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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

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 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)

Global

Cardiovascular

Circulation

(electric network model)

Multi-scale modeling of the carotid artery Several submodels of different length- and timescalesCarotid bifurcation

Fluid dynamics

Wall dynamics

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:

Computational fluid dynamics: ComFloThe 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

Modeling the wall as a mass-spring systemSimple boundary conditions:

Dynamic boundary conditions: Deriving boundary conditions from lumped parameter models, i.e. modeling the cardiovascular circulation as an electric network (ODE)

Boundary conditionsOutflow

Outflow

Inflow

Coupling the submodels

Carotid bifurcation

Weak coupling between

fluid equations (PDE)

and wall equations (ODE)

Weak coupling between

local and global

hemodynamic submodels

Future work: Numerical stability

Fluid dynamics

PDE

wall motion

pressure

Wall dynamics

ODE

Boundary conditions

Global

Cardiovascular

Circulation

ODE

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

Qin

Qout

P, V

P

Qin

Qout

C

Flow in tubes Compliance due to the elasticity of the wallP: Pressure in tube

V: Volume of tube

V0: Unstressed volume

Qin: Inflow

Qout: Outflow

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

Pin

Q

Pout

Q

Pin

Pout

R

Flow in tubesResistance due to fluid viscosityPin: 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

Pout

Pin

Q

Flow in tubesResistance due to inertiaPin: Inflow pressure

Pout: Outflow pressure

Q: Volume flux

Pin

Q

Pout

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

Qin

P

Qout

C(t)

1/C’(t)

-V0(t)/C(t)

The ventricle modelElastic sphere with time-dependent complianceP: Pressure in sphere

V: Volume of sphere

V0: Unstressed volume

P, V

Use the EDPVR and the ESPVR

from the PV diagram of the left ventricle

Assume a linear ESPVR and EDPVR with slopes Ees and Eed and unstressed volumes V0,es and V0,ed:

Clinical applicationParameterization of the ventricle model: the PV diagramEjection

Relaxation

Contraction

Filling

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):

Clinical applicationParameterization of the ventricle model: the driver function e(t)Differentiate the time-dependent PV relation

and use conservation of mass to obtain the

ventricle model:

with

C(t): Time-dependent compliance, function of Ees and Eed

M(t): Voltage generator, can be left out when assuming V0,es = V0,ed = 0

Qin

P

Qout

C(t)

1/C’(t)

M(t)

Clinical applicationParameterization of the ventricle model: electric analogMinimal electrical modelSimple ventricle model

Peripheral resistance

Carotid

Artery

Input resistance

Ventricle model

Minimal electrical modelParallel systemic loop, internal/external carotid peripheral elements

Carotid

Bifurcation

Structure of the model

Red: Arterial compartments

Blue: Venous compartments

Green: Capillaries

Carotid

Bifurcation

A simulation is performed to see if the model can capture global physiological flow properties:

Parameter values are not yet realistic

Simulation exampleSimulated flow rate for two cycles

Simulation example

- Left ventricle simulation results show global correspondence to real data (Wiggers diagram)

Aortic valve closes

Aortic valve opens

Pressure in left ventricle (solid)

Pressure in aorta (dash)

Volume in left ventricle

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?

Future work
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