Blood Flow and Pressure Change Simulation in the Aorta with the Model

Generated from CT Data

Nobuhiko Mukai, Yuhei Okamoto, Kazuhiro Aoyama and Youngha Chang

Computer Science, Tokyo City University, 1-28-1 Tamazutsumi, Setagaya, Tokyo, Japan

Keywords:

Computer Graphics, Physics based Simulation, Particle Method, Medical Application, Aorta.

Abstract:

We have performed the blood ﬂow and the pressure change simulation in the aorta with the model generated

from CT (Computerized Tomography) data. There have been some previous researches related to the aortic

valve and the blood ﬂow in the aorta. Some works simulated the aortic valve behavior with artiﬁcial models,

and others investigated the blood ﬂow in the aorta with models generated from MRI (Magnetic Resonance

Imaging) data. In this paper, we demonstrate the simulation of the blood ﬂow and the pressure change in the

aorta with a model generated from CT data, which model includes not only the aorta but also the left ventricle.

In the simulation, blood ﬂows into the left ventricle through the mitral valve, the pressure increases according

to the blood ﬂow that moves into the left ventricle through the mitral valve, and the aortic valve opens by the

pressure increase in the left ventricle. Finally, we have conﬁrmed that the pressure change in the left ventricle

corresponds to a literature value.

1 INTRODUCTION AND

RELATED WORKS

There is a valve called “aortic valve” between the

aorta and the left ventricle in our hearts, and some

kinds of surgeries are performed if the valve falls into

malfunction. There are mainly two types of surgeries:

AVR (Aortic Valvular Replacement) and AVP (Aortic

ValvuloPasty). AVR replaces the dysfunctional live

valve with a prosthetic one. The surgery is not so dif-

ﬁcult; however, taking warfarin is necessary to pre-

vent blood from coagulating. On the other hand, AVP

retrieves the valvular function by repairing the dys-

functional live valve. Taking wafarin is not necessary;

however, the surgery is very difﬁcult so that the preop-

erative computer simulation is needed. For the simu-

lations, there have been some previous researches re-

lated to the aortic valve and the aorta.

(Hart et al., 2000) presented two-dimensional

ﬂuid-structure interaction model, and (Hart et al.,

2003) expanded the model into three-dimension, and

visualized the maximum principle Cauchy stresses in

the leaﬂets of the aortic valve. On the other hand,

(Cheng et al., 2004) investigatedthe ﬂuid velocity dis-

tribution and the wall shear stress on a bileaﬂet me-

chanical heart valve. In the simulation, there are two

types of materials: blood and aorta. Blood is ﬂuid

and the aorta is a solid body so that ﬂuid-solid in-

teraction should be considered. (Loon et al., 2005)

used Navier-Stokes equation for the blood ﬂow and

hyperelastic Neo-Hookean model for the solid defor-

mation. (Carmody et al., 2006) used FEM (Finite El-

ement Method) for the simulation of ﬂuid-structure

interaction. (Mukai et al., 2014) employed a parti-

cle method to simulate the aortic valve behavior by

considering heart’s pulsation. (Hsu et al., 2014) and

(Hsu et al., 2015) used Lagrangian-Eulerian methods

for ﬂuid-structure interaction. They created artiﬁcial

models for the simulations, and the models were not

generated from medical data such as CT or MRI.

On the other hand, (Seo et al., 2011) generated a

simulation model from CT images, and simulated the

ﬂow characteristic in the aortic arch. (Wendell et al.,

2013) generated an aorta model from MRI to inves-

tigate the behavior of the aortic valve. (Mukai et al.,

2016) also generated the simulation model from CT

data. These aortic models used for the simulations

were realistic because they were constructed with a

real data; however, they did not have the left ventri-

cle part. Then, (Le and Sotiropoulos, 2013) used a

simulation model including the left ventricle for the

simulation of ﬂuid-structure interaction between the

blood ﬂow and a mechanical heart valve.

