The document describes a simulation comparing stress and deformation in healthy vs. calcified aortic heart valve leaflets. A model of a simplified elliptic paraboloid leaflet was created in COMSOL. The healthy leaflet was modeled with lower elastic modulus and density than the calcified leaflet. For linear material assumptions, the calcified leaflet had higher maximum stress and larger displacement than the healthy leaflet. But for hyperelastic assumptions, the calcified leaflet had higher maximum stress but smaller displacement. The simulation provides a starting point for more complex and realistic models of valve biomechanics.

1. Stress and Deformation Analysis on a Simplified Elliptic Paraboloid
Model of an Aortic Heart Valve Leaflet: Healthy vs. Calcified
Emma Manzano, Jin Tanizaki, and Qianhe Zhang
Students of the Department of Bioengineering, University of California, Berkeley
Abstract
healthy aortic heart valve can experience calcification and calcified aortic stenosis is one of the most common diseases of aortic valve. In
this paper, a simulation model of an ideal leaflet of an aortic valve was built and the stress and deformation distribution on a healthy leaflet
was compared with the distribution on a calcified leaflet under diastolic pressure. The healthy valve and the calcified valve were assumed to
be made of isotropic and linear material and have the same ideal symmetric geometry, but they have different Elastic Modulus, Poisson’s ratios, and
densities. Another simulation was conducted where the material was assumed to be hyperelastic and the pressure was scaled down by a factor of 10,
meanwhile the other parameters stayed the same. The calcified valve was assumed to have a coaptation pattern of calcification. The results showed
that the calcified valve had a larger maximum stress as well as a larger displacement compared to the healthy valve for a linear assumption
(healthy=6.35 x 103
Pa, dhealthy=0.156mm; calcified=4.29 x 104
Pa, dcalcified=0.48 mm; healthy<calcified, dhealthy<dcalcified), but a larger maximum stress and
a smaller displacement for a hyperelastic assumption (healthy=2.96 x 104
Pa, dhealthy=0.815 mm; calcified=2.11 x 105
Pa, dcalcified=0.07 mm;
healthy<calcified, dhealthy>dcalcified). This simulation can be the starting point for more complex models that are more realistic and make closer
approximations to the actual values.
Keywords aortic heart valve | heart valve diseases | COMSOL modeling | cardiovascular biomechanics
Introduction
The aortic valve is a crucial part to the everyday function of the
heart and maintains regular blood flow. This valve allows for blood
to flow from the left ventricle to the aortic artery in the heart. It also
prevents the blood from flowing backwards and causing less blood to
be pumped to the rest of the body as efficiently. There is a common
cause of decreased aortic valve function: calcification. A healthy
aortic valve can become calcified over time due to calcium deposit
build up, which stiffens the valve and leads to its inability to open
properly (Weinberg, et al.). High areas of stress within a calcified
heart valve can lead to tissue failure, so it is a very serious heart
condition that many people face.
A healthy aortic valve can have varying geometry between
leaflets. Usually no two leaflets are of the same size (Figure 1a). A
calcified aortic valve has calcified deposits that tend to occur in
specific patterns (Figure 1b). In this project a coaptation pattern is
used, where calcification primarily occurs in the region closest to the
wall and around the coaptation edge, or the fixed edge in the models
(Figure 1c).
(a) (b) (c)
Figure 1. (a) Healthy aortic heart valve ("What is Severe Aortic
Stenosis?", 2016) (b) calcified aortic heart valve (Rajamannan, 2010),
and (c) the coaptation pattern(Thubrika et al., 1986).
In this project, a simulation model of a single leaflet of an ideal
tri-leaflet valve was built using COMSOL. The static stress and strain
distribution on a healthy aortic valve leaflet and a calcified leaflet
under diastolic pressure were then compared. The reason for
choosing the diastolic pressure is that the valve experiences the
maximum amount of stress during diastole, and it would be when the
valve mostly likely fails. It is hypothesized that the stress remains the
same because the geometry of the valve does not change, but the
strain becomes smaller for the calcified valve because it is stiffer. The
result of the project can suggest the possible site of tissue failure and
be the first step to analyze the need of a prosthetic replacement for a
patient.
Methods
General Assumptions
1. The aortic valve was treated as perfectly symmetrical for
simplicity. The inner wall of the aortic valve would appear as
circular in our analysis if it is looked at along the axis of the aorta.
2. All leaflets are identical in both shape and size and contain no
deformities.
