Stress Analysis of the Aortic Valve Using Finite Element Modelling Software
Toumar A.J. and Pang S.D.
Engineering Science Programme, National University of Singapore
Kent Ridge Road, Singapore
ABSTRACT
This project aims to create a finite element model of the healthy aortic valve in the human
heart, and analyse the stresses on the valve tissue during one part of the cardiac cycle. The aortic
valve is modelled geometrically and analysis is performed in the finite element software package
ABAQUS/CAE. Changes in various parameters that could be associated with real life valve or
circulation problems are then examined with respect to their influence on the maximum stresses
of the valve tissue. The positioning of the maximum stresses is also examined and conclusions
are drawn from the results of the analysis. Recommendations are made towards further
investigation and model refinement.
The results of this analysis can be used to identify areas of the aortic valve under greatest
stress, which may be more vulnerable to rupture or damage in the case of trauma or disease than
other areas. The results of the parameter analyses may provide substantial insight into how
stresses are influenced by the deviation from common properties of healthy aortic valves. Thus,
the study hopes to provide vital information for the prediction of possible problems with the
aortic valve, and to aid in the development and implementation of prosthetic heart valves.
INTRODUCTION
Many studies have been made into the structure and function of the valves found in the
human heart. Such studies seek to understand with greater accuracy the properties of the valves,
and how these can be affected by various factors leading to valvular heart disease. The results of
such studies are paramount in the design of more efficient, better functioning, economical
replacement valves, which are longer lasting and safer for the individual.
The aortic valve has been of particular interest with researchers, as it connects the left
ventricle to the aorta, bringing oxygenated blood from the heart to the body’s largest artery,
from which it is delivered to the rest of the body. The healthy aortic valve prevents regurgitation
into the left ventricle, maintaining the efficiency of blood transfer.
Normal wear and tear on this valve, as well as irregularities such as scars or deposits on the
valve tissue, impair its function and the efficiency with which blood can be transferred through
it. Calcium deposits or scarring of the valve can lead to aortic valve stenosis, which increases the
workload of the heart and thus leads to a risk of heart failure (Badash, 2007). Such problems are
generally treated by surgical implantation of a ball or disc prosthetic valve, bioprosthesis or an
animal (porcine) aortic valve (Arcidiacono, Corvi, Severi, 2004). Manmade plastic prostheses
and those made of biomaterials must therefore be designed to imitate and maintain the function
of the healthy aortic valve.
Techniques varying from magnetic resonance imaging (Vesely, Eickmeier, Rutt, &
Campbell, 1991) to echocardiography (Shim et al., 2007) are used to give information about the
geometry and material properties of the valve and its surrounding region in order to derive
information for better prosthesis design. Other information, such as the influence of changes to
various material parameters on the stresses within the valve, is less straightforward to obtain,
though also very important. For example, the calcification of prostheses causes increased
rigidity and impaired valve function, and occurs commonly at areas of highest stress and strain
(Knierbein, Rosarius, Unger, Reul, & Rau, 1992). Therefore, it is important to examine the
regions of the valve in which maximum stresses occur and what factors or parameters may
influence the magnitude of such stresses. Such studies can be done with relative ease and
efficiency by utilising finite element software.
Scope
There are many factors concerning the blood and valve itself, which may influence
parameters within a computer model and also the magnitude of stresses on the valve tissue.
Deposits on the valve walls, for example, may be approximated by a change in the coefficient of
friction between the blood and valve interior. Calcification and scarring changes the way the
valve can be stretched, which can be simulated by changes in the elasticity properties of the
computer model. Blood consistency may change due to a disease of the blood, or the use of
artificial blood in blood transfusions (Winslow, 2006). This would in turn change properties
such as the viscosity or shear modulus of the finite element blood model. Changing these
parameters within a controlled computer model requires much less effort and time expenditure
than observing their effects in nature, and is less invasive than removing and examining animal
heart valves. In addition, a computer model allows just one parameter to be changed at a time,
thus avoiding confounding variables that would more likely be present in in vivo observations.
This study examines the effect of changes in valve elasticity and blood viscosity on the
stresses in the aortic valve tissue. Once the results of this analysis on the open valve are verified,
this model can be coupled with that of the closed valve in order to simulate an entire cardiac
cycle. As well as aiding in understanding of what influences the stresses on the heart valve, the
results of this study may lead to the improvement of prosthesis designs.
