Novel Reconstruction and Quantification Methods for Oxygen-17 Magnetic Resonance Imaging at Clinical Field Strengths

Inaugural-Dissertation zur Erlangung des Doktorgrades der Fakultät für Mathematik und Physik der Albert-Ludwigs-Universität Freiburg im Breisgau

vorgelegt von M.Sc. Dmitry Kurzhunov aus Kasan, Russland Juli 2017

This thesis is dedicated to my beloved parents: to my father, a physicist by profession and vocation, who opened the fascinating world of physics to me, guided me and helped me develop my potentials; and to my mother, who always supported me in my endeavors and motivated me to become happy and successful.

Посвящается моим любимым родителям: моему папе, физику по профессии и по призванию, который открыл для меня захватывающий мир физики, а также направлял меня и помогал мне в раскрытии и развитии моего потенциала; и моей маме, которая всегда поддерживала меня в моих начинаниях и мотивировала мои стремления к счастью и успеху.

Everybody has a capacity for a happy life Lev Landau

Supervisor: Prof. Dr. Michael Bock Referee: Prof. Dr. Günter Reiter Examiners: Prof. Dr. Jens Timmer Prof. Dr. Oskar von der Lühe Examination day 19.09.1989

ABSTRACT

Oxygen metabolism, which is altered by many neurodegenerative diseases such as Alzheimer’s disease or Parkinson’s disease and in brain tumor regions, is quantified by the cerebral metabolic rate of oxygen consumption (CMRO2). Positron emission tomography (PET) with the oxygen 15 isotope O is considered the gold standard for CMRO2 mapping in humans; however, it is rarely used due to the short isotope half-life of only 2 min, which requires costly on-site production. An alternative method is a direct dynamic 17O magnetic resonance imaging (MRI) with inhalation of 17 17 17 isotopically enriched O gas. In O MRI, signal changes from H2 O molecules are observed during and after inhalation of 17O gas, and the pharmacokinetic model is applied to quantify 17 CMRO2. CMRO2 measurements with O MRI are challenging due to low natural abundance of 17 * O isotope of 0.037 %, fast effective transversal relaxation (T2 ), and low signal-to-noise ratio (SNR) of 17O magnetic resonance (MR) images. Thus, it was previously only possible at ultra-high field (UHF; B0 ≥ 4 T) MRI systems. The aim of this work is to develop novel reconstruction, 17 quantification, and simulation methods to enable CMRO2 measurement with O MRI in a clinical 3 T MRI system.

First, a flexible simulation framework with two types of phantoms (analytical brain tumor and numerical brain phantoms) was developed to find optimal imaging parameters such as readout bandwidth, and spatial (훥푥) and temporal (훥푡) resolutions. Optimal acquisition parameters found in the simulations were consistent with those previously used in 17O MRI at UHF. Therefore, 150 ≤ BW≤ 175 Hz/pixel, 8.0 ≤ 훥푥 ≤ 10 mm, and 훥푡 = 60 sec were used in four 17O MRI experiments with 17O gas inhalation. In these experiments, a rebreathing circuit which allows for re- inhalation of the stored 17O gas in subsequent inhalation cycles was implemented for efficient usage of rare and expensive 17O gas.

To analyze the influence of the different model parameters on the identifiability of CMRO2, a profile likelihood (PL) analysis was performed for different settings of the model parameters. In particular, the 17O enrichment fraction of the inhaled 17O gas, 훼, was investigated assuming constant and linearly varying models. Identifiability was analyzed for white matter (WM) and grey matter (GM) brain regions separately, and the dependency on different priors was studied. Prior knowledge about only one 훼-related parameter was sufficient to resolve the CMRO2 non- identifiability, and the CMRO2 values (0.72 – 0.99 µmol/gtissue/min in WM, 1.02 – 1.78 15 µmol/gtissue/min in GM) were in good agreement with the results of O-PET studies. In particular, the proposed advanced quantification model with non-constant α values significantly improved model fitting. The profile likelihood analysis showed that CMRO2 can be measured reliably in a 17O gas MRI experiment if the 17O enrichment fraction is used as prior information for the model calculations.

A new method called DIrect Estimation of 17O ImageS (DIESIS) was suggested to correct for partial volume effects (PVEs), which are present in 17O MR images due low spatial resolution of * 8 to 10 mm and blurring from fast T2 decay of about 2 ms. DIESIS avoids full reconstruction of

Abstract

MR images and directly estimates the 17O MR images from the measured data based on parcellation (i.e., segmentation of the 3D image to a predefined number of volumes) of the 3D 1H MR image. It is an alternative to conventional Kaiser-Bessel (KB) gridding method, where 17O signal has to be averaged over large brain regions to get the SNR sufficient for CMRO2 quantification 17 by the model fitting of the O signal-time curves. DIESIS provided correction of PVEs: CMRO2 decreased by 6 – 19% in WM and increased by 29 – 46 % in GM compared to KB gridding, thus 15 getting close to the CMRO2 values from O-PET studies.

To test the feasibility of pixel-wise CMRO2 quantification (mapping) in a clinical 3 T MRI system, a new iterative reconstruction was proposed, which uses the edge information contained in a coregistered 1H MR image to construct a non-homogeneous anisotropic diffusion (AD) filter. AD- constrained reconstruction of 17O MR images was compared to conventional KB with and without Hanning filtering, and to iterative reconstruction with a total variation constraint. AD- constrained reconstruction provided 17O images with improved resolution of fine brain structures both in the numerical brain phantom and in two in vivo data sets of one healthy volunteer, and it resulted in higher SNR and provided CMRO2 maps, which were comparable with maps acquired at 9.4 T.

The results of this work show feasibility of 17O MRI in clinical 3 T MRI systems and provide a 17 solid basis for clinical translation of O MRI for non-invasive CMRO2 quantification in tumor patients.

ZUSAMMENFASSUNG

Hirntumore und neurodegenerative Erkrankungen wie z.B. Alzheimer oder Parkinson zeigen einen veränderten Sauerstoffmetabolismus. Die Positronenemissionstomographie (15O-PET) mit dem Sauerstoffisotop 15O stellt den Goldstandard zur ortsaufgelösten Bestimmung der zerebralen 15 Sauerstoffumsatzrate (CMRO2) im Menschen dar. Aufgrund der kurzen Halbwertszeit von O (ca. 2 min) muss das Radiopharmakon vor Ort mit einem Zyklotron produziert werden, so dass die Verwendung von 15O-PET heute nur auf wenige Forschungszentren beschränkt ist. Eine alternative Methode zu PET bietet die Magnetresonanztomographie (MRT) mit dem nicht radioak- 17 17 17 tiven Sauerstoffisotop O. Bei der O-MRT werden die Signaländerungen von H2 O detektiert, welches nach der Gabe von angereichertem 17O-Gas im Zuge des Sauerstoffmetabolismus bzw. der oxidativen Phosphorylierung entsteht. Durch Anpassen eines pharmakokinetischen Modells 17 an den H2 O-Signalverlauf erhält man abschließend die CMRO2-Werte. Aufgrund der niedrigen natürlichen Häufigkeit des 17O-Isotops von 0,037% und dessen schneller effektiver transversaler * Relaxation (T2 ) ist das Signal-Rausch-Verhältnis (engl., SNR) klein und somit die Sensitivität der 17O-MRT im Vergleich zur 1H-MRT um fünf Größenordnungen geringer. Deswegen wurde 17O-

MRT bisher nur an MR-Tomographen mit ultrahohen Magnetfeldern (B0 ≥ 4 T) durchgeführt.

Das Ziel dieser Arbeit ist die Entwicklung eines Simulationsframeworks sowie neuartiger Quanti- fizierungsmethoden und Rekonstruktionsalgorithmen, um CMRO2 verlässlich an einem klinischen 3 T MR-Tomographen bestimmen zu können.

Zur Bestimmung der optimalen Aufnahmebandbreite (BW) sowie der zeitlichen (훥푡) und räumli- chen (훥푥) Auflösung wurde zunächst ein flexibles Simulationsframework entwickelt, bestehend aus zwei unterschiedlichen Phantomtypen (einem analytischen Hirntumorphantom und einem numerischen Hirnphantom). Die optimierten Simulationsparameter stimmten mit den Parame- tern der Ultrahoch-Messungen überein, daher wurden für alle vier durchgeführten 17O- Inhalationsexperimenten die folgenden Parameter gewählt: 150 ≤ BW ≤ 175 Hz/Pixel, 8,0 ≤ 훥푥 ≤ 10 mm und 훥푡 = 60 s, unter Verwendung des entwickelten Rückatemsystems.

Um den Einfluss unterschiedlicher Modelparameter auf die Identifizierbarkeit des CMRO2 zu untersuchen, wurde die Profile-Likelihood-Analyse für unterschiedliche Parametereinstellungen durchgeführt, wobei der 17O-Anreicherungsgrad 훼 in zwei unterschiedlichen Modellen als kon- stant bzw. linear-verändert angenommen wurde. In Abhängigkeit unterschiedlichen Vorwissens wurde die Identifizierbarkeit der CMRO2-Bestimmung in grauen (GM) und weißen (WM) Hirn- regionen evaluiert. Es wurde gezeigt, dass Vorwissen von nur einem 훼-abhängigen Parameters notwendig ist, um CMRO2 verlässlich zu bestimmen. Die quantifizierten CMRO2-Raten

(0,72 – 0,99 µmol/gtissue/min im WM, 1,02 – 1,78 µmol/gtissue/min im GM) waren in guter Über- einstimmung mit den Ergebnissen von 15O-PET Untersuchungen. Dabei zeigte das Modell die beste Übereinstimmung mit den gemessenen Werten, bei dem ein nichtkonstanter Verlauf von 훼 angenommen wurde. Die Profile-Likelihood-Analyse hat gezeigt, dass der CMRO2 mit Hilfe von 17O-MRT Inhalationsmessungen zuverlässig bestimmt werden kann.

iii

Zusammenfassung

Zur Korrektur von Partialvolumeneffekten in den 17O-MR-Bildern, die aufgrund der geringen * räumlichen Auflösung von 8 bzw. 10 mm und kurzer T2 -Zeiten von ca. 2 ms vorhanden sind, wurde die neuartige Methode DIrect Estimation of 17O ImageS (DIESIS) implementiert. Bei dieser Methode erhält man die 17O-MR-Bilder direkt aus dem gemessenen 17O MR Signal und mit Hilfe der Parzellierung von 3D 1H-MR-Bildern. Dabei stellt DIESIS eine Alternative zur konven- tionellen Kaiser-Bessel (KB) Gridding-Methode dar, bei welcher das 17O-Signal für eine verlässli- che CMRO2-Quantifizierung über große Hirnregionen gemittelt werden muss. Mit DIESIS waren die CMRO2-Werte im Vergleich zu der KB-Gridding-Methode in der WM um 6 % bis19 % ver- mindert, und in der GM um 29 % bis 46 % erhöht, womit eine noch bessere Übereinstimmung 15 mit CMRO2-Werten aus O-PET-Studien erreicht und die Partialvolumeneffekte korrigiert wurden.

Zur pixelweisen (ortaufgelösten) Darstellung des CMRO2, wurde ein neuer iterativer Rekonstruk- tionsalgorithmus entwickelt, welcher die Kanteninformation koregistrierte 1H-MR-Bilder ver- wendet, um einen inhomogenen anisotropen Diffusionsfilter (AD) zu konstruieren. Diese AD- beschränkte Rekonstruktion von 17O-MR-Bildern wurde mit herkömmlichem KB-Gridding mit und ohne Hanning-Filterung und mit einer iterativen Rekonstruktion mit einer Totalvariationsbe- schränkung verglichen. Für das numerische Hirnphantom sowie für zwei in vivo-Datensätzen eines gesunden Probanden lieferte die AD-beschränkte Rekonstruktion eine bessere Auflösung feiner Gehirnstrukturen. Zudem wurde CMRO2-Karten mit verbessertem SNR erzielt, welche mit Karten vorheriger Messungen bei 9,4 T vergleichbar sind.

17 Die Ergebnisse dieser Arbeit zeigen die prinzipielle Durchführbarkeit der O-MRT für CMRO2- Quantifizierung an einem klinischen 3 T MR-Tomographen und bieten eine solide Basis für die nicht-invasive Messung des Sauerstoffmetabolismus bei Tumorpatienten.

TABLE OF CONTENTS

1. INTRODUCTION ...... 1 2. THEORY AND BACKGROUND ...... 5

2.1. Nuclear magnetic resonance ...... 5

2.1.1. Nuclear spin and magnetic moment ...... 5 2.1.2. Quadrupole interactions ...... 7 2.1.3. Macroscopic magnetization ...... 9 2.1.4. Motion of the magnetization ...... 10 2.1.5. Effect of alternating magnetic fields ...... 10 2.1.6. Relaxation of nuclear magnetization ...... 12 2.1.7. Relaxation of quadrupolar nuclei ...... 13 2.1.8. NMR signal detection and sensitivity ...... 14

2.2. Magnetic resonance imaging ...... 15

2.2.1. Spatial encoding ...... 15 2.2.2. Infinite discrete k-space sampling ...... 17 2.2.3. Discrete finite k-space sampling ...... 18 2.2.4. Spatial resolution ...... 19 2.2.5. MRI pulse sequences ...... 20 2.2.6. Signal-to-noise ratio ...... 22

2.3. Respiration and cellular metabolism ...... 23

2.3.1. External respiration ...... 23 2.3.2. Internal respiration and cellular metabolism ...... 24

2.4. Quantification of the cerebral metabolic rate of oxygen consumption ...... 27

2.4.1. Fick principle ...... 27 15 2.4.2. CMRO2 quantification with O-PET ...... 28 1 2.4.3. CMRO2 quantification with H MRI ...... 28 17 2.4.4. CMRO2 quantification with O MRI...... 30 2.4.5. Pharmacokinetic model for 17O MRI ...... 32 2.4.6. Profile likelihood ...... 34

3. MATERIAL AND METHODS ...... 37

3.1. Experimental setup ...... 37

Table of contents

3.1.1. MRI system ...... 37 3.1.2. 17O gas breathing setup ...... 39 3.1.3. 17O MRI inhalation experiments ...... 40

3.2. Simulation framework ...... 42

3.2.1. ACROBAT phantom ...... 42 3.2.2. Numerical phantom ...... 45 3.2.3. MR reconstruction quality assessment ...... 45 3.2.4. Dynamic framework ...... 47

3.3. Image acquisition ...... 48

1 3.3.1. H MRI: T1-weighted MPRAGE sequence ...... 48 3.3.2. 17O MRI: density adapted acquisition ...... 48 3.3.3. 17O MRI: 3D radial sampling ...... 50

3.4. Image reconstruction ...... 52

3.4.1. Kaiser-Bessel gridding ...... 52 3.4.2. DIrect Estimation of 17O ImageS ...... 55 3.4.3. Iterative reconstruction ...... 57 3.4.4. Anisotropic diffusion constraint ...... 58 3.4.5. Conjugate gradient algorithm ...... 59 3.4.6. Comparison of MR reconstruction methods ...... 60

3.5. Advances in CMRO2 quantification analysis ...... 61

3.5.1. Extraction of the 17O signal-time curves ...... 61 3.5.2. Advanced pharmacokinetic model...... 61 3.5.3. Profile likelihood analysis ...... 63

3.5.4. 3D CMRO2 mapping ...... 63

4. RESULTS ...... 65

4.1. Optimization of acquisition parameters with 17O MRI phantoms ...... 65

4.1.1. ACROBAT phantom ...... 65 4.1.2. Numerical phantom ...... 66

4.2. CMRO2 quantification and profile likelihood analysis ...... 67 4.3. DIrect Estimation of 17O ImageS ...... 73

4.4. Comparison of reconstruction methods and 3D CMRO2 mapping ...... 75 vi

Table of contents

4.4.1. Reconstruction comparison: phantom ...... 75 4.4.2. Reconstruction comparison: in vivo data ...... 75

4.4.3. In vivo CMRO2 mapping ...... 79

5. DISCUSSION ...... 83

5.1. Optimization of acquisition parameters with 17O MRI phantoms ...... 83

5.2. Profile likelihood analysis of the CMRO2 pharmacokinetic model ...... 83 5.3. Comparison of MR images from different reconstruction methods ...... 86

5.4. In vivo CMRO2 quantification ...... 87 5.5. DIrect Estimation of 17O ImageS ...... 88

5.6. In vivo CMRO2 mapping ...... 89

6. CONCLUSIONS AND OUTLOOK ...... 91 REFERENCES ...... 95 LIST OF SYMBOLS ...... 107 LIST OF ABBREVIATIONS ...... 111 LIST OF PUBLICATIONS ...... 113

Journal publications ...... 113 Conference Contributions ...... 113 Awards ...... 115

ACKNOWLEDGEMENTS ...... 117

vii

viii

1. INTRODUCTION

Magnetic resonance imaging (MRI) is a powerful medical imaging technique for non-invasive investigation of the human brain and body. MRI is used for detection, diagnosis, and treatment monitoring of various diseases, in particular cancer and stroke, because it offers high spatial resolution and an excellent soft-tissue contrast. A significant advantage of MRI over other imaging modalities such as computed tomography (CT) or positron emission tomography (PET) is that it uses neither ionizing radiation nor radioactive materials. Thus, MRI examinations do not pose a hazard to patients and healthcare workers, which is especially important for patients who require long-term monitoring after treatment. Moreover, MRI is widely used in neuroscience and clinical research, as the lack of side effects allows experiments on healthy volunteers. Instead of providing only a single contrast, as for example in x-ray imaging, MRI allows mapping of diverse physical and functional properties, including proton density, tissue relaxation, diffusion, perfusion, blood flow, or brain activation.

MRI is recognized as one of the most important advances in healthcare technology in the last century. It emerged from several fundamental discoveries in physics in the 20th century and expe- rienced an immense advancement in the last four decades. Development of quantum theory in the beginning of last century was useful for understanding of absorption and emission spectra of atoms (Bohr, 1913a, 1913b, 1913c), Nobel Prize in Physics awarded to Niels Bohr in 1922). These spectra had discrete lines which could be quantitatively described only by quantum physics. In the early 1920s several experimental studies led to an establishment of concepts of electron spin and the magnetic moment of electrons. In particular, in a Stern-Gerlach experiment, the beams of silver atoms were separated in an inhomogeneous magnetic field, according to orienta- tion of the electron magnetic moment, which confirmed the basis of quantum theory (Gerlach and Stern, 1922). With increased sensitivity of the molecular beam apparatus, a beam of hydrogen molecules was detected in the Stern-Gerlach experiment in 1933, which allowed detection of a nuclear magnetic moment (Estermann and Stern, 1933; Frisch and Stern, 1933). For these discoveries the Nobel Prize in Physics was awarded to Otto Stern in 1943.

The first observation of the nuclear magnetic resonance (NMR) was carried out in 1938 by Isidor I. Rabi under high vacuum in a molecular beam, where the deflection of the beam served to detect the resonance (Rabi et al., 1938). In this experiment, a stream of hydrogen molecules was sent through a uniform magnetic field, where they were exposed to radio frequency (RF) electromagnetic energy. This energy was absorbed by the molecules at a specific frequency yielding a measurable deflection of the beam. Yevgeny Zavoisky, who discovered electron paramagnetic resonance (Zavoisky, 1945), observed NMR signal in solids in 1941, but he could not detect a reproducible signal, because the strict requirement of the homogeneous magnetic field was not met (Altshuler and Kozyrev, 1971). In 1946 Felix Bloch detected stable NMR signals in water. In his experiments, after RF excitation, the precessing magnetization induced electrical signals in the detection coil (Bloch, 1946). Soon thereafter, Edward M. Purcell measured the absorption of the RF energy by the proton magnetic moment in a bulk sample of paraffin (Purcell et al., 1946). The 1

Introduction

Nobel Prize in Physics was awarded to Isidor I. Rabi in 1944 for recording the magnetic properties of atomic nuclei, and to Felix Bloch and Edward M. Purcell for the discovery of NMR phe- nomena in 1952. In 1950, Erwin L. Hahn proposed a modified approach of measuring the NMR signal based on finite RF pulses (Hahn, 1950), which has become the basis for many NMR and MRI applications. Further development of the physics of NMR, electromagnetic technology, and advanced electronics boosted NMR studies in chemistry, biology, and material science.

The initial concept of the medical application of NMR dates back to 1971, when Raymond Damadian proposed that magnetic resonance (MR) relaxation times could be used to distinguish cancer from healthy tissue (Damadian, 1971). In 1973 Paul Lauterbur suggested using magnetic field gradients to differentiate NMR signals originating from different locations (Lauterbur, 1973). Multiple projections, acquired along different gradient directions were combined to reconstruct a two-dimensional image. Shortly afterwards, Peter Mansfield and his colleagues introduced critical methods for efficient image generation, such as slice selective RF excitation (Garroway et al., 1974) and fast acquisition schemes, where 2D images could be obtained in a few tens of milli- seconds (Mansfield and Maudsley, 1977). In 1975 Richard Ernst showed that 2D and 3D MRI images can be obtained by applying magnetic field gradient during MR signal acquisition and using the Fourier transform methods for MR image reconstruction (Ernst and Anderson, 1966; Kumar et al., 1975). For these discoveries concerning MRI, Richard Ernst was awarded the 1991 Nobel Prize in Chemistry, and Paul Lauterbur and Peter Mansfield were jointly awarded the No- bel Prize in Physiology or Medicine in 2003. The MR technology was rapidly developing in 1980s; in particular due to invention of fast MR imaging methods (Haase et al., 1986; Hennig et al., 1986; Mugler and Brookeman, 1990) and application of superconducting magnets in MRI systems that enabled critical improvements in image resolution for medical diagnosis. Finally, MRI has become routine in medical practice with more than 30,000 operational MRI systems world- wide and more than 100 million examinations performed every year (Hayden and Nacher, 2016).

In clinical routine, today only proton (1H) MRI is used. There are several reasons for this: 1H has the highest gyromagnetic ratio among all stable isotopes, a natural abundance of 100%, and the highest concentration in the human body. The vast majority of the 1H MR signal comes from the hydrogen nuclei of water molecules, which make up about 50 – 65% / 75% of the total body / brain mass. As a result, both signal-to-noise ratio (SNR) and spatial resolution of 1H MRI are high. However, other nuclei with a nonzero nuclear spin (e.g., 23Na, 13C, 31P, or 17O), called X- nuclei, are present in the human body. The feasibility of clinical X-nuclear MRI is limited by the lower gyromagnetic ratio, the lower concentration and the lower natural abundance of these nuclei. However, X-nuclei MRI could provide important functional and molecular information of living tissue that cannot be obtained with 1H MRI, such as cell viability, energy balance, or oxygen metabolism.

The oxygen metabolism is altered by many neurodegenerative diseases such as Alzheimer’s disease, Parkinson’s disease, and Huntington's disease (Beal, 1992; Frackowiak et al., 1988; Fukuyama et al., 1994; Ishii et al., 1996; Maurer et al., 2001, 2000; Wallace, 1992), or in brain tu-

Introduction

mors (Ito et al., 1982; Thulborn et al., 1999; Tyler et al., 1987). For a quantitative analysis of the metabolism of these diseases, an imaging method would be desirable that can map the cerebral 15 metabolic rate of oxygen consumption (CMRO2). PET with the oxygen isotope O is considered the gold standard for CMRO2 mapping (Fukuyama et al., 1994; Ishii et al., 1996; Ito et al., 1982; Leenders et al., 1990; Lehnert et al., 2012; Mintun et al., 1984; Ter-Pogossian et al., 1970); however, it is rarely used due to the short isotope half-life of only 2 min, which requires costly on-site 1 production. As an alternative, several indirect methods for CMRO2 imaging with H MRI have been proposed (Bolar et al., 2011; Davis et al., 1998; Jain et al., 2010; Lu and Ge, 2008; Mellon et al., 2010; Reddy et al., 1995; Regatte et al., 2003; Xu et al., 2009). Yet, the detection of the MR 17 accessible stable oxygen isotope O is preferable for CMRO2 quantification as it can directly de- 17 tect the metabolic end product H2 O. To calculate metabolic rates of oxygen consumption in 17 17 humans, the O MRI signal changes from H2 O molecules are observed during and after inhala- 17 tion of isotope-enriched O gas, and the pharmacokinetic model is applied to quantify CMRO2 (Atkinson and Thulborn, 2010; Hoffmann et al., 2011).

17O MR spectroscopy (MRS) and MRI were originally performed mostly in small animals (Lu et al., 2013; Zhang et al., 2004; Zhu and Chen, 2011; Zhu et al., 2002, 2001) and at ultra-high fields 17 (UHF; B0 ≥ 4 T). UHFs are advantageous for O MRI as they partly compensate for the low MR sensitivity of the 17O isotope which is only about 1.1·10-5 of 1H due to the low natural abundance of 17O nucleus of 0.037% and approximately sevenfold lower gyromagnetic ratio.

In 2010, a pharmacokinetic model for CMRO2 quantification in humans was proposed, which allowed CMRO2 mapping of human brain at 9.4 T (Atkinson and Thulborn, 2010). Later, a pulsed 17O gas delivery system combined with a rebreathing (RB) circuit was suggested for inhalation experiments at 7 T to quantify CMRO2 in large white matter (WM) and gray matter (GM) regions (Hoffmann et al., 2011). Dynamic 17O MRI at 7 T was then used in a glioblastoma tumor patient to assess the difference in oxygen metabolization in the necrotic center, the contrast- enhancing rim and perifocal edema regions of the tumor (Hoffmann et al., 2014).

There are several limitations of the previous in vivo 17O MRI studies which prevent their wide 17 clinical implementation. First, CMRO2 quantification with O MRI was never done in clinical MRI systems - UHF MRI systems are not widely available and are not yet used in clinical routine.

Second, CMRO2 identifiability (i.e., how well, if at all, particular model parameter can be determined, which depends on choice of model and amount of experimental data) was not shown; the necessary amount of assumptions about the other model parameters and their effect on the pre- 17 cision of CMRO2 were not analyzed. Third, the high cost of the enriched O gas is one of the major limitations of 17O MRI. The 17O inhalation experiment and the pharmacokinetic model were not optimized to save rare and costly 17O gas. Fourth, only conventional gridding image reconstruction method was used, which do not provide sufficient SNR for pixel-wise CMRO2 quantification. Moreover, a homogeneous Hanning filter that was always applied to the measured MR data to increase SNR caused additional partial volume effects (PVEs) that were already present due to fast transverse relaxation of 17O nucleus.

Introduction

The aim of this work is to develop novel simulation, reconstruction and quantification methods 17 to enable CMRO2 measurements with O MRI in a clinical 3 T MRI system. For this, the above mentioned limitations were addressed as following:

1) A simulation framework was developed to optimize the acquisition parameters, and two 17O MRI brain phantoms, analytical and numerical, were included in the framework. The numerical phantom was also used for quantitative comparison of the MR reconstruction methods.

2) A new method called DIrect Estimation of 17O ImageS (DIESIS) was suggested to correct for PVEs. DIESIS avoids full reconstruction of MR images and directly estimates the 17O MR images based on parcellation (segmentation of the 3D image to a predefined number of volumes) of a co-registered 1H MR image.

3) To determine CMRO2 identifiability and its confidence intervals in a nonlinear CMRO2 quantification model, the method of profile likelihood (PL) was used (Raue et al., 2015, 2009). Because the this method does not require an analytical solution of differential equa-

tions for the CMRO2 values, a more complex and realistic input function was suggested for the 17O enrichment fraction. Based on the parameter profiles of the amended pharmacokinetic model, the amount of prior information is analyzed that is crucial for the identifiability

of CMRO2. In addition, the dependence of the CMRO2 uncertainty on the certainty of the prior information was calculated.

4) To enable pixel-wise 3D CMRO2 mapping in a human brain, an iterative reconstruction technique is proposed with a proton-based constraint from co-registered 1H MR data of higher spatial resolution. The edge information contained in a 1H gradient image was used to construct a non-homogeneous anisotropic diffusion (AD) filter. AD-constrained reconstruction method were compared to conventional Kaiser-Bessel (KB) gridding and iterative reconstruction with the total variation (TV) constraint, which has been successfully used in image restoration and iterative reconstructions. Reconstructions were compared in a numerical 17O MRI brain phantom and two dynamic in vivo 17O MR data sets.

This thesis begins with theoretical and background information, which is important for understanding of the work (Chapter 2). In Chapter 3 the performed 17O MRI inhalation experiments, the proposed MR image reconstruction methods, analysis of the advanced pharmacokinetic model, and simulation framework are described. The results are presented in Chapter 4 and discussed in Chapter 5. At the end, the major finding of the work are summarized in Chapter 6, where also limitations and further prospective of 17O MRI are addressed.

2. THEORY AND BACKGROUND

This chapter includes theoretical and background information that is relevant for this work and essential for understanding the utilized methods and their underlying principles. At first, physical and mathematical basics of the nuclear NMR are introduced and are specified for the case of 17O nucleus. It is followed by the description of the MRI, which is used to produce anatomical and functional images of the brain and body. In the third section, the basics of respiration and cellular oxygen metabolism are described, and the biomarker of brain oxygenation, CMRO2, is introduced. In the fourth section, several imaging modalities, such as 15O-PET, 1H MRI and 17O MRI, which have been used for CMRO2 quantification, are presented. Finally, the method of PL analy- 17 sis is presented, which was used for modelling of the O MR signal change for CMRO2 quantification.

2.1. Nuclear magnetic resonance

In NMR, atomic nuclei in a static magnetic field 푩ퟎ absorb and emit electromagnetic energy at specific resonance frequencies. For this to happen, the nuclei with an odd number of protons and/or neutrons must be exposed to a static external magnetic field. Then, an additional time- varying magnetic field 푩ퟏ must be applied at the resonance frequency of the isotope perpendicular to the static field. In this case, the magnetization rotates away from its alignment along the static magnetic field into the perpendicular plane where it can be detected. The resonance frequency depends on the strength of the external magnetic field and the magnetic properties of the nucleus. The frequency, amplitude and the time evolution of the MR signal also depend on the local environment of the nucleus, providing valuable information about the structure of various compounds as well as biochemical reactions and pathways.

This section starts with the introduction of the nuclear spin, nuclear magnetic moment and their interactions (Subsections 2.1.1 and 2.1.2), followed by the description of the macroscopic magnetization (Subsection 2.1.3), its precessional motion (Subsection 2.1.4) and the effect of a per- turbing RF field applied perpendicular to the static field (Subsection 2.1.5). Then, relaxation properties of the nuclear magnetization are discussed (Subsection 2.1.6), including the influence of quadrupole interactions (Subsection 2.1.7), which are relevant for the isotopes with a nuclear spin higher than ½. Finally, Subsection 2.1.8 deals with the basics of NMR signal detection and MR sensitivity of nuclei. More detailed information can be found in (Abragam, 1961), (Slichter, 1989) and (Haacke et al., 1999); the theoretical background of 17O NMR is presented in (Gerothanassis, 2010a, 2010b).

2.1.1. Nuclear spin and magnetic moment

Atomic nuclei are made up of protons and neutrons (collectively called nucleons) bound together by the residual strong force (nuclear force). While positively charged protons repel each other,

2.1 Nuclear magnetic resonance

electrically neutral neutrons stabilize the nucleus in two ways. First, their presence increases the distance between the protons, hence reducing electrostatic repulsion between the protons. Sec- ond, they provide additional attractive nuclear force to bind all the nucleons that is stronger than the repulsive Coulomb force between protons at short distances (less than about 1.7·10-15 m).

Protons, neutrons as well as any composite particle made of an odd number of these subatomic particles, such as many atoms and nuclei, are called fermions. Isotopes that contain an odd number of protons and/or neutrons have a nonzero vector sum of all nuclear spins and, hence, the isotope carries a net nuclear spin 푰 (or what is often referred to as intrinsic angular momentum), which is associated with the nuclear magnetic moment

푔퐼휇푛 흁 = 훾푰 = 푰. (2.1) 퐈 ℏ The proportionality constant 훾 is called the gyromagnetic ratio 훾. It is a characteristic constant for each nucleus and determines their interaction with magnetic fields (Table 2.1). The gyromagnetic ratio is quantified by the dimensionless proportionality constant called g-factor 푔퐼, the reduced Planck constant ℏ and the nuclear magneton 휇 : 푛 푒ℏ −27 2 휇푛 = = 5.05 ∙ 10 Am , (2.2) 2푚푝 where 푒 is the elementary charge and 푚푝 is the proton rest mass.

The nuclear spin obeys commutation relations analogous to those of the orbital angular momentum

[퐼̂푖, 퐼̂푗] = 휖푖푗푘 ∙ 푖ℏ퐼̂푘, (2.3) 2 [푰̂ , 퐼̂푖] = 0, (2.4) where the Levi-Civita symbol 휖푖푗푘 represents the sign of a permutation of the natural numbers.If the z-axis is a quantification axis, the eigenvectors of the operators 푰̂2 and 퐼̂ are expressed as 푧 푰̂2|퐼,푚⟩ = 퐼(퐼 + 1) ∙ ℏퟐ|퐼,푚⟩, (2.5)

퐼̂푧|퐼,푚⟩ = 푚푠ℏ|퐼,푚⟩, (2.6) with 2퐼 + 1 values

푚 = −퐼, −퐼 + 1, … , +퐼. (2.7) The eigenvalues 퐼, which are half-integer or integer numbers, are termed spin quantum numbers, and 푚 is referred to magnetic quantum numbers. Both of them characterize the orthogonal ei- genfunctions |퐼,푚⟩. In the field-free space, the eigenstate |퐼,푚⟩ does not depend on 푚 and, thus, is (2퐼 + 1)-fold degenerated. In the presence of a static external magnetic field 푩ퟎ the energy levels split (Zeeman effect). In this case the Hamiltonian, which is the operator corresponding to the total energy of the system, can be expressed as the sum of the unperturbed Hamiltonian 퐻̂0 and the perturbation due to the magnetic field 퐻̂ : 푍 퐻̂ = 퐻̂0 + 퐻̂푍, (2.8)

2. Theory and background

Gyromagnetic Natural abun- Relative MR Spin, 푰 ratio, 휸⁄ퟐ흅 dance, % sensitivity 1H 1/2 42.58 99.99 1.00 13C 1/2 10.71 1.11 1.70∙10-4 17O 5/2 -5.77 0.037 1.08∙10-5 19F 1/2 40.08 100 0.83 23Na 3/2 11.27 100 9.27∙10-2 31P 1/2 17.25 100 6.65∙10-2

Table 2.1: NMR properties and natural abundance of NMR detectable nuclei present in human body (Harris et al., 2002). MR sensitivity is relative to 1H isotope and was calculated from equation (2.44).

with

퐻̂푍 = −흁퐈푩ퟎ = −훾푰̂푩ퟎ. (2.9)

If the static magnetic field is applied along the z-direction (푩ퟎ = (0,0, 퐵0)), the equation (2.9) simplifies to

퐻̂푍 = −훾퐼̂푧퐵0. (2.10)

The energy levels 퐸푚 in the external magnetic field can be obtained using the equation (2.10) and the time-independent Schrödinger equation

퐻̂푍|퐼,푚⟩ = 퐸푚|퐼,푚⟩, (2.11) which results in

퐸푚 = −훾푚ℏ퐵0. (2.12) In this case neighboring energy levels are equidistant with the difference in the secondary spin quantum number of ∆푚 = ±1 and the energy difference of

∆퐸 = 훾ℏ퐵0 = ℏ휔0. (2.13)

Here, 휔0 is the Larmor frequency that corresponds to this transition energy. In the presence of a static external magnetic field of 3 T the ground state of the proton (퐼1퐻 = 1/2, 훾 = 2π·42.58 MHz) splits into two levels with the Larmor frequency of 127.7 MHz. In contrast to the proton, the oxygen isotope 17O, which consist of 8 protons and 9 neutrons, has a spin of 5/2, thus, the ground state splits into six levels with the transition frequency of 16.7 MHz, as shown in Figure 2.1. Furthermore, nuclei with 퐼 > ½ possess a nuclear electric quadrupole moment, which causes the quadrupole interaction which acts as a second-order perturbation to the Zeeman interaction.

