MATHEMATICS AND CARDIOLOGY: PARTNERS FOR THE FUTURE
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MATHEMATICS AND CARDIOLOGY: PARTNERS FOR THE FUTURE
Suncica CanicDepartment of Mathematics University of Houston
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LVAD
ComplianceChamber 1
ComplianceChamber 2
RESERVOIR
LVAD DRIVING CONSOLE
INLET VALVE
OUTLET VALVE
CATHETER
PRESSURETRANSDUCERS
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LVAD
ComplianceChamber 1
ComplianceChamber 2
RESERVOIR
LVAD DRIVING CONSOLE
INLET VALVE
OUTLET VALVE
CATHETER
PRESSURETRANSDUCERS
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LVAD
ComplianceChamber 1
ComplianceChamber 2
RESERVOIR
LVAD DRIVING CONSOLE
INLET VALVE
OUTLET VALVE
CATHETER
PRESSURETRANSDUCERS
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• aortic abdominal aneurysm (AAA) repair
• coronary artery disease (CAD) repair.
PROBLEM
FLUID-STRUCTURE INTERACTION BETWEEN BLOOD FLOW AND ARTERIAL WALLS IN HEALTHY AND DISEASED STATES
1. Help predict initiation of disease
2. Help improve treatment of disease
Prostheses design for non-surgical treatment of AAA and CAD
ANALYSIS OF FLUID-STRUCTURE INTERACTION CAN:
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DIFFICULT PROBLEM TO STUDY: MULTI-PHYSICS AND MULTI-SCALE NATURE
• 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
Plasma
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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
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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)
TRADITIONAL SOFTWARE FOR BLOOD FLOW SIMULATION• ASSUMES FIXED VESSEL WALLS
ACTIVE AREA OF RESEARCH in the years to come
CHALLENGES: - 3D simulations of larger sections of cardiovasc. sys. - complicated geometries - complicated tissue models
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• Design of a numerical algorithm (“kinematically coupled”) with a novel operator splitting approach (hyperbolic/parabolic) with improved stability properties.
• Fundamental properties of the interaction and of the solution.
• Derivation of new closed, effective models.
COMPREHENSIVE STUDY OF FLUID-STRUCTURE INTERACTION IN BLOOD FLOW
(medium-to-large arteries: laminar flow and Re away from the turbulent regime)
• Models allowing two different structures (stent modeling).
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• Application to AAA repair and coronary angioplasty with stenting.
• Experimental validation.TEXAS MEDICAL CENTERHOUSTON
ANALYSIS
COMPUTATION
VALIDATION AND TREATMENT
• Fluid-cell-structure interaction algorithm
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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
*
**
*
*
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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.
THE TEXAS HEART INSTITUTE Dr. Denton Cooley: Founder of THI Pioneer of Heart Transplants
Dr. DeBakey Dr. Cooley
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
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PROJECTSAortic 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
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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
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METHODS
• EXPERIMENTAL MEASUREMENTS OF PROSTHESES MECHANICAL PROPERTIES (Ravi-Chandar, UT Austin)
• MATHEMATICAL MODELING OF PROSTHESES MECHANICS AND
DYNAMICS
• COMPUTER SIMULATIONS
• EXPERIMENTAL VALIDATION
STUDY OPTIMAL PROSTHESIS DESIGN FOR AAA REPAIR
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RESULTS LEAD TO NEW STENT-GRAFT DESIGN
•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 COMPARISON MOVIE
NON-UNIFORM STIFNESS MINIMIZES STRESS AT ANCHORING
[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 II Journal of Appl. Mechanics, (2004.)[4] SIAM News, Vol. 37 No. 4 (2004) Dana McKennzie
[1] Canic, Krajcer, Lapin, Endovascular Today (2006)
MODELING AND COMPUTATION PRODUCED:
Next slide
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AAA Walstent (compliant)
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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 limbs movie
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
[1] Canic,Krajcer,Lapin, Endovascular Today (Cover Story) May 2006. Show paper
• RESULTS FOR BIFURCATED STENT-GRAFT DESIGN
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INFLUENCING STENT-GRAFT INDUSTRY
AneuRx Stent-Graft NEW Endologix Stent-Graft (2007)
Our results showthis geometry will have lower limbthrombosis rates.
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MATHEMATICAL MODELING AND COMPUTATION
DETECT DEVICE’S STRUCTURAL DEFICIENCIES
SUGGEST IMPROVED DEVICE DESIGN
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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
MEDICAL PROBLEM
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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
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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.)
