Fluid mechanical performance of ureteral stents: The role of side hole and lumen size

Abstract Ureteral stents are indispensable devices in urological practice to maintain and reinstate the drainage of urine in the upper urinary tract. Most ureteral stents feature openings in the stent wall, referred to as side holes (SHs), which are designed to facilitate urine flux in and out of the stent lumen. However, systematic discussions on the role of SH and stent lumen size in regulating flux and shear stress levels are still lacking. In this study, we leveraged both experimental and numerical methods, using microscopic‐Particle Image Velocimetry and Computational Fluid Dynamic models, respectively, to explore the influence of varying SH and lumen diameters. Our results showed that by reducing the SH diameter from 1.1 to 0.4mm the median wall shear stress levels of the SHs near the ureteropelvic junction and ureterovesical junction increased by over 150%, even though the flux magnitudes through these SH decreased by about 40%. All other SHs were associated with low flux and low shear stress levels. Reducing the stent lumen diameter significantly impeded the luminal flow and the flux through SHs. By means of zero‐dimensional models and scaling relations, we summarized previous findings on the subject and argued that the design of stent inlet/outlet is key in regulating the flow characteristics described above. Finally, we offered some clinically relevant input in terms of choosing the right stent for the right patient.


| INTRODUCTION
Ureters are long collapsible tubes that transport urine from the kidneys to the bladder. The efficacy of transport, however, can be impeded by a range of congenital and acquired pathological obstructions. 1 To restore the flow, ureteral stents are inserted to relax and widen the ureter.
The ureteral stents are thin polymer tubes typically 20-30 cm in length, with circular cross sections of outer diameters ranging between 1:6 mm (4:8 Fr) and 2:67 mm (8 Fr). Modern ureteral stents also feature a curl at each end of the stent, known as pigtails, to prevent dislocation of the stent after placement. Annually, there are over 1.5 million stents used worldwide; yet, more than 80% of the patients suffer from stent-related complications, posing significant threats to the quality-of-life of patients and creating economic burdens for the health care system. 2,3 Among all complications of indwelling stents, the frequent development of biofilm and encrustation remains a key limiting factor of stent efficacy. They are aggregates of conditioning films, bacterial colonies and crystals that start to accumulate on the stent surfaces once in contact with urine. Depending on the actual physicochemical and microbiological environment, the composition may vary, but over time the growing volume will cause secondary stent obstructions that compromise the drainage capacity, and the bacterial content will significantly increase the chance of urinary tract infections. 4,5 Although many efforts have been made to fight against biofilm and encrustation by means of material design and functional surface coatings, it remains one of the primary challenges in stent development. 2 Previous clinical observations have drawn attention to the side holes (SHs), which are small openings in the stent wall to facilitate exchange of fluids between the luminal (in the stent lumen) and extraluminal (between stent and ureter walls) spaces, that often end up heavily encrusted or completely occluded. [6][7][8] Once a SH becomes occluded, flow cavities in the vicinity also aggravate bacterial attachment, 9 further increasing the risk of infection.
The causal relation between SH and encrustation was investigated recently 10 using microfluidic chips, and the low shear stress level near the SH was considered the main reason for the accumulation of micro-particles. The authors 10 proposed a "streamlined" SH architecture with optimized wall thickness (0:3 mm) and vertex angle (45 ) that reduced the encrustation rate by approximately 90%. Prior to that, several numerical models exists that examined urine flows in full-scale ureter models under the impact of different stent geometries and ureter shapes, with and without local obstruction. [11][12][13] One of the common conclusions was that the fluxes through SHs were strongest near the ureteropelvic junction (UPJ) and the ureterovesical junction (UVJ). Other SHs underwent fluxes only when a local obstruction was present in the vicinity. 14 A recent study 15 linked these observations to the inhomogeneous shear stress patterns along the stent and further showed that the shear stress level was lower in the proximal region (close to the UPJ), where more encrustations have been reportedly observed in clinical studies. It seems that the interplay between the large-scale flow characteristics (e.g., flow rates through SHs) and the small-scale quantities (e.g., flow patterns near the SH and associated shear stresses) is key to understand the dynamics of the encrustation process.
In this study, we present the first experimental setup of a stented ureter model in full scale, which allows microscopic particle image velocimetry (μ-PIV) measurements near the SHs at various streamwise locations. Numerical counterparts of the experiments are exploited to investigate further variations of the SH and lumen diameter of the stent. As such, we hypothesize that the flow behaviors in stented ureters in vivo can be explored by means of in vitro studies using simplified models, given that the baseline fluid mechanical principles stay the same, and that the flow characteristics can be manipulated in favor of the stent efficacy by varying the SH and lumen diameter. In Section 3, fluxes through the SHs and the associated shear stresses are presented, and the link between these quantities is discussed. In the discussion, we use simple zero-dimensional (0D0D) models and scaling relations to summarize the previous conclusions on large-scale flow characteristics and briefly evaluate the impact of varying parameters. The objective of the current study is to offer a glimpse into the underlining principle of fluid mechanics in the stented ureteric system by means of experimental, numerical, and theoretical results, trying to offer some clinically relevant input in terms of choosing the right stent for the right patient and a practical reference for future stent development.

