CFD investigation of multiple peristaltic waves in a 3D unobstructed ureter

Ureters are essential components of the urinary system and play a crucial role in the transportation of urine from the kidneys to the bladder. In the current study, a three-dimensional ureter is modelled. A series of peristaltic waves are made to travel on the ureter wall to analyse and measure parameter effects such as pressure, velocity, gradient pressure, and wall shear at different time steps. The flow dynamics in the ureters are thoroughly analysed using the commercially available ANSYS-CFX software. The maximum pressure is found in the triple wave at the ureteropelvic junction and maximum velocity is observed in the single and double wave motion due to the contraction produced by the peristalsis motion. The pressure gradient is maximum at the inlet of the ureter during the single bolus motion. The contraction produces a high jet of velocity due to neck formation and also helps in urine trapping in the form of a bolus, which leads to the formation of reverse flow. Due to the reduction in area, shear stress builds on the ureter wall. The high shear stress may rupture the junctions in the ureter.


Introduction
The urinary system, also referred to as the renal system or urinary tract, plays a crucial role in maintaining homeostasis within the body.This intricate system is responsible for regulating not only the volume and pressure of blood but also ensuring proper electrolyte balance and waste removal.Also to maintain the chemical balance and, eliminate the waste from the body.The urinary system is divided into the upper and lower urinary tract [1,2].The upper urinary tract consists of the kidney and ureter.The main function of the upper tract is to collect urine and propel urine to the bladder, where urine is stored temporarily and periodically to eliminate waste from the body [3,4].The pacemaker plays a crucial role in the urinary system by generating the peristalsis wave in the ureter.This remarkable mechanism is responsible for propelling urine from the kidney to the bladder [5,6].
During peristalsis, smooth muscle contraction plays a crucial role in propelling urine through the ureters.This rhythmic and coordinated movement aids in moving urine efficiently as it forms into boluslike masses.It is fascinating to note that various factors can influence this intricate mechanism [7,8].One such factor is the flow rate of urine.When there is an increase in urine flow rate, which may occur due to hydration or other physiological reasons, it can have an impact on peristalsis [9,10].As more urine enters the ureters at a faster pace, it triggers changes within the smooth muscle fibres responsible for contraction.In response to an elevated flow rate, there tends to be an increase in the frequency of contractions during peristalsis [11,12].This heightened frequency allows for more rapid propulsion of urine through the ureters towards its ultimate destination -either the bladder or further along in the excretory pathway.The intensified contractions work harmoniously with each other, creating a wave-like motion that effectively moves larger volumes of fluid [13][14][15].Kill et al, [16] delved into the intricate dynamics of the ureteral peristaltic rate.He observed that in peristalsis rate tends to rise in response to increased urine flow.However, he also noted that there are instances where the peristaltic rate remains unaltered or even decreases.Such findings highlight the complex nature of ureteral response patterns and suggest that there is no definitive regularity in urine flow.The peristalsis wave is responsible for propelling urine from the kidneys to the bladder and travels at varying speeds along the length of the ureter.This wave-like contraction and relaxation movement ensures a smooth flow of urine in the ureter.Depending on factors such as hydration levels and individual physiology, the speed of this peristalsis wave can range from 20 mm sec −1 to 60 mm s −1 .Usually, a typical ureter length in humans varies from 220 mm to 300 mm [17,18].In adults, the peristalsis happens about 60 to 480 times per second and averages about 180 times per second [19,20].In recent years, there has been a growing interest among researchers in the field of biomechanics to better understand peristaltic behaviour.Both theoretical and experimental analyses have been extensively conducted to gain deeper insights into this phenomenon [21][22][23][24][25]. Li et al, [26] studied the peristalsis transport in circular cylindrical tubes.A long-wave approximation was adopted to show that the axial pressure gradient does not vary in the radial direction of the tube, and reflux is observed near the axis of