Magnetoacoustic Tomography with Magnetic Induction

Magnetoacoustic Tomography with Magnetic Induction (MAT-MI) is a hybrid imaging modality that combines electromagnetic induction and ultrasound detection to map the electrical conductivity distribution of biological tissues. A time‐varying magnetic field induces eddy currents in conductive regions; these currents interact with an applied static magnetic field to generate Lorentz forces. The resulting mechanical vibrations propagate as acoustic waves, which are subsequently detected by ultrasound transducers.

Ultrasound transducers positioned around the subject detect the emitted acoustic signals, recording time‐of‐flight and amplitude information. A tomographic reconstruction algorithm then back‐projects these signals to yield high‐contrast images of conductivity variations. MAT‐MI is non‐ionizing and leverages the high sensitivity of conductivity contrast to pathological changes, making it promising for early disease detection and tissue characterization.

1. Magnetic excitation

   A pulsed RF field

   $$\mathbf{B}_1(\mathbf{r},t) = \mathbf{B}_1(\mathbf{r})f(t)$$

   induces an electric field

   $$\mathbf{E}(\mathbf{r},t)=\mathbf{E}(\mathbf{r})f’(t).$$

2. Eddy currents

   In a conductor with conductivity $\sigma(\mathbf{r})$, Ohm’s law gives the approximate equations for the current

   $$\mathbf{J}(\mathbf{r},t)=\sigma(\mathbf{r})\mathbf{E}(\mathbf{r},t).$$

   Decomposing the current into a rotational and a non-rotational part gives

   $$\mathbf{J} = \mathbf{J}_c + ∇θ,$$

   where only the rotational part $\mathbf{J}_c$ drives the acoustic source.

3. Lorentz force & source term

   With a (quasi-static) static field $\mathbf{B}_0$, the Lorentz force is

   $$\mathbf{F} = \mathbf{J}\times\mathbf{B}_0.$$

   Its divergence gives the acoustic source

   $$A_s(\mathbf{r},t)=∇·\mathbf{F} \approx ∇·(\mathbf{J}_c\times\mathbf{B}_0).$$

4. Acoustic wave equation

   The pressure $p(\mathbf{r},t)$ satisfies

   $$∇^2p - \frac{1}{c_s^2},\frac{∂^2p}{∂t^2} = A_s(\mathbf{r},t),$$

   where $c_s$ is the sound speed.

5. Received pressure (Green’s integral)

   For detectors at $\mathbf{r}$ and sources in object region $\Omega$:

   $$p(\mathbf{r},t) = \frac{1}{4\pi}\int_{\Omega} \mathbf{F}_d(\mathbf{r}’)·∇_{r’},G(\mathbf{r},\mathbf{r}’,t)d^3r’,$$

   with $\mathbf{F}_d=\mathbf{J}_c×\mathbf{B}_0$,

   $$G(\mathbf{r},\mathbf{r}’,t) = \frac{\delta\bigl(t - \tfrac{|\mathbf{r}-\mathbf{r}’|}{c_s}\bigr)} {|\mathbf{r}-\mathbf{r}’|}.$$