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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 University of Groningen Computational Mechanics & Numerical Mathematics Area of interest distal Atherosclerosis in the carotid arteries is a major cause of ischemic strokes! proximal ACI: internal carotid artery ACE: external carotid artery ACC: common carotid artery University of Groningen Computational Mechanics & Numerical Mathematics Multi-scale modeling of the carotid artery Several submodels of different length- and timescales • 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 Carotid bifurcation Fluid dynamics Wall dynamics • A model for the global cardiovascular circulation outside the region of interest (better boundary conditions) Global Cardiovascular Circulation (electric network model) University of Groningen Computational Mechanics & Numerical Mathematics 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: University of Groningen Computational Mechanics & Numerical Mathematics 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 University of Groningen Computational Mechanics & Numerical Mathematics Boundary conditions • Simple boundary conditions: Outflow Outflow Inflow • Dynamic boundary conditions: Deriving boundary conditions from lumped parameter models, i.e. modeling the cardiovascular circulation as an electric network (ODE) University of Groningen Computational Mechanics & Numerical Mathematics Coupling the submodels Carotid bifurcation Fluid dynamics PDE wall motion pressure Weak coupling between fluid equations (PDE) and wall equations (ODE) Wall dynamics ODE Boundary conditions Global Cardiovascular Circulation ODE University of Groningen Computational Mechanics & Numerical Mathematics Weak coupling between local and global hemodynamic submodels Future work: Numerical stability Global cardiovascular circulation model Carotid Bifurcation Electric Hydraulic Current Voltage Flow rate Q Pressure P University of Groningen Computational Mechanics & Numerical Mathematics Flow in tubes Compliance due to the elasticity of the wall Qin P, V Qout P: Pressure in tube V: Volume of tube V0: Unstressed volume Qin: Inflow Qout: Outflow • Consider an elastic tube, with internal pressure P and volume V The linearized pressure-volume relation is given by • Differentiate the PV relation and use conservation of mass to obtain C: Compliance of the tube • Electric analog: Capacitor Q: Current, P: Voltage P Qin C Qout University of Groningen Computational Mechanics & Numerical Mathematics Flow in tubes Resistance due to fluid viscosity Pin: Inflow pressure Pout: Outflow pressure Q: Volume flux Pin Q Pout • Consider stationary Poiseuille flow (parabolic velocity profile) Conservation of momentum is given by: R: Resistance due to fluid viscosity • Electric analog: Resistor Q: Current, P: Voltage Q Pin R Pout University of Groningen Computational Mechanics & Numerical Mathematics Flow in tubes Resistance due to inertia Pin Q Pout Pin: Inflow pressure Pout: Outflow pressure Q: Volume flux • Consider inviscous potential flow (flat velocity profile) Conservation of momentum is given by (Newton’s law): L: Resistance due to inertia (mass) • Electric analog: inductor Q: Current, P: Voltage L Pin Q Pout University of Groningen Computational Mechanics & Numerical Mathematics The ventricle model Elastic sphere with time-dependent compliance • Linearized pressure-volume relation for elastic sphere P: Pressure in sphere V: Volume of sphere V0: Unstressed volume P, V • Include heart action by making the compliance C time-dependent Qin P Qout C(t): Time-dependent compliance of the ventricle • Differentiate the time-dependent PV relation and use conservation of mass to obtain C(t) 1/C’(t) -V0(t)/C(t) University of Groningen Computational Mechanics & Numerical Mathematics Clinical application Parameterization of the ventricle model: the PV diagram • Use the EDPVR and the ESPVR from the PV diagram of the left ventricle Ejection Relaxation Filling Contraction • Assume a linear ESPVR and EDPVR with slopes Ees and Eed and unstressed volumes V0,es and V0,ed: University of Groningen Computational Mechanics & Numerical Mathematics Clinical application Parameterization of the ventricle model: the driver function e(t) • Construct PV relations for intermediate times by moving between the ESPVR and EDPVR according to a driver function e(t) between 0 and 1: • Example of a driver function e(t): University of Groningen Computational Mechanics & Numerical Mathematics Clinical application Parameterization of the ventricle model: electric analog • Differentiate the time-dependent PV relation and use conservation of mass to obtain the ventricle model: Qin P Qout C(t) 1/C’(t) with M(t) C(t): Time-dependent compliance, function of E es and Eed M(t): Voltage generator, can be left out when assuming V0,es = V0,ed = 0 University of Groningen Computational Mechanics & Numerical Mathematics Minimal electrical model Simple ventricle model Peripheral resistance Carotid Artery Input resistance Ventricle model University of Groningen Computational Mechanics & Numerical Mathematics Minimal electrical model Heart valves modeled by diodes Carotid Artery University of Groningen Computational Mechanics & Numerical Mathematics Minimal electrical model Input/output compliance, resistance around ventricle Carotid Artery University of Groningen Computational Mechanics & Numerical Mathematics Minimal electrical model Compliance in peripheral element Carotid Artery University of Groningen Computational Mechanics & Numerical Mathematics Minimal electrical model Parallel systemic loop, internal/external carotid peripheral elements Carotid Bifurcation University of Groningen Computational Mechanics & Numerical Mathematics Structure of the model Carotid Bifurcation Red: Arterial compartments Blue: Venous compartments Green: Capillaries University of Groningen Computational Mechanics & Numerical Mathematics Simulation example • A simulation is performed to see if the model can capture global physiological flow properties: Simulated flow rate for two cycles • Parameter values are not yet realistic University of Groningen Computational Mechanics & Numerical Mathematics Simulation example • Left ventricle simulation results show global correspondence to real data (Wiggers diagram) Aortic valve opens Aortic valve closes Pressure in left ventricle (solid) Pressure in aorta (dash) Volume in left ventricle University of Groningen Computational Mechanics & Numerical Mathematics Future work • Parameterization of the electric network model (resistors, inductors, capacitors): linking the model to clinical measurements • Coupling of the electric network model to the 3D carotid bifurcation model • Multi-scale simulations for individual patients? University of Groningen Computational Mechanics & Numerical Mathematics