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Title: Mathematical Modeling of the CARDIOVASCULAR SYSTEM


1
Alfio Quarteroni EPFL - Lausanne,
Switzerland and MOX - Politecnico di Milano, Italy
Mathematical Modeling of the CARDIOVASCULAR
SYSTEM
2
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
3
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)
4
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
5
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
6
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)
7
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)
8
MATHEMATICAL MODEL
  • Local flow analysis
  • 2. Fluid-structure interaction
  • 3. Geometric multiscale model
  • 4. A global scenario

9
MATHEMATICAL MODEL
Local Flow Analysis
10
MATHEMATICAL MODEL
From global to local
Blood is a suspension of red cells, leukocytes
and platelets on a liquid called plasma
11
Representation Framework Neither Lagrangian
MATHEMATICAL MODEL
12
nor Eulerian
MATHEMATICAL MODEL
13
ALE!
MATHEMATICAL MODEL
14
MATHEMATICAL MODEL
ALE framework an abstract setting
The moving control domain
15
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
16
Velocity profiles in the carotid bifurcation
(rigid boundaries, Newtonian)
MATHEMATICAL MODEL
(M.Prosi)
17
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)
18
Shape Optimization New optimized shapes and
new materials for less invasive surgery
19
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.
20
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
21
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
22
Some Generalized Non-Newtonians Models
MATHEMATICAL MODEL
(Y.I.Cho and K.R.Kensey, Biorheology, 1991)
23
MATHEMATICAL MODEL
Fluid-Structure Coupling
24
Model of the arterial vessel
MATHEMATICAL MODEL
Mechanical interaction(Fluid-wall coupling)
Biochemical interactions(Mass-transfer
processesmacromolecules, drug delivery,
Oxygen,)
25
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
26
The coupled fluid-structure problem
MATHEMATICAL MODEL
Equations for the geometry
Equations for the fluid
Equations for the structure
27
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)
28
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
29
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)
30
FSI for carotid bifurcation wall deformation
MATHEMATICAL MODEL
(G.Fourestey)
31
Flowfield
MATHEMATICAL MODEL
(G.Fourestey)
32
Spurious reflections with free-stress outflow
conditions
OUTFLOW CONDITIONS
33
MATHEMATICAL MODEL
Geometric Multiscale Models
34
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)

35
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
36
Dimensional reduction by geometric multiscale
MATHEMATICAL MODEL

37
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.
38
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).
39
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)

40
One-dimensional models
MATHEMATICAL MODEL
Junction of three arteries (stented abdominal
aorta)
Stenosed artery
Network of 55 arteries
41
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
42
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)
43
A 0D model of the whole circulation
MATHEMATICAL MODEL
Continuity of fluxes and pressure yields the DAE
system
44
A full geometric multiscale model 0D-1D-2D (or
3D) coupling
MATHEMATICAL MODEL
Monitoring Station
45
3D - 1D - 0D
MATHEMATICAL MODEL
46
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
47
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
48
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
49
3D-1D for a cylindrical artery pressure pulse
MATHEMATICAL MODEL
3D model (spurious reflections)
3D-1D coupled model
(A.Moura)
50
3D-1D for the carotid velocity field
MATHEMATICAL MODEL
(A.Moura)
51
3D-1D for the carotid pressure pulse
MATHEMATICAL MODEL
(A.Moura)
52
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)
53
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
54
APPLICATIONS
Cerebral Aneurysms The ANEURISK Project Siemens
Italia Niguarda Hospital, Milan Lab of Biological
Structures, Politecnico of Milan
55
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
56
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

57
THE ANEURISK PROJECT
Morphological Analysis
  • Length
  • Curvature
  • Torsion
  • Tortuosity
  • Mean Radius
  • Min Radius
  • Max Radius

Radius
58
From geometric reconstruction to numerical
simulations
THE ANEURISK PROJECT
Reconstruction of the aneurisms geometry
Pressure field
Velocity streamlines
59
Particle tracing in an aneurysm during a full
cardiac pulse
THE ANEURISK PROJECT
60
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
61
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
62
APPLICATIONS
Drug Eluting Stents Haemodel EU Project, 6th
Framework MIUR, Italian Ministry of Research and
University FNS, Swiss National Funds Fondazione
Cariplo
63
Stenosys in the carotid bifurcation
DRUG ELUTING STENTS
Angiography after stent placement
64
DRUG ELUTING STENTS
Stent deployment
65
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

66
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)

67
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
68
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)
69
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)
70
Uniform vs multilayered coating release dynamics
DRUG ELUTING STENTS
2 days
3 days
1 day
Uniform
Multi-layered
(M.Prosi)
71
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.
72
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
73
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
74
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.

75
Equation for uk and pk
Generalized Stokes Problem
76
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
77
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
78
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