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Fluid and Deformable-Structure Interactions in Bio-Mechanical Systems Lucy Zhang Department of Mechanical, Aerospace, and Nuclear Engineering Rensselaer 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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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

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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

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IFEM nomenclature domain densitySpatial coordinatesvelocityCauchy stress Solid s s x s (X s,t)vsvs s Fluid f f xv f NO-SLIP BOUNDARY CONDITION Solid is completely Immersed in the fluid

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Equations of motion Principle of virtual work: s

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Solid: in s fluid: in Overlapping s

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Interpolations at the interface Force distribution Velocity interpolation solid node Influence domain Surrounding fluid nodes Uniform spacing

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Read solid & fluid Geometries Apply initial conditions Distribute F onto the fluid F FSI,s -> F FSI Update solids positions d solid =V solid *dt Interpolate v fluid onto solids V solid v fluid ->V solid Fluid analysis (N-S) Solve for v fluid Structure analysis Solve for F FSI,s Algorithm

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Validations Flow past a cylinder Soft disk falling in a channel Leaflet driven by fluid flow 3 rigid spheres dropping in a channel

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Particle (elastic): Density= 3,000 kg/m 3 Young modulus: E = 1,000 N/m 2 Poisson ratio: 0.3 Gravity: 9.81 m/s 2 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/m 3 Fluid viscosity = 0.1 N/s.m 2 Fluid initially at rest Fluid mesh: 2121 Nodes and 2000 Elements A soft disk falling in a viscous fluid

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Fluid recirculation around the soft disk

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Pressure distribution

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t = 0.0 s t = 1.1 s t = 2.2 s t = 3.3 s t = 4.35 s Stress distribution on the soft disk

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

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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 , 2007.An iterative matrix-free method in implicit immersed boundary/continuum methods,

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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 Red Blood Cell aggregation Heart modeling - left atrium 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 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 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. A F D E C B

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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

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Left atrium geometry Courtesy of Dr. A. CRISTOFORETTI, 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, ; Pulmonary veins Left atrium Left atrial appendage Pulmonary veins Mitral valve Left atrium Blood clots

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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, (8): p Ernst G., et al., Morphology of the left atrial appendage. The Anatomical Record, : p Left atrium Left atrial appendage Pulmonary veins Left atrium geometry 77mm 28mm 20mm 17mm 56mm

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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/s Blood density: 1055 kg/m 3 Blood viscosity: 3.5X10 -3 N/s.m 2 Fluid mesh: 28,212Nodes, 163,662 Elements Solid 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

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WpWp WaWa a 1 = a 2 = a 3 = a 4 = a 5 = a 6 = a 7 = a 8 = a 9 = a 10 = a 11 = a 12 = a 13 = From W. Xie and R. Perucchio, Computational procedures for the mechanical modeling of trabeculated embryonic myocardium, Bioengineering Conference, ASME 2001, BED-Vol. 50, pp Wall muscle constitutive equation Strain energy Passive strain during diastole Active strain during systole Second Piola-Kirchhoff stress Green-Lagrange strain First Piola-Kirchhoff stress

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Pressure distribution at the center of the atrium during a diastole and systole cycle Transmitral velocity during diastole Left atrium with appendage

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Rigid wall

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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, E A Left atrium (comparison with clinical data) 5 Pressure (mm hg) 2Time (s)

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Transmitral velocity during one cardiac cycle (with and without the appendage) Velocity inside the appendage during one cardiac cycle Influence of the appendage

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Transmitral velocity during one cardiac cycle (with and without the appendage) Velocity inside the appendage during one cardiac cycle Influence of the appendage

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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 systole Atrial diastole Atrial systole ECG LA Volume Min LA volume Max LA volume 75% of max LA volume 1 Atrial contraction 2 Isovolumetric contraction 3 Rapid ejection 4 Reduced ejection 5 Isovolumetric relaxation 6 Rapid ventricular filling 7 Reduced ventricular filling atrial volume

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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 RBC FEM RBC model From Dennis Kunkel at 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 -3 dyn/cm Bending modulus: 1.8* dyn/cm. Property of inner cytoplasm Incompressible Newtonian fluid empirical function

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The shear rate dependence of normal human blood viscoelasticity at 2 Hz and 22 °C (reproduced from Bulk aggregatesDiscrete cells Cell layers Red blood cells and blood

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

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log (m) biomaterial How to link all these together? platelet protein red blood cell vessel heart Shear induced

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Micro-air vehicles 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 Gross Weight (Lbs)

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MAVs Features: improved efficiency, more lift, high maneuverability, reduced noise. 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

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Bio-inspired flapping wings muscle contraction

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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?

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IFEM: Governing Equations Navier-Stokes equation for incompressible fluid Governing equation of structure Force distribution Velocity interpolation

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IFEM: Fluid solving algorithm Petrov-Galerkin Weak Form and discretization 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. With τ m and τ c as stabilization parameters depending on the grid size

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IFEM: Solid Force Calculation 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 Internal Forces: hyperelastic material description (Mooney-Rivlin material). S – 2 nd Piola Kirchhoff stress tensor ε - Green Lagrangian strain tensor Total Lagrangian Formulation

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Solve for velocity using the Navier-Stokes equation Eq. (III) The interaction force f FSI,s is distributed to the fluid domain via RKPM delta function. The fluid velocity is interpolated onto the solid domain via RKPM delta function in The interaction force is calculated with Eq. (I) I. IV. III. II. P 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

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Structure Analysis- hyperelastic material Mooney-Rivlin material Elastic energy potential: 2 nd Piola Kirchhoff stress S: Green-Lagrangian strain : Cauchy stress : Internal force f k : Deformation gradient, F: Cauchy deformation tensor, C:

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Red blood cell model RBC From Dennis Kunkel at Shear rate dependence of normal human blood viscoelasticity at 2 Hz and 22 °C (reproduced from Bulk aggregatesDiscrete cells Cell layers

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Fluid: H = 1.0 cm L = 4.0 cm U=1cm/s Density= 1.0 g/cm 3 Viscosity = 10.0 dynes/s.cm 2 Re= Nodes and 2626 Elements Leaflet (linear elastic): = 0.8 cm t = cm Density= 6.0 g/cm 3 Young modulus: E = 10 7 dynes/cm 2 Poisson ratio: Nodes and 575 Elements Rigid leaflet driven by a uniform fluid flow

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Fluid flow around a rigid leaflet Re = 10 INSERT MOVIE1.AVI

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Pressure field around a rigid leaflet Re = 10

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Leaflet (linear elastic):Fluid: = 0.8 cm H = 1.0 cm t = cmL = 4.0 cm Density= 6.0 g/cm 3 Density= 1.0 g/cm 3 Young modulus: E = 10 7 dynes/cm 2 Viscosity = 1.0 dynes/s.cm 2 Poisson ratio: 0.5Fluid initially at rest 456 Nodes and 575 Elements2500 Nodes and 2626 Elements Leaflet driven by a sinusoidal fluid flow

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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: Projection operator for the scale a a: dilation parameter Wavelet function: Complementary projection operator: low scale + high scale

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