Models of ischemic stroke

E. Grenier

UMPA, Crns Umr 5669

Ecole Normale Supérieure de Lyon

In collaboration with

JP Boissel, MA Dronne (pharmacology, Lyon I)

M. Hommel (Grenoble hospital)

S. Descombres, G. Chapuisat (ENS Lyon)

G. Bricca (pharmacology, Lyon I)

D. Bresch (mathematics, Grenoble)

T. Dumont (mathematics, Lyon I)

Contents

• Introduction: what is an ischemic stroke ?

• Ionic models

• Spreading depressions and related mathematical models

• Global models

Introduction : medical aspects

• One of the main cause of mortality in developped countries.

• No satisfactory therapeutic solutions !

• Good therapeutics for Rat.

• Difference between Rat / Human

• Try to make mathematical models to understand better the situation

• Lots of different phenomena: ionic motions, oedema, blood flow, anatomy, apoptosis and necrosis …

Clinical aspects

• Cerebral artery get blocked:– Various causes

– Temporary or definitive

– Partial or total

– Various localisations

• Clinical manifestations:- Loss of mobility (arms, legs, …)

- Loss of language (aphasia), cecity

- Evolution within a few hours

- Finished in 6 – 12 hours

Medical aspects

• Imagery:– Angiography (arteries map)

– Oedema (cell swelling: fraction of extracellular space)

– Blood flow (with large errors)

• Drugs:- None !

- Except thrombolysis (reopening of the blocked artery)

- Only valid in 10 % of the cases

- Risk: oedema

Stroke developpement

• Three zones:– Ischemic core: blood flow is very low, all the cells die (loss of

ionic balance, cell swelling and explosion by necrosis)

– Penumbra: cell viability is borderline. Part of them die of necosis, part of apoptosis.

– Rest of the brain

• Phenomena:- Ionic exchanges leading to oedema

- Necrosis and apoptosis (programmed cell death)

- Spreading depression: progressive waves (Rat)

- Risk: oedema

Scanner X – IRM

37 years old female: hemiplegia and aphasia. Scanner 4h30 after stroke

IRM diffusion + ARM

FLAIRFLAIR DWIDWI ARMARM

ROI sylvien profond

ROI sylvien superficiel

IRMBlood flow images

Recuperation of aphasia 4 days after stroke

FLAIRFLAIR DWIDWI ARMARM

J4

Penumbra = diffusion - perfusion

I. Models of ionic exchanges

• Main ions:– K+, Na+, Cl-, Ca2+, glutamate– Difference between intra and extracellular

concentrations:• K+: extra 4 mM/l, intra: 140 mM/l• Na+: extra 120 mM/l; intra: 12 mM/l• Ca2+: extra 1mM/l, intra: < 1 micromol / l

– Membrane potential is different from 0: about -50 mV to -60 mV

– Energy is needed to maintain these gradients of concentrations.

• Ionic motions:– Through voltage dependent channels, which

open and close, depending on the various stimuli: KDR, NaT, …

– Through exchangers– Through pumps:

• ATP dependent

• Ions move against their electrochemical gradient.

– Very complex system !– During stroke: pumps are not efficient > ions

follow their gradients > depolarization of the cell > cell swelling (oedema) ….

Grey and white matters

• Grey matter: neurons centers, glial cells

• White matter: glial cells, axons of neurons

Models of ionic

exchanges in

Grey matter

Neuron(soma)

Astrocyte

Extracellularspace

3Na+

2K+

Ca2+Cl-

pump Ca2+

pump Cl-

Cl-Ca2+

pump Ca2+

pump Cl-

Ca2+Ca2+ voltage-gated channel (CaHVA)

Na+Na+ voltage-gated

channel (NaP)

K+ K+ voltage-gated channel (KDR, BK)

Ca2+Ca2+ voltage-gated channel (CaHVA)

Na+

Na+ voltage-gatedchannel (NaP)

K+

3Na+

Ca2+

exchanger Na+/Ca2+

Ca2+

3Na+ exchanger Na+/Ca2+

K+

gluNa+glu

Na+

K+ glutamate transporter

Na+

2Cl-K+

contransporterNa+/K+/Cl-

Cl-HCO3

-exchanger Cl-/HCO3

-

Cl-exchanger Cl-/HCO3

-HCO3

-

K+

Na+receptor

AMPAK+

Ca2+

Na+

K+

Na+

receptor NMDA

receptor AMPA

K+ voltage-gated channel (KDR, BK, Kir)

glutamate transporter

pump Na+/K+

3Na+

2K+pump Na+/K+

glu glu

Cl- Cl-extra currents extra currents

ATP ATP

Grey matter

gap-junctions

Difficulties• Very large number of components

• Very large number of parameters ( ~ 100)

• Very large indetermination on the parameters:– Difficulty to measure them in vivo

– Difference in vivo / in vitro

– Difference from one species to another

– Difference from one type of cell to another

• Models of channels depend on the author

• Some parts of the models come from thermodynamics, some don’t

• Conductivities vary much !

