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Multi-scale modeling of the caro

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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