- Email: [email protected]
Lorentz-force-induced motion in conductive media
Magnetic Resonance Imaging 23 (2005) 647 – 651
Lorentz-force-induced motion in conductive media Alexandra T. Basforda, Jeffrey R. Basfordb,T, Jennifer Kugelc, Richard L. Ehmanc a
Department of Neuroscience, University of Minnesota, Minneapolis, MN 55455, USA Department of Physical Medicine and Rehabilitation, Mayo Clinic and Foundation, Rochester, MN 55905, USA c Magnetic Resonance Imaging Laboratory, Department of Radiology, Mayo Clinic and Foundation, Rochester, MN 55905, USA Received 11 August 2004; accepted 21 February 2005 b
Abstract This project was designed to assess whether MRI imaging could detect Lorentz-force-induced motion in conductive samples. Experiments were performed by applying alternating voltages across 2% agar and 18% bovine gels placed in the field of a 1.5-T MRI scanner. Motion-sensitized time-gated MRI images that were obtained and analyzed with custom-developed software used in previous studies revealed the production of movement in both agar and gel samples. Motion was most pronounced in the plane vertical to the sample and had the greatest amplitude when the current path was perpendicular to the scanner’s magnetic field. These findings are compatible with the vector cross product nature of the Lorentz force and suggest that the imaging of Lorentz-force-induced motion in conductive samples is feasible. Whether this approach can be extended to study electrically active tissues such as the peripheral nerves, brain and heart remains to be seen. D 2005 Elsevier Inc. All rights reserved. Keywords: MRI; Lorentz force; Imaging; Electrical current
1. Introduction Magnetic resonance imaging (MRI) provides detailed information about tissue structure but little insight into its physiological properties. Electrical currents, however, are ubiquitous in living tissue, and visualization of their interactions with the body could conceivably provide insights into the function of tissues such as the nerves, brain and heart. The Lorentz forces that these currents experience as they flow in the field of a MRI scanner may provide such an avenue. The forces produced by this interaction are proportional to the vector cross product jB (where j is the current density and B the scanner’s magnetic field) and might be expected to produce distortions in the tissue medium in which they move. There is some precedence for this approach. For example, magnetic resonance elastography (MRE) is a MRI technique that uses motion-sensitized time-gated phase contrast imaging to image mechanically induced cyclic displacements as small as 0.1 Am in a sample [1]. The approach
T Corresponding author. E-mail address: [email protected] (J.R. Basford). 0730-725X/$ – see front matter D 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.mri.2005.02.014
proposed for this study would be similar to that used in MRE with the exception that the distortions would be produced by Lorentz forces acting on a current flowing in a medium rather than by mechanical means. The goal of this study was to determine whether Lorentz-force-induced displacements could be generated and imaged in conductive samples.
2. Methods 2.1. General Samples were placed in the field of a 1.5-T MRI GE Medical Systems Sigma Scanner (General Electric Medical Systems, Milwaukee, WI) and imaged as an alternating voltage was applied across them. 2.2. Data analysis Data analysis of the motion-sensitized time-gated MRI imaging utilized specialized software developed in previous MRE research [1– 4]. Displacements were calculated by rearranging the equation describing the observed phase shift in the NMR signal [maximum displacement e is p//[2cNTG cos(kr+h)], where r(t) is the position vector, k is the wave number, h is an initial phase offset, c is the gyromagnetic ratio (4257 Hz/Gauss), N is the number of gradient cycles,
648
A.T. Basford et al. / Magnetic Resonance Imaging 23 (2005) 647– 651
with the nonconductive agar. In two additional experiments, the channel was lined with thin plastic film to prevent diffusion of NaCl from the channel into the nonconductive agar. A fifth experiment also involved a plastic film barrier. However, in this case, a 1-mm-wide air gap served as a barrier to transmission of mechanical energy from the channel to one side of the nonconductive agar. 3. Results
Fig. 1. Plexiglas mold. Main chamber: 12.7510.254 cm. Extensions: 1.522.754 cm.
