
 
Silny J. Intraluminal Multiple Electric Impedance Procedure for Measurement of Gastrointestinal Motility // Journal of Gastrointestinal Motility 1991; 3(3):151162.
Intraluminal Multiple Electric Impedance Procedure for Measurement of Gastrointestinal MotilityJiri Silny HelmholtzInstitute for Biomedical Engineering, University of Technology, Aachen, Germany A new catheterrelated procedure for highresolution measurements of gastrointestinal motility is presented. The method is based on simultaneous acquisition of the electric impedance in the surrounding body volume conductor from a number of annular electrodes, successively arranged on the catheter. The impedance of a volume conductor around the catheter, consisting of a bolus, the organ wall, body fluids, and so forth, has a characteristic value for each segment and phase of contraction wave, as theoretical and experimental investigations revealed. A calculation on the basis of a simplified model reveals that the impedance changes as a logarithmic function of the bolus thickness, in which the highest sensitivity is advantageously obtained at a small lumen size and utilizing unipolar or bipolar electrode setups. The high resolution in time, space, and amplitude of the changes in the bolus shape allow us to use this impedance method for evaluation of functional stages in each measuring segment and of the beginning, end, and type of contraction wave, as well as their characteristics. The unique mechanical properties of the catheter (thinner than 3 mm, several meters long, flexible, closed surface) and the ability to distribute more than 32 measuring segments of different lengths on the catheter make this procedure suitable even for longterm physiologic and pathologic studies of gastrointestinal motility. (Journal of Gastrointestinal Motility 1991; 3(3):151162)Introduction One of the most important functions of the gastrointestinal tract (GIT) is the transport of food and fermented material from one digestive or resorption stage to another. This is based on propulsive mechanisms distributed along the whole length of the organ. In this way, one or several contracted segments of varying lengths push the pulpy or liquid food through a muscular tube, leading to the mixture and movement of its contents. These contraction waves are produced by smooth and striated muscles embedded circularly and longitudinally in the organ wall. The organs of the GIT are separated by sphincters, which allow flow in only one direction under physiologic conditions. The bolus in an organ can be segmented, mixed, or moved in different directions depending on the degree of the wall contraction and the direction of the contraction movement. The parameters of the transport and mixing processes in the organs of the GIT differ considerably in adaptation to the graded digestion in the alimentary tract. For example, just one parameter, the average transport velocity, comes to 3 to 4 cm/s in the main transport organ, the esophagus (1), whereas it is slower than 1 to 2 cm/s in the intestine (2). The transport and mixing processes in each organ of the GIT also differ in time and space and are subject to numerous physiologic and psychological influences. The resulting peristaltic activity can therefore be characterized as a time and spacedependent, nonstationary process. These contraction patterns have been used for diagnosis of dysfunctions of the GIT. The following general measurement rules should be followed to obtain accurate results.
Catheters with semiconductor pressure microtransducers do show excellent dynamic characteristics, but they are costly and relatively fragile (5). Therefore it is necessary to look for new and better procedures for the measurement of peristaltic waves in the organs of the GIT. In this context a new procedure is presented here in which the peristaltic wave is simultaneously measured by intraluminal electric impedance changes near many annular electrodes placed sequentially on the catheter. Intracorporal and extracorporal applications of the electric impedance technique for the characterization of different physiologic processes and anatomic structures have already been suggested and partly proved many times in experiments (6, 7). A theoretical approach to the determination of the motility in the organs of the GIT by means of intraluminal electric impedance measurement, however, has not yet been presented systematically in the literature. The impedance measurement of the intracardiac volume and related problems have been dealt with extensively (6, 8, 9). But, as the transport processes in the GIT are basically different from the heart pump function, the deduced principle cannot be translated to this problem. Therefore this paper deals not only with an explanation and the experimental proof of this procedure in the esophagus and