The potential induced in anisotropic tissue by the ... - Springer Link
Med Bio Eng Comput (2008) 46:195–197 DOI 10.1007/s11517-007-0292-9
SHORT COMMUNICATION
The potential induced in anisotropic tissue by the ultrasonically-induced Lorentz force Nancy Tseng Æ Bradley J. Roth
Received: 12 September 2007 / Accepted: 20 November 2007 / Published online: 7 December 2007 Ó International Federation for Medical and Biological Engineering 2007
Abstract In the presence of a magnetic field, an ultrasonic wave propagating through tissue will induce Lorentz forces on the ions, resulting in an electrical current. If the electrical conductivity is anisotropic, this current is tilted toward the fiber direction, causing charge to accumulate between half-wavelengths: positive charge where the current vectors converge and negative where the current vectors diverge. This charge produces an electric field in the direction of propagation that is associated with an electrical potential, and this electric field causes an additional current that is also tilted by the anisotropy. The final result is the total current pointing perpendicular to the direction of propagation and a charging of the tissue every half wavelength. The potential has a greater magnitude than that obtained from colloidal suspensions or ionic solutions (ultrasonic vibration potentials) and may be used as the basis of a technique to image conductivity.
conductivity in tissue. Electrical impedance tomography is the traditional approach to imaging conductivity [2]. A new method, which is an alternative to electrical impedance tomography, is based on electrical effects arising when an ultrasonic wave propagates through tissue in the presence of a magnetic field [3–6]. The acoustic wave causes ions to move, inducing Lorentz forces that create an electrical current. The detected current can be used to deduce the conductivity distribution. In this rapid communication, we describe a novel feature of the ultrasonically-induced Lorentz force in anisotropic tissue: an oscillating electrical potential propagates along with the ultrasonic wave. This potential is analogous to ultrasonic vibration potentials measured in colloidal suspensions or ionic solutions [1], but it is caused by a different mechanism.
2 Methods and results Keywords Lorentz force Anisotropy Ultrasound Electrical impedance tomography Ultrasonic vibration potential
1 Introduction An important goal of biomedical engineering is to develop techniques to image the distribution of electrical
Consider a uniform sheet of tissue having anisotropic conductivity (Fig. 1). An ultrasonic wave propagates in the x direction such that the tissue velocity, u, varies as ^ 0 cos ð2px=kÞ; u ¼ iu
ð1Þ
where u0 is the amplitude and k is the wavelength. A static, uniform magnetic field of strength B is in the y direction (out of the paper). B ¼ ^jB:
ð2Þ
The Lorentz force per unit charge is equal to the cross product u 9 B, which is in the z direction,
B. J. Roth (&) Deparment of Physics, Oakland University, Rochester, MI 48309, USA e-mail: [email protected]
u B ¼ ^ku0 B cos ð2px=kÞ:
N. Tseng University of Michigan, Ann Arbor, MI 48109, USA
The electrical conductivity of the tissue is anisotropic with the direction of highest conductivity (the ‘‘fiber direction’’)
ð3Þ
123
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Med Bio Eng Comput (2008) 46:195–197
rxz k u0 B sin ð2px=kÞ rxx 2p ðrL rT Þ cos h sin h k u0 B sin ð2px=kÞ: ¼ rL cos2 h þ rT sin2 h 2p
V¼
Fig. 1 Three half-wavelengths of an ultrasonic wave propagating to the right
at an angle h with respect to the x axis. The conductivity ~ is tensor r rxx rxz ~¼ r rxz rzz rL cos2 h þ rT sin2 h ðrL rT Þ cos h sin h ¼ ; ð4Þ ðrL rT Þ cos h sin h rL sin2 h þ rT cos2 h where rL and rT are the conductivities parallel to and perpendicular to the fiber axis, respectively. The Lorentz force will induce a current density JLorentz equal to ~ðu BÞ: If the tissue were isotropic, so that the r conductivity was simply a scalar r, this current density would be entirely in the z direction and would equal JLorentz ¼ ^ kru0 B cos ð2px=kÞ: In anisotropic tissue, however, the current has a component in the x direction parallel to the direction of wave propagation, JLorentz ¼ ^irxz þ ^ krzz u0 B cos ð2px=kÞ: ð5Þ The current density J in biological tissue is continuous, implying that rJ = 0. (Here we assume the current is quasistatic. We could account for capacitive effects using the continuity equation, r J ¼ oq=ot; where q is the charge density, but we ignore capacitive effects in our analysis). The expression for JLorentz in isotropic tissue has zero divergence, but the expression in anisotropic tissue does not. This means that charge accumulates in anisotropic tissue, creating an electrical potential V. We can write the current density as the sum of two terms ~ðrV þ u BÞ: J ¼ Jcharge þ JLorentz ¼ r
