Ondes de dépression corticales envahissantes

Migraine et AVC

E. Grenier

Biological aspects

• 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

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

K + , glutamate Ca 2+

transporteur du glutamate

inversé

glutamate K +

canal K + voltage-

dépendant

récepteur AMPA

Ca 2+ Ca 2+

Ca 2+

récepteur NMDA

glutamate

transporteur du glutamate

inversé

K +

récepteur purinergique

ATP

gap - junction

canal K + voltage-

dépendant

neurone

astrocyte

propagation de l’onde de dépolarisation

Spreading depression in a cat brain

Mathematical models

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 – ν∂Δ 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)

Spreading depressions in migraine with aura

The domain of propagation

Numerical domain of propagation

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 !

Numerical computation: Rolando sulcus

Simulation for Rolando sulcus

Discussion

• Topography of Rolando sulcus by itself may explain that spreading depressions always stop there during migraine with aura

Spreading depressions in stroke:Global models

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 zone Spreading depressions

Influence of diffusion / apoptosis / toxicity

Influence of diffusion / apoptosis / toxicity

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 !