intestine, but also with calculations of the electric fields and impedance conditions around the catheter before, during, and after a contraction wave. Phases of the Motility Wave The mechanical transports as well as the mixing of food and fermented material take place in all the organs of the GIT with comparable mechanisms (5). Narrowing or complete constriction of neighboring segments in an organ lead to segmentation and mixing of the contents. A contracted segment that moves at a velocity, V, pushes the bolus in front of it. A contraction wave at a given time can be divided in a simplified way into four typical phases, as shown schematically in Figure 1.Fig 1. Phases IIV of mechanical peristaltic activity in the esophagus, 1 = saliva film, 2 = organ wall, 3 = bolus, 4 = body During the resting phase I the wall muscles are relaxed with the exception of the sphincter regions. In the resting state the organ walls lie smooth or folded together and swim on a thin fluid film. In phase II the organ wall is extended by the bolus. It remains to be seen how far this extension is affected by the force of the pressed bolus or by active opening mechanisms. A fully contracted segment that pushes the bolus is associated with phase III. In phase IV the muscular tube is relaxed again and the organ walls can touch each other. The fluid film, which is transported away with the bolus, is slowly built up again. Distribution of different physical characteristics, such as wall tension, lumen size, or pressure in the muscle tube, can be used for the quantitative description of the contraction wave. In phases I and IV the wall tension and pressure are clearly at their lowest values, whereas the lumen size can differ from its minimum value (phase III). Phase II is marked by increased wall tension and maximal lumen size. In this phase the pressure shows only a very small increase, which is frequently difficult to measure. The wall tension and pressure are at their maximum values in phase III, whereas the lumen size reaches its minimum here. Measurement of wall tension and lumen size would yield the best distinction between the different phases of the contraction wave. Unfortunately, however, there are no sensors that can directly register the wall tension in situ without injuring the wall. By measuring changes of pressure during a contraction wave we can recognize only phase III, in which the organ is totally contracted. Most procedures widely used in clinical practice for the determination of motility, such as perfusion catheters or solidstate transducers (5), give the wall pressure of only a small wall segment. The change in the lumen size during the contraction wave could be the second most important parameter besides wall tension, as it changes from phase to phase. A catheterrelated electric impedance technique adapted to this special problem could represent the change in the lumen size in a simple way. Results of a rough estimation of the impedance around an annular measurement electrode and its relationship to the phases of the contraction wave and different parameters are presented below. Impedance Measurements: Theoretical Approach For measuring impedance a plastic catheter is placed in the esophagus or intestine. Metallic cylinder electrodes with a radius of p_{0} and length of L = 2l are attached at a distance of d on the catheter (Fig. 2). The ring electrodes are connected with terminals outside the body by means of thin wires which run inside the catheter. This setup can be used to measure an impedance Z_{B} between two electrodes on the catheter (bipolar measurement) and impedance values Z_{u} between one electrode on the catheter and one large reference electrode on the body surface (unipolar arrangement).Fig 2. Intraluminal electric impedance catheter in the esophageal tube during deglutition with characteristics of the catheter and of the volume conductor, and electric conductivities of the body σ_{B}, the wall σ_{w}, and the bolus σ_{F}. 1 = saliva film, 2 = organ wall, 3 = bolus, 4 = body, 5 = air, 6 = impedance catheter electrode The question is how these impedances change depending on different parameters in bipolar or unipolar configuration when a contraction wave runs along the catheter. For the calculations in the Appendix, the following simplifications are made:
