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Ann Thorac Surg 2002;73:1122-1129
© 2002 The Society of Thoracic Surgeons

Original article: cardiovascular

Dynamic analysis of the aortic valve using a finite element model

Ramakrishna Gnyaneshwar, BTecha, Ramarathnam Krishna Kumar, PhDa, Komarakshi R. Balakrishnan, MBBS, MS*b

a Department of Mechanical Engineering, Indian Institute of Technology, Madras, Chennai, India
b Department of Cardiothoracic Surgery, Ramachandra Medical College, Porur, Chennai, India

Accepted for publication October 17, 2001.

* Address reprint requests to Dr Balakrishnan, Professor and Head, Department of Cardiothoracic Surgery, Ramachandra Medical College, Porur, Chennai 600 116, India
e-mail: cvskrb{at}giasmd01.vsnl.net.in


   Abstract
Top
 Abstract
Introduction
 Material and methods
 Results
 Comment
 Acknowledgments
 References
 
Background. The major aim of this study was to examine the leaflet/aortic root interaction during the cardiac cycle, including the stresses developed during the interaction.

Methods. Dynamic finite element analysis was used along with a geometrically accurate model of the aortic valve and the sinuses. Shell elements along with proper contact conditions were also used in the model. Pressure patterns during the cardiac cycle were given as an input, and a linear elastic model was assumed for the material.

Results. We found that aortic root dilation starts before the opening of the leaflet and is substantial by the time leaflet opens. Dilation of the root alone helps in opening the leaflet to about 20%. The equivalent stress pattern shows an instantaneous increase in stress at the coaptation surface during closure. Stresses increase as the point of attachment is approached from the free surface.

Conclusions. The complex interplay of the geometry of the valve system can be effectively analyzed using a sophisticated dynamic finite element model. Results not previously brought out by the earlier static analysis shed new light on the root/valve interaction.


   Introduction
Top
 Abstract
 Introduction
Material and methods
 Results
 Comment
 Acknowledgments
 References
 
The aortic valve has been extensively studied over the last few decades [1]. However, with increasing surgical sophistication in aortic root surgery, including valve sparing root replacements, the simplistic assumption that the valve consists of leaflets that open and close due to pressure differentials on either side is inadequate.

Sophisticated analysis of the complex interplay between the aortic root and the leaflets, as well as the development of stresses and strains in the leaflet during opening and closing, is now possible using powerful computer techniques. This may aid in a better understanding of the dynamics of the aortic valve complex and hopefully will help in designing better substitutes for replacing the diseased aortic valve. More importantly, these might prove invaluable in predicting patterns of failure of replacement devices or repair techniques in the workbench setting rather than experiencing it in the clinical situation, with unfortunate consequences.

There have been several attempts to understand the mechanics and function of the aortic valve and its surrounding geometric structures through finite element analysis, a powerful mathematical tool [1–3]. These models have used static finite element analysis. Dynamic analysis has been used to study stresses in an artificial leaflet by Thornton [4]; but only a small part of the cardiac cycle was used for the study, which does not bring out the dynamic aspects of the leaflet/aortic root interplay. As pointed out by Thubrikar in the discussion of work by David and colleagues [5], there is a need to understand the geometry of the valve in a dynamic state. Also, no other author has studied the aortic valve using a geometrically accurate system of leaflet and the sinuses with a dynamic finite element model. The work undertaken in this study brings out certain important features that are not captured by the static finite element model.


   Material and methods
Top
 Abstract
 Introduction
 Material and methods
Results
 Comment
 Acknowledgments
 References
 
An accurate surface model of the valve was constructed using a computer aided design software SDRC/IDEAS (EDS, Plano, Texas, USA). Earlier work on the aortic valve has been well documented along with substantial details of the geometry [1]. Such dimensions have been used in recent studies [3]. The procedure illustrated by Thubrikar [1] has been used to create the current model. The dimensions are given in Table 1 [1]. The thickness variation of the leaflet, as given in [1], has also been incorporated into the model.


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  		Table 1. Dimensions of Valve Used in Study

 
The surface model was used to develop a shell finite element model using the same package. A total of 5,480 shell elements were generated. The model is shown in Figure 1. The mesh was further refined manually to reduce distortion and warping. Symmetry has not been exploited in this model because of the type of analysis to be performed.



