Incorporating accurate statistical modeling in PET reconstruction for whole-body imaging

228 Pages·2007·2.07 MB·English

UNIVERSIDADE DE LISBOA
FACULDADE DE CIÊNCIAS
DEPARTAMENTO DE FÍSICA

Incorporating accurate statistical modeling in PET
reconstruction for whole-body imaging

Luis Manuel de Almeida Soares Janeiro

Thesis supervised by Professor Doutor Pedro Almeida, Faculdade de Ciências
da Universidade de Lisboa, and by Doctor Claude Comtat, Service Hospitalier
Frédéric Joliot, C.E.A., Orsay, France

Doutoramento em Biofísica
2007

Para a minha família

Table of contents i

TABLE OF CONTENTS

Table of contents ............................................................................................................. i
Acknowledgments ………………………………………………………………………… v
Abstract ……………………………………………………………………………………. vii
Resumo ……………………………………………………………………………………. ix
Abbreviations …………………………………………………………………………….... xv

Introduction …………………………………………………………………………..…… 1

1 – An introduction to Positron Emission Tomography (PET) ………………………... 5
1.1 – A brief history of PET …………………………………………....………………. 6
1.2 – Positron Emission Tomography overview ………………………………………... 8
1.2.1 – Physical constraints and the three-gamma annihilation …...……………….... 10
1.2.1.1 – Positron range ………………………………………………………. 10
1.2.1.2 – Annihilation photons non-colinearity ………..………………………. 12
1.2.1.3 – Three photons positron annihilation ………………………………... 13
1.2.2 – Signal detection in PET ….………………………………………………… 14
2321.2F.2i.l1 – DdBetecktors j……i ………………………………………………………. 14
1.2.2.2 – Photomultiplier Tubes (PMTs) ……………………………………… 17
1.2.2.3 –Block detector readout ……………………………………………….. 18
1.3 – Data acquisition …………………………………………………………………... 19
1.3.1 – 2D and 3D mode for data acquisition ……………………………………… 20
1.3.2 – Radial sampling ……………………………………………………………. 21
1.3.3 – The Michelogram ………………………………………………………….. 22
1.3.4 – Sinograms and projections …………………………………………………. 24
1.3.5 – Detected events in PET ……………………………………………………. 26
1.4 – Data corrections ………………………………………………………………….. 28
1.4.1 – Attenuation correction ……………………………………………………... 30
1.4.2 – Correction for random coincidences ……………………………………….. 32
1.4.3 – Dead time correction ………………………………………………………. 33
1.4.4 – Normalization …………………………………………………………….... 34
1.4.4.1 – Effects contributing to differences in the sensitivity ………………… 35
1.4.4.2 – Normalization procedures ………………………………………….... 36
ii Table of contents

1.4.5 – Scatter correction …………………………………………………………... 37
1.5 – PET camera performance ……………………………………………………….... 40
1.5.1 – Spatial resolution …………………………………………………………... 40
1.5.2 – Energy resolution ………………………………………………………….. 43
1.5.3 – Sensitivity ………………………………………………………………….. 44
1.5.4 – Noise Equivalent Count (NEC) ……………………………………………. 45

2 – Image reconstruction algorithms in PET ………………………………………..….. 47
2.1 – The Radon transform in PET …………………………………………………….. 48
2.2 – A primary distinction between reconstruction algorithms ………………………… 50
2.3 – Analytical reconstruction …………………………………………………………. 52
2.3.1 – Direct Fourier Methods ……………………………………………………. 53
22.33.22 – FFiillteredd BBackkprojjectiion …………………………………………………….. 55
2.3.3 – The 3DRP algorithm ………………………………………………………. 58
2.4 – Algebraic reconstruction ………………………………………………………….. 61
2.4.1 – Five components of an algebraic reconstruction method …………………... 64
2.4.1.1 – A finite parameterization of the image ………………………………. 64
2.4.1.2 – System matrix ……………………………………………………….. 65
2.4.1.3 – A model of the measurement uncertainty …………………………..... 67
2.4.1.4 – Objective function …………………………………………………... 70
2.4.1.5 – Numerical algorithm ………………………………………………… 72
2.4.2 – Algebraic Reconstruction Technique (ART) ……………………………….. 74
2.4.3 - Maximum Likelihood – Expectation Maximization (ML-EM) ……………… 76
2.4.4 – Ordered Subsets – Expectation Maximization (OSEM) ……………………. 81
2.4.4.1 – Block-iterative methods ……………………………………………... 85
2.4.4.2 – Raw-action methods ………………………………………………… 86
2.4.5 – Bayesian reconstruction ……………………………………………………. 87
2.5 – Rebinning algorithms ……………………………………………………………... 92
2.5.1 – FORE (Fourier Rebinning) ………………………………………….……... 93
2.5.2 – Other rebinning techniques ………………………………………………... 96

