Influence of Skin Effect on Current Flow Through Electrodes of Electro-Surgical Instruments and Biological Tissue Volodymyr Sydorets Department of Gas Discharge Physics and Plasma Devices, E.O. Paton Electric Welding Institute of the NAS of Ukraine, Kyiv, Ukraine Andrii Dubko Department of Welding and Related Technologies in Medicine and Ecology, E.O. Paton Electric Welding Institute of the NAS of Ukraine, Kyiv, Ukraine Oleksandr Bondarenko Industrial Electronics Department, National Technical University of Ukraine “Kyiv Polytechnic Institute”, Kyiv, Ukraine Roman Kosenko Power Electronics Group, Department of Electrical Engineering, Tallinn University of Technology, Tallinn, Estonia

E.O. Paton Electric Welding Institute in cooperation with the Institute of Surgery and Health Ministry of Ukraine developed advanced medical technology of joining soft tissues without using sutures, metal staples, adhesive-solders or other materials extraneous for the body.

The new welding technology was recognized to be novel, which certified by patents of Ukraine, US and other countries.

A flow of alternating current through a conducting medium develops its non-uniform distribution (skin effect). Many authors do not consider this effect in development of the mathematical models of current flow through the electrodes of electro-surgical instruments and biological tissue. An equation of electromagnetic field is used for the analysis of electromagnetic processes taking place in the conductors at alternating current effect (1) where ∇ is the operator nabla; E is the electric field intensity; ω is the angular frequency; j is the imaginary unit; σ is the specific conductivity; ε is the dielectric permeability; μ is the magnetic permeability of conducting medium.

Main aim of the investigation lies in analysis of current density distribution in the two-phase medium containing copper electrode and biological tissue taking into account skin effect. A geometry model of the two-phase medium is considered in cylindrical system of coordinates. It consist of cylindrical copper electrode and biological tissue cylindrical sample (Figure. 1). Fig. 1. Cylindrical model of two-phase medium (1- electrode, 2 - biological object) The axis of electrode and biological tissue agree with coordinate z. In this case, the current density in any point of the model depends only on coordinates x and y. Current density is the same in any point of the model located on distance r from the axis. Isosurfaces of current density distribution are the cylindrical surfaces coaxially located to the cylindrical model surface. There is a relationship between radial coordinate r and x and y coordinates

Total height of the model equals Zmax = 1 mm. Such a geometry model allows reducing the problem to axially symmetric case. Figure 2 show a semi-longitudinal section of the geometry model electrode – biological tissue. Fig. 2. Cylindrical semi- longitudinal axially symmetric model (1- electrode, 2 - biological tissue) Total height of the model equals Zmax = 1 mm. It consists of electrode of Re=0,5 mm radius and Ze = Zmax-Zt =0,5 mm height and biological tissues (radius Rt=1 mm and height Zt=0,5 mm).

Equation (1) for two-dimensional current density distribution in the cylindrical system of coordinates takes on form where r, z – coordinates. To solve this problem we apply the method of finite differences.

The boundary conditions of the first order (Dirichlet): The boundary conditions of the first order (Dirichlet) G3(r,z), G4(r,z), G5(r,z) on the outer boundaries of electrode and biological tissue as well as the symmetry conditions of the second order (Neumann) G1(r,z), G2(r,z), G6(r,z) are shown in Figure 3. Fig. 3. Boundary conditions The boundary conditions of the first order (Dirichlet): where Js is the current density on the outer boundaries of electrode and biological tissue. The boundary conditions of the second order (Neumann)

A numerical experiment was realized in mathematical package Matlab for copper electrode and biological tissue. Frequency was 440 kHz and current density at the outer boundaries of considered materials Js = 1 A/m2. Table 1 gives the specific conductivity of electrode materials and biological tissue. TABLE I. PARAMETERS OF MATERIALS (a) - average value for internal organs (liver, kidney)

A region of problem solution is located on uniform grid of 100x100 A region of problem solution is located on uniform grid of 100x100. Figure 4 shows amount and positioning of the non-zero elements of sparse matrix in the solution region of given model. Fig. 4. Mirror view of non-zero elements of model sparse matrix (nz - non-zero elements)

Matrix non-zero elements of the coefficients are shown in Figure 5. Equation (3) and boundary conditions (4, 5) set up a system of equations, which, as a result of approximation of the partial derivatives to the corresponding finite-differences, is transformed in a system of linear algebraic equations (SLAE). Matrix non-zero elements of the coefficients are shown in Figure 5. Figure 6 shows non-zero elements of a column of absolute terms. Fig. 5. Matrix of coefficients (nz - amount of non-zero elements) Fig. 6. Vector of absolute terms column

A value of vector of unknowns was found as a result SLAE solution and this vector was transformed in a matrix of electric field distribution E (Figure 7). Fig. 7. Distribution of electric field in two-phase medium (electrode-biological tissue)

The distribution of current density in the two-phase medium was found by multiplying the elements of matrix E, which are related to areas of electrode or biological tissue, by specific electric conductance of the corresponding. Figure 8 shows the result of mathematical modeling of current density distribution in two-phase medium (electrode of electrosurgical instrument and biological tissue). Fig. 8. Distribution of current density in two-phase medium (frequency 440 kHz)

Mathematical modeling with 66 kHz frequency and using the same materials (Figure 9) was carried in order to compare frequency effect on current density distribution in two-phase medium. Fig. 9. Distribution of current density in two-phase medium (frequency 66 kHz)

The results of modeling of current density distribution in the two-phase medium were experimentally verified. Figure 10 shows the result of retina to choroid welding. Influence of skin effect is clearly defined. In welding the biological tissues are subjected to the maximum coagulation on electrode outer perimeter. Figure 11 shows the electrosurgical instrument for retina welding (electrode 0.66 mm diameter, frequency of 66 kHz). Fig. 10. Coagulation ring on retina after its welding choroid Fig. 11. Electrosurgical instrument for retina welding

The developed two-dimensional stationary mathematical model for determination of current density distribution in two-phase environment can be applied to cylindrical electrodes of electrosurgical instruments with different conductivity (copper, gold, silver, brass, steel and others) and biological tissues (vessels, liver, kidneys, lungs and others) in a wide frequency range from 66 kHz to top limit of operation of high frequency electrosurgical power supplies – 4 MHz. The model shows that current density has a non-uniform distribution in a place electrode-biological tissue contact. The maximum current density concentrate nearby the outer part of the electrode and its contact surface. Development of the mathematical models of spatial heat sources, caused by current flow through the conducting media will become a further evolution of this investigation.

CONCLUSIONS ACKNOWLEDGMENT The mathematical model was developed which allow for analyzing the electromagnetic processes taking place in two-phase media (electrodes of electrosurgical instruments and biological tissues). This model is a two-dimensional elliptic problem for modeling skin effect in the cylindrical coordinate system. Influence of this effect should be considered in development of the new efficient electrosurgical instruments. Variation of current frequency and shape of electrodes can effect the width of zone of biological tissue coagulation. ACKNOWLEDGMENT The authors express their thanks to Prof. E.A. Nastenko, Dr. of Biological Sciences, head of chair of Medical Cybernetics of NTUU «KPI», for possibility of mathematical of modeling in Matlab package. This work was co-financed by the Estonian Ministry of Education and Research under Project SF0140016s11 and by the Estonian Centre of Excellence in Zero Energy and Resource Efficient Smart Buildings and Districts under Grant 2014-2020.4.01.15-0016 funded by the European Regional Development Fund.

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