The second layer {outer infinite conductor) represents the body in phase III (σ_{2 }= σ_{B}); an equivalent conductor for the wall and the body in phase II (σ_{2} = σ_{B}/σ_{w} + σ_{B}); and the body in phase II, if σ_{F} ≈ σ_{w} (σ_{2} = σ_{B}). The most important results of the calculations in the Appendix are visualized in Figures 3 to 9. Fig. 3. Normalized impedance in the unipolar measuring arrangement as a function of the bolus layer thickness D for different ratios σ_{1}/σ_{2} and a given electrode geometry (ρ_{o} = 1.5 mm, l = 2 mm) The resistances R_{u} and R_{B} in Equations (16) and (18) (see Appendix) are in a logarithmic relationship to the thickness D of layer I (see Fig. 14), and they depend on the conductivities σ_{1} and σ_{2}. Figures 3 and 4 depict the connection between the products σ_{1}R_{u}, σ_{1}R_{B}, and D for a realistic variation of thickness D and the ratio of conductivities σ_{1}/σ_{2}, as well as for practicable dimensions of the catheter. The radius po of the catheter and length L of each electrode as well as the distance between the electrodes should be determined for the construction of the catheter. Practice has shown that catheters with ρ_{o}  1.5 mm, L = 34 mm, and d = 1 2 cm can be constructed relatively easily, that they have good mechanical properties, and that they provide a precise enough solution of motility in esophagus, stomach, and intestines. The effect on organ function of a catheter with a diameter of 3 mm can also be tolerated for most sections of the GIT. Therefore, in all graphs parameters within these ranges are introduced as examples. For the chosen configurations, the bipolar setup gives a nearly double resistance value R_{B} (Figs. 3 and 4). Fig. 4. Normalized impedance in the bipolar measuring arrangement as a function of the bolus layer thickness D for different ratios σ_{1}/σ_{2} and a given electrode geometry (ρ_{o} = 1.5 mm, l = 2 mm, d = 10 mm) Here the sensitivity ΔR/ΔD, as a ratio between the resistance change and a small variation of the thickness D, decreases with the thickness D in both the unipolar and the bipolar arrangements. Furthermore, the sensitivity is higher in the bipolar arrangement than in the unipolar array. The rate of increase depends on the electrode size, the interelectrode spacing, and the thickness D. In Figures 3 and 4, we can distinguish between two regions with σ_{1 }>_{ }σ_{2} and σ_{1} < σ_{2}. The courses of the resistances are symmetrical about the axis σ_{1} =_{ }σ_{2} for different reciprocal values of σ_{1}/σ_{2} in both setups. These graphs allow explanation of the change in resistance in the individual phases of the contraction wave in the esophagus and intestine, as discussed below. Figure 5 plots the resistance dependency of the length / of the half electrode at a predefined thickness D and conductivity ratios σ_{1}/σ_{2} as well as a possible catheter configuration. The resistance diminishes slowly with l especially in the unipolar configuration. Fig. 5. Changes of normalized impedance as a function of the electrode length in the unipolar and bipolar arrangements for different ratios σ_{1}/σ_{2} and a predefined electrode geometry (ρ_{o} = 15 mm, d = 2 cm) The variation of the interelectrode spacing d in the bipolar setup causes a strong change in the resistance R_{B} only at very short distances (Fig. 6). In the range of realistic interelectrode spacing d > 0.5 cm, only a weak increase of the resistance can be noticed. Figures 7 and 8 show the normalized longitudinal distribution of the normal component of the current density within the boundary ρ = ρ_{1} (see Fig. 14) for both unipolar and bipolar arrays arid different parameters of the catheter and conductors, chosen as examples (see Eq. [20] in Appendix). Both plots demonstrate the longitudinal sensitivity (see Eq. [22] in Appendix) of the unipolar and bipolar setups for given example parameters. Fig. 6. Changes of normalized impedance as a function of the electrode distance in the bipolar array for different ratios σ_{1}/σ_{2}and for a given geometry of the measuring setup (ρ_{o }= 1.5 mm, l = 1.5 mm) Fig. 7. Longitudinal selectivity of the unipolar arrangement derived from the normal component of the current density within the boundary ρ = ρ_{1} for given electrode geometry (ρ_{0} = 1.5 mm, l=2 mm) and different thicknesses of the bolus layer (a, D = 1 mm; b, D = 5 mm) as well as for given conductivities (σ_{1} = 4 mS/cm, σ_{2} = 0.4 mS/cm) Fig. 8. Longitudinal selectivity of the bipolar arrangement derived from the normal component of the current density in the boundary p = Pi for a given electrode radius p_{0} = 1.5 mm, different electrode distances, and chosen thicknesses of the bolus (a, D = 1 mm, d = 10 mm; b, D = 1 mm, d = 5 mm; c, D = 5 mm, d = 10 mm) at given conductivities (σ_{1} = 4 mS/cm, σ_{2} = 0.4 mS/cm) Figure 9 presents the notated longitudinal bandwidth for 90% of the total current (see Eq. [21] in Appendix) and selectivity as a function of thickness D for unipolar and bipolar measurements. In both cases a reduction of the longitudinal bandwidth and an increase in selectivity follows the decrease of the layer thickness. This comparison shows that the bipolar array has a greater bandwidth and a much higher selectivity. The sudden loss of sensitivity in the middle of the bipolar arrangement (Fig. 8) could