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Finite element simulation of the impedance response of a vascular segment as a function of changes in electrode configuration

M. Amini
M. Amini
, 
H. Kalvøy
H. Kalvøy
 and 
Ø.G. Martinsen
Ø.G. Martinsen
   | Dec 31, 2020
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Published Online: Dec 31, 2020
Page range: 112 - 131
Received: Nov 20, 2020
DOI: https://doi.org/10.2478/joeb-2020-0017
Keywords
© 2020 M. Amini et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Figure 1

Different constituents of the COMSOL 3D finite element model.
Different constituents of the COMSOL 3D finite element model.

Figure 2

Extra fine mesh created for the model
Extra fine mesh created for the model

Figure 3

Diagram of the complex impedance illustrating the relationship between the real and imaginary parts of the impedance, where Zreal is the real or resistive component R of the impedance, whereas Zimag is the imaginary or capacitive X component. |Z| and ϕ define the impedance magnitude and phase shift respectively.
Diagram of the complex impedance illustrating the relationship between the real and imaginary parts of the impedance, where Zreal is the real or resistive component R of the impedance, whereas Zimag is the imaginary or capacitive X component. |Z| and ϕ define the impedance magnitude and phase shift respectively.

Figure 4

Y–Z Slice plots of a) electrical potential, b) current density, c) sensitivity and d) volume impedance density for the model with 28 surface electrodes simulated with and without the vein at the frequency of 1 MHz where all the surface electrodes were excited as terminals simultaneously (radius of the ground electrode = 0.5 mm, radius of the terminal electrodes = 1 mm).
Y–Z Slice plots of a) electrical potential, b) current density, c) sensitivity and d) volume impedance density for the model with 28 surface electrodes simulated with and without the vein at the frequency of 1 MHz where all the surface electrodes were excited as terminals simultaneously (radius of the ground electrode = 0.5 mm, radius of the terminal electrodes = 1 mm).

Figure 5

Y–Z Slice plots of a) electrical potential, b) current density, c) sensitivity and d) volume impedance density for the model with 28 surface electrodes simulated with and without the vein at the frequency of 1 MHz where only the middle electrode was excited as the terminal (radius of the ground electrode = 0.5 mm, radius of the terminal electrode = 1 mm).
Y–Z Slice plots of a) electrical potential, b) current density, c) sensitivity and d) volume impedance density for the model with 28 surface electrodes simulated with and without the vein at the frequency of 1 MHz where only the middle electrode was excited as the terminal (radius of the ground electrode = 0.5 mm, radius of the terminal electrode = 1 mm).

Simulations results including the percentage of the volumetric power loss density and volume integration of volume impedance density for vein and with two different surface and ground electrode dimensions (rt = 1 mm and rg = 0.5 mm versus rt = 2.5 mm and rg = 0.5 mm), regardless of the number of surface electrodes (28, 32, 44 and 48) and with only one surface electrode excited as the terminal (middle electrode, electrode at the first end close to the ground electrode output or electrode at the second end)

Electrodes dimensions (mm) Location of the terminal electrode Volumetric power loss density in vein (%) Volume impedance density for vein (%)
rt=1.0 mmrg= 0.5 mm Middle 95.98% 95.98%
rt=1.0 mmrg= 0.5 mm First end close to the ground electrode 95.97% 95.97%
rt=1.0 mmrg= 0.5 mm Second end 95.96% 95.96%
rt=2.5 mmrg= 0.5 mm Middle 89.93% 89.93%
rt=2.5 mmrg= 0.5 mm First end close to the ground electrode 89.87% 89.87%
rt=2.5 mmrg= 0.5 mm Second end 89.82% 89.82%

Impedance measurements at 1 MHz with one of the 28 surface electrodes as terminal (7 electrodes in 4 rows)

Surface electrode radius (mm) Ground electrode radius (mm) Impedance between surface and ground electrode (Ω)
2.5(electrode at the end closer to the ground electrode) 0.5 2.6984E2−3.3115E-1i
2.5(electrode in the middle) 0.5 2.6967E2−3.3335E-1i
2.5(electrode at the other end) 0.5 2.7026E2−3.4495E-1i

Impedance measurement at 1MHz with all the 32 surface electrodes excited simultaneously as terminals (8 electrodes in 4 rows)

