Patient-Specific Computational Simulation of the Mitral Valve Function Using Three-Dimensional Echocardiography

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1 Patient-Specific Computational Simulation of the Mitral Valve Function Using Three-Dimensional Echocardiography Y. Rim 1, S. T. Laing 1, P. Kee 1, K. B. Chandran 2, D. D. McPherson 1 and H. Kim 1 1 Department of Internal Medicine, The University of Texas Health Science Center at Houston, Houston, Texas, USA 2 Department of Biomedical Engineering, The University of Iowa, Iowa City, Iowa, USA Abstract - Abnormal mitral valve (MV) morphology may lead to high stress generation. Computational simulation of the MV apparatus may help us better understand and characterize the biomechanics and physiology of MV function and disease-related alterations. The MV apparatus geometry was identified using three-dimensional (3D) transesophageal echocardiographic (TEE) patient data. These data were converted into a 3D computational MV model. Computational simulation of the MV function was performed using finite element analysis with an experimentally-determined material model. Functional and morphological characteristics were evaluated and compared to the 3D TEE data. The MV simulation demonstrated the complex 3D MV shape with good agreement to 3D TEE, and determined regions of stress concentration in the MV components. Asymmetric stress distribution was clearly displayed on the MV leaflets indicating realistic and physiologic alteration of MV morphology and function. This novel computational simulation strategy may provide a powerful tool for patientspecific structural evaluation of the MV. Keywords: Mitral valve, Finite element analysis, Threedimensional echocardiography, Computational modeling, Patient-specific, Simulation 1 Introduction The mitral valve (MV) apparatus has a complex anatomical structure consisting of two asymmetric leaflets, a saddle-shaped annulus, chordae tendineae, and papillary muscles [1]. It is important to assess the physiologic characteristics of the MV apparatus. Abnormally increased stress distribution in the mitral leaflets may play a major role in pathological alteration, improper remodeling and malfunctions of the MV apparatus. Three-dimensional (3D) echocardiography can provide detailed morphology of the MV leaflets and annulus contributing to our understanding of MV function and anatomy. It is more suitable to conduct 3D assessment for the evaluation of MV function because of the complex MV structure [2]. Surgeons are now using 3D transesophageal echocardiography (TEE) prior to MV repair to evaluate MV geometry and focus the surgical procedure, particularly for MV prolapse. Current clinical 3D echocardiography can demonstrate excellent volumetric morphology of the MV apparatus and provide information on the regurgitated flow jet across the MV leaflets using Doppler ultrasound in real time allowing evaluation of mitral regurgitation [3]. However, biomechanical information such as high stress concentration and abnormal bending curvature within the MV apparatus structure is not available from 3D echocardiography alone. Numerical evaluation methods such as finite element analysis can be utilized to assess biomechanical characteristics of the MV apparatus. Finite element analysis is an effective method for morphologic evaluation and stress determination of native aortic and mitral valves [4-6] as well as bioprosthetic valves [7, 8]. This is due in part to our understanding that localized concentration of mechanical stress and large flexural deformation are closely related to tissue degeneration and calcification in heart valve diseases [9-11]. Specifically for MV pathology, annular shape and geometric distribution of chordae tendineae play an important role with respect to functional valvular abnormalities [12-14]. The combination of 3D echocardiography and computational simulation can provide a powerful tool to evaluate complex structural and functional information of the MV apparatus. In this study, we develop an integrated modeling platform to create a precise MV geometry model using patient 3D TEE data, and perform computational evaluation of the MV function. This computational evaluation strategy, with rigorous validation, may help us better understand the dynamics of MV function and provide a comprehensive noninvasive imaging and evaluation techniques potentially improving the diagnosis and treatment of MV pathology. 2 Materials and Methods The MV apparatus including the anterior and posterior leaflets, the mitral annulus and the location of the papillary muscles was identified using 3D TEE data of a patient. This study was approved by the Institutional Review Board of The University of Texas Health Science Center at Houston. 2.1 Finite element modeling of the MV A Philips ie33 ultrasound unit (Philips Medical Systems, Bothell, WA) with a 3D TEE transducer was