MODELLING
Geometry of the Aortic Valve
As stated previously, the aortic valve joins the aorta to the left ventricle. It is asymmetric, and
consists of three leaflets and three sinuses (cavities behind the leaflets). The sinuses can be
viewed as approximately ellipsoid dilations of the aorta, to which leaflets are connected at the
base of the valve (Thubrikar, 1990). During the cardiac cycle, the valve leaflets open and are
pushed out into the sinuses by the flow of blood. It is in this position that the leaflets were
modelled in this investigation, according to the modelling method suggested by Thubrikar and
dimensions typical of a healthy human aortic valve (Gnyaneshwar, Kumar, & Balakrishnan,
2002).
The finite element software package ABAQUS/CAE was used to create the geometric model
and to run the analysis. The three leaflets of the aortic valve were modelled with their region of
attachment to the base of the aorta. The aortic valve was modelled in the fully open position,
with the three leaflets in the fully expanded position within the sinuses (not included in the
model).
Table 1. Dimensions of Aortic Valve Model Used in Investigation
Dimensions of Aortic Valve Model
Radius of Commissures
Radius of Aortic Base
Height of Valve
Height of Commissures
Angle of free edge of leaflet to plane passing
through commissures
Angle of undersurface of leaflet to plane
passing through commissures
12.0 mm
12.0 mm
17.0 mm
8.52 mm
32 degrees
22 degrees
Modelling of the Open Aortic Valve
The three leaflets of the aortic valve were modelled as cylindrical shell elements stemming
from a circular shell base. The leaflet and base tissues were modelled initially with an isotropic,
linear elastic material with a Young’s Modulus of 2.0 MPa (Ranga, Mongrain, Biadilah, &
Cartier, 2007) and a Poisson’s ratio of 0.3 (Gnyaneshwar et al., 2002). The thickness of the
leaflets in the human aortic valve was found to vary between approximately 0.25 and 1.33 mm
throughout each leaflet, however in this model, a uniform thickness of 1 mm was used
(Thubrikar, 1990). Further modelling and analysis could take into account the variation in
thickness of the valve elements, however for this model, a simplified representation was created.
The valve material model would be improved by the use of a hyperelastic material, instead of an
elastic material, however for the elastic model created, parameters were chosen within the
measures of healthy human aortic valves in order to simulate the valve tissue as accurately as
possible.
Valve Model
Commissural height
8.52 mm
Blood Model
Valve height
17 0 mm
4 mm
Base Radius
12.0 mm
Figure 1: Features of the Blood and Open Valve Model in Viewport of ABAQUS/CAE
Blood flow was modelled by the procession of a 4 mm deep section of ‘blood’ through the
open valve model. This section was modelled to match the internal geometry of the valve model
at the base of the leaflets, so that contact could be established at the initial incremental time-step
of its motion. As the modelling and analysis were conducted in ABAQUC/CAE, the fluid blood
section was modelled as a Neo-Hookean hyperelastic solid with a low initial shear modulus.
Further studies can improve on the model by utilising fluid and solid interaction software, and
more accurately depicting the viscosity and consistency of blood flowing from the heart.
Contact was established between the blood section and interior walls of the valve and
leaflets. Several coefficients of friction between the blood section and leaflets were tested
against the maximum Mises equivalent stresses on the valve tissue. Significant changes in the
friction coefficient were found to have little effect on the magnitude of the stresses within the
model, with maximum stresses varying as little as 0.1% for doubling the reference friction
coefficient; hence an arbitrary, small coefficient of friction was used for all subsequent analyses.
These analyses were used to test the effects of changing the elasticity of the valve, and the blood
consistency, on the Mises equivalent stresses in the valve.
Meshing
The solid part representing a section of blood fluid was meshed with 10-node modified
tetrahedral (C3D10MH) elements, while the homogeneous shell structure was meshed with
triangular, general-purpose shell (S3) elements. For initial studies, a small number of elements
were used for both parts due to restrictions of the student edition of ABAQUS 6.7.
ANALYSIS OF RESULTS
The regions of maximum stress were examined visually throughout the blood motion. The
analysis was then repeated to test the influence of a change in the valve elasticity and blood
viscosity against the reference model of a ‘healthy’, open human aortic valve.
Position of Maximum Stresses
The movement of the blood section lead to a range of stresses, apparent on the valve model
shown in Figure 2 below. Maximum Mises equivalent stresses were found to be at the
attachment of the leaflets to the base of the aorta (highlighted in the top left image), as can be
seen from these images of four stages of the blood motion.
