Biaxial Mechanical Behavior of Swine Pelvic Floor Ligaments: Experiments and Modeling

Biaxial Mechanical Behavior of Swine Pelvic Floor Ligaments: Experiments and Modeling Winston Reynolds Becker Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Master of Science in Engineering Mechanics Raffaella De Vita, Chair Sunghwan Jung John J. Socha May 6, 2014 Blacksburg, Virginia Keywords: Cardinal Ligament, Uterosacral Ligament, Pelvic Floor Disorders, Viscoelasticity, Stress Relaxation

Biaxial Mechanical Behavior of Swine Pelvic Floor Ligaments: Experiments and Modeling Winston Reynolds Becker ABSTRACT Although mechanical alterations to pelvic floor ligaments, such as the cardinal and uterosacral ligaments, are one contributing factor to the development and progression of pelvic floor disorders, very little research has examined their mechanical properties. In this study, the first biaxial elastic and viscoelastic tests were performed on uterosacral and cardinal ligament complexes harvested from adult female swine. Biaxial elastic testing revealed that the ligaments undergo large strains and are anisotropic. The direction normal to the upper vagina was typically stiffer than the transverse direction. Stress relaxation tests showed that the relaxation was the same in both directions, and that more relaxation occurred when the tissue was stretched to lower initial strains. In order to describe the experimental findings, a threedimensional constitutive model based on the Pipkin-Rogers integral series was formulated and the parameters of such model were determined by fitting the model to the experimental data. In formulating the model, it was assumed that the tissues consist of a ground substance with two embedded families of fibers oriented in two directions and that the ligaments are incompressible. The model accounts for finite strains, anisotropy, and strain-dependent stress relaxation behavior. This study provides information about the mechanical behav-

ior of female pelvic floor ligaments, which should be considered in the development of new treatment methods for pelvic floor disorders. iii

Acknowledgments I would like to thank my advisor, Dr. Raffaella De Vita, for her guidance and mentorship during my undergraduate and graduate studies. I am grateful to Dr. Jung and Dr. Socha for serving on my committee and providing feedback on my work. I would also like to thank the other members of the Mechanics of Soft Biological Systems Laboratory, Matt Webster and Ting Tan, for all of their help over the past few years. This material is based upon work supported by the National Science Foundation under CAREER Grant No. 1150397 and the National Science Foundation Graduate Research Fellowship Program. iv

Contents 1 Introduction 1 2 Methods 6 2.1 Sample Preparation................................ 6 2.2 Biaxial Testing.................................. 7 2.3 Biaxial Test Protocol............................... 8 3 Theoretical Formulation 11 3.1 Constitutive Model................................ 11 3.2 Biaxial elastic deformation............................ 15 3.3 Biaxial stress relaxation response........................ 16 3.4 Identification of Model Parameters....................... 16 v

4 Results 18 4.1 Biaxial Elastic Properties............................ 18 4.2 Biaxial Viscoelastic Properties.......................... 19 4.3 Modeling...................................... 21 5 Discussion 27 5.1 Experimental Results............................... 27 5.2 Modeling...................................... 31 6 Conclusions 34 6.1 Summary and Future Directions......................... 34 Bibliography 36 vi

List of Figures 2.1 a) Bath of Instron planar biaxial tensile testing system containing load cells, actuator arms, and specimen. b) Orientation at which samples were cut from swine uterosacral cardinal ligament complexes.................. 9 2.2 Protocol used during biaxial mechanical testing of pelvic floor ligaments. The displacement was controlled and was always the same in both directions... 10 4.1 Stress-stretch curves and normalized relaxation curves of 9 samples taken from a single sow. The locations the specimens were taken from are indicated by the schematic in the top left of the top two graphs. The stress relaxation curves represent the same samples presented in the stress-stretch curves. In both the relaxation curves and the stress-stretch curves, the filled symbols represent data collected from stretching in the direction normal to the upper vagina/cervix while the unfilled symbols denote data collected from stretching in the direction transverse to the upper vagina/cervix.............. 20 vii

4.2 Stress relaxation response for two specimens. Three tests were performed on each specimen with a 1 hour resting period between each test. a) The specimen was stretch to λ 1 = 1.046 and λ 2 = 1.071 during the first relaxation test, λ 1 = 1.074 and λ 2 = 1.101 during the second relaxation test, and λ 1 = 1.095 and λ 2 = 1.140 during the third relaxation test. b) The specimen was stretch to λ 1 = 1.178 and λ 2 = 1.101 during the first relaxation test, λ 1 = 1.148 and λ 2 = 1.070 during the second relaxation test, and λ 1 = 1.110 and λ 2 = 1.045 during the third relaxation test.................... 22 4.3 Continuum model fit to experimental data of 13 specimens taken from 3 sows. Filled symbols represent data collected from stretching in the direction normal to the upper vagina/cervix, the unfilled symbols denote data collected from stretching in the direction transverse to the upper vagina/cervix, and the solid lines indicate the constitutive model....................... 23 4.4 Dependency of α 1 (Left) and α 2 (Right), which are measures of the percent relaxation of the specimens, on the first invariant. The lines represent the functions α 1 (I 1 (τ)) = a 1 e a 2(I 1 3) and α 2 (I 1 (τ)) = a 3 e a 4(I 1 3) with a 1 = 1.078, a 2 = 2.303, a 3 = 2.175, and a 4 = 4.915.................. 24 4.5 Predicted stress relaxation behavior of ligaments stretched to different levels based on fitting the model to all 22 data sets simultaneously. The predictions were computed by fitting the constitutive model to all 22 sets of relaxation data simultaneously................................ 26 viii

