Hemodynamic Characterization of Localized Aortic Valve

Calcifications

Reza Daryani

, Emre Cenk Ersan

and M. Serdar Çelebi

Faculty of Mechanical Engineering, Istanbul Technical University, Istanbul, Turkey

Informatics Institute, Department of Computational Science & Eng., Istanbul Technical University, Istanbul, Turkey

Keywords: Calcification, Aortic Valve, FSI Simulation, Hemodynamic Characterization, Blood Flow.

Abstract: Different hemodynamic characteristics of the blood flow can be studied by numerical simulations of the blood

flow around the heart valves, which are significantly useful in various fields such as recognition and prediction

of cardiovascular diseases, valve surgery, replacement, and advanced design of patient-specific prosthetic

valves. One of these common valvular diseases is aortic valve stenosis, which mainly occurs due to the

decreased orifice area between the valves’ leaflets and leads to insufficient blood pumping. In the aortic valve,

calcification is the main reason for stenosis in which calcium deposits on the leaflets increase their rigidity

and consequently prevent them from fully opening and closing. Severe cases of this disease lead to morbidity

and mortality. In this work, different localized calcifications of the aortic valve are studied for several grades

of this disease and compared with the healthy case. For this purpose, single-phase FSI simulations of blood

flow are performed for various degrees of localized calcification patterns and pertinent hemodynamic

parameters are obtained. Critical flow parameters, transvalvular indexes and Wall Shear Stress (WSS) based

indexes are discussed in detail.

1 INTRODUCTION

One of the most frequent Valvular Heart Diseases

(VHD) among the elderly population is Aortic

Stenosis (AS) which accounts for 43% of VHD

(Pandya, 2012). According to Nudel (2015), 25% of

adults over the age of 65 are affected by this disease.

The occurrence of this pathological condition is

mainly due to the calcium deposition on the aortic

valve leaflets and it is characterized by narrowed

orifice area of the leaflets. The initiation phase

originates in the form of a nodulus which grows over

time (Halevi et al., 2016). This stage is associated

with endothelial damage of the leaflets as a result of

higher flow shear and mechanical stresses (Freeman

et al., 2005; Otto, 2008). AS is classified into mild,

moderate, and severe levels, which are based on the

measured Aortic Valve Area (AVA) using

Transthoracic Doppler Echocardiography (TDE)

analysis (Kappetein et al., 2013). Severe AS cases are

followed by morbidity and mortality complications

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and its treatment requires aortic valve replacement

(Luraghi et al., 2020; Halevi et al., 2018).

Luraghi et al. (2020) worked on the potential

complication associated with calcification patterns in

Transcatheter Aortic Valve Replacement (TAVR)

utilizing FSI models. Sarbandi et al. (2021) developed

2D and 3D FSI models to investigate the role of the

bio-transport process in calcification and thrombosis

for aortic valves. Their results revealed that there is a

close relationship among wall shear stress, flow

vortices, and concentration patterns near and far from

the leaflets. However, there is a lack of information

about the process of 3D calcification formation,

flutter dynamics of leaflets and other pertinent

hemodynamic factors. So far, numerous studies

regarding FSI modelling of the aortic valve

calcification have attempted to incorporate the effect

of calcification as an increase in the thickness of the

valve with the assumption of uniform distribution or

the stiffness of the leaflets (Gilmanov et al., 2019;

Oks et al., 2022). Gilmanov et al. (2019) conducted a

Daryani, R., Ersan, C. and Ã

Gelebi, M.

Hemodynamic Characterization of Localized Aortic Valve Calciﬁcations.

DOI: 10.5220/0012054100003546

In Proceedings of the 13th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2023), pages 37-48

ISBN: 978-989-758-668-2; ISSN: 2184-2841

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

patient-specific FSI simulation to study the effect of

varying leaflet stiffness in transvalvular

hemodynamics.

It is reported that the calcification pattern and its

growth over time are not random. In an examination

of a group of patients with calcified aortic leaflets, it

was found that the spatial distribution of calcium

deposits has two identified patterns (Thubrikar et al.,

1986). The first one is located along the coaptation

line of the leaflets and is named the coaptation

pattern. The other one is the radial pattern which is

seen along the attachment line of the leaflets. Halevi

et al. (2015) implemented a new reconstructing

technique of 3D-geometry of calcification initiation

and growth which confirms the work of Thubrikar et

al. (1986). They found that calcification patterns

strongly affect the mechanical flexibility of the

leaflets and also their orifice area. Halevi et al. (2016)

analyzed the hemodynamic effect of 3D patient-

specific calcification patterns of the aortic valve by

using Fluid-Structure Interaction (FSI) simulations.