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200x100x
Day 2
400x
• Cardiovascular Surgery Research Lab– Texas Heart Institute • Marie Ng• Boniface Magesa• Doreen Rosenstrauch• Arash Tadbiri
Day 3
100x 200x 400x
STENT COATING
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optimize initial seeding for fast complete coverage of stent
study initial cell loss under flow conditions (pre-conditioning) (cell rolling and detachment)
RESULTS:
Show results
USE MATHEMATICS AND COMPUTATION
TO OPTIMIZE THE PRODUCTION OF CELL-COATED STENTS
TO REDUCE THE EXTENT OF EXPERIMENTAL INVESTIGATION
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PRE-CONDITIONING PHASE: CELL ROLLING AND DETACHMENT
Fluid velocity=const.
Fluid velocity=0
Period boundary conditions
No-slip boundary condition
t = 0
Fluid velocity=const.
Fluid velocity=0
t > 0
MATHEMATICAL AND COMPUTATIONAL ALGORITHM
DYNAMIC ADHESION ALGORITHM Hammer and Apte, Biophys.J. (1992)
FLUID-PARTICLE INTERACTION ALGORITHMGlowinski,Pan et al., J. Comp. Phys. (2001)
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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 (/s) Detachment %
0.01 100 0
0.01 200 25
0.05 5 00.05 8 100.05 9 30
(blood:0.03 ; 100 in dog’s coronaries)
Adhesion Algorithm coupled with Fluid-Particle Interaction Algorithm
Number of cells = 80 Mesh size h for the velocity=0.1 m (using P1 element)Cell size (ellipsas)= 2 x 1.6 m Mesh size h for the pressure=0.2 m (using P1 elements)Channel length=400 m Each cell occupies 20x16 mesh blocks.
Dual core AMD Opteron 275 @ 2.2 GHz : 11h 30min 4 sec (not parallelized)
NEXT Optimize pre-conditioning by varying shear rate and fluid viscosity
Click inside the picture to run the movie:
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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
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MATHEMATICAL PROBLEM
FLUID (BLOOD)
Newtonian, incompressible fluid
Unsteady
Incompressible Navier-Stokes
COMPLIANT WALLS [SIAP ‘06, SIAMMS ’05, Annals of Bimed Eng ’05,CRAS ’04, SIADS ’03, CRAS
‘02]
Linearly ELASTIC and linearly VISCOELASTIC
Koiter SHELL model (Koiter, Ciarlet et al.)
Linearly ELASTIC and linearly VISCOELASTIC MEMBRANE model
NONLINEARLY ELASTIC MEMBRANE
FLUID-CELL-STRUCTURE INTERACTION
CELLS
Auricular chondrocytes
Cell adhesion and detachment
Hammer’s adhesion dynamics algorithm
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MATHEMATICAL FLUID-STRUCTURE
INTERACTION IN BLOOD FLOW
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MODELS
FLUID (BLOOD)Newtonian, incompressible fluid
Unsteady
COMPLIANT WALLS
Medium arteries Large arteries
Incompressible Stokes eqns.
Incompressible Navier-Stokes
FLUID-STRUCTURE INTERACTION BETWEEN
Linearly elastic membrane
Linearly elastic Koiter shell
Linearly viscoelastic Koiter shell
Linearly viscoelastic membrane
Nonlinearly elastic membrane
3D Linearly Elastic Pre-Stressed aaaaaaTHICK-WALL Tube
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LARGE ARTERIES & MODERATE Re
0
)(
u
upuut
u Fluid:
Structure: (Long. displ. neglig) z
R0
t
Through the kinematic and dynamic lateral boundary conditions :
(1) Continuity of the velocity
(2) Balance of contact forces: Fstructure = -Ffuid
Coupling:
),0(),(t
uu rz
t
)1( 222
2
Rp
R
Eh
thF refwr
(membrane)tR
hC
Rp
R
Eh
thF refwr
)1( 2222
2(viscoelastic membrane)
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• 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:
BENCHMARK PROBLEM IN BLOOD FLOW
)(
conditionDirichlet :
)(
1
ref1
t
u
ptPp
0 v,0
t
LztzRrzrrt 0),,(,sin ,cos
• The initial conditions:
)(
conditionDirichlet :
)(
2
ref2
t
u
ptPp
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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)
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A PRIORI SOLUTION ESTIMATES
DERIVATION OF A CLOSED, REDUCED, EFFECTIVE MODEL WHEN =R/L << 1
RESULT: small (coronary) arteries (Stokes equations, linear coupling) SIAM Appl. Dyn Sys. 2003.