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  		Fig 1. (A) Finite element model of the aortic valve with sinuses. (B) Cut section of the finite element model showing the leaflet in open position.

 
To ensure impenetrability between the leaflets, detailed contact conditions were defined. Self-contact due to warping of the leaflet during analysis and leaflet-to-leaflet contact were invoked. Although prior models have used contact [2, 3], no self-contact was considered. Also, contact condition has been defined between two deforming bodies. This contact condition allows sliding of the contacting bodies. The model thus created was exported to the nonlinear finite element package of ABAQUS (Hibbitt, Karlsson, and Sorensen, Inc., Pawtucket, Rhode Island, USA). A nonlinear shell formulation takes care of the inherent geometric nonlinearities of the leaflets [6]. All finite element calculations in this study were done using this package.

One of the major issues in modeling an aortic valve (or any other biological organ) is the definition of the stress-free state. As pointed out by Xie and colleagues [7], an unloaded organ (in their study, a vein) is not in a stress-free state. Clearly, further work is required to study the stress-free state or to quantify the residual stress state of an aortic leaflet. The major effect of the residual stress state is the actual material behavior, which is highly nonlinear. In this work, the leaflets were assumed to be stress free in the open position.

Although the finite element model is complete and detailed, the material used in this study is linear elastic. The actual material model ideally to be used is hyperelastic, taking into account the anisotropy. Though ABAQUS supports hyperelastic material model, the published data [1], essentially uniaxial in nature is not accurate enough to fit an Ogden model or Mooney-Rivlin model [6]. A proper material characterization requires biaxial tension results as well. This aspect may be the main limitation of this work. However, in this study, the modulus has been chosen to mimic the actual behavior as closely as possible.

The elastic modulus in this work has been taken to be 1 Mpa for the leaflet [1]. The stress-strain relation is essentially bilinear in nature, with a transition region [1, 8]. The modulus varies by a factor of nearly 150 between the pre- and posttransition strains [1]. The range of the modulus is also very high even within a region. Grande and colleagues [8] have assumed the leaflets to be in the posttransition region and have used a corresponding modulus value. However, within a leaflet there are areas where the strains are in the transition region or even in the pretransition region. Hence it is difficult to choose a value of modulus for the analysis, and from the data given in [1], 1 MPa appears reasonable. The modulus of the sinus was taken to be 2 MPa. Experiments conducted on porcine aortic root wall [9] show that the behavior of the aortic root is highly anisotropic. Young’s modulus values measured are about 1 to 2 MPa. Hence, in this work 2 MPa has been assumed.

Such assumptions will not have a serious effect on stress, as the deformation is controlled by pressure acting on the aortic system. However, the calculated displacements will be conservative for a higher modulus. A Poisson’s ratio of 0.3 was chosen to overcome the numerical difficulty encountered at values near 0.5. The density of the leaflets was taken to be 1.1 g/mL and that of the sinus 2.0 g/mL.

Finite element analysis requires restraint conditions and pressure to be given as inputs for the problem. The following pressure-time relationship is assumed in this work [10].

The surfaces were classified as aortic surface, ventricular surface, and leaflet surface. The pressure on the aortic surface was modeled as two ramps, with the first ramp accounting for a pressure drop from 120 mm Hg to 80 mm Hg and the second ramp indicating an increase of pressure from 80 to 120 mm Hg. The complete cycle is shown in Figure 2. The pressure applied to the ventricular region is also shown. Again, the pressure variations are modeled as a number of linear ramp loadings. The pressure applied in the leaflet is, in principle, the pressure difference between the ventricle and the aorta. Figure 2 also brings out this pressure difference. The pressure cycle followed in this work is explained with respect to Figure 2. The start of the analysis, T0 indicates a particular stage in the diastolic cycle. T1 indicates the end of the diastolic cycle. The pressure in the left ventricle increases from this time. At T2, which equals approximately 0.4 seconds, the leaflet pressure attains a positive value and the leaflet opens. At T3 (approximately 0.47 seconds), the aortic pressure overcomes the ventricular pressure and the leaflets start closing. Although the entire cardiac cycle is for 0.83 seconds, in the actual execution of this program, the time of closure is reduced to compress the time required to run the problem.



