Highly Accelerated Fmri Using Non-Cartesian Trajectories: Enhanced Data Acquisition and Enabling Real-Time Reconstruction Disser

Highly Accelerated fMRI Using Non-Cartesian Trajectories: Enhanced Data Acquisition and Enabling Real-Time Reconstruction

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

vorgelegt von Bruno Thomas Riemenschneider geboren am 17.12.1984 in Tübingen

Dezember 2019 Dekan: Prof. Dr. Wolfgang Soergel Leiter der Arbeit: Prof. Dr. Dr. h.c. Jürgen Hennig Referent: Prof. Dr. Dr. h.c. Jürgen Hennig Koreferent: Prof. Dr. Oliver Waldmann Prüfer: Prof. Dr. Günter Reiter Prof. Dr. Thomas Filk

Datum der mündlichen Prüfung: 06.02.2020

Contents

1 Introduction1

2 Physical Background5 2.1 Particle with Spin ...... 6 2.2 Ensemble of Spins in a Magnetic Field ...... 7 2.2.1 Density Operator ...... 7 2.2.2 Magnetization at Thermal Equilibrium ...... 7 2.2.3 Bloch Equations Without Relaxation ...... 9 2.2.4 Resonant Absorption of Energy - Radiofrequency Pulse ...... 10 2.2.5 Relaxation ...... 12 2.2.6 Solution of the Bloch Equations ...... 16 2.3 MR Signal Formation ...... 17 2.3.1 Faraday’s Law in Magnetization Dependence ...... 17 2.3.2 Induction by the Spin Ensemble ...... 18

3 Magnetic Resonance Imaging 19 3.1 Signal Demodulation and Signal Equation ...... 20 3.2 Encoding Image Information ...... 21 3.2.1 Fourier Imaging and k-Space ...... 22 3.2.2 Field Inhomogeneities and Local Macroscopic Eﬀects ...... 23 3.3 k-Space Information in Unaccelerated MRI ...... 25 3.3.1 Fourier Transform ...... 25 3.3.2 Nyquist Sampling Criterion ...... 28 3.3.3 MRI as iDFT of truncated Fourier Series Coeﬃcients ...... 28 3.4 General Image Reconstruction via Pseudoinverse ...... 30 3.4.1 Matrix Formulation of the Signal Equation ...... 31 3.4.2 Pseudoinverse Method ...... 32 3.4.3 Point Spread Function ...... 34 3.5 Measurement vs. Reconstruction Model ...... 35 3.6 Parallel Imaging ...... 36 3.7 Signal-to-Noise Ratio ...... 37 3.7.1 Thermal Noise Propagation ...... 38 3.7.2 Temporal Noise ...... 40 3.8 MR Imaging Sequence ...... 40 3.8.1 Spectral Saturation ...... 41 3.8.2 Volume-Selective RF Pulses ...... 42 3.8.3 Steady-State Magnetization ...... 42 3.8.4 Gradient Spoiling ...... 43 3.8.5 Gradient-Echo (fMRI) Sequence ...... 43 3.8.6 Physical Constraints ...... 45 3.9 Functional MRI ...... 46 3.9.1 Hemodynamic Response Function ...... 46 3.9.2 Functional Connectivity ...... 47 vi Contents

3.9.3 Advanced fMRI Techniques ...... 48

4 Own Contributions 51

5 Investigation of Off-Resonance-Gradient Vulnerability due to Variable-Density Sampling 53 5.1 Motivation ...... 54 5.2 Theory ...... 54 5.2.1 Off-Resonance ...... 54 5.2.2 Variable-Density Sampling ...... 55 5.3 Methods ...... 57 5.3.1 Off-resonance Model ...... 57 5.3.2 Trajectories ...... 58 5.3.3 Simulation Model and Image Reconstruction ...... 59 5.4 Results ...... 60 5.5 Discussion ...... 63 5.6 Conclusion ...... 64

6 Trading Oﬀ Spatio-Temporal Properties in 3D High-Speed fMRI Using Inter- leaved Stack-of-Spirals Trajectories 65 6.1 Motivation ...... 66 6.2 Material and Methods ...... 67 6.2.1 Trajectory Design ...... 67 6.2.2 Data Acquisition ...... 68 6.2.3 Signal Simulation and Image Reconstruction ...... 70 6.2.4 Time Series Processing ...... 70 6.2.5 Thermal Noise Propagation ...... 71 6.2.6 tSNR ...... 71 6.2.7 Functional Characterization ...... 71 6.3 Results ...... 72 6.3.1 Image Quality ...... 72 6.3.2 Thermal Noise Propagation ...... 73 6.3.3 tSNR ...... 76 6.3.4 Functional Characterization ...... 77 6.4 Discussion ...... 80 6.4.1 Image Quality ...... 80 6.4.2 SNR ...... 82 6.4.3 Functional Characterization ...... 83 6.5 Conclusion ...... 83

7 Targeted Partial Reconstruction for High-Speed fMRI Neurofeedback 85 7.1 Motivation ...... 86 7.2 Theory ...... 86 7.2.1 Voxel-Wise Reconstruction ...... 87 7.2.2 Volume-wise Reconstruction ...... 88 7.2.3 Eﬃciently Approximated Reduced-FOV Reconstruction ...... 91 Contents vii

7.3 Methods ...... 94 7.3.1 Data Acquisition ...... 94 7.3.2 General Reconstruction Details ...... 94 7.3.3 Voxel-wise Reconstruction ...... 95 7.3.4 Volume-wise Reconstruction ...... 95 7.3.5 Efficiently Approximated Reduced-FOV Reconstruction ...... 96 7.3.6 Feedback Scan Workflow ...... 97 7.4 Results ...... 98 7.4.1 Voxel-wise Reconstruction ...... 98 7.4.2 Volume-wise Reconstruction ...... 98 7.4.3 Efficiently Approximated Reduced-FOV Reconstruction ...... 102 7.4.4 Feedback Scan Implementation Performance ...... 103 7.5 Discussion ...... 105 7.5.1 Workflow and Implementation ...... 105 7.5.2 Reconstruction quality ...... 106 7.5.3 Reconstruction phase correction ...... 106 7.6 Conclusion ...... 107

8 Conclusion and Outlook 109 8.1 Conclusion ...... 109 8.2 Outlook ...... 110

A Appendix 111 A.1 Symbols and Deﬁnitions ...... 111 A.2 Abbreviations ...... 115

Journal Publications 117

Bibliography 119

Acknowledgments 131

1 Introduction

While the first effects of electron and even nuclear spin were measured already in the 1880s by Michelson,1, 2 the fine and hyper fine structures visible in the spectra obtained with the Michelson Interferometer were not yet understood. The idea of quantization of subatomic particle properties was still unborn. Also the first direct experimental evidence of quantized spins of particles that was published in 1922 by Stern and Gerlach3 was still wrongly interpreted as a quantized atomic angular moment at the time. Only 3 years later, in an effort to explain the fine structure of the hydrogen spectrum, Uhlenbeck and Goudsmit4 were the first to publish the hypothesis of a "quantized electron rotation". Interestingly, various (nowadays famous) physicists had discarded the idea as unrealistic in the year before this publication.5 6 After Pauli’s formulation of the Schrödinger equation for spin-1⁄2 particles in 1927, Dirac derived the famous Dirac equation by factorizing the relativistic energy momentum equation in 1928.7 The Dirac equation is counted as one of the greatest successes of theoretical physics, being consistent with both quantum mechanics and special relativity, describing all massive spin-1⁄2 particles with the Dirac Spinors that arise along the derivation. Stern perfected the particle beam method in the years after the Stern-Gerlach experiment, and, eventually, in 1933 presented the first experimental data on deflecting a hydrogen beam8 in an inhomogeneous magnet field together with Frisch. This represented the first direct evidence of the proton spin, and moreover contradicted the gyromagnetic values predicted by Dirac at the time, leading to further discussion of the g-factors that Landé introduced already in 1921 to explain the anomalous Zeeman effect.9 A former student of Stern’s and his molecular beam method, Rabi conducted the first direct measurement of nuclear spin resonance (NMR) in 1938, manipulating the nuclear spins in a molecular beam.10 NMR experiments in solids and liquids followed by Bloch and Purcell, independently from each other, in 1946.11, 12 The first medical and physiological application of NMR to biologic tissue samples was performed by Odeblad in 1955.13 Interestingly, he had proceeded on his own after Bloch had denied a cooperation with him in 1952,14 arguing that NMR is a tool for physicists only, and not for research into physiology, medicine, or biology. Until the early 1970’s, however, all NMR measurements lacked spatial encoding of the measured samples in more than one dimension along a constant magnet field gradient. Lauterbur introduced the idea of using spatially varying magnetic field gradients to encode multiple dimensions in 1973,15 originally calculating a 2-dimensional image from multiple 2 Introduction

1-dimensional Larmor frequency projections. In the same year, Mansfield published the idea of "NMR diffraction",16 which also implied multi-dimensional feature detection by NMR, while not formulated as explicitly as Lauterbur. This new paradigm led to the expression "NMR imaging", nowadays referred to as "MR imaging" (MRI) in the medical context, in order to prevent distrust of patients to the method17 by omitting "nuclear". Soon after the inception of multi-dimensional MRI, the "application of a sequence of pulsed orthogonal linear field gradients to the sample during the FID" and image reconstruction via Fourier transform were described. This imaging method, which would become a standard method until today, was conceived shortly after Lauterbur’s contribution and was published in 1975 by Kumar, Welti, and Ernst.18 The numerous advances in hardware and imaging sequence design that followed the innovation of MRI are too many to account for here. To become a useful tool in the radiologist’s repertoire, however, MRI still needed fast and feasible imaging techniques tailored to the temporal constraints that arise in the clinical environment. In the frame of this outline, the basis set of today’s clinical imaging sequences ought to be mentioned; three sequence developments contributed considerably, each in its own way, and many more applications arose from combinations of these: (1) In 1977, Mansfield introduced the idea of concatenating multiple or all Fourier-space lines after a single RF excitation in a multi-dimensional gradient (multiline-)echo read-out, i.e., echo planar imaging (EPI).19 The first movie-like MRI time series using this technique were published already 5 years later.20 Still being the fastest method to ∗ form a T2 -weighted image, EPI and its derivatives constitute the standard method for functional imaging until today. (2) In 1986, Hennig applied the concept of an echo train after a single excitation to spin echo imaging with "Rapid Acquisition with Relaxation Enhancement" (RARE),21 where each line is concatenated to the next by a refocusing pulse in order to form spin and

not gradient echoes. This method became the standard for fast T2-weighted imaging. (3) Also in 1986, Haase and Frahm published results using the "Fast Low-Angle SHot" (FLASH)22, 23 sequence. This advancement incorporated the build-up of a pseudo steady-state of the non-relaxed spin ensemble with fast repetitions of low ﬂip angle excitations, enabling a rapid read-out of arbitrarily many gradient echoes of that steady-state. This method and its derivatives enabled weighting dependent on sequence

parameters, including T1- and proton density weighting. 3

Innovation continues to be made in all aspects of MRI: more powerful and eﬃcient hardware, novel image reconstruction algorithms, or the inception of methodologies to apply MRI as a tool for analysis of an increasing number of pathologies. The research of NMR/MRI fundamentals spawned 6 Nobel prizes to 8 awardees, counting Stern’s and Rabi’s prizes, which were related to the physical foundation of NMR: Otto Stern, USA: Nobel Prize in Physics 1943 "for his contribution to the development of molecular ray method and his discovery of the magnetic moment of the proton" Isidor I. Rabi, USA: Nobel Prize in Physics 1944 "for his resonance method for recording the magnetic properties of atomic nuclei" Felix Bloch, USA and Edward M. Purcell, USA: Nobel Prize in Physics 1952 "for their discovery of new methods for nuclear magnetic precision measurements and discoveries in connection therewith" Richard R. Ernst, Switzerland: Nobel Prize in Chemistry 1991 "for his contributions to the development of the methodology of high resolution nuclear magnetic resonance (NMR) spectroscopy" Kurt Wüthrich, Switzerland: Nobel Prize in Chemistry 2002 "for his development of nuclear magnetic resonance spectroscopy for determining the three-dimensional structure of biological macromolecules in solution" Paul C. Lauterbur, USA and Peter Mansﬁeld, United Kingdom: Nobel Prize in Physiology or Medicine 2003 "for their discoveries concerning magnetic resonance imaging"

Functional MRI (fMRI) for mapping of physiological function of the brain by time series acquisition and exploiting the blood oxygen level dependent (BOLD) effect was first shown in vivo in 1990.24 The method of fMRI has since become an integral tool in neuroscience, and much effort has been dedicated to enhancing the image quality and resolution in space and time. Hennig et al. introduced the term "MR-Encephalography" (MREG) in 2007 in a publication25 that initiated an effort to push the limits of temporal resolution in fMRI beyond the constraints of gradient encoded MRI. The original idea incorporated abandoning gradient encoding altogether and retrieve spatial information solely from the arrangement of receiver coil arrays, analogous to electroencephalography (EEG), allowing for a TR of 20 ms or less. In the following line of work,26–29 decreasingly sparse gradient encoding was added to the concept, which evolved into 3D-encoded single-shot whole-brain gradient-echo fMRI. The presented thesis extends the line of research around the method of MR-Encephalography, and aims at enabling and demonstrating the flexibility of the method beyond the capabilities of previously published fMRI methods. The methods of accelerated constant-density sampling fMRI and ultra-highspeed non-Cartesian fMRI are conceptionally connected, and, moreover, the (ultra-)high-speed method is made accessible for real-time fMRI (rtfMRI) and neurofeedback.