T is the period of mechanical excitation (T =2p/x), G is the gradient strength (1.76 Gauss/cm) and / is the experimentally measured phase shift) [1]. 2.3. Apparatus and samples A 12.7510.254-cm acrylic mold was used in the experiments. Elongated extensions (1.522.754 cm) were attached to ensure that the electrodes were not within the scanner’s field of view (Fig. 1). Fourteen-karat gold leaf electrodes (13.8 cm) were fastened to the ends of the extensions with double-sided cellophane tape, and the molds were filled with the medium of interest at room temperature. Two sets of experiments were performed. The first set involved the application of a current across two homogenous and uniformly conductive mediums that differed in stiffness and hence in their ability to attenuate mechanical waves. Specifically, 2% agar and 18% bovine gelatin phantoms were used to represent relatively nonattenuating and attenuating environments, respectively, as 2% agar is less degrading to displacement waves than 18% bovine gelatin. Conductivity effects were studied by altering saline concentrations. The second set of experiments involved samples that contained a central 1.54.0 cm 0.9% NaCl 2% agar conductive channel that was imbedded within nonconductive 2% agar (0% NaCl) (Fig. 2). In two of these latter experiments, the conductive channel was in direct contact
Fig. 2. Conductive channel. Note that the central conductive channel (B) may be electrically (with plastic film) or mechanically (with an air gap) isolated from the surrounding nonconductive agar (A). The thick black lines indicate electrode position.
Fig. 3 presents a time-ordered sequence (advancing in time from the upper left to the lower right panels) of four images representative of the findings obtained in the first set of experiments performed with homogeneous 2% agar and 18% bovine gelatin NaCl phantoms. In this case, the figure represents the case of a 25% NaCl 2% agar medium in which the current flow is perpendicular to the magnetic field. These samples had impedances of about 11 V, and, as is apparent in the figure, distinct wave-like displacements with amplitudes of about 65 Am were observed. For example, the location marked by the arrow in the first (upper left) image shows the agar next to the wall at a maximally upwards (bright red-yellow) displaced position. In the next frame, the medium is at a more intermediate position (red). In the third frame, it is in a strongly downward displacement (blue-white), and in the last image
Fig. 3. Homogeneous medium. Representative series of gradient echo images (advancing in time from the upper left to the lower right panels) sensitized to motion perpendicular to the plane of the 2% agar sample (current parameters: 50 V, 100 Hz). Note the displacement of the sample at the location next to the wall of the mold noted by the arrow and displayed with pseudocolors (bright yellow-red, maximal upwards displacement; bright white-blue, maximal downwards displacement). In the first image, it is at its maximum upwards displacement (bright red-yellow); in the second, it is in an intermediate position (blackish); in the third, it is at its maximum displacement downwards (intense blue-white); and in the fourth, it is a downwards but more intermediate position.
A.T. Basford et al. / Magnetic Resonance Imaging 23 (2005) 647– 651
it has returned to a more intermediate (bluish) position. Displacements produced in the relatively attenuating 18% bovine gelatin samples followed similar patterns but had lower amplitudes that further lessened as the distance from the main current path increased. Experiments (images not shown) in which the current was 0, the electrodes shunted or the current flow parallel to the scanner’s field revealed only random displacements. As noted above, a number of experiments were also performed with a conductive channel of 0.9% NaCl
649
surrounded with nonconductive (0% NaCl ) agar. These experiments revealed a pattern of alternating, up-and-down displacements perpendicular to the plane of the sample when the current flow was perpendicular to the magnetic field. As was true in the previous studies, only random motion was observed when the current flow was absent or parallel to the magnetic field. Fig. 4 displays a time-ordered sequence (advancing in time from the upper left to the lower right panels) of eight images representative of the results of a bconductive channelQ experiment in which there was an air gap on the left and a plastic film–agar interface on the right. Consider the waveform indicated by the arrows in Fig. 4. A maximum upwards (red-yellow) displacement of about 75 Am is apparent in the first (upper left) image that then appears to progress through the next four panels. The beginning of a new wave cycle is apparent at the end of the cycle in the last two images at the bottom of the figure. Note that this experiment involved a conductive channel with an air gap on the left. Waves, therefore, were generated in the channel but were able to mechanically propagate only to the right. As was true in the homogeneous medium studies, displacement amplitudes were largest in the direction perpendicular to the plane of the sample and when the current flow was at right angles to the scanner’s B field. 4. Discussion
Fig. 4. Representative conductive channel experiment (with air and plastic wrap separations) (current parameters: 40 V, 100 Hz). Images advance in time from the upper left to the lower right panels and are sensitized to motion perpendicular to the plane of the sample. The nonconductive agar on the left was electrically and mechanically isolated from the conductive channel by a plastic film and an air gap (the line of background color); the nonconductive agar on the right was separated from the channel with the plastic film only. Displacements are displayed with pseudocolors (bright yellow-red, maximal upwards displacement; bright white-blue, maximal downwards displacement). Note the apparent propagation of the red-yellow maximum upwards displacement (advancing in time from the upper left to the lower right panels) of the wave (yellow-red) from the center of the first image outward throughout the next five images. Images 7 and 8 show the pattern beginning to repeat.
