MATHEMATICAL METHODS FOR CARDIOVASCULAR STENTING

Suncica Canic Department of Mathematics University of Houston

COLLABORATORS Prof. Josip Tambaca (U of Zagreb, CRO) Dr. Craig Hartley (Baylor), Dr. David Paniagua (THI&St.Luke’s) Mate Kosor (grad student UH & U Zagreb)

• Fundamental properties of the interaction and of the solution• Derivation of new closed, effective FSI models (existence,uniqueness&stability)

(thin,thick,elastic,viscoelastic structure)• Effective model for Taylor dispersion in compliant/excited vessels (intravascular drug delivery)• Derivation of reduced, effective models for endovascular stent modeling

• Fluid-cell-structure interaction and adhesion algorithm for cell coating of coronary stents

• FSI: monolithic scheme vs. design of a novel loosely coupled scheme (“kinematically coupled scheme”) with a novel operator splitting approach exhibiting superb stability properties.

COMPREHENSIVE STUDY OF FLUID-STRUCTURE INTERACTION IN BLOOD FLOW

(with O. Boiarkine, R. Glowinski, G. Guidoboni, D. Kuzmin, A. Mikelic, J. Tambaca, A. Quaini)

• Models allowing two different structures

• Coronary angioplasty with stenting• AAA repair• Echocardiographic assessment of mitral valve regurgitation

ANALYSIS

COMPUTATION

EXPERIMENTAL VALIDATION AND TREATMENT (WITH TEXAS MED. CENTER)with Drs. Zoghbi, Little, Hartley, Fish, Paniagua, Rosenenstrauch

numericsflow chamber experiment

STENT

• MESH TUBE THAT IS INSERTED INTO A NATURAL CONDUITOF THE BODY TO PREVENT OR COUNTERACT A DISEASE-INDUCEDLOCALIZED FLOW CONSTRICTION

• USED IN THE CARDIOVASCULAR SYSTEM, TRACHEOBRONCHIAL,BILIARY AND UROGENITAL SYSTEM

• STENTS PLAY A CRUCIAL ROLE IN THE TREATMENT OF CORONARYARTERY DISEASE

• WHICH STENT IS APPROPRIATE FOR A GIVEN LESION?

• WHICH STENTS ARE APPROPRIATE FOR TORTUOUS GEOMETRIES?

• WHAT IS THE OPTIMAL STENT DESIGN FOR THE AORTIC VALVE STENT PLACEMENT?

• WHAT IS THE STENT’S LONGITUDINAL STRAIGHTENING (BENDING RIGIDITY) AND HOW DOES IT DEPEND ON ITS GEOMETRY?

• LONGITUDINAL EXTENSION/SHORTENING DURING PULSATION?

• BIOCOMPATIBILITY and RESTENOSIS

MANY OPEN QUESTIONS

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BIOCOMPATIBILITY and RESTENOSIS

LARGE CARDIOVASCULAR LITERATURE• CASE REPORTS• Zarins, Mehta, Gyongyosi, Rieu, Sainsous,Ormiston, Webster, Dixon, Post, Kuntz, Mirkovitch, Sigwart, Garasic, Edelman, Rogers, Kastrati, Sigwart, Dyet, Watts, Ettles, Schomig, Rogers, Tseng, Edelman, Squire, Gruntzig, Mayler, Hanna,…

MODERATLY LARGE ENGINEERING LITERATURE• SIMULATIONS USE 3D COMMERCIAL SOFTWARE• Moore, Timmins, Berry, Dumoulin, Taylor, Bedoya, Schmidt, Behrens, Cochelin, Holzapfel, Gasser, Stadler, Magliavacca, Petrini, Colombo, Auricchio, Hoang, …

DRAWBACKS:• 3D simulation of each stent strut is computationally very expensive• thin and long structure: need extremely fine mesh to achieve reasonable accuracy• commercial software uses “black box” approach: do not know which models are used• computationally prohibitive to include dynamic 3D stent modeling in a fluid-structure

interaction solver

MECHANICAL PROPERTIES OF STENTS

HELPED UNDERSTAND MANY STENT PERFORMANCE FEATURES!!!

OUR APPROACH: DIMENSION REDUCTION AND MULTI-SCALE MODELING

• STENT STRUTS MATHEMATICAL THEORY OF 1D CURVED RODS*

• STENT 3D MESH OF 1D CURVED RODS SATISFYING CERTAIN GEOMETRIC AND CONTACT CONDITIONS AT VERTICES

• SPEEDS UP CALCULATION BY SEVERAL ORDERS OF MAGNITUDE STENT DESIGN OPTIMIZATION AND COUPLING WITH FLUID

• EFFECTIVE PRESSURE-DISPLACEMENT RELATIONSHIP FROM LEADING-ORDER ENERGY FORMULATION • CAN BE USED IN NUMERICAL FLUID-STRUCTURE INTERACT. STUDIES

COMPARISON WITH 3D MODEL

3D simulation: freeFEM++, P2 elements, computational mesh (h=1/10,…,1/40)

30x displacement magnification

meshdiff

h=1/1019705 nodes

h=1/2024868 nodes

h=1/3051549 nodes

h=1/4064826 nodes

h=1/50123921 nodes

h=1/60211337 nodes

0.07563 0.046564 0.0381521 0.0314075 0.0277053 0.0258485

3D simulations converge with mesh refinement to 1D solution with #3D nodes: 211337 v.s. #1D nodes: 474 for 2.7% diff.

