Models of ischemic stroke

1
Models of ischemic stroke

• E. Grenier
• Unité de Mathématiques Pures et Appliquées, Cnrs
  Umr 5669
• Ecole Normale Supérieure de Lyon
• ANR Biosys 2006 AVC in silico
• JP Boissel, MA Dronne (Pharmacology department,
  Lyon I)
• M. Hommel, A. Jaillard (Grenoble hospital)
• S. Descombes, G. Chapuisat (ENS Lyon)
• D. Bresch (Mathematical Department, Chambéry)
• T. Dumont (Mathematics, Lyon I)

2
Contents

• Introduction what is an ischemic stroke ?
• A qualitative approach
• Discussion
• Models and submodels
• Ionic channels and blockers
• Going further into the details

3
Introduction medical aspects
4
Introduction medical aspects

• Cerebral artery get blocked
• Various causes
• Temporary or definitive
• Partial or total
• Various territories
• Clinical manifestations
• Loss of mobility (arms, legs, )
• Loss of language (aphasia), cecity
• Evolution within a few hours
• Finished in 6 12 hours
• One of the main cause of death in developped
  countries.

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Medical aspects

• Imagery
• Angiography (arteries map)
• Oedema (cell swelling fraction of extracellular
  space)
• Blood flow (with large errors)
• Drugs
• None ! (only one positive clinical test in 2006
  )
• Except thrombolysis (reopening of the blocked
  artery)
• Only valid in 10 of the cases
• Risk blood coagulation

6
A clinical case 37 years old woman suffering
from hemiplegia and aphasia. Scanner 4h30 after
stroke
FLAIR
DWI
ARM
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Blood flow
8
4 days later recuperation of aphasia
FLAIR
DWI
ARM
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Qualitative models of stroke
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Qualitative models

• Energy
• Blood flow brings energy to neurons and glial
  cells.
• Used for various purposes homeoastis, self
  repair, metabolism, electric activity
• Entropy
• State of the cell.
• Increases through degradation and stress.
• Decreases thanks to self repair
• Homeostasis
• Ionic equilibrium
• Passive ionic motions destroy homeostasis
• Active, energy consuming motions restore
  homeostasis
• Toxicity
• In necrosis, cells explode and liberate
  toxins.

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Energy

• Input
• Blood flow brings oxygen to cells.
• Neurons have no reserves.
• First step input proportional to the blood flow.
• Ouput
• Output lt Input
• Output is the sum of the energy used for
• Eself self repair
• Eapopt apoptosis
• Ehomeo homeostasis
• Eself Eapopto Ehomeo lt Input

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Entropy

• Increase
• Spontaneous (second principle)
• Stress (calcium entry)
• Decrease
• Self repair
• Energy dependent
• Evolution
• dS/dt C0 C1 Eself stress
• Life and death
• S0 best state
• S1 death
• S0 lt S1

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Homeostasis

• Index
• 0 u 1 describes ionic homeostasis
• u 0 homeostasis, u 1 total desequilibrium.
• Evolution
• Extracellular diffusion
• Passive ionic motions fpassive
• Active ionic motions factive
• du/dt - µ?u fpassive - factive
• Passive motions
• fpassive is a function of u
• Describes the various motions through voltage
  dependent gates, ionic exchangers
• Active motions
• factive depends on u, energy and entropy.

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Homeostasis factive

• Reaction of the cell
• Depends on the energy of the cell Ehomeo
• Depends on the state of the cell S.
• Depends on homeostasis u
• Ehomeo F(u) / S
• Reaction of the tissue
• Delayed reaction of the tissue
• Increased blood flow
• Proportionnal to Ehomeo and ?, index of
  recuperation
• 0 ? 1
• ? increases if u gt 0.6
• ? decreases if u lt 0.1

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Apoptosis

• Definition
• Cell suicide, programmed cell death.
• A cell decides to suicide itself if its state is
  too bad to survive and multiply without
  mutations.
• Fragmentation of DNA, formation of small
  fragments.
• Apoptosis needs energy.
• Evolution
• a measures the evolution of apoptosis.
• a 0 sane cell
• a 1 death by apoptosis
• Apoptosis begins when S exceeds S2 with S0 lt S2 lt
  S1
• da/dt C2 Eapopt

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Typical simulation 1D Rat brain
Blood flow after stroke
Space
1D is sufficient
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Influence of diffusion of u and of apoptosis
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Influence of diffusion of u and of apoptosis
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Discussion
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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

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Spreading depression

• In cat 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

22
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.

23
Thresholds

• In 1D, various sizes of stroke and various blood
  flow profiles

The value of blood flow on the border of the dead
zone is constant threshold The final size of
the dead zone only depends on the local blood
flow, despite propagation of waves !
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Simulation of antiapoptic drugs

• Simulation of a drug that blocks completly
  apoptosis.
• Effect decreases with the size of the stroke
• Effect depends on the profile of the blood flow

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Anatomy
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The domain of propagation
Grey
White
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Spreading depression a simple model

• Simple model through a bistable reaction
  diffusion equation
• ?t u ?? u f(u)
• In grey substance f bistable
• f(u) a u (1 u) (u u0)
• u state variable
• u 0 in normal state
• u 1 in completly depressed state
• u proportionnal to extracellular K or to
  membrane potential.
• Parameters ? (diffusion), a (strength of
  nonlinearity), u0 (0 lt u0 lt 1).
• In white substance buffering effect
• f(u) - b u
• Parameter b constant (reuptake of K).