List of Tables 4.1 Mean and standard deviation of the parameters of all 22 specimens considered in this study.................................... 24 ix

Chapter 1 Introduction Recent research predicts that the number of American women with pelvic floor disorders (PFDs), including fecal incontinence, urinary incontinence, and pelvic organ prolapse (POP), will increase from 28.1 million in 2010 to 43.8 million in 2050 (38). Studies have suggested that approximately 11% of women will undergo surgery for POP in their lifetime, and that around 30% of these women will require additional procedures due to recurrence (22). Alhough the causes of PFDs are not completely understood, mechanical changes to pelvic floor ligaments appear to contribute to their development and progression (21). In particular, PFDs may result from damage to the cardinal (CL) and uterosacral ligaments (USL), which provide apical support to the uterus and upper vagina (6; 18). The CL and USL are visceral ligaments that connect the upper vagina or cervix to the pelvic sidewall and provide support to the internal pelvic organs. These visceral ligaments, which 1

Winston R. Becker Chapter 1. Introduction 2 connect an organ to the body wall, are different from skeletal ligaments, which connect bone to bone, in both structure and composition. The CL and USL form a thin, membranelike complex that is composed of smooth muscle, blood vessels, nerve fibers, collagen, and elastin. This structure allows the ligaments to provide support to the pelvic organs while accommodating for highly mobile structures, such as the uterus (29). The loading conditions experienced by these ligaments in vivo are likely biaxial, with larger forces occurring along the axis normal to the upper vagina/cervix. Although mechanical damage to pelvic floor connective tissue is closely linked to the development of PFDs, no experiments have attempted to investigate the mechanical response of the CL, and only a few have described the mechanical properties of the USL. To this point, the stress relaxation behavior of USLs in the cynomolgus monkey (Macaca fascicularis) (35; 32) and the tensile properties of USLs taken from cadavers (17) have been examined. While these tests provided the first description of the mechanical properties of these tissues, many things remain unknown about their mechanical behavior. To the authors knowledge, these studies were the first to show that the USL and CL undergo large strains. This finding could be expanded by employing more accurate strain measurement techniques. This includes the use of optical methods to measure strain as well as not assuming that the strain is linear. Optical measurements are preferred to grip-to-grip measurement because they do not include boundary effects near the grips and they measure the strain of the actual specimen. Furthermore, assuming linearity when computing the strain can lead to errors in strain calculations, particularly at the high strains that the CL and

Winston R. Becker Chapter 1. Introduction 3 USL experienced in the previous studies. The studies on the viscoelastic properties of the USL were the first to examine the relaxation behavior of these tissues (35; 32). The time-depedent mechanical properties of the CL and USL are highly relevant to the physiology because they experience loading for long periods of time in vivo. In the previous experiments, stress relaxation was performed multiple times on each specimen without any time for the specimen to recover between tests. This revealed that significant relaxation occurs in these tissues, but many details about the relaxation behavior of the CL and USL have not been studied. The effect of the mechanical history on the relaxation behavior, which has been shown to alter the viscoelastic properties of other ligaments (34), has not been studied. It is also unknown if the stress relaxation of the CL and USL is nonlinear and dependent on strain, which is the case in many articular ligaments (24; 14). Another important contribution of the previous mechanical work on the USL was the quantification of the uniaxial mechanical properties. However, because these ligaments support complex, biaxial loads in vivo, additional information about their physiologic mechanical behavior could be collected with biaxial mechanical testing (13; 36; 11; 20; 39). This would reveal if the CL and USL are anisotropic, and show how they respond to biaxial loading conditions. Although a few studies have examined the mechanical properties of the CL and USL, no attempt has been made to model the elastic or viscoelastic behavior of these ligaments. Mechanical models for these tissues should account for the biaxial loads that they can ex-

Winston R. Becker Chapter 1. Introduction 4 perience in vivo, as well as the experimentally observed stress relaxation behavior. Many constitutive theories have been applied to the relaxation behavior of other biological tissues. For cases where the stress relaxation behavior is not strain dependent, quasi linear viscoelasticity (QLV) (10) is applicable. Models similar in form to QLV have been employed to model the biaxial relaxation of epicardium (1), the urinary bladder wall (20), and aortic elastin (39). For cases where relaxation is strain dependent, modifications to QLV have been applied to model the one dimensional relaxation behavior of collagen gels (26). Other constitutive theories, such as nonlinear superposition (9) and the Pipkin-Rogers integral series (23), also capture the strain-dependent viscoelastic behavior commonly demonstrated by biological specimens. These formulations have been used to successfully model the uniaxial nonlinear relaxation behavior of ligaments and tendons (24; 14; 7; 8; 4). A few studies have considered biaxial loading conditions with these models. In particular, Rajagopal and Wineman (28) discuss the application of the Pipkin-Rogers integral series to a variety of loading conditions without specifying a particular material, and Davis and De Vita (5) formulated a three-dimensional model based on the Pipkin-Rogers integral series for articular ligaments. However, these models have not been applied to materials for which biaxial mechanical data is available. In this study, biaxial elastic and stress relaxation tests were performed on CLs and USLs taken from a swine model. The strain dependency of stress relaxation was studied by stretching specimens to different levels of strain. A three-dimensional constitutive model accounting for anisotropy and nonlinear stress relaxation was formulated and the parameters were de-

Winston R. Becker Chapter 1. Introduction 5 termined by fitting the model to the experimental data.

