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MATHEMATICS AND CARDIOLOGY: PARTNERS FOR THE FUTURE

MATHEMATICS AND CARDIOLOGY: PARTNERS FOR THE FUTURE. Suncica Canic Department of Mathematics University of Houston. RESERVOIR. Compliance Chamber 1. LVAD. Compliance Chamber 2. PRESSURE TRANSDUCERS. LVAD DRIVING CONSOLE. INLET VALVE. CATHETER. OUTLET VALVE.

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MATHEMATICS AND CARDIOLOGY: PARTNERS FOR THE FUTURE

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  1. MATHEMATICS AND CARDIOLOGY: PARTNERS FOR THE FUTURE Suncica Canic Department of Mathematics University of Houston

  2. RESERVOIR Compliance Chamber 1 LVAD Compliance Chamber 2 PRESSURE TRANSDUCERS LVAD DRIVING CONSOLE INLET VALVE CATHETER OUTLET VALVE

  3. RESERVOIR Compliance Chamber 1 LVAD Compliance Chamber 2 PRESSURE TRANSDUCERS LVAD DRIVING CONSOLE INLET VALVE CATHETER OUTLET VALVE

  4. RESERVOIR Compliance Chamber 1 LVAD Compliance Chamber 2 PRESSURE TRANSDUCERS LVAD DRIVING CONSOLE INLET VALVE CATHETER OUTLET VALVE

  5. PROBLEM FLUID-STRUCTURE INTERACTION BETWEEN BLOOD FLOW AND ARTERIAL WALLS IN HEALTHY AND DISEASED STATES ANALYSIS OF FLUID-STRUCTURE INTERACTION CAN: 1.Help predict initiation of disease 2.Helpimprove treatment of disease Prostheses design for non-surgical treatment of AAA and CAD • aortic abdominal aneurysm (AAA) repair • coronary artery disease (CAD) repair. 33

  6. DIFFICULT PROBLEM TO STUDY: MULTI-PHYSICS AND MULTI-SCALE NATURE Plasma • BLOOD has complicated rheology: red blood cells, white blood • cells and platelets in plasma (relevant at small scales) • VESSEL WALLS have complex structure: intima, media, adventitia • (+ smaller scales layers); different mech. char. • Challenging to model. • INTERACTION (COUPLING) exceedingly complicated. Red Blood Cells Platelets White Blood Cells

  7. COUPLING BETWEEN BLOOD FLOW AND VESSEL WALL MOTION • NONLINEAR COUPLING: density of the arterial walls is roughly the same as • density of blood • TWO TIME SCALES: fast traveling waves in arterial walls and • slow bulk blood flow velocity • COMPETITION BETWEEN “HYPERBOLIC” AND “PARABOLIC” EFFECTS • (wave propagation vs. diffusion) • algorithms developed for other applications, e.g. aeroelasticity, UNSTABLE; • novel ideas and algorithms needed • resolving both scales accurately requires sophisticated methods • resolving the two different effects requires different techniques

  8. TRADITIONAL SOFTWARE FOR BLOOD FLOW SIMULATION • ASSUMES FIXED VESSEL WALLS • PAST 10 YEARS: intensified activity in fluid-structure interaction studies due to development of new mathematical tools (beginning with earlier work of Peskin (1989).) • CURRENT METHODS (far from optimal): • computationally expensive (implicit, monolithic schemes, commercial software) • OR • suffer from stability problems (explicit, loosely coupled algorithms) CHALLENGES: -3D simulations of larger sections of cardiovasc. sys. - complicated geometries - complicated tissue models ACTIVE AREA OF RESEARCH in the years to come

  9. COMPREHENSIVE STUDY OF FLUID-STRUCTURE INTERACTION IN BLOOD FLOW(medium-to-large arteries: laminar flow and Re away from the turbulent regime) ANALYSIS • Fundamental properties of the interaction and of the solution. • Derivation of new closed, effective models. COMPUTATION • Design of a numerical algorithm (“kinematically coupled”) with a novel operator splitting approach (hyperbolic/parabolic) with improved stability properties. • Models allowing two different structures (stent modeling). • Fluid-cell-structure interaction algorithm VALIDATION AND TREATMENT TEXAS MEDICAL CENTER HOUSTON • Experimental validation. • Application to AAA repair and coronary angioplasty with stenting. 33

