградский просп, д. 49
Ким Роман Павлович, аспирант кафедры информационной безопасности автоматизированных систем, kimrp1988@gmail.com, https://orcid.org/0000-0002-7986-5810
Корчагин Сергей Алексеевич, кандидат физико-математических наук, доцент департамента анализа данных, принятия решений и финансовых технологий, korchaginser@gmail.com, https://orcid.org/ 0000-0001-8042-4089
Аннотация. В статье приводятся математические и компьютерные модели, позволяющие исследовать электрофизические свойства (диэлектрическую проницаемость, импеданс) многоклеточной структуры при воздействии наносекундных электрических импульсов. В работе предлагается подход моделирования, включающий в себя комплексное использование классической теории описания электродинамических свойств дисперсных систем и теории эффективной среды. Для описания геометрии клеток используются формулы Джилиса, которые позволяют учитывать неправильные формы клеточных мембран. Проведен вычислительный эксперимент с моделями клеток по исследованию частотных зависимостей диэлектрической проницаемости и импеданса при воздействии наносекундных электрических импульсов. Изучено влияние мембранной пористости на проводимость и диэлектрическую проницаемость клетки. Проведено компьютерное моделирование механизма электропора-ции плазматической мембраны. Полученные результаты будут полезны для более глубокого понимания фундаментальных процессов, происходящих в клеточной мембране при импульсном электрическом воздействии, и могут использоваться в различных практических приложениях, таких как адресная доставка лекарств, включения генов ДНК и РНК в бактериальные клетки и клетки млекопитающих, а также избирательном уничтожении раковых клеток.
Ключевые слова: математическое моделирование, клеточная мембрана, электропорация, импеданс, диэлектрическая проницаемость
Для цитирования: Km R. P., Korchagin S. A. Mathematical and computer simulation of the electrophysical properties of a multicellular structure exposed to nanosecond electrical pulses [Ким Р. П., Корчагин С. А. Математическое и компьютерное моделирование электрофизических свойств многоклеточной структуры при воздействии наносекундных электрических импульсов] // Известия Саратовского университета. Новая серия. Серия: Математика. Механика. Информатика. 2021. Т. 21, вып. 2. С. 259-266. https://doi.org/10.18500/1816-9791-2021-21-2-259-266
Статья опубликована на условиях лицензии Creative Commons Attribution License (CC-BY 4.0)
Introduction
Field experiments which are carried out to study the cell membrane electroporation mechanism are time-consuming, expensive and require high-precision calibration of the equipment, as well as careful preparation of the object of study [1]. The main task during the experiment is to obtain the most reliable data at minimum cost. We find dependencies between factors as a part of the study, construct an approximation of the response function of various orders, perform a sensitivity analysis and determine the probability of one kind or other event.
Analysis of scientific literature [2-5] shows that short rectangular pulses are used in the overwhelming number of studies. For instance, Maxwell's equation is used to calculate the transmembrane electric potential induced by an external electric field in a spherical cell. As a consequence of the electric shock, temporary pores are formed in the cell membrane, thus increasing its permittivity. Electroporation conditions change depending on the type of substance introduced into the cells. The electroporation method [6], awarded the Nobel Prize in Chemistry 2003, is a needle-free alternative to classical mesotherapy. Seriousness of the method is confirmed by its active use in the field of medicine. Nowadays electroporation is the only non-injection effective method of transporting an active substance to skin cells while maintaining the maximum possible concentration — more than 90% [7].
Mathematical and computer simulation is a powerful tool for theoretical research in biophysics [8]. Development of mathematical models opens up a wide range of possibilities for a multimethod research of electroporation mechanism. It happens because parameter structure of mathematical models physically corresponds to the objects of study. Computer simulation as a research method denotes the concept of an iterative paradigm of the computational experiment, since we define a mathematical model more precisely, improve the computational algorithm and, in some cases, revise computational process organization in the experiment.
Analysis of such models allows to predict the most favorable conditions for their subsequent experimental study. In addition, it also allows to acquire new fundamental knowledge about dynamics of the cell membrane exposed to electromagnetic pulses. It helps to apply practical knowledge to medicine, cosmetology and other fields of science and technology in the future. Mathematical and computer modeling to study the effect of electrical pulses on biological objects is a relevant field of research, which allows to obtain detailed information about the object of study as well as data for further full-scale experiment and prediction of experiment results.
