A Multiscale Finite Element Simulation of Human Aortic Heart Valve

Shahrokh Shahi1, a, Soheil Mohammadi1, b 1School of Civil Engineering, University of Tehran, Tehran, Iran

ashahi.shahrokh@ut.ac.ir, bsmoham@ut.ac.ir

Keywords: Biomechanical simulations, Multiscale, Finite element, Aortic valve, Tissue engineering.

Abstract. Some of the heart valve diseases can be treated by surgical replacement with either a mechanical or bioprosthetic heart valve (BHV). Recently, tissue-engineered heart valves (TEHVs) have been proposed to be the ultimate solution for treating valvular heart disease. In order to improve the durability and design of artificial heart valves, recent studies have focused on quantifying the biomechanical interaction between the organ, tissue, and cellular –level components in native heart valves. Such data is considered fundamental to designing improved BHVs. Mechanical communication from the larger scales affects active biomechanical processes. For instance any organ-scale motion deforms the tissue, which in turn deforms the interstitial cells (ICs). Therefore, a multiscale solution is required to study the behavior of human aortic valve and to predict local cell deformations. The proposed multiscale finite element approach takes into account large deformations and nonlinear anisotropic hyperelastic material models. In this simulation, the organ scale motion is computed, from which the tissue scale deformation will be extracted. Similarly, the tissue deformation will be transformed into the cell scale. Finally, each simulation is verified against a number of experimental measures.

Introduction

The aortic heart valve is a one-way valve located between the left ventricle and the aorta, preventing blood from flowing backwards into the heart. Distinct mechanical behaviors govern this vital biomechanical structure at different length scales in the aortic valve [1], A mechanical communication can be observed between the scales in a way that behavior at one scale can influence another [2].

While each simulation is performed across a distinct set of length scale, new techniques need to be developed to link these scales[2]. To the best of our knowledge, such links have only been performed in a few cases, such as embedding the valve organ-scale motion to the larger organ-scale motion of the left ventricle [3] and linking the cell-scale model to the pressure applied at the organ-scale by using an analytical approach [4]. Weinberg et al. perform a comprehensive multiscale simulation of heart valve mechanics. They introduced a system of multiple reference configurations set up at different cell-, tissue- and organ-scale with a linkage between scales from top to bottoms. In this approach element deformations were tracked in some particular points at upper scales for mapping to the lower scales as boundary conditions [2].

In this work, a hierarchical simulation of the aortic valve has been developed to investigate how mechanical stimuli in the organ scale are transmitted to interstitial cells in the cell scale. For this purpose, a simple one-way coupling is utilized to relate different scale simulations by passing data from the largest scale to the smallest one. First, the organ-level simulation is performed. The results of element deformations at the organ-level simulation are applied, as boundary conditions, to a representative volume element (RVE) of tissue. Then, deformations from the tissue-level simulation are passed similarly to the cell-level simulation and the cell aspect ratio will be extracted. The aortic valve leaflets undergo large displacements through the blood pressure and due to material constituents, known as anisotropic hyperelastic materials [2, 4-7]. Accordingly, a Matlab code has been developed to predict the highly nonlinear and transversely isotropic material behaviors based on the total lagrangian procedure of large deformation theory [8] within a multilevel finite element method.

Applied Mechanics and Materials Vol. 367 (2013) pp 275-279© (2013) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMM.367.275

All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP,www.ttp.net. (ID: 151.239.98.143-20/06/13,18:32:29)

CC

For each simulation, an initial reference configuration is chosen and the geometry and constitutive models are described at that configuration. The results will be compared with a set of published experimental data [4, 5, 9, 10].

Organ scale simulation

Geometry of the aortic valve (AV) was created in SolidWorks (Fig. 1a). One separate leaflet is considered as an organ-level shape and created by sweep and loft features in SolidWorks. Dimensions were determined from collected measurements [7]. The leaflet is positioned in the unloaded configuration. The 8-node brick meshes of the domain were created in the ABAQUS CAE preprocessor. Finally, the meshed geometry was imported to the developed Matlab program.

