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
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!
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)
A multi

scale computational model: Several
submodels of different length

and timescales:
Carotid bifurcation
Fluid dynamics
Wall dynamics
Global
Cardiovascular
Circulation
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:
Computational Mechanics & Numerical Mathematics
University of Groningen
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)
Inflow
Outflow
Outflow
Computational Mechanics & Numerical Mathematics
University of Groningen
The wall dynamics (1)
•
Simple algebraic law:
•
Independent rings model:
w
r
(z,t) and w
z
(z,t):
displacement of vessel
wall in radial and
longitudinal direction
Elasticity
Pressure
Pressure
Elasticity
Inertia
Computational Mechanics & Numerical Mathematics
University of Groningen
•
Generalized string model:
•
Navier equations:
Wall dynamics (2)
Elasticity
Pressure
Inertia
Damping
Shear
Elasticity
Pressure
Shear
Inertia
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
Computational Mechanics & Numerical Mathematics
University of Groningen
The mass

spring system compared to the
(simplified) Navier equations
•
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
Elasticity
Damping
Pressure
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
pressure
wall motion
Boundary conditions
Computational Mechanics & Numerical Mathematics
University of Groningen
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
No
Yes
Yes
Peripheral resistance
No
Yes
No
Yes
Computational Mechanics & Numerical Mathematics
University of Groningen
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
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