## 2 Materials and Methods  
A CFD model for the numerical analysis was composed of a ureter connected to a renal pelvis and a bladder, and a double-J stent (Figure 1). The ureter was made based on the two-dimensional axial computed tomography image data of human anatomies, including the kidney, ureter, and bladder, of 19 males into 15 equal sections perpendicular to the direction in which the patient was lying (z-axis), as shown in Figure 1. The beginning and end points of the ureter were separated into the cross-section of the renal pelvis and the ureter and that of the ureter and the bladder. Figure 1 shows the ureter shape of each section of Patient 1. The area of each separated cross-section was measured to calculate the diameter when the cross-section was assumed to be circular. This method was repeated for 18 additional patients. S. 1-1' is segment 1-1'. The mean diameter of each section from the 19 patients are shown in Table 1. The STDEV function calculated the standard deviation based on the entire population given as arguments [14]. The standard deviation is a measure of how widely the values are dispersed from the average value. The distance between the center points of each cross-section was measured similarly to the method mentioned above. Table 2 presents the measured distance for Patient 1. S. 1-2 is the distance between segments 1-1' and 2-2'. Similarly, measurements were made on the x, y, and z axes between sections of the other 18 patients. According to the measured information, a ureter model was constructed by averaging the data of the ureteral lengths and diameters from the patients. The model was curved and 226.21-mm long, and the proximal end of the ureter was connected to the renal pelvis, which has a conical shape. The distal end of the ureter was connected to the bladder, which has a dome shape. The base of the renal pelvis and the bladder was round and 40 mm in diameter. The height of the renal pelvis and bladder was 15.5 mm. The ureter was tubular and 4.57 mm in diameter. Another ureter model was made with a 5.39-mm ureter diameter in view of the 8-Fr stents. A double-J stent (DJS) model inserted into the ureter using the ureteral data collected fromthe patients was developed using SolidWorks software (Dassault Systemes SE, Velizy-Villacoublay, France) and ICEM CFD software (version 16.2; Ansys, Inc., Canonsburg, PA, USA). The DJS model was based on the Cook Medical stent (Cook Group Inc., Bloomington, IN, USA). This DJS model consists of two coils with a J-shape and a shaft. The shaft has several side holes that provide a flow diversion from the stent to the ureter or vice versa, and the coiled ends have various side holes which play a role in urine flow in and out of the stent in the kidney and the bladder, as shown in Figure 2.  
Figure 1: Cross-sectional segments of the ureter between the renal pelvis and the bladder (Patient 1). Figure 2: Schematic of the DJS model. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Case**} & \multicolumn{1}{c}{**1**} & \multicolumn{1}{c}{**2**} & \multicolumn{1}{c}{**3**} & \multicolumn{1}{c}{**4**} & \multicolumn{1}{c}{**5**} & \multicolumn{1}{c}{**6**} & \multicolumn{1}{c}{**7**} & \multicolumn{1}{c}{**8**} & \multicolumn{1}{c}{**9**} & \multicolumn{1}{c}{**10**} & \multicolumn{1}{c}{**11**} & \multicolumn{1}{c}{**12**} & \multicolumn{1}{c}{**13**} & \multicolumn{1}{c}{**14**} & \multicolumn{1}{c}{**15**} & \multicolumn{1}{c}{**16**} & \multicolumn{1}{c}{**17**} & \multicolumn{1}{c}{**18**} & \multicolumn{1}{c}{**19**} & \multicolumn{1}{c}{**Mean**} & \multicolumn{1}{c}{**STDEVP**} \\ \hline S.1-1 & 4.88 & 4.20 & 6.19 & 5.13 & 5.18 & 5.46 & 4.79 & 6.89 & 4.99 & 6.07 & 4.70 & 6.31 & 7.04 & 6.85 & 7.09 &  
The DJS model included a shaft in the middle that was as long as the ureter, at 226.21 mm. It had two coils in the ends that were round and 10 mm in diameter.