Chapter 2 Methods 2.1 Sample Preparation All animal protocols were approved by the Institutional Animal Care and Use Committee (IACUC) at Virginia Polytechnic Institute and State University. Four adult female domestic swine (Sus scrofa domesticus) from a slaughterhouse were dissected and the pelvic floor ligaments were extracted. The swine model was selected because its pelvic floor tissues are histologically similar to humans and because prolapse can naturally occur in swine (12). The CL and USL were carefully cleaned of extraneous fat and muscle tissue, cut into approximately 3 cm squares, and frozen (orientation of specimens shown in Fig. 2.1). Previous research has shown that the proper freezing of ligaments has little or no effect on their mechanical properties (37). Prior to testing, the specimens were thawed at room temperature. 6

Winston R. Becker Chapter 2. Methods 7 The length and width of each specimen was determined optically by analyzing pictures taken with a CMOS camera (DCC1645C, Thor Labs), and the thickness of each specimen was determined by averaging four thickness measurements made with calipers containing a force gauge (accuracy ± 0.05 mm, Mitutoyo ABSOLUTE Low Force Calipers Series 573, Japan) by applying a 50 gf compressive load. 2.2 Biaxial Testing Four bent safety pins connected to fishing line (4 lb Extra Tough Trilene, Berkley Fishing Company, USA) were inserted into each side of each specimen. The specimens were then mounted into an Instron Planar Biaxial Tensile Testing Device (Partially shown in Fig. 2.1, Instron, Norwood, MA, USA) by wrapping the fishing line around custom grips. These grips consisted of two pulleys and a bearing of negligible friction that could rotate to ensure that the tensions in each of the four lines attached to each side of the specimen were equivalent. During testing, the specimens were submerged in a bath containing phosphate buffered saline (PBS, 0.5 M, ph 7.4) at 37 o C. Load was recorded with four 20 N load cells (accuracy 0.25% or better) simultaneously during mechanical testing. The force along each axis was taken as the average of the loads recorded by the two load cells located along each axis (which were approximately equal). Stress was calculated by dividing the average force measured along each axis by the cross-sectional area of the sample along the given axis. Prior to testing, four poppy seeds were glued to the surface of the ligaments to produce

Winston R. Becker Chapter 2. Methods 8 suitable contrast for non-contact strain measurements. Images were captured with a CMOS camera (DCC1645C, Thor Labs) and a 25 mm fixed focal length lens (25 mm compact fixed focal length lens, TECHSPEC). The displacements of the poppy seeds were tracked with ProAnalyst Software (Xcitex Inc. Woburn, MA, USA). The displacement gradient was estimated using an interpolation method originally introduced by Humphrey (16) and later implemented by other researchers (30). From the displacement gradient, the deformation gradient tensor, F, and the right Cauchy-Green deformation tensor, C, were calculated as follows: F = u X + I (2.1) C = F T F (2.2) where u is the displacement vector, X is the position vector of the undeformed configuration, and I is the identity tensor. 2.3 Biaxial Test Protocol A schematic of the testing protocol is presented in Fig. 2.2. All specimens (n=22) were preconditioned with the same protocol: First the specimens were preloaded to 0.04 N, which was taken to be the point of zero load for the test. Then the specimens were loaded from 0.1N to 0.6N for 10 cycles at a displacement rate of 0.1 mm/s in both directions. Following preconditioning, the specimens were displaced at a constant rate (0.1 mm/s) in both directions until the force along one axis reached a force between 2 and 12 N that was selected

Winston R. Becker Chapter 2. Methods 9 a) b) RECTUM E2 E1 V A G I N A Figure 2.1: a) Bath of Instron planar biaxial tensile testing system containing load cells, actuator arms, and specimen. b) Orientation at which samples were cut from swine uterosacral cardinal ligament complexes. prior to the test. More specifically, 9 specimens were displaced to a maximum load between 2 and 4.9N, 7 specimens were displaced to a maximum load between 5 and 7.9 N, and 6 specimens were displaced to a maximum load between 8 and 12 N. The data from these ramps were recorded and used to determine the elastic properties of the specimens. Once the desired force was reached, the ligaments were allowed to relax at a constant displacement for at least 50 minutes while the force and time data were recorded. The testing device controlled the displacement during these tests, and stopped the displacement once a preset load was reached. By displacing the specimens to a range of forces from 2 to 12 N, we were able to ensure that a range of maximum displacements would be considered so that we could examine the strain dependent relaxation behavior of the ligaments. The displacement rate was selected to ensure that the elastic response would be quasi static,

Winston R. Becker Chapter 2. Methods 10 Displacement Ramp (Elastic Response) Preconditioning Relaxation Additional Relaxation (7 Samples) Time Figure 2.2: Protocol used during biaxial mechanical testing of pelvic floor ligaments. The displacement was controlled and was always the same in both directions. because similar displacement rates have been employed by others for biaxial stress relaxation tests on collagenous tissues (1), and because it is unlikely that high strain rates would be experienced by these tissues in vivo. Multiple stress relaxation tests at different displacements were performed on 7 of the samples in this study. For each sample, a relaxation test was performed as described above, followed by two additional relaxation tests immediately (n=2 samples), after a 1 hour (n=3 samples) resting period between each test, or after a 12 hour (n=2 samples) resting period between each test. For half of the specimens, three initial displacement levels were applied in increasing order (e.g. displacement at 3 N, displacement at 6 N, displacement at 9 N), and for the other half of the specimens three initial displacement levels were applied in decreasing order (e.g. displacement at 9 N, displacement at 6 N, displacement at 3 N).