Figure 2: Maximum stresses on the valve model (shown circled) in four stages of blood motion.
Maximum Mises equivalent stresses were apparent throughout the analysis at the regions of
attachment of the leaflets to the base of the aorta. These ‘hinge’ regions would thereby need to
withstand the highest stresses during blood flow, and must be reinforced during the design for
prostheses. This positioning of the maximum stresses on the valve tissue is consistent with the
thickness of the leaflets of natural valves, which tend to become thicker and thus more resistant
to tearing at their bases (Thubrikar, 1990).
Valve Elasticity
In natural as well as prosthetic valves, calcium deposits on the valves or changes in the tissue
associated with aging or scarring around the valve (Kuehn et al., 2008) influence the elasticity of
the leaflets and sinuses, making them more stiff and rigid. It is known that such changes may
impair the functionality of the valve and demand extra pressure from the heart, increasing the
risk of heart failure in the long term. It is not as widely investigated what such a change in
elasticity does to the mechanical stresses within the valve itself. In theory, an increase in the
magnitude of stresses on the valve would increase the risk of further damage, and this factor was
investigated in this study.
The valve model was tested with an increasing Young’s modulus, and the stresses caused by
the blood flow in the stiffer valve were compared to the reference values. The Mises equivalent
stresses were evaluated within an incremental analysis step with 100 increments, and data from
three representative increments (coinciding with 3 positions of the blood section within the
valve) were recorded. The stages chosen were at Increment 40, 60 and 80 to avoid integration
problems that may have occurred at the beginning and end of analysis. Figure 2 shows the plot
of the normalised stress data against the relative increase in Young’s Modulus.
Table 2: Normalised Maximum Stresses at Increments 40, 60 and 80
Normalised Normalised Maximum Stress at
Young’s
Modulus
Increment 40
Increment 60 Increment 80
1
1.000
1.000
1.000
1.25
1.162
1.128
1.091
1.5
1.311
1.238
1.166
1.75
1.449
1.336
1.229
2
1.579
1.424
1.283
2.25
1.700
1.503
1.331
2.5
1.814
1.576
1.353
2.75
1.921
1.643
1.394
3
2.023
1.704
1.431
3.25
2.119
1.762
1.464
Increment 40
Increment 60
Increment 80
Normalised Maximum Stress
(MPa/MPa)
Change in Valve Elasticity versus Maximum Mises
Equivalent Stress
2.500E+00
2.000E+00
Increment 40
1.500E+00
Increment 60
1.000E+00
Increment 80
5.000E-01
0.000E+00
0
1
2
3
4
Norm alised Young's Modulus
(MPa/MPa)
Figure 3: Maximum Mises Equivalent Stress vs. Change in Valve Elasticity at Various
Increments of Analysis
An increase in stiffness of the leaflets and aortic root lead to an increase in the magnitude of
stresses measured during contact between the blood section and the valve. Since higher stress
areas tend to accumulate calcium deposits, and thus cause further stiffening in the valve, it is
important that prostheses are designed to mimic the elastic flexibility of natural, healthy aortic
valves. The stresses increase most prominently at Increment 40, which corresponds to the lower
part of the valve model, where leaflets are attached to the aortic base. The increase in stresses
appears to become less prominent as higher Young’s Moduli are tested. This is indicative of a
non-linear relationship between the stress and Young’s Modulus of the tissue. Stress and
Young’s Modulus in an isotropic object can be related thus:
E
 stress

 strain
The trends observed are fitting with this equation, as the stress is proportional to both the
increased Young’s Modulus and the decreasing strain. The three differing trends observed at the
three positions in the valve model may be attributed to the differing geometry throughout the
valve, as the leaflets (at the top of the valve shown in Table 2) are free to move while the base of
the aortic valve is fixed with displacement constraints.
Blood Consistency
Diseases in the blood or the use of blood substitutes in patients influences the viscosity of the
fluid passing through the aortic valve during the cardiac cycle (Winslow, 2006). The change in
viscosity and its effect on the stress in the aortic valve were investigated in this study.
The section of blood flowing through the valve model was modelled as a Neo Hookean
hyperelastic solid. An increase in blood viscosity was therefore approximated by increasing the
value of the initial shear modulus (μ0 = 2C10), by changing the temperature dependent parameter
C10 in the material model (Dassault Systèmes, 2007). Stresses caused by this more ‘viscous’
blood material were compared with reference values at three increments of the blood motion
step. Figure 4 shows the plot of the normalised stress data against the relative increase in C10.
