Fluid and DeformableStructure Interactions in BioMechanical Systems

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Fluid and Deformable-Structure Interactions in
Bio-Mechanical Systems

• Lucy Zhang
• Department of Mechanical, Aerospace, and Nuclear
  EngineeringRensselaer Polytechnic Institute
• Troy, NY

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Numerical methods for fluid-structure interactions

• Commercial softwares (ABAQUS, ANSYS, FLUENT)
• Explicit coupling technique - generate numerical
  instabilities (oscillations), diverged solutions
• Arbitrary Lagrangian Eulerian (ALE)
• limited to small mesh deformations
• requires frequent re-meshing or mesh update

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• 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

• 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)

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Finite element based approach for
Fluid-deformable structure interactions
t0

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

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IFEM nomenclature
NO-SLIP BOUNDARY CONDITION
Solid is completely Immersed in the fluid
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Equations of motion
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Interpolations at the interface
Force distribution
Velocity interpolation
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Algorithm
Update solids positions dsolidVsoliddt
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Validations
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A soft disk falling in a viscous fluid
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
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Fluid recirculation around the soft disk
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Pressure distribution
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Stress distribution on the soft disk
t 0.0 s t 1.1 s t 2.2 s t 3.3 s t
4.35 s
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Terminal velocity of the soft disk
Comparison between the soft sphere and the
analytical solution of a same-sized rigid sphere
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3 rigid spheres dropping in a tube
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3 rigid spheres dropping in a tube
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• 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.
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• 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.

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Biomechanical applications
Heart modeling - left atrium
Red Blood Cell aggregation
Deployment of angioplasty stent
Venous valves
Large deformation (flexible)
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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.
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Cardiovascular system
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.
F
A
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.
C
D
B
E
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Atrial fibrillation and blood flow
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.
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Left atrium geometry
Courtesy of Dr. A. CRISTOFORETTI, ale_at_science.unit
n.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
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Left atrium geometry
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. 1349-1357
Ernst G., et al., Morphology of the left atrial
appendage. The Anatomical Record, 1995. 242 p.
553-561.
Left atrium
Left atrial appendage
Pulmonary veins
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Left atrium with pulmonary veins
During diastole (relaxes, 0.06s lt t lt 0.43s) , no
flow through the mitral valve (v0) During
systole (contracts, 0.43s lt t lt 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/s Blood density 1055 kg/m3 Blood
viscosity 3.5X10-3 N/s.m2 Fluid mesh
28,212Nodes, 163,662 Elements Solid mesh 12,292
Nodes, 36,427 Elements
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.
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Wall muscle constitutive equation
Passive strain during diastole
Strain energy
Active strain during systole
Green-Lagrange strain
Second Piola-Kirchhoff stress
First Piola-Kirchhoff stress
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
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Left atrium with appendage
Pressure distribution at the center of the atrium
during a diastole and systole cycle
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Rigid wall
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Left atrium (comparison with clinical data)
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
Pressure distribution at the center of the atrium
during one cardiac cycle
A
E
Transmitral velocity during one cardiac cycle
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Influence of the appendage
Transmitral velocity during one cardiac cycle
(with and without the appendage)
Velocity inside the appendage during one cardiac
cycle
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Influence of the appendage
Transmitral velocity during one cardiac cycle
(with and without the appendage)
Velocity inside the appendage during one cardiac
cycle
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Then what?

• Use realistic atrial geometry
• How?
• Medical School (Computed Tomography CT scan), but
  the device is ruined due to Katrina
• Help from Dr. A. Cristoforetti, University of
  Trento, Italy

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atrial volume
1 Atrial contraction 2 Isovolumetric
contraction 3 Rapid ejection 4 Reduced ejection 5
Isovolumetric relaxation 6 Rapid ventricular
filling 7 Reduced ventricular filling
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Left atrium and fluid mesh (II)
Fluid, left atrium and inlet fluid velocity
inside the pulmonary veins
Left atrium and inlet fluid velocity inside the
pulmonary veins
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Red blood cells and blood
FEM RBC model
RBC
empirical function
From Dennis Kunkel at http//www.denniskunkel.com/

• 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 4.210-3dyn/cm
• Bending modulus 1.810-12 dyn/cm.

• Property of inner cytoplasm
• Incompressible Newtonian fluid

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Red blood cells and blood
The shear rate dependence of normal human blood
viscoelasticity at 2 Hz and 22 C (reproduced
from http//www.vilastic.com/tech10.html)
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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
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heart
How to link all these together?
vessel
red blood cell
platelet
protein
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Micro-air vehicles

• three types of MAVs
• airplane-like fixed wing model,
• helicopter-like rotating wing model,
• bird-or insect-like flapping wing model.