QUANTITATIVE DIFFERENCE BETWEEN 1D and 3D DISPLACEMENT FOR 2 ZIG-ZAGS

1D SIMULATION

3D SIMULATION

• NON-UNIFORM GEOMETRY INFLUENCES STENTS’ MECHANICAL RESPONSE: SMALLER DIAMONDS IMPLY HIGHER STIFFNESS

• MAXIMALLY EXPANDED STENTS ARE STIFFER:

Stent expanded radius R; max displ=15% Stent expanded radius 0.8 R; max displ=23.5%

THIS INDICATES THAT POST-DILATATION PRACTICE IS HIGHLY ADVISABLE

Max dipl=0.8cm Max dipl=0.56cm

USED IN OPTIMAL STENT DESIGNFOR PERCUTANEOUS AORTICVALVE REPLACEMENT (PRODUCEDBY A PRIVATE CONSORTIUM IN HOUSTON with A FACTORY IN NJ)

APPLICATION(with Drs. Paniagua (THI and VA Hospital), Fish (THI & St. Luke’s Hospital))

• CORONARY STENT RESPONSE TO COMPRESSION AND TO BENDING

STENTS CONSIDERED:

Palmaz by Cordis

Express by Boston Sci.

Cypher by Cordis

Xience by Abbott

Express-like stent mesh

Palmaz-like stent mesh

Cypher-like stent mesh

Xience-like stent mesh

Movie Gallery of Coronary Stents Exposed to Compression and Bending

Express-like stent; open cell design (strut thickness 132 m)

Compression

Cypher-like stent (strut thickness 140 m & 140/3 m)

Xience-like stent; open cell design (strut thickness 80 microns; CoCr)

Uniform (Palmaz-like) stent (strut thickness 80 microns)

Bending

*Gyongyosi et al. Longitudinal straightening effect of stents is an additional predictorfor major adverse cardiac events. J American College of Cardiology 35 (2000)

CONCLUSIONS• Palmaz-like stent is by far the hardest stent with respect to bothcompression and bending (should not be clinically applied in tortuous geometries [*])

• open-cell design provides more flexibility to bending(important since longitudinal straightening effect of rigid stent has been clinically associated with increased incidence of major adverse cardiovascular events [*])

• Express-like stent has high flexibility (bending) while keeping high radial strength (radial displacement: 0.24%) (important to avoid buckling of bent stents)

• Xience-like stent has the smallest longitudinal extension under cyclic loading (“in phase” circumferential rings; not “opposing”) (clinically important when landing a stent in an “angle” area formed by a native artery)

• New design: more flexibility than Express with higher radial strength: Cypher-like stent with open-cell design

Xience-like

Express-like

pre post

Cypher-like

Computer-generated Cypher-like stent with open cell design

STENT BIOCOMPATIBILITY

movie

• re-stonosis; development of neo-intimal hyperplasia = scar tissue in response to mechanical intervention with material of poor biocompatibility

SOLUTION METHODS

• endothelial cells (optimal lining but not easily accessible, harvested or isolated)• genetically engineered smooth muscle cells (similar)

• genetically engineered auricular chondrocytes (Dr. Doreen Rosenstrauch THI)

- genetically engineered to produce NO - easily accessible: minimally invasive harvesting - superior adhearance (collagen) - good results with LVADs Scott-Burdent, Rosenstrauch et al.)

Cardiovascular Surgery Research Lab– Texas Heart Institute (Marie Ng, Boniface Magesa, Doreen Rosenstrauch, Arash Tadbiri)

Day 3

100x 200x 400x

STENT COATING

CELL COATING OF ARTIFICIAL SURFACESwith J. Hao, R. Glowinski, T.W. Pan, Drs. D. Rosenstrauch, C. Hartley

• OPTIMIZE INITIAL SEEDING FOR FAST COMPLETE COVERAGE (Canic and Rosenstrauch: Use of auricular chondrocytes in lining of artificial surfaces: A mathematical model. IEEE Transactions of Nanobioscience Vol 7(3) 2008, 240-245.)