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Spreading depression existence issues

• Simplified case cylinder
• ?t u ?? u f(u,z)
• where f(u,z) a u (1 u) (u u0) 1G b u 1B
• Grey substance cylinder, direction x, radius r
• White substance complementary
• Existence of progressive waves is well known
• In cylinders
• In cylinders, with transport terms
• In domains with periodic holes
• Bidimensionnal flames
• References Fife, McLeod, Berestycki, Nirenberg,
  
• Problems here
• completly multidimensionnal profile
• no idea of the velocity
• Maximum principle based approaches seem to fail

White matter
r
Grey matter
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Numerical illustration
Including a recovery mechanism creation of two
progressive waves Case of large r
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Numerical illustration
Small r no ignition

• Diffusion of u to the white substance where it
  is damped
• Creation of u through f(u) is not sufficient in
  grey substance

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Non existence for small radius

• Theorem (G. Chapuisat, E.G.)
• There exists a Rc gt 0 such that if r lt Rc there
  exists no progressive wave solution of
• ?t u ?? u f(u,z)
• where f(u,z) a u (1 u) (u u0) 1G b u 1B
• Proof in 2D
• Let U(x-ct,z) being such a progressive wave
• Let w(z) U(-8,z) be the profile at -8 then
• ? w f(w,z)
• Phase plane analysistry to put together
• z lt r and z gt r
• Gives non existence for small r

32
May the geometry stop waves ?

• Propagation of progressive waves in a cylinder
  with variable radius
• O (x,y) y lt R(x)
• with for instance
• R(x) R si x lt 0 and R(x) R si x gt 0
• Propagation in a fold ? open question

33
May the topography stop waves ? Yes

• Case R(x) R for x lt 0 and R for x gt 0, Neuman
  boudary conditions
• Theorem (G. Chapuisat, E. G.)
• For some sets of coefficients, travelling waves
  coming from -8 are stopped near x 0. They do not
  go to 8 as time goes to 8
• Proof
• One possibility investigate the existence of a
  non trivial solution to
• ?? v f(v)
• in the domain

v 1
Neuman
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May the topography stop the waves ? Yes

• Sketch of proof
• Decompose u in a
• linear part u to lift the boundary conditions
• a non linear part u
• Evaluate u gt u0/2 using explicit formulas
  (Green functions)
• Evaluate u gt u0/2 by variation of constant
• Show that
• u gt u0 is included in u gt u0/2
• under a smallness assumption

u lt u0 / 2
u gt u0 / 2
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A real geometry Rolando sulcus

• Migraine with aura
• May lead to aphasia area 3
• Never leads to motion problems areas 4
• Question spreading depression stops before 3a
  and 4p because of
• a change in the nature of neural tissue ?
• geometry severe fold ?
• A first numerical experiment may we find
  numerical parameters such that a spreading wave
  created at
• takes the first turn and goes to 2 ?
• stops after 3a ?
• severe test the two folds are almost identical

36
Simulation for Rolando sulcus
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Spreading depressions

• In Rat, spreading depressions are observed in
  vivo
• Important in the progression of the dead core
• Size of the stroke proportionnal to the number of
  spreading depressions
• Some therapeutics aim at stopping them
• In human, no spreading depressions are observed
  in vivo
• Coherent with previous section
• spreading depressions may be blocked by folds.
• They may propagate on small lengths difficult to
  observe
• Explains failures of some therapeutics ?
• Existence of spreading depressions for very small
  strokes ?
• Stroke in young men
• Trace of the propagations of spreading
  depressions ?

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Discussion

• Topography of grey matter may explain by itself
  that spreading depression do not propagate in the
  whole brain during migraine with aura
• It may be stop by sulci (folds)
• Precise geometry of folds changes from one
  patient to another
• Precise symptoms change from one patient to
  another
• Migraine with aura are mainly genetic
• Defect in some Ca voltage dependent channel
• Different parameter from one patient to another
• To do
• Shoud be verified on larger 2D cuts of brain
• Should be verified in 3D (difficult numerical
  challenge !)
• Should be verified on a more complete ionic
  model.

