## 3 Materials and Methods
### Microessel Thickness Measurement
#### 3.5.2 Microvessel Thickness Measurement  
In the microvessel object shown in Figure 20, the red, solid line represents the microvessel center axis, and the blue points represent the pixels along the center axis. For example, the pixels along the center axis are indicated by \(1,\ldots,m\), where \(P_{1}\) and \(P_{m}\) represent the end points of the microvessel object.  
Figure 17: Branch point patterns.  
Figure 18: End point patterns.  
The microvessel thickness at point \(P_{n}\) is calculated using the angle \(\theta_{n}\) defined by \(P_{n}\) and \(P_{n+1}\). In Figure 11(b), the microvessel thickness is given by the length of the straight line passing through \(\theta_{n}+90^{\circ}\) and \(\theta_{n}-90^{\circ}\). To calculate the angle \(\theta_{n}\), the points of each center axis are arranged by distance. Thus, the distance from \(P_{1}\) to \(P_{m}\) along the center axis of the microvessel can be calculated using Equation (27). In this study, the Euclidean distance method was applied to calculate the distance.  
\[\text{Distance}=\sqrt{\left(x_{n}-x_{n+1}\right)^{2}+\left(y_{n}-y_{n+1} \right)^{2}} \tag{27}\]  
Assuming that the adjacent points \(P_{n}\) and \(P_{n+1}\) exist among the points arranged around the center point, \(\theta_{n}\) is defined as shown in Figure 22 and can be calculated using Equation (28).  
\[\theta_{n}=\tan^{-1}\frac{y_{n}-y_{n+1}}{x_{n}-x_{n+1}} \tag{28}\]  
Figure 21: Thickness measurement method using the angle \(\theta_{n}\) defined by \(P_{n}\) and \(P_{n+1}\).  
Figure 20: Schematic view of a microvessel.  
Figure 22: Angle between any two points.  
The microvessel thickness measurement method using \(\theta_{n}\) is illustrated in Figure 23. The green line in Figure 23 represents \(D\), which is the length to be measured at \(\theta_{n}+90^{\circ}\), while the yellow line represents \(D_{2}\) which is the line to be measured at \(\theta_{n}-90^{\circ}\).  
\(T(x,\ y)\) is the coordinate of the point at a distance \(D\) from \(P_{n}\), and \(T^{\prime}(x^{\prime},\ y^{\prime})\) is the coordinate of the point at a distance \(D^{\prime}\) from \(P_{n}\). \(D\) and \(D^{\prime}\) are increased by 1 until the microvessel wall is encountered. \(T(x,\ y)\) and \(T^{\prime}(x^{\prime},\ y^{\prime})\) can be obtained using Equation (29).  
\[\begin{array}{ll}x=cos(\theta_{n}+90^{\circ})*D&x^{\prime}=cos(\theta_{n}-9 0^{\circ})*D^{\prime}\\ y=sin(\theta_{n}+90^{\circ})*D&y^{\prime}=sin(\theta_{n}-90^{\circ})*D^{\prime }\end{array} \tag{29}\]  
The thickness measurement method shown in Figure 23 is described in Table 2.  
The thickness measurement results are shown in Figure 24. The red points represent the pixels along the center axis of the microvessel. The green and yellow lines indicate the length measurements at \(\theta+90^{\circ}\) and \(\theta-90^{\circ}\) from the pixels along the microvessel center axis, respectively.  
\begin{table}
\begin{tabular}{c l} \hline \hline Step 1 & Set the initial position at a distance \(D\) from \(P_{n}\), defined as \(T(x,\ y)\). \\ \hline Step 2 & Check if the \(T(x,\ y)\) value encounters the microvessel wall. \\ \hline Step 3 & If \(T(x,\ y)\) reaches the microvessel wall, stop searching and proceed to Step 4 to \\  & end the algorithm or increase the length of D by 1 in Step 1 and repeat Step 2. \\ \hline Step 4 & End. \\ \hline \hline \end{tabular}
\end{table}
Table 2: Flowchart of the thickness measurement method.  
Figure 23: Microvessel thickness measurement method.  
Figure 24: Microvessel thickness measurement results.