Their results revealed that the presence of calcific

deposits dramatically influences the flow shear

stresses. Although it has paved the way to study the

hemodynamic effect of calcification patterns, there is

little focus on other hemodynamic parameters, such

as wall shear stress-based analysis.

In this study, we developed computational 3D

models of the calcified aortic valve using the FSI

approach within an Immersed Boundary (IB)

framework combined with Finite Element (IBFE)

(Griffith and Luo, 2017). Simulations were

established based on 3D geometries for different

grades of deposition accumulation, which can be

classified as Grades 1-6. The results are obtained and

analyzed for a single aortic flow condition where

aortic pressure fluctuates around 120-80 mmHg for

the healthy case. For graded cases, while outflow

pressure is identical to the healthy case, inflow

pressure increases with the severity of the

calcification grade. Quantification of the critical

parameters in blood flow is performed by using

numerous transvalvular hemodynamic and WSS-

based indexes. Additionally, the effect of deposition

patterns on the vortical structures of the flow field is

investigated. The influence of calcification severity

on aortic valves is studied by using transvalvular

hemodynamic indexes such as energy loss, aortic jet

velocity, kinetic energy and the average magnitude of

the velocity. To the authors’ best knowledge, this is

the first attempt to discuss the comprehensive

hemodynamic characterization of localized aortic

valve calcification using a range of calcification

grades.

2 MATERIAL AND METHODS

The mathematical model of our hemodynamic FSI

coupling algorithm will be discussed first, then model

geometry and further details in the numerical scheme

and boundary conditions will be presented.

2.1 Mathematical Modeling

The FSI approach we used is based on the IBFE

formulation proposed by Griffith and Luo (2017).

Deformation, elasticity and stresses of the immersed

structure are described by a Lagrangian form, while

the Eulerian form is used to describe the

incompressible Navier-Stokes equations. Let 𝛺∈ℝ



and 𝐱𝑥,𝑦,𝑧 ∈𝛺 denote the physical domain for

the coupled fluid-structure system and the

corresponding Cartesian coordinates, respectively.

Let 𝑈∈ℝ



denote the initial coordinate system for

the structure, represented by the curvilinear

Lagrangian coordinates



𝑞,𝑟,𝑠



∈𝑈. At time 𝑡,

𝛘𝐗,𝑡



describes the position of the material point 𝐗;

then, the structure and the fluid occupy the regions

𝛘𝑈,𝑡



𝛺



and 𝛺





𝑡



𝛺\𝛺



at time 𝑡,

respectively.

The equations of motion for the coupled fluid-

structure system in IB form are:

𝜌

𝜕𝐮

𝜕𝑡



𝐱,𝑡



𝐮



𝐱,𝑡



∙∇𝐮



𝐱,𝑡





∇𝑝



𝐱,𝑡



𝜇∇



𝐮



𝐱,𝑡



𝐟



𝐱,𝑡



(1)

∇∙𝐮



𝐱,𝑡



0 (2)

𝐟



𝐱,𝑡



𝐅



𝐗,𝑡



𝛿𝐱

𝛘



𝐗, 𝑡



𝑑𝐗



(3)

𝜕

𝛘

𝜕𝑡



𝐗,𝑡



𝐮



𝛘

𝐗,𝑡



,𝑡

𝐮



𝐱,𝑡



𝛿𝐱

𝛘



𝐗, 𝑡



𝑑𝐱



(4)

Eqs. 1 and 2 are the incompressible momentum

and continuity equations, where 𝜌 and 𝜇 are the

density and the dynamic viscosity, 𝐮



𝐱,𝑡



and 𝑝



𝐱,𝑡



are the Eulerian velocity and pressure fields, and

𝐟



𝐱,𝑡



is a body force term representing the Eulerian

elastic force density which is applied to the fluid by

the structure. Eqs. 3 and 4 describe the interaction

between the Lagrangian and Eulerian domains by

integral transformation with a Dirac delta

function 𝛿



𝐱



𝛿





𝛿





𝛿





in three dimensions.

