Chapter 21: Photoacoustic Tomography

Tomography so far has either shone radiation through the body from outside (CT) or measured radiation emitted by an internal tracer (PET/SPECT). Photoacoustic tomography (PAT) is neither. It shines light to create sound inside the body, then listens to that sound to form an image. Straddling two physics, it changes the very geometry of measurement, and reconstruction shifts from "lines" to "arcs."

Heat with light, listen with sound

A short pulse of laser light on tissue instantly warms the molecules that absorb light well (hemoglobin in blood, melanin, and so on). Warming means expansion, and expansion launches a pressure wave — ultrasound. This is the photoacoustic effect. The ultrasound travels through the tissue and reaches detectors surrounding the object.

The appeal is that it takes the best of light and sound. Light has enormously high absorption contrast (it carries molecular-level information, such as distinguishing oxygenated from deoxygenated hemoglobin) but scatters strongly in tissue and blurs quickly. Ultrasound scatters weakly and travels straight to depth, but has poor contrast of its own. Photoacoustic imaging uses light to create a high-contrast acoustic source and receives that sound as weakly scattered ultrasound, carrying light's molecular information out with ultrasound's depth penetration.

The measurement is an integral over a circle

For reconstruction, the decisive point is what the measured signal is an integral of. Consider one detector. With sound speed $c$, the signal arriving at time $t$ came from radius $r = ct$ away from that detector. In two dimensions, every source on the circle of radius $r$ centered on the detector arrives at the same time, superimposed. So the time series is an integral of the source over concentric circles centered on the detector.

$$g(\mathbf{d}, t) \;\propto\; \int_{|\mathbf{x} - \mathbf{d}| = ct} p_0(\mathbf{x})\, dl$$

Where CT's measurement was a line integral along a straight line (the Radon transform), photoacoustic is an integral along an arc. This is the circular Radon transform (the 2D version of the spherical mean transform).

Simulation: circular Radon transform and universal backprojection

A vessel-like source is "recorded" by a full ring of detectors (the circular Radon transform generates the time series). Painting it back along circles centered on each detector makes the vessels emerge. Narrow the detector arc and arc-shaped streaks remain in the background, blurring the vessels. In particular, structure "sideways" to the reduced detector set — oriented so the sound flies along the line of detectors — is lost first. This is the "some directions are invisible" problem of Chapter 20's limited angle, now appearing in the arc geometry.

Differences from CT, and what is shared

Photoacoustic resembles PET/SPECT in that the source is inside the body (emission tomography). But where PET integrated along straight LORs, photoacoustic integrates along arcs, so the reconstruction geometry is fundamentally different. At the same time, the idea of raising the source by backprojection, the limited-view problem when detectors cannot ring the whole object, and the spread of research filling the gaps with compressed sensing and deep learning are all shared entirely with CT and MRI. Different physics, a different integral shape — yet the same skeleton as an inverse problem. Here too, tomography shows itself as one piece of mathematics wearing many faces.

Key points

Photoacoustic tomography creates ultrasound sources inside the body with pulsed light and images them with surrounding detectors. Its advantage is carrying light's high absorption contrast out with the depth penetration of weakly scattered ultrasound. The measured signal is an integral over circles centered on the detector (the circular Radon transform), a different geometry from CT's line integral, and reconstruction becomes universal backprojection along arcs. Yet "raise it by backprojection," "it breaks without full coverage," and "fill the gap with prior knowledge" are all shared with CT. The next chapter treats the missing wedge that arises when a specimen cannot be tilted far, in electron tomography and in three dimensions — Chapter 20's wedge, now solid.

References

• Xu M, Wang LV. Universal back-projection algorithm for photoacoustic computed tomography. Physical Review E 71, 016706 (2005).
• Wang LV, Hu S. Photoacoustic tomography: in vivo imaging from organelles to organs. Science 335, 1458–1462 (2012).
• Kuchment P, Kunyansky L. Mathematics of thermoacoustic tomography. European Journal of Applied Mathematics 19, 191–224 (2008).