2 Physical Background

As already apparent from the introduction, the core physical property that is exploited for nuclear magnetic resonance imaging is the nuclear spin. In this chapter, the MR signal equation is derived from the quantum mechanical description of spins in a magnet ﬁeld.

Contents

2.1 Particle with Spin ...... 6 2.2 Ensemble of Spins in a Magnetic Field ...... 7 2.2.1 Density Operator ...... 7 2.2.2 Magnetization at Thermal Equilibrium ...... 7 2.2.3 Bloch Equations Without Relaxation ...... 9 2.2.4 Resonant Absorption of Energy - Radiofrequency Pulse ...... 10 2.2.5 Relaxation ...... 12 2.2.6 Solution of the Bloch Equations ...... 16 2.3 MR Signal Formation ...... 17 2.3.1 Faraday’s Law in Magnetization Dependence ...... 17 2.3.2 Induction by the Spin Ensemble ...... 18 6 Physical Background

2.1 Particle with Spin

Subatomic particles exhibit the quantum mechanical property of spin. The spin is a constant inherent property of a particle, and has no general imaginable correspondence in classical physics. In the case of the descriptive representation of the particle as a charge density as can be made with nucleons, spin can be grasped with the classical analog of a rotational angular momentum of the particle. Like the classical analog, a spin angular momentum T Sˆ = Sˆx, Sˆy, Sˆz induces a magnetic moment µˆS:

µˆS = γSˆ, (2.1) with the particle-specific gyromagnetic ratio γ. The hat denotes an observable in the mathematical formalism of quantum mechanics. This formalism identifies physical measurements with the application of observable operators to quantum states, and stipulates the application of Sî to the state of a particle with spin. T When the spin particle is exposed to a magnetic field B = (Bx,By,Bz) , its Hamiltonian

X Hˆ = −µˆB = −γ SˆjBj (2.2) j expresses the energy shift induced by the magnetic moment µˆ. In the context of nuclear magnetism, the assumption of no orbital magnetic moment is made, i.e., µˆ = µˆS. The spin angular momentum is a quantized property, meaning that quantum mechanical eigenstates of the corresponding observables lead to distinct eigenvalues, which represent the possible distinct measurement results. The eigenvalue equation for a particle with spin quantum number s in the state |s, msi with eigenvalue ms, the spin projection quantum number, reads: ˆ Si |s, msi = ~ms |s, msi . (2.3)

The possible measurement results, however, are not to be confused with naturally occurring states |ψi of particles before measurement, which are represented by superpositions of the eigenstates with complex weightings cms : X |ψi = cms |s, msi . (2.4) ms

The spin angular momentum operators in the above introduced basis can be expressed by the (generalized) Pauli matrices σi, Sˆ = ~σ , (2.5) i 2 i Ensemble of Spins in a Magnetic Field 7 which follow the later used relations

Tr {σi} = 0, (2.6)

[σi, σj] = 2iijkσk, (2.7)

σiσj = δij1 + iijkσk, (2.8) using the Levi-Civita symbol ijk.

2.2 Ensemble of Spins in a Magnetic Field

2.2.1 Density Operator

The density operator of N quantum systems with states |ψni describes a statistical quantum state: 1 X ρˆ ≡ |ψni hψn| (2.9) N n is the average of the projections onto the individual quantum states, and it allows an elegant formulation of the macroscopic expectation value of observable Aˆ: n o < Aˆ >= Tr ρˆAˆ . (2.10)

The explicit matrix representation of the density operator in a certain basis is called the density matrix. The time evolution of the density operator in the Schrödinger picture, i.e., the von Neumann equation, reads dˆρ i h i = − H,ˆ ρˆ . (2.11) dt ~ The collection of > 1022 nuclear spins in every MR-imaged voxel volume allows for a virtually perfect validation of a statistical description via the density operator formalism.

2.2.2 Magnetization at Thermal Equilibrium

For ease of notation, the derivation of the magnetization in a magnet ﬁeld at thermal 1 equilibrium is performed for the case of spin s = 2 and B = Bez, ez being the canonical basis vector of the 3-dimensional Euclidian space in z-direction. However, the derivation is not restricted to this case.

The individual spin systems with same spin s occupy the energy eigenstates |φii = |s, ms(i)i 8 Physical Background

with energies Ei: Hˆ |φii = Ei |φii . (2.12)

Statistical considerations lead to the following two properties of the density matrix at thermal equilibrium ρêq: (1) All off-diagonal elements, called coherences, are zero. This follows from the assumption n P n n that the complex coefficients ci of all the spin systems expressed by |ψni = i ci |φi i are statistically distributed, and such

eq X n∗ n ρˆlm = cl cm = 0. (2.13) n

(2) The diagonal elements, which represent the state populations, follow the Boltzmann distribution: eq X n 2 exp(−Em/kBT ) ρˆmm = |cm| = P (2.14) n s exp(−Es/kBT ) −23 −1 where kB = 1.38066 · 10 JK is the Boltzmann constant, and T the temperature of the sample. The avaliable thermal energy at room temperature is ≈ 4 · 10−21 J, and the energy of 7 −1 −1 the spin-1/2 eigenstates of protons (γ = 4.258 · 10 s T ) at a ﬁeld strength of 3 T is −26 ∓~γB/2 ≈ ±4.2 · 10 J, such that the exponentials of the Boltzmann distribution can be approximated by exp(−Em/kBT ) ≈ 1 ± ~γB/2kBT . With these considerations, the density matrix becomes eq 1 ~γB −~γB 1 ~γB ρˆ = · 1 + diag , = · 1 + σz. (2.15) 2 4kBT 4kBT 2 4kBT

The macroscopic magnetization density with spin density n(r) at equilibrium is given by

M(r) = n(r) · hµˆi = n(r) · Tr {ρeqµˆ} , (2.16) and with use of (2.1), (2.5), (2.6), (2.8), and (2.15), it becomes

" 2 2 # ~γ ~ γ B Mi(r) = n(r) · Tr {σi} + Tr {σ3σi} 4 8kBT (2.17) !T 2γ2 ⇔ M(r) = 0, 0, n(r) ~ B . 4kBT

The result (2.17) reﬂects the occurrence of nuclear paramagnetism, in which the magnet ﬁeld evokes a macroscopic magnetization of the nuclear spins in the same direction of the magnet Ensemble of Spins in a Magnetic Field 9

ﬁeld. This result can also be written in the form1 1 M = χB, (2.18) µ0

217 2 2 with scalar magnetic susceptibility χ = µ0n~ γ /4kBT > 0, and magnetic constant µ0. The inverse proportionality of the magnetic susceptibility to temperature for the case of spin paramagnetism is called Curie law, after renowned physicist Pierre Curie. Generally, the total spin quantum number of an atomic nucleus is determined by combination of the −1/2- and 1/2-states of the nucleons in the encountered energy state of the nucleus. In the ground states of the main elemental isotopes across the periodic system, this leads to values in the range of [0, 8] in steps of 1/2. The general formula for the paramagnetic susceptibility of spin s is330

2 2 χ = µ0ns(s + 1)~ γ /3kBT. (2.19)

2.2.3 Bloch Equations Without Relaxation

While the state evolution of individual spins is governed by the quantum mechanical framework, the time evolution of the resulting macroscopic nuclear magnetization density M(t) in a magnet field B is described by the Bloch equations, also called the equations of motion of nuclear magnetization, introduced by Bloch in 1946.11 Whereas, in an alternative context, the Bloch equations describe the polarization of an ensemble of two-level quantum systems in an oscillating electrical field, in this case of spins in the magnetic field they describe the Larmor precession of the nuclear magnetization, with Larmor frequency ω0 = −γB, B = |B|. The derivation of the Bloch equations follows subsequently, with the assumption of non- interacting spins expressed by the used single-spin Hamiltonian in (2.2). Again, the statistical description of the macroscopic state is made with the density operator formalism. To this end, the temporal derivative of the macroscopic magnetization density

d d M(r, t) = n(r) · hµˆi (t) (2.20) dt dt is formed under the assumption of constant spin density. Using Relations and Formulas

1This notation of magnetization in direct proportionality of the magnetic ﬁeld B is in line with MRI textbooks.17, 30 This is an alternative formulation to M = χH, with magnetic ﬁeld strength H [A/m], and B = µ0 (H + M). 2The magnetic susceptibility of most crystals is a tensor of 2nd order, expressing anisotropic magnetic susceptibility. 3With a ca. 658-fold higher gyromagnetic ratio of the electron with respect to the proton, and a γ2- dependency of χ, it becomes clear that unpaired electrons lead to much stronger spin paramagnetism than nucleons. 10 Physical Background

(2.1), (2.2), (2.5), (2.7), (2.10), and (2.11), it becomes, in component-wise formulation

d d i h i Mi(r, t) = n(r) · Tr {σˆµî} = n(r) · Tr − H,ˆ σˆ µî dt dt ~    i X  = n(r) · Tr [ˆµjBj, σˆ]µ î ~ j     X  = n(r) · Tr − ijkσˆµˆkγBj  j,k  (2.21) X = − ijkγBjMk j,k

= (M × γB)i

d ⇒ M(r, t) = M(r, t) × γB(r). dt The commutator relation [A, B] C = [A, BC] − B [A, C] was used in the third line of (2.21), where the trace of a commutator, i.e., the first term on the right side of the relation when inserted in (2.21), vanishes. In the derivation of the Bloch equations in (2.21) no relaxation is implied, as the assumption of completely non-interacting particles was made. The effect of spin interactions as a perturbation that lead to relaxation of any macroscopic magnetization state back to thermal equilibrium is discussed in Section 2.2.5. The Bloch equations constitute the foundation of MRI, and the later discussed spin ensemble manipulations via magnetic field gradients, which are very slowly varying compared to the precession, are a consequence of these equations, such that no need for further quantum mechanical approaches occur in that case. The magnetization tipping due to RF fields that represent fast changing electromagnetic fields perpendicular to the static B-field are discussed separately in the next section (Section 2.2.4), however, because the derivation of the Bloch equations was made for a time-independent Hamiltonian.

2.2.4 Resonant Absorption of Energy - Radiofrequency Pulse

A temporally confined oscillating external field, i.e., a pulse, in the radiofrequency (RF) range 4 at or close to resonance, i.e., with Larmor frequency ωRF ≈ ω0 = −γB, can be applied to tip the spins of the precessing magnetized ensemble away from the B-axis with a nutation. For this, the RF pulse needs to have oscillating field components perpendicular to the main field. No relaxation needs to be incorporated into this derivation, yet, as the time-scale of relaxation is about 3 orders of magnitude larger than that the length of a typical RF pulse. As the magnetic energy of the magnetized ensemble in the magnetic field is proportional to −M · B, magnetization tipping constitutes a transfer of energy from the applied RF field

4Local micro- and macroscopic field inhomogeneities due to molecular field structures and macroscopic magnetism always occur in MRI. Ensemble of Spins in a Magnetic Field 11 into the spin ensemble. This energy will be released during relaxation (see Section 2.2.5) back to thermal equilibrium after the pulse, and can meanwhile be detected (see Section 2.3) with its imprinted tissue-specific relaxation signature. Strictly speaking, the energy transfer is not explained by the semi-classical approach with non-quantized magnetic field used in this section. However, the aim of describing a large ensemble in a - from quantum perspective very strong - oscillating magnetic field justifies a Hamiltonian of spin systems in a classical field.

As in Section 2.2.2, the coordinate system is defined with B = Bez. For convenience of the mathematical description of the oscillating RF field BRF with amplitude BRF , w.l.o.g. 5 oscillating along the x-axis with phase φRF , a decomposition of the oscillation into two rotating field components is made:

BRF (t) = BRF cos(ωRF t + φRF )ex 1 = B {cos(ω t + φ )e + sin(ω t + φ )e } + RF 2 RF RF x RF RF y (2.22) 1 + {cos(ω t + φ )e − sin(ω t + φ )e } . 2 RF RF x RF RF y

The circular arrows are a graphic interpretation of the corresponding line in the x-y-plane, assuming positive γ. The upper line corresponds to the on-resonant component of the RF

field, and the lower line is off-resonant by (≈)2ω0, and can thus be assumed to have no relevant influence on the spin evolution and be neglected. The magnetic field strength of the RF pulse is several orders of magnitude smaller than the constant B-field, and can as such be regarded as a perturbation that does not change the Larmor precession of the system. The comparatively small field strength of the pulse also implies that imperfect incidence of the pulse with small Bz component can be neglected. Again, for convenience of mathematical description, a time-dependent coordinate transformation is applied, in which the coordinate system rotates with ωRF around the main magnet field. This transforms a time-dependent problem into a time-independent one. The rotation of the magnet field about the z-axis can be expressed by a rotation matrix

Rz(ΦRF ) = Rz(ωRF t + φRF ), and the rotation of the coordinate system with the same frequency by Rz(Φframe) = Rz(ωRF t + φframe):

1 B (t) = B R (Φ )e , RF 2 RF z RF x 1 ⇒ B˜ (t) = B R (Φ )R (−Φ )e0 (2.23) RF 2 RF z RF z frame x 1 = B R (φ)e0 , 2 RF z x with B˜ RF representing the oscillating ﬁeld after coordinate transformation, and φ = φRF − φframe. This coordinate transformation is commonly called "rotating-frame"transformation.