be considered a disadvantage. No essential information, however, is lost through this characteristic if the distance between the electrodes is shorter than the length of the bolus. This condition can be fulfilled in practice. Fig. 9. Longitudinal bandwidth and selectivity of the unipolar and bipolar arrangements in the boundary ρ = ρ_{1} as a function of the bolus layer thickness D in unipolar and bipolar setups (ρ_{0} = 1.5 mm, l=2 mm, d = 10 mm, σ_{1} = 4 mS/cm, (σ_{2} = 0.4 mS/cm) Experimental Proof MethodsA setup as shown in Figure 10 was used in the first experimental investigations of the impedance method in the esophagus and intestine. The essential parts of the equipment are an impedance catheter (10) and impedancevoltage transducers for eight channels. There are nine thin metal ring electrodes mounted on the lower part of a 2mlong plastic catheter with a diameter of 3 mm. The center distance between each 0.4cmlong electrode is 2 cm. Each electrode is connected with a thin wire that runs inside the catheter over a connector to the rows of a crossbar distributor (see the horizontal lines in Fig. 10).Fig. 10. Setup for electrical impedance measurement: (•) connected; (O) unconnected The tenth row is contacted with a reference electrode 200 cm^{2} in size. This metal electrode is attached to the surface of the thorax using a conductive gel. In the vertical direction of the crossbar distributor eight impedancevoltage transducers are connected with their virtual inputs. Depending on the wiring of the distributor we can obtain a bipolar (channel 1 in Fig. 10) or unipolar (channel 8 in Fig. 10) setup for each channel. The specifications of each impedancevoltage transducer are as follows:
Impedans of food As theoretical consideration has shown, the electric conductivity σ_{F} of the organ contents has a great influence on the measured impedance. Especially in the esophagus, the transported bolus can assume very different conductivities depending on the kind of food. Therefore we carried out a preliminary study to determine the conductivity for some soft foods, liquids, and saliva. The fourelectrode measurements (11) are applied in an airtight measurement container (area 1 cm^{2}, length 1 cm) at 1 kHz frequency and a current of 6 μA. Table 1 gives a set of measured conductivities and their standard deviations, both related to a preset temperature, measuring frequency, and some additional data from the literature. Table 1. Conductivities of Different Tissues, Fluids, and Soft Foods
Measurement in the Esophagus Figure 11 presents three typical courses of the impedance change measured simultaneously in different parts of the esophagus before, during, and after an arbitrarily chosen swallowing of yoghurt. The upper signal marks with the arrow S the beginning of the swallowing process in the region of the larynx. The three impedance curves Z_{1}, Z_{7}, and Z_{8} can be subdivided into four characteristic phases following the sequences of the contraction wave (Figs. 1 and 2). To check these courses we consider first the constraint sequence of the functional phases and the conformity with the theoretically derived connections. Moreover, experimental investigations with combined catheters (bipolarly measured impedance and manometry), the results of which will soon be published, confirm these theoretically derived connections. In this respect phase III (Fig. 11) promises to give the best confirmation of the theoretical approach, as here only the organ wall with a low conductivity is located between the catheter and the highly conductive volume conductor of the body. This phase corresponds to an entirely contracted segment, and therefore the impedance has in each segment a characteristic and reproducible value of С independent of the swallowed food.Fig. 11. Impedance changes in the bipolar setup in the esophagus during deglutition of 15 ml of yoghurt: phases I, II, III, and IV of a peristaltic contraction A = bolus front, b = maximal bolus diameter, С = constriction, S = beginning of swallowing, T = time difference; 1 = saliva, 2 = wall of muscular tube, 3 = bolus The impedance values in points С (Fig. 11) reach about 1.5 kΩ in all three curves Z_{1}, Z_{7}, and Z_{8}. From this value the average conductivity of the esophageal wall σ_{W} can be estimated. For example, a realistic value of σ_{W} = σ_{1} = 0.35 mS/cm results from Figure 4, point 1, considering the wall thickness to be 3 mm and the conductivity of the surrounding body to be σ_{B} = σ_{2} = 7 mS/cm, as some in vitro measurements with pig esophagus have indicated. Phase II corresponds to the filling of each organ segment with food. By choosing appropriate food with a conductivity σ_{F} » σ_{W} the bolus of a known conductivity σ_{F} = σ_{1} is modeled by layer I (see Fig. 14). The infinite conductor II represents