Particle-based Simulation on Aortic Valve Behavior with CG Model

Generated from CT

Nobuhiko Mukai, Tomofumi Takahashi 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 tried to simulate the aortic valve behavior with a particle method. The simulation model has been

generated from CT data; however, the model was a little bit distorted due to some diseases. The distorted

model is not appropriate for the simulation because we have to simulate the aortic valve behavior for various

kinds of conditions. Then, we have created the normal model by modifying the generated model. In order to

simulate the aortic valve behavior, blood should be ﬂown in the aortic model and particle methods are suitable

for ﬂuid simulation. On the other hand, FEM (Finite Element Method) is usually used for the simulation

of elastic body such as the aortic wall and the aortic valve. However, it is difﬁcult to calculate the mutual

interaction between ﬂuid and elastic body if two different kinds of methods such as particle method and FEM

are adopted for one simulation. Therefore, we have applied a particle method for both of ﬂuid and elastic

body. As the result of the simulation, we have been able to visualize the blood ﬂow and the pressure inside the

aorta, and the stress distribution on the aortic valve.

1 INTRODUCTION

Computer graphics and virtual reality technologies

have been progressed rapidly and their related sys-

tems are being developed especially in medical ﬁelds

such as preoperative planning, surgical simulation,

intraoperative navigation, and so on. Among them,

some systems use robotic manipulators (Yeniaras

et al., 2011). For the purpose of surgical training us-

ing virtual reality, the target organs are necessary and

they are generated based on CT (Computed Tomog-

raphy), MRI (Magnetic Resonance Imaging), MRA

(Magnetic Resonance Angiography), PET (Positron

Emission Tomography), and so forth. In order to

generate the computer model used for the simula-

tion or training, segmentation is needed in advance.

There are mainly three types of segmentations: re-

gion model based level set segmentation, hierarchi-

cal segmentation, and hybrid segmentation of the two

methods (Song et al., 2011).

In addition, there are many medical systems us-

ing computer graphics and virtual reality technolo-

gies, and they treat some organs such as brain, lung,

stomach, liver, and so on. On the other hand, there

are many blood vessels in our body and blood cir-

culation is very important to keep our good condi-

tions. Then, there are so many research works re-

lated to blood vessels (Kirbas and Quek, 2004). Blood

vessels are also divided into some categories such as

aorta, artery, vein, capillary, and so forth. Among

them, the largest and the most important blood ves-

sel is the aorta, which sends necessary nutrition from

the heart to everywhere in our body through artery

and capillary. Then, there are also some studies re-

lated to the aorta and the heart such as aortic di-

ameter estimation using mono-static radar (Solberg

et al., 2010), heart reconstruction based on volumetric

imaging (Bajaj and Goswami, 2008), 3D volumetric

shape reconstruction (Mukai et al., 2011), and blood

stream simulation (Nakagawa et al., 2011b; Seo et al.,

2011; Wendell et al., 2013).

Inside the aorta, there is a valve called the aor-

tic valve, and the blood does not ﬂow correctly if the

aortic valvular dysfunction occurs due to some dis-

eases. One disease is aortic valvular stenosis that nar-

rows the aortic valvular port and makes blood ﬂow

unsmooth, which causes the high pressure difference

between the left ventricle and the aorta. Another dis-

ease is aortic valvular insufﬁciencythat closes the aor-

tic valve incorrectly, which causes blood ﬂow back

from the aorta to the left ventricle. As a result, the aor-

tic pressure remains lower even after the aortic valve

closes. (Izawa, 2009; Levick, 2011; Klabunde, 2012;

Silbernagl and Despopoulos, 2009).

248

Mukai, N., Takahashi, T. and Chang, Y.

Particle-based Simulation on Aortic Valve Behavior with CG Model Generated from CT.

