NUMERICAL INVESTIGATIONS of UNOBSTRUCTED and OBSTRUCTED HUMAN URETER PERISTALSIS a Thesis Presented to the Graduate Faculty of T

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NUMERICAL INVESTIGATIONS OF UNOBSTRUCTED AND OBSTRUCTED

HUMAN URETER PERISTALSIS

A Thesis

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulﬁllment

of the Requirements for the Degree

Master of Science

Ahmed Tasnub Takaddus

April, 2017 NUMERICAL INVESTIGATIONS OF UNOBSTRUCTED AND OBSTRUCTED

HUMAN URETER PERISTALSIS

Ahmed Tasnub Takaddus

Thesis

Approved: Accepted:

Advisor Interim Dean of the College Dr. Abhilash J. Chandy Dr. Donald P. Visco Jr.

Faculty Reader Executive Dean of the Graduate School Dr. Ajay Mahajan Dr. Chand K. Midha

Faculty Reader Date Dr. Xiaosheng Gao

Department Chair Dr. Sergio D. Felicelli

ii ABSTRACT

Urine transported through the ureter from the kidney to the urinary bladder by peristaltic mechanism. Some of the problems related to urine transportation in ureter is back-ﬂow, severe pain or in some rare cases, rupture of the ureter wall in an obstructed ureter. Ureter

ﬂow can get obstructed due to mineral deposits, from highly concentrated urine in the kidney and these deposits also known as kidney stones can then travel through the urinary tract. A number of numerical analyses have been conducted on the ureter to understand the ﬂow mechanics as well as to understand the wall properties. But very few studies exist focusing on an obstructed ureter. To fully understand the effect of obstruction in a ureter, a comparative study of obstructed and unobstructed human ureter is required.

So in this paper, the ﬁrst 2D axisymmetric study has been carried out on a human ureter, where the ureteral wall has been modelled as an isotropic hyper-elastic material, based on curve ﬁt-data from previous literature. The peristalsis waves are created using contractile forces on the ureter wall. In addition, 2D axisymmetric studies have been conducted on both obstructed and unobstructed ureters, where the constitutive material for the ureter is assumed to be anisotropic hyperelastic based on a previous experiments on real human ureters. Here, urine is transported by the isolated bolus created by the expansive force acting on the outer surface of the ureter wall. Finally, a 3D comparative study has

iii been done on ureter with 50%, 70% obstruction and no obstruction. The material properties and wall mechanics for peristalsis are consistent with the previous cases. In all of these studies, both the ﬂuid and structural domains are solved using a ﬁnite-element based

fluid-structure interaction (FSI) approach between ureter wall and urine. The coupling between the domains is two-way, so both fluid and structure influence each other. While all the 2D simulations are solved in fully coupled monolithic way, for the 3D cases, a segregated approach is used with different nonlinear solvers for fluid and structure. For all the simulations, the fluid domain is considered to be incompressible laminar flow.

From the analysis of the results, it is observed that peristalsis ﬂow has a slight tendency of generating backﬂow from the bladder to the outlet even without any obstruction.

So failure of the one-way valve like junctions, for instance the junction between bladder and ureter, can cause urine reﬂux. As the size of the obstruction increases, amount of urine backﬂow also increases. Also, high peaks are observed in pressure gradient and wall shear stress values near the location of the obstruction, which may lead to severe pain and even in some rare cases rupture of the ureter wall. The main objective of this study is to quantify the effect of ureteral obstruction, which will help physicians to understand and assist in the treatment procedure of an obstructed ureter.

iv ACKNOWLEDGEMENTS

At ﬁrst, I would like to thank my advisor Dr. Abhilash J. Chandy. This work is able to see the daylight because of his vision and guidance. It happened countless times, that I ran into a problem, and asked him what to do. His door was always open, and his advices always showed me a way.

I would like to thank Dr. Ajay Mahajan and Dr. Xiaosheng Gao for their time to be in the committee, and truly grateful for their valuable and insightful comments.

At last, I would like to thank my parents, as because of heir continuous support, I am where I am today.

v TABLE OF CONTENTS

Page

LIST OF TABLES ...... viii

LIST OF FIGURES ...... ix

CHAPTER

I. INTRODUCTION ...... 1

II. PROBLEM DESCRIPTION ...... 11

2.1 Geometry of the model ...... 11

2.2 Material Properties ...... 12

III. MATHEMATICAL FORMULATION ...... 16

3.1 Fluid ...... 16

3.2 Solid ...... 16

3.3 Fluid-Structure Interaction...... 17

IV. METHODOLOGY ...... 19

4.1 Numerical Details ...... 19

4.2 Computational Mesh ...... 21

V. RESULTS ...... 30

5.1 2D axisymmetric unobstructed ureter with isotropic wall properties . . . . 30

5.2 2D axisymmetric unobstructed ureter with anisotropic wall properties . . . 38

vi 5.3 2D axisymmetric obstructed ureter with anisotropic wall properties . . . . 50

5.4 3D axisymmetric obstructed and unobstructed ureter with anisotropic wall properties ...... 63 5.5 Conclusions ...... 73

VI. SUMMARY ...... 75

BIBLIOGRAPHY ...... 78

vii LIST OF TABLES

Table Page

4.1 Number of mesh elements for the different cases presented in this study. . . . 25

viii LIST OF FIGURES

Figure Page

1.1 A cross-sectional picture of ureter showing it’s three layers [1]...... 8

1.2 Urine reﬂux due to ureterovesical junction valves failure [2]...... 9

1.3 Luminar change of ureter during peristalsis [3]...... 10

2.1 Stress-strain curve fitting using five parameter money-rivlin model in circumferential direction...... 14 4.1 Geometry and mesh of the fluid and structural domain. (a) mesh before deformation and (b) mesh after deformation for 2D axisymmetric unobstructed ureter with isotropic wall properties...... 22 4.2 Computational mesh of the fluid and structural domains for 2D axisymmetric unobstructed ureter with anisotropic wall properties...... 23 4.3 von Mises stress at a point 15 cm away from the ureter vs. time for 2D axisymmetric unobstructed ureter with anisotropic wall properties...... 26 4.4 Geometry and mesh of the fluid and structural domain for 2D axisymmetric obstructed ureter with anisotropic wall properties...... 27 4.5 Mesh convergence study for three different element number for 2D axisymmetric obstructed ureter with anisotropic wall properties...... 28 4.6 Geometry and mesh of the fluid and structural domain for 3D axisymmetric obstructed and unobstructed ureter with anisotropic wall properties. . . 29 5.1 Urine flow rate at ureter inlet for 2D axisymmetric unobstructed ureter with isotropic wall properties...... 31

ix 5.2 Contour of von-misses stress for structure and streamline of velocity vector overlapped with velocity vector arrows for flow field at time, t/4 = 25.5s for 2D axisymmetric unobstructed ureter with isotropic wall properties...... 32 5.3 Velocity profile at ureter inlet for 2D axisymmetric unobstructed ureter with isotropic wall properties...... 35 5.4 Pressure gradient along the ureter wall for 2D axisymmetric unobstructed ureter with isotropic wall properties...... 36 5.5 Wall shear stress along the ureter wall for 2D axisymmetric unobstructed ureter with isotropic wall properties...... 37 5.6 Pressure contours of the flow filed for 2D axisymmetric unobstructed ureter with anisotropic wall properties...... 39 5.7 Streamlines colored by velocity magnitude superimposed with velocity vectors of the flow filed for 2D axisymmetric unobstructed ureter with anisotropic wall properties...... 40 5.8 Urine flow rate at various cross-sections along the length of the ureter for four different times for 2D axisymmetric unobstructed ureter with anisotropic wall properties...... 42 5.9 Velocity profile at inlet or four different times for 2D axisymmetric unobstructed ureter with anisotropic wall properties...... 43 5.10 Pressure gradient along the ureter wall for four different times for 2D axisymmetric unobstructed ureter with anisotropic wall properties...... 44 5.11 Wall shear stress along the ureter wall or four different times for 2D axisymmetric unobstructed ureter with anisotropic wall properties...... 45 5.12 von Mises Stress plot in the ureter wall both for linear and hyperelastic material...... 48 5.13 von Mises stress (top) and displacement (bottom) along the ureter wall at time, t = 6.5 s for 2D axisymmetric unobstructed ureter with anisotropic wall properties...... 49 5.14 Volume flow rate along the ureter length at 6 different time instants for 2D axisymmetric obstructed ureter with anisotropic wall properties...... 51 5.15 Velocity profile at ureter inlet for 2D axisymmetric obstructed ureter with anisotropic wall properties...... 52

x 5.16 Percentage of cross-sectional area blockage with time at the the stone center for 2D axisymmetric obstructed ureter with anisotropic wall properties. 53 5.17 Pressure gradient along ureter wall for 2D axisymmetric obstructed ureter with anisotropic wall properties...... 55 5.18 Wall shear stress along ureter wall for 2D axisymmetric obstructed ureter with anisotropic wall properties...... 56 5.19 Contour of von mises stress for the structure, along with streamlines overlapped with velocity vectors for the fluid flow for 2D axisymmetric obstructed ureter with anisotropic wall properties...... 62 5.20 Inlet mass flow rate at four different time instances, t = 2.25 s, t = 2.75 s, t = 3.25 s, and t = 3.4 s, for the Unobstructed and 50% and 70% obstructions. 64 5.21 Pressure gradient along the length of the ureter at four different time instances, t = 2.25 s, t = 2.75 s, t = 3.25 s, and t = 3.4 s, for the Unob- structed and 50% and 70% obstructions...... 65 5.22 Wall shear stress along the length of ureter at four different time instances, t = 2.25 s, t = 2.75 s, t = 3.25 s, and t = 3.4 s, for the Unob- structed and 50% and 70% obstructions...... 66 5.23 Velocity vector of urine coloured with velocity magnitude and ureter wall deformation for 70% blockage ...... 67 5.24 Pressure and von Mises stress contour in the ureter lengthwise cross- section for 70% blockage...... 68 5.25 Pressure and von Mises stress contour at circumferential cross-section in the middle of the 70% circular blockage...... 70 5.26 Wall velocity in material frame in ureter wall...... 71