5 Other axes deﬁnitions resulting in ωRF t + φRF → ωRF t can be chosen, and consequently φRF represents nutation about diﬀerent axes, as will become more obvious in the following. However, the phase φRF is in general a (slowly) time-dependent value during the RF pulse, and is therefore taken explicitly into account. 12 Physical Background

ˆ The Hamiltonian in the rotating frame for a spin in the main magnetic ﬁeld (H˜B) during the ˆ RF pulse (H˜RF ), using the coordinate transformation rules for matrices, and the rotating- frame Hamiltonian6, then becomes:

ˆ ˆ ˆ H˜ = H˜B + H˜RF 1 H˜ˆ = −γBR (−Φ )Sˆ R (Φ ) − ω Sˆ − γB R (φ)Sˆ R (−φ) z frame z z frame RF z 2 RF z x z = (ω0 − ωRF )Sˆz + ωnut cos(φRF )Sˆx + sin(φRF )Sˆy (2.24)     ωnutcos(φRF ) Sˆx    ˆ  =  ωnutsin(φRF )  ·  Sy  , ω0 − ωRF Sˆz

1 with ωnut = 2 γBRF , and the degree of freedom in the phase of the rotating frame was used 7 for φframe = π. It is now straightforward to conclude the time evolution of the magnetization in the rotating frame from this Hamiltonian. When recalling Section 2.2.3, one can conclude that a Hamiltonian of Hˆ = −γBSˆ = ωSˆ leads to a rotation of the magnetization about the ω-axis. This matter of fact results in a rotation by the RF pulse about the axis T ωeff ≡ (ωnutcos(φRF ), ωnutsin(φRF ), ω0 − ωRF ) with rotation frequency 2 21/2 |ωeff | = ωnut + (ω0 − ωRF ) . Further, one can conclude from this that RF excitation far from the resonance frequency leads to only a rotation approximately about the main field, and as such has no magnetization tipping effect. On resonance, on the other hand, the magnetization is tipped about an axis in the x-y-plane, defined by φRF , with frequency ωnut. On a side note: as mentioned above, this formulation implies an energy transfer only indirectly. If quantum field theory was applied, the tipping of the spins away from equilibrium would go along with increasing probability of photon absorptions from the field, which in the ensemble again leads to the same magnetization result as shown here. Ensemble tipping with an increasing angle measured from the equilibrium state, meaning an increase in energy of the ensemble in the main field, corresponds to dominating photon absorption. On the other hand, decreasing angles measured from the equilibrium state due to an external RF field, meaning a decrease in energy of the ensemble in the main field, correspond to dominating stimulated photon emission.

2.2.5 Relaxation

In NMR(I), it can be observed that the spin ensemble starts to return to thermal equilibrium right after RF excitation. However, the magnetization equations of motion, which were derived in Section 2.2.3 under the assumption of non-interacting spins in an external magnetic

6 ˆ A complete derivation of H˜B including the term −ωRF Sˆz, which results from the rotating frame of reference, is found in "spin dynamics".17 7 This degree of freedom is used to obtain a positive ωnut, assuming positive γ. Ensemble of Spins in a Magnetic Field 13

field, lead to the solution of a perfectly stable Larmor precession, periodic from any starting condition. Furthermore, it can be argued that spontaneous emission is negligible at frequencies found in NMR, and also induced emission is not a mechanism capable of bringing the spin ensemble into equilibrium.31 In fact, the evolution of the spin ensemble is perturbed by interactions between the nuclei and between the nuclei and fields within the nuclei-containing molecules. The effects of the interactions, being very small compared to the effect of the main field, lead to a remarkably slow relaxation of the magnetization back to thermal equilibrium, i.e., in the range of seconds in the case of water in a clinical MRI scanner.8 In general, there are a number of possible interactions of a spin-carrying nucleus with its environment, other than with external magnet fields. In liquids of molecules with spin-1/2 nuclei, e.g., the largest perturbations arise from the dipole-dipole coupling of the nuclei within a molecule, and anisotropy of the chemical shift of the Larmor frequency, which is evoked by the electronic structure of the molecule.17 Both perturbations are characterized by the molecular motion in the magnet field, and are thus fluctuating in time, which leads to the destabilization of the precession and the slow return to equilibrium. The explicit analysis of interaction effects and induced relaxation mechanisms is substance-dependent, though, and complex - which is why this chapter is restricted to a phenomenological description, for details please refer to NMR literature.17, 31 Furthermore, medical MRI is performed on mixtures of many substances, where the origin of the relaxation mechanisms is not of primary interest, and the various relaxation constants are not resolved. The fundamental property of the diverse tissues that are imaged in the medical context is that a number of pathologically affected tissues show different relaxation times than healthy tissue. This circumstance is exploited by generating images which are weighted by the relaxation times, as discussed in the next chapter (Section 3.5), to give image contrast to pathologically affected tissue. For contrast generation and imaging sequence design, it is still important to acknowledge that there are fundamentally different types of relaxation:

Longitudinal Relaxation - T1 The longitudinal relaxation describes the evolution of the magnetization component which is oriented longitudinal to the main ﬁeld, i.e., its return to the thermal equilibrium state. In terms of the density operator matrix, it describes the evolution of the populations. It is also called the spin-lattice relaxation, because it can be derived as a consequence of spins interacting with a quantum mechanical system with a very large number of degrees of freedom. The latter can be regarded as an external energy reservoir with quasi-continuous energy spectrum, called the lattice for historical reasons. Under the assumption of one time constant for the decay of the diﬀerence of the magnetization component from it equilibrium value, the longitudinal magnetization component reads

8Nuclear relaxation times in NMR greatly depend on the sample and its temperature, and can vary between 10−5 and 105 s.31 For protons in human MRI, however, relaxation times back to thermal equilibrium vary from a few hundred milliseconds to a few seconds. 14 Physical Background

(in the rotating frame):

d M (t) = (M − M (t)) /T dt z 0 z 1 (2.25)

−t/T1 ⇒ Mz(t) = M0 − (M0 − Mz(0)) · e ,

where M0 is the magnitude of the magnetization at thermal equilibrium, t = 0 represents the start of the relaxation, and R1 = 1/T1 is the longitudinal relaxation rate.

Transverse Relaxation - T2 The transverse relaxation describes the evolution of the magnetization component which lies in the transverse plane to the main ﬁeld. The transverse component exhibits a steady loss of magnitude due to interaction-induced loss of phase coherence, represented by the coherences of the density operator matrix. This loss of coherence does not imply a change of the energy state of the ensemble. It represents an increase in entropy of the ensemble, and is thus irreversible. It is also called the spin-spin relaxation for historical reasons, which should not imply that it is caused purely by spin-spin interactions. Under the assumption of one time constant for the loss of coherence, the transverse components read (in the rotating frame):

d M (t) = −M (t)/T dt x,y x,y 2 (2.26)

−t/T2 ⇒ Mx,y(t) = Mx,y(0) · e .

Again, t = 0 represents the start of the relaxation, and R2 = 1/T2 is the transverse relaxation rate. The assumption of mono-exponential decay is generally not justified. It would only be true for one substance representing one dominating relaxation time constant. However, as described above, the complexity of the decay is not necessarily of interest in the frame of medical MRI, and the assumption of one dominating effect in one image voxel is generally made. As the interactions which lead to transverse relaxation effects include the interactions that lead to longitudinal relaxation, one can phenomenologically state that

T2 < T1.

This is a very general relation9, which becomes more pronounced as the correlation times of the interaction-induced local ﬂuctuations increase.17 In conclusion, the phenomenological Bloch equations including relaxation, along a magnetic

9 For protons in human MRI, T2 values range in the order of tens of milliseconds for most tissues, while T1 values range from a few hundred milliseconds to a few seconds. Ensemble of Spins in a Magnetic Field 15

ﬁeld in z-direction, read (in the laboratory frame):

 M (r, t)  x /T2(r) d M (r, t) M(r, t) = M(r, t) × γB(r) −  y /T2(r)  . (2.27) dt   (M (r, t) − M (r)) z 0 /T1(r)

Additionally to these fundamental substance-specific relaxation components, there is one more aspect to cover, which manifests itself in a relaxation-like signal behavior of the transverse component: 0 Transverse Relaxation-Like Decay - T2 Due to the heterogeneous structure of the "samples" (i.e., parts of a human subject) that are scanned in medical MRI, the susceptibility (including para- and diamagnetism from all parts of all molecules in the subject) of the total magnetization to the main field induces field inhomogeneities of microscopic and macroscopic structure.10 One way to characterize the inhomogeneities within a volume is the spectral magnetization density m(ω), which depends on the occurring resonance values ω. While the macroscopic effects of field inhomogeneities on the signal are greatly dependent on the spectral magnetization density, it can be easily shown32 that the spin dephasing caused by a Cauchy-distributed magnetization density during a signal relaxation leads to a mono-exponential signal decay. Thus, under the assumption of a Cauchy-distributed magnetization density, one can 0 0 define a local transverse decay rate, usually named R2 = 1/T2. Summing up the decay rates, the total transverse decay during signal relaxation within a finite volume then becomes 1 1 1 1 ⇒ ∗ = + 0 . (2.28) T2 T2 T2 T2 It is important to note that the field inhomogeneity-induced decay is a reversible effect, and can be minimized, or even zeroed for certain time points, by an appropriate imaging sequence design when desired. 0 The effect of T2 is essentially a consequence of not being able to resolve the spin precession at the microscopic level. Moreover, whenever discretization, i.e., signal 0 summation, of an imaged region plays a role, the apparent T2-decay becomes more pronounced due to not only microscopic, but also macroscopic field inhomogeneities. In the following parts of this thesis that mathematically describe the signal behavior ∗ in a spatially continuous fashion, the formulation already implies T2 , when referring to the microscopic field inhomogeneities only. The formulation still adheres to the time ∗ constant T2 after discretization and introduction of macroscopic field-inhomogeneities in the signal equation, as this nomenclature is common in MRI. ∗ As an outlook: the image-weighting with T2 -decay of the fastest MR imaging sequences, ∗ 11 together with the fact that T2 is an intrinsic marker for blood oxygenation , has

10The main field divergence evoked by physical limitations in the realization of a homogeneous magnet field are generally slowly varying in space, and do not play a role on the order of MRI voxel size. 11Oxyhemoglobin is diamagnetic, while desoxyhemoglobin is paramagnetic. Moreover, the dipole-dipole interactions of both molecules show different strength. This circumstance leads to a pronounced change in ∗ T2, and even stronger in T2 , depending on the blood oxygen concentration of the blood in the imaged 16 Physical Background

led to the successful application of blood oxygen level dependent (BOLD) functional MR imaging (fMRI) in neuroscience. BOLD-fMRI for the analysis of brain function, ∗ realized by highly repetitive T2 -scans of the brain during oxygen-consuming neuronal activation, is one of the main applications of the results presented in this thesis.

2.2.6 Solution of the Bloch Equations

(Quasi-)Static Field In Section 2.2.5, the rotating frame solutions of the single components with relaxation were already solved. These solutions are adequate under the assumption of frame rotation at exact constant Larmor frequency. In the laboratory frame, the Bloch equations with relaxation, as already formulated in Relation (2.27), give rise to the general solution

 −t/T ∗(r)  (Mx(r, 0) cos(ω0(r)t) + My(r, 0) sin(ω0(r)t)) · e 2 ∗  −t/T2 (r)  M(r, t) =  (My(r, 0) cos(ω0(r)t) − Mx(r, 0) sin(ω0(r)t)) · e  −t/T (r) M0(r) − (M0(r) − Mz(r, 0)) · e 1  iφ (r,t) −t/T ∗(r)  (2.29) M⊥(r) Re e M · e 2  iφ (r,t) −t/T ∗(r)  =  M (r) Im e M · e 2  ,  ⊥  M0(r) − (M0(r) − Mz(r, 0)) · e−t/T1(r)

with the phase of the magnetization density in the transverse plane φM (r, t) = −ω0(r)t+ 1/2 2 2 φM (r, 0), and the transverse magnetization component M⊥ = Mx + My . This solution represents the spatially dependent Larmor precession which decays in the transverse plane and relaxes back to thermal equilibrium. RF Excitation - Small-Tip Approximation During the RF excitation, which lasts typically a few ms in MRI, i.e., 2-3 orders of magnitude shorter than the typical relaxation constants, relaxation can be ignored. The Bloch equations in the rotating frame then become, as derived in Section 2.2.4:

d M(r, t) = M(r, t) × ω (r, t), (2.30) dt eff

T with ωeff ≡ (γ/2 BRF cos(φRF ), γ/2 BRF sin(φRF ), δω0) , δω0 = ω0 − ωRF .