| Ureter model
Previous studies in humans 16   In the human ureter, the physiological Reynolds number can be estimated as Re Ã ¼ U Ã c D Ã =ν Ã ≈ 10:55, where the ureter diameter D Ã ¼ 4 mm and viscosity ν Ã ¼ 1:005 Â 10 À6 m 2 =s, and the centerline velocity U Ã c ¼ 2:65 mm=s was calculated assuming laminar pipe flow with rigid walls (Poiseuille flow) and a flow rate of Q Ã ¼ 1 ml=min. All values were taken from physiological values of the human urinary system. 18 To match the Reynolds number, the flow rate of the experiment was chosen to be Q exp ¼ 10:03 ml=min as the closest approximation permitted by the pump.

| PIV setup
To measure the velocity field, a μ-PIV system was built. The system consisted of a Nd:YAG continuous laser with 5 W maximum energy Stk ¼ τ p U c =D Ã ¼ 1:52 Â 10 À5 , which was much smaller than 0.1, suggesting a good tracing fidelity. 20 The acrylic box shown in Figure  After calibrating the imaging system using an ex situ procedure (see Supporting Information S1 for further details with code), the RMS error of the re-projected image was $ 0:2 px on average (range 0:1-0:3 px), which incorporated both the calibration error and the residual error caused by the refractive index matching.
During acquisition, the camera was operated at 500 Hz, and 20,000 images were acquired at each SH location. After removing the background, an image-overlapping algorithm was used to enhance the particle image density (see Supporting Information S1 for more

| Numerical simulations
Numerical counterparts of the same experiments were performed to cross validate and to investigate further stent geometries. In a preliminary study, a full three-dimensional (3D) numerical simulation was performed and the results showed that velocities in the z direction were negligible on the median plane. As a result, two-dimensional (2D) models on the median plane of the ureter model were constructed ( Figure 1c). The geometry was discretized using second-order where e U ¼ U=U c , and U Ã is the theoretical Poiseuille profile. For experimental results, we have ϵ exp U ¼ 2:33%, and ϵ exp V ¼ 0:13%, whereas for  were SH3 and SH6 at larger D SH and therefore should not be interpreted as active SHs.

| Wall shear stress
We continue by presenting the wall shear stress at the SHs with different D SH using the CFD data ( Figure 4). The shear stresses at the SH walls (r À0:5, 0 ½ mm) are defined as where the derivatives are calculated from the velocity field using the one-sided second-order finite difference scheme. The violin plots in Figure 4 give the distribution of τ wall and their median (empty circles) for each case. As D SH decreases, the distribution of τ wall varies such that their values at the two ends of the stent (SH1