  10. Treats more than 5.5 million patient visits annually; 73,600 employees 37 million sq feet of space 46 institutions (hospitals, educational, service) http://www.texmedctr.tmc.edu/root/en/GetToKnow/FactsandFigures/FactsAndFigures.htm * * * * *

  11. THE TEXAS HEART INSTITUTE Dr. Denton Cooley: Founder of THI Pioneer of Heart Transplants HOUSTON (July 13, 2007)U.S. News & World Report ranked the Texas Heart Institute among the nation's top ten heart centers for the 17th consecutive year. Michael Ellis DeBakey born SEPTEMBER 7, 1908. • pioneer in the field of cardiovascular surgery • pioneer in surgical treatment of AAA • 2006 (age 97): Dr. DeBakey treated for AAA; his procedure • oldest patient ever to undergo this treatment • hospital recovery lasted 8 months Dr. DeBakey Dr. Cooley

  12. PROJECTS • Aortic Abdominal Aneurysm (AAA) • Optimal stent design for non-surgical treatment of AAA • Compliancy • Geometry • Graft Permeability • Experiments • Coronary Artery Disease (CAD) • Tissue engineered stents for coronary angioplasty • Auricular chondrocytes lining of artificial surfaces • Stent Design for CAD and heart valve replacement

  13. What is abdominal aneurysm? Aneurysm: dilatation of an artery Mortality: 90% for out-of-hospital rupture (Experimental) Nonsurgical Procedure: - Developed for high-risk patients - Performed using catheterization Complications: - Stent and stent graft migration (20.2%) - Change in shape (56%) - Formation of new aneurysms near anchoring - Graft limb thrombosis - Permeable grafts-> aneurysm growth

  14. STUDY OPTIMAL PROSTHESIS DESIGN FOR AAA REPAIR • METHODS • EXPERIMENTAL MEASUREMENTS OF PROSTHESES • MECHANICAL PROPERTIES (Ravi-Chandar, UT Austin) • MATHEMATICAL MODELING OF PROSTHESES MECHANICS AND • DYNAMICS • COMPUTER SIMULATIONS • EXPERIMENTAL VALIDATION

  15. RESULTS LEAD TO NEW STENT-GRAFT DESIGN MODELING AND COMPUTATION PRODUCED: • RESULTS FOR FLEXIBLE bare Wallstent. • Wallstent 10 times more elastic than aorta: large radial displacements ANGIO • large stresses and strains near anchoring (possibility of migration)PLAY MOVIE • POOR PERFORMANCE NO LONGER USED • RESULTS FOR FABRIC-COVERED STENT-GRAFTS • graft is stiff; elastic exoskeleton tends to pulsate: possibility for suture breakage • stiff graft: elevated local transmural pressure COMPARISONMOVIE • NON-UNIFORM STIFNESS MINIMIZES STRESS AT ANCHORING Next slide [1] Canic, Krajcer, Lapin, Endovascular Today (2006) [2] Canic, Krajcer, Chandar, Mirkovic, Lapin, Texas Heart Institute Journal (2005) [3] R. Wang and K. Ravi-Chandar, Mechanical response of an aortic stent I and IIJournal of Appl. Mechanics, (2004.) [4] SIAM News, Vol. 37 No. 4 (2004) Dana McKennzie

  16. AAA Walstent (compliant)

  17. Suggested Opt. Design in[1] (NEW) Variable stiffness Limbs diameter around 0.7 of main body diameter Larger main body diameter Longer main body Low shear stress rates in the limbsmovie AneuRx Stent-Graft (OLD) Uniformly stiff Limbs diameter less than 0.5 of main body High shear stress rates in the limbs Small limb diameter implies high SSR refs • RESULTS FOR BIFURCATED STENT-GRAFT DESIGN [1] Canic,Krajcer,Lapin, Endovascular Today (Cover Story) May 2006. Show paper

  18. INFLUENCING STENT-GRAFT INDUSTRY NEW Endologix Stent-Graft (2007) AneuRx Stent-Graft Our results show this geometry will have lower limb thrombosis rates.