2.1.2. Quadrupole interactions

The quadrupole interaction is the electrostatic interaction between the nuclear quadrupolar electric moment 푄, which arises from the non-spherical nuclear charge distribution, and the electric

2.1 Nuclear magnetic resonance

Figure 2.1. The Zeeman splitting of the energy levels of 1H (a) and 17O (b) nuclei. The allowed energy transitions are illustrated by two sided arrows. In the presence of a static external magnetic field the (2I + 1)-fold degradation is removed by the Zeeman effect. It is interesting to note that the transition energy ħw0 of 1H is 7.4 times higher than that of 17O as the gyromagnetic ratio of 1H is 7.4 higher compared to that of 17O.

field gradients which result from the particular distribution of the electrons around the nucleus (Lucken, 1969). The Hamiltonian associated with this interaction is given by

1 퐻̂ = ∑ 퐸퐹퐺 ∙ 푄 , 푄 6 푘,푗 푘,푗 (2.14) 푘,푗 where the indices k and j refer to the x-, y- and z-axes, 퐸퐹퐺푘,푗 is the electric field-gradient (EFG) tensor, and the nuclear quadrupole moment 퐸퐹퐺푘,푗 for 푁푛푢푐 nucleons, which are either protons or neutrons, is given by

푁푛푢푐 2 푄푘,푗 = 푒 ∑ (3푥푘,푛 ∙ 푥푗,푛 − 푟푛 ), (2.15) 푛 where 푟푛 is the distance of the nth nucleons from the center of the atomic nucleus.

Transformation into the principal axis system is often useful. In this case, the EFG tensor is di- agonal with components 퐸퐹퐺푥푥, 퐸퐹퐺푦푦 and 퐸퐹퐺푧푧, which satisfy the Laplace's equation 퐸퐹퐺푥푥 + 퐸퐹퐺푦푦 + 퐸퐹퐺푧푧 = 0. Moreover, the EFG tensor can be described by two independent parameters. The first is the main component of the EFG tensor

푒푞 = 퐸퐹퐺푧푧, (2.16) and the second is the asymmetry parameter component

퐸퐹퐺푥푥 + 퐸퐹퐺푦푦 퐸퐹퐺푥푥 + 퐸퐹퐺푦푦 휂 = = . (2.17) 퐸퐹퐺푧푧 푒푞 The quadrupole interaction is mainly caused by intramolecular interactions. The external magnetic field is typically large enough (퐻̂푍 ≫ 퐻̂푄) that the quadrupolar interactions can be treated as a

2. Theory and background

perturbation of the Zeeman energy levels. Thus, 퐻̂푄 can be added as an additional term to the Hamiltonian of the nucleus introduced in the equation (2.8)

퐻̂ = 퐻̂0 + 퐻̂푍 + 퐻̂푄. (2.18) The quadrupole interaction causes the 푚2-dependent shift of 2퐼 + 1 energy levels. This means that the level with the quantum number +푚 has the same energy shift with respect to the ground energy as the level with the quantum number −푚. In case of the matter with lattice structure (휂 = 0) it leads to 2퐼 resonance lines and the central resonance line remains unchanged, while the other lines are symmetrically shifted with respect to it. For isotropic fluids the quadrupole interaction is averaged out and only a single resonance line is observed.

2.1.3. Macroscopic magnetization

In NMR and MRI applications not a single spin but spin ensembles are considered, and signals from macroscopic samples is detected. Consisting of particles with half-integer spins, such spin ensembles obey the Fermi-Dirac statistics, which describes a distribution of particles over energy states. In the Fermi-Dirac statistics the system consists of many identical particles that obey the Pauli exclusion principle, i.e. two or more identical fermions cannot simultaneously occupy the same quantum state. When a sample is placed in a static magnetic field 퐵0 = 3 T at room temperature (푇 = 300 K), the high temperature limit 푘퐵푇 ≫ 훾ℏ퐵0 is valid and the Fermi-Dirac distribution reduces to the Boltzmann distribution, which describes the distribution of spins among the available energy levels 퐸 according to 푚 퐸 훾푚ħ퐵 1 − 푚 1 − 0 푝 = 푒 푘퐵푇 = 푒 푘퐵푇 , (2.19) 푚 푍 푍 where 푝푚 is the probability of finding the particle in the state with energy 퐸푚, 푘퐵 is the Boltz- mann’s constant, and 푍 is the partition function given by

퐼 퐸 − 푚 푍 = ∑ 푒 푘퐵푇. (2.20) 푚=−퐼 The above distribution shows that the states with lower energy will always have a higher probability of being occupied than the states with higher energy. The ratio of Boltzmann distributions for two states 푠푡_훼 and 푠푡_훽 of a spin ½ nucleus is known as the Boltzmann factor, which only depends on the energy difference ∆퐸 = 훾ℏ퐵 : 0 훾ℏ퐵 푁푠푡_훼 − 0 = 푒 푘퐵푇 , (2.21) 푁푠푡_훽 where 푁푠푡_훼 and 푁푠푡_훽 denote the occupation of the spin states 푠푡_훼 and 푠푡_훽. At high temperature limit any two levels are therefore nearly equally populated and the first order approximation can be used in the Taylor expansion of the equation (2.19). In this case, the difference in occupation numbers is

2.1 Nuclear magnetic resonance

∆푁 |푁푠푡_훼 − 푁푠푡_훽| 훾ℏ퐵0 = ≈ , (2.22) 푁푠푡_훼+푠푡_훽 푁푠푡_훼+푠푡_훽 2푘퐵푇

17 where 푁푠푡_훼+푠푡_훽 = 푁푠푡_훼 + 푁푠푡_훽 is the total number of spins. In case of O nucleus at 3 T, only about one out of 106 spins contributes to the population difference between two levels, and, thus, to the detected signal.

The macroscopic magnetization 푴 of the ensemble with 푁푠 spins is given by the sum of the expectation values of the magnetic moment 푁 푁 퐼 푴 = ∑〈흁̂푖〉 = ∑ 훾〈푰̂푖〉 = 푁푠 ∑ 푝푚훾ℏ푚. (2.23) 푖=1 푖=1 푚=−퐼 Only the components that are aligned with the external magnetic field contribute to the macroscopic magnetization. In general, when particles have an arbitrary spin and not only ½, the magnetization within a volume element 푉 follows Curie's law in the high temperature limit, with

2 2 푀 푁푠훾 ℏ 퐼(퐼 + 1) = 퐵0. (2.24) 푉 3푘퐵푇

This means that the magnetization density (푀/푉) is proportional to the number of the spins 푁푠, 17 the external magnetic field 퐵0 and the square of the gyromagnetic ratio 훾. O NMR sensitivity is described later in Subsection 2.1.8.

2.1.4. Motion of the magnetization

In the Heisenberg picture of quantum mechanics, the time evolution of expectation values of the nuclear magnetic moment is given by

푑 푖 〈흁̂ 〉 = − 〈[흁̂ , 퐻̂]〉, (2.25) 푑푡 푖 ℏ 푖 where 퐻̂ is the Hamiltonian and [•,•] denotes the commutator of two operators. Using the commutation relations from the equations (2.3) and (2.4), the time evolution of the magnetization introduced in the equation (2.23) is given by

푑 푴 = 훾푴 × 푩. (2.26) 푑푡

In the presence of a static external magnetic field 푩ퟎ = (0,0, 퐵0) the magnetization tends to align with the magnetic field 푴 ∥ 푩ퟎ, which is the state of minimum magnetic potential energy. The magnetization will rotate, or precess, with the Larmor frequency 휔0 = 훾퐵 around the z-axis if it is displaced out of its equilibrium alignment by an external energy injection coming from a second magnetic field perpendicular to 푩ퟎ as described in the following subsection.

2.1.5. Effect of alternating magnetic fields

The Boltzmann distribution at thermal equilibrium (equations (2.19) - (2.22)) can be perturbed if an energy quantum of the RF field, a photon, induces a transition from the state with lower ener- 10

2. Theory and background

Figure 2.2. Time evolution of the magnetization vector (yellow) in the rotating system of coordinates. In resonance (ωRF = ω0), the magnetization Mres sees only B1 applied in the x’-direction and rotates along the green ellipse around B1. In the non-resonance case, the magnetization vector Mnonres rotates on the blue cone around the effective magnetic field Beff.

gy into the state with higher energy. The projection of the spin of the nucleus on the z-axis 퐼̂푧, which is defined by increments at intervals of ℏ (equation (2.6)), needs to be compensated by the spin angular momentum of light carried by a photon. Only photons propagating along the z-axis and that have a negative helicity (left-hand circular polarization) will be absorbed, because they have the required projection of the spin on the z-axis of ℏ. Since the magnetic and electric fields are perpendicular to the direction of propagation of the photon, the alternating magnetic field

푩ퟏ(푡) has to be perpendicular to the static magnetic field 푩ퟎ for any absorption to take place. The behavior of the macroscopic magnetization can be explained by classical physics according to the correspondence principle that states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers (i.e. quantum calculations must agree with classical calculations). In particular, the magnetization is rotated by applying 푩ퟏ(푡) perpendicular to 푩ퟎ = (0,0, 퐵0), as given by

퐵1,푥(푡) 퐵1cos(휔RF푡) 푩ퟏ(푡) = [퐵1,푦(푡)] = [퐵1sin(휔RF푡)]. (2.27) 0 0 Thus, two external magnetic fields contribute to the effective magnetic field (Figure 2.2)

푩풆풇풇 = 푩ퟎ + 푩ퟏ. (2.28) The time evolution of the magnetization from the equation (2.26) is then given by

푑 푴 = 훾푴 × (퐵 cos(휔 푡), 퐵 sin(휔 푡), 퐵 )푇 (2.29) 푑푡 1 RF 1 RF 0

2.1 Nuclear magnetic resonance

Equation (2.29) can be simplified if the system of coordinates rotating with 휔RF around z-axis is considered (x’, y’, z’: z’ = z, x’ ∥ 푩ퟏ). In this case the time-dependent component of the RF excitation vanishes

푇 푑 휔RF 푴′ = 훾푴 × (퐵 , 0, 퐵 − ) . (2.30) 푑푡 1 0 훾 At resonance, the frequency of the alternating field is equal to the Larmor frequency, which is the natural resonance frequency of a spin system, and the z-component of 푩 disappears and 풆풇풇 휔RF = 휔0 = 2휋푓0. (2.31)

17 For a O nucleus at 퐵0 = 3 T the Larmor frequency has the value of 푓0 = 16.7 MHz. The time evolution of the magnetization at resonance is given by

′ 푇 푀 (푡) = 푀0(0, sin(훾퐵1푡) , cos(훾퐵1푡)) . (2.32) This corresponds to a rotation of the magnetization around the x’-axis in the rotating system with the resonance frequency 휔RF (Figure 2.2). In particular, an RF excitation of a duration 휏 causes rotation of the magnetization by an angle 휃푀, which is called the flip angle, defined by

휏 휃푀 = 훾 ∫ 퐵1(푡)푑푡 = 훾휏퐵1. (2.33) 0 퐵1 = 푐표푛푠푡 For example, a 90° pulse is required to rotate the magnetization from its alignment along the static magnetic field into the transverse (x’-y’) plane.

2.1.6. Relaxation of nuclear magnetization

Various interactions of nuclear spins with each other and with their micro-environment cause the excited spin ensemble to return back to the thermal equilibrium. This process is described phe- nomenologically by two relaxation times. The spin-lattice relaxation time 푇1 describes the rate at which the macroscopic magnetization relaxes back in the z-direction and distributes its excess energy to the local environment. The spin-spin relaxation time 푇2 describes the loss of the phase coherence between the excited spins and leads to the decrease of the transverse magnetization 푀 , which can be expressed in complex notation as ⊥ 푀⊥ = 푀푥 + 푖푀푦. (2.34) The Bloch equations describe the time evolution of the magnetization and account for these two relaxation processes (Bloch, 1946)

푑푀⊥ 푀⊥ = 훾(푴 × 푩)⊥ − (2.35) 푑푡 푇2

푑푀푧 푀푧 − 푀0 = 훾(푴 × 푩)푧 − (2.36) 푑푡 푇1 The solution to the equations (2.35) and (2.36) in rotating frame is given by

−푡/푇2 푀⊥(푡) = 푀⊥(0)푒 (2.37)

2. Theory and background

−푡/푇1 −푡/푇1 푀푧(푡) = 푀푧(0)푒 + 푀0(1 − 푒 ) (2.38) The observable NMR signal generated by an exponential decay of the transverse magnetization in equation (2.37) after an RF excitation is called free induction decay (FID).

Besides the magnetic field fluctuations, which are caused by the motion and vibration of the charged particles and lead to 푇2 decay, there are local changes in the static magnetic field de- ′ scribed by the relaxation time 푇2. They are mainly caused by the susceptibility differences between various tissues and air. These field fluctuations cause an additional signal decay that can be combined with the spin-spin relaxation time 푇 to yield the effective transverse relaxation time 푇∗ 2 2 1 1 1 ∗ = + ′. (2.39) 푇2 푇2 푇2

The spin dephasing (푇2 decay) leads to an irreversible loss of the transverse magnetization. Addi- tional losses due to the local changes in 퐵0 can be restored by applying an additional 180° pulse in a method called spin echo (cf. Subsection 2.2.5). Ensembles of nuclear magnetic moments are then refocused to produce electromagnetic echo signals (Hahn, 1950).

2.1.7. Relaxation of quadrupolar nuclei

All nuclei with 퐼 >1/2 (e.g., 23Na or 17O) possess a nuclear electric quadrupole moment and par- ticipate in quadrupole interactions (Subsection 2.1.2), which are an additional strong pathway for the relaxation of the magnetization. The strength of quadrupole interactions is several orders of magnitude higher than the strength of magnetic interactions, thus, the magnetic relaxation can be neglected in the description of the relaxation properties of quadrupolar nuclei. For isotropic systems in the absence of chemical exchange (e.g., in liquid samples) the extreme narrowing condition is valid

휔0휏퐶 ≪ 1, (2.40) where 휏퐶 is the correlation time, which is time taken for the molecule to rotate by roughly 1 radi- 1 17 an. In this case, neglecting a scale coupling between H and O nuclei the 푇1 and 푇2 relaxation times are equal and are given by

2 1 1 3 2퐼 + 1 휂2 푒2푞푄 (2.41) = = 2 (1 + ) ( ) 휏퐶, 푇2 푇1 40 퐼 (2퐼 − 1) 3 ℏ where 휂 is the asymmetry parameter as described in equation (2.17), and 푒2푞푄⁄ℏ is the coupling constant of the quadrupole interaction (Abragam, 1961). Unlike for 1H nucleus, for nuclei with

퐼 >1/2 the 푇1 and 푇2 relaxation times depend only on 휏퐶 and are not dependent on the external magnetic field 푩ퟎ (Borowiak et al., 2014a; Lu et al., 2013; Thelwall et al., 2003; Zhu et al., 2001).

17 17 1 17 For O nuclei in the H2 O water molecule there is a scalar coupling between H and O nuclei (Rauben et al., 1962) as well as a proton exchange (Meiboom, 1961). These two processes lead to the decrease in 푇2 values without significant influence on 푇1 values (Richardson, 1989).

2.1 Nuclear magnetic resonance

2.1.8. NMR signal detection and sensitivity

Once the magnetization has acquired a transverse component 푀⊥, an MR signal can be detected. The precession of 푀⊥ causes a change in magnetic flux Φ that induces an electromotive force

푈푖푛푑 in the detection coil (Faraday’s law of induction)

푑Φ 푈 = − with Φ = ∫ 푩푑푨, 푖푛푑 푑푡 (2.42) 푐표푖푙 푎푟푒푎 where 푑푨 has a magnitude 푑퐴, which is an element of the coil surface area, and is normal to the differential area in the direction chosen for the definition of positive flux. Using the principle of reciprocity (Carson, 1924), where the roles of the magnetization source and the detection coil are reversed, equation (2.42) can be written as

푑 푈 (푡) = − ∫ 푴(풓, 푡)푩 (풓)푑3푟, 푖푛푑 푑푡 푟푒푐푒푖푣푒 (2.43) 푠푎푚푝푙푒 where 푩푟푒푐푒푖푣푒(풓) is the receive field produced by the detection coil for a hypothetical direct current of unit amplitude flowing in the coil.

The signal in an MR experiment is the amplified voltage 푈푖푛푑(푡) in equation (2.43). If the relaxation effects are neglected, equations (2.24) and (2.43) can be combined to express the detected signal

3 2 푆 ∝ 푐푁퐴훾 퐼(퐼 + 1)퐵0 푉, (2.44) where the natural abundance of the isotope 푐푁퐴 is used instead of the total number of the spins. 2 The detected signal is therefore proportional to the sample volume and to 퐵0 . However, in a general analysis electronic and sampling noise must also be considered (cf. Subsection (2.2.6)).

MR sensitivities of NMR detectable nuclei present in the human body (relative to 1H) are summarized in Table 2.1. For 17O isotope the relative MR sensitivity 푆푒푛푠 is given by

3 푆푒푛푠17 푐 17 훾17 퐼17 (퐼17 + 1) 푂 푁퐴, 푂 푂 푂 푂 −5 = ∙ 3 ∙ = 1.08 ∙ 10 . (2.45) 푆푒푛푠1퐻 푐푁퐴,1퐻 훾1 퐼1 (퐼1 + 1) 퐻 퐻 퐻 For in vivo applications, where the signal from water molecules is detected, this ratio is smaller by a factor of two. Compared to the other MR detectable isotopes the oxygen content in the human body is high, but, due to very low natural abundance of 0.037%, the 17O MR sensitivity is low.

2. Theory and background

2.2. Magnetic resonance imaging

In the previous section, the formation and acquisition of the MR signal from a whole sample was described. The goal of MR imaging, however, is not only to detect the signal of a particular isotope, but also to determine its spatial distribution within the sample. Identical isotopes precess at different frequencies if the magnetic field varies with position. This phenomenon is the core principle of MRI and is used for quantification of the local distribution of the isotopes. During an MRI experiment, well-defined magnetic field variations are superimposed on the homogenous static magnetic field (Subsection 2.2.1). The spatial encoding for the ideal case of infinite discrete sampling is described in Subsection 2.2.2, and for the realistic case of finite discrete sampling in Subsection 2.2.3. Later, the explanation of the spatial resolution (Subsection 2.2.4) and the description of a basic MR pulse sequence (Subsection 2.2.5) are presented. The MR pulse sequence is designed to acquire an MR image with the desired properties (image contrast) by controlling the magnetic field gradients and the RF excitation pulses on the MRI system. At the end of the section, the signal-to-noise ratio is defined and method for estimating it from the acquired MR images are explained (Subsection 2.2.6). More information about MRI can be found in (Bernstein et al., 2004) and (Haacke et al., 1999).

2.2.1. Spatial encoding

For localized and spatially-resolved MR signal detection, in addition to the external static magnetic field 퐵 , time- and spatially-dependent gradient fields 푮(푡) are applied: 0 휕퐵푧(푡) 휕퐵푧(푡) 휕퐵푧(푡) 푮(푡) = 풆 + 풆 + 풆 = 퐺 (푡)풆 + 퐺 (푡)풆 + 퐺 (푡)풆 , (2.46) 휕푥 풙 휕푦 풚 휕푧 풛 푥 풙 푦 풚 푧 풛 where 풆풙, 풆풚 and 풆풛 are the unit vectors in x-, y- and z-directions. The gradient fields are created by gradient coils that are located between the static permanent magnet and the imaging volume and induce a position dependent Larmor frequency

휔(풓, 푡) = 훾퐵푧(풓, 푡) = 훾(퐵0 + 풓 ∙ 푮(푡)) = 훾 (퐵0 + 푥 ∙ 퐺푥(푡) + 푦 ∙ 퐺푦(푡) + 푧 ∙ 퐺푧(푡)). (2.47)

In the presence of the gradient fields, the Bloch equations (2.35) and (2.36) in the rotating coordinate system become

푑푀⊥ 푀⊥ = 훾풓푮(푡)푀⊥ − (2.48) 푑푡 푇2

푑푀푧 푀0 − 푀⊥ = (2.49) 푑푡 푇1 Integration of the equations (2.48) and (2.49) leads to the expression for the transverse and longitudinal magnetization

푡 − −푖휑(풓,푡) 푇 (2.50) 푀⊥(푡) = 푀⊥(0)푒 푒 2, 푡 − 푇 (2.51) 푀푧(푡) = 푀0 − (푀0 − 푀푧(0))푒 1.

2.2 Magnetic resonance imaging

Figure 2.3. k-Space signal and the corresponding magnitude MR images reconstructed from: (a) all k- space data points, (b) the center of k-space only and (c) from the edges of k-space only.

Here, 휑(풓, 푡) is the phase accumulated during precession in the presence of the gradient fields 푡 휑(풓, 푡) = 훾 ∫ 푮(푡′)푑푡′풓 + 휑(0) = 2휋풌(푡)풓 + 휑(0), (2.52) 0 and 풌 is the wave vector

푡 훾 풌(푡) = ∫ 푮(푡′)푑푡′. (2.53) 2휋 0 The measured signal 푠(풌(푡), 푡) is an integral of the transverse magnetization (equation (2.50)) over the volume 푉 where the signal is acquired. If relaxation processes are neglected and the spin density 휌푠(풓), which represents the amount of the detectable spins (cf. Subsection 2.1.3) is used instead of 푀 , then using equation (2.53) the measured signal is given by ⊥ −푖2휋풌(푡)풓 3 푠(풌(푡), 푡) ∝ ∫ 휌푠(풓)푒 푑 푟. (2.54)

This equation corresponds to the Fourier transform (FT) of the spin density 휌푠(풓). The inverse FT can be used to calculate 휌(풓) from the measured signal 푠(풌(푡), 푡)

+푖2휋풌(푡)풓 3 휌푠(풓) ∝ ∫ 푆(풌(푡), 푡)푒 푑 푘. (2.55)

2. Theory and background

Figure 2.4. Illustration of the 2D Cartesian sampling scheme (left) and 2D radial center-out sampling scheme (right).

To reconstruct an MR image from the measured signal 푠(풌(푡), 푡), the Fourier space (so-called k- space) is sampled (encoded) by sequentially applying gradient fields of varying strength that determine the k-space trajectories (equation (2.53)). The signal acquired near the center of k-space contains the signal to noise and contrast information of the image, whereas peripheral regions of k-space contain the information determining the image resolution (Figure 2.3). The reconstructed image is complex and includes phase and magnitude information. Currently, most diagnostic applications are based on magnitude information.

Most conventional MRI techniques perform a line-by-line sampling of k-space, whereas arbitrary sampling schemes can be employed without loss of generality (Figure 2.4). For the detection of the MR signal with short relaxation times radial acquisition schemes are preferred. Here, the k- space data is sampled along spokes (also called projections) originating from the center of k- space. This will be described in more detail in Section 3.3.

2.2.2. Infinite discrete k-space sampling

In real MR experiments, the k-space signal is always acquired in discretized form. In general, the sampling points are equidistantly acquired on a Cartesian grid (Figure 2.4, left) in order to efficiently reconstruct the image with the fast Fourier transformation (FFT) algorithm (Cooley and Tukey, 1965). In the following, all relaxation effects are neglected, which means that the reconstructed image is a direct representation of the underlying physical spin density.

2.2 Magnetic resonance imaging

Mathematically, the k-space sampling is described as a multiplication of the continuously measured signal with the sampling function 푢(풌) constructed from a sum of the Dirac delta functions called Dirac comb

∞ 푢(풌) = ∆푘 ∑ 훿(풌 − 푝∆푘). (2.56) 푝=−∞ Taking into account the convolution theorem, which states that the FT of the product of two functions is the convolution of the FT of each function, the reconstructed image 휌̂∞(풓) can be represented as the convolution of the original image 휌(풓) with the inverse FT of the sampling function 푈(풓)

휌̂∞(풓) = 휌(풓) ∗ 푈(풓) (2.57) The FT of the Dirac comb (2.56) is also a Dirac comb with spacing 1⁄∆푘. This means that the FT of the infinite discrete sampled signal is a periodic repetition of the original image. The spatial interval over which the reconstructed image repeats itself is called the field-of-view (FOV)

1 퐹푂푉 ≡ . (2.58) ∆푘 If the FOV is larger than the length of the object 퐿 in a particular dimension, i.e.,

1 퐹푂푉 > 퐿 ↔ ∆푘 < , (2.59) 퐿 the reconstructed images do not overlap. Equation (2.59) is referred to as the Nyquist sampling criterion (Nyquist, 1928). If ∆푘 is chosen too large (i.e. the FOV that is too small), the Nyquist condition is violated and the images overlap. This type of artifact is called aliasing.

2.2.3. Discrete finite k-space sampling

In the previous subsection it was assumed that an infinite number of discrete sample points are obtained from the continuous k-space signal. In practice, sampling is performed only for a limited time, which means that the data points are truncated, and a finite number of discrete k-space sampling points 푁푠푝 are encoded up to the maximum value 푘푚푎푥.

Mathematically, this can be expressed as the multiplication of the infinite discrete sampling points with the filter function 푤(풌). In image space this corresponds to the convolution of the image

휌̂∞(풓) obtained by the infinite discrete sampling (2.57) with the inverse FT of the filter function 푊(풓)

휌̂(풓) = 휌̂∞(풓) ∗ 푊(풓) = 휌(풓) ∗ 푈(풓) ∗ 푊(풓). (2.60) The function 푊(풓) is defined to be the point spread function (PSF). In the case of Cartesian sampling, 푤(풌) corresponds to the FT of a rectangular window function, which is equals to sin(푥)⁄푥 = 푠푖푛푐 function. The image 휌̂∞(풓) that is already replicated due to discrete sampling, is additionally convolved with 푊(풓). This leads to blurring and Gibb’s ringing artifacts in the final reconstruction 휌̂(풓). 18

2. Theory and background

Gibb’s ringing artifacts can be reduced by additional filtering of the k-space data. For example, a Hanning filter can be used to suppress the sidelobes of the sinc function

2휋풌 1 + cos ( ) 푊퐻푎푛푛푖푛푔 2 휋풌 (2.61) 푤퐻푎푛푛푖푛푔(풌) = = cos ( ), 2 푊퐻푎푛푛푖푛푔 with the width of 푊퐻푎푛푛푖푛푔 = 2푘푚푎푥. The FT of the Hanning filter corresponds to an averaging in the image domain

1 1 푊 (풓) = 훿(풓) + (훿(풓 − ∆풓) + 훿(풓 + ∆풓)), (2.62) 퐻푎푛푛푖푛푔 2 4 where ∆풓 is the distance between the image pixels.

2.2.4. Spatial resolution

Generally, the spatial resolution of an imaging method is the smallest resolvable distance between two different objects, or two different features of the same object (Haacke et al., 1999). There are several mathematical definitions of the spatial resolution, but the most common is based on the PSF, which for an ideal experiment with infinite amount of data is described by a delta function, and given by

1 ∞ ∆푥 = ∫ 푃푆퐹(푥)푑푥. (2.63) 푃푆퐹(0) −∞ The nominal resolution in MRI is defined as the smallest separation between two objects which can be determined if no k-space filtering is applied (i.e., the filter function 푤(풌) of the discrete sampling pattern is also not considered). For Cartesian sampling it is given by

퐹푂푉푥 1 ∆푥 = = . (2.64) 2푁푥 2푘푚푎푥,푥 Analogously, it is defined for the other directions (y, z). For 3D radial center-out k-space sampling the nominal resolution is isotropic

퐹푂푉 1 ∆푥 = = . (2.65) 푁푠푝 2푘푚푎푥 where 푁푠푝 is the number of the sampling points along one spoke. For comparison of different acquisition and reconstruction schemes the theoretical full width at half maximum (FWHM) of the PSF is used as a measure. In case of 3D radial acquisition FWHM is equal to 1.59, which is 31% higher than FWHM of the sinc function used in Cartesian sampling (Rahmer et al., 2006). * The decay of the MR signal during the data sampling due to T2 relaxation is an inherent filter to the MR experiment leading to an increase of FWHM for the density adapted 3D radial sampling (Nagel et al., 2009) (cf. Subsection 3.3.2). Finally, a Hanning filter (equation (2.61)) causes an additional, stronger FWHM increase by an amount of 0.85. The increase of the FWHM of the PSF

2.2 Magnetic resonance imaging

Figure 2.5. Basic gradient-echo MR imaging sequence consisting of the RF excitation pulse, gradient fields in x-,y- and z-direction, and data acquisition. Gx gradient applied during the RF excitation is called the slice-selective gradient; Gy gradient, which is used for a line-by-line acquisition of k-space, is called the phase-encoding gradient; and Gz gradient, which is rephased during the signal acquisition along a line in k-space, is called the frequency-encoding gradient. Refer to Figure 2.4 (left) for the illustration of line- by-line Cartesian sampling.

and, consequently, the decrease of the spatial resolution of the MR images can be visually seen as image blurring.

2.2.5. MRI pulse sequences

An MRI pulse sequence is a set of semi-repetitive NMR experiments (cycles) with which the magnetization is driven out of equilibrium by short resonance RF pulses and by spatially encoding gradient fields, as a result of which an image is acquired. In Figure 2.5 a schematic representation of a pulse cycle of a gradient echo MR sequence is shown. The duration of each pulse sequence cycle is called the repetition time (TR). Each cycle of the pulse sequence begins with an

RF excitation pulse with a duration of Tpulse and the signal acquisition is performed later at the echo time (TE). For Cartesian or a 2D radial sampling patterns, this time is used to apply the slice-selective gradient fields. A 3D center-out radial acquisition scheme, however, does not require the slice-selective gradients, and in each TR the signal acquisition starts in the center of k- pace (푘 = 0). The MR signal is recorded as soon as possible after the RF excitation and signal loss 17 due to T2 decay is minimized. Such MR sequence, which was used in this work for O MRI experiments, is called ultrashort TE (UTE) sequence and will be described in Subsection 3.3.3.

The number of RF pulses in a pulse sequence cycle determines the type of signal that is generated. In the simplest case only one RF pulse is applied and so an FID is acquired (equation (2.37)). In the spin echo sequence two RF pulses are used, a 90° excitation pulse followed by a 180° refo- cusing pulse, to generate spin-echo signals in which the dephasing effects of static magnetic field

2. Theory and background

inhomogeneities have been compensated. The stimulated gradient-echo pulse sequence uses three RF pulses of 90°, or less than 90° for rapid acquisition. The signal evolution for a particular MRI sequence can be described by the signal equation, which depends on the sequence parameters and relaxation properties of the tissue in the imaging volume.

The Fast Low Angle SHot (FLASH) MRI sequence (Haase et al., 1986) is a basic measuring principle for fast MRI. At first, a low flip angle RF excitation (휃푀 ≈ 5°- 15°≪ 90°) is applied, leaving unused longitudinal magnetization available for the next excitation. Second, it takes advantage of the gradient echo acquisition, which does not need a further RF pulse that would affect the residual longitudinal magnetization. The combination of both features allows for the rapid repetition of the pulse sequence cycles (i.e. short TR) and results in about a 100-fold reduction of the total measurement time. In a FLASH sequence a steady-state of the longitudinal magnetization 푀SS is established after several pulse cycles

푇푅 − 1 − 푒 푇1 푀SS = 푀0 푇푅 (2.66) − 푇 1 − cos휃푀 ∙ 푒 1 Taking into account that the signal detection is delayed by TE after the signal excitation, and that the MR signal is modulated by the T * relaxation, the FLASH signal equation is given by 2 푇푅 − 푇퐸 푇 푇퐸 − ∗ 1 − 푒 1 − ∗ 푇2 푇2 푆 = 푀SS ∙ sin휃푀 ∙ 푒 = 푀0 푇푅 ∙ sin휃푀 ∙ 푒 (2.67) − 푇 1 − cos휃푀 ∙ 푒 1 The maximum signal occurs for an angle 휃 E 푇푅 − 푇 휃E = arccos (푒 1 ), (2.68) which is called the Ernst angle (Ernst and Anderson, 1966).

During the MR signal acquisition the readout gradient is switched on, and an analog to digital converter (ADC) samples the signal. The ADC converts the analog MR signal to digital values and writes them to the memory at equidistant time intervals ∆휏 (dwell time). The readout time (also called sampling time) 푇 is then given by 푅푂 푇푅푂 = 푁푠푝 ∙ ∆휏, (2.69) The receiver bandwidth has units of Hz/pixel and defined as

퐵푊 = 1/푇푅푂. (2.70) At the end of the pulse sequence cycle the spoiling gradient can be applied. Spoiling is achieved by switching on the gradients after the ADC to destroy the remaining transverse coherences that may persist from one cycle to the next. For the same purpose, RF-spoiling can be used in which the phase of the RF pulse is incremented from one TR to another.

