2. MODELING OF GASTRIC ELECTRICAL ACTIVITY
2.1. MAIN CONCEPTS IN MODELING OF GASTRIC ELECTRICAL ACTIVITY.
In the last decade there has been a substantial interest in non-invasive electrogastrography (EGG) as a possible diagnostic tool in gastroenterology. Cutaneous EGG signals from the stomach have been recorded since 1922 (3), but the poor quality of the recordings and the difference between the electrical parameters of cutaneous EGG and those of the internal gastric electrical activity (GEA) complicated clinical applications of the method (10).
Mathematical modeling of the electrical phenomena that occur in the human stomach can solve some of the above problems. There are two general approaches towards this type of modeling:
(a) "passive" models (34 ,35, 36, 37), which try to represent a precise picture (map) of voltage and current distribution on the stomach surface usually by means of coupled relaxation oscillators;
(b) "active" models (11, 32, 33) in which dipole theory is applied to describe mathematically vector characteristics of the electrical field produced by the stomach. Using the characteristics of gastric electrical field a theoretical value for the electrical potential can be obtained at any point of interest around the stomach.
Active models can be divided into two main streams themselves: models that try to represent the electrical phenomena in the stomach by means of current dipoles and models that represent the above phenomena electrostatically as a distribution of electric charge in dielectric medium. According to Plonsey and Barr (38) these two approaches could be considered dualistic.
Passive models, although reasonably accurate in representing voltages and currents on the stomach surface, cannot simulate recordings from the areas surrounding the stomach including on the abdominal surface. Therefore, for the purpose of EGG applications, active models are far more valuable. There are two known active models: "cylindrical" (11), which is the first model to use the dipole theory, and "conical" (32, 33), which applies the "annular band" concept when describing the dipole distribution and movement.
Of the two active models the latter is more related to the electrophysiological phenomena that occur in the human stomach. Some of the main concepts of this model that are widely accepted as fundamental are (32, 33) :
(a) electrical activity originates in the mid corpus of the stomach and is present throughout the terminal antrum but is not found in the orad corpus and fundus;
(b) the component of the stomach electrical activity known as Electrical Control Activity (ECA) is always present, i.e. it is not intermittent in time;
(c) ECA is associated with the distal movement of a d-wide annular band polarized by electrical dipoles pointed perpendicularly to the stomach axis;
(d) the dipoles in the rest of the stomach have random polarization and generate null potential at every point of space;
(e) the fundamental frequency of ECA is about 3 cycles per minute (cpm);
(f) the propagation velocity is slow in the corpus but increases distally;
(g) the ECA recorded from different points of the stomach wall shows a phase lag in the distal direction;
(h) the abdominal surface can be represented by means of a surface a and is separated from the stomach by a dielectric.
Despite these important assumptions the conical model has two major disadvantages which are avoided in the proposed conoidal model. The propagation velocity along the greater and the lesser curvature of the stomach is different, which explains the pattern of electrical coupling between different parts of the stomach (39, 40, 41). This difference could not be taken into account by the conical model. The conical model is developed in Cartesian coordinate system, which complicates the model equations and much simplification is required to make these equations solvable. Eventually because of these simplifications the model becomes rather cylindrical (32, 33). Taking into account the above facts the following new concepts are considered in the most recent conoidal model of gastric electrical activity:
(i) the waveshape of gastric electrical activity as recorded in experiments in vivo depends on the position and configuration of the recording electrodes and is not, in general, the same as the waveshape of intracellular ECA;
(k) normally the velocity of propagation of ECA is different along the greater and the lesser curvature of the stomach, so that the depolarization waves in the greater and lesser curvatures spread simultaneously through any radial plane perpendicular to the stomach axis despite the different route that they have to pass, i.e., there is full electrical coupling between different parts of the normal stomach;
(l) for the purpose of this modeling, the form and the shape of the stomach is most closely represented by a truncated conoid with a finite length;
(m) the modeling is to be done in spherical system of coordinates with no transformations into Cartesian coordinates, thus avoiding any unwanted mathematical simplifications;
(n) using the model, researchers should be able to do fast and efficient simulation of experiments with both invasive (attached on the stomach wall) and non-invasive (placed on the abdominal wall) electrodes, and obtain a plot of the potential (when monopolar recording is simulated) or the potential difference (in the case of bipolar recording simulation) versus time;
(o) Electrical Response Activity (ERA) could be responsible for some power changes in cutaneous EGG signals, but in general it has no significant effect on the spatial and temporal organization, frequency, velocity of propagation, waveform, phase lock, and coupling of the signals and therefore it can be ignored when modeling ECA.
The conoidal dipole model of gastric electrical activity has been incorporated into a computer program (TURBOC++, Borland, 1990) that shows a picture of a truncated conoid (the stomach) on the screen and allows the user to place monopolar or bipolar electrodes simulating real experiments. The output of the program MODEL plots the dynamics of the potential or the potential difference (depending on the chosen electrode configuration) calculated after the simulation. When doing cutaneous simulations the abdomen is represented as a plane on which the stomach is projected. Actual recordings of gastric electrical activity obtained with implanted electrodes were compared with the results from the simulation with the program MODEL initially for mid-corpus electrode pairs and after that for antral pairs. Comparing the results from the tests with those from the computer model it can be suggested that the model represented relatively well stomach anatomy and electrophysiology and could be used as a tool for better understanding the parameters of human electrogastrographic signals.