Eq. 3 defines the conversion of Eulerian elastic force

SIMULTECH 2023 - 13th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

density 𝐟



𝐱,𝑡



into the Lagrangian elastic force

density 𝐅



𝐗,𝑡



, while Eq. 4 specifies the no-slip

condition of the viscous fluid on the fluid-structure

interface.

2.2 Model Geometry

The geometry of the healthy aortic valve is illustrated

in Figure 1. The primary CAD model is obtained from

work of Wang (2015) and the ascending aorta section

is added as a straight tube for the sake of simplicity.

The housing height in Figure 1 is 0.116 m with an

inlet diameter of 0.025 m and a wall thickness of

0.001 m. The height of the aortic valve is 0.015 m and

its uniform thickness is 0.0005 m.

As stated in the work of Thubrikar et al. (1986),

the locations of deposits are not random and they

gradually develop along the coaptation line or near to

the aortic root. Based on the work of Bahler et al.

(1999), the severity of the calcification can be

classified into several grades, which are described as:

 Grade 1 represents no calcification,

 Grade 2 represents the localized area of

increased reflectivity but no areas of dense

calcification,

 Grade 3 represents markedly increased

reflectivity (calcification) in one leaflet but

equal to or less than Grade 2 changes in

other leaflets,

Figure 1: Reference geometry of the aortic valve for all

calcification models.

 Grade 4 represents markedly increased

reflectivity in two leaflets but equal to or less

than Grade 2 changes in the third leaflet,

 Grade 5 represents moderately increased

reflectivity in all leaflets,

 Grade 6 represents severely increased

reflectivity in all leaflets.

Grade 3 Grade 4 Grade 5 Grade 6

Top view

Bottom view

Isometric view

Figure 2: Geometries of the calcification grades merged with the reference aortic valve geometry.

Hemodynamic Characterization of Localized Aortic Valve Calciﬁcations





Figure 3: Illustration of the unstructured mesh of a) housing, b-e) each valve of Grades 3-6, respectively.

Due to the lower impact of Grade 2 calcification, we

ignored this grade in our simulations and the

geometries were created for Grades 3-6. Calcification

geometries are adopted from the 3D geometries in the

work of Lavon et al. (2019) and they were merged

with our reference geometry of the aortic valve

model. Model geometries are generated based on the

patterns using medical classifications presented in the

work of Bahler et al. (1999) for several grades, which

are illustrated in Figure 2. It should be noted that the

geometry of Grade 6 is the only case where the

calcific deposition is visible from the bottom view

(see in Grade 6 in Fig. 2).

2.3 Numerical Schemes and Boundary

Conditions

The fluid mesh treatment of our IB method uses a

staggered, block-structured fixed Cartesian grid to

discretize the Eulerian variables, velocity, pressure

and force density (Griffith, 2017). The divergence,

gradient, and Laplace operators are approximated

using standard second-order accurate finite difference

methods (Griffith, 2009). A version of the piecewise

parabolic method (PPM) is used to discretize the

convective term in momentum equation (Colella et

al., 2017). The Lagrangian variables associated with

the immersed structure are discretized using an

unstructured finite element mesh.

2.3.1 Structure Mesh

Figure 3a represents the reference housing and in

Figures 3b-e, the structure meshes of each grade are

illustrated. For whole structure mesh, unstructured

first-order tetrahedral elements are used. The total

number of structure mesh elements are 1.89, 1.88,

1.93 and 2.12 million for Grades 3-6, respectively.

For healthy case, the number of mesh elements is 1.86

million.

Figure 4: Computational mesh for the calcified aortic valve

model, a) Isometric view, b) Cross-diagonal section view.

)

SIMULTECH 2023 - 13th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

2.3.2 Fluid Mesh

The fluid domain is considered as a Cartesian box and

discretized by N × 2N × N mesh elements in x, y, and

z directions, respectively. A static mesh refinement

level with a refinement ratio of 2 is used near the

immersed interface, as shown in Figure 4. In the

simulations, N is taken as 64, which leads to a total

fluid mesh of about 2.15 million mesh elements with

the dimensions of 0.058 m × 0.116 m × 0.058 m.

