(2) Modelling and Analysis of the Supraspinatus Tendon: A Finite Element Approach Alejandro Celemín Pardo B.Sc. Mech. Eng Juan Carlos Briceño Triana Ph.D. Universidad de los Andes Advisor Daniel Ricardo Suárez Venegas Ph.D. Pontificia Universidad Javeriana Co Advisor Juan Carlos González Gómez M.D. M.Sc. Fundación Santa Fe de Bogotá Co Advisor Department of Mechanical Engineering School of Engineering Universidad de los Andes Bogotá, Colombia 2011. Key words: tendon, supraspinatus, hyperelastic, anisotropy, failure, finite element method. 1.

(3) Abstract Supraspinatus tendon tearing is one of the most common injuries of the rotator cuff. This tearing produces pain and reduces the mechanical performance of the tendon; therefore, it reduces the functionality of the shoulder. A finite element model of the supraspinatus tendon, modelled as a transversely isotropic hyperelastic material, allows quantifying the effect of a tear on the load-bearing capacity of the tendon by means of a strength index. Comparing the strength index value with a given physiological force, it is possible to calculate a critical tear size that works a clinical criterion for surgical prescription. The critical tear size is determined when the insertion can no longer withstand the forces that are applied during normal activities and, therefore, the effect of the variation of the physiological force on the critical tear size is investigated..

(4) Contents 1 Introduction 2 Methodology 2.1 Constitutive Model . . 2.2 Finite Element Model 2.3 Failure Analysis . . . . 2.4 Strength Index . . . .. 5. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 8 . 8 . 9 . 11 . 14. 3 Results 3.1 Strength Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Critical Tear Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Physiological Force Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15 15 15 16. 4 Discussion. 18. 5 Conclusions. 21. Bibliography. 21. A Clinical Background A.1 Rotator Cuff Basic Anatomy . . . A.2 Rotator Cuff Physiology . . . . . . A.3 Osteotendinous Insertion Histology A.4 Tendon Structure . . . . . . . . . . A.5 Rotator Cuff Pathology . . . . . . A.6 Biological Processes . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 26 26 27 28 28 29 31. B Constitutive Model B.1 Large Strain Mechanics . . . . . . . . . . . . . B.2 Thermodynamics of Elastic Deformations . . . B.3 Isotropic Hyperelasticity . . . . . . . . . . . . . B.4 Transversely Isotropic Hyperelasticity . . . . . B.5 Stress-Stretch Curve Fitting . . . . . . . . . . . B.6 Jacobian Elasticity Tensor . . . . . . . . . . . . B.6.1 Volumetric Part of the Elasticity Tensor B.6.2 Deviatoric Part of the Elasticity Tensor B.6.3 Tendon Stiffness Example . . . . . . . . B.6.4 Tensor Calculus . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. 33 33 34 35 36 37 40 41 41 42 43. . . . . . .. . . . . . .. . . . . . .. . . . . . .. C Mesh Convergence Analysis. . . . . . .. . . . . . .. 44. 1.

(5) List of Figures 2.1 2.2 2.3 2.4 2.5 2.6. 3.1 3.2. 3.3 3.4. Supraspinatus simplified geometry. Modified from Itoi (1995). . . . . . . . . . . . . . . . Supraspinatus model mesh and boundary conditions. . . . . . . . . . . . . . . . . . . . . Crescent-shaped tear in the rotator cuff with much retraction [1]. . . . . . . . . . . . . . Failure of the tear edge fibre bundle once tendon ultimate strain εut is reached. . . . . . Tensile test on a sheep plantaris tendon with a slit. The material fails in shear where the crack tip was and the faces of the slit separate almost paralel between each other [2]. Scheme of the strength index behaviour as function of the tear size for a given ultimate stress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Failure force as function of tear size for model with parameters of tendons A and B. . Strength index for the hyperelastic parameters of tendon A with an ultimate stress of (a) 1.6 MPa and (b) 12.6 MPa, and for hyperelastic parameters of tendon B with an ultimate stress of (c) 1.6 MPa and (d) 12.6 MPa. . . . . . . . . . . . . . . . . . . . . . Critical tear size as function of the ultimate stress for the hyperelastic parameters of tendons A and B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of an increased physiological force on the critical tear size for (a) the hyperelatic parameters of tendon A, and (b) the hyperelastic parameters of tendon B. . . . . . . .. A.1 Anterior view of the human rotator cuff [3]. . . . . . . . . . . . . . . . . . . . . . . . . A.2 Posterior view of the human rotator cuff [3]. . . . . . . . . . . . . . . . . . . . . . . . . A.3 Four osteotendinous insertion zones: dense fibrous connective tissue (CT), uncalcified fibrocartilage (UF), calcified fibrocartilage (CF), and bone (B). The calcified and uncalcified fibrocartilage are separated by a tidemark (TM) that is continuous with a similar tidemark in the adjacent articular cartilage (AC) [4]. . . . . . . . . . . . . . . . . . . . A.4 Hierarchical structure of tendons [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Superior views of three types of rotator cuff tears according to Burkhart: (a) crescentshaped, (b) U-shaped, and (c) acute L-shaped [1]. . . . . . . . . . . . . . . . . . . . . . A.6 Rotator cuff tear patterns according to Sallay et al : (a) transverse, (b) anterior Lshaped, (c) posterior L-shaped, (d) tongue-shaped, (e) V-shaped, and (f) U-shaped [6]. A.7 Rotator cuff patterns according to Davidson et al. (2010): (a) crescent-shaped, (b) longitudinal U-shaped, (c) longitudinal L-shaped, and (d) massive contracted [7]. . . . A.8 Negative correlation between the score of degeneration and the ultimate tensile stress of the supraspinatus tendon [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10 11 12 13 13 14. . 15. . 16 . 16 . 17 . 27 . 27. . 28 . 29 . 30 . 30 . 31 . 32. B.1 Experimental stress-stretch data with its fit for a) tendon A and b) tendon B . . . . . . 40 B.2 Longitudinal strain field in mm/mmm for tendons A and B with an ultimate stress of 12.6 MPa in presence of a 10 mm tear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 C.1 Failure force convergence analysis for (a) the hyperelastic parameters of (a) tendon A and (b) tendon B. Both models have an utimate stress of 12.6 MPa for tear sizes of 10 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44. 2.

(6) List of Tables 2.1 2.2 2.3 4.1. Hyperelastic anisotropic parameters estimated for two human supraspinatus tendons. . . 10 Ultimate tensile engineering stress in the longitudinal direction of the human supraspinatus tendon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Ultimate strain in mm/mm for the set of hyperelastic parameters of tendons A and B. . 12 Percentage reduction of the critical tear size due to a 20% increase on the reference physiological force Fphysio =117 N. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19. B.1 Hyperelastic Anisotropic Parameters of Human Supraspinatus Tendons. . . . . . . . . . 40 B.2 Longitudinal instantaneous elastic modulus for tendons A and B. . . . . . . . . . . . . . 42. 3.

(7) List of Algorithms B.1 Transverse and longitudinal stress-stretch fitting algorithim for MATLAB. . . . . . . . . 39. 4.