• STUDY CELL LOSS, ROLLING AND ADHESION IN PRECONDITIONING (UNDER CONTROLLED FLOW CONDITIONS IN A FLOW LOOP) (J. Hao, T.W. Pan, S. Canic, R. Glowinski, D. Rosenstrauch. A Fluid-Cell Interaction and Adhesion Algorithm for Tissue-Coating of Cardiovascular Implants. SIAM J Multiscale Modeling and Simulation 7(4) 1669-1694 (2009)

USE MATHEMATICS AND COMPUTATION

TO OPTIMIZE CELL COATING OFARTIFICIALCARDIOVASCULAR SURFACES

TO REDUCE THE EXTENT OF EXPERIMENTAL INVESTIGATION

FLUID-PARTICLE INTERACTION AND ADHESION ALGORITHM

Fluid velocity=const.

Fluid velocity=0

Periodic boundary conditions

No-slip boundary condition

t = 0

Fluid velocity=const.

Fluid velocity=0

t > 0

DYNAMIC ADHESION ALGORITHM Hammer and Apte, Biophys.J. (1992)

FLUID-PARTICLE INTERACTION ALGORITHMGlowinski,Pan et al., J. Comp. Phys. (2001)

J. Hao, T.W. Pan, S. Canic, R. Glowinski, D. Rosenstrauch. A Fluid-Cell Interaction and Adhesion Algorithm for Tissue-Coating of Cardiovascular Implants. SIAM J Multiscale Modeling and Simulation 7(4) 1669-1694 (2009)

CELL ADHESION modeled viarandomly distributed adhesionmolecules (Hookean springs)and stochastic bond dynamics.

RESULTS Cell detachment in the pre-conditioning stage (stochastic bond dynamics)

• observed chondrocyte sliding in simulations (experimentally verified!!)• captured cell detachment (initial linear growth experimentally verified)

Viscosity(g/cm s) Shear rate (1/s) Detachment %

0.01 100 00.01 200 250.05 5 00.05 8 100.05 9 30

(blood:0.03 ; 100 in dog’s coronaries)

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)

USE OF OUR COMPUTATIONAL MODELas a start to study cell-detachment, cell adhesion, and formation of stable cartilage by varying shear rate and fluid viscosity for a given cell type

J. Hao, T.W. Pan, S. Canic, R. Glowinski, D. Rosenstrauch. A Fluid-Cell Interaction and Adhesion Algorithm for Tissue-Coating of Cardiovascular Implants. SIAM J Multiscale Modeling and Simulation 7(4) 1669-1694 (2009)

REFERENCES (selected)

[1] J. Tambaca, M. Kosor, S. Canic, and D. Paniagua, Mathematical Modeling of Vascular Stents. SIAM J Applied Mathematics Volume 70 (6) pp. 1922-1952 (2010).

[2] J. Tambaca, S. Canic and D. Paniagua, A Novel Approach to Modeling Coronary Stents using a Slender Curved Rod Model: A Comparison between Fractured Xience-like and Palmaz-like Stents. Applied and Numerical PDEs: Scientific Computing, Simulation, Optimisation and Control (eds Fitzgibbon, Kuznetov, Neittanamitski, Periaux, Pironneau). Springer pp. 41-58 (2010).

[3] J. Tambaca, S. Canic, M. Kosor, D. Paniagua, and D. Fish. Mechanical Properties of Commercially Available Coronary Stents in Their Expanded State. J Am. College of Cardiology (under revision)

[4] J. Hao, T.W. Pan, S. Canic, R. Glowinski, D. Rosenstrauch. A Fluid-Cell Interaction and Adhesion Algorithm for Tissue-Coating of Cardiovascular Implants. SIAM J Multiscale Modeling and Simulation 7(4) 1669-1694 (2009)

[5] G. Guidoboni, R. Glowinski, N. Cavallini, S. Canic. Stable loosely-coupled-type algorithm for fluid-structure interaction in blood flow. Journal of Computational Physics Vol. 228, Issue 18 6916-6937 (2009).

[6] S. Canic and D. Rosenstrauch. Use of auricular chondrocytes in lining of artificial surfaces: A mathematical model. IEEE Transactions of Nanobioscience Vol 7(3) 2008, 240-245.

[7] S. Canic, Z. Krajcer, and S. Lapin. Design of Optimal Prostheses Using Mathematical Modeling. Endovascular Today (Cover Story). May Issue (2006) 48-50.

University of Houston, MD Anderson Library

THANKS: The National Science Foundation

The National Institutes of Health (joint with NSF: NIGMS program)

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.

COMPARISON WITH 3D MODEL

3D simulation: freeFEM++, P2 elements, computational mesh (h=1/10,…,1/40)

1D SIMULATION 3D SIMULATION30x displacement magnification

meshdiff

h=1/10 h=1/20 h=1/30 h=1/40 h=1/50 h=1/60

0.07563 0.046564 0.0381521 0.0314075 0.0277053 0.0258485

3D simulations converge to 1D solution for h=1/10,…,1/60#3D nodes: TODO #1D nodes: 474

QUANTITATIVE DIFFERENCE BETWEEN 1D and 3D DISPLACEMENTOVERALL DISPLACEMENT COMPARSION