Inverse, in-vivo kinematic field calculation of abdominal aortic aneurysms Towards a novel method

105 Pages·2016·19.27 MB·English

Inverse, in-vivo kinematic field calculation
of abdominal aortic aneurysms
Towards a novel method for estimating rupture risk
Thesis submitted to the
Vásárhelyi Pál Doctoral School of Civil Engineering and Earth Sciences
in partial fulfillment of the requirements of
Doctor of Philosophy
in the field of
Civil Engineering
by
Róbert Nagy
supervised by
Imre Bojtár
Budapest University of Technology and Economics,
Department of Structural Mechanics
September 13, 2016
Contents
Declaration of Authenticity iv
Acknowledgment v
Abstract vi
Összefoglalás vii
List of Figures vii
1 Introduction 1
1.1 Research background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Research history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Current collaboration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Abdominal aneurysms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 AAA management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Discussion of the existing approaches . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Purpose of the research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.6 Research questions and improvements . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6.1 Adaptation of a new imaging modality . . . . . . . . . . . . . . . . . . . 7
1.6.2 Automatic image segmentation . . . . . . . . . . . . . . . . . . . . . . . 7
1.6.3 Spline representation of branching topologies . . . . . . . . . . . . . . . . 7
1.6.4 Spline surface fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6.5 Motion tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6.6 Finite element shell formulation . . . . . . . . . . . . . . . . . . . . . . . 9
i
CONTENTS ii
1.7 Outline of the text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 In-silico aneurysm model 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Experimental setup and image acquisition . . . . . . . . . . . . . . . . . . . . . 12
2.3 Branching pipeline representation . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Parameter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 Surface representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.3 Continuity at the irregular point . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Surface fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.1 Bilevel optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.2 Direct surface fitting algorithm . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 AAA Phantom geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6 Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6.1 Displacement by projection . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6.2 Comparison to FEM simulation . . . . . . . . . . . . . . . . . . . . . . . 36
2.7 Theses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 In-vivo strain measurement 43
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Image acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Parametric surface representation . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.1 Center line fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.2 Reference surface fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 Reference surface kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5.1 Auxiliary assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5.2 Isogeometric analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5.3 Reference surface motion function . . . . . . . . . . . . . . . . . . . . . . 52
3.6 Material point kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6.1 Cosserat surface with one director . . . . . . . . . . . . . . . . . . . . . . 53
3.6.2 Amendments to the Kirchhoff–Love formulation . . . . . . . . . . . . . . 56
3.7 Results of the AAA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
CONTENTS iii
3.7.1 Displacement, velocity, acceleration . . . . . . . . . . . . . . . . . . . . . 57
3.7.2 Strain field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.8 Comparison to a calcificated artery . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.9 Theses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 Further developments 64
4.1 Extension of the presented procedures . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 Load field determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3 Material parameter identification . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 Degree of vessel wall degeneration . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5 Experimental validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 Theses 67
6 Tézisek 71
A Spline interpolation I
A.1 Spline functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
A.1.1 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
A.1.2 Tensor product surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . II
A.2 Cubic Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III
Bibliography V
Publications of the author XIV
Pertaining to the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIV
Journal papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIV
Papers in edited book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV
Conference papers and presentations . . . . . . . . . . . . . . . . . . . . . . . . XV
Not pertaining to the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVIII
Journal papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVIII
Papers in edited book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVIII
Conference papers and presentations . . . . . . . . . . . . . . . . . . . . . . . XIX
Declaration of Authenticity
I declare that all material presented in this dissertation are either my own work or fully and
specifically acknowledged wherever adapted from other sources.
In Hungarian:
Kijelentem, hogy ezt a doktori értekezést magam készítettem és abban csak a megadott for-
rásokat használtam fel. Minden olyan részt, amelyet szó szerint vagy azonos tartalomban, de
átfogalmazva más forrásból átvettem, egyértelműen, a forrás megadásával megjelöltem.
Budapest, September 13, 2016
Róbert Nagy
iv
Acknowledgment
Iwouldliketoexpressmysinceregratitudetomysupervisor,Prof. ImreBojtár,aboveallforhis
patience, continuous support and guidance throughout my research in the field of biomechanics
he together with Dr. István Nyáry introduced me to.