2.2 Magnetic resonance imaging

2.2.6. Signal-to-noise ratio

Aside from the induction signal, noise affects a measurement, which is characterized by the SNR. The SNR is a key parameter determining the effectiveness of any given imaging experiment and providing information on whether the signal from the object in the voxel can be distinguished from noise. For 3D imaging the SNR dependence on imaging parameters can be expressed as

푆푁푅 ∝ ∆푥 ∙ ∆푦 ∙ ∆푧 ∙ √푇푅푂 ∙ √푁푎푐푞, (2.71) where 푁푎푐푞 is the number of acquisitions and ∆푥, ∆푦 and ∆푧 are the spatial resolution in x,y and z directions, correspondingly.

17 As O relaxation times are B0-independent (cf. Subsection 2.1.7), the MR signal intensity is pro- 2 portional to 퐵0 (equation (2.44)). The noise intensity depends linearly on 퐵0 when sample losses are predominant, which is mostly the case for human subjects at resonance frequencies ω0 > 1

MHz. Thus, the SNR can be assumed to be proportional to 퐵0 (Hoult and Lauterbur, 1979). Combining equations (2.44) and (2.71), the SNR in a voxel with dimensions ∆푥 is given by

2 3 푆푁푅 ∝ 푐푁퐴훾 퐼(퐼 + 1)(∆푥) 퐵0√푇푅푂 ∙ √푁푎푐푞, (2.72) where 푐푁퐴 is the natural abundance of the nucleus. This means, for example, that doubling 푁푎푐푞 gives a factor of √2 improvement in SNR, and a two-fold decrease of the imaging voxel size ∆푥 will cause an 8-fold degradation in SNR. Moreover, a higher 퐵0 provides a higher SNR, which in turn can be used to decrease the imaging voxel size.

The measured signal intensity 푆푚푒푎푠 is a combination of the noise-free signal intensity 푆푟푒푎푙 and the standard deviation of the noise 휎 푛표푖푠푒 2 2 푆푚푒푎푠 = √푆푟푒푎푙 + 휎푛표푖푠푒. (2.73)

However, for the SNR of five or higher the mean of the magnitude image is approximately equal to the noise-free image (Gudbjartsson and Patz, 1995). For SNR quantification the noise level or the standard deviation of the signal intensity must be known. In complex MR images the noise is Gaussian distributed and in the magnitude images it is Rician distributed, but for an SNR of five or higher it can be assumed to be Gaussian distributed. In the regions of the magnitude image that do not contain any MR signal, the Gaussian distribution is not valid. However, the mean value of the noise 푆푛표푖푠푒 and its standard deviation 휎푛표푖푠푒 are related to the standard deviation of the Gaussian distribution 휎 (Gudbjartsson and Patz, 1995) 퐺푎푢푠 휋 휋 푆 = 휎 √ and 휎 = 휎 √2 − . (2.74) 푛표푖푠푒 퐺푎푢푠푠 2 푛표푖푠푒 퐺푎푢푠푠 2 Equation (2.74) allows for SNR quantification from the magnitude MR images as the ratio of the signal intensity within the imaging object and the noise intensity or its standard deviation outside the object (e.g. in the noise region). This was the method used in this work for the determination of the SNR.

2. Theory and background

2.3. Respiration and cellular metabolism

This section describes the processes of respiration: the first is external respiration, such as that corresponding to absorption of O2 and removal of CO2 from the body (Subsection 2.3.1). The second is internal respiration, i.e. gas exchange between the cells and their fluid medium, utilization of oxygen and production of carbon dioxide in the cells, called cellular metabolism (Subsec- tion 2.3.2). For a more detailed description, refer to (Barrett et al., 2010), (Schmidt et al., 2011), (LaManna et al., 2008) and (Atkinson, 1977).

2.3.1. External respiration

The respiratory system consists of a gas-exchanging organ (the lungs) and a “pump” that venti- lates the lungs. The “pump” is made up of the chest wall (the respiratory muscles that increase and decrease the size of the thoracic cavity), the areas in the brain that control the muscles, and the tracts and nerves that connect the brain to the muscles. The inspired air is first passed through the nasal passages and pharynx, where it is warmed and takes up water vapor. Then, it passes down the trachea and through the bronchioles, respiratory bronchioles, and alveolar ducts down to the alveoli, where ultimately gas exchange occurs (Figure 2.6).

At rest, the normal human breathes 10 to 15 times per minute, which is referred to as the respiratory rate (RR): at about 500 mL of air per breath, a total of about 6 L is inspired and expired per minute. In the lungs, the inhaled air mixes with the gas in the alveoli, and by diffusion, O2 enters the blood in the pulmonary capillaries while CO2 enters the alveoli. In this manner, 250 mL of O2 enter the body per minute and 200 mL of CO2 is excreted (Barrett et al., 2010). Consequently, the oxygen fraction in the exhaled gas (14-16%) is smaller compared to the inhaled air (21%).

The amount of air that passes into the lungs with each inspiration is called the tidal volume

(푉푡푖푑푎푙, typical value is 500 mL). Because gaseous exchange in the respiratory system occurs only in the alveoli, the gas that occupies the rest of the respiratory system is not available for gas exchange with pulmonary capillary blood. Normally, the volume of this dead space 푉푑푒푎푑 is about 150 mL. Thus, with each expiration, the first 150 mL expired is the gas that occupied the dead space, and only the last 350 mL is the gas coming from the alveoli. This phenomenon is quantified by the alveoli ventilation (AV) which is the amount of air reaching the alveoli per minute

AV = (푉푡푖푑푎푙 − 푉푑푒푎푑) · RR (2.75) Considering that the amount of air inspired per minute (normally, about 6 L) and the anatomical dead space are both constant, the alveoli ventilation will decrease with an increasing respiratory rate.

2.3 Respiration and cellular metabolism

Figure 2.6 Structure of the respiratory system: (A) The respiratory system is illustrated with a transparent lung to emphasize the flow of air into and out of the system. (B) Enlargement of the boxed area from (A) shows transition from the conducting airway to the respiratory airway, with emphasis on the anatomy of the alveoli. Red and blue represent oxygenated and deoxygenated blood, respectively. Figure from (Barrett et al., 2010).

2.3.2. Internal respiration and cellular metabolism

Although the human brain is only about 2% of the body’s weight, it consumes about 20% of body’s total oxygen supply (Rolfe and Brown, 1997). The major part of energy consumption is used to meet the high energy requirements of the electrophysiological activities of the neurons responsible for inter-neuron transmission and communication in the central nervous system (Attwell and Laughlin, 2001; Zhu and Chen, 2011). Neuronal activity must be highly dynamic to account for wide range of physiological conditions of the brain energy demands. Thus, a timely efficient balance between energy demand and supply is required, which is regulated by various biochemical reactions associated with brain metabolism.

Internal respiration refers to a metabolic process in which oxygen is released to tissues or living cells and carbon dioxide is absorbed by the blood. Once inside the cell, the oxygen is used for

2. Theory and background

Figure 2.7. Schematic representation of key processes of cellular metabolism. Oxygen and glucose are delivered via blood flow through the blood-brain barrier into the brain cells. ATP molecules store the energy and the end product, water, is formed via oxidative phosphorylation, where the electrons are transferred to oxygen.

producing energy in the form of adenosine triphosphate (ATP), which is referred to as the energy “currency” of the cell.

Key processes of cellular metabolism are schematically illustrated in Figure 2.7. Oxygen is transported in blood by hemoglobin, which is made up of four symmetrical subunits and four heme groups that contribute to the red color found in muscles and blood. Each heme group contains an iron atom that is able to bind one oxygen O2 molecule. Oxygen and glucose, which is free to dissolve in the blood, are the major fuels for brain metabolism and are continuously supplied by the circulating blood flow from capillaries. Glucose is transported into brain cells through the blood-brain barrier and converted to two pyruvate molecules in the cytosol via glycolysis. Py- ruvate is then converted to the molecule called acetyl-Co A in the mitochondrion ( a double membrane-bound organelle within the cell), and oxidized via the Krebs cycle to generate high energy electron carriers. These electron carriers enter the electron transport chain and generate an electrochemical proton gradient across the mitochondrial inner membrane that drives the synthe- sis of ATP. Here, energy in form of hydrogen ions moving down an electrochemical gradient is used by the enzyme ATP synthase to create ATP from adenosine diphosphate (ADP) and inorganic phosphate (Pi). In ATP, the energy is stored chemically in the form of highly strained bonds and is later released during oxidative phosphorylation where the electrons are finally transferred to oxygen and, with the addition of two protons, the final product of water is formed.

A quantitative measure of metabolic brain activity is the cerebral metabolic rate of oxygen consumption (CMRO2). Oxygen metabolism is altered in stroke (Baron et al., 1981; Sobesky et al.,

2.3 Respiration and cellular metabolism

2005) and brain tumors (Hoffmann et al., 2014; Ito et al., 1982; Mineura et al., 1986; Rhodes et al., 1983; Thulborn et al., 1999; Tyler et al., 1987) as well as in many neurodegenerative diseases such as Alzheimer’s disease, Parkinson’s disease and Huntington’s disease (Beal, 1992; Frackowiak et al., 1988; Fukuyama et al., 1994; Ishii et al., 1996; Maurer et al., 2001, 2000;

Wallace, 1999, 1992; Wong-Riley et al., 1997). There are several techniques to quantify CMRO2 in the human brain, which are discussed in more detail in the next chapter .

2.4. Quantification of the cerebral metabolic rate of oxygen consumption

CMRO2 represents a key measure of brain function, because the brain almost entirely depends on oxidative metabolism to fulfill its significant energy demands. The Fick principle of arteriovenous difference in oxygen content, introduced in Subsection 2.4.1, is the basis for CMRO2 quantification with PET using the oxygen isotope 15O (cf. Subsection 2.4.2) and 1H MRI (cf. Subsection 15 1 2.4.3). So far, O-PET is considered to be the gold standard for CMRO2 quantification. H MRI 1 can be used for CMRO2 quantification, as several H MR contrasts are affected by the change of the deoxyhemoglobin (dHb) concentration in blood. Yet, the detection of the MR accessible sta- 17 ble oxygen isotope O is preferable for CMRO2 quantification, as it can directly detect the meta- 17 17 bolic end product H2 O. O MRI is introduced in Subsection 2.4.4, followed by the description 17 of the pharmacokinetic model used for CMRO2 quantification with O MRI at UHFs (Subsec- tion 2.4.5). Finally, the method of the profile likelihood analysis is introduced, which provides information about the model parameter identifiability (Subsection 2.4.6). More information about the CMRO2 quantification methods can be found in (Atkinson and Thulborn, 2010; Hoffmann, 2011; Rodgers et al., 2016; Shukla and Kumar, 2006; Zhu and Chen, 2011) and references therein.

2.4.1. Fick principle

(Kety and Schmidt, 1948) proposed the use of Fick principle for non-localized quantification of

CMRO2 and the cerebral blood flow (CBF), the amount of blood passing through the brain tissue. Under assumption that all oxygen extracted from the blood is used for ATP production, the Fick principle of arteriovenous difference in oxygen content states that CMRO can be written as 2 퐶푀푅푂2 = 퐶퐻푏 ∙ 퐶퐵퐹 ∙ (푌푎 − 푌푣), (2.76) where 푌푎 and 푌푣 are the arterial and venous oxygen saturation (in %), CBF and CMRO2 are in µmol per 100g of brain tissue per minute, 퐶퐻푏 is the proportionality factor, which depends on the hemoglobin concentration and can be obtained from hematocrit measurement. 푌푎 can be measured continuously with a digital pulse oximeter, while 푌푣 and CBF are typically measured with 15O-PET or 1H MRI. Brain oxygen extraction is typically determined as the oxygen extraction fraction (OEF) given by

푌푎 − 푌푣 푂퐸퐹 = . (2.77) 푌푎

In the first work on CMRO2 quantification with the Fick principle (Kety and Schmidt, 1948), 푌푎 and 푌푣 were measured directly with co-oximetry of arterial and venous blood. 푌푎 and 푌푣 have typical values of 1 and 0.7, which yields OEF values of about 0.3. CBF was quantified during continuous inhalation of nitrous oxide (N2O) gas by integrating the differential concentration of

N2O in arterial and venous blood. This technique is highly invasive and provides only one non- localized measure of CBF and CMRO2. Nevertheless, the Fick principle became the basis for 15 1 development of CMRO2 quantification methods using O-PET and H MRI.

2.4 Quantification of the cerebral metabolic rate of oxygen consumption

15 2.4.2. CMRO2 quantification with O-PET

The first measurements of the regional cerebral oxygen utilization in humans with 15O-PET was done in 1970s (Raichle et al., 1976; Ter-Pogossian et al., 1970). 15O signal was locally measured in various brain regions using local detectors (scintillation counters). Later, a new tracer, inhaled 15O 15 gas, was introduced in addition to the intravenous injection of H2 O (Mintun et al., 1984). Fur- ther technological improvements of the detection precision made 15O-PET the gold standard method for pixel-wise CMRO2 mapping.

In PET, positron-emitting radioactive tracers with specific chemical properties are injected or/and inhaled. Two photons with an energy of 511 keV are produced after annihilation of the emitted positrons with surrounding electrons. The photons are emitted in opposite directions and detected by scintillation detectors of the PET system. This process results in detection of a coincidence event, which localizes an annihilation event along the line joining the two detectors. A set of such coincidence lines provides the spatial distribution of the tracer.

15 CBF is quantified using intravenous injection of H2 O water, while OEF is measured in a sepa- 15 rate experiment with inhalation of radioactive O gas. Later, CMRO2 is calculated from CBF and OEF using equations (2.76) and (2.77). There are several significant limitations that prevent a wide spread use of 15O-PET. First, it is invasive, requiring intravenous injections and involves patient exposure to ionizing radiation. Second, it is costly and the short half-life of only 2 min requires onsite isotope production with a cyclotron, which few sites around the world are equipped with.

1 2.4.3. CMRO2 quantification with H MRI

Unlike 15O-PET, MRI is inherently non-invasive and commonly available in modern medical 1 1 centers. H MRI methods have been designed for CMRO2 quantification as several H MR con- * trasts are affected by the change of the dHb concentration in blood. T2 and T2 time constants are typically used as 1H MR contrasts. Similar to 15O-PET, the Fick Principle is utilized to calculate

CMRO2 from CBF (e.g., using arterial spin labelling (ASL) (Williams et al., 1992)) and the measured OEF. Whole brain CMRO2 values can be reliably quantified using the T2-relaxation-under- spin-tagging (TRUST) method (Lu and Ge, 2008; Xu et al., 2009) in which venous blood signal is isolated via the application of spin tagging, combined with phase-contrast MRI (PC-MRI)

(Moran, 1982) for CBF measurement. Another method for whole brain CMRO2 quantification is a simultaneous measurement of venous oxygen saturation with MR susceptometry-based oximetry, which exploits the intrinsic susceptibility of dHb, and the average blood inflow rate with PC- MRI (Jain et al., 2010).

2. Theory and background

* For voxel-wise CMRO2 quantification, one method is the T2 -based calibrated blood-oxygen-level dependent (BOLD) contrast imaging (Blockley et al., 2013; Davis et al., 1998), as both CMRO2 and CBF are locally increased during neuronal activation. Overall, the CBF increase dominates, resulting in a reduction of an extravascular-modelled dHb and a concomitant increase in BOLD signal intensity. Another voxel-wise CMRO2 quantification method is the T2-based QUantitative Imaging of eXtraction of Oxygen and TIssie Consumption (QUIXOTIC) (Bolar et al., 2011), where a combination of velocity-selective excitation pulses (Wong et al., 2006) and blood tagging is used to isolate signal from venous blood in the post-capillary venous compartment for pixel- wise OEF measurement. All of these CMRO2 quantification techniques are indirect and require assumptions that are not necessarily valid under pathological conditions.

1 Several other H MRI-based methods have been developed for CMRO2 quantification in animals based on the influence of 17O-1H scalar coupling on 1H relaxation of water. Such coupling be- 1 17 16 tween H and O spins is modulated by the fast proton chemical exchange between H2 O and 17 H2 O molecules (Meiboom, 1961), which shortens proton T2 or the rotating-frame spin-lattice 17 relaxation time T1ρ. The proton T2 relaxation time was shown to be linearly correlated with H2 O 17 concentration (Hopkins and Barr, 1987). Thus, the signal changes after H2 O injection detected 1 with T2-weighted H MRI were used for CBF and CMRO2 quantification in the early studies (Hopkins et al., 1991; Kwong et al., 1991). Later, this method was improved by introducing 17O 17 17 decoupling, and the difference of proton signals of H2 O in the presence and absence of O 17 decoupling was used to estimate H2 O content (Ronen et al., 1998, 1997).

1 H MRI with T1ρ-dispersion imaging method (Reddy et al., 1995) uses a single proton RF channel 1 for acquiring T1ρ-weighted H MR images with two different spin locking RF powers. Despite the lack of understanding of the T1ρ contrast in biological tissues and difficulty of obtaining prior information on the intrinsic T1ρ dispersion, this method was applied for CMRO2 estimation in the animal brain (Mellon et al., 2010, 2009). The advantage 1H MR-based for detection of metabo- 17 lized H2 O in brain tissue is the high sensitivity of the proton signal and the ability to apply conventional MRI hardware and software. For example, in T1ρ-weighted MRI only proton receiver coils are required, which are widely available. The major drawbacks of 1H MRI methods using 17 1 17 O- H scalar coupling are the difficulties of absolute H2 O quantification and that the effect in signal change, which is in the range of 1%, is likely to be contaminated by signal fluctuations caused by the physiological noise (e.g. respiration or pulsation) or instability of the MRI system

(Zhu and Chen, 2011; Zhu et al., 2005). To the best of my knowledge, neither proton T2 nor proton T1ρ MRI methods have been used for CMRO2 quantification in humans.

2.4 Quantification of the cerebral metabolic rate of oxygen consumption

17 2.4.4. CMRO2 quantification with O MRI

Oxygen-17 is one out of three natural abundant stable oxygen isotopes: 16O, 17O, and 18O. It is the only MR-visible isotope with 퐼 = 5/2. 17O nucleus, which is chemically identical to 16O, has a low MR sensitivity, which is only about 1.1·10-5 of 1H due to the low natural abundance of 17O nucleus of 0.037% and the approximately sevenfold lower gyromagnetic ratio (cf. Table 2.1 and equation (2.45)). In 17O MRI experiments, 17O is inhaled as isotope-enriched 17O gas, the meta- 17 17 bolic end product H2 O is detected with dynamic O MRI and CMRO2 is quantified using a pharmacokinetic model (Atkinson and Thulborn, 2010; Hoffmann, 2011; Hoffmann et al., 2011). A more detailed description of the 17O inhalation experiment can be found in Section 3.1.

15 15 A drawback of O-PET is that the PET signals emitted from O2 molecules and from the meta- 15 17 bolically generated H2 O molecules cannot be differentiated. In contrast, O MRI exclusively 17 17 detects the signal from the metabolically generated H2 O molecules after inhalation of O gas. 17 The resonance peak of O2 molecules bound to the large hemoglobin protein in the blood is extremely broad due to its very slow rotational motion. The 휏퐶 values for the rotational motion of the hemoglobin molecules are at the order of 10-6 s to 10-8 s (Cassoly, 1982), which is approximately 106 times slower than that of free water (Zhu and Chen, 2011). Thus, according to equa- 17 tion (2.41) T2 relaxation times are very fast and not MR detectable. O2 molecules in gas form or dissolved in water possess two unpaired electrons and are paramagnetic, causing strong dipolar coupling between electrons and 17O and hence, cannot be detected in clinical MRI systems. This 17 17 unique selectivity for the metabolized H2 O detection makes O MRI the only direct CMRO2 measuring modality.

Direct 17O MR spectroscopy and MRI were mostly performed in small animals and UHFs to partially compensate for the low in vivo concentration of 17O. First investigations were done in early 1990s (Fiat and Kang, 1993, 1992; Pekar et al., 1991). Later, the complete model for 17 CMRO2 quantification within a few minutes of O2 inhalation in small animals was established (Lu et al., 2013; Zhang et al., 2004; Zhu and Chen, 2011; Zhu et al., 2002, 2001). However,

CMRO2 calculation in humans remains challenging (Zhu and Chen, 2011). Respiration rate, lung capacity, body size and blood circulation speed differ substantially between humans and small animals. In particular, in a human lung the exchange between 16O and inhaled 17O has much slower and longer blood circulation time through the human body further slowing down the binding of 17O to hemoglobin. Thus, 17O enrichment fraction of the inhaled gas is not constant and the CMRO2 quantification model that worked well for the small animal brain, failed to provide acceptable CMRO2 values in human brains (Zhu et al., 2006).

Recently, a pharmacokinetic model for CMRO2 quantification in humans was proposed

(Atkinson and Thulborn, 2010) that allowed CMRO2 mapping of the human brain at 9.4 T. Later, a pulsed 17O gas delivery system, combined with a RB circuit was introduced and tested in inhalation experiments at 7 T to quantify CMRO2 in large WM and GM brain regions (Hoffmann et al., 17 2011). The obtained tissue-specific H2 O signal-time curves are shown in Figure 2.8.

2. Theory and background

Figure 2.8. H217O signal-time curves (left) averaged over large gray (shown in red on the right) and white (shown in blue on the right) brain regions. They were obtained in the dynamic 17O MRI experiment with inhalation of isotope-enriched 17O gas at 7 T MRI system. 17O2 administration via the pulsed gas delivery (DODS phase) is shown with the dark blue bar. The breathing phase, during which the volunteer was breathing the gas stored in the RB circuit, is shown with light blue bar. Figure taken from (Hoffmann et al., 2011).

The dynamic 17O MRI imaging experiment with the RB circuit is divided into four subsequent phases (Hoffmann et al., 2011):

1) the baseline phase, when 17O MR signal was acquired at natural abundance; 2) the 17O-inhalation phase, wherein 17O-enriched gas was delivered in pulses. This phase is also called the DODS phase if 17O gas is delivered in pulses with the demand oxygen delivery system (DODS); 3) the rebreathing phase, where the volunteer was breathing the gas stored in the RB circuit (which contains the exhaled 17O gas that is then reused to increase the 17O MR signal); 4) the wash-out phase, where the RB circuit was opened and the volunteer was breathing room air.

Detailed explanation of the dynamic 17O MRI experiment can be found in Section 3.1. Clinical importance of dynamic 17O MRI was demonstrated in a glioblastoma tumor patient at 7 T (Hoffmann et al., 2014). In particular, the difference in oxygen metabolization in the necrotic center, the contrast-enhancing rim and perifocal edema regions of the tumor was presented. In Figure 2.9 the relative 17O signal increase after administration of 17O gas is compared to the anatomical 1H MR image after contrast media administration. The central necrotic region of the tumor coincides well with the area of the lowest 17O signal increase of about 2%, whereas the highest gain in intensity of about 20% was found in regions with a high gray matter fraction.

2.4 Quantification of the cerebral metabolic rate of oxygen consumption

Figure 2.9. (a) Anatomical T1-weighted 1H MR image after contrast media administration with typical ring enhancement. (b) Parameter map of the relative 17O signal increase after 17O2 inhalation in the corresponding slice. (c) Colored overlay of the 17O MR signal increase and anatomical 1H reference image. Figure taken from (Hoffmann et al., 2014).

2.4.5. Pharmacokinetic model for 17O MRI

17 Following the principle of mass conservation, the change of the H2 O concentration within a given volume can be caused either by water creation and conversion to other intermediates in the volume, or inward and outward diffusion to or from neighboring volumes. Therefore, the

CMRO2 quantification model, as proposed by (Atkinson and Thulborn, 2010), describes the 17 change of H 17O concentration 푀퐻2 푂(푡) in the given volume as 2 푑 17 17 17 17 퐻 푂 푂 퐻 푂 퐻 푂 푀 2 (푡) = ⏟2 ∙ 퐶푀푅푂 2 ∙ 퐴 ( 푡) − ⏟퐾퐿 ∙ 푀 2 ( 푡) + ⏟퐾퐺 ∙ 퐵 2 ( 푡), (2.78) 푑푡 17 17 17 퐻2 푂 푚푒푡푎푏표푙푖푠푚 퐻2 푂 푙표푠푠 퐻2 푂 푔푎푖푛 where the rate constant 퐾퐿 reflects the loss by diffusion to blood and chemical conversion to other intermediates, and 퐾퐺 represents the gain by diffusion from blood. The factor of 2 is in- 17 cluded because 2 mol of water are produced from one mole of oxygen. 퐴 푂(푡) denotes the fraction of 17O-labelled arterial oxygen gas and for the four phase 17O inhalation experiment (Hoffmann, 2011) is given by

0 푡 ≤ 푇퐴 17 푘 (훼 − 퐴 푂(푡)) 푇 ≤ 푡 ≤ 푇 푑 17 표푥 퐴 퐵 푂 퐴 (푡) = 17 , (2.79) 푂 푑푡 푘표푥 (훽 − 퐴 (푡)) 푇퐵 ≤ 푡 ≤ 푇퐶

17푂 { −푘표푥퐴 (푡) 푇퐶 < 푡

17 where the time points 푇퐴, 푇퐵 and 푇퐶 determine the different phases of the dynamic O MRI experiment (Figure 2.8), which are introduced in Subchapter 2.4.4 and described in Subchapter -1 17 3.1.2. Here, 휌표푥 = 0.75 min is the rate at which fresh O binds to hemoglobin in the pulmonary (Atkinson and Thulborn, 2010), and 훼 and 훽 are the 17O enrichment fractions of the inhaled gas above natural abundance in the DODS phase and the RB phase, correspondingly.

17 퐵퐻2 푂(푡) is the relative amount of H 17O in blood (both in excess of natural abundance) 2 32

2. Theory and background

푑 17 17 퐵퐻2 푂(푡) = 퐴 푂(푡). (2.80) 푑푡 The analytical solution of the equation (2.78) for the four phase model was recently found (Hoffmann, 2011)

푐푁퐴

−푘 (푡−푇 ) 퐾퐺 −퐾 (푡−푇 ) 푐6 + 훼푐1푒 표푥 퐴 + 훼 (푡 − 푇퐴) − 푐9푒 퐿 퐴 + 푐푁퐴 17 퐾퐿 퐻2 푂 푀 (푡) = 퐾퐺 (2.81) −푘표푥(푡−푇퐵) −퐾퐿(푡−푇퐵) 푐7 + 푐5푐1푒 + 훽 (푡 − 푇퐵) − 푐10푒 + 푐푁퐴 퐾퐿 −푘표푥(푡−푇퐶) −퐾퐿(푡−푇퐶) {푐8 − 푐3푐1푒 + 푐11푒 + 푐푁퐴 The constants from the equation (2.81) are given by

1 퐾퐺 푐1 = ( − 2 ∙ 퐶푀푅푂2) 퐾퐿 − 푘표푥 푘표푥

−푘표푥(푇퐵−푇퐴) 푐2 = 훽 − 훼(1 − 푒 )

−푘표푥(푇퐶−푇퐵) 푐3 = 훽 − 푐2푒 훼 −푘표푥(푇퐵−푇퐴) 푐4 = 훼(푇퐵 − 푇퐴) − (1 − 푒 ) 푘표푥 푐2 −푘표푥(푇퐶−푇퐵) 푐5 = 푐4 + 훽(푇퐶 − 푇퐵) − (1 − 푒 ) 푘표푥

훼 퐾퐺 훼 훼 푐6 = 2 ∙ 퐶푀푅푂2 − ( + ) 퐾퐿 퐾퐿 푘표푥 퐾퐿 (2.82)

훽 퐾퐺 푐2 훽 푐7 = 2 ∙ 퐶푀푅푂2 − ( + − 푐4) 퐾퐿 퐾퐿 푘표푥 퐾퐿

퐾퐺 푐3 푐8 = (푐5 + ) 퐾퐿 푘표푥

푐9 = 푐6 + 훼푐1

퐾퐺 −푘표푥(푇퐵−푇퐴) −퐾퐿(푇퐵−푇퐴) 푐10 = 푐6 − 푐7 + 훼푐1푒 − 푐1푐5 + 훼 (푇퐵 − 푇퐴) − 푐9푒 퐾퐿

퐾퐺 −푘표푥(푇퐶−푇퐵) −퐾퐿(푇퐶−푇퐵) 푐11 = −푐8 − 푐7 + 푐1푐2 + 푐1푐5푒 + 훽 (푇퐶 − 푇퐵) − 푐10푒 퐾퐿

2. Theory and background

2.4.6. Profile likelihood

17 17 퐻2 푂 17 The temporal behavior of the H2 O concentration 푀 (푡), in the O MRI inhalation experiment introduced in equation (2.78), can be summarized in the underlying ordinary differential equation (ODE) 푑 17 17 푀퐻2 푂(푡, 푢 (푡), 휃) = 푓(푀퐻2 푂(푡), 푢 (푡), 휃), (2.83) 푑푡 푠 푠 which depends on initial values and kinetic rate parameters contained in 휃 and an externally pro- 17 vided stimulus 푢푠(푡). The model components are linked to the measured magnitude of O MR signal 푆 (푡), by an observational function 푔 푚푒푎푠 17 퐻2 푂 푆푚푒푎푠(푡) = 푔(푀 (푡, 푢푠(푡), 휃)) + 휖(푡), (2.84) with the assumption of Gaussian errors 휖 ~ 푁(0, 휎2), which is valid if the SNR of the MR images satisfies at least SNR > 2 (Gudbjartsson and Patz, 1995). To compare the model response to the measured data, the scaled log-likelihood is calculated via

2 퐻17푂 푆푚푒푎푠,푖 − 푔 (푀 2 (푡푖, 푢푠(푡푖), 휃)) −2 log(픏) = 휒2(휃) = ∑ ( ) + const. (2.85) 휎푖 푖

The optimal parameter set 휃̂ is estimated through minimization of 휒2(휃). To estimate parameter uncertainties, the PL approach is utilized (Raue et al., 2009; Venzon and Moolgavkar, 1988). Herein, the PL of parameter 휃 is defined as 푗 2 푃퐿(휃푗) = min 휒 (휃). (2.86) 휃푖≠푗

The CI of parameter 휃푗 is then given by all parameter values for which the corresponding likeli- 2 hood value does not exceed the threshold denoted by ∆퐶퐿, the 휒 distribution with one degree of freedom and confidence level 퐶퐿 (Venzon and Moolgavkar, 1988)

2 ̂ 퐶퐼휃푗,퐶퐿 = {휃푗│푃퐿(휃푗) ≤ 휒 (휃) + ∆퐶퐿}. (2.87) From the CI, fundamental information about the identifiability of parameters 휃 can be derived and results in an infinite CI (Figure 2.10). A flat parameter profile renders the particular parameter as structurally non-identifiable. In this case, either no information about the parameter is contained in the measurements, or the other parameters can fully compensate if the parameter value is fixed (i.e. the parameters are correlated). On the other hand, a parameter profile which exceeds the threshold given by ∆퐶퐿 in maximal one direction, renders the parameter practically non- identifiable (Raue et al., 2009). Here, the data possesses insufficient information to restrict the parameter to a finite CI. Whereas structural non-identifiabilities can be resolved by fixing model parameters, e.g. through prior knowledge, elimination of practical non-identifiabilities needs additional information through new experiments.

Once the quantification model is modified, the improvement over the original model needs to be determined. For this, the statistical significance of the model change is quantified by the likeli-

2.4 Quantification of the cerebral metabolic rate of oxygen consumption

Figure 2.10. Schematic representation of the PL profiles of parameters θ1, θ2 and θ3 of a nonlinear model. Optimal parameter values are indicated by asterisks, with the likelihood value indicated by blue line; thresholds for 95% and 67% CIs are indicated by red dashed lines. Flat CI indicates structurally non- identifiable parameter θ1 (a), practically non-identifiable parameter θ2 has a profile that exceeds the 95% threshold in maximal one direction renders (b), identifiable parameter θ3 has finite CI.

hood-ratio (LR) test (Neyman and Pearson, 1933). Therein, a null hypothesis Hyp1: 휃0 is compared to an alternative hypothesis Hyp1: 휃1, with 휃0 ∈ 휃1. As the negative log-likelihood in equation (2.87) is minimized, the LR is transformed into differences of 휒2 values. The test statistic reads

2 2 푅 = 휒 (휃̂0) − 휒 (휃̂1). (2.88) Similar to parameter profiles, the test statistic is asymptotically 휒2-distributed (Wilks, 1938) with

푛푑표푓 degrees of freedom according to the difference in dimensionality of both parameter sets: 2 푛푑표푓 = 푛휃1 − 푛휃0. Based on 휒푛푑표푓, a P-value can be assigned, and the new model is rated an improvement over the original one, if P < 0.05.

3. MATERIAL AND METHODS

This chapter covers the description of the utilized hardware, constructed breathing system, implemented MRI sequence, proposed MR image reconstruction techniques and simulation framework, as well as pharmacokinetic model analysis and development. First, the experimental setup is introduced, which consists of the MRI system and the oxygen delivery system with a RB circuit. Then, the simulation framework for optimization of acquisition parameters is shown. It is followed by the description of the implemented MRI pulse sequence, which is adapted for imag- * ing of isotopes with fast T2 relaxation. Later, several MR reconstruction methods are characterized: a conventional gridding method and proposed iterative 17O reconstruction technique, which uses the edge information from the coregistered 1H MR data of higher spatial resolution. In the last section, the method of profile likelihood analysis is discussed, which was used to analyze the identifiability of the parameters of the pharmacokinetic model for CMRO2 quantification. At the 17 end, the modification of the pharmacokinetic model was proposed to properly model the H2 O signal changes and to account for efficient usage of rare and costly 17O gas.

3.1. Experimental setup

In Subsection 3.1.1 the basic components of the MRI system are presented. In particular, for 17O MRI in addition to the clinical MRI system the broad-band RF receiver to detect the MR signal at 17O resonance frequency and a custom-built transmit/receive 17O volume coil are required. In Subsection 3.1.2 the pulsed oxygen delivery system with the RB circuit is presented, which is needed for dynamic imaging of 17O MR signal change before and after 17O gas inhalation. At the end, the acquisition and inhalation parameters of four 17O MRI experiments performed at 3 T are summarized (Subsection 3.1.3). More information on the MRI equipment can be found in (McRobbie et al., 2003) and (Collins, 2016).