Chapter 3 Theoretical Formulation It was assumed that the cardinal and uterosacral ligaments consist of an isotropic matrix of elastin and proteoglycans as well as a fibrous component and smooth muscle cells oriented both transversely and orthogonally to the loading direction. For this reason, the ligaments are assumed to be orthotropic. Due to the high water content of ligaments, they are assumed to be incompressible. 3.1 Constitutive Model The integral series formulation, first introduced by Pipkin and Rogers (23) was employed to model the three-dimensional stress relaxation behavior of these ligaments. The first term of the integral series is used, resulting in a Piola-Kirschoff Stress tensor, P(t), of the form (28) 11

Winston R. Becker Chapter 3. Theoretical Formulation 12 ( t P(t) = p(t)f T (t) + F(t) R[C(t), 0] + R[C(τ), t τ] ) (t τ) (3.1) where F(t) is the deformation gradient, C(t) is the right Cauchy-Green deformation tensor, R[C(τ), t τ] is the tensorial relaxation function, p(t) is the Lagrange multiplier that accounts for incompressibility, and t is time. Because the ligaments are taken to be orthotropic, the tensorial relaxation function can be written as R[C(τ), t τ] = k 1 1 + k 2 C(τ) + k 3 M M + k 4 [M (C(τ)M) + (C(τ)M) M] + k 5 N N + k 6 [N (C(τ)N) + (C(τ)N) N] (3.2) where M is the unit vector that defines the direction normal to the upper vagina/cervix, N is the unit vector that defines the direction transverse to the upper vagina/cervix, and k 1, k 2, k 3, k 4, k 5, and k 6 are constitutive functions that depend on the invariants of C and t τ. The invariants of C are defined as I 1 (τ) = tr(c(τ)), I 2 (τ) = 1 2 (I 1(τ) 2 tr(c(τ) 2 ), I 4 (τ) = M C(τ)M I 5 (τ) = M C(τ) 2 M, I 6 (τ) = N C(τ)N. (3.3) The relaxation function is assumed to consist of components accounting for the contribution of the isotropic matrix, the family of fibers normal to the upper vagina/cervix, and the family of fibers transverse to the upper vagina/cervix. This can be written in the following form:

Winston R. Becker Chapter 3. Theoretical Formulation 13 R[C(τ), t τ] = R gs [C(τ), t τ] + R cfn [C(τ), t τ] + R cft [C(τ), t τ] (3.4) where R gs is the relaxation function that accounts for the contribution of the ground substance, R cfn is the relaxation function that accounts for the contribution of the family of fibers normal to the upper vagina/cervix, and R cft is the relaxation function that accounts for the contribution of the family of fibers transverse to the upper vagina/cervix. Now, it is assumed that k 1 is a function of I 1 (τ) and t τ, k 3 is a function of I 4 (τ) and t τ, k 5 is a function of I 6 (τ) and t τ, and k 2, k 4, and k 6 are zero. These assumptions allow k 1 to account for the behavior of the isotropic matrix, k 3 to account for the behavior of the fibers normal to the upper vagina/cervix, and k 5 to account for the behavior of the fibers transverse to the upper vagina/cervix. This leads to relaxation functions, R gs, R cfn, and R cft of the form: R gs = k 1 1, (3.5) R cfn = k 3 M M, (3.6) R cft = k 5 N N. (3.7) Finally, the constitutive functions, k 1, k 3, and k 5, were selected to capture the behavior of these ligaments: k 1 = c 1 (3.8)

Winston R. Becker Chapter 3. Theoretical Formulation 14 c 2 (e c 3(I 4 (τ) 1) 1)r 1 (I 1 (τ), t τ), I 4 (τ) > 1 k 3 = 0, I 4 (τ) 1 c 4 (e c 5(I 6 (τ) 1) 1)r 1 (I 1 (τ), t τ), I 6 (τ) > 1 k 5 = 0, I 6 (τ) 1 (3.9) (3.10) where c 1, c 2, c 3, c 4, and c 5 are constant parameters. It is assumed that the fibrous component does not support compressive loads in the definition of k 3 and k 5. The relaxation function, r 1, was selected to model the isotropic stress relaxation response of the ligament: r 1 (I 1 (τ), t τ) = 1 + α 1(I 1 (τ))e (t τ)β 1 + α 2 (I 1 (τ))e (t τ)β 2 1 + α 1 (I 1 (τ)) + α 2 (I 1 (τ)) (3.11) where α 1 (I 1 (τ)) and α 2 (I 1 (τ)) are functions accounting for the strain dependent viscoelastic behavior of the ligaments and β 1 and β 2 are constant parameters. Note that r 1 goes to 1 when t = τ. The functions α 1 (I 1 (τ)) and α 2 (I 1 (τ)) were defined as: α 1 (I 1 (τ)) = a 1 e a 2(I 1 3) (3.12) α 2 (I 1 (τ)) = a 3 e a 4(I 1 3) (3.13) where a 1, a 2, a 3, and a 4 are constant parameters.

