General equation

Given the heating function ${\displaystyle H({\vec {r}},t)}$, the generation and propagation of photoacoustic wave pressure ${\displaystyle p({\vec {r}},t)}$ in an acoustically homogeneous inviscid medium is governed by

${\displaystyle \nabla ^{2}p({\vec {r}},t)-{\frac {1}{v_{s}^{2}}}{\frac {\partial ^{2}}{\partial {t^{2}}}}p({\vec {r}},t)=-{\frac {\beta }{C_{p}}}{\frac {\partial }{\partial t}}H({\vec {r}},t)\qquad \qquad \quad \quad (1),}$

where ${\displaystyle v_{s}}$ is the speed of sound in medium, ${\displaystyle \beta }$ is the thermal expansion coefficient, and ${\displaystyle C_{p}}$ is the specific heat capacity at constant pressure. Eq. (1) holds under thermal confinement to ensure that heat conduction is negligible during the laser pulse excitation. The thermal confinement occurs when the laser pulsewidth is much shorter than the thermal relaxation time.^[4]

The forward solution of Eq. (1) is given by

${\displaystyle \left.p({\vec {r}},t)={\frac {\beta }{4\pi C_{p}}}\int {\frac {d{\vec {r'}}}{|{\vec {r}}-{\vec {r'}}|}}{\frac {\partial H({\vec {r'}},t')}{\partial t'}}\right|_{t'=t-|{\vec {r}}-{\vec {r'}}|/v_{s}}\qquad \quad \,\,\,\,(2).}$

In stress confinement, which occurs when the laser pulsewidth is much shorter than the stress relaxation time,^[4] Eq. (2) can be further derived as

${\displaystyle p({\vec {r}},t)={\frac {1}{4\pi v_{s}^{2}}}{\frac {\partial }{\partial t}}\left[{\frac {1}{v_{s}t}}\int d{\vec {r'}}p_{0}({\vec {r'}})\delta \left(t-{\frac {|{\vec {r}}-{\vec {r'}}|}{v_{s}}}\right)\right]\qquad \,(3),}$

where ${\displaystyle p_{0}}$ is the initial photoacoustic pressure.

Universal reconstruction algorithm

In a PAT system, the acoustic pressure is detected by scanning an ultrasonic transducer over a surface that encloses the photoacoustic source. To reconstruct the internal source distribution, we need to solve the inverse problem of equation (3) (i.e. to obtain ${\displaystyle p_{0}}$). A representative method applied for PAT reconstruction is known as the universal backprojection algorithm.^[5] This method is suitable for three imaging geometries: planar, spherical, and cylindrical surfaces.

The universal back projection formula is

${\displaystyle \left.p_{0}({\vec {r}})=\int _{\Omega _{0}}{\frac {d\Omega _{0}}{\Omega _{0}}}\left[2p({\vec {r_{0}}},v_{s}t)-2v_{s}t{\frac {\partial p({\vec {r_{0}}},v_{s}t)}{\partial (v_{s}t)}}\right]\right|_{t=|{\vec {r}}-{\vec {r_{0}}}|/v_{s}},\qquad \quad (4),}$

where ${\displaystyle \Omega _{0}}$ is the solid angle subtended by the entire surface ${\displaystyle S_{0}}$ with respect to the reconstruction point ${\displaystyle {\vec {r}}}$ inside ${\displaystyle S_{0}}$, and

${\displaystyle d\Omega _{0}={\frac {dS_{0}}{|{\vec {r}}-{\vec {r_{0}}}|^{2}}}{\frac {{\hat {n}}_{0}^{s}.({\vec {r}}-{\vec {r_{0}}})}{|{\vec {r}}-{\vec {r_{0}}}|}}.}$

Simple system

A simple PAT/TAT/OAT system is shown in the left part of Fig. 3. The laser beam is expanded and diffused to cover the whole region of interest. Photoacoustic waves are generated proportional to the distribution of optical absorption in the target, and are detected by a single scanned ultrasonic transducer. A TAT/OAT system is the same as PAT except that it uses a microwave excitation source instead of a laser. Although single-element transducers have been employed in these two systems, the detection scheme can be extended to use ultrasound arrays as well.

Biomedical applications

Intrinsic optical or microwave absorption contrast and diffraction-limited high spatial resolution of ultrasound make PAT and TAT promising imaging modalities for wide biomedical applications: