N UMERICAL M ODELING of S OME P ROBLEMS R ELATED TO C EREBRAL H AEMODYNAMICS

1. mox.polimi.it Innsbruck, October 13th 2005 Workshop: Computational Life Sciences NUMERICAL MODELING of SOME PROBLEMS RELATED TO CEREBRAL HAEMODYNAMICS Alessandro VENEZIANI Modeling and Scientific Computing (MOX) Department of Mathematics “F. Brioschi” Politecnico MILANO, ITALY

2. Vasa meningem tenuem intertexentia sunt arteriae et venae. Arteriae sunt quatuor, due carotides et due vertebrales.Ex utroque latere infundibuli arteriarum carotidum extremitates abscissae prostant, quarum trunci sursum ascendentes, statim ex utroquelatere in ramum anteriorem & posteriorem diffunduntur.(T. Willis) Arteries: 5mm diameter Arterioles: 10-90 mm Capillaries: 8 mm The BBB is semi-permeable; it allows some materials to cross, but prevents others from crossing. In most parts of the body, the capillaries, are lined with endothelial cells featuring small spaces so substances can move readily between the inside and the outside of the vessel.In the brain, the endothelial cells fit tightly together and substances cannot pass out of the bloodstream. Some molecules, such as glucose, are transported out of the blood by active mechanisms. INTRODUCTION Brain is 2% in weight of the human body It receives 15% of the total cardiac flow rate: 100 g of cerebral tissue receive 57ml/min blood and need 3.5ml/min Oxygen, 5.5mg/min glocose INGREDIENTS OF CEREBRAL CIRCULATION 1 Willis Circle 2 Cerebral Microcirculation 3 Blood-Brain Barrier (BBB)

3. OUTLINE These features are quite specific and must be considered in mathematical/numerical modeling of cerebral hemodynamics pathologies TWO CASES Focal Ischemia Aneurysms Biochemics (BBB)-Haemodynamics GOAL:Numerical model of cerebral infarction Morphology-Haemodynamics: ANEURISK Project GOAL: Numerical/Statistical investigation of development/rupture risks

4. 1st Case: Numerical Model for Focal Ischemia STROKE STROKE Cerebral Hemorragy (brokening of cerebral capillaries) Cerebral Ischemia Cerebral Ischemia (occlusion of cerebral incoming arteries) global (blood supply is inhibited to the whole brain) focal focal (blood supply reduction involves a part of the brain) J. w. with E. Agostoni, S. Gerardo Hospital, Monza, A. Di Matteo, M. Perego (MOX)

5. Physiolgical Conditions (each artery works) 1 2 Arterial Occlusion (a cerebral ground is not supplied) 1 2 • compensatory circulation (Willis circle) • tissue degeneration at different levels: 3 4 Ischemic umbra: total degeneration 3 4 Ischemic penumbra: partial degeneration, reversible damages

6. A reasonable therapy (working for instance in the coronaries): FIBRINOLYSIS 1 2 1 2 Opening of the stenosed artery DRAWBACK: Blood leakage (cerebral infarction) 3 4 3 4 Large molecules pass through the BBB

7. A PHENOMENOLOGICAL MODEL The BLOOD-BRAIN BARRIER Physiology Small Inter-Cellular Spaces, Filtering of Molecules 2 nm Pathology Missing oxygen supply (hypoxia) induces endothelial degeneration 30 nm Opening of the Cells Junctions BBB filtering reduction Red Cells, Proteins, etc. leakage

8. BIOCHEMICS of ISCHEMIA ECS ICS Potassium increases OUTSIDEthe cells Membrane depolarization K+ K+ ATP Ca++ Calcium increase INSIDE the cell Potassium diffusion in ECS Ca++ ATP Spreading Depression: Brain electrical activity depression propagating in different zones Toxicity (Lipase,Protease) Tissue Damage Energetic stress for tissues

9. A Complete Model Biochemics* Haemodynamics Accounts for the ionic disorder Induced by the reduced/absent Blood supply Haemodynamics in Cerebral Microcirculation yields yields Potassium (K+) and Calcium (Ca++) concentration Cells energetic stores (E) Tissue integrity (I) Flow rate (q) * Ruppin, Reggia Computers in Biology and Medicine (1999)

10. HAEMODYNAMICS Brain described as a POROUS MEDIUM Flow rate Blood velocity Piezometric potential Blood pressure Darcy Law: Hydraulic conducibility Blood incompressibility

11. (a,p) J(K,Ca) are functions of E,I BIOCHEMICS Passive ionic flux Active ionic flux Potassium volumic concentration: EC concentration tortuosity IC concentration Calcium volumic concentration: metabolism Energy stores: Tissue Integrity: ATP/vol Critical value of Energy Store Critical value of Calcium

12. THE COMPLETE MODEL Haemodynamics Biochemics Vector form of biochemical problem

13. A FEW WORDS ON THE WELL POSEDNESS ANALYSIS Application of the Schauder fixed point Theorem: T1 Theorem T is a compact operator fulfilling the requirements of the Schauder Theorem* T2 T = T2 o T1 *E. Agostoni, M. Perego, S. Salsa, A.V., in preparation, 2005 Innsbruck, October 13th 2005 13

14. NUMERICAL RESULTS BIOCHEMICS: Galerkin Finite Elements HAEMODYNAMICS (DARCY): Mixed Finite Elements Code: LifeV (www.lifev.org) FIRST TEST CASE: Biochemics Spherical domain (r=2cm) with no flux at the centre Physiological initial and boundary conditions

15. BIOCHEMICAL MODEL EC Potassium Concentration IC Potassium Concentration In the ischemic umbra the active mechanisms are inhibited t = 2 min. t = 2,5 min. Increase of EC Potassium in the penumbra In the non damaged zones the EC potassium is reassorbed t = 3 min. t = 3,5 min. Spreading Depression physiology: 140 mM/l physiology: 2 mM/l Pathology 30 mM/l Pathology: 135 mM/l

16. BIOCHEMICAL MODEL Energy Stores E t = 2 min. t = 3,5 min. E and I vanish at the centre (umbra) Tissue Intactness I t = 2 min. The degradated zone expands t = 3,5 min.

17. SECOND TEST CASE: Haemodynamics Physiology Geometry:cylinder with arteries and veins Physiological conditions:Arterial Pressure: 100 mmHgVenous Pressure: 10 mmHg Boundary Conditions:Dirichlet along the vessels Neumann on the boundaries Piezometric Potenzial F Flow rate q

18. SECOND TEST CASE: Haemodynamics Pathology Central artery is occluded: Flow rate vanishes The neighborhood is not correctly perfused Piezometric Potential F Flow rate q

19. SECOND TEST CASE: Haemodynamics Fibrinolithic Therapy:Reperfusion High hydraulic conductivity near the occluded zone K Artery perfusion Blood Leakage