Winston R. Becker Chapter 3. Theoretical Formulation 15 3.2 Biaxial elastic deformation Let {E 1, E 2, E 3 } and {e 1, e 2, e 3 } be two sets of unit vectors that define the reference and deformed configurations, respectively. The ligament is assumed to undergo an isochoric deformation described by x 1 = λ 1 (t)x 1, x 2 = λ 2 (t)x 2, x 3 = 1 λ 1 (t)λ 2 (t) X 3. (3.14) This deformation was selected because a negligible amount of shear was observed in the biaxial experiments. It follows that the deformation gradient tensor, F(t), and the right Cauchy-Green strain tensor, C(t), are given by: F(t) = λ 1 (t)e 1 E 1 + λ 2 (t)e 2 E 2 + C(t) = λ 1 (t) 2 e 1 E 1 + λ 2 (t) 2 e 2 E 2 + 1 λ 1 (t)λ 2 (t) e 3 E 3, 1 λ 1 (t) 2 λ 2 (t) 2 e 3 E 3. (3.15) To derive the first-piola Kirchhoff stress tensor that defines the instantaneous elastic response, Eq. 3.1 is evaluated at t = τ by substituting Eqs. 3.2-3.11 and Eq. 3.15. It is assumed that the surface of the ligament is traction-free so that P 33 = 0. Enforcing this boundary condition to solve for the Lagrange multiplier, p, P 11 and P 22 are found to be: P 11 = c 2 ( 1 + e c 3(λ 2 1 1) )λ 1 + c 1 (λ 1 1 ) λ 3 1λ 2 2 (3.16) P 22 = c 4 ( 1 + e c 5(λ 2 2 1) )λ 2 + c 1 (λ 2 1 ) λ 3 2λ 2 1 (3.17)

Winston R. Becker Chapter 3. Theoretical Formulation 16 3.3 Biaxial stress relaxation response To model the stress relaxation response, constant stretches, λ 1 (t) = λ 1 and λ 2 (t) = λ 1, are applied. The stress relaxation response is then given by solving Eq. 3.1 by substituting Eqs. 3.2-3.11, Eq. 3.15, and setting P 33 = 0. This results in P 11 and P 22 of the form: P 11 = c 1 (λ 1 1 )+ λ 3 1λ 2 2 c 2 λ 1 ( 1 + e c 3(λ 2 1 1) ) 1 + α 1(I 1 (τ))e (t τ)β 1 + α 2 (I 1 (τ))e (t τ)β 2 1 + α 1 (I 1 (τ)) + α 2 (I 1 (τ)) (3.18) P 22 = c 1 (λ 2 1 )+ λ 3 2λ 2 1 c 4 λ 2 ( 1 + e c 5(λ 2 2 1) ) 1 + α 1(I 1 (τ))e (t τ)β 1 + α 2 (I 1 (τ))e (t τ)β 2. (3.19) 1 + α 1 (I 1 (τ)) + α 2 (I 1 (τ)) 3.4 Identification of Model Parameters The built in minimization function in MATLAB, fmincon, was utilized to compute the model parameters by minimizing the sum of the squares function for the error between the model and the data. To determine the elastic parameters c 1 -c 5, the elastic experimental data in both directions was reduced and Eqs. 3.16 and 3.17 were simultaneously fit to the data. Although a small amount of shear was observed in all tests, the specimens were assumed to undergo no shear when calculating the model parameters. To compute the viscoelastic parameters, the experimental data from each direction of the relaxation tests were reduced and the elastic parameters were held constant. First, β 1, β 2, α 1, and α 2 were computed by

Winston R. Becker Chapter 3. Theoretical Formulation 17 minimizing the sum of the squares function for the error between one set of biaxial relaxation data of a sample stretched to an intermediate stretch level and Eqs. 3.18 and 3.19. Then, the parameters β 1 and β 2 were held constant while α 1 and α 2 were computed for each individual data set by fitting Eqs. 3.18 and 3.19 to each data set. Next, plots of α 1 and α 2 versus I 1 were generated and Eqs. 3.12 and 3.13 were fit to the data to determine the parameters a 1 -a 4. All parameters were constrained to be greater than zero. An alternative approach to fitting the model was also considered for the relaxation. In this case the relaxation function was defined a priori and the constants a 1, a 2, a 3, a 4, β 1, and β 2 were computed by fitting the model to the data collected from all 22 relaxation tests simultaneously.

Chapter 4 Results 4.1 Biaxial Elastic Properties Nonlinear, anisotropic mechanical behavior was observed during the ramp portion of the mechanical tests. The ligaments experienced large strains as high as 75% during the tests conducted. While anisotropy was typically observed, it was much more pronounced in some cases than others. The results of the biaxial elastic tests for a single sow are presented in Fig. 4.1 with the anatomical locations that the specimens were taken from indicated in the inset in the top corner of each plot. This figure presents 9 of the 22 specimens tested in this study. The specimens are separated into two plots based on stress level to make it easier to see each individual curve. As shown in the figure, the specimens exhibited a greater stiffness in the direction normal to the upper vagina/cervix. It is also clear that a large amount 18

Winston R. Becker Chapter 4. Results 19 of variability was observed, even though the specimens were taken from a single sow. The results suggest that the mechanical behavior of the tissue may be location dependent. In particular, it appears that USL specimens and specimens located closer to the USL within the USL-CL complex are stiffer in the direction normal to the upper vagina/cervix than the specimens further from the USL are in the same direction. 4.2 Biaxial Viscoelastic Properties Multiple stress relaxation tests were performed on 7 of the 22 specimens considered in this study. In each case, regardless of whether the highest or lowest stretch was applied first, the first stress relaxation test on a single specimen always resulted in the greatest amount of relaxation (Fig. 4.2). The same result was observed when the resting time between relaxation tests was 0 hours, 1 hour, or 12 hours. These tests revealed that the strain history significantly effects the relaxation behavior of the CL and USL. Therefore, the second and third relaxation tests on these specimens were not considered to determine the relaxation behavior of these ligaments. After selecting only the first relaxation tests on the specimens tested multiple times and conducting 15 single relaxation tests on 15 samples, a total of 22 relaxation tests from 22 samples were obtained. Relaxation tests for 9 specimens taken from a single sow are presented in Fig. 4.1. The 9 relaxation tests in this figure correspond to the 9 stress-stretch curves. The end point for each of the stress-stretch curves is the starting point for each relaxation

Stress (MPa/MPa) Stress (MPa/MPa) Winston R. Becker Chapter 4. Results 20 a) c) b) 1 d) 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 500 1000 1500 2000 2500 3000 Time (s) 0 0 500 1000 1500 2000 2500 3000 Time (s) Figure 4.1: Stress-stretch curves and normalized relaxation curves of 9 samples taken from a single sow. The locations the specimens were taken from are indicated by the schematic in the top left of the top two graphs. The stress relaxation curves represent the same samples presented in the stress-stretch curves. In both the relaxation curves and the stress-stretch curves, the filled symbols represent data collected from stretching in the direction normal to the upper vagina/cervix while the unfilled symbols denote data collected from stretching in the direction transverse to the upper vagina/cervix.

