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PPT – Mathematical Modeling of the CARDIOVASCULAR SYSTEM PowerPoint presentation | free to download - id: 25e45b-Zjg0N

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Alfio Quarteroni EPFL - Lausanne,

Switzerland and MOX - Politecnico di Milano, Italy

Mathematical Modeling of the CARDIOVASCULAR

SYSTEM

Patients real data

- uncertainty
- sensitivity

Mathematical Model

Experimental Model

PROBLEM Analysis of the cardiovascular system

Geometry

In Vivo

PDES and analysis

In Vitro

Numerical methods

Feedback

Literature benchmark

Computer simulation

Post-processing

Comparison with experiments, validation

3D visualization of results

GEOMETRIC PRE-PROCESSING

Geometric Pre-Processing

Extraction of 3D geometric model from medical

images (anatomy)

Statistical analysis and classification

(according to clinical protocols)

Generation of boundary and initial conditions

(physiology)

Generation of computational mesh for surfaces and

volumes (2D and 3D)

Extracting geometry from medical images

GEOMETRIC PRE-PROCESSING

MR (Magnetic Resonance)

Contour extraction by segmentation (using

B-Splines)

Stack of images from MRI (1mm)

Sample points on extracted geometry

Building the surface S from sample points

GEOMETRIC PRE-PROCESSING

Implicit definition

Radial basis expansion

or

Two possible choices

Extracting information from the surface

Normalized Hessian

Allows computation of curvature

Generating a computational mesh

GEOMETRIC PRE-PROCESSING

Constrained optimization procedures are needed to

maximize a suitable measure of the grid quality

(to avoid triangle distorsion) while keeping the

desired accuracy of surface representation

Optimized grid (J. Peiro et al, 2006)

Splines on sections

Original grid (marching cube algorithm,

J.Bloomenthal,1994)

Volume-grid generation (carotid artery)

GEOMETRIC PRE-PROCESSING

A good surface mesh is a key factor for the

generation of a 3D volume grid for the numerical

simulation of blood flow

(A.Veneziani)

MATHEMATICAL MODEL

- Local flow analysis
- 2. Fluid-structure interaction
- 3. Geometric multiscale model
- 4. A global scenario

MATHEMATICAL MODEL

Local Flow Analysis

MATHEMATICAL MODEL

From global to local

Blood is a suspension of red cells, leukocytes

and platelets on a liquid called plasma

Representation Framework Neither Lagrangian

MATHEMATICAL MODEL

nor Eulerian

MATHEMATICAL MODEL

ALE!

MATHEMATICAL MODEL

MATHEMATICAL MODEL

ALE framework an abstract setting

The moving control domain

Fluid equations

MATHEMATICAL MODEL

G(t)

- Assumptions on the fluid (in large arteries)
- Homogeneous
- Newtonian

Gf,N

Gf,D

Of(t)

Cauchy stress tensor

Strain rate tensor

Incompressible Navier-Stokes equations in ALE

conservation form

Velocity profiles in the carotid bifurcation

(rigid boundaries, Newtonian)

MATHEMATICAL MODEL

(M.Prosi)

WSS (Wall Shear Stress) - an indicator of

atherosclerosis

MATHEMATICAL MODEL

- velocity field

normal and tangential unit vectors to the

vessel wall

WSS on coronaries (M.Prosi, K.Perktold TU-Graz)

WSS pulmonary artery (congenital heart disease)

Shape Optimization New optimized shapes and

new materials for less invasive surgery

Local Boundary Variations

Original configuration

Vorticity Reduction 15

Vorticity Reduction 25

Vorticity Reduction 45

Vorticity Reduction 35

Taylor Patch given by Optimal Control on

Vorticity.

Feedback using higher complex models and indexes.