In the previous researches, some used artiﬁcial

models to simulate the aortic valve behavior and oth-

ers used simulation models generated from real data;

392

Mukai, N., Okamoto, Y., Aoyama, K. and Chang, Y.

Blood Flow and Pressure Change Simulation in the Aorta with the Model Generated from CT Data.

DOI: 10.5220/0006479403920397

In Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2017), pages 392-397

ISBN: 978-989-758-265-3

however, the target valves for the simulation were

prosthetic. Therefore, in this paper, we demonstrate

the blood ﬂow and the pressure change simulation in

the aorta with a particle simulation model generated

from CT data, and conﬁrm that a live aortic valve

opens by the pressure increase in the left ventricle.

In addition, we investigate if the pressure change in

the left ventricle corresponds to a literature value.

2 MODEL GENERATION

In the simulation, there are two types of materials

to be treated: blood and the aorta. Blood is ﬂuid

and the aorta and the aortic vavle are solid bodies

so that we have to consider the interaction between

them. Especially, the topology of blood changes ac-

cording to the opening and the closing of the aortic

valve. For the topological change, particle methods

are suitable rather than FEM. Then, we employ a par-

ticle method for the simulation, and the simulation

model constructed with particles should be generated

from a medical data. Figure 1 shows the CT images

of a heart used for the generation of the simulation

model. The data is composed of 114 images, and the

images are numbered from the top to the bottom. The

format of the image is “bitmap” and the resolusion is

512×512.

Figure 1: CT image data of a heart.

On the other hand, Figure 2 shows a vertical cross

section image of the heart, which explains the loca-

tions of the aorta, the aortic wall, the aortic valve, the

Valsalva’s sinus, and the left ventricle. Pseudo color

is mapped to each part for easy recognition.

In order to generate the particle model of the

aorta, it is the best way to extract the target region

from the volume data shown in Figure 1; however,

Figure 2: Vertical image of the heart.

the model size would be about 30M voxels because

the image resolution is 512×512 and the number is

114. 30M particles are too much for a normal PC to

handle in the main memory so that the data reduction

is necessary. On the other hand, some breaks or holes

would happen in the aorta model if the image data

is simply reduced. In addition, in particle methods,

extra particles are needed outside of the model since a

high speed particle might jump out of the aortic wall

if the aortic wall is thin. Then, the model generation

algorithm is as follows.

1. The resolution of the CT image data is reduced to

64×64 from 512×512.

2. Every four image plane is extracted for the model

generation.

3. The reduced image is binarized and the target vox-

els are extracted as a closed region by manual.

4. The closed region is ﬁlled with voxels and the two

outer voxels are extracted as the aortic wall ele-

ments.

5. The two extra voxels are added outside of the aor-

tic wall model as dummy voxels for the simula-

tion.

6. The particle model is generated by combining all

voxel data.

The particle model is generated in three dimen-

sion, and Figure 3 shows a cross section of the par-

ticle model generated from the CT data. The right

lower part is the left ventricle and the upper left part

is the aorta. The central part that has a different color

is the aortic valve, and the hole in the left ventricle is

the mitral valve. Figure 4 is another cross section of

the model viewed from a different angle. The lower

part is the left ventricle and the upper part is the aorta.

The right part next to the left ventricle is the mitral

valve. Here, the aortic valve is not so clear on the CT

image so that the aortic valve model is generated by

Blood Flow and Pressure Change Simulation in the Aorta with the Model Generated from CT Data

393

manual from the shape of the valsalva’s sinus. Figure

5 shows the particle model of the aortic valve, which

leaﬂets are differentiated by different colors.

Figure 3: Particle model of the aorta and the left ventricle.

Figure 4: Particle model viewed from a different angle.

Figure 5: Particle model of the aortic valve.

3 SIMULATION METHOD

In this simulation, a particle method is used be-

cause the topology of blood changes often accord-

ing to the opening and the closing of the aortic valve.

