Title: Multi-scale modeling of the carotid artery
1Multi-scale modeling of the carotid artery
G. Rozema, A.E.P. Veldman, N.M.
Maurits University of Groningen, University
Medical Center Groningen The Netherlands
2Area 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
3Multi-scale modeling of the carotid artery
Several submodels of different length- and
timescales
Carotid bifurcation
- 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)
Fluid dynamics
Wall dynamics
4Computational 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
5Modeling 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
6Boundary 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)
Outflow
Outflow
Inflow
7Coupling 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
pressure
wall motion
Wall dynamics ODE
Boundary conditions
Global Cardiovascular Circulation ODE
8Global cardiovascular circulation model
Carotid Bifurcation
Electric Hydraulic
Current Flow rate Q
Voltage Pressure P
9Flow in tubes Compliance due to the elasticity
of the wall
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
10Flow in tubesResistance due to fluid viscosity
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
11Flow in tubesResistance due to inertia
Pin Inflow pressure Pout Outflow pressure Q
Volume flux
- Consider inviscous potential flow (flat velocity
profile) - Conservation of momentum is given by (Newtons
law) - L Resistance due to inertia (mass)
- Electric analog inductor
- Q Current, P Voltage
Q
Pin
Pout
12The ventricle modelElastic 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 Pressure in sphere V Volume of sphere V0
Unstressed volume
P, V
13Clinical applicationParameterization 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 Ees
and Eed and unstressed volumes V0,es and V0,ed
Ejection
Relaxation
Contraction
Filling
14Clinical applicationParameterization 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)
15Clinical applicationParameterization 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 Ees
and Eed - M(t) Voltage generator, can be left out when
assuming V0,es V0,ed 0
16Minimal electrical modelSimple ventricle model
Peripheral resistance
Carotid Artery
Input resistance
Ventricle model
17Minimal electrical modelHeart valves modeled by
diodes
Carotid Artery
18Minimal electrical modelInput/output compliance,
resistance around ventricle
Carotid Artery
19Minimal electrical modelCompliance in peripheral
element
Carotid Artery
20Minimal electrical modelParallel systemic loop,
internal/external carotid peripheral elements
Carotid Bifurcation
21Structure of the model
Red Arterial compartments Blue Venous
compartments Green Capillaries
Carotid Bifurcation
22Simulation example
- A simulation is performed to see if the model can
capture global physiological flow properties - Parameter values are not yet realistic
Simulated flow rate for two cycles
23Simulation 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
24Future 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?