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Anatomy and small strokes
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Models and submodels
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(No Transcript)
42
Models of ionic channel blockers
43
Ionic exchanges during acute phase of stroke
44
Clinical trials of ionic channel blockers

• Na voltage dependent channel blocker
• Fosphénytoine (Pulsinelli, 1999)
• Ca voltage dependent channel blocker
• 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)
• Positive results in Rat
• Negative results in human a long serie of
  clinical failures

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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
• Ca extra 1mM/l, intra lt 1 micromol / l
• Membrane potential is different from 0 about
  -50 mV to -60 mV
• Passive motions following electrochemical
  gradient voltage dependent channels, exchangers,
  
• Energy consuming motions pumps
• During acute phase of stroke
• Energy is lacking
• Pumps no longuer efficient
• Large ionic motions through channels and
  exchangers following electrochemical gradient
• Large changes in concentrations which tend to
  equilibrate

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Grey and white matters

• Grey matter neurons centers, glial cells
• White matter glial cells, axons of neurons

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pump Ca2
pump Cl-
pump Ca2
pump Cl-
2K
2K
pump Na/K
pump Na/K
Grey matter
Cl-
Ca2
Cl-
Ca2
3Na
3Na
3Na
exchanger Na/Ca2
3Na
exchanger Na/Ca2
ATP
ATP
Ca2
Ca2
Na voltage-gated channel (NaP)
Na voltage-gated channel (NaP)
Na
Na
Ca2 voltage-gated channel (CaHVA)
Ca2 voltage-gated channel (CaHVA)
Ca2
Ca2
Neuron (soma)
Astrocyte
K
K voltage-gated channel (KDR, BK, Kir)
K voltage-gated channel (KDR, BK)
K
K
glu Na
K
glutamate transporter
glutamate transporter
glu Na
gap- junctions
K
receptor AMPA
K
Na
Ca2
receptor NMDA
Na
Na 2Cl-
contransporter Na/K/Cl-
K
K
receptor AMPA
Na
HCO3-
exchanger Cl-/HCO3-
Extracellular space
Cl-
exchanger Cl-/HCO3-
HCO3-
Cl-
glu
glu
extra currents
Cl-
Cl-
extra currents
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pump Ca2
pump Cl-
pump Ca2
pump Cl-
2K
2K
pump Na/K
pump Na/K
White matter
Cl-
Ca2
Cl-
Ca2
3Na
3Na
3Na
exchanger Na/Ca2
ATP
ATP
Ca2
3Na
exchanger Na/Ca2
Ca2
Na voltage-gated channel (NaP)
Na
Na voltage-gated channel (NaP)
Na
Ca2 voltage-gated channel (CaHVA)
Ca2
Neuron (axon)
Oligo- dendrocyte
Ca2 voltage-gated channel (CaHVA)
Ca2
K
K voltage-gated channel (KDR, BK, Kir)
K
K voltage-gated channel (KDR, BK)
K
glu Na
glutamate transporter
K
receptor AMPA
Na
K
glutamate transporter
glu Na
Ca2
Na 2Cl-
contransporter Na/K/Cl-
K
exchanger Cl-/HCO3-
HCO3-
HCO3-
exchanger Cl-/HCO3-
Cl-
Extracellular space
Cl-
glu
glu
extra currents
Cl-
Cl-
extra currents
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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 (up to a factor 4)
• Difference from one species to another (up to a
  factor 4)
• Difference from one type of cell to another
• Models of channels depend on the author (up to a
  factor 100)
• Some parts of the models come from
  thermodynamics, some dont
• Conductivities vary much (factor 10 to 100)

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Strategy looking for parameters
Collect the various equations
Collect the various ranges for the parameters
Choose at random parameters
Satisfied
Not Satisfied
Check basic properties Equilibrium, stability,
general behavior
Keep the parameters
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Strategy testing an hypothesis
Formulate the hypothesis
Test all the parameters found in the precedent
phase
Some tests negative
All tests positive
Hypothesis is not consistent with the model, or
the models needs further studies to refine the
parameters
Hypothesis is coherent with the model and the
parameters
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Simulation of a stroke
Strong attack
Moderate stoke
? dead core
? penumbra
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Evolution of the ionic concentrations
Strong attack
? Coherent with experimental results
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Simulation of the action of a NaP channel blocker
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
55
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

Grey
White
? White matter is more resistant
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Simulation of the action of a NaP channel blocker
Values of rADCw with and without blocker as a
function of residual ATP production (Rat)
without
with
? Positive effet, for any residual ATP.
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Rat versus Man
Comparison of the benefit of NaP channel blocker
in grey human substance and in rat as a function
of residual ATP
human
Definition of benefit
rat
rADCwb  rADCw with blocker rADCws  rADCw
without blocker tm 1 hour

• Benefit is much larger in Rat than in human
• Different ration astrocytes / neurons may be an
  explanation

58
Effets of other pharmacological agents
? Results are coherent with experimental
observations
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Conclusion

• The failure of all these ionic channel blockers
  may have been predicted !
• The difference in the ratio astrocytes / neurons
  may by itself explain these failures.
• 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.

60
Going further
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Integration of submodels

• Metabolism
• Inflammation
• Present in all cerebral deseases
• Apoptosis
• Major role in cancer
• Major role in Parkinson
• Free radicals
• Cerebral anatomy
• Grey and white matter
• Fonctionnal territories
• Vascular territories

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Example Metabolism
(A. Aubert et R. Costalat)
63
Example functionnal areas
(Talairach)
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A LEGO of submodels

• Metabolism, ionic motions, inflammation,
  apoptosis, necrosis, free radicals, anatomy
• common to
• Stroke
• Parkinson
• Migraine
• And many other neurological diseases !