Computer Graphics & Geometry

Inverse Problem in Geometrical Modeling of the Sources of Medical Signals

B. Lvov
e-mail: blvov@yu.edu.jo

H. Al-Nashash

A. Al-Zaben

Department of Electronic Engineering
Yarmouk university, Irbid, Jordan
Telephone: 962 2 7271100 ext. 3134

Contents

Abstract

Some application of geometrical modeling for biomedical signal processing are discussed. A new model of the heart nervous conductive system, used previously for direct electrogram problem is used now to solve the inverse problem. Geometrical modeling can give an approximate estimation of the source position and strength even if the available amount of data is not sufficient for a correct mathematical solution It can be useful in ECG and EEG inverse problems as well as in creating a model of human torso with non-uniform conductivity.

Key words: His bundle electrogram, inverse problem, geometrical modeling

1. Introduction

Localization of the sources of electrical activity inside a human body is important for various applications. Our work is concentrated on two of them, which are the heart activity and the brain activity. In the first case the electrical signal (electrocardiogram or ECG) comes from a set of depolarizing cells and the source of the signal is changing in time its location and amplitude. In the second case the signal is called electroencephalogram or EEG and the sources are multiple but immovable. In both cases there exist models of the sources and simulation programs resulting in modeled ECG and EEG signals . Those signals can be compared with the experimental results for normal or for some pathological cases . But the most interesting and the most important problem, which is a direct problem in medicine, is in fact an inverse problem in modeling. This problem is the location of sources using the EEG or ECG signals . First we shall analyze one of the important parts of the ECG signal called His bundle electrogram (HBE). We’ll apply various methods , including geometrical modeling , to the solution of the inverse problem in this case . Second, we’ll show a simple geometrical modification of a human torso model if the conductivity of the body differs from one point to another. Finally we’ll try to illustrate the application of geometrical modeling to the analysis of EEG signals. Our aim is to help medical specialists in their everyday practical analysis of biomedical signals, so the proposed methods are as simple as possible and are oriented to applications in clinics and medical centers using small personal computers.

2. Inverse problem for His bundle electrograms (HBE)

Many heart defects occur not in the myocardium, but in the specialized conduction system of the heart - His- Purkinje system (HPS) [1]. The excitation of HPS takes place during the P-R interval of the electrocardiogram (ECG). In 1969, Scherlag et al. [2] introduced an invasive catheterization measurement technique to record the electrical activity from the bundle of His and was called the “His bundle electrogram” (HBE). The limitations of catheterization have prompted various research studies into possible non-invasive measurement techniques [3-7], The major problem which faced researchers was that potentials which originate from the HPS as seen from the body surface are in the range of microvolts corrupted by muscular random noise of the same amplitude level and frequency bandwidth. High amplification and ensemble or space averaging was first used [3,4]. Al-Nashash et al., and others have used other recording techniques based on beat to beat basis [5,6]. Interest in the exact morphology of the HPS signals on the body surface associated with various conduction defects has lead to a number of modeling studies [7-9]. Despite the fact that these models produced acceptable results by showing that the simulated ramp signal is similar to the experimental results, still however, there are a number of deficiencies in these models which include:

1.The 2-D model is not accurate since the heart is a 3-D system.

2. Most models consider the HPS as a single dipole source at a given moment of time.

3.These models did not take the right bundle branch into consideration.

4. Assumption of a constant velocity throughout the HPS.

Earlier [10] we presented a modified model for the HPS which avoids the above four defects. It is based on the transformation of the two dimensional HPS sheet onto a three dimensional curvilinear system which resembles the ventricles in shape and dimensions. Computation of the surface potential is carried out using classical volume conductor theory . To compare simulation results with those obtained experimentally, the standard orthogonal X, Y, Z leads are used. As the same model helped us in solving the inverse problem, we’ll explain the main features of the model. The detailed description can be found in [10].