3. Coaptation surfaces, or the surfaces of the aortic leaflet that are in
contact with other leaflets, were not incorporated into the model.
4. Leaflet material for both normal and calcified aortic valve was
considered to be nearly incompressible and isotropic.
5. The Poisson’s ratio for normal aortic valve material was assumed
to be 0.3 as was done in the study by Hamid et al. (1985).
6. Aortic valve annulus deformation was ignored similar to Hamid et
al. (1985).
Leaflet Geometry
Elliptic paraboloids were determined to be adequate to represent
a simplified, single aortic heart valve leaflet (Hamid et al., 1985)
(Gould et al., 1973). A study by Swanson and Clark determined that
the diameter of the ventricular tract (the area on the side of the left
ventricle directly underneath the aortic valve) changed the least over
the course of systole and diastole and a single value for the
ventricular diameter could be used in models and simulations (1974).
In order to determine the ventricular diameter for our model, an
average of the diameters in the varying pressure ranges was taken.
The ventricular diameter used in our model is 26mm. The geometry
is that of a twenty-nine-year-old female (series 4 in the study).
Swanson and Clark also determined the angle from the cross
sectional plane of the artery to the bottom edge of the coaptation
surface (surface of leaflet that contacts other leaflets), with the root of
the leaflet as the vertex. Again, an average of the angles, 21.25°,
provided for the specific heart valve was used (Swanson and Clark,
1974). Here it was assumed that the length of the leaflet would be
half of the diameter so that it would reach from the wall of the valve
to the center of the aortic valve, and that its width would span the
distance of
𝑑 𝑣√3
2
, the secant distance of a circle with radius 𝑑 𝑣 from
0° to 120°. This would yield us an equation of:
𝑧2
25.0721
+
𝑥2
33.4294
= 𝑦
This equation is defined where 𝑦 ≤ 5.05542 and 𝑥 ≥ 0 when axis 𝑥
describes the length towards the center of the valve, 𝑦 is the height
along the length of the aorta, and 𝑧 is the width along the wall
surface. This equation was used to describe the bottom surface of the
simplified aortic heart valve leaflet model.
Leaflet Thickness
A

2. Hamid, Sabbah, and Stein (1985) created a topographic map of
the thickness of the aortic heart valve using data from a study by
Clark and Finke (1974). Thickness data of the edge fixed to the wall
was ignored. The total height of the leaflet was broken into 11 equal
sized sections as done in the map, and the midpoint of each section
was assigned its respective thickness as determined from the map. A
quartic regression was used to fit the curve, as shown in Figure 2.
The curve describing the equation is as follows.
t(y) = 0.0062y4 + 0.0518y3 + 0.1616y2 + 0.4195y + 1.1035
Note that both thickness and height units are millimeters,
and that the height is measured from the top of the leaflet, or the
coaptation edge as also seen in the figure. In order to fit this curve
onto the surface of the leaflet, it is necessary to define the thickness
in terms of distance from edge of valve, 𝑥. It was also assumed that
the thickness function would not describe the absolute thickness of
the valve, but rather the vertical thickness because the surface of the
valve is reasonably flat. The thickness function superimposed on the
leaflet surface on the 𝑥𝑦 plane is as follows:
𝑡(𝑥) = [
𝑥2
33.4294
− 5.554] + 0.0062 ∗ (
𝑥2
33.4294
− 5.05542)
4
+
0.0518 ∗ (
𝑥2
33.4294
− 5.05542)
3
+ 0.1616 ∗ (
𝑥2
33.4294
− 5.05542)
2
+
0.4195 ∗ (
𝑥2
33.4294
− 5.05542) + 1.1035
This equation was used to guide a lofted cut of the 3D model.
Figure 2. Thickness of leaflet. Data from topographic map made by
(Hamid, Sabbah, and Stein). Note that for this graph only, height is
measured from the top of the leaflet.
Figure 3. Left. Isometric view of leaflet designed in Solidworks. Right.
Cross-sectional view of leaflet along xy plane. Note that the thickness
differs along the entire leaflet according to the equation. The blue
arrow points to the coaptation edge and the green line outlines
where the leaflet attaches to the wall.
Material Properties
It was determined that aortic valve leaflets are best modeled as
hyperelastic materials and that at low (1%) strains, the normal aortic
valve has an Elastic Modulus of 300 kPa (Hamid et al., 1985) and it
was also assumed as in previous studies that the Poisson’s ratio was
0.3 (Hamid et al, 1985) (Yeh et al., 2014). Density of normal aortic
valve material was assumed to be 2116 kg/m3
(Yeh et al., 2014).