The goal of this experiment was to establish whether Lorentz-force-induced motion could be produced, imaged and quantitated in conductive samples. The findings are encouraging in that the approach produced quantifiable motion in the homogenous and conductive channel models (Figs. 3 and 4). In addition, this induced motion was consistent with the jB cross-product nature of the Lorentz force in that it was maximal when the current flow was perpendicular to the scanner’s magnetic field and was absent when the current flow was absent or parallel to the magnetic field. These experiments also revealed that the waves generated in a conductive channel could propagate into a mechanically coupled contiguous nonconductive medium and what can be interpreted as a standing wave pattern with stationary locations of maximal upward and downward displacements. A comparison of the results of this experiment with those predicted from theory may provide a more quantitative assessment of the role of Lorentz forces in the findings. Exact determinations depend on the elasticity and density of the medium as its shape. However, reasonable estimates suitable for the purpose of discussion are relatively easy to establish. Consider the case of the 12.7510.254.0-cm agar block that was discussed first (Fig. 3). In this experiment, the conduction pathway (l) may be estimated as the 12.75-cm length of the sample, the magnetic field (B) 1.5 T,
650
A.T. Basford et al. / Magnetic Resonance Imaging 23 (2005) 647– 651
and, as was true for many of the experiments, the electrical current was 4.4 A. The Lorentz equation, the force on a current flowing in a magnetic field, is: F ¼ Il B ¼ IlB sinh
ð1Þ
Substituting in these values as well as the fact that the magnetic field and current flow were perpendicular (i.e., sin h =1) and that the mass of the sample, assuming that the agar has the density of water, was 0.522 kg, acceleration in this estimate is thus: a ¼ F=m ¼ 0:8415=0:522 ¼ 1:612 m=s2
ð2Þ
The applied currents were sinusoidal, and, as a result, acceleration ranged from 0 to maximum in a quarter cycle. In these experiments, typical frequencies were about 50 Hz, and the duration of a quarter cycle was, therefore, 0.005 s (i.e., (1/50)0.25). Thus: 2
displacement ¼ 1=2 at ¼14:2 10
6
m¼20:15 Am
ð3Þ
These displacements, despite the simplifications of the model, are on the order of the 60 –75 Am observed in the experiments discussed above (Figs. 3 and 4). It may also be interesting to estimate the magnitude of the current densities ( j) in terms of the total currents and cross-sectional area of the sample analyzed above. On this basis:
technique of this experiment. Other significant limitations of CDR include frequent reliance on expensive magnetically shielded rooms and the fact that current density estimations involve complicated modeling and calculations [5–7]. Current density imaging (CDI) [8] is another technique that uses MR imaging to interpret the effects of an electric current passing through a sample. However, this approach has been used to study biochemical properties of tissue such as chemical reactions and ion mobility [9–14] and, thus, provides a limited amount of information directly relevant to this experiment. Magnetic resonance imaging has also been used to investigate the displacements produced in a gel phantom by passing a current through a copper wire [15]. This approach overlaps ours to some extent but has the significant limitation that it is invasive. 5. Conclusions This research demonstrated that it is possible to produce, image and quantitate Lorentz-force-induced mechanical waves in conductive material. At present, these findings are little more than a curiosity. However, they are intriguing and may support the idea that this technique can be extended to study the physiological processes and physical properties of soft tissue. Further study seems warranted.
