Flowing Matter – COST Action MP1305, , Sofia NUMERICAL ANALYSIS OF THE STENOSIS EFFECTS ON THE BLOOD FLOW IN THE COMMON CAROTID ARTERY BIFURCATION FOR UNSTEADY FLOWS N. Antonova a, X. Dong b, I.Velcheva c, E. Kaliviotis d, P. Tosheva a a Department of Biomechanics, Institute of Mechanics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria, b School of Civil Engineering, Tianjin University, China. c Department of Neurology, University Hospital of Neurology and Psychiatry “St. Naum”, Medical University. Sofia, Bulgaria d Faculty of Engineering and Technology, Department of Mechanical Engineering and Materials Science and Engineering, Limassol,Cyprus

Flowing Matter – COST Action MP1305, , Sofia AIM The aim of the study is to investigate the hemodynamics in the carotid artery bifurcation with stenoses by means of the 3D numerical analysis of blood flow based on the numerical solution of the Navier- Stokes and continuity equations.

Flowing Matter – COST Action MP1305, , Sofia METHODS 3.1. Navier-Stokes Equations The blood flows in the arteries are treated as incompressible viscous flow. Fluid motion is modeled by solving numerically the system of equations of the continuity and the equations of motion using the finite volume method. The Navier-Stokes equations for incompressible viscous flows read: where is the vector of fluid velocities, is the pressure normalized with the fluid density and is a body force term.

Flowing Matter – COST Action MP1305, , Sofia Entrance boundary condition scheme

Flowing Matter – COST Action MP1305, , Sofia Pulse wave (pressure wave) To model the development of the blood flow induced by a pulse wave (pressure wave) time dependent boundary conditions are used.

Flowing Matter – COST Action MP1305, , Sofia Shape of the computational area To create the complicated shape of the computation area a CAD system is applied

Flowing Matter – COST Action MP1305, , Sofia Methods To create the complicated shape of the computation area a CAD system is applied with tetrahedral elements. A fluid solver was developed using unstructured mesh. It is a three-dimensional fluid solver with second order accuracy in both time and space and is based on a finite volume formulation. For time-stepping, a Crank-Nicolson scheme was used for the diffusion term and the convection term is discretised using an explicit two-step Adams-Bashforth scheme.

Flowing Matter – COST Action MP1305, , Sofia Surface mesh for the common carotid bifurcation without stenosis and with one, two and three stenoses Tetrahedral elements are adopted for its flexibility for complex geometries. The average mesh size is around 0.05 cm and the total amount of elements is around 250, 000 for each case. The time step for the simulation is dt = s, yielding a CFL number of around 0.1.

Flowing Matter – COST Action MP1305, , Sofia RESULTS

Flowing Matter – COST Action MP1305, , Sofia Distribution of axial velocity in the flow around carotid bifurcation without stenoses at T = 0s, 0.1s, 0.2s

Flowing Matter – COST Action MP1305, , Sofia Distribution of axial velocity the flow around carotid bifurcation without stenoses at T=0.3s, 0.4s,0.5s

Flowing Matter – COST Action MP1305, , Sofia Velocity magnitude contour and wall shear stress distribution in the carotid artery bifurcation without stenoses. Left: velocity magnitude contour unit: m/s Right: Wall shear stress distribution unit: Pa

Flowing Matter – COST Action MP1305, , Sofia Velocity magnitude contour and wall shear stress distribution in the carotid artery bifurcation with one stenosis. Left: velocity magnitude contour unit: m/s Right: Wall shear stress distribution unit: Pa

Flowing Matter – COST Action MP1305, , Sofia Velocity magnitude contour and wall shear stress distribution in the carotid artery bifurcation with two stenoses. Left: velocity magnitude contour unit: m/s Right: Wall shear stress distribution unit: Pa

Flowing Matter – COST Action MP1305, , Sofia Velocity magnitude contour and wall shear stress distribution in the carotid artery bifurcation with three stenoses. Left: velocity magnitude contour unit: m/s Right: Wall shear stress distribution unit: Pa

Flowing Matter – COST Action MP1305, , Sofia Comparison of peak wall shear stress for different cases