5.27 Comparison of the values of wall shear stress and pressure gradient between the full 3D and 2D axisymmetric cases when the peristaltic wave is front of the obstruction in the 70 % obstruction case...... 72

xi CHAPTER I

INTRODUCTION

The ureter is an approximately 25 − 35 cm long tube that carries urine from the kidney to the bladder [4]. Urine transportation through the ureter happens due to peristaltic motion of the ureteral wall [5]. The ureter begins at the renal pelvis and enters into the bladder obliquely, forming a one-way valve that prevents backflow of urine from the bladder at the ureterovesical junction. The wall of the ureter is made of three layers: mucosa, muscularis and adventitia (see Fig. 1.1). A peristaltic wave is generated by the contraction of the muscularis layer, stimulated due to the distention of the ureter by the urine entering it. One of the abnormalities related to the peristaltic transport is the retrograde flow of urine from the bladder to the ureter known as vesicoureteric reflux [6]. Ureterovesical reflux can cause urinary tract infection, and result in the formation of stones in the ureter [7, 8]. The reason for this may be the congenital defect in ureterovesical junction in infants and in adults due to the increased pressure in urethra during voiding [6]. Previously, another reason for the backflow of the urine was thought to be urinary tract infection [9, 10]. But studies carried out later reported that infection and reflux are independent of each other and happen mainly due to defects in the ureterovesical junction [11, 12].

Other than the complication related to ureter junctions, another problem is obstruction in the ureter ﬂow path. Such an obstruction can be because of formation of renal

1 calculi or kidney stone. Obstruction in the urine ﬂow path can also cause urine back-

ﬂow [13], which is an unwanted phenomenon and can lead upto various complications i.e. urinary tract infections [7]. Another problem related to renal caculi or ureterolithia- sis is the rupture of the ureter wall and as a result in some cases, surgical interventions are needed [14]. Chances of developing kidney stone in humans in a lifetime range from

10% to 25% of the population, depending on the geographical region [15]. The prevalence of kidney stone around the world is on a rise [16]. In the United States, about 5% women and 12% men develop kidney stone sometime in their life [17]. A high concentra- tion of calcium, urate, phosphate, oxalate magnesium salts present in the urine, crystallizes and forms renal calculi or kidney stones, as the salts precipitate due to small variation in pH [18]. Urolithiasis occurs due to different environmental and metabolic factors, including acidity in urine, urinary tract infection (struvite stones) and genetic disorder (cystine stones) [19]. Stones smaller than a diameter of 5 mm normally passes through the urinary tract spontaneously [20]. Also, according to [20], for stones of diameter smaller than 1 cm in the proximal ureter, shock wave lithotripsy is a preferable procedure before moving to a more invasive surgical procedure. But larger stones can cause pain by blocking the ureter, which can lead to an increase in internal pressure while severe pain is felt when the ureter comes in contact with the sharp calculi during the peristalsis contraction [21]. For removal of distal ureteral calculi, another effective method is ureteroscopy [22]. To maximize the stone removal, especially in the kidney, a more invasive and effective surgical procedure other than the ureteroscopy and Extracorporeal Shock Wave Lithotripsy (ESWL) is percutaneous nephrolithotomy [23]. Kidney stone disease, also known as nephrolithiasis, is

2 diagnosed based on patient history, physical examination, urine analysis and abdominal radiography [24]. Studies have been carried out previously to find a relationship between the size and location of renal calculi with its spontaneous passage through the urinary tract [25]. On examining unenhanced computerized tomography (CT) scans, it was found that the rate of passing the stone without any surgical intervention varied with its size and position. Hubner¨ et al. [26] investigated different treatment modalities and also made similar observations. A case study by Drake et al. [27] showed that, as the kidney was obstructed by the ureteric stone, the ureter became distended, and with the peristaltic wave superimposed on this distended ureter periodically, the severity of the pain felt by the patient also followed the same periodicity. Further affirmation of the effect of renal calculi on the ureter can be obtained if an in situ experiment, which incorporates the peristaltic mechanics of the ureter and ureteric stone can be conducted. But an experiment like that of Rose et al. [28], which was performed on dogs, is difficult to perform on a living human body, as it requires surgical procedure to quantify critical parameters i.e. wall tension, intraluminal pressure etc. So, a detailed numerical study to quantify these obstructions in a ureter is necessary, which will help to determine the effects of it and assist physicians decide upon the treatment procedure based on actual numbers, rather then some abstract concept such as pain felt by the patient.

As the ureter peristalsis itself is a complex biological phenomenon, various numerical studies have been conducted on this topic. One of the earliest studies was carried out by Burns and Parkes [29], where peristaltic motion was simulated with no pressure gradient and a sinusoidal varying cross-section with pressure gradient. Later, there were other

3 papers that also modeled ureter peristalsis as sinusoidal waves (See for instance [30–33], and most recently [34]). Najafi [35] performed a numerical study on an obstructed ureter and modeled peristalsis as sinusoidal waves. Lykoudis et al. [36] presented a peristaltic wave not as a sine wave, but rather as an algebraic expression in the form of x ∼ hm, where, h and x were the radial and longitudinal distances, respectively, along which the peristaltic wave travels. They also used Fourier analysis to completely define the wave shape of the ureter. Their results on pressure distribution had good agreement with experimental studies for both algebraic and Fourier analysis of the wave. Studies of Griffiths [37, 38] have investigated numerically peristaltic motion in a highly distensible tube. The 1989 study of Griffiths [38] considered ureter peristalsis as a series of compression waves at constant speed. All of these studies solved only for the fluid domain in the context of either a 2D channel or an axisymmetric geometry.

Several studies have been conducted to understand the morphology of ureter peristalsis. Non-enhanced helical CT was used to obtain information about the external shape of the ureter through imaging [39]. Woodburne et al. [3] conducted experiments on dog ureter lumen during peristalsis (see Fig. 1.3). Ureteric lumen is normally square or diamond- shaped, but as it is distended, the lumen becomes circular. As the contractile wave moves forward it creates a slight distention in the ureter [5]. Morphologic and volumetric studies on an isolated bolus were also conducted by Ohlson [40].

The muscularis layer of the ureter wall is made of smooth muscles and is responsible for generating the peristaltic wave [41]. Uniaxial tensile tests have been carried out on an isolated ureteral segment by Yin et al. [42]. They observed anisotropic and non-linear

4 behavior of ureteral wall from the stress-strain curve. Their study also included stress relaxation, creep under constant load and isometric contraction. Studies including [43, 44] focused on experiments on pig’s ureter wall properties. Sokolis [45] carried out biaxial tests on a rabbit ureter and reported that it is a nonlinear anisotropic material. He used a four parameter Fung-type strain energy function [46] to model the material. More recently, biaxial tests on human ureter were conducted by Rassoli et al. [47]. They developed an anisotropic constitutive model of the ureter using a four parameter Fung-type model and a modiﬁed Mooney-Rivlin model [48]. The material model used here is based on the study done by Rassoli et al. [47].

In order to fully understand the ureteral morphology during peristalsis, the interaction between the ﬂow of the urine and the ureteral wall mechanics need to be incorporated in a study. Therefore, a numerical study incorporating fully coupled two-way ﬂuid- structure interaction is necessary. One of the very few studies with such a focus involves

3D numerical calculations using a fully two-way coupled FSI [49]. In that study, the ﬂuid and solid domains were solved using boundary immersed and discrete element methods, respectively. The material was assumed to be linear, elastic, and homogeneous. A two- way FSI calculation of ureter peristalsis on a 2D axisymmetric geometry was conducted by Vahidi et al. [50], where the wall was assumed to be linear and elastic, and a rigid contact surface was used to propagate the contraction waves. In a later study, Vahidi et al. [51] modeled the ureter as a hyperelaslastic material based upon the uniaxial tests on segmented mammalian ureters by Yin et al. [42]. The peristaltic mechanism in [51] was represented as a series of isolated boluses. It should also be noted that, though their study [51] included

5 data from experiments [42] on a mammalian ureter, the mammals used were rabbit, dog, guinea pig and human fetus, but not a ureter from full grown human being.

There have been very few investigations in the past with regard of an obstructed ureter. Studies of peristaltic pumping of microscopic solid particles in viscoelastic fluid showed that viscoelasticity of the fluid did not affect the particle transport [52]. Najafi [35] carried out a 2D axisymmetric numerical simulation of ureter peristalsis with different shapes and sizes of stones. Later, Najafi et al. [13] conducted 3D numerical simulations of ureteral peristalsis with and without stones, and in both these studies, only the fluid domain was considered and peristalsis is described as a series of sinusoidal waves.

To our knowledge, there have not been any numerical studies incorporating a two- way FSI methodology in the context of an obstructed ureter. Also, none of the above- mentioned studies considered constitutive material modeling from experimental data based on bi-axial tests of human ureter. Previously, Takaddus et al. [53] studied ureter peristalsis

, where ureter wall was modeled as an isotropic hyperelastic material and the peristalsis mechanism was modeled as a series of compression waves. In that study, the peristalsis on a human ureter was simulated using the material model from the study by Rassoli et al. [47], anisotropic hyperelastic material. Studies are organized here in the following order: 2D axisymmetric study with isotropic hyperelastic material on unobstructed ureter, followed by a study with the same model but an anisotropic hyperelastic material on unobstructed ureter, next a study is conducted with the same material on an obstructed ureter and ﬁnally, a 3D comparative study on both obstructed and unobstructed ureters is presented. In all the cases the obstruction is modeled as a circular stone for simplicity. This study can shed some

6 lights on the peristaltic mechanics and it’s ﬂow pattern as well as the effect of obstruction in the ureter. It can also help in understanding and developing the treatment procedure for patients with kidney stones in the ureter.