Winston R. Becker Chapter 4. Results 21 test. For example, the endpoint of the stress-stetch curves denoted by green squares in Fig. 4.1a (λ 1 = 1.08 and λ 2 = 1.14) represents the constant stretch for the relaxation test denoted by green squares in Fig. 4.1b. Substantial stress relaxation was observed in all specimens tested. In general, the relaxation was very fast for the first few hundred seconds, and the samples continued to relax slowly through the end of the test. The specimens that relaxed a greater percentage, typically had greater rates of relaxation throughout the tests. A wide range of relaxation behavior was observed in the tests conducted. For example, some samples retained about 65% of the initial load at 3000 seconds (red squares in Fig. 4.1), while others maintained as little as 20% of the initial load at 3000 seconds (green squares in Fig. 4.1). The observed differences in relaxation rate and percent relaxation were weakly dependent on the magnitude of the stretch applied to the specimens. In general, relaxation was higher for specimens stretched to lower strain levels when compared to specimens stretched to higher strain levels. Interestingly, in all of the stress relaxation tests performed, the normalized stress relaxation was identical in both directions even if the elastic behavior was anisotropic. This can be seen in Fig. 4.1, where many of the relaxation curves in the two different orientations overlap (i.e. filled purple triangles overlap unfilled purple triangles). 4.3 Modeling Representative results of the model fitting to the biaxial data are presented for thirteen specimens taken from three sows in Fig. 4.3. The model was able to capture the elastic

Normalized Stress (MPa/MPa) Normalized Stress (MPa/MPa) Winston R. Becker Chapter 4. Results 22 a) 1 b) 1 0.8 0.8 0.6 0.6 0.4 0.2 0 1.046 1.071 1.074 1.101 1.095 1.140 0 500 1000 1500 2000 Time (s) 0.4 0.2 0 1.178 1.101 1.148 1.070 1.110 1.045 0 500 1000 1500 2000 Time (s) Figure 4.2: Stress relaxation response for two specimens. Three tests were performed on each specimen with a 1 hour resting period between each test. a) The specimen was stretch to λ 1 = 1.046 and λ 2 = 1.071 during the first relaxation test, λ 1 = 1.074 and λ 2 = 1.101 during the second relaxation test, and λ 1 = 1.095 and λ 2 = 1.140 during the third relaxation test. b) The specimen was stretch to λ 1 = 1.178 and λ 2 = 1.101 during the first relaxation test, λ 1 = 1.148 and λ 2 = 1.070 during the second relaxation test, and λ 1 = 1.110 and λ 2 = 1.045 during the third relaxation test.

Stress (MPa) Normalized Stress (MPa/MPa) Stress (MPa) Normalized Stress (MPa/MPa) Stress (MPa) Normalized Stress (MPa/MPa) Winston R. Becker Chapter 4. Results 23 a) 1.6 b) 1 1.4 1.2 0.8 1 0.8 0.6 0.6 0.4 0.4 0.2 0 1 1.05 1.1 1.15 1.2 1.25 1.3 Stretch 0.2 0 500 1000 1500 2000 2500 3000 Time (s) c) 1.6 d) 1 1.4 1.2 1 0.9 0.8 0.8 0.7 0.6 0.4 0.2 0.6 0.5 0 1 1.05 1.1 1.15 1.2 1.25 Stretch 0.4 0 500 1000 1500 2000 2500 3000 Time (s) e) 2.5 f) 1 2 0.8 1.5 0.6 1 0.5 0.4 0 1 1.05 1.1 1.15 1.2 1.25 Stretch 0.2 0 500 1000 1500 2000 2500 3000 Time (s) Figure 4.3: Continuum model fit to experimental data of 13 specimens taken from 3 sows. Filled symbols represent data collected from stretching in the direction normal to the upper vagina/cervix, the unfilled symbols denote data collected from stretching in the direction transverse to the upper vagina/cervix, and the solid lines indicate the constitutive model.

α2 α1 Winston R. Becker Chapter 4. Results 24 Parameter Mean Standard Deviation c 1 (Pa) 722 2220 c 2 (Pa) 3.79 10 5 6.07 10 5 c 3 9.05 7.18 c 4 (Pa) 1.89 10 5 2.73 10 5 c 5 8.76 7.00 Table 4.1: Mean and standard deviation of the parameters of all 22 specimens considered in this study. Figure 4.4: Dependency of α 1 (Left) and α 2 (Right), which are measures of the percent relaxation of the specimens, on the first invariant. The lines represent the functions α 1 (I 1 (τ)) = a 1 e a 2(I 1 3) and α 2 (I 1 (τ)) = a 3 e a 4(I 1 3) with a 1 = 1.078, a 2 = 2.303, a 3 = 2.175, and a 4 = 4.915.