This equation with general RF waveform BRF (t) has no analytic solution. However, when small-tip excitation12 is assumed, Equation (2.30) becomes

d M˜ (r, t) = iγM (r)B (t) − iδω (r)M˜ (r, t), (2.31) dt ⊥ z 1 0 ⊥

with the complex representation of the transverse magnetization M˜ ⊥ ≡ Mx(r, t) + iMy(r, t), and B1(t) ≡ 1/2BRF (t)(cos(φRF ) + i sin(φRF )).

tissue. 12 This term is used in MRI for the assumption of negligible Mz evolution when tipping the magnetization by small angles from the main ﬁeld direction, i.e., Mz ≈ M0 = const.. MR Signal Formation 17

Solving the ﬁrst order diﬀerential equation with initial condition M˜ ⊥(r, 0) = 0 leads to

Z T −i δω0(r)·(T −t) M˜ ⊥(r,T ) = iγMz(r) B1(t)e dt 0 (2.32) Z T ˜ i δω0(r)t −1 ⇒ M⊥(r,T ) ∝ B1(t)e dt = FB (δω0(r)), 0 1

where F −1(δω ) is the inverse Fourier transform of the RF pulse that is played out in B1 0 t ∈ [0,T ]. The small-tip approximation holds well quantitatively for angles up to 30◦, however, the qualitative behavior remains for larger angles up to 90◦. The important result here is that the resulting transverse magnetization after an RF pulse is dependent on the local Larmor frequency of the sample, and its value is governed by the inverse Fourier transform of the pulse.

2.3 MR Signal Formation

This chapter derives a voltage oscillation in a receive coil from a magnetized volume within reach of the coil. To this end, a dependence of the voltage from the magnetization density precession is derived using Faraday’s law. Some of the formulas and argumentation of this chapter follow closely chapter 7 of "Magnetic Resonance Imaging - Physical Principles and Sequence Design".30 For more detailed derivations please refer to the book.

2.3.1 Faraday’s Law in Magnetization Dependence

Faraday’s law describes the voltage along a closed loop, H E(r) dl, with electric ﬁeld E(r), induced by temporal changes in the magnetic ﬂux Φ through the surface spanned by the d R loop, dt B(r) dS. First, it is used to express the voltage in a receive coil by the magnetic vector potential A, where Stokes’ theorem is applied to the integral about A:

I d Z d E(r) dl = − B(r) dS = − Φ dt dt d Z = − ∇ × A(r) dS (2.33) dt d I = − A(r) dl dt

The magnetic vector potential A can be derived from an (eﬀective) source current density J: µ Z J(r0) A(r) = 0 . (2.34) 4π |r − r0|

Next, a so-called receive field B = B/I = ∇ × A/I is formulated, which represents the magnetic field per unit current that would be induced by a current I in the coil. It will be 18 Physical Background used to represent its reciprocal effect, however, i.e., for the current in the coil that is induced by a magnetization volume. This formulation enables the expression of Φ by integrating over the magnetization volume, not the flux surface. With the effective current density of a magnetization J = ∇×M, inserted into Relation (2.34), again inserted into the flux relation Φ = H A(r) dl, the vector identity a·(b×c) = −(a×c)·b, and the principle of reciprocity, the magnetic flux can written as Z Φ(t) = B(r) · M(r, t) dr, (2.35) where B is a property of the coil, and M(r, t) was given in Section 2.2.6.

2.3.2 Induction by the Spin Ensemble

When taking into account the temporal derivative of Relation (2.35) and the temporal dynamics of the solution of the Bloch equations, Equation (2.29), it becomes clear that the temporal derivative of the z-component is negligible with respect to that of the x- and y-components. It also has to be noted that the signal of the precessing spin ensemble is quite weak, and thus proper tuning of the receive coils to resonance with the spin ensemble precession is critical. In other words, only the transverse component of the magnetization is measurable. The receive ﬁeld, being a time-constant vector at any r, can be written in terms of its magnitude B⊥ and phase φB, i.e., as Bx = B⊥cos(φB) and By = B⊥sin(φB). Furthermore, as with the z-component, temporal derivations of relaxation terms can be neglected. Then, combining (2.29) and (2.35) in Faraday’s law leads to

I d E(r) dl = − Φ dt Z d −t/T ∗(r) h iφ (r,t) iφ (r,t)i = − M (r)e 2 B (r)Re e M + B (r)Im e M dr dt ⊥ x y Z ∗ h i −t/T (r) iφM (r,t) iφM (r,t) = M⊥(r)B⊥(r)e 2 ω0(r) cos(φB)iRe e + sin(φB)iIm e dr Z −t/T ∗(r) = M⊥(r)B⊥(r)e 2 ω0(r) sin(ω0(r)t − φM (r, 0) + φB(r)) dr, (2.36) with the trigonometric identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b). This derivation (in fact, most of this chapter) gives rise to a very intuitive expression for the measured MR signal: the induced signal voltage is an oscillation that is composed of all the microscopic magnetization precessions with their spatially dependend Larmor frequencies and magnetization density magnitudes, and every magnetization-induced oscillation decays with its local transverse relaxation rate. A weighting factor with the Larmor frequency ω0(r) can be neglected and merged into a proportionality factor, as the present frequencies are sharply concentrated around the mean external main ﬁeld frequency, on an absolute scale. Lastly, coil-dependent local phase and magnitude are imposed on the signal. 3 Magnetic Resonance Imaging

Contents

3.1 Signal Demodulation and Signal Equation ...... 20 3.2 Encoding Image Information ...... 21 3.2.1 Fourier Imaging and k-Space ...... 22 3.2.2 Field Inhomogeneities and Local Macroscopic Eﬀects ...... 23 3.3 k-Space Information in Unaccelerated MRI ...... 25 3.3.1 Fourier Transform ...... 25 3.3.2 Nyquist Sampling Criterion ...... 28 3.3.3 MRI as iDFT of truncated Fourier Series Coeﬃcients ...... 28 3.4 General Image Reconstruction via Pseudoinverse ...... 30 3.4.1 Matrix Formulation of the Signal Equation ...... 31 3.4.2 Pseudoinverse Method ...... 32 3.4.3 Point Spread Function ...... 34 3.5 Measurement vs. Reconstruction Model ...... 35 3.6 Parallel Imaging ...... 36 3.7 Signal-to-Noise Ratio ...... 37 3.7.1 Thermal Noise Propagation ...... 38 3.7.2 Temporal Noise ...... 40 3.8 MR Imaging Sequence ...... 40 3.8.1 Spectral Saturation ...... 41 3.8.2 Volume-Selective RF Pulses ...... 42 3.8.3 Steady-State Magnetization ...... 42 3.8.4 Gradient Spoiling ...... 43 3.8.5 Gradient-Echo (fMRI) Sequence ...... 43 3.8.6 Physical Constraints ...... 45 3.9 Functional MRI ...... 46 3.9.1 Hemodynamic Response Function ...... 46 3.9.2 Functional Connectivity ...... 47 3.9.3 Advanced fMRI Techniques ...... 48 20 Magnetic Resonance Imaging

As apparent from the previous chapter, i.e., from Formulas (2.18), (2.19), and (2.36), the MR signal generated at position r at ﬁeld strength B, is proportional to s(s + 1) and γ3:

signal(r) ∝ n(r)s(s + 1)γ3B2, (3.1) where the spin density n(r) involves the natural abundance of the imaged nucleus in the human body. When taking the natural abundances in the human body and the quite high γ of the proton (relative to most other nuclei) into account, the choice of 1H hydrogen nuclei for MRI is obvious. Other nuclei are used as well in clinical MRI, however, when these can be used as natural markers of clinically relevant physiological processes. The ﬁrst choice for anatomical and most functional imaging remains 1H, and this is the nucleus used in this thesis. The Larmor frequency of 1H at 3 T is ∼128 MHz, which corresponds to a wave length of ∼26 cm in the body33 and ∼2.35 m in air. Diﬀraction limits the resolution of optical imaging systems to the order of magnitude of the wavelength, meaning the localization of NMR in the human body necessitates another image encoding concept. This chapter covers the basics of MR image encoding and reconstruction, mostly restricted to understanding the concepts that were used for the main part of this dissertation, i.e., Chapters 5 to 7.

3.1 Signal Demodulation and Signal Equation

In continuation of the last chapter’s conclusion, i.e., the explicit formulation of the MR signal, a necessary intermediary step is explained before moving onto the MR imaging framework. Recalling that the MR signal consist of an integral over a spectrum of oscillations with different frequencies close to the Larmor frequency, i.e., almost 128 MHz in this thesis, one can conclude that a reduction of this signal in some form is necessary for efficient digitization of the signal. This step is realized by demodulation of the nominal (main field) Larmor frequency ω0B from the signal. In the signal equation, Equation (2.36), the signal oscillation is characterized by local signal sources contributing a sinusoid with the two parameters frequency and phase, resulting in a real-valued signal. As obvious from the derivation of the signal equation, the local signals arise from the projection of circular rotating vectors in the x-y-plane, representing the spin ensembles’ precession. Reflecting the nature of the MR signal, the oscillations can also be written as i(ω0(r)t−φM (r,0)+φB(r)) sin ((ω0(r)t − φM (r, 0) + φB(r))) = Re (−i)e . (3.2)

With this in mind, the MR signal induced in a coil Scoil is multiplied with a complex rotating unit vector1:

Sunﬁltered ≡ Scoil · [sin(ω0Bt) − i cos(ω0Bt)] . (3.3)

1The MRI scanner hardware takes care of this by treating real and imaginary parts separately. Encoding Image Information 21

This rotating unit vector would represent the rotating frame transformation (see last chapter, Section 2.2.4) of the complex signal sources in Equation (3.2). However, when applied to only the real part of the complex oscillations, as apparent in the measured MR signal, a second band of frequencies at ω0(r) + ω0B arises in addition to the oscillations in the rotating 2 frame at ω0(r) − ω0B. This can be derived from the trigonometric identities 1 sin a sin b = (cos(a − b) − cos(a + b)) 2 1 sin a cos b = (sin(a − b) − sin(a + b)). 2

After ﬁltering the high frequency band from Sunﬁltered, which is well separated from its base band due to relatively very small Larmor frequency variations in ω0(r), the complex MR signal S becomes:

Z ∗ iφB(r) −iφM (r,0) −t/T (r) iδω0(r)t S(t) = B⊥(r)e ω0(r)M⊥(r)e e 2 e dr (3.4) Z ∗ −t/T (r) iδω0(r)t = c(r)ρ⊥(r)e 2 e dr,

iφ (r) −iφ (r,0) with the complex coil map c(r) = B⊥(r)e B , the quantity ρ⊥(r) = ω0(r)M⊥(r)e M , in the following (incorrectly3) referred to as transversal magnetization density, and the

Larmor frequency deviation δω0(r) = ω0B − ω0(r). In conclusion, a real-valued signal with high-frequency modulation was demodulated by a rotating frame transformation. In the same process, its imaginary part was restored by ﬁltering a high frequency band that can be assumed to be non-physical from consideration of the signal properties. The resulting signal on the left of Equation (3.4) is digitized and constitutes the data that will be dealt with in the following. The resulting Equation (3.4) and its derivates, using the expression from the second line, will be referred to as the signal equation in the following. In the frame of image reconstruction, discretizations of this −t/T ∗(r) equation are solved for approximations of ρ⊥(r)e 2 , i.e., so-called relaxation-weighted approximations of ρ⊥(r), in Section 3.4.

3.2 Encoding Image Information

In Section 2.2.4 it was discussed how a transverse magnetization component can be evoked in a spin ensemble. For now, it is just assumed that ρ⊥(r) is not zero. Its generation during an MRI scan will be discussed more thoroughly in Section 3.8.

2This can be done oscillation-component-wise, on the components shown in Equation (3.2), and is left to the reader. The interesting part is that [sin(ω0t + ζ) + i cos(ω0t + ζ)] · [sin(ω0B t) − i cos(ω0B t)] cancels the high-frequency terms, as can be easily reproduced by the Euler formula. On the other hand, sin(ω0t + ζ) · [sin(ω0B t) − i cos(ω0B t)] is left with high-frequency parts. 3In the special case of a preceding RF-pulse on the thermal equilibrium ensemble, and neglecting the spatial dependency of ω0, ρ⊥ becomes directly proportional to the transverse component of the magnetization density. The assumption of approximately constant ω0 is justiﬁed, though, and a constant phase does not interfere with the imaging framework. In general, this quantity is highly dependent on the spin history created by the imaging sequence, and also introduces T1-dependence. 22 Magnetic Resonance Imaging

3.2.1 Fourier Imaging and k-Space

As outlined in the introduction of this and the last chapter, modern MRI uses the concept of using Larmor frequency modulation along several directions of a volume to calculate an image from the resulting signal. When linear magnetic field gradients are used for the frequency modulation, and enough image information is gathered, a Fourier transformation can be used for the (approximative) solution of the signal equation, which is shown in this section. The Fourier encoding concept is the standard image read-out method until today, even though it is modified for acceleration in most cases. Some of the acceleration methods are part of this thesis and will also be covered in Section 3.6. Fourier encoding makes use of linear spatial encoding gradients, added to the main field, which alter the Larmor frequency and thus signal information in space in a linear dependent manner. It has to be noted that this section isolates the effects of encoding fields from the effects of susceptibility-induced field inhomogeneities, which will be added to the framework in the next section. The Larmor frequency modulation δω0(r) that is induced by a linear encoding gradient G reads: enc δω0 (r) = −γG · r. (3.5)

In the next step, the Larmor frequency modulation has to be considered time-dependent, to be able to create a time-dependent signal modulation for image encoding. However, the derivation of the signal equation did not incorporate time-dependent Larmor frequencies. The same steps in the signal derivation in Chapter 2 with relatively small time-dependence of ω0(r, t) can be repeated, which leads to the same expression with the replacement R t 0 0 exp (iδω0(r)t) ⇒ exp i 0 δω0(r, t )dt :

Z Z t −t/T ∗(r) 0 0 S(t) = c(r)ρ⊥(r)e 2 exp i δω0(r, t )dt dr, (3.6) 0 when the minimal temporal dependence in the contribution ω0(r, t) to ρ⊥(r) is neglected (see Equation (3.4)). Using Relation 3.5 with a time-dependent gradient G(t), and the deﬁnition

Z t k(t) = γG(t0)dt0, (3.7) 0 Equation (3.6) becomes: Z −t/T ∗(r) −ik(t)·r S(t) = c(r)ρ⊥(r)e 2 e dr. (3.8)

−t/T ∗(r) Under neglect of the relaxation term e 2 , for now, a mapping relation of one k-space 3/2 point per time point t(k), proper data ordering, and f(r) = (2π) c(r)ρ⊥(r), the equivalence of the signal to the 3D Fourier transform becomes apparent: Z F(k) = (2π)−3/2 f(r) e−ik·r dr. (3.9) Encoding Image Information 23

Under the assumptions that (i) enough image information in the form of k-space data points is gathered by suitable imaging gradient evolution (see Section 3.3.3), and (ii) the relaxation term introduces benign eﬀects (see Section 3.5), an image of the (relaxation- weighed) transverse magnetization can be reconstructed by inverse DFT. This simple image reconstruction is not used in this thesis. However, the k-space formalism is kept for its elegance of the description of image information read-out. The latter is called the k-space trajectory.