7 Figure 1.1: A cross-sectional picture of ureter showing it’s three layers [1].

8 Figure 1.2: Urine reﬂux due to ureterovesical junction valves failure [2].

9 Figure 1.3: Luminar change of ureter during peristalsis [3].

10 CHAPTER II

PROBLEM DESCRIPTION

2.1 Geometry of the model

As ureter is highly deformable and arbitrary shaped channel [6], a simplicaﬁcation is necessary to model this numerically. First Boyarsky et al. [54] suggested to simplify the ureter as a 30 cm long tube of diamter 10 mm, which is then followed by a lot of authors [13, 34, 35, 50, 51]. For the 2D axisymetric studies here, the ureter is modeled as a

30 cm long tube, with its fluid flow channel having a diameter of 8 mm. Peristalsis wave are generated by a pressure force with Gaussian distribution along the length of the ureter wall’s outer surface. In the first study, where the material is considered to be isotropic, a compressive force is given; which is very similar to Vahidi et al. [50], in all other cases the force is given is expansion force, which creates a bolus like Vhaidi el al’s [51] later work.

So, the result is a deformation near 1 mm in radial direction inward and outward resectively with a span of 10 mm lengthwise and follows the suggestion made my Boyarksy et al. [54].

In 3D studies, to reduce computation expense, the channel is shorted to 3 cm long, with both end ﬁxed boundary condition. The stone is placed at the middle of the tube, that is

1.5 cm away from the inlet. The stone in 2D obstructed case is considered to be a circle of 3.6 mm of radius with its center located 3 cm away from the inlet, and resulting in a 81

11 % blockage in the undeformed ureter’s ﬂuid ﬂow path. For the 3D case, the blockage is a circular obstruction of diameter 2.82 mm and 3.4 mm creating 50% and 70% obstruction in

flow path respectively. So the stone is a void in the fluid and does not move depending on the urine flow and fixed at the middle.

2.2 Material Properties

For all the case studies, urine material properties are the same. In the ﬁrst 2D study ureter wall is modeled as isotropic hyperelatic for simplicity. For the rest of the studies ureter wall is modeled as anisotropic hyperelastic.

2.2.1 Fluid

As mentioned earlier, urine is considered to be an incompressible liquid with a constant density of 1050 kg m−3, and viscosity of 1.3×10−3 Pa-s [13,35,50,51]. A no slip boundary condition at the FSI wall is also employed here. Reference pressure for urine is taken as

130 Pa. The inlet and outlet boundary condition for ﬂuid domain are as follow:

2.2.1.1 Pressure Graident

The ﬂuid domain has an inlet pressure of 130 Pa and an outlet pressure of 129.7 Pa [50].

So the 0.3 Pa here helps the peristaltic force to drive the ﬂuid from inlet to the outlet. This boundary condition is used for the ﬁrst 2D axisymmetric study.

12 2.2.1.2 No Pressure Graident

For the next simulations, both unobstructed and obstructed case, the inlet and outlet pressure is set to 130 Pa. So, only the peristaltic force drives the ﬂuid.

2.2.2 Structure

2.2.2.1 Isotropic Hyperelastic Material

The ureter wall is modeled here as a non-linear isotropic hyperelastic material. The stress and strain in circumferential direction from a biaxial test of ureter sample [47] has been curve ﬁtted using a ﬁve parameter Mooney-Rivlin model [55]. Here, the strain energy is:

2 2 Ws = C10(I¯1 − 3) +C01(I¯2 − 3) +C20(I¯1 − 3) +C02(I¯2 − 3) (2.1) 1 +C (I¯ − 3)(I¯ − 3) + κ(J − 1)2 11 1 2 2 The solid is considered to be a nearly incompressible material [56], such that the initial bulk modulus, κ, is chosen as 1 GPa, which is several orders of magnitude higher then the other ﬁve material constants to incorporate the incompressibility.

2.2.2.2 Anisotropic Hyperelastic Material

A nonlinear anisotropic hyperelastic material is considered here for the ureter wall, as determined by the experimental study of Rassoli et al. [47]. To describe the material property, a modiﬁed version of Mooney-rivlin model [55] is employed, where the strain energy density function is deﬁned as follows [48]:

k1 2 Ws = C1(I1 − 3) + D1[exp(D2(I1 − 3))] + [exp(k2(I4 − 1) − 1)] (2.2) 2k2

13 Figure 2.1: Stress-strain curve ﬁtting using ﬁve parameter money-rivlin model in circumferential direction.

Here, I4 is an invariant calculated according to [57]:

I4 = Ci j(nc)i(nc) j. (2.3)

nc is a normal vector acting on the circumferential direction of the ureter. The structural domain is considered as a nearly incompressible material [56]. So, the volumetric strain energy density is given by: κ W = (J − 1)2 (2.4) vol 2 14 where, κ is the initial bulk modulus with magnitude of 1 GPa, which quite high compared to the other material constants, thus ensuring the nearly incomprehensibility of the material.

For 3D, this value is increased to 105 GPa, to get a better convergence.

15 CHAPTER III

MATHEMATICAL FORMULATION

In all the studies, both the ﬂuid and solid domain solved incorporating the ﬂuid-structure interaction between the ureter wall and urine. So, the same set of governing equations are solved for all.

3.1 Fluid

Urine is modeled as an incompressible liquid. The governing ﬂuid equations are the incompressible Navier-Stokes equations given by the continuity:

∇ · u f luid = 0 (3.1) and momentum:

∂u f luid T ρ + ρu · ∇u = −∇p + ∇ · µ ∇u + ∇u (3.2) ∂t f luid f luid f luid f luid

3.2 Solid

The local equilibrium equation of solid for a transient situation is given by:

2 ∂ usolid ρ − ∇ · σ = Fv, (3.3) ∂t2

16 Here, σ and Fv are the Cauchy Stress tensor and body force, respectively. The Cauchy stress is calculated from the second Piola-Kirchhoff stress, S [58] as follows:

σ = J−1 FSFT (3.4)

The deformation gradient, F, can be expressed in terms of the gradient of displacement vector usolid as follows:

F = (I + ∇usolid), (3.5) where I is the identity matrix, and J is the Jacobian of the deformation and is given by:

J = det(F). (3.6)

By deﬁnition of the hyperelastic material [56], the second Piola-Kirchhoff stress is calculated from the derivative of the strain energy density function with respect to Lagrangian strain tensor as follows: ∂W S = s , (3.7) ∂ε where, the Lagrangian strain, ε, is calculated from:

1 ε = (∇u )T + ∇u + (∇u )T ∇u (3.8) 2 solid solid solid solid

3.3 Fluid-Structure Interaction.

The ﬂuid and solid domains are solved separately in coordinate systems based on Eule- rian and Lagrangian formulations, respectively. The arbitrary Lagrangian-Eulerian (ALE) method is used to couple the ﬂuid and the structural solutions. The total force acting on the

17 ﬂuid-solid boundary due to ﬂuid domain is given by:

h T i fr = n · pI + µ ∇u f luid + ∇u f luid , (3.9) where, n is the normal acting outward at the boundary. The force at boundary of the structure is given by:

Fr = σ · n (3.10)

So to couple these forces respectively in spatial and material co-ordinate system, a force transformation is required. This is carried out using the ALE method as follows:

dv F = f · . (3.11) r r dV

In the above coupling equation, the mesh element scale factors are dv and dV for the ﬂuid and the material frames, respectively. The second coupling relates the structural velocity of the moving wall with the ﬂuid velocity as follows:

u f luid = uw, (3.12) where, the rate of change of the displacement of the solid is deﬁned as the structural velocity.

∂usolid uw = (3.13) ∂t

18 CHAPTER IV

METHODOLOGY

4.1 Numerical Details

All the analysis here are performed using a commercial ﬁnite element based solver, COM-

SOL Multiphyiscs.

4.1.1 2D Axisymmetric Unobstructed Ureter with Isotropic Wall Properties

The coupling is two-way, with the fluid flow affected by the structural deformation and vice versa. The governing equations for the fluid and structure are solved simultaneously by a single monolithic solver. Hence, the fluid and structure boundary is strongly coupled and implicit in nature. The problem is first solved using a steady state-based Newton-

Raphson iterative method, with three Gaussian pulses on the outer wall of the ureter, 12 cm apart. Using the steady-state solution as the initial condition, an unsteady solution is then obtained, with each Gaussian pulse, moving along the ureter outer wall with a velocity of

2 cm/s. The simulation is carried out for a period of 30 s, with a backward difference time stepping scheme.

4.1.2 2D Axisymmetric Unobstructed Ureter with Anisotropic Wall Properties

Simulations in this case is also done incorporating a two-way fully coupled fluid-structure interaction between the fluid and structural domain. The finite element-based discretization 19 is used, with first order accurate linear elements for the fluid domain, and second order accurate quadratic elements for the structural domain. For the time discretization, a second order accurate backward difference scheme is used. The expansion wave in the problem is generated using a force with a Gaussian distribution, acting on the outer boundary of the ureter wall. The maximum magnitude of the force is 2.55×105 N/m2, with a span of 3 cm.

As a ﬁrst step, the solution is initialized using a steady state simulation with the center of the expansive force at a distance of 3 cm from the ureter inlet. In the next step using this steady simulation, a transient simulation is conducted for 20 s, where the expansion force starts to move towards the outlet at a velocity of 2 cm/s.

4.1.3 2D Axisymmetric Obstructed Ureter with Anisotropic Wall Properties

The ﬂuid and the solid domains are fully coupled in an implicit fashion, using a monolithic approach, like previous cases. So, the interaction between the ﬂuid and structure is two- way, with both the solutions being dependent on each other. The numerical procedure for this case is exactly the same as the previous case.

