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:

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)

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.

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…

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

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

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

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

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

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,…)

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

Carotid deformation and velocity field

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

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…

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

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.

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).

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)

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

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