📂Tomography

Principles of Photoacoustic Tomography

Photoacoustic Effect

The photoacoustic (or optoacoustic) effect is a physical phenomenon discovered¹ by Alexander Graham Bell, who is erroneously known² as the inventor of the telephone in 1880. When a material is exposed to light (electromagnetic waves), it absorbs the light, leading to an increase in temperature and thermal expansion. Once the radiation ceases, the material cools and contracts. This expansion and contraction of the material result in pressure changes, thereby emitting acoustic waves. This is referred to as the photoacoustic effect.

Photoacoustic Tomography

PhotoAcoustic Tomography (PAT) signifies a nondestructive testing method that utilizes the photoacoustic effect. The principle is as follows:

1. Light is directed at the object of interest.
2. Due to the photoacoustic effect, the object emits sound waves, which are measured by detectors placed around the object.
3. A series of algorithms reconstructs an internal image of the target from these measured signals.

Mathematical Model

Assumption: The speed of sound is constant

Sound waves refer to the transmission of pressure changes through a medium. Let’s denote the initial pressure generated by the photoacoustic effect as $f(x)$.

$$ \begin{equation} \begin{aligned} \partial_{t}^{2} p(x,t) =&\ \Delta p(x,t) & (x, t) \in \mathbb{R}^{n} \times [0, \infty) \\ p(x,0) =&\ f(x) \\ \partial_{t}p(x,0) =&\ 0 \end{aligned} \end{equation} $$

Let the region where the detector exists be denoted as $\Omega$. Then, the sound waves measured by the detector are the restriction $p|_{\Omega}$ to the detector space of the solution to the initial value problem of the wave equation given initial value $f$. As the solution $p|_{\Omega}$ is determined whenever an initial value $f$ is provided, let’s define this mapping as the wave forward operator $\mathcal{W}$.

$$ \begin{align*} \mathcal{W} : f &\mapsto p|_{\Omega}, & p \text{ is solution of } (1) \\ \mathcal{W}f &= p|_{\Omega} \end{align*} $$

Hence, the goal in PAT is to reconstruct the initial value $f$ from the given data $\mathcal{W}f$, in other words, to find $\mathcal{W}^{-1}$.