Fatigue Modeling of Collagenous Soft Tissue

Transcription

1 Proceedings of the SEM Annual Conference Mohegan Sun, Uncasville, Connecticut, USA, June 13-16, Society for Experimental Mechanics Fatigue Modeling of Collagenous Soft Tissue Caitlin Martin and Wei Sun Tissue Mechanics Laboratory Biomedical Engineering and Mechanical Engineering University of Connecticut, Storrs ABSTRACT In this study, a phenomenological tissue damage model has been developed to describe the fatigue-induced stress softening and permanent set of biological tissues. Since damage evolution is an irreversible dissipative process, following thermodynamic principles, an equivalent strain proportional to the strain energy of the material is employed as the damage criterion. The maximum equivalent strain represents the value necessary to cause complete sample failure during one loading cycle, while the minimum equivalent strain is the value required to elicit the onset of fatigue damage. The damage parameter evolves from zero (below minimum equivalent strain) to one at maximum equivalent strain as a function of both the equivalent strain and number of loading cycles. The permanent set evolves as a function of the peak strain in the principal directions. The damage model is implemented into ABAQUS via a user defined material (UMAT) in conjunction with the nonlinear orthotropic Fungelastic model. For the purpose of this study, glutaraldehyde-treated bovine pericardium (GLBP), a collagenous tissue traditionally used for fabricating bio-prosthetic heart valve (BHV) leaflets, is utilized as a representative collagenous tissue due to its limited durability in BHV applications. INTRODUCTION Biologically derived, chemically-treated collagenous tissues are extensively utilized in a broad spectrum of medical applications including cardiovascular grafts and bio-prosthetic heart valves (BHV), as well as ligament, tendon, cartilage, sclera, and hernia repair and replacement. However, despite their widespread use, similar to devices made of metallic and polymeric materials, fatigue-induced degradation, wear and tearing have been identified as some of the major problems associated with collagenous tissue implant failure. Fatigue is an especially critical issue for the application of BHVs. Today, the only effective, long-term treatment for valvular heart disease (VHD) is open-chest cardiac valve repair or replacement surgery. BHVs fabricated from glutaraldehyde-treated bovine pericardium (GLBP) have been used to treat VHD for over three decades 1, and continue to be one of the dominant replacement valve modalities, either as a conventional prosthetic valve design or more recently for minimally invasive percutaneous delivery 2,3. BHVs display superior hemodynamics to mechanical valves, and they eliminate the need for anticoagulant therapy. Regardless of the specific design, long-term fatigue resilience remains the major limitation in the durability of the GLBP tissue leaflets (1-15 years) and the mechanisms governing this process are largely unknown. There are many challenges associated with fatigue testing of BHV materials. The experimental methods can be very time consuming, involving complex testing instruments. The current FDA requirement mandates new BHV designs to be tested up to 2 million cycles to evaluate the fatigue performance using accelerated wear testers 4. Although, accelerated wear testers can approximate the in vivo hemodynamics, it remains difficult to determine the specific effects of different fatigue modalities (tensile, compressive, bending, etc.) on the leaflet material properties when using this technique. To address this drawback, several groups have conducted isolated material tests in order to determine for instance, the effects of uniaxial cyclic loading on GLBP material properties 5,6 and collagen fiber orientation 6. Others have investigated the effects of cyclic bending fatigue on the leaflet flexural rigidity 4. These studies, however, are limited to less than 1 millions of cycles of fatigue. 1