Table 3: Normalised Maximum Stresses at Increments 40, 60 and 80
Normalised Normalised Maximum Stress at
Material
Parameter C10 Increment 40 Increment 60 Increment 80
0.5
8.068E-01
7.339E-01
6.537E-01
0.75
9.195E-01
8.880E-01
8.555E-01
1
1.000
1.000
1.000
1.25
1.063
1.086
1.116
1.5
1.113
1.154
1.211
1.75
1.156
1.210
1.291
2
1.198
1.328
1.359
2.25
1.225
1.379
1.418
2.5
1.254
1.423
1.470
2.75
1.279
1.462
1.516
3
1.380
1.524
1.558
3.5
1.615
1.759
1.630
4
1.829
1.994
1.689
Increment 40
Increment 60
Increment 80
Normalised Maximum Stress
(MPa/MPa)
Change in Initial Shear Modulus (C10) versus
Maximum Mises Equivalent Stress
2.500E+00
2.000E+00
1.500E+00
Increment 40
Increment 60
1.000E+00
Increment 80
5.000E-01
0.000E+00
0
1
2
3
4
5
Normalised Initial Shear Modulus (MPa/MPa)
Figure 4: Maximum Mises Equivalent Stress vs. Change in Initial Shear Modulus at Various
Increments of Analysis
Predictably, an increase in the viscosity or ‘solidity’ of the blood passing through the heart
valve increases the magnitude of stresses in the valve itself. The blood shears less readily,
exerting a higher force on the valve. For all three stages of blood flow examined, the
relationship between the initial shear modulus of the blood and the maximum Mises equivalent
stress in the valve tissue appears to be non-linear. At increments 40 and 60, the relationship
between increase in C10 and increase in stress seen in Figure 4 appears to be consistent with that
at increment 80, but this tends to break down at higher values of C10. The initial shape of the
curves suggests a logarithmic relationship, after which the values at 40 and 60 appear to become
linear. This inconsistency may be due to the particular geometry of the valve itself. As can be
seen in Table 3 above, the open valve is significantly wider at the free edge than at its base.
Inconsistency in results at different increments may also be due to inadequately refined mesh,
which was limited by the use of the student version of the software. The more powerful,
professional version should be used in future investigations to refine the mesh.
Since blood substitutes are used in blood transfusions, which may be necessary during
surgery (Winslow, 2006), it is important that their viscosity be monitored and maintained at a
level close to that of natural blood, to decrease the stress on the heart valves, which may lead to
extra demand on the heart itself
Recommendations for Future Investigations
There are many ways in which this simulation can be expanded and improved in order to
achieve more plausible and useful results. Several aspects of the geometry of the model can be
refined, such as the variation in thickness within the valve base and leaflets, the addition of
sinuses and muscular reinforcements surrounding the valve and the slight asymmetry of the
three leaflets observed in nature. The geometry of the closed and opening valve can be included
in the model in order to widen the scope of the model to that of the entire cardiac cycle, in order
to analyse the stresses on the valve leaflets during the opening or closing of the valve, where
they may suffer the largest mechanical stresses.
Another area of improvement is the material model of the valve tissue. An isotropic, linear
elastic material model is not entirely reflective of the valve tissues anisotropic, hyperelastic
nature. A more accurate model would be one that would deform nonlinearly – a hyperelastic
model. This would require additional data in order to model in ABAQUS, including biaxial
tension data, which may be found in literature or through experiment (Gnyaneshwar et al, 2002).
It is recommended that further investigations use a more refined mesh, in order to utilise the
valve geometry more accurately during analysis.
CONCLUSIONS
The development of computer software has allowed for the nature and effectiveness of
biological systems and their prostheses to be modelled and investigated without the use of
invasive or time consuming observational techniques. The heart’s four main experience
changing pressure, stresses and wear continually, and this, together with forms of valvular or
blood disease, scar tissue from previous surgeries or calcium deposits in higher stress areas, can
change the material properties of the valve tissue. The aortic valve, connecting the heart to the
body’s largest artery, is of particular importance and the subject of much study.
The effects of wear and tear and other influences discussed in this study are difficult to
calibrate and observe in nature, but can be better understood and approximated with relative ease
when modelled and analysed appropriately using finite element software. These modelling
techniques allow for a greater range of conceptual testing for prostheses, and allow material and
shape properties of prosthetic aortic valves to be optimised in the design stage, leading to faster
evolving prosthesis designs with improved functionality and thereby an improved outlook for
the patients who require them.
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