2.3.3 Constraints and Boundary Conditions

The housing structure is anchored to the top and

bottom surfaces of the physical domain. This is done

by assigning integer numbers to each surface in

ICEM CFD® software during the meshing process,

then imposing zero displacement constraints on the

top and bottom surfaces of Figure 4. Wall boundary

conditions are applied on the bottom and top

boundaries. For the side boundaries of the physical

domain, zero tangential velocity and zero normal

velocity gradient conditions are employed. In Figure

5, the velocity profile is used as the inflow and the

pressure profile is implemented as the outflow

boundary condition on the circular domains located at

the bottom and top walls of the computational box,

respectively.

Figure 5: The inflow and outflow boundary conditions for

the calcified aortic valve model, a) Inlet velocity profile

(Spühler et al., 2018), b) Outlet pressure profile

(Rubenstein et al., 2015).

2.3.4 Numerical Implementation

An explicit version of Crank Nicolson–Adams

Bashforth scheme is employed for time stepping with

a step size of ∆𝑡 1.0  10

-5

s. Blood is modeled as

a single-phase Newtonian fluid with a dynamic

viscosity and density of 0.0035 Pa∙s and 1056 kg/m

respectively. The flow is transient and the simulations

are performed initially for one cardiac cycle (0.8 s).

We used the open source IBAMR software

framework (https://ibamr.github.io/) for our models.

Simulations run on The National Center of High-

Performance Computing of Turkey (UHeM). We

used InfiniBand®-based (FDR, 56 Gbps) computing

cluster with two compute nodes, each consisting of an

Intel® 2.4 GHz Xeon® E5-2680 v4 processor with 28

cores and 128 GB of memory. We observed a parallel

speedup factor of nearly 1.5 by doubling the number

of processors in up to two nodes (56 cores). However,

by employing three nodes, the parallel performance

reduces due to network communication and high

frequency parallel I/O. So we decided to employ two

compute nodes due to the performance/cost ratio.

Each simulation took approximately 200 hours of

wall clock time. The typical total CPU hours for each

simulation was around 11500.

2.3.5 Material Model

As the material model of the aortic valve, a Neo-

Hookean hyper-elastic model is used and its strain

energy distribution function is given as

𝐖

𝜇



𝐼



3



𝜇ln



det



𝐹





𝜆



ln det



𝐹





(5)

where λ, 𝜇 are the Lame constants of linear

elasticity and 𝐹 is the deformation gradient. In our

model, the aortic valve and the housing are

considered to be incompressible



det



𝐹



1



Piola–Kirchhoff stress tensor can be written as

ℙ𝜇𝐹



𝜆ln



det



𝐹



𝜇



𝐹



(6)

The housing is modeled as a semi-rigid tube.

Calcified deposits are modeled as a linear elastic

material with Young's modulus of 2 MPa and

Poisson's ratio of 0.45. 1

Piola–Kirchhoff stress

tensor for a linear elastic model is

ℙ𝜇𝐹𝐹



2𝐼𝜆tr



𝐹𝐼



𝐼

(7)

The list of material parameters is given in Table 1.

Table 1: Material properties for the calcified aortic valve

models.

Part Material model Parameters

Aortic valve Neo-Hookean

𝜇140 kPa

𝜆17 MPa

Calcifications Linear elastic

𝐸2 MPa

𝜈0.45

Hemodynamic Characterization of Localized Aortic Valve Calciﬁcations

3 NUMERICAL RESULTS AND

HEMODYNAMIC

CHARACTERIZATION OF

CALCIFIED AORTIC VALVE

The calcification of leaflets can result in their

deterioration and remodeling, which in turn leads to a

shorter life span of the Bioprosthetic Heart Valve

(BHV) and, in general, reduces the functionality of a

valve. Subsequent changes in transverse pressure

gradients and shear forces may lead to further

complications, which make localized calcifications

necessary to be investigated. Thus, hemodynamic

alteration in the aortic valve region due to

calcification plays a key role in problems associated

with VHD. In the following sections, the flow

patterns for various calcification grades and

underlying hemodynamic parameters divided into

Transvalvular hemodynamic and Wall Shear Stress

(WSS) based indexes will be discussed.

3.1 Analysis of Flow Pattern

For a better understanding of the calcification effects

on valve leaflets, at first, velocity vectors of flow

around healthy aortic valve will be compared with the

calcified cases shown in Figure 6. Initially, the

leaflets remain closed until the pressures are

equalized on the ventricular and aortic sides. When

the pressure increases on the ventricular side, the

valve starts to open around 𝑡0.02 s. The valve

opens in a form with a fully circular orifice area

before the velocity reaches its peak at the systole

phase. The blood moves through the valve into the

ascending aorta without any significant flow

deformation until this time instant. After the peak

systole, when the jet velocity decreases, several

vortices are generated due to the flow separation from

leaflet tips. When the jet velocity approaches zero

during the late systole, the leaflet closure occurs due

to the backflow into the Valsalva sinus region filling

the valve packets as a result of the hemostatic effect.

