# Image Human Thorax Using Ultrasound Traveltime Tomography with Supervised Descent Method
## 4 Numerical Experiments
### Numerical Experiments
#### 4.3.2 SDM Traveltime Inversion of Thorax on Different States  
Some respiratory system diseases might change the amount of gas and fluid in the thorax, which will cause changes in acoustic velocity because acoustic velocity in air is significantly smaller than in water. In this section, SDM traveltime tomography will be used to test whether these changes can be detected. Here, we take pleural effusion and pneumothorax as examples. Patients with pleural effusion tend to have more fluid in the thorax [59]. Although patients with pneumothorax will have more gas in the thorax [60].  
Firstly, a pleural effusion patient's thorax acoustic velocity distribution is reconstructed, see the first column of Figure 12. We assume that the acoustic velocity of lungs when breathing in air will decrease to 1200 m/s (lower than average tissue acoustic velocity of 1540 m/s and higher than air acoustic velocity of 340 m/s). We suppose that the velocity distribution of a pleural effusion patient when breathing in air is like Figure 12a. The lower part of the left lung is the pleural effusion area. We suppose that the velocity of this area is 1540 m/s, same as the average tissue acoustic velocity. We want to reconstruct this high-velocity area near the low-velocity area through SDM traveltime tomography. The reconstruction result is shown in Figure 12c. The reconstruction result shows that both lungs' boundaries are clear and the left lung is smaller than the right one because the effusion area's velocity is higher than the normal lung area. For comparison, the result of the Gauss-Newton method is shown in Figure 12e, which has more artifacts. This example verifies that the change of acoustic velocity resulting from pleural effusion can be detected by SDM traveltime inversion.  
Then, thorax acoustic velocity distribution of a pneumothorax patient is reconstructed, see the second column of Figure 12. We suppose that the velocity distribution of a pneumothorax patient when breathing out air is like Figure 12b. We suppose that the pneumothorax area is also located in the lower-left part of the thorax. The acoustic velocity of the pneumothorax area is assumed to be 1200 m/s, the same as the normal thorax region when breathing in air. In this example, a low-velocity region among high-velocity region will be reconstructed. The reconstruction result is shown in Figure 12d. The result shows that the boundaries of both lungs are clear and the pneumothorax area is reconstructed. The lower part of the left lung is darker than the lower part of the right lung. For comparison, the result of the Gauss-Newton method is shown in Figure 12f, where the boundary between normal area and pneumothorax area is not clear. This example verifies that the change of acoustic velocity resulting from pneumothorax can be detected by SDM traveltime inversion. In addition, the generalization ability of the method can also be demonstrated in these experiments. The learned descent direction can be applied to thorax under different conditions.  
Figure 12: Cont.