2015 16th International Conference on Computational Problems of Electrical Engineering (CPEE)
Lviv, Ukraine
Mathematical Modeling of the Current Density
Distribution in a High-Frequency Electrosurgery
Volodymyr Sydorets, Alexei Lebedev, Andrey Dubko
E.O. Paton Electric Welding Institute of the NAS of Ukraine
Kyiv, Ukraine
sydorvn@gmail.com, biowelding@mail.ru, andreyies17@gmail.com
mechanically held by living tissue and pass through them an
alternating current of high frequency. Fig. 2 shows a bipolar
electrosurgical clamp. The welded artery is shown in the Fig. 3.
Abstract—An important goal of surgical exposure is a fusion
of living tissues. The electric welding Institute (PWI) developed
the method of welding of soft tissues. Method of welding provides
reliability of connection without the use of suture. The developed
mathematical model can be applied for rectangular conductors
with different conductivity in a wide frequency range.
Keywords—Electric welding; mathematical model; MatLab;
surgical; electro coagulator ЕКВЗ-300; living tissues; artery
I. INTRODUCTION
An important goal of surgical exposure is a fusion of living
tissues. Connection method largely determines the time of
surgery, the postoperative period, recovery of physiological
functions of the operated organs. The electric welding Institute
(PWI) in cooperation with doctors of the Ministry of Health
developed the method of a bipolar high-frequency welding of
soft living tissues of animals and humans, which allows to
obtaine a welded joint of pre-cut organs and tissues with the
restoration of their structure and functions. The idea to solve
the problem of the connection of living soft tissues by means of
electric welding belongs to academician Boris Paton. The
hypothesis of the mechanism of connection of the sections of
living tissue using RF welding have been proposed by
academics Boris Paton and Vladimir Lebedev [1].
Fig. 1. High frequency welding electro coagulator ЕКВЗ-300
Method of high frequency electric welding of soft living
tissues provides: reliability of connection without usage of
suture materials (threads, adhesives, brackets, etc.); the absence
of necrosis and foreign bodies in the wound; tight joints; less
blood loss; reduced time of operation; no suppuration; reliable
hemostasis; simplification of the operation; reduction of the
postoperative period; no smoke during operation. The device
for this method developed in PWI. High frequency welding
electro coagulator ЕКВЗ-300 (Fig. 1) has passed State
registration and certification in Ukraine.
(a)
This device is used in various areas of general surgery,
cardiology,
ophthalmology,
urology,
otolaryngology,
gynecology, oncology etc. [2-7].
(b)
Dozens of types of electrosurgical instruments were
designed and tested [8, 9]. Important components of the
electrosurgical instruments are electrodes, which are
Fig. 2. Bipolar electrosurgical clamp: (a) – General view; (b) – electrodes
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2015 16th International Conference on Computational Problems of Electrical Engineering (CPEE)
Lviv, Ukraine
1

  E   ( j  2) E  0,



(1)
where:  is the nabla operator; E is the electromagnetic field
intensity; ω – angular frequency; j is the imaginary unit [10].
This equation is derived from Maxwell's equations.
Consider the two-dimensional stationary problem of the
current density distribution over the cross section of the
rectangular copper electrodes. The problem can be solved
using finite difference method.
Let write equation (1) for isotropic and two-dimensional
stationary problem in the Cartesian coordinate system:
1   2 E ( x, y )  2 E ( x , y ) 
2
 

  ( j   ) E ( x, y )  0,
  x 2
y 2 
Fig. 3. Welded artery
(2)
where x, y – coordinate; E(x, y) – function that is searched with
boundary conditions of first kind (Dirichlet) at the boundaries
х = хmin, х = хmax, у = уmin, у = уmax.
J(x,y) = E(x,y)∙σ,
(3)
where J – the current density.
A system of linear algebraic equations was compiled as a
result of approximation of the partial derivatives. We solved
this system of equations together with (3) and obtained the
distribution of the electromagnetic field E and the distribution
of the current density.
Vectors define a rectangular area on a two-dimensional
uniform grid
Fig. 4. Incision brewed artery: 1 – adventitia; 2 – media; 3 – intima
G={( хі = iΔx, уk = kΔу), і=1, 2, …, n; k=1, 2, …, m}. (4)
Fig. 4 shows a section of the artery, which differ in the
three layers of the vessel (adventitia, media and intima). The
outer layer is involved in the formation of connection. The
inner and middle layers are squeezed out from the seam during
welding.
Boundary conditions of first kind (Dirichlet) in this
problem:
Е(x1,y)=g1(y);
E(xn,y)=g2(y);
Е(x,y1)=g3(x);
Е(x,ym)=g4(x). Where x1, xn – coordinates of the boundary
points of the хmin, хmax area; y1, ym – coordinates of the boundary
points of the уmin, уmax area; g1(y), g2(y), g3(x), g4(x) – is a
functions of x and y coordinates. In our case
g1(y)=g2(y)=g3(x)=g4(x)=J/σ.
II. THE ANALYSIS OF THE DISTRIBUTION OF CURRENT DENSITY
IN THE E LECTRODES
The analysis of the distribution of current density in the
electrodes with a rectangular cross section, which are widely
used in welding of live tissues by means of mathematical
modeling of physical processes was the aim of this work.
Numerical calculations of high-frequency current flow in the
electrodes of the electrosurgical instruments were conducted in
the environment of mathematical package MatLab using finite
difference method.
The result of mathematical modeling in MatLab of the
current density distribution in the copper electrode of
rectangular cross section 1×1 mm2 is shown in the Fig. 5. The
main inputs of the model are as follows: frequency 440 kHz,
σ = 57·106 S/m.
Fig. 6 shows the result of mathematical simulation of
current density distribution in the copper electrode of
rectangular cross section 1×1 mm2 (frequency 1.5 MHz;
σ = 57·106 S/m).
Electro physical characteristics of the electrodes are
determined by their specific electrical conductivity σ,
permittivity ε and magnetic permeability μ. An alternating
electromagnetic field is described by the equation [10].
Increasing the frequency reduces the thickness of the
surface layer of the AC current (Fig. 5-6).
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2015 16th International Conference on Computational Problems of Electrical Engineering (CPEE)
Lviv, Ukraine
III. CONCLUSION
The mathematical model can be applied for rectangular
conductors with different conductivity in a wide frequency
range. It is necessary to consider the skin effect in the
development of tools for high frequency connections living
tissues. To increase the area that conducts current is possible by
increasing the total length of the outer perimeter of the
electrode by changing the shape of the cross section due to the
removal of individual sections of the electrode. Uniform
distribution of current density in a rectangular electrode is
possible with the use of a set of isolated from each other
electrode.
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Fig. 5. The distribution of current density in the electrode (frequency 440
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Fig. 6. The distribution of current density in the electrode (frequency
1.5MHz)
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