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Electrical conductivity can be measured using the ultrasonically-induced Lorentz force. An ultrasonic wave is passed through tissue in the presence of a magnetic field. Moving charges in a magnetic field are subject to the Lorentz force, which acts as the source of current and potential. This paper shows that ultrasonically-induced Lorentz force imaging can be formulated in a way that makes it similar to tomography: an image can be reconstructed using waves propagating in various directions. More specifically, measuring the dipole strength for a particular direction and wavelength is equivalent to measuring the Fourier transform of the conductivity distribution at one point in frequency space. Measurements at a variety of wavelengths and directions are equivalent to mapping the Fourier transform of the conductivity distribution. The conductivity can then be found by an inverse Fourier transform.
Wen et al.  and Montalibet et al. [3,4] developed a technique to image electrical conductivity using the ultrasonically-induced Lorentz force. An ultrasonic wave is passed through tissue in the presence of a magnetic field. The wave causes the tissue--and the charged ions within the tissue--to move. A moving charge in a magnetic field is subject to the Lorentz force . This force causes positive and negative charge to separate, and acts as the source of current and potential. Montalibet et al.  were able to detect the potential produced by a 1.5 MPa, 500 kHz ultrasonic wave in a 0.35 T magnetic field. They used these measurements to produce a one-dimensional image of conductivity gradients in biological tissue.
The goal of this paper is to analyze and extend this method to two or three dimensions, and to perform a preliminary calculation to prove that the method works in principle. We show that ultrasonically-induced Lorentz force imaging can be formulated in a way that makes it similar to tomography: reconstructing an image using waves propagating in various directions. More specifically, measuring the dipole strength for a particular direction and wavelength is equivalent to measuring the Fourier transform of the conductivity distribution at one point in frequency space. Measurements at a variety of wavelengths and directions are equivalent to mapping the Fourier transform of the conductivity distribution throughout frequency space, allowing us to determine the conductivity as a function of position by an inverse Fourier transform.
Let an ultrasound planar wave pass through tissue in the direction x. At one instant of time, the velocity u(x) of the tissue in the direction of propagation is
where uo is the amplitude of the wave and k is the spatial frequency. Assume the tissue has a conductivity distribution given by
where σo is a uniform background conductivity and σ′ (x, y, z) specifies a localized spatial variation about this background. A uniform, steady magnetic field is in the z-direction, B = Bo k^ . The movement of charge in the magnetic field will give rise to a Lorentz force, producing a current density J L , which by the right-hand-rule is directed in the y-direction
If separation of charge occurs, an Ohmic current density J C is also present
where V is the induced voltage. Continuity of total current, J L + J C , implies
In a completely homogeneous conductivity distribution (σ′ = 0), there would be currents J L inside the object having a strength that is proportional to the conductivity (as opposed to the gradient of the conductivity). These strictly solenoidal currents (zero divergence) have nonzero curl ( ×). In an inhomogeneous conductivity distribution, charge accumulates where the gradient of the conductivity is non-zero. The electric field produced by these charges is purely irrotational (has zero curl), but does have a divergence ( ·), leading to Eq. 5.
If the deviations of the conductivity distribution are small compared to the background, then we can ignore σ′ compared to σo on the left-hand-side of Eq. 5, obtaining Poisson's equation
Following a standard multipole expansion for the potential , the dipole term is given by
where the dipole is
If we integrate by parts in the y-direction and assume that σ′ is localized, so it goes to zero at large distances, then the y-component of the dipole strength becomes
In this paper, we consider only a 2-dimensional slab of tissue that is uniform and of thickness δ in the z direction, giving
The integral in x has the form of a Fourier transform.
The two-dimensional Fourier transform of the conductivity distribution is defined as
If we let ky = 0, this becomes
Thus, comparing Eqs. 10 and 12, we find that measurement of py(k) provides information about the two-dimensional Fourier transform of the conductivity distribution at the point in frequency space kx = k and ky = 0.