Arantius nodules and coaptation lines are observed to

be well aligned spatially and temporally. As expected,

no flow is observed behind the leaflets during the

opening phase. However, vortical flow patterns can

be seen within the packets when the valve closes. In

the final figure at 𝑡 0.30 s, some regurgitation is

observed in the middle area close to the Arantius

nodules of all three leaflets.

As stated in Section 2.2 before, Grade 3

represents the case that only one leaflet is involved

with calcification. Due to relatively increased valve

stiffness as a result of calcification, leaflets undergo a

sudden opening with a delay. However, in healthy

case valves open earlier and the slope of the velocity

increment is smoother. Therefore, at the beginning of

the systole blood velocity is higher in healthy case.

Blood flow passes the valve orifice in a spatially

asymmetric manner due to unstable displacement of

one leaflet. This asymmetry shifts the flow towards

the ascending aorta wall and diverges from symmetric

flow which the healthy case has. As a result of this

deflection, flow circulates toward the calcified leaflet

which can be seen in the plots of 𝑡0.15, 0.185 s.

This flow circulation contributes to an earlier closure

of the calcified leaflet that is shown in plots of 𝑡

0.27, 0.30 s. In the healthy aortic valve, the jet flow

passing the orifice retains its uniform shape; however,

the constrained leaflet disturbs the flow and thus the

jet flow becomes chaotic. While healthy aortic valve

closes by the end of 𝑡 0.30 s, Grade 3 calcified

valve closes with a slight delay.

In Grade 4 calcified aortic valve, two leaflets have

notable calcium deposits on their upper surface facing

to ascending aorta. More intense calcification leads to

more valve resistance which results in a lower orifice

area and increased jet velocity. Furthermore, the jet

becomes narrower and centric whereas it is more

uniformly distributed for the healthy case. In contrast

to Grade 3, in the Grade 4, the flow is more centric in

the systole phase, but due to the earlier closure of the

strongly calcified two leaflets, the flow deviates. This

shifts the blood flow toward the third leaflet which

can be observed in figures of 𝑡0.27, 0.35 s. This

flow deviation creates a strong vortex in the

ascending aorta region, and this leads to a higher

energy dissipation.

By comparing the results of the velocity vectors

for the blood flow of Grade 5 with previous cases, it

is seen that increasing calcification severity leads to a

stronger jet flow and unstable closure of leaflets.

Also, due to the reduced orifice area, the jet flow

width becomes smaller. In Grade 5 case, deposit

distribution is almost symmetric and we observe an

almost symmetric blood flow comparing to Grade 4.

The leaflet with no deposit, takes a concave form in

early systole phase and during the valve closure in the

late systole phase, it deviates the flow toward the

ascending aorta wall. The resulting collision creates

vortices in the ascending aorta after valve closure.

Grade 6 calcification case is the most severe one and

as expected the strongest jet flow is observed. In

comparison with the Grade 3 case, there is an increase

of about 25% in the jet velocity magnitude. The

length and duration of the aortic jet are measured

maximum due to the smallest Geometric Orifice Area

SIMULTECH 2023 - 13th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

 

Healthy



Grade 3



Grade 4



Grade 5



Grade 6



Figure 6: Illustration of flow vectors [m/s] at different times over a cardiac cycle for various calcification cases.

𝑡0.02 s

𝑡0.065 s

𝑡0.15 s

𝑡0.27 s

𝑡0.30 s

Hemodynamic Characterization of Localized Aortic Valve Calciﬁcations

(GOA) where jet width is minimum. The relatively

symmetric distribution of the deposits on all three

leaflets, their opening and closure is almost

symmetrical. However, the latest valve opening is

observed with the smallest GOA generating the

strongest vortex structures. Because of the narrower

jet flow, the strongest backflow circulations toward

the leaflets are observed in the ascending aorta.

3.2 Transvalvular Hemodynamic Index

Analysis

Disturbed hemodynamics of unsteady aortic flow in

the presence of localized calcifications can be

characterized by some hemodynamic parameters that

are utilized in the early detection of valvular heart

diseases. Calcification affects the aortic valve

kinematics and reduces its functionality by

decreasing the amount of blood passed through the

leaflets.

