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Nonlinear Optics and Quantum Optics, Vol. 39, pp. 137–144
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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
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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
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OVECM for Electrical Analysis of Biological Tissues
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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.
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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.
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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.
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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
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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.
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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.
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