Fluid and Deformable-Structure Interactions in Bio-Mechanical Systems

Lucy Zhang

Department of Mechanical, Aerospace, and Nuclear Engineering

Rensselaer Polytechnic InstituteTroy, NY

• Goals:

• accurate (interpolations at the fluid-structure interface)

• efficient (less/no mesh updating required)

• flexible (deformable and rigid structures, boundary conditions)

• extensibility (multi-phase flows, various applications)

• Immersed Boundary Method (Peskin) - flexible solid immersed in fluid• structures are modeled with elastic fibers• finite difference fluid solver with uniform grid

•Arbitrary Lagrangian Eulerian (ALE)• limited to small mesh deformations

• requires frequent re-meshing or mesh update

Finite element based approach for: Fluid-deformable structure interactions

t=0

Assumptions:

• No-slip boundary condition at the fluid-solid interface• Solid is completely immersed in the fluid• Fluid is everywhere in the domain

solid

t = t1

solid

Equations of motion

ext,d

dijij

i ft

v

Principle of virtual work:

0d

d ext,

ijij

ii f

t

vv

1 2 3

ssis

isif

iif

i dt

vvd

t

vvd

t

vv

ss

d

d

d

d

d

d 1

ssjiji

sfjiji

fjiji ss

dvdvdv ,,, 2

si

si

si

fii

fi dgvdgvdgv

ss

3

s

f iFSI ,s f s dv i

s

dt ij, j

s ij, jf s f giSolid: in s

f (v i

t v jv i, j ) ij, j

f f iFSI(x, t)

v i,i 0

fluid: in

Overlapping

s

Interpolations at the interface

d)(),(),( sFSI,FSI ss tt xxXfxf

vs(Xs, t) v(x, t)(x x s)d

Force distribution

Velocity interpolation

solid nodeInfluence domainSurrounding fluid nodes

Uniform spacing

Read solid & fluid Geometries

Apply initial conditions

Distribute F onto the fluidFFSI,s -> FFSI

Update solids positionsdsolid=Vsolid*dt

Interpolate vfluid onto solids Vsolid

vfluid->Vsolid

Fluid analysis (N-S)Solve for vfluid

Structure analysis Solve for FFSI,s

Algorithm

Validations

Flow past a cylinder

Soft disk falling in a channel

Leaflet driven by fluid flow

3 rigid spheres dropping in a channel

Particle (elastic):

Density= 3,000 kg/m3

Young modulus: E = 1,000 N/m2

Poisson ratio: 0.3

Gravity: 9.81 m/s2

Particle mesh: 447 Nodes and 414 Elements

Fluid:

Tube diameter, D = 4d =2 cm

Tube height, H = 10 cm

Particle diameter, d = 0.5 cm

Density= 1,000 kg/m3

Fluid viscosity = 0.1 N/s.m2

Fluid initially at rest

Fluid mesh: 2121 Nodes and 2000 Elements

A soft disk falling in a viscous fluid

Fluid recirculation around the soft disk

Pressure distribution

yy xy

t = 0.0 s

t = 1.1 s

t = 2.2 s

t = 3.3 s

t = 4.35 s

xx

Stress distribution on the soft disk

Comparison between the soft sphere and the analytical solution of a same-sized rigid sphere

Terminal velocity of the soft disk

ut s gr2

4ln

L

r

0.91571.7244

r

L

2

1.7302r

L

4

3 rigid spheres dropping in a tube

3 rigid spheres dropping in a tube

• Why is it unique?

• fluid- deformable structure interactions

• two-way coupling, higher order interpolation function

• Limitations?

• time step constraint

• rigid solid case

• Possible expansions?

• compressible system

• multiphase flow

• Usefulness?

• numerous applications!

X. Wang - " An iterative matrix-free method in implicit immersed boundary/continuum methods, " Computers & Structures, 85, pp. 739-748, 2007.

Use numerical methods to understand and study cardiovascular diseases.

Find non-invasive means to predict physical behaviors and seek remedies for diseases Simulate the responses of blood flow (pressure and velocities) under different physiologic conditions. Compare our results (qualitatively) with published clinical data and analyze the results.

Biomechanical applications

Red Blood Cell aggregationHeart modeling - left atrium

Deployment of angioplasty stent

Venous valves

Large deformation (flexible)

Why heart?

Cardiovascular diseases are one of the leading causes of death in the western world.

Cardiovascular diseases (CVD) accounted for 38.0 percent of all deaths or 1 of every 2.6 deaths in the United States in 2002. It accounts for nearly 25% of the deaths in the word.

In 2005 the estimated direct and indirect cost of CVD is$393.5 billion.

Cardiovascular system

D: The oxygen-poor blood (blue) from the superior vena cava and inferior vena cava fills the right atrium.

E: The oxygen-poor blood in the right atrium fills the right ventricle via tricuspid valve.

F: The right ventricle contracts and sends the oxygen-poor blood via pulmonary valve and pulmonary artery to the pulmonary circulation.