(8) Chapter 1. Introduction Rotator cuff syndrome is a range of acute and chronic pathologies of the shoulder, affecting each rotator cuff component separately. Individuals with this condition may present pain and functional deterioration of the shoulder [9–11]. This syndrome is one of the most common musculoskeletal disorders among adult population [12]. It is estimated that 16% of the population suffer shoulder pain or disability [3]. Only in the United Kingdom, around 20% of the shoulder surgical referrals present rotator cuff tears [3]. In the United States 18 million people reported shoulder pain in 2005 and it was estimated that 39% of the population presented some type of rotator cuff tear [12]. In Colombia, the number of individuals reporting rotator cuff syndrome is increasing since 2002 [9]. Chronic rotator cuff pathologies are more common in middle-aged workers, athletes and a substantial portion of the elderly population [9, 11, 12]. Non-operative management of rotator cuff syndrome is usually successful, although, its outcome strongly depends upon factors such as age, injury size, and functional demand of the shoulder [13], unfortunately some injuries may excessively grow during non-surgical management, resulting in a more difficult or even unviable treatment [10]. Surgical management of the rotator cuff syndrome presents good to excellent results in both function and pain relief [13]. Andarawis-Puri et al stated that in an ideal scenario, surgical management should only be indicated for rotator cuff injuries with high risk of progression, however in clinical practice it is still unknown which injuries are likely to propagate and which are not [10]. Surgical management is prescribed for patients experiencing painful shoulder conditions or for patients with no improved outcome after three months of non-operative management [9]. Clinical evidence indicates that the most common rotator cuff injuries initiate as a partial-thickness tear in the osteotendinous junction of the supraspinatus tendon [3, 8, 10, 14]. In its early stages the partial-thickness tear propagates from the articular to the bursal surface of the tendon to become a full-thickness tear, then it propagates to the anterior and posterior edges of the insertion and in its final stages the tear propagates to subscapularis or infraspinatus tendons [3,8,10,14]. Geometrical and dimensional classifications of rotator cuff tears have been established [1,6,7], however, these categories do not provide a criterion for surgery prescription yet. In the absence of an objective criterion for rotator cuff syndrome treatment, a deeper understanding of the mechanical behaviour and the failure mechanisms of the involved tissues is required. This criterion would assist orthopaedists during diagnosis and treatment prescription for rotator cuff injuries by answering the following question: what is the critical tear size for which surgical treatment is recommended? From an engineering perspective, a tear in tendinous tissues acts as a discontinuity that concentrates stresses and reduces the tendon load-bearing capacity. To quantify how a tear affects the tendon loadbearing capacity three aspects must be taken into account: rotator cuff and tear geometries, mechanical behaviour of the tissues, and forces applied to the tissues of interest. Wakabayashi et al considered these aspects in a two-dimensional finite element model of the supraspinatus osteotendinous insertion. 5.

(9) without tear to evaluate its mechanical environment as function of shoulder abduction angle [15]. Sano et al used a similar computational model including three different types of partial thickness tears in the supraspinatus. Mechanical environment was estimated for each tear type as function of shoulder abduction angle [16]. Seki et al used a three-dimensional finite element model of a supraspinatus tendon without tear in order to evaluate the mechanical environment of the tendon considering a muscle force gradient [17] purposed by Itoi et al [14]. Suárez quantified supraspinatus tendon mechanical behaviour and geometry in a simplified finite element model that allowed estimating the safety factor of the tendon as function of tear size [18]. Espinosa used a three-dimensional model of the scapula-supraspinatushumerus complex without tear from magnetic resonance images. Finite element analysis of the model provided the stress distribution in the supraspinatus tendon, which was assumed to be hyperelastic [19]. This study aims at assessing the mechanical behaviour of the supraspinatus tendon and its loadbearing capacity under a static uniaxial tensile load in the presence of a full-thickness tear. This study proposes the use of a single value parameter, the strength index, that varies as a function of tear size, ultimate stress, and physiological force. It is hypothesized that such strength index would give insight about if a tendon is able to withstand its physiological tensile force.. 6.

(10) Objective The objective of this study is to determine the critical tear size in a supraspinatus tendon for which a surgical treatment is recommended. The specific objectives are the following: • Implementation of a non-linear transversely isotropic constitutive model • Implementation of a supraspinatus tendon model with a full thickness tear • Proposal and implementation of a strength index for load-bearing capacity analysis. 7.

(11) Chapter 2. Methodology A finite element model of a simplified supraspinatus tendon under static longitudinal tension is used to determine the load state and the strength index of the tendon. Comparing the results with the physiological force developed by the supraspinatus muscle it is possible to calculate the tear size that produces a failure strain in the osteotendinous insertion when a physiological force is applied. The analysis is performed under the following assumptions: tendon is considered homogeneous [20, 21], perfectly symmetric [22], transversely isotropic [4, 20, 21, 23, 24], and finally, its mechanical response until failure is non-linearly elastic [20, 21, 24]. The finite element model provides the tensile load magnitude applied to the proximal edge of the supraspinatus tendon. The tensile load magnitude that allows quantifying the strength index, and therefore, to establish the supraspinatus status for a given tear size. This section describes the constitutive model of the tendon, the simplified geometry, the material parameters from the constitutive model, the specifications of the finite element model, the failure mechanics of tendinous tissues, and the definition of the strength index that leads to the quantification of a critical tear size criterion.. 2.1. Constitutive Model. Biological and living tissues are complex and some assumptions are needed to derive a model that describes their stress-strain state. Four mayor assumptions regarding supraspinatus behaviour are used in this study: nearly incompressible behaviour, homogeneity, orthotropy, and strain-rate insensitivity. Tendinous tissues are assumed to be nearly incompressible since their water content represents 50% to 60% of their total weight [20, 21, 25–29]. Supraspinatus tendon is a flat tissue that undergoes complex loading modes such as tension, compression and shear. A combined loading mode produces a complex microanatomy consisting of longitudinal, oblique and transverse collagen fibres [25]. Moreover, compression and friction of the tissue with the humeral head lead to inhomogeneities between articular and bursal surfaces [11]. For simplicity, tendinous and ligament tissues are often assumed to be homogeneous with all their collagen fibres aligned in the longitudinal (load) direction [20, 21]. Tensile tests have shown that tendons are relatively strain-rate insensitive within their physiological strain-rate range [20,30], especially when compared to bone [25]. Therefore, viscous and dynamic effects are neglected in the model. The mathematical model that describes the stress-strain state of the tendon is built upon symmetric and invariant expressions of stress and strain relative to rigid rotations [31] such as the second PiolaKirchhoff stress tensor (S) and the Cauchy-Green strain tensor (C). For a tendon, as an hyperelastic material [20,24], the stress-strain relation is defined by means of the Clausius–Duhem inequality [20,24] as follows:. 8.

