## 2 Materials and Methods
### Simulation Study  
To understand the relation between the quality of the reconstructed images and number of transducers in the 3D HA-PACT, we have modeled a hemispherical cap in k-wave toolbox [47]. The diameter of the hemisphere was chosen to be 12 cm based on the consideration of neonatal brain imaging as a potential application of the proposed system. Typically, the neonatal head circumference is between 20 cm to 35 cm [48]; we chose the maximum diameter and considered room for a thin layer of coupling medium inside the hemisphere. Based on the diameter of the transducer and the thickness of the designed hemisphere, 50 transducers are the maximum number that can be placed. Any further number of transducers would result in hollow space overlap in the hemisphere model. Additionally, we have simulated 14, 26, 38, and 50 equidistant transducer elements on the hemisphere and measured the quality of the reconstructed image. We modeled directivity, frequency range, size, and the arrangement of transducers in our simulations. In all the above configurations, transducers with a diameter of 12.7 mm, frequency of 3.5 MHz, 63% \(-\)6 dB fractional bandwidth, with an equidistant fashion from each other, center to center, were used. The reason that the transducers used in the simulations were 3.5 MHz  
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline  
(lower than what was used in the experiment, i.e., 5 MHz and 63% bandwidth) was because we did not have sufficient computational power to run the simulations with spatial sampling below the Nyquist limit. The difference between the simulation and experimental setup will not affect the justification of using the maximum number of transducer on the hemisphere, because the presented results show the relative difference in different numbers of transducers; using a higher frequency transducer will only improve the resolution of the resultant image. To model the element size of the transducers, each point transducer was expanded to a neighborhood of 4000 omnidirectional point transducers to mimic the size of 12.7 mm diameter. Transducer directivity was realized using spatial averaging on the surface of each single element transducer consistent of the 4000 point transducers. To place the transducers equidistant, we used the algorithm presented in [49]. The algorithm for the equidistant placement of transducers on a surface of the hemisphere was derived by first plotting a 2D curve of \(n\) points where the coordinates are determined using: \((x,y)=\big{(}s\cdot a,\pi/2\times\text{signum}(s)\times\big{(}1-\sqrt{1-|s|} \big{)}\big{)}\) where \(s\) can be ranged from \(-1+1/(n-1)\) to \(-1-1/(n-1)\) in steps of \((2-2/(n-1))/(n-1)\), \(a=0.1+1.2n\), and signum is a function that determines the sign of \(s\). These 2D cartesian points are then converted into 3D Cartesian points using \((x^{\prime},y^{\prime},z^{\prime})=\left\{\cos(x)\cos(y),\sin(x)\cos(y),\cos(y )\sin(y)\right\}\). The arrangement of the highest number (50) of equidistant elements on the hemisphere is shown in Figure 1a. For the generation of simulation data in a hemisphere with a diameter of 12 cm, a k-wave volume of \(12\times 12\times 12\) cm\({}^{3}\) was created using \(750\times 750\times 750\) voxels with 160 \(\upmu\)m width in every direction. Our simulation provides two spatial samples per wavelength and meets [PERSON]'s limit. Since k-wave is a pseudospectral method, for a completely accurate simulation it requires at least 10 grid points per wavelength, which is beyond our computational power. We, therefore, inspected the contrast to noise ratio (CNR) of reconstructions of a simple triangle phantom with differing grid-sizes (i.e., \(300\times 300\times 300\), \(400\times 400\times 400\), \(500\times 500\times 500\), \(750\times 750\times 750\) voxels).