# Image Human Thorax Using Ultrasound Traveltime Tomography with Supervised Descent Method
## 4 Numerical Experiments
### Description of the Test Domain and Forward Modeling  
The size of the test domain is 28.5 cm in length and 19 cm in width, and we divide it into 57 \(\times\) 38 elements with length of 5 mm. The thorax model is shown in Figure 2, containing lung, heart, rib, spine, etc. [55]. The acoustic velocity of each tissue can be checked in [56]. The velocity of primary tissues in the thorax model is shown in Table 1. The blue background area is set to be water. The 22 black points mark the positions of the transducer. The transducer is labeled counterclockwise and the top-middle transducer in Figure 2 is the first one. Each transducer serves as the transmitter in turn and others serve as receivers. Totally \(22\times 21=462\) sets of traveltime data are obtained. \begin{table}
\begin{tabular}{c c} \hline \hline
**Tissue** & **Acoustic Velocity (m/s)** \\ \hline Heart & 1561 \\ Lung (deflated) & 1500 \\ Bone (cancellous) & 2117.5 \\ Bone Marrow (yellow) & 1371.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The acoustic velocity of primary tissues in the thorax model. The number of nodes \(n_{nodes}\) on each edge of the element during the offline training process and online predicting process is chosen to be 3. \(n_{nodes}\) will influence the accuracy of the forward modeling and the time it takes. Increasing the number of nodes will improve the accuracy but at the same time increase the computation time. We calculate the traveltime data with \(n_{nodes}=1,2,\ldots,7\). The computation time for the forward modeling is shown in Figure 3a. We choose the traveltime obtained with \(n_{nodes}=7\) as the standard and calculate the relative error. Here, we plot the relative error of the traveltime data obtained from transmitter No. 6 in Figure 3b. When \(n_{nodes}=3\), the relative error of each data point is almost smaller than 0.01, which is small enough. If we keep increasing \(n_{nodes}\), the decrease in the relative error is small, while the increase in the computation time is large. The computation time of \(n_{nodes}=4\) is almost twice as long as \(n_{nodes}=3\). In consideration of both efficiency and accuracy,  
Figure 3: (**a**) Computation time of the forward modeling versus \(n_{nodes}\). (**b**) The relative error of the traveltime data from transmitter No.6. The standard is the traveltime data obtained with \(n_{nodes}=7\). Figure 2: The thorax model in [55], containing lung, heart, rib, spine, etc. It is 28.5 cm in length and 19 cm in width. The blue background is set to be water. The black points around the thorax are where the transducers placed. The transducer is labeled counterclockwise and the top-middle transducer is the No. 1 transducer. we choose \(n_{nodes}=3\). In addition, in order to avoid inverse crime, the traveltime data used for inversion in the following numerical experiments computed by SPR method is obtained from the grid with \(n_{nodes}=5\), because increasing \(n_{nodes}\) is equivalent to using a higher-resolution grid for forward modeling. The distribution of the traveltime data obtained from normal thorax is shown in Figure 4. Maximum traveltime data are less than 180 \(\upmu\)s. The median is 101.58 \(\upmu\)s and its position is marked by the red line. To further avoid inverse crime, zero-mean white Gaussian noise with \(\sigma=2\)\(\upmu\)s is added to the data. Then, we compare the traveltime data obtained from SPR method with the data obtained from k-Wave simulation [57]. We check tissues' density and attenuation coefficient in [56]. The thorax model is upsampled to 114 \(\times\) 76 and put into a 128 \(\times\) 128 domain with grid size of 2.5 mm. The timestep is set to be 0.25 \(\upmu\)s and the simulation lasts for 500 \(\upmu\)s. Gaussian waveform input signal is assigned to the transmitter and the rest transducers serve as receivers. We apply the Akaike information criteria (AIC) method to extract traveltime from the received waveform [58]. The transmitted waveform from transmitter No.