| Stent lumen size
We complemented our study by changing the stent lumen size D s to evaluate the luminal flux.
at the inlet of the stent, and the magnitude of all transverse fluxes were also probed to represent the stent's overall capacity to exchange to the UPJ and UVJ, 11,13 and local obstruction activates the SHs directly upstream and downstream to the site. 14,19 To continue the discussion, we first show that these large-scale flow characteristics can be derived from 0D models and scaling relations using simple fluid mechanical principals. If we approximate a stented ureter model as a straight coaxial annulus (ignoring the luminal space of stent for a moment), the bulk flow rate is given analytically by where D is the diameter of the ureter, and D o is the outer diameter of the stent. The pressure gradient term À∂p=∂x can be approximated by P 0 À P L ð Þ =L, where P 0 > 0 ð Þ is the relative pressure at the inlet of the ureter, P L is the pressure loss, and L is the ureter length. The pressure loss due to viscosity and changes of cross-sectional shape (such as contraction or expansion of the ureter) can be approximated by 23 where f is the friction coefficient of the surface, L is the pipe length, K is the loss coefficient, and M is the number of minor losses present in the system. In the laminar flow regime, the friction coefficient can be derived as where Re D h is the Reynolds number based on the hydraulic diameter Assuming D ¼ 4 mm with constant ∂p=∂x and U in Equations (5) and (7), we show that by increasing the outer diameter of the stent D o from 1 to 3 mm, the viscous loss coefficient increases by 200%, and the flow rate decreases by over 90% (Figure 8), quantifying the previous conclusion that larger stent size causes smaller total flow rate. 12 The effect of ureter shape, such as those described as tapered or undulated in the literature, 19 can be evaluated by the term P minor in Equation (6), where any change in cross-sectional shape of the ureter, either contraction or expansion, will increase the minor loss and result in a smaller Q. This is consistent with the previous conclusion that straight ureter gives the highest total flow rate compared to the tapered and undulated shapes when other parameters are kept the same. 19 If we consider SHs in Equation (6), more SHs are equivalent to smaller length of the wall L due to the reduced surface area, and consequently smaller P viscosity . Meanwhile, the impact of SHs on P minor might be negligible since most of the SHs are inactive (Figure 6c,d).
The angular rotation of the SHs does not change the pressure loss as the equivalent L and number of minor losses M are both independent of the rotation angle. That is to say, the more SHs the smaller the P L , the larger the total flow rate Q, and that the angular rotation does not affect the total flow rate. This is again consistent with the previous conclusions from previous conclusions. 11,19 For analysis on the luminal flow rate within the stent, the inlet and outlet of the stent lumen are of primary interest. The loss coefficient K for sudden contraction (K c at stent inlet) and sudden expansion (K e at stent outlet) is given by where D s is the lumen diameter of the stent (D s < D), and c is an empirical constant. 23 It can be inferred that the larger the D s , the smaller the K c and K e . Consequently, larger D s leads to less pressure loss (Equation 6) and encourages higher flow rates in the stent lumen, which explains the larger Q i for D s ¼ 1 mm (Figure 6a). The pressure loss at the stent inlet leads to the pressure imbalance between the luminal and the extraluminal spaces, which ultimately drives the activation of the SHs.
In this study, we showed that only SH1 and SH8 were always active.
Since they were close to the inlet and outlet of the stent, the interluminal pressure difference re-balances through these SHs, leaving the majority of the SHs inactive in the middle of the stent, characterized by low shear stress and low flux (Figure 7). This conclusion agrees with previous findings using 3D numerical simulations, 11,13 and perhaps explains why SHs are often heavily encrusted in clinical observations. [6][7][8] The fact that only SHs close to the UPJ and UVJ are active is determined by the pressure loss at the stent inlet/outlet. The rest of the SHs can only be activated where local obstruction is present. In the ideal scenario, stents should perhaps differentiate for patients with proximal/distal obstructions, as categorized in the clinic, so that more SHs are placed near the obstruction site to improve the drainage.
If we extend the stent geometry in this study to include longer pigtails, the loss coefficient at the inlet can be scaled as K $ s=D s , where s is the length of the protrusion part of the stent. 23 Therefore, longer pigtails (s=D s ) 1) create higher pressure loss at the inlet that further impede the luminal flow and cause larger pressure difference between the luminal and the extraluminal spaces.