(12) ∂W (C) , (2.1) ∂C where W is the free energy function or strain energy function. The strain energy for nearly incompressible materials can be decoupled into its volumetric function f (U ) and deviatoric W components. Similarly, the deviatoric component of anisotropic materials can g be decoupled into its isotropic W m and transversely isotropic (Wf ) components [26]. The general form of the strain energy function for the supraspinatus tendon would be: S=2. g W =U +W m + Wf .. (2.2). The volumetric component of the strain energy function has the form: κv 2 (J − 1) , (2.3) 2 where κv is the bulk modulus of the tendon and J is the volume ratio defined as the square root of the determinant of the right Cauchy-Green tensor [31]. Due to its water content, tendon bulk modulus is assumed to be 2 GPa which is the same of the water [20]. The isotropic component of the deviatoric strain energy can have the form purposed by Yeoh [32], expanded to two parameters: U=. 2 g e e W , m = C10 I1 − 3 + C20 I1 − 3. (2.4). where Ie1 is the first modified invariant of the right Cauchy-Green tensor, and C10 , C20 are model parameters. Finally, the transversely isotropic component of the strain energy function can be written as: Wf =. C3 (exp (η (I4 − 1)) − η (I4 − 1) − 1) , η. (2.5). where C3 and η are model parameters [26]. Exponential parameter η in particular, indicates the stiffness of the tendon for large strain states. Variable I4 corresponds to the fourth invariant of the right Cauchy-Green tensor, which is written as: I4 = a0 · Ca0 ,. (2.6). where a0 is a unit vector parallel to collagen fibres direction [20, 24, 31]. For a more detailed description of the model refer to Appendix B.. 2.2. Finite Element Model. The finite element model is developed by considering four aspects: a simplified geometry, the material parameters according to the constitutive model, and the finite elements and boundary conditions specifications.. Simplified Geometry Supraspinatus tendon is a flat tissue that surrounds the upper surface of the humeral head. The model considers a uniform thickness that allows assuming laterally symmetry. Figure 2.1 presents the simplified geometry of the model in which its dimensions are taken from Suárez and Itoi et al [14, 18].. 9.

(13) Figure 2.1: Supraspinatus simplified geometry. Modified from Itoi (1995). The computational model of the supraspinatus was built and solved in Comsol Multiphysics 4.2 (Comsol, Stockholm, Sweden). The two-dimensional model with the dimensions presented in Figure 2.1 has a plane-stress approximation in order to consider thickness change as the tendon is pulled. Thickness is set to 5 mm according to Suárez [18].. Material Parameters Thie constitutive model presented in this study, represents the mechanical contribution of the collagen fibres and their tendinous matrix, thus, considering transverse isotropy and incompressibility assumptions, it is possible to derive analytical expressions that relate stress and stretch for both longitudinal and transverse directions of the tendon. Parameters C10 , C20 , C3 , and η are estimated by fitting two sets of experimental stress-stretch data reported by Espinosa [19] to analytical stress-stretch expressions. Data fitting is performed with MATLAB R2010a (MathWorks, Natik MA, U.S.) by means of a built-in non-linear least-squares algorithm. Fitting equations and MATLAB algorithm are included in Appendix B.5. Parameters are presented in Table 2.1. Table 2.1: Hyperelastic anisotropic parameters estimated for two human supraspinatus tendons. Sample C10 C20 C3 η κv Tendon A Tendon B. 12.18 kPa 20.01 kPa. 153.50 kPa 159.70 kPa. 23.81×10−3 kPa −1. 15.06×10. kPa. 31.83. 2 GPa. 12.42. 2 GPa. It must be remarked the difference of the exponential parameter η between tendon A and B; since tendon A has an η value about three times greater than tendon B, it is expected that for large strains, tendon A is much stiffer than tendon B. Further details on this analysis are given in Appendix B.6.3. In this study each set of model parameters is used independently and then compared.. Finite Elements and Boundary Conditions A mapped mesh of quadrilateral quadratic elements is used [33]. Element size is set to 0.2 mm approximately for a total of 5772 elements in the model. See Appendix C for convergence analysis details. 10.

(14) The osteotendinous insertion is modelled as a fixed boundary condition in the lower edge of the tendon. The anterior-to-posterior width of the tear is controlled by releasing proper nodes at the insertion (lower) edge. This boundary condition assumes that bone is much stiffer than tendon as it has been stated in previous studies [15–17]. Lateral symmetry is enforced by restraining the nodes along the left-hand boundary of the model from displacing horizontally. Since the model attempts to represent an uniaxial tensile load test, an uniform displacement condition is applied in the upper boundary of the model. This boundary condition allows controlling the longitudinal strain of the fibre located at the tear edge. Figure 2.2 presents the model and its boundary conditions. Uniform displacement Force monitored. Figure 2.2: Supraspinatus model mesh and boundary conditions.. 2.3. Failure Analysis. Failure can mean separation between two or more pieces, permanent distortion of a component, reliability reduction, or functional compromise [34]. In this study, failure refers to the loosening of the fibres from the humeral head’s greater tuberosity. This particular failure has been defined as a crescentshaped tear according to geometrical classifications of rotator cuff tears [1, 6, 7]. Crescent-shaped tear, also known as transverse tear, is the most common type of tear of the rotator cuff [6]. It is relatively short and wide and its medial-to-lateral length is usually less than its anterior-to-posterior width as seen in Figure 2.3 [1, 6, 7].. 11.

(15) Supraspinatus. Humerus. Figure 2.3: Crescent-shaped tear in the rotator cuff with much retraction [1]. The size of this particular tear has been classified according its anterior-to-posterior width into four categories: small from 0 to 10 mm, medium from 10 to 30 mm, large from 30 to 50 mm, and massive from 50 mm and larger [3, 6, 35]. According to clincial findings reported by Sallay et al, the mean size of a crescent-shaped tear is 20 mm and only five out of seventy five crescent-shaped tears exceed 50 mm [6]. According to tendon dimensions presented in Figure 2.1, this study focuses on tears ranging from 0.4 to 19.6 mm. For a more detailed description of the clinical background refer to Appendix A. To understand supraspinatus tendon failure it is necessary to measure its failure properties. Several researchers have performed tensile tests on human supraspinatus tendons for strength measurement [8, 14, 18, 19]. These tests were performed by pulling the tendon in its longitudinal direction until it was torn. Table 2.2 presents the ultimate tensile engineering stress reported in the available literature [8, 14, 18, 19]. Table 2.2: Ultimate tensile engineering stress in the longitudinal direction of the human supraspinatus tendon. Reference Ultimate tensile engineering stress Itoi et al. (1995) [14]. 8.87±3.67 MPa. Sano et al. (1997) [36]. ∼1 to 7 MPa. Suárez (2001) [18]. 4.5±0.4 MPa. Espinosa (2009) [19]. ∼3.5 to 6.5 MPa. Since the ultimate stress of tendons A and B from Table 2.1 is unknown, this property is treated as an independent variable that ranges from the minimum to the maximum values found in the available literature (1.6 to 12.6 MPa respectively), see Table 2.2. Since failure criterion is defined upon strain, it is necessary to express the ultimate engineering stress as ultimate strain. This is done by using the constitutive model presented in section 2.1 that allows calculating the strain for a given stress state as presented in Table 2.3. Table 2.3: Ultimate strain in mm/mm for the set of hyperelastic parameters of tendons A and B. Ultimate stress. 1.6 MPa. 2.6 MPa. 4.6 MPa. 6.6 MPa. 8.6 MPa. 10.6 MPa. 12.6 MPa. Tendon A. 0.150. 0.157. 0.164. 0.169. 0.173. 0.175. 0.178. Tendon B. 0.219. 0.235. 0.253. 0.265. 0.273. 0.279. 0.285. 12.