Viscosity depends on shear rate and vessel

radius

Blood is a suspension of red cells, leukocytes

and platelets on the plasma

Rouleax aggregation

Fahraeus-Lindquist effect

In small vessels (below 1mm radii) red blood

cells move toward the central part of the vessel,

whence blood viscosity shifts toward plasma

viscosity (much lower)

Red blood cells aggregate as in stack of coins

Non-Newtonian Models

MATHEMATICAL MODEL

Cauchy stress tensor

Generalized Newtonian model

( Rate of deformation, or shear rate)

POWER LAW model

Shear thinning if nlt1, is a decreasing

function of

Some Generalized Non-Newtonians Models

MATHEMATICAL MODEL

(Y.I.Cho and K.R.Kensey, Biorheology, 1991)

MATHEMATICAL MODEL

Fluid-Structure Coupling

Model of the arterial vessel

MATHEMATICAL MODEL

Mechanical interaction(Fluid-wall coupling)

Biochemical interactions(Mass-transfer

processesmacromolecules, drug delivery,

Oxygen,)

Mechanical interaction equations for the solid

wall

MATHEMATICAL MODEL

The momentum conservation (elastodynamic)

equation (Lagrangian approach)

density in reference configuration

deformation gradient

Jacobian

second Piola-Kirchoff tensor

is a density of elastic energy

(hyper-elastic material)

Green-Lagrange strain tensor

The coupled fluid-structure problem

MATHEMATICAL MODEL

Equations for the geometry

Equations for the fluid

Equations for the structure

Some references

MATHEMATICAL MODEL

(Existence of strong or weak solutions, control,

stability of time-discretizations in

time-dependent domains) Le Tallec and Mouro

(95) Beirao da Veiga (04), Desjardin and Esteban

(99), Osses and Puel (99), Grandmont and Maday

(00-02), J.L.Lions and Zuazua (95), Zhang and

Zuazua (04-06), Murea and Vazquez (05), Cheng,

Coutand and Shkoller (06) L.Formaggia and

F.Nobile (99-04) D.Boffi and L.Gastaldi (04)

Some delicate numerical issues

1. For time-discretization, geometric

conservation laws (GCL) can be a concern for

stability. This is a general issue for evolution

equations in changing domains.

2. Density of structure density of fluid makes

implicit FSI schemes ideal as they guarantee

energy conservation (strong coupling matching

conditions satisfied exactly at each

time-step) 3. Numerical instability observed

(and even proven theoretically) for weakly (or

loosely) coupled schemes 4.Possible spurious

reflections caused by outflow conditions

FSI ALGORITHMS

DD Formulation Preconditioned Iterations

1. Compute the residual stress from a given

displacement

2. Apply the inverse of the preconditioner to the

stress

recover displacement

3. Update displacement

(S. Deparis, M. Discacciati, G.Fourestey and A.Q.

2004)

FSI for carotid bifurcation wall deformation

MATHEMATICAL MODEL

(G.Fourestey)

Flowfield

MATHEMATICAL MODEL

(G.Fourestey)

Spurious reflections with free-stress outflow

conditions

OUTFLOW CONDITIONS

MATHEMATICAL MODEL

Geometric Multiscale Models

Representative fluid dynamics values

MATHEMATICAL MODEL

- Geometrical and mechanical parameters of blood

vessels vary highly from the arterial scale to

the capillary one - Customarily, the flow has a

laminar regime

- Full scale turbulence (high Re) can develop in a

few cases only - High cardiac output (exercise)
- Stenoses
- Low blood density (for example anemia)

A local-to-global approach

MATHEMATICAL MODEL

Local (level1) 3D flow model Global (level 2)

1D network of major arteries and veins Global

(level 3) 0D capillary network

Dimensional reduction by geometric multiscale

MATHEMATICAL MODEL

Geometric multiscale models

MATHEMATICAL MODEL

3D Navier-Stokes (F) 3D ElastoDynamics (V-W)

- Assume that
- uz gtgt ux ,uy
- uz has a prescribed steady profile
- average over axial sections
- static equilibrium for the vessel

Then we obtain a 1D problem.