3.2.2 Field Inhomogeneities and Local Macroscopic Eﬀects

The last section made the assumption of no main ﬁeld inhomogeneities, i.e., oﬀ-resonances. 32, 34–36 inhom When the formalism is generalized to incorporate these, denoted as ωor ≡ δω0 , Equation (3.5) becomes:

enc inhom δω0(r, t) = δω (r, t) + δω (r) 0 0 (3.10) = −γG(t) · r + ωor(r), assuming static ﬁeld inhomogeneities. Inserted in Equation (3.4), the full version of (3.8) becomes Z ∗ −t/T (r) −i(k(t)·r+ωor(r)·t) S(t) = c(r)ρ⊥(r)e 2 e dr. (3.11)

This equation is generally much harder to handle than (3.8), and one approach of its inversion will be discussed later in Section 3.4.1. For most instances of MRI, however, very short read-outs are used, such that t takes only small values and the additional term does not play a signiﬁcant role.4 Consequently, the signal equation (3.8) is usually assumed for image reconstruction. In BOLD-fMRI, on the other hand, which makes use of long read-outs, the additional term does play a role. To break down the behavior of the full signal equation (3.11) to some extent, at ﬁrst the linearity of the signal acquisition is pointed out:

Z ∗ X −t/T (r) −i(k(t)·r+ωor(r)·t) S(t) = c(r)ρ⊥(r)e 2 e dr, (3.12) n Vn i.e., the MR signal is a sum of signals from arbitrarily divided contiguous subvolumes Vn.

In the next step, a Taylor expansion is made to the oﬀ-resonances at an arbitrary point rn within Vn: ∞ X ωor(r) = ωor(rn) + ∇ωor(rn) · (r − rn) + Ti(ωor(r), rn) i=2 (3.13) ∞ = ωor(rn) + ∇ωor(rn) · (r − rn) + T2 (r), with polynomial terms of order i, evaluated at position rn, denoted by Ti(ωor(r), rn), and

4 inhom Most of the time, δω0 (r) is very smooth and takes values of below 100 Hz. This means that typical read-out times of a few milliseconds lead to an additional phase accrual of well below 1 rad along the read-out, with almost no intra-voxel phase accumulation. 24 Magnetic Resonance Imaging

∞ their sum of orders 2 to inﬁnity T2 (r). Inserting the Taylor expansion into the phase-carrying exponent of the signal equation leads to

Z t 0 0 i δω0(r, t )dt = −i (k(t) · r + ωor(r) · t) 0 h ∞ i = −i k(t) · r + (ωor(rn) + ∇ωor(rn) · (r − rn) + T2 (r)) · t h (3.14) = −i k(t) · rn + ωor(rn) · t

+ (k(t) + ∇ωor(rn) · t) · (r − rn) ∞ i + T2 (r) · t

When this relation, evaluated at all support points rn, is inserted into Equation (3.12), the subdivided signal equation becomes5

Z ∗ ∞ X −i(k(t)·rn+ωor(rn)t) −t/T (r) −i(k(t)+∇ωor(rn)·t)·(r−rn) −iT (r)·t S(t) = e c(r)ρ⊥(r)e 2 e e 2 dr n Vn + Z ∗ X −t/T (rn) −i(k(t)·rn+ωor(rn)t) −t/T (r) −ikn(t)·(r−rn) ≈ e 2 e c(r)ρ⊥(r)e 2 e dr, | {z } V | {z } n global enc. n local enc. (3.15) + with the newly introduced constant T2 and the following considerations: (1) The encoding is split in global and local parts. The global encoding is the part of the

phase modulations that can be described on a discretized grid with grid points rn. The local encoding represents additional macroscopic phase modulations within the

volumes Vn. (2) The additional local encoding is defined to contain first order off-resonance effects, and

is described by a local k-space within each Vn: kn(t) ≡ k(t) + ∇ωor(rn) · t. (3) Higher order macroscopic off-resonances within a volume lead to more complex and case- dependent effects. In MRI, it is common to incorporate all effect of local microscopic ∗ and macroscopic field inhomogeneities, starting from the first order, into T2 (see last chapter, Section 2.2.5). This nomenclature is refrained from in Equation (3.15), because the first-order macroscopic fields are isolated, and the higher-order macroscopic field 0 + effects are discriminated from T2 by the newly introduced constant T2 . For the rest of this thesis, however, this detail is omitted in accordance to common + practice in MRI. This means that T2 , mostly also incorporating effects of first order ∗−1 −1 0−1 +−1 macroscopic field inhomogeneities, is incorporated into T2 = T2 + T2 + T2 . In conclusion, the description of field inhomogeneities in the context of image encoding led to the formulation of a more complex signal equation. The formulation of Equation (3.15) constitutes an intermediary step between the derivation of the continuous form of the signal equation (3.11) and its discretization (see Section 3.4), which is necessary for image reconstruction. The only discretization approximation that was made so far is the introduction of volume- and field-dependent transverse decay due to reversible spin dephasing.

5 This formalism assumes that all Vn are small and not excessively oblong, i.e., that the global structure of the oﬀ-resonances on the spatial scale of the imaged object is well captured by the gradients within Vn. k-Space Information in Unaccelerated MRI 25

Within the formal division of the to-be-measured object into subvolumes, a discrimination between first order and higher order field inhomogeneity effects was made. This discrimination led to the description of local k-space, which essentially implies location-dependent image information encoding. For the rest of the introductory chapter on MRI, macroscopic first-order field inhomogeneities are assumed to be negligible, for conceptional and notational ease. The concept of local k-space is picked up again in Chapters 5 and 6.

3.3 k-Space Information in Unaccelerated MRI

The encoding of image data is fundamentally constrained by physical and technological boundaries. Under this limitation, only a finite number of discrete k-space points can be acquired, which naturally also implies the reconstruction of a discrete approximated image of a continuous magnetization density. Section 3.2 established the analogy of Larmor frequency modulations and image information encoding in Fourier space. In consequence, under the assumption of negligible transverse decay (for decay effects, see Section 3.5) and field inhomogeneities (see Section 3.5 and Chapter 5), the properties of the Fourier transform apply to the image reconstruction from k-space data. For ease of notation in both chapters, Section 3.2 used a different k-definition than this chapter. Now, within Section 3.3, the transition k → k/2π is made. This chapter describes equidistant k-space sampling for a conceptional introduction, usually referred to as Cartesian sampling. Non-Cartesian sampling is used in this thesis, however. The transition between the two concepts is discussed in Section 3.4. Moreover, this chapter ignores the inherent spatial encoding information of the coil maps c(r) when incorporating several receive coils. Accelerated imaging by exploitation of coils maps is discussed in Section 3.6. Lastly, this chapter is formulated for the 1-dimensional case for notational ease. Multi- dimensional Fourier transforms are performed by subsequent Fourier transform along the multiple dimensions. The Fourier transforms are associative, and their effects do not interfere across dimensions.

3.3.1 Fourier Transform

The general term "Fourier transform" encompasses a family of transformations that can be classiﬁed into 4 types:

Continuous Fourier Transform The continuous Fourier transform (FT) is the most general form, from which the other 3 can be derived. It is a unitary transformation that relates an integrable function f : R → C over an inﬁnite continuous x-space to its Fourier transform F over an 26 Magnetic Resonance Imaging

inﬁnite continuous k-space:

Z ∞ −i2πkx Ff (k) = f(x) e dx −∞ Z ∞ (3.16) i2πkx f(x) = Ff (k) e dk. −∞

Fourier Series The Fourier series describes a FT of a periodic function. It relates a continuous x-space to a discrete k-space.

The transform of a function fL on an interval of length L, with x ∈ [−L/2, L/2], contains all Fourier coeﬃcients that are needed to express that of its periodic reproduction

fL∞. To derive the Fourier series of fL∞ from the continuous Fourier transform, it is expressed in terms of fL: fL∞(x) = (XL ∗ fL)(x), (3.17) P∞ with Dirac comb function XL(x) = n=−∞ δ(x − nL), Dirac delta distribution δ(x), and convolution operator ∗. Due to the convolution theorem of the (continuous) Fourier transform, it follows:

FfL∞ (k) = FXL (k) ·FfL (k) 1 (3.18) = X (k) ·F (k), L 1/L fL with the Fourier transformation property of the Dirac comb. This implies that the k-space of a periodic function consists of a discrete set of frequencies, i.e., S 1 n F [n] = Ff ( ) fL∞ L L L (3.19)

= ∆k FfL (n ∆k), with n ∈ Z and ∆k = 1/L.

The term "Fourier series" arises from the series expansion of fL∞ that is the inverse Fourier transform of the discrete k-space:

∞ f (x) = X F S [n] ei2πnx/L L∞ fL∞ n=−∞ ∞ (3.20) X i2πn∆k x = ∆k FfL (n∆k) e . n=−∞

Discrete-Time Fourier Transform The discrete-time Fourier transform (DTFT) relates a set of discrete x-space points to a continuous periodic k-space. The term "discrete-time" comes from its origin in Fourier analysis of discretely sampled time series. Consequently, it can be derived from the continuous Fourier transform by multiplying a Fourier comb to the x-domain. The transform mirrors the relations of the Fouries series (3.19) and (3.20) with inverse k-Space Information in Unaccelerated MRI 27

role of x- and k-space. This results in the same relations with interchanging roles:

S fL∞(x) ↔ F [n] fL∞ (3.21) S ⇒ f [m] ↔ Ff S (k),

with m being the x- and n being the k-discretization index. With the Fourier transform being unitary, this transformation is fully equivalent to the Fourier series transformation. Discrete Fourier Transform The discrete Fourier transform (DFT) relates a discrete x-space to a discrete k-space. It results from the same consideration that led to the discretization of k-space in the transition from continuous transform to Fourier series, i.e., periodicity. The periodicity is now assumed not only for the x-space (in case of the Fourier series), but also for the discretized k-space of the Fourier series. As a result, a periodic discretized x-space is transformed into a periodic discretized k-space. Assuming k-space periodicity of N, where each sample is spaced by ∆k = 1/L, Equation (3.19) becomes: F S [n] = (∆k F ∗ )(n ∆k), (3.22) fL∞ fL XN/L

where FfL is non-zero only on the interval [−N/2L, N/2L). Due to the convolution theorem, Equation (3.20), the inverse transform, then becomes

N/2−1 X i2πn∆k m∆x fL∞(m∆x) = ∆x ∆k FfL (n∆k) e n=−N/2 (3.23) N/2−1 −1 X i2πnm/N = N FfL (n∆k) e , n=−N/2

where ∆x = L/N, and image space is sampled at positions x = m · ∆x. The known form of the DFT/iDFT:

N−1 X −i2πmn/N Ff [n] = a f[m] e m=0 (3.24) N−1 X i2πmn/N f[m] = b Ff [n] e , n=0

with normalization factors ab = N −1, is equivalent and follows from consideration about the underlying periodicity. In this thesis, the normalization (a = 1, b = N −1) is kept, as it follows from the derivation in MRI, see Section 3.3.3. An important detail to note: during the derivation, periodicity in both domains was implied. Both domains have the same number of non-periodic coeﬃcients, N.

The DFT is a very important transformation, as it is the only class of Fourier transforms than can be handled by digital devices. It is important to understand the underlying implication of N-periodicity, which can lead to so-called aliasing eﬀects in digitization of continuous - or also downsampling of discrete - signals, as discussed in the next section. 28 Magnetic Resonance Imaging

3.3.2 Nyquist Sampling Criterion

The Fourier series describes the implications of a periodic x-domain, i.e., the multiplication of k-space with a Dirac comb. Describing the inverse effect, the DTFT as the mathematical dual of the Fourier series describes the implications of discrete equidistant sampling in the continuous x-space, i.e., a periodic k-space. No difference in the effects described here arises, when the x- and k-domains are inverted, as in MRI, and a continuous k-space is sampled discretely. As implied by the inversion of the Fourier series relations (3.17)-(3.19), the rate of k-space sampling 1/L leads to a periodicity of L in the x-domain, induced by a convolution with a Dirac comb. Consequently, when the k-space of an object that is larger than L˜ is sampled with 1/L˜, the portions of fL that exceed the interval [−L/˜ 2, L/˜ 2] are replicated into the far side of the interval. This replication of fL that appears on the far side of the exceeded interval is linearly added onto the accurate portions of fL. This foldover effect, induced by the convolution, is called aliasing. The Nyquist sampling criterion formulates the minimal sampling rate that is necessary to avoid aliasing. For an object of size L the Nyquist rate is consequently 1/L.