4.1.4 3D Axisymmetric Obstructed and Unobstructed Ureter with Anisotropic Wall Prop-

erties

All the simulations presented here are performed using a commercial ﬁnite element based solver, COMSOL Multiphysics. The ﬂuid and solid domains are two-way coupled and the

ﬂuid-structure interaction simulation is solved with a segregated solver. For the structural solver, a second-order accurate quadratic element is used, while for the the velocity and pressure, a ﬁrst-order accurate linear element is used. To accelerate convergence, a steady-

20 state solution is obtained without any force acting on the boundary and is later used as an initial condition for the transient solution. For the structural part of the solver, a parallel sparse direct solver MUMPS is used, while the ﬂuid domain is solved using an iterative solver with a generalized minimal residual method (GMRES) and a geometric multi-grid approach with 2 levels of meshes. The time-dependent solution is obtained for a period of

5 s.

In order to generate the peristaltic wave, a Gaussian pulse of pressure force in the outward direction from the center of the ureter is provided on the ureter wall with a maximum value of 2.1×105 N/m2 with a width of 10 mm. The pulse is incorporated 2 cm from the inlet and it starts to move inside the domain towards the outlet with a velocity of

2 cm/s. As the force enters the domain gradually, it helps in attaining a better convergence.

For time discretization, a second order backward scheme is used. The ﬂuid and structural domains are solved using sparse direct solvers, MUMPS and PARDISO, respectively.

4.2 Computational Mesh

4.2.1 2D Axisymmetric Unobstructed Ureter with Isotropic Wall Properties

A 2D mapped mesh is used for both the ﬂuid and solid domains. Three different cases are simulated by increasing the number of grid points until the results in terms of the calculated inlet velocity proﬁle, are grid independent. The mesh count for these cases for solid and

ﬂuid domains are 2316 and 8492, 7200 and 24000, 12800 and 40000, respectively. So the results shown in this paper use a total mesh count of 31200. For the deforming mesh

21 Figure 4.1: Geometry and mesh of the ﬂuid and structural domain. (a) mesh before deformation and (b) mesh after deformation for 2D axisymmetric unobstructed ureter with isotropic wall properties.

smoothing type, a Winslow smoothing technique is used [59]. The maximum deformation in the radial direction is set at 0.95 mm at the middle of the compression wave, while in front of the expansion wave (bolus) it is 0.15 mm. as shown in Fig. 4.1.

4.2.2 2D Axisymmetric Unobstructed Ureter with Anisotropic Wall Properties

For the fluid domain, mostly triangular elements are used for meshing, except near the boundary layer, which has quadrilateral elements. First order accurate linear elements are used for both fluid and pressure field discretization. All the meshes in the structural domain consists of triangular elements and they are quadratic by nature with second order accuracy.

Three different simulations using different number of grid points were conducted to show

22 Figure 4.2: Computational mesh of the ﬂuid and structural domains for 2D axisymmetric unobstructed ureter with anisotropic wall properties.

that results were mesh independent, and the number of elements in those three cases were

45,701, 80,528 and 462,176. Mass ﬂow rate at the inlet and the von Mises stress at a point at the middle of the inner ureter wall is plotted with time in Figure 4.3 for these three different mesh sizes. It can be seen that, all the plots coincide over each other for Figures 4.3a and

4.3b, thus proving the mesh independence of the results.

4.2.3 2D Axisymmetric Obstructed Ureter with Anisotropic Wall Properties

A 2D triangular element dominant mesh is used (Fig. 4.4) for the simulations here. Quadri- lateral elements are present only in ﬂuid domain at the boundary layer near the ﬂuid-solid interface. For the mesh independent study, simulations are carried out for three different meshes. The mesh for the different cases consists of 191349, 80174 and 45659 elements.

Temporal evolution of the mass ﬂow rate is plotted for three different meshes in Fig. 4.5a.

Also, von Mises stress at the ureter wall along the length of the ureter is plotted in Fig. 4.5b

23 at time 3s, when the expansion wave is closest to the stone center. As it can be seen from

Fig. 4.5, for the three different mesh element cases, the lines almost perfectly coincide with each other.

4.2.4 3D Axisymmetric Obstructed and Unobstructed Ureter with Anisotropic Wall Prop-

erties

This section presents the computational mesh for the simulations presented here. Note that grid-dependent studies were carried out for the unobstructed case to ensure that the results are validated and are independent of the mesh that is employed. For the unobstructed ureter, a structured quad map mesh is swept throughout the domains, both ﬂuid and solid.

In the obstructed case, the mesh is divided into 3 different regions. As can be seen from the

Fig. 4.6, the outer layer is built with a structured mesh, which represents the ureter wall. For the fluid domain, there are two layers. The thinner outer layer is meshed with tetrahedral elements arranged in a structured manner, while the inner most part is unstructured and it contains the obstruction as a circular void within the fluid domain. Only the thin outer layer of fluid deforms with the deformation of the structure, while the inner cylinder containing the blockage remains in place. This also aids in retaining a good mesh quality, since only the semi-structured tetrahedral elements are deformed. A 2D axisymmetric case is also presented here, whose results are mesh-independent as well. This case is also presented in order to at least qualitatively validate the more sophisticated 3D model with a more simple one, like the 2D axisymmetric model. The number of grid points associated with the fluid and structural domains for all the four cases are given in Table 4.1.

24 Case Fluid Structural Total

Unobstructed 3D 200,000 180,000 380,000

50% Obstructed 3D 850,000 140,000 990,000

70% Obstructed 3D 920,000 140,000 1,060,000

70% Obstructed 2D 8,670 1,032 9,702

Table 4.1: Number of mesh elements for the different cases presented in this study.

25 (a) Ureter inlet ﬂow rate vs. time for 2D axisymmetric unobstructed ureter with anisotropic wall properties.

(b) von Mises stress along the length of the ureter at time 3s for 2D axisymmetric unobstructed ureter with anisotropic wall properties.

Figure 4.3: von Mises stress at a point 15 cm away from the ureter vs. time for 2D axisymmetric unobstructed ureter with anisotropic wall properties.

26 Figure 4.4: Geometry and mesh of the ﬂuid and structural domain for 2D axisymmetric obstructed ureter with anisotropic wall properties.

27 (a) Flow rate vs time at the ureter inlet for 2D axisymmetric obstructed ureter with anisotropic wall properties.

(b) von Mises stress along the length of the ureter at time 3s for 2D axisymmetric obstructed ureter with anisotropic wall properties.

Figure 4.5: Mesh convergence study for three different element number for 2D axisymmetric obstructed ureter with anisotropic wall properties.

28 Figure 4.6: Geometry and mesh of the ﬂuid and structural domain for 3D axisymmetric obstructed and unobstructed ureter with anisotropic wall properties.

29 CHAPTER V

RESULTS

5.1 2D axisymmetric unobstructed ureter with isotropic wall properties

The objective of this study is to understand the peristalsis in human ureter as well as to investigate and quantify the backﬂow of urine from the ureter to the kidney. Due to the traveling contraction waves, the reaction force on the ureter wall is also an important parameter to analyze. The contraction waves are 6 s apart, thus producing similar characteristics during 6 s intervals. Hence, the time period, T, for analysis, is 6 s. Results are obtained at time instants of 25.5 s, 27 s, 28.5 s and 30 s which corresponds to T/4,T/2,3T/4 and

5.1.1 Flow Rate and Velocity Proﬁle at Inlet

The ﬂow rate at the inlet of the ureter is calculated using the following equation (Eqn. (5.1)).

Z V˙in = − 2πr(n · u)ds (5.1) Sin

The ﬂow rate at four different times is plotted in Fig. 5.1. It is observed that at time

T/4, when the contraction in the ureter wall due to peristalsis, moves from the inlet to the downstream, backflow occurs. The same flow characteristic of urine reflux is also seen in

Fig. 5.3 in the radial velocity proﬁle. In Fig. 5.3, it can be seen that due to the no-slip boundary conditions at the wall, the velocity is zero. The magnitude increases towards 30 Figure 5.1: Urine ﬂow rate at ureter inlet for 2D axisymmetric unobstructed ureter with isotropic wall properties.

the axis attaining maximum at the center, except for T/4, when the magnitude is still at its maximum at this time, but with the ﬂow now reversed in direction. So, there is a tendency for chronic urine reﬂux while the peristalsis wave is near the ureter and renal pelvis junction.

31 Figure 5.2: Contour of von-misses stress for structure and streamline of velocity vector overlapped with velocity vector arrows for ﬂow ﬁeld at time, t/4 = 25.5s for 2D axisymmetric unobstructed ureter with isotropic wall properties.

5.1.2 Pressure Gradient and Wall Shear Stress

The pressure gradient is plotted along the length of the ureter wall in Fig 5.4. Peaks are observed in the pressure gradient corresponding to the time and location of the contraction in the ureter wall. This also means that these peaks shift with the peristaltic wave, along the length of the ureter. These sudden pressure jumps occur in the compression zones due to peristaltic waves, and are responsible for the recirculation and local retrograde ﬂow of the urine. Similarly, peaks are also visible in wall shear stresses plotted in Fig. 5.5. On further observing the behavior of wall shear stress, a comparatively average value is evident at the bolus, which is then followed by abrupt jumps in wall shear stress towards both maximum and minimum direction at the contractile waves.

