5th International Conference on Computational and Mathematical Biomedical Engineering – CMBE2017 10-12 April 2017, United States P. Nithiarasu, A.M. Robertson (Eds.)
DYNAMIC SIMULATION OF AORTIC VALVES: COMPARISON BETWEEN ISOTROPIC AND ANISOTROPIC MATERIAL MODEL Rana Zakerzadeh1, Michael C. H. Wu2, Ming-Chen Hsu2, and Michael S. Sacks1 1 Institute for Computational Engineering & Science, The University of Texas at Austin
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[email protected] 2 Department of Mechanical Engineering, Iowa State University
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[email protected] SUMMARY This study considers numerical simulations of bioprosthetic heart valve dynamics modeled by means of both nonlinear isotropic and anisotropic material constitutive laws. In particular, it focuses on an anisotropic model to enhance the realism of the simulations by characterizing the activation of collagen fibers at finite strains. In this framework, we investigate the influence of the material model on the numerical results and, in the case of the anisotropic model, the importance of the collagen fibers on the overall mechanical behavior of the tissue during the cardiac cycle. The methodology presented here will be particularly useful for studies of mechanobiological processes in the healthy and diseased heart valves. Key words: heart valve simulations, Fung-type hyperelastic model, anisotropic hyperelastic model 1 INTRODUCTION The aortic valve regulates the flow between the left ventricle of the heart and the ascending aorta. Diseases affecting the aortic valves result in either obstruction to forward flow (stenosis) or reversal of flow across an incompetent valve (regurgitation). In either case, valve replacement is a major treatment option for the patients suffering from heart valve diseases. Replacement heart valves fabricated from biologically derived materials are referred to as bioprosthetic heart valves (BHVs). While these devices have provided great benefits for many patients, device failure continues to be the result of leaflet structural deterioration mediated by fatigue and/or tissue mineralization. There is thus a profound need for the development of the simulation technology for simulating BHV in the design context. One conspicuous shortcoming of some earlier work in this area is the relatively simple material model of the valve leaflets. The influence of leaflet material model on its mechanics has been studied in several previous numerical works. The effects of using different leaflet material models have been investigated in [1] by comparing linear and nonlinear isotropic models of leaflets. In [2] the influence of anisotropy of a pericardial heart valve has been studied using orthotropic materials, showing that even a small amount of orthotropy can significantly affect the mechanical behavior of the valve. Wei Sun et al. [3] adopted a nonlinear fiber-based structural model to evaluate the impact of leaflet properties on the stress distribution of the BHV. However, this study has been performed using the quasi-static approach only, by applying a uniform pressure to the aortic side of the leaflet. Quasi-static simulation ignores the dynamic moving leaflets, especially during the opening/closing phases of the cardiac cycle when leaflet flexure occurs that can contribute to the leaflet damage. A systematic parametric study that determines the role of leaflets material properties in the valve’s dynamics has not yet been carried out. Built upon the computational framework presented in [4, 5], in this study, we utilize a nonlinear anisotropic material model for the leaflets to enhance the physical realism of the BHV dynamics simulation and evaluate the impact of leaflet properties on the deformation and dynamics of the BHV. This anisotropic behavior arises because of the distribution and mechanical contribution of the collagen fibers to the overall mechanical behavior of the tissue. In addition, to address the influence of the constitutive model on dynamics simulation; we present a comparison by considering both nonlinear isotropic and anisotropic constitutive models.