Winston R. Becker Chapter 4. Results 25 and stress relaxation behavior of the ligaments. The mean and standard deviation of the elastic parameters computed for 22 data sets are presented in Table 4.1. In many cases, the elastic constant c 1 in Eqs. 3.16-3.17, which represents the contribution of the ground substance, approached zero when fitting the data. This suggests that the ground substance had a relatively small mechanical contribution when compared to the fibrous components of the tissues. The elastic constants c 2 and c 3, which account for the contribution of the fibers normal to the upper vagina/cervix, were typically larger than the constants c 4 and c 5, which account for the contribution of the fibers transverse to the upper vagina/cervix. This is a result of the specimens being stiffer in the direction normal to the upper vagina/cervix. Differences in these parameters allowed the model to capture the anisotropic behavior of the specimens. The values of α 1 and α 2 versus I 1 are presented in Fig. 4.4. In general, the magnitude of α 1 and α 2 were higher for lower levels of stretch. Incorporating the dependency of α 1 and α 2 on I 1 allowed the model to capture the strain dependent relaxation behavior of the tissue. When the alternative approach of fitting the model to all 22 data sets simultaneously was considered, the model predicts the general trend in the relaxation, but cannot fit every data set because of the noise in the data. The predicted stress relaxation behavior of ligaments stretched to different initial stretches when the model parameters are determined with this method is presented in Fig. 4.5.

Normalized Stress (MPa/MPa) Winston R. Becker Chapter 4. Results 26 1 0.8 0.6 0.4 0.2 0 lx=ly=1.05 lx=ly=1.1 lx=ly=1.15 lx=ly=1.2 0 500 1000 1500 2000 2500 3000 Time (s) Figure 4.5: Predicted stress relaxation behavior of ligaments stretched to different levels based on fitting the model to all 22 data sets simultaneously. The predictions were computed by fitting the constitutive model to all 22 sets of relaxation data simultaneously.

Chapter 5 Discussion 5.1 Experimental Results The anisotropic mechanical response with the direction normal to the upper vagina/cervix being stiffer is not surprising considering that the majority of the physiological loading on these ligaments likely occurs in this direction. It is noted that the anisotropy was less pronounced for a few of the specimens in this study. This suggests that in some cases the collagen fibers may be less organized, or that there may be a similar quantity of collagen fibers oriented in each direction. The viscoelastic tests on these tissues revealed the amount that they relax, if they relax different amounts in each direction, and if the stress relaxation is nonlinear (i.e. strain-dependent). Significant stress relaxation was observed in this study, which is similar to the results of the 27

Winston R. Becker Chapter 5. Discussion 28 previous stress relaxation tests. In the final step of Vardy et al s (2005) representative USL load response, the stress relaxed to about 70% of its initial value at just over 2000 seconds. Although this was after relaxing multiple times, the percent relaxation is similar to the relaxation observed in some of the tests in this study. The relaxation tests also showed that, even though the elastic behavior was anisotropic, the normalized stress relaxation response was always the same in both directions. It is unclear why this was observed; however, it may be because all of the individual collagen fibers relaxed approximately the same amount. While the amount of collagen fibers oriented in each direction is likely different, the amount that each individual fiber relaxes may be the same. In this case, normalizing the relaxation by dividing by the initial stress would normalize each direction for the concentration of collagen fibers and result in identical normalized stress relaxation in each direction. The idea that relaxation takes place within individual collagen fibers has been speculated by previous studies that have shown that the spacing and orientation of collagen fibers do not change during stress relaxation (27). Alternatively, the observed relaxation behavior may be due to relaxation taking place within the isotropic matrix rather than within the collagen fibers. If this is the case the model would need to be reformulated to include a relaxation term in the term accounting for the ground substance. This mechanism may be less likely because collagen is stiffer than the other components of the tissue, and likely supports more force than the matrix of the tissue. Considering that the stress has decreased to as little as 20% of the initial stress, it is likely that the force supported by the collagen fibers must decrease for this much relaxation to occur.

Winston R. Becker Chapter 5. Discussion 29 In order to determine the dependeny of relaxation on the initial strain, multiple stress relaxation tests were first completed on 7 of the specimens. Performing multiple tests on one sample overcomes the problems with identifying a trend that may result from the variability between different samples. Furthermore, stress relaxation tests at multiple stretch levels have previously been performed for this ligament (35). The problem with this method is that the tissue has a mechanical history, which can affect the specimens viscoelastic properties (34). In this study, the order of the stretch levels was varied to determine if the same trend in the normalized stress relaxation would be observed regardless of which stretch level was applied first. This revealed that the mechanical history is important for determining the relaxation behavior, which aided in the design and interpretation of this study. One limiting factor that may have affected the relaxation results in these experiments is that the hydration level of the tissue cannot be controlled. The longer each test and resting period was, the longer the specimen remained submerged in fluid, which could potentially alter the mechanical properties. It is possible that the change in hydration over this time period affected the result as much or more than the mechanical history. In particular, if a resting period of 12 hours is actually needed so that the mechanical history does not effect the relaxation results, it may be necessary to find a different way to preserve the specimen rather than submerging it in a bath during the resting period. After determining that multiple stress relaxation tests could not be performed on individual samples, single stress relaxation tests on different samples were compared to determine the relationship between relaxation rate and the initial strain. The rate of stress relaxation de-

