The statistical physics of embolic stroke

1. The statistical physics of embolic stroke Jim Hague Department of physics and astronomy

2. Thanks With the collaboration of Emma Chung, Kumar Ramnarine, Manu Katsogridakis, Jonathon Littler,Richard Everley, Caroline Banham, Rizwan Patel, Mathew Martin, David Evans at Leicester Royal Infirmary Marie-Ann Chanrion Leicester Royal Infirmary / University of Lyon Jim Hague

3. Coming up… • Motivation for studying the problem of embolic stroke • Sources of multiple embolisation • The cerebral vasculature • A minimal model for embolic stroke as a many body problem • The critical behaviour of embolic stroke • Clinical application Jim Hague

4. Why study embolic stroke? An embolus is a clot, bubble or fatty deposit that causes a blockage • Embolic stroke is a significant cause of death and disability. 10% of the people in this room could be affected • Aim to develop a model to forecast the risk of embolic stroke from multiple embolisation. First computational model. E.M.L.Chung, J.P.Hague and D.H.Evans. Physics in Medicine and Biology 52 7153-7166 (2007) J.P.Hague and E.M.L.Chung. Phys. Rev. E, 80 (2009) 051912 J.P.Hague and E.M.L.Chung, Int. J. Mod. Phys. B, 23 4150 (2009) E.M.L.Chung, J.P. Hague, M-A. Chanrion, K.V. Ramnarine, E. Katsogridakis and D.H. Evans,Accepted for publication in Stroke. Jim Hague

5. A source of multiple embolisation • Multiple emboli form during and after operation to clear plaque from carotid artery. (Society for vascular surgery) Jim Hague

6. Another source of multiple emboli (mainly gas bubbles) • Open heart surgery: ~10000 small gaseous emboli from e.g. clearing air from the heart (source National Institutes of Health) Jim Hague

7. Destination of the emboli J.R.Cebral et al. “Blood-flow models of the circle of Willis from magnetic resonance data”. J. Eng. Math. 47:369-386 (2003). • Downstream from the circle of Willis, the arteries look like a bifurcating (fractal) tree  Can’t usefully solve fluid dynamics to get destination ~ 106 vessels and initial position sensitivity Jim Hague

8. A B Root node Minimal model of the arterial tree Capillary mesh ~ 6 m • Trajectory weighted according to the distribution of embolic blockages • Monte-Carlo simulation used to determine average behaviour of ensemble of emboli entering tree (sensitivity to initial conditions) • Spherical emboli dissolving proportionally to surface area • (or following experimental parameterization) 12 m n = 5 n = 3 n = 13 β= 2.8 E.M.L.Chung, J.P.Hague and D.H.Evans. Physics in Medicine and Biology 52 7153-7166 (2007) PA= 1/3 PB= 2/3 n = 1 β= 2.3 1 mm Major arteries > 1 mm Embolus blocks node Jim Hague

9. There is reasonable jusfification for linear flow weighting in the model – majority of pressure dropped over smallest arterioles f = fA / (fA + fB) X = nA / (nA + nB) fA is flow in A direction and nA is number of free end nodes Jim Hague J.P.Hague and E.M.L.Chung. Phys. Rev. E, 80 (2009) 051912

10. Smaller emboli are more likely to obey linear flow weighting • We’re working on more sophisticated parameterizations (r is the proportion of emboli moving in A direction) D. Bushi, Y. Grad, S. Einav, O. Yodfat, and B. Nishri, Stroke 36, 2696 2005. Jim Hague

11. Following the emboli Dense emboli block same capiliaries Small number of emboli avoid other emboli Life of one embolus Representation of capiliary flow. Dark patches are blocked. J.P.Hague and E.M.L.Chung. Phys. Rev. E, 80 (2009) 051912 Jim Hague

12. J.P.Hague and E.M.L.Chung. Phys. Rev. E, 80 (2009) 051912 Time correlator • Correlation timescale large at intermediate embolus size. Indicator of phase transition? M=20, g=1.736, t−1 =0.1 s−1, and a=3x10−4 mm/s Jim Hague