  19. MATHEMATICAL MODELING AND COMPUTATION SUGGEST IMPROVED DEVICE DESIGN DETECT DEVICE’S STRUCTURAL DEFICIENCIES

  20. MEDICAL PROBLEM • Coronary stenosis • constriction or narrowing of coronary arteries • coronary arteries supply oxygenated blood to the heart. • 12,600,000 Americans suffer from CAD • 515,000 die from heart attacks • caused by CAD each year (NHLBI) • Treatment: • coronary angioplasty

  21. COMPLICATION: RESTENOSIS • related to the development of neo-intimal hyperplasia • scar tissue in response to mechanical intervention with material of poor biocompatibility • 35% after angioplasty without stent • 19 % with stent (R. Kurnik) movie

  22. Biocompatibility/hemocompatibility • (Dr. Doreen Rosenstrauch) • endothelial cells • optimal lining but not easily accessible, harvested or isolated • genetically engineered smooth muscle cells (similar) • auricular chondrocytes (ear cartilage) (with Dr. Rosenstrauch) • - genetically engineered to produce NO • - easily accessible: minimally invasive harvesting • - superior adhearance (collagen) • - good results with LVADs (Dr. Rosenstrauch, Scott-Burden et al.)

  23. STENT COATING Day 2 100x 200x 400x Day 3 100x 200x 400x • Cardiovascular Surgery Research Lab– Texas Heart Institute • Marie Ng • Boniface Magesa • Doreen Rosenstrauch • Arash Tadbiri

  24. RESULTS: • optimize initial seeding for fast complete coverage of stent • study initial cell loss under flow conditions (pre-conditioning) (cell rolling • and detachment) Show results USE MATHEMATICS AND COMPUTATION TO REDUCE THE EXTENT OF EXPERIMENTAL INVESTIGATION TO OPTIMIZE THE PRODUCTION OF CELL-COATED STENTS

  25. PRE-CONDITIONING PHASE: CELL ROLLING AND DETACHMENT Fluid velocity=const. t = 0 Period boundary conditions Fluid velocity=0 No-slip boundary condition Fluid velocity=const. t > 0 Fluid velocity=0 MATHEMATICAL AND COMPUTATIONAL ALGORITHM FLUID-PARTICLE INTERACTION ALGORITHM Glowinski,Pan et al., J. Comp. Phys. (2001) DYNAMIC ADHESION ALGORITHM Hammer and Apte, Biophys.J. (1992)

  26. Number of cells = 80 Mesh size h for the velocity=0.1 mm (using P1 element) Cell size (ellipsas)= 2 x 1.6 mm Mesh size h for the pressure=0.2 mm (using P1 elements) Channel length=400 mm Each cell occupies 20x16 mesh blocks. Dual core AMD Opteron 275 @ 2.2 GHz : 11h 30min 4 sec (not parallelized) Adhesion Algorithm coupled with Fluid-Particle Interaction Algorithm • RESULTS • Cell detachment in the pre-conditioning stage (stochastic bond dynamics) Shear rate (1/s) Viscosity(g/cm s) Detachment % • observed chondrocyte sliding in simulations • (experimentally verified!!) • captured cell detachment (initial linear growth • experimentally verified) 0.01 100 0 0.01 200 25 0.05 5 0 0.05 8 10 0.05 9 30 Click inside the picture to run the movie: (blood:0.03 ; 100 in dog’s coronaries) • NEXT • Optimize pre-conditioning by varying shear rate and fluid viscosity