j ¼ I=area ¼ 4:4=ð10:25 4:0Þ ¼ 0:107 A=cm2
Acknowledgments
The displacements measured in the experiments were on the order of 70 Am (Figs. 3 and 4), while MRE techniques have demonstrated the ability to distinguish displacements on the order of 0.1 Am [1]. On this basis, it is possible that the effects of current densities as much as 700 times smaller (i.e., 0.15 mA/cm2 or 1.5 AA/mm2) could, in theory, be detected. This value compares favorably with the 2–10 AA/mm2 current densities of the human heart established from current density reconstruction (CDR) modeling [5]. These calculations suggest that it may be possible to use this technique to image electrically active tissues in the body. A potential area assessment may be to study whether the currents involved in peripheral nerve, brain, or cardiac muscle can be imaged. Another area of potential interest is whether the forces on the individual current elements, and hence the Young’s modulus, can be determined from the observed distortions and first principles. The literature contains some relevant reports. Current density reconstruction approaches, for example, have been developed to the point where they permit the localization of cardiac conduction deficits within 1–2 cm and, as noted above, permit the estimation of cardiac current densities [5–7]. Current density reconstruction, however, suffers from a number of limitations that currently limit its utility. Among these is that while localizations of 1 to 2 cm are possible, they are relatively imprecise relative to the submicron localizations [1] potentially possible from the
We would like to thank Phillip Rossman, the Magnetic Resonance Research Group and the Mayo Summer Undergraduate Research Fund for the support and insights that made this work possible. References [1] Muthupillai R, Lomas DJ, Rossman PJ, Greenleaf JF, Manduca A, Ehman RL. Magnetic resonance elastography by direct visualization of propagating acoustic strain waves. Science 1995;269(5232): 1854 – 7. [2] Dresner A, Rose GH, Rossman PJ, Muthupillai R, Manduca A, Ehman RL. Magnetic resonance elastography of skeletal muscle. J Magn Reson Imaging 2001;13:269 – 76. [3] Kruse SA, Smith JA, Lawrence AJ, Dresner A, Manduca A, Greenleaf JF, et al. Tissue characterization using magnetic resonance elastography: preliminary results. Phys Med Biol 2000;45:1579 – 90. [4] Wu T, Felmlee JP, Greenleaf JF, Riederer SJ, Ehman RL. Assessment of thermal tissue ablation with MR elastography. Magn Reson Med 2001;45:80 – 187. [5] Muller HP, Godde P, Czerski K, Agrawal R, Feilcke G, Reither K, et al. Localization of a ventricular tachycardia — focus with multichannel magnetocardiography and three-dimensional current density reconstruction. J Med Eng Technol 1999;23(3):108 – 15. [6] Tavarozzi I, Comani S, DelGratta C, DiLuzio S, Romani GL, Gallina S, et al. Magnetocardiography: current status and perspectives: Part II. Clinical applications. Ital Heart J 2002;3(3):151 – 65. [7] Tavarozzi I, Comani S, DelGratta C, Romani GL, DiLuzio S, Brisinda D, et al. Magnetocardiography: current status and perspectives: Part I. Physical principles and instrumentation. Ital Heart J 2003;3(2):75 – 85.
A.T. Basford et al. / Magnetic Resonance Imaging 23 (2005) 647– 651 [8] Scott GC, Joy LG, Armstrong RL, Henkelman M. Sensitivity of magnetic-resonance current-density imaging. J Magn Reson 1992; 97:235 – 54. [9] Yoon RS, Czaya A, Kwan HC, Joy ML. Changes in the complex permittivity during spreading depression in rat cortex. IEEE Trans Biomed Eng 1999;46(11):1330 – 8. [10] Yoon RS, Demonte TP, Jorgenson D, Joy ML. Study of current pathways in porcine heart using current density imaging. Proc Int Soc Magn Reson Med 2001;9. [11] Joy GS, Henkelman M. In vivo detection of applied electric currents by magnetic resonance imaging. Magn Reson Imaging 1989;7:89 – 94.
651
[12] Joy ML, Levedev VP, Gati JS. Imaging of current density and current pathways in rabbit brain during transcranial electrostimulation. IEEE Trans Biomed Eng 1999;40(9):1139 – 49. [13] Mikac U, Desmar A, Sersa I, Desmar F. Electric current density imaging of chemical reactions. Proc Int Soc Magn Reson Med 2001;9. [14] Beravs K, Desmar A, Desmar F. Magnetic resonance current density imaging of chemical processes and reactions. J Magn Reson 1999;137:253 – 7. [15] Song AW, Takahashi AM. Displacement encoded imagine of small electric current. Proc Int Soc Mag Reson Med 2000;8.