3 – The weighted OSEM approach and the adopted implementation …..…………….. 98
3.1 – Weighted OSEM …………...…………………………………………………….. 99
3.1.1 – NEC transform and NEC-OSEM …………….…………………………… 102
3.2 – The effects of FORE on the data variance …………………………………..……. 104
3.3 – The adopted OSEM implementation ……………………………………………... 105
232 Fil dB k j i
Table of contents iii

4 – Numerical observer studies comparing different OSEM approaches for 3D whole-
body PET ……………………………………………………..………………………. 110
4.1 – Data simulation …...………...…………………………………………………….. 111
4.1.1 – The anthropomorphic phantom …………....……….……………………… 111
4.1.2 – The analytical simualator (ASIM) ……………………………………..……. 112
4.1.3 – Simulaion conditions ……………………………...………………………... 114
2324.1F.3i.l1 – CdaBlculkationj ofi the slice variance reduction factors ……………….…... 119
4.2 – Image reconstruction ……………………...…………………………………….... 120
4.3 – Image quality assessment ……………………...………………………………….. 1239
4.3.1 – NPWMF observer study …………………………………………………… 123
4.3.2 – Study of contrast versus noise …………………………………………….... 125
4.3.3 – Methodology to fix the amount of post-smoothing and the number of
iterations …………………………………………………………………. 125
4.4 – Results …………………………………………………………………………… 126
4.4.1 – Preliminary results for fixing the amount of post-smoothing and the number
of iterations ………………………………………………………………... 126
4.4.2 – Study of contrast versus noise …………………………………………….... 130
4.4.3 – NPWMF observer study …………………………………………………… 132
4.5 – Discussion ………………………………………………………………….…….. 136

5 – Comparing NEC-based OSEM with other weighted OSEM approaches for
clinical 3D whole-body PET imaging ……………………………………………….. 139
5.1 – Data acquisition and description ………………….…...………………………….. 141
5.2 – Data pre-processing ……..…………...…………....……….……………………… 141
5.3 – Data corrections and the estimation of the mean and the variance ……………….. 144
5.3.1 – The estimation of the mean ………………………………………………... 145
5.3.2 – The estimation of the variance ……………………………………………... 147
5.4 – FORE and the weighting factors …………………………………………………. 152
5.5 – Image reconstruction and final results ……………………………………………. 156
5.5.1 – Image reconstruction ………………………………………………………. 156
5.5.2 – Reconstructed images ……………………………………………………… 157
5.6 – Discussion and conclusions …….……………………………………………….... 161

Final discussion and conclusion ………………………………….……………………… 163

Appendix
A.1 – Coordinate systems is 2D and 3D ……………….……………………………….. 171
iv Table of contents

A.1.1 – 2D case ………………………………………….………………………… 171
A.1.2 – 3D case ……………………………………………………………………. 171
A.2 – The Radon Transform …...………………………………………………………. 174
A.3 –2 T3h2e BFacilkprodjeBctiokn Opjeraitor ...………………………………………………….. 176
A.4 – Central Section Theorem ...………………………………………………………. 177
A.5 – The Likelihood Function ……..………………………………………………….. 180
A.6 – A reconstruction techniques diagram …………………………………………….. 182
A.7 – The Non-Prewhitening Matched Filter (NPWMF) numerical observer …………... 183
A.8 – The EXACT HR+ PET scanner ………………………………………………… 185