 3. Model of the His-Purkinje system.

Each ventricle of the heart is modeled as a part of a three-dimensional ellipsoidal cavity. The position of any point of the endocardium can then easily be defined in a spheroidal system of coordinates x , h , j , formed by a family of hyperboloids

(1)

and a family of ellipsoids

(2)

x = r cosj ; (3)

z = r sinj ; (4)

where,

x, y, and z are the Cartesian coordinates with the origin in the center of the ellipsoid,

y is directed along its major axis,

x is supposed to lie in the frontal plane of the torso and

c is the distance between the upper focus and the lower focus of the ellipsoid .

Equations (1), (2), (3) and (4) can be solved into:

(5)

 Each branch of the HPS is first modeled with a two-dimensional matrix. The central element of each 2-D matrix is placed in the lowest point of the corresponding ellipsoid (i.e. h = -1).

If i and j are the row’s and column’s indices respectively, then for any (i , j) element, the coordinates h and j are defined by:

(6)

(7)

 where (i0 , j0) are the indices of the central element.

Let X, Y and Z be the Cartesian coordinates of each element within the laboratory system connected to the torso with the origin in the middle of the heart. These coordinates are calculated through the corresponding coordinates given in equation (5) taking into consideration the position and orientation of the ventricles within the torso. Normal position of the ventricles requires to turn the ellipsoid around the z axis to an angle b so that

x = Xcosb +Ysinb +x0; (8)

y = Ycosb -Xsinb +y0; (9)

 where x0 and y0 are the coordinates of the center of the ellipsoid cavity.

The excitation and transmission process of the impulse from depolarized to polarized fibres through the His Purkinje System is modeled as the step by step depolarization of the neighboring polarized elements. At any step of the excitation process, an active element may excite the four neighbouring elements provided that these new elements belong to the His-Purkinje System and are not already active. The coordinates and the dipole moments for each active element were computed for every 0.5 millisecond for the complete excitation process of the HPS.

According to Plonsey [11] and others[12],[13], the human torso is mostly a conductive medium with small reactance, so the electrode potentials are defined mainly by the equivalent current sources of the active elements. To calculate the electric potential j at any electrode at the body surface, each active element is considered to be a source in Poisson’s equation:

(10)

 If the medium is conductive with conductivity s , then using Ohm’s law-s Ñ j = j and the equation of continuity , equation (10) can be reduced to:

(11)

where j is an equivalent current source density of the element during its activation and r is the electrical charge density. Equation (11) was solved numerically starting from the potential j 0 in the close vicinity of the equivalent current source. For each active element, j 0 was defined using dipole approximation.

(12)

The equivalent current source j for each active element was assumed to be of the same magnitude, though different source strength can easily be introduced in the initial HPS matrix. This can be achieved through the user interactive program which was designed to create the two dimensional HPS matrix. The direction of a source was defined by the configuration of HPS and by the propagation of excitation wave. It was computed automatically while calculating the set of active elements at a certain moment of time. The total potential at the electrode position was defined by the summation of the solutions of Poisson’s equation for each active element. The torso is modeled in the form of an ellipsoidal cylinder with the dimensions 35 by 30 by 20 cm in x ,y and z directions correspondingly. We use the standard system of coordinates where y is vertical, x is horizontal in the frontal plane and z goes horizontally from the back to the front of the body. The origin of the coordinates lies in the centre of the ellipsoidal cavity. The standard orthogonal X,Y,Z surface leads system described by Berbari [4] were used. In this system the x, y and z electrodes have the coordinates:x-electrodes:

x-electrodes: x=0.15m and -0.20m, y=0,z=0.

y-electrodes : x=0, y= -0.15m and + 0.15m, z=0.

z-electrodes: x=0, y=0, z= 0.07m and -0.14m.

At the present stage, the torso is assumed to be a uniform conductive medium with conductivity s and the solution for each element simply has the form of (12). Nevertheless the program gives the possibility to generalize the problem introducing secondary sources at the boundaries of the regions with different conductivity [8], [11], [13].