In the study we use a complex approach to carry out mathematical simulation of the electrophysical properties of cell membranes exposed to nanosecond electrical pulses. The approach includes the classical theory of describing the electrodynamic properties of dispersed systems and the effective medium theory [9].
1. Mathematical simulation of the electrophysical properties of a cell
The object of study is a multicellular structure exposed to uniform pulses of an electric field and consisted of arbitrarily shaped cells (Fig. 1), each of which has a plasma membrane and an intracellular organelle.
Every cell is described by the following values: ec — complex permittivity of a cell; ac — conductivity of a cell; ev — complex permittivity of a cell membrane; ap — conductivity of a cell membrane, eorg — conductivity of a cell organelle, epor — complex permittivity of an organelle membrane. The following parameters simulate cell geomet-
ry: Tij, Xij, yi,j — position vector and rectangular coordinates, which describe lateral view of a cell, 0i,j — polar angle, which characterizes a local coordinate system of a membrane; rm,n, xm,n, ym,n — position vector and rectangular coordinates, which describe lateral view of an organelle, 9m,n — polar angle, which characterizes a local coordinate system of an organelle, hc — thickness of a plasma membrane, horg — thickness of an organelle membrane.
We use Maxwell's equations in dispersed media to describe electrophysical properties of a multi-cell structure:
Vx H = 6o § + § + aE, (1)
Vx E = Mo f. (2)
The simulation approach is quite common and used in many studies, e.g. [10,11]. The system of equations is rather difficult to solve for realistic multicellular structures. However, due to the small size of a cell, we can neglect the change in the magnetic field over time, hence V x E = 0. An electric field E can be derived from the scalar potential 0 using the equation:
E = -V0. (3)
Substituting (3) for (1), we obtain the following partial differential equation:
V ■ (so^ + aV0 - f) = 0, (4)
where a is static ionic conductivity. The following boundary conditions are imposed to the mathematical model to satisfy the continuity equation V x J = 0:
n • (J|r+ - J|r-) = 0, (5)
where ( )
J = (a + sof) E + f. (6)
r+ and r- are outer and inner boundaries of cell membranes, respectively, n is normal vector, which is taken outside the membrane boundaries. The electric potential for each cell is calculated by root-finding algorithm (4). The transmembrane voltage Ui,j is calculated as the difference between the electrical potential at the outer and inner boundaries (0-j and 0+j respectively) of each cell membrane:
Ui,j = - 0+j. (7)
We propose to use the effective medium theory to simulate complex permittivity of a multicellular structure. The study considers cells from 10 to 100 micrometers, frequency of the electromagnetic radiation wave is 3-30 GHz, the wavelength of the electromagnetic field is by an order of magnitude more than the cell size, so the simulation method is acceptable for use.
We can use the Rayleigh equation to describe complex permittivity of a multicellular structure [12]
S<m - Sc
1 +
3V
£h^£m V - 1.31 £h -£3m33 V
£h-£m £h + ö £m
(8)
where gm is complex permittivity of a multicellular structure, gh is complex permittivity of the matrix, V is cell volume fraction in the medium. Permittivity of a cell is calculated by modifying the equations which were obtained in [13]:
^Sorg (3gg + (a - 1) (gg + 2£org)) - £c (3£org + (a - 1) (gg + 2£org)) +
Pi a2gc ((a - 1) gg + 2 (a + 1) gorg) + gorg ((a + 2) gg + 2 (a - 1) gorg)
+ (1 - Pi a) gh = 0, gorg (3gg + (a - 1) (gg + 2gorg)) - gc (3gorg + (a - 1) (gg + 2gorg)) ,
P2 a
2gc ((a - 1) gg + 2 (a + 1) gorg) + gorg ((a + 2) gg + 2 (a - 1) gorg)
+ (1 - P2a) = 0,
gg + gorg
(9)
where gg is permittivity of a cell nucleus, p membrane to the total cell volume, p2 =
(r i,j +hc)
r3
j-3 is volume fraction of the plasma
(rm,n +horg )
_ „3 „.-3
■3 is volume fraction of the organelle
membrane to the total organelle volume, a = rfjr_fn. Cell geometry is simulated by Gillis equations:
R«,7 №,7 )
- )C0S 5
^«,7 — B«,7 R«,7 №,7 )sin ^«,7 5
cos(
mi,2,-i Oij
a« ,27—1
ni, 2 j — 1
+
sin(
mi,2j Oía
a
:«,27
ni, 2 j
(10) (11)
(12)
3
r
i
)
)
i
A
A
where i = 1,...,p and j = 1,...,q and M = pq is a total number of cells and membranes, Qiyj e [—n; n] is a polar angle, which characterizes a local coordinate system, mi)2j_i, , ni)2j_i, ni)2<7- and bi)<7- e R+ (positive real numbers), ai)2<7-_i, ai)2<7- e R+ (strictly positive integers), , are scale parameters, Rj is a position vector of the corresponding cell profile.
