Modelling of blood flow in aneurysm geometry

64 Pages·2015·1.84 MB·English

by  Tereza Zábojníková

Univerzita Karlova v Praze
Matematicko - fyzika´ln´ı fakulta
´ ´
DIPLOMOVA PRACE
Tereza Z´abojn´ıkov´a
Modelov´an´ı proudˇen´ı krve v geometrii
aneuryzma
Matematicky´ u´stav
Vedouc´ı diplomov´e pr´ace: RNDr. Jaroslav Hron, Ph.D.
Studijn´ı program: Fyzika
ˇ
Studijn´ı obor: Uˇcitelstv´ı fyziky pro SS v kombinaci s odbornou fyzikou
(Matematick´e modelov´an´ı ve fyzice a technice)
Praha 2015
Charles University in Prague
Faculty of Mathematics and Physics
MASTER THESIS
Tereza Z´abojn´ıkov´a
Modelling of blood flow in aneurysm
geometry
Mathematical Institute
Supervisor of the master thesis: RNDr. Jaroslav Hron, Ph.D.
Study programme: Physics
Specialization: Mathematical Modelling in Physics and Technology
Prague 2015
´
I thank to Miguel Angel Ferna´ndez, my consultant from Universit´e Pierre et
Marie Curie, Paris 6, who has spent a lot of time consulting my work during
my Erasmus in Paris and who has gave me the original idea of this thesis, my
supervisor Jaroslav Hron who helped me to finish my work started in Paris and
my parents who have supported me during my studies.
I declare that I carried out this master thesis independently, and only with the
cited sources, literature and other professional sources.
I understand that my work relates to the rights and obligations under the Act
No. 121/2000 Coll., the Copyright Act, as amended, in particular the fact that
the Charles University in Prague has the right to conclude a license agreement
on the use of this work as a school work pursuant to Section 60 paragraph 1 of
the Copyright Act.
In ...................... on ............. Author signature
Na´zev pra´ce: Modelova´n´ı proudˇen´ı krve v geometrii aneurysma
Autor: Tereza Za´bojn´ıkova´
Katedra: Matematicky´ u´stav UK
Vedouc´ı diplomov´e pra´ce: RNDr. Ing. Jaroslav Hron, Ph.D., Matematicky´ stav
UK
Abstrakt: C´ılem t´eto pra´ce bylo nal´ezt stabiln´ı sch´ema, kter´e by ˇreˇsilo Stokes˚uv
probl´em tekutiny, ve kter´e je ponoˇrena´ elasticka´ pevn´a la´tka. Narozd´ıl od vˇetˇsiny
sch´emat ˇreˇs´ıc´ıch interakci pevn´e l´atky s tekutinou, naˇse sch´ema nevyˇzaduje, aby
na sebe s´ıtˇe pevn´e la´tky a tekutiny navazovaly. Omezili jsme se na dvoudi-
menziona´ln´ı oblast pro tekutinu, ve kter´e je ponoˇrena jednodimenziona´ln´ı ela-
sticka´ pevn´a la´tka. Pro popis interakce jsme pouˇzili metodu vnoˇren´e hranice
(Immersed boundary method). Na zaˇca´tku jsme povaˇzovali pevnou la´tku za
nehmotnou. Upravili jsme jiˇz existuj´ıc´ı sch´ema ˇreˇs´ıc´ı takovy´to probl´em tak, aby
bylo nepodm´ınˇenˇe stabiln´ı, coˇz jsme matematicky doka´zali a numericky otesto-
vali. Pot´e jsme navrhli modifikaci sch´ematu tak, aby pevna´ la´tka jiˇz mˇela nˇejakou
hmotnost, a t´eˇz dok´azali jeho nepodm´ınˇenou stabilitu. Navrˇzen´a sch´emata jsme
implementovali v programu Freefem++ a otestovali jejich chova´n´ı na geometrii
podobn´eaneurysma. Vyzkouˇselijsmetak´echova´n´ınavrˇzeny´chsch´ematvpˇr´ıpadˇe,
kdy se rostouc´ı aneurysma dotkne pˇreka´ˇzky, napˇr´ıklad kosti (s no-slip podm´ınkou
na okraji).
Kl´ıˇcov´a slova: interakce tekutiny s pevnou l´atkou, metoda ponoˇren´e hranice,
proudˇen´ı krve
Title: Blood flow in geometry of aneurysm
Author: Tereza Za´bojn´ıkov´a
Department: Mathematical Institute of Charles University
Supervisor of the master thesis: RNDr. Jaroslav Hron, Ph.D., Mathematical
Institute of Charles University
Abstract: The aim of this work is to find a stable scheme which would solve
the Stokes problem of the fluid flow, in which an elastic structure is immersed.
Unlike most of the schemes solving fluid-structure interaction problems, in our
scheme meshes of fluid and structure do not have to coincide. We have restricted
ourselves to two-dimensional domain occupied by fluid with one-dimensional im-
mersed structure. To describe a fluid-structure interaction, we have used an
Immersed boundary method. At first we consider the strucure to be massless.
We have modified an existing scheme and made it unconditionally stable, which
was mathematically proven and numerically tested. Then we have proposed a
modification where the structure is not massless and also proved the uncondi-
tional stability in this case. The proposed schemes were implemented using the
Freefem++ software and tested on aneurysm-like geometry. We have tested the
behavior of our scheme in case when the qrowing aneurysm touches an obstacle,
for example a bone (with no-slip condition on the bone boundary).