Aortic valve leaflets exhibit typical soft tissue mechanical behavior such as incompressibility and strong anisotropy, which leads to a much stiffer reaction in the circumferential direction [5, 7]. A transversely isotropic hyperelastic constitutive model can best describe most prominent aspects of leaflet soft tissue mechanics [2, 4, 7]. In order to describe the anisotropic hyperelastic response of the aortic valve leaflets, the following form is employed for the strain energy function ψ [11]:

(1)

where in the volumetric part, which taken into account the incompressibility, κ is a positive penalty parameter [11, 12]. The isochoric part of the strain energy function is written as the sum of two exponential terms, where the second term accounts for transverse isotropy [7],

(2) ,)(ψ)(ψ),(ψ 4141 IIII +=

in which

(3)

)1(2

)(ψ )3(

11

121

111 −= −Ice

c

cI

,0,)1(

2

0,0)(ψ

4)1(

41

42

4

42

441

≥−

<= −

Iec

c

I

I Ic

where c11, c12, c41 and c42 are the material parameters, set to 4.7483, 27.3732 kPa, 80.4291 and 11.9752 kPa, respectively. Ī1 is the first invariant of the modified right Cauchy-Green tensor and Ī4=a0. .a0 is known as the pseudo-invariant. {a0}={0 cos(θ) sin(θ)}T is a unit vector representing the orientation of collagen fiber aligned with the angle of θ. As can be seen in Fig. 1(d), the leaflet is partitioned into strip sections with the angle of collagen fibers determined based on experimental measurements (see Table 1) [4].

In this simulation, constant physiological pressures of 1, 2, 4, 60, and 90 mmHg were applied on the ventricular side of the leaflet, as shown in Fig. 1(e), and the deformations of specific elements were mapped to the tissue-level RVE (Fig. 2).

Table 1 Angle of collagen fibers in separate strips of the aortic valve leaflet

Strip ID 1L 2L 3L 4L 2M 3M 4M

Fiber angle (θ) 5˚ 5˚ 10˚ 10˚ 0˚ 0˚ 0˚

,)1(2

1),(ψ),,(ψψ

:)(

2

part isochoric

4141

����������

partvolumetricJU

JIIJII −+== κ

276 Mechanics, Simulation and Control III

Fig 1. (a) CAD geometry of the whole valve, (b) coordinate directions of the AV, (c) coordinate direction of the Leaflets, (d) leaflet model with strip section identification, and (e) uniform pressure loading on ventricular side of the AV leaflet

Tissue Scale Simulation

At the tissue scale, the aortic valve leaflet is comprised of three distinct layers with the overall thickness of about 0.2-2.0 mm: the ventricularis, spongiosa, and the fibrosa. The fibrosa and ventricularis have embedded families of aligned collagen fibers, the fibrosa is highly undulated, and the spongiosa is gel-like. The organ level simulation does not predict the local tissue deformations, Therefore a multi-layered model of tissue should be created with continuum models for each layer.

Similar to the organ level model, the geometry was created in SolidWorks and the mesh generated in ABAQUS CAE was imported to the developed Matlab code.

In this scale the fibrous layers are considered to be composed of an isotropic matrix reinforced with one fiber family in the circumferential tissue direction. Hence the isochoric part of the strain energy function for the fibrosa and ventricularis are written as the sum of two exponential forms. Additionally, the existing bending data for these layers exhibit an initial modulus that it can be represented by a single-term Mooney-Rivlin formulation. The complete strain energy function for a fibrous layer can be written as [2]:

where C1m, C2m, C1f, C2f, and C1 are experimental constants. The values of these constants for fibrosa are set to 0.25 Pa, 16.06, 102.51 Pa, 303.71, and 2e3 Pa, respectively, and 2.5e5 Pa, 10.95, 0.04 Pa, 5.40, and 2e3 Pa for ventricularis. Spongiosa is modeled by a single-term Mooney-Rivlin strain energy function with C1=20 Pa.