  27. NEXT: • study behavior of cell-coated stent inserted in a compliant vessel • (latex tube; in vitro testing) complex hemodynamics conditions: • MODELING: Fluid-Cell-Structure Interaction Algorithm

  28. MATHEMATICAL PROBLEM FLUID-CELL-STRUCTURE INTERACTION CELLS Auricular chondrocytes Cell adhesion and detachment Hammer’s adhesion dynamics algorithm

  29. MATHEMATICAL FLUID-STRUCTURE INTERACTION IN BLOOD FLOW

  30. MODELS FLUID-STRUCTURE INTERACTION BETWEEN COMPLIANT WALLS Linearly elastic membrane Linearly elastic Koiter shell Linearly viscoelastic membrane Linearly viscoelastic Koiter shell Nonlinearly elastic membrane Large arteries Medium arteries 3D Linearly Elastic Pre-Stressed aaaaaaTHICK-WALL Tube Incompressible Stokes eqns. Incompressible Navier-Stokes

  31. (membrane) (viscoelastic membrane) LARGE ARTERIES & MODERATE Re Fluid: S(t) h Structure: (Long. displ. neglig) R0 W(t) z Coupling: Through the kinematic and dynamic lateral boundary conditions : (1) Continuity of the velocity (2) Balance of contact forces: Fstructure = -Ffuid

  32. BENCHMARK PROBLEM IN BLOOD FLOW • The fluid equations (incompressible, viscous, Navier-Stokes) • on the domain with a moving boundary • The structure equations (viscoelastic membrane/shell) • The lateral boundary conditions (coupling) • The inlet and outlet boundary conditions: • The initial conditions:

  33. REVIEW • Bioengineering/math hemodynamics literature (numerical methods): • Groups: EPFL & Milano (Quarteroni et al.), NYU (Peskin and McQueen), Stanford (Taylor&Hughes(UT)), U of Pitt (Robertson), New Zeland (Hunter, Pullan), Eindhoven (de Haar), UC-San Diego (YC Fung), Graz University of Technology (Holzapfel), Cambridge (TJ Pedley), University of Trieste (Pedrizzetti), Technical University Graz (Perktold, Rappitsch), WPI(Tang) • METHODS: Immersed Boundary, ALE, Fictitious Domain, Lattice Boltzman, Coupled Momentum Method, … Commercial Software (ANSYS,ADINA,…) • Many issues remain open • Mathematical fluid-structure interaction (existence/stability proofs): • H.B. daVeiga; Esteban, Chambolle, Desjardins, Grandmont; LeTallec; M. Padula,V. Solonnikov. • Existence for 3D benchmark problem with physiological data remains open • S.Canic, T. Kim, G. Guidoboni: existence for an effective model (2007)

  34. DERIVATION OF A CLOSED, REDUCED, EFFECTIVE MODEL WHEN e=R/L << 1 ANALYSIS NUMERICAL SIMULATION WEAK FORMULATION ENERGY ESTIMATE A PRIORI SOLUTION ESTIMATES EXPERIMENTAL VALIDATION ASYMPTOTIC EXPANSIONS HOMOGENIZATION Nonlinear EXISTENCE REDUCED (EFFECTIVE) EQUATIONS Moving boundary New information CONVERGENCE || SOLUTION – solutione|| ? ERROR ESTIMATES RESULT: small (coronary) arteries (Stokes equations, linear coupling) SIAM Appl. Dyn Sys. 2003.

  35. 10% of R ENERGY EQUALITY where A PRIORI ESTIMATES (when inertial forces dominate viscous forces)

  36. Show movie Transport of R+h with average fluid velocity Fluid diffusion is dominant in the r-direction THE REDUCED EQUATIONS 0th-order approximation: NOTE: nonlinearity due to the fluid-structure coupling dominates the nonlinearity of fluid advection. INITIAL and BOUNDARY DATA Dominanat smoothing well-posedness novel numerical algorithm for benchmark problem COMING SOON: Kinematically coupled scheme