Variation of the GOA in healthy and calcified

aortic valves for a time range of half a cardiac cycle

is given in Figure 7. Based on the published clinical

data by Thubrikar (1990), the Aortic Valve Area of a

healthy aortic valve is within the range of 3.9 ± 1.2

and our results for the healthy case agree and

align with this range.

Figure 7: Geometric orifice area (GOA) variation for the

healthy and calcified aortic valve in a half-cardiac cycle.

It can be observed that calcification greatly affects

leaflet openings and the resulting orifice area

reduction is remarkable. Results show that, with the

increased calcification, openings of the valve leaflets

are delayed. In the healthy aortic valve, leaflets

opening occurs around 0.02 s; however, for the

diseased cases opening time is about 0.05 s. The main

reason for this is the increased resistance of calcified

leaflets. In the healthy aortic valve case, the GOA

profile follows the trend of the inlet profile where the

orifice area increases until the peak systole and

decreases until valve closure. In contrast, the trend of

the calcified cases is completely different and there is

a sudden reduction at nearly 0.13 s. Additionally, by

comparing the GOA curves of the calcified cases, it

can be observed that by increasing the calcification

intensity, leaflets closure occurs more gradually.

Similar to the delay in the opening phase, the delay in

the closure of the leaflets is observable for the

calcified cases. Furthermore, in these cases, there are

increments in the GOA after the valve closure

referring to a severe delay in the closing of coaptation

lines. In this situation, severe regurgitation occurs due

to the closing delay of the valve leaflets, resulting in

insufficient blood flow to the ascending aorta. In the

diseased cases, the increased stiffness of the leaflets

leads to increasing ventricle pressure resulting in

small inflation at the bottom section of the model

wall. As stated, by looking at the GOA curves of the

calcified cases, there is an increment after about 0.36

s. One of the reasons for this bump might be the

depletion of the stored fluid volume in the inflated

section of the model wall even after the early diastole.

Another transvalvular hemodynamic parameter is

the maximum velocity, 𝑉



, of the jet passing

through the aortic orifice area. By using this

parameter, pressure drop across the valve can be

calculated using the simplified Bernoulli equation. In

Figure 8, the maximum velocity of flow passing

through the orifice area for healthy and calcified cases

is given. Results show that increasing the

calcification severity directly raises the jet velocities

due to the decreased orifice area to maintain the flow

rate.

One important observation is the early drop of jet

velocity near t = 0.2 s at Grade 4 where two leaflets

are constrained by deposit and the third one is free to

move. This result reveals that calcification severity is

not the only reason for the velocity increment and

divergence but also pattern asymmetry plays an

important role.

Figure 8: The variation of the maximum jet velocity 𝑉



for healthy and calcified valves in a half cardiac cycle.

The third transvalvular hemodynamic parameter

is the rate of energy dissipation. This parameter

determines the instantaneous power loss of blood

flow. It is measured in Watts and can be calculated by

𝐸







𝑡





















𝛷𝑑𝑉. In this equation, 𝛷

SIMULTECH 2023 - 13th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

𝜏⨂𝛻𝜈 represents a dissipation function where 𝜏 is the

viscous shear stress tensor, and 𝐸





𝑡 is the power

loss in the Valsalva sinus region and the following

ascending aorta. This parameter explains the shear-

related energy dissipation due to the viscous blood

flow and the presence of aortic stenosis. The variation

of this parameter over a half-cardiac cycle is

presented in Figure 9.

Figure 9: The variation of rate of energy loss for healthy

and calcified valve models in a half cardiac cycle.

These results prove that the increasing severity of

the stenosis as a result of aortic valve calcification

leads to a higher energy dissipation of the blood flow.

The value of energy dissipation for the healthy case,

in general, is minimum among all cases due to the

least resistance to blood flow. For Grade 3 at peak

systole, an increase of 10% in energy dissipation is

seen in comparison with the healthy case. For Grades

4 and 5 the results are close to each other which

implies that the rate of energy loss for higher

calcification on two leaflets is similar to the rate of

moderate calcification on the three leaflets. The

sudden increase in the late systole for Grade 4 might

be due to the higher flutter of the leaflets during their

closure phase and consequently the creation of the

strong vortices that lead to a higher energy

dissipation. The highest energy dissipation belongs to

the Grade 6 case which is the severe stenosis disease

and energy dissipation is doubled in comparison with

the healthy aortic valve.