There are mainly two types of particle methods: SPH

(Smoothed Particle Hydrodynamics) and MPS (Mov-

ing Particle Semi-implicit). In general, SPH is used

for compressible ﬂuid, while MPS is used for incom-

pressible ﬂuid. Blood is generally treated as incom-

pressible ﬂuid so that we employ MPS (Koshizuka,

2005) for the simulation.

Two kinds of governing equations are used for the

continuous body simulation: Cauchy’s equation of

motion and equation of continuity, which are written

as the following (Eqs. (1) and (2)).

= ∇· σ+ b (1)

Dρ

+ ρ∇ · v = 0 (2)

where, ρ is density, v is velocity, t is time, σ is stress

tensor, and b is body force acceleration such as grav-

ity.

In addition, the constitutive equation of elastic

body is described as follows (Eqs. (3) and (4)).

= λtr(ε)I+ 2µε (3)

ε =

∇u+ (∇u)

(4)

where, σ

is stress of elastic body, ε is strain tensor,

I is unit tensor, u is displacement, λ and µ are lame

constants, which are expressed as follows (Eqs. (5)

and (6)).

λ =

νE

(1+ ν)(1− 2ν)

(5)

µ =

2(1+ ν)

(6)

where, ν is Poisson’s ratio and E is Young’s module.

By substituting Eqs.(3) and (4) for Cauchy’sequa-

tion (Eq.(1)), the next Cauchy-Navier equation (Eq.

(7)) is obtained, which equation is applied to analyze

the behavior of the aortic valve.

= (λ+ µ)∇(∇ · u)µ∇

u+ b (7)

On the other hand, the constitutive equation of ﬂuid is

written as the following (Eqs. (8) and (9)).

= −pI+ 2ηD (8)

D =

∇v+ (∇v)

(9)

where, σ

is stress of ﬂuid, p is pressure, I is unit

tensor, η is viscosity, D is tensor of strain velocity,

and v is velocity. By substituting Eqs.(8) and (9) for

Eq.(1), Navier-Stokes equation (Eq. (10)) is obtained

as follows, which is applied to analyze the behavior

of blood.

= −∇p+ η∇

v+ b (10)

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394

4 SIMULATION RESULTS

The simulation was performed with a normal PC (Per-

sonal Computer), which has i7-3770K CPU (Central

Processing Unit) and GeForce GTX570 GPU (Graph-

ics Processing Unit). The simulation time for 1[step]

corresponds to 0.1[ms] in real time, and the particle

radius was 2 [mm]. Figure 6 shows the visualization

of the pressure inside the left ventricle and the aorta.

Figure 6: Visualization of the pressure in the left ventricle

and the aorta.

At ﬁrst, particles are ﬂown into the left ventricle

through the mitral valve (See Figure 6 (a)). Then,

the pressure in the left ventricle is gradually increased

(See Figure 6 (b) and (c)). When the pressure differ-

ence between the left ventricle and the aorta is high,

the aortic valve opens and some particles (blood) ﬂow

into the aorta (See Figure 6 (d) and (e))). When the

pressure in the left ventricle becomes high, many par-

ticles ﬂow rapidly into the aorta (See Figure 6 (f)).

On the other hand, Figure 7 and 8 show the pres-

sure change of the left ventricle and the aorta, respec-

tively. As particles ﬂow into the left ventricle through

the mitral valve, the pressure in the left ventricle in-

creases. The pressure in the aorta also increases a lit-

tle bit later after the aortic valve opens by the pressure

in the left ventricle. There are some variances of the

pressure at the maximum level because particles ﬂow

from the left ventricle to the aorta, and the pressure is

unsteady at the time.

Figure 7: Pressure change in the left ventricle.

Figure 8: Pressure change in the aorta.

