## 2 Materials and Methods
### Spatial Sampling of the Scan  
Our scanning scheme combines motion of the US transducer in translation and in rotation. Therefore, both the linear and the angular spatial sampling periods need to be set. To improve upon the approach proposed for PA tomography in [20], we sought to determine the sampling periods offering the best compromise between image quality and acquisition speed. Indeed, increasing the number of tomographic positions usually leads to higher image quality (contrast, homogeneity) but will result in longer acquisition time. The minimal sampling periods required to avoid grating lobes and other aliasing artifacts on the image depend both on the transducer geometrical parameters (Figure 1a) and the reconstruction method. The reconstruction algorithm for 3-D imaging is detailed in Appendix A. Our reconstruction algorithm is based on a basic delay-and-sum process. In the 2-D configuration, the transducer is assumed to be a point transducer located at O\({}_{\text{a}}\) (Figure 1a), which emits circular waves (cylindrical in 3-D) and has an isotropic detection pattern. The spatial impulse response of the transducer is not modeled, but its directivity is taken into account with an amplitude mask. For a given position of the transducer, the pulse-echo signal only contributes to the image in a slice of width \(2\times d_{\text{u}}\) (Figure 1b) around the transducer axis. The distance \(d_{u}\) is a distance to the transducer axis in the elevation direction. In this slice, an apodization window (20% cosine taper) was used to avoid discontinuities. The pulse-echo (two-way) \(-12\) dB full width of the ultrasound beam is \(1.2\times H\)[21] where \(H=\lambda_{\text{c}}\times\text{f-number}\), with \(\lambda_{\text{c}}\) the wavelength at the center frequency \(f_{\text{c}}\) of the transducer. So, to account for the focusing of the transducer, we set \(d_{\text{u}}=0.6\times H\). The linear sampling period \(\Delta I\) is defined as the distance between the transducer axis for two successive translation positions at a constant angle \(\alpha_{rot}\) (Figure 1c). The angular sampling period, \(\Delta\alpha\), corresponds to the angular change between two successive translations across the imaged area. Performing Nyquist-Shannon sampling in translation and in rotation [22], when considering point detectors in the reconstruction, would require a large number of positions and would lead to a prohibitive scan duration. To obtain more efficient scanning, we reduced the synthetic array to a rotation array: For each pixel in the imaged area, we coherently added signals corresponding a single position in translation for each rotation angle \(\alpha_{rot}\). Thereby, we avoided generating a synthetic array with the successive translation positions at a given \(\alpha_{rot}\). The selection of the translation positions was made based on the amplitude mask. We should have \(\Delta I\geq 2\times d_{\text{u}}\) so that \(\Delta I\geq 1.2\times H\). On the other hand, an effective coverage of the imaged area would require \(\Delta I\leq H\), because \(H\) is the two-way full-width half-maximum (FWHM) of the ultrasound beam [23]. We chose to set \(\Delta I=H\), as a compromise justified by the 20% cosine taper of the apodization mask. One can note that \(\Delta I\) and \(d_{u}\) are strongly linked because of the reduction of the synthetic array to a rotation array. For the translation-only scan used for comparison, a synthetic translation array was formed. Therefore, the volumes probed by successive translation positions should overlap and then \(\Delta I_{translation-only}\leq 2\times d_{\text{u}}\). We set \(\Delta I_{translation-only}=H/8\)[24] (see also Appendix B) while \(d_{\text{u}}\) was still defined by \(d_{u}=0.6\times H\) to account for the transducer directivity. Angular sampling in our scanning geometry has not yet been studied to the best of our knowledge. From studies performed for other geometries, we found (Appendix B) that the minimal angular sampling period is \(\Delta\alpha=\beta\).