Download Un - KTH

Plats för
ev
logotype
Simulation of Blood Flow in the Left Ventricle
Using Moving Geometries Based on Ultrasound Measurements
INTRODUCTION
Cardiac illness is a major
cause of health problems
and the number one cause
of death in the world. Many
of the cardiac diseases lead
to deformations of the
geometry of the heart and
thereby also to changes in
the qualitative and
quantitative properties of the
blood flow in the heart. Our
goal is to apply numerical
simulation to study such
effects of geometry changes
on the blood flow in the left
ventricle (LV) of the heart.
The purpose of such studies
is to gain understanding of
the effects of different
pathologies and furthermore
to possibly study the effects
of treatment. Our approach
[1] is based on a complete
process including patient
specific measurements,
automatic geometry
modeling based on the
measurements, meshing
and simulation of the
Navier-Stokes equations.
From a patient specific ultra sound measurement to the geometric
model of the heart
In our approach the blood in the LV is driven by the prescribed movement of the
inner heart wall.
The model geometry of the LV is based on ultrasound measurements of the
position of the inner wall at three different levels at twelve specific time points
during the cardiac cycle, as the movement of the heart wall during one heart cycle
which lasts about one second is not uniform. From these measured points surface
meshes of the chamber are constructed for each sample time.
We then build a three dimensional mesh of tetrahedrons at the initial time, apply
Hermite interpolation to allocate the position of all the boundary nodes at every time
step and use a Laplace mesh smoothing algorithm to deform the mesh to fit the
given surface meshes during the cardiac cycle.
Results
Quantities which are interesting for medical examination and can as well be used
for the evaluation of the model are the mean velocity at the outflow and inflow and
the pressure.
Remember that the mean velocity at the inflow is mainly given by the prescribed
inflow profile and that the calculation of the discrete pressure corresponds to the
Poisson equation.
Figure 3a: Mean velocity at the inflow
calculated during the simulation
Figure 3b: Velocity at the inflow derived with
help of ultrasound measurements
Figure 4a: Mean velocity at the outflow
calculated during the simulation
Figure 4b: Velocity at the outflow derived with
help of ultrasound measurements
Figure 1: Specific points of time when the samples are taken for one heart cycle
Mathematical Model
We simulate the blood flow with the incompressible Navier-Stokes equations. To
take into account the movement of the mesh the Arbitrary Lagrangian-Eulerian
description [3] is used where u denotes the velocity, p the pressure and w the mesh
velocity:
u̇u−w ∇  u−  u∇ p= f
METHODOLOGY
We simulate the blood flow
with the incompressible
Navier-Stokes equations
using a weighted standard
Galerkin/streamline diffusion
method.
CONCLUSION
With our approach we are
able to calculate the blood
flow in the left ventricle
without the complex
mechanical and electrical
model of the heart wall.
The results for the validation
of the model are satisfying
and encouraging to
complete the model with
properties as the aortic
valves and a
Windkesselmodel. Different
collaborations have started
to further derive interesting
properties from the existing
data and to make the model
more applicable for various
fields.
∇⋅u=0
By defining different boundary conditions we divide the cardiac cycle into the basic
stages of diastole, systole, isovolumetric relaxation and isovolumetric contraction.
A no-slip boundary condition on the wall and closed valves is applied and the
pressure is prescribed by a standard curve to model the inflow through the mitral
valve and the outflow through the aortic valve. A flat inflow profile for the velocity is
given as well.
−3 kg
=2.7⋅10
Considering the size of the LV, the dynamic viscosity
and density
m⋅s
kg
=1.06⋅103 3 the Reynolds number is about 20'000.
m
Figure 5: Mean pressure in the LV
Software
The simulations are done in DOLFIN, a differential equation problem solving
environment, and UNICORN, a unified continuum mechanics solver, which are
developed as a part of the software project FEniCS and where we are an active
contributor.
See https://launchpad.net/fenics and https://launchpad.net/unicorn
Current and future work
Our present efforts as shown in Fig. 5 are focused on parallelizing our code for
making it possible to run the calculation on multi-cores [4].
Further, as the next major step in our project we would like to extend and improve
our model by including the aortic valves. To secure, develop and make our model
applicable the collaboration, discussions, dialogue and interaction with other
universities, institutes and research areas are carried out and striven for.
Figure 2:Snapshot of the pressure and velocity during systole
CONTACT INFORMATION
Jeannette Spühler
Johan Hoffman
Johan Jansson
Computational Technology Laboratory
School of Computer Science and Communication
Royal Institute of Technology KTH
(spuhler, jhoffman, jjan)@csc.kth.se
Ulf Gustafsson
Department of Public Health and
Clinical Medicine
Umeå University
[email protected]
Mikael Brommé
School of Technology and Health
Royal Institute of Technology KTH
and
ECMO Center, Karolinska University
Hospital
Karolinska Institutet
[email protected]
Mats Larsson
Per Vesterlund
Computational Mathematics Laboratory
Department of Mathematics and
Mathematical Statistics
Umeå University
(mats.larson,per.vesterlund)@math.umu.se
Jonas Forslund
Eva-Lotta Sallnäs Pysander
Human-Computer Interaction
Royal Institute of Technology KTH
(evalotta, [email protected])
Alex Olwal
School of Computer Science and Communication
Royal Institute of Technology KTH
[email protected])
Galerkin finite element method with least-squares stabilization
We apply the G2 cG(1)cG(1) Finite Element Method with piecewise continuous
linear solution in time and space for solving the governing equation [2].
For the standard FEM formulation of the model we only have stability
of the discrete solution U but not of its spatial derivatives. This means the solution
can be oscillatory, causing inefficiency by introducing unnecessary
error. We therefore choose a weighted least-squares stabilized Galerkin method.
The weak solution to be solved on the computational mesh is thus as following:
Find the discrete solution U =[U , P ] such that
n
U −U
n−1
−1
n
n
 k  U −W
n−1
n
n
⋅∇ U , v  2   U  , v
n
n
n
n
n
n
− P , ∇ v ∇⋅U , qSD  U , P ; v , q= f , v  ∀ v , q∈V 0 ×Z
and the stabilization term is defined as:
n
n
n
SD  U , P ; v , q=1  U −W
n−1
n
n
n
⋅∇ U ∇ P − f  ,U −W
n−1
n
n
n−1
where [v , q ] are test functions and U =1/ 2 U U  .
n
⋅∇ v∇ q2 ∇⋅U , ∇⋅v
Figure 5: Simulation run on several cores
Figure 6: The interactive HEART demonstrator
in collaboration with CSC/HCI within SimVisInt
[1] M. Aechtner, Arbitrary Lagrangian-Eulerian Finite Element Modelling of the Human Heart},
Master thesis, {TRITA-CSC-E} 2009:022, 2009.
[2] J. Hoffman and C. Johnson, Computational Turbulent Incompressible Flow: Applied
Mathematics Body and Soul Vol 4, Springer-Verlag Publishing, 2006.
[3] J. Donea, A. Huerta, J.-Ph. Ponthot and A. Rodríguez-Ferran Arbitrary Lagrangian-Eulerian
Methods, p. 413-437, Wiley, 2004.
[4] N. Jansson, J. Hoffman and J. Jansson, Parallel Adaptive FEM CFD,
KTH-CTL-4008, http://www.publ.kth.se/trita/ctl-4/008/, 2010.