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  		Fig 2. Pressure boundary condition with time. a = aortic pressure; b = left ventricular pressure; c = leaflet pressure.

 
The model was run using the ABAQUS/EXPLICIT code. The dynamic analysis takes into account the inertial forces of the structure [11]. One of the major problems of explicit technique is the stability of the time marching scheme. The stable time step size, {Delta}t is given by the Courant conditions as: {Delta}t <= Lmin /Cd, where Lmin is the smallest element dimension in the mesh and Cd is the dilatational wave speed in the material. Because extremely fine mesh has been used in this study, the stable step size is very small. Nevertheless, no artificial increase in density has been resorted to increase {Delta}t.

The program was run in a twin processor SGI/ORIGIN 200 workstation (Silicon Graphics, Inc., Mountain View, CA, USA). Typical processor time is about 20 Cpu hours.


   Results
Top
 Abstract
 Introduction
 Material and methods
 Results
Comment
 Acknowledgments
 References
 
We focused our attention on two aspects of aortic valve function: (1) aortic root and leaflet deformation typically occurring in one cardiac cycle, and (2) stresses developing in the various parts of the leaflet during one cardiac cycle.

Aortic root and leaflet deformation
The deformation of the leaflet and aortic wall present an interesting spectacle. Figure 3 provides a guide for understanding the pattern of deformation.



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  		Fig 3. Positon of three locations (nodes A, B, and C) in the model.

 
Node A indicates the nodule of Arantius; node B is situated in the aortic wall at the height of the commissures; and node C is at the aortic root in the midpoint of the attachment of the aortic leaflet. The displacement pattern for node A is shown in Figure 4. Point 2 is the fully open position and is the reference point for the model. The displacement at this position is zero. A displacement of about 16 mm very early in the cycle indicates the movement of the leaflet from an open position (from a previous cycle not shown in the figure) to a closed position at the beginning of the graph. Until 0.4 seconds, the leaflet remains closed. In fact, the actual opening starts at about 0.37 seconds; this is highlighted within the box marked "1" in Figure 4. A change of slope is clearly visible at 0.37 seconds. This change in slope is conceived as the beginning of the opening cycle (see [5] and discussions therein). It can be concluded that the aortic root dilatation explained below may precede and, in fact, may be responsible for, the opening of the valve leaflet even before the advent of positive pressure on it. The opening is rapid after 0.41 seconds and is aided by the difference in pressure on either side of the leaflet and this opening continues till the point 2 is reached. The closure is again complete at 0.47 seconds. This figure also indicates the need for a geometric nonlinear analysis, as the displacement is very large. The corresponding deformation of node B is also shown in Figure 4. The figure clearly shows that the aortic root dilatation starts before the opening of the leaflet. As seen in the figure, there is an exponential increase in the dilatation of the root from 0.3 seconds and reaches a very high value of 1.14 mm at 0.4 seconds. At that time, the leaflet opens and this results in a violent shudder of the root. This shudder continues till the leaflet closes and the deformation of this portion approaches zero from then on. Although the behavior can be explained by the pressure pattern shown in Figure 2, the interesting result is the exponential increase in deformation, which may be due to the complex geometry of the system. Deformation at node C in Figure 5 is similar to that at B, although the magnitudes are much less. The violent shudder is evident during opening and closing.



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  		Fig 4. Displacement pattern of nodes A and B with time.

 


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  		Fig 5. Radial deformation at location C with time.

 
One of the important questions raised in the discussion in several papers [2, 5] is the effect of the geometry of the sinus in the efficiency of opening of the aortic valve. To verify this effect, a pseudo-analysis was carried out. In this analysis, no pressure was applied as load to the leaflets, but other pressure patterns in the aortic segments were maintained. Hence, the leaflet opening is due purely to the dilatation of the aortic wall. The deformed leaflet resulted in an opening of 3 mm, nearly 20% of the total displacement of 16 mm (Fig 6). Also of interest is the fact that there is a substantial increase in the aortic diameter at the level of the commissures before the leaflets actually starts to open. This is captured quite graphically in this study.



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  		Fig 6. Leaflet deformation due to the geometry of the aortic system.