2 There is a lack of constitutive models for describing the damage process in soft biological tissues, which can arise from the plastic deformation or fracture of the fibers or tearing of the matrix 7-9. In 1987, Simo 7 introduced the concept of a fully three-dimensional (3D) finite-strain viscoelastic damage model, based on irreversible thermodynamics with internal variables. This pioneering work served as the basis for several soft tissue models 8-1. Each of these models can be utilized to model damage induced by cyclic stretching with increasing magnitude; however, damage in these models is dependent only upon the strain history and not on the number of loading cycles, which limits the applicability of such models to study long-term fatigue damage in BHVs. In this study, we propose a phenomenological damage model to describe the overall fatigue damage of GLBP due to the strain history and number of loading cycles for the purpose of investigating the fatigue damage process of BHVs. The model incorporates descriptions of the fatigue-induced stress softening and permanent set of biological tissues. The damage model is implemented into ABAQUS via a user defined material (UMAT) in conjunction with the nonlinear orthotropic Fung-elastic model, and is utilized in a tissue biaxial fatigue loading simulation and a BHV fatigue simulation. METHODS Un-fatigued State Tissue Properties Glutaraldehyde-treated bovine pericardium is assumed to be anisotropic, nonlinear, hyperelastic material per Sun et al. 11. Therefore, the in-plane second Piola Kirchhoff stress (S ) at the un-fatigued state can be computed by Eq. 1, where E represents the Green-Lagrangian strain tensor and W is a strain energy function. S ij = W E ij i, j = 1,2 (1) Furthermore, because this tissue is assumed to be incompressible and planar, the generalized Fung-type strain energy function for the planar biaxial response of soft biological tissues given by the following equations was used for subsequent calculations 12 : where c and A i are material parameters. The elasticity tensor is given by Eq. 4. Damage Evolution W = c 2 (eq 1) (2) Q = A 1 E A 2 E A 3 E 11 E 22 + A 4 E A 5 E 11 E A 6 E 22 E 12 (3) = S ij E ij, i, j = 1,2 (4) The stress-strain response of collagenous soft tissues have 3 distinct regions: the toe region where the collagen fibrils extend, the quasi-linear region as the collagen fibrils uncrimp as they bear load causing the tissues stiffness to greatly increase, and the damage region as the strain exceeds a specific limit and collagen fibrils and interfibrillar bonds progressively break down, and the material stiffness decreases until the tissue fails 13. Here it is assumed that material damage is related to the maximum distortional energy, independent of hydrostatic pressure 7. In order to establish the law of tissue damage evolution, we first define an equivalent strain Ξ t 7, a scalar quantity proportional to the strain energy at time t, Ξ t (E (t)) = 2W (E (t)) (5) where E (t) is the deviatoric Green strain tensor at time t [, T] and W is the isometric strain energy of the unfatigued tissue. In an effort to reduce computational time, the damage criterion were only evaluated at the peak of each sinusoidal loading cycle. Therefore, we introduce a peak equivalent strain Ξ n peak Ξ peak n = Ξ 1 t= n+, n =, 1, 2, 3,, N (6) f 2 where f is the loading frequency and n is the number of loading cycles. 2

3 Fatigue Induced Stress Softening The stress softening caused by cyclic fatigue damage is a function of both equivalent strain and the number of loading cycles. The parameters ψ min and ψ max are employed to describe the accumulation of damage. A specimen cyclically loaded to an equivalent strain of ψ max will completely fail during the first loading cycle. A specimen loaded to an equivalent strain of ψ min will be able to endure a maximum number,n max, of loading cycles before complete failure. Therefore, ψ min represents the minimum equivalent strain to elicit the accumulation of fatigue damage, while ψ max represents the tissue equivalent strain limit. Here we chose n max = 368 million loading cycles as an upper bound because this translates to approximately 1 patient years considering a heart rate of 7 beats per minute. Therefore, the number of cycles until failure (n tot ) is given by the following novel equation: n tot Ξ peak n = β(n max 1) peak Ξ n ψmin +β peak Ξ α 1 n ψmax 1 exp 1 exp α 1 ψ min ψmax + 1 if Ξ n peak < ψ min if ψmin Ξ n peak ψ max 1 if Ξ max n > ψ max where n max represents the number of cycles required to completely damage a specimen loaded successively to an equivalent strain of ψ min, while α and β represent the rate of damage accumulation as a function of Ξ peak n. These parameters should be measured experimentally. Assuming further that at a set equivalent strain, an equal amount of damage is accumulated during each loading cycle, the rate of damage accumulation is given by dd 1 = 1, if ψ dn n min Ξ peak n ψ max. (8) tot Therefore the total amount of damage after N tensile loading cycles is given by if Ξ peak n < ψ min D 1 Ξ peak N n = n if ψ n min Ξ peak n ψ max. (9) n=1 tot 1 if Ξ peak n > ψ max Fatigue Induced Permanent Set Tissue damage is associated with permanent tissue elasticity loss. Therefore, when the elastic limits of the tissue are exceeded, a permanent set is exhibited upon unloading. A negative stress at zero deformation is equivalent to a permanent set (deformation) at zero stress; therefore, we employ a plastic stress S P to ensure this condition 14. Sp ij = (1 D 2 ) N E ij (1) D 2 = peak J+KΞ n tanh Ξ t peak Ξ n tanh(1) where D 2 is a function of the equivalent strain which governs the contribution of S P to the overall tissue response throughout loading and unloading and N is a pseudo-energy function following the theory of pseudo-elasticity of Dorfmann and Ogden 14. We chose N in the form of a modified Fung energy function, which is dependent upon the maximum strain achieved in each principal direction. This modified Fung-type strain energy function enforces a negative stress at zero deformation which is necessary to capture the permanent set. Thus, N is given by the following relations (7) (11) N = C 2 2 (eq 2 1) (12) Q 2 = B 1 (E 11p E 11 ) 2 + B 2 (E 22p E 22 ) 2 + 2B 3 E 11p E 11 (E 22p E 22 ) (13) 3