symmetry of the cylindrical tube.In their study on ureter dynamics, Burns and Parkes et al [20] employed the utilization of sinusoidal waves to induce peristaltic motion on the ureter wall.Their objective was to investigate and compare the effects of pressure gradients on symmetric pipes with and without such gradients.By manipulating the sinusoidal wave, they were able to simulate peristalsis in a controlled environment, allowing for a comprehensive analysis of its impact under different conditions [26,27].In a study conducted by Bruijnes et al [28], the pressure profile on the dogs' ureters was thoroughly analyzed using a polyethene catheter.The researchers made an interesting discovery regarding the pressure variation along the mid-ureter, as well as the consistent pressure observed at the junction of the ureter and pelvis.This finding sheds light on different sections of the ureter with effect of pressure.Kundo et al [13] conducted research to investigate the dynamics of pressure changes in the ureter after systolic contraction.Their findings revealed that there is a rapid and immediate formation of adverse pressure in both human and dog ureters.Shafik et al [29] discovered that there is an increase in the pressure within the ureter during peristalsis, specifically when comparing it to both normal and pathologic ureter pressure profiles.This rise in pressure occurs from the basal level to high levels of pressure.The findings suggest that peristalsis plays a significant role in altering the dynamics of ureteric pressure, highlighting its importance in the overall functioning of the urinary system.The numerical simulation of urine transportation in the ureter lumen is a challenging task due to its inherent complexity.As a result, researchers have primarily relied on simple two-dimensional modeling and boundary conditions to study this intricate process [30].The mechanical properties of the ureter play a crucial role in the analysis of fluid-structure interaction [31][32][33][34].In a study conducted by Rassoli et al [35], the Mooney-Rivlin model was employed to accurately predict and analyze the behaviour of the human ureter during peristalsis.The biaxial study allowed them to explore various mechanical aspects of the ureter's behaviour to develop predictive models.They discovered that the ureter wall is anisotropic and can be modelled as hyperplastic material.In their study, Takaddus et al, [36] focused on investigating the impact of flow on the wall by examining a circular tube.Two-way fluidstructure interaction modelling is used on the twodimensional axisymmetric model, considering the ureter as a non-linear hyperelastic model.They showed the failure of the UPJ due to urine reflux.Vahidi et al [30] took an innovative approach by utilizing the actual ureter model to understand the complex flow dynamics within this vital organ.By employing this realistic ureter model, they were able to investigate and analyze various aspects related to the urine movement through the ureter.Najafi et al [34,37] conducted experiments using both two-dimensional and three-dimensional ureter models to investigate the impact of obstruction on flow behaviour.They examined conditions with clogged and passable obstructions in order to study the factors that affect the flow dynamics within the ureter.In recent years, there has been a significant increase in research efforts dedicated to studying the intricate flow dynamics within the ureter by adopting a single propagative wave.These can indeed help a surgeon to understand the physiology behind the peristalsis in the human organs with the advancement in computational techniques [38][39][40].The functioning of human organs can be effectively studied using the finite element method, a computational technique that allows for detailed analysis and simulation.This method provides researchers with valuable insights into the behaviour and performance of various organs, enabling them to understand their complex dynamics in depth [41,42].In this work, three contraction models namely single, double, and triple peristaltic waves are used to analyze pressure, velocity, gradient pressure, and wall shear during the peristalsis in the ureter.Usually, three waves per minute are generated in the ureter [43,44].In the current work, the same is used to evaluate the effect of peristaltic waves.Multiple peristalsis waves in the ureter are essential for moving urine efficiently from the kidneys to the bladder.Its effects on the ureter wall is crucial for diagnosing and treating conditions related to urinary flow and preventing complications like urinary reflux.This works helps to understand the multiple peristalsis wave effects on ureter wall using CFD.

