NUMERICAL MODELING of SOME PROBLEMS RELATED TO CEREBRAL HAEMODYNAMICS

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

• mox.polimi.it

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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)
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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
GOALNumerical model of cerebral infarction
Morphology-Haemodynamics ANEURISK Project
GOAL Numerical/Statistical investigation of
development/rupture risks
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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)
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Physiolgical Conditions (each artery works)
1
2
Arterial Occlusion (a cerebral ground is
not supplied)
2
1

• compensatory circulation (Willis circle)

• tissue degeneration at different levels

3
4
Ischemic umbra total degeneration
3
4
Ischemic penumbra partial degeneration, reversibl
e damages
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A reasonable therapy (working for instance in
the coronaries) FIBRINOLYSIS
1
2
2
1
Opening of the stenosed artery
DRAWBACK Blood leakage (cerebral infarction)
3
4
3
4
Large molecules pass through the BBB
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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
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BIOCHEMICS of ISCHEMIA
ICS
ECS
Potassium increases OUTSIDE the cells Membrane
depolarization
K
K
ATP
Ca
Calcium increase INSIDE the cell
Ca
Potassium diffusion in ECS
ATP
Spreading Depression Brain electrical activity
depression propagating in different zones
Toxicity (Lipase,Protease)
Tissue Damage
Energetic stress for tissues
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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)
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HAEMODYNAMICS
Brain described as a POROUS MEDIUM
Flow rate
Blood velocity
Piezometric potential
Blood pressure
Darcy Law
Hydraulic conducibility
Blood incompressibility
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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
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THE COMPLETE MODEL
Haemodynamics
Biochemics
Vector form of biochemical problem
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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
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NUMERICAL RESULTS
BIOCHEMICS Galerkin Finite Elements
HAEMODYNAMICS (DARCY) Mixed Finite Elements
Code LifeV (www.lifev.org)
FIRST TEST CASE Biochemics
Spherical domain (r2cm) with no flux at the
centre
Physiological initial and boundary conditions
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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 3,5 min.
t 3 min.
Spreading Depression
physiology 2 mM/l
Pathology 30 mM/l
physiology 140 mM/l
Pathology 135 mM/l
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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.
t 3,5 min.
The degradated zone expands
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SECOND TEST CASE Haemodynamics
Physiology
Geometrycylinder with arteries and veins
Physiological conditionsArterial Pressure 100
mmHgVenous Pressure 10 mmHg
Boundary ConditionsDirichlet along the vessels
Neumann on the boundaries
Piezometric Potenzial F
Flow rate q
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SECOND TEST CASE Haemodynamics
Pathology
Central artery is occluded Flow rate vanishes
The neighborhood is not correctly perfused
Piezometric Potential F
Flow rate q
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SECOND TEST CASE Haemodynamics
Fibrinolithic TherapyReperfusion
High hydraulic conductivity near the occluded zone
K
Artery perfusion
Blood Leakage