An Analysis of The Single Moving Dipole Source for Electrocardiography Inverse Problem

Dang Thanh Trung Department of Software Engineering,

Faculty of Information Technology, Hanoi National University of Education,

Ha noi, Viet nam Email : trungdt@hnue.edu.vn

Pham Tuan Minh Department of Computer Technic and Network,

Faculty of Information Technology, Hanoi National University of Education,

Ha noi, Viet nam Email : minhpt@hnue.edu.vn

Abstract - The electrocardiogram inverse problem is a non-linear and ill-posed problem for which it is very difficult to find an exact solution. In this paper, we presented an efficient and robust method to solve this problem. The heart activity is modeled by a single moving dipole and the human body is considered as finite volume conductor constructed based on an anatomic atlas. For solution method, finite element method is applied to the forward problem. The volume conductor is meshed into arbitrary triangular elements. The efficiency of the methods stems from the employment of genetic algorithm for minimization of difference between the measured potentials and the calculated potentials generated from a predicted dipole source. Finally, the algorithm is successfully tested with simulation model and used to estimate dipole source for real data obtained from 31 electrodes on body surface. As a result, the equivalent dipole always moves clockwise and its direction changes counterclockwise.

Keywords - Electrocardiography (ECG), Inverse problem, Finite Element Method, Genetic Algorithm

I. INTRODUCTION

The objective of electrocardiography (ECG) in general is the qualitative and quantitative representation of the heart’s electrical activity exploiting the information provide by the potential fields at the body surface. The electrocardiography inverse problem in particular refers to a predetermined modeling of the cardiac activity by a variety of equivalent electric sources as a rotating dipole, a single moving and rotating electric dipole, multi-pole, multiple dipoles, e.g [1].

The aim of the electrocardiography inverse problem is to reconstruct the heart activity from a given set of body surface potentials. This problem is non-linear and ill-posed in the Hadamard sense [2]. The solution does not depend continuously on the data, and small errors in the measurement of the torso potentials or thorax geometry can yield unbounded errors. However, the solution is very importance and useful for the cardiologists in diagnosing and determining the abnormal activity in the heart. Because of these reasons, it has attracted numerous researchers to devote themselves to it in the past decades (Rudy, Mesinger and Rapport 1988, Huiskamp and van Oosterom 1989, Jaakko Malmivuo and Robert Plonsey 1995,

Furukawa et la 1989, Gulrajani et la 1988 [3],[4] , T. Musha et la 1998 [5],[6], etc).

A common characteristic of these inverse problems is their ill-posed nature, where this difficulty becomes worst when the number of parameters is increased. Namely, the problem is best conditioned when source is considered as a single moving-rotating dipole lying within entirely a finite, inhomogeneous volume conductor representing the whole body.

For solution, we firstly use a numerical method to solve the forward problem (section II). Because our discussion is limited to inverse problem in inhomogeneous volume conductor, finite element method (FEM) is exclusively used in this problem [7],[8]. By this approach, the volume conductor (solution domain) is discretized into a number of finite elements that are connected via nodes. At each node, the governing differential equation is approximated by an algebraic expression, called interpolation function. These interpolation functions are then substituted into the integral equation, integrated and combined with the results from the solution domain to obtain a system of equations. Finally, the system is solved for unknown variables.

Next, Genetic Algorithm (GA) [13],[14] is applied to compute the parameters of location, orientation and magnitude of a set of dipoles whose potential fields best match to measurement potentials in a least squares sense (section IV). Mathematically, it is a very difficult problem because the geometry of the model is inhomogeneous and its objective function is very complex. GA is a relatively effective approach for this problem. Although GA might not find the best solution, it can find a near perfect solution with acceptable time.

To estimate the accuracy of our approach, a two dimension simulation model is constructed from a classical anatomic atlas [15] for numerical testing (section V). The trajectory of dipole is assumed to be a circle inside the heart. The magnitude and orientation of dipole also change by time. The program is carried out three times to confirm the stability. The simulation result shows that our approach is efficient and robust.

It should be pointed out that while the present effort is focused on the localization of a single moving dipole in two dimension volume conductor, the method is general and can be

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extended more general source distributions as moving multi-pole, multiple moving dipoles and three dimension model of body.

Moreover, this approach is used in estimation of moving dipole source with real data obtained from 31 electrodes on body surface (section VI). The volume conductor is a cross section of body created by stereo matching technique [17], [18]. A series of successfully estimated moving dipoles tracking the hearts temporal behavior will be presented. As a result, the equivalent dipole always moves clockwise and its direction change counterclockwise in a normal heart.