3.3.3 MRI as iDFT of truncated Fourier Series Coeﬃcients

The general MRI signal equation before digitization, as derived in this thesis until Section 3.2.1, is fully continuous in x- and k-space. Equation (3.9) as a result of Equation (3.8) conceptionally represents the continuous FT that leads to an image. Keeping the assumption of negligible relaxation and oﬀ-resonance eﬀects on the reconstruction, for now, the process of digitized/discrete image formation can be summarized by 3 concepts:

Signal Discretization The measured signal S(t) is sampled at discrete time points, which correspond to discrete k-points. This represents a transition from FT to Fourier series, and imposes periodicity on the resulting image in x-space. Due to the Nyquist sampling criterion,

an object with size LO needs to be sampled with maximally 1/LO-distant k-space data points. This step implies one more pitfall concerning the discrete sampling of the oscillating signal S(t), when looking back at Equations (3.4) and (3.5): the maximum bandwidth 6 of frequencies in the signal, with maximum gradient amplitude Gmax, is γGmaxLO. This consequently constitutes the minimum sampling rate, again implied by the Nyquist sampling criterion, applied on the conjugate variable pair frequency/time. Thus, the −1 maximum dwell time per data point is tdwell = (γGmaxLO) . k-Space Truncation Due to physical limitations, only a ﬁnite number of discrete k-space points of the

6This consideration needs to be done over all spatial directions. k-Space Information in Unaccelerated MRI 29

Fourier series can be sampled. This can be formulated as

S S,w 1 n n F [n] ⇒ F [n] = Ff ( ) · w( ), (3.25) fL∞ fL∞ L L L L

with k-space window function w.7 Due to the general k-space properties of to-be- measured objects and signal-to-noise considerations in MRI, this window is chosen centered around the k-space origin. Inverting the transformation from Equation (3.17) to Equation (3.19), the resulting image is, due to the Fourier convolution theorem:

w −1 f (x) = ((XL ∗ fL) ∗ F )(x), L∞ w (3.26) −1 = (XL ∗ (fL ∗ Fw ))(x),

i.e., the periodic image is additionally convolved with the inverse Fourier transform of −1 the window function Fw . In non-accelerated MRI with Cartesian sampling, the function w is chosen as a rect- function spanning at least 64 k-space data points,8 such that the additional convolution −1 kernel Fw is a narrow sinc-function. Consequently, it can be assumed that the periodicity-producing Dirac comb and the truncation-induced sinc-function act on different scales and do not interfere. This is an important result: the k-space windowing results in a window-dependent blurring-like convolution.9 This step defines the resolution in MRI, where generally −1 the distance of the first zero-crossing of Fw from its center is considered the spatial resolution of the measurement. However, it needs to be reminded that a continuous blurring of the object underlies the later reconstructed image. In general, the window-function is multi-dimensional, and thus the imaging properties depend on its 2- or 3-dimensional shape.10 iDFT Reconstruction While the truncation of k-space was considered as blurring-inducing (as a side effect), −1 it also enables the transition to the (Fw -convolved) image-domain via inverse DFT. This is achieved by the theoretical implication of a N-periodic k-space. The truncated Fourier series

N/2−1 f w (x) = X F S [n] ei2πnx/L, (3.27) L∞ fL∞ n=−N/2

7 1 In general, k-space points are not sampled equidistantly. The sampled k-space is then L FfL (k(n)) · w(k(n)). This leads to a more complex framework, though, and this section sticks to the equidistantly spaced k-space. At a later point, non-equidistantly spaced k-space samples and their projection onto a equidistant grid will be discussed. 8 n Due to the mostly used FFT algorithm operating on 2 , n ∈ N, data points, the usual sampling windows in MRI contain 64, 128, 256, ... sampling points. 9 Additionally to the blurring, one more prominent eﬀect is created: edges in the object fL(x) lead to oscillations due to the sinc-convolution. This eﬀect is called Gibbs-ringing. 10Usually, MRI makes use of Cartesian sampling patterns with rectangular shapes. This thesis, however, later implements a 3-dimensional sphere for w. 30 Magnetic Resonance Imaging

evaluated at x = m · L/N, becomes the iDFT

N/2−1 w −1 X i2πnm/N fL∞[m] = N FfL (n∆k) e , (3.28) n=−N/2

analog to (3.23). When an x-space of length/periodicity L is sampled at N k-space points, the iDFT results in image voxel support points that are spaced by L/N. This spacing also coin- cides with the resolution-deﬁning spacing of the ﬁrst (and subsequent) zero-crossing(s) of the sinc-function.11

3.4 General Image Reconstruction via Pseudoinverse

The last section (3.3) made the assumption of equidistant k-space sampling, which led to the direct applicability of an iDFT onto properly ordered data for image reconstruction. This assumption was made for conceptional ease, and also reﬂects the historically most used way of MRI sampling, but it is not generally necessary for image creation. In this section, the k-space deﬁnition of Section 3.2, i.e., scaled by 2π, is used again. Recalling the signal equation (3.8), its full discretization in both domains, following the argumentation from Section 3.3.3, reads Z −t/T ∗(r) −ik(t)·r S(t) = c(r)ρ⊥(r)e 2 e dr

∗ X −tn/T¯ −ik(tn)·rm ⇒ S(tn) ≈ cmρ¯me 2 m e , (3.29) m ∗ X −t(kn)/T¯ −ikn·rm ⇔ Sn ≈ cmρ¯me 2 m e , m

−1 with Fw -convolved discretized transverse magnetization density ρ¯, ρ¯m = ρ¯(rm), and cm = −1 −1 c(rm). The underlying step in the discretization Fw ∗ (c(r) · ρ⊥(r)) ≈ c(r) · (Fw ∗ ρ⊥(r)) holds under the assumption of slowly varying and approximately constant coil maps on the −1 12 length scale of relevant components of Fw . Another approximation is introduced by the ¯∗ discretized relaxation constant T2 (see Section 3.2.2). Under the assumption of spatially slowly varying oﬀ-resonances, the same argumentation

11This fact sometimes leads to the wrong assumption that the rect-shaped windowing function does not induce contamination between neighboring voxel signals. This assumption stems from simpliﬁed applications of the concept of the point-spread function (see Section 3.4.3), and does not consider the underlying continuous convolution in Equation (3.26). 12This assumption implicitly underlies implementations of the "weak approach" of the SENSE37-formalism. This becomes important in Section 3.6, but could also be omitted at this point by incorporating the single coil map into the image intensity. The approximation becomes apparent when the relation is examined in the k-domain. The assumption that the sampling window w is considerably larger than the support region of Fc is usually justiﬁed. General Image Reconstruction via Pseudoinverse 31 holds for the general signal equation (3.11):

Z ∗ −t/T (r) −i(k(t)·r+ωor(r)·t) S(t) = c(r)ρ⊥(r)e 2 e dr

∗ or X −tn/T¯ −i(k(tn)·rm+ω ·tn) ⇒ S(tn) ≈ cmρ¯me 2 m e m , (3.30) m ∗ or X −t(kn)/T¯ −i(kn·rm+ω ·tn) ⇔ Sn ≈ cmρ¯me 2 m e m , m with variable indexing analogous to Equation (3.29). For non-uniformly sampled k-space data, the formalism is left intact, but additional considerations need to be made. In the signal equation (3.8), no assumption of equidistant sampling was made, yet, such that it holds for any k(t). For the line of argumentation that led to the discretization in (3.29), however, the assumption has to be made that enough data is available to reproduce the information of a uniformly sampled k-space from the non-uniform samples. Generally, this might not be the case, which can lead to image artifacts in the reconstructions of (relaxation-weighted) ρ¯m.

3.4.1 Matrix Formulation of the Signal Equation

For the discretized signal equation (3.29) (or (3.30)) to be solved for ρ¯m, an invertible and computation-eﬃcient formulation of these equations is necessary. As the measurement process is a linear operation, the equations can be formulated as a forward problem in matrix form: s = Eρ,¯ (3.31) with signal s = (S1, ..., SN ), ρ¯ = {ρ¯1, ..., ρ¯M }, and forward operator of the measurement N×M E ∈ C . The full formulation and explicit inversion of the matrix E (especially later in Section 3.6) is generally impossible due to computational constraints. This implies that the operation E should be expressed using a DFT, which can then implement an FFT algorithm. One consequence is that the non-Cartesian k-space data is interpolated from a Cartesian grid in (3.31). This step is called gridding, and is a part of the non-uniform FFT (nuFFT38, 39). For technical details of the gridding operation please refer to the existing literature.38

Time-Independent When the explicit temporal dependencies (relaxation and oﬀ-resonances) in the signal equation are neglected, the forward operator of a single-coil measurement can be formulated as: E = GF SC, (3.32)

13 with coil matrix C = diag {c1, ..., cM }, diagonal scaling matrix S , DFT matrix or FFT N×M operation F , and gridding matrix G ∈ C . In Cartesian imaging, the operations G and S are omitted, while in non-Cartesian imaging the operation GF S represents the

13This is a design parameter for minimization of grid interpolation errors.38 32 Magnetic Resonance Imaging

nu(F)FT14.

Time-Segmented or There are two challenges in formulating (3.30): (i) correct model parameters ωm are needed, and (ii) the explicit time-dependence of E needs to be expressed eﬃciently. Model parameters are usually approximated by an additional dedicated MRI scan. The eﬃcient expression in terms of (multiple) FFTs results from the assumption

that temporal dependencies are suﬃciently described by interpolation in between NT support points τ in time.40 The time-dependency is approximated by:

NT −iβmtn X −iβmτa e ≈ ηa(tn)e , (3.33) a=1

with interpolation ﬁlters ηa(tn) and βm ∈ C containing relaxation and/or oﬀ-resonance values. The measurement operation including explicit time-dependency can then be approxi-

mated by NT nuFFTs: N XT E = GaFSCTa, (3.34) a=1 n o −iβ τa −iβ τa with temporal support weighting Ta = diag e 1 , ..., e M , and k-space-ﬁltered gridding operators Ga = diag {ηa(t1), ..., ηa(tN )} G.

3.4.2 Pseudoinverse Method

In general, no statement can be made that the measurement operation E is invertible, even if ρ is considered discretized. After the application of E∗ to Equation (3.31), leading to the normal equation E∗s = E∗Eρ,¯ (3.35) the formal left inverse (E∗E)−1E∗ can be applied under the assumption of suﬃcient gradient encoding. Considering the latter, the Hermitian matrix E∗E can be assumed to be positive- deﬁnite (see below for exceptions) and invertible:

E†s = (E∗E)−1E∗s =ρ ¯LS. (3.36)

The left inverse is a special case of the pseudoinverse41 E†. ρ¯LS is the solution in the ∗ ∗ least-squares sense, i.e., the minimization of the term kE s − (E E)¯ρk2. As the data in MRI is noise contaminated, this is an important property. If k-space is (locally) undersampled15, however, the matrix E∗E is singular and not invertible.

14This is a problem referred to as type 2, in which a uniform space is mapped onto a non-uniform space. Other problem formulations, e.g., type 1, which maps a non-uniform space onto a uniform one, also need a density compensation for the mapped data. 15In case of sensitivity encoding (see Section 3.6): "If gradient encoding and coil encoding are not suﬃcient for determination of the matrix equation". General Image Reconstruction via Pseudoinverse 33

In this case, the pseudoinverse of E∗E can be formed:

E†s = (E∗E)†E∗s =ρ ¯PI , (3.37) where ρ¯PI is the result of the underdetermined pseudoinversion, again equivalent to mini- ∗ ∗ mization of kE s − (E E)¯ρk2. The operator on both left sides of Equations (3.36) and (3.37) is the pseudoinverse of E: E† = (E∗E)†E∗ (3.38) E† = (E∗E)−1E∗, where the pseudoinverse (E∗E)† becomes the inverse (E∗E)−1 if the inverted matrix E∗E is non-singular. The pseudoinverse of a matrix X can be understood as an inversion of the parts that were not projected into the null space of X, i.e., all encoded information. For example, one method to compute the pseudoinverse is an inversion by singular value decomposition UΣV ∗, where only the non-zero singular values in Σ are inverted and zero-values remain zero-valued.