3.1.1. MRI system

An MRI system consists of a strong superconductivity magnet to create a stable magnetic field, gradient coils for spatial localization and an RF transmit/receive coil for 푩ퟏ excitation and MR signal detection (Figure 3.1). The MRI system is placed in an RF-shielding enclosure (Faraday cage) to avoid interference of external electromagnetic fields with MR signals, which may lead to the signal distortion and image artefacts. For RF shielding, the magnet room which houses the MRI system is lined on all six sides with copper, and wire embedded glass is used for the win- dows. A special penetration panel is available for electrical connections between the magnet system and the control units outside the magnet room. For other physical connections such as liquid or gas pipelines, the penetration panel is equipped with waveguides that can filter out external interference effectively. 17O gas was also delivered from the control room to the volunteer inside the MRI system through one of these waveguides (cf. Subsection 3.1.2).

3.1 Experimental setup

Figure 3.1. (a) Schematic illustration of the MRI system, where the main components are indicated. (b) Configuration of the gradient coils used for spatial encoding in all three dimensions. Transceiver indicates the RF system comprising a transmitter, coil and receiver (cf. description in the main text). Figures taken from (Coyne, 2012).

In this work, the MRI measurements were performed at a 3 T Siemens MAGNETOM TRIO MRI system equipped with a broadband transmit/receive RF system, enabling MRI measurement 17 at the Larmor frequency of the O nucleus (푓0 = 16.7 MHz). The system has a superconducting magnet of cylindrical shape where the volunteer/patient is placed inside the magnet bore (diameter: 60 cm, system length: 213 cm) (Figure 3.1a). The head of the volunteer, which was studied in this work, was positioned at the center of the magnet, where the static magnetic field has the highest homogeneity. Technically, it is not possible to produce a perfectly homogeneous magnet due to construction limits, but the field homogeneity is improved from about 100 ppm to less than 1 ppm using so-called passive shimming. For this, small pieces of metal pellets are fixed at the magnet bore. However, the field is destroyed as the patient is placed inside the MRI system. The field homogeneity can be then improved by active shimming, in which the current running through the shim coils, similar in design to gradient coils, is adjusted to optimize magnetic field homogeneity over the volunteer’s head.

Three magnetic gradient fields, 퐺푥, 퐺푦 and 퐺푧 are required for spatial location in MRI introduced in Subsection 2.2.1. These fields are generated by the gradient coils mounted on a cylindrical former inside the magnet bore (Figure 3.1b). At the center of the gradient coils, which is also the center of the magnet (isocenter), all the gradient fields are zero and linearly increase in all three directions with the increase of the distance to the isocenter (cf. equation (2.47)). The maximum strength of the gradient of the utilized MRI system is 40 mT/m and the maximum gradient ramp-up (slew rate) is 180 mT/m/ms.

The third component of the MRI system is the RF system, which consists of a transmitter, RF coils and a receiver. The transmitter generates the RF pulses at the resonance frequency with the

3. Material and Methods

desired bandwidth, phase, amplitude and shape. A transmit coil is used to excite spines. For 1H MRI measurements, the body coil built-in volume coil at the magnet bore (body coil) is used, which generate a 푩ퟏ field. The MR signal is detected with one or more receive coils. For optimal SNR, it is necessary to minimize the coil dimensions so that the coil volume is filled as much as possible by the sample and as close as possible to it. Thus, a commercial volume head coil was used for 1H MRI signal detection instead of the body coil. The detected MR signal is typically five orders of magnitude smaller than the excitation signal. It is increased by a low-noise preamplifier before entering the receiver to make the MR signal compatible with the ADC level and to make the best use of the available dynamic range of the digitalization offered by the ADC.

For 17O MRI, a transmit/receive 17O volume head coil was used. The 17O coil was constructed and tuned to the 17O Larmor frequency of 16.7 MHz (at 3 T) by Robert Borowiak from the Med- ical Physics department of the University Medical Center Freiburg (Borowiak, 2017). In the first two out of four 17O MRI inhalation experiments a prototype coil driven in a linear mode was used (Borowiak et al., 2014b), and for the last two experiments the coil was operated in quadrature mode (a detailed description of the inhalation experiments is presented in the following subsections). A quadrature driven mode is achieved by supplying the coil with 90° out of phase RF currents at two separate ports. During detection, the signals from the two quadrature modes add constructively, resulting in maximal improvement by a factor of √2 (i.e. about 40%) in SNR compared to the coil driven in linear mode, as well as higher B1 homogeneity in our setting.

3.1.2. 17O gas breathing setup

The first setup for 17O gas delivered in pulses was proposed by (Hoffmann et al., 2011). In that breathing system, three pneumatic valves were designed for switching between room air and oxygen or nitrogen supply. The pressurized nitrogen was used to flush the remaining 17O gas from the tubing and to change the membrane positions in the valves. In our group a similar 17O breathing setup was developed, in which neither pneumatic valves nor nitrogen gas was used. Thus, the system was built only from standard breathing equipment, providing easy handling of the oxygen gas. The quantification of the breathing system is presented in (Wagner, 2014).

The positioning of the oxygen breathing system on the volunteer during the 17O MRI inhalation experiment and the constituents of the setup are shown in Figure 3.2. A combination of a commercially available DODS (Oxytron3, Weinmann, Hamburg, Germany) with a custom-built RB circuit was used to deliver 70% isotope-enriched 17O gas to the volunteer lying in the MRI system. The DODS delivers small pulses of 17O gas with a volume of 40 - 50 mL via a nasal cannula through thin tubing (with the length of 7.5 m, the inner diameter of 4 mm and the dead volume of 94 ml) to the alveoli. The release of the 17O pulses is triggered in the DODS by the detection of small pressure changes at the nasal cannula at the beginning of each inhalation. During the experiments, the number of 17O pulses was counted so that the administered volume of 17O gas could be calculated. The DODS system and the cylinder with 17O gas (NUKEM Isotopes Imag- ing, Alzenau, Germany) were placed outside the magnet room because they are not MR safe (i.e. 39

3.1 Experimental setup

Figure 3.2. (a) Positioning of the breathing system on the volunteer in the 17O MRI inhalation experiment. (b) Constituents of the oxygen breathing setup. 17O gas stored in the gas cylinder is delivered in pulses via DODS into the nasal cannula attached to the breathing mask. A rebreathing circuit is connected to the mask and consists of a CO2 filter and two reservoir bags.

they contain magnetic components). They were connected via a two-stage pressure regulator (Oxytron WM 30171, Weinmann, Hamburg, Germany) to provide the required DODS input pressure of 1.5 bar.

The RB circuit (the bottom of Figure 3.2b) was implemented for efficient usage of rare and expensive 17O gas. The circuit allows for re-inhalation of the stored 17O gas in subsequent inhalation cycles. During the two phases of the inhalation experiment, the DODS phase and the RB phase, (cf. Subchapter 2.4.5) the circuit was closed, whereas before (the baseline phase) and after (wash- out phase) 17O inhalation it was open and the patient was breathing room air. The circuit consists of 22 mm diameter tubing connecting the CO2 absorption filter, two reservoir bags (each of the volume of 2 L) and an adult double port sleep apnea (CPAP) breathing mask (Intersurgical, Berk- shire, United Kingdom). The thin tubing was put through the mask to deliver 17O gas to the nasal cannula. It is worth noting, that since the RB circuit was closed during 17O gas delivery in small pulses, the rest of the tidal volume was filled by the exhaled gas stored in the RB circuit.

3.1.3. 17O MRI inhalation experiments

Four dynamic 17O MRI experiments with inhalation of 17O gas were performed on a single volunteer (male, 50 years) using the pulsed delivery system and the RB circuit described in the previous subsection. In the experiments referred to as Exp1, Exp2, Exp3 and Exp4 the 3D radial acquisition with the density adaptation was used (cf. Section 3.3). All experiments consisted of four subsequent phases: the baseline phase, DODS phase, RB phase and wash-out phase (cf. Subchapter 2.4.5). The experiments differ in the spatial resolution, number of radial projections, duration and amount of the inhaled 17O gas as well as some other parameters listed in Table 3.1. The following parameters were the same for all the experiments: temporal resolution ∆푡 = 1 min, duration of the non-selective RF excitation pulse Tpulse = 0.8 ms, TE = 0.52 ms, 1 average, 128 sampling 40

3. Material and Methods

Exp1 Exp2 Exp3 Exp4 Inhaled 70%-enriched 17O gas, L 2.7 2.5 3.0 1.5 17 Volume of O2 pulse, mL 40 50 50 50 Number of 17O pulses 68 50 60 30 Number of spokes 7500 8570 7500 7922 per measurement Spatial resolution, mm 10 8 10 8 TR, ms 8.00 7.00 8.00 7.58 BW, Hz/pixel 150 175 150 175

TRO, ms 6.7 5.7 6.7 5.7 Duration of the baseline phase, 11.8 10.8 10.5 9.2 min Duration of the DODS phase, min 5.0 4.2 6.7 4.1 Duration of RB phase, min 7.5 5.5 5.6 5.0 Duration of wash-out phase, min 22.0 25.1 20.9 20.1 Total duration, min 45 44 45 40 16O gas in the RB phase no no no yes

Table 3.1: Inhalation and acquisition parameters of four 17O MRI experiments. In this table only the parameters which differ between individual experiments are listed. The parameters that were the same for all experiments are listed in the main text.

points per projection, RF spoiling was applied. Furthermore, in Exp1 and Exp2 a prototype RF coil was operated in linear mode, whereas in Exp3 and Exp4 it was operated in quadrature mode (cf. Subsection 3.1.1).

In Exp4 the supply was switched to 16O gas after 17O gas inhalation, which solved the problem of oxygen shortage at the end of the RB phase and increased patient comfort. In total, 1.35 L of 16O gas with purity > 99.5% was delivered in 27 pulses with the volume of 50 mL. Additionally, only 1.5 L of 17O gas was administered in Exp4.

To ensure volunteer safety, temperature measurements of the RF-induced heating with Bowman temperature probes (Bowman, 1976) were performed prior to the 17O experiments and the same MR sequence in an agar gel phantom, and no increase in temperature was observed.

3.2. Simulation framework

To optimize the acquisition parameters and to perform a quantitative comparison of the MR reconstruction algorithms in this work, a simulation framework was developed (Kurzhunov et al., 2016a). Two types of 3D 17O MRI brain phantoms, an analytical tumor and a numerical phantom were constructed in image and k-space domains and used to simulate the dynamic 17O MRI inhalation experiment. In particular, a wide range of acquisition bandwidths, spatial and temporal resolutions were investigated to find out which of the parameters give the best correspondence between the model input and output CMRO2 values. The numerical phantom was used for a comparison of gridding and iterative reconstruction techniques, presented in Section 3.4, as its original representation provided the ground truth that is required for a quantitative comparison of the image quality.

The analytical phantom takes advantage of the analytical representation of k-space signal intensities. The phantom can consist of any combination of 3D objects with known analytical expressions of their FT representation (e.g., boxes, cylinders, or ellipsoids). As a consequence, it enables bypassing the need for interpolation in the Fourier domain so that a more objective comparison of the influence of different acquisition parameters on CMRO2 precision can be tested. The advantage of this numerical phantom is that more complex brain structures that are present in human brain can be analyzed, which is important for a realistic comparison of image-reconstruction methods.

3.2.1. ACROBAT phantom

An AnalytiCally Represented Oxygen-17 BrAin Tumor (ACROBAT) phantom was first constructed in image domain as a combination of several ellipsoids; each of them corresponds to a specific human brain or tumor tissue (Figure 3.3). For this, the 3D Shepp-Logan phantom 17 (Shepp and Logan, 1974; Shepp, 1980) was modified to account for H2 O contrast, and a heterogeneous tumor region was added (cf. Table 3.2). In total, six tissues were included in the AC- ROBAT phantom: GM, WM and CSF brain regions as well as three tumor glioblastoma regions: contrast-enhancing rim (CE), necrotic tumor center (NE) and perifocal edema (PE). The 3D image is then represented as a linear combination of 푁 ellipsoids 퐼푚 (푥, 푦, 푧): 푒푙 푒푙,푖 푁푒푙 퐼푚 17 (푥, 푦, 푧) = ∑ 퐼푚 (푥, 푦, 푧), (3.1) 퐻2 푂 푒푙,푖 푖푒푙=1 with

2 2 2 푝푒푙,푥 푝푒푙,푦 푝푒푙,푧 휌푒푙,푖 for ( ) + ( ) + ( ) ≤ 1 푎푖 푏푖 푐푖 퐼푚푒푙,푖(푥, 푦, 푧) = 2 2 2 , (3.2) 푝푒푙,푥 푝푒푙,푦 푝푒푙,푧 0 for ( ) + ( ) + ( ) > 1 { 푎푒푙,푖 푏푒푙,푖 푐푒푙,푖 and

3. Material and Methods

Figure 3.3. ACROBAT phantom consisting of GM, WM, CSF and three tumor glioblastoma regions: contrast-enhancing rim (CE), necrotic tumor center (NE) and perifocal edema (PE). The color intensities correspond to the H217O concentration. Please refer to Table 3.2 for a detailed specification. The matrix size is 128×128×128.

푝푒푙,푥 푥 − 훿푥,푖 −1 (푝푒푙,푦) = 푹풐풕풊(휙1, 휙2, 휙3) (푦 − 훿푦,푖) (3.3) 푝 푒푙,푧 푧 − 훿푧,푖

17 where 휌푒푙,푖 is the signal intensity within the ith ellipsoid that correspond to the H2 O concentration in a corresponding tissue, 푎푒푙,푖, 푏푒푙,푖 and 푐푒푙,푖 are half the length of the principal axes, and

훿푥,푖, 훿푦,푖, 훿푧,푖 describe the translation of the ith ellipsoid from the center of coordinate system. The 17 specification for the ACROBAT phantom is summarized in Table 3.2. The H2 O concentration 17 was calculated based on the H2 O natural abundance of 20.56 µmol/gwater, water partition coefficients [0.71 g/mL for WM, 0.83 g/mL for GM and 0.99 g/mL for CSF (Whittall et al., 1997); 0.95 g/mL for NE, 0.72 g/mL for CE and 0.90 g/mL for PE (Hoffmann et al., 2014)], and averaged density of brain tissue of 1.038 g/mL (Takagi et al., 1981). 푹풐풕풊 is the rotation matrix which can be expressed in terms of the Euler angles:

푹풐풕풊(휙1, 휙2, 휙3) = 푹풐풕풙,풊(휙1)푹풐풕풚,풊(휙2)푹풐풕풛,풊(휙3) (3.4) with

1 0 0 푹풐풕풙,풊(휙1) = (0 cos(휙1) − sin(휙1)), 0 sin(휙1) cos(휙1)

cos(휙2) 0 sin(휙2) 푹풐풕풚,풊(휙2) = ( 0 1 0 ), (3.5) − sin(휙2) 0 cos(휙2)

cos(휙3) −sin(휙3) 0 푹풐풕풙,풊(휙3) = (sin(휙3) cos(휙3) 0). 0 0 1

퐼푚 17 (푥, 푦, 푧) Taking the linearity of the FT into account, the 3D FT of 퐻2 푂 can be written as

3.2 Simulation framework

17 Coordinates Euler H2 O Ellipsoid of the Axis length angle 흓ퟏ, concentration, tissue center grad µmol/L 1 (0, -0.018,0) (0.663, 0.874, 0.880) 0 20.35 CSF 2 (0, -0.018,0) (0.652, 0.864, 0.870) 0 17.06 GM 3 (0, -0.018,0) (0.633, 0.838, 0.844) 0 14.60 WM 4 (0, 0.55,0) (0.200, 0.200, 0.200) 0 17.06 GM 5 (0.120, 0,0) (0.077, 0.217, 0.154) 18° 20.35 CSF 6 (0.120, 0,0) (0.085, 0.239, 0.169) 18° 17.06 GM 7 (0, -0.018,0) (0.150, 0150, 0.150) 0 19.59 NE (-0.25, - (0.200, 0.200, 0.200) 0 8 14.70 CE 0.27,0) (-0.32, - (0.250, 0.400, 0.250) 18° 9 18.50 PE 0.25,0)

Table 3.2: Specification for the ACROBAT phantom consisting of 9 ellipsoids which represent three brain tissues (GM, WM and CSF) and three glioblastoma tumor regions (contrast-enhancing rim (CE), necrotic tumor center (NE) and perifocal edema (PE)). Coordinates of the center of the ellipsoids and their axis length are normalized to one. In the simulation they were multiplied by the FOV/2= 120 mm and FOV= 240 mm for the coordinates and the axis lengths, correspondingly. In this phantom, only ϕ 1 angle was used, the other two Euler angles, ϕ 2 and ϕ 3, were set to zero. Calculation of the H217O concentration is described in the main text.

푁푒푙

푆푘푖푑푒푎푙(푘푥, 푘푦, 푘푧) = ∑ 푆푘푖푑푒푎푙,푖(푘푥, 푘푦, 푘푧), (3.6)

푖푒푙=1 with

∞ ∞ ∞ −푖2휋(푘푥푥+푘푦푦+푘푧푧) 푆푘푖푑푒푎푙,푖(푘푥, 푘푦, 푘푧) = ∫ ∫ ∫ 퐼푚푒푙,푖(푝푥, 푝푦, 푝푧)푒 푑푥푑푦푑푧. (3.7) −∞ −∞ −∞ Using the analytical expression of an ellipsoid (Koay et al., 2007), the FT in the equation (3.7) can be expressed as:

푇 sin(2휋퐾푖) − 2휋퐾cos(2휋퐾푖) 푆푘 (푘 , 푘 , 푘 ) = 휌 푎 푏 푐 푒−푖2휋풌 휹푖 [ ], (3.8) 푖푑푒푎푙,푖 푥 푦 푧 푒푙,푖 푖 푖 푖 2 3 2휋 퐾푖 where

2 2 2 ̂ −1 퐾푖 = √(푎푖푘̂푥) + (푏푖푘̂푦) + (푐푖푘̂푧) and 풌 = 푹풐풕풊 풌. (3.9)

Thus, the k-space representation of the phantom can be calculated directly.

3. Material and Methods

Figure 3.4. Three orthogonal views of the numerical 17O MRI brain phantom constructed from WG, GM and CSF brain regions (obtained from the segmented 1H MRI data set) and using tissue-specific H217O concentration. The matrix size is 128×128×128.

3.2.2. Numerical phantom

To create analytically more precise phantom, the software tool Statistical Parametric Mapping Package (SPM8) (Ashburner and Friston, 2005, 2000) was used to segment GM, WM and CSF brain regions in 1H MPRAGE image (cf. Subsection 3.3.1). These segmented regions with initial 17 signal intensities normalized to 1 were then multiplied with the tissue-specific H2 O concentration (cf. Subsection 3.2.1). The numerical 17O MRI brain phantom was constructed as the sum of those three components, interpolated into the same matrix as the ACROBAT phantom (128×128×128)

퐼푚 17 (푥, 푦, 푧) = 휌 퐼푚 (푥, 푦, 푧) + 휌 퐼푚 (푥, 푦, 푧) + 휌 퐼푚 (푥, 푦, 푧). 퐻2 푂 퐺푀 퐺푀 푊푀 푊푀 퐶푆퐹 퐶푆퐹 (3.10)

To calculate the k-space representation of the phantom, a NUFFT operator 푨푛푢푓푓푡 (Fessler and Sutton, 2003) was constructed, which represents 17O MRI measurement using parameters of a single acquisition from Exp1 (cf. Subsection 3.1.3):

푆푘 (푘 , 푘 , 푘 ) = 푨 ∙ 퐼푚 17 (푥, 푦, 푧). 푖푑푒푎푙 푥 푦 푧 푛푢푓푓푡 퐻2 푂 (3.11)

3.2.3. MR reconstruction quality assessment

The signal intensity in k-space estimated in equation (3.11) represents the ideal case that does not account for the measurement noise and tissue relaxation. To simulate realistic 17O MR images of * * the human brain at 3 T, a T2 decay with transverse relaxation time of T2 = 2 ms (Borowiak et al., 2014a) was included, which acts as an exponential filter along each radial spoke in k-space. Addi- tionally, complex Gaussian noise was added to obtain the experimentally observed SNR of 7 for 17O MRI with radial sampling and the nominal spatial resolution ∆푥 =10 mm (cf. Exp1 in Sub- section 3.1.3). The realistic k-space MR signal 푆푘 (풌, 푡̃) can be then expressed as 푟푒푎푙 −(푇퐸+푡̃)⁄푇∗ 푆푘푟푒푎푙(풌, 푡̃) = 푆푘푖푑푒푎푙(풌, 푡̃)푒 2 + 퐴푛표푖푠푒(푟푎푛푑푛 + 푗 ∙ 푟푎푛푑푛), (3.12)

3.2 Dynamic framework

where 푡̃ indicates the sampling time for a certain position 풌 = (푘푥, 푘푦, 푘푧) along each radial spoke in k-space and the second term represents zero-mean complex Gaussian noise with the amplitude 퐴푛표푖푠푒, which was adjusted to get the desired SNR in the resulted MR image.

The numerical 17O MRI phantom was used to quantitatively compare different reconstruction techniques which are described later in Section 3.4. Reconstruction error of the complete phantom image was obtained by comparison of the reconstructed image 퐼푚표푢푡 with the original image 퐼푚 17 (equation (3.10)), which was used as a ground truth. First, the root-mean-square error 퐻2 푂 (RMSE) was calculated for all image pixels 푁푝푥 within the brain phantom

1 2 푅푀푆퐸(퐼푚 , 퐼푚 17 ) = ∑ ∑ ∑(퐼푚 (푥 , 푦 , 푧 ) − 퐼푚 17 (푥 , 푦 , 푧 )) 표푢푡 퐻2 푂 √ 표푢푡 푖 푗 푘 퐻2 푂 푖 푗 푘 (3.13) 푁푝푥 푖 푗 푘

Second, the structural similarity (SSIM) quality assessment index was calculated. This alternative measure is a more native way of image comparison, because it is based on the assumption that human visual perception is highly adapted for extracting structural information from a scene (Wang et al., 2004, 2003). SSIM considers image degradation as perceived change in structural information and is based on the computation of three terms: the luminance term 푙푆푆퐼푀, the contrast term 푐푆푆퐼푀and the structural term 푠푆푆퐼푀. The overall index is a multiplicative combination of the three terms

1 푆푆퐼푀(퐼푚 , 퐼푚 17 ) = ∑ ∑ ∑ 푆푆퐼푀(푥 , 푦 , 푧 ), 표푢푡 퐻2 푂 푖 푗 푘 (3.14) 푁푝푥 푖 푗 푘 with

훾1 훾2 훾3 푆푆퐼푀(푥푖, 푦푗, 푧푘) = 푙푆푆퐼푀(푥푖, 푦푗, 푧푘) ∙ 푐푆푆퐼푀(푥푖, 푦푗, 푧푘) ∙ 푠푆푆퐼푀(푥푖, 푦푗, 푧푘) (3.15) and

2휇 (푥 , 푦 , 푧 )휇 17 (푥 , 푦 , 푧 ) + 퐶 표푢푡 푖 푗 푘 퐻2 푂 푖 푗 푘 1 푙푆푆퐼푀(푥푖, 푦푗, 푧푘) = 2 2 , 휇 (푥 , 푦 , 푧 ) + 휇 17 (푥 , 푦 , 푧 ) + 퐶 표푢푡 푖 푗 푘 퐻2 푂 푖 푗 푘 1

2휎 (푥 , 푦 , 푧 )휎 17 (푥 , 푦 , 푧 ) + 퐶 표푢푡 푖 푗 푘 퐻2 푂 푖 푗 푘 2 푐푆푆퐼푀(푥푖, 푦푗, 푧푘) = 2 2 , (3.16) 휎 (푥 , 푦 , 푧 ) + 휎 17 (푥 , 푦 , 푧 ) + 퐶 표푢푡 푖 푗 푘 퐻2 푂 푖 푗 푘 2

휎 17 (푥 , 푦 , 푧 ) + 퐶 표푢푡,퐻2 푂 푖 푗 푘 3 푠푆푆퐼푀(푥푖, 푦푗, 푧푘) = . 휎 (푥 , 푦 , 푧 )휎 17 (푥 , 푦 , 푧 ) + 퐶 표푢푡 푖 푗 푘 퐻2 푂 푖 푗 푘 3 where 휇 17 , 휇 , 휎 17 , 휇 and 휎 17 are the local means, standard deviations, and 퐻2 푂 표푢푡 퐻2 푂 표푢푡 표푢푡,퐻2 푂 cross-covariance of the original and reconstructed images. The coefficients were chosen as in the -4 -4 original publication (Wang et al., 2004): 훾1 = 훾2 = 훾3 = 1, 퐶1 = 10 , 퐶2 = 9·10 , and 퐶3 = 퐶2/2.

With increasing correspondence between the original 17O image and the reconstructed image, RMSE values decrease and SSIM values increase.

3. Material and Methods

WM GM CSF CE NE PE CMRO , 2 1.59 0.62 0 0.62 0.56 0.52 µmol/gtissue/min

Table 3.3: The input CMRO2 values (in µmol/gtissue/min) from 15O-PET studies (Ito et al., 1982; Leenders et al., 1990) that were used in the dynamic framework.

3.2.4. Dynamic framework

To optimize the acquisition parameters of the 17O MRI experiments, the signal intensities of the ellipsoids of the ACROBAT phantom (equations (3.1), (3.2), and (3.12)) and brain regions of the 17 numerical phantom (equations (3.11) and (3.12)) were modeled according to the expected H2 O concentration change during and after inhalation of 17O gas (cf. Figure 2.8). For this, the input 15 CMRO2 values were used (Table 3.3) which are based on the results of O-PET studies (Ito et al., 1982; Leenders et al., 1990). Later, a series of 45 phantom images representing the dynamic 17 O MRI experiment were reconstructed; the output CMRO2 values were quantified in various brain and tumor regions and compared to the input CMRO2 values. In particular, the deviations of the output from the input CMRO2 values were calculated for spatial resolution over a range of 5 mm to 15 mm, temporal resolution from 30 s to 300 s and readout bandwidth of 50 Hz/pixel to 1000 Hz/pixel. Finally, 2D scatter plots of the CMRO2 deviations were created for the pairs of either spatial resolution and readout bandwidth, or spatial and temporal resolutions. For the AC- ROBAT phantom, the dynamic simulation of each parameter set was done 10 times in order to determine the standard deviation of the output CMRO2 values.

3.3. Image acquisition

This chapter describes the MR pulse sequences that were used for signal acquisition in 1H and 17 1 O MRI experiments. In Subsection 3.3.1 a T1-weighted H MPRAGE sequence is presented, which was used to construct the numerical brain phantom and compare different MR reconstruction techniques. In 17O MRI, UTE acquisitions with FID signal detection are required to account 17 * for rapid O MR signal decay due to a fast T2 relaxation. In addition, the density adaptation was implemented that provides a more homogeneous sampling of k-space and higher SNR (Subsec- tion 3.3.2). 3D coverage with the radial center-out sampling pattern is described in Subsection 3.3.3.

1 3.3.1. H MRI: T1-weighted MPRAGE sequence

1 T1-weighted H Magnetization Prepared RApid Gradient Echo (MPRAGE) sequence (Mugler and Brookeman, 1990) was used to acquire high-resolution anatomical image. The T1-weighted contrast of the MPRAGE data was needed for brain tissue segmentation and construction of the numerical 17O MRI phantom, which was used for optimization of 17O MRI acquisition parameters (cf. Subsection 3.2.2) and comparison of the reconstruction methods (cf. Subsection 3.4.6). Furthermore, high tissue contrast provided by the MPRAGE sequence is required to extract the tissue border information for the novel 17O reconstruction method, which uses the edge information from the gradient of the proton image to improve the SNR and quality of 17O MRI images (cf. Subsection 3.4.4).

The MPRAGE sequence uses a magnetization preparation with a 180° inversion pulse to encode the longitudinal magnetization with a T1-dependent contrast. After an inversion decay TI, a rapid gradient echo acquisition is used to sample the prepared magnetization using low flip angles. At the end, a magnetization recovery occurs. This preparation-acquisition-recovery cycle is repeated until the desired 3D k-space volume is covered. The sequence parameters used in this study are: TE = 2.86 ms, TR = 2300 ms, duration of the inversion time TI = 1100 ms, 3 BW = 130 Hz/pixel, flip angle 휃푀=12°, spatial resolution (0.6 × 0.6 × 1) mm , total duration 8:36 min. More information about the basics of the MRI pulse sequence can be found in Subsec- tion 2.2.5.

3.3.2. 17O MRI: density adapted acquisition

* 23 17 For nuclei with fast T2 relaxation (e.g., Na or O) it is advantageous to start the signal acquisition immediately after the RF excitation. For this, 3D radial center-out acquisitions of the FID signal are typically used, in which the k-space points are sampled equidistantly as soon the gradients are ramped up to. A disadvantage of this acquisition scheme is that the density of the sampling points is high, close to the center of k-space, and low in the outer part. More homogeneous coverage of k-space can be achieved by keeping the uniform sampling density in the outer part of

3. Material and Methods

Figure 3.5. (a): Schematic representation of the DAPR sequence. Unlike conventional radial acquisition with constant gradient amplitude Grad, the readout gradient in DAPR sequence GDAPR is divided into three parts: the ramp up sampling (A), the part with the constant gradient amplitude G0 (B), and the density-adapted part which starts at t = t0 (C). The echo time (TE) is defined as the interval between the middle of the RF excitation pulse and the beginning of readout. (b): Illustration of the 3D center-out radial sampling of k-space (Courtesy of Siemens Healthcare).

k-space, as was proposed in (Nagel et al., 2009). Such a sampling, called the density adapted projection reconstruction (DAPR) technique, was shown to provide higher SNR and spatial resolution in 23Na MRI. In this word, it was implemented for the 17O MRI acquisitions.

The sampling density for radial k-space trajectories with the gradient strength 퐺(푘) is given by

1 퐷 (푘) = . (3.17) 푟푎푑 4휋푘2퐺(푘) For uniform sampling density the gradient shapes are designed in such a way that in each spherical shell of k-space, the averaged sampling density is constant. Due to hardware restrictions, this is impossible for the inner sphere of k-space with a certain radius 푘0 (cf. equation (3.21)), however, the uniform sampling density can be achieved for k-space points outside this inner sphere by requiring

1 1 2 = 2 . (3.18) 4휋푘0 퐺(푘0) 4휋푘 퐺(푘) The k-space trajectories that fulfill this requirement are expressed as

3 2 3 푘(푡) = √3(훾⁄2휋)푘0퐺(푡 − 푡0) + 푘0 for 푡 > 푡0, (3.19) where 푡0 is the time point at which 푘0 is acquired. According to the equation (2.53) the gradient strength is proportional to the derivative of equation (3.19)

2 2 2 3 − (3.20) 퐺퐷퐴푃푅(푡) = 푘0 퐺0(3(훾⁄2휋)푘0퐺(푡 − 푡0) + 푘0) 3 for 푡 > 푡0.

3.3 Image acquisition

The gradient shape of the DAPR sequence is illustrated in Figure 3.5a. The ramp up part (A) in which the gradient strength is linearly increasing is followed by a constant gradient strength of 퐺0 (B). Finally, at 푡 > 푡0 (C) the gradient is shaped according to the equation (3.20). In practice, the minimal values of 푘0 are limited by the maximum slew rate (Nagel et al., 2009), which is largest at the transition between part B and C and given by

4∆푥훾 푆퐵→퐶 = 2 , (3.21) 2휋퐺0 푝퐷퐴푃푅 where ∆푥 is the nominal spatial resolution and the fraction 푝 is defined as 퐷퐴푃푅 푘0 푝퐷퐴푃푅 = . (3.22) 푘푚푎푥

17 In all O MRI experiments described in Subsection 3.1.3 푝퐷퐴푃푅 = 0.3 was used. According to the equation (2.53) the non-constant gradient shape in the density-adapted part yields a non- equidistant separation of the sampled k-space points ∆푘. To prevent aliasing artefacts, the distance between two subsequent k-space points must be smaller than the inverse of the FOV (equation (2.59)).

3.3.3. 17O MRI: 3D radial sampling

For the 3D k-space coverage, the density-adapted gradient must be applied in all different directions as it is illustrated in Figure 3.5b. For this, the x-,y- and z-components of the gradients are given by

퐺퐷퐴푃푅,푥 = 퐺퐷퐴푃푅sin휓1cos휓2,

퐺퐷퐴푃푅,푦 = 퐺퐷퐴푃푅sin휓1sin휓2, (3.23)

퐺퐷퐴푃푅,푧 = 퐺퐷퐴푃푅cos휓1, where 휓1 ∈ [0, 휋) and 휓2 ∈ [0,2휋) are polar and azimuthal angles in the spherical coordinates, correspondingly. The homogeneous distribution of 푁푠푝 spokes in the 3D k-space can be calculated according to (Rakhmanov et al., 1994)

휓2,푛 = arccos(휁푛), 0 for 푛 = 1 (3.24) 휓 = 4휋 1,푛 (휓 + √ ) (푚표푑 2휋) for 푛 > 1 1,푛−1 푁 (1 − ℎ2) { 푠푝 푛 with

2(푛 − 1) 휁푛 = −1 + (3.25) 푁푠푝 − 1 In this case the distance between k-space points acquired in two subsequent spokes is equal in both polar and azimuthal directions.

3. Material and Methods

Due to the radial type of acquisition, the inner k-space points are sampled more compared to the points in the outer part of k-space. In this case, in order to fulfill the Nyquist criterion (equation (2.59)) the number of spokes 푁 must be 푠푝 2 푁푝푥 푁 = 4휋 ( ) , (3.26) 푠푝 2 where 푁푝푥 is the number of pixels along one direction in the reconstructed MR image. For example, for a 64×64×64 matrix, 12868 spokes are required, whereas for a 128×128×128, matrix 푁푠푝 must be increased to 51472. Furthermore, as radial sampling is non-Cartesian, either gridding to a Cartesian grid before computing the FT or iterative reconstruction of the MR data is needed. Both reconstruction methods are described in the following Section.

3.4. Image reconstruction

The standard reconstruction method of MR images acquired with radial sampling is the Kaiser- Bessel gridding presented in Subsection 3.4.1. For correction of PVEs, which are present in 17O * MR images due to low spatial resolution of 8 to 10 mm and fast T2 of about 2 ms, a new method for 17O MR image estimation called DIESIS was proposed in Subsection 3.4.2 (Kurzhunov et al., 2017b). Alternatively, reconstruction can be represented as a constrained optimization problem and images are obtained in an iterative procedure (Subsection 3.4.3). Two iterative reconstructions with two different constraints were implemented. First, the total variation was used as a constraint, and secondly, an anisotropic diffusion constraint was proposed, which uses the edge information from 1H MPRAGE data (Kurzhunov et al., 2017c, 2017d, 2016b, 2015) (Subsection 3.4.4). The conjugate gradient algorithm, which was used for iterative image reconstruction, is described in Subsection 3.4.5. The numerical brain phantom and in vivo data from Exp3 and Exp4 were used to compare gridding and iterative reconstruction methods as described in Subsection 3.4.6. Additional information about the MR image reconstruction can be found in (Bernstein et al., 2004; Block, 2008; Block et al., 2007; Jackson et al., 1991; Pipe, 2000).