(16) The equivalence between ultimate stress and ultimate strain makes possible to compare the results from experimental data presented in Table 2.2 with a computational model that provides the strain of the fibres bundles of the tendon. Ultimate strain εut is calculated in the computational model by measuring the relative displacement between two points of the same bundle of fibres [4]. Since the tear is expected to grow along the insertion, the bundle of interest is the one located at the edge of the tear as shown in Figure 2.4. Loaded tendon. Original bundle length. Deformed bundle length. Unloaded tendon. Failure at the edge of the tear once ultimate strain εut is reached. Tear. Osteotendinous insertion. Figure 2.4: Failure of the tear edge fibre bundle once tendon ultimate strain εut is reached. The failure of a single fibre bundle in the edge of the tear is an assumption based on the fact that tendinous tissues have a negligible load transmission between fibre bundles [2]. This phenomenon has already been identified by Ker who performed tensile tests on tendon samples with artificial cracks [2] as shown in Figure 2.5.. Figure 2.5: Tensile test on a sheep plantaris tendon with a slit. The material fails in shear where the crack tip was and the faces of the slit separate almost paralel between each other [2].. 13.

(17) 2.4. Strength Index. Load-bearing capacity of the supraspinatus tendon is quantified by means of a strength index (S.I.) defined as the ratio between the force Ff ail, tear required to produce a failure in the edge of the tear as shown in Figure 2.4 and the maximum force Ff ail, healthy that a tendon without tear can withstand: S.I. =. Ff ail, tear . Ff ail, healthy. (2.7). Forces Ff ail, tear and Ff ail, healthy are calculated by integration of the longitudinal stress field over the transverse area of the upper boundary of the finite element model presented in Figure 2.2. Each tear size has a particular strength index associated and it varies from zero for a tear size equal to the anterior-to-posterior width of the insertion, to 1 for a tear size of 0 mm. In other words, the larger the tear size is, the smaller the strength index. When the force Ff ail, tear equals a physiological force Fphysio for a given tear size and ultimate stress, the supraspinatus tendon reaches a critical tear size for which the tendon can no longer withstand its physiological mechanical demand. Therefore, critical tear size is a quantity that can be used as one criterion for surgical treatment prescription. Figure 2.6 presents a scheme of the strength index behaviour and the critical the tear size for a given ultimate stress. Safety region. Risk region. Strength index for the physiological force: Ffail, tear=Fphysio. Critical tear size. Strength index. Ffail, tear=Ffail, healthy. Tear size. Figure 2.6: Scheme of the strength index behaviour as function of the tear size for a given ultimate stress. In Figure 2.6 it is possible to see that, if the force required to produce failure (Ff ail, tear ) is greater than the physiological force (Fphysio ), the tendon is within a safety region in which its tear has not reached the critical size, thus, the tendon can still withstand physiological forces exerted by the muscle. On the contrary, if the force Ff ail, tear is smaller than Fphysio , the tendon is within a risk region in which its tear size is greater than the critical value. For the later scenario, a physiological force can produce failure of the tissue, hence, a surgical treatment of the injury may be recommended. According to Hughes and An, the maximum physiological force Fphysio developed by the supraspinatus during an abduction movement of the arm is 117 N [37], hence, the critical tear size is the one for which a reference force of 117 N produces the failure of the tendon as defined in section 2.3. Nevertheless, the maximum physiological force of the supraspinatus reported by Hughes and An, was measured under specific conditions and it is likely that its magnitude is higher for realistic scenarios. Forces 10% and 20% higher than the reference force are studied to quantify the effect of the physiological force on the critical tear size of the supraspinatus tendon.. 14.

(18) Chapter 3. Results Five major results are obtained from the supraspinatus model: failure force, strength index, critical tear size as function of ultimate tensile stress, and physiological force influence on critical tear size. Failure Force The force required to produce a failure strain of the fibre bundle at the edge of the tear Ff ail tear depends upon tear size and ultimate tensile stress. Figure 3.1 presents the results. 1500. 1500. 1.6 MPa. 1.6 MPa. 2.6 MPa. 2.6 MPa 1200. 4.6 MPa. Fuerza de Falla [N]. Fuerza de Falla [N]. 1200. 6.6 MPa 900. 8.6 MPa 10.6 MPa. 600. 12.6 MPa 300. 6.6 MPa 900. 8.6 MPa 10.6 MPa. 600. 12.6 MPa. 300. 0. 0 0. (a). 4.6 MPa. 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. 0. (b). Tamaño de Desgarro [mm]. 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. Tamaño de Desgarro [mm]. Figure 3.1: Failure force as function of tear size for model with parameters of tendons A and B.. 3.1. Strength Index. Supraspinatus load-bearing capacity for the hyperelastic parameters of tendons A and B is quantified by means of the strength index defined in section 3.2. As an example, Figure 3.2 presents the results for ultimate stresses of 1.6 and 12.6 MPa. The continuous flat line in these plots indicates the scenario in which the force required to produce failure of the tissue is equivalent to the physiological supraspinatus force.. 3.2. Critical Tear Size. As presented in Figure 2.6, from the strength index plots presented in section 3.1 it is possible to calculate the critical tear size as function of the ultimate stress as shown in Figure 3.3.. 15.

(19) 1. 1. (a). (b). Strength index for the physiological force 0,8. Strength index. Strength index. 0,8 0,6 0,4 0,2. 0,6 0,4 0,2. 0. Strength index for the physiological force. 0 0. 5. 10. 15. 20. 25. 0. 5. 10. Tear size [mm] 1. 20. 1. (c). 25. (d). 0,8. Strength index. 0,8. Strength index. 15. Tear size [mm]. 0,6 0,4 0,2. 0,6 0,4 0,2. 0. 0 0. 5. 10. 15. 20. 25. 0. 5. 10. Tear size [mm]. 15. 20. 25. Tear size [mm]. Figure 3.2: Strength index for the hyperelastic parameters of tendon A with an ultimate stress of (a) 1.6 MPa and (b) 12.6 MPa, and for hyperelastic parameters of tendon B with an ultimate stress of (c) 1.6 MPa and (d) 12.6 MPa.. 20. Critical tear size [mm]. 16. 12. 8 Parameters of tendon A. 4. Parameters of tendon B 0 0. 2. 4. 6. 8. 10. 12. 14. Ultimate stress [MPa]. Figure 3.3: Critical tear size as function of the ultimate stress for the hyperelastic parameters of tendons A and B.. 3.3. Physiological Force Influence. Since the critical tear size depends on the assigned value of the physiological force (Fphysio ), it is useful to quantify the effect of its variation. As stated in section 3.1, the maximum physiological force exerted by the supraspinatus is 117 N [37], nevertheless, this force might be higher depending on the individual. Figure 3.4 presents the effect of a 10% and a 20% increase of the physiological force on the critical tear size for models with the hyperelastic parameters of tendons A and B respectively.. 16.

