# Vascular Structure Identification in Intraoperative 3D Contrast-Enhanced Ultrasound Data
## Materials and Methods
### Vascular Structure Segmentation  
A common solution to distinguish the blood vessels in the 3D-iCEUS data is their extraction by using segmentation methods. However, the segmentation of vascular structures in ultrasound image data is generally a complex task because of the speckle and the blood vessel diameters less than 2 millimeters (Figure 1b,d). Model-based techniques, which include a priori knowledge of the object shape or its data representation, are required to guide the segmentation process and improve the success of algorithms [25; 26]. In this work, the extraction of blood vessels is performed in two steps: the computation of a Hessian based vesselness measure for enhancing the vascular structures, followed by the Otsu thresholding algorithm [27]. Moreover, the segmentation is performed in limited target regions of interest in the 3D-iCEUS data, as it will be explained in Section 2.3. The vesselness measure, which is used, was introduced by [PERSON] [28; 29] to describe tubular structures within an image using the Hessian matrix. The Hessian matrix \(H\) of an image \(I\) is defined by:  
\[H=\begin{bmatrix}\frac{\partial^{2}I}{\partial x^{2}}&\frac{\partial^{2}I}{ \partial x\partial y}&\frac{\partial^{2}I}{\partial x\partial z}\\ \frac{\partial^{2}I}{\partial x\partial y}&\frac{\partial^{2}I}{\partial y \partial z}&\frac{\partial^{2}I}{\partial y\partial z}\\ \frac{\partial^{2}I}{\partial x\partial z}&\frac{\partial^{2}I}{\partial y \partial z}&\frac{\partial^{2}I}{\partial z^{2}}\end{bmatrix} \tag{1}\]  
The second order derivative of the image \(I\) is computed from convolution with a Gaussian kernel of standard deviation \(\sigma\).  
\[I_{\sigma}=I*G_{\sigma} \tag{2}\]where \(G_{\sigma}\) is a Gaussian function with a standard deviation \(\sigma\).  
\[G_{\sigma}=\frac{1}{\sqrt{(2\pi\sigma^{2})^{3}}}\exp(-\frac{x^{2}+y^{2}+z^{2}}{2 \sigma^{2}}) \tag{3}\]  
Then, the multiscale Hessian matrix will be computed as:  
\[H_{\sigma}=\
abla^{2}I_{\sigma} \tag{4}\]  
Based on experimental analysis of an ideal tube model, [PERSON] _et al._[28; 30; 31] proposed a vessel enhancement filter function given by  
\[v(\sigma)=\left\{\begin{array}{ll}0&\mbox{if }\lambda_{c}=0\\ \exp{(\frac{-\lambda_{1}^{2}}{2(\alpha_{1}\lambda_{c})^{2}}).\lambda_{c}}& \mbox{if }\lambda_{1}\leq 0,\lambda_{c}\
eq 0\\ \exp{(\frac{-\lambda_{1}^{2}}{2(\alpha_{2}\lambda_{c})^{2}}).\lambda_{c}}& \mbox{if }\lambda_{1}>0,\lambda_{c}\
eq 0\end{array}\right. \tag{5}\]  
where \(\alpha_{1}<\alpha_{2}\) and \(\lambda_{c}=\min(-\lambda_{2},-\lambda_{3})\). The values of \(\alpha_{1}\) and \(\alpha_{2}\) are 0.5 and 2, respectively; and \(\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\) (eingenvalues of the Hessian matrix).  
Moreover, it has been shown that the variation of sigma values \(\sigma\) enables to describe vascular structures with different radii sizes. Therefore, \(v(\sigma)\) is computed at each voxel position in the image for different values of the standard deviations \(\sigma_{min}\leq\sigma\leq\sigma_{max}\) and the maximum response kept \(v_{I}\) is described by:  
\[v_{I}=\max_{\sigma_{min}\leq\sigma\leq\sigma_{max}}v(\sigma) \tag{6}\]  
Finally the obtained image of vesselness responses is thresholded using the Otsu method [27] in order to extract the vascular structures. Figure 2 presents an illustration of blood vessel segmentation process from cT1 MR, 3D-iCEUS\({}_{start}\) and 3D-iCEUS\({}_{end}\). An additional step of 3D representation is added here.  
Figure 2: Illustration of vascular structure segmentation in cT1 MR, 3D-iCEUS\({}_{start}\) and 3D-iCEUS\({}_{end}\).