In fact, we show an auxiliary case study in the Supporting Information S1, where we added a pigtail to the stent model with D SH ¼ 1 mm and D s ¼ 1 mm. The resulting Q i showed a reduction by 60%, and the flux through SH1 increased by more than 100% in response to the escalated interluminal pressure difference. The smaller Q i in stent lumen is likely to cause local flow stasis between the inlet/outlet and the first SH in the vicinity, which promotes microparticle aggregations and bacterial attachments. 9,10 This perhaps explains the heavy encrustations on stent pigtails observed in several clinical studies. 6,[24][25][26] Recent microfluidic studies 9,10 demonstrated that, by optimizing the shape of SHs to increase the local wall shear stress levels, both micro-particle aggregation and bacterial attachment were reduced.
Nonetheless, the actual shear stress level around SH largely depends F I G U R E 7 Transverse fluxes through the SHs (Q SH ) versus the median wall shear stress levels at the SHs (τ wall ) with different stent geometries. The orange arrows mark the direction of increasing D SH . Legends are the same as Figure 6, except that SH2-7 are filled in gray to distinguish from SH1 (on the left, negative fluxes) and SH8 (on the right, positive fluxes).
F I G U R E 8 Variation of Q (solid, black) and f (dashed, blue) at various stent size D o relative to the case of D o ¼ 1 mm. on the regional flow pattern (e.g., Figure 5) that varies with the SH diameter and its longitudinal location. In Figure 7, we showed that for the same stent lumen size D s (blue or green) the larger the SH diameter D SH , the larger the flux Q SH , and the smaller the median wall shear stress τ wall , which was explained by the flow pattern within the SH ( Figure 5). It demonstrated the importance of full scale simulations with the focus to resolve detailed flow characteristics to understand the interactions between scales.
In terms of lumen diameter of the stent, larger D s promoted both luminal flow rate and the interluminal exchange of fluid ( Figure 6). The latter was mainly contributed by the active SHs. Based on Equation (9), the larger lumen diameter helped alleviate the pressure loss at the inlet/outlet of the stent and thus encouraged more luminal flux.
Nonetheless, the luminal flux only accounted for up to 9% of the total flux in our study and further reduced down to $ 3% when the pigtail was included (see Supporting Information S1). In this regard, further studies should focus on evaluating stent pigtails with different inlet designs and lumen diameters, balancing the antidislocation function and its impact on the stent performance for different types of patient.
In summary, our results suggest that larger SH and larger lumen size should be chosen where internal obstruction is the primary concern since they promote better drainage capability and encourage interluminal exchange of fluids. For long-term stenting (e.g., during pregnancy) where encrustation is the primary concern, smaller SH seems to be a better choice since more SHs are activated and shear stress levels on the SHs close to the UPJ and UVJ are higher. Smaller SH also means better tensile strength against radial compression so seems to suit patients with extrinsic obstructions. Further comparative evaluations with other stent designs such as those of noncircular cross sections 27,28 might be of interest.
To close the discussion, we acknowledge the major limitations of the current study. First, even though the role of pigtail was inferred in the discussion and briefly visited in the Supporting Information S1, full documentation on its impact on the flow characteristics especially with different design parameters is still desirable. Second, the ureter was modeled as a straight rigid tube in this study with constant crosssectional geometry, where the elasticity and tapering of real ureters are not modeled. These factors can be evaluated using previous results on reduced order models of the urinary tract. 18 Further, the vesicoureteral reflux, which is the flux of urine from bladder to the ureter(s) during bladder contraction, was not investigated in this study. Such retrograde flow will impose dynamic changes to the flow characteristics near the UVJ. Although, the relevant changes caused by this pressure rise should be alleviated by including more SHs in the distal part of the stent according to our discussions. The stented ureter poses a complex multidisciplinary system, and patient-specific physiochemical environment is an important factor to address.
Nevertheless, the results from current study offered a glimpse into the baseline fluid mechanical principles of the system and could be used to guide further studies of the system with more complex physiochemical conditions.
In short, the "perfect stent" should be more patient oriented, not necessarily individualized but should be designed to address the primary need in each type of patient.

CONFLICT OF INTEREST
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available on request from the corresponding author. The calibration code used in the experiment can be found on Github at https://github.com/zhengsk/. Further information and supplementary data can be found in Supporting Information.