Nowadays, the extraction of intraoperative stone fragments has increasingly become an essential component of FURL using SNP-UAS, representing a critical aspect of its surgical protocol [17, 18]. The SNP-UAS-assisted FURL technique greatly improves the chances of achieving a nearly stone-free status after surgery compared to traditional methods using a straight sheath or no sheath [19]. This advancement reduces the burden of stone expulsion and the risk of recurrence, leading to its widespread adoption. In addition, advancements in laser performance have increased the allowable stone size for FURL procedures to 2.00–3.00 cm or larger, necessitating the removal of more stone fragments [20, 21]. Consequently, alongside laser lithotripsy [22] and ureteral sheath placement, the process of stone removal is increasingly becoming a significant factor influencing the duration of surgical procedures.

The stone fragments primarily comprise dusts (< 0.02 cm), grits (0.02–0.10 cm), and gravels (0.10–0.30 cm). Dusts and grits can be consistently removed from the gap between the FURS and SNP-UAS during laser lithography. It has been documented that smaller RESD and larger FR facilitate this removal process [23]. For gravels, unless powdered into dusts and grits [24, 25], they must be removed from the SNP-UAS using a continuous water flow via the pull-back movement of FURS.

Injection pump and NPSS enhance the velocity of water and gravel flow towards the tail-end of the SNP-UAS through powerful suction [26]. At this juncture, if the FURS are pulled back to create a space, the gravel will gradually move towards the space until it is extracted at the tail-end of SNP-UAS. However, if the FR is insufficient or the PBS is excessively rapid, the gravel may fail to follow and subsequently descends. The FR and RESD, along with the size and weight of the gravel, can directly or indirectly affect this process. To the best of our knowledge, no other studies have been documented concerning the efficiency of gravel removal on FR, RESD, and PBS, with a particle size of 0.10 cm to 0.30 cm, apart from our own research [27, 28]. This study aims to investigate this phenomenon by focusing on these specific variables.

Therefore, we first developed in vitro experiments to explore the relationship between FR, RESD, and PBS. The SNP-UAS model based on the male urinary tract was divided into four parts: a steep and slow segment in the kidney’s pelvis and calyx, a flat segment in the ureter, a steep segment in the bladder and urethra before the pubic junction, and a flat segment from the pubic junction to the tail-end of the SNP-UAS (Figs. 1b). The short and straight female urethra allows for a gentle slope and combination of the third and fourth SNP-UAS segments, while stone removal in the third segment is more challenging in male. Thus, male FURL was selected as the representative of the in vitro experiments.

Capturing max-PBS is challenging, therefore, we developed three strategies to address this objective. First, the senior physician rehearsed three times before each experiment to align with current parameters. Second, we ensured at least a 30% failure and success rates to approach critical max-PBS. Furthermore, we utilize the central tendency of the data set as the max-PBS to approximate the critical threshold as closely as possible.

In the first sub-study, FR of 20.00, 30.00, and 40.00 ml/min were insufficient for the ball removal at a RESD of 0.73. An effective max-PBS was achieved as FR increased to 50.00 ml/min, suggesting that an FR exceeding 50.00 ml/min is appropriate for in vitro conditions. The max-PBS increased with higher FR when RESD was 0.73, 0.79 and 0.87. Furthermore, for a given FR, increasing RESD corresponded to a significant growth of max-PBS. This indicates that, for an individual stone sample, both an increase in FR and RESD can enhance the max-PBS, which refers to the speed of a single pull-back movement (Table 1).

In the second sub-study, the circle movement with varying FR was initially assessed using CFD analyses, with an RESD of 0.73 and a PBS of 0.00 cm/s. The findings indicated that at FR ranging from 50.00 to 100.00 ml/min, the circle tended to hover near the head-end of FURS. Conversely, at FR of 40.00 ml/min or lower, although the circle exhibited upward movement, the lift force was insufficient to propel it near the head-end of FURS. Given that the FURS remained static during this observation, it can be inferred that the circle would not follow with FURS. This finding aligned with in vitro experiments, indicating that a weak FR prevents a stone from following FURS.

Using an FR of 60.00 ml/min (with a RESD of 0.73 and a PBS of 0.00 cm/s) as a representative example, we can categorize the circle movement into three distinct phases: acceleration, uniform speed, and deceleration. During the acceleration phase, the vortex formed around the circle without significant pressure differences. Notably, vortex beneath the circle moved upward to support it, while the one above moved tangentially. This analysis indicates that the circle’s upward movement is not directly attributable to pressure differentials but rather to the dynamics of water flow. During uniform speed, vortices surrounding the circle aligned tangentially, producing a net force of zero, thereby maintaining the circle’s uniform motion. In the deceleration phase, the bottom vortex remained tangential while the top vortex pointed downward, producing a net downward force that slowed and eventually stopped the circle (Fig. 5).

Based on the findings from in vitro experiments of the first sub-study, the max-PBS for the three RESD fluctuated around 5.00 cm/s under an FR of 50.00 ml/min, and increased with larger RESD. Consequently, when PBS is maintained at 5.00 cm/s, this relationship should be evident in CFD experiments, which was indeed confirmed. The second sub-study demonstrated that at a PBS of 5.00 cm/s and an FR of 50.00 ml/min, the circle (stone) with larger RESD moved faster and further, thereby indicating easier circle (stone) removal.

This study acknowledges some limitations. Primarily, due to the necessity for reproducible quality control in in vitro experiments and the computational demands of CFD analysis, this study was confined to a single stone with an approximately regular shape. Future research should address scenarios involving irregularly shaped stones and the simultaneous movement of multiple gravels. Additionally, aluminum balls were utilized to ensure experimental repeatability in the event that actual stones were compromised during repeated movements. Second, due to the greater length and steeper angle of the male urinary tract compared to the female’s, employing it as a reference for the SNP-UAS model in vitro could potentially result in more precise experimental outcomes. However, while these findings may offer some degree of representation, they may not be entirely applicable to females. Consequently, further evidence is needed for female models. Third, to enhance the feasibility of CFD analysis, the model has been simplified, particularly by changing the gravel removal path from an inclined curve to a vertical line. Although this modification deviates from the actual environmental conditions, it does not impede the validity of our observations. However, future analyses should strive to more accurately replicate in-vivo conditions.