MATHEMATICS AND CARDIOLOGY: PARTNERS FOR THE FUTURE

Suncica CanicDepartment of Mathematics University of Houston

LVAD

ComplianceChamber 1

ComplianceChamber 2

RESERVOIR

LVAD DRIVING CONSOLE

INLET VALVE

OUTLET VALVE

CATHETER

PRESSURETRANSDUCERS

LVAD

ComplianceChamber 1

ComplianceChamber 2

RESERVOIR

LVAD DRIVING CONSOLE

INLET VALVE

OUTLET VALVE

CATHETER

PRESSURETRANSDUCERS

LVAD

ComplianceChamber 1

ComplianceChamber 2

RESERVOIR

LVAD DRIVING CONSOLE

INLET VALVE

OUTLET VALVE

CATHETER

PRESSURETRANSDUCERS

33

• 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:

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

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

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

• 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).

33

• Application to AAA repair and coronary angioplasty with stenting.

• Experimental validation.TEXAS MEDICAL CENTERHOUSTON

ANALYSIS

COMPUTATION

VALIDATION AND TREATMENT

• Fluid-cell-structure interaction algorithm

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

*

**

*

*

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

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

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

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

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

AAA Walstent (compliant)

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

INFLUENCING STENT-GRAFT INDUSTRY

AneuRx Stent-Graft NEW Endologix Stent-Graft (2007)

Our results showthis geometry will have lower limbthrombosis rates.

MATHEMATICAL MODELING AND COMPUTATION

DETECT DEVICE’S STRUCTURAL DEFICIENCIES

SUGGEST IMPROVED DEVICE DESIGN

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

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

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.)

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

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

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)

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:

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

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

MATHEMATICAL FLUID-STRUCTURE

INTERACTION IN BLOOD FLOW

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

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)

• 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

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)

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

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

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

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)

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

LVAD

ComplianceChamber 1

ComplianceChamber 2

RESERVOIR

LVAD DRIVING CONSOLE

INLET VALVE

OUTLET VALVE

CATHETER

PRESSURETRANSDUCERS

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

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:

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

““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