“We have developed a new approach called magnetoacoustic tomography with magnetic induction (MAT-MI) by combining ultrasound and magnetism. In this method, the object is in a static magnetic field and a time-varying (μs) magnetic field... The time-varying magnetic field induces an eddy current in the object. Consequently, the object will emit ultrasonic waves through the Lorentz force produced by the combination of the eddy current and the static magnetic field. The ultrasonic waves are then collected by the detectors located around the object for image reconstruction. MAT-MI combines the good contrast of EIT [electrical impedance tomography] with the good spatial resolution of sonography.”One nice feature of MAT-MI is that it fits so well into the 4th edition of Intermediate Physics for Medicine and Biology, in which Russ Hobbie and I analyze both eddy currents caused by Faraday induction (Chapter 8) and ultrasound imaging (Chapter 13). Another characteristic of MAT-MI is that the physics is simple enough that it can be summarized in a homework problem. So, dear reader, here is a new problem that will help you understand MAT-MI.

Section 8.6

Problem 25 ½Assume a sheet of tissue having conductivity σ is placed perpendicular to a uniform, strong, static magnetic fieldB_{0}, and a weaker spatially uniform but temporally oscillating magnetic fieldB_{1}(t).

(a) Derive an expression for the electric fieldEinduced by the oscillating magnetic field. It will depend on the distance r from the center of the sheet and the rate of change of the magnetic field.

(b) Determine an expression for the current densityJby multiplying the electric field by the conductivity.

(c) The force per unit volume,F, is given by the Lorentz force,J×B_{0}(ignore the weakB_{1}). Find an expression forF.

(d) The source of the ultrasonic pressure waves can be expressed as the divergence of the Lorentz Force. Derive an expression for∇ ·F.

(e) Draw a picture showing the directions ofJ,B_{0}, andF.

**F**, and therefore influence the radiated ultrasonic wave. Thus, information about the conductivity distribution

*σ(x,y)*is contained in the pressure. Subsequent papers by He and his colleagues explore methods for extracting

*σ(x,y)*from the ultrasonic signal. Potential applications include using MAT-MI to image breast cancer tumors.

I’ve worked on MAT-MI a little bit. University of Michigan student Kayt Brinker and I published a paper describing MAT-MI in anisotropic tissue like skeletal muscle, where the conductivity is much higher parallel to the muscle fibers than perpendicular to them [Brinker, K. and B. J. Roth (2008) The effect of electrical anisotropy during magneto-acoustic tomography with magnetic induction,

*IEEE Transactions on Biomedical Engineering*,

**55**: 1637-1639]. For some reason the figures published by the journal were not of high quality, so here I reproduce a better version of Figure 6, which shows the pressure wave produced during MAT-MI.

Fig. 6. Pressure at 20, 40, 60, and 80 μs in isotropic and anisotropic tissue. Each panel represents a 400 mm by 400 mm area.

In isotropic tissue, the wave propagates outward, the same in all directions. In electrically anisotropic tissue, the pressure is greater in the direction perpendicular to the fiber axis (vertical) than parallel to it (horizontal). The main difference between our calculation and that in the new homework problem given above is that Kayt and I restricted the oscillating magnetic field

**B**

_{1}to a small region (40 mm radius) at the center of the tissue sheet.

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