5.1.3 von Mises Stress and Flow Field

Shown in Fig. 5.2, are the velocity vectors overlapped on streamlines. The streamlines are colored by the velocity magnitude. It can be seen that, at the contraction wave, recirculation 32 occurs in the flow field. This explains the reason for the abrupt jump in the pressure gradient seen in Fig. 5.4. Due to this recirculation zone, urine backflow occurs, as is evident from the direction of arrows in the velocity vector representation. Contour plots for von Mises stresses are also shown in Fig. 5.2. The highest value of these stresses are near the center of the contraction wave, while the value decreases towards the directions of the boluses that follow. It should also be noted that at the bolus, the von Mises stresses are not zero, and they are consistent with the wall shear stress behavior in Fig 5.5.

5.1.4 Discussion and Conclusions

The analysis presented above demonstrated that even with a positive pressure gradient from the inlet to the outlet, reflux in urine can still occur at the ureteropelvic junction, when the peristaltic wave is near. Also, local backflow of urine at the outlet (ureterovesical junction) can occur in this case. However, due to the pressure boundary condition and the flow velocity in front of following peristalsis wave traveling upstream, this backflow moves out of the ureter.

Flow velocity proﬁles reported in this study are consistent with previous investigations [50,51]. Backﬂow at the inlet observed in Fig. 5.3 was similar to Vahidi et al. [51].

Average ﬂow rate was calculated to be 0.223 ml/min, again quite close to the value in the numerical study performed by Vahidi et al. [51], and also within the range of values suggested by a 1989 experimental study [40]. The peaks in pressure gradient observed in

Fig. 5.4, is in accordance with the study of Weinberg [60]. Maximum pressure occurs at the compression zone which is just behind bolus as Weinberg reported. Also, similar pat-

33 terns of pressure gradient and shear stress were found in the numerical study by Vahidi et al. [50].

Furthermore, recirculation and backflow which were shown to occur at the peristaltic wave in Fig. 5.2 were also reported in some studies [50,51]. To our knowledge, this is the first study that incorporates data from the actual human ureter. For instance, some of the previous notable works like that of Vahidi et al. [50] used a linear elastic material in their study. Later the same authors used a non-linear hyper elastic material form uniaxial test [42,51]. In our paper, data is extracted from a biaxial test on human ureter [47], which defines the material behavior more completely then an uniaxial test. To model the data, the material is assumed to be non-linear isotropic hyperelastic. According to the study of

Rassoli et al. [47], material shows some anisotropic characteristics towards circumferential and longitudinal directions. In the future, a material model can be incorporated that includes an anisotropic model.

To study the reﬂux phenomenon in the ureter, in this case a pressure difference from inlet to the outlet of 0.3 Pa is taken so that pressure developed at the kidney and ureter junction is higher then the pressure developed at other end, bladder and ureter junction.

It is observed that retrograde ﬂow of urine happens when the peristaltic wave is near the ureteropelvic junction. In addition, study on a more pressurized bladder during urination or an obstructed ureter due to the formation of ureter stone will further address the issues of urine reﬂux.

34 Figure 5.3: Velocity proﬁle at ureter inlet for 2D axisymmetric unobstructed ureter with isotropic wall properties.

35 Figure 5.4: Pressure gradient along the ureter wall for 2D axisymmetric unobstructed ureter with isotropic wall properties.

36 Figure 5.5: Wall shear stress along the ureter wall for 2D axisymmetric unobstructed ureter with isotropic wall properties.

37 5.2 2D axisymmetric unobstructed ureter with anisotropic wall properties

An expansion wave on the ureter wall traveling from the direction of kidney to bladder at a speed of 2 cm/s transports the urine. Flow and structural (ureter wall) variables are reported at times 1 s, 6.5 s, 12 s and 17 s, which correspond to the instances when the expansion force has just entered into the domain, is 1/3rd of the total distance from the inlet, 2/3rd of the total distance from the inlet and near the outlet, respectively.

5.2.1 Flow Field

Pressure contours in the domain are shown at three different times in Figure 5.6. At the time of 1 s, when the bolus is near inlet, a pressure field is generated in the fluid, due to the expansion wave on the wall. As the bolus moves toward the outlet, it carries with itself the high pressure wave in flow field, which is evident from the figure at time 6.5 s. Finally, as the expansive wave starts to leave the domain at 17 s, the pressure wave in the flow field leaves the domain as well.

Streamlines colored by velocity magnitude and superimposed with velocity vectors shown in Figure 5.7, for the same three time instances as presented earlier (in Fig- ure 5.6). From the streamlines, it is clear that the highest velocity is observed near the expansion wave for all time instances. Also, the velocity vectors are directed towards the outlet near the expansion wave, but elsewhere they are directed backwards, i.e. towards the inlet. From Figure 5.7 and 5.6, it can be summarized that, a high pressure zone is created due to the expansion wave on the ureter wall. As the bolus moves from the inlet to the outlet, it carries the urine with itself. As will be seen in the ﬁgures later, results are also

38 Figure 5.6: Pressure contours of the ﬂow ﬁled for 2D axisymmetric unobstructed ureter with anisotropic wall properties.

shown at an additional time instant of t = 12 s. At both the time instances of 6.5 s and

12 s, the bolus of urine moves along the ureter length from left to the right, with the same shapes. This makes the contour plots shown in Figures 5.6 and 5.7 identical at 6.5 s and 12 s. Hence contours at t = 12 s are omitted here.

5.2.2 Volume Flow Rate and Velocity Proﬁle at Inlet

The flow rate is plotted in Figure 5.8 at 30 different cross-sections, 10 mm apart from each other at 4 different times: 1 s, 6.5 s, 12 s and 17 s. At time 1 s, urine flow rate is highest at inlet. That is because the expansion wave is near the inlet at that time. Peaks in the flow rate follow the expansion wave. Also, at times 6.5 s and 12 s, the peak in the flow rate

39 Figure 5.7: Streamlines colored by velocity magnitude superimposed with velocity vectors of the ﬂow ﬁled for 2D axisymmetric unobstructed ureter with anisotropic wall properties.

40 Figure 5.8: Urine ﬂow rate at various cross-sections along the length of the ureter for four different times for 2D axisymmetric unobstructed ureter with anisotropic wall properties.

is observed in the middle of the expansion wave. This observation is consistent with the streamline plot in Figure 5.7, where the highest value was seen in the middle of the bolus.

The ﬂow rate is positive only near the bolus region, and elsewhere it is negative, which again is consistent with the direction of the velocity vectors presented in Figure 5.7.

Furthermore, the maximum value of the ﬂow rate observed here is around 0.6 ml/min, which is similar to the values in the study by Vahidi et al. [51], and also falls within the range 0.265 ml/min and 2.45 ml/min of volume ﬂow rate as reported by Ohlson [40].

41 Figure 5.9: Velocity proﬁle at inlet or four different times for 2D axisymmetric unobstructed ureter with anisotropic wall properties.

42 The inlet velocity profile is also plotted at the times corresponding to 1 s, 6.5 s, 12 s and 17 s. All the four curves have a zero velocity at the wall, satisfying no-slip velocity boundary condition. The value of velocity magnitude increases or decreases with distance from the ureter wall, with the highest value being accomplished at center of the ureter at time 1 s, when the bolus is near the inlet. At times 6.5 and 12 s, there is a tendency for backflow in inlet. This backflow tendency corresponds to what was observed in the streamlines and velocity vector plots shown in Figure 5.7. A back-flow tendency was also seen in the inlet velocity profiles shown in Vahidi et al.’s investigation [51].

5.2.3 Pressure Gradient and Wall Shear Stress

The pressure gradient is plotted along the ureter wall length in Figure 5.10. Again, peaks in the pressure gradient follow the expansion wave, and it moves from the inlet to the outlet as expansion wave moves. The expansion bolus generates a high pressure zone near it, as was shown in Figure 5.6, which explains the peak values of pressure gradient on ureter wall near the expansion bolus. A similar pattern is observed in the shear stress behavior as demonstrated in Figure 5.11. At a time of 1 s, the highest value of shear stress is near the inlet. As the expansion wave moves to the right, these peaks in the wall shear stress also move with it, as seen for the times 6.5, 12.5 and 17 s.

5.2.4 von Mises Stress and Displacement

A second simulation is also carried out here with a linear material properties of the ureter wall. The von Mises stresses on the ureter wall for both the linear and hyperelastic material are plotted in Fig. 5.12 at 4 s. The material properties for the ureter are chosen such that

43 Figure 5.10: Pressure gradient along the ureter wall for four different times for 2D axisymmetric unobstructed ureter with anisotropic wall properties.

44 Figure 5.11: Wall shear stress along the ureter wall or four different times for 2D axisymmetric unobstructed ureter with anisotropic wall properties.

45 the elastic modulus is 0.5 MPa [50] and the Poisson’s ratio for the nearly incompressible material is 0.49. The force is manipulated in such a way that, for both the materials, an identical displacement is determined and employed in the simulations here. The highest value of displacement in the radial direction is 1 mm. So, for equal displacement, the highest value of force for the linear material is 0.0325 MPa, whereas for the hyperelastic material, it is 0.255 MPa. From the stress distribution presented here on the ureter wall, it can be seen that the value is much lower for the linear material than the hyperelastic material. This characteristic of higher strength in the circumferential direction corresponds with Rassoli et al.’s study [47], as they used a fourth strain invariant I4 to model this anisotropic tendency between the longitudinal and circumferential directions.

Displacement and von Mises stress in the ureter wall for the hyperelastic material is plotted at 6.5 s in Figure 5.13. The expansion wave is created using an external force with a Gaussian distribution. The same bell shaped behavior is observed in the contours of displacement on the ureter wall with a maximum displacement of 1 mm at the middle of the wave. Also, the von Mises stress is highest in the middle of expansion wave. This highest value of von Mises stress corresponds with the location where the highest value of wall shear stress is observed in the ﬂuid, i.e. in the middle of expansive wave in Figure 5.11.

The values of the von Misses stresses are within the range of stress-strain curves from the experiment of Rassoli et al. [47]. Also note that, the maximum value of the deformation in the radial direction is 1 mm and the length of the bolus along the ureter wall is 6 cm, which correspond with the suggestions made by Boyarsky et al. [54] for ureteral peristalsis dimensions for bio-engineering modeling.