II. FORMULATION OF FORWARD PROBLEM

Mathematically, most bioelectric field problems can be formulated in terms of Poisson’s equation as following [7].

∇.σ∇Φ = Iv in Ω. (1)

Where, Φ is the potential field, σ is the electrical conductivity tensor, Iv, is the current per unit volume within the volume conductor Ω and “∇.” is the divergence operator. The forward problem would be solved (1) for Φ with given a known description of IV, and the Neumann boundary condition (2) and the Dirichlet condition (3):

σ∇Φ.n = 0 on ΓT. (2)

Φ = Φ0 on Σ⊆ΓT. (3)

Because of the complex geometries and inhomogeneity of volume conductor, it is very difficult to find an exact solution of this problem. Most numerical techniques for solving this kind of problem require that the continuous domain (volume conductor) be broken up into discrete elements, called mesh or grid, and then solutions of the volume conductor problem can only be obtained by employing a particular numerical approximation method.

In our study, we applied the finite element method [10]-[12] to solve this problem. The primary idea of this method is discretizing the solution domain into elements which are characterized by interpolation function. The governing differential equation is approximated by a system of equations.

KΦ = H. (4)

The matrix K contains all geometry and conductivity information of the model and vector H describes the distribution of source. Because the source is consider as a single dipole including one source and one sink, the basic function is nonzero only in a few intervals and the number of nodes in mesh is great, the vector H is sparse and large. Solving linear equation system (4) with the specific boundary conditions, we can calculate the potential at every node within volume conductor.

There are several methods to solve a linear equation system. Because we use the solution of forward problem in conjunction with genetic algorithm to solve the inverse problem, this equation is solved repeatedly with the same matrix K. To speed up the process, the inverse matrix, K-1, will be calculated

availably by Gauss–Jordan elimination method. Then, the solution of forward can be calculated by formulation:

Φ = K-1 H. (5)

As a matter of fact, some potential fields at electrode sites are interested in computing the inverse problem. Only corresponding elements of potential vector, Φ, and source vector, H, and a part of inverse matrix, K-1, is stored to reduce the memory and execution time.

III. SIMULATION MODEL

A. Torso model

A two dimension torso model is created from an image of thorax obtained from the visible human viewer project [15]. In our study, the body can be defined as a domain of volume conductor which has different conductivities (thorax, lung, heart), in which resides a current source. The conductivities of body tissue in this study are shown in Table I [19].

Table I. Conductivities of body tissues

Element Conductivity (S/m)

Thorax 0.21

Lung 0.04

Heart 0.10

There are several different strategies for discretizing the geometry into fundamental elements, such as: the divide and conquer (or subsequent subdivision) strategy, Delaunay triangulation strategy, etc. For sake of simplicity, we use triangular elements which are produced from Delaunay triangulation strategy [16]. A finite element model of volume conductor is represented in Fig. 1. Because the electric field around the source becomes very strong, the mesh points required for FEM are concentrated near source to increase the accuracy in calculation.

Figure 1. Finite element model of body

B. Dipole Model

The two dimension model of equivalent electric dipole for

x(mm)

y(m

m)

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cardiac electric activity consist of four independent parameters [2]: two parameters for dipole position, r(x,y), and two other parameters for the dipole moment, P(Px,Py) with an arbitrary direction, α.

Equivalently, a dipole moment, P md= , is considered as the source exciting volume conductor mode. Therefore, the dipole can be represented by a point current source, m, and point sink providing, -m and separated by a small distance, d (Fig. 2).

Figure 2. Dipole model

IV. SOLUTION METHOD

A. Calculation of Equivalent Dipole

The equivalent current dipole is expressed with two co-ordinates r(xc, yc) (Fig. 2) and two components of its moment P(Px, Py). The best-fitting dipole is found by optimal algorithm such that mean squared difference between the dipole-calculated potentials Φcal = (Φ1,cal, Φ2,cal, …, ΦN,cal) and the observed potentials Φobs = (Φ1, obs, Φ2, obs, …, ΦN, obs) at N electrode sites is minimized. Since the potentials are proportional to the dipole moment P, one has:

Φobs = A(r).P (6)

Where A(r) is a transfer matrix from a given unit dipole at r onto the torso potentials at the electrode sites; this matrix depends on the dipole location and the torso model. P and r are the dipole moment and location, respectively. To calculate Φcal,the expected value of Φobs, in an iterative fashion, an initial moment p and position r are arbitrarily selected:

Φcal = A(r).P (7)

The squared error between Φobs and Φcal is defined as:

S(r,p) = (Φobs-Φcal)t(Φobs - Φcal) (8)

An equivalent dipole which best fit Φobs is obtained by minimizing S(r,P). For a given dipole location r, the squared error is minimized for an optimal dipole moment given as

Popt = [A(r)tA(r)]-1A(r)t Φobs (9)

Substituting the formula for popt into S(r,p) gives

S(r,Popt) = (Φobs-Φcal)t (Φobs - Φcal)

= ΦobstE-A(r)[A(r)tA(r)]-1A(r)tΦobs (10)

Where, E is a N-D unit matrix. Because Popt is a function of r, S(r,Popt) depends on r only and it is gain minimized by a optimal algorithm.