  37. linear elasticity viscoelasticity COMPARISON WITH EXPERIMENTS (human femoral artery) Measured pressure-diameter response (Armetano et al.*): Numerical simulation using the reduced (Biot) model nonlinear elasticity in-vitro measurement LINEARLY VISCOELASTIC CYLINDRICAL MEMBRANE [1] SIAM J Multiscale Modeling and Simulation 3(3) 2005. [2] Annals of Biomedical Engineering Vol. 34, 2006. [3] SIAM J Applied Mathematics 67(1) 2006. coming soon: user-friendly software posted onwww.math.uh.edu/~canic(Tambaca&Kosor) *[1] Armentano R.L., J.G. Barra, J. Levenson, A. Simon, R.H. Pichel. Arterial wall mechanics in conscious dogs: assessment of viscous,inertial,and elastic moduli to characterize aortic wall behavior. Circ. Res. 76: 1995. * [2] Armentano R.L., J.L. Megnien, A. Simon, F. Bellenfant, J.G. Barra, J. Levenson. Effects of hypertension on viscoelasticity of carotid and femoral arteries in humans. Hypertension 26:48--54, 1995.

  38. VELOCITY MEASUREMENTS ANDCOMPARISON WITH NUMERICAL SIMULATIONS (S. Canic, Dr. C. Hartley, Dr. D. Rosenstrauch, J. Chavez, H. Khalil, B. Stanley ) Research Laboratory at THI • mock flow loop with compliant walls and pulsatile flow pump • pulsatile flow pump: HeartMate LVAD • compliant tubbing: custom made latex (Kent Elastomer Inc.) • ultrasonic imaging and Doppler methods: measure velocity & displacement • high frequency (20 MHz) crystal probe was used • non-dairy coffee creamer dispersed in water to enable reflection

  39. RESERVOIR Compliance Chamber 1 LVAD DRIVING CONSOLE LVAD PRESSURE TRANSDUCERS Compliance Chamber 2 INLET VALVE CATHETER OUTLET VALVE

  40. CONCLUSIONS • made progress in understanding fluid-structure • interaction in blood flow; in the design of numerical • methods to capture the interaction, and in the design of • stents and stent-grafts for CAD and AAA treatment • sophisticated mathematics can help improve • design of vascular devices, and give an insight • into the hemodynamics of cardiovascular interventions • problems arising in cardiovascular interventions • can drive the development of sophisticated mathematics

  41. COLLABORATORS Cardiologists: Mathematicians: Dr. Z. Krajcer, M.D., THI Dr. D. Rosenstrauch, M.D. THI Dr. A. Mikelic, U of Lyon 1, FR Dr. J. Tambaca, U of Zagreb, HR Dr. G. Guidoboni, U of H, U of Ferrara, IT Engineering/Measurements Math/Sci. Computing: Dr. K. Ravi-Chandar, UT Austin Dr. R. Glowinski, U of Houston Dr. T.-W. Pan , U of Houston Dr. D. Mirkovic, MD Anderson Cancer Center Dr. C. Hartley, Baylor College of Medicine Students: J. Hao,S. Lapin, T.B. Kim, B. Stanley, M. Kosor, T. Josef (Rice), J. Gill (Rice), K. Mosavardi (UT Health Sci. Houston), J. Chavez, H. Khalil, K. Vo, R. Patel, C. Chmielewski (UH&NCState), H. Melder, A. Young (Penn State), D. Roy,Y. Barlas, K. Buss (UH), E. Delavaud & J. Coulon (U of Lyon1), D. Lamponi (EPFL)

  42. University of Houston, MD Anderson Library • THANKS: • The National Science Foundation • The National Institutes of Health (joint with NSF: NIGMS) • Roderick Duncan MacDonald Research Grant at St. Luke’s Episcopal Hospital, Houston • Texas Higher Education Board (ATP Mathematics) • Kent Elastomer Products Inc. • UH Mathematics Department Summer Research Grant • Medtronic Inc. Final note: For the first, the FDA might require the use of math modeling and numerical simulations for peripheral prostheses design FDA

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