List of figures ……………………………………………………………………………… 189
List of tables ………………………………………………………………………………. 193
Bibliography ………………………………………………………………………………. 195
Acknowledgments v

Acknowledgments

My first word goes to the teaching staff of the Institute of Biophysics and Biomedical
Engineering (IBEB), Faculty of Sciences of the University of Lisbon. Prof. Pedro Miranda,
Prof. Eduardo Ducla Soares and Prof. Alexandre Andrade have always been generous, attentive,
patient and I have always had their helpful comments and suggestions whenever they were
pertinent. Their doors were always opened and this gave me additional motivation and
responsibility. I also do not forget their personal support and how important it was.
I am also grateful to all my colleagues of the IBEB, for the mood within the Institute.
This is priceless and makes IBEB a perfect place for working and studying. In special, I would
like to thank Nuno Matela and Mónica Martins for all the work we have developed together. We
began together and we had many fruitful discussions along the journey. Technical discussions,
of course, but not solely… I thank them for their support, their comments, their generosity,
their altruism and, specially, for their personal support. I felt it.
My gratitude goes also to the administrative staff of the Institute. Beatriz and Ana were
always there. They know each one of us, and have a special way to handle, and solve, our
problems. For those who are abroad, this is, again, priceless… and most of the times they were
the first persons I met when I was back to the Institute. Their support was enormous.
I would also like to acknowledge my colleagues and all the persons with whom I had the
priviledge to work with at the Service Hospitalier Frédéric Joliot (SHFJ), namely Doctor Régine
Trébossen, Charles Pautrot, Frédéric Bataille and Sébastian Jan, for their comments and helpful
suggestions. I am specially in debt to Doctor Maria João Ribeiro, for all the support and for the
kindness of having made possible the study we have done with real clinical data. Her expertness
in the field was essential.
For all the comments, insights and shared knowledge, I am also in debt to Prof. Paul
Kinahan, of the University of Washington, in Seattle. My gratitude to Prof. Paul Kinahan is
extended for having accepted, and partially financed, my stay in Seattle, within the Imaging
Research Laboratory. I felt home many thousand miles away and I met an extraordinary group
of persons. Thank you to Adam Alessio for the fruitful discussions and his very clear way of
exposing ideas.
vi Acknowledgments

My gratitude to Doctor Fernando Tomé and Doctor Maria Leonor Arsénio Nunes is
enormous: for their personal support, their experience, their knowledge and culture. In Paris
their presence was essential and I grew up with their example.
Doctor Claude Comtat was my supervisor in Orsay. I have no words to acknowledge
him for everything. From the timeless discussions in Paris, no matter where – and many times
dans “une terrasse”! - to the complete support at the SHFJ. Their critical comments,
suggestions, perspectives and solutions have always astonished me. And still do. Many things
I’ve learned, I’ve learned from him. I still keep the schematics he drew to make things easier,
and everything matches. I remember to rewrite some texts many times, because there was
always a better and clear way to put the things. Fortunately, I also keep some old versions, so I
can compare. He was perfectly right! In any case, I have always believed that Claude would be
there in case I would be unable to find the solution. And I was right.
He also introduced me many thing in Paris. We shared opinions about many, many
different things. He often told me: “take your time”. And how necessary it was! From a
personal point of view, I had his entire support. My debt to Doctor Claude Comtat is complete.
Finally, and most of all, I would like to thank Doctor Pedro Almeida. Once was the
Master, now is the PhD. He has always been my supervisor. In every situation I have always felt
his support. The most important thing, nevertheless, was the motivation he gave me. It was
fundamental and, probably, the key for concluding all this work. He knew exactly what I was
doing at any time, and helped me to solve each and every problem I had. I am not only referring
to technical problems. I am, above all, referring to other more essential questions we have
deeply discussed. I believe that, for reaching the end, technical and scientific hindrances pose
minor problems and difficulties.
Fortunately, or in consequence, our friendship goes very beyond the scope of this work.
I admire his example, and I hope to follow it.

This PhD work was made possible by the support of the fellowship
BD/SFRH/4989/2001 from Fundação para a Ciência e Tecnologia (FCT). I would like to
clearly state my gratitude to FCT.