Surface electrode radius (mm) Ground electrode radius (mm) Impedance between surface and ground electrode (Ω)
2.5 0.5 1.0653E1−1.0485E-2i
2.5 1.0 9.8828E0−9.6378E-3i
2.0 0.5 1.3421E1−1.3592E-2i
2.0 1.0 1.2610E1−1.2638E-2i
1.0 0.5 2.8212E1−2.9215E-2i
1.0 1.0 2.7457E1−2.8411E-2i

Impedance measurements at 1 MHz with all the 28 surface electrodes excited simultaneously as terminals (7 electrodes in 4 rows)

Surface electrode radius (mm) Ground electrode radius (mm) Impedance between surface and ground electrode (Ω)
2.5 0.5 1.2119E1−1.1656E-2i
2.5 1.0 1.1050E1−1.0267E-2i
2.0 0.5 1.5423E1−1.5411E-2i
2.0 1.0 1.4601E1−1.4506E-2i
1.0 0.5 3.2653E1−3.4107E-2i
1.0 1.0 3.1826E1−3.3228E-2i

Volume integration of Volume Impedance Density (VID) measurements at 1MHz with one of the 32 surface electrodes as terminal (8 electrodes in 4 rows)

Surface electrode radius (mm) Ground electrode radius (mm) VID for vein (Ω) VID for saline (Ω) VID for vein + saline (Ω) VID for the nonconductive parts (Ω) VID for vein (%) VID for saline (%)
1.0 (electrode at the end closer to the ground electrode) 0.5 1.1343E-3+4.9177E-7i 4.7536E-5+1.5713E-8i 1.1819E-3+5.0749E-7i −1.0513E-6+1.5073E-10i 95.97% 4.02%
1.0 (electrode in the middle) 0.5 1.1329E-3+4.9362E-7i 4.7381E-5+1.3227E-8i 1.1803E-3+5.0685E-7i −1.0588E-6+7.3337E-10i 95.98% 4.01%
1.0 (electrode at the other end) 0.5 1.1339E-3+5.0746E-7i 4.7651E-5 1.1815E-3+5.0733E-7i −1.0620E-6+1.0793E-9i 95.96% 4.03%

Simulation results including the percentage of the volumetric power loss density and volume integration of volume impedance density for vein and with the electrode dimensions of rt = 1 mm and rg = 1 mm where all the surface electrodes were excited simultaneously as terminals.

Number of electrodes Volumetric power loss density in vein (%) Volume impedance density for vein (%)
28 92.80% 92.81%
32 91.48% 91.48%
44 89.28% 89.28%
48 89.06% 89.06%

Volumetric power loss density measurements at 1MHz with both all and one of the 48 surface electrodes as terminal (12 electrodes in 4 rows)

Surface electrode radius (mm) Ground electrode radius (mm) Power loss density in vein (W) Power loss density in saline (W) Power loss density in the whole model (W) Power loss density in vein (%) Power loss density in saline (%)
1.0(all electrodes) 1.0 2.2788E-2 2.7982E-3 2.5587E-2 89.06% 10.94%
1.0(all electrodes) 0.5 2.1003E-2 3.5551E-3 2.4561E-2 85.51% 14.47%
1.0(electrode at the end closer to the ground electrode) 0.5 5.6653E-4 2.4003E-5 5.9053E-4 95.93% 4.06%
1.0(electrode in the middle) 0.5 5.6885E-4 2.3795E-5 5.9265E-4 95.98% 4.01%
1.0(electrode at the other end) 0.5 5.6191E-4 2.4793E-5 5.8671E-4 95.77% 4.22%

Volumetric power loss density measurements at 1 MHz with all 28 surface electrodes excited simultaneously as terminals (7 electrodes in 4 rows)

Surface electrode radius (mm) Ground electrode radius (mm) Power loss density in vein (W) Power loss density in saline (W) Power loss density in the whole model (W) Power loss density in vein (%) Power loss density in saline (%)
2.5 0.5 3.1263E-2 9.9870E-3 4.1259E-2 75.77% 24.20%
2.5 1.0 3.6575E-2 8.6698E-3 4.5248E-2 80.83% 19.16%
2.0 0.5 2.6133E-2 6.2797E-3 3.2418E-2 80.61% 19.37%
2.0 1.0 2.9147E-2 5.0954E-3 3.4244E-2 85.11% 14.87%
1.0 0.5 1.3855E-2 1.4566E-3 1.5313E-2 90.47% 9.51%
1.0 1.0 1.4582E-2 1.1284E-3 1.5710E-2 92.80% 7.18%