(20) 20. Critical tear size [mm]. Critical tear size [mm]. 20. 15. 10 117 N 5. 128.7 N. 15. 10 117 N. 5. 128.7 N. 140.4 N. 140.4 N. 0. 0 0. (a). 2. 4. 6. 8. 10. 12. 14. 0. (b). Ultimate stress [MPa]. 2. 4. 6. 8. 10. 12. 14. Ultimate stress [MPa]. Figure 3.4: Effect of an increased physiological force on the critical tear size for (a) the hyperelatic parameters of tendon A, and (b) the hyperelastic parameters of tendon B.. 17.

(21) Chapter 4. Discussion As far as the constitutive model concerns, hyperelastic and transversely isotropic models for connective tissues such as tendons have already been implemented [20, 21, 26]. Weiss etal have implemented a g fully incompressible model in which the isotropic strain energy function W m follows a neo-Hookean formulation and the transversely isotropic function (Wf ) is defined by parts according to the fibres stretch state [21]. As proposed by Weiss, the model includes one isotropic constant C1 and four transversely isotropic constants: λ∗ is the stretch at which the collagen fibres are straightened, C3 is a linear parameter, C4 is the rate of uncrimping of the collagen fibres, and C5 is the elastic modulus of the straightened fibres [21]. However, the author does not indicate experimental evidence of the relationship between the model parameters and the physical quantities indicated above. Natali el at proposed a nearly incompressible hyperelastic model with a volumetric energy (U ) of the strain g form presented in equation (2.3). The deviatoric isotropic strain energy W followed the Mooneym Rivlin formulation [38] and the transversely istropic function (Wf ) had the form presented in equation (2.5) [20, 26]. This model uses five parameters in which one of them, the bulk modulus κv , is assumed to be equivalent to the bulk modulus of the water. As required by the Mooney-Rivlin formulation, the deviatoric isotropic function includes two parameters C10 , C20 , and the transversely isotropic function requires two parameters: C3 , which is a linear scale factor and η, which is an exponential factor [20,26]. In this study, the deviatoric isotropic component of the model proposed by Natali et al, is replaced by a Yeoh hyperelastic model [32] expanded to two parameters. This modification intended to avoid non-physical behaviour resulting from negative values for parameter C20 as reported by Weiss [23]. Concerning the finite element model of the supraspinatus tendon, previous numerical studies by Wakabayashi et al [15], Seki et al [17], and Sano et al [16] elucidated the stress state of the supraspinatus tendon under several scenarios, however, the overall effect of a tear on supraspinatus load-bearing capacity has not been studied yet. The main contribution of this study is the quantification of the load-bearing capacity of the supraspinatus tendon through the strength index which, when compared to a given physiological force, provides a critical tear size that works as a clinical criterion for surgical treatment recommendation. Strength index was determined by means of a finite element model that allowed varying geometry, constitutive model parameters and boundary conditions. As shown in Figure 3.1, the force required to produce failure (Ff ail tear ) increases as tendon strength increases as it would be expected. Furthermore, failure force decreases as tear size increases. Using to the maximum ultimate tensile stress reported in the available literature [14], the maximum force required to produce tendon failure is about 1.37 and 1.43 kN depending on tendon properties. Experimental studies performed by Itoi et al (1995) found that the failure force of the supraspinatus tendon ranges from 372.7 to 929.5 N [14]. According to these results, the healthy supraspinatus model without tear is much more stronger than normal supraspinatus tendons, however, Itoi et al tested segments of the tendon instead of complete samples, which might reduced the strength of the sample segments.. 18.

(22) From Figure 3.1 it is also perceptible that models with hyperelastic parameters of tendon B require greater forces than models with parameters of tendon A. This finding can be observed in Figure 3.2when comparing the magnitude of the strength index between each set of hyperelastic parameters; models with parameters of tendon A have smaller strength indexes than models with hyperelastic parameters of tendon B. From the definition of the strength index in equation (2.7), a smaller strength index is an indication of reduced load-bearing capacity. Therefore, since tendon A is stiffer than tendon B for large (ultimate) strains as shown in Table B.2, it is possible to state that for a given ultimate stress, compliant tendons such as tendon B can withstand greater loads than stiffer tendons such as tendon A in presence of crescent-shaped tear of the same size. It can be observed that the strength index calculated from a physiological force Fphysio , showed as a flat continuous line, significantly changes its magnitude depending on the ultimate tensile stress of the tendon. In Figure 3.2, the strength index for a 117 N physiological force in models with parameters of tendon A decreases from 0.82 at 1.6 MPa, to 0.10 at 12.6 MPa, whilst in models with parameters of tendon B, the strength index decreases from 0.66 at 1.6 MPa, to 0.08 at 12.6 MPa. These drastic reductions on the load-bearing capacity relative to tendon ultimate stress are due to the exponential character of the longitudinal stress-strain behaviour defined in the constitutive model. As described in Figure 2.6, the critical tear size depends on the strength index calculated for a given physiological force. Hence, critical tear size varies strongly upon tendon ultimate stress as shown in Figure 3.3. It is remarkable the rapid reduction of the critical tear size as ultimate stress reduces. Quantitatively, within an ultimate stress range of 2.6 to 1.6 MPa, the critical tear size might be reduced from 12.5 to 1.9 mm. This is an indication of the high risk of supraspinatus tear progression in individuals with some degree of tissue degeneration [8]. In clinical practice, the result presented in Figure 3.3 would give a quantitative assessment of a particular supraspinatus tear. For instance, if it would be possible to known that a certain patient has a supraspinatus tendon with the properties of tendon A and an ultimate tensile stress of 4 MPa, surgical treatment would be recommended if the tear is above 15 mm approximately. As expected, the challenge relies on the measurement or knowledge of the tendon mechanical behaviour, and particularly the ultimate tensile stress in vivo. The effect of tendon stiffness on load-bearing capacity stated before is also perceptible regarding critical tear size as shown in Figure 3.3, in which the critical tear size for a given ultimate stress, is greater for the set of parameters of tendon B in comparison to the set of parameters of tendon A. This means that compliant tendons, such as tendon B, allow greater tear sizes than stiff tendons, such as tendon A. Regarding the effect of the physiological force, Figures 3.4 (a) and (b) show that as the physiological force increases the critical tear size decreases. This means that individuals with higher mechanical demand of the shoulder are more susceptible to supraspinatus tear growing as it would be expected. Table 4.1 presents the percentage reduction of the critical tear size for the ultimate stress range given in section 2.3 when the physiological force is incremented by 20% relative to the reference force. Table 4.1: Percentage reduction of the critical tear size due to a 20% increase on the reference physiological force Fphysio =117 N. Ultimate stress. 1.6 MPa. 2.6 MPa. 4.6 MPa. 6.6 MPa. 8.6 MPa. 10.6 MPa. 12.6 MPa. Tendon A. 86.5%. 29.2%. 8.3%. 4.8%. 3.5%. 2.8%. 4.9%. Tendon B. 69.9%. 18.8%. 7.4%. 4.5%. 3.2%. 1.7%. 0.8%. It should be noted that the reduction of the critical tear size is more severe for tendons with reduced strength, And again, as seen on Figure 3.3, stiffer tendons such as tendon A present major loses regarding their critical tear size in comparison to compliant tendons such as tendon B. The finite element model presented allows analysing supraspinatus tendon status in order to obtain. 19.