I am indebted to my closest collaborators Dr. Péter Sótonyi, Dr. Csaba Csobay-Novák,
Attila Lovas, and Dr. Brigitta Szilágyi. They inspired a great part of my research with their
expertise and never-ceasing enthusiasm.
I also appreciate the help of Prof. Ender A. Finol for providing the raw data of the in-silico
measurement.
I would like to extend my gratitude to the whole staff of the Department of Structural
Mechanics especially to Dr. Zoltán Bocskai and Dr. Gergely Molnár for the inspirative con-
versations on the most diverse issues; to Prof Zsolt Gáspár for his insightful comments and his
guidance in research cases we did together; and to all the members I have been teaching and
working with.
Last but not least, I would like to thank my whole family for their unconditional support
and sacrifices, to Judit Szántó for her devoted kindness, endurance, and spiritual support she
is probably unaware of. It means a lot to me.
v
Abstract
As the initial step towards the long-term goal of my research to identify the in-vivo material
parameterdistributionoftheaortaformechanicallyaccurateaneurysmruptureriskassessment,
in this thesis work, I present the first milestone, the displacement field determination.
Ishowedthatthenewgenerationofelectrocardiography-gatedcomputedtomography(ECG-
gated CT) scanners is adaptable for capturing the aorta to determine its motion during a
stabilized cardiac cycle.
I adopted a reproducible method with minimal user interaction based on a novel Markov-
chainmodel, yieldingthepointcloudrepresentingthelumensurfaceinthemeasuredload-steps.
I developed a method to represent an arbitrary branching topology by a tensor product
spline surface augmented by interpolating blending functions around the extraordinary nodes
( C1), readily applicable to model the aneurysmatic aorta together with the neighboring iliac
∈
bifurcation and renal branches.
I presented a novel robust and highly efficient algorithm to fit a spline surface of arbitrary
topology to the measured error prone point cloud of a corresponding surface with the same
topology. It is a reformulation of the complex and resource-consuming bi-level optimization
problemtoasimpledirectiteration, whichalsoovercomestheissueofapossibleself-intersection
during the steps.
Assuming that the displacement of a reference surface point aligns with the current surface
normal vector, the ill-posed motion tracking problem becomes well defined. I showed that with
numerically increased load-stage resolution the accuracy of this procedure is sufficient for the
aneurysm model.
Completing these ingredients with a novel incompressible Kirchhoff–Love shell formulation,
Iconstructedthealgorithmenvisagedinthetitle, whichIvalidatedwithin-silicomeasurements.
Finally, I present the application and the results of two sample studies of (1) an aneurysmatic
and (2) an undilated atherosclerotic aorta segment.
vi
Összefoglalás
Doktori értekezésemben a kutatásom hosszútávú céljaként megfogalmazott hasi aneurizmák
felhasadási kockázatának mechanikailag megalapozott számítását lehetővé tevő inverz in-vivo
anyagparaméter azonosítás első lépését, a fal kinematikai állapotjelzőinek meghatározását mu-
tatom be.
Rávilágítottam, hogy a számítógépes tomográfia elektrokardiogram-kapuzott új változata
alkalmazható a nagy artériákon, lehetővé téve a fal mozgásának meghatározását az állandósult
szívciklus alatt.
Egységes rendszerré formáltam egy Markov-láncon alapuló szegmentáló eljárást, mely meg-
ismételhető, minimális felhasználói beavatkozást igényel és a kontrasztanyaggal megfestett vér-
áram külső felületét adja meg teherlépcsőnként egy-egy ponthalmaz formájában.
Példával illusztrált módszert adtam egy tetszőleges topológiájú elágazásos csőhálózat fe-
lületét leíró, deriváltjaival együtt folytonos függvény létrehozására. Ennek alapja a mindkét
változójában harmadfokú splinefelület, melyet lokálisan egy ötszög tartományon értelmezett
simító interpolációs függvények egészítenek ki a négytől különböző fokszámú pontok körül.
A térbeli felületillesztés többszintű optimalizációs problémáját egy közvetlen iterációvá re-
dukálva létrehoztam egy stabil (a felület összemetsződését kiküszöbölő és a kiindulási alakra az
eddigieknél kevésbé érzékeny) és hatékony numerikus eljárást, mely tetszőleges splinefüggvényt
illeszt egy a kiindulásival azonos topológiájú felületet leíró zajjal terhelt ponthalmazra.