Winston R. Becker Chapter 5. Discussion 30 creased with increasing strains, which is consistent with previous studies on other ligaments, such as the medial collateral ligament (14; 25). In each of those studies specimens stretched to lower initial strains relaxed more than specimens stretched to higher initial strains. The underlying cause of this trend in stress relaxation behavior is unknown (14); however, it has been speculated that it may result from water loss (25). Alternatively, others have speculated that relaxation may occur within individual collagen fibers (27), suggesting that this trend may have to do with molecular behavior within individual collagen fibers. Tests aimed at understanding the molecular changes that occur during stress relaxation in these tissues could advance our understanding of this experimental result. Understanding the mechanical behavior of these tissues is especially important to disease pathology because many other researchers have noted that significant remodeling takes place in these tissues. In particular, collagen fibers become disorganized under certain physiologic conditions, especially during the development of PFDs. For example, CL samples from 8/10 women with uterine prolapse had loosely arranged connective tissue and less dense fiber bundles (31), and samples of the USL had significantly reduced organization in individuals who underwent hysterectomies (3). Other studies have determined that diminished collagen levels are associated with the development of PFDs (2). These studies suggest that the mechanical properties of the tissues likely change as the tissue becomes diseased, which should be examined in future studies. One important limitation that must be considered in the context of the experimental results is the limited information available for the history of the sows. Because they were obtained

Winston R. Becker Chapter 5. Discussion 31 from a slaughterhouse, we do not have detailed information on their health and history. Nevertheless, within the same sow, we are confident of our findings. 5.2 Modeling After collecting data on the elastic and relaxation behavior of the CL and USL, a threedimensional finite strain nonlinear viscoelastic constitutive relation based on the Pipkin- Rogers integral series was formulated to model their mechanical behavior. The model was capable of describing the elastic and relaxation responses quite well. A large amount of variability was observed in the elastic parameters, c 1 -c 5. For c 1, the values of the mean and standard deviations are not representative of the overall data set. In all but three cases, the value of c 1 was much less than one meaning that the term containing c 1 did not contribute to the relaxation function. In the three cases where the value of c 1 was larger, the term containing c 1 likely had a slightly more significant contribution, but it was still much less than the contribution of the other components. While the values of c 2 -c 5 were all approximately the same order of magnitude, the large standard deviations are representative of the significant variability in the shape of the stress-strain curves. The function r 1, which was selected to be identical in both fiber directions, depends only on the first invariant because the stress relaxation response was isotropic for all ligaments tested. If the parameters depending on the strain invariant, I 1, in the relaxation function are taken to be constant, our model collapses to quasi-linear viscoelasticity. Simplifying the model to one

Winston R. Becker Chapter 5. Discussion 32 dimension would result in a relaxation function similar in form to the nonlinear viscoelastic constitutive model proposed by Pryse el at (26). In particular, they employed a functional form, A i (ɛ)e t/τi(ɛ), that is similar to the form of r 1 (I 1 (τ), t τ) (Eq. 3.11) to account for changes in relaxation due to the magnitude of the initial strain. The model presented here is highly versatile and could be employed in the modeling of many biological tissues. In this case, it was used to capture the behavior of a tissue with two primary collagen fibers orientations. However, the model could easily be extended to model the behavior of other collagenous tissues with fibers embedded at multiple orientations. Other extensions, such as altering the relaxation function to account for the active mechanics of biological tissues, could be done following the work of previous studies (19; 33), and may allow this model to provide additional information. This extension is particularly relevant when studying pelvic floor ligaments because they include a significant amount of smooth muscle and their smooth muscle content changes during disease development. The modeling of the CL and USL remains limited because only biaxial experimental data can be collected. While biaxial mechanical tests are useful for characterizing the anisotropic behavior of soft tissues, it is not possible to fully characterize the three-dimensional elastic behavior of anisotropic materials with biaxial mechanical tests because the mechanical behavior in the third direction is not characterized (15). In the tests performed, it was also impossible to completely prevent a small amount of shear from occurring. Finally, the test could be improved if we could determine the orientation of the fibers within the specimen prior to and during testing. This would aid us in explaining why we see anisotropy in the

Winston R. Becker Chapter 5. Discussion 33 mechanical response.

Chapter 6 Conclusions 6.1 Summary and Future Directions This work presents the first biaxial elastic and viscoelastic tests on the CL and USL. The results revealed that the specimens are anisotropic, and stiffer in the direction normal to the upper vagina/cervix. The normalized relaxation behavior of each specimen was the same in both directions and the amount of relaxation was dependent on the initial strain. Specimens stretched to lower initial strains relaxed more than specimens stretched to higher strains. A novel three dimensional constitutive model for multiple fibrous groups embedded in a matrix that accounts for anisotropy and strain dependent stress relaxation was formulated to describe the experimental observations. It was assumed that the specimens were composed of two families of fibers oriented perpendicular to each other and that the 34

Winston R. Becker Chapter 6. Conclusions 35 tissues were incompressible. The resulting model effectively fit the elastic and viscoelastic data collected through mechanical experiemnts. The results of this study provide detailed mechanical information about pelvic floor ligaments, which should aid in the development of new treatment methods for PFDs in the future. This study provides the basis for additional work on pelvic floor connective tissues. First, the condition of the sow and the effects of parity on the mechanical properties of pelvic floor connective tissue should be carefully studied. Second, this study would benefit from collecting the structural information of the tissues. In particular, the orientations of the fibrous components should be quantified before and during mechanical testing. This would reveal the unloaded molecular configuration as well as identify how this configuration changes during loading and relaxation. Performing these studies would aid in the connection of many of the mechanical observations to their underlying structural causes. For example, this may explain why the elastic response was isotropic while the relaxation response was anisotropic. Finally, future studies should consider the active mechanical properties of these tissues. Previous histological examinations have revealed that a significant amount of smooth muscle is present in these ligaments. Computing the active mechanical properties of these tissues would reveal the physiological role of this smooth muscle.

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