Modelling and meshing
It is important to note that the length of the ureter can vary within a range of 220 mm to 300 mm [14,45].This variation in length is significant as it plays a crucial role in understanding and studying the functionality of this vital anatomical structure.For our specific analysis, we have chosen to focus on a ureter with a length of 275 mm.Using ANSYS 2019-R2 a three-dimensional ureter is modelled.The ureter model is considered linear isotropic with a diameter of 3 mm [37].The wall thickness of 1 mm is considered as shown in figure 1.For the analysis, we have utilized a computational dynamics package called ANSYS-CFX.This commercially available software is designed to provide accurate and reliable results in various engineering applications.With its comprehensive range of features and capabilities, ANSYS-CFX offers an extensive suite of tools for simulating fluid flow, heat transfer, and other related phenomena.
The second-order upwind scheme, a widely used numerical method in fluid dynamics, is adopted for both the momentum and pressure calculation [15].For mesh movement, diffusion-based smoothing is used.Figure 2 represents the meshed model of the ureter model.A total of 2,790,000 elements and 2,823,381 nodes are generated.For the meshed model, ten inflation layers are added with a growth rate of 1.2.For the analysis, a total time of 18 s is considered at a time set of 0.01s for the wave propagation from the inlet to the outlet of the ureter model.The grid dependency study was carried out for an inlet pressure of 0.3 Pa and at 0.01 mesh size, no variation in pressure values was observed [46].

Boundary condition and governing equation
A constant boundary condition of pressure difference of 0.3 Pa is applied at the inlet and outlet to ensure a steady flow of urine through the ureter.Additionally, no slip and no penetration are considered between the ureter and urine, meaning that there is no relative motion or leakage between these two entities [46,47].In the ureter wall, a moving mesh technique is implemented using equation (3).
Urine is commonly described as a homogenous substance with a viscous and incompressible nature.Its properties can be further analyzed using the Navier-Stokes equation, which takes into account its density of 1050 kg m −3 and viscosity of 1.3 cP [48].By utilizing an adopted model based on both continuity and momentum equations, we can predict variables such as velocity distribution, pressure gradients, and overall flow rates.Equations (1) and (2) are the continuity and momentum equations respectively.
Where the velocity vector represents the direction and magnitude of fluid flow denoted by u .f Additionally, p, which stands for pressure, influences the movement and distribution of fluids.Another significant factor to consider is urine density denoted by ρ.Lastly, μ represents fluid dynamic viscosity.The SIMPLE algorithm specifically focuses on handling the pressure terms present in momentum equations governing fluid flow.SIMPLE algorithm lies in its iterative procedure used to obtain solutions for discretized equations derived from these fundamental principles [49].The physiological velocity range of the ureter varies from 20 to 60 mm s −1 [50].In our current work, we have chosen to apply a peristaltic wave in the form of a sinusoidal waveform on the ureter wall.This peristaltic wave helps propel fluids through the ureter and maintain its functionality.Specifically, we have set the velocity of this peristaltic wave at 20 mm s −1 .
The displacement of the ureter wall is an important factor to consider in various medical scenarios.It can be quantified using equation (3), which takes into account several variables such as the pressure gradient, fluid flow rate, and the compliance of the ureter wall itself.This information plays a crucial role in diagnosing conditions like kidney stones or urinary tract infections.
Where a represents the amplitude of displacement, k is the wave number and ω represents the frequency.For the analysis, the a = 1 mm, k = 2, 4, and 6 are considered.

The pressure profile analysis
To conduct the analysis, a consistent pressure input condition of 0.3 Pa is applied.Figure 3 shows the pressure profile for the three contraction models a single wave, double wave, and triple wave.The analysis takes into account a complete wave cycle lasting 18 s, featuring a wave velocity of 20 mm s −1 .
In his research, Wienberg [51] employed a single contraction wave motion and observed the presence of a maximum pressure behind the advancing urine bolus.In the current study, we examined the pressure profiles of all three wave contraction models, noting the maximum pressure located behind the bolus, as depicted in figures 3(a)-(c).These findings are in alignment with Wienberg's [51] earlier investigation, which also focused on a single wave motion scenario.
In the context of triple wave motion, the centre bolus exhibits the lowest pressure when compared to the first and last boluses, as illustrated in figure 3(c).Additionally, the contraction of peristalsis in the   wave propagation, where the maximum pressure magnitude of 1.06 Pa is found at T/2 s.This is attributed to the production of only one wave at this particular time step, resulting in the recorded pressure value.
In the case of the triple wave, as depicted in figure 4(c), the analysis reveals that the maximum pressure of 1.581 Pa is observed precisely at T/4 s.The negative pressure observed in the neck region leads to reflux formation in the ureter.
According to the findings of Vahidi et al [52], it was reported that reflux at the kidney is triggered during wave onset.However, as the wave propagates further, the reflux diminishes notably at the UPJ.Takaddus et al [36] concluded that the maximum pressure generated at the UPJ compared to the UVJ.A similar analysis is observed in our simulation result as shown in figure 4, for all the contraction models.
In this study, it has been observed that the maximum pressure is recorded precisely at the time T/4 s, indicating that the bolus reaches the UPJ at that moment.This finding aligns with the observations made by Eytan et al, who reported that pressure distribution along the axis would deteriorate as it approaches the outlet.In the current work, it is demonstrated that, for all the wave models, at T/4 s, the maximum pressure is observed in the first bolus.However, as the wave propagates towards the outlet, at time T s, the pressure gradually decreases.This pattern of pressure distribution supports the notion that pressure tends to diminish as the wave advances along the ureter.

