* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Overview on the Equivalent Circuit Method for Electrical Analysis of
Electrochemistry wikipedia , lookup
High voltage wikipedia , lookup
National Electrical Code wikipedia , lookup
Network analyzer (AC power) wikipedia , lookup
Lorentz force wikipedia , lookup
History of electromagnetic theory wikipedia , lookup
Topology (electrical circuits) wikipedia , lookup
Residual-current device wikipedia , lookup
Nanofluidic circuitry wikipedia , lookup
Maxwell's equations wikipedia , lookup
Faraday paradox wikipedia , lookup
Scanning SQUID microscope wikipedia , lookup
Earthing system wikipedia , lookup
Static electricity wikipedia , lookup
Electric current wikipedia , lookup
History of electrochemistry wikipedia , lookup
Electric charge wikipedia , lookup
Electricity wikipedia , lookup
Electrostatics wikipedia , lookup
Nonlinear Optics and Quantum Optics, Vol. 39, pp. 137–144 Reprints available directly from the publisher Photocopying permitted by license only ©2009 Old City Publishing, Inc. Published by license under the OCP Science imprint, a member of the Old City Publishing Group Overview on the Equivalent Circuit Method for Electrical Analysis of Biological Tissues Anca-Iulia Nicu and Radu Ciupa Technical University of Cluj Napoca, Electrical Engineering Department, 15, C. Daicoviciu str. 400020, Cluj Napoca, Romania E-mail: [email protected] This paper is an overview of one of the models developed in literature [1–3] for numerical analysis of the electric field distribution, based on the Equivalent Circuit Method (ECM). In the paper only the cell scale model presented in [1–3] in 3D is investigated with comments and proper observations in two dimensions. The author’s contribution is the developed analytical method for solving the matrix model from [1]. Keywords: Equivalent circuit method, biological tissue, bioelectromagnetism, cell scale model. 1 INTRODUCTION Electrical properties of biological tissues are very complex. Cell shapes are irregular and biological tissues present strong dispersive behaviour of their electric properties from very low to microwave frequencies. An important feature of the studied method is the facility of boundaries conditions modelling, by simplifying circuit elements which are connected with the adjacent media [1–3]. The authors present in this paper only the two dimensional case. 2 EQUIVALENT CIRCUIT METHOD It is a method of electrical analysis for numerical simulation based on an electric circuit, or more simple – equivalent circuit method. With this method it can be obtained an iterative process for spatial distributions of electrical fields, currents and charge densities in an excitated media for electrical potentials. In 137 “NLOQO” — “nloqo-bio-04” — 2009/8/19 — 17:46 — page 137 — #1 138 A.-I. Nicu and R. Ciupa FIGURE 1 Continuous problem (discretized in order to be solved with ECM). order to understand this method it is very important to understand at first the system discretization. 2.1 System discretization Is briefly explained the analytical method which proposes exact solutions, in order to give a correct answer for electromagnetic applications. The proposed configurations that make continuous changes to the boundary conditions are easier than in other methods. For biological systems with anisotropy and nonlinearities, an analytical method could not be implemented, and so it was necessary to use numerical methods which give approximate solutions. In order to use numerical methods is necessary to search for a numerical finite solution in sub-regions (elements), and this type of division is called discretization. An application of potentials in biological systems it is a continuous problem and it is discretized in order to obtain an approximate solution (Figure 1). Spatial discretization it is one of the most important phases in definition of the system that is to be solved. Increasing the number of blocks in order to improve the definition of the system will make the processing time to long, but decreasing the number of blocks can give errors in calculations of potentials, fields, currents and charge densities. For the biological systems, we can limit the preoccupation for the discretization of the boundaries for isolating membranes, because this charge accumulation often produces intense spatial variations in the electrical current field. This numerical calculus is very difficult to be solved. The biological systems are very complex, and to represent interstitial spaces between two neighbour cells is very difficult because these spaces are too small. A representation with few blocks will give a major error for the field calculus. The first step is to define the system which is not necessary to be square and it can be rectangular. 