## 3 Combining PAI and LUI in Biomedicine
### Combined PA and LU Speed-of-Sound Imaging  
LU speed-of-sound (SoS) measurement method found wide applications in material characterization, due to its high accuracy. SoS can be used to determine local elastic moduli, porosity, and residual stresses. Depending on composition of the biological tissue, the SoS can vary between 1350 m/s and 1700 m/s [99]. The quality of PA images reconstructed, without taking into account SoS variations, may be degraded. The short duration and aperiodic waveform of LU probe pulse makes it possible to make accurate transmission mode measurements of the SoS in various media [70], and use the reconstructed SoS map to correct PA images. In the approximation of geometrical acoustics, the total travel time of a probe pulse along the ray path is  
\[\mathcal{T}=\int_{l(\mathbf{r})}\frac{dl}{\widetilde{c}(\mathbf{r})},\]  
where \(\widetilde{c}(\mathbf{r})\) is the inhomogeneous speed of sound, and \(l(\mathbf{r})\) is the ray path. Generally, \(l(\mathbf{r})\) depends on \(\widetilde{c}(\mathbf{r})\), and the problem is non-linear. However, if the variations in the speed of sound are small (weakly refractive tissues), the problem can be linearized. If \(\mathcal{T}_{0}\) is the travel time and \(l(\tau_{0})\) is a straight path in a homogeneous medium with a speed of sound \(c\), we get  
\[\mathcal{T}=\mathcal{T}_{0}+\delta\mathcal{T}\approx\mathcal{T}_{0}+\int_{l( \tau_{0})}\left(\frac{1}{\widetilde{c}(\mathbf{r})}-\frac{1}{c}\right)dl.\]  
Since the integration paths are now straight, we have a form of a classical Radon transform with well-established solution techniques. [PERSON] et al. was the first to use a carbon fiber or a horsetail hair placed in the path of laser light as a PA generator, and a US array as a receiver in fan-beam fashioned (Figure 4) SoS imaging [99; 100; 101; 51]. Such a PA generator (which the authors refer to as a \"passive element\"--PE) acted as a point PA source in the image plane. The laser light was not blocked completely by the PE 250 m in diameter, and the remaining major part of laser light was used for illumination of the sample and PA imaging. A cross-correlation of the recorded waveforms with the reference waveforms (without a sample) was used to obtain the rough estimate of travel time, and the phase difference \(\delta\psi\) at a specific frequency \(f\) was used to calculate \(\delta\mathcal{T}=\delta\psi/(2\pi f)\) in [51], because frequency-dependent acoustic attenuation and SoS dispersion in the tissue could cause the pulse shape to degenerate. The setup was later modified to perform the hybrid \"PA + SoS + Acoustic attenuation\" imaging [100]. The time-domain travel-time estimation method employed iterative Gauss-Newton optimization for a non-linear problem: \([\mathcal{T},\mathrm{A}]=\mathrm{arg}_{[\tau,\mathrm{A}]}\mathrm{min}\| \mathrm{p}(\mathrm{t})-\mathrm{Ah}(\mathrm{t}+\tau)\|^{2}\), where \(\mathrm{p}(\mathrm{t})\) is the recorded waveform, and \(\mathrm{h}(\mathrm{t})\) is the signal from the PA generator. The acoustic attenuation was estimated in the frequency domain by minimization accounting for frequency dependence and dispersion. Later, multiple PEs, illuminated all at once by a single laser pulse, were introduced in [101] to speed up data acquisition. Nine horsetail hairs were placed so that signals from individual PEs did not overlap. Although on phantoms, the measurement time was reduced eightfold, the image quality slightly degraded when compared with a single PE. Finally, in [99], the authors employed an iterative TV-regularized reconstruction algorithm that used Eikonal Equation for both SoS and PA imaging: \(\left|\
abla\mathcal{T}\right|^{2}=\widetilde{\text{c}}^{-2}\), where \(\
abla\mathcal{T}\) is the gradient of the wavefront. The modified fast marching method was used to calculate the first arrival time, accounting for ray bending according to [PERSON]'s law. The SoS and PA images of a kidney of a mouse embedded in an agar matrix were presented, although SoS map did not detail the inner structure of the organ.