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2 METHODOLOGY We consider the pressure-driven structure dynamics of a deformable valve representing a BHV replacing an aortic heart valve, which regulates flow between the left ventricle of the heart and the aorta. We model the leaflets as a thin shell structure using different constitutive models for the valve structural model: isotropic and anisotropic. To model the anisotropic properties of the leaflets, we use the strain energy function W presented in [6]. The added nonlinear Neo-Hookean term represents the low-strain responses and is needed also for numerical stability [7]. Moreover, the exponential part captures the important exponential nature of the soft tissue behavior. therefore, strain energy is a function of two invariants as follows: 𝑐0 2 4 (𝐼 − 3) + 𝑐1 (𝑒 𝑐2 (𝐼1 −3) +𝑐3 (√𝐼4 −1) − 1) 2 1 Where 𝐼1 is the first invariant of the left Cauchy–Green deformation tensor C, and 𝐼4 is the fourth invariant for a transversely isotropic material. and M is the unit vector defining the collagen fiber direction in the undeformed configuration. 𝑊(𝐼1 , 𝐼4 ) =
𝐼1 = 𝑡𝑟 𝐶 𝐼4 = 𝑀 ∙ 𝐶 ∙ 𝑀 The values of the coefficients c1 c2 and c3 are obtained from the data presented in [6]. The coefficient of the Neo-Hookean term is set to 100 kPa. For the isotropic model we use a simplified isotropic approximation to the true anisotropic leaflet behaviors by combining an isotropic Fung model of collagen fiber stiffness with a neo-Hookean model of cross-linked ground matrix stiffness to obtain the following strain energy functional used previously in [4]: 𝑐0 𝑐1 2 (𝐼1 − 3) + (𝑒 𝑐2 (𝐼1 −3) − 1) 2 2 c0 c1 and c2 are material parameters. c0 denotes small-strain bending stiffness while c1 and c2 define tensile stiffness of the BHV leaflets. Since we assume that the structure is incompressible, both anisotropic and isotropic models are combined with the incompressibility condition. In order to approximate this problem, we use a thin shell isogeometric structure dynamic finite element solver developed in [5] for the simulation of an aortic bioprosthetic heart valve. We apply a physiological transvalvular pressure load to the leaflets to enforce the pressure difference between left ventricle and aorta (Figure 1). The pressure data taken from [8] where the duration of a single cardiac cycle is 0.76 s. Defining contact between leaflets is important to prevent the leakage back into the left ventricle during the closing phase. We use the contact algorithm discussed in [5]. 𝑊(𝐼1 ) =
Figure 1. Transvalvular pressure applied to the leaflets as a function of time.
Directions of collagen fibers are modeled as unit vectors in the tangent space of a reference configuration for the tissue and it is assumed that initially the fibers are aligned in the circumferential direction through the leaflet. We use the valve geometry discussed in [5]. The thickness of the leaflets is 0.0386 cm and the density is 1.0 g/cm3. The time-step size used in the dynamic simulation is 0.0001 s and the clamped boundary condition is applied to the leaflet attachment edge. 3 RESULTS AND DISCUSSION The deformation and strain distribution of the leaflets at several points in the cardiac cycle (after reaching a periodic solution) is illustrated in Figure 2, compared also with the results from our 619
isotropic model. We observe a noticeable influence of the material model; namely isotropic vs anisotropic on the overall tissue deformation. Anisotropic
Isotropic
Figure 2. Comparison between different material models. Deformation of the valve at several points during the cardiac cycle predicted from the structural dynamics computations colored by Maximum in-plane principal Green-Lagrange strain, the largest eigenvalue of E.
To investigate the effect of alteration of the tissue mechanical properties, we also simulate the valve deformations with various perturbations of the tissue properties including the collagenous fiber orientation, as well as the fiber and matrix stiffness. These results can provide guidelines for designing leaflet tissues to improve valve durability. The sensitivity results for different values of fiber stiffness for several points of a cardiac cycle is presented in Figure 3. The chosen time points are consistent with Figure 2. We observe that perturbing fiber stiffness coefficient, has a significant influence on the valve deformation only at the closing stage. The reason is that in dynamic simulation of heart valve in a full cardiac cycle, the opening phase is in the small strain regime, when tissue does not behave exponentially and the Neo-Hookean term dominates. However, when the valve is closed, the exponential terms dominate that representing the fiber action. The goal is to provide a framework which enables us to use more physically realistic soft tissue material model in heart valve simulations. Starting from the microstructural material model presented in [9] we are currently developing a phenomenological model that covers the interaction between the tissue’s collagen fibers and its ground matrix as well as the interaction between different fibers. We use this computationally tractable material model to facilitate the integration of the model into our FSI solver [5]. 4 ACKNOWLEDGMENTS This work has been supported by NHBLI grant R01 HL129077. HPC resources that have contributed to the results reported in this paper are provided by the Texas Advanced Computing Center (TACC).
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0.1 c3
c3
10 c3
Figure 3. Comparison between different values for fiber stiffness parameter in anisotropic model, at several points during the cardiac cycle; opening of the valve (first row), fully opened valve (middle row), and closing of the valve (third row)
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