Flowing Matter – COST Action MP1305, , Sofia The radial velocity (left) and vorticity (right) magnitude in the CAB without stenosis

Flowing Matter – COST Action MP1305, , Sofia The radial velocity (left) and vorticity (right) magnitude in the CAB with one stenosis

Flowing Matter – COST Action MP1305, , Sofia The radial velocity (left) and vorticity (right) magnitude in the CAB with two stenoses

Flowing Matter – COST Action MP1305, , Sofia The radial velocity (left) and vorticity (right) magnitude in the CAB with three stenoses

Flowing Matter – COST Action MP1305, , Sofia Conclusion Different models of the carotid bifurcation were examined. The results of computational simulations may supplement MRI, CT, Doppler and other in vivo diagnostic techniques to provide an accurate picture of the hemodynamics in particular vessels, which may help demonstrate the risks of embolism or plaque rupture posed by particular plaque deposits. The use of imaging investigation with mapping of wall shear stress distribution (WSS) in the carotid arteries in parallel with numerical simulation could help for detection of the vessel sites where atherosclerotic plaques in the separate individuals would develop. The results from the changes in the hemodynamic profile could also prompt the therapeutic behaviour in the examined patients. Based on MRI, image segmentation, geometry reconstruction and mesh generation could be done.

Flowing Matter – COST Action MP1305, , Sofia The case studies are based on different anatomies presented by the left, right or common carotid vessel, various degrees of geometrical non-planarity, and variation in severity of stenosis in carotid arteries for a group of patients. The physiological geometry can be imported into a CFD solver. As such, this technique can be applied non-invasively at arterial sites where vascular anatomy typically exhibits substantial inter-individual variability. The results play an important role in the formation, growth, rupture and prognosis of damage of the vessel wall and may be a practical tool for planning treatment and follow-up of patients after neurosurgical or endovascular interventions with 3D angiography. The results present the potential of using medical imaging and numerical simulation to provide existing clinical prerequisites for diagnosis and therapeutic treatment.

Flowing Matter – COST Action MP1305, , Sofia Collaborators N. Antonova, P. Tosheva - Department of Biomechanics, Institute of Mechanics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria D. Xu - School of Civil Engineering, Tianjin University, China I.Velcheva - Department of Neurology, University Hospital of Neurology and Psychiatry “St. Naum”, Medical University. Sofia, Bulgaria E. Kaliviotis - Department of Engineering, University College London, London, UK

Publications Antonova N., D. Xu, I. Velcheva, E. Kaliviotis, P. Tosheva,. Stenosis effects on the fluid mechanics of the common carotid artery bifurcation for unsteady flows. J. Mechanics in Medicine and Biology, 15(2), 2015, (ISSN: ; IF: 0.803) 2015, (ISSN: ; IF: 0.803) Antonova N., X. Dong, P. Tosheva, E. Kaliviotis, I. Velcheva. Numerical analysis of 3D blood flow and common carotid artery hemodynamics in the carotid artery bifurcation with stenosis. J. Clinical Hemorheology and Microcirculation, 57, 2014, (ISSN: ; IF: 2.215) 2014, (ISSN: ; IF: 2.215) Antonova N., X. Dong, P. Tosheva, I.Velcheva. Numerical analysis of 3D blood flow and common carotid artery hemodynamics in the carotid artery bifurcation without stenoses. J. Series on Biomechanics, vol.28 (3-4), Avangard, Sofia, 2013, (ISSN: Antonova N., P. Tosheva, I.Velcheva. Numerical analysis of 3D blood flow and common carotid artery hemodynamics in the carotid artery bifurcation with stenosis, J. Series on Biomechanics, vol.27 (3-4), Avangard, Sofia, 2012, (ISSN: )58) Flowing Matter – COST Action MP1305, , Sofia

Acknowledgements The Ministry of Education, Youth and Science of Bulgaria supported the study - funded by the Operational Programme "Human Resources Development" within the Project № BG051PO /0001 regimen «Science-Business" under Grant No. ДО-803/2012 is gratefully acknowledged.