Volume integration of Volume Impedance Density (VID) measurements at 1MHz with all the 32 surface electrodes excited simultaneously as terminals (8 electrodes in 4 rows)

Surface electrode radius (mm) Ground electrode radius (mm) VID for vein (Ω) VID for saline (Ω) VID for vein + saline (Ω) VID for the nonconductive parts (Ω) VID for vein (%) VID for saline (%)
2.5 0.5 6.8068E-2+1.1734E-5i 2.5775E-2+1.9988E-5i 9.3843E-2+3.1721E-5i −6.7451E-5+1.2849E-6i 72.53% 27.46%
2.5 1.0 7.9045E-2+1.7764E-5i 2.2133E-2+1.8676E-5i 1.0118E-1+3.6440E-5i −8.0453E-5+1.0100E-6i 78.12% 21.87%
2.0 0.5 5.8027E-2+1.2993E-5i 1.6467E-2+1.3744E-5i 7.4495E-2+2.6737E-5i −6.2039E-5+5.5682E-7i 77.89% 22.10%
2.0 1.0 6.5491E-2+1.7554E-5i 1.3809E-2+1.2381E-5i 7.9300E-2+2.9935E-5i −6.4510E-5+6.7324E-7i 82.58% 17.41%
1.0 0.5 3.1567E-2+1.0773E-5i 3.8756E-3+3.4894E-6i 3.5443E-2+1.4263E-5i −2.9575E-5+1.1871E-7i 89.06% 10.93%
1.0 1.0 3.3318E-2+1.2025E-5i 3.1017E-3+2.9791E-6i 3.6419E-2+1.4999E-5i −3.0186E-5+1.3771E-7i 91.48% 8.51%

Volumetric power loss density measurements at 1MHz with both all and one of the 44 surface electrodes as terminal (11 electrodes in 4 rows)

Surface electrode radius (mm) Ground electrode radius (mm) Power loss density in vein (W) Power loss density in saline (W) Power loss density in the whole model (W) Power loss density in vein (%) Power loss density in saline (%)
1.0(all electrodes) 1.0 2.1008E-2 2.5198E-3 2.3529E-2 89.28% 10.71%
1.0(all electrodes) 0.5 1.9396E-2 3.2042E-3 2.2603E-2 85.81% 14.17%
1.0(electrode at the end closer to the ground electrode) 0.5 5.6628E-4 2.3644E-5 5.8992E-4 95.99% 4.01%
1.0(electrode in the middle) 0.5 5.6795E-4 2.3717E-5 5.9167E-4 95.99% 4.01%
1.0(electrode at the other end) 0.5 5.6727E-4 2.3802E-5 5.9107E-4 95.97% 4.02%

Volumetric power loss density measurement at 1MHz with one of the 32 surface electrodes as terminal (8 electrodes in 4 rows)

Surface electrode radius (mm) Ground electrode radius (mm) Power loss density in vein (W) Power loss density in saline (W) Power loss density in the whole model (W) Power loss density in vein (%) Power loss density in saline (%)
1.0(electrode at the end closer to the ground electrode) 0.5 5.6717E-4 2.3768E-5 5.9093E-4 95.97% 4.02%
1.0(electrode in the middle) 0.5 5.6644E-4 2.3691E-5 5.9013E-4 95.98% 4.01%
1.0(electrode at the other end) 0.5 5.6695E-4 2.3827E-5 5.9077E-4 95.96% 4.03%

Volumetric power loss density measurements at 1 MHz with one of the 28 surface electrodes as terminal (7 electrodes in 4 rows)

Surface electrode radius (mm) Ground electrode radius (mm) Power loss density in vein (W) Power loss density in saline (W) Power loss density in the whole model (W) Power loss density in vein (%) Power loss density in saline (%)
2.5(electrode at the end closer to the ground electrode) 0.5 1.6653E-3 1.8764E-4 1.8529E-3 89.87% 10.12%
2.5(electrode in the middle) 0.5 1.6674E-3 1.8673E-4 1.8541E-3 89.93% 10.07%
2.5(electrode at the other end) 0.5 1.6618E-3 1.8819E-4 1.8500E-3 89.82% 10.17%

Simulations results including the percentage of the volumetric power loss density and volume integration of volume impedance density for vein and with the electrode dimensions of rt = 1 mm and rg = 0.5 mm where all the surface electrodes were excited simultaneously as terminals.