Hopeless ?• Putting together various pieces of models from various

authors completly fail !

• Indetermination on the coefficients by a factor 4 or more !

• What can we expect from numerical simulations in these conditions ?

• In many published models, coefficients are laking: impossible to check the models !

Strategy: looking for parametersCollect the various equations

Collect the various domains for the parameters

Choose at random parameters

Check basic properties:

Equilibrium, stability, general behavior

Not Satisfied

Satisfied

Keep the parameters

Strategy: testing an hypothesisFormulate the hypothesis

Test all the parameters found in the precedent phase

All tests positive

Hypothesis is coherent with the model and the parameters

Some tests negative

Hypothesis is not consistant with the model, or the models needs further studies to refine the parameters

0.2 0.4 0.6 0.8 1temps

0.2

0.4

0.6

0.8

1

pATP

0.2 0.4 0.6 0.8 1temps

0.2

0.4

0.6

0.8

1

pATP

10 20 30 40 50 60t

-100

-80

-60

-40

-20

nVm

10 20 30 40 50 60t

-100

-80

-60

-40

-20

aVm

10 20 30 40 50 60t

0.6

0.7

0.8

0.9

rADCw

10 20 30 40 50 60t

-100

-80

-60

-40

-20

nVm

10 20 30 40 50 60t

-100

-80

-60

-40

-20

aVm

10 20 30 40 50 60t

0.75

0.8

0.85

0.9

0.95

rADCw

Strong attack Moderate stoke

dead core penumbra

Simulation of a stroke

Evolution of the ionic concentrations

Strong attack

10 20 30 40 50 60t

100105110115120125130135

Kn

10 20 30 40 50 60t

0.10.20.30.40.50.6

Can

10 20 30 40 50 60t

2025303540455055Nan

10 20 30 40 50 60t

20

25

30

35

Cln

10 20 30 40 50 60t1.75

22.252.52.75

33.253.5glun

10 20 30 40 50 60t

100105110115120125130135

Ka

10 20 30 40 50 60t

0.10.20.30.40.50.6

Caa

10 20 30 40 50 60t

2025303540455055Naa

10 20 30 40 50 60t

20

25

30

35

Cla

10 20 30 40 50 60t1.75

22.252.52.75

33.253.5glua

10 20 30 40 50 60t

20406080100120

Ke

10 20 30 40 50 60t

0.51

1.52

2.5

Cae

10 20 30 40 50 60t

20406080100120140Nae

10 20 30 40 50 60t

146147148149150Cle

10 20 30 40 50 60t

0.51

1.52

2.53

3.54glue

Coherent with experimental results

Study of the action of various neuroprotectors

• NaP channel blockers– Fosphénytoine (Pulsinelli, 1999)

• CaHVA channels blockers– Nimodipine (VENUS, Horn et al., 2001)– Flunarizine (FIST, Franke et al., 1996)

• NMDA receptors antagonists– Selfotel (Morris et al., 1999)– Aptiganel (Albers et al., 2001)

Good results on rats, but no results (even toxicity) for Man Clinical studies have been stopped

Simulation of the action of a NaP channel blocker

10 20 30 40 50 60t

-100

-80

-60

-40

-20

nVm

10 20 30 40 50 60t

-100

-80

-60

-40

-20

aVm

10 20 30 40 50 60t

0.8

0.85

0.9

0.95

rADCw

10 20 30 40 50 60t

-100

-80

-60

-40

-20

nVm

10 20 30 40 50 60t

-100

-80

-60

-40

-20

aVm

10 20 30 40 50 60t

0.75

0.8

0.85

0.9

0.95

rADCw

fig. 1 : potential and rADCW without neuroprotector

fig. 2 : values with a blocker introduced at t = 20 min

Positive effect (Man and animal) in a moderate stroke

Simulation of the action of a NaP channel blocker

Positive effet, for any residual ATP.

Values of rADCw with and without blcoker as a function of residual ATP production (Rat)

0.2 0.4 0.6 0.8 1pATP

0.5

0.6

0.7

0.8

0.9

1

rADCw

avec bloq NaP

sans bloqueur

Comparison human/animal

Values of rADCw (1h after stroke and addition of a NaP channel blocker at t = 20 min) as a function of residual ATP production.

Effect is more important in Rat that in human, whatever the residual ATP production is.