the organ wall and the surrounding body in this example. Thereby the conductivity of the wall is decisive for the introduced equivalent conductivity σ_{2}, as its value is several times lower than the conductivity of the body (if σ_{B }» σ_{W} then σ_{2} ≈ σ_{W}) (see Eqs. [16] and [18] in Appendix). A growing diameter of yoghurt bolus with a conductivity σ_{F} = σ_{1} = 3.2 mS/ cm (Table 1) and σ_{1} ≈ σ_{F }= 0.35 mS/cm (σ_{1/}σ_{2}) causes a decrease of impedance (upper part of Fig. 4). The functional relation between Z and D (Fig. 4) allows an assessment of the thickness D from the measured impedance in phase II (Fig. 2). The minimal impedance value of 270 Ω in point В (Fig. 11) results from Figure 4, point 2, a realistic maximum bolus layer thickness of D = 4 mm. From the delay between two arbitrary points A_{M} and A_{N} or С_{M} and C_{N} and the central distance between the channels M and N the average velocity of the bolus front V`_{MN} and of tne contraction wave V_{MN} can be calculated. Furthermore, the time difference T_{AC} between the points A_{M} and C_{M}, the velocities V`_{MN} and V_{mn} and the longitudinal bandwidth B, (see Appendix) allow us to estimate the bolus length for each measuring segment M: in front of the bolus BL = Т_{AC} • V_{M}  В_{9}, and Small intestine measurments In these investigations the impedance catheter (Fig. 10) is inserted through the nose, esophagus, and stomach into different parts of the small intestine. The final position of the catheter is checked by xray, whereafter the catheter is attached to the nose. Figure 13 demonstrates a record of the motility with eight impedance channels from a 16cmlong region of the proximal jejunum in a volunteer 20 minutes after a meal, for example. In all traces recurrent resting (R) and active (A) phases appear simultaneously. The impedances in the resting states have a similar value of 214 ± 17 Ω with each trace as well in the interchanneling comparison. The active phases are manifested by oscillations of the impedance around the resting level. On the average the maximum of the impedance amounts to 300 Ω and the minimum to about 100 Ω in this phase. Table 1 shows that the electrical conductance of the bolus in the small intestine generally has a high electric conductivity of about 8 to 12 mS/cm, which is comparable to the conductivity of the body fluids (11). In the volume conductor considered, consisting of the bolus, the organ wall, and the body fluids, the impedance along the catheter can change in connection with the transit of a bolus only if the conductivity of the intestine wall is lower than the conductivity of the body and of the bolus. These conclusions coincide with common experience, that tissues like muscles or epithelia have essentially lower conductances than blood or body fluids (11). The conductivity of the intestine wall, σ_{W} = σ_{}_{1} =2.1 mS/cm, can be estimated for the maximum impedance of 300 Ω under the assumption of the conductivity, σ_{B} = σ_{2} = 7 mS/cm, in the remaining body and a layer thickness of the organ wall of D = 3 mm, as shown in Figure 4, point 4. The plot for σ_{F} = 8 mS/cm and the ratio σ_{1}/σ_{2} = σ_{F}/σ_{W} = 4 in Figure 4 characterizes the change of the impedance as a function of the thickness D. Consequently, the maximum impedance of 300 Ω in the traces in Figure 13 corresponds to the constriction D = 0 in the measuring segment, and the minimum of 100 Ω indicates a bolus layer thickness of D = 4.5 mm (point 5 in Fig. 4). If the catheter radius is 1.5 mm, the outside diameter of the bolus could maximally reach 12 mm. In the same way, for the resting state with an impedance of 214 Ω only a thin bolus layer (D < 1 mm) can be derived. The impedance patterns in Figure 13 demonstrate the typical contraction activity of the small intestine, known as the migrating motor complex (2). In the active phase the frequency varies between 10 and 12 contractions per minute. Only some strong contraction waves fully occlude or dilate the intestinal tube. These intensive peristaltic waves are propulsive in most cases. An average transit rate of 1.5 cm/s results from the time and space comparison between channels one and eight. Fig. 13. Impedance tracings from the jejunum recorded with the bipolar setup shown in Figure 10. Discussion A new approach for intraluminal evaluation of gastrointestinal motility, based on multiple electric impedance measurements, has been verified in theoretical and experimental studies.In the theoretical considerations several simplifications discussed below have been made to describe the measuring problem analytically. At a frequency of 1 kHz, the impedance of the tissue can be replaced by a resistance without any essential error, as the capacitive reactance is low (11). The annular electrodes are modeled by concentric finitelength