A: The oxygen-rich blood (red) from the pulmonary vein fills the left atrium.

B: The oxygen-rich blood in the left atrium fills the left ventricle via the mitra valve.

C: The left ventricle contracts and sends the oxygen-rich blood via aortic valve and aorta to the systemic circulation.

AF

D

E

CB

During Atrial Fibrillation (a particular form of an irregular or abnormal heartbeat):

The left atrium does not contract effectively and is not able to empty efficiently.

Sluggish blood flow may come inside the atrium.

Blood clots may form inside the atrium. 

Blood clots may break up

Result in embolism.

Result in stroke.

Atrial fibrillation and blood flow

Without blood clots

with a blood clot

Left atrial appendage

Left atrium geometry

Courtesy of Dr. A. CRISTOFORETTI,ale@science.unitn.it

University of Trento, Italia

G. Nollo, A. Cristoforetti, L. Faes, A. Centonze, M. Del Greco, R. Antolini, F. Ravelli: 'Registration and Fusion of Segmented Left Atrium CT Images with CARTO Electrical Maps for the Ablative Treatment of Atrial Fibrillation', Computers in Cardiology 2004, volume 31, 345-348;

Pulmonary veins

Pulmonary veins

Left atrium

Left atrial appendage

Pulmonary veins

Pulmonary veins

Mitral valveLeft

atriumBlood clots

From Schwartzman D., Lacomis J., and Wigginton W.G., Characterization of left atrium and distal pulmonary vein morphology using

multidimensional computed tomography. Journal of the American College of Cardiology, 2003.

41(8): p. 1349-1357Ernst G., et al., Morphology of the

left atrial appendage. The Anatomical Record, 1995. 242: p.

553-561. Left atrium

Left atrial appendage

Pulmonary veins

Left atrium geometry

77mm

28mm

20mm

17mm 56mm

During diastole (relaxes, 0.06s < t < 0.43s) , no flow through the mitral valve (v=0)

During systole (contracts, 0.43s < t < 1.06s), blood flow is allowed through the mitral valve (free flow)

Blood is assumed to be Newtonian fluid, homogenous and incompressible. Maximum inlet velocity: 45 cm/sBlood density: 1055 kg/m3

Blood viscosity: 3.5X10-3 N/s.m2

Fluid mesh: 28,212Nodes, 163,662 ElementsSolid mesh: 12,292 Nodes, 36,427 Elements

Left atrium with pulmonary veins

Klein AL and Tajik AJ. Doppler assessment of pulmonary venous flow in healthy subjects and in patients with heart disease. Journal of the American Society of Echocardiography, 1991, Vol.4, pp.379-392.

W p a1 exp a2xx2 a3yy

2 a4zz2 2a5xxyy 2a6yyzz 2a7xxzz 1

Wa I1 3 a8xx2 a9yy

2 a10zz2 2a11xxyy 2a12yyzz 2a13xxzz

a

p

W

WW

Wp Wa

a1=0.095034

a2=0.674883

a3=1.120781

a4=1.488016

a5=0.419720

a6=0.703426

a7=0.515748

a8=0.151188

a9=0.239858

a10=0.192101

a11=0.092228

a12=0.091987

a13=0.067475

From W. Xie and R. Perucchio, “Computational procedures for the mechanical modeling of trabeculated embryonic myocardium”, Bioengineering Conference, ASME 2001, BED-Vol. 50, pp. 133-134

Wall muscle constitutive equation

Strain energyPassive strain during diastole

Active strain during systole

ijij E

WS

)(2

1ijmjmiij FFE

jkikij FSP

Second Piola-Kirchhoff stress

Green-Lagrange strain

First Piola-Kirchhoff stress

Pressure distribution at the center of the atrium during a diastole and systole cycle

Transmitral velocity during diastole

Left atrium with appendage

Rigid wall

Pressure distribution at the

center of the atrium during

one cardiac cycle

Transmitral velocity during

one cardiac cycle

Kuecherer H.F., Muhiudeen I.A., Kusumoto F.M., Lee E., Moulinier L.E., Cahalan M.K. and Schiller N.B., Estimation of

mean left atrial pressure from transesophageal pulsed Doppler echocardiography of pulmonary venous flow

Circulation, 1990, Vol 82, 1127-1139

E

A

Left atrium (comparison with clinical data)

5

Pressure (mm hg)

2 Time (s)

1.51

0

Transmitral velocity during one cardiac cycle (with and without the appendage)

Velocity inside the appendage during one cardiac cycle

Influence of the appendage

Red blood cells and blood

RBC FEM RBC model

From Dennis Kunkel at http://www.denniskunkel.com/

10μm

2μm

Property of membrane•Thickness of RBC membrane: 7.5 to 10 nm•Density of blood in 45% of hematocrit: 1.07 g/ml•Dilation modulus: 500 dyn/cm•Shear modulus for RBC membrane: 4.2*10-3dyn/cm•Bending modulus: 1.8*10-12 dyn/cm.