Calcified properties were also obtained in similar manner from
research. The Elastic Modulus of a calcified section of the aortic
heart valve was determined to be 871 kPa at low strains with a
density of 1600 kg/m3
and a Poisson’s ratio of 0.495 (Loree et al.,
1994).
As we were planning to conduct a COMSOL analysis with both
linear elastic materials and hyperelastic materials to analyze the
difference between the two simulations, the material properties
assigned earlier were assigned to both the linear and hyperelastic
materials.
Calcified Structure
A study by Thubrikar, Aouad, and Nolan identified three
common types of calcification patterns found in severely calcified
aortic valves (Thubrikar et al., 1986). In this study we decided to
model a coaptation pattern which is described as a pattern where
calcification occurs mainly along the edge where the leaflet attaches
to the wall, and also begins from the coaptation surface and
progresses towards the center of the leaflet (Thubrikar et al., 1986).
In order to model this calcification pattern, we decided to partition
the model into three sections based off of the distance from the wall
surface, or its distance in x. We chose the first section closest to the
wall, defined as 0𝑚𝑚 ≤ 𝑥 < 2.52𝑚𝑚 to be made out of calcified
material. This is the roughly the first fifth of the entire length of the
leaflet. It was also assumed based off readings that 80% calcification
would be a good baseline for severe calcification in order to
determine potential sites of tissue failure. Calcified properties were
applied to the third section of the partitioned leaflet, ending at the tip
of the leaflet and beginning in such a way that roughly 80%
calcification by vertical area could be achieved. The third section, or
the boundary 4.58𝑚𝑚 < 𝑥 < 13𝑚𝑚, was given calcified
properties, and the middle section was assigned normal aortic valve
material properties.
Finite Element Analysis in COMSOL
After obtaining all geometry and material properties, finite
element analysis was conducted. The thin vertical surface that would
be attached to the wall of the artery shown in Figure 3 as a green line
was held as fixed. Also, the outermost edge of the coaptation surface,
pointed at by a blue arrow in Figure 3, was fixed in such a way that
all points on the edge would only move in the y direction, or in the
direction of the axis of the artery. The edge of the coaptation surface
would be in contact with coaptation edges of the other leaflets, and
under our symmetry assumption, there would be no movement of this
edge in the x or z direction. Free tetrahedral mesh elements were used
to conduct this calculation. Four iterations of calculations were
conducted in COMSOL.
As this study concentrates on the stresses and strains during
diastole, aortic and ventricular pressures during diastole were needed.
A pressure of 120mmHg was placed on the aortic surface of the
model and the coaptation surface, and a pressure of 5mmHg was
placed on the ventricular surface of the model (Mitchell and Wang,
2014). These two pressures were chosen at the instance the valve
shuts after systole, and are like those used in the study by Hamid et
al.
To conduct analysis with linear materials, the normal heart
valve geometry without partitions was assigned the Young’s Modulus
of 300kPa, Poisson’s ratio of 0.3 and density of 2116 kg/m3 as was
determined in the Material Properties section. The boundaries
specified earlier were held as fixed or constrained in one direction. In
order to simulate the calcified conditions, the model was partitioned
in COMSOL with the sections as previously discussed, and the outer
two sections were given calcified linear properties while the middle
section was given normal aortic valve linear properties. All other
conditions remained the same throughout the stationary study.

3. To conduct further analysis with hyperelastic materials, the
pressures were both reduced by a factor of 10, thus the new assigned
aortic pressure was 12mmHg and the new assigned ventricular
pressure was 0.5mmHg. This would allow us to work with COMSOL
analysis without extremely large deformations that would prevent the
application from generating correct calculations. The same Elastic
Modulus, Poisson’s ratio, and density values used in the linear
materials were used for each respective hyperelastic. All other fixed
surfaces, edges, meshes, the domains where the material properties
were applied to were maintained the same.
Results/Discussion
General Bar Model
To begin our research and gain a general idea of what should
be expected based on the linear material properties of a healthy aortic
heart valve and calcified aortic heart valve under diastolic pressure.
The general 2D bar model is attached to a wall to exemplify the fixed
edge of the leaflet. The equations we got from these calculations
were:
(Healthy)
(Calcified)
Based on these final equations, the healthy aortic valve should
have a greater displacement than the calcified valve. So, from here,
we could compare our COMSOL results to these general
expectations.