(23) a critical tear size criterion, however, the model is limited in terms of its load conditions and geometry since it is two-dimensional. Also, the model is limited regarding the constitutive model since only two sets of hyperelastic parameters were available.. 20.

(24) Chapter 5. Conclusions From the previous analysis it is possible to conclude that: • The critical tear size of the supraspinatus osteotendinous insertion is highly sensitive to the ultimate stress below 6 MPa. • Compliant tendons can withstand higher tear sizes than stiff tendons. • A 20% increase on physiological force acting on the insertion, may reduce the critical tear size up to 86% depending on tendon properties.. Future Work Further studies shall be focused on the following aspects among others: • Characterization of supraspinatus tendon histology with focus on collagen fibres orientation within the tissue. • Characterization of the tendon mechanical behaviour in its elastic domain for several tendon samples. This characterization shall include a careful measurement of stress-strain curves in both longitudinal and transverse directions for further statistical treatment of the data. • Characterization of the fracture behaviour of the tendon in presence of artificial tears for further validation the results presented in this study. • Development of a three-dimensional model of the supraspinatus osteotendinous insertion to study the effect of the humeral head and other load modes on the load bearing-capacity of the insertion.. 21.

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(29) Appendix A. Clinical Background Given the clinical nature of this study, a brief review of the clinical aspects associated to the rotator cuff syndrome is described. The chapter begins with the description of the basic anatomy of the rotator cuff, then its physiological and biological features.. A.1. Rotator Cuff Basic Anatomy. The rotator cuff is a complex mechanism comprising four muscles and tendons. Each muscle arises from a specific site in the scapula and extends towards the lateral direction where they connect with the humerus through a corresponding set of tendons. As rotator cuff tendons approach their insertion, they blend between each other forming a continuous cuff around the humeral head [3, 39]. Muscles of the rotator cuff are the teres minor, subscapularis, infraspinatus, and the supraspinatus [39]. Teres minor is the smallest cuff muscle. It emerges from the lower lateral scapular border and its tendon inserts into de greater tuberosity of the humeral head. Subscapularis is the most powerful of the rotator cuff muscles. It extends from the subscapular fossa to the lesser tuberosity of the humerus. Its insertion is wide-spread and it interacts with the shoulder capsule and the glenohumeral ligaments as well [3]. Infraspinatus arises from the infraspinatus fossa of the scapula and its tendon inserts into the posterior area of the greater tuberosity [3]. Supraspinatus muscle emerges from the suprascapular fossa, extends laterally beneath the coracoacromial arch and its tendons inserts in to the greater tuberosity of the humerus. The space defined by the lower surface of the acromion and the upper surface of the humeral head is known as the supraspinatus outlet, through which supraspinatus tendon must pass. Some researchers have hypothesized that supraspinatus tendon might be compressed and damaged by the surfaces of the supraspinatus in a process called impingement [3]. Figures A.1 and A.2 present the anterior and posterior views of the rotator cuff respectively [3].. 26.

(30) Supraspinatus. Coracoid. Facet for articulation with the clavicle Acromion. Superior angle. Capsule of the shoulder joint Transverse humeral ligament Tendon of long head of biceps brachii. Subscapularis. Interior angle. Figure A.1: Anterior view of the human rotator cuff [3].. Superior angle Supraspinatus. Spine of scapula. Infraspinatus. Teres minor. Teres major Inferior angle. Figure A.2: Posterior view of the human rotator cuff [3].. A.2. Rotator Cuff Physiology. Rotator cuff has a key role in the function and mechanics of the shoulder [3]. Nevertheless, three main functions may be identified; first, it provides stability to the glenohumeral joint in all shoulder positions by compressing the humeral head into the glenoid fossa [3,39], second, it rotates the humeral head with respect to the scapula, and third, it counteracts dislocating forces from muscles such as deltoid or latissimus dorsi [3]. Supraspinatus tendon enables the muscle force to be transmitted across the shoulder to the humerus. Its flexibility allows it to bend over the humeral head and accommodate a wide range of shoulder movements [40].. 27.

(31) A.3. Osteotendinous Insertion Histology. Supraspinatus osteotendinous insertion is a direct insertion. Direct insertions present fibrocartilaginous tissues and they are typically located at the epiphyses of bones [4]. Fibrocartilaginous insertions are divided into four zones: tendon, uncalcified fibrocartilage, calcified fibrocartilage, and bone as shown in Figure A.3 [25].. Figure A.3: Four osteotendinous insertion zones: dense fibrous connective tissue (CT), uncalcified fibrocartilage (UF), calcified fibrocartilage (CF), and bone (B). The calcified and uncalcified fibrocartilage are separated by a tidemark (TM) that is continuous with a similar tidemark in the adjacent articular cartilage (AC) [4]. Collagen fibres of the tendon pass directly through the fibrocartilaginous zones and enter the bone matrix directly [4]. This gradual change from tendon to bone tissue occurs within 1 mm approximately [4] and it is thought to dissipate the load at the insertion ensuring that tendon fibres bend gradually with joint motions [25].. A.4. Tendon Structure. From the gross anatomy to molecular level, tendinous tissues such as the supraspinatus are constituted by almost six different hierarchy levels: fascicles, fibres, fibrils, and molecular microfibrils [5]. Collagen fibrils have a diameter from 10 to 150 nm approximately [20]. Figure A.4 presents a scheme of the hierarchical structure of tendons [5].. 28.

(32) Figure A.4: Hierarchical structure of tendons [5]. Each fascicle or bundle is constituted by fibres, fibres are constituted by fibrils, and fibrils are constituted by several types of collagen molecules held together by covalent cross-links [2,4,5]. Fibrillar collagen type I is the most common collagen found in biological tissues [41]. Approximately 95% of the tendinous tissues consist of type I collagen [25]. Collagen type III, associated to elastic tissues such as skin, accounts for less than 5% in tendinous tissues [25]. As connective tissue becomes fibrocartilage, the amount of collagen type II increases [4]. Collagen type II is associated to cartilaginous tissues [25, 41].. A.5. Rotator Cuff Pathology. Rotator cuff syndrome typically involves the supraspinatus tendon and the posterior cuff in a very variable degree [3]. Supraspinatus tearing and avulsion are very common diseases that occur in the weakest link of the muscle-tendon-bone complex. Most sound tendons are tougher than muscle and bone, therefore, tendons are able to withstand physiological forces better than bones; according to this, failure through bone or muscle is more likely to happen than failure through the midsubstance of the tendon [25]. However, clinical findings indicate that midsubstance disruptions of the supraspinatus tendon occur in weakened tissues with pre-existing disease [8, 25, 36]. Rotator cuff syndrome is characterized by the presence of tears in the osteotendinous insertion of the supraspinatus. These tears may be vertical, horizontal or combined, and they present a wide range of sizes. These widespread differences have lead to a lack of a universal classification of the rotator cuff disease. However, the depth of the tear from articular to bursal surfaces of the tendon differentiates partial from full-thickness tears [3]. Regarding full-thickness tears, several geometrical patterns have been identified and classified [1, 6, 7]. Burkhart purposed surgical techniques for three types of rotator cuff tears [1]: crescent-shaped, U-shaped, and L-shaped. Figure A.5 presents a scheme for each tear type.. 29.