Az érfal elmozdulásaira tett lokális megszorítással, a korrigált normalitás feltételezésével
határozottátettemazoptikaielmozdulásmérésalulhatározottproblémáját, ésmegállapítottam,
hogy ezáltal a mért felületek splinemegadás által lehetővé tett teherlépcső sűrítésével együtt
megfelelő pontosság érhető el az elmozdulások meghatározásában.
Ezen összetevőket kiegészítve az újonnan bevezetetett összenyomhatatlan Kirchhoff–Love-
héjleírással állítottam össze a címben foglalt módszert, melyet egy szilikon anyagú mintán
igazoltam, és két kiértékelt, valós esethez tartozó példán keresztül a dolgozatban be is mutatok.
vii
List of Figures
1.1 The five cornerstones of the research of in-vivo material parameter identification. 6
2.1 Experimental setup of the silicone abdominal aortic aneurysm phantom. . . . . . 13
2.2 The three-dimensional sample initial geometry for the surface fit with two bi-
furcations located on the main branch (extraordinary nodes of valence five and
elements surrounding them are colored green); and the corresponding parameter
space with the parameters of the discretization. . . . . . . . . . . . . . . . . . . 14
2.3 The three types of uniform tensor product bi-cubic spline nodal basis functions
necessary for a surface periodic in one direction and open, natural (vanishing
2nd and 3rd derivatives prescribed at the boundaries) in the other. The basis
functions are attached to a control point (0,0) a) at a general location, S ; b)
0
next to the boundary, S ; and c) at the boundary, S . . . . . . . . . . . . . . . 16
1 2
2.4 Control net topology around the irregular point . . . . . . . . . . . . . . . . . . 17
2.5 Pentagonal support of the blending function and the constructors of the local
coordinate system attached to it. . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6 Blue vicinity arrows stretch from the green sampling points of the fitting surface
to the closest measured point indicated by blue color among the brown others.
Only the sampling points corresponding to the middle control point are shown,
the red displacement vector of which is the weighted average of the blue vectors. 25
2.7 Direct algorithm for arbitrary spline surface fitting . . . . . . . . . . . . . . . . 26
2.8 Flowchart of the direct algorithm for arbitrary spline surface fitting . . . . . . . 27
2.9 Proof of robustness in two-dimensions with the general, a) and two degenerate,
b) and c) cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.10 Proof of robustness in three-dimensions . . . . . . . . . . . . . . . . . . . . . . . 30
viii
LIST OF FIGURES ix
2.11 Resolution of the measurement equipment presented through the histogram of
the distribution of the distance of each neighboring measured points . . . . . . . 32
2.12 Stepsofthesplinesurfacefit: a)topologygeneration; b)controlmeshsmoothing;
c) distance of the measured points from the initial surface; d) distance of the
measured points from the final surface; e) control mesh of the fitted surface. . . 33
2.13 The first 3 steps of the direct iteration showing the updated control net (left).
The convergence measures of the algorithm. Shaded regions indicate the fulfilled
convergence criteria (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.14 Mesh convergence study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.15 Spatial distribution of total displacements (from 0 to 150 Hgmm) in the case of
the projection algorithm (left) and the FE simulation (right) alongside with the
corresponding histograms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.16 Maximal displacement of the reference surface as the function of the internal
overpressure in the case of FE simulation and the projection algorithm. . . . . . 40
2.17 Simulated maximal displacements against the variation of the model parameters 40
3.1 Flowchart of the in-vivo strain measurement. . . . . . . . . . . . . . . . . . . . . 44
3.2 The geometry (left) is transformed so that the main axis (c(Z)) coincides with
the Z axis (center), the unfolded skin surface in parametric coordinates (right).
The color code represents the distance (r) of a point from the axis. The largest
dilatation occurs in the frontal part (ϕ = 0) of the aorta indicated by solid black
line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Natural and periodic cubic uniform univariate basis function constituting the
bi-cubic uniform spline surface. The parameter mesh and the P control point
ij
values are shown on the left, and an illustrative physical mesh on the right. . . . 48
3.4 Radial correction convergence of the smoothing surface (top left). Statistical
distribution in case of e = 4 mm (bottom left), where the shaded blue area lies
outside the voxel size ( d ) containing 4.756 10 3% of the sampling points,
v −
± ·
the red line shows the fitted normal distribution (µ = 0, σ = 0.0619). Spatial
distribution of the correction error (right). . . . . . . . . . . . . . . . . . . . . . 49