2 utilized to obtain the 3D geometry of the MV and its apparatus. Following our standard 3D TEEE echo protocols, 3D TEE was performed to acquire the best MV geometry. The 3D TEE data was stored in digital format and converted into a 3D computational MV model. Fig. 1 demonstrates the algorithms to convert 3D echocardiographic data of the MV apparatus into a finite element model followed by computational dynamic simulation. The 3D coordinate data of the MV apparatus structure was digitized, and the MV leaflets and annulus geometry segmented and traced in a multiple number of sagittal binary images of the MV structure. The 3D geometry of the leaflets was created using a B-spline surface modeling method. All the algorithms to create 3D MV geometry were developed using MATLAB software ( The Mathworks Inc., Natick, MA). The 3D MV leaflets geometry was imported to ABAQUS (SIMULIA, Providence, RI), and meshed using S4R element type (4- node shell elements). The main and two secondary chordae tendineae weree added to the MV model by connecting the location of the papillary muscles and the edge of two leaflets [15]. Each chordae tendineae was modeled using a series of 3D line elements (T3D2 element type). Figure 1. Flowchart of MV modeling and finite element dynamic simulation 2.2 Material modeling of the MV The MV leaflets were modeled as a hyperelastic material. The nonlinear hyperelastic mechanical behavior of the leaflet tissue was modeled using a Fung-elastic constitutive model [16]. Material parameters were determined by fitting the biaxial mechanical test data of the anteriorr and posterior leaflet tissue from a previous study [17]. The Levenberg-Marquardt nonlinear least squares algorithm was used for the curve fitting. Elastic properties of the MV leaflets were defined by nearly incompressible, nonlinear and anisotropic strain energy function W with four parameters (c, A1, A2 and A3) as the shear components are small enough to be negligible [17]. The strain energy function used in this study is as follow. c W = [ 1], 2 e Q 2 2 Q = A1 E11 + A 2E A3 E11E (1) 22 In nearly incompressible hyperelastic material, the relationship between the Cauchy stress and the Green- Lagrange strain is given by σ c = ( 2E ) cexp( Q )( A1 E11 + A3 3E 22) (2) σ r = ( 2E ) cexp( Q )( A3 E11 + A2 2E 22) (3) The stress-strain relationships were definedd along the circumferential (σ c ) and radial (σ r ) directions, and the parameters obtained by fitting the previously reported experimental data [17]. The leaflet thickness was set to 0.69 mm for the anterior leaflet and 0.51 mm for the posterior leaflet. The chordae tendineae were modelled using a linear elastic and isotropic material model with a Young s modulus of 470 MPa and a Poisson s ratio of 0.48 [18]. Density and cross-sectional area of the chordae tendineae were set to 1,100 kg/m 3 and 0.4 mmm 2, respectively [18]. Validation studies pertaining to implementation of these material properties into ABAQUS were performed by biaxial test simulation with a single element model. Computer-predicted stress-strain relationship from the single element biaxial test simulation was compared with the experimental data. Following the vigorous validation studies, we performed dynamic simulation of the MV apparatus model created from a patientt 3D TEE data. 2.3 Boundary conditions for MV simulation Boundary conditions were assigned to the finite element MV apparatus model such that all the degrees of freedom were restricted along the saddle-shaped mitral annulus. The locations of the papillary muscles were also fixed. Pressure change in the left ventricle and left atrium is shown in Fig. 2 [19]. Time-varying physiological pressure difference across the MV was applied on the anterior and posterior leaflet surface as pressure loads.

3 PL AL PL AL (a) 3D TEE dataa of the MV when closed and open Figure 2. Pressure change across the MV during a cardiac cycle During the closing phase, coaptation of the MV leaflets was modeled using the surface-to-surface contact algorithm with the penalty contact method in ABAQUS. 3 Resultss 3.1 Finite element modeling of the MV The 3D echocardiographic image data were successfully converted to a 3D finite element MV model. Fig. 3a depicts the extracted 3D volumetric MV structure when the valve is closed and open. The mitral annulus and two leaflets are clearly visible from the ventricular view. A 3D surface model of the MV apparatus was created with the user-defined semi-automated algorithm, and then imported to ABAQUS followed by meshing and adding the chordae tendineae. Fig. 3b and 3c demonstrate the consequent structural model of the MV apparatus with a total of 656 shell elements for the leaflets and 296 line elements for the chordae tendineae. Anomaly of the MV leaflet morphology was clearly observed in both 3D TEE image data and the finite element model. Surface smoothnesss of the leaflets was excellent allowing successful surface mesh subsequently. No mesh distortion was observed. 3.2 Validation studies of the material modeling The anterior and posterior leaflets have different material characteristics. We have implemented the experimentally determined material properties of the anterior and posterior leaflets from a previous study [17] into the finite element model. The posterior leaflet is more extensible than the anterior eaflet. Validation studies of the material modeling of the MV leaflets were performed. In Fig. 4, x- and y-axis represent the Green-Lagrange strain ( E) and the Cauchy stress (σ), respectively. Nonlinear (b) Finite element model of the MV apparatus (c) Components of the 3D finite element MV model Figure 3. Computational MV apparatus model created by using patient 3D TEE data regression curves (solid lines) fitted to the Fung-elastic constitutive model [Eq. (1)-(3)] showed good agreement to the experimental stress-strain relationships dataa for both anteriorr and posterior leaflets (r ). The open circles and triangles demonstrate the previously reported biaxial mechanical test data [17], and the computer-predicted data obtained from single element biaxial test simulations using the implemented Fung-elastic constitutive model and the material parameters are shown in solid circles and triangles. These validation data indicate that the Fung-elastic constitutive model has been successfully implemented in the ABAQUS platform and this model can be utilized for simulations of the complex MV function.