(33) Figure A.5: Superior views of three types of rotator cuff tears according to Burkhart: (a) crescentshaped, (b) U-shaped, and (c) acute L-shaped [1]. Sallay et al identified the frequency of various patterns of full-thickness tears of the rotator cuff [6]. Tear patterns follow a morphologic classification based on Ellman’s classification [42]. The classification includes six tear patterns: transverse or crescent-shaped, anterior L-shaped, posterior L-shaped, tongue-shaped, V-shaped, and U-shaped. Figure A.6 presents a scheme of each tear type.. (a). (b). (c). (d). (e). (f). Figure A.6: Rotator cuff tear patterns according to Sallay et al : (a) transverse, (b) anterior L-shaped, (c) posterior L-shaped, (d) tongue-shaped, (e) V-shaped, and (f) U-shaped [6]. Davidson et al purposed four types of rotator cuff tears: crescent-shaped, longitudinal U-shaped or L-shaped, and massive contracted tears [7]. Figure A.7 presents a scheme of each tear type.. 30.

(34) (a). (b). (c). (d). Figure A.7: Rotator cuff patterns according to Davidson et al. (2010): (a) crescent-shaped, (b) longitudinal U-shaped, (c) longitudinal L-shaped, and (d) massive contracted [7]. Massive contracted correspond to large injuries in which the osteotendinous insertion has failed completely. Crescent-shaped, U-shaped, L-shaped, V-shaped, and tongue-shape tears are injuries in which failure is not complete and surgical management has good to excellent prognosis [1, 6, 7].. A.6. Biological Processes. Tendons present regenerative and degenerative processes simultaneously; therefore, tendon health depends on a balance between damage and repair within the tissue [2]. It has been postulated that injury mechanisms and degenerative processes are either intrinsic or extrinsic [3]. Intrinsic mechanisms are associated to collagen degeneration and thinning, poor vascularity at the osteotendinous junction, fatty infiltrations, and calcium crystal deposition [3,25,39]. It has been hypothesized that biological degenerative processes of tendinous tissues are caused by high intramuscular pressure that reduces blood microflow to these tissues, leading to inflammation and collagen quality reduction [9]. Sano et al studied the effect of three kinds of tendon degeneration on supraspinatus strength [8]. Degree of degeneration was scored by histological observations in which presence of fibres thinning, granular tissue, or partial-thickness tears was determined. According to the author, collagen fibres thinning reflects the degree of tendon atrophy, presence of granulated tissue (calcium crystal deposition) at the insertion leads to tendon disruption, and partial-thickness tears is evidence of fibres discontinuities [8]. Figure A.8 presents the engineering ultimate tensile stress of the human supraspinatus tendon relative to the score of degeneration that ranges from 0 no degeneration, to 12 complete degeneration of the tissue according to Sano et al [8].. 31.

(35) Figure A.8: Negative correlation between the score of degeneration and the ultimate tensile stress of the supraspinatus tendon [8]. Extrinsic degenerative mechanisms on the other hand, are associated to the impingement of specific shapes of the acromion over the bursal surface of the tendon [3, 9, 25], excessive mechanical loading, and overuse. All three mechanisms lead to tendon inflammation, edema, and further degeneration of the tissue [28]. The osteotendinous junction is particularly susceptible to overuse injuries known as enthesopathies. Enthesopathies are characterized by metabolically active junctions in which the extracellular matrix composition of the tendon is altered, leading to intrinsic injury mechanisms such as bundle loosening, lipids accumulation, and microcalcification [28]. Healing process of tendons can be divided into three major phases: the inflammatory, repairing, and remodelling phases [43]. During the inflammatory phase, several inflammatory migrate to the wound to clean the site of necrotic materials. Simultaneously, these cells release chemical substances that recruit fibroblasts to begin collagen synthesis. During the repairing phase, abundant collagen and proteoglycans are produced. Later, during the remodelling phase, collagen and proteoglycan synthesis is decreased due to decreased cellularity. The repaired tissue changes to a scar-like fibrous material. In the final stages of the remodelling phase, bonding between collagen fibres takes place, which results in a stiffer and tougher tissue [28, 43].. 32.

(36) Appendix B. Constitutive Model Since tendinous and biological tissue display complex mechanical behaviour, three mathematical topics are explained in order to support the development of a procedure to estimate the parameters of a constitutive model for the supraspinatus tendon. These topics are large strain mechanics, thermodynamics of elastic deformations, and hyperelastic anisotropy.. B.1. Large Strain Mechanics. Movement within a body might be defined in terms of a deformed position vector x with respect to a reference position vector X [24, 27, 31, 44, 45]. Both vectors are related by linear operator F of the form: x = FX.. (B.1). Linear operator F is known as the deformation gradient tensor and it is independent of the reference configuration X of the body. This tensor might be expressed as: dx dx dx F = ∇x = . dX dy dX dz dX. dY dy dY dz dY. dZ dy dZ dz dZ. .. (B.2). This tensor can be decomposed into two tensors of pure distortion and one tensor of rigid rotation. This multiplicative operation is known as polar decomposition and it is written as: F = RU = VR,. (B.3). where R is the pure rotational orthogonal tensor and U, V are the right and left stretch tensors respectively. In order to obtain a symmetric and positive definite expression of strain, right C and left B Cauchy-Green tensors are defined as: U2 = FT F = C,. (B.4). V2 = FFT = B.. (B.5). Right Cauchy-Green tensor C is one of the most common measures of large deformation and it is the basis of several constitutive equations. Eigen values of this tensor are calculated through the expression: det (C − αI) = 0, 33.

(37) where α is an arbitrary scalar and I is the identity matrix. Moreover, the determinant can be written as: −α3 + I1 α2 − I2 α + I3 = 0, where coefficients I1 , I2 , and I3 of this polinomial are independent of any rigid rotation of the tensor C, therefore, these coefficients are invariant to reference frame motion. The invariants of the right Cauchy-Green tensor have the form: I1 = tr (C) , I2 =. 1 tr2 (C) − tr C2 , 2. I3 = det (C) = J 2 ,. (B.6) (B.7) (B.8). where J is the volume ratio which describes the compressibility of the body. Often, material models are decoupled into volumetric and deviatoric parts [21]. This decoupling begins with the volumetric normalization of the deformation gradient tensor F: e = J − 13 F. F This normalization modifies the right Cauchy-Green tensor and its invariants I1 and I2 , which are written as: e = J − 32 C, C. (B.9). 2 Ie1 = J − 3 I1 ,. (B.10). 4 Ie2 = J − 3 I2 ,. (B.11). where the third modified invariant Ie3 = 1. The constitutive model used in this study is a function of the modified invariants Ie1 and Ie2 , and volume ratio J which define the strain state of the body.. B.2. Thermodynamics of Elastic Deformations. Constitutive models of elastic materials are built over a thermodynamic balance between specific internal energy ξ, specific entropy S, heat transfer Q, stress σ, and strain ε [24, 30, 45]. From the first law of thermodynamics, the change of internal energy has the form: 1 dξ = dQ + σdε, ρ. (B.12). where ρ describes the density of the body. The second law of thermodynamics states that entropy and heat transfer are related by the expression: dQ = T dS.. (B.13). When assumed that internal energy ξ and entropy S are functions of strain, stress for a fixed temperature T can be written as:. 34.