10-4 10-3 10-2 10-1 1 10
102 103 104 105 106
Gross Weight (Lbs)
potential military and surveillance use
http//www.fas.org/irp/program/collect/docs/image1
.gif
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MAVs
Loitering wings high span and a large surface
area Fast wings a low wing span and a small
area Flying efficiently at high speed small,
perhaps, swept wings Flying at slow speed for
long periods long narrow wings

• Features
• improved efficiency,
• more lift,
• high maneuverability,
• reduced noise.

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Bio-inspired flapping wings
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Future work

• Link IFEM to multiscale numerical approach
• Enhance numerical methods for interfacial
  problems (multiphase)
• Identify and solve good engineering problems

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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)

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Where do we go from here?

• Advance current numerical approaches
• Collaborate with experimentalists/physicians to
  investigate various applications
• Future plans
• thrombosis hemostasis (protein dynamics, cell
  mechanics, bio-material, microfluidics)
• surface interaction - droplet on nanopatterned
  surfaces (molecular dynamics, contact angle)

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What can you do?
Eat Healthy!
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IFEM Governing Equations
Governing equation of structure
Force distribution
Navier-Stokes equation for incompressible fluid
Velocity interpolation
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IFEM Fluid solving algorithm

• Petrov-Galerkin Weak Form and discretization

With tm and tc as stabilization parameters
depending on the grid size

• Newton Iteration solve for the 4 unknowns per
  node u, v, w, p (three velocity components
  pressure)
• Matrix-free formulation is solved by the
  Generalized Minimum Residual Method (GMRES)
• Note that the force exerted from the structure is
  not updated during the Newton Iteration,
  therefore the coupling is explicit.

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IFEM Solid Force Calculation
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IFEM Governing Equations
II.
Insert this inhomogeneous fluid force field into
the N-S eqn.
Distribution of interaction force
The interaction force fFSI,s is distributed to
the fluid domain via RKPM delta function.
I.
III.
Solve for velocity using the Navier-Stokes
equation Eq. (III)
The interaction force is calculated with
Eq. (I)
IV.
P and v unknowns are solved by minimizing
residual vectors (derived from their weak forms)
Update solid displacement with solid velocity
The fluid velocity is interpolated onto the solid
domain via RKPM delta function
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Structure Analysis-hyperelastic material
Mooney-Rivlin material
Elastic energy potential
Cauchy deformation tensor, C
Deformation gradient, F
2nd Piola Kirchhoff stress S
Green-Lagrangian strain ?
Cauchy stress ?
Internal force fk
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Red blood cell model
RBC
From Dennis Kunkel at http//www.denniskunkel.com/
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Rigid leaflet driven by a uniform fluid flow
Fluid H 1.0 cm L 4.0 cm U1cm/s Density
1.0 g/cm3 Viscosity 10.0 dynes/s.cm2 Re10 2500
Nodes and 2626 Elements Leaflet (linear
elastic) ? 0.8 cm t 0.0212 cm Density 6.0
g/cm3 Young modulus E 107 dynes/cm2 Poisson
ratio 0.5 456 Nodes and 575 Elements
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Fluid flow around a rigid leaflet
INSERT MOVIE1.AVI
Re 10
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Pressure field around a rigid leaflet
Re 10
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Leaflet driven by a sinusoidal fluid flow
Leaflet (linear elastic) Fluid l 0.8 cm
H 1.0 cm t 0.0212 cm L 4.0
cm Density 6.0 g/cm3 Density 1.0 g/cm3 Young
modulus E 107 dynes/cm2 Viscosity 1.0
dynes/s.cm2 Poisson ratio 0.5 Fluid initially
at rest 456 Nodes and 575 Elements 2500
Nodes and 2626 Elements
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Leaflet motion and fluid flow
Re 1.0 and St 0.5
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Tip velocity and displacement (I)
Re 1.0 and St 0.5
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Tip displacement (II)
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Venous Valve
Courtesy of H.F. Janssen, Texas Tech University.

• 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

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Venous Valve Simulation
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Venous Valve
Comparison between experiment and simulation at 4
different time steps
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Multi-resolution analysis

• Window function with a dilation parameter

a dilation parameter

• Projection operator for the scale a

• Wavelet function

• Complementary projection operator

low scale high scale