2. Computer simulation of the electroporation mechanism
We develop a computer model and carry out the computational experiment to study the electroporation mechanism of a multicellular structure. We use an algorithm for interpreting a full-scale experiment with an "ex vivo" method. Figure 2 shows a capacitor discharge circuit for generation of an exponentially decaying electric field pulse of a multicellular structure simulated using NI Multisim software package.
VDC
Oscilloscope
C
it
PC
Electrodes
Multicellular structure Fig. 2. Computer model of a capacitor discharge circuit for generation of an exponentially decaying electric field pulse to a multicellular structure
We use a multicellular structure with dimensions of 30x30x3 mm in the computational experiment. We use values in the range of 4-5 kOhm as the initial data of the impedance value of the object of study. The values are obtained as a result of a full-scale experiment in the research paper [14]. A computer model allows to study 160 samples, which are divided into 4 groups depending on the voltage applied to the object of study (230, 550, 750, and 1000 V for each
group, respectively). If the electric properties change, the current which flows through a multicellular structure is calculated by the voltage change in the electrodes. We use a model of a 4-channel digital oscilloscope with the following characteristics: bandwidth (3.5 GHz), sampling frequency (10 GHz) in the computational experiment.
3. Results and discussions
Figure 3, a shows a dependency graph of voltage on the time of exposure to an electric field pulse for each sample group. The graph shows that voltage increases significantly in the range of 25-30 ns until it reaches its maximum value and then exponentially decreases. It means that the electrophysical properties of a multicellular structure can significantly change under the influence of electrical nanopulses. Figure 3, b shows dependence of the impedance of a multicellular structure on the volume fraction of cells in the medium.
U, V 1000
800
600
400
200
0
7a group 4
group 3
group 2
group 1
z, kOhm 7 б 5 4 3
0 10 20 30 40 50 60 70 80 90 t, ns a
200
400 б00 b
800 1000 U, V
Fig. 3. Dependence of voltage on the time of exposure to an electric field pulse (a) and dependence of the impedance of a multicellular structure on the volume fraction of cells in
the medium (b)
0
0
The computational experiment shows that impedance drops significantly near the peak of the electric field pulse and then changes insignificantly. High intensity of electrical pulses results in slower impedance recovery. Hence, time values of electrical pulses of 40-100 ns approximately have a greater effect on the cell membrane recovery.
Figure 4 shows dependence of permittivity of a multicellular structure on the wavelength of external factors. Simulation results show resonant bursts which can be connected to relaxation phenomena of the nuclear membrane and plasma membrane polarization.
100
_l_I_I_I_I__I_I_L
200 300 400 500 600 700 800 900 nm
Fig. 4. Dependence of permittivity of a multicellular structure on the wavelength of external factors
Conclusion
The paper presents a simulation approach that includes complex use of the classical theory of describing the electrodynamic properties of dispersed systems and the effective medium theory. Gielis equations provide a wide range of possibilities for simulation of multicellular systems of various geometric configurations. As a result of the study, we find out that the electrophysical properties of a multicellular structure exposed to electrical nanopulses can significantly change. We study the dynamics of the electrophysical properties of a multicellular structure against the frequency of external force and the duration of the electric field pulses as well. We set time values of electrical impulses which have a greater effect on the cell membrane recovery. The obtained results will help to understand better the fundamental processes which occur in the cell membrane exposed to electrical pulses and can be used in various practical applications, such as targeted drug delivery, the incorporation of DNA and RNA genes into bacterial and mammalian cells, as well as the selective destruction of cancer cells.