4 where E 11p and E 22p are the strains along the principal axes at the peak of the loading curve, and C 2, B 1, B 2, and B 3 are material parameters. Fatigued State Tissue Properties The inclusion of D 1 and D 2 provides a means of changing the form of the energy function, which is no longer elastic, as a response to cyclic loading. Assuming that D 1 and D 2 are independent of E, the second Piola-Kirchoff stress tensor for the matrix may be expressed in the following form: S(n) = (1 D 1 ) W E + (1 D 2) N E. (14) Therefore, the deviatoric elasticity tensor in the material description may be expressed in the following form: (n) = (1 D 1 ) 2 W +(1 D E 2) 2 N 2 E2. (15) RESULTS Determination of Damage Parameters The model parameters for the un-damaged GLBP tissue (Eqn. (2)) were obtained using the biaxial mechanical testing data reported previously by Sun et al. 15. Typical equi-biaxial testing data of GLBP is illustrated in Figure 1a. The damage criterion, ψ min and ψ max, were chosen based upon the biaxial data of GLBP. Illustrated in Figure 1b, is the contour plot of the equivalent strain. In this study, we assumed that ψ min = 9 and ψ max =, within which marks the damage evolution region. The maximum number of loading cycles, n max, was set to 368 million cycles. However, to reduce computational cost, in the simulation, n max, was proportionally scaled from 368 million to 36 cycles, reflecting an accelerated accumulation of damage. Stress (S) S 12 S 11 Strain (E) S E E11 Fig.1 - Contour plot of equivalent strain with ψ min = 9 and ψ max = marking the damage evolution region. The effects of varying α and β in Eqn. (9) on the tissue damage factor are illustrated in Figure 2. It can be seen (Fig. 2a) that an increase in α results in a more gradual accumulation of damage, whereas an increase in β accelerates the rate of damage (Fig. 2b). The corresponding effects of varying α and β on the number of cycles to failure are illustrated in Fig. 2c&d

5 a. b. Damage Factor alpha Damage Factor beta Number of Cycles to Failure Total Number of Cycles to Failure x 1 8 x alpha=2 c. beta=1 alpha= d. 3 alpha=22 alpha=23 3 alpha= alpha= alpha Fig.2 - Illustration of the effects of parameters alpha and beta on the (a&b) rate of damage evolution and the (c&d) number of cycles to failure as a function of equivalent strain. Fatigue biaxial loading simulation alp ha Number of Cycles to Failure The fatigue damage model of Eqn. (14) was incorporated in the finite element software ABAQUS via the user subroutine UMAT. To evaluate model effects, a finite element model of a biaxial test specimen was constructed and subjected to simulated equi-biaxial cyclic loading. The biaxial testing simulation model consisted of 4 plane stress elements to allow for test specimen geometry of 25 mm x 25 mm x.4 mm (Fig. 3a). Four evenly spaced node forces, with 5 mm between two adjacent nodes and 2.5 mm inside the specimen edge, were imposed on each side (Fig. 3a). Each node force was 2.5 N, imposing a net total 1 MPa Lagrangian stress on each edge. Similar to the actual biaxial testing experiments 15,16, only the central region was used for stress strain measurements. This was accomplished by averaging the stress and strain tensor components for sixteen elements located in the center of the FE model, delimiting a 5 mm x 5 mm region. Details of the constitutive model implementation into ABAQUS have been previously presented 11, 17. The biaxial test fatigue simulation demonstrated the altered mechanical properties of the tissue specimen in response to cyclic loading. As illustrated in Fig. 3c, the equi-biaixal stress-strain curves exhibited progressive tissue softening and increased permanent deformations throughout the cycles. A redistribution of the maximum principal stress throughout the central 16 elements of the tissue specimen model is evident during the progression of damage (Fig. 3d-f) beta alpha=2 beta=1 beta=2 beta=5 beta=1 beta=5 beta=1 5