ð6Þ
If we set the divergence of the current density to zero, we find that the potential obeys Poisson’s equation, ~rV ¼ r JLorentz : rr
ð7Þ
Ignoring effects at the tissue boundaries, we can solve this equation for V,
123
ð8Þ
For isotropic tissue, rL equals rT, so V = 0. In anisotropic tissue, if h equals 0 or 90°, the potential vanishes. For anisotropic tissue with angles between 0 and 90°, the potential is out of phase with the velocity (where u = 0, V has its largest amplitude). The potential is proportional to the magnetic field strength, the amplitude of the ultrasound wave, and the wavelength. It oscillates sinusoidally in x, creating ‘‘stripes’’ of positive and negative potential. Figure 1 shows this behavior graphically. Three halfwavelengths of the ultrasonic wave are shown with the wave propagating to the right and the magnetic field coming out of the paper. The direction of the tissue motion and the Lorentz force alternate in each half-wavelength. The current density due to the Lorentz force is tilted from the z direction by the fiber angle. This causes charge to accumulate: positive charge where the JLorentz vectors converge and negative where the JLorentz vectors diverge. This charge produces an electric field E that is in the x direction and is associated with the potential V. This electric field causes a current Jcharge that is also tilted by the anisotropy. The final result is the total current J pointing in the z direction (so that the total current has no divergence) and a charging of the tissue every half wavelength.
3 Discussion Montalibet et al. [4] have measured currents caused by the ultrasonically-induced Lorentz force, and we use parameters from their experiment to estimate the magnitude of the potential. The frequency of the ultrasound pulse was 500 kHz, the peak pressure was 1.5 MPa, and the magnetic field was B = 0.35 T. If the speed of sound is 1,500 m/s and the acoustic impedance is 1.5 MPa s/m, the wavelength of the ultrasound wave is k = 3 mm and the speed of the tissue is u0 = 1 m/s. Assume the measurement is performed in skeletal muscle, which has an anisotropy ratio rL =rT of about 3, and take the angle to be h = 45°. Equation (8) indicates that in this case the amplitude of the potential is 0.17 mV. Potentials of this size are easily measurable for imaging, although they probably have little biological effect. If the pressure and magnetic field were larger, for instance during lithotripsy monitored by magnetic resonance imaging, the potential might become large enough to have a significant biological effect. Our analysis was for a homogeneous unbounded tissue. If the tissue is bounded, or if the conductivity is heterogeneous, additional effects occur. In fact, Montalibet et al.
Med Bio Eng Comput (2008) 46:195–197
proposed their technique as a way to image conductivity gradients because isotropic tissue with uniform conductivity produces zero potential. In fibrous tissue, such as muscle or nerve, both heterogeneity and anisotropy must be included in order to interpret conductivity images produced using the ultrasonically-induced Lorentz force. Our analysis describes the potential in a macroscopically anisotropic tissue. Macroscopic anisotropy arises from oriented microscopic heterogeneities. The significance of the resulting microscopic voltage fluctuations will depend on the recording electrode. If the electrode is large enough that it averages the potential over many heterogeneities, only the macroscopic effects remain and our results should hold. If it records the potential variation over the spatial scale of the heterogeneities, our results may represent only one part of the signal. Equation (8) gives the voltage measured with respect to some distant ground. If the voltage difference is measured between two electrodes aligned in the x-direction, the result depends crucially on the electrode separation. If the electrodes are separated by an integral number of wavelengths, no signal is measured. If they are separated by a halfintegral number of wavelengths, the signal oscillates with twice its normal amplitude. Vibration potentials in ionic solutions or colloidal suspensions appear similar to the potential we calculate, but the mechanism is different. In vibration potentials, the positive and negative ions move by different amounts in response to an ultrasonic wave because of the different inertia and mobility of the two ions (or, for a colloid suspension, the different inertia and mobility of the charged colloidal particle and the surrounding counterions) [1]. No magnetic field is required to produce vibration potentials. Our effect, on the other hand, is proportional to the magnetic field and is independent of the inertia and mobility of the charge carriers. Vibration potentials in muscle, measured by Beveridge et al., are