46 Figure 5.12: von Mises Stress plot in the ureter wall both for linear and hyperelastic material.

47 Figure 5.13: von Mises stress (top) and displacement (bottom) along the ureter wall at time, t = 6.5 s for 2D axisymmetric unobstructed ureter with anisotropic wall properties.

5.2.5 Discussion and Conclusion

To understand and quantify ureter peristalsis, different flow and structural variables are analyzed in the current study. COMSOL FSI simulations of ureter peristalsis are conducted here where an external force is employed for the deformation of the ureteral wall and the response of the fluid is captured through solutions of the fluid dynamics equations. In addition, the fluid solution is coupled to the structure response, thus leading to a fully coupled multi-physics model.

A mesh convergence study was presented as well, in order to validate the model and the results. One of the main goals of this study was to understand the backflow phenomena of the ureter peristalsis. For this purpose, flow rate along the ureter length and inlet velocity profiles were analyzed, both of which showed a tendency for backflow. The urine was primarily carried by the isolated bolus. As the bolus of the urine movesd forward, a

48 back-flow was created behind it, which was also a feature of the velocity vectors in the domain. In the inlet velocity profile, the back flow tendency was observed in all the time instances, except when the bolus was near the inlet or outlet. The reason for the retrograde

flow was explained through the pressure contours, which showed that, as the urine bolus generated a high pressure zone in the fluid domain and moved forward, it pushed the urine from both sides away from it, thus creating a trail of backflow behind it. A backflow of urine was also created in front of the bolus, but the upstream urine flow blocked this, and eventually urine was transported through the outlet as the bolus moved towards it.

49 5.3 2D axisymmetric obstructed ureter with anisotropic wall properties

As mentioned earlier, an in-vivo experimental study such as Rose et al. [28] in an obstructed human ureter is extremely difficult to perform and to our knowledge, there have been no such investigations in the past. The only numerical study available was carried out by Najafi et al. [13,35], which was a comparative study with and without obstructed ureter cases. The study only considered the fluid domain and used a sinusoidal function to define the peristalsis. The current study differs from [13, 35], with a fully coupled two-way FSI employed to take into account the structural deformations.

5.3.1 Volume Flow Rate

The volume ﬂow rate has been calculated on 30 different cross-sections along the length of the ureter using the following equation:

Z V˙in = − 2πr(n · u)ds (5.2) Sin

In Fig. 5.14, 6 different times are chosen to display the flow rates. When the bolus is at the inlet, the flow rate at the inlet is also high at that time. As the bolus moves along the length of the ureter, the volume flow rate also increases. At the time instances of 2 s and 3 s, the bolus is near the stone. At these times, the flow rate curve is ’M-shaped’ near the bolus.

As the bolus moves away from the stone, the shape of the curve changes into a bell-shaped pattern, following the deformation of the bolus. Also, the peak value increases as the bolus moves away from the stone. Besides, from the ﬁgure it is also clear that, the peaks in the

ﬂow rate are lower at t = 1 s, when the peristaltic expansion wave is not fully within the

50 Figure 5.14: Volume ﬂow rate along the ureter length at 6 different time instants for 2D axisymmetric obstructed ureter with anisotropic wall properties.

domain yet, and at 2 s and 3 s, when the wave is near the stone, compared to t = 6 s and t = 8.8 s. These lower peak values in the flow rate at 2 s and 3 s are an indication of the obstruction created by the stone in the flow field.

5.3.2 Velocity Proﬁle at Inlet

Fig. 5.15 shows the radial velocity at the inlet at four different times, t = 1 s, 3 s, 8.8 s and

17 s. It is clear from the ﬁgure that the velocity magnitude at the inlet is highest at time

51 Figure 5.15: Velocity proﬁle at ureter inlet for 2D axisymmetric obstructed ureter with anisotropic wall properties.

t = 1 s. This is because at this time the bolus is near the inlet carrying the urine flow. For all the times, at the ureter wall, the velocity is 0, thereby satisfying the no-slip boundary condition. When the bolus is near the stone, i.e. at time t = 3 s, there is a tendency for back-flow, as indicated by the negative values of velocity magnitude, and near 2 mm from the ureter wall, the flow is again in the positive direction. This tendency of back-flow near the wall at t = 3 s is due to the presence of the obstruction. At t = 8.8 s, the bolus is in the

52 Figure 5.16: Percentage of cross-sectional area blockage with time at the the stone center for 2D axisymmetric obstructed ureter with anisotropic wall properties.

middle of the ureter, and there is almost no ﬂow at the ureter inlet. Compared to 8.8 s, the

ﬂow has a slightly higher magnitude at time 17s, when the bolus is exiting the ureter, but still lower than the 1s and 3 s instances.

5.3.3 Blockage

The circular stone considered in this study, with a diameter of 3.6 mm, and located at

3 cm from the inlet, creates a blockage of 81% with respect to the cross-sectional area.

53 In Fig. 5.16, the blockage in terms of the cross-sectional area due to the stone is plotted against time. As the bolus moves from left to right, the percentage of blockage area changes accordingly following the deformation pattern in the ureter wall. This blockage is lowest at time t = 3 s, at which the bolus is near the center of the stone. After 5 s, the blockage maintains a steady value, and reduces after 17 s, when the bolus is near the outlet and starts to leave the domain. Because of the expansion force wave and due to the material property of the ureter wall, there is a slight compression far away from the bolus. For that reason, the maximum blockage is around 82%, which drops down to 81% as the expansion wave starts to leave the domain. So, the expansion bolus on the ureter wall due to the material properties, creates a slight compression zone ahead of it. As discussed earlier, while the opposite, a compression zone fails to create this expansion bolus in the ureter [50, 53, 61].

Pressure Gradient and Wall Shear Stress

The pressure gradient magnitude and wall shear stress are plotted in Fig. 5.17 and Fig. 5.18, respectively. Both of the curves show similar patterns, with their highest values always being near the stone. With regard to the timing, the highest values in these curves occur at different times. So in the case of pressure gradient, the maximum occurs near the stone at t = 1 s, whereas for the wall shear stress it occurs at t = 3 s. As the bolus moves further away from the stone, these peaks subside. Later, for both the curves, small increases can be observed at the center and near the outlet of the ureter, at t = 8.8 s and t = 17 s, respectively.

Furthermore, as mentioned above, in both the pressure gradient and wall shear stress plots (in Figs. 5.17 and 5.18), there is sharp peak near the center of the stone, and

54 Figure 5.17: Pressure gradient along ureter wall for 2D axisymmetric obstructed ureter with anisotropic wall properties.

secondary peaks are near the center of the peristaltic wave. These aspects are similar to what were observed by Nafaji et al. [13, 35] in their study. This kind of sharp jump in pressure gradient and wall shear stress near the obstruction in peristaltic transport is common in other biomechanical activities. For instance, a jump in wall shear stress in an arterial wall near a blood clot was reported by Vahidi et al. [61].

55 Figure 5.18: Wall shear stress along ureter wall for 2D axisymmetric obstructed ureter with anisotropic wall properties.

5.3.4 von Mises Stress and Flow Field von Mises stresses on the solid structure along with velocity streamlines overlapped with velocity vectors are plotted for 5 different times in Fig. 5.19. The times include t = 1 s and t = 2 s, when the peristaltic wave is near the inlet, at t = 3 s, when the wave is near the center of the stone, at t = 8.8 s when the wave at the middle of the ureter, and ﬁnally at t = 17 s, when the wave is at the outlet of the ureter. At the time instances of 1 s and 2 s, the velocity

56 vectors before the stone demonstrated that the flow velocities are forward-directed, while after the stone, the vectors are backwards-directed. When the wave is near the middle of the stone at 3s, all velocity vectors are forward directed, except some backflow occurring near the inlet and also further downstream of the wave. At t = 8.8 s, there is no local backflow observed near the peristalsis wave or anywhere else in the ureter. On the other hand, when the wave reaches near the outlet of the ureter at t = 17 s, except near the outlet, velocity vectors show backflow everywhere. Velocity streamlines indicate a vortex being created near the center of the peristalsis wave for all time instances. The highest value of velocity magnitude is also near the center of the peristalsis wave. By observing the von

Mises stresses contours, the values are highest at times t = 3 s and t = 8.8 s, while the wave is fully within the domain, and the highest value of the stress is at the center of peristalsis wave center.

5.3.5 Discussion and Conclusion

In this paper, for the first time, an investigation of an obstructed human ureter that incorporates a two-way fluid-structure interaction between ureter wall and fluid domain, is presented. In the simulations carried out as part of this effort, a 2D axisymmetric model is chosen, and the material considered is non-linear anisotropic and hyperelastic. The viscoelasticity of the material is not considered, which neglects the effect of stress-relaxation, a common phenomenon in biomaterials. To quantify the effect of the ureter obstruction, a change in the different field variables like flow field, velocity magnitude, von Mises stresses, pressure gradient and wall shear stress at the wall is investigated.

57 The average volumetric flow rate for one isolated bolus found in the current study is 1.98 mil/min. This is consistent with previous studies [38, 40]. According to the study performed by Griffiths [38], the maximum flow rate in steady peristalsis should be in the order of 2 ml/min. Also, Ohlson [40] carried out flow rate studies on a human ureter and reported that the volumetric flow rate due to isolated bolus should be within the range of

0.265 mil/min to 2.45 mil/min. In addition, mesh convergence results presented in Fig. 4.5 validated the formulation as well. Due to the obstruction near the inlet in the ureter, a back-ﬂow of urine was observed when the peristaltic wave was near the obstruction at 3 s.