This analysis is very similar to the method of two-dimensional image reconstruction from projections by Fourier transform, often used to reconstruct computed tomography or magnetic resonance images . By analogy to CT, there is nothing special about the direction that we chose for the ultrasonic wave. Applying the ultrasonic wave in another direction and measuring the dipole moment perpendicular to that direction will provide information about the conductivity distribution along another line in frequency space. By taking measurements along many directions and using different wavelengths, we can map out the entire frequency plane. Once this information is known, a two-dimensional inverse Fourier transform will yield an image of the conductivity distribution.
We tested this algorithm by a four-step procedure: 1) calculate V(r,) numerically using Eq. 6 for various values of k and θ, 2) estimate the dipole strength p(k,θ) from the calculated V in each case, 3) use a bilinear interpolation to obtain p(kx, ky) from p(k,θ), and 4) perform a 2D inverse Fourier transform of p(kx, ky) to obtain the conductivity σ′(x,y).
We assume the tissue is circular, with radius 20 mm (Fig. 1a). The outer boundary of the tissue is sealed. The potential is solved using a finite difference approximation of Eq. 6 in polar coordinates (r, ), and overrelaxation. The space step is Δr = 0.2 mm and Δ = 2° (200 points in the r-direction, and 180 points in the -direction).
where C = 1 S m−1, yo = 3 mm, and a = 3 mm. Figure 1c shows the real part of the source term for k =1.18 mm−1 (corresponding to a wavelength of 5.33 mm) and θ = 50°. This source term results in the potential distribution in Fig. 1d.
In order to provide a more challenging test of the algorithm, we consider a conductivity distribution σ′(x,y) that is the sum of five Gaussians
where the constants Cn, an, xn, and yn are given in Table 1. Figure 2a plots σ′(x,y). We calculate the potential for 16 spatial frequencies ki = (i−1) Δk, where Δk = 0.393 mm−1, in 18 directions θj = (j−1) Δθ, where Δθ = 10°.
The potential distribution in Fig. 1d has two contributions: a part that has a spatial distribution very similar to the source function, and a part more distant from the source that has approximately a dipole pattern. Our algorithm requires determination of the dipole strength. We obtain a measure of the relative strength of the dipole by measuring the potential difference between opposite sides of the circular tissue, perpendicular to the direction of propagation of the ultrasonic wave (the dots in Fig. 1a). These measurements yield dipole values p(k, θ) at each k and θ.
In order to perform an inverse Fourier transform, we need the dipole strength for a regular array of kx and ky values. Therefore, we use a bilinear interpolation to obtain p(kx, ky) from p(k,θ), for 32 values (for both positive and negative spatial frequencies) using Δkx = Δky = 0.393 mm−1. Values of kx and ky corresponding togreater than 5.89 mm−1 are outside the range measured and are therefore set to zero.
The qualitative agreement between Figs. 2a and 2b is proof that in principle our method can be used to image the conductivity distribution. The method would be subject to many problems that arise when using Fourier techniques: aliasing, ringing at sharp discontinuities of conductivity, limitations on the spatial resolution caused by an upper limit of the values of the spatial frequency, etc. These problems are not immediately apparent in Fig. 2, because the conductivity distribution we consider is smooth, and therefore does not contain a lot of high frequency information.
Proof in principle does not imply that the method will be practical. We foresee several limitations of this method, some evident in our calculation and some not.
Despite these limitations, we believe that our analysis provides a valuable proof-of-principle that ultrasonically-induced Lorentz force tomography is possible. Wen et al.  and Montalibet et al. [3,4] showed that the method is feasible experimentally, and our computations (Fig. 2) show that under a somewhat idealized geometry the method works. Further research is necessary to determine if this new type of tomography will be useful in practice and what are the relative strengths and weaknesses of the method compared to other techniques to measure electrical conductivity.
This research was supported by the National Institutes of Health grant R01EB008421, and by the SMaRT program at Oakland University, a Research Experience for Undergraduates (REU) that was funded by NSF grant DMR-055 2779.