197
smaller than 0.02 lV [1]. The potential we calculate arising from the ultrasonically-induced Lorentz force is 10,000 times larger than this. In fact, traditional measurements of vibration potentials in anisotropic tissue could be corrupted by effects arising from the ultrasonically-induced Lorentz force in the presence of the earth’s magnetic field (about 0.05 mT). In conclusion, our analysis predicts a new type of potential induced when an ultrasonic wave propagates in anisotropic tissue in the presence of a magnetic field. Both the ultrasonically-induced Lorentz force and ultrasonic vibration potentials have been proposed as ways to obtain images of conductivity in biological tissue. Our results suggest that algorithms to image the conductivity of muscle and nerve must account for tissue anisotropy if they are to produce reliable images. We hope that our results will motivate experimentalists to test our predictions in the laboratory. Acknowledgments This research was supported by the SMaRT program at Oakland University, a Research Experience for Undergraduates (REU) and was funded by NSF grant DMR-055 2779.
References 1. Beveridge AC, Wang S, Diebold GJ (2004) Imaging based on the ultrasonic vibration potential. Appl Phys Lett 85:5466–5468 2. Holder DS (2004) Electrical impedance tomography: methods, history and applications. Institute of Physics Pub, Philadelphia 3. Montalibet A, Jossinet J, Matias A (2001) Scanning electric conductivity gradients with ultrasonically-induced Lorentz force. Ultrason Imaging 23:117–132 4. Montalibet A, Jossinet J, Matias A, Cathignol D (2001) Electric current generated by ultrasonically induced Lorentz force in biological media. Med Biol Eng Comput 39:15–20 5. Wen H (1999) Volumetric hall effect tomography—a feasibility study. Utrason Imaging 21:186–200 6. Wen H, Shah J, Balaban RS (1998) Hall effect imaging. IEEE Trans Biomed Eng 45:119–124
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SHORT COMMUNICATION
The potential induced in anisotropic tissue by the ultrasonically-induced Lorentz force Nancy Tseng Æ Bradley J. Roth
Received: 12 September 2007 / Accepted: 20 November 2007 / Published online: 7 December 2007 Ó International Federation for Medical and Biological Engineering 2007
Abstract In the presence of a magnetic field, an ultrasonic wave propagating through tissue will induce Lorentz forces on the ions, resulting in an electrical current. If the electrical conductivity is anisotropic, this current is tilted toward the fiber direction, causing charge to accumulate between half-wavelengths: positive charge where the current vectors converge and negative where the current vectors diverge. This charge produces an electric field in the direction of propagation that is associated with an electrical potential, and this electric field causes an additional current that is also tilted by the anisotropy. The final result is the total current pointing perpendicular to the direction of propagation and a charging of the tissue every half wavelength. The potential has a greater magnitude than that obtained from colloidal suspensions or ionic solutions (ultrasonic vibration potentials) and may be used as the basis of a technique to image conductivity.
conductivity in tissue. Electrical impedance tomography is the traditional approach to imaging conductivity [2]. A new method, which is an alternative to electrical impedance tomography, is based on electrical effects arising when an ultrasonic wave propagates through tissue in the presence of a magnetic field [3–6]. The acoustic wave causes ions to move, inducing Lorentz forces that create an electrical current. The detected current can be used to deduce the conductivity distribution. In this rapid communication, we describe a novel feature of the ultrasonically-induced Lorentz force in anisotropic tissue: an oscillating electrical potential propagates along with the ultrasonic wave. This potential is analogous to ultrasonic vibration potentials measured in colloidal suspensions or ionic solutions [1], but it is caused by a different mechanism.