DOI: 10.5220/0005766402460251

In Proceedings of the 11th Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2016) - Volume 1: GRAPP, pages 248-253

ISBN: 978-989-758-175-5

There are mainly two types of surgeries to cure

the preceding diseases. One is aortic valvular re-

placement (AVR), which replaces the dysfunctional

live valve with an artiﬁcial one. The other surgery

is aortic valvuloplasty (AVP), which retrieves the

valvular function by repairing the dysfunctional live

valve. The surgeries are very difﬁcult so that preoper-

ative planning and simulation are necessary, and aor-

tic valve simulations have been performed based on

FEM (Hart et al., 2003; van Loon et al., 2005; Le

and Sotiropoulos, 2013; Hsu et al., 2014; Hsu et al.,

2015). In this method, FSI (Fluid-Structure Interac-

tion) should be analized with two different models:

Navier-Stokes equation for ﬂuid and Neo-Hookean

model for solid. Then, we have simulated the aor-

tic valve behavior based on particle model for both

ﬂuid and elastic body (Nakagawa et al., 2012; Mukai

et al., 2013; Mukai et al., 2014a; Mukai et al., 2014b;

Mukai et al., 2015).

The model used in the previous research, however,

was generated by referring to some medical books just

for the purpose of general blood ﬂow simulation with

a normal aorta. Therefore, we report the generation

of the simulation model based on CT data, and the

visualization of the blood ﬂow, the pressure inside the

aorta and the stress distribution on the aortic valve.

2 MODEL GENERATION

2.1 Aorta Model

The simulation model of the aorta is generated from

CT data. The CT image data of the heart is shown

in Figure 1, which is composed of 114 images, and

is numbered from the top to the bottom. The image

format is “bitmap” and the size is 512×512.

On the other hand, Figure 2 shows the vertical sec-

tion image of the heart, which explains the locations

of the aorta, the aortic wall, the aortic valve, the Val-

salva’s sinus, and the left ventricle.

Figure 3 is the polygon model generated from the

CT data. In order to simulate the aortic valve behav-

ior, the model surroundedby the blue rectangle is nec-

essary. In addition, blood should be ﬂown in the aorta

for the simulation, and some extended simulation ar-

eas are necessary above and below the target model

because blood comes from the left ventricle and goes

to the aorta with some velocity.

Figure 4 (a) and (b) show the extracted model

from Figure 3 and the aorta model generated by

adding cylinders above and below the extracted model

for the simulation of blood ﬂow, respectively. How-

ever, the added cylinders are not on the same line

Figure 1: CT image of the heart.

Figure 2: Vertical image of the heart.

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

since the extracted model is a little bit distorted. If

this generated aorta model is used for the simulation,

blood does not ﬂow smoothly in the aorta so that we

cannot investigate correctly the aortic valve behavior

caused by the blood ﬂow.

On the other hand, the Valsalva’s sinus is com-

posed of three cusps. Then, we have decided to select

one cusp from the Valsalva’s sinus, and generate the

simulation model by combining three cusps copied

from one cusp. Figure 5 shows the generated simula-

tion model. Red parts are selected and copied cusps,

and gray parts are connection part and cylinders lo-

cated above and below the target area. Two cylinders

Particle-based Simulation on Aortic Valve Behavior with CG Model Generated from CT

249

Figure 4: Extracted model and generated aorta model.

Figure 5: Simulation model.

above and below the target area are on the same line.

2.2 Aortic Valve Model

It was difﬁcult to generate the aortic valvemodel from

CT data because the valve image is not so clear com-

pared with the aortic wall that we have decided to gen-

erate the aortic valve model by referring to a medical

book (Arai, 2003). Figure 6 shows the polygon model

of the aortic valve, which is composed of three cusps.