13. Non-equilibrium order parameter? 18 levels, g=2, =0.3m/s, -1=0.1emb/s • There is some indication of a ordered phase below d0=0.18mm with order parameter: (humps indicate that d0 is not exactly same size as node) J.P.Hague and E.M.L.Chung. Phys. Rev. E, 80 (2009) 051912 Jim Hague

14. Finite size scaling • Unusual finite size scaling – should make contiguous layer as similar in size as possible to make tree infinite, i.e. g->large (strictly at realistic g=3, the behaviour is crossover) Jim Hague

15. PATIENT b a (Ultrasound) embolus detection Clinical intervention to preserve cells and reduce embolization Virtual patient simulation c d Computational forecast of embolic burden The future: Towards the virtual patient Jim Hague

16. Summary and challenges • Model of embolic stroke – simulation via Monte Carlo. Analytics via continuum steady-state model. • Rapid change from free flowing to blocked state -> non-equilibrium criticality. Order parameter is overlap between blockages. • Challenge 1: Fluid dynamical analysis of embolus flow at bifurcation • Challenge 2: Sufficiently realistic fluid dynamics and structure of tree • Challenge 3: Integration of major arteries and arterial tree Jim Hague

17. Analytics: Steady state analysis Nn = Number of emboli blocking level n (N.B. dmin,dmax = d0 assumed to coincide with one of the nodes) J.P.Hague and E.M.L.Chung. Phys. Rev. E, 80 (2009) 051912 Jim Hague

18. Dilute limit No overlap between blockages (Mbig is the largest level with blockages in) Note in passing: Analytics indicate volume conserving breakup reduces blockage for g>2 J.P.Hague and E.M.L.Chung. Phys. Rev. E, 80 (2009) 051912 Jim Hague

19. Mutually exclusive Dense limit All g=1 in dense limit • Deviation from mutually exclusive results indicates overlap is an order parameter Dense (deviation because of fluctuations) J.P.Hague and E.M.L.Chung. Phys. Rev. E, 80 (2009) 051912 Jim Hague

20. Problems of clinical significance Volume conserving break up of emboli reduces total blockage E.M.L.Chung, J.P.Hague and D.H.Evans. Physics in Medicine and Biology 52 7153-7166 (2007) Jim Hague

21. Percolation models Kind of dynamic directed percolation. Model differs from conventional model as flow carries bond breaking particles, and bonds reform after characteristic time. See e.g. H. Hinrichsen, Advances in Physics, 49, 815 (2000) for directed percolation review. Jim Hague

22. Percolation problems Percolation models are useful in a wide variety of different applications: • chromatography, • toilet paper, • oil exploration • + many others Jim Hague

23. Mutually exclusive Percolation threshold • Deviation from mutually exclusive results indicates overlap is order parameter Dense (deviation because of fluctuations) = percolation threshold J.P.Hague and E.M.L.Chung. “The Physics of Embolic Stroke”. arXiv:0808.1075 Jim Hague

24. Finite size scaling • Appropriate scaling (increase levels, but keep arterioles fixed size and increase embolization rate by 2 1/g leads to sharper transition J.P.Hague and E.M.L.Chung. “Similarities between embolic stroke and percolation problems”. Under consideration, proceedings of CMT32 Jim Hague

25. Experimental justification of directional weighting In vitro test of embolus choice of path using y-shaped bifurcation and blood mimic shows linear weighting to be reasonable zeroth approximation J. P. Littler, R. Everley, K. V. Ramnarine, J. P. Hague, and E. M. L. Chung. In preparation. Jim Hague

26. Fluid dynamics and probabilities Jonathan Littler, MSc Dissertation Jim Hague

27. Parameters of the model • Rate of embolisation t-1 • Initial embolus size distribution: dmax, dmin • Dissolve rate of embolus, a (also more realistic parameterization for air bubbles) • Number of levels, Ml • bifurcation ratio, g Parameters are varied to match clinical scenarios ) ( Jim Hague

28. Clinical Scenarios and average arteriole blockage Rapid change from flowing to blocked state E.M.L.Chung, J.P.Hague and D.H.Evans. Physics in Medicine and Biology 52 7153-7166 (2007) Jim Hague