The velocity profile analysis
The velocity vector for the different contraction models at the beginning of the peristalsis is shown in figure 5.As the wave propagates in the single bolus wave model, the analysis reveals the occurrence of reverse flow at both the neck and the inlet regions, as depicted in figure 5(a).A similar velocity vector is found in the double and triple bolus as shown in figures 5(b) and (c) respectively.The high velocity is observed in the neck region due to the contraction formed by the peristalsis.This will produce the highvelocity jet flow at the neck.In their research, Hosseini et al [53] observed fluid dynamics in the ureter.They reported that the phenomenon of fluid inertia results in the maximum flow occurring at the contracted region.This contraction in the flow creates a trapping effect on the bolus, leading to the formation of reverse flow in certain regions.Vahidi et al [15], and Eyten et al [54] have documented the phenomenon of trapping in their respective studies.The trapping phenomenon manifests in the form of stagnant regions observed within the boluses, especially when a higher number of waves are present.However, as the waves travel towards the outlet, the adverse flow at the inlet of the ureter diminishes significantly.
Figure 6 shows the velocity profile plot for the three contraction models.It is found that the maximum velocity was recorded at T/4 s as shown in figures 6(a) and (b).The maximum velocity magnitude of 0.039 m s −1 was observed in single-wave  remaining time step, the velocity will not have any significant effect on the flow dynamics in all the models that are considered.

The pressure gradient analysis
The pumping efficiency is determined by the pressure increase over a specific length (dp/dz).illustrates the pressure gradient for the three contraction models at various time intervals along the ureter axis.
In figure 7(a), it is evident that during a single wave, the maximum pressure gradient magnitude along the ureter axis occurs around the contraction of the wall caused by peristalsis.This results in peaks in the pressure plot, with a high-pressure gradient of 431.57Pa m −1 observed at T/4 s near the inlet of the ureter.As the wave progresses towards the outlet, there is a decrease in the pressure gradient due to the transition in urine flow.
A similar pattern is observed in figure 7(b) during double wave propagation, where the maximum pressure gradient magnitude of 412.71 Pa m −1 is recorded at T/4 s, and the pressure variation remains relatively constant in the subsequent time steps.
In the case of triple wave propagation, as shown in figure 7, the maximum pressure gradient of 331.126Pa m −1 is recorded in the T s of the flow due to incomplete wave propagation.
Throughout all the contraction models shown in figure 7, there is a consistent observation that the pressure gradient is higher in the neck region due to a reduction in the area.The research by Vahidi et al [55] further supports this finding, showing that the propagation of the peristalsis wave leads to an increase in the pressure gradient value caused by the contraction of the ureter wall.These results are consistent across all the models analyzed in this study.