 4. Inverse Problem

The inverse problem consists of the source modeling using the set of potentials at every electrode as a set of data. If we assume that the source is a single dipole, than we need only six independent electrodes to locate the source and define the vector of the equivalent dipole p. We solve numerically the system of equations

(13)

 where n is the number of the electrode ( from 1 to 6),

(14)

 xn , yn , zn - coordinates of the electrode, while x, y, z - coordinates of the dipole. The solution gives us three coordinates of the dipole x, y, z and three components of the dipole vector px , py , pz .

The single dipole approximation is not full enough for many cases, and the real system of sources consists of a moving string with a distributed set of dipoles. Exact calculation for this case requires multiple electrodes and it is not practical. Geometrical modeling gives for this case an approximate position of the sources, including the defects in the heart conductive system. The results can be presented as a three dimensional image which is very practical for medical purposes.

Fig1. Location of the defect

Fig.2 HBE signal, used for reconstruction

 

Fig.3 Location of the reconstructed defect in 3D picture

Fig.1 shows the defect called right bundle branch block and left anterior hemiblock. The defect is reconstructed from the HBE signal shown in Fig.2. For the reconstruction the above described model of the nervous conducting system was used. The series of strings of active elements was modeled according to the timing of the signal . For every string the average dipole value and position was calculated by direct integration of elementary dipoles. The results were compared with those obtained from the signal in Fig.2 solving the system of equations for the equivalent dipole parameters. Near the defect, the comparison shows a big difference between the full string dipole and the dipole found from the signal analysis. This is the first indicator of the defect position. Estimating the dipole strength p from Fig.2 shows that only a half of the string is active at the defect region. Taking the position of the dipole from Fig.2 we can reconstruct the 3-D defect (Fig3.)

Unfortunately, for real experimental (not simulated) HBE signals the situation is more difficult. First, every heart has its individual size and shape, so the timing of the signal is approximate. In [10] we showed some possibilities to improve the accuracy of the timing. Even more discouraging is the absence of full data. Real signals are full of noises, and only the differences of electrode voltages are used practically after compensation of noises in a differential amplifier. A usual 6-electrode system gives three independent signals which can help to find the equivalent dipole components, but not the exact location of the dipole.

Nevertheless, some important conclusions can be made using those insufficient data. In Fig.4,5 we present the defect, reconstructed from experimental data taken from [4] and corresponding to the right branch block and left branch hemiblock. The procedure was the same as the one used for simulated signals, but a certain ambiguity in the defect position along the string comes from the absence of data which can define the exact position of the equivalent dipole. In this situation the dipole coordinates were supposed to be equal zero. Simulated results show that the dipole coordinates are always close to zero, compared with the electrode coordinates, but the real validity of this approximation should be checked by multiple experiments with 12-lead systems, giving 6 independent signals.

Fig.4. Approximate reconstruction of an HPS defect, using insufficient experimental data.

Fig.5 Location of the reconstructed defect in 3D picture

The inverse problem in the case of full ECG signals analysis is even more difficult than in the case of HBE signals. The sources are distributed all over the heart surface and their number and position depend on time. The amount of data received from cardiograph’s electrodes is definitely insufficient to localize all the sources. A number of simplifying models were developed [14-16]. The simple ones present the set of sources as a combination of 6…10 rigid dipoles in on-off operation. For this approximation a set of voltages from 12 points on the torso surface is enough for a reasonable estimation of the situation.

A standard inverse problem solution consists of the following steps.

First , the sum of potentials from every active dipole is defined at every electrode.

 Then the difference between the simulated sum of potentials and the measured electrode potential Vj is calculated , squared and summed over all electrodes.

(15)

 Next one of the methods like the steepest descent method is usually applied to minimize this sum and to find the best combination of active dipoles for a given set of the electrode potentials.

For a 9 or 10- dipole model and 12 electrodes the amount of calculation is reasonable, but the simulated potentials cannot repeat exactly the real ECG signal as a function of time[14 ].More elaborated models like 74 zone time dependent models, where the dipoles are switched on and off not instantly but according to a certain function of time give closer approximation to the experiment, dipole model require rather long time for calculation and often give a number of close solutions (local minima)[15]. Those models require the data from several hundred points on the torso (268 points in [14] ). Such a detailed mapping of the surface potentials is difficult in clinical practice.