The proposed schemes were implemented using the Freefem++ software and
tested on aneurysm-like geometry
Keywords: fluid-structure interaction, immersed boundary method, blood flow
Contents
1 Introduction 3
1.1 Aneurysms and computational hemodynamics . . . . . . . . . . . 3
1.2 Blood flow modelling . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Fluid-Structure interaction . . . . . . . . . . . . . . . . . . 6
1.2.2 Arbitrary Lagrangian Eulerian formulation . . . . . . . . . 6
1.2.3 Immersed boundary . . . . . . . . . . . . . . . . . . . . . . 7
2 Basic Scheme 9
2.1 Massless membrane . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Model with simplified elasticity . . . . . . . . . . . . . . . . . . . 11
2.3 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Finite elements discretisation . . . . . . . . . . . . . . . . . . . . 12
2.4.1 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.2 Pseudo-compressibility method . . . . . . . . . . . . . . . 14
2.4.3 Force-term adjustment . . . . . . . . . . . . . . . . . . . . 15
2.5 Time discretisation and linearization . . . . . . . . . . . . . . . . 16
3 Scheme Modification 18
3.1 Problem 5: Structure update modification . . . . . . . . . . . . . 18
3.2 Problem 6 with inertial part . . . . . . . . . . . . . . . . . . . . . 19
3.3 Stability of scheme 6 . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 Stability of scheme 5 . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.5 Membrane with fixed borders . . . . . . . . . . . . . . . . . . . . 24
4 Implementation 26
4.1 Force assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Fluid problem matrix definition . . . . . . . . . . . . . . . . . . . 27
4.3 Integration of inertial member . . . . . . . . . . . . . . . . . . . . 27
4.4 Algorithm Complexity . . . . . . . . . . . . . . . . . . . . . . . . 33
5 Numerical tests 34
5.1 Steady state - analytical solution . . . . . . . . . . . . . . . . . . 34
5.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3 Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.4 Unsteady state - Ellipse . . . . . . . . . . . . . . . . . . . . . . . 43
5.4.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.4.2 Comparison of geometries of results using problem 4-6 . . 44
5.5 Test - tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.6 Geometry of aneurysma . . . . . . . . . . . . . . . . . . . . . . . 48
1
6 Conclusion 52
Bibliography 54
List of Figures 56
List of Tables 58
2
Chapter 1
Introduction
1.1 Aneurysms and computational hemodynam-
ics
Themotivationtostudycomputationalhemodynamics(ascienceaboutblood
flow) is clear. Almost half of all deceases in Europe is caused by vascular deseases
and 2/3 of all dispenses of the public health care are spent because of them.
Figure 1.1: vascular deseases: aneurysme and atherosclerose [8]
In recent years, the blood flow modelling has matured greately. One reason is
the development of imaging techniques, the second one is the huge development
in the field of scientific computation (by that we mean the bigger performance of
the computational clusters and also agorithmic progress).
The aim is to develop a computational method which would be quick (in
some cases of vascular deseases the time is really important and we simply can’t
3
wait one month for the cluster to show us the patient-specific result), stable and
accurate. Once we would have it, we could scan the problematic place in the
patient’s vascular system, and the information about pressure and flow could
help us to decide what kind of treatment we should use.
For example in case of aneurysm, there are two possibilities how to exclude
the aneurysm from a vascular system and prevent it from bursting. We can either
fill the space of aneurysm by coil, or clip the neck of aneurysm by a steel clip to
seal it off from the parent vessel.
Figure 1.2: Two possibilities to treat aneurysms: clipping and coiling [1]
Decidingwhichtreatmentismoreconvenientinspecificcasecouldbesimlified
by running a simulation on patient-specific model.
Over the years not only the development of computers and clusters shortened
the time needed to obtain a result. In 11 years, thanks to various algorithmic
modifications, the computational time has reduced 50 times (solving the same
problem on the same computer, but with some algorithmic modifications)! (see
[8])
Figure 1.3: Progress of computational hemodynamics in INRIA, presented in
´
course of Ecole Polytechnique and Universit´e Paris 6 [8]
4