2 Methods and results Keywords Lorentz force Anisotropy Ultrasound Electrical impedance tomography Ultrasonic vibration potential
1 Introduction An important goal of biomedical engineering is to develop techniques to image the distribution of electrical
Consider a uniform sheet of tissue having anisotropic conductivity (Fig. 1). An ultrasonic wave propagates in the x direction such that the tissue velocity, u, varies as ^ 0 cos ð2px=kÞ; u ¼ iu
ð1Þ
where u0 is the amplitude and k is the wavelength. A static, uniform magnetic field of strength B is in the y direction (out of the paper). B ¼ ^jB:
ð2Þ
The Lorentz force per unit charge is equal to the cross product u 9 B, which is in the z direction,
B. J. Roth (&) Deparment of Physics, Oakland University, Rochester, MI 48309, USA e-mail: [email protected]
u B ¼ ^ku0 B cos ð2px=kÞ:
N. Tseng University of Michigan, Ann Arbor, MI 48109, USA
The electrical conductivity of the tissue is anisotropic with the direction of highest conductivity (the ‘‘fiber direction’’)
ð3Þ
123
196
Med Bio Eng Comput (2008) 46:195–197
rxz k u0 B sin ð2px=kÞ rxx 2p ðrL rT Þ cos h sin h k u0 B sin ð2px=kÞ: ¼ rL cos2 h þ rT sin2 h 2p
V¼
Fig. 1 Three half-wavelengths of an ultrasonic wave propagating to the right
at an angle h with respect to the x axis. The conductivity ~ is tensor r rxx rxz ~¼ r rxz rzz rL cos2 h þ rT sin2 h ðrL rT Þ cos h sin h ¼ ; ð4Þ ðrL rT Þ cos h sin h rL sin2 h þ rT cos2 h where rL and rT are the conductivities parallel to and perpendicular to the fiber axis, respectively. The Lorentz force will induce a current density JLorentz equal to ~ðu BÞ: If the tissue were isotropic, so that the r conductivity was simply a scalar r, this current density would be entirely in the z direction and would equal JLorentz ¼ ^ kru0 B cos ð2px=kÞ: In anisotropic tissue, however, the current has a component in the x direction parallel to the direction of wave propagation, JLorentz ¼ ^irxz þ ^ krzz u0 B cos ð2px=kÞ: ð5Þ The current density J in biological tissue is continuous, implying that rJ = 0. (Here we assume the current is quasistatic. We could account for capacitive effects using the continuity equation, r J ¼ oq=ot; where q is the charge density, but we ignore capacitive effects in our analysis). The expression for JLorentz in isotropic tissue has zero divergence, but the expression in anisotropic tissue does not. This means that charge accumulates in anisotropic tissue, creating an electrical potential V. We can write the current density as the sum of two terms ~ðrV þ u BÞ: J ¼ Jcharge þ JLorentz ¼ r
ð6Þ
If we set the divergence of the current density to zero, we find that the potential obeys Poisson’s equation, ~rV ¼ r JLorentz : rr
ð7Þ
Ignoring effects at the tissue boundaries, we can solve this equation for V,
123
ð8Þ
For isotropic tissue, rL equals rT, so V = 0. In anisotropic tissue, if h equals 0 or 90°, the potential vanishes. For anisotropic tissue with angles between 0 and 90°, the potential is out of phase with the velocity (where u = 0, V has its largest amplitude). The potential is proportional to the magnetic field strength, the amplitude of the ultrasound wave, and the wavelength. It oscillates sinusoidally in x, creating ‘‘stripes’’ of positive and negative potential. Figure 1 shows this behavior graphically. Three halfwavelengths of the ultrasonic wave are shown with the wave propagating to the right and the magnetic field coming out of the paper. The direction of the tissue motion and the Lorentz force alternate in each half-wavelength. The current density due to the Lorentz force is tilted from the z direction by the fiber angle. This causes charge to accumulate: positive charge where the JLorentz vectors converge and negative where the JLorentz vectors diverge. This charge produces an electric field E that is in the x direction and is associated with the potential V. This electric field causes a current Jcharge that is also tilted by the anisotropy. The final result is the total current J pointing in the z direction (so that the total current has no divergence) and a charging of the tissue every half wavelength.