Geometrical modeling is not as effective here as in the case of HBE, but it can improve the results obtained using the 10-dipole model and make them closer to reality applying to it the sample based interpolation method, proposed in [17]. For the interpolation one cane suppose that only the centre of each zone is fully on or fully off but the periphery of a zone has the dipole strength density defined through a minimum energy function [17]. Small amount of samples results in a small time of calculation which is necessary to find the smooth approximation of the scattered data. This method can be useful as well to create a full mapping of the body potentials from the scattered data of the electrode potentials. Later in this paper we discuss the implementation of this procedure for reconstruction of the EEG signal sources.

5. Non-uniform conductivity of the torso.

Usually in heart activity models an approximation of the uniform conductive torso is used. Now some good 3-D models of human torso are created (see, for example, [8,12,13]). One can introduce different conductivity to different part of the body, which is closer to reality, especially when we need both conductivity of lungs and that of muscles. In our calculation of HBE potentials and in the inverse problem we used the uniform torso conductivity model, but the potentials were calculated through step by step integration to keep the way open for the application of different conductivity for every step. However, 3-D models of the torso and iterative integration take extra time and memory. Geometrical modeling can significantly simplify this problem. The potential at a distance R from an elementary equivalent dipole in a uniform medium with conductivity s is equal (see equation (13)

 For a non-uniform medium we can introduce the effective length

similar to the optical length for a dielectric. Then we can simply redraw the torso model using the effective dimensions of every part instead of the physical ones. Such a distorted figure nevertheless is very useful for the potential calculations. Once created, this model of the torso can be used to find the effective distance from electrodes to any point inside the torso. Electrode potentials then are simply calculated as

(16)

 Fig.6 shows a torso model with an internal region with lower conductivity. Fig. 7 shows the modified model where all the dimensions are changed into effective ones. To make a smooth contact of two surfaces we used the R-function method, developed in [18] .

Fig.6. Human torso model with real dimensions of conductivity.

Fig.7 Equivalent dimensions of the uniform conductivity model

 6. Inverse problem in EEG.

For the case of EEG signals the important sources are brain defects (tumors, for example). The sources are immovable, but the system of dipoles is distributed over a surface, which makes the exact calculation even more difficult, than for the case of ECG signals. Usually methods of the inverse problem solution in this case are similar to those in ECG case [19 ], but the amount of calculations makes it necessary to use simplified models of the sources. Even for simplified models the correct definition of the source location using the minimum difference method is difficult because experimental EEG signals are noisy and the number of electrodes is limited. Geometrical modeling again can help showing the approximate position of the source in the form of an image. We tried to find a graphical solution of the inverse problem for the set of experimental data taken from [20]. The method of the solution is similar to the method used for HBE signals, the brain region was modeled as a uniformly conducting sphere. Signals only from eight electrodes were available, so the interpolation method [17] was used to find the approximate potential mapping on the brain surface. The source was modeled as a single sphere with uniform density of the dipole strength over volume.

Fig.8. Approximate reconstruction of a brain defect, using insufficient experimental data.

The coordinates of the centre and the radius of the sphere were defined and shown in Fig.8. Subsequent redistribution of the electrodes can improve the detailed description of the approximately located source.

7. Conclusion

Geometrical modeling is a promising new approach in biomedical signal processing. It can simplify the solution of the inverse problem of the electric source localization, the information can be presented in the form of images, which make possible a visual analysis of the situation. Very important for clinical application is the fact that some information about the sources of biomedical signals can be obtained even if the amount of data is not sufficient for the full solution of the inverse problem using traditional analytical or numerical methods. Further development of computers and computer software without any doubt can transform geometrical modeling in a powerful instrument for the analysis and processing of biomedical signals.

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Computer Graphics & Geometry