(38) σ=ρ. ∂ξ ∂S −T ∂ε ∂ε. .. (B.14). T. Isentropic processes are ideal and, even under laboratory conditions, are difficult to guarantee. That is not the case for isothermal processes because temperature is quite easy to control; therefore, temperature might be treated as an independent variable [30]. Then, considering entropy change, it is possible to define a new state variable W of the form: W = ξ − T S,. (B.15). where W is the Helmholtz free energy [20], also called strain energy function [30, 31]. This new variable allows to rewrite equation (B.12) as: dW = dξ − T dS − SdT =. 1 σdε. ρ. Thus, stress for a given entropy may be written as: ∂W ∂T . +S σ=ρ ∂ε ∂ε S. (B.16). (B.17). If temperature is considered to be independent and constant during the process, then the stress depends exclusively on the change of the Helmholtz free energy W relative to strain: ∂W . (B.18) ∂ε An alternative demonstration of equation (B.18) starts from the Clausius-Duhem inequality [24]. This inequality is an expression of the second law of thermodynamics in which an irreversible process presents energy dissipation. The inequality in terms of the internal energy has the form: ρ ξ˙ − T Ṡ − σ ε̇ ≤ 0, (B.19) σ=ρ. where ξ˙ is the first time derivative of the internal energy, Ṡ is the first time derivative of the entropy, and ε̇ is the strain rate of the body. Rewriting equation (B.19) properly leads to the constitutive equation (B.18) [20, 24]. Equation B.18 allows deriving the stress state from the strain state by knowing the value of the free energy function that, as it will be shown in section B.3, also depends on the strain state defined by the strain invariants derived in section B.1.. B.3. Isotropic Hyperelasticity. Strain energy function W shall be unique and independent of any rigid rotation of the body [27, 31]. This is the reason why strain energy is expressed in terms of the right Cauchy-Green tensor C. Rewriting equation (B.18) in terms of tensor C leads to: ∂W (C) , (B.20) ∂C where S is a measure of stress known as second Piola-Kirchhoff stress. Strain energy W , as well as tensor C, describes deviatoric and volumetric deformations simultaneously. For nearly incompressible materials, a deviatoric/volumetric split can be done by using equation (B.9): e J . W (C) = W C, (B.21) S=2. 35.

(39) f , alIt is not possible to determine the exact form of the deviatoric strain energy function W though, a Taylor expansion of two terms around J = 1 may give a reasonable approach to the deviatoric/volumetric split [31]: 2f e e 1 f C, ∂W C, 1 2 ∂ W e 1 + f C, W =W , (B.22) + ∂J 2 ∂J 2 where equals J − 1, corresponds to the volumetric dilatation of the body. In equation (B.22), the f relative to J corresponds to the hydrostatic stress p. Furthermore, second first partial derivative of W f partial derivative of W relative to J corresponds to the slope of hydrostatic stress change relative to volume change [24, 31]. This quantity is identified as the bulk modulus κv of the material. Hence, if no hydrostatic stress is applied to the body, strain energy function takes the form: e + U (J) , f C W =W where κv 2 (J − 1) . (B.23) 2 The term U (J) is known as the volumetric energy function and it describes the compressibility of the material. e , may take several forms according to material behaviour. f C Deviatoric strain energy function, W One of the formulations used for isotropic materials is the Mooney-Rivlin expansion [38]: U (J) =. g W m =. ∞ X. i j Cij Ie1 − 3 Ie2 − 3 .. (B.24). i,j=0. This function is an infinite polynomial expansion of the modified strain invariants where the constant C00 is equal to zero so that the body does not store energy in its reference configuration . However, Weiss and Cowin showed that the two parameter expansion of the Mooney-Rivlin function for soft biological tissues leads to non-physical behaviour under compression conditions [23]. This behaviour is a result of the negative value of the C01 parameter, which defines the positive concavity of the stress-stretch curve in tension, typically found in biological tissues. g e Previous works have suggested that ∂ W m /∂ I2 is numerically close to zero, this will make possible to formulate an alternative hyperelastic form in which the second invariant Ie2 is neglected [24]. This model was purposed by Yeoh [32] and its deviatoric strain energy function is a polynomial expansion of the first modified invariant of the Cauchy-Green tensor Ie1 exclusively: g W m =. ∞ X. i Ci0 Ie1 − 3 ,. (B.25). i=0. with C00 = 0. Yeoh’s hyperelastic model expanded to two parameters is able to produce a stress-stretch curve with positive concavity under tensile conditions without producing misleading responses under compression conditions.. B.4. Transversely Isotropic Hyperelasticity. Biological tissues reinforced with collagen fibres are usually transversely isotropic, this means that mechanical response is isotropic in the plane orthogonal to fibres direction exclusively [21, 27, 31, 45]. In order to model this behaviour, strain energy function W must be modified to include two new. 36.

(40) invariants of tensor C. These new invariants depend on a unit vector a0 aligned with the reinforcement fibres: I4 = a0 · Ca0 ,. (B.26). I5 = a0 · C2 a0 .. (B.27). Invariant I4 corresponds to the square of the stretch ratio in the fibres direction. A physical demonstration of this invariant begins by defining the square of the stretch ratio parallel to a0 [24]: λ2a0 = λa0 · λa0 , where vector λa0 may be written as: λa0 = Fa0 , hence, λ2a0 = a0 · FT Fa0 . Invariant I5 corresponds to the fourth power of the stretch ratio in the fibres direction, which is related to shear effects within the fibres [26]. Usually, shear strain effects might be neglected without much change in the overall results [20, 27], therefore, strain energy function takes the general form: g e e W =W m I1 , I2 + U (J) + Wf (I4 ) + Wmf (I1 , I2 , I3 , I4 ) . Term Wf represents mechanical contribution of the collagen fibres, while Wmf represents the interaction between fibres and ground substances. Term Wmf is frequently omitted since it does not represent a significant contribution to the model [26]. Under these assumptions, mechanical behaviour g e e of hyperelastic transversally isotropic materials can be well modelled using terms Wm I1 , I2 , U (J), and Wf (I4 ) [20, 27]. Typically, tensile tests in the fibre -longitudinal- direction of biological tissues exhibit exponential stress-strain curves [20,21,26,30]. This has lead to linear-exponential forms of function Wf such as [20]: Wf =. C3 (exp (η (I4 − 1)) − η (I4 − 1) − 1) , η. (B.28). where C3 is the linear fit parameter and η is the exponential fit parameter.. B.5. Stress-Stretch Curve Fitting. Equations in section B.3 can be directly implemented in finite element packages [21]. However, the application of these equations to real mechanical tensile tests requires rewriting these expressions in terms of measureable quantities. The deformation gradient tensor F defined in equation (B.2) states that deformation in each direction is determined by the derivative of the position vector x relative to the reference position vector X. Physically, this means that partial derivatives of tensor F correspond to the stretch ratios in the orthogonal directions i = 1, 2, 3 [24, 30]: ∂y ∂z ∂x = λ1 , = λ2 , = λ3 , ∂X ∂Y ∂Z where stretch ratio λ is related to strain ε as:. 37. (B.29).