|| SOLUTION – solution
ENERGY ESTIMATE
ASYMPTOTIC EXPANSIONS
REDUCED (EFFECTIVE) EQUATIONS
CONVERGENCE
ERROR ESTIMATES
WEAK FORMULATION
HOMOGENIZATION
EXISTENCENonlinear Moving boundary
New information
ANALYSIS NUMERICAL SIMULATION
EXPERIMENTAL VALIDATION
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ENERGY EQUALITY
(when inertial forces dominate viscous forces)
22
2
2
2222
22222
1)1(4)(
1
1)1(4)(
1
2
2
PBhE
Rtu
LR
PBEh
RRt
L
L
L
0012.0)(1
),0(2 LL
tL
10% of R
A PRIORI ESTIMATES
).()()()()(22 0
22
)(
tWtWtVtVtEdt
ddz
tdt
dh
Rdxv
dt
doutinfs
L
s
t
R
Lzzout
R
zzin
tLf
L
s
L
rdrvtPtWrdrvtPtW
vDtVdzzt
Dzt
Dt
DRtVdzz
Cz
CCR
tE
0
2
001
2
))((0
2
2
3
2
22
1
2
0
0
2
2
2
2
2
10
.)()( ,)()(
,)(2)( ,)( ,2
)( 2
where
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0th-order approximation:
membrane) (elastic ~
~1
02 0
Cp
z
p
r
vr
rrt
v
drrvzt
zz
R
z
0)0( ,)(
)( ,)(
)0(
0)0( ,0)( ,)0(
0000
0000
tC
tPLz
C
tPz
tvRrvboundedrv
L
zzz
)1(
22
R
hECwhere
INIT
IAL
and
BO
UN
DA
RY
DA
TA
satisfy ),( and 0 zr vv
vv
NTDISPLACEME
memb.)tic (viscoelas t
D
THE REDUCED EQUATIONS 1L
R
NOTE: nonlinearity dueto the fluid-structure couplingdominates the nonlinearity of fluid advection.
Transport of R+ with average fluid velocity
Show movie
Fluid diffusion is dominant in the r-direction
Dominanat smoothing
well-posedness
novel numerical algorithm for benchmark problem
COMING SOON:Kinematically coupled scheme
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linear elasticity
viscoelasticity
Measured pressure-diameter response(Armetano et al.*):
*[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.
nonlinear elasticity
LINEARLY VISCOELASTIC CYLINDRICAL MEMBRANE
Numerical simulation using the reduced (Biot) model
in-vitro measurement
(human femoral artery)
COMPARISON WITH EXPERIMENTS
[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 on www.math.uh.edu/~canic (Tambaca&Kosor)
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VELOCITY MEASUREMENTS AND COMPARISON 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
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LVAD
ComplianceChamber 1
ComplianceChamber 2
RESERVOIR
LVAD DRIVING CONSOLE
INLET VALVE
OUTLET VALVE
CATHETER
PRESSURETRANSDUCERS
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CONCLUSIONS
• sophisticated mathematics can help improvedesign of vascular devices, and give an insightinto the hemodynamics of cardiovascular interventions
• problems arising in cardiovascular interventionscan drive the development of sophisticated mathematics
• 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
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COLLABORATORS
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)
Dr. Z. Krajcer, M.D., THI
Dr. D. Rosenstrauch, M.D. THI
Mathematicians:
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 Dr. R. Glowinski, U of Houston
Dr. T.-W. Pan , U of Houston
Dr. D. Mirkovic, MD Anderson Cancer Center
Engineering/Measurements
Dr. K. Ravi-Chandar, UT Austin
Dr. C. Hartley, Baylor College of Medicine
Cardiologists:
Math/Sci. Computing:
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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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““Peripheral vascular stents: Peripheral vascular stents:
–Computer models of human Computer models of human physiology are necessary to physiology are necessary to test and predict failure test and predict failure (before animal and human (before animal and human studies)”,studies)”, May 2005.May 2005.
Dan G. Schultz, MD
Director, CDRH
AdvaMed Submissions WorkshopAdvaMed Submissions Workshop