B. Source Localization by Genetic Algorithm

Genetic Algorithms (GA) are capable of searching for global in functions which cause difficulty for gradient based methods. Principal advantages of GA are domain independence, non-linearity and robustness. Because of these characteristics, GA is suitable for this problem.

In optimization, that is source localization, the aim of our problem is finding the best equivalent dipole, so objective function, f, has the following equation:

( , )= =

Φ − ΦΦ Φ

obs cal

obs obs

fS r P

(11)

In GA, a solution is called a chromosome. Structure of chromosome is built from the unknown parameters of dipole (x,y) and is encoded by a string of bits 0 and 1 (Fig. 3). The length of string depends on the required precision and solution region is bounded surround the heart.

Figure 3. Representation of solution and solution region in GA

Figure 4. Scheme of source localization

x

y

0101011 1101010

x y

A chromosome

17 bits

17 b

its

Solution Region

Measured Potentials

Mean square error

Genetic Algorithm

Predicted Dipole • Position (x,y) • Direction α• Magnitude m

FEM

Calculated Potentials

-

+

x

y

xc

ycα

m

-m

d P md=

Px

Py

α

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The scheme of source localization by GA can be summarized as following (Fig. 4):

• Generating an initial set of random different dipoles in solution region. Each dipole is characterized as a chromosome.

• The potentials of these dipoles are calculated by FEM. The objective function in GA is calculated from calculated potentials and measured potentials in mean square sense.

• By using GA operators as selection (tournament selection), crossover (uniform crossover), and mutation, a new population of solution is created to replace the old one.

• The generational process is repeated until a termination condition has been reached. In our program, the process will be finished after a finite number of iterations.

• The solution of inverse problem is the chromosome minimizing the evaluation function in the current population of the last generation.

The conventional GA, however, has a very poor local performance because of the random search. To get a good solution, great computational costs are inevitable. Some improvements are necessary to enhance performance of GA. In our program, we used some important modifications that can dramatically improve the performance of the conventional GA such as: elitist strategy, hybrid algorithm.

V. NUMERICAL RESULTS

We assumed the actual trajectory of dipole to be a circle which has radius, r = 25mm, and center point, (x0, y0) = (270, 85). The formulation of trajectory, orientation and magnitude of dipole in respect to time, t, are represented by equations:

x(t) = r*cos(2πt) + x0;y(t) = r*sin(2πt) + y0;α(t) = 4πt;m(t) = 100*cos2(2πt) + 100.

Figure 5. Assumed dipoles

In the experiment, we discretized the trajectory of dipole at 100 time instants, t = 0, 0.01, 0.02, …, 0.09, corresponding to 100 different dipoles (Fig. 5). The length of dipoles is a

constant (d = 1mm). Some parameters for FEM and GA used for solution localization are shown in Table II. The program was carried out at least three times to confirm stability.

Table II. Some parameters in FEM and GA

Finite element method

Node Element Electrode

734014397

31

Genetic algorithm

Precision Bit length Population Iteration Crossover Probability Mutation Probability

10-7

173001000.95 0.015

The experiments reported here were executed on the IBM server system. The machines are interconnected via a 100-Mbps fast Ethernet switch. The operating system that our code was executed on was Red Had Linux 7.2. The code was written in ANSI C with Message Passing Interface (MPI) constructs to execute in parallel. This algorithm was previously parallelized to allow for interoperability on different parallel platforms. In particular, the parallelization centered on input data area. Synchronous MPI communication calls were utilized to conduct communications between machines.

Figure 6. Simulation results

40200 320

140

85

270

• calculated parameters - assumed parameters

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With the parameters of FEM and GA as in table II, the execution time of our program (in elapsed CPU seconds) is approximately 0.8s for each dipole. The results are shown in Fig. 6. The solid line represents the assumed parameters, and the dot line represents calculated parameters. The result is very good, two lines are fairly concur.

The average of error of parameters is shown in Table III. The mean residual error is very small, about 4.7*10-6, this value proves that our approach is stable and good enough to estimate for real data.