Number of electrodes Volumetric power loss density in vein (%) Volume impedance density for vein (%)
28 90.47% 90.48%
32 89.05% 89.06%
44 85.81% 85.82%
48 85.51% 85.59%

Impedance measurements at 1MHz with both all and one of the 44 surface electrodes as terminals (11 electrodes in 4 rows)

Surface electrode radius (mm) Ground electrode radius (mm) Impedance between surface and ground electrode (Ω)
1.0(all electrodes) 1.0 2.1251E1−2.2560E-2i
1.0(all electrodes) 0.5 2.2121E1−2.3553E-2i
1.0(electrode at the end closer to the ground electrode) 0.5 8.4757E2−9.3724E-1i
1.0(middle electrode) 0.5 8.4507E2−9.3706E-1i
1.0(electrode at the other end) 0.5 8.4592E2−9.4144E-1i

Volume integration of Volume Impedance Density (VID) measurements at 1MHz with both all and one of the 48 surface electrodes as terminal (12 electrodes in 4 rows)

Surface electrode radius (mm) Ground electrode radius (mm) VID for vein (Ω) VID for saline (Ω) VID for vein + saline (Ω) VID for the nonconductive parts (Ω) VID for vein (%) VID for saline (%)
1.0(all electrodes) 1.0 4.5575E-2+1.6240E-5i 5.5963E-3+4.3527E-6i 5.1171E-2+2.0593E-5i −3.6781E-5+4.2610E-8i 89.06% 10.94%
1.0(all electrodes) 0.5 4.1652E-2+1.4285E-5i 7.0078E-3+4.6332E-6i 4.8660E-2+1.8918E-5i −3.5480E-5+2.6543E-8i 85.59% 14.40%
1.0(electrode atthe end closerto the groundelectrode) 0.5 1.1331E-3+5.2890E-7i 4.8006E-5−2.1807E-8i 1.1811E-3+5.0709E-7i −1.0724E-6+2.4808E-9i 95.93% 4.06%
1.0(one middleelectrode) 0.5 1.1377E-3+4.9692E-7i 4.7590E-5+1.2089E-8i 1.1853E-3+5.0901E-7i −1.0576E-6+7.7056E-10i 95.98% 4.01%
1.0(electrode atthe other end) 0.5 1.1238E-3+5.5811E-7i 4.9585E-5−5.4704E-8i 1.1734E-3+5.0341E-7i −1.1555E-6+8.2855E-9i 95.77% 4.22%

Volume integration of Volume Impedance Density (VID) measurements at 1MHz with both all and one of the 44 surface electrodes as terminal (11 electrodes in 4 rows)

Surface electrode radius (mm) Ground electrode radius (mm) VID for vein (Ω) VID for saline (Ω) VID for vein + saline (Ω) VID for the nonconductive parts (Ω) VID for vein (%) VID for saline (%)
1.0(all electrodes) 1.0 4.2016E-2+1.4290E-5i 5.0395E-3+4.7016E-6i 4.7056E-2+1.8991E-5i −4.0449E-5+3.4149E-7i 89.28% 10.71%
1.0(all electrodes) 0.5 3.8792E-2+1.2009E-5i 6.4084E-3+5.6141E-6i 4.5200E-2+1.7622E-5i −3.9316E-5+2.9663E-7i 85.82% 14.17%
1.0(electrode at the end closer to the ground electrode) 0.5 1.1326E-3+4.9125E-7i 4.7288E-5+1.5428E-8i 1.1798E-3+5.0668E-7i −1.0507E-6+1.6192E-10i 95.99% 4.01%
1.0(one middle electrode) 0.5 1.1359E-3+4.9613E-7i 4.7434E-5+1.2071E-8i 1.1833E-3+5.0820E-7i −1.0576E-6+7.5986E-10i 95.99% 4.01%
1.0(electrode at the other end) 0.5 1.1345E-3+5.0853E-7i 4.7603E-5−9.1060E-10i 1.1821E-3+5.0762E-7i −1.0618E-6+1.1214E-9i 95.97% 4.02%

Volume integration of Volume Impedance Density (VID) measurements at 1 MHz with one of the 28 surface electrodes as terminal (7 electrodes in 4 rows)

