Multi-scale modeling of the carotid artery G. Rozema, A.E.P. Veldman, N.M. Maurits University of Groningen, University Medical Center Groningen The Netherlands

Area of interest Atherosclerosis in the carotid arteries is a major cause of ischemic strokes! distal proximal ACI: internal carotid artery ACE: external carotid artery ACC: common carotid artery

A multi-scale computational model: Several submodels of different length- and timescales: Carotid bifurcation A model for the local blood flow in the region of interest: A model for the fluid dynamics: ComFlo A model for the wall dynamics A model for the global cardiovascular circulation outside the region of interest (better boundary conditions) Fluid dynamics Wall dynamics Global Cardiovascular Circulation

Computational fluid dynamics: ComFlo Finite-volume discretization of Navier-Stokes equations Cartesian Cut Cells method Domain covered with Cartesian grid Elastic wall moves freely through grid Discretization using apertures in cut cells Example: Continuity equation  Conservation of mass:

Boundary conditions Simple boundary conditions: Future work: Deriving boundary conditions from lumped parameter models, i.e. modeling the cardiovascular circulation as an electric network (ODE) Outflow Outflow Inflow

The wall dynamics (1) Simple algebraic law: Independent rings model: wr(z,t) and wz(z,t): displacement of vessel wall in radial and longitudinal direction Simple algebraic law: Independent rings model: Elasticity Pressure Inertia Elasticity Pressure

Wall dynamics (2) Generalized string model: Navier equations: Inertia Shear Elasticity Damping Pressure Pressure Shear Inertia Elasticity

Modeling the wall as a mass-spring system The wall is covered with pointmasses (markers) The markers are connected with springs For each marker a momentum equation is applied x: the vector of marker positions

The mass-spring system compared to the (simplified) Navier equations Material points move in radial and longitudinal direction only Generalized string model Material points move in radial direction only Mass-spring system Material points (markers) are completely free: Conservation of momentum in all directions: Inertia Shear Pressure Elasticity Damping

Coupling the submodels Carotid bifurcation Weak coupling between fluid equations (PDE) and wall equations (ODE) local and global hemodynamic submodels Future work: Numerical stability Fluid dynamics PDE wall motion pressure Wall dynamics ODE Boundary conditions Global Cardiovascular Circulation ODE

Results: clinical data and CFD Example: Doppler flow wave form. Model variations: Rigid wall / elastic wall, Traction-free outflow / peripheral resistance A B C D Elastic wall No Yes Peripheral resistance

Results (2): Conclusion Both elasticity and peripheral resistance must be taken into account to obtain a close resemblance between measured and calculated flow wave forms Future work: Clinical follow-up data 3D ultrasound Patient specific modeling