Now, we have the both models of the aortic wall

and the aortic valve; however, those models are com-

posed of polygons. The aortic wall and the aortic

valve are elastic body so that FEM is usually used for

the simulation. However, our purpose is the aortic

valve behavior caused by blood ﬂow. Blood ﬂow is

ﬂuid and the topology is changed by the opening and

closing of the aortic valve. In general, particle method

is used for the simulation of ﬂuid. Here, there is one

problem, that is the collision detection and the mutual

interaction between the ﬂuid and the elastic body. If

two different methods are adopted for the simulation,

Figure 6: Aortic valve model.

it is difﬁcult to detect the collision between the ﬂuid

and the elastic body and to convey the force from the

ﬂuid to the elastic body. Therefore, we have decided

to use a particle method for both materials: ﬂuid and

elastic body. Then, we have to transform the polygon

model to a particle model with depth peeling method

(Nakagawa et al., 2011a). The particle models trans-

formed from the polygon models are shown in Figure

Figure 7: Particle model of the aortic wall and valve.

3 SIMULATION

In this simulation, a particle method is used. There

are two types of particle methods: SPH (Smoothed

Particle Hydrodynamics) and MPS (Moving Particle

Semi-implicit). In general, blood is treated as incom-

GRAPP 2016 - International Conference on Computer Graphics Theory and Applications

250

pressible ﬂuid so that we use MPS (Koshizuka, 2005)

for the simulation.

3.1 Governing Equations

Two kinds of governing equations are used for ﬂuid

simulation: Cauchy’s equation of motion and equa-

tion of continuity, which are written as the following.

= ∇ · 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.

= 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.

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’s equa-

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

tained, which equation is applied to analyze the be-

havior of the aortic wall and valve.

= (λ + µ)∇(∇· u)µ∇

u+ b (7)

On the other hand, the constitutive equation of ﬂuid is

written as the following.

= −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 is obtained as follows,

which is applied to analyze the behavior of blood.

= −∇p+ η∇

v+ b (10)

3.2 Simulation Results

The simulation was performed with a normal PC,

which has i7-3770K CPU and GeForce GTX570

GPU. The simulation took about 1[s], and the sim-

ulaiton time for 1[step] corresponds to 0.1[ms] in real

time.

The number of particles for the aortic wall and

valve were about 9k and 900, respectively. On the

other hand, the number of blood changes from about

26k at the beginning to 42k at the end of the simula-

tion, because some particles are added as the particle

in the left ventricle becomes shorter. The added parti-

cles are provided from solid particles constructing the

cylinder below the target area. The numbers of solid

particles that are located above and below the target

area are 25k and 52k for the aortic wall and blood,

respectively.

In the real heart, there is heart pulsation, which

causes the blood ﬂow from the left ventricle to the

aorta; however, the particles are ﬂown with sinusoidal

velocity in the simulation.

Figures 8 shows the visualization of the pressure

inside the aorta. At the initial state, the pressures in

the aorta (top side) and the left ventricle (bottom side)

are the same. As blood ﬂows from the left ventricle to

the aorta, the pressure in the left ventricle is a little bit

higher than the aorta. By the pressure difference, the

aortic valve starts to open. During the opening of the

aortic valve, the pressure in the left ventricle is higher

than the aorta. However, the pressure difference be-

comes gradually small and the aortic valve starts to

close.

On the other hand, Figure 9 shows the visualiza-

tion of the stress distribution on the aortic valve. The

scale is different from that of the pressure to use the

dynamic range efﬁciently. The opening and closing

mechanisms are the same as that of the pressure; how-

ever, the stress value is higher than the pressure. High

stresses occur at the root of the cusps especially when

the aortic valve is opening and closing.

4 CONCLUSIONS

In this paper,we have reported the particle-based sim-

ulation on the aortic valve behavior. The simulation

model was generated from CT data; however, the gen-

erated model was a little bit distorted due to some dis-

eases, and the model was not suitable for the general

purpose of the simulation so that we have modiﬁed

Particle-based Simulation on Aortic Valve Behavior with CG Model Generated from CT

251

Figure 8: Pressure inside the aorta.

