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Matematical models for cerebral circulation

Matematical models for cerebral circulation

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Matematical models for cerebral circulation

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  1. Matematical models forcerebral circulation Bergamo, September 25th 2006 Tiziano Passerini Modeling and Scientific Computing (MOX) Department of Mathematics “F. Brioschi” Politecnico MILANO, ITALY

  2. ACKNOWLEDGMENTS: THE “ANEURISK” PROJECT (2005-2007) Medical Solutions Italy (Ing. M. Fiorani) S.Bacigaluppi, E. Boccardi, L. Antiga, B. Ene-Iordache, M. Piccinelli, A. Remuzzi, G. Dubini, F. Migliavacca, L. Socci, P. Secchi, S. Vantini, M.R. De Luca, A. Veneziani

  3. STRUCTURE OF THE PROJECT 3D reconstruction and semi-automatic detection of relevant morphological features Clinical Data (DICOM) GEOMETRICAL ANALYSIS DATA-BASE NUMERICAL MODELING STATISTICAL ANALYSIS Correlation, Functional Data Analysis 3D Simulations, FSI, WSS Computation,…

  4. Assessing cerebral aneuryms development / rupture risk • Can we identify vascular geometrical features associated to the risk of rupture? MORPHOLOGICAL + FLUID DYNAMICAL CLASSIFICATION • Is local fluid dynamics affected from upstream/downstream flow conditions? MODELS FOR CEREBRAL ARTERIAL NETWORK

  5. SIDE WALL DB/DA≥90% B A B A SIDE WALL WITH BRANCHING VESSEL B B DB/DA<90% A END WALL Reference bibliography: MORPHOLOGICAL CLASSIFICATION Possible classification of aneurysmsshape (Hassan et al, J Neurosurg,2005)

  6. Flow impingement region • Neck • Body • Dome • Lobulation • Changing LOCATION (IR) + SIZE (IS) • Large/Small Flow Pattern (FP) • Complexity • Stability Size of the Inflow Jet (JS) • Large/Small Reference bibliography: HEMODYNAMIC CHARACTERIZATION Parameters possibly related to aneurysmsrupture (Cebral et al., Am J Neurorad, 2005)

  7. ANEURISK: WHICH CLASSIFICATION? • Morphological classification (following Hassan) PROBLEMS: • ambiguous geometrical features (also in medical images) • does not consider dimensions • Flow-related classification • High flow rates/High-shear aneurysms • Low flow rates/High-pressure aneurysms Correlation with rupture? High shear aneurysms feature blebs prone to the rupture High pressure aneurysms exhibit blood flow high residence time and volume increasing

  8. ANEURISK CFD SIMULATIONS: first results Materials and methods • Blood = Newtonian fluid • Unsteady incompressible Navier-Stokes equations • Rigid vessel walls: no-slip boundary conditions • Pulsatile velocity boundary conditions at inlet: flat velocity profile + flow extensions • Physiological velocity flow rate • NS numerical solver with IP stabilization (Burman et al.) • Code LifeV C++ (C. Prud’homme, J.F. Gerbeau, L. Formaggia et al.)

  9. A NON-LOCAL PERSPECTIVE: the Willis Circle • Anatomical Variations: • Anterior part • Posterior part How does this relate with the risk of aneurysm development? Are there any preferential location for aneurysm development?

  10. 1D REPRESENTATIONS OF THE WILLIS CIRCLE What is the effect of a localized pathology (aneurysm)?

  11. Dirichlet Ω1 Ω2 Neumann Ω1 GEOMETRICAL MULTISCALE= DOMAIN SPLITTING + MODEL REDUCTION Ω Quarteroni, Veneziani 97 Hughes,Taylor 98 Formaggia, Nobile, Quarteroni, Veneziani,99 Laganà et al, 00 Formaggia, Gerbeau, Nobile, Quarteroni 00,01 Quarteroni, Veneziani 02 Fernandez, Milisic, Quarteroni, 03 Veneziani.,Vergara 04,06 NUMERICAL PROBLEMS 1 Defective boundary conditions matching 2 Iterative numerical coupling: convergence

  12. 3D – 1D COUPLING pressure 1D 3D flow rate • Exchange boundary conditions at the interface in order to guarantee • Continuity of flow rate • Continuity of total pressure Fixed point iterative algorithm

  13. 3D – 1D COUPLING: fixed point iterative algorythm Convergence? Solution at time n Solution at time n+1 NO YES Boundary Condition 1D Solve 3D problem INTERFACE PRESSURE INTERFACE FLOW RATE Solve 1D problem Boundary Condition 3D Continuity Condition 1 Continuity Condition 2

  14. 3D (flow extension) 3D + 1D 1D Geometrical multiscale approach • Realistic boundary conditions • No need for flow extensions • First step towards wider arterial networks 0D

  15. Future developments • more refined morphological / fluid dynamical classification: WSS, OSI, Vorticity, RT, impingement region • perform “full” coupled 3D – 1D simulations: whole circle of Willis giving inflow + outflow BC • include 3D vessel wall mechanics: FSI • integrate CFD parameters into statistical analysis