2.2 Cellular level analysis Based on the theory developed in [1–3], our aim is to describe two major facilities of this method. Initially, applying the Ampere law to an elementary volume represented in Figure 2 we obtain: ∂ E (1) ∂t Applying a divergence operator in both members of (1), we obtain: ∇ · (∇ × H ) = 0, which is valid for any vectorial field. Applying a surface integral to ∇ × H = Jq + ε “NLOQO” — “nloqo-bio-04” — 2009/8/19 — 17:46 — page 138 — #2 OVECM for Electrical Analysis of Biological Tissues 139 FIGURE 2 Elementary volume in rectangular coordinates. the right member of the equation (1) is obtained three distinct currents which are crossing the surface: the diffusion current Idif , the conduction current Icond and the displacement current Idispl . The conduction current is defined: Icondx = µx ρx A V L (2) Where x – identifies every charge type; µx – the ionic mobility; ρxm = (ρxo +ρxx )/2 – media charge density between the discretized elements. A – area between the faces of two volume elements; L – distance between the 2 centre of the volume elements; V = V0 − Vx – difference of potentials between two adjacent elements, see Figure 3: The equivalent circuit method proposed by Ramos solves two types of models: cell scale model treated in this paper and tissue scale model. In order to obtain the equivalent electric circuit of a media in the cell scale model the volume under analysis should be divided in a large number of small parallelepiped blocks, as shown in Figure 4. Each block constitutes a node in an equivalent circuit and communicates with its neighbours by a set of parallel circuit elements: a conductance and a current source for each kind of electric charge carrier in the media, representing the conduction and diffusion currents, and a capacitance representing the displacement current. FIGURE 3 Two adjacent discretized elements. “NLOQO” — “nloqo-bio-04” — 2009/8/19 — 17:46 — page 139 — #3 140 A.-I. Nicu and R. Ciupa FIGURE 4 Space discretization of the parallelepiped blocks. The calculation of each element is based on the dimension of the block and the transport properties (electric) in that point of space. The total current leaving each node of the circuit in this discretized space is given by the following differential equation: I= (gnx x V + knx x ρn ) + (gny y V + kny y ρn ) n + Cx δ(y V ) δ(x V ) + Cy δt δt (3) where n – identifies each type of charge carriers in the medium and x, y are directions in space. V – electric potential, ρ – charge density inside a block; and δ – indicate the differential operator in space and time. The parameters g, k and C are the conductance, diffusion coefficient and capacitance of the block, respectively, and they are given by: Ax Lx Ax knx = fnx Dnx Lx Ax Cx = εx Lx gnx = ρn µnx (4) (5) (6) where A and L are the area and length at the connection of two adjacent blocks and fn is a voltage-dependent factor from the solution of the one-dimensional Nerst–Planck equation between two nodes. The same equation could be written for the y direction too. The currents, charge distribution and electric potential can be obtained by solving the resulting equivalent electric circuit. “NLOQO” — “nloqo-bio-04” — 2009/8/19 — 17:46 — page 140 — #4 OVECM for Electrical Analysis of Biological Tissues 141 Defining Iqn as a part of the total current due to the conduction and diffusion of the n type charge carriers and applying Kirchhoff’s Current Law to the central node (node O) in Figure 4 is obtained: CON ∂(VO − VN ) ∂(VO − VS ) IqnON + COS IqnOS + + ∂t ∂t n n + COW ∂(VO − VW ) ∂(VO − VE ) IqnOW + COE + IqnOE = 0 + ∂t ∂t n n (7) The equation above can be rewritten simpler: (CON + COS + COW + COE )VO − CON VN − COS VS − COW VW − COE VE = QO (8) where QO is the total charge in the volume of the central block, which is calculated in each time step: former Qactual = QO − IqnOM ∂t (9) O M n where M indicates each neighbour node connected to the central node through the branch OM. When leaving the node, currents are considered positive. The charge density for each type of carrier can be obtained: 1 former − IqnOM ∂t (10) ρnactual = ρn ν M where ν is the volume of the block. An electrical equivalent circuit between adjacent nodes is proposed in the next figure based on the model created based on equations (3) and (4). The ECM in the cell scale model consists in obtaining the equivalent circuit of the media and to solve the system of equations as (7), one for each node of the equivalent circuit, in time steps, for the given boundary and initial conditions, and to update the total charge and charge densities in each step, according to equations (9) and (10). This set of node equations (7) can be rewritten using matrix notations: CV = Q + F (11) where C is a capacitance matrix N × N (N is the number of nodes of the circuit) containing the coefficients of node potentials. In order to understand this iterative method a cellular level flux diagram of ECM is presented: 1. Initial values Vn = 0, Qn = 0, ρn = 0. “NLOQO” — “nloqo-bio-04” — 2009/8/19 — 17:46 — page 141 — #5 142 A.