1. 1 Mathematical Assessment of Arterio-Venous Malformations Embolisation 1 Moscow Institute of Physics and Technology 2 Novosibirsk State University 3 Institute of Hydrodynamics SD RAS Simakov 1 S.S., Gorodnova 1 N.O., Chupakhin 2,3 A.P., Khe 2,3 A.K. Workshop on mathematical models and numerical methods in biomathematics 2013

2. 2 Cardiovascular Loop

3. 3 Motivation AVM embolisation: Mortality (1.5-3%) Disability (3.8-7%) Postoperative risk (5% during the next 5 years) Without treatment: Stroke at the age of 30-40 years

4. 4 Motivation

5. 5

6. 6 1D Network Mathematical model of circulation

7. 7 Global blood flow 1) Mass balance 2) Momentum balance 3) Boundary conditions at junctions 3.1 3.2 Compatibility conditions along outgoing characteristics (finite difference discretisaztion) 3.3 equations Kholodov 2001,et. al.

8. 8 Boundary conditions: vessels junction Equations set

9. 9 Vessel wall elasticity Pedley, Luo, 1998 Analytic approximation

10. 10 Vessel elasticity modeling

11. 11 AVM Haemodynamics with 1D Approach

12. 12 AVM

13. 13 1D AVM scheme

14. 14 Pressure distribution input output left route right route input left route output input left route occluded

15. 15 Velocity distribution right route output input left route occluded

16. 16 Relative measures of embolisation quality Pressure embolisation quality Velocity embolisation quality

17. 17 1D AVM scheme

18. 18 Pressure embolisation quality

19. 19 Velocity embolisation quality

20. 20 P-U diagrams before and after surgical embolisation

21. 21 P-U diagrams before and after surgical embolisation

22. 22 Q-E diagrams before and after surgical embolisation

23. 23 Q-E diagrams before and after surgical embolisation

24. 24 Arteries Before surgery Before After V-P Q-E

25. 25 Arteries After surgery Before After V-P Q-E

26. 26 Thank You!