Table III. Mean error of simulation results

Parameter Value

x(mm) 0.1349

y(mm) 0.1479

α(rad) 4.1*10-5

m(µCm/s) 0.0017

Residual Error 4.7*10-6

VI. ANALYSIS OF REAL DATA

The proposed algorithm was applied for the localization of the dipole source for measured data sets at different electrode sites and satisfactory results were obtained.

The potential field data were obtained from 31 electrodes at body surface by a special device, Polymate AP1532 in 1 second (corresponding to 2000 time instants). The object is a 27 year-old male person and data were recorded while he was sitting.

The electrode configuration is shown in Fig. 7. A reference electrode is placed at left chain bone and 31 other electrodes are arranged clockwise surround on the body.

Figure 7. The electrode configuration

A volume conductor was created from a set of pictures of electrode configuration with stereo matching technique [17], [18] (Fig. 8). By this technique, we took 16 pictures from different directions around body, in which they can view all 31 electrodes.

Figure 8. A cross section of body created from electrodes

(a) The measured potentials

(b) The calculated potentials

Figure 9. ECG signal at 9 first electrodes

Fig. 9 illustrates the measured potentials and calculated potentials at 9 first electrodes (ch1, ch2, …, ch9). As we can see, the calculated potentials have exactly the same form as the measured potentials with small variations in their values. The condition of measurement is shown in Table IV.

… …

Reference electrode

Front Back

Ch1 Ch17 Ch18 Ch31

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Table IV. Condition of measurement

Device Polymate AP1532

Electrode 31

Sampling Frequency 2kHz

Low Interception Frequency 1Hz

High Interception Frequency 30Hz

Figure 10. Results of calculated dipole

Some estimated results of inverse problem are described in Fig. 10. The mean residual error is satisfactory, about 0.07.

Figure 11. The estimated trajectory of dipole in the heart

Figure 12. Vectorcardiogram

The potential of the first electrode, ch1, is used for reference. According to the change of potential, we analyzed the estimated dipole in four intervals as: [0.00s - 0.25s], [0.25s - 0.50s], [0.50s - 0.75s] and [0.75s - 1s].

Fig. 11 illustrates the trajectory of the calculated dipole in four intervals, respectively. The vectorcardiogram depicting the change of direction and magnitude of dipole is shown in Fig. 12.

• In the first interval, [0.00s – 0.25s], the potential wave is small; the displacement of dipole is narrow. Dipole moves mainly in the right of heart. Magnitude of dipole alters linearly to the variation of potential.

• In the second interval, [0.25s – 0.50s], the potential looks like a line. The dipole moves inside a small area. Also, the magnitude and orientation of dipole vary a little.

• In the third interval, [0.50s – 0.75s], the potential wave is large, the increasing and decreasing of potential is strongest. The dipole source moves almost whole heart. In this case, the magnitude of the equivalent dipole varies largely.

• In the last interval, [0.75s – 1s], the change of potential nearly is the same as the first interval, but the dipole move mainly in the left of heart.

From these analyses, it is obvious that any change in the potentials can alter the trajectory, magnitude and direction of dipole source, that is, all heart temporal behaviors can be characteristic by a moving equivalent dipole. For this reason, this is a good approach to detect and estimate all serious abnormalities of the heart.

As a result, with a normal heart, the equivalent dipole source always moves clockwise (Fig. 11) and direction of dipole change counterclockwise (Fig. 12).

0.00s

0.25s

0. 50s

0.25s

0. 50s

0. 75s

1.00s

0.75s

x(mm)

y(m

m)

0.25 0.50 0.75

0.00s

0.25s

0.25s

0.50s

0.75s

0.50s

0.75s

1.00s

Px (µCm/s)

P y (µ

Cm

/s)

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VII. CONCLUSION

In this paper, we represented an efficient and robust method to solve the ECG inverse problem in bioelectric field by finite element method in conjunction with genetic algorithm. We have first introduced finite element method to solve the forward problem. The solution of inverse problem was solved by genetic algorithm. We also introduced some additional features to overcome the disadvantages of conventional genetic algorithm. The performance of our approach was evaluated based on a two dimension realistic torso model. The results are as good as what we expected.

Also, we used our approach to analysis the real data obtaining from 31 electrodes on body surface. The residual error is acceptable and a series of successfully estimated moving dipole depicting the heart’s activity were presented. As the result, dipole always moves clockwise and direction of dipole change counterclockwise. Further work along these lines is ongoing.

ACKNOWLEDGMENT

This study was supported in part by the 21st Century COE (Center of Excellence) Program "Global Renaissance by Green Energy Revolution" and the Grant-in-Aid for Scientific Research (15300070) from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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