Geometric multiscale model

MATHEMATICAL MODEL

1D Euler(F) Algebraic pressure law

- Assume to
- linearize 1D equations
- consider average internal variables
- relate interface values to averaged ones

Then we obtain a 0D problem (ODE).

Geometric multiscale model

MATHEMATICAL MODEL

0D Lumped parameters (system of linear ODEs)

- RLC circuits model large arteries
- RC circuits account for capillary bed
- Can describe compartments
- (such as peripheral circulation)

One-dimensional models

MATHEMATICAL MODEL

Junction of three arteries (stented abdominal

aorta)

Stenosed artery

Network of 55 arteries

A lumped 0D model for a single artery

MATHEMATICAL MODEL

By choosing the state variables and the forcing

terms we can select the quantities exchanged at

interfaces

A lumped 0D model for the heart

MATHEMATICAL MODEL

A model for the evolution of the pressure p (for

a given flux) in left ventricle may be obtained

by assuming istantaneous equilibrium, accounting

for the time evolution of properties of the

muscle fibers compliance C(t) , activation

function M(t)

A 0D model of the whole circulation

MATHEMATICAL MODEL

Continuity of fluxes and pressure yields the DAE

system

A full geometric multiscale model 0D-1D-2D (or

3D) coupling

MATHEMATICAL MODEL

Monitoring Station

3D - 1D - 0D

MATHEMATICAL MODEL

MATHEMATICAL MODEL

How do models communicate?

Exchange of boundary conditions at the interface

Warning Reduced model only deals with average

quantities

Mean pressure

Flow rate

Average values not sufficient to make the 3D

problem well posed

MATHEMATICAL MODEL

Mathematical treatment of defective boundary

conditions

- Lagrange multipliers for the flow rate problem
- (L.Formaggia, J.F.Gerbeau, F.Nobile and

A.Q., 2002 and A.Veneziani, C.Vergara, 2005) - Flow rate condition is treated as a

constraint on the solution, introducing a

Lagrange multiplier LL(t) and adding to the

variational formulation

2) Control problem approach ( L.Formaggia,

A.Veneziani, C.Vergara, 2006) Search for

the (a priori unknown) Neumann boundary data k(x)

s.t.

then pretending that either of the following

functionals is minimized

Flow-rate problem

Mean-pressure problem

Coupling 3D and 1D models

MATHEMATICAL MODEL

Continuity of total pressure and flow-rate at

interface (does not yield continuity of

cross-sectional area) A-priori estimates hold,

yielding numerical stability

3D-1D for a cylindrical artery pressure pulse

MATHEMATICAL MODEL

3D model (spurious reflections)

3D-1D coupled model

(A.Moura)

3D-1D for the carotid velocity field

MATHEMATICAL MODEL

(A.Moura)

3D-1D for the carotid pressure pulse

MATHEMATICAL MODEL

(A.Moura)

Some references on the 1D system and geometric

multiscale

MATHEMATICAL MODEL

L.Euler, Principia pro motu sanguinis per

arteria determinando, 1775 Continuous dependence

of 1D L.Formaggia. J.F. Gerbeau, F. Nobile, A.

Q., 2001 Existence of local-in-time regular

solution for in the half-space for 1D S. Canic,

E.H. Kim, 2003, S.Canic and A.Mikelic,

2004 Asymptotic analysis for 1D-0D coupling

M.Fernandez, V.Milisic and A.Q., 2004 Existence

of regular global solution on bounded domains

without source term and special b.c D.Amadori,

S. Ferrari and L.Formaggia, 2006

Treatment of interfaces between models of

different dimension (A.Q. and A.