Figure 10: The variation of the kinetic energy (𝐸



) of the

blood flow in a half cardiac cycle for a healthy and calcified

aortic valve.

The kinetic energy of the blood flow is the other

important hemodynamic parameter and can be

determined by 𝐸





𝑡









𝜌





𝜐





𝑑𝑉 which is

directly integrated over the aorta’s volume. 𝐸



describes the mechanical energy of the blood flow,

which is exchanged with pressure and the viscous

dissipation terms. In Figure 10, this parameter is

illustrated for healthy and calcified cases. Similar to

the previous hemodynamic parameters, increasing the

severity of the calcification tends to increase the

amount of kinetic energy which leads to an increased

energy dissipation and decreased pressure drop.

The last transvalvular hemodynamic index is the

average magnitude of the vorticity which is measured

in the volume of aorta. The magnitude of vorticity can

be quantified using 𝛺





𝑡















𝛺

𝑑𝑉,



where

𝛺





𝑤





𝑤





𝑤





, 𝛺



𝛻v



. Figure 11

shows that the vorticity structure of the blood flow is

amplified by increasing the severity of the

calcification. Similar to the other parameters Grade 6

has the highest values except opening and closing

phases of the valve leaflets. In other words, high

amount of calcification results in the generation of the

strong vortices which leads to a larger energy

dissipation rate at time interval t = 0.05 - 0.30 s. In

comparison with healthy case in which the vorticity

magnitude increases until peak systole and decreases

after it, the value of this parameter changes within a

narrow range after valve opening until the beginning

of late systole phase at about 𝑡0.30 s. One main

reason for this is that the valve leaflets start to flutter

at this time range and generate strong vortices at their

tip region.

Figure 11: History of vorticity (𝛺



) calculated for healthy

and different calcified valves over a half cardiac cycle.

3.3 WSS Based Hemodynamic Index

Analysis

In general, hemodynamic parameters and/or indexes

can be used for quantifying some cardiovascular

diseases. An important group of these parameters can

be classified as the Wall Shear Stress (WSS) based

hemodynamic indexes. The viscous stresses exerted on

the wall surface by the fluid is explained by WSS,

Hemodynamic Characterization of Localized Aortic Valve Calciﬁcations

WSS

𝑛



.𝜏



where 𝑛



is surface normal vector and

𝜏



is viscous stress tensor. In this work, we will be

discussing the results of Time Averaged Wall Shear

Stress (TAWSS), Oscillatory Shear Index (OSI),

Relative Residence Time (RRT), and Transverse Wall

Shear Stress (transWSS) indexes for different cases of

the healthy and calcified aortic valves.

Comparing the results of healthy and calcified

aortic valve cases in Figure 12, it is observed that the

growth of the calcification deposits increases the value

of TAWSS both at the edges of the leaflets and the

radial region. Furthermore, the intensity of the

calcification also leads to an increase in TAWSS. By

investigation of the results in Grade 3, it can be

observed that TAWSS reaches to higher values at the

tip of the non-calcified leaflets. This is caused by the

shear loads as a result of the extra transverse pressure

gradient fluctuations and by the fluttering movement as

a result of the limited opening of the calcified leaflets.

In other grade cases, TAWSS values are slightly

increased with the larger volume of deposits and

spatially span more area. One of the reasons for this is

that the shear stress rises at the tips of the leaflets as a

result of the higher jet velocity and sharper pressure

fluctuations which occur due to the narrowing of the

orifice area.

OSI index has a scalar ranging between 0 and 0.5,

and it is used to determine whether a flow is laminar

(unidirectional) or oscillating (turbulent). The critical

point to know is that it does not give any information

about the actual magnitude of the oscillation. Figure

12 shows that there is a significant WSS oscillation

on the surfaces of the valve leaflets. Additionally,

relatively lower OSI values can be observed on

surfaces with large calcification deposits. On the

other hand, OSI values are relatively lower in the root

regions of healthy case.

Relative RRT represents the relative duration that

blood resides close to a wall in one cardiac cycle.