Lorentz-force-induced motion in conductive media Alexandra T. Basforda, Jeffrey R. Basfordb,T, Jennifer Kugelc, Richard L. Ehmanc a
Department of Neuroscience, University of Minnesota, Minneapolis, MN 55455, USA Department of Physical Medicine and Rehabilitation, Mayo Clinic and Foundation, Rochester, MN 55905, USA c Magnetic Resonance Imaging Laboratory, Department of Radiology, Mayo Clinic and Foundation, Rochester, MN 55905, USA Received 11 August 2004; accepted 21 February 2005 b
Abstract This project was designed to assess whether MRI imaging could detect Lorentz-force-induced motion in conductive samples. Experiments were performed by applying alternating voltages across 2% agar and 18% bovine gels placed in the field of a 1.5-T MRI scanner. Motion-sensitized time-gated MRI images that were obtained and analyzed with custom-developed software used in previous studies revealed the production of movement in both agar and gel samples. Motion was most pronounced in the plane vertical to the sample and had the greatest amplitude when the current path was perpendicular to the scanner’s magnetic field. These findings are compatible with the vector cross product nature of the Lorentz force and suggest that the imaging of Lorentz-force-induced motion in conductive samples is feasible. Whether this approach can be extended to study electrically active tissues such as the peripheral nerves, brain and heart remains to be seen. D 2005 Elsevier Inc. All rights reserved. Keywords: MRI; Lorentz force; Imaging; Electrical current
1. Introduction Magnetic resonance imaging (MRI) provides detailed information about tissue structure but little insight into its physiological properties. Electrical currents, however, are ubiquitous in living tissue, and visualization of their interactions with the body could conceivably provide insights into the function of tissues such as the nerves, brain and heart. The Lorentz forces that these currents experience as they flow in the field of a MRI scanner may provide such an avenue. The forces produced by this interaction are proportional to the vector cross product jB (where j is the current density and B the scanner’s magnetic field) and might be expected to produce distortions in the tissue medium in which they move. There is some precedence for this approach. For example, magnetic resonance elastography (MRE) is a MRI technique that uses motion-sensitized time-gated phase contrast imaging to image mechanically induced cyclic displacements as small as 0.1 Am in a sample [1]. The approach
T Corresponding author. E-mail address: [email protected] (J.R. Basford). 0730-725X/$ – see front matter D 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.mri.2005.02.014
proposed for this study would be similar to that used in MRE with the exception that the distortions would be produced by Lorentz forces acting on a current flowing in a medium rather than by mechanical means. The goal of this study was to determine whether Lorentz-force-induced displacements could be generated and imaged in conductive samples.
2. Methods 2.1. General Samples were placed in the field of a 1.5-T MRI GE Medical Systems Sigma Scanner (General Electric Medical Systems, Milwaukee, WI) and imaged as an alternating voltage was applied across them. 2.2. Data analysis Data analysis of the motion-sensitized time-gated MRI imaging utilized specialized software developed in previous MRE research [1– 4]. Displacements were calculated by rearranging the equation describing the observed phase shift in the NMR signal [maximum displacement e is p//[2cNTG cos(kr+h)], where r(t) is the position vector, k is the wave number, h is an initial phase offset, c is the gyromagnetic ratio (4257 Hz/Gauss), N is the number of gradient cycles,
648
A.T. Basford et al. / Magnetic Resonance Imaging 23 (2005) 647– 651
with the nonconductive agar. In two additional experiments, the channel was lined with thin plastic film to prevent diffusion of NaCl from the channel into the nonconductive agar. A fifth experiment also involved a plastic film barrier. However, in this case, a 1-mm-wide air gap served as a barrier to transmission of mechanical energy from the channel to one side of the nonconductive agar. 3. Results
Fig. 1. Plexiglas mold. Main chamber: 12.7510.254 cm. Extensions: 1.522.754 cm.
