# Contrast-Enhanced Ultrasound Imaging Based on Bubble Region Detection
## 2 Materials and Methods
### Spatial Domain
#### 2.2.1 Eigenvalue  
In the bubble region detection algorithm, the signal intensity is the most important characteristic. In the theory of ESBMV, the maximum eigenvalue belongs to the mainlobe signals [24]. The signal subspace is comprised of eigenvectors corresponding to the largest few eigenvalues, and the noise subspace is constructed by eigenvectors according to lower eigenvalues. The small eigenvalues tend to represent the noise field and the large eigenvalues represent the signal field [25]. Furthermore, the maximum eigenvalue gives an estimation of the signal power. [PERSON] et al. proposed a method to detect the signal based on the eigenvalues [26; 27; 28], which proves that it is practicable to distinguish the UCA and tissue region according to their eigenvalues from the side. Based on the observation of the experiment, we found that the amplitude of the maximum eigenvalue in the UCA region is obviously higher than that in the tissue region. The eigenvalues are calculated as follows:  
The RF signals were divided into several overlapping subarrays. The covariance matrix was computed after spatial smoothing and diagonal loading as follows:  
\[R(k)=\frac{1}{M-L+1}\sum_{p=1}^{M-L+1}x_{d}{}^{p}(k)x_{d}{}^{p}(k)^{H} \tag{4}\]  
where \(R\) is the covariance matrix; \(k\) is the time index; \(M\) is the transducer elements; \(M-L+1\) denotes the overlapping subarray number; \(L\) is the length of subarray; \(p\) is the subarray number; \(x_{d}{}^{p}(k)\) is the signal after delay; and \((\cdot)^{H}\) is the transposition. The eigen-decomposition of the covariance matrix was required:  
\[R=U\Lambda U^{H} \tag{5}\]  
where \(\Lambda=diag(\lambda_{1},\lambda_{2},\ldots\lambda_{L})\) is the square matrix of eigenvalues in the descending order and \(U\) is the eigenvector corresponding to the eigenvalue. By taking a phantom study as an example, the bubble detection ability of the eigenvalue is explained. The scheme of the phantom experiment is shown in Figure 1. The experiment system includes an ultrasonic research platform Verasonics Vantage 128 (Verasonics, Inc., Kirkland, WA, USA), a linear array transducer (L11-4v), a homemade gelatin phantom, a medical syringe, a tube, a beaker, a computer, Sonovue microbubble (Bracco Suisse SA, Switzerland), and fresh pork. The phantom is 11 cm in length, 11 cm in width, 6 cm in height, and is made of gelatin. The process of making the phantom is as follows: (1) mix 50 g gelatin granules with 700 mL purified water; (2) heat the mixture for about 10 min until the gelatin particles are completely dissolved in water; (3) glue a plastic tube (whose diameter is 0.5 mm) to the sides of the plastic mold; (4) pour the mixture into the mold; (5) refrigerate the mixture in the freezer for about 3 h; (5) remove the mixture from the mold after the mixture is solidified; (6) draw the plastic pipe out of the phantom. A phantom with a wall-less tube is completed. The fresh pork (the belly of a pig, 15 mm in thickness, 40 mm in length, and 25 mm in width) was bought from a local meat market on the day of the experiment. Verasonics was used to excite the ultrasound wave and record the RF data. Matlab was used to analyze the RF signal offline. The flowing Sonovue solution (diluted by 1000 times with 0.9% physiological saline) was injected into the pipe by a medical syringe, and the used solution flowed into the beaker. We placed a piece of fresh pork over the phantom. The ultrasonic coupling agent was applied between the pork and the phantom to ensure the signal transmission. The phantom image beamformed with DAS is shown in Figure 2, where the depth ranging from 0-15 mm is the pork area and that ranging from 30.5-35.5 mm is the bubble area. Taking the area at the width of 15 mm as an example, the maximum eigenvalue curve under different depths is shown in Figure 3. The area enclosed with the red rectangle represents the UCA region. Its maximum eigenvalue is quite large due to the existence of strong scattering signals produced by the UCA.