This was evident both in the ﬂow rates and in the inlet velocity proﬁle, (See Fig. 5.14 and

Fig. 5.15). This phenomenon is undesirable [7,8]. Back flow of urine from the bladder can damage the kidney, and result in something known as reflux nephropathy [62]. Long lasting vesicoureteral reflux or the back flow of urine from the bladder to the kidney can cause renal scarring and urinary tract infection [63]. High peaks in the pressure gradient and wall shear stress value were found near the stone (See Fig. 5.17 and Fig. 5.18). High peaks in the pressure gradient create a pressure zone near the obstruction. This high pressure zone due to the obstruction in the flow path is responsible for the back flow in the ureter. Peaks in the wall shear stress values show the effect of the obstruction on the ureter wall due to flow domain. Hence, it can be concluded that these increases in the values of pressure gradient and wall shear stress help to quantify the effect of obstruction in a ureter.

One of the more important parameters related to the structural behavior is the von

Mises stress. Material fails as the von Mises stress exceeds the yield strength. In order to assess this feature, the von Mises stresses are plotted on the ureter wall in Fig. 5.19.

58 The highest value is observed at the middle of the peristaltic wave. Also, high peaks in the values of wall shear stress and pressure gradient are observed near the obstruction in

Fig. 5.17 and Fig. 5.18. These high values of wall shear stress and pressure gradient have very little effect on the values of the von Mises stress. So, a rupture due the presence of obstruction is very much unlikely or rare, which is consistent with the ﬁndings of Liu et al. [14]. They found that a spontaneous rupture due obstruction such as renal calculi is a very rare urological disorder and reported only two such cases. It can be indeed be conclusded here that with the kind of obstruction size used in this study, the value of the von Mieses stresses indicate that such a type of rupture impossible.

The pain felt by patient is a major parameter for diagnostics and the treatment procedure of renal calculi. A case study by Drake et al. [27] showed that, as the kidney was obstructed by the ureteric stone, the ureter become distended, and with the peristaltic wave superimposed on this distended ureter periodically, the severity of the pain felt by the patient also followed the same periodicity. The analysis in the study showed that the deformation in the ureter wall combined with the obstruction in the flow creates these high peaks in the wall shear stress and pressure gradient distribution, which in turn can correspond to the pain felt according to the case study of Drake et al.. So indeed these two parameter scan be used to quantify the severity of blockage. As the material is pre-stressed, and is not a viscoelastic material, a direct correlation of the peaks in the fluid pressure field with the von Mises stresses are difficult to quantify on the ureter wall. However, from

Fig. 5.19, it can still be observed that wall shear stress is the highest at the center of the peristaltic wave. So, the affect of obstruction on ureter wall will be highest at the time when

59 the peristaltic wave passes over the obstruction, thus possibly leading to the patient feeling pain [27]. In the future, an aspect of a pain index can further be explored, where the ureter wall is modelled with stress relaxation properties, and the affect of the wall shear stress and pressure jumps can be correlated with the von Mises stress values. It should be noted that a FSI study between the ureter wall and fluid domain is crucial, because only such an approach unlike a traditional fluid calculation, can numerically simulate the coupling between the obstructed flow and the peristaltic ureter wall.

Furthermore, the analysis presented here introduces for the ﬁrst time, a high-

ﬁdelity approach such as FSI to evaluate the peristaltic mechanism in the obstructed ureter, which could further be extended to 3D modeling and eventually investigate the effect of various shapes of obstruction on ureter ﬂow. Such studies have the potential to improve our understanding of the effect of the kidney stones on the ureteral wall, thereby improving the treatment method of the same. As an example for kidney stone removal methods, ureteroscopy is a minimally invasive and effective procedure. However, even this technique can lead to complications in basketing and also injury to the ureter in some cases [64, 65].

The current authors are involved in an effort towards developing an intelligent kidney stone extraction basket, which measure excessive forces on the ureter while extracting stones using a force sensor [66–68]. The device records safe and unsafe zones using experimental values of force during perforation and avulsion. The major limitation though for such a device is the fact that the safe and unsafe zones are based on experimental work with spherical stones in pig ureters. A high-ﬁdelity FSI model such as the one developed here for obstructed ureters could be eventually used for investigating the forces on the ureter for

60 stones of various shapes and sizes. The knowledge gained through such analyses can then be used to create dynamic scales for the basket, leading to a much more accurate description of the safe and unsafe zones recorded on the basket. In this paper, for the first time, an investigation of an obstructed human ureter that incorporates a two-way fluid-structure interaction between ureter wall and fluid domain, is presented. In the simulations carried out as part of this effort, a 2D axisymmetric model is chosen, and the material considered is non-linear anisotropic and hyperelastic. Viscoelasticity of the material is not considered, which neglects the effect of stress-relaxation, a common phenomenon in biomaterials. To quantify the effect of the ureter obstruction, a change in the different field variables like

flow field, velocity magnitude, von Mises Stress, pressure gradient and wall shear stress at the wall is investigated. Due to the obstruction in the flow field, a tendency of urine back-flow is observed both in the volume flow filed and velocity profile at the inlet. This retrograde flow of urine is harmful [7, 8].

61 Figure 5.19: Contour of von mises stress for the structure, along with streamlines overlapped with velocity vectors for the ﬂuid ﬂow for 2D axisymmetric obstructed ureter with anisotropic wall properties.

62 5.4 3D axisymmetric obstructed and unobstructed ureter with anisotropic wall properties

FSI-based simulations of obstructed and unobstructed ureters are presented here. Two different obstructed cases are analyzed, one with a 50% and the other with a 70% obstruction.

The obstruction is considered to a be a smooth sphere with its center aligned with the axis of the ureter tube. The peristaltic wave is a Gaussian pressure distribution that propagates through the ureter wall at a velocity of 2 cm/s velocity, with its center initially at 2 cm upstream of the inlet (outside the computational domain) and headed towards the outlet.

Four time instances have been chosen to assess the effect of the level of obstruction on the

flow and structural characteristics. They include t = 2.25 s, when the wave is slightly in front of the stone, t = 2.75 s, when the wave center coincides with the center of the circular obstruction, t = 3.25 s, when the wave is slightly behind the obstruction and finally t = 3.4 s, when the wave is quite further away from the obstruction. The results are divided into three separate sections. The analysis begins with comparisons of some over specific results related to the effects of different levels of obstruction. Later, the effect of a 70% obstruction on the flow dynamics is investigated in detail. Finally, 2D axisymmetric simulations are compared to 3D simulations in order to assess the axisymmetric nature of the flow in this problem.

5.4.1 Comparisons between 50%, 70% and no obstruction cases

Figure 5.20 shows the mass flow rate at the four different time instances for the 0% (unobstructed), 50% and 70% blockage cases. Firstly from these figures it can be seen that as expected, the unobstructed case has the highest mass flow rate compared to the obstructed

63 Figure 5.20: Inlet mass ﬂow rate at four different time instances, t = 2.25 s, t = 2.75 s, t = 3.25 s, and t = 3.4 s, for the Unobstructed and 50% and 70% obstructions.

ones, and larger the obstruction, larger the mass flow rate. Secondly, the unobstructed case displays almost no backflow whereas the other two cases do show backflow and in fact the backflow increase with increase in obstruction.

Wall shear stress and pressure gradient are plotted along the length of the ureter for the 3 cases in Figures 5.21 and 5.22, respectively. For the unobstructed case, as the wave moves forward, the pressure gradient does not change signiﬁcantly with time. However

64 Figure 5.21: Pressure gradient along the length of the ureter at four different time instances, t = 2.25 s, t = 2.75 s, t = 3.25 s, and t = 3.4 s, for the Unobstructed and 50% and 70% obstructions..

with the obstruction present in the ureter, a peak appears at the location of the obstruction, showing an increase in the pressure gradient. In addition, the peak value is higher with an increasing level of obstruction.

A similar behavior is observed with wall shear stress as well. The distribution of wall shear stress is quite different for the obstructed cases in comparison to the unobstructed case. Again, there is a sudden rise in wall shear stress values near the obstruction and like the pressure gradient, they are higher with higher obstructions. Another interesting feature that is evident in the wall shear stress is the fact that values ”post-stone” are also higher for majority of the time instances relative to the unobstructed case. These high values in the

65 Figure 5.22: Wall shear stress along the length of ureter at four different time instances, t = 2.25 s, t = 2.75 s, t = 3.25 s, and t = 3.4 s, for the Unobstructed and 50% and 70% obstructions..

shear stress and the pressure gradient might be an indication of the pain felt by a patient in the case of renal calculi present in the ureter or in some rare cases in damage of the ureter wall and urine leaking [14].

5.4.2 70% obstructed ureter ﬂow

This section presents detailed results of the maximum obstruction case, i.e. 70%. The objective here is to present the effects of extreme obstruction on the ﬂow and structural dynamics. Qualitatively, the 50% obstruction case is not signiﬁcantly different from the

70% case, except for the fact that the effects of the obstruction are felt to a lesser extent in the former. Figure 5.23 shows the velocity vectors in the ﬂuid domain colored by velocity

66 Figure 5.23: Velocity vector of urine coloured with velocity magnitude and ureter wall deformation for 70% blockage .

magnitude at four different time instances, t = 2.25 s, t = 2.75 s, t = 3.25 s, and t = 3.4 s,. Also shown in the figure is the ureter wall colored by the deformation at that specific instant in time. At t = 2.25 s, the peristaltic wave is just in front of the circular obstruction and the bolus created due to the peristaltic force, which acts outward from the center of the ureter, carries the urine with itself. This can be observed from the direction of the velocity vectors in the figure. At t = 2.75 s, the wave is in the middle of the ureter tube,

67 Figure 5.24: Pressure and von Mises stress contour in the ureter lengthwise cross-section for 70% blockage.

which means it is right on top of the obstruction, and clearly the flow velocity vectors are impacted because of this. A recirculation zone is formed and further downstream, backflow starts to occur. High-velocity fluid goes around the stone and collides with the back flow coming from behind the obstruction. This backflow propagates towards the front the stone as seen during the time instances, t = 3.25 s and t − 3.4 s, whereas behind the stone, the bolus generated by the peristaltic wave continues to carry the urine forward.