line sources. In this way, an ellipsoid forms the fictitious electrode surface with a desirable nonuniform distribution of current density. On the other hand, the surface area of such a model is dependent on the surrounding volume conductor, and this area is generally greater than the surface of the modeled cylindrical electrode. Underestimation of the impedance, especially in the case of a thin bolus, is the consequence of this simplification. The volume conductor around the electrodes is simulated by two concentric cylindrical layer conductors of different conductivities and infinite extension of both layers in the longitudinal direction and of the outer conductor in the radial direction. To discuss these points one must consider the electric field around the electrode. The current density decreases rapidly with increasing distance from the electrode. Within a radius of about 2.5 times the length of a slim electrode the current density comes down to a few percent. This means that only a small pickup area of the volume conductor around a short electrode with a small radius is responsible for the measured impedance. In the case of such a limited area it is possible to decide on the advantageous division into two representative layers without making an essential mistake in the simulation. The small pickup area also allows the assumption of an infinite layer extension, as both the bolus and the investigated organ are much longer. Despite these inaccuracies, the results offer important information about the influence of various parameters on the impedance. Essential parameters are, for example, geometry of the electrodes as well as thickness and conductivity of the bolus in unipolar or bipolar measuring arrangements. According to the calculations the impedance is a reciprocal function of the logarithm of the bolus layer thickness D in both the unipolar and bipolar setups. Consequently, a change in a thin bolus advantageously produces a much stronger impedance alteration than the same thickness change would in the case of a thick bolus. The sensitivity of these changes is about twice as high in the bipolar arrangement as in the unipolar setup. The procedure measures an average impedance from one interval, which is denoted as a longitudinal bandwidth. The longitudinal bandwidth grows with the bolus thickness in both arrangements. Moreover, it is also determined by the electrode length in both arrangements, and by the distance of the electrodes in the bipolar array. The longitudinal bandwidth of the bipolar array, and thus the distance between the electrode pair, should be shorter than the expected bolus length. Otherwise, due to the characteristics of the bipolar setup, the transit of the bolus will cause two responses. At an appropriate electrode distance the bipolar setup supplies a much better longitudinal selectivity than the unipolar array. This advantage is very important if the measuring segments are arranged successively. Experimental investigations have been carried out in healthy volunteers to verify the theoretical approach. The intraluminal multiple impedance measurements in the esophagus and in the small intestine yielded traces that characterized the mechanical peristaltic activity in detail. In each trace, which represents one organ segment of predefined length, it is possible to distinguish between the resting states, the bolus transit, and the contraction of the wall. This applies to both the unipolar and the bipolar arrangements. The features of the unipolar and bipolar impedance measurements are compared in a separate esophageal study. Both arrangements directly supply distinct motility curves without breathing artifacts or other disturbances. The sensitivity and selectivity are evidently higher in the bipolar than in the unipolar setup, as was predicted theoretically. The advantage of the unipolar arrangement could be the requirement for only half the number of measuring electrodes on the catheter for the same number of channels if the interchannel spacing is greater than the distance between the measuring electrode pair in the bipolar setup. From each impedance trace a number of parameters can be derived, which describe an average activity in a defined organ segment. The starting and final time or the duration of the bolus transit as well as the extrema of impedance are important characteristics of each channel. On the basis of several impedance tracings the contractile patterns can be classified as segmenting, propulsive, or repropulsive functions. For the propulsive or repropulsive activities the average transit rate of the contraction wave and the bolus front can be calculated from the time delay of two events in different traces and the longitudinal distance between these channels. Moreover, on the basis of all parameters, the estimation of the