Property of inner cytoplasm •Incompressible Newtonian fluid

empirical function

The shear rate dependence of normal human blood viscoelasticity at 2 Hz and 22 °C (reproduced from http://www.vilastic.com/tech10.html)

Bulk aggregates Discrete cells Cell layers

Red blood cells and blood

Shear of a RBCs Aggregate

The shear of 4 RBCs at low shear rate

The RBCs rotates as a bulk

The shear of 4 RBCs at high shear rate

The RBCs are totally separated and arranged at parallel layers

The shear of 4 RBCs at medium shear rate

The RBCs are partially separated

RBC-RBC protein dynamic force is coupled with IFEM (NS Solver)C-Cf

log (m)

-2

-7

-6 -5

-4 -3

-8

biomaterial

How to link all these together?

platelet

protein

red blood cell

vessel

heart

Shear induce

d

Micro-air vehicles

http://www.fas.org/irp/program/collect/docs/image1.gif

three types of MAVs:1. airplane-like fixed wing model, 2. helicopter-like rotating wing model, 3. bird-or insect-like flapping wing model.

potential military and surveillance use

10-4

10

-3

10

-2

1

0-1

1

1

0

10

2

10

3

10

4

1

05

10

6

Gro

ss W

eig

ht

(Lb

s)

Bio-inspired flapping wings

muscle contraction

Acknowledgement

•Graduate students: • Mickael Gay, Yili Gu

• Collaborators: • Dr. Holger Salazar (Cardiology Department, Tulane University)• Dr. A. Cristoforetti (University of Trento, Italy)

• Funding agencies: NSF, NIH, Louisiana BOR

• Computing resources: •Center for Computational Sciences (CCS) - Tulane• SCOREC (RPI)

What can you do?

IFEM: Governing Equations

s 2us

t 2

s f sur sg on s

f f sur(X)(x X)dV s

f vt vv

? f on

v0

dX /dt v(x)(x X)dV

Navier-Stokes equation for incompressible fluid

Governing equation of structure

Force distribution

Velocity interpolation

s

IFEM: Solid Force Calculationextsurint fffMa

gf )(ext sVol

External Forces: External forces can be arbitrary forces from diverse force fields (e.g. gravity, buoyancy force, electro-magnetic fields).

g – acceleration due to gravity

VSs

I

pqpqI dint

X

f

Internal Forces: hyperelastic material description (Mooney-Rivlin material).

S – 2nd Piola Kirchhoff stress tensor

ε - Green Lagrangian strain tensorTotal Lagrangian Formulation

vt vv f f FSI

v0

Solve for velocity using the Navier-Stokes equation Eq. (III)

The interaction force fFSI,s is distributed to the fluid domain via RKPM delta function.

The fluid velocity is interpolated onto the solid domain via RKPM delta function

s

d)(),(),( sss tt xxxvXv

d)(),(),( sFSI,FSI ss tt xxXfxf

vs

t ( s

x f

x)g

sin in

),(FSI txf),(sFSI, tsXf

tt

ss

vu

2

2

The interaction force is calculated with Eq. (I)

sFSI,f

I.

IV.

III.

II.

),( txvP and v unknowns are solved

by minimizing residual vectors (derived from their

weak forms)

Distribution of interaction force

Insert this inhomogeneous fluid force field into the N-S eqn.

Update solid displacement with

solid velocity

IFEM Governing Equations

sFSI,f

Red blood cell model

RBC

From Dennis Kunkel at http://www.denniskunkel.com/

2μm

Shear rate dependence of normal human blood viscoelasticity at 2 Hz and 22 °C (reproduced from http://www.vilastic.com/tech10.html)

Bulk aggregates Discrete cells Cell layers

Venous Valve

Courtesy of H.F. Janssen, Texas Tech University.

QuickTime™ and aMicrosoft Video 1 decompressorare needed to see this picture.

QuickTime™ and aBMP decompressor

are needed to see this picture.

• Site of deep venous thrombosis formation• Prevents retrograde venous flow (reflux) • Site of sluggish blood flow• Decreased fibrinolytic activity• Muscle contraction prevents venous stasis:

– Increases venous flow velocity– Compresses veins

• Immobilization promotes venous stasis

Venous Valve Simulation

QuickTime™ and aMicrosoft Video 1 decompressorare needed to see this picture.

Venous Valve

Comparison between experiment and simulation at 4 different time steps

Multi-resolution analysis

j

j

jja xa

xx

xaxx 1

)(

• Window function with a dilation parameter:

NNP

jjjaja

Ra xxxxxuxuPxu1

);()()()(

• Projection operator for the scale a

a: dilation parameter

• Wavelet function:);();();( 22 jajaja xxxxxxxxx

• Complementary projection operator:

NNP

jjjaja xxxxxuxuQ

122 );()()(

)()()( 22 xuQxuPxuP aaa low scale + high scale