3.4.1. Kaiser-Bessel gridding

In 3D radial sampling k-space is nonuniformly sampled, and the data must be resampled onto a uniform rectilinear grid before FFT. The simplest and fastest method is a nearest-neighbor interpolation (Oesterle et al., 1999), where each measured k-space point is interpolated to the nearest point on the rectilinear grid. However, the k-space points which are in between several grid points result in high level of artefacts. Another method for reconstruction of radially sampled k- space data is based on the projection reconstruction algorithm (Bracewell and Riddle, 1967; Smith et al., 1973). This method has two major drawbacks. First, it has intrinsically low SNR, because a filter is used that suppresses in the MR signal near the center of k-space. Second, in the DAPR sequence the sampling is not uniform along the projection (cf. Subsection 3.3.2), which results in image distortion and artefacts.

In gridding reconstruction, the k-space data is resampled onto a Cartesian grid after being convolved with a smooth function (Jackson et al., 1991; O’Sullivan, 1985). The sinc function, which is used in the reconstruction of the Cartesian data (cf. Subsection 2.2.3) has an infinite extend in k- space and therefore relatively long computational time is required for multiplication of the sinc function by all measured k-space points. To reduce the reconstruction time, the sinc function in the equation (2.60) is replaced by the gridding kernel, which is a compactly supported function. The gridding kernel must be a separable function with the known analytical expression of its FT. In the gridding reconstruction, the convolution of the k-space data with the gridding kernel is equivalent to the image multiplication by the FT of the kernel. It results in image shading which is lowering of the intensity in the outer part of the image, and can be removed by deapodization (i.e., dividing the image by the FT of the kernel). Furthermore, aliasing from the original sampling is only attenuated but not eliminated, because the FT of the kernel has an infinite extent in the

3. Material and Methods

Figure 3.6. (a): Schematic representation of the gridding algorithm. Intensity of each measured k-space point (red) is distributed to the neighboring uniform rectilinear grid points (blue) that lie within the region with the width WKB (gray). The interpolation of the measured signal intensity Sk(krad) to the grid points kgrid is weighted by the distance dependent gridding function CKB(|krad - kgrid|). (b): Two dimensional schematic representation of the density compensation function. To calculate this function, an area element with borders shown in blue is assigned to each sampled point (red). Similarly, the density compensation is done for 3D sampling.

image space. The aliasing can be removed by k-space oversampling to increase the FOV in which the aliased replicates are pushed further apart in the image, and then discarding the excess FOV after the FT.

A KB function was chosen as the gridding kernel, because it provides the best approximation the ideal image (Jackson et al., 1991). The KB function (Kaiser, 1966) is given by

퐶퐾퐵(푘푥, 푘푦, 푘푧) = 퐶퐾퐵(푘푥) ∙ 퐶퐾퐵(푘푦) ∙ 퐶퐾퐵(푘푧), (3.27) with

2 1 2푘푥 퐼퐵푒푠푠푒푙,0 ∙ (훽퐾퐵√1 − ( ) ) for |∆푘푥| ≤ 푊퐾퐵 퐶퐾퐵(푘푥) = 푊 푊 , (3.28) 퐾퐵 퐾퐵 { 0 elsewhere where 퐼퐵푒푠푠푒푙,0 is the zero-order modified Bessel function of the first kind, and 푊퐾퐵 is the width of the kernel (i.e., the intensity of the measured k-space points are interpolated only to the neighboring grid points with the distance |∆푘| ≤ 푊퐾퐵). Equation (3.28) is analogous for 푘푦 and 푘푧.

The optimal scaling parameter 훽퐾퐵 can be found for each the k-space oversampling 훼푂푆 and 푊퐾퐵 (Beatty et al., 2005):

푊2 퐾퐵 2 (3.29) 훽퐾퐵 = 휋√ 2 (훼푂푆 − 0.5) − 0.8. 훼푂푆

3.4 Image reconstruction

In this work the oversampling of 훼푂푆 = 2 was used, which is optimal in the sense that the relative amount of aliasing energy is minimal (Jackson et al., 1991). The width of the kernel was chosen to be 푊퐾퐵 = 3, because 푊퐾퐵 = 4 increases the computation time by a factor of 2.4 without significant increase in image quality. The FT of the KB function (equation (3.29)) is given by

2 2 2 2 sin (√휋 푊퐾퐵푥 − 훽퐾퐵) 푐퐾퐵(푥) = . (3.30) 2 2 2 2 √휋 푊퐾퐵푥 − 훽퐾퐵 In non-Cartesian acquisition, data is sampled more densely in some parts of k-space than in oth- ers. Several methods have been proposed for sampling density compensation for arbitrary k- space trajectories (Pipe, 2000). For simple k-space trajectories such as radial acquisition, where the density is higher near the center of k-space that in the periphery, density compensation can be calculated analytically. For this, a volume 푉푖푠푝푝 associated with each sampling point 푖푠푝푝 is calculated as the spherical shell volume divided by the number of the symmetric radial spokes 푁 and can be expressed as 푠푝표푘푒푠 (푘1 + 푘2) for 푖푠푝푝 = 1 푁푠푝표푘푒푠 푉 = (3.31) 푖푠푝푝 (푘 + 푘 )3 − (푘 + 푘 )3 푖푠푝푝 푖푠푝푝+1 푖푠푝푝 푖푠푝푝−1 for 푖푠푝푝 > 1 { 푁푠푝표푘푒푠 where 푘푖푠푝푝 is the distance of the 푖th sampling point along the radial spoke from the center of k- space. The schematic representation of the density compensation for the 2D radial acquisition is shown in Figure 3.6b. It is worth nothing, that the sampling points do not need to be equally distributed along the spoke, which means that the equation (3.31) is valid for both the ramp-up and trapezoidal parts of the DAPR sequence (cf. Figure 3.5a), whereas the density-adapted part has the density compensation per se (equation (3.18)).

In addition, radially sampled k-space data can be filtered with the Hanning filter (equation (2.61)) to increase SNR and to compensate for the Gibb’s ringing artifacts (cf. Subsection 2.2.3). In this work both KB reconstruction with and without Hanning filter are used and referred to as KBwH and KBnH, correspondingly. All data were reconstructed onto a 128×128×128 matrix.

The KB gridding algorithm, which was implemented in MATLAB (version 8.1.0.604; Math- Works, Natick, Massachusetts, USA), can be summarized as following:

1) For density compensation (Figure 3.6b), the signal intensity of each measured k-space data point is divided by the corresponding density calculated according to the equation (3.31). 2) A Hanning filter (equation (2.61)) can be applied in k-space if desired. 3) For each sampling point the neighboring rectilinear grid points are found that lie within the

gridding compact support distance 푊퐾퐵 (shown as shaded region in Figure 3.6a). 4) For each sampling point, and each of the neighboring rectilinear grid points the KB grid-

ding kernels 퐶퐾퐵(∆푘푥, ∆푘푦, ∆푘푧) are calculated according to the equations (3.27) and (3.28).

3. Material and Methods

5) The signal intensities are interpolated from the radially sampled points onto the rectilinear grid (Figure 3.6a) using the KB kernel as a weighting function. At each rectilinear grid point the signal intensity is summed from the neighboring sampling points. 6) FFT is applied to the gridded data. 7) Deapodization of the image is done by dividing the image by the FT of the KB kernel given in the equation (3.30). 8) The center portion of the image corresponding to the desired FOV is extracted if the k- space oversampling was done to reduce aliasing artefacts.

3.4.2. DIrect Estimation of 17O ImageS

Radially acquired images are typically reconstructed by KB gridding followed by FT as described in the previous Subsection. Due to low MR sensitivity of 17O nucleus (Subsection 2.1.8), signal intensities have to be averaged over large brain regions to get signal-time curves with acceptable for fitting noise level (Figure 2.8). In other words, large amount of information from 40 - 45 3D 17 O images is reduced to two CMRO2 values in the GM in WM regions. However, CMRO2 values ∗ are prone to PVEs because of low spatial resolution of 8 to 10 mm and fast 푇2 decay of about 2 ms (Subsection 2.2.4). To address these issues, a new method called DIESIS was suggested in this work. DIESIS avoids full reconstruction of MR images (Fourier encoding) and directly estimates the 17O MR images based on parcellation (i.e., segmentation of the 3D image to a predefined number of volumes) of a co-registered 1H image. A similar method was originally introduced in functional MRI for direct imaging of functional networks (Wong, 2014), where a few hundred of predefined parcels were used. Here, three parcels for WM, GM and CSF regions in the brain and 180 parcels outside the brain (including regions outside the head) were defined to minimize the influence of susceptibility artifacts, volunteer motion under radial sampling and coil inhomogeneities.

DIESIS models the MR image as the sum over parcel’s masks 푀푎푠푘푖(풓) weighted by the signal intensity values 휚 i # 푝푎푟푐푒푙푠 퐼푚 17 (풓) = ∑ 휚 ∙ 푀푎푠푘 (풓). (3.32) 퐻2 푂 푖 푖 푖 The k-space signal for one spoke (풔풑) in radial acquisition of ith parcel is then given by

−푡 푠푝 푖∙풓̅∙풔풑̅̅̅̅∙훾∙푡 ⁄푇 ∗ 푆푘 푖(푡) = ∫ 푑풓 ∙ 푀푎푠푘푖(풓) ∙ 푒 푒 2 , (3.33)

* where the last T2 term leads to a blurring of the data in image space, and thus to PVEs. Using ∗ ∗ experimentally determined 푇2 = 2 ms for GM and WM and 푇2 = 3 ms for CSF (Borowiak et al., 2014a; Hoffmann, 2011), an implicit PVE correction is possible. This calculation of the k-space representation of the parcels was done using NUFFT operator (Fessler and Sutton, 2003), which includes information about the k-space trajectory used in 17O MRI experiments.

3.4 Image reconstruction

Figure 3.7. Conventional gridding reconstruction and DIESIS method for quantification of CMRO2 maps in human brain in dynamic 17O MR experiment. In the DIESIS method, no Fourier reconstruction is applied, and direct least-squares estimation of the measured 17O signal is used, which includes explicit correction for PVEs.

17 The measured O MR signal in k-space (푆푘푚푒푎푠) is expressed as a linear combination of the calculated k-space signals of the parcels (푆푘 ) 푒푠푡,푖 # 푝푎푟푐푒푙푠

푆푘푚푒푎푠(푡) = ∑ 휚푖 ∙ 푆푘푒푠푡,푖(푡), (3.34) 푖

The intensity values 휚i, which are to be mapped back to image space (equation (3.32)), are then obtained by direct least squares estimation without Fourier encoding, by minimization of:

# 푝푎푟푐푒푙푠 2

퐿(휚) = ‖푆푘푚푒푎푠(푡) − ∑ 휚푖 ∙ 푆푘푒푠푡,푖(푡)‖ . (3.35) 푖 2 DIESIS was applied to each raw data set of four 17O MR experiments (Subsection 3.1.3), and

CMRO2 values were extracted from the signal dynamics using the proposed flexible pharmacokinetic model (Subsection 3.5.2).

3. Material and Methods

3.4.3. Iterative reconstruction

In the iterative algorithms, MR data reconstruction is formulated as a constrained optimization problem (Block et al., 2007) and the image is obtained by minimizing the objective function 퐽(흆) consisting of the data consistency and regularization terms

2 퐽(흆) = ‖푨 ∙ 흆 − 푺풌풎풆풂풔‖2 + 휆 ∙ 푹, (3.36) where 푨 denotes the system matrix that maps the image 흆 to the corresponding raw data 푺풌풎풆풂풔, λ is the weighting factor of the regularization term 푹 that allows for switching the preference from matching the image to the measured data to satisfy a prior knowledge. The data consistency term is a so-called 푙2-norm (least squares, in which the sum of the square of the differences between the target value and the estimated values is minimized).

The system matrix 푨 is a mathematical description of the MRI process. In case of utilized non- Cartesian radial sampling, the matrix 푨 consists of FFT 퓕 , k-space interpolation 퓖 on the radial trajectory (like in the gridding approach) and image deapodization using the FT of the gridding kernel 퓔: 푨 = 퓖퓕퓔. (3.37) Consistent with the traditional gridding reconstruction, KB kernel (equations (3.27) and (3.28)) was also chosen in iterative reconstructions.

The first regularization term that was used in this work is the TV constraint, which has been successfully utilized in image restoration and in iterative reconstructions (Block et al., 2007; Gnahm, 2014; Lustig et al., 2007; Rudin et al., 1992). The TV penalty term is based on the assumption that the image consists of regions with constant or slightly varying intensities, which applies well for 17O MR images. If the image is piecewise constant, then the optimal solution of the optimization problem given in equation (3.36) should be the one with the lowest derivatives at all pixel position. In this case, the TV regularization term 푹TV is so-called 푙1-norm (sum of the absolute values) and is given by

푹 (흆) = ‖훥 흆‖ + ‖훥 흆‖ + ‖훥 흆‖ , TV 푥 1 푦 1 푧 1 (3.38) where 훥푥, 훥푦 and 훥푧 denote the derivatives in x, y and z direction, respectively and are calculated from the finite difference between neighboring pixels

휕푥흆(푙, 푚, 푛) = [흆(푙 + 1, 푚, 푛) − 흆(푙, 푚, 푛)],

휕푦흆(푙, 푚, 푛) = [흆(푙, 푚 + 1, 푛) − 흆(푙, 푚, 푛)], (3.39)

휕푧흆(푙, 푚, 푛) = [흆(푙, 푚, 푛 + 1) − 흆(푙, 푚, 푛)] .

3.4 Image reconstruction

3.4.4. Anisotropic diffusion constraint

The nonlinear anisotropic smoothing filter for removal of background noise in images was originally proposed by (Perona and Malik, 1990). This method is a well-accepted image filtering technique because of its algorithmic simplicity and computational efficiency (Alvarez et al., 1992; Landini et al., 2005; Weickert, 1998). It was applied to 2D and 3D MRI data (Gerig et al., 1992), where it was shown to overcome the major drawbacks of conventional filtering methods, such as blurring of object boundaries and suppression of fine structural details. Considering intrinsically low spatial resolution of 17O MR images, it was proposed in this work to use the edge information from the gradient of the proton image of high spatial resolution by interpolating both 17O and 1H images onto a 128×128×128 matrix. It is important to point out that not the intensities of proton image, but only the information about the borders between the tissues was used, which are basically the same in both images.

In the proposed AD constraint, information from a high resolution 1H prior was used for reconstruction of 17O MR images. Therefore, 1H MPRAGE data was coregistered to the time-averaged 17O MR image (reconstructed with KBnH) using mutual information-based rigid image registration in a 3D Slicer software platform (version 4.3.1; www.slicer.org, (Fedorov et al., 2012)). From the coregistered 1H MPRAGE data, a gradient operator 품 was calculated for the regularization term 푹 in the AD constraint AD

푹AD = ∫ 흆훻(푫훻흆), (3.40) with

훻(퐃훻흆) = ∑ 휕푖D푖푗(휕푗흆) = ∑(D푖푗휕푖휕푗흆 + (휕푖D푖푗)(휕푗흆)), (3.41) 푖,푗∈(푥,푦,푧) 푖푗 and using an anisotropic non-homogeneous diffusion operator

품 ∙ 품푇 품2 푫 = 1 − (1 − 1⁄√1 + ). (3.42) |품|2 푎2

The first order derivative 휕푥 at the pixel position (푙, 푚, 푛) is calculated as finite difference operator

1 휕 흆(푙, 푚, 푛) = [흆(푙 + 1, 푚, 푛) − 흆(푙 − 1, 푚, 푛)] , (3.43) 푥 2 and analogous expressions are used for 휕푦 and 휕푧. The second order derivatives at the pixel position (푙, 푚, 푛) are calculated as exemplary shown for 휕2 and 휕 휕 푥 푥 푦 2 휕푥 흆(푙, 푚, 푛) = 흆(푙 + 1, 푚, 푛) − 2흆(푙, 푚, 푛) + 흆(푙 + 1, 푚, 푛), 1 휕 휕 흆(푙, 푚, 푛) = [흆(푙 + 1, 푚 + 1, 푛) + 흆(푙 − 1, 푚 − 1, 푛) (3.44) 푥 푦 4 −흆(푙 + 1, 푚 − 1, 푛) − 흆(푙 − 1, 푚 + 1, 푛)].

3. Material and Methods

The objective function 퐽(흆) in the equation (3.36) was minimized using a linear conjugate gradient (CG) algorithm (Hestenes and Stiefel, 1952) for the AD constraint and a non-linear CG algorithm (Hager and Zhang, 2005) for the TV constraint. The CG algorithms, described in the following Subsection, were implemented in MATLAB. The number of iterations required to reach convergence (i.e., a 10-5 convergence tolerance) was less than 100.

3.4.5. Conjugate gradient algorithm

Due to the large size of MRI data in image and k-space and non-Cartesian sampling (in this case the system matrix 푨 in cannot be directly inverted), an efficient optimization method is required to find a solution to equation (3.36). The solution is the reconstructed MR image that complies with the measured data and prior knowledge. CG, which was initially proposed for the solution of linear equations (Hestenes and Stiefel, 1952), is a suitable approach, and has been widely used for iterative MRI reconstructions (Block et al., 2007; Gnahm, 2014; Pruessmann et al., 2001).

The CG method is an iterative two-step procedure, which is repeated until the solution is found. In the first step, the search direction is calculated from the gradient of the objective function for the current estimate. Instead of using this gradient directly, as it is done in the steepest descent method (Curry, 1944), the search direction is obtained from the conjugate gradient. All conjugate gradients are orthogonal to each other making minimization efficient and guaranteeing the conju- gacy of successive search directions. In the steepest descent method, however, each direction can be utilized many times, which is inefficient.

The gradient of data consistency term in equation (3.37) is given by

2 † † ∇(‖푨 ∙ 흆 − 풚‖2) = 2(푨 푨흆 − 푨 풚), (3.45) where 푨† denotes the adjoint to matrix 푨, which is the transposed matrix with each entry replaced by its complex conjugate. It is worth noting that 푨† is not the inverse matrix to 푨, because the forward operation 푨 (equation (3.37)) is in general non-invertible due to the projection of the Fourier transform to the spokes.

The gradient of the AD regularization term is

∇푹AD = 훻(푫훻흆). (3.46)

The 푙1-norm used in the TV constraint is the magnitude function, which is neither a smooth function nor differentiable at 0 and can only be approximated by a smooth differentiable function (Lustig et al., 2007):

|푥| = √푥∗푥 + 휇, (3.47) where 휇 is a positive smoothing parameter set to 10-9; and the derivative of the function is:

푑|푥| 푥 = . (3.48) 푑푥 √푥∗푥 + 휇

3.4 Image reconstruction

In the second step of CG, a line search into that direction is performed until the minimum of the cost function in this direction is identified (i.e., a new image is found that yields in smaller value of the objective function). If the objective function 퐽(흆) is not a convex quadratic, as it is the case for the TV constraint, the linear CG can be replaced by more time-consuming nonlinear CG method (Hager and Zhang, 2005).

3.4.6. Comparison of MR reconstruction methods

The performance of the different reconstruction methods was first compared using the numerical 17O MRI phantom (cf. Subsection 3.2.2 and Figure 3.4). For this, 17O MRI phantom images were reconstructed with different weighting factors for TV / AD constraints (휆푇푉 / 휆퐴퐷) over the range from 10-3 to 105 to identify the values which give the best correspondence with the original phantom image. RMSE, SSIM, and SNR values were used for comparison of quality and precision of different reconstructions (cf. Subsection 3.2.3).

For reconstruction of the in vivo 17O MR images of two 17O MRI experiments Exp3 and Exp4, the optimal 휆푇푉 from the phantom analysis was taken, whereas 휆퐴퐷 was chosen based on a good visual image quality regarding the noise level. In particular, signal intensity within the eyes, which 17 1 are bright in O image due to high water content and dark in H T1-weighted MPRAGE image, was taken as a quantitative measure. Noise level decreases with increasing 휆퐴퐷, but at a given threshold overregularisation sets in and the reconstructed 17O signal intensity starts to decrease due to the influence of the proton information. Thus, 휆퐴퐷 was set close to the overregularization threshold to minimize noise.

3.5. Advances in CMRO2 quantification analysis

17 The pharmacokinetic model for CMRO2 quantification from dynamic O MRI that has been used so far is presented in the Subsection 2.4.5. The inhalation experiment consists of three or four phases: the baseline phase, DODS phase, wash-out phase and an additional RB phase can be 17 17 17 included to gain higher H2 O signal increase after O gas inhalation (Figure 2.8). Pulsed O gas delivery and usage of the RB circuit (cf. Subsection 3.1.2) lead to uncertainties in the determination of the 17O enrichment fraction of the inhaled gas, which in turn can lead to systematic errors in the quantities derived from this enrichment fraction. To address this, in this Section the implementation of the profile likelihood analysis method to determine parameter identifiability and their confidence intervals in a nonlinear CMRO2 quantification model are described (Kurzhunov et al., 2017a). Furthermore, an advanced pharmacokinetic model was developed to address the 17 time dependency of the enrichment fraction of O gas. At the end, pixel-wised CMRO2 quantification with the proposed model is discussed.

3.5.1. Extraction of the 17O signal-time curves

To exploit the method of PL analysis and compare the simplified pharmacokinetic model (cf. Subsection 2.4.5) with the advanced model (Subsection 3.5.2), the 17O MR data sets of Exp1 and 1 Exp2 were used. To extract the tissue-specific signal-time curves, a T1-weighted 3D H MR image was used.

First, both 17O and 1H MR images were interpolated onto the same grid (128×128×128). Second, the 1H MR data was coregistered to the time-averaged 17O MR image (reconstructed with KBnH) using mutual information-based rigid image registration in a 3D Slicer software platform (version 4.3.1; www.slicer.org, (Fedorov et al., 2012)). Rigid transformation consisting of translation and rotation in three directions was performed and a transformation matrix for 1H images was obtained. Later, the software tool Statistical Parametric Mapping Package (SPM8) (Ashburner and Friston, 2005, 2000) was used to segment GM and WM brain regions in the 1H MPRAGE data. After this, the transformation matrix was applied to the 3D binary masks of WM and GM regions, the masks were then applied to the co-registered 17O MR images. Finally, averaged 17O MR signals were calculated for each tissue to obtain tissue-specific 17O signal-time curves, which were 17 normalized to H2 O content in a particular tissue as described in the Subsection 3.2.1.

3.5.2. Advanced pharmacokinetic model

The expected 17O signal change in WM brain region is shown in Figure 3.8a. A key parameter of 17 the CMRO2 quantification model is the O enrichment fraction 훼 (cf. Subsection 2.4.5), which 17 푉17 can be calculated as the ratio of the volume of the inhaled O gas 푂 to the total volume of the inhaled oxygen gas 푉16 17 : 푂+ 푂

3.5 Advances in CMRO2 quantification analysis

Figure 3.8. (a) Expected 17O signal change during MR examination with inhalation of 17O gas based on the model parameters reported by (Hoffmann, 2011) for WM region. Four phases of the experiment are indicated. (b) Time evolution of the 17O enrichment fraction α for the advanced CMRO2 quantification model. It assumes a non-constant enrichment fraction and includes contributions from DODS pulses (αDODS) and 17O gas stored in the RB circuit, which is described during the DODS phase by s1 and during the RB phase by αRB and s2 (c) Simplified quantification model, which assumes constant α values.

푉17푂 푉17푂 푝푢푙푠푒 ∙ 퐶17푂,푏표푡푡푙푒 훼 = = , (3.49) 푉16 17 17 17 17 푂+ 푂 (푉푡푖푑푎푙 − 푉푑푒푎푑 − 푉 푂 푝푢푙푠푒) ∙ 퐶푂,푎푖푟 + 푉 푂 푝푢푙푠푒 ∙ 퐶 푂,푏표푡푡푙푒

푉 푉 푉17 where 푡푖푑푎푙 and 푑푒푎푑 are the tidal and dead volumes described in Subsection 2.3.1, 푂 푝푢푙푠푒 is the volume of the pulse delivered by DODS (cf. Subsection 3.1.2), 퐶푂,푎푖푟= 0.21 is the oxygen 17 퐶17 content in the air and 푂,푏표푡푡푙푒= 0.7 is the O isotope enrichment of oxygen stored in the gas 푉17 훼 cylinder. For 푂 푝푢푙푠푒 = 40 ml / 50 ml equation (3.49) yields = 0.27 / 0.31. However, this assumption of a constant 훼 is not valid when the RB circuit is closed during 17O pulse delivery, since part of the exhaled 17O gas will be stored in the RB circuit.

Figure 3.8b shows the proposed realistic time evolution of 훼(푡) for the experiment with the RB circuit and a pulsed supply of 17O gas. This curve takes into account that a small fraction of the 17 17 O2 gas is exhaled as it did not reach the alveoli. Exhaled O gas is stored in the RB circuit and used up in subsequent inhalation cycles. Thus, during the DODS phase, in addition to the 17O 17 gas being delivered by DODS pulses (훼퐷푂퐷푆), re-inhalation of the exhaled O gas occurs leading to a linear increase in 훼. Therefore, a linear increasing model with a slope 푠1 was introduced. Sim- ilarly, in the RB phase, 훼 was assumed to be linearly decreasing (푠2) to 훼푅퐵. The CMRO2 quantification model can be simplified by setting both slopes 푠1 and 푠2 to zero (Figure 3.8c), i.e. assuming constant 훼 values as it was previously done (Atkinson and Thulborn, 2010; Hoffmann, 2011), but it might lead to a reduced model fit quality. The advanced model with slopes 푠1 and 푠2 is later compared to the simplified model with constant 훼 values using the LR test (equation (2.88)).

3. Material and Methods

3.5.3. Profile likelihood analysis

Parameters of the pharmacokinetic model for CMRO2 quantification were calibrated according to the equation (2.85). The numerical optimization was conducted using the trust region-based optimization algorithm lsqnonlin implemented in MATLAB (Coleman and Y., 1996). In a nonlinear setting, multiple local optima are often present. Thus, a deterministic multi-start was performed in order to find the global optimum (Raue et al., 2013). All model analysis, optimization and uncertainty calculations were performed within the open-source and freely available MATLAB- based framework D2D (Raue et al., 2013). Therein, the ODE solver CVODES from the SUN- DIALS suite is used for ODE integration (Hindmarsh et al., 2005). Following the model calibration, parameter uncertainties of the pharmacokinetic model were calculated. If the target parameter CMRO2 was non-identifiable, prior information based on estimation of the other model parameters was included to resolve the non-identifiability. Thereby, a small amount of additional prior information was desired, because the measurement of the models prior is complex and im- plies additional sources of errors. The influence of the uncertainty of the prior information on 2 ̂ the optimal CMRO2 values, estimated through minimization of 휒 (휃) (equation (2.86)), and on the calculated CIs (equation (2.87)) was also investigated. In this case, relative CIs of CMRO2, which are the CIs of CMRO2 divided by the optimal CMRO2 values, were considered to account for different CMRO2 values in various brain tissues.

To analyze the prediction capability of the advanced pharmacokinetic model for CMRO2, the following approach was taken: at first, the model parameters CMRO2, 퐾퐿, 퐾퐺, 훼퐷푂퐷푆, 훼푅퐵, 푠1 and

푠2 were allowed to vary (flat prior) within boundaries of -5 to 3 in log-space. If CMRO2 was non- identifiable, 훼퐷푂퐷푆 was fixed to 0.27 / 0.31 (for Exp1 / Exp2) based on the estimated amount of 17 O2 inhaled with a single DODS pulse (equation (3.49)). If CMRO2 was still non-identifiable, an averaged value of 훼 during the DODS phase, which constrains both 훼퐷푂퐷푆 and 푠1, was used as prior information. This averaged 17O enrichment fraction was calculated based on the total amount of delivered 17O gas and the total duration of DODS phase (0.27 / 0.31 for

Exp1 / Exp2). For the simplified pharmacokinetic model, 훼퐷푂퐷푆 was fixed to the same values. A 10% uncertainty was assumed for both 훼퐷푂퐷푆 and the averaged 훼. For Exp3 and Exp4 the averaged 17O enrichment fraction, i.e. the total amount of the inhaled 17O gas divided by the total amount of the inhaled oxygen gas, was taken as a prior knowledge. Lower and upper boundaries of the CIs of CMRO2, which include the optimum CMRO2 value, were calculated using (equation

(2.87)) for the confidence level 퐶퐿 = 0.33 and were used to present the calculated CMRO2 values.

It is worth noting that in this work the CMRO2 values are presented for each dataset separately, and not as the range among several MR examinations, unless otherwise stated.

3.5.4. 3D CMRO2 mapping

The series of MR images from Exp3 and Exp4 reconstructed with different methods (cf. Section

3.4) were used for CMRO2 mapping within the brain region. In each pixel the signal dynamics

3.5 Advances in CMRO2 quantification analysis

17 was extracted and normalized to the H2 O content of a particular pixel, which was obtained from the numerical 17O brain phantom (cf. Subsection 3.2.2).

The percentage of pixels 푃푛푓 was calculated, for which CMRO2 mapping failed due to low SNR. As a failure criterion, the PL analysis was performed on a pixel-by-pixel basis which provided information about the principle identifiability of the fitting parameter CMRO2. Finally, the

CMRO2 values calculated in the whole brain WM and GM regions were compared to the literature results of 15O-PET studies (Leenders et al., 1990), 17O MRI studies at UHFs (Atkinson and Thulborn, 2010; Hoffmann et al., 2011) and 1H MRI QUIXOTIC method (Bolar et al., 2011).

4. RESULTS

This chapter begins with the results of the optimization of acquisition parameters with the analytical and numerical phantoms (Section 4.1). Secondly, the profile likelihood analysis of the

CMRO2 identifiability and the development of the flexible and realistic pharmacokinetic model are presented (Section 4.2). It follows by CMRO2 quantification with the DIESIS method (Sec- tion 4.3). Then, gridding and iterative MR image reconstruction methods are compared for nu- 17 merical O brain phantom and in vivo data (Section 4.4). Finally, pixel-wise CMRO2 mapping from the dynamic 17O MRI data sets is compared for different image reconstruction methods.

4.1. Optimization of acquisition parameters with 17O MRI phantoms

The rare and costly 17O gas makes it difficult to perform in vivo optimization of the acquisition parameters; thus, their optimization was performed in a simulation framework with two types of human brain phantoms. In the ACROBAT phantom the k-space representation was analytically calculated and three tumor regions were included, whereas in the numerical phantom the k-space representation was numerically calculated using NUFFT operator and segmented brain regions obtained from 1H image.

4.1.1. ACROBAT phantom

The analytical brain tumor phantom called ACROBAT (cf. Subsection 3.2.1) consists of three normal brain regions and three brain tumor tissues (Figure 3.3). In the dynamic simulation framework (Subsection 3.2.4), which simulates the dynamic 3D 17O MRI experiment with 17O gas inhalation, the ACROBAT phantom facilitated testing of wide range of MR parameters. In this work, the influence of the readout bandwidth BW, spatial resolution ∆푥 and temporal resolution

∆푡 on CMRO2 precision was investigated.

Figure 4.1 shows CMRO2 differences in GM, WM and CE tumor region as a function of ∆푥 and BW. For WM, small deviations of 2 % are seen at ∆푥 ≥ 8mm and at all BWs, since the simulated

WM region is large (Figure 3.3). For ∆푥 ≤ 8mm, CMRO2 is underestimated due to nonlinear signal behavior at low SNR. If only 8 or less out of 10 simulation runs for the parameter pair were successful, then they are not plotted, otherwise a blue / black error bar is shown for 9 / 10 successful runs. As seen in Figure 4.1, BW ≤ 250 Hz/pixel and ∆푥 ≥ 8 mm are required for reproducible CMRO2 quantification in CE tumor region. Results for different temporal and spatial resolution are more heterogeneous: CMRO2 values change smoothly for a temporal resolution 75 ≤ ∆푡 ≤ 240 sec, whereas they vary strongly otherwise. Thus, these intermediate parameter settings seem unfavorable. For CMRO2 quantification in the CE region 150 ≤ BW ≤ 250 Hz/pixel, and ∆푥 ≥ 7mm are required.

4.1 Optimization of acquisition parameters with 17O MRI phantoms

Figure 4.1 Deviation of the simulated CMRO2 values from the input CMRO2 values in gray and white matter regions and contrast enhancing rim tumor region of the ACROBAT phantom in the dependence of spatial resolution and readout bandwidth (upper part), and in the dependence of spatial and temporal resolution (in the middle). For the numerical brain phantom: deviation of the simulated CMRO2 values in white and gray matter regions is in the dependence of spatial resolution and readout bandwidth (bottom part). Color scale represents deviation in %.

4.1.2. Numerical phantom

Results of the numerical phantom (Figure 3.4) show that in GM CMRO2 (1.12 ± 0.1

µmol/gtissue/min) is underestimated by about 30%, but depends neither on BW nor on ∆푥, whereas for WM CMRO2 values are overestimated. The best corresponding of the output

CMRO2 values with the model input CMRO2 values were found at BW = 150 Hz/pixel and 7 ≤ ∆푥 ≤ 9mm, and for BW = 250 Hz/pixel and 8 ≤ ∆푥 ≤ 12mm, with mean

CMRO2= 0.74 ± 0.03 µmol/gtissue/min and 20% overestimation.

4. Results

4.2. CMRO2 quantification and profile likelihood analysis

In the PL analysis (Subsection 2.4.6) the advanced pharmacokinetic model (Subsection 3.5.2) was 17 17 used to investigate the identifiability of CMRO2 in O MRI experiments with O gas inhalation. The identifiability analysis as well as model prior assumptions are presented in Subsection 3.5.3. For the analysis, dynamic 17O MR data were obtained from Exp1 and Exp2 (Section 3.1), reconstructed with KB method without any additional filtering, and tissue-specific signal time curves were extracted as described in Subsection 3.5.1.