4 (a) Atrial view (b) Commissural view Anterior leaflet (a) Anterior leaflet Posterior leaflet (c) Oblique view Figure 5. Von Mises stress distribution in the MV apparatus at the closed position (b) Posterior leaflet Figure 4. Stress-strain relationships of the anterior and posterior leaflets 3.3 Dynamic simulation of the MV function We performed dynamic simulation of the MV function over the cardiac cycle. The von Mises stress distributions in the MV leaflets and annulus at the closed and open positions are shown in Figs. 5 and 6. The largest stress values were distributed in the vicinity of the commissures at the fully closed position (Fig. 5). For this particular patient data, the highest stress value was observed near the regions with extremely high curvature where self-contact (i.e. contact between the elements in the same leaflet) occurred in both the anterior and posterior leaflets. The maximum stress value was located at the selfcontact region in the anterior leaflet (1.17 kpa). The stress distribution was centralized at the belly region and spread toward the annulus. It was clear that the regions with relatively high stress values corresponded to the highly stretched regions or the self-contact regions with large curvature for both the leaflets. It is noteworthy that the Posterior leaflet (a) Atrial view (c) Oblique view (b) Commissural view Anterior leaflet Figure 6. Von Mises stress distribution in the MV apparatus at the open position

5 anterior and posterior leaflets showed different stress distribution patterns and values. At the fully open position, relatively large stress values ( kpa) were distributed near the commissural areas where the two leaflets were connected (Fig. 6). The posterior leaflet was under relatively larger stress than the anterior leaflet at this position. The stress distribution had a similar pattern as the strain distribution in both the leaflets. 4 Discussion With the integrated modeling platform developed in this study, we acquired 3D TEE data of a patient including the anterior and posterior leaflets, the annulus and the location of the papillary muscles to create a 3D virtual MV apparatus model. The MV leaflets geometry was segmented in a multiple number of sagittal binary images, threedimensionally reconstructed, and meshed. The chordae tendineae were added to complete the finite element modeling of the MV apparatus. The material characteristics of the anterior and posterior leaflets were modeled as a nonlinear hyperelastic material, and successfully implemented into ABAQUS and validated by single element biaxial mechanical test simulations. Computational simulation of the MV function was performed using finite element dynamic analysis. Even though many studies have been reported in computational simulations of the MV, most of the previous studies performed simulations with geometrically simplified models due to the complexity of the MV apparatus [18, 20-22]. Precise 3D geometry of the MV apparatus acquired from 3D echocardiography can be utilized for computational evaluation of MV functional characteristics. Computational simulation combined with 3D echocardiography can help us better understand the features of dynamic morphologic and structural alterations of the MV apparatus capturing the consequences of valve function in microseconds. Moreover, the use of patient-specific MV apparatus geometry may be a crucial factor to determine the extent and severity of abnormality of each structural component, such as extreme local stress concentration in the leaflets, annulus, or chordae tendineae during valve function. The integrated modeling platform developed in this study allows us to create a geometrical MV apparatus model from patient 3D TEE data for computationally accurate and physiologic evaluation of the MV function. Abnormally deformed MV leaflet morphology was clearly observed in both the 3D echocardiogram and the corresponding 3D structural MV model. Validation studies of the material modeling of the MV leaflets showed good agreement between the experimental data, curve fitting data, and finite element-predicted data. Using these MV modeling techniques and material models, we were able to perform finite element dynamic analysis of the MV function of a patient. The dynamic simulation demonstrated different stress distribution patterns between the anterior and posterior leaflets. In addition, the largest stress values were observed around the self-contact regions in both the leaflets indicating that abnormal distortion of leaflet morphology could induce extremely high stress in the leaflet. There are several limitations in the present study. Material parameters for the Fung-elastic constitutive model were determined by fitting the porcine MV tissue data. However it is not feasible to obtain material properties of patient MV tissue at present. In the future studies, we will employ improved material properties for modeling of the MV leaflets and chordae tendineae when experimental data of human tissue are available. To our knowledge, this is the first study to conduct a computational simulation using 3D echocardiographic image data of a patient to evaluate biomechanical characteristics of the MV structure over the cardiac cycle. The presented modeling algorithm and simulation techniques can help us understand the dynamics of MV function. This computational MV evaluation strategy may provide comprehensive noninvasive imaging with valuable biomechanical information of the MV during valve function potentially improving the diagnosis and treatment of MV pathology. This methodology has the potential to quantitate the extent of disease-related functional alterations and restoration towards normal valvular function following MV repair. 5 References [1] S. Y. Ho, "Anatomy of the mitral valve," Heart, vol. 88 Suppl 4, pp. iv5-10, Nov, [2] J. Hung, R. Lang, F. Flachskampf, S. K. Shernan, M. L. McCulloch, D. B. Adams, J. Thomas, M. Vannan, and T. Ryan, "3D echocardiography: a review of the current status and future directions," J Am Soc Echocardiogr, vol. 20, pp , Mar, [3] J. Kwak, M. Andrawes, S. Garvin, and M. N. 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