3 Discussion Montalibet et al. [4] have measured currents caused by the ultrasonically-induced Lorentz force, and we use parameters from their experiment to estimate the magnitude of the potential. The frequency of the ultrasound pulse was 500 kHz, the peak pressure was 1.5 MPa, and the magnetic field was B = 0.35 T. If the speed of sound is 1,500 m/s and the acoustic impedance is 1.5 MPa s/m, the wavelength of the ultrasound wave is k = 3 mm and the speed of the tissue is u0 = 1 m/s. Assume the measurement is performed in skeletal muscle, which has an anisotropy ratio rL =rT of about 3, and take the angle to be h = 45°. Equation (8) indicates that in this case the amplitude of the potential is 0.17 mV. Potentials of this size are easily measurable for imaging, although they probably have little biological effect. If the pressure and magnetic field were larger, for instance during lithotripsy monitored by magnetic resonance imaging, the potential might become large enough to have a significant biological effect. Our analysis was for a homogeneous unbounded tissue. If the tissue is bounded, or if the conductivity is heterogeneous, additional effects occur. In fact, Montalibet et al.
Med Bio Eng Comput (2008) 46:195–197
proposed their technique as a way to image conductivity gradients because isotropic tissue with uniform conductivity produces zero potential. In fibrous tissue, such as muscle or nerve, both heterogeneity and anisotropy must be included in order to interpret conductivity images produced using the ultrasonically-induced Lorentz force. Our analysis describes the potential in a macroscopically anisotropic tissue. Macroscopic anisotropy arises from oriented microscopic heterogeneities. The significance of the resulting microscopic voltage fluctuations will depend on the recording electrode. If the electrode is large enough that it averages the potential over many heterogeneities, only the macroscopic effects remain and our results should hold. If it records the potential variation over the spatial scale of the heterogeneities, our results may represent only one part of the signal. Equation (8) gives the voltage measured with respect to some distant ground. If the voltage difference is measured between two electrodes aligned in the x-direction, the result depends crucially on the electrode separation. If the electrodes are separated by an integral number of wavelengths, no signal is measured. If they are separated by a halfintegral number of wavelengths, the signal oscillates with twice its normal amplitude. Vibration potentials in ionic solutions or colloidal suspensions appear similar to the potential we calculate, but the mechanism is different. In vibration potentials, the positive and negative ions move by different amounts in response to an ultrasonic wave because of the different inertia and mobility of the two ions (or, for a colloid suspension, the different inertia and mobility of the charged colloidal particle and the surrounding counterions) [1]. No magnetic field is required to produce vibration potentials. Our effect, on the other hand, is proportional to the magnetic field and is independent of the inertia and mobility of the charge carriers. Vibration potentials in muscle, measured by Beveridge et al., are
197
smaller than 0.02 lV [1]. The potential we calculate arising from the ultrasonically-induced Lorentz force is 10,000 times larger than this. In fact, traditional measurements of vibration potentials in anisotropic tissue could be corrupted by effects arising from the ultrasonically-induced Lorentz force in the presence of the earth’s magnetic field (about 0.05 mT). In conclusion, our analysis predicts a new type of potential induced when an ultrasonic wave propagates in anisotropic tissue in the presence of a magnetic field. Both the ultrasonically-induced Lorentz force and ultrasonic vibration potentials have been proposed as ways to obtain images of conductivity in biological tissue. Our results suggest that algorithms to image the conductivity of muscle and nerve must account for tissue anisotropy if they are to produce reliable images. We hope that our results will motivate experimentalists to test our predictions in the laboratory. Acknowledgments This research was supported by the SMaRT program at Oakland University, a Research Experience for Undergraduates (REU) and was funded by NSF grant DMR-055 2779.
References 1. Beveridge AC, Wang S, Diebold GJ (2004) Imaging based on the ultrasonic vibration potential. Appl Phys Lett 85:5466–5468 2. Holder DS (2004) Electrical impedance tomography: methods, history and applications. Institute of Physics Pub, Philadelphia 3. Montalibet A, Jossinet J, Matias A (2001) Scanning electric conductivity gradients with ultrasonically-induced Lorentz force. Ultrason Imaging 23:117–132 4. Montalibet A, Jossinet J, Matias A, Cathignol D (2001) Electric current generated by ultrasonically induced Lorentz force in biological media. Med Biol Eng Comput 39:15–20 5. Wen H (1999) Volumetric hall effect tomography—a feasibility study. Utrason Imaging 21:186–200 6. Wen H, Shah J, Balaban RS (1998) Hall effect imaging. IEEE Trans Biomed Eng 45:119–124
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