An example of a 3D 17O MRI data set with the co-registered 1H MPRAGE image as well as WM and GM masks is shown in Figure 4.2. The contour of the brain, which has higher water content than the rest of the head, as well as the eyes are clearly visible on 17O MR images. The transmit/receive coil was driven in the linear mode, which had an inhomogeneous excitation profile, resulting in L-R asymmetries as seen in the posterior parts of the brain. These asymmetries, how- 17 ever, have only minor effects on CMRO2 quantification, because H2 O signal-time curves were obtained from large WM and GM regions and were normalized to the baseline before 17O gas inhalation. The SNR of the MR images acquired within 1 min in the baseline phase were 6 / 4 for Exp1 / Exp2, thus the noise pattern can be assumed to be Gaussian (Gudbjartsson and Patz, 1995) as is presumed in equation (2.84).

Figure 4.3 shows the calculated PLs of the parameters of the advanced CMRO2 quantification model (in log-space), where all model parameters were set with a flat prior, i.e. no prior knowledge about 훼 was assumed. With a flat prior, the only quantifiable parameter was 퐾퐿, since 17 17 it determines the decay constant in the wash-out phase, when 퐴 푂 is zero and 퐵퐻2 푂 is constant

(equation (2.78)). The target parameter CMRO2 and the other model parameters were structurally non-identifiable. Even after assuming a constant value of 훼퐷푂퐷푆, CMRO2 remained non- identifiable.

When the averaged 훼 value during the DODS phase was taken as a constraint, the structural non- identifiability of parameters CMRO2, 퐾퐺, 훼푅퐵 was resolved (Figure 4.4). However, either 훼퐷푂퐷푆 or 푠1 are practically non-identifiable for each of the presented data sets. The profile of the parameter 푠2 shows that a value of zero (i.e. 훼 is constant during the RB phase) is consistent with the model without impairing the 휒2. The cyan dashed line represents the contribution of the chosen prior to the respective parameter profile. For example, the optimal CMRO2 value is slightly left from the minimum of the chosen prior, and the CI of the parameters CMRO2 and 퐾퐺, are domi- nated by the uncertainty of the prior. In contrast, the prior uncertainty has a much smaller impact on the CI of 퐾퐿 and 푠2.

Model fits for WM and GM regions are shown in Figure 4.5, in which prior knowledge about the mean 훼 value during the DODS phase was used. Here, CMRO2 rates were 0.80 – 0.99 / 0.72 –

0.95 µmol/gtissue/min in WM and 1.02 – 1.27 / 1.21 – 1.78 µmol/gtissue/min in GM for 15 Exp1 / Exp2. Compared to the results of O-PET studies (Leenders et al., 1990), CMRO2 rates were 34% – 42% overestimated in WM and 9% – 28% underestimated in GM (Table 4.1). If

4.2 CMRO2 quantification and profile likelihood analysis

Figure 4.2. (a) Different orientations of an 17O MR image from Exp1, averaged over the whole MR examination. (b) Coregistered T1-weighted 1H MR image. (c) Transversal slice of WM mask. (d) Trans- versal slice of GM mask.

one / two of the first DODS pulses are disregarded to account for the 94 ml dead volume of the cable connecting DODS with nasal cannula (i.e., a later signal onset is assumed), 3 – 4% / 7 – 8% higher CMRO2 values were found.

In Figure 4.6 the comparison of model calibration with the advanced and the simplified models is presented. The simplified model shows a stronger deviation from the data in the DODS phase than the advanced model, which is also reflected in the 휒2 values: 휒2 = 40.0 (simplified) and 휒2 = 55.6 (advanced). Using these data, the LR test, described by equation (2.88), was calculated (푃 = 4.1·10-4 < 0.05)showing a significant improvement with the advanced model. Moreover, the simplified model in the GM region yields in CMRO2 of 1.11-1.59 µmol/gtissue/min, which is 9% and 18% underestimated compared to the advanced model and to the results of 15O-PET studies (Leenders et al., 1990).

Figure 4.7 shows how the 훼-uncertainty affects the relative CIs of CMRO2. In this case, CI, which is the difference between upper and lower boundaries of the calculated parameter, represents two standard deviations. These dependencies are well represented by a quadratic polynomial, but the CIs are specific for each data set. For example, a 10% uncertainty in 훼 leads to relative

CIs of CMRO2 of 0.22 / 0.28 for WM region, and 0.23 / 0.40 for GM region in Exp1 / Exp2.

4. Results

Figure 4.3. Exploiting the PL of the parameters of the advanced CMRO2 quantification model in WM (a, c) and GM (b, d) regions from Exp1 (a, b) and Exp2 (c, d). All presented model parameters were set with a flat prior in log-space (i.e., no prior knowledge about the 17O enrichment fraction α was assumed). Optimal parameter values are indicated by asterisks, with the likelihood value indicated by the blue line; thresholds for 95% and 67% CIs are indicated by red dashed lines. Flat CIs indicate structurally non- identifiable parameters. CMRO2, KG and KL have units of µmol/gtissue/min, s1 and s2 – min-1, αDODS and αRB are dimensionless.

4.2 CMRO2 quantification and profile likelihood analysis

Figure 4.4. Exploiting the PL of the parameters of the advanced CMRO2 quantification model in WM (a, c) and GM (b, d) regions from Exp1 (a, b) and Exp2 (c, d). The averaged α value during the DODS phase was implemented as prior knowledge. Optimal parameter values are indicated by asterisks, with the likelihood value indicated by the blue line; thresholds for 95% and 67% CIs are indicated by red dashed lines. Cyan dashed lines represent the contribution of the chosen prior to the respective parameter profile. CMRO2, KG and KL have units of µmol/gtissue/min, s1 and s2 – min-1, αDODS and αRB are dimensionless.

4. Results

Figure 4.5. H217O signal-time curves obtained in two 17O MR experiments (Exp1 and Exp2) with 17O gas inhalation in WM and GM brain regions (black squares). Data fit with the advanced pharmacokinetic model is presented by blue lines. For all data, additional information on the 17O enrichment fraction was taken into account.

Figure 4.6. H217O signal-time curve obtained in the 17O MR experiment (Exp1) in the GM region (black circles). Data points are fitted with the simplified model with constant α values (dashed blue line) and the advanced CMRO2 quantification model (red line). Additional information on the 17O enrichment fraction was taken into account.

4.2 CMRO2 quantification and profile likelihood analysis

Figure 4.7 Relative CIs of CMRO2 as a function of the uncertainty of the estimated averaged 17O enrichment fraction during the DODS phase(α) for WM (left) and GM (right) brain regions. Calculated values are fitted with a quadratic polynomial for two MR experiments (Exp1 and Exp2).

White Matter Gray matter

Exp1 0.80 – 0.99 1.02 – 1.27

Exp2 0.72 – 0.95 1.21 – 1.78

Exp3 1.26 – 1.55 1.63 – 2.01

Exp4 0.78 – 1.02 1.03 – 1.37

17O MRI at 7 T 0.50 – 0.89 0.80 – 1.61

17O MRI at 9.4 T 0.64 – 0.86 1.37 – 1.47 1H MRI - 1.10 – 1.40 QUIXOTIC 15O-PET 0.52 – 0.72 1.36 – 1.82

Table 4.1: Comparison of CMRO2 values in WM and GM regions of human brain (in µmol/gtissue/min) quantified with direct 17O MRI in four experiments (Exp1, Exp2, Exp3, and Exp4) to the literature values from 17O MRI at high field strengths (Atkinson and Thulborn, 2010; Hoffmann et al., 2011), 1H MRI- based QUIXOTIC method (Bolar et al., 2011) and 15O-PET (Leenders et al., 1990). With QUIXOTIC, CMRO2 quantification in WM was not possible due to low amount of blood capillaries.

4. Results

Figure 4.8 Axial slices of CMRO2 maps in human brain of one volunteer from four 17O MRI experiments obtained by DIESIS method with correction of PVEs. CMRO2 are very distinctive in GM, WM and CSF regions.

4.3. DIrect Estimation of 17O ImageS

The DIESIS method without direct reconstruction of 17O images allows correction of PVEs

(Subsection 3.4.2). The PL analysis showed identifiability of CMRO2 obtained with the DIESIS 17 method, and CMRO2 maps in the brain of the volunteer from four O MRI experiments are shown in Figure 4.8. Good separation of GM, WM and SCF regions was observed. The highest

CMRO2 values are found in GM, whereas WM values almost twice smaller. In CSF, the CMRO2 values are consistent with the expected value of zero. A quantitative comparison of CMRO2 values from DIESIS with the conventional KB gridding is presented in Table 4.3. In WM, DIESES reduces CMRO2 from 0.97 ± 0.25 µmol/gtissue/min with KB gridding to 0.85 ± 0.21

µmol/gtissue/min, whereas in GM it increases from 1.37 ± 0.31 µmol/gtissue/min to 1.86 ± 0.36 15 µmol/gtissue/min. In both cases the CMRO2 are getting closer to the literature values from O- PET studies.

The separation between GM and WM can be described by the ratio of CMRO2 in the GM region to CMRO2 in the WM region. In KB reconstruction without any PVEs correction, the CMRO2 ratio is 1.42 ± 0.22. With DIESIS with PVE correction, the ratio of 2. 23 ± 0.51 is about 60% higher, which is in good agreement with the results of 15O-PET studies (ratio: 2.56).

4.3 DIrect Estimation of 17O ImageS

Exp KB gridding DIESIS 15O-PET

Exp1 0.86 (0.80 – 0.99) 0.77 (0.67 – 0.88)

Exp2 0.83 (0.72 – 0.95) 0.67 (0.32 – 0.99)

WM Exp3 1.34 (1.21 – 1.50) 1.15 (1.00 – 1.33)

Exp4 0.86 (0.72 – 1.01) 0.81 (0.71 – 1.00)

mean 0.97 ± 0.25 0.85 ± 0.21 0.62 ± 0.10

Exp1 1.14 (1.02 – 1.27) 1.67 (1.42 – 1.93)

Exp2 1.45 (1.21 – 1.78) 2.00 (1.29 – 3.13)

GM Exp3 1.77 (1.60 – 1.97) 2.29 (1.94 – 2.75)

Exp4 1.11 (0.94 – 1.31) 1.47 (1.20 – 1.79)

mean 1.37 ± 0.31 1.86 ± 0.36 1.59 ± 0.23

Exp1 1.32 2.17

Exp2 1.75 2.96

GM/WM Exp3 1.32 1.99

Exp4 1.29 1.81

mean 1.42 ± 0.22 2.23 ± 0.51 2.56

Table 4.2: Comparison of CMRO2 values in WM and GM (in µmol/gtissue/min) quantified with direct 17O MRI in four experiments (Exp1, Exp2, Exp3, and Exp4) using conventional KB gridding and DIESIS method, to the literature values from 15O-PET (Leenders et al., 1990). For each experiment, optimal parameter values and 67% CIs quantified with PL analysis are presented.

4. Results

4.4. Comparison of reconstruction methods and 3D CMRO2 mapping

In this Section, gridding and two iterative reconstruction methods introduced in Section 3.4 are compared for the numerical brain phantom and two in vivo 17O MRI data sets from Exp3 and Exp4. Original image of the 17O phantom served as the ground truth for quantitative image quality assessment of the reconstructed data sets (Subsection 4.4.1). In particular, SNR, RMSE, and SSIM values were calculated (cf. Subsections 0 and 3.4.6). Secondly, comparison of image quality and SNR of the reconstructed in vivo data was performed (Subsection 4.4.2). Finally, pharmacokinetic model was fitted to these data sets, and pixel-wise CMRO2 maps were quantified (Subsec- tion 4.4.3).

4.4.1. Reconstruction comparison: phantom

With KBnH gridding reconstruction, no internal structural information is visible in the reconstructed phantom images (Figure 4.9). The application of a Hanning filter led to a slight blurring. In the KBwH data, CSF regions become weakly visible, SNR is increased 3-fold, SSIM had a 20% increase and RMSE is decreased by 2.5% compared to KBnH (Table 4.3). TV-constrained reconstruction (휆푇푉 = 1) resulted in a blocky image appearance with a 70% / 20% increase in SNR / SSIM and a 7% decrease in RMSE. In the image reconstructed with the AD constraint

(휆퐴퐷 = 100) brain structures are better visible than in all other reconstructions. The ventricles, in particular, have about 12% higher signal intensity and are clearly separated from the surrounding WM region; in the peripheral GM regions of the brain, the signal intensity is about 5% higher than in the WM. Compared to KBnH / KBwH, SNR values increased 11-fold / 2.5-fold, SSIM values increased by 103% / 70% and RMSE values were 9% / 12% higher, respectively.

4.4.2. Reconstruction comparison: in vivo data

In vivo 17O MR images reconstructed with four methods are shown in Figure 4.10 and Figure 4.11 for Exp3 and Exp4, correspondingly. In 17O images averaged over the whole MR examination of 40 – 45 min, regions with high water content such as the eyes and the ventricles are clearly distin- guishable. In Exp4, where the nominal spatial resolution was 25% higher than in Exp3, the eyes and the ventricles are less blurred and their shapes in 17O and 1H images are similar.Images acquired in 1 min with KBnH are contaminated by noise (SNR = 7 / 5 for Exp3 / Exp4). KBwH resulted in a 3-fold increase in SNR (SNR = 23 for Exp3 and SNR = 14 for Exp4), but the images appear more blurred, which is particularly visible in the eyes. TV-constrained reconstruction resulted in SNRs of 11 / 5 for Exp3 / Exp4 with a blocky image appearance as it was seen in the phantom images. AD-constrained reconstruction (휆퐴퐷 = 1000 for Exp3 and 휆퐴퐷 = 3000 for Exp4) yields a substantial SNR increase (SNR = 48 for Exp3, SNR = 31 for Exp4) which made it possible to define the ventricular shape and even some fine GM structures.

4.4 Comparison of reconstruction methods and 3D CMRO2 mapping

Figure 4.9. Axial slices of the 1H MPRAGE data, which was segmented to construct a 17O MRI numerical brain phantom in image space and k-space. Measured SNR and T2* values were taken into account and the acquisition scheme of Exp1 was used. Phantom images were reconstructed with Kaiser-Bessel gridding without/with Hanning filter (KBnH / KBwH) and in iterative reconstruction procedure with either TV or AD constraint. Quantitative comparison is presented in Table 4.3.

4. Results

Figure 4.10. 1H MPRAGE data, coregistered to the 17O MR image from Exp3, averaged over the total 45 min exam (reconstructed with KBnH); single 17O MR images (TA = 1 min, Exp3) reconstructed with Kaiser-Bessel gridding without/with Hanning filter (KBnH / KBwH) and with iterative reconstruction with either TV or AD constraint. SNR is higher and structural information is better visualized in KBwH and AD-constrained reconstructions.

4.4 Comparison of reconstruction methods and 3D CMRO2 mapping

Figure 4.11. 1H MPRAGE data, coregistered to the 17O MR image from Exp4, averaged over the total 40 min examine (reconstructed with KBnH); single 17O MR images (TA = 1 min, Exp4) reconstructed with Kaiser-Bessel gridding without/with Hanning filter (KBnH / KBwH) and with iterative reconstruction with either TV or AD constraint. Spatial resolution in Exp4 is higher than in Exp3.

4. Results

RMSE SSIM SNR KBnH 0.158 0.35 7 KBwH 0.154 0.42 23 TV-constrained 0.147 0.42 12 AD-constrained 0.172 0.71 80

Table 4.3: Root-mean-square errors (RMSE), structural similarity (SSIM) and SNR values of the 17O MRI numerical phantom images reconstructed with the different methods shown in Figure 4.9. For the calculation only values within the brain phantom were considered. With increasing correspondence with the original 17O MRI image RMSE values decrease and SSIM values increase.

Exp3 Exp4 KBnH 27.1% 51.0 % KBwH 9.1 % 29.7 % TV-constrained 24.6 % 45.2 % AD-constrained 6.4 % 10.7 %

Table 4.4: The percentage of the pixels Pnf, for which CMRO2 mapping failed due to low SNR, calculated for KBnH and KBwH gridding as well as for iterative reconstruction with either TV or AD constraint.

4.4.3. In vivo CMRO2 mapping

17 H2 O signal dynamics in single pixels show that for KBnH and TV-constrained reconstructions the signal-time curves are highly contaminated by noise (Figure 4.12), and PL analysis revealed that CMRO2 was practically non-identifiable. For KBwH, CMRO2 values were 1.37 – 1.93

/ 1.10 – 1.95 µmol/gtissue/min in GM/WM, and for the AD-constrained reconstruction CMRO2 values were 1.47 – 1.95 / 1.09 – 1.41 µmol/gtissue/min. The PL-based CIs for the AD-constrained reconstructions were 0.48/0.32 µmol/gtissue/min, which was 14% / 35% smaller than for KBwH.

17 In H2 O signal-time curves averaged over the whole WM and GM regions (Figure 4.13) about 2.5-fold signal increase was observed in Exp3 compared to Exp4, which correlates with the two- 17 fold higher amount of O gas in Exp3. Quantified CMRO2 values of WM / GM regions were

1.26 – 1.55 / 1.63 – 2.01 µmol/gtissue/min for Exp3 and 0.78 – 1.02 / 1.03 – 1.37 µmol/gtissue/min for Exp4 (Table 4.1).

4.4 Comparison of reconstruction methods and 3D CMRO2 mapping

Figure 4.12. H217O signal-time curves in a single WM (a) and GM pixel (b) obtained in Exp3 with four different reconstructions. Signal dynamics from KBnH and TV-constrained reconstructions have a higher noise contamination than KBwH and AD-constrained methods. Four phases of the 17O MRI inhalation experiment are indicated.

Figure 4.13. White and grey matter averaged H217O signal-time curves from Exp3 and Exp4 and data fit with the pharmacokinetic model. As expected, in Exp4 an about two-fold smaller signal increase is observed compared to Exp3, where twice the amount of 17O gas was used. Four phases of the 17O MRI inhalation experiments are indicated. 80

4. Results

Figure 4.14. Axial slice of CMRO2 maps of human brain from Exp3 (upper row) and Exp4 (bottom row) reconstructed with KBnH gridding (a, e), KBwH gridding (b, f), and with the iterative reconstruction procedure using the TV constraint (c, g) and the AD constraint (d, h). Black pixels denote locations where CMRO2 quantification failed due to low SNR. The amount of pixels with successful CMRO2 quantification is higher in KBwH and AD-constrained reconstruction compared to KBnH and TV- constrained reconstruction.

Figure 4.15. CMRO2 map of the whole brain obtained with the AD-constrained reconstruction in Exp3. In gray matter regions CMRO2 is about 30% higher than in white matter region.

In Figure 4.14 axial CMRO2 maps of different reconstruction techniques from Exp3 and Exp4

are presented. KBnH reconstruction had the percentage of the pixels, for which CMRO2 mapping failed due to low SNR, of 푃푛푓 = 27.1% / 51.0%, and for the TV-constrained method,

푃푛푓 decreased only by 9% / 11% for Exp1 / Exp2 (Table 4.4). In KBwH gridding, 푃푛푓 decreased by 66% / 42% for Exp3 / Exp4 compared to KBnH, whereas AD-constrained reconstruction

had 푃푛푓 = 6.4% / 10.7%, which is 76% / 79% smaller then KBnH and 30% / 64% smaller then

KBwH. 3D CMRO2 maps with AD-constrained reconstruction (Figure 4.15) show the expected

CMRO2 differences in brain tissues – for example, CMRO2 in WM and CSF is smaller than in GM regions.

5. DISCUSSION

In this Chapter, first, optimization of acquisition parameters with 17O MRI phantoms is addressed (Section 5.1). Secondly, the proposed CMRO2 quantification model and the influence of model parameters on the CMRO2 precision are evaluated with the profile likelihood analysis (Sec- tion 5.2). Then, the differences of simulated and in vivo MR images reconstructed with gridding and iterative methods are discussed (Section 5.3). Later, quantified at 3 T CMRO2 values in whole 17 brain GM and WM regions as well as 3D CMRO2 maps are compared to the results of O MRI at UHFs, 15O-PET, and 1H-MRI based methods (Sections 5.4, 5.5 and 5.6).

5.1. Optimization of acquisition parameters with 17O MRI phantoms

Two types of the simulated 17O MRI phantoms, ACROBAT and numerical, allowed testing a wide range of acquisition parameters that is otherwise not possible. Due to analytical calculation of the k-space representation of the ACROBAT phantom (Figure 3.3), unnecessary interpolation in k-space was eliminated that facilitated more precise simulation of the 17O MRI measurement. The flexible nature of the ACROBAT phantom and simulation framework, together with linearity of the FT, provided a straightforward way of adding the brain tumor regions. Contrarily, the numerical phantom (Figure 3.4) represents realistic anatomical brain structures (e.g., accounted for increased tissue boundaries). Thus, the simplicity of the ACROBAT phantom structure might be the reason why calculated underestimation of the CMRO2 values is different in two phantoms.

However, the regions of the most precise CMRO2 values are similar: 150 ≤ BW ≤ 250 Hz/pixel and 8 ≤ 훥푥 ≤ 10 mm for brain and tumor regions for 17O MRI at 3 T (Figure 4.1).

In 17O MRI inhalation experiments at UHFs, 175 ≤ BW ≤ 200 Hz/pixel, 8.0 ≤ 훥푥 ≤ 9.4 mm, and 42 ≤ 훥푡 ≤ 50 sec were used (Atkinson and Thulborn, 2010; Hoffmann et al., 2014, 2011). These parameters are in good agreement with the parameters obtained from the simulation framework, thus, in 17O MRI inhalation experiments 150 ≤ BW ≤ 175 Hz/pixel, 8.0 ≤ 훥푥 ≤ 10 mm, and 훥푡 = 60 sec were used.

5.2. Profile likelihood analysis of the CMRO2 pharmacokinetic model

Altered oxygenation is found in brain tumors and neurodegenerative diseases. Thus, it is of high clinical interest to map oxygen metabolism in clinical routine. With the recent implementation of 17 O MRI (Atkinson and Thulborn, 2010; Hoffmann et al., 2014, 2011), clinical CMRO2 quantification and oxygen metabolism mapping might become feasible; however, the different parameters in the numerical description of the oxygen uptake are often not identifiable from time- resolved measurements alone. In this part of the work, the PL method was used to identify those parameters in the CMRO2 quantification model that require prior knowledge for a unique identi- fication.

5.2 Profile likelihood analysis of the CMRO2 pharmacokinetic model

PL is an established method used to assess parameter uncertainties in nonlinear settings (Becker et al., 2010; Raue et al., 2013, 2009), where asymptotic CIs based on Fisher information are typically inappropriate (Raue et al., 2009). The latter are exact if the solution of the model is linear in the parameters and are a good approximation for a large amount of data and low measurement noise. If these conditions are not fulfilled, the Fisher information is underestimating the true CI and cannot capture the non-linearity effects outside the region near the optimum. In the 17O MRI experiments, the SNR of the acquired MR images was SNR = 4 – 7, leading to a relatively high measurement noise. Because the CMRO2 quantification model is nonlinear, asymptotic CIs, which are commonly used in the least-square fitting algorithms, might be misleading.

Initially, the RB circuit was proposed in (Hoffmann et al., 2011), where a potential 17O signal during RB phase was excluded from the data analysis due to complexity of the analytical solution of equation (2.78). Later, it was solved for constant 훼 values (Hoffmann, 2011); here, a numerical integration is used instead of solving the ODEs analytically. This numerical solution is beneficial 17 for modelling of H2 O signal-time curves, as it is more flexible and can be used for more elabo- rate 17O MRI experiments that, for example, use a lower amount of the rare 17O gas.

Both the DODS and the RB circuit were used to efficiently deliver and use the rare and costly 17 17 O gas. Decreasing the amount of the O gas required for a single patient experiment would directly reduce the total cost of 17O MR examination, which might be an important aspect for clinical studies. The DODS is internally triggered by the patient’s inhalation and efficiently delivers a precise and well-defined amount of 17O gas for each inhalation. As originally discussed Hoffmann et al. (Hoffmann et al., 2011), the risk of gas leakage and accidents in gas handling are minimized, because no transfer of the rare gas from the cylinder is required, and standard, clinically approved breathing components can be used. The PL analysis showed that only one 훼- related parameter needs to be estimated in a more complex CMRO2 quantification model for the MR examination with DODS and RB phases, as it is the case for the experiment with the RB phase only (Atkinson and Thulborn, 2010). The parameters of the advanced model, 푠1 and 푠2, are practically non-identifiable, but have no influence on either precision nor uncertainty of the target parameter CMRO2.

In this work, the advanced pharmacokinetic model for CMRO2 quantification was used, which accounts for linearly varying enrichment fraction of the inhaled 17O gas (Figure 3.8). If all model parameters were initialized without prior information, the values of CMRO2 and 훼 were 17 non-identifiable (Figure 4.3). Additional prior information about 훼퐷푂퐷푆, which represents the O enrichment from a DODS pulse, was not sufficient to quantify CMRO2. However, if the averaged 훼 during the DODS phase was taken as prior, the structural non-identifiability of CMRO2 was resolved (Figure 4.4). The identifiability is achieved because both α퐷푂퐷푆, which describes the 17 inhaled portion of O gas from a fresh DODS pulse, and 푠1, which describes the additional amount of 17O gas from the RB circuit, are constrained. This leads to an improved description of 17 the nonlinear increase of 푀퐻2 푂(푡) through the change from a constant α to a more realistic in- 17 put shape. In particular, the influence of the H2 O gain via metabolism and diffusion from

5. Discussion

blood, which are both positive in equation (2.78), can be distinguished by the model and lead to a better description of the data. From an analysis of the parameter profiles, 푠2 could be excluded from the model without affecting either the optima or the CIs of the other model parameters. A constant 훼 value during the RB phase is equivalent to a closed RB circuit in which 17O gas is ho- mogeneously distributed after the DODS phase and in which the 17O fraction remains constant, although the total amount of oxygen decreases after each inhalation.

As can be seen in Figure 4.4, either 훼퐷푂퐷푆 or 푠1 are practically non-identifiable; however, this non-identifiability does not affect the identifiability of the target parameter CMRO2 and the other model parameters. If 푠1 is set to zero, the practical non-identifiability of 훼퐷푂퐷푆 is resolved. How- ever, this simplified pharmacokinetic model with constant α values, which was previously used in (Hoffmann, 2011; Hoffmann et al., 2014, 2011), led to a significant decrease of 휒2 of the model up to 28% (Figure 4.6), whereas CI and optimal values of CMRO2 were underestimated by up to 9%. Thus, the use of the advanced model is beneficial even if not all model parameters can be fully identified.

In the simulation of different uncertainties for the prior information contained in 훼, a quadratic dependency between the relative CI of CMRO2 and the uncertainty of 훼 was found, as shown in Figure 4.7. Precision of 훼 can be increased by measuring the tidal volume and the dead volume of the lungs. The tidal volume can be measured with spirometry, which is a standard pulmonary function test. In addition, when the CO2 concentration of the exhaled air is measured, the dead space can be calculated. The increase of the 훼 precision from 10% to 5% would lead to an increase in CMRO2 precision of 20% – 39% / 10% – 33% for WM / GM regions.

17 Another model parameter that affects the CMRO2 value is the arrival time of the first O gas pulse at the alveoli. The 17O gas bottle and DODS, neither of which are MR safe, were placed outside the MR magnet room, and the tube delivering 17O gas into nasal cannula had a dead volume of 94 ml. Thus, the first two DODS pulses (volume: 40 ml / 50 ml in Exp1 / Exp2) had a 17 lower O2 concentration than the remaining pulses due to mixing with the dead volume. In a worst-case scenario this would lead to 7 – 8% underestimation of CMRO2. In our experiments, DODS was tested with 17O gas pulses before the actual MR examination, so that the tube was 17 17 well filled with O gas. Thus, this time delay potentially only affected the first O2 pulse, which would lead to a systematic error in the CMRO2 quantification of no more than 1 – 2%, which is much smaller than the calculated CIs.

In contrast to the models used previously (Atkinson and Thulborn, 2010; Hoffmann et al., 2011), profile likelihood allows addressing the amount of prior knowledge needed for robust CMRO2 quantification. As shown by the structural identifiability in Figure 4.3, an arbitrary value of

CMRO2 may be determined if no sufficient amount and quality of data is available, or if the chosen model renders the parameter non-identifiable. Whenever a nonlinear model is utilized, the method of profile likelihood should be used to assess accurate uncertainties of the parameters. The practical advantage of the advanced model is the flexibility it gives for 17O MR experiments.

5.3 Comparison of MR images from different reconstruction methods

In particular, in Exp4 the problem of the oxygen shortage at the end of the RB phase was solved by switching to a 16O gas supply during the RB phase, which was included in the advanced quantification model.

5.3. Comparison of MR images from different reconstruction methods

A synthetic numerical 17O MRI brain phantom was constructed from the segmented 1H MPRAGE data for the quantitative comparison of the reconstruction techniques (Figure 4.9). In the 17O MR images reconstructed with KBnH, no anatomical structures can be differentiated due to low SNR, whereas in KBwH and TV-constrained reconstructions the ventricles are faintly visible due to an increased SNR. RMSE values, however, did not change significantly (< 9% difference relative to KBnH, cf. Table 4.3), as RMSE does not correspond well to human image quality perception, which is highly adapted for extracting structural information from an image (Wang et al., 2004). On the other hand, SSIM considers image degradation as a perceived change in structural information while incorporating image luminance, contrast and structure comparison functions (Wang et al., 2004, 2003), and was used for comparison of reconstruction techniques in MRI (Anand and Sahambi, 2010; Gnahm and Nagel, 2015; Zhu et al., 2013). In this work, SSIM increased from KBnH to KBwH and TV-constrained reconstruction by 20%, and by 103% in the AD-constrained reconstruction. This corresponds to a visual improvement of 17O image quality: the ventricles obtained a shape similar to the original phantom image, and were clearly distin- guishable from the surrounding WM region of lower signal intensity.

For in vivo 17O MRI, the increased spatial resolution in Exp4 compared to Exp3 is evident (cf. time-averaged data in Figure 4.10 and Figure 4.11); however, for quantification of oxygen metabolism, the dynamic information contained in the single 1-min 17O images is analyzed. Images from Exp3 reconstructed with KBnH are already noisy (SNR = 7) due to the low natural abundance of 17O nucleus (0.037%). In Exp4, SNR decreased additionally by about 40%, and the brain is barely seen in KBnH images. For both experiments, application of a Hanning filter led to a 3-fold increase in SNR at the cost of image blurring. The TV constraint slightly increased the SNR of the images but led to a more blocky appearance compared to KBnH, which is common for the TV constraint. On the other hand, the AD constraint, which used the edge information from a coregistered 1H image, enabled a 6-7-fold SNR increase compared to KBnH, and made it possible to define the ventricular shape and even some fine GM structures. Moreover, AD- constrained reconstruction led to about a 2-fold SNR increase compared to KBwH. The AD constraint smoothed data among pixels with similar intensity (mostly within one brain tissue component) and preserved borders within different tissue compartments, which is favorable compared to isotropic homogeneous Hanning filtering.

It is not to be expected that AD filtering causes artefactual edges in the CMRO2 maps, because this filter essentially only introduces a directionality in the local signal filter, so that spatial filtering is not performed across the edges. In particular, if the 1H gradient image would contain information about the tissue borders which are not present in 17O image, it would cause only less fil-

5. Discussion

tering in that particular direction, but no additional artifacts in 17O images. On the other hand, if signal changes are present in the 17O image, but not in the 1H gradient image (e.g., in a heterogeneous tumor region), additional 1H information should be added, for example, by combining various 1H MRI contrasts to obtain the necessary edge information. The quantitative analysis of the numerical brain phantom revealed that the structural information was preserved in the AD- 1 constrained reconstruction (cf. Table 4.3). In principle, other H contrast (e.g., T2-weighting or proton density) can be used to extract the edge-information, but T1-weighted MPRAGE data is a natural choice because of high GM / WM tissue contrast and high resolution in all three directions. Contrarily, the 17O image averaged over the whole examination of 40 – 45 min cannot be considered as a good alternative due to high image blurring.

Recently, proton-constrained iterative reconstructions were shown to reduce partial volume effects and to increase the SNR of 23Na MR images (Gnahm and Nagel, 2015; Gnahm et al., 2014). In 23Na MRI, either the TV constraint in combination with a 1H binary mask prior (Gnahm et al., 2014) or an anatomically-weighted second-order TV constraint (Gnahm and Nagel, 2015) based on the idea of anatomical weighting for regularization (Haldar et al., 2008) were used. These methods provided lower SNR increase as KBwH compared to KBnH, whereas AD-constrained reconstruction led to an about 2-fold SNR increase compared to KBwH. A non-linear CG for L1- norm minimization was used in these studies (here, for TV-constrained reconstruction), which is about 10 times more computationally intensive compared to a linear CG used for AD constraint, which can be a practical limitation for 17O MRI, where about 40 – 45 datasets needed to be reconstructed.

5.4. In vivo CMRO2 quantification

17 CMRO2 rates obtained in all four O MRI experiments are in good agreement with previously reported results of 15O-PET studies (Ito et al., 1982; Leenders et al., 1990; Mineura et al., 1986; Rhodes et al., 1983) and 17O MRI at UHFs (Atkinson and Thulborn, 2010; Hoffmann, 2011;

Hoffmann et al., 2014, 2011). Yet, CMRO2 values of GM and WM regions are closer to each other compared to the results of 15O-PET studies (Table 4.1). This is mainly caused by the partial 17 * volume effects due to the low spatial resolution of O MR images and short T2 . As described in Subsection 2.2.4, the FWHM of the PSF, which determined spatial resolution, is about 1.79-1.95 (depending on the readout bandwidth). Hanning filtering causes an additional, stronger FWHM increase by 0.85, thus being unfavorable for 17O MRI applications. A partial volume correction can be applied using, for example, a “geometric transfer matrix” algorithm (Hoffmann et al., 2014; Rousset et al., 1998).

17 17 For the O nucleus, T1 is about 6 ms (Hoffmann, 2011); thus, the O images in this study are T1- weighted. Nevertheless, the 17O MR signal increase (10% – 20%) can be assumed to be propor- 17 tional to the increase of the H2 O concentration since the signal-time curves are normalized to the baseline, so that the T1-weighting effect is cancelled. The normalization to the baseline before 17O gas inhalation assumes a linearity of signal in the magnitude images – this linearity assump-

5.5 DIrect Estimation of 17O ImageS

tion is not valid for low SNR, but for SNR of five and higher (as it was for the simulated and in vivo 17O MRI data) the Rician noise distribution can be approximated as a Gaussian distribution, and the mean of the magnitude image is equal to the noise-free image (Gudbjartsson and Patz, 1995).