0.2 0.4 0.6 0.8 1pATP

0.5

0.6

0.7

0.8

0.9

1

rADCw

homme

animal

Simulation of the action of a KDR blocker

Values of rADCw with and without a NaP channel blocker, as a function of residual ATP production.

Negative effet of any KDR channel blocker.

0.2 0.4 0.6 0.8 1pATP

0.5

0.6

0.7

0.8

0.9

1

rADCw

avec bloq KDR

sans bloqueur

Effets of other pharmacological agents

blocker type Effect

KDR channel blocker -

BK channel blocker =

Kir channel blocker =

NaP channel blocker +

CaHVA channel blocker +

Na/Ca exchanger +

Glutamate transport +/-

Na/K/Cl transport +

NMDA receptor +

Results are coherent with experimental observations

Hints for new drugs

Drugs that may reduce ischemic damages in grey matter:

– blocker of the inversion of Na/Ca exchanger

– blocker of the inversion of the glutamate transport

– blocker of transporteur Na/K/Cl transport

Some of these agents are currently under test.

White matter

Neuron(axon)

Oligo-dendrocyte

Extracellularspace

3Na+

2K+

Ca2+Cl-

pump Ca2+

pump Cl-

Cl-Ca2+

pump Ca2+

pump Cl-

Ca2+Ca2+ voltage-gated channel (CaHVA)

Na+Na+ voltage-gated

channel (NaP)

K+ K+ voltage-gated channel (KDR, BK)

Ca2+Ca2+ voltage-gated channel (CaHVA)

Na+

Na+ voltage-gatedchannel (NaP)

K+

3Na+

Ca2+

exchanger Na+/Ca2+

Ca2+

3Na+ exchanger Na+/Ca2+

K+

gluNa+

gluNa+

K+ glutamate transporter

Na+

2Cl-K+

contransporterNa+/K+/Cl-

Cl-HCO3

-exchanger Cl-/HCO3

-Cl-

exchanger Cl-/HCO3

-HCO3

-

K+

Na+receptor

AMPA

K+ voltage-gated channel (KDR, BK, Kir)

glutamate transporter

pump Na+/K+

3Na+

2K+pump Na+/K+

glu glu

Cl- Cl-extra currents extra currents

White matter

ATP ATP

Ca2+

Comparison of a stroke in white and grey matters

Values of rADCw 1 hour after stroke as a function of residual production of ATP in grey and white matter

0.2 0.4 0.6 0.8 1pATP

0.5

0.6

0.7

0.8

0.9

1

rADCw

SB

SG

White matter is more resistant

II. Spreading depressions

• 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.

Spreading depression: simple model

Simple model through a bistable reaction diffusion equation

∂t u – ν∂Δ u = f(u)

with f bistable

f(u) = a u (1 – u) (u – u0)

u state variable:

• u = 0 in normal state

• u = 1 in completly depressed state

Parameters: ν (diffusion), a (strength of nonlinearity), u0

(0 < u0 < 1).

Spreading depression: classical questions

For such bistable reaction diffusion equations

∂t u – ν∂Δ u = f(u)

f(u) = a u (1 – u) (u – u0)

• Existence of progressive waves is well known:– In cylinders

– In cylinders, with transport terms ..;

• Behavior in domains with holes (Beresycki, …)

Spreading depression: grey substance

Here bistable f(u) only takes place in grey matter

∂t u – ν∂Δ u = f(u)

where: in grey matter

f(u) = a u (1 – u) (u – u0)

in white matter

f(u) = - b u

And the domain Ω is

The domain Ω

May the topography stop waves ?

• Propagation of progressive waves in a cylinder with variable radius

Ω = { (x,y) | || y || < R(x) }

with for instance

R(x) = R si x < 0

R(x) = R’ si x > 0

or

R(x) = R + R’ sin(x)

• Propagation in real geometries ?

May the topography stop waves ? Yes

Work with G. Chapuisat (to appear in C.P.D.E.).

Case R(x) = R for x < 0 and R’ for x > 0

Theorem: for some sets of coefficients, travelling waves coming from - ∞ are stopped near x = 0. They do not

go to + ∞ as time goes to + ∞

Proof relies on careful construction of supersolutions.

The domain Ω in ischemic stroke

The domain Ω for migraine with aura

Numerical computation: Rolando sulcus

Simulation for Rolando sulcus

Discussion

• Topography of grey matter may explain by itself that spreading depression do not propagate in the whole brain during migraine with aura– Shoud be verified on larger 2D cuts of brain

– Should be verified in 3D (difficult numerical challenge !)

– Should be verified on more complete ionic models (link with Ca channels)

• 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 !

III. 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 !