Here, Figure 9 shows the diagram of the pressure

change of the left ventricle and the aorta in one heart

pulsation (Izawa, 2009; Levick, 2011; Klabunde,

2012; Silbernagl and Despopoulos, 2009). At the

atrial systole stage, the pressure of the aorta is higher

Blood Flow and Pressure Change Simulation in the Aorta with the Model Generated from CT Data

395

than the left ventricle, and the aortic valve closes. At

the isovolumetric contraction stage, blood ﬂows into

the left ventricle and it is ﬁlled with blood. In ad-

dition, the left ventricle shrinks isovlumetrically so

that the pressure of the left ventricle rapidly increases

to the same level as the aorta. Actually at the rapid

ejection stage, the pressure of the left ventricle be-

comes slightly higher than the pressure of the aorta.

Therefore, the aortic valve opens. While blood ﬂows

from the left ventricle to the aorta, the pressure of the

left ventricle is almost the same level as the pressure

of the aorta although the pressure of the left ventri-

cle is slightly higher than the pressure of the aorta.

After some blood has ﬂowen from the left ventricle

to the aorta, the pressure of the left ventricle gradu-

ally decreases at the reduced ejection stage. Thus, the

aortic valve closes. After the aortic valve closes, the

left ventricle expands isovolumetrically at the isovol-

umetric relaxation stage, and the pressure of the left

ventricle rapidly decreases. If the aortic valve closes

correctly, no blood ﬂows back from the aorta to the

left ventricle and the pressure difference between the

aorta and the left ventricle increases. At the rapid ﬁll-

ing stage, blood ﬂows into the left ventricle so that it

is ﬁlled with blood again at the reduced ﬁlling stage.

On the other hand, Figure 10 shows the diagram

that has the simulation results overlaid on Figure 9,

where the pressure value is changed to SI unit. From

the ﬁgure, the pressure of the left ventricle corre-

sponds well to the literature value; however, the pres-

sure of the aorta does not correspond to the literature

value. One reason is that particles ﬂown into the aorta

from the left ventricle spread out because the aorta

does not form a closed region. The other is that the

initial pressure of the aorta is zero although the aorta

in the actual heart has some blood from the beginning.

Figure 9: Diagram of the pressure change in one heart pul-

sation.

Figure 10: Pressure comparison between the simulation re-

sults and a literature value.

5 CONCLUSIONS

In this paper, we have demonstrated the blood ﬂow

and the pressure change in the aorta with the particle

model generated from CT data. We employed a parti-

cle method for the simulation because particle meth-

ods are useful for the topology change of the blood by

the opening and the closing of the aortic valve. In or-

der to perform the simulation, we had to generate the

model with particles and a precise model needs too

many memories to handle with a normal PC. Then,

we have constructed the particle model by reducing

the original CT image data and also by attaching ad-

ditional particles outside the model to prevent particle

explosion.

In the simulation, there are two types of ma-

terials to be handled so that two types of equa-

tions, Cauchy-Navier and Navier-Stokes equations,

should be solved. The simulation results were visu-

alized with particles. In the visualization, we have

conﬁrmed that particles ﬂew into the left ventricle

through the mitral value, the aortic valve opened by

the pressure of the left ventricle, and ﬁnally particles

ﬂew into the aorta.

In the comparison of the simulation results with a

literature value, the pressure change in the left ven-

tricle corresponded well to the literature value, while

the pressure change in the aorta did not. This is due

to the openness of the aorta and the emptiness of the

blood in the aorta at the beginning.

Then, we haveto try the simulation with the model

having a closed region of the aorta and conﬁrm that

the pressure change in the aorta also corresponds to

the literature value. In addition, we have treated the

aortic valve as an elastic body; however, the aortic

wall was treated as a solid body instead of an elastic

body in this simulation. In the future, we plan to per-

SIMULTECH 2017 - 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

396

form the simulation by treating the aortic wall as an

elastic body.

ACKNOWLEDGEMENTS

We greatly appreciate Dr. Shuichiro Takanashi, who

is a chief director of Sakakibara Heart Institute, for

providing us the CT data and some useful advices.

This work has also been supported by JSPS KAK-

ENHI Grant Number 15K00176.

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