6 a. T 22 c. 12 X1 fatigued state X2 fatigued state X1 unfatigued state X2 unfatigued state x 2 x 1 T11 1 b. Stress (kpa) Strain d. e. f. Fig.3 - (a) Loading conditions for simulated biaxial test specimen, and (b) the max principal stress contour plot for the simulated specimen under maximum tension. (c) The average stress-strain response of the central 16 elements throughout loading cycles. Fatigue response of simulated biaxial test specimen subjected to cyclic loading cycles. The max principal stress at peak loading of the 16 central elements at the (d) un-fatigued state, the (e) moderately fatigued state, and the (f) highly fatigued state. Fatigue BHV simulation The BHV finite element model was developed previously by Sun et al. 18. Briefly, the tri-leaflet valve is fabricated from GLBP sheets that are die-cut to form leaflets, which are then mounted onto a metal frame. In the finite element model the wireframe was modeled using beam elements. The nominal material properties of the wireframe were taken from the material properties of Elgiloy with a Young s modulus of 2.33x1 7 kpa and Poisson ratio of.3. The leaflets were attached to the wire frame and modeled using large strain shell elements. Each leaflet had its own local coordinate system for material property definitions that are fully defined by the constitutive law Eq. (14). It was assumed that each leaflet can be modeled with one set of material parameters 18. The leaflets were also assumed to have uniform thickness, which has been verified for pericardial tissue that is selected for valve fabrication. The valve model was cyclically loaded from a stress-free position to a fully loaded configuration by applying a uniform pressure to the aortic side of the leaflet. The contact between two leaflets was modeled using a master-slave contact pair (an option in ABAQUS). The leaflet that was stiffer in the radial direction was assigned as the master surface while the other was specified as the slave surface. 6

7 a. b. c. d. e. Fig.4 - (a) Damage contour plot and (b) residual strain contour plot of an unloaded (zero stress) BHV model after cyclic loading. Strain contour plot of a BHV leaflet at the un-fatigued state, the (b) moderately fatigued state, and the (c) highly fatigued state under ~12mmHg of pressure. The valve simulation demonstrated the model capability to describe the fatigue-induced effects of GLBP leaflets subjected to cyclic loading cycles of 12mmHg pressure. A contour plot of the damage factor (Fig.4a) reveals high damage in the belly region of the leaflets, and a contour plot of the logarithmic strain (Fig.4b) shows residual strains in the damaged regions at the zero-stress state. As the leaflets become more damaged, they achieve higher strains at loading as evident from the strain contour plots of a GLBP leaflet at different fatigue states in Fig.3c-e. DISCUSSION In this study, we developed a theoretical and computational framework to model soft tissue damage that incorporates the long-term fatigued-induced stress softening and permanent set phenomena exhibited by biological tissues under cyclic loading. We demonstrate the feasibility of this approach by simulating tissue fatigue damage under biaxial loading conditions as well as for BHV applications. The distinguishing characteristic of our approach with respect to that of others 7-9,19, is that our focus is on long-term fatigue effects. Unlike previous damage models for fibered composites 7-9,19 we considered an incremental accumulation of damage at a set equivalent strain, whereas other models considered an accumulation of damage only when the previous maximum equivalent strain had been exceeded. Our approach was also unique to other models that consider residual strain 14,2, because our focus was not to model the hysteresis effect upon each unloading cycle, rather the progressive permanent change in specimen geometry due to long-term loading cycles, and thus our D 1 and D 2 functions are active during both loading and unloading cycles at the onset of damage. Therefore, our framework is not intended for the accurate fitting of each loading and unloading curve, rather we assume uniform damage accumulation to model the transition from a zero-state to a fatigued state with long-term (1+ million) loading cycles. One of the limitations of this study is that we assume a linear progression of damage at a given equivalent strain as a function of loading cycles, whereas the damage may be more dramatic during the initial cycles. Also, the damage parameters used in simulations may not be representative of actual experimental fatigue data of GLBP tissue. 7

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