1. Computational Mechanics & Numerical Mathematics University of Groningen 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

2. Computational Mechanics & Numerical Mathematics University of Groningen ACC: common carotid artery ACE: external carotid artery ACI: internal carotid artery distal proximal Area of interest Atherosclerosis in the carotid arteries is a major cause of ischemic strokes!

3. Computational Mechanics & Numerical Mathematics University of Groningen 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) Multi-scale modeling of the carotid artery Several submodels of different length- and timescales Carotid bifurcation Fluid dynamics Wall dynamics Global Cardiovascular Circulation (electric network model)

4. Computational Mechanics & Numerical Mathematics University of Groningen 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:

5. Computational Mechanics & Numerical Mathematics University of Groningen 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

6. Computational Mechanics & Numerical Mathematics University of Groningen Boundary conditions Simple boundary conditions: Dynamic boundary conditions: Deriving boundary conditions from lumped parameter models, i.e. modeling the cardiovascular circulation as an electric network (ODE) Inflow Outflow

7. Computational Mechanics & Numerical Mathematics University of Groningen Weak coupling between fluid equations (PDE) and wall equations (ODE) Weak coupling between local and global hemodynamic submodels Future work: Numerical stability Coupling the submodels Carotid bifurcation Fluid dynamics PDE Wall dynamics ODE Global Cardiovascular Circulation ODE pressurewall motion Boundary conditions

8. Computational Mechanics & Numerical Mathematics University of Groningen Global cardiovascular circulation model Carotid Bifurcation ElectricHydraulic CurrentFlow rate Q VoltagePressure P

9. Computational Mechanics & Numerical Mathematics University of Groningen Flow in tubes Compliance due to the elasticity of the wall 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 Q out Q in C Q out P, V P: Pressure in tube V: Volume of tube V 0 : Unstressed volume Q in : Inflow Q out : Outflow

10. Computational Mechanics & Numerical Mathematics University of Groningen Flow in tubes Resistance due to fluid viscosity 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 P in P out R QP in P out P in : Inflow pressure P out : Outflow pressure Q: Volume flux

11. Computational Mechanics & Numerical Mathematics University of Groningen Flow in tubes Resistance due to inertia 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 Q L P in P out QP in P out P in : Inflow pressure P out : Outflow pressure Q: Volume flux

12. Computational Mechanics & Numerical Mathematics University of Groningen The ventricle model Elastic sphere with time-dependent compliance Linearized pressure-volume relation for elastic sphere Include heart action by making the compliance C time-dependent C(t): Time-dependent compliance of the ventricle Differentiate the time-dependent PV relation and use conservation of mass to obtain P, V P: Pressure in sphere V: Volume of sphere V 0 : Unstressed volume C(t) 1/C’(t) -V 0 (t)/C(t) PQ in Q out

13. Computational Mechanics & Numerical Mathematics University of Groningen Clinical application Parameterization of the ventricle model: the PV diagram Use the EDPVR and the ESPVR from the PV diagram of the left ventricle Assume a linear ESPVR and EDPVR with slopes E es and E ed and unstressed volumes V 0,es and V 0,ed : Relaxation Contraction Filling Ejection

14. Computational Mechanics & Numerical Mathematics University of Groningen 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):

15. Computational Mechanics & Numerical Mathematics University of Groningen 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: with C(t): Time-dependent compliance, function of E es and E ed M(t): Voltage generator, can be left out when assuming V 0,es = V 0,ed = 0 C(t) 1/C’(t) M(t) PQ in Q out

16. Computational Mechanics & Numerical Mathematics University of Groningen Minimal electrical model Simple ventricle model Ventricle model Input resistance Peripheral resistance Carotid Artery

17. Computational Mechanics & Numerical Mathematics University of Groningen Minimal electrical model Heart valves modeled by diodes Carotid Artery

18. Computational Mechanics & Numerical Mathematics University of Groningen Minimal electrical model Input/output compliance, resistance around ventricle Carotid Artery

19. Computational Mechanics & Numerical Mathematics University of Groningen Minimal electrical model Compliance in peripheral element Carotid Artery

20. Computational Mechanics & Numerical Mathematics University of Groningen Minimal electrical model Parallel systemic loop, internal/external carotid peripheral elements Carotid Bifurcation

21. Computational Mechanics & Numerical Mathematics University of Groningen Red: Arterial compartments Blue: Venous compartments Green: Capillaries Structure of the model Carotid Bifurcation

22. Computational Mechanics & Numerical Mathematics University of Groningen A simulation is performed to see if the model can capture global physiological flow properties: Parameter values are not yet realistic Simulation example Simulated flow rate for two cycles

23. Computational Mechanics & Numerical Mathematics University of Groningen Simulation example Left ventricle simulation results show global correspondence to real data (Wiggers diagram) Pressure in left ventricle (solid) Pressure in aorta (dash) Volume in left ventricle Aortic valve opens Aortic valve closes

24. Computational Mechanics & Numerical Mathematics University of Groningen 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?