Figure 9: Stress distribution on the aortic valve.

the model by combining three copied cusps.

In addition, the simulation has to treat two differ-

ent materials: ﬂuid and elastic body. Particle method

is suitable for ﬂuid simulation, while FEM is suit-

able for elastic body simulation. However, it is dif-

ﬁcult to treat mutual interaction between two materi-

als if we use two different methods. Then, we have

adopted a particle method in this simulation so that

the simulation models were transformed from poly-

gons to particles for the particle based simulation. We

have also used two different but uniﬁed equations for

the simulation: Cauchy-Navier equation and Navier-

Stokes equation for elastic body and ﬂuid, respec-

tively, which are introduced from Cauchy’s equation

of motion by changing the constitutive equation ac-

cording to the materials: elastic body and ﬂuid, re-

spectively.

Finally, we have been able to perform the simu-

lation, and visualize the pressure inside the aorta and

the stress distribution on the aortic valve. We have

also found that high stresses occur at the root of the

cusps when the aortic valve is opening and closing.

In the real heart, the heart pulsation causes the

blood ﬂow; however, we did not consider the heart

pulsation in this simulation. Therefore, we have to

calculate the blood velocity based on the heart pulsa-

tion. In addition, we have to validate our method by

comparing the simulation results with the real video,

and also expand the simulation method to perform

many kinds of simulations for the planning of real

surgeries.

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 also has been supported by Grant-in-Aid

for Scientic Research on Innovative Areas from the

Ministery of Education, Culture, Sports, Science and

Technology of Japan.

REFERENCES

Arai, T. (2003). Surgery of the Cardiac Valvulopathy.

Igaku-shoin, Tokyo, 2nd edition.

GRAPP 2016 - International Conference on Computer Graphics Theory and Applications

252

Bajaj, C. and Goswami, S. (2008). Multi-component heart

reconstruction from volumetric imaging. In The 2008

ACM symposium on Solid and Physical Modeling,

pages 193–202.

Hart, J. D., Peters, G. W. M., Schreurs, P. J. G., and Baai-

jens, F. P. T. (2003). A three-dimensional computa-

tional analysis of ﬂuid-structure interaction in the aor-

tic valve. Journal of Biomechanics, 36:103–112.

Hsu, M. C., Kamensky, D., Bazilevs, Y., Sacks, M. S., and

Hughes, T. J. R. (2014). Fluid-structure interaction

analysis of bioprosthetic heart valves: Signiﬁcance of

arterial wall deformation. Computational Mechanics,

54:1055–1071.

Hsu, M. C., Kamensky, D., Xu, F., Kiendl, J., Wang,

C., Wu, M. C. H., Mineroff, J., Reali, A., Bazilevs,

Y., and Sacks, M. S. (2015). Dynamic and ﬂuid-

structure interaction simulations of bioprosthetic heart

valves using parametric design with t-splines and

fung-type material models. Computational Mechan-

ics, 55:1211–1225.

Izawa, Y. (2009). Medical Note: Cardiovascular Disease.

Nishimura, Tokyo.

Kirbas, C. and Quek, F. (2004). A review of vessel extrac-

tion techniques and algorithms. Computing Surveys,

36(2):81–121.

Klabunde, R. E. (2012). Color Atlas of Physiology. Lippin-

cott Williams & Wilkins, Baltimore, 2nd edition.

Koshizuka, S. (2005). Particle Method. Maruzen, Tokyo.

Le, T. B. and Sotiropoulos, F. (2013). Fluid-structure inter-

action of an aortic heart valve prosthesis driven by an

animated anatomic left ventricle. Journal of Compu-

tational Physics, 244:41–62.

Levick, J. R. (2011). An Introduction to Cardiovascular

Physiology. Medical Science International, Tokyo.

Mukai, N., Abe, Y., Chang, Y., Niki, K., and Takanashi,

S. (2014a). Aortic valve simulation by considering

heart’s pulsation and axial ﬂow. In NICOGRAPH In-

ternational 2014, pages 18–22.