T is the period of mechanical excitation (T =2p/x), G is the gradient strength (1.76 Gauss/cm) and / is the experimentally measured phase shift) [1]. 2.3. Apparatus and samples A 12.7510.254-cm acrylic mold was used in the experiments. Elongated extensions (1.522.754 cm) were attached to ensure that the electrodes were not within the scanner’s field of view (Fig. 1). Fourteen-karat gold leaf electrodes (13.8 cm) were fastened to the ends of the extensions with double-sided cellophane tape, and the molds were filled with the medium of interest at room temperature. Two sets of experiments were performed. The first set involved the application of a current across two homogenous and uniformly conductive mediums that differed in stiffness and hence in their ability to attenuate mechanical waves. Specifically, 2% agar and 18% bovine gelatin phantoms were used to represent relatively nonattenuating and attenuating environments, respectively, as 2% agar is less degrading to displacement waves than 18% bovine gelatin. Conductivity effects were studied by altering saline concentrations. The second set of experiments involved samples that contained a central 1.54.0 cm 0.9% NaCl 2% agar conductive channel that was imbedded within nonconductive 2% agar (0% NaCl) (Fig. 2). In two of these latter experiments, the conductive channel was in direct contact
Fig. 2. Conductive channel. Note that the central conductive channel (B) may be electrically (with plastic film) or mechanically (with an air gap) isolated from the surrounding nonconductive agar (A). The thick black lines indicate electrode position.
Fig. 3 presents a time-ordered sequence (advancing in time from the upper left to the lower right panels) of four images representative of the findings obtained in the first set of experiments performed with homogeneous 2% agar and 18% bovine gelatin NaCl phantoms. In this case, the figure represents the case of a 25% NaCl 2% agar medium in which the current flow is perpendicular to the magnetic field. These samples had impedances of about 11 V, and, as is apparent in the figure, distinct wave-like displacements with amplitudes of about 65 Am were observed. For example, the location marked by the arrow in the first (upper left) image shows the agar next to the wall at a maximally upwards (bright red-yellow) displaced position. In the next frame, the medium is at a more intermediate position (red). In the third frame, it is in a strongly downward displacement (blue-white), and in the last image
Fig. 3. Homogeneous medium. Representative series of gradient echo images (advancing in time from the upper left to the lower right panels) sensitized to motion perpendicular to the plane of the 2% agar sample (current parameters: 50 V, 100 Hz). Note the displacement of the sample at the location next to the wall of the mold noted by the arrow and displayed with pseudocolors (bright yellow-red, maximal upwards displacement; bright white-blue, maximal downwards displacement). In the first image, it is at its maximum upwards displacement (bright red-yellow); in the second, it is in an intermediate position (blackish); in the third, it is at its maximum displacement downwards (intense blue-white); and in the fourth, it is a downwards but more intermediate position.
A.T. Basford et al. / Magnetic Resonance Imaging 23 (2005) 647– 651
it has returned to a more intermediate (bluish) position. Displacements produced in the relatively attenuating 18% bovine gelatin samples followed similar patterns but had lower amplitudes that further lessened as the distance from the main current path increased. Experiments (images not shown) in which the current was 0, the electrodes shunted or the current flow parallel to the scanner’s field revealed only random displacements. As noted above, a number of experiments were also performed with a conductive channel of 0.9% NaCl
649
surrounded with nonconductive (0% NaCl ) agar. These experiments revealed a pattern of alternating, up-and-down displacements perpendicular to the plane of the sample when the current flow was perpendicular to the magnetic field. As was true in the previous studies, only random motion was observed when the current flow was absent or parallel to the magnetic field. Fig. 4 displays a time-ordered sequence (advancing in time from the upper left to the lower right panels) of eight images representative of the results of a bconductive channelQ experiment in which there was an air gap on the left and a plastic film–agar interface on the right. Consider the waveform indicated by the arrows in Fig. 4. A maximum upwards (red-yellow) displacement of about 75 Am is apparent in the first (upper left) image that then appears to progress through the next four panels. The beginning of a new wave cycle is apparent at the end of the cycle in the last two images at the bottom of the figure. Note that this experiment involved a conductive channel with an air gap on the left. Waves, therefore, were generated in the channel but were able to mechanically propagate only to the right. As was true in the homogeneous medium studies, displacement amplitudes were largest in the direction perpendicular to the plane of the sample and when the current flow was at right angles to the scanner’s B field. 4. Discussion
Fig. 4. Representative conductive channel experiment (with air and plastic wrap separations) (current parameters: 40 V, 100 Hz). Images advance in time from the upper left to the lower right panels and are sensitized to motion perpendicular to the plane of the sample. The nonconductive agar on the left was electrically and mechanically isolated from the conductive channel by a plastic film and an air gap (the line of background color); the nonconductive agar on the right was separated from the channel with the plastic film only. Displacements are displayed with pseudocolors (bright yellow-red, maximal upwards displacement; bright white-blue, maximal downwards displacement). Note the apparent propagation of the red-yellow maximum upwards displacement (advancing in time from the upper left to the lower right panels) of the wave (yellow-red) from the center of the first image outward throughout the next five images. Images 7 and 8 show the pattern beginning to repeat.