Cell Scale Simulation

The cell-level RVE consists of a single IC surrounded by extracellular matrix (ECM), either fibrosa or ventricularis. The geometric data are extracted from existing experimental measurements [4]. The steps for geometry and mesh creation are the same as the upper scale simulations. While constitutive models for the ECM are considered to be identical to the related tissue-level model, a

(4)

( ) [ ]( )�����

��� ���� ��

��� ���� ��

initial

fiber

242

matrix

12

ψ

11

ψ

)3(

2

1

ψ

)3(141 )3(1

21),(ψ −+−+−×= −−

ICeC

CeCII

IC

f

fIC

mfm

(a)

(b)

(c) (d)

(e)

Applied Mechanics and Materials Vol. 367 277

single-term Mooney-Rivlin material with the value C1=400 Pa is employed to model the cell. The cell aspect ratio (CAR) is obtained from the cell-level simulation and they are compared with available set of experimental data [4].

Org

an l

evel

sim

ulat

ion

Tis

sue

leve

l si

mul

atio

n

Cel

l le

vel

sim

ulat

ion

Fig 2. Linking of AV simulations at different scales and the predicted results

Results

Fig. 2 illustrates an overall description of the proposed approach and the result of each level simulation for a diastolic pressure of 90mmHg. In each level simulation, the patterns and magnitudes of predicted deformation and stress are consistent with those of other AV models reported in the literature [9, 10]. As a result of this multi-level simulation, the predicted CARs for the applied physiological pressures are compared with the experimental measurements in Fig. 3, where a good agreement can be observed.

circumferential

radial

radial circumferential

278 Mechanics, Simulation and Control III

Fig 3. Predicted cellular aspect ratios compared to experimental measurements

Conclusion

A multiscale finite element approach has been developed to predict the local deformations of interstitial cells in the aortic valve leaflets through linking a set of models at different length scales.

While the models represent a simplified version of the native AV, a good agreement with experimental data can be observed. Therefore the proposed approach can be used in the study of the mechanical behavior of heart valves and such a procedure may also be regarded as a predictive multiscale modeling in computational biomechanics. The developed code can also be employed not only for analysis of native biomechanical systems but also in the design of tissue engineered constructs, such as TEHVs.

References

[1] M.S. Sacks, and A.P. Yoganathan: Philosophical Transactions of the Royal Society B: Biological Sciences Vol. 362 (2007), p. 1369

[2] E.J. Weinberg, and M.R. Kaazempur Mofrad: Cardiovascular Engineering Vol. 7 (2007), p. 140

[3] C.J. Carmody, G. Burriesci, I.C. Howard, and E.A. Patterson: Journal of Biomechanics Vol. 39 (2006), p. 158

[4] H.Y.S. Huang: Micromechanical simulations of heart valve tissues, University of Pittsburgh (2005)

[5] K. Billiar, and M. Sacks: Transactions-American Society of Mechanical Engineers Journal of Biomechanical Engineering Vol. 122 (2000), p. 23

[6] E.J. Weinberg, and M.R. Kaazempur Mofrad: Journal of Biomechanics Vol. 41 (2008), p. 3482

[7] T. Koch, B. Reddy, P. Zilla, and T. Franz: Computer Methods in Biomechanics and Biomedical Engineering Vol. 13 (2010), p. 225

[8] K.J. Bathe: Finite element procedures, (Prentice Hall, 1996).

[9] A. Cataloglu, R.E. Clark, and P.L. Gould: Journal of Biomechanics Vol. 10 (1977), p. 153

[10] K.J. Grande, R.P. Cochran, P.G. Reinhall, and K.S. Kunzelman: Annals of Biomedical Engineering Vol. 26 (1998), p. 534

[11] G.A. Holzapfel: Nonlinear solid mechanics: a continuum approach for engineering, (Wiley, 2000).

[12] J. Bonet, and R.D. Wood: Nonlinear continuum mechanics for finite element analysis, (Cambridge University Press, 2008).

Applied Mechanics and Materials Vol. 367 279