5.5. DIrect Estimation of 17O ImageS

In this work, the new method for 17O MR image estimation called DIESIS was proposed (Sub- section 3.4.2). For 17O MRI, DIESIS is an alternative to the standard gridding reconstruction (Figure 3.7), where 17O signal has be averaged over large brain regions to get the SNR sufficient 17 for CMRO2 quantification by the model fitting of the O signal-time curves (Figure 2.8). In DIESIS, 17O MR images were estimated after parcellation of the 3D imaging volume, which is based on the segmentation of the coregistered 1H MR data. Correction of PVEs was done by including the blurring effect of the radial acquisition scheme and the exponential tissue-specific * * T2 term (based on measured T2 values) into the calculation of the k-space representation of each parcel. It is worth noting, that the radial sampling scheme, which had to be used for the acquisition of the fast-decaying 17O MR signal as soon as possible after the RF excitation, causes more * PVEs (FWHM of the PSF for the radial sampling is 1.59) compared to the exponential T2 term, which causes additional increase of the FWHM of the PSF of less than 20%. In other words, * 17 considering the T2 term only is not sufficient for meaningful correction of PVEs. Measured O MR signal in k-space was represented as a linear combination of the estimated k-space signals of the parcels. The linear coefficients were then calculated for each 3D set of the dynamic 17O MR experiments, and obtained parcel-specific signal-time curves were fitted with the advanced

CMRO2 quantification model. The PL analysis showed identifiability of CMRO2 obtained with the DIESIS method and provided PL-based CIs.

15 CMRO2 values obtained with O-PET, which is considered as the gold standard, are well separated in GM and WM brain regions - the ratio of the CMRO2 values in GM and WM regions is about 2.5 (Leenders et al., 1990). In 17O MRI, spatial resolution roughly two times lower and PVE cause mixing of brain tissues. If the standard KBnH reconstruction was used for 17O MRI, this ratio was only 1.42 ± 0.22. With the DIESIS method, the CMRO2 decreased by 6 – 19% in WM and increased by 29 – 46% in GM compared to KBnH, therefore getting close to the 15 CMRO2 values from O-PET (Table 4.2 and Figure 4.8). As a consequence, the ratio of the

CMRO2 values in GM and WM increased to 2.23±0.51. These results show that DIESIS provided successful corrections of PVEs with the total computational time of about 5 times smaller compared to KBnH reconstruction.

DIESIS can also be used to obtain CMRO2 in glioblastoma tumor patients if regions of the heterogeneous tumor (Figure 2.9) can be represented as additional parcels. Moreover, PVEs can be corrected on sub-parcel level if some parcels are considered to consist of several tissues. Howev- er, the applicability of DIESIS requires parcellation to either single homogenous regions or to a known combination of those, which is not always be possible to get from 1H MR data. Addition-

5. Discussion

ally, it should be tested if SNR in small tumor parcels will be sufficient to solve a set of linear equations and to reliably quantify CMRO2 values.

5.6. In vivo CMRO2 mapping

PL analysis showed that the high noise level in the signal-time curves obtained from KBnH and

TV-constrained reconstructions (Figure 4.12) often led to practical non-identifiability of CMRO2.

The noise contribution also resulted in a high amount of pixels in the CMRO2 maps that could not be evaluated (Figure 4.14 and Table 4.4). Comparison of the CMRO2 maps of different reconstruction techniques showed that only AD-constrained and KBwH reconstructions provide

CMRO2 maps with sufficient quality– the other methods had Pnf > 24% (Table 4.4). 3D CMRO2 maps with AD-constrained reconstruction had Pnf < 11% and show the expected CMRO2 differences in brain tissues – for example, CMRO2 in WM is smaller than in GM regions (Figure 4.15).

The quality of the CMRO2 maps from Exp4 is lower than that of Exp3, and the CMRO2 values are systematically underestimated. This underestimation in the pixel-wise calculations can be 17 caused by the smaller H2 O signal increase in Exp4 which makes the signals more prone to noise contamination. Therefore, with the present set-up, 1.5 L of 17O gas use in Exp4 cannot be considered sufficient, and a higher amount of gas should be inhaled. CMRO2 calculation on a 64×64×64 matrix was also tested, but the signal-time curves had high noise contamination pre- venting adequate quality of the CMRO2 maps.

3 T CMRO2 maps have lower spatial resolution and are more heterogeneous (as they were at 9.4 T (Atkinson and Thulborn, 2010)) compared to the maps from 15O PET or 1H MRI-based studies. It might be explained by low SNR within small pixels and by imperfections in the RF excitation profiles of the home-built volume coil (Borowiak, 2017). For CMRO2 mapping at 9.4 T, a 23 Na data was used to extract tissue water content, which is required for CMRO2 quantification. However, this indirect measure is confounded by the fact that not all 23Na signal is captured due to short relaxation times, which can lead to inaccurate values in tumors with altered water content, as brain tumor patients often show an increased water content in WM up to 20% (Andersen, 1997). In this work, a segmented 1H MPRAGE data was used to create a water content map based on known WM, GM and CSF water partition coefficients (Whittall et al., 1997). Thus, the results are not directly applicable to tumor regions, but might be adapted using fast 1H MR-based methods for absolute water mapping (Neeb et al., 2008, 2006).

1 Over the last decade several promising pixel-wise CMRO2 quantification techniques using H MRI have been developed (Blockley et al., 2013; Bolar et al., 2011; Davis et al., 1998). Despite the 1 high spatial resolution provided by H MRI, and the potential to acquire CMRO2 information in only 5-10 minutes, these methods have several severe limitations. Firstly, CBF cannot be reliably measured in WM regions due to low amount of blood capillaries, which led to a CMRO2 non- 17 15 identifiability in WM and to distortion of CMRO2 maps as compared to O MRI and O-PET. Second, an assumption of fully oxygen saturated arterial blood is not valid, e.g. for hypoxic patients causing overestimation of OEF and, thus, CMRO2 values. CSF signal contamination is also

5.6 In vivo CMRO2 mapping

a concern and may result in an underestimation of CMRO2 (Rodgers et al., 2016) as compared to the results of 15O-PET studies (Leenders et al., 1990). Calibrated BOLD (Davis et al., 1998) has very short acquisition times; however, many inherent physiological assumptions in the signal modelling, and calibration procedures are challenging (Rodgers et al., 2016). Moreover, the mutual dependence between oxygen saturation and blood deoxygenation makes it difficult to differentiate between the cases when both parameters are high or both are small (Sohlin and Schad, 2011).

6. CONCLUSIONS AND OUTLOOK

In this thesis, a comprehensive study was performed to show feasibility of 17O MRI in a clinical MRI system for direct and non-invasive quantification of oxygen metabolism in the human brain. For this, novel simulation, image reconstruction and pharmacokinetic model analysis methods 17 were implemented to optimize O MRI examination, show CMRO2 identifiability, increase SNR of 17O MR images and correct for PVEs.

As an initial step, the flexible simulation framework was proposed to optimize acquisition parameters of the dynamic 17O MRI experiment. Two types of the phantoms were included into the framework to find optimal BW, spatial and temporal resolutions. The first one was the ACRO- BAT phantom, in which k-space representation was analytically calculated and three tumor regions were included. The second one was the numerical phantom with more complex brain structures, which are present in human brain, and was also used for a realistic comparison of image-reconstruction methods. Acquisition parameters, which were previously used in 17O MRI at UHFs were within the range of optimal parameters found with the simulation framework for both phantoms. Thus, 150 ≤ BW ≤ 175 Hz/pixel, 8.0 ≤ 훥푥 ≤ 10 mm, and 훥푡 = 60 sec were used in the 17O MRI experiments.

Four dynamic 17O MRI measurements with 70% isotope-enriched 17O gas inhalation were performed on a clinical 3 T MRI system. The breathing system was built from standard breathing equipment and provided efficient usage of rare and expensive 17O gas via the RB circuit, in which the stored 17O gas was re-inhaled in subsequent inhalation cycles.

17 The PL analysis showed that CMRO2 can be measured reliably in a O MRI experiment if the 17O enrichment fraction is estimated based on the experimental system and introduced as prior information into the model calculations. This model parameter has the highest impact on the precision of CMRO2 and, thus, has to be precisely estimated. Realistic estimation of the time- variant 17O enrichment fraction was feasible in the suggested advanced pharmacokinetic model, because the PL framework does not require an analytical solution of the model differential equation. It yielded not only in a better fit of the experimental data, but also solved the problem of the oxygen shortage at the end of the RB phase in Exp4 by switching to a 16O gas supply after 17O gas inhalation, which was included into the model. Uncertainty of another model parameter, the 17 arrival time of the first O gas pulse at the alveoli, originates from the dead volume of the breathing system, but was shown to have negligible influence on CMRO2. The quantified CMRO2 values of 0.72 – 0.99 µmol/gtissue/min in WM and 1.02 – 1.78 µmol/gtissue/min in GM brain regions were in good agreement with the results of 15O-PET and 17O MRI at UHFs.

The new method for 17O MR image estimation called DIESIS was proposed, which is an alternative to standard gridding methods, where 17O signal has to be averaged over large brain regions to 17 get the SNR sufficient for CMRO2 quantification by the model fitting of the O signal-time curves. It provided correction of PVEs, which are present in 17O MR images due to low spatial

6. Conclusions and outlook

* resolution and fast T2 decay. With the DIESIS method, the CMRO2 decreased by 6 – 19% in WM and increased by 29 – 46% in GM compared to KB gridding, therefore getting close to the 15 1 CMRO2 values from O-PET studies. If H MR data can provide segmentation of a heterogeneous glioblastoma tumor region into several homogenous regions, they can be easily included into the parcellation and provide information about tumor oxygen metabolism.

Finally, to address the key challenge of 17O MRI, low SNR – a novel iterative reconstruction method was suggested, which uses prior information from 1H MR data. AD constraint smoothes data among pixels with similar intensity (mostly within one brain tissue component) and pre- serves borders within different tissue compartments (edge-preservation), which is favorable compared to isotropic homogeneous Hanning filtering. AD-constrained reconstruction improves the quality of 17O MR images, as was demonstrated with simulated and experimental 17O MR data of the human brain. Providing higher SNR, AD-constrained reconstruction enabled quantification 17 of 3D CMRO2 maps in dynamic O MRI experiments at 3 T, which are comparable with the maps acquired at 9.4 T On the contrary, TV-constrained reconstruction did not provide a substantial SNR increase. In the experiments with a nominal spatial resolution of 10 mm, 3 liters of 17 70%-enriched O gas was required for CMRO2 mapping at 3 T.

17 17 Despite the advances in experimental set-up (effective pulsed O2 gas delivery), O MR signal acquisition (density adapted UTE sampling), image reconstruction (AD constraint) and modelling of the signal dynamics (flexible CMRO2 quantification model), the quality of CMRO2 maps from 17O MRI at 3 T could not yet compete with 15O-PET and 1H MRI. Further methodological improvements are needed to test whether the spatial resolution can be increased from 10 mm of 17O MRI to a typical 5 mm for 15O-PET. For 3D imaging this would require an 8-fold SNR improvement which is currently very challenging. SNR increase can either be achieved by through improved RF coil hardware. Recently, a 30 channel receive coil for 17O MRI at 7 T was designed (Meise et al., 2011), and this concept can also be applied at 3 T to locally increase SNR by up to a factor of 2 – 3. Furthermore, so-called “low-rank plus sparse matrix decomposition” model can be applied in reconstruction of dynamic 17O MR images as a superposition of static (natural abundance) and time-variant components, as this model provided higher SNR and improved image quality in cardiovascular 1H MRI (Otazo et al., 2015). In addition, higher amounts of 17O gas can be applied which is expected to increase the relative signal change accordingly. This work was intended to use only a minimal amount of the costly 17O gas, however, with an increasing demand for 17O MRI studies it is to be expected that the cost for the gas will decrease so that higher amounts can be used.

Currently, the total measurement time of 17O examinations is about 40 – 45 min, which can be decreased in clinical practice by about a factor of 2, as the wash-out phase does not need to be measured for 20 - 25 min. Another obstacle to a wide spread use of 17O MRI currently is the limited availability of the non-proton MR equipment (broadband RF transmitter, 17O MR coil); however, all this equipment is commercially available. In general, more 17O MRI examinations need to be performed to analyze the robustness and applicability in patient studies. In particular, the per-

6. Conclusions and outlook

formance of the AD-constrained reconstruction should be tested in 17O MRI experiments on patients with brain tumors (e.g., glioblastoma) to make sure that the quantification of metabolism of different tumor regions is not affected by 1H prior knowledge. In general, UHF MRI systems are advantageous for 17O MRI investigations because they provide 17 higher SNR values. As O relaxation times are B0 independent, MR signal intensity is propor- 2 tional to B0 and the noise intensity depends linearly on B0. According to this estimation, 7 T / 9.4 T MRI systems provide 2.3 / 3.1-fold higher SNR values than measurements at 3 T. Howev- er, UHF MRI systems are limited to few research centers and are often not used in clinical routine – if they become clinically more available, the feasibility of 17O MRI investigations would also increase.

Recently, high temporal resolution 17O MRS in combination with 17O-labeled glucose was used to study cellular glycolysis in mouse brains (Borowiak et al., 2017). In particular, the dynamics of labeled oxygen in the anomeric 1-OH and 6-CH2OH groups was detected after suppression of water peak, and the apparent cerebral metabolic rate of 17O-labelled glucose was extracted from a 17 17 pharmacokinetic model fit of the H2 O concentration-time course. Therefore, dynamic O-MRS of highly labeled 17O-glucose could provide additional information about glucose metabolism to 17O MRI with 17O gas inhalation, which might be important in the assessment of brain patholo- gies.

The high cost of the 17O gas (currently, about 2,000 $ for 1 L of 70% isotope-enriched gas) con- stitutes a severe limitation of 17O MRI. These costs are expected to decrease as more centers perform 17O measurements and more enriched 17O gas is produced. 17O MRI-based examinations would be easier compared to 15O-PET, because 17O is a stable isotope and much more conven- ient for production and distribution. Chemical safety and absence of radioactivity make 17O gas a safer alternative to radioactive PET tracers.

REFERENCES

Abragam, A., 1961. The Peinciples of Nuclear Magnetism. Oxford University Press, Oxford. Altshuler, S.A., Kozyrev, B.M., 1971. On the History of Discovery of Electron Paramagnetic Resonance, in: Paramagnitny Resonans. 1944-1969. Nauka, Moscow, pp. 25–31. Alvarez, L., Lions, P.L., Morel, J.M., 1992. Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Number. Anal. 29, 845–866. Anand, C.S., Sahambi, J.S., 2010. Wavelet domain non-linear filtering for MRI denoising. Magn. Reson. Imaging 28, 842–861. Andersen, C., 1997. In vivo estimation of water content in cerebral white matter of brain tumour patients and normal individuals: Towards a quantitative brain oedema definition. Acta Neurochir. (Wien). 139, 249–256. Ashburner, J., Friston, K.J., 2000. Voxel-Based Morphometry - The Methods. Neuroimage 11, 805–821. Ashburner, J., Friston, K.J., 2005. Unified segmentation. Neuroimage 26, 839–851. Atkinson, D., 1977. Cellular Energy Metabolism and its Regulation. Academic Press, New York. Atkinson, I.C., Thulborn, K.R., 2010. Feasibility of mapping the tissue mass corrected bioscale of cerebral metabolic rate of oxygen consumption using 17-oxygen and 23-sodium MR imaging in a human brain at 9.4 T. Neuroimage 51, 723–733. Attwell, D., Laughlin, S.B., 2001. An energy budget for signaling in the grey matter of the brain. J. Cereb. blood flow Metab. 21, 1133–1145. Baron, J.C., Bousser, M.G., Rey, A., Guillard, A., Comar, D., Castaigne, P., 1981. Reversal of focal “misery-perfusion syndrome” by extra- intracranial arterial bypass in haemodynamic cerebral ischaemia. Stroke 12, 454–459. Barrett, K., Brooks, H., Boitano, S., Barman, S., 2010. Ganong’s Review of Medical Physiology, 23rd ed. The McGraw-Hill Companies, New York. Beal, M.F., 1992. Does impairment of energy metabolism result in excitotoxic neuronal death in neurodegenerative illnesses? Ann. Neurol. 31, 119–130. Beatty, P.J., Nishimura, D.G., Pauly, J.M., 2005. Rapid Gridding Reconstruction With a Minimal Oversampling Ratio. IEEE Trans. Med. Imaging 24, 799–808. Becker, V., Schilling, M., Bachmann, J., Baumann, U., Raue, A., Maiwald, T., Timmer, J., Klingmüller, U., 2010. Covering a broad dynamic range: information processing at the erythropoietin receptor. Science. 328, 1404–1408. Bernstein, M.A., King, K.F., Zhou, X.J., 2004. Handbook of MRI Pulse Sequences. Elsevier, London. Bloch, F., 1946. Nuclear induction.‏ Phys. Rev. 70, 460–474. Block, K.T., 2008. Advanced Methods for Radial Data Sampling in Magnetic Resonance Imaging. PhD Thesis, University of Göttingen. Block, K.T., Uecker, M., Frahm, J., 2007. Undersampled radial MRI with multiple coils. Iterative image

References

reconstruction using a total variation constraint. Magn. Reson. Med. 57, 1086–98. Blockley, N.P., Griffeth, V.E.M., Simon, A.B., Buxton, Richard B. Buxtona, C., 2013. A review of calibrated blood oxygenation level-dependent (BOLD) methods for the measurement of task- induced changes in brain oxygen metabolism. NMR Biomed. 26, 987–1003. Bohr, N., 1913a. On the constitution of atoms and molecules. Philos. Mag. 26, 1–24. Bohr, N., 1913b. On the Constitution of Atoms and Molecules. Part II – Systems containing only a Single Nucleus. Philos. Mag. 26, 476–502. Bohr, N., 1913c. On the constitution of atoms and molecules. Part III – Systems containing several nuclei. Philos. Mag. 26, 857–875. Bolar, D.S., Rosen, B.R., Sorensen, A.G., Adalsteinsson, E., 2011. QUantitative Imaging of eXtraction of oxygen and TIssue consumption (QUIXOTIC) using venular-targeted velocity-selective spin labeling. Magn. Reson. Med. 66, 1550–62. Borowiak, R., 2017. Neue Methoden zur Untersuchung metabolischer Prozesse mittels spektroskopischer und bildgebender 17O-Kernspinresonanz. University of Freiburg, PhD thesis. Borowiak, R., Groebner, J., Haas, M., Hennig, J., Bock, M., 2014a. Direct cerebral and cardiac 17O-MRI at 3 Tesla: initial results at natural abundance. MAGMA 27, 95–99. Borowiak, R., Groebner, J., Kurzhunov, D., Fischer, E., Dragonu, I., Bock, M., 2014b. Direct cerebral 17O-MRI at a clinical field strenght of 3 Tesla using a Tx/Rx head coil. Proc. 22nd Annu. Meet. ISMRM; Milan, Italy 22, 4088. Borowiak, R., Reichardt, W., Kurzhunov, D., Schuch, C., Leupold, J., Krafft, A.J., Reisert, M., Lange, T., Fischer, E., Bock, M., 2017. Initial investigation of glucose metabolism in mouse brain using enriched 17O-glucose and dynamic 17O-MRS. NMR Biomed. , in print. Bowman, R.R., 1976. A probe for measuring temperature in radiofrequency-heated material. IEEE T Microw Theory 24, 43–45. Bracewell, R.N., Riddle, A.C., 1967. Inversion of fan-beam scans in radio astronomy. Astrophys. J. 150, 427–434. Carson, J.R., 1924. A Generalization of the Reciprocal Theorem. Bell Syst. Tech. J. 3, 393–399. Cassoly, R., 1982. Interaction of hemoglobin with the red blood cell membrane. A saturation transfer electron paramagnetic resonance study. Biochim. Biophys. Acta, 689, 203–209. Coleman, T.F., Y., L., 1996. An interior, trust region approach for nonlinear minimization subject to bounds. SIAM J. Optim. 6, 418–445. Collins, C.M., 2016. Electromagnetics in Magnetic Resonance Imaging. Morgan & Claypool Publishers, San Rafael, CA. Cooley, J.W., Tukey, J.W., 1965. An Algorithm for Machine Calculation of Complex Fourier Series. Math. Comput. 19, 297–301. Coyne, K., 2012. MRI: A Guided Tour [WWW Document]. Natl. high Magn. F. Lab. URL https://nationalmaglab.org/education/magnet-academy/learn-the-basics/stories/mri-a-guided-tour

References

Curry, H.B., 1944. The method of steepest descent for non-linear minimization problems. Quart. Appl. Math. 2, 258–261. Damadian, R., 1971. Tumor Detection by Nuclear Magnetic Resonance. Science. 171, 1151–1153. Davis, T.L., Kwong, K.K., Weisskoff, R.M., Rosen, B.R., 1998. Calibrated functional MRI: mapping the dynamics of oxidative metabolism. Proc. Natl. Acad. Sci. USA 95, 1834–1839. Ernst, R.R., Anderson, W.A., 1966. Application of fourier transform spectroscopy to magnetic resonance. Rev. Sci. Instrum. 37, 93–102. Estermann, I., Stern, O., 1933. Über die magnetische Ablenkung von Wasserstoffmolekülen und das magnetische Moment des Protons. II. Zeitschrift für Phys. 85, 17–24. Fedorov, A., Beichel, R., Kalpathy-Cramer, J., Finet, J., Fillion-Robin, J.C., Pujol, S., Bauer, C., Jennings, D., Fennessy, F., Sonka, M., Buatti, J., Aylward, S., Miller, J. V., Pieper, S., Kikinis, R., 2012. 3D Slicer as an image computing platform for the Quantitative Imaging Network. Magn. Reson. Imaging 30, 1323–1341. Fessler, J.A., Sutton, B.P., 2003. Nonuniform Fast Fourier Transforms Using Min-Max Interpolation. IEEE Trans. signal Process. 51, 560–574. Fiat, D., Kang, S., 1992. Determination of the rate of cerebral oxygen consumption and regional cerebral blood flow by non-invasive 17O in vivo NMR spectroscopy and magnetic resonance imaging: Part 1. Theory and data analysis methods. J. Neurol. Res. 14, 303–311. Fiat, D., Kang, S., 1993. Determination of the rate of cerebral oxygen consumption and regional cerebral blood flow by non-invasive 17O in vivo NMR spectroscopy and magnetic resonance imaging. Part 2.

Determination of CMRO2 for the rat by 17O NMR. J. Neurol. Res. 15, 7–22. Frackowiak, R.S.J., Herold, S., Petty, R.K.H., Morgan-Hughes, J.K., 1988. The cerebral metabolism of glucose and oxygen measured with positron tomography in patients with mitochondrial diseases. Brain 111, 1009–1024. Frisch, R., Stern, O., 1933. Über die magnetische Ablenkung von Wasserstoffmolekülen und das magnetische Moment des Protons. I. Zeitschrift für Phys. 85, 4–16. Fukuyama, H., Ogawa, M., Yamauchi, H., Yamaguchi, S., Kimura, J., Yonekura, Y., Konishi, J., 1994. Altered Cerebral Energy Metabolism in Alzheimer’s Disease : A PET Study. J. Nucl. Med. 35, 1–6. Garroway, A.N., Grannell, P.K., Mansfield, P., 1974. Image formation in NMR by a selective irradiative process. J. Phys. C Solid State Phys. 7, L457–L462. Gerig, G., Kubler, O., Kikinis, R., Jolesz, F. a, 1992. Nonlinear anisotrophic filtering of MRI data. IEEE Trans. Med. Imaging 11, 221–232. Gerlach, W., Stern, O., 1922. Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld. Zeitschrift für Phys. 9, 349–352. Gerothanassis, I.P., 2010a. Oxygen-17 NMR spectroscopy: basic principles and applications (part I). Prog. Nucl. Magn. Reson. Spectrosc. 56, 95–197. Gerothanassis, I.P., 2010b. Oxygen-17 NMR spectroscopy: basic principles and applications. Part II. Prog.

References

Nucl. Magn. Reson. Spectrosc. 57, 1–110. Gnahm, C., 2014. Development of an iterative reconstruction algorithmincluding anatomical pri- or information for 23Na magnetic resonance tomography In. Universität Heidelberg. Gnahm, C., Bock, M., Bachert, P., Semmler, W., Behl, N.G.R., Nagel, A.M., 2014. Iterative 3D projection reconstruction of 23Na data with an 1H MRI constraint. Magn. Reson. Med. 71, 1720–1732. Gnahm, C., Nagel, A.M., 2015. Anatomically weighted second-order total variation reconstruction of 23Na MRI using prior information from 1H MRI. Neuroimage 105, 452–61. Gudbjartsson, H., Patz, S., 1995. The Rician Distribution of Noisy MRI Data. Magn. Reson. Med. 34, 910–914. Haacke, E.M., Brown, R.W., Thompson, M.R., Venkatesan, R., 1999. Magnetic Resonance imaging. Physical Principles and Sequence Design. John Wiley & Sons, New York. Haase, A., Frahm, J., Matthaei, D., Hanicke, W., Merboldt, K.D., 1986. FLASH imaging. Rapid NMR imaging using low flip-angle pulses. J. Magn. Reson. 67, 258–266. Hager, W.W., Zhang, H., 2005. A New Conjugate Gradient Method with Guaranteed Descent and an Efficient Line Search. SIAM J. Optim. 16, 170–192. Hahn, E.L., 1950. Spin echoes. Phys. Rev. 80, 580–594. Haldar, J.P., Hernando, D., Song, S.-K., Liang, Z.-P., 2008. Anatomically constrained reconstruction from noisy data. Magn. Reson. Med. 59, 810–818. Harris, R.K., Becker, E.D., Cabral de Menezes, S.M., Goodfellow, R., Granger, P., 2002. NMR Nomenclature: Nuclear Spin Properties and Conventions for Chemical Shifts. Solid State Nucl. Magn. Reson. 22, 458–483. Hayden, M.E., Nacher, P.-J., 2016. History and physical principles of MRI, in: Saba, L. (Ed.), Magnetic Resonance Imaging Handbook. CRC Press, Boca Raton, pp. 2–25. Hennig, J., Nauerth, A., Friedburg, H., 1986. RARE imaging: a fast imaging method for clinical MR. Magn Reson Med 3, 823–33. Hestenes, M.R., Stiefel, E., 1952. Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49, 409–436. Hindmarsh, A.C., Brown, P.N., Grant, K.E., Lee, S.L., Serban, R., Shumaker, D.E., Woodward, C.S., 2005. SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers. ACM Trans. Math. Softw. 31, 363–396. Hoffmann, S.H., 2011. Localized quantification of the cerebral metabolic rate of oxygen consumption

CMRO2 with 17 O magnetic resonance tomograghy. PhD Thesis, University of Heidelberg. Hoffmann, S.H., Begovatz, P., Nagel, A.M., Umathum, R., Schommer, K., Bachert, P., Bock, M., 2011. A measurement setup for direct 17O MRI at 7 T. Magn. Reson. Med. 66, 1109–1115. Hoffmann, S.H., Radbruch, A., Bock, M., Semmler, W., Nagel, A.M., 2014. Direct 17O MRI with partial volume correction: first experiences in a glioblastoma patient. MAGMA 27, 579–87. Hopkins, A.L., Barr, R.G., 1987. Oxygen-17 Compounds as Potential NMR T2 Contrast Agents:

References

Enrichment Effects of H217O on Protein Solutions and Living Tissues. Magn. Reson. Med. 4, 399– 403. Hopkins, A.L., Lust, W.D., Haacke, E.M., Wielopolski, P., Barr, R.G., Bratton, C.B., 1991. The stability of

proton T2 effects of oxygen-17 water in experimental cerebral ischemia. Magn Reson Med 22, 167– 74. Hoult, D.I., Lauterbur, P.C., 1979. Sensitivity of the Zeugmatographic Experiment Involving Human Samples. J. Magn. Reson. 34, 425–433. Ishii, K., Kitagaki, H., Kono, M., Mori, E., 1996. Decreased Medial Temporal Oxygen Metabolism in Alzheimer’ s Disease Shown by PET. J. Nucl. Med. 37, 1159–1165. Ito, M., Lammertsma, A.A., Wise, R.J.S., Bernardi, S., Frackowiak, R.S.J., Heather, J.D., McKenzie, C.G., Thomas, D.G.T., Jones, T., 1982. Measurement of regional cerebral blood flow and oxygen utilisation in patients with cerebral tumours using 15O and positron emission tomography: analytical techniques and preliminary results. Neuroradiology 23, 63–74. Jackson, J.I., Meyer, C.H., Nishimura, D.G., Macovski, A., 1991. Selection of a convolution function for fourier inversion using gridding. IEEE Trans. Med. Imaging 10, 473–478. Jain, V., Langham, M.C., Wehrli, F.W., 2010. MRI estimation of global brain oxygen consumption rate. J. Cereb. blood flow Metab. 30, 1598–1607. Kaiser, J.F., 1966. Digital Filters, in: Kuo, F.F., Kaiser, J.F. (Eds.), System Analysis by Digital Computers. John Wiley & Sons, New York, pp. 218–285. Kety, S.S., Schmidt, C.F., 1948. The nitrous oxide method for the quantitative determination of cerebral blood flow in man: theory, procedure and normal values. J. Clin. Investig. 27, 476–483. Koay, C.G., Sarlls, J.E., Ozarslan, E., 2007. Three-dimensional analytical magnetic resonance imaging phantom in the Fourier domain. Magn. Reson. Med. 58, 430–6. Kumar, A., Welti, D., Ernst, R.R., 1975. NMR Fourier zeugmatography. J. Magn. Reson. 18, 69–83. Kurzhunov, D., Borowiak, R., Hass, H., Wagner, P., Krafft, A.J., Timmer, J., Bock, M., 2017a. Quantification of oxygen metabolic rates in human brain with dynamic 17O MRI: profile likelihood analysis. Magn. Reson. Med. , in print. Kurzhunov, D., Borowiak, R., Krafft, A.J., Bock, M., 2016a. The framework and AnalytiCally Represented

Oxygen-17 BrAin Tumor (ACROBAT) phantom for optimization of CMRO2 quantification protocols, in: In Proceedings of 24th Annual Meeting of ISMRM; Singapure. p. 1029. Kurzhunov, D., Borowiak, R., Reisert, M., Krafft, A.J., Bock, M., 2017b. DIrect EStimation of 17O MR

ImageS (DIESIS) for CMRO2 Quantification in the Human Brain with Partial Volume Correction, in: In Proceedings of 25th Annual Meeting of ISMRM; Honolulu, Hawaii, USA. p. 0488.

Kurzhunov, D., Borowiak, R., Reisert, M., Krafft, A.J., Özen, A.C., Bock, M., 2017c. 3D CMRO2 mapping in human brain with direct 17O MRI: comparison of conventional and proton-constrained reconstructions. Neuroimage 155, 612–624.

Kurzhunov, D., Borowiak, R., Reisert, M., Krafft, J.A., Özen, A.C., Bock, M., 2017d. 3D CMRO2

References

Mapping in Human Brain with Direct 17O-MRI: Comparisonof Methods for Image Reconstruction and Partial Volume Correction, in: In Proceedings of 25th Annual Meeting of ISMRM; Honolulu, Hawaii, USA. p. 3602.

Kurzhunov, D., Borowiak, R., Wagner, P., Reisert, M., Bock, M., 2015. Proton-Constrained CMRO2 Quantification with Direct 17O-MRI at 3 Tesla, in: In Proceedings of 23rd Annual Meeting of ISMRM; Toronto, Ontario, Canada. p. 2450.

Kurzhunov, D., Borowiak, R., Wagner, P., Reisert, M., Krafft, A.J., Bock, M., 2016b. 3D CMRO2 mapping in human brain with direct 17O-MRI and proton-constrained iterative reconstructions, in: In Proceedings of 24th Annual Meeting of ISMRM; Singapure. p. 1470. Kwong, K.K., Hopkins, A.L., Belliveau, J.W., Chesler, D.A., Porkka, L.M., McKinstry, R.C., Finelli, D.A., Hunter, G.J., Moore, J.B., Rosen, B.R., 1991. Proton NMR imaging of cerebral blood flow using

H217O. Magn. Reson. Med. 22, 154–158. LaManna, J.C., Puchowicz, M.A., Xu, K., Harrison, D.K., Bruley, D.F., 2008. Oxygen Transport to Tissue, Advances i. ed. Springer, New York. Landini, L., Positano, V., Santarelli, M.F., 2005. Signal Processing and Communications, Magnetic Resonance Imaging. Taylor & Francis, London. Lauterbur, P.C., 1973. Image formation by induced local interactions. Nature 242, 190–191. Leenders, K.L., Perani, D., Lammertsma, A.A., Heather, J.D., Buckingham, P., Healy, M.J., Gibbs, J.M., Wise, R.J., Hatazawa, J., Herold, S., 1990. Cerebral blood flow, blood volume and oxygen utilization. Normal values and effect of age. Brain 113, 27–47. Lehnert, W., Gregoire, M.-C., Reilhac, A., Meikle, S.R., 2012. Characterisation of partial volume effect and region-based correction in small animal positron emission tomography (PET) of the rat brain. Neuroimage 60, 2144–2157. Lu, H., Ge, Y., 2008. Quantitative evaluation of oxygenation in venous vessels using T2-relaxation-under- spin-tagging MRI. Magn. Reson. Med. 60, 357–363. Lu, M., Zhang, Y., Ugurbil, K., Chen, W., Zhu, X.-H., 2013. In vitro and in vivo studies of 17O NMR sensitivity at 9.4 and 16.4 T. Magn. Reson. Med. 69, 1523–1527. Lucken, E.A.C., 1969. Lucken_1969. Academic Press, London. Lustig, M., Donoho, D., Pauly, J.M., 2007. Sparse MRI: The application of compressed sensing for rapid MR imaging. Magn. Reson. Med. 58, 1182–1195. Mansfield, P., Maudsley, A.A., 1977. Planar spin imaging by NMR. J. Magn. Reson. 27, 101–119. Maurer, I., Zierz, S., Möller, H.J., 2000. A selective defect of cytochrome c oxidase is present in brain of Alzheimer disease patients. Neurobiol. Aging 21, 455–462. Maurer, I., Zierz, S., Möller, H.J., 2001. Evidence for a mitochondrial oxidative phosphorylation defect in brains from patients with schizophrenia. Schizophr. Res. 48, 125–136. McRobbie, D.W., Moore, E.A., Graves, M.J., Prince, M.R., 2003. MRI From Picture to Proton, second. ed. Cambridge University Press.