bolus or lumen shape along the measuring catheter also seems to be possible. These connections as well as the experimental comparisons of the impedance procedure with other established manometric methods would go beyond the scope of this article, however, and will therefore be treated in a forthcoming paper. Acknowledgments For the continuous support of this work I want to thank the director of the HelraholtzInstitute, Prof. Dr. rer. nat. G. Rau, Aachen.Appendix: Resistance of Ring Electrodes in an Infinite Volume Conductor In the first analytical approach the fields of the electrodes are simulated by centric current line sources with a length L (13,14). The problem is assumed to be quasistatic. In the cylindrical coordinate system the distance between one point P and a point of the line sources e_{1} or e_{2} is r_{1}(ξ) or r_{2}(ξ) (Fig. 14).Fig. 14. Line sources in an infinite twolayer conductor in cylindrical coordinates for simulation of the measuring electrodes and the electric volume conductor of the body around the catheter in unipolar and bipolar setups. Only the upper half of the symmetrical plane is depicted (left, unipolar setup; right bipolar setup) In a homogeneous and isotropic infinitely extended volume conductor with a conductivity σ every point of the line sources (Fig. 14) with a current (1) builds up a cylindersymmetrical potential contribution(2) in an arbitrary point P(p, z) (12). The potential of the whole line source (Fig. 2) in a point P(p, z)(3) is given by the addition of all contributions.The unipolar resistance R_{UH} of the electrode e_{1}(ρ = ρ_{0}, z = 0) or e_{2}(ρ = ρ_{0}, z = d) in an infinite homogeneous volume conductor is given from Eq. (3) (4) For the bipolar setup (Fig. 14), it follows in the same manner:(5) where(6) represents the mutual influence of the line source e_{1} and sink e_{2}.As in Eq. (3) and (5) (7) in Eq. (5)(8) have to be valid.Inhomogeneous volume conductor In this part the electric volume conductor around the electrodes in each of the four phases of the contraction wave (Fig. 2) is simplified by one cylindrical layer I with a various conductivity σ_{1} and thickness D, which encircles the catheter and lies in a homogeneous, infinitely extended volume conductor II with a conductivity σ_{2} (Fig. 14). In the simulation of phase III (Fig. 2) the volume conductor II represents, for example, the conducting medium of the body fluids. Such a problem may be solved by the method of images (5) in a direct manner following the treatment in Eq. (1) to Eq. (8). The influence of layer II on the potential distribution of a current line source e_{1} (Fig. 14) in layer I can be considered by an additional image line sink e_{3}, which lies symmetrical to the axis in a distance ρ_{2} and supplies the current I'. Thus the contribution of a point of the line source (Fig. 14) is given in layer I by (9) Accordingly, we replace in в\ a new line source with the current I" to obtain the potential distribution in layer II of a source point(10) Equations (9) and (10) must satisfy Laplace's equation, which for surface of discontinuity demands this boundary condition:(11) These demands can be fulfilled if(12) Substituting Equations (1) and (12) into (9) and (10) and integration of the resulting equations according to Equation (3) gives, with the notation(13) the potential function in layer II from two electrodes(14) and in layer I(15) The resistance of an electrode in Figure 2 (unipolar arrangement) results from Equations (15), (9), and (4) in(16) where D = ρ_{1}  ρ_{0} is the thickness of layer I (Fig. 14). Accordingly, the resistance of a bipolar set up R_{B} in Figure 2 can be expressed as(17) Substituting Equations (6), (15), and (16) into (17) gives(18) The last term in Equation (18) depicts the mutual influence of the image source e_{1} and e_{3} (R_{13}). Only a part of R_{U} (Eq. 16) and R_{13} changes with D.The distribution of the current density in the boundary ρ = ρ_{1}, (19) can be used advantageously to compare some characteristics of the unipolar and bipolar arrangements in reference to the longitudinal extension of the measured region. Only the current density component in the direction of the normal(20) resulting from Equations (19), (13), and (14) is important for the change of Ru and R_{B} depending on the thickness D.We introduce segments of length B_{5} and B_{9} that comply with (21) where B_{5} and B_{9} define longitudinal bandwidth in which 50% or 90% of the total current I_{t}(ρ_{1}) flows.In other words, the part of resistance R_{U} or R_{B} that characterized the change of the thickness D in layer I is determined to 50% or 90% in a longitudinal segment that is B_{5} or B_{9} wide. Finally, for the longitudinal selectivity the notation (22) is adopted.References
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