Conjugate Gradient Method In all pseudo-inversions that were executed in this thesis, the conjugate gradient method was used to solve Equation (3.35) for ρ¯LS or ρ¯PI , without denoting the super-scripted specification of the result ρ¯. The conjugate gradient method is an iterative algorithm for numerical solution of the system of equation Ax = y for x, i.e., A−1y, where A has to be Hermitian, and is generally required to be positive-definite. In the case of Equation (3.35): A = (E∗E), and y = E∗s. If A is positive-indefinite, though, the conjugate gradient method with initial guess x = 0 leads to the pseudoinversion of the problem, i.e., xPI = A†y.42

Tikhonov Regularization If the inversion problem is not well conditioned, i.e., the maximal and minimal eigenvalues of E∗E diﬀer by a large factor, the convergence of the algorithm can be sped up by regularization16. Tikhonov regularization was incorporated in all reconstructions in this thesis, meaning that the inversion of E∗E is replaced by an inversion of E∗E + Γ∗Γ:

⇒ (E∗E + Γ∗Γ)−1E∗s ≈ ρ¯LS/P I , (3.39)

with Tikhonov matrix Γ = λ1, λ ∈ R. λ has to be chosen small enough that the image estimation is not substantially inﬂuenced17, while fast convergence of the algorithm is ensured. The eﬀect on the condition number of E∗E by the regularization becomes apparent

16Convergence speed in general, though, is dependent on two factors: (i) condition number, and (ii) eigenvalue concentration, which is beyond the scope of this chapter. 17The process of finding a suitable λ was done by increasing the parameter until a visible change in image properties was observed. The values that led to visible change were then discarded. While regularization parameter optimization is a complex topic, in the case of noisy and artifact-affected MRI reconstructions, the visual inspection is a cost-effective alternative with equivalent outcome. 34 Magnetic Resonance Imaging

when analyzing the eigenvalues of the problem. When the singular values of E are i, then the eigenvalues of the Tikhonov-regularized inversion matrix (E∗E + Γ∗Γ)−1 are 2 2 −1 (i + λ ) . The modification of eigenvalues implies another property of the Tikhonov regularization. It enables to enforce positive-definiteness of (E∗E + Γ∗Γ), when (E∗E) is indefinite due to lack of data sampling.

The pseudo-inversion (or regularized inversion) presented here is used in highly undersampled MREG reconstructions,27–29 where a decrease of image ﬁdelity is traded oﬀ against minimization of sampling time by omitting k-space samples.

3.4.3 Point Spread Function

In MRI, the point spread function PSF of the imaging process deﬁnes the imaged response 43 of the system to a point source at position r0 in the object domain. It can be simulated by considering the vector of partial data in the k-domain s, stemming from a source at r0:

full s(r0) = E δ(r − r0), (3.40) where, formally, the measurement operation Efull represents the full measurement operation of discrete k-space samples in continuous30 spatial domain.18 The imaged response from a point source at r0, is then

full∗ full † full∗ full PSF (r0, r) = (E E ) E E δ(r − r0), (3.41) or, with Tikhonov regularized reconstruction:

full∗ full ∗ −1 full∗ full PSF (r0, r) = (E E + Γ Γ) E E δ(r − r0) (3.42)

The concept of the image artifact characterization by a point spread function stems from considering the linearity of the imaging process, i.e., the image at position r0 can be regarded as superposition of the object-weighed PSF -influence from all positions: Z ρ¯(r0) = ρ(r)PSF (r0, r) dr full∗ full † full∗ full (3.43) = (E E ) E E ρ(r0) † = Efull s, when s = Efullρ. This characterization requires a linear image reconstruction process. ∗ If the PSF is simulated to contain effects of explicit temporal dependencies (T2 and off- resonance) in the measurement Etemp., which cannot be accounted for in the reconstruction

18Due to necessity in the simulation, the continuous spatial domain can be replaced by a grid that is considerably ﬁner than the image resolution. The simulation grid should be ﬁne enough to capture the sub-voxel structure of the point spread function. Measurement vs. Reconstruction Model 35 with Estatic, then (3.41) becomes

static∗ static † static∗ temp. PSF (r0, r) = (E E ) E E δ(r − r0). (3.44)

CAVE: The point spread function is sometimes simulated from a point source that lies on the reconstruction grid (which is implied by reconstructing the PSF from a k-space pattern ﬁlled with the values 143). When, moreover, the grid size of the reconstruction is used, and that reconstruction reﬂects the resolution of the k-space sampling, that simulation does not consider the sub-voxel structure of the PSF and the resulting leakage artifacts which arise from the continuous-to-discrete mapping. These always emerge due to the underlying convolution in the image ρ¯ (see Section 3.3.3). E.g., for Cartesian imaging, this method would result in a Kronecker-delta, while the Dirichlet kernel that is the PSF of (ideal) Cartesian imaging does not become evident. Later in this thesis (Chapter 6), the point spread function is simulated on the reconstructed grid size due to computational constraints. However, there is no evidence for the existence of a pathological situation as found with the Dirichlet kernel in Cartesian imaging.

3.5 Measurement vs. Reconstruction Model

Most of the previous reconstruction framework neglected the explicit intrinsic time dependency βt ∗ of the measurement process, i.e., the term e with β = −1/T2 + iωor in the signal equations. This is a common approach in MRI, which leads to benign errors in the image domain under certain circumstances, as shown below. While intra-voxel off-resonance effects on encoding were covered partly in Section 3.2.2, and will be discussed further in Chapter 5, the macroscopic off-resonances and relaxation were ignored so far. In general, image reconstruction under neglect of the explicit time dependencies can be formalized as ρ¯rec = Erec†Emeasρ, (3.45) with the underlying object magnetization at the start of the read-out ρ, the approximated reconstruction of the convolved object ρ¯rec, real measurement operation Emeas, and the modeled operation for reconstruction, Erec. The reconstruction ρ¯rec differs in two ways from the underlying object: (i) by the convolution due to k-space truncation; (ii) by the inversion errors due to ignoring explicit time dependency in the measurement. When assessing the signal equation including time dependencies, Equation (3.11),

Z ∗ −t/T (r) −i(k(t)·r+ωor(r)·t) S(t) = c(r)ρ⊥(r)e 2 e dr Z (3.46) β(r)t −ik(t)·r = c(r)ρ⊥(r)e e dr, it can be seen that using small read-out times (on the scale of β) at t ∈ [t0 − δt, t0 + δt], β(r)t −t /T ∗(r) iω (r)t the time dependent term becomes approximately the constant e 0 = e 0 2 e or 0 . ∗ This constant then represents the T2 (r) weighting of the image with oﬀ-resonance induced 36 Magnetic Resonance Imaging phase. At longer read-out times, the introduced error becomes highly trajectory-dependent.19 Due to the linearity of the reconstruction, the time dependent term can be understood as a spatially dependent k-space ﬁlter function:

eβ(r)t e−ik(t)·r ⇒ K(k(t), r) e−ik(t)·r, (3.47) meaning that the Fourier reconstruction of the signal from a point in space is convolved with the inverse Fourier transform of the ﬁlter at that point, K(k, r).

Benign behavior under neglect of explicit time dependency is a deﬁning constraint for k-space trajectory design, since the values of β are (i) costly to determine, and (ii) determination is subject to various errors. One important concept of benign relaxation and oﬀ-resonance behavior is implementing a trajectory which imposes the time dependencies linearly along 20 one k-space dimension j1 :

−ik·r −ikj1 ·rj1 −ikj2 ·rj2 −ikj3 ·rj3 K(k, r) e ⇒ K(kj1 , r) e e , (3.48) with dimensions j1, j2, and j3.

The image reconstruction via inverse Fourier transform along that direction j1 then becomes −1 convolved with the Fourier transform of the weighting FK . In case of a linear read-out, the Fourier transform of an exponential decay with phase evolution, i.e., a shifted Lorentzian −1 function, is convolved along that direction with the image. FK is then shifted proportionally to off-resonance, and shows Lorentzian width and attenuation in dependence of transverse relaxation. The other dimensions of the image remain unaffected by the weighting convolution in this case. The same effect approximately appears when a multidimensional trajectory has an overall slowest encoding direction in k-space, and the other dimensions are encoded fast relative to the time scale of β.

3.6 Parallel Imaging

The expression "parallel imaging" describes the method of using more than one receive coil for parallel acquisition of gradient-encoded MRI data. Using multiple receive coils can add spatial information21 to the measurement by incorporating the coil maps c(r) of diﬀerent coils in signal equations (3.4) and the discretized (3.29). When L coils with diﬀerent coil maps cl(r), l ∈ [1,L], are incorporated into the (discretized)

19The error that is convolved onto the image can be simulated by the point spread function, as described in Section 3.4.3. 20E.g., by reading out lines in k-space, where one line is recorded during one read out, and all lines are read in the same k-direction. 21Increase in signal-to-noise ratio using to parallel acquisition44 is not the subject of this thesis. Signal-to-Noise Ratio 37 signal equation(s) X ¯∗ l l −t(kn)/T2 m −ikn·rm Sn ≈ cmρ¯me e (3.49) m T for L signals Sl, the measurement operator for the concatenated signal vector S = S1, ..., SL becomes:  GF SC1    E =  ...  , (3.50) GF SCL

l n l l o with C = diag c1, ..., cM . For a note on discretization of these equations see footnote 12 in Section 3.4.

Under the assumption of knowledge of the receiver coil maps, the image reconstruction of the parallel imaging problem can be done exactly as already described in Section 3.4.2. This is the method that was used in this thesis, where the coil maps were estimated45 from a low-resolution prescan right before the main acquisition. This reconstruction problem is known as the weak approach of the SENSitivity Encoded (SENSE)37, 46 reconstruction. The additional information from the spatial receiver sensitivity encoding can be used to omit some of the k-space encoding and consequently save time during the acquisition. Phenomenologically, parallel imaging permits to reduce k-space sampling density due to the low-frequency information that is superimposed by the coil map profiles: the multiplication of the low-frequency coil map for every channel in the image space can be understood as a convolution of the k-space data with different narrow functions; this additionally convolved information is implicitly deconvolved by the image reconstruction and thus enables to reduce k-space sampling density. In the SENSE formalism, mathematically speaking, range(E) is increased when adding acquisition channels with different cl(r), using the same read-out pattern.

Apart from the SENSE-framework, there is the method of reconstructing missing k-space samples for every coil from the implicitly convolved information encoded in the multi-coil data, and use the coil-wise reconstructed full Cartesian k-spaces for joint image reconstruction. For this framework, i.e., GRAPPA (GeneRalized Autocalibrating Partially Parallel Acquisitions) and its derivates,47, 48 please refer to the literature.

3.7 Signal-to-Noise Ratio

The expression "signal-to-noise ratio" (SNR) usually reflects the ratio of image intensity to the thermal noise-induced standard deviation of image intensity. The image intensity in a voxel is proportional to the voxel size, assuming constant signal in the voxel. This follows from Equation (3.4) and the linearity of the imaging problem. Apart from relaxation time constants, the signal strength is governed by imaging sequence parameters, spin density, field strength, etc., which greatly influence the magnitude of M⊥(r) 38 Magnetic Resonance Imaging in Equation (3.4). Moreover, the reconstructed image intensity depends on the acquired number of data points.

3.7.1 Thermal Noise Propagation

All thermal noise sources (in body, coil, and electronics) can be assumed to be zero-mean independent Gaussians, which means their sum is a zero-mean Gaussian. Thus, a nor- mally distributed noise vector νk of length N is added to the object signal sobj during measurement: s = sobj + νk. (3.51)

Characterization of noise propagation into image noise νr is based on the linearity of the reconstructed image: νr = E†νk. (3.52)

2 2 In a single-coil measurement, the vector of variances in image space σ , i.e., variance σm in the voxel indexed by m, becomes:

2 r † k σm = var (νm) = var (E ν )m X X † † ∗ k k = EmiEjm cov νi , νj i j X X † † ∗ 2 = EmiEjm δijσk i j (3.53) 2 X † † ∗ = σk EmiEim i

2 2 n † †∗o ⇒ σ = σk diag E E ,

2 under the assumption of white k-space noise with variance σk. In the multi-coil case, noise is correlated amongst the channels, i.e.,

k,l k,l0 2 cov νi , νj = σk,ll0 δij, (3.54) with l and l0 being channel indices. As shown before in the SENSE literature,37 the image noise variance σ2 becomes n ∗o σ2 = diag E†ΨE† , (3.55)

1N×N 2 with Ψ = V ⊗ , Vll0 = σk,ll0 , being the receiver noise correlation matrix acting channel- wise. For a measurement operation E with full rank, and the noise-minimizing reconstruction (E∗Ψ−1E)−1E∗Ψ−1 (using the weak approach), this reduces to

−1 σ2 = diag E∗Ψ−1E . (3.56)

In parallel imaging, the resulting image noise becomes spatially dependent and colored. Signal-to-Noise Ratio 39

Single Coil Cartesian Measurement In a single coil measurement with Cartesian trajectory, E† becomes an iDFT matrix (with normalization factor 1/N, see Section 3.3), and thus E†E†∗ = 1/N · 1 In other words, νr becomes again a white noise vector due to the properties of the Fourier transform, and the resulting variance in image space is then inversely proportional to the length of the Fourier transform22. As an example, increasing the resolution by a factor of D along a dimension while holding the ﬁeld of view and gradients constant results in a loss of SNR by a factor of √ D (loss of signal per voxel by a factor of D, and decrease of noise standard deviation √ by a factor of D).

