Thermoacoustic Computed Tomography with large planar receivers

Thermoacoustic tomography

Current research demonstrates that thermoacoustic computed tomography (TCT) is a promising hybrid imaging technique for nondestructive evaluation and medical imaging [3,4,5]. It combines the advantages of purely optical imaging (high contrast) and ultrasound imaging (high resolution).

Fig
16:
THERMOELASTIC EFFECT. The absorbed electromagnetic energy within the illuminated part of the fluid causes thermal expansion and a subsequent pressure field.
\includegraphics[width = 20em]{kuvat/heating.eps}

When an absorbing sample is illuminated by a pulsed electromagnetic wave, the absorbed power produces heat and results in mechanical expansion of the absorber (see figure 16). This thermoelastic expansion induces an acoustic pressure field

$\displaystyle p(\mathbf{r},t) = \frac{d}{dt} (t \, \mathbf{M}f ) (\mathbf{r},t)\,.
$

Here $ \mathbf{M}f $ is the spherical mean operator

$\displaystyle (\mathbf{M}f) (\mathbf{r},t) := \frac{1}{4 \pi} \int_{S^2}
f(\mathbf{r}+ t\omega) d\Omega(\omega)\;,
$

and $ f$ is the normalized energy deposition function.

Fig
17:
STANDARD SCANNING TECHNIC. Based on (idealized) point measurement data $ p$ one has to recover $ f$ form its spherical means.
\includegraphics[height = 11em]{kuvat/tctaufbau.eps}

TCT is concerned with the inverse problem of recovering an unknown energy deposition function $ f$ from temporal measurement data of the thermoacoustic pressure field $ p$ taken on a surface outside the illuminated sample. Potential application of thermoacoustic imaging reach from non-destructive evaluation to medical applications [3,4,5]. In 1998 Kruger et. al. [5] developed a prototype of the world's first thermoacoustic breast scanner.

Existing reconstruction algorithms utilize the fact that a small receiver measures an acoustic signal that at a given time approximates the integral of the energy density function over the surface of a sphere with the detector in the center (see figure 17). To generate images with high spatial resolution throughout the three-dimensional object this would require the use of ultrasound detectors that are much smaller than the imaged object.

Fig
:
NOVEL SCANNING TECHNIC. The planar receiver measures the Radon transform $ \mathbf{R}f(\mathbf{r},\cdot)$ of the energy deposition function $ f$ over parallel planes.
\includegraphics[height = 13em]{kuvat/tctlarge.eps}

Large planar receivers

In [1] we suggest to use large planar detectors to collect thermoacoustic data. In mathematical terms this data is given by the integral

$\displaystyle P (\mathbf{r},t) := \int_{E_\mathbf{r}} p(\mathbf{r}+ y,t) \,dE_\mathbf{r}(y)\;,
$

of the thermoacoustic pressure field over the planar detector surface $ E_\mathbf{r}$ . In practical experiments, the data required for imaging can be collected with relatively large piezo foils (see figure 19).

We have shown that $ P$ is closely connected to the standard Radon transform of the energy deposition function $ f$ [1]. This fact can be utilized for efficient numerical algorithms for thermoacoustic imaging.

Since in our technique the finite size of the receivers is explicitly included the limitation of the spatial resolution is only determined by the frequency bandwidth of the detector. A spatial resolution less than one micrometer should be possible with high bandwidth receiver [2].

Experiments with large planar receivers have been performed by the Upper Austrian Research GmbH (http://www.uar.at) in cooperation with G. Paltauf (Uni Graz). Future joint work will be done to develop and characterize high frequency detectors with the aim to develop a thermoacoustic microscope.

Fig
19:
EXPERIMENTAL SETUP. In the experiment a cylindrically symmetric object consisting of three cylindrical absorbers with diameters two, three respective four millimeters was mounted on a frame and was rotated around the central axis. Every $ 5^\circ $ the sample is illuminated by a short laser pulse of $ \hat{t} = 20 $ ps (picoseconds) and the generated thermoacoustic pressure signal was recorded with a 25 $ \mu $ m (micrometers) thick film of piezoelectric PVDF (Polyvinylidene Fluoride), which generated an electric signal proportional to the total pressure acting on the piezoelectric film.
\includegraphics[height = 10em]{kuvat/set.eps}
Fig
20:
Measured thermoacoustic pressure signal (time versus angle) for the experiment shown in figure 19.
\includegraphics[height = 12em, width = 16em]{kuvat/p_real.eps}
Fig
21:
Reconstruction from real measurement data.
\includegraphics[width = 16em]{kuvat/rec_real.eps}

Numerical results

For the numerical reconstruction we use a filtered backprojection (FBP) algorithm based on the reconstruction formula (see [1])

$\displaystyle f(x) =
\frac{i}{\pi} \int_0^{\pi}
\left( \mathbf{H} \, \frac{ \pa...
...
\bigl( \theta(\alpha), 1 - \langle x,\theta(\alpha)
\rangle \bigr) d\alpha\;,
$

valid for objects with cylindrical symmetry. Here $ \mathbf{H}$ is the Hilbert transform and $ \theta(\alpha)= ( \cos(\alpha),\sin(\alpha), 0)$ is the unit vector orthogonal to the planar receiver.

The numerical reconstruction from the measurement data gathered with the set-up shown in figure 19 is plotted in figure 21.

Acknowledgment

This work of M.H. is supported by the FWF (Austrian Science Foundation), grant Y-123 INF.

Bibliography

1
M. Haltmeier, O. Scherzer, P. Burgholzer, and G. Paltauf. Thermoacoustic computed tomography with large planar receivers. Inverse Problems 20(5), pp. 1663-1673, 2004.

2
P. Burgholzer, C. Hofer, G. Paltauf, M. Haltmeier, and O. Scherzer. Thermoacoustic tomography with integrating area and line detectors. Submitted to IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control, 2004.

3
X. Wang, Y. Pang, G. Ku, X. Xie, G. Stoica, and L.-H. Wang Non-invasive laser-induced photoacoustic tomography for structural and functional imaging of the brain in vivo. Nature Biotechnology 21 (7), pp. 803-806, 2003.

4
D. Finch, S. K. Patch, and Rakesh. Determining a function from its mean values over a family of spheres. SIAM J. Math. Anal., Vol. 35, pp. 1213-1240. 2004.

5
R.A. Kruger, K.M. Stantz, and W.L. Kiser. Thermoacoustic CT of the Breast. Proc. SPIE 4682, pp. 521-525. 2002.