References
1. Lv Y., Yao C., Rubinsky B. A. Conceivable mechanism responsible for the synergy of high and low voltage irreversible electroporation pulses. Annals of Biomedical Engineering, 2019, vol. 47, no. 7, pp. 1552-1563. https://doi.org/10.1007/s10439-019-02258-5
2. Gupta R., Rai B. Electroporation of skin stratum corneum lipid bilayer and molecular mechanism of drug transport: A molecular dynamics study. Langmuir, 2018, vol. 34, no. 20, pp. 5860-5870. https://doi.org/10.1007/s10439-019-02258-5
3. van Veldhuisen E., Vogel J. A., Klaessens J. H., Verdaasdonk R. M. Thermal Effects of Irreversible Electroporation. In: M. Meijerink, H. Scheffer, G. Narayanan, eds. Irreversible Electroporation in Clinical Practice. Springer, Cham, 2018, pp. 121-136. https://doi.org/10.1007/978-3-319-55113-5_9
4. Yao C., Liu H., Zhao Y., Mi Y., Dong S., Lv Y. Analysis of dynamic processes in single-cell electroporation and their effects on parameter selection based on the finite-element model. IEEE Transactions on Plasma Science, 2017, vol. 45, iss. 5, pp. 889-900. https://doi.org/10.1109/TPS.2017.2681433
5. Rolong A., Davalos R. V., Rubinsky B. History of Electroporation. In: Meijerink M., Scheffer H., Narayanan G., eds. Irreversible Electroporation in Clinical Practice. Springer, Cham, 2018, pp. 13-37. https://doi.org/10.1007/978-3-319-55113-5_2
6. Royer H. D. Centenary Nobel prize in physiology or medicine for the cell cycle. Journal of Molecular Medicine, 2001, vol. 79, pp. 683-685. https://doi.org/10.1007/s00109-001-0303-5
7. Cao Y., Enbo Ma E., Cestellos-Blanco S., Zhang B., Qiu R., Su Y., Doudna J. A., Yang P. Nontoxic nanopore electroporation for effective intracellular delivery of biological macro-molecules. Proceedings of the National Academy of Sciences USA, 2019, vol. 116, no. 16, pp. 7899-7904. https://doi.org/10.1073/pnas.1818553116
8. Korchagin S. A., Terin D. V. Research electrodynamic properties of layered composite the fractal structure. 2016 International Conference on Actual Problems of Electron Devices Engineering (APEDE). Saratov, 2016, pp. 1-4. https://doi.org/10.1109/APEDE.2016.7879012
9. Kim R. P., Romanchuk S. P., Terin D. V., Korchagin S. A. The use of a genetic algorithm in modeling the electrophysical properties of a layered nanocomposite. Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics, 2019, vol. 19, iss. 2, pp. 217-225. https://doi.org/10.18500/1816-9791-2019-19-2-217-225
10. Mescia L., Chiapperino M. A., Bia P., Lamacchia C. M., Gielis J., Caratelli D. Design of electroporation process in irregularly shaped multicellular systems. Electronics, 2019, vol. 8, no. 1, pp. 37. https://doi.org/10.3390/electronics8010037
11. Sack M., Mueller G. Scaled design of PEF treatment reactors for electroporation-assisted extraction processes. Innovative Food Science and Emerging Technologies, 2016, vol. 37, pt. C, pp. 400-406. https://doi.org/10.1016/j-.ifset.2016.09.005
12. Terin D. V., Korchagin S. A., Romancuk S. P., Onosov I. A. Influence of the depth of fractal on the frequency dependence of inpedance in constructing models of composite materials. 2014 International Conference on Actual Problems of Electron Devices Engineering (APEDE). Saratov, 2014, pp. 258-259. https://doi.org/10.1109/APEDE.2014.6958756
13. Romanchuk S. P., Terin D. V., Klinayev Yu. V., Katz A. M. Mathematical modelling of structures and interaction processes of electromagnetic radiation with core-shell nanoob-jects. Vestnik Saratov State Technical University, 2011, vol. 4, iss. 2 (60), pp. 98-102 (in Russian).
14. Warindi, Hadi S. P., Berahim H., Suharyanto. Impedance measurement system of a biological material undergoing pulsed electric field exposed. Procedia Engineering, 2017, vol. 170, pp. 410-415. https://doi.org/10.1016/j-.proeng.2017.03.066
Поступила в редакцию / Received 14.05.2020 Принята к публикации / Accepted 12.10.2020 Опубликована / Published 31.05.2021
