1 1D-0D coupled algorithms for haemodynamical modeling Sergey Simakov Timur Gamliov Moscow Institute of Physics and Technology Moscow, INM, th Russian Workshop on Mathematical Models and Numerical Methods in Biomathematics and 4-th International workshop on the multiscale modeling and methods in biology and medicine

2 Review 1D+0D coupled models Vessel wall elasticity modeling remark Blood flow + Heart Respiratory flow + alveolar volume

3 Multiscale approaches in haemodynamics 0D Electric circuits3D flows 1D-3D

4 Global blood flow + heart

5 1D vascular network

6 Global blood flow 1) Mass balance 2) Momentum balance 3) Boundary conditions at junctions Compatibility conditions discretisation along outgoing characteristics 3.3 equations

7 Boundary conditions at junctions

8

9 equations

10 Heart model Isovolumetric contraction (0.08 s), Ejection (0.293 s), Isovolumetric relaxation (0.067 s), Ventricles filling (0.56 s) Mass conservation Volume averaged chamber motion Left auricle Left ventricle

11 Boundary conditions at heart junctions Arteries: Veins: Discretisation of compatibility conditions

12 Next step with 1D Boundary conditions at heart junctions

13 Respiratory flow + alveolar volume

14 Thanks to Yura Ivanov 1D trachea-bronchial tree

15 Junction: Nasopharynx: Gray’s Anatomy 1D trachea-bronchial tree Discretisation of compatibility conditions

16 Coupling with alveolar volume

17 1D+0D potential 1) Aneurisms modeling in global circulation: 1D + 0D + 1D 2) 1D – 3D junction: 1D + 0D + 3D + 0D + 1D

18 Vessel wall elasticity modeling remark

19 Boundary conditions at junctions

20 Elasticity: analytic approximation Kholodov Quarteroni, Formaggia Olufsen, Peskin Toro, Mueller Favorskii, Mukhin Pedley, Luo …

21 Elasticity: qualitative analysis and modeling Pedley, Luo Holzapfel, Gasser Multilayer elasticity simulations -> S-like curve Collapsible tubes study in lab -> S-like curve Still not included: viscoelasticity, autoregulation …

22 Elasticity modeling 1) Tension in deformable fiber 2) Density of elasticity force 3) Tansmural pressure for collagen fibers otherwise Peskin, Rosar 2001Implemented by Vassilevski, Ivanov, Salamatova 2011

23 Elasticity modeling Vassilevski, Ivanov, Salamatova

24 Thank You!