Diastolic Pressure and Linear Material in
COMSOL
Our first scenario analyzes the effects of diastolic pressure on a
healthy and calcified aortic heart valve that has linear material
properties.
The stress in the healthy aortic valve was concentrated towards
the center, bottom closest to the edge that is fixed. There is also
concentrated stress at the tip of the coaptation edge and this is where
the maximum stress of 6.35 x 103
Pa. All of this stress is seen on both
the arterial and ventricular side of the leaflet. The maximum
displacement this leaflet underwent was 0.156 mm.
For the calcified aortic valve, recall that it was split into three
sections, where the top and bottom regions were calcified and the
middle region had the healthy valve properties. The stress was
concentrated, in this case, in the bottom calcified region and is also
where the maximum stress of 4.29 x 104
Pa is seen. The maximum
stress is only visible on the ventricular side of the leaflet, but there
was a little bit of a larger stress on the arterial side that is quite a bit
less than the maximum. Lastly, the maximum displacement in this
case was 0.48 mm.
In the end, the results for this situation was not what was
expected. The healthy valve had a smaller maximum stress, as well as
a smaller displacement compared to the calcified valve.
Figure 3a
Figure 3b
Figure 3. Stress and displacement of (a) the healthy aortic valve and
(b) the calcified aortic valve under diastolic pressure and with linear
material properties. These images were made using COMSOL. It is
really important to recognize that the color legend on the right side
of each image is unique to that image so that the stress
concentration is visible for each scenario. Each range for the color
legend is different, so make sure to keep that in mind as they are
analyzed and pay attention to the maximum and minimum values. A
final important note is that the healthy valve displacement is in
meters while the calcified displacement is in mm in figures 3a and
3b. This is due to the imported geometry we used that made this
unable to be changed.
Lower Pressure and Hyperelastic Material in
COMSOL
Next, we analyzed the healthy and calcified arterial valves
under a lower pressure (a factor of 10 less than diastolic pressure) and
with hyperelastic material properties.
For the healthy aortic valve, the stress was concentrated at base
towards the side of the leaflet where the leaflet is fixed, as well as on
the sides of the coaptation edge. The stress was only seen on the side
of the leaflet facing the ventricle. The maximum stress the leaflet
underwent was about 2.96 x 104
Pa and was seen at both the base and
side locations. The maximum displacement the valve experience in
this situation was 0.815 mm.
For the calcified aortic valve, the stress was concentrated at the
corners where the calcified portion runs into the healthy valve portion
in the upper calcified region, as well as throughout the center of the
bottom calcified region. The stress was seen on both the ventricular
and arterial faces of the leaflet, therefore, the stress runs all the way
through the leaflet. In this situation, the maximum stress value was
about 2.11 x 105
Pa in the corner locations previously described and
the maximum displacement the valve experienced was 0.07 mm.
In the end, we found what we were expecting to find, the
healthy valve had a smaller maximum stress and a significantly larger
displacement than the calcified model.
Figure 4a
Figure 4b
Figure 4. Stress and displacement of (a) the healthy aortic valve and
(b) the calcified aortic valve under a factor of ten less than diastolic
pressure and with hyperelastic material properties. These images
were made using COMSOL. It is really important to recognize that
the color legend on the right side of each image is unique to that
image so that the stress concentration is visible for each scenario.
Each range for the color legend is different, so make sure to keep
that in mind as they are analyzed and pay attention to the maximum
and minimum values. A final important note is that the healthy valve
displacement is in meters while the calcified displacement is in mm in
figures 4a and 4b. This is due to the imported geometry we used that
made this unable to be changed.
Setbacks and Future Research
One direction for future research that can be done based on the
research conducted thus far would be to look into why the healthy

4. leaflet under diastolic pressure and with linear material properties had
a smaller displacement than the calcified leaflet.
We tried to make our COMSOL model as a hyperelastic
material under diastolic pressure, however, when we tried to do this,
the equations did not converge. We found this limitation in
COMSOL after trying multiple paths to get around this problem, so
we decided to do one model that has the diastolic pressure and one
that has the hyperelastic material properties. This way we were able
to simplify the equations enough that COMSOL could calculate the
stress and displacement for our models. However, in the future,
another research opportunity would be to use a different program that
can take these extremely complex equations and converge them so
we get a more realistic model of the leaflet.
