Ionic exchanges : reaction term Ions diffuse in extracellular space Ions diffuse through « gap junctions » (small holes in the membranes of cells). Reaction diffusion equations in the center of the model Are there travelling waves ? YES: spreading depressions observed in various species: rat, chicken, … observed during stroke in rats conjectured in man during migraine with aura Spreading depression
In Rat cortex Injection of KCl in some part of the brain At injection point, depolarization of the cells Depolarization propagates 2 – 4 mm / min Recovery after depolarization Progressive wave: depolarization wave Two waves do not cross
Spreading depression Occurs in Migraine with aura –Starts in visual areas –Stop at different locations, depending of the patients –Speed of a few mm / min Strokes in rat –Created at the border of the dying area –Propagate in the penumbra –Exhausts cells in the penumbra –Final size of the dead zone is proportionnal to the number of spreading depressions which propagate. No evidence during stroke in human.
Principaux mécanismes impliqués dans la propagation des ondes de dépolarisation de la zone infarcie vers la zone saine par un processus de diffusion des ions K+ et du glutamate dans l'espace extracellulaire et par un processus de propagation d'ondes calciques dans les astrocytes
Spreading depression: simple model State described by a variable u(t,x) 0 <= u <= 1: 0 normal state / 1 fully depressed state. Evolution of u under –Diffusion –Local reaction of the tissu f(u) t u – νΔ u = f(u) –In grey matter: F(u) is negative if u is small (normal state is stable) F(u) is positive if u is large (fully depressed state is stable, up to recovery phenomena) f(u) = a u (1 – u) (u – u0) –In white matter: recovery f(u) = - b u
Spreading depression: ionic model Introduce all extra and intracellular ionic concentrations: K+, Na+, Ca2+, Glu, Cl-, … Extracellular ionic concentrations diffuse in extracellular space Some species diffuse from glial cells to glial cells through gap junctions Reaction term is given by all local ionic exchanges through voltage dependent gates, exchangers, pumps … Very complex system ! Still to be studied completly !
Spreading depression: ionic model Simulation of a spreading depression following Shapiro (2001)
Discussion Topography of grey matter may explain by itself that spreading depression do not propagate in the whole brain during stroke Topography of grey matter may explain why spreading depressions have never been observed during stroke –Should also come from experimental difficulties –Observed in vitro on small cuts of grey substance (coherent) Big difference with Rat, where the final size of dead area is proportional to the number of spreading depressions !
Global models of stroke Very large models, combining Ionic models: –Simple bistable equations –Complete ionic model of the first section Oedema models Blood flow Death of cells (apoptosis / necrosis) –Programmed cell death : a kind of cell suicide Energy management Topography Toxicity
Typical simulation Dead zoneSpreading depressions
Spreading depressions In Rat, spreading depressions are observed in vivo –Important in the progression of the dead core –Important to try to block them In human, no spreading depressions are observed at large scales –Coherent with previous section –Remains to be checked numerically on the whole model –Explains failures of some therapeutics ? Existence of spreading depressions for very small strokes ? –Stroke in young men –Trace of the propagations of spreading depressions ?
Perspectives To complete model –Include complete ionic model –Add free radicals –Realistic 2D geometries (in progress) –Realistic 3D geometries (very challenging) To compare with clinical cases –Basis of clinical images already set up Numerical challenges –Very different time scales (from 1ms to 12h) –Very complex topography (already in 2D, … 3D …) –Very expensive !
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