100

References

Meiboom, S., 1961. Nuclear Magnetic Resonance Study of the Proton Transfer in Water. J. Chem. Phys. 34, 375–388. Meise, F.M., Groebner, J., Nagel, A.M., Umathum, R., Stark, H., Hoffmann, S.H., Semmler, W., Bock, M., 2011. A 30-Channel Phased Array for Oxygen-17 Brain MRI at 7 Tesla, in: In Proceedings of 19th Annual Meeting of ISMRM; Montreal, Quebec, Canada. p. 472. Mellon, E.A., Beesam, R.S., Baumgardner, J.E., Borthakur, A., Witschey, W.R., Reddy, R., 2009. Estimation of the regional cerebral metabolic rate of oxygen consumption with proton detected 17O

MRI during precision 17O2 inhalation in swine. J. Neurosci. Methods 179, 29–39. Mellon, E.A., Beesam, R.S., Elliott, M.A., Reddy, R., 2010. Mapping of cerebral oxidative metabolism with MRI. Proc. Natl. Acad. Sci. USA 107, 11787–11792. Mineura, K., Yasuda, T., Kowada, M., Shishido, F., Ogawa, T., Uemura, K., 1986. Positron emission tomographic evaluation of histological malignancy in gliomas using oxygen-15 and fluorine-18- fluorodeoxyglucose. J. Neurol. Res. 8, 164–168. Mintun, M.A., Raichle, M.E., Martin, W.R.W., Herscovitch, P., 1984. Brain Oxygen Utilization Measured with O-15 Radiotracers and Positron Emission Tomography. J. Nucl. Med. 25, 177–187. Moran, P.R., 1982. A flow velocity zeugmatographic interlace for NMR imaging in humans. Magn. Reson. Imaging 1, 197–203. Mugler, J.P., Brookeman, J.R., 1990. Three-dimensional magnetization-prepared rapid gradient-echo imaging (3D MP RAGE). Magn. Reson. Med. 15, 152–157. Nagel, A.M., Laun, F.B., Weber, M.-A., Matthies, C., Semmler, W., Schad, L.R., 2009. Sodium MRI using a density-adapted 3D radial acquisition technique. Magn. Reson. Med. 62, 1565–1573. Neeb, H., Ermer, V., Stocker, T., Shah, N.J., 2008. Fast quantitative mapping of absolute water content with full brain coverage. Neuroimage 42, 1094–1109. Neeb, H., Zilles, K., Shah, N.J., 2006. Fully-automated detection of cerebral water content changes: Study of age- and gender-related H2O patterns with quantitative MRI. Neuroimage 29, 910–922. Neyman, J., Pearson, E.S., 1933. On the Problem of the Most Efficient Tests of Statistical Hypotheses. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 231, 289–337. Nyquist, H., 1928. Thermal agitation of electric charge in conductors. Phys Rev 32, 110–113. O’Sullivan, J.D., 1985. A Fast Sinc Function Gridding Algorithm for Fourier Inversion in Computer Tomography. IEEE Trans. Med. Imaging 4, 200–207. Oesterle, C., Markl, M., Strecker, R., Kraemer, F.M., Hennig, J., 1999. Spiral reconstruction by regridding to a large rectilinear matrix: A practical solution for routine systems. J. Magn. Reson. Imaging 10, 84–92. Otazo, R., Candès, E., Sodickson, D.K., 2015. Low-rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components. Magn. Reson. Med. 73, 1125–36. doi:10.1002/mrm.25240 Pekar, J., Ligeti, L., Ruttner, Z., Lyon, R.C., Sinnwell, T.M., van Gelderen, P., Fiat, D., Moonen, C.T.W.,

101

References

McLaughlin, A.C., 1991. In Vivo measurement of cerebral oxygen consumption and blood flow using 17O MRI. Magn. Reson. Med. 21, 313–319. Perona, P., Malik, J., 1990. Scale-Space and Edge Detection Using Anisotropic Diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639. Pipe, J.G., 2000. Reconstructing MR images from undersampled data: data-weighting considerations. Magn. Reson. Med. 43, 867–75. Pruessmann, K.P., Weiger, M., Börnert, P., Boesiger, P., 2001. Advances in Sensitivity Encoding With Arbitrary k-Space Trajectories. Magn. Reson. Med. 651, 638–651. Purcell, E., Torrey, H., Pound, R., 1946. Resonance Absorption by Nuclear Magnetic Moments in a Solid. Phys. Rev. 69, 37–38. Rabi, I.I., Zacharias, J., Millman, S., Kusch, P., 1938. A New Method of Measuring Nuclear magnetic Moment. Phys. Rev. 53, 318. Rahmer, J., Börnert, P., Groen, J., Bos, C., 2006. Three-dimensional radial ultrashort echo-time imaging

with T2 adapted sampling. Magn. Reson. Med. 55, 1075–1082. Raichle, M.E., Grubb, R.L., Eichling, J.O., Ter-Pogossian, M.M., 1976. Measurement of brain oxygen utilization with radioactive oxygen-15: experimental verification. J Appl Physiol 40, 638–640. Rakhmanov, E.A., Saff, E.B., Zhou, Y.M., 1994. Minimal Discrete Energy on the Sphere. Math. Res. Lett. 1, 647–662. Rauben, J., Samuel, D., Tzalmona, A., 1962. The direct measurement of the 17O-proton spin-spin coupling in water. Proc Chem Soc 353. Raue, A., Kreutz, C., Maiwald, T., Bachmann, J., Schilling, M., Klingmüller, U., Timmer, J., 2009. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 1923–1929. Raue, A., Schilling, M., Bachmann, J., Matteson, A., Schelker, M., Kaschek, D., Hug, S., Kreutz, C., Harms, B.D., Theis, F.J., Klingmüller, U., Timmer, J., 2013. Lessons learned from quantitative dynamical modeling in systems biology. PLoS One 8, e74335. Raue, A., Steiert, B., Schelker, M., Kreutz, C., Maiwald, T., Hass, H., Vanlier, J., Tönsing, C., Adlung, L., Engesser, R., Mader, W., Heinemann, T., Hasenauer, J., Schilling, M., Höfer, T., Klipp, E., Theis, F., Klingmüller, U., Schöberl, B., Timmer, J., 2015. Data2Dynamics: A modeling environment tailored to parameter estimation in dynamical systems. Bioinformatics 31, 3558–3560.

Reddy, R., Stolpen, A.H., Leigh, J.S., 1995. Detection of 17O by Proton T1rho Dispersion Imaging. J. Magn. Reson. 108, 276–279. Regatte, R.R., Akella, S.V.S., Borthakur, A., Reddy, R., 2003. Proton spin-lock ratio imaging for quantitation of glycosaminoglycans in articular cartilage. J. Magn. Reson. imaging 17, 114–121. Rhodes, C.G., Wise, R.J.S., Gibbs, J.M., Frackowiak, R.S.J., Hatazawa, J., Palmer, A.J., Thomas, D.G., Jones, T., 1983. In vivo disturbance of the oxidative metabolism of glucose in human cerebral gliomas. Ann. Neurol. 14, 614–626.

102

References

Richardson, S.J., 1989. Contribution of proton exchange to the oxygen-17 nuclear magnetic resonance transverse relaxation rate in water and starch-water systems. Cereal Chem. 66, 244–246. Rodgers, Z.B., Detre, J.A., Wehrli, F.W., 2016. MRI-based methods for quantification of the cerebral metabolic rate of oxygen. J. Cereb. Blood Flow Metab. 36, 1165–1185. Rolfe, D.F., Brown, G.C., 1997. Cellular Energy Utilization and Molecular Origin of Standard Metabolic Rate in Mammals. Physiol. Rev. 77, 731–758.

Ronen, I., Lee, J.-H., Merkle, H., Ugurbil, K., Navon, G., 1997. Imaging H217O distribution in a phantom

and measurement of metabolically produced H217O in live mice by proton NMR. NMR Biomed. 10, 333–340.

Ronen, I., Merkle, H., Ugurbil, K., Navon, G., 1998. Imaging of H217O distribution in the brain of a live rat by using proton-detected 17O MRI. Proc. Natl. Acad. Sci. USA 95, 12934–12939. Rousset, O.G., Ma, Y., Evans, A.C., 1998. Correction for partial volume effects in PET: principle and validation. J. Nucl. Med. 39, 904–911. Rudin, L.I., Osher, S., Fatemi, E., 1992. Nonlinear total variation based noise removal algorithms. Phys. D 60, 259–268. Schmidt, R.F., Lang, F., Heckmann, M., 2011. Physiologie des Menschen, 31st ed. Springer, Heidelberg. Shepp, L.A., 1980. Computerized Tomography and Nucler Magnetic Resonance. J. Comput. Assised Tomogr. 4, 94–107. Shepp, L.A., Logan, B.F., 1974. The Fourier reconstruction of a head section. IEEE Trans. Nucl. Sci. NS- 21, 21–43. Shukla, A.K., Kumar, U., 2006. Positron emission tomography: An overview. J Med Phys 31, 13–21. Slichter, C.P., 1989. Principles of Magnetic Resonance. Springer, Berlin. Smith, P.R., Peters, T.M., Bates, R.H.T., 1973. Image reconstruction from finite numbers of projections. J. Phys. A Math. Nucl. Gen. 6, 361–382. Sobesky, J., Weber, O.Z., Lehnhardt, F.G., Hesselmann, V., Neveling, M., Jacobs, A., Heiss, W.D., 2005. Does the mismatch match the penumbra? Magnetic resonance imaging and positron emission tomography in early ischemic stroke. Stroke 36, 980–985. Sohlin, M.C., Schad, L.R., 2011. Susceptibility-related MR signal dephasing under nonstatic conditions: Experimental verification and consequences for qBOLD measurements. J. Magn. Reson. Imaging 33, 417–425. Takagi, H., Shapiro, K., Marmarou, A., Wisoff, H., 1981. Microgravimetric analysis of human brain tissue: Correlation with computerized tomography scanning. J Neurosurg 54, 797–801. Ter-Pogossian, M.M., Eichling, J.O., Davis, D.O., Welch, M.J., 1970. The measure in vivo of regional cerebral oxygen utilization by means of oxyhemoglobin labeled with radioactive oxygen-15. J. Clin. Invest. 49, 381–391.

Thelwall, P.E., Blackband, S.J., Chen, W., 2003. Field dependence of 17O T1, T2 and SNR - in vitro and in vivo studies at 4.7, 11 and 17.6 Tesla, in: In Proceedings of 11th Annual Meeting of ISMRM;

103

References

Toronto, Ontario, Canada. p. 504. Thulborn, K.R., Davis, D., Adams, H., Gindin, T., Zhou, J., 1999. Quantitative tissue sodium concentration mapping of the growth of focal cerebral tumors with sodium magnetic resonance imaging. Magn. Reson. Med. 41, 351–359. Tyler, J.L., Diksic, M., Villemure, J.G., Evans, A.C., Meyer, E., Yamamoto, Y.L., Feindel, W., 1987. Metabolic and hemodynamic evaluation of gliomas using positron emission tomography. J. Nucl. Med. 28, 1123–1133. Venzon, D.Z., Moolgavkar, S.H., 1988. A method for computing profile-likelihood-based confidence intervals. Appl. Stat. 37, 87–94. Wagner, P., 2014. Aufbau und Charakterisierung eines Atemsystems für die 17O Bildgebung im MRT. Bachelor Thesis, University of Freiburg. Wallace, D.C., 1992. Mitochondrial genetics: a paradigm for aging and degenerative diseases? Science. 256, 628–632. Wallace, D.C., 1999. Mitochondrial Diseases in Man and Mouse. Science. 283, 1482–1488. Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P., 2004. Image quality assessment: From error visibility to structural similarity. IEEE Trans. Image Process. 13, 600–612. Wang, Z., Simoncelli, E.P., Bovik, A.C., 2003. Multi-Scale Structural Similarity For Image Quality Assessment. Proc. IEEE Asilomar Conf. Signals, Syst. Comput. 37, 1398–1402. Weickert, J., 1998. Anisotropic Diffusion in Image Processing. B.G. Teubner, Stuttgart. Whittall, K.P., MacKay, A.L., Graeb, D.A., Nugent, R.A., Li, D.K.B., Paty, D.W., 1997. In vivo

measurement of T2 distributions and water contents in normal human brain. Magn. Reson. Med. 37, 34–43. Wilks, S.S., 1938. The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses. Ann. Math. Stat. 9, 60–62. Williams, D.S., Detre, J.A., Leigh, J.S., Koretsky, A.P., 1992. Magnetic resonance imaging of perfusion using spin inversion of arterial water. Proc. Natl. Acad. Sci. USA 89, 212–6. Wong, E.C., 2014. Direct imaging of functional networks. Brain Connect. 4, 481–6. Wong, E.C., Cronin, M., Wu, W.-C., Inglis, B., Frank, L.R., Liu, T.T., 2006. Velocity-selective arterial spin labeling. Magn. Reson. Med. 55, 1334–1341. Wong-Riley, M., Antuono, P., Ho, K.C., Egan, R., Hevner, R., Liebl, W., Huang, Z., Rachel, R., Jones, J., 1997. Cytochrome oxidase in Alzheimer’s disease: biochemical, histochemical, and immunohistochemical analyses of the visual and other systems. Vision Res. 37, 3593–3608. Xu, F., Ge, Y., Lu, H., 2009. Noninvasive quantification of whole-brain cerebral metabolic rate of oxygen

(CMRO2) by MRI. Magn. Reson. Med. 62, 141–148. Zavoisky, E., 1945. Spin-magnetic resonance in paramagnetics. Fiz. Zhurnal 9, 211–245. Zhang, N., Zhu, X.-H., Lei, H., Ugurbil, K., Chen, W., 2004. Simplified methods for calculating cerebral metabolic rate of oxygen based on 17O magnetic resonance spectroscopic imaging measurement

104

References

during a short 17O2 inhalation. J. Cereb. blood flow Metab. 24, 840–848. Zhu, X.-H., Chen, W., 2011. In vivo oxygen-17 NMR for imaging brain oxygen metabolism at high field. Prog. Nucl. Magn. Reson. Spectrosc. 59, 319–335. Zhu, X.-H., Merkle, H., Kwag, J.H., Ugurbil, K., Chen, W., 2001. 17O relaxation time and NMR sensitivity of cerebral water and their field dependence. Magn. Reson. Med. 45, 543–9. Zhu, X.-H., Zhang, N., Zhang, Y., Zhang, X., Ugurbil, K., Chen, W., 2005. In vivo 17O NMR approaches for brain study at high field. NMR Biomed. 18, 83–103. Zhu, X.-H., Zhang, X., Zhang, N., Zhang, Y., Strupp, J., Ugurbil, K., Chen, W., 2006. High- field 17O

Study of 3D CMRO2 Imaging in human visual cortex, in: Proc. Intl. Soc. Mag. Reson. Med. Seattle, p. 409. Zhu, X.-H., Zhang, Y., Tian, R.-X., Lei, H., Zhang, N., Zhang, X., Merkle, H., Ugurbil, K., Chen, W., 2002. Development of 17O NMR approach for fast imaging of cerebral metabolic rate of oxygen in rat brain at high field. Proc. Natl. Acad. Sci. USA 99, 13194–13199. Zhu, Z., Wahid, K., Babyn, P., Yang, R., 2013. Compressed sensing-based MRI reconstruction using complex double-density dual-tree DWT. Int. J. Biomed. Imaging.

105

106

LIST OF SYMBOLS

17 퐴 푂 fraction of 17O-labelled arterial oxygen gas (in excess of natural abundance) 퐻17푂 퐵 2 relative amount of H217O in blood (in excess of natural abundance) 퐶17 17 푂,푏표푡푡푙푒 O isotope enrichment of oxygen stored in the gas cylinder

퐶퐾퐵 Keiser-Bessel function 퐶푂,푎푖푟 oxygen content in the air

퐷푟푎푑 sampling density for radial k-space trajectories

퐸푚 energy levels in external magnetic field 퐻푏푡표푡푎푙 total concentration of hemoglobin in blood 퐼 nuclear quantum number 퐼퐵푒푠푠푒푙,0 zero-order modified Bessel function of the first kind 퐼푚 17 17 퐻2 푂 simulated O MR image 17 퐼푚푒푙 simulated O MR image of an ellipsoid 17 퐾퐺 rate constant that reflects H2 O gain by diffusion from blood 17 퐾퐿 rate constant that reflects H2 O loss by diffusion to blood 퐿 length of the object 퐻17푂 푀 2 H217O concentration

푀⊥ transverse magnetization 푀푆푆 steady-state of longitudinal magnetization

푀푎푠푘푖 mask of ith parcel 푁푆 number of spin in spin ensemble 푁푎푐푞 number of acquisitions

푁푒푙 number of ellipsoids 푁푛푢푐 number of nucleons 푁푝푥 number of pixels

푁푠푝 number of sampling points along one spoke (projection) 푁푠푝표푘푒푠 number of spokes 푁푠푡_훼 and 푁푠푡_훽 occupation of spin states 푠푡_훼 and 푠푡_훽

푁푠푡_훼+푠푡_훽 total number of spins in two states 푠푡_훼 and 푠푡_훽

푃푛푓 percentage of pixels, for which CMRO2 mapping failed due to low SNR

푄 nuclear quadrupole moment

푆퐵→퐶 maximum slew rate of the DAPR sequence 푆푒푛푠17 푆푒푛푠1 17 1 푂 and 퐻 MR sensitivity of O and H nuclei, correspondingly

푆푘푒푠푡 estimated k-space signal 17 푆푘푖푑푒푎푙 FT of simulated O MR image 푆푘푚푒푎푠 measured k-space signal * 푆푘푟푒푎푙 푆푘푖푑푒푎푙 with added Gaussian noise and T2 effect

푆푚푒푎푠 measured signal intensity 푆푛표푖푠푒 amplitude of noise 푆푟푒푎푙 noise-free signal intensity

107

List of symbols

17 푇퐴, 푇퐵 and 푇퐶 time points that determine phases of the dynamic O MRI experiment

푈 (풓) FT of sampling function

푈푖푛푑 electromotive force

푉 volume element 푉푑푒푎푑 dead space volume of the respiratory system 푉푡푖푑푎푙 tidal volume 푊 (풓) FT of the filter function

푊퐾퐵 width of the KB kernel 푌푎 and 푌푣 arterial and venous oxygen saturation

푍 partition function of the Boltzmann distribution 푎푒푙,푖, 푏푒푙,푖 and 푐푒푙,푖 half the length of the principal axes of the ith ellipsoid

푐퐾퐵 FT of the Keiser-Bessel function

푐푁퐴 natural abundance of the isotope 푑 푨푐 coil element 푒 elementary charge

푒푞 main component of EFG tensor 푓0 Larmor frequency (in Hz) 푔퐼 dimensionless magnetic moment (g-factor)

푘0 k-space value, from which density adaptation is applied 푘퐵 Boltzmann’s constant 17 푘표푥 rate at which fresh O binds to hemoglobin 푙푆푆퐼푀, 푐푆푆퐼푀 and 푠푆푆퐼푀 luminance, contrast and structural terms of structural similarity

푚 magnetic quantum number 푚푝 proton rest mass

푛푑표푓 number of degrees of freedom

푝퐷퐴푃푅 DAPR fraction 푝푚 probability of finding the particle in the state with energy 퐸푚 17 푠1 slope of the O enrichment change during the DODS phase 17 푠2 slope of the O enrichment change during the RB phase 푢 (풓) sampling function

푢푠(푡) externally provided stimulus 푤 (풓) filter function

푨 system matrix that maps image to the corresponding raw data 푨푛푢푓푓푡 NUFFT operator

푩1 alternating magnetic field 푩푟푒푐푒푖푣푒 receive field produced by detection coil

푩ퟎ static magnetic field 푫 anisotropic non-homogeneous diffusion operator 푮 gradient fields

푰 net nuclear spin 푴 macroscopic magnetization 푹 regularization term

108

List of symbols

푹풐풕 rotation matrix 품 gradient operator

풌 wave vector

퓔 gridding kernel 퓕−ퟏ inverse FFT

퓖 k-space interpolation

훥퐶퐿 threshold for the likelihood value 훷 magnetic flux 17 훼 O enrichment fractions of inhaled gas in the DODS phase 17 17 훼퐷푂퐷푆 O enrichment originated from O pulses delivered during the DODS phase 훼푂푆 k-space oversampling 17 훼푅퐵 O enrichment at the end of the RB phase 17 훽 O enrichment fractions of the inhaled gas in the RB phase 훽퐾퐵 optimal scaling parameter of the KB function 훾 gyromagnetic ratio 훿푥,푖, 훿푦,푖 and 훿푧,푖 translation from the center of coordinate system of ith ellipsoid

휂 asymmetry parameter of the EFG tensor 휃 model parameters 휃̂ optimal model parameters

휃퐸 Ernst angle 휃푀 rotation angle of magnetization (flip angle)

휆 weighting factor of the regularization term 휇푛 nuclear magneton 휌 (풓) or 흆 MR image

휌퐺푀, 휌푊푀 and 휌퐶푆퐹 signal intensity within GM, WM and CSF brain regions

휌푆 spin density 휌푒푙,푖 signal intensity within the ith ellipsoid

휎퐺푎푢푠푠 standard deviation of the Gaussian distribution 휎푛표푖푠푒 standard deviation of noise 휏퐶 correlation time 휑 (풓, 푡) phase accumulated during precession in the presence of gradient fields 휓1 and 휓2 polar and azimuthal angles in the spherical coordinates

휔0 Larmor frequency (in radians per second)

휔푅퐹 frequency of the alternating field 휖푖푗푘 Levi-Civita symbol

휙1, 휙2, 휙3 Euler angles

휚푖 signal intensity of ith parcel ∆푡 temporal resolution ∆푥 spatial resolution ∆휏 dwell time

109

110

LIST OF ABBREVIATIONS

ACROBAT AnalytiCally Represented Oxygen-17 BrAin Tumor (phantom) AD anisotropic diffusion ADC analog-digital-converter ADP adenosine diphosphate ASL arterial spin labelling ATP adenosine triphosphate AV alveoli ventilation BOLD blood-oxygen-level dependent (MR constrast) BW receiver bandwidth CBF cerebral blood flow CE contrast-enhancing rim of the tumor CG conjugate gradient CI confidence interval

CMRO2 cerebral metabolic rate of oxygen consumption CT computed tomography DAPR density adapted projection reconstruction dHb deoxyhemoglobin DIESIS DIrect Estimation of 17O ImageS DODS demand oxygen delivery system EFG electric field-gradient FFT fast Fourier transformation FID free induction decay FLASH Fast Low Angle SHot (MR sequence) FOV field-of-view FT Fourier transform FWHM full width at half maximum GM gray (brain) matter Ĥ Hamiltonian operator ħ reduced Planck constant Hyp hypothesis KB Kaiser-Bessel (gridding) KBnH Kaiser-Bessel gridding without any filtering KBwH Kaiser-Bessel gridding with Hanning filter k-space Fourier space (spatial frequencies in the MR image) LR likelihood-ratio MPRAGE Magnetization Prepared RApid Gradient Echo (MR sequence) MR magnetic resonance

111

List of abbreviations

MRI magnetic resonance imaging NE necrotic center of the tumor NMR nuclear magnetic resonance NUFFT non-uniform fast Fourier transform ODE ordinary differential equation OEF oxygen extraction fraction PC-MRI phase-contrast MRI PE perifocal edema of the tumor PET positron emission tomography Pi inorganic phosphate PL profile likelihood PSF point spread function PVEs partial volume effects QUIXOTIC QUantitative Imaging of eXtraction of Oxygen and TIssie Consumption RB rebreathing RF radio frequency RMSE root-mean-square error RR respiratory rate SNR signal-to-noise ratio SSIM structural similarity

T1 spin-lattice relaxation time

T1ρ spin-lattice relaxation time in the rotating frame

T2 spin-spin relaxation time

T2* effective spin-spin relaxation time

T2’ decay time due to local magnetic field inhomogeneities TE echo time TI inversion decay time

Tpulse duration of the RF excitation pulse TR repetition time of the MR pulse sequence cycle

TRO readout (sampling) time

TRUST T2-Relaxation-Under-Spin-Tagging (MR sequence) TV total variation UHF ultra-high field UTE ultrashort TE WM white (brain) matter X-nuclei non-proton nuclei with a nonzero nuclear spin

112

LIST OF PUBLICATIONS

Journal publications

Kurzhunov, D., Borowiak, R., Hass, H., Wagner, P., Krafft, A.J., Timmer, J., Bock, M., 2017. Quantification of oxygen metabolic rates in human brain with dynamic 17O MRI: profile likelihood analysis. Magnetic Resonance in Medicine, in print, doi: 10.1002/mrm.26476.

Kurzhunov, D., Borowiak, R., Reisert, M., Krafft, A.J., Özen, A.C., Bock, M., 2017. 3D CMRO2 mapping in human brain with direct 17O MRI: comparison of conventional and proton-constrained reconstructions. NeuroImage 155, 612-624, doi: 10.1016/j.neuroimage.2017.05.029.

Borowiak, R., Reichardt, W., Kurzhunov, D., Schuch, C., Leupold, J., Krafft, A.J., Reisert, M., Lange, T., Fischer, E., Bock, M., 2017. Initial investigation of glucose metabolism in mouse brain using enriched 17O-glucose and dynamic 17O-MRS. NMR in Biomedicine, in print, doi: 10.1002/nbm.3724.

Conference Contributions

Kurzhunov, D., Borowiak, R., Reisert, M., Krafft, A.J., Bock, M., 2017. Direct estimation of 17O-

MRI images (DIESIS) for CMRO2 quantification in the human brain with partial volume correction. In Proceedings of the 25th annual meeting of the International Society for Magnetic Resonance in Medicine. Honolulu, USA.

Kurzhunov, D., Borowiak, R., Reisert, M., Krafft, A.J., Özen, A.C., Bock, M., 2017. 3D CMRO2 mapping in human brain with direct 17O MRI: Comparison of methods for image reconstruction and partial volume correction. In Proceedings of the 25th annual meeting of the International Society for Magnetic Resonance in Medicine. Honolulu, USA.

Kurzhunov, D., Borowiak, R., Özen, A.C., Bock, M., 2017. CMRO2 quantification in human brain with direct 17O MRI: profile likelihood analysis for protocol optimization. In Proceedings of the 25th annual meeting of the International Society for Magnetic Resonance in Medicine. Honolulu, USA. Borowiak, R., Reichardt, W., Kurzhunov, D., Schuch, C., Görling, B., Leibfritz, D., Leupold, J., Lange, T., Haas, H., Timmer, J., Bock, M., 2017. Quantification of cerebral metabolic rates of 17O-labelled glucose in mouse brain with dynamics 17O-MRS. In Proceedings of the 25th annual meeting of the International Society for Magnetic Resonance in Medicine. Honolulu, USA. Fischr, J., Özen, A.C., Kurzhunov, D., Reisert, M.,Tesfai, A., Rühli, F., Ludwig, U., Bock, M., 2017. Cross-modality MR image reconstruction: CT-constrained anisotropic diffusion to preserve edge information in MRI of an ancient mummified hand. In Proceedings of the

113

List of publications

25th annual meeting of the International Society for Magnetic Resonance in Medicine. Honolulu, USA. Kurzhunov, D., Borowiak, R., Reisert, M., Bock, M., 2017. Proton-constrained iterative 17 reconstructinos of O-MRI images for CMRO2 quantification. In Proceedings of the 4th International Biomedical and Astronomical Signal Processing (BASP) Frontiers workshop. Villars-sur-Ollon, Switzerland. Kurzhunov, D., Borowiak, R., Krafft, A.J., Bock, M., 2016. The Framework and analytically

represented oxygen-17 brain tumor (ACROBAT) phantom for optimization of CMRO2 quantication protocols in dynamic 17O-MRI. In Proceedings of the 24th annual meeting of the International Society for Magnetic Resonance in Medicine. Singapore.

Kurzhunov, D., Borowiak, R., Wagner, P., Reisert, M., Krafft, A.J., Bock, M., 2016. 3D CMRO2 mapping in human brain with direct 17O-MRI and proton-constrained iterative reconstructions. In Proceedings of the 24th annual meeting of the International Society for Magnetic Resonance in Medicine. Singapore. Borowiak, R., Reichardt, W., Kurzhunov, D., Schuch, C., Leupold, J., Krafft, A.J., Reisert, M., Lange, T., Fischer, E., Bock, M., 2016. Initial investigation of glucose metabolism inmouse brain using enriched 17O-glucose and dynamic 17O-MRS. In Proceedings of the 24th annual meeting of the International Society for Magnetic Resonance in Medicine. Singapore. Kurzhunov, D., Borowiak, R., Reisert, M., Krafft, A.J., Bock, M., 2016. Direct estimation of 17O-

MRI images (DIESIS) for CMRO2 quantification in the human brain with partial volume correction. In Proceedings of 47th annual meeting of the German Society of Medical Physics and 19th annual meeting of the German Chapter of the International Society for Magnetic Resonance in Medicine. Würtzburg, Germany. Kurzhunov, D., Borowiak, R., Reisert, M., Wagner, P., Bock, M., 2015. Proton-constrained 17 CMRO2 quantification with direct O-MRI at 3 Tesla. In Proceedings of the 23rd annual meeting of the International Society for Magnetic Resonance in Medicine. Toronto, Canada. Borowiak, R., Kurzhunov, D., Wagner, P., Reisert, M., Bock, M., 2015. Dynamic 17O-MRI at 3

Tesla for in vivo CMRO2 quantification. In Proceedings of the 23rd annual meeting of the International Society for Magnetic Resonance in Medicine. Toronto, Canada. Kurzhunov, D., Borowiak, R., Bock, M., 2015. An analytical 17O MRI head tumor phantom for

optimization of CMRO2 quantification protocols. In Proceedings of the 18th annual meeting of the German Chapter of the International Society for Magnetic Resonance in Medicine. Münster, Germany. Kurzhunov, D., Borowiak, R., Bock, M., 2014. Simulation analysis of noise and BW on the 17 precision of CMRO2 quantification in direct O MRI. In Proceedings of the 17th annual meeting of the German Chapter of the International Society for Magnetic Resonance in Medicine. Jena, Germany. Borowiak, R., Gröbner, J., Kurzhunov, D., Fischer, E., Dragonu, I., Bock, M., 2014. Direct cerebral 17O-MRI at a clinical field strength of 3 Tesla using a TxRx head coil. In Proceedings of the 22nd annual meeting, of the International Society for Magnetic Resonance in Medicine. Milan, Italy.

114

Awards

1. Magna Cum Laude Merit Award of the International Society for Magnetic Resonance in Medicine (ISMRM) was awarded to Dmitry Kurzhunov et al. for their work entitled 17 “CMRO2 quantification in human brain with direct O MRI: profile likelihood analysis for protocol optimization” presented at the 25th annual meeting of ISMRM, Honolulu, USA, 22-27 April 2017.

2. Magna Cum Laude Merit Award of the International Society for Magnetic Resonance in Medicine (ISMRM) was awarded to Dmitry Kurzhunov et al. for their work entitled 17 “Direct estimation of O-MRI images (DIESIS) for CMRO2 quantification in the human brain with partial volume correction” presented at the 25th Annual Meeting of ISMRM, Honolulu, USA, 22-27 April 2017.

3. Magna Cum Laude Merit Award of the International Society for Magnetic Resonance in Medicine (ISMRM) was awarded to Dmitry Kurzhunov et al. for their work entitled “The Framework and analytically represented oxygen-17 brain tumor (ACROBAT) phantom for 17 optimization of CMRO2 quantication protocols in dynamic O-MRI” presented at the 24th Annual Meeting of ISMRM, Singapore, 07-13 May 2016.

4. Certificate of Merit. Department of Radiology - Medical Physics, University Medical Center Freiburg, 15 December 2015, Freiburg, Germany.

5. First author publication "Kurzhunov D, Borowiak R, Hass H, Wagner P, Krafft AJ, Tim- mer J, Bock M. Quantification of oxygen metabolic rates in Human brain with dynamic 17O MRI: Profile likelihood analysis. Magn Reson Med. 2017;78(3):1157-1167. http://doi.org/10.1002/mrm.26476" was selected by the International Society for Magnet- ic Resonance in Medicine (ISMRM) as the ISMRM Highlight of the Month in September 2017.

115

116

ACKNOWLEDGEMENTS

I would like to thank several people for all their support, effort, and motivation that made the completion of this thesis possible.

First and foremost, I am very grateful to my supervisor Prof. Dr. Michael Bock for his excessive guidance, feedback and assistance on a day-to-day basis during the entire process of my research practice and preparation of this thesis, which have been absolutely invaluable. He has set an example of excellence as a researcher, mentor, and instructor.

I am very thankful to:

 Prof. Dr. Jürgen Hennig for warmly welcoming me and the opportunity to be a part of the Medical Physics department, best MRI research group in the world, with its unique environment and full of intelligent and supportive colleagues  Robsen Borowiak, Altmeister, with whom we prepared and did our 17O MRI experiments, and who opened the world of 17O MRI for me with his immense passion for the science and helped me broaden my horizons extensively  Prof. Dr. Jens Timmer and Helge Hass for their assistance with implementation of the

profile likelihood analysis for the CMRO2 pharmacokinetic model  Dr. Marco Reisert for his brilliant ideas and tremendous support in implementation of the image reconstruction techniques  Dr. Ali Caglar Özen for his strong support on the hardware side of the 17O MRI project  Prof. Dr. Günter Reiter, Prof. Dr. Oskar von der Lühe and Prof. Dr. Jens Timmer for their support as referees and examines of my PhD defense  Dr. José Solera, Johannes Fischer and Dr. Ali Caglar Özen for proofreading this thesis.  colleagues from the Experimental Radiology group and the Medical Physics department for helpful, friendly, and intellectual atmosphere

My special thanks to my wife, Anya. Her support, encouragement, quiet patience and unwavering love are of great importance for me.

Finally, and most importantly, I want to thank my beloved parents: to my father, a physicist by profession and vocation, who opened the fascinating world of physics to me, guided me and helped me develop my potentials; and to my mother, who always supported me in my endeavors and motivated me to become happy and successful.

117