Mukai, N., Abe, Y., Chang, Y., Niki, K., and Takanashi, S.

(2014b). Particle based simulation of the aortic valve

by considering heart’s pulsation. In Medicine Meets

Virtual Reality, pages 285–289. IOS Press.

Mukai, N., Abe, Y., Chang, Y., Niki, K., and Takanashi, S.

(2015). Visualization of pressure and stress distribu-

tions in aortic valve simulation by considering heart’s

pulsation and axial ﬂow. The Journal of the Society

for Art and Science, 14(1):1–8.

Mukai, N., Nakagawa, M., Abe, Y., Chang, Y., Niki, K., and

Takanashi, S. (2013). Simulation of the aortic valve

deformation by considering blood ﬂow reﬂection. In

Medicine Meets Virtual Reality, pages 286–292. IOS

Press.

Mukai, N., Tatefuku, Y., Nakagawa, M., Niki, K., and

Takanashi, S. (2011). Construction of 3d volumetric

shape model from 2d us images. In The 19th Interna-

tional Congress on Modelling and Simulation, pages

179–185.

Nakagawa, M., Mukai, N., and Kosugi, M. (2011a). A fast

collision detection method by using modiﬁed depth

peeling. Journal of The Institute of Image Informa-

tion and Television Engineers, 65(6):825–832.

Nakagawa, M., Mukai, N., Niki, K., and Takanashi, S.

(2011b). A bloodstream simulation based on parti-

cle method. In Medicine Meets Virtual Reality, pages

389–393. IOS Press.

Nakagawa, M., Mukai, N., Niki, K., and Takanashi, S.

(2012). Particle based visualization of stress distri-

bution caused by the aortic valve deformation. In

The 2012 International Workshop on Advanced Image

Technology, pages 40–45.

Seo, T., Jeong, S. H., Kim, D. H., and Seo, D. (2011).

The blood ﬂow simulation of human aortic arch model

with major branches. In International Conference on

Biomedical Engineering and Informatics, pages 923–

926.

Silbernagl, S. and Despopoulos, A. (2009). Color Atlas of

Physiology. Georg Thieme Verlag, Stuttgart, 6th edi-

tion.

Solberg, L. E., Balasingham, I., and Hamran, S. E. (2010).

Candidate estimators for aorta diameter estimation us-

ing monostatic rader. In The Fifth Internaional Con-

ference on Body Area Networks, pages 124–130.

Song, Y., Luboz, V., Din, N., King, D., Gould, D., Bello,

F., and Bulpitt, A. (2011). Segmentation of 3d vas-

culatures for interventional radiology simulation. In

Medicine Meets Virtual Reality, pages 599–605. IOS

Press.

van Loon, R., Anderson, P. D., Baaijens, F. P. T., and

van de Vosse, F. N. (2005). A three-dimensional

ﬂuid-structure interaction method for heart valve mod-

elling. C.R.Mecanique, 333:856–866.

Wendell, D. C., Samyn, M. M., Cava, J. R., Ellwein, L. M.,

Krolikowski, M. M., Gandy, K. L., Pelech, A. M.,

Shadden, S. C., and LaDisaJr., J. F. (2013). Includ-

ing aortic valve morphology in computational ﬂuid

dynamics simulations: Initial ﬁndings and application

to aortic coarctation. Medical Engineering & Physics,

35:723–735.

Yeniaras, E., Deng, Z., Syed, M. A., Davies, M. G., and

Tsekos, N. V. (2011). A novel virtual reality envi-

ronment for preoperative planning and simulation of

image guided intracardiac surgeries with robotic ma-

nipulators. In Medicine Meets Virtual Reality, pages

716–722. IOS Press.

Particle-based Simulation on Aortic Valve Behavior with CG Model Generated from CT

253