Veneziani, MMS SIAM, 2004 (3D-0D, Shauder fixed

point) L.Formaggia, J.F.Gerbeau, F.Nobile

and A.Q., 2002 A.Veneziani, C.Vergara,

2006, L.Formaggia, A.Veneziani, C.Vergara, 2006

(by either Lagrange multipliers or optimal

control)

A Global Scenario An Outlook

MATHEMATICAL MODEL

Irrorated compartments

Heart model

Respiratory system

Nervous system

chemoreflex

Circulation model

Systemic Resistance

Flow Rate Arterial Concentration

Venous Concentrations

Pressures

baroreflex

APPLICATIONS

Cerebral Aneurysms The ANEURISK Project Siemens

Italia Niguarda Hospital, Milan Lab of Biological

Structures, Politecnico of Milan

Project description

THE ANEURISK PROJECT

CEREBRAL ANEURYSMS are lesions arising on

cerebral vessels characterized by a bulge of the

vessel wall. Quite often they are subject to

rupture, yielding dangerous cerebral haemorrhage.

It is estimated that 5 of the population has

some type of aneurysm in the brain. The incidence

of ruptured aneurysm is approximately 10 out of

100,000 people per year. About 10 of patients

who have one aneurysm will have at least one

more. National Library of Medicine, NIH US,

http//www.nlm.nih.gov

PROJECT GOAL To highlight the possible

relationships between vascular morphology and

risk of development and rupture of aneurysms

METHODS Integration of extensive data analysis

and numerical simulations

THE ANEURISK PROJECT

Morphological Analysis

1

2

3

5

6

4

- Model
- Centerlines
- Maximal Inscribed Sphere Radius

- Bifurcations Identification
- Centerlines of each branch
- Branch Identifications

THE ANEURISK PROJECT

Morphological Analysis

- Length
- Curvature
- Torsion
- Tortuosity
- Mean Radius
- Min Radius
- Max Radius

Radius

From geometric reconstruction to numerical

simulations

THE ANEURISK PROJECT

Reconstruction of the aneurisms geometry

Pressure field

Velocity streamlines

Particle tracing in an aneurysm during a full

cardiac pulse

THE ANEURISK PROJECT

APPLICATION 2 THE ANEURISK PROJECT

Morphological Data of Internal Carotid Artery

(ICA) for 5 Classes of Patients

VV Patients with End Wall Aneurysm at the

bifurcation

AA Patients with an Aneurysm downstream the ICA

CC Patients with Side Wall in the ICA

NN Patients without Aneurysms

TT Patients with Side Wall in the ICA with a

Branching Vessel

Average

StdDev

1-Classes introduced in Hassan et al., J.

Neurosurgery, Nov. 2005

THE ANEURISK PROJECT

Statistical analysis and CFD on 65 patients

Peak in ICA aneurysms density is slightly

downstream the peak of vessel curvature,

suggesting a correlation with fluid dynamics

Curvature

Curvilinear Abscissa

Occurrence of ICA Aneurysms Histogram of

Aneurysmslocation shows that ICA aneurysms occur

essentially in two sites

Classes introduced in Hassan et al., J.

Neurosurgery, 2005

APPLICATIONS

Drug Eluting Stents Haemodel EU Project, 6th

Framework MIUR, Italian Ministry of Research and

University FNS, Swiss National Funds Fondazione

Cariplo

Stenosys in the carotid bifurcation

DRUG ELUTING STENTS

Angiography after stent placement

DRUG ELUTING STENTS

Stent deployment

DRUG ELUTING STENTS

Four commercial coronary stents

Different stent design may affect the local drug

distribution across the arterial wall

The final configuration reached after the stent

deployment has to be taken into account an

incorrect expansion may cause sites of toxic dose

DRUG ELUTING STENTS

Mathematical Model

Arterial-wall thickness 0.4 1.0 mm

Coating thickness 5 m

Modelled with three phases

- Effective solid phase (drug bound to the polymer)
- Virtual solid phase (polymer swelled free

interface) - Liquid phase (drug dissolved in plasma)