2.2. STUDY OF THE ELECTRODE DISTANCE IN BIPOLAR RECORDINGS.
A simple example of the usefulness of the model could be a study of the effects of electrode distance on GEA recordings. Figure 2.1 represents a simulation of an experiment with implanted bipolar electrodes in the antral area with a distance varying from 5 mm to 2 cm.
Figure 2.1. Effect of gradual increment of the distance between serosal bipolar electrodes in simulated experiments. The initial inter electrode distance of 0.5 cm (Channel 1) was successively increased with 0.5 cm. In all recordings positive electrode was placed in the proximal antrum, while negative electrode was moved towards terminal antrum.
Increasing the electrode distance increased the gap between the two potentials changing the obtained voltage from the well-known biphasic signal to a signal with relatively narrow frequency spectrum. The closer the electrode was placed to the terminal antrum, the higher the amplitude of the obtained potential was; this also contributed to the waveshape change.
2.3. STUDY OF THE ELECTRODE SURFACE AREA.
In the previous section the potential V at a point of interest Q was investigated. Converted into a real experiment this means that the measuring electrode had an extremely small surface area, which could be a reasonable assumption, provided serosal needle electrodes were used. When an experiment with wider electrodes was modeled, one more integration over the electrode surface area was required. Figure 2.2 shows the impact of increased electrode surface area on simulated experiments with antral long distance bipolar (LDB) electrodes. The electrode diameter was increased gradually from 2.5 mm to 1 cm. The greater the electrode surface, the closer to a sine wave the signal became, i.e. the spectrum of the signal became narrower. Increment of the electrode surface area increased the averaging effect upon the obtained potentials, and changed the waveshape of the obtained signal gradually transforming it to a signal with significantly fewer frequency components.
Figure 2.2. Effect of gradual increment of electrode surface on GEA wave shape in simulated experiments (A). Electrode diameter was increased from 2.5 mm (Channel 1) to 6.5 mm (Channel 2), and to 1.0 cm (Channel 3). Distance between the electrode centers was kept 4 cm. The signal in Channel 3 resembled the wave form of cutaneous EGG (B).
2.4. EFFECT OF INCREASED DISTANCE BETWEEN AN ELECTRODE AND THE FIELD SOURCE.
The electrostatic approach (38) suggests that the following two factors would influence the signal when the distance between the electrode and the source of the electrical field changes: (a) the cosine of the angle between the vector of the equivalent dipole moment P and the vector distance r from the infinitesimal area dS of the d-wide band of polarized cells to the recording electrode; (b) the magnitude of the vector of the distance r .
Considering that vector P is always perpendicular to the radial plane of the current ring of depolarized cells, one can easily see that moving the electrode (the point of interest) away from the serosa (or mucosa) would bring the cosine of the angle (P, r) close to zero. In addition, the increment of magnitude of vector r contributes to a further parabolic reduction of the potential at the point of interest Q. Therefore, when measuring the EGG the best approach would be to reduce the distance between the serosa and the abdomen. This distance is less in thin people, and that is possibly why some success was reported in recording time shifts (phase lags) between different cutaneous EGG channels from thin volunteers (41). Another approach to reduce the distance r and the angle (P, r) would be to distend the stomach. Unfortunately, stomach distention changes the conditions under which the stomach operates and this affects the parameters of cutaneous gastric electrical activity (42). However, stomach distention may be the only way to get reliable information from cutaneous EGG signals, because a distended stomach would be closer to the recording abdominal electrodes. This could help to overcome the negative effect of obesity on EGG.
Although always questionable, modeling of bioelectrical phenomena, if done properly, can be valuable not only in theoretical studies but also as a tool that replaces the need for expensive and some times very uncomfortable experiments. Mirrizzi et al. (1985, 1986; 32, 33) made a significant step in modeling of the stomach electrical field, proposing their "truncated cone" model. Unfortunately no further improvements or clinical tests of this model were published.
The methodology followed in this study was not only to create the new, "truncated conoid" model based on the electrostatic approach, but also to implement it for theoretical assessment of the correlation between the obtained biovoltages, different electrode positions and different electrode surfaces. In that aspect this type of modeling is quite convenient, because with the recent computer technology it allows almost immediate results after the electrode parameters and positions are determined by the investigator.
The following main conclusions could be suggested as a result from this study.
Different electrode configurations have profound impact on GEA signals. Increment of the distance between individual electrodes in bipolar recordings and increment of the electrode surface area make the spectra of the recorded signals narrower. Moving electrodes away from the stomach dramatically reduces the signal-to-noise ratio and one way to improve it is with gastric distention. Additional simulations can be performed with the model to examine quantitatively this effect, but it can be estimated that the decay in the amplitude is reversely proportional to the square of the distance between the electrodes and the stomach (providing the electrodes are with one and the same active surface area). A high quality GEA amplifier with flexible frequency characteristics is required when attempts are made to obtain the cutaneous EGG, especially when electrodes with smaller active surface area are to be used.
The proposed model can be a useful testbed for many other simulations. For example, uncoupling or irregularities can be simulated by changing the way the depolarization ring propagates in order to study how these two phenomena are recorded with different electrode configurations.