Surface electrode radius (mm) Ground electrode radius (mm) VID for vein (Ω) VID for saline (Ω) VID for vein + saline (Ω) VID for the nonconductive parts (Ω) VID for vein (%) VID for saline (%)
2.5(electrode at the end closer to the ground electrode) 0.5 3.3306E-3+1.4766E-6i 3.7528E-4+2.6723E-8i 3.7059E-3+1.5034E-6i −3.7961E-6+5.9637E-9i 89.87% 10.12%
2.5(electrode in the middle) 0.5 3.3347E-3+1.4305E-6i 3.7346E-4+7.4700E-8i 3.7082E-3+1.5052E-6i −3.8315E-6+9.1808E-9i 89.93% 10.07%
2.5 (electrode at other end) 0.5 3.3236E-3+1.5898E-6i 3.7638E-4−8.8830E-8i 3.7000E-3+1.5010E-6i −3.9716E-6+2.1288E-8i 89.82% 10.17%

Volume integration of Volume Impedance Density (VID) measurements at 1 MHz with all the 28 surface electrodes excited simultaneously as terminals (7 electrodes in 4 rows)

Surface electrode radius (mm) Ground electrode radius (mm) VID for vein (Ω) VID for saline (Ω) VID for vein + saline (Ω) VID for the nonconductive parts (Ω) VID for vein (%) VID for saline (%)
2.5 0.5 6.2527E-2+1.4138E-5i 1.9974E-2+1.4785E-5i 8.2501E-2+2.8923E-5i −6.4902E-5+2.8425E-7i 75.78 % 24.21 %
2.5 1.0 7.3150E-2+1.9518E-5i 1.7339E-2+1.4010E-5i 9.0490E-2+3.3528E-5i −6.6240E-5+4.2870E-7i 80.83 % 19.16 %
2.0 0.5 5.2266E-2+1.3515E-5i 1.2559E-2+1.0436E-5i 6.4826E-2+2.3951E-5i −5.2804E-5+2.3455E-7i 80.62 % 19.37 %
2.0 1.0 5.8294E-2+1.7023E-5i 1.0191E-2+9.4960E-6i 6.8485E-2+2.6519E-5i −5.4264E-5+3.4159E-7i 85.11 % 14.88 %
1.0 0.5 2.7710E-2+9.7267E-6i 2.9132E-3+2.7658E-6i 3.0623E-2+1.2493E-5i −2.5621E-5+6.7807E-8i 90.48 % 9.51%
1.0 1.0 2.9163E-2+1.0749E-5i 2.2568E-3+2.3572E-6i 3.1420E-2+1.3107E-5i −2.6169E-5+8.2649E-8i 92.81 % 7.18%

Impedance measurement at 1MHz with one of the 32 surface electrodes as terminal (8 electrodes in 4 rows)

Surface electrode radius (mm) Ground electrode radius (mm) Impedance between surface and ground electrode (Ω)
1.0(electrode at the end closer to the ground electrode) 0.5 8.4612E2−9.3477E-1i
1.0(electrode in the middle) 0.5 8.4727E2−9.4202E-1i
1.0(electrode at the other end) 0.5 8.4635E2−9.4246E-1i

Impedance measurements at 1MHz with both all and one of the 48 surface electrodes as terminals (12 electrodes in 4 rows)

Surface electrode radius (mm) Ground electrode radius (mm) Impedance between surface and ground electrode (Ω)
1.0(all electrodes) 1.0 1.9541E1−1.8887E-2i
1.0(all electrodes) 0.5 2.0357E1−1.9872E-2i
1.0(electrode at the end closer to the ground electrode) 0.5 8.4669E2−9.6656E-1i
1.0(middle electrode) 0.5 8.4367E2−9.3392E-1i
1.0(electrode at the other end) 0.5 8.5221E2−1.0653E0i

Volumetric power loss density measurement at 1MHz with all the 32 surface electrodes excited simultaneously as terminals (8 electrodes in 4 rows)

Surface electrode radius (mm) Ground electrode radius (mm) Power loss density in vein (W) Power loss density in saline (W) Power loss density in the whole model (W) Power loss density in vein (%) Power loss density in saline (%)
2.5 0.5 3.4034E-2 1.2890E-2 4.6933E-2 72.51% 27.46%
2.5 1.0 3.9523E-2 1.1067E-2 5.0593E-2 78.11% 21.87%
2.0 0.5 2.9014E-2 8.2354E-3 3.7255E-2 77.87% 22.10%
2.0 1.0 3.2746E-2 6.9049E-3 3.9652E-2 82.58% 17.41%
1.0 0.5 1.5784E-2 1.9382E-3 1.7723E-2 89.05% 10.93%
1.0 1.0 1.6659E-2 1.5510E-3 1.8210E-2 91.48% 8.50%
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