68 Figure 5.24 shows the pressure contours along an axial cross-section in the middle of the domain and also the ureter wall with the von Mises stress contours displayed as well. At time t = 2.25 s, the highest value of von Mises stresses is observed at the middle of the peristaltic wave and the highest value of the pressure is seen in the center of the bolus as well. This merely implies that as the peristaltic wave moves forward, it creates a pressure gradient which drives the ﬂow forward. This is the case for the three time instances, t =2.25, 3.25 and 3.4 s, even when there is backﬂow such as at times, t =3.25 and

3.4 s. At t = 2.75 s, the peristaltic pulse’s center is at the center of the circular obstruction.

However, from the contours in the ﬁgure, it can be seen that the maximum pressure is slightly in front of the obstruction. This is a result of the recirculation zone formed at this time in front of the stone, which then results in backﬂow at the later times (t =3.25 and 3.4 s) as seen in Figures 5.23 and 5.20.

Pressure and von Mises stress contours are also plotted at fours different time instances on a circumferential cross-section located on the center of the circular obstruction in Figure 5.25. Of all the times displayed in this figure, the pressure is the highest at time t = 2.25 s. At time, t = 2.75 s, when the center of the expansive forces is aligned with the plane shown, the highest value in the von Mises stress is observed. Also, a gradient is observed in the pressure contours of the flow field with the higher values on the outer side near the ureter inner wall and lower values near the stone. In the von Mises stress contours, the gradient is oriented in the opposite direction, where the higher value is near the ureter inner wall; as if the high pressure values in the fluid are being translated to the structural domain in the form of von Mises stresses. As the force leaves the plane location and goes

69 Figure 5.25: Pressure and von Mises stress contour at circumferential cross-section in the middle of the 70% circular blockage.

behind the obstruction, both the pressure and von Mises stress values reduce in the next time instances.

Wall velocity vectors are plotted on the material frame at the inner surface of the ureter wall in Figure 5.26. It shows how the structural velocity or the rate of change of displacement interacts with the moving peristaltic wave. The highest value in the wall velocity is observed in the middle of the peristaltic wave and the direction of the velocity

70 Figure 5.26: Wall velocity in material frame in ureter wall.

vectors is at the center. Just in front of the wave, velocity vectors are directed forward, while behind the wave this velocity vectors are in backward direction, although the magnitude is lower. The bolus created by the peristaltic wave mainly carries the urine forward, but it does create a small amount of backﬂow in its trailing zone as was even observed for the unobstructed case in Figure 5.20. This characteristic of the the peristaltic motion observed here mainly originates from the structural displacement rate. As the ﬂuid velocity and structural displacement are coupled together, this picture explains how the peristaltic wave

71 Figure 5.27: Comparison of the values of wall shear stress and pressure gradient between the full 3D and 2D axisymmetric cases when the peristaltic wave is front of the obstruction in the 70 % obstruction case.

in the structure transfers its velocity in the ﬂuid domain and carries the urine forward.

Inward facing higher values of wall velocity in the material frame mainly generates the higher ﬂuid velocity and high pressure zones at the bolus center.

5.4.3 Comparison between 2D Axisymmetric and Full 3D model

In order to assess the axisymmetric nature of the ﬂow, comparisons are made between 3D simulations presented earlier and new 2D axisymmetric calculations of the 70% obstructed case. This also serves as a justiﬁcation for using simpler and cheaper 2D axisymmetric models in the future, provided the obtstruction itself is symmetric. Wall shear stress and pressure gradient along the length of the ureter are plotted at time t = 2.75 s in Figure 5.27,

72 for both the 2D axisymmetric and full 3D cases. Analysis of the full 3D case revealed that one of the most important parameters that quantiﬁed the effect of obstruction was the wall shear stress and pressure gradient. From the results, it can be seen that both the curves almost coincide with each other, with very minor deviations. Since the full 3D case can be computationally expensive, it can be argued from these comparisons that a cheaper 2D axisymmetric computation can shed sufﬁcient light in terms of quantifying the obstruction in ureter at least for symmetric obstructions.

5.5 Conclusions

The commercial solver Comsol Multiphysics is used to simulate 3D obstructed and unobstructed ureters using an FSI approach. To our knowledge this is the ﬁrst time, where 3D

FSI has been applied between a ureter wall and the flow of urine for comparative numerical studies in obstructed and unobstructed ureters. Previously, although Najafi et al. [13] did carry out a detailed comparative study on ureters with and without obstructions, they only considered the fluid domain. In this paper, the flow is primarily driven due to the deformation in the structural domain. As seen from Figure 5.26, the rate of deformation in the ureter wall, which is an anisotropic hyperelastic material, is transferred to the fluid domain due to the FSI coupling, and generates a complex flow pattern in the bolus, which could not be generated if only the fluid domain were considered or the ureter wall was modeled as a rigid surface. Also, the values of von Mises stress plotted in Figures 5.24 and 5.25 showed that the maximum value of the von Mises stresses was found to be around 2.5 MPa, which is very well below the maximum stress of 4.1, Pa, a human ureter can withstand before 73 failure as reported by Shilo et al. [69]. For improving convergence, the initial bulk modulus was chosen to be very high, thereby modeling it as a nearly incompressible material, a common practice for these kinds of non-linear FSI simulations [48, 56, 70].

High shear stresses and pressure gradient values were observed near the location of the stone. Previous studies also demonstrated the same kind of behavior [13,35,61]. There was a slight increase in the calculated volume ﬂow rate compared to 2D axisymmetric studies conducted previously [50, 51], including the previous works of the current authors

[53], though the average ﬂow rate was within the range. The mass ﬂow rate reported in

Figure 5.20, shows that backflow tendencies and mass flow rate values at inlet were similar to the work of Najafi et al. [13].

Due to the presence of the obstruction, there is a peak in the wall shear stress and pressure gradient values, which can potentially quantify the effect of renal calculi present in the ureter. Varying the size of the obstruction has varying effects on the ureter wall, and creates varying amounts of backflow. Even in the unobstructed case, there is a very low amount of back flow, but this is typically suppressed by the ureteropelvic junction or junction between kidney and ureter. However, it can be concerning that the larger the size of the obstruction, higher the amount of backflow. This study can potentially contribute towards the treatment of kidney stones in the ureter flow passage by helping physicians to determine the treatment procedure. The latter can be based upon specific quantifiable parameters, such as the size of the stone and its effects on the ureter wall and backflow produced.

74 CHAPTER VI

SUMMARY

In the ﬁrst case, numerical simulations of unobstructed ureter ﬂows are presented here.

A transient 2D axisymmetric calculation of ureteral wall peristalsis and urine flow is performed with a fully, two-way coupled monolithic solver using an ALE method, as part of an FSI approach. A 2D mapped finite element mesh is used for both the fluid and structural domain. Displacement due to peristalsis on ureteral wall is achieved using compressive forces with a Gaussian variation over the ureter length, at a certain wavelength based on previous measurements. Several quantities such as flow rates, pressure gradients, wall shear stresses, velocity vectors and von Mises stresses are analyzed in order to understand better the flow phenomena, in particular the reflux and retrograde flow that could potentially occur in the ureter in the presence of a wall peristalsis. This study is a first step towards understanding an obstructed ureter flow which can result in various complications in such a biological system.

The next study models the ureter as an anisotropic material using experimental data found from the previous literature. Also, the peristalsis is modeled with an expansive force creating a bolus of urine that moves forward from the inlet to the outlet of the ureter.

Here, the ﬂow of urine only occurs due to the peristalsis as the inlet and outlet pressures are kept the same. Also, no valve mechanism exists, and both the ends on the kidney and the

75 bladder sides are kept open. In summary, peristalsis mechanism shows backﬂow of urine, if no pressure boundary conditions are considered. Therefore, an understanding of the valve mechanisms in both the ureteropelvic (ureter-kidney) and ureterovesical (ureter-bladder) junctions are important, because a failure in these can cause urine reﬂux.

In the next case study, Ureter blockage is studied placing a circular object in the

flow path of the ureter. Pressure gradient and wall shear stress values are very large near the blockage, compared to which values in unobstructed ureter are negligible. Also, the stone creates a back-flow at the inlet of the ureter, which is unexpected. High peaks in the pressure gradient and wall shear stress value are found near the stone. This increase in the value of pressure gradient and wall shear stress helps to quantify the effect of obstruction in a ureter. The type of study presented here can help understand the treatment procedure for a patient with an obstructed ureter, better relating and quantifying the pain felt with the change in flow variables.

Finally, a 3D portion of ureter is simulated for both obstructed and unobstructed cases. Due to the obstruction, there is a significant increase in back-flow. Also, wall shear stress and pressure gradient values are higher at the location of the blockage, where in an unobstructed case this values are negligible. As the size of the blockage increase, this detrimental parameters’ values also keep increasing. Wall velocity in the ureter is plotted in the material co-ordinate, which also shows how the material properties change with the direction of the velocity vectors. These vectors in the ureter wall drive the urine flow inside the ureter interacting with one another. So, it can be concluded that not only an FSI, but also a full 3D numerical study is required to fully capture this complex mechanism.

76 This is the first time a numerical study has been carried out using FSI on a ureter wall and urine in an obstructed ureter. Flow field variables related some of the ureter related complicacy such as back-flow and damage to the ureter wall is analyzed. Further study in this field can be done to improve our understanding on a blocked ureter where stone position is not fixed but depends on urine flow. Furthermore, experimental and theoretical studies can be carried out to better model the ureter wall in order to capture it’s compressibility effect.

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