DRUG ELUTING STENTS

A Multi-Domain/Multi-Phase Problem

Macroscale, mm (in the arterial wall)

Macroscale, µm (in the coating matrix)

LIQUID PHASE

VIRTUAL SOLID PHASE (free interface)

EFFECTIVE SOLID PHASE (dynamics of polymer

concentration)

Depend on polymer characteristics (porosity,

tortuosity,...) Determined by stochastic models

DRUG ELUTING STENTS

Numerical strategy

Dont consider the coating as a 3D domain,

rather approximate the transient flux at the

interface to the arterial wall

Grid around the stent

in stent coating in

the wall Elements 965.081

1.018.475 (many more for realistic

geometries)

DRUG ELUTING STENTS

Heparin release from stent coating

Concentration around a simplified geometry

Effective time 1 day (uniform coating)

Blood plasma pressure distribution for a

realistic model (deployed stent)

Simulation of stent expansion and drug release

(M.Prosi)

Uniform vs multilayered coating release dynamics

DRUG ELUTING STENTS

2 days

3 days

1 day

Uniform

Multi-layered

(M.Prosi)

Model-oriented design from global (coarse) to

local level

P2DE Sens.Analysis

1-Outer level (coarse) Reduced-Basis Method on

qualitative geometrical configurations

(multi-parameters).

H

q

Gout

Gw

H

?

t

D

Gw

L

S

Simple 2-D Qualitative-Configuration

2-Medium level (fine) Flow Control and Local

Shape Optimization (on the upper incoming

branch) lower fidelity method

1) Local (Nodal) Boundary Variation (LBV)

2) Linearized Design by small Perturbations (SP)

3-Inner level Feedback using a higher fidelity

model on previously optimized configurations.

Optimal Shape Design on fine Geometrical

configuration.

Shape Design using Small Perturbation

- Mathematical Modelling and
- Numerical simulation (CFD) of physiological flows

allow better understanding of phenomena - involved in bypass implanting
- procedures
- (Oscillatory shear index, recirculation zones,

disturbed flows, abnormal flow pattern,

restenosis). - Optimal Shape Design may
- improve arterial bypass graft,
- provide new design indications and
- reduce post-surgical failure.

Starting Configuration

f0(x)

steady

Optimized Configuration

Velocity 10(-2) m/s)

f(x)fo(x)? f1(x)

unsteady version

Reduced Basis Techniques for Bypass Configuration

Computational method for accurate and efficient

evaluation of input-output (geometry/design

quantities) relationship governed by

parameterized PDEs in the context of Design and

Optimization.

Input

Real time indications on different

configurations for surgery by outputs indexes.

Output

An Optimal Control approach by Small Perturbations

From a shape design problem in real domain to an

optimal control problem on coefficients

(containing shape functions).

- Mapping to reference domain.
- Coordinates Transformation.
- f(x,e) models the upper shape,
- ( f0(x) is the unperturbed shape).
- HypSolution u and p infinitely differentiable

w.r. to e. - Using SP theory we can derive equations for uk e

pk.

Equation for uk and pk

Generalized Stokes Problem

CONCLUSIONS/OUTCOME

Better understanding of physiological processes

(basic research)

Assessment of risk indicators for pathological

uprises (clinical diagnosys)

Tool for therapeutic/surgical planning

(optimization)

NEW MATHEMATICAL DEVELOPMENTS

ACKNOWLEDGMENTS

L. Formaggia, A.Moura, F.Nobile, C.Passerini, M.

Prosi, P.Secchi, S.Vantini, A. Veneziani, P.

Zunino, G.Aloe, L.Lo Curto, L.Paglieri

C.DAngelo, G. Fourestey, C.Vergara

CHUV University Hospital (Lausanne), Great

Hormond Street Hospital (London), Niguarda

Hospital (Milan), Haemodel EU Project, Siemens

(Milan), Laboratory of Biological Structures

(Politecnico of Milan)

External Collaborations

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