The goal of this experiment was to establish whether Lorentz-force-induced motion could be produced, imaged and quantitated in conductive samples. The findings are encouraging in that the approach produced quantifiable motion in the homogenous and conductive channel models (Figs. 3 and 4). In addition, this induced motion was consistent with the jB cross-product nature of the Lorentz force in that it was maximal when the current flow was perpendicular to the scanner’s magnetic field and was absent when the current flow was absent or parallel to the magnetic field. These experiments also revealed that the waves generated in a conductive channel could propagate into a mechanically coupled contiguous nonconductive medium and what can be interpreted as a standing wave pattern with stationary locations of maximal upward and downward displacements. A comparison of the results of this experiment with those predicted from theory may provide a more quantitative assessment of the role of Lorentz forces in the findings. Exact determinations depend on the elasticity and density of the medium as its shape. However, reasonable estimates suitable for the purpose of discussion are relatively easy to establish. Consider the case of the 12.7510.254.0-cm agar block that was discussed first (Fig. 3). In this experiment, the conduction pathway (l) may be estimated as the 12.75-cm length of the sample, the magnetic field (B) 1.5 T,
650
A.T. Basford et al. / Magnetic Resonance Imaging 23 (2005) 647– 651
and, as was true for many of the experiments, the electrical current was 4.4 A. The Lorentz equation, the force on a current flowing in a magnetic field, is: F ¼ Il B ¼ IlB sinh
ð1Þ
Substituting in these values as well as the fact that the magnetic field and current flow were perpendicular (i.e., sin h =1) and that the mass of the sample, assuming that the agar has the density of water, was 0.522 kg, acceleration in this estimate is thus: a ¼ F=m ¼ 0:8415=0:522 ¼ 1:612 m=s2
ð2Þ
The applied currents were sinusoidal, and, as a result, acceleration ranged from 0 to maximum in a quarter cycle. In these experiments, typical frequencies were about 50 Hz, and the duration of a quarter cycle was, therefore, 0.005 s (i.e., (1/50)0.25). Thus: 2
displacement ¼ 1=2 at ¼14:2 10
6
m¼20:15 Am
ð3Þ
These displacements, despite the simplifications of the model, are on the order of the 60 –75 Am observed in the experiments discussed above (Figs. 3 and 4). It may also be interesting to estimate the magnitude of the current densities ( j) in terms of the total currents and cross-sectional area of the sample analyzed above. On this basis:
technique of this experiment. Other significant limitations of CDR include frequent reliance on expensive magnetically shielded rooms and the fact that current density estimations involve complicated modeling and calculations [5–7]. Current density imaging (CDI) [8] is another technique that uses MR imaging to interpret the effects of an electric current passing through a sample. However, this approach has been used to study biochemical properties of tissue such as chemical reactions and ion mobility [9–14] and, thus, provides a limited amount of information directly relevant to this experiment. Magnetic resonance imaging has also been used to investigate the displacements produced in a gel phantom by passing a current through a copper wire [15]. This approach overlaps ours to some extent but has the significant limitation that it is invasive. 5. Conclusions This research demonstrated that it is possible to produce, image and quantitate Lorentz-force-induced mechanical waves in conductive material. At present, these findings are little more than a curiosity. However, they are intriguing and may support the idea that this technique can be extended to study the physiological processes and physical properties of soft tissue. Further study seems warranted.