 
Figures 7 and 8 show the deformation patterns typically observed. The red outline of the mesh is the original open position of the leaflet. The wavelike nature of the deformation of the leaflet has so far not been observed in previous simulations [2, 3] are brought out in these figures.



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  		Fig 7. Deformation of the leaflet after 0.409 seconds.

 


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  		Fig 8. Deformation of the leaflet after 0.444 seconds.

 
von-Mises stress in the leaflets
The other aspect studied was the development of stress in the leaflets with time. The equivalent stress (called von-Mises stress), with evolution of time at a number of locations (Fig 9), is presented in Figure 10A through 10H and Figure 11A to 11C.



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  		Fig 9. Locations in the leaflet for monitoring stress.

 


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  		Fig 10. (A through H) von-Mises stress variation at locations A through H (shown in Fig 9), respectively.

 


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  		Fig 11. von-Mises stress contours at (A) 0.009429 seconds, (B) 0.189 seconds, and (C) 0.444 seconds.

 
The striking finding that emerges from the analysis of these figures is that the stresses during closure increase as the point of attachment of the leaflet to the root is approached radially from the free surface. For example, maximum stress in points A, B, and C (which lie almost in the same line) are 0.1 MPa, 0.2 Mpa, and 0.4 Mpa, respectively. Such a trend is due to the cantilever effect when the leaflets are closed and are subject to pressure. A similar trend can be seen in the circumferential direction by comparing the maximum stresses at point B and E. The other impressive finding is the sudden, almost instantaneous increase in the stress at the coaptation surface due to impact during closure. Locations A and F in the coaptation region experience such an increase. On the other hand, C, at the bottom of the leaflet is not affected by the impact. Figure 10G and 10H show that the edges of the leaflet are not affected by closure. However, position G is subjected to a very high stress level during the closure of the leaflet.


   Comment
Top
 Abstract
 Introduction
 Material and methods
 Results
 Comment
Acknowledgments
 References
 
The aortic valve complex has been the subject of intense investigation by several workers using a variety of techniques, including mathematical modeling with finite element analysis [1–4].

Grande-Allen and colleagues [2] used a finite element model to study the effect of sinuses on the stresses developing in leaflets in valve-sparing root replacements. There have been other studies to elucidate various aspects of the functioning of aortic valve [8, 12]. Cacciola and colleagues [3] used similar techniques to show that a stentless trileaflet valve has 75% lower stress levels in the leaflet as compared with stented valves. However, all of these studies have been carried out using static finite element model. We used a dynamic finite element model of the aortic valve, and this has brought out certain features not previously captured, especially with regard to the complex interplay between the aortic root and the leaflets. We studied two main aspects of aortic valve function in a typical cycle, namely, aortic root and leaflet deformation; and stresses in the various parts of the leaflet.

Aortic root and leaflet deformation
The most striking finding was that the leaflets begin to open even before any positive pressure is applied, due primarily to the effect of aortic root dilatation. In fact, substantial increase in diameter during systole occurs before the actual opening of the aortic valve. This finding is striking and was confirmed by a separate pseudo-analysis in which no pressure was applied to the leaflets but only to the aortic root. This type of parametric testing, which is easily possible, is indeed one of the strengths of this technology. Interestingly, this finding of the effect of aortic root dilatation in aiding of the leaflet opening, has been confirmed independently using animal models and sonomicrometry [5, 13]. Although the complex interplay between the aortic leaflets and the aortic root has long been obvious, this is the first time that this relationship has been strikingly demonstrated in a computer simulation model. In fact, when the compliance of the wall is increased during the parametric study, the leaflets begin to prolapse—an event that is known to happen in patients with annuloaortic ectasia causing aortic incompetence. It is thus not altogether unexpected to encounter instances of leaflet thickening following the David technique of including the aortic valve in a rigid cylinder [2]. The other striking finding is the violent shudder in the leaflets during opening and closing. It is therefore not surprising to find leaflet substitutes made of disparate material failing over a period of time, inside the aortic root, when used as replacement devices.

von-Mises stress in the leaflets
The other aspect that was studied was the development of stress in various parts of the leaflets. The important observation is that there is an instantaneous increase in the stress at the coaptation surface during closure. Also, there is an increase in stress as the point of attachment of the leaflet to the root is approached from the free surface (Fig 11). Although it is not possible to make a direct comparison with the work of Grande-Allen and colleagues [2] and Grande and associates [12], the stress states are in the same range as reported in those works.

