1. Building the surface S from sample points
GEOMETRIC PRE-PROCESSING
Building the surface S from sample points
Implicit definition
Radial basis expansion or
Two possible choices:
Extracting information from the surface :
Normalized Hessian
Allows computation of curvature:

2. Generating a computational mesh
GEOMETRIC PRE-PROCESSING
Generating a computational mesh
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
Splines on sections
Original grid (marching cube algorithm, J.Bloomenthal,1994)
Optimized grid (J. Peiro et al, 2006)

3. Image enhancement
In dealing with the problem of building representations of the geometry of blood vessels, we have assumed that the location of their surface was known.
How to actually derive such information from medical images?
Medical images can be affected by noise and artefacts that may interfere with the segmentation process, or the image data may be provided in a form not suitable for the segmentation method of choice, as in the case of anisotropic resolutions, for which the spacing of the imaging grid along different axes is different.
In order to alleviate these problems, an imaging enhancement step can be performed prior to segmentation. The introduction of such a step can substantially alter the information contained in an image and potentially play a role in the outcome of haemodynamic modelling.

4. MATHEMATICAL MODELLING
Preso da Berlino Completo. Ho quasi uniformato lo stile, anche se in alcuni punti resta un po` “misto”. Sulla versione finale ridotta metto tutto coerente…

5. MODELLING PERSPECTIVE
From global to local
Blood is a suspension of
red cells, leukocytes
and platelets on a liquid
called plasma

6. Viscosity depends on shear rate and vessel radius
MATHEMATICAL MODEL
Viscosity depends on shear rate and vessel radius
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

7. MATHEMATICAL MODEL Non-Newtonian Models Cauchy stress tensor
Generalized Newtonian model:
( Rate of deformation, or shear rate)
POWER LAW model:
Shear thinning if n<1, is a decreasing function of

8. Some Generalized Non-Newtonians Models
MATHEMATICAL MODEL
Some Generalized Non-Newtonians Models
(Y.I.Cho and K.R.Kensey, Biorheology, 1991)

9. Velocity profiles in the carotid bifurcation (rigid boundaries, Newtonian)
(M.Prosi)

10. Mechanical interaction (Fluid-wall coupling)
Mechanical model of the arterial vessel: linear or non-linear elasticity in Lagrangian formulation: hyperelastic or Kirchoff-St Venant material
Mechanical interaction (Fluid-wall coupling)
INTIMA
MEDIA
ADVENTITIA
Biochemical interactions (Mass-transfer processes: macromolecules, drug delivery, Oxygen,…)

11. The coupled fluid-structure problem
Equations for the geometry:
Equations for the fluid:
Equations for the structure:

12. Carotid deformation and velocity field

13. Geometrical multiscale modelling
Global features have influence
on the local fluid dynamics
Local changes in geometry or
material properties (e.g. due to
surgery, aging, stenosis, …) may
induce pressure waves reflections
 global effects
Modelling strategy
use the expensive 3D model only in the region of interest
couple with network models with peripheral impedances to account for global effects

14. GEOMETRICAL MULTISCALING
Preso da Berlino Completo. Ho quasi uniformato lo stile, anche se in alcuni punti resta un po` “misto”. Sulla versione finale ridotta metto tutto coerente…

15. A local-to-global approach
Local (level1): 3D flow model
Global (level 2): 1D network of major arteries and veins
Global (level 3): 0D capillary network

16. Geometric multiscale models
MATHEMATICAL MODEL
Geometric multiscale models
3D Navier-Stokes (F) + 3D ElastoDynamics (V-W)
Assume that:
uz >> ux ,uy
uz has a prescribed steady profile
average over axial sections
static equilibrium for the vessel
Then we obtain a 1D problem.

17. Geometric multiscale 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).

18. Geometric multiscale model
0D Lumped parameters (system of linear ODE’s)
The analogy
Fluid dynamics
Electrical circuits
Pressure
Voltage
Flow rate
Current
Blood viscosity
Resistance R
Blood inertia
Inductance L
Wall compliance
Capacitance C
RLC circuits model “large” arteries
RC circuits account for capillary bed
Can describe compartments
(such as peripheral circulation)

19. 3D-1D for the carotid: pressure pulse
MATHEMATICAL MODEL
3D-1D for the carotid: pressure pulse
(A.Moura)

20. 3D-1D for the carotid: velocity field
MATHEMATICAL MODEL
3D-1D for the carotid: velocity field
(A.Moura)