-I. Nicu and R. Ciupa FIGURE 5 Cellular level flux diagram of ECM. FIGURE 6 Equivalent circuit for cell scale model. 2. Potentials calculus V = C −1 (Q + F ) where it is a matrix formed with the term F = cfn Vfn , and cfn - the capacitance and Vfn – the applied potential. 3. Conductance’s calculus made with the equations of gx and kx . 4. Currents calculus 5. The densities and charge actualization. 6. The results are saved and returned to step 2 for a new iteration. Based on the mathematical model presented above, an equivalent circuit between adjacent nodes is proposed in the next Figure 6: [3] The letters O and X indicate nodes of the circuit. The numbers 1 and 2 indicate different charge carrier types First, in a general form, an ECM consists from obtaining a media equivalent circuit and solving an equation system like equation (3), one equation for each equivalent circuit, every time step, for initial and limit conditions. The “NLOQO” — “nloqo-bio-04” — 2009/8/19 — 17:46 — page 142 — #6 OVECM for Electrical Analysis of Biological Tissues 143 actualization of the total charge and charge densities is made at every step. The conductance for every charge carrier is also actualized using the newest values of the charge densities. The vectors V and Q are vectors of node potential and total charge and F is the excitation vector, defined by: CF 1 VF 1 C F 2 VF 2 F = ··· CFN VFN (12) where CFN is the capacitance connecting the source electrode to the medium and VFN is the source potential in the node n. So, having the system of nonlinear matrix equation CV = Q + F , it can be solved analytically in MathCad program with an iterative approximate method given below: [4] V mas( C Q F) N m 20 ¢0² m m last B for i 0 m B i i m0 for j 0 m B i j § Ci j · if j z i ¨ Ci i © ¹ m ¨ C for i 0 m ( Q F) D m i i C i i D ¢0² x mD for k 0 N ¢k 1² ¢k² x m B x D x ¢ N² x Naturally, since geometry and permittivity of the medium do not change during processing, C is constant and its inverse matrix can be obtained once at the beginning of the calculation. On the other hand, Q and F should be updated in each iteration. “NLOQO” — “nloqo-bio-04” — 2009/8/19 — 17:46 — page 143 — #7 144 A.-I. Nicu and R. Ciupa 3 CONCLUSIONS In this paper the authors presents some comments and observations regarding the models developed in literature [1–3] for numerical analysis of the electric field distribution in 2D, based on the Equivalent Circuit Method (ECM). The authors have developed an analytical method for solving the matrix model from deduced by the ECM. An important feature of the studied method is the facility of boundaries conditions modelling, by simplifying circuit elements which are connected with the adjacent media [1–3] but the authors present in this paper only the two dimensional case. The ECM in the cell scale model consists in obtaining the equivalent circuit of the media and to solve the system of equations one for each node of the equivalent circuit, in time steps, for the given boundary and initial conditions, and to update the total charge and charge densities in each step, according to the presented MathCad program [4]. ACKNOWLEDGMENTS Acknowledgments for Professor A. Ramos and eng. J. Marques for all the received technical materials and for all the interesting discussion in this topic. REFERENCES [1] A. Ramos, A. Raizer and J. L. Marques. A new computational approach for electrical analysis of biological tissues. Bioelectrochemistry 5758 (2003), 1–12. [2] A. Ramos, A. Raizer and J. L. Marques. Modelling the Electric Field and Ionic Charge Distribution in Biological Tissue through the Equivalent Circuit Method. 3rd International Conference on Bioelectromagnetism, 8–12 October, 2000, Bled, Slovenia, pp. 41–42. [3] A. Ramos, A. Raizer, J.L. Marques. New Method for Field Calculation Using the Equivalent Electric Circuit. In 13th Compumag, 2–5 July, Evian, France pp. 232–233, 2001. [4] D.D.Micu. Numerical methods for electromagnetic interference calculations. Mediamira, Cluj - Napoca, Romania, 2005. “NLOQO” — “nloqo-bio-04” — 2009/8/19 — 17:46 — page 144 — #8