Comparing the RRT results, it is seen that maximum

values appear at the roots of the leaflets and minimum

values are at the edges of the leaflets due to the short

residence time of blood in this region. Additionally,

increasing the calcification level leads to an increase in

the RRT values and the maximum values are observed

in the Grade 6 case. For each case, RRT values appear

to be particularly higher in the root regions of the

calcified leaflets. This means that the blood stays for a

relatively long time behind the calcification deposits.

The final WSS based hemodynamic index studied

is the transWSS, which is defined as the average of

WSS components perpendicular to the temporal mean

WSS vector over a cardiac cycle. By observing the top

views given in the last column of Figure 12, the

presence of calcification slightly increases the

transWSS values. For example, in Grade 3, transWSS

values on calcified leaflets are slightly higher

compared to the healthy leaflets. As it is known,

transWSS basically looks at the variation of stresses in

lateral directions perpendicular to the main flow

direction. Therefore, increase in transWSS occurs due

to the greater lateral load components as a result of

temporally and spatially unstable fluctuations in cases

with increasing generally asymmetric patterned

calcifications.

4 DISCUSSION

In this study, we developed an FSI model for the

aortic valve and investigated the effect of several

grades of localized calcifications. For the first time in

the literature, a comprehensive hemodynamic

characterization is conducted for an aortic flow in the

presence of medically graded localized calcifications.

We conducted three stage analysis: analysis of flow

pattern, study of transvalvular hemodynamic indexes

and WSS based index analysis.

Our initial results showed that increasing

calcification severity leads to a stronger jet flow and

unstable closure of leaflets. However, we also

observed that asymmetric pattern of the localized

calcifications may lead to a spatially and temporally

unstable behavior of the valve leaflets even if

calcification volume is not larger. Due to the reduced

orifice area, the jet flow width becomes smaller and

blood flow passes the valve orifice as spatially

asymmetric flow due to unstable displacement of

leaflets. This asymmetry shifts the flow towards the

ascending aorta wall and exposes the wall to a higher

wall shear stress for a limited period of time. But this

periodical WSS impact may lead to erode of

endothelial surface of ascending aorta with time. With

the increasing calcification grades opening of the

valves are delayed, flow velocities significantly

increased due to the limited GOA and severe chaotic

flow in ascending aorta are observed at the late systole

phase. This flow regime generates a significant back

flow and recirculation to a Valsalva sinus and root

region of the valve. Another potential effect of this

flow is the redirection of the feeding flow at the right

and left coroner artery entries. It is clear that there is a

need for further analysis of this back flow effect locally

on the right and left coroner entry zones whether the

rate of blood flow is reduced or not. Another important

finding in this study is to observe a higher energy

dissipation of blood flow due to the increasing severity

of aortic valve calcification.

SIMULTECH 2023 - 13th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

TAWSS [Pa] OSI

RRT [Pa

-1

]

transWSS [Pa]

Healthy



Grade 3



Grade 4



Grade 5



Grade 6



Figure 12: WSS based hemodynamic indexes for healthy and different grade calcification cases.

The variation of the maximum jet velocity 𝑉



also align with the energy dissipation behaviour of

blood flow which is exposed to a higher resistance.

Similarly, increasing severity of the calcification

tends to increase both the amount of kinetic energy

and the vorticity structure of the blood flow which

lead to an increased energy dissipation and decreased

pressure drop. TAWSS and OSI indexes show that

growth in localized calcification leads to an increase

of these two indexes. WSS values are mostly

increased at the edges of the leaflets and flow

oscillations occur everywhere in the Valsalva sinus

compared to healthy case. Moreover, residence time

of the blood slightly increased at the root region but

significantly reduced at the edges of the leaflets.

When we combine low TAWSS and transWSS with

high RRT, we can conclude that the root region may

be a good candidate for further formation of the stasis

which is the initial stage of thrombosis according to

the low shear stress theory. Similarly, combination of

high TAWSS, OSI and transWSS may lead to a

thrombosis formation at the edges of the valves based

on the high shear stress theory.

For a future work, two and more phase FSI blood

flow models are needed to observe RBC and platelet

Hemodynamic Characterization of Localized Aortic Valve Calciﬁcations

aggregations around both back region of the aortic

valve and edges of the leaflets.

ACKNOWLEDGEMENTS

This work is funded by the Scientific and

Technological Research Council of Turkey

(TUBITAK) as ARDEB 1001 project under the grand

number 120M671. Computing resources used in this

work were provided by the National Center for High-

Performance Computing of Turkey (UHeM) under

grant number 5010662021.

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