1. 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)

2. Contents Introduction: what is an ischemic stroke ? Ionic models Spreading depressions and related mathematical models Global models

3. 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 …

4. 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

5. 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

6. 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

7. Scanner X – IRM 37 years old female: hemiplegia and aphasia. Scanner 4h30 after stroke

8. IRM diffusion + ARM FLAIR DWI ARM

9. ROI sylvien profond ROI sylvien superficiel IRM Blood flow images

10. Recuperation of aphasia 4 days after stroke FLAIR DWI ARM

11. J4

12. Penumbra = diffusion - perfusion

13. 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.

14. 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) ….

15. Grey and white matters Grey matter: neurons centers, glial cells White matter: glial cells, axons of neurons

16. Models of ionic exchanges in Grey matter

17. Neuron (soma) Astrocyte Extracellular space 3Na + 2K + Ca 2+ Cl - pump Ca 2+ pump Cl - Ca 2+ pump Ca 2+ pump Cl - Ca 2+ Ca 2+ voltage-gated channel (CaHVA) Na + Na + voltage-gated channel (NaP) K+K+ K + voltage-gated channel (KDR, BK) Ca 2+ Ca 2+ voltage-gated channel (CaHVA) Na + Na + voltage-gated channel (NaP) K+K+ 3Na + Ca 2+ exchanger Na + /Ca 2+ Ca 2+ 3Na + exchanger Na + /Ca 2+ K+K+ glu Na + glu Na + K+K+ glutamate transporter Na + 2Cl - K+K+ contransporter Na + /K + /Cl - Cl - HCO 3 - exchanger Cl - /HCO 3 - Cl - exchanger Cl - /HCO 3 - HCO 3 - K + Na + receptor AMPA K+K+ Ca 2+ Na + K+K+ receptor NMDA receptor AMPA K + voltage-gated channel (KDR, BK, Kir) glutamate transporter pump Na + /K + 3Na + 2K + pump Na + /K + glu Cl - extra currents ATP Grey matter gap- junctions

18. 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 !

19. 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 !

20. Strategy: looking for parameters Collect 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

21. Strategy: testing an hypothesis Formulate 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

22. Strong attack Moderate stoke  dead core  penumbra Simulation of a stroke

23. Evolution of the ionic concentrations Strong attack  Coherent with experimental results

24. 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

25. 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

26. 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)

27. 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.

28. 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.

29. Effets of other pharmacological agents blocker typeEffect 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

30. 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.

31. White matter

32. Neuron (axon) Oligo- dendrocyte Extracellular space 3Na + 2K + Ca 2+ Cl - pump Ca 2+ pump Cl - Ca 2+ pump Ca 2+ pump Cl - Ca 2+ Ca 2+ voltage-gated channel (CaHVA) Na + Na + voltage-gated channel (NaP) K+K+ K + voltage-gated channel (KDR, BK) Ca 2+ Ca 2+ voltage-gated channel (CaHVA) Na + Na + voltage-gated channel (NaP) K+K+ 3Na + Ca 2+ exchanger Na + /Ca 2+ Ca 2+ 3Na + exchanger Na + /Ca 2+ K+K+ glu Na + glu Na + K+K+ glutamate transporter Na + 2Cl - K+K+ contransporter Na + /K + /Cl - Cl - HCO 3 - exchanger Cl - /HCO 3 - Cl - exchanger Cl - /HCO 3 - HCO 3 - K + Na + receptor AMPA K + voltage-gated channel (KDR, BK, Kir) glutamate transporter pump Na + /K + 3Na + 2K + pump Na + /K + glu Cl - extra currents White matter ATP Ca 2+

33. 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  White matter is more resistant

34. 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

35. 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

36. 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.

37. 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).

38. 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, …)

39. 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

40. The domain Ω

41. 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 ?

42. May the topography stop waves ? Yes Work with G. Chapuisat (to appear in C.P.D.E.). Case R(x) = 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.

43. The domain Ω in ischemic stroke

44. The domain Ω for migraine with aura

45. Numerical computation: Rolando sulcus

46. Simulation for Rolando sulcus

47. 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 !

48. 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

49. Typical simulation Dead zoneSpreading depressions

50. Influence of diffusion / apoptosis / toxicity

52. 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 ?

53. 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 !