Conclusions
With increasing sophistication in the surgical treatment of aortic root diseases, including in the application of valve-sparing root replacements and aortic valve repairs, there is clearly a need for a precise understanding of the complex interplay of the various components of the aortic root. Historically, surgical treatment of valvular heart disease has been based on designing a strategy or a valve substitute, applying it in a clinical setting, and then waiting (sometimes for years) to see if it will work. Some of the earlier failures of replacement devices could have been predicted if sophisticated techniques to test them had been available. With the widespread availability of powerful computers and supportive software, precise mathematical modeling using techniques such as finite element analysis make it possible to model and study the aortic valve in a dynamic state. These techniques also allow one to observe the most minute changes taking place in the leaflets and root, as well as the stresses developing in the leaflets in a manner that is not possible using animal or other models. Use of these techniques will enable the testing of newer devices, development of new leaflet materials, as well as newer types of stents for stent-mounted valves and repair techniques at the computer workbench before clinical application, so that better long-term results can be expected.


   Acknowledgments
Top
 Abstract
 Introduction
 Material and methods
 Results
 Comment
 Acknowledgments
References
 
The authors thank Andrew V. Denise for help during the course of this work.


   References
Top
 Abstract
 Introduction
 Material and methods
 Results
 Comment
 Acknowledgments
 References
 

  1. Thubrikar M.J. Geometry of the aortic valve. The aortic valve. Boca Raton, FL: CRC Press, 1990:1-19.
  2. Grande-Allen J., Cochran R.P., Reinhall P.G., Kunzelman K.S. Recreation of sinuses is important for sparing the aortic valve: a finite element study. J Thorac Cardiovasc Surg 2000;119:753-763.[Abstract/Free Full Text]
  3. Cacciola G., Peters G.W.M., Schreurs P.J.G. A three-dimensional mechanical. analysis of a stentless fibre-reinforced aortic valve prosthesis. J Biomech 2000;33:521-530.[Medline]
  4. Thornton M.A., Howard I.C., Patterson E.A. Three-dimensional stress analysis of polypropylene leaflets for prosthetic heart valves. Med Eng Phys 1997;19:388-397.
  5. Pang D.C., Choo S.J., Luo H.H., et al. Significant increase of aortic root volume and commissural area occurs prior to aortic valve opening. J Heart Valve Dis 2000;9:9-15.[Medline]
  6. ABAQUS theory manual. Pawtucket, Rhode Island: Hibbitt, Karlsson, Sorenson, Inc, 1999.
  7. Xie J.P., Liu S.Q., Yang R.F., Fung Y.C. The zero-stress state of rate viens and vena cava. Trans ASME J Biomech Eng 1991;113:36-41.
  8. Grande K.J., Cocharan R.P., Reinhall P.G., Kunzelman K.S. Stress variations in the human aortic root and valve: the role of anatomic asymmetry. Ann Biomed Eng 1998;26:534-545.[Medline]
  9. Ferraresi C., Bertetto A.M., Mazza L., Maffiodo D., Franco W. One dimensional experimental mechanical characterisation of procine aortic root wall. Med Bio Eng Comput 1999;37:202-208.
  10. Zannoli R., Schiereck P., Celletti F., Branzi A., Magnani B. Effects of wave reflection timing on left ventricular mechanics. J Biomech 1999;32:249-254.[Medline]
  11. Cook R.D., Malkus D.S., Plesha M.E. Concepts, and applications of finite element analysis. New York: Wiley, 1989:367-417.
  12. Grande K.J., Cochran R.P., Reinhall P.G., Kunzelman K.S. Mechanics of aortic valve incompetence. Finte element modelling of aortic root dilatation. Ann Thorac Surg 2000;69:1851-1857.[Abstract/Free Full Text]
  13. Vesely I. Aortic root dilation prior to valve opening explained by passive hemodynamics. J Heart Valve Dis 2000;9:16-20.[Medline]



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