The wall shear stress analysis
In figure 8, the wall shear developed in the ureter during the bolus movement from the inlet to the outlet is depicted for the three different contraction conditions.In figures 8(a) and (b) depicting single and double wave motion, respectively, it was observed that the maximum shear stress occurs at the inlet of the UPJ.Specifically, at T/4 s, a magnitude of 0.095 Pa is recorded during single peristalsis wave motion, while a magnitude of 0.090 Pa is observed at the UPJ for double wave motion in figure 8(b).Subsequently, as the wave propagates, a constant shear is observed along the wall for both single and double-wave conditions.Figure 8(c) demonstrates that during triple contraction wave propagation, the maximum magnitude of 0.073 Pa is attained at the UVJ.This outcome is a result of incomplete wave propagation at the end of the cycle.The maximum magnitude was found in all the contractions developed during the peristalsis.Hosseini et al [53] and Vahidi et al [56] presented research findings indicating that shear stress is notably high around the contraction region.Additionally, at the UPJ, elevated shear stresses are consistently observed.As the peristalsis wave advances towards the bladder, the wall shears behind the trapped region and the maximum shear around the stream of the moving contracted wall gradually decrease.

Limitations of the present work
The study demonstrates that peristalsis-induced contractions result in the formation of reflux at the ureter's inlet.The velocity considered in the analysis is 20 mm s −1 .However, it is worth noting that the literature reports variations in velocity ranging from 20 mm s −1 to 60 mm s −1 .In the current research, the minimum velocity of 20 mm s −1 is employed.Yet, exploring different velocities can provide valuable insights into their impact on the ureter wall when multiple peristaltic waves are involved.
Moreover, the study highlights the possibility of investigating the effects of varying the length and diameter of the ureter.By modifying the ureter's length and diameter, the research can gain a deeper understanding of how these factors influence its behaviour during peristalsis and reflux formation.The FSI study can be further conducted to know the Ureteral wall elasticity.

Conclusion
The study focused on analyzing the peristalsis wave motion generated by the urine bolus in the ureter.The investigation considered single, double, and triple wave motions and examined their effects on the ureter wall across four time-step intervals.The findings reveal that the maximum pressure is generated at the UPJ in comparison to the UVJ.The research indicates that during triple-wave motion, the UPJ experiences a maximum pressure of 1.581 Pa at the time T/4 s.Additionally, in single-wave motion, a maximum velocity of magnitude 0.039 m s −1 is observed, which is higher than the velocity recorded in double-wave motion.Moreover, it was observed that the velocity for the other time steps does not significantly affect the flow behaviour.The study identifies that high velocity is observed in the neck region of the ureter due to the contraction caused by peristalsis.This contraction contributes to the trapping of the bolus, leading to the formation of reverse flow.The reverse flow, in turn, creates a risk of urinary tract infection as bacteria and toxins from the ureter may enter the renal pelvis and kidneys.Over time, this condition can potentially lead to serious kidney problems.
The study highlights that the contraction in the ureter wall results in increased pressure gradient values in the flow.Specifically, during single-wave motion, a pressure gradient of 431.57Pa is recorded at time T/4 s, which exceeds that of double and triple waves.This rise in pressure gradient is attributed to the reduction in the area, leading to the accumulation of shear stress on the ureter wall.In the case of single peristalsis wave motion, a magnitude of 0.095 Pa is found at the UPJ at time T/4 s.Similarly, during triple contraction wave propagation, the maximum magnitude of 0.073 Pa is observed at the UVJ.This discrepancy is attributed to the unfinished wave propagation at the end of the cycle at the UVJ.Indeed, the high shear stress observed in the ureter may pose a risk of rupturing the junctions within its structure.To gain a more comprehensive understanding of the impact of shear stresses on constrictions in the ureter, further experimental work can be undertaken.One approach could involve manufacturing ureter tubes using suitable rapid prototyping techniques.These techniques can help simulate the physiological conditions more accurately and allow researchers to study the behaviour of the ureter under various shear stress conditions.This experimental work can provide valuable insights into the mechanical response of the ureter and aid in designing appropriate measures to prevent potential complications arising from high shear stresses.

Figure 2 .
Figure 2. Cross-section of the meshed structure of the ureter.

Figure 5 .
Figure 5. Velocity vector for (a) Single peristalsis wave motion (b) Double peristalsis wave motion (c) Triple peristalsis wave motion.