Transcript Document

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
• A model for the global cardiovascular
circulation outside the region of interest
(better boundary conditions)
Carotid bifurcation
Fluid dynamics
Wall dynamics
Global
Cardiovascular
Circulation
(electric network model)
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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:
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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
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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)
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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
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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
Flow rate Q
Voltage
Pressure P
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Flow in tubes
Compliance due to the elasticity of the wall
Qin
Qout
P, V
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
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C
Qout
Flow in tubes
Resistance due to fluid viscosity
Pin
Q
Pout
Pin: Inflow pressure
Pout: Outflow pressure
Q: Volume flux
• 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
Pout
R
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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
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Computational Mechanics & Numerical Mathematics
Q
Pout
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)
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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
Contraction
Filling
• Assume a linear ESPVR and EDPVR with slopes Ees and Eed and
unstressed volumes V0,es and V0,ed:
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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
C(t)
P
Qout
1/C’(t)
with
M(t)
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
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Computational Mechanics & Numerical Mathematics
Minimal electrical model
Simple ventricle model
Peripheral resistance
Carotid
Artery
Input resistance
Ventricle model
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Minimal electrical model
Heart valves modeled by diodes
Carotid
Artery
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Computational Mechanics & Numerical Mathematics
Minimal electrical model
Input/output compliance, resistance around ventricle
Carotid
Artery
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Computational Mechanics & Numerical Mathematics
Minimal electrical model
Compliance in peripheral element
Carotid
Artery
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Computational Mechanics & Numerical Mathematics
Minimal electrical model
Parallel systemic loop, internal/external carotid peripheral elements
Carotid
Bifurcation
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Computational Mechanics & Numerical Mathematics
Structure of the model
Carotid
Bifurcation
Red: Arterial compartments
Blue: Venous compartments
Green: Capillaries
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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
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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