Conclusion
In this project, the static stress and strain distribution on an
aortic heart valve during diastole was simulated, and our models
provided a general idea of how calcification of the aortic valve might
affect its ability to function. The outcome of the simulation model did
not meet all of our hypothesis, yet the unexpected result did suggest
the significant impact that different assumptions might cause. As
shown in our project, depending on whether linearity or
hyperelasticity was assumed as the material property, opposite results
were achieved for the displacement of the valve. One of the messages
is that when various assumptions are made to simplify a model, the
accuracy is sacrificed. We should be aware of the possible effects of
such simplifications and find the appropriate balance between
accuracy and complexity.
Future studies can be done with fewer simplified assumptions,
such as simulating a fluid mechanics model or considering the
geometry change upon calcification, which would give more
complicated models but will give results that are closer to real-life
cases.
Acknowledgments
We want to thank our biomechanics professor Mohammad
R.K. Mofrad as well as the GSI for the class Ali Madani for their help
in formulating the project idea and providing guidance as the project
proceeded.
References
Clark, R.E., Finke, E.H., (1974). Scanning and light microscopy in
human aortic leaflets in stressed and relaxed states. Journal of
thoracic and cardiovascular surgery, 67 (5). Retrieved from
https://www.ncbi.nlm.nih.gov/pubmed/4823314
Gould, P.L., Cataloglu, A., Dhatt, G., Chattopadhyay, A., & Clark,
R.E. (1973). Stress Analysis of the Human Aortic Valve.
Computers and Structures, 3, 377-384. Retrieved from
http://www.sciencedirect.com/science/article/pii/004579497390
0242
Hamid, M.S., Sabbah, H.N., Stein, P.D., (1985). Large-Deformation
Analysis of Aortic Valve Leaflets During Diastole. Engineering
Fracture Mechanics, 22 (5), 773-385. Retrieved from
http://www.sciencedirect.com/science/article/pii/001379448590
1079
Loree, H.M., Grodzinsky, A.J., Park, S.Y., Gibson, L.J., and Lee,
R.T., (1994). Static Circumfrential Tangential Modulus of
Human Atheroscerotic Tissue. Journal of Biomechanics 27(2),
195-204. Retrieved from
https://www.ncbi.nlm.nih.gov/pubmed/8132688
Mitchell, J.R., Wang, J., (June 2014). Expanding application of the
Wiggers diagram to teach cardiovascular physiology. Advances
in Physiology Education, 38(2), 170-175. Doi:
10.1152/advan.00123.2013 Retrieved from
http://advan.physiology.org/content/38/2/170
N.A., (2016). What is Severe Aortic Stenosis?. CoreValve:
Transcatheter Aortic Valve Replacement (TAVR) Platform.
Retrieved from http://www.corevalve.com/us/what-severe-
aortic-stenosis/index.htm
Swanson, W.M., Clark, R.E., (December 1974). Dimensions and
Geometric Relationships of the Human Aortic Valve as a
Function of Pressure. Circulation Research, 35, 871-882.
Retrieved from
http://circres.ahajournals.org/content/circresaha/35/6/871.full.pd
f
Thubrikar, M. J., Aouad, J., & Nolan, S. P. (1986). Patterns of
calcific deposits in operatively excised stenotic or purely
regurgitant aortic valves and their relation to mechanical stress.
The American Journal of Cardiology, 58(3), 304-308.
doi:10.1016/0002-9149(86)90067-6.
https://www.ncbi.nlm.nih.gov/pubmed/3739919
Rajamannan, Nalini M., (2010). Mechanisms of aortic valve
calcification: the LDL-density-radius theory: a translation from
cell signaling to physiology. American Journal of Physiology –
Heart and Circulatory Physiology, 298 (1). Retrieved from
http://ajpheart.physiology.org/content/298/1/H5
Weinberg EJ, Schoen FJ, Mofrad MRK (2009) A Computational
Model of Aging and Calcification in the Aortic Heart Valve.
PLoS ONE 4(6): e5960.doi:10.1371/journal.pone.0005960.
Retrieved from
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2693668/pdf/po
ne.0005960.pdf
Yeh, H.H., Grecov, D., Karri, S., (2014). Computational Modelling
of Bileaflet Mechanical Valves Using Fluid-Structure
Interaction Approach. Journal of Medical and Biomedical
Engineering, 34(5), 482-486. doi: 10.5405/jmbe.1699 Retrieved
from http://www.jmbe.org.tw/files/2524/public/2524-6352-1-
PB.pdf