j ¼ I=area ¼ 4:4=ð10:25 4:0Þ ¼ 0:107 A=cm2
Acknowledgments
The displacements measured in the experiments were on the order of 70 Am (Figs. 3 and 4), while MRE techniques have demonstrated the ability to distinguish displacements on the order of 0.1 Am [1]. On this basis, it is possible that the effects of current densities as much as 700 times smaller (i.e., 0.15 mA/cm2 or 1.5 AA/mm2) could, in theory, be detected. This value compares favorably with the 2–10 AA/mm2 current densities of the human heart established from current density reconstruction (CDR) modeling [5]. These calculations suggest that it may be possible to use this technique to image electrically active tissues in the body. A potential area assessment may be to study whether the currents involved in peripheral nerve, brain, or cardiac muscle can be imaged. Another area of potential interest is whether the forces on the individual current elements, and hence the Young’s modulus, can be determined from the observed distortions and first principles. The literature contains some relevant reports. Current density reconstruction approaches, for example, have been developed to the point where they permit the localization of cardiac conduction deficits within 1–2 cm and, as noted above, permit the estimation of cardiac current densities [5–7]. Current density reconstruction, however, suffers from a number of limitations that currently limit its utility. Among these is that while localizations of 1 to 2 cm are possible, they are relatively imprecise relative to the submicron localizations [1] potentially possible from the
We would like to thank Phillip Rossman, the Magnetic Resonance Research Group and the Mayo Summer Undergraduate Research Fund for the support and insights that made this work possible. References [1] Muthupillai R, Lomas DJ, Rossman PJ, Greenleaf JF, Manduca A, Ehman RL. Magnetic resonance elastography by direct visualization of propagating acoustic strain waves. Science 1995;269(5232): 1854 – 7. [2] Dresner A, Rose GH, Rossman PJ, Muthupillai R, Manduca A, Ehman RL. Magnetic resonance elastography of skeletal muscle. J Magn Reson Imaging 2001;13:269 – 76. [3] Kruse SA, Smith JA, Lawrence AJ, Dresner A, Manduca A, Greenleaf JF, et al. Tissue characterization using magnetic resonance elastography: preliminary results. Phys Med Biol 2000;45:1579 – 90. [4] Wu T, Felmlee JP, Greenleaf JF, Riederer SJ, Ehman RL. Assessment of thermal tissue ablation with MR elastography. Magn Reson Med 2001;45:80 – 187. [5] Muller HP, Godde P, Czerski K, Agrawal R, Feilcke G, Reither K, et al. Localization of a ventricular tachycardia — focus with multichannel magnetocardiography and three-dimensional current density reconstruction. J Med Eng Technol 1999;23(3):108 – 15. [6] Tavarozzi I, Comani S, DelGratta C, DiLuzio S, Romani GL, Gallina S, et al. Magnetocardiography: current status and perspectives: Part II. Clinical applications. Ital Heart J 2002;3(3):151 – 65. [7] Tavarozzi I, Comani S, DelGratta C, Romani GL, DiLuzio S, Brisinda D, et al. Magnetocardiography: current status and perspectives: Part I. Physical principles and instrumentation. Ital Heart J 2003;3(2):75 – 85.
A.T. Basford et al. / Magnetic Resonance Imaging 23 (2005) 647– 651 [8] Scott GC, Joy LG, Armstrong RL, Henkelman M. Sensitivity of magnetic-resonance current-density imaging. J Magn Reson 1992; 97:235 – 54. [9] Yoon RS, Czaya A, Kwan HC, Joy ML. Changes in the complex permittivity during spreading depression in rat cortex. IEEE Trans Biomed Eng 1999;46(11):1330 – 8. [10] Yoon RS, Demonte TP, Jorgenson D, Joy ML. Study of current pathways in porcine heart using current density imaging. Proc Int Soc Magn Reson Med 2001;9. [11] Joy GS, Henkelman M. In vivo detection of applied electric currents by magnetic resonance imaging. Magn Reson Imaging 1989;7:89 – 94.
651
[12] Joy ML, Levedev VP, Gati JS. Imaging of current density and current pathways in rabbit brain during transcranial electrostimulation. IEEE Trans Biomed Eng 1999;40(9):1139 – 49. [13] Mikac U, Desmar A, Sersa I, Desmar F. Electric current density imaging of chemical reactions. Proc Int Soc Magn Reson Med 2001;9. [14] Beravs K, Desmar A, Desmar F. Magnetic resonance current density imaging of chemical processes and reactions. J Magn Reson 1999;137:253 – 7. [15] Song AW, Takahashi AM. Displacement encoded imagine of small electric current. Proc Int Soc Mag Reson Med 2000;8.