Single Coil Non-Cartesian Measurement SNR behavior with arbitrary trajectories is case dependent. E.g., in the case of a gridding reconstruction with density compensation (instead of the pseudoinverse E†), the noise propagation in a 2D single coil measurement follows49, 50 as Z 2 2 σm = σk 1/D(kx, ky) dkxdky, (3.57) K

with sampled k-space area K, and k-space sampling density D(kx, ky). In the case of variable sampling density D(k), no general quantitative statements can be made about the resulting SNR. However, Equation (3.57) reveals increasing noise in the case of decreasing k-space sampling density. Note that the density compensated gridding reconstruction is not equal to the pseudoinverse, which can be considered noise-minimizing as it is the least squares reconstruction. Cartesian SENSE Imaging For Cartesian undersampling in SENSE imaging, i.e., sampling of only the Rth line in Cartesian k-space, it can be shown37 that the SNR of the reduced k-space at position m, SNRred, is always smaller than that of the full k-space SNRfull by at least a factor √ m m of R: q full Xred √ SNRm m red = q = gm R, (3.58) SNR full m Xm

with the coil conﬁguration-dependent geometry factor gm ≥ 1, Non-Cartesian SENSE Imaging Like in the single-coil case, SNR relations become case dependent with variable density sampling. The derivation of a simple reduction factor R like in Equation (3.58) is not possible anymore. In cases of reconstruction matrices E† that are too large to be explicitly calculated, like in this thesis, and when only an iterative solution method is an option, noise propagation has to be simulated by a Monte Carlo Method.51 To this end, a high number of coil-correlated noise realizations were reconstructed (see Chapter 6) for diﬀerent trajectories, and the resulting simulated SNR maps were divided point-wise

22In multiple dimensions, the resulting variance is inversely proportional to the product of Fourier transform lengths in all dimensions. 40 Magnetic Resonance Imaging

for a characterization of noise propagation enhancement P:

full SNRm Pm ≡ red . (3.59) SNRm

3.7.2 Temporal Noise

In MRI time series that consist of repeated measurements of the same object, such as performed in fMRI, each image exhibits a particular noise realization. In the case of purely thermal noise, the noise in the time series would correspond to the results of the previous section. However, time series of living subjects usually contain pink noise, which is induced by image ﬂuctuations due to physiological processes. The relative inﬂuence of physiological noise over thermal noise increases as SNR increases, because physiological noise is part of the image signal. The combined thermal and physiological noise apparent in time series are integrated in the measure temporal SNR (tSNR). This is the measure that is ultimately decisive for the analysis of the time series.

3.8 MR Imaging Sequence

The basic ideas of image encoding in k-space were established in Section 3.2, and the information content of the encoding in Sections 3.3 and 3.4. It is now left to introduce the data acquisition sequence, which is an interplay of playing out (i) RF pulses for generation of transverse magnetization, and (ii) field gradients for optimal signal read-out. To gather enough image information, in most MRI scans a high number of excitations/read- outs is necessary. In a typical MRI scan, however, there is not enough time to wait for complete relaxation of the spin ensemble in between two RF excitations, and read-out is not possible during the whole length of relaxation due to SNR considerations. The repeated re-excitation of a non-relaxed spin ensemble can create complex phase shapes of ρ⊥(r) due to spin-history effects. The general challenge of sequence design is finding the best compromise of: • Sufficient transverse magnetization during read-out (maximize SNR) • Adequate relaxation-weighting of the signal for the imaging task (imaging specificity) • Benign effects of time dependencies that cannot be adequately handled by the image reconstruction framework, see Section 3.5 (minimize image artifacts) • Shortest possible scan-time (patient comfort, motion artifact reduction, depiction of fast processes, and minimization of operating costs) • Non-detrimental spin-history23 effects (minimize image artifacts and maximize signal-

23A speciﬁc type of gradients, the so-called crusher gradients, that are involved in selecting diﬀerent spin- MR Imaging Sequence 41

to-noise ratio) Only a small aspect of these points can be covered in this thesis. Being focused on BOLD- ∗ fMRI (see Section 3.9), a fast T2 -weighed read-out was used for this thesis, and within the scope of this instance the above points are discussed in the following. Before the imaging sequence that was used for this thesis is introduced, its building blocks and main aspects are explained in the temporal order in which they are applied in the sequence.

The time between two RF excitations is commonly referred to as repetition time TR, and the time at which the k-space center is acquired in a gradient-echo sequence is referred to as 24 echo time TE.

3.8.1 Spectral Saturation

The great majority of 1H-nuclei that are excited in MRI are bound to either fat or water. Fat- and water-bound protons have slightly different Larmor frequencies due to the different mean molecular fields they are exposed to. This effect is called chemical shift. The chemical shift between fat and water is about ∼143 Hz T−1. In MRI of the head at 3 T, even when considering the main field inhomogeneities, the spectral histogram of δω0(r) without applied gradients shows two distinguishable components centered around the water and fat Larmor frequencies.25 The problem arising from the chemical shift is that fat and water signal become mixed during the acquisition, and the reconstruction suffers from (trajectory-dependent) misregistration as outlined in Section 3.5. Moreover, fat has a shorter T1, meaning faster signal recovery, which can lead to relatively enhanced fat signal in short TR measurements. With the result from Equation (2.32) from the last chapter, Section 2.2.6, one can excite the spin ensemble with a spectrally selective RF pulse that excites only the fat frequencies in the volume that is covered by the RF coil. For fat saturation, this pulse is followed by spin dephasing with a large enough gradient moment. When done immediately before the next excitation, the following excitation will lead to coherent tipping of only the remaining water-bound protons. One limitation of this method is the frequency overlap between fat and water protons in volumes with highly inhomogeneous main field.

history pathways are not necessary in the applied imaging sequence, and are out of scope of this thesis. 24 The nature of the spin echo is not discussed in this thesis. The time of the spin echo is also called TE . 25 The δω0(r)-histogram always depends on the subject and imaged area, and especially on the shim gradients, which are a speciﬁc gradient system for the compensation of main ﬁeld inhomogeneities. 42 Magnetic Resonance Imaging

3.8.2 Volume-Selective RF Pulses

A very important concept in MRI is the slice- or slab-selective excitation. This type of excitation gives means of precise confinement of the signal origin, and as such enables efficient encoding strategies. Again, using Equation (2.32) from Section 2.2.6, the frequency selective RF pulse can be tranformed into a spatially selective one by applying a gradient during the excitation: a field gradient along one direction leads to each Larmor frequency being exhibited (approximately) on a plane that lies perpendicular to the gradient.

3.8.3 Steady-State Magnetization

When the repetition time TR that separates two excitations becomes smaller than T1, the still relaxing z-magnetization Mz that is tipped again is smaller than in the thermal equilibrium. In consequence, the transverse magnetization after the excitation is smaller, too. It is assumed here that no transverse magnetization is left before re-excitation (see next Section 3.8.4).

Under the assumption of constant ﬂip angle and TR, after a number of excitations the regrowth of Mz within one period TR counterbalances exactly the loss of Mz due to the excitation. This state is called a steady state, as the spin ensemble is found in identical state after/before each excitation. The absence of transverse magnetization before the re-excitation leads to a so-called incoherent steady state. The discussion of coherent steady states, which result when transverse magnetization remains before excitation, are out of the scope of this thesis. With these prerequisites, the steady state magnetization evolution ρ¯ss after the excitation with ﬂip angle α can be derived30 as

−TR/T1 ss 1 − e −t/T ∗ ρ¯ (α, t) =ρ ¯sin α e 2 (3.60) 1 − e−TR/T1 cos α with t ∈ (0,TR). The angle αE that maximizes this expression is called the Ernst angle, and it is found by minimizing the derivative of (3.60):

−TR/T1 αE = arccos(e ). (3.61)

To determine the magnitude and weighting of the steady state using the Ernst angle, αE is inserted in the weighting term of Equation (3.60), evaluated right after the excitation:

!1/2 1 − e−TR/T1 1 − e−TR/T1 1 − e−TR/T1 sin α = = , (3.62) E −T /T −2T /T 1/2 −T /T 1 − e R 1 cos αE (1 − e R 1 ) 1 + e R 1 √ 2 using sin(arccos(x)) = 1 − x . For small ratios of TR/T1 the T1-weighting as well as the magnitude vanishes, while for larger ratios the magnitude becomes attenuated in dependence MR Imaging Sequence 43

of T1.

3.8.4 Gradient Spoiling

When TR is in the range of magnitude of T2 or smaller, the remaining transverse magnetization at the end of the read-out can be dephased by a large gradient spoiler, right before the next sequence cycle, to produce an incoherent steady state. The direction of the gradient is not important, as long as the same direction is chosen before every excitation to avoid partial refocusing of not yet decayed fractions of the re-excited transverse magnetization. It was shown52 for gradient-echo fMRI sequences, that a large gradient spoiler, of as much as 10 times the gradient moment that is necessary for 2π-dephasing of a voxel, is necessary to attenuate all time series fluctuation that is induced by main field fluctuation and spin history.

3.8.5 Gradient-Echo (fMRI) Sequence

The sequence diagram and k-space trajectory of a generic gradient-echo (GRE) sequence as can be used for anatomical imaging is depicted in Figure 3.1. The GRE-fMRI sequence as used in this thesis and the resulting single-shot trajectory29 from the shown diagram are depicted in Figures 3.2 and 3.3, respectively. The sequences consists of 4 elements, which are color-coded along the time axis in the ﬁgures: • Fat saturation (cyan): transversally dephase all fat-bound protons after a spectrally selective excitation by 90◦ • RF excitation (red): excite a slab of the (remaining) water-bound protons with ﬂip angle α • Read-out (blue): apply gradient encoding during the relaxation of the spin ensemble, and record signal • Gradient spoiling (green): completely dephase the remaining transverse relaxation with large gradient moments before the next excitation ⇒ Repeat until enough k-space information is gathered.

While the GRE sequence shown in Figure 3.1 leads to only minimal image distortion in the x-direction due to its short read-outs (see Section 3.5), one excitation per k-space line leads to considerably higher total acquisition time than applying a gradient navigation through greater portions of k-space after each excitation. The sequence shown in Figure 3.2, as used in parts of this thesis, represents the extreme case of 3D single-shot acquisition after only one excitation. Single-shot acquisition can also be performed using Cartesian trajectories, such as in applications using echo planar imaging (slice-wise, 2D) or echo volumar imaging (3D). 44 Magnetic Resonance Imaging

Figure 3.1: The sequence diagram of a generic GRE sequence with read-out of one k-space line per excitation is shown on the left. A single repetition interval is shown. The gray and black gradient forms in the Gy-diagram represent diﬀerent repetition cycles of the sequence, corresponding to diﬀerent lines in k-space. This example is a 2D sequence, where z-gradients are only applied for slice selection (negative lobe) and return to kz = 0 (positive lobe). No fat saturation pulse is executed in this example. On the right side, the integrated gradients, i.e., the trajectory, in the kx-ky-projection are shown. Blue points schematically represent the sampled k-space points (in the k-space diagram on the right) or the corresponding read-out times (on the time axis on the left). The time axis of the sequence diagram also shows time of excitation (red) and gradient spoiling in the kx-direction (green). Time is measured from the center of the excitation pulse, and the shown sequence has TE = 4.1 ms. The sequence parameters were chosen for demonstration purpose only, and not for generation of a clinically relevant image.

The drawback of a long trajectory after the RF pulse is the accumulation of signal model inconsistencies with time, as described in Section 3.5, and the image distortions and artifacts that arise in consequence thereof.26 As shown before,29 the non-Cartesian stack-of-spirals trajectory leads to blurring and voxel signal shift in the presence of off-resonance, but is a highly efficient read-out scheme. In comparison with other highly efficient non-Cartesian trajectories,27, 28 it also reduces signal drop-out induced by off-resonance gradients.

The TR values used in this thesis lie in the range of 80 to 100 ms, which introduces some T1-weighting (see Section 3.8.3), considering the T1-values in the range of 800 to 1600 ms 53 at 3 T in white and gray matter, respectively. The T1-weighting appears in the value of transverse magnetization density ρ in the signal equation, Equation (3.11). However,

considering the used values of TR being a fraction of T1-values in the brain, according to ∗ Equation (3.11) the sequence imposes primarily T2 -weighting.

26Also, SNR becomes worse at later times of the read-out due to the signal decay. However, sequence SNR is a complex topic that depends on many parameters, and cannot be fully discussed here. MR Imaging Sequence 45

Figure 3.2: Sequence diagram of a 3D single-shot GRE-fMRI sequence with a spherical stack-of- spirals trajectory, as used in Chapter 6. A slab of 150 mm is excited with a ﬂip angle of 5◦ for demonstration purpose. The four building blocks are marked by diﬀerent colors: fat saturation (cyan), RF excitation (red), read-out (blue), and gradient spoiling (green). The resulting trajectory is shown in Figure 3.3

Figure 3.3: The integration of the gradients shown in the sequence diagram in Figure 3.2 leads to the depicted spherical stack-of-spirals trajectory. The time after the excitation is color-coded along the trajectory.

3.8.6 Physical Constraints

The gradient system consists of 3 coils for field gradients in the x-, y-, and z-direction in standard contemporary MRI systems, besides the gradient power supply. Due to physical limitations and life cycle management of the gradient system, the maximum gradient amplitude and its temporal derivative, the slew rate, are subject to vendor-provided hard limits. Both maximum gradient amplitude and slew rate are crucial parameters, especially 46 Magnetic Resonance Imaging for trajectory design. Constraints of the RF system, i.e., maximum field strength, do not play an essential role in the used sequence. In sequences that are dominated by gradient variation, such as the shown stack-of-spirals GRE sequence in Figure 3.2, the peripheral nerve stimulation induced in the subject by strong gradient variation is the limiting factor for read-out speed. The nerve stimulation is dependent on the gradient field shapes, and the scanned subject’s condition and position,54 and manifests itself in muscle twitching.

3.9 Functional MRI

While not all findings of this thesis are restricted to it, the main application that was investigated and is referred to in Chapters 5 to 7 is functional MRI. Functional MRI aims at mapping physiological function of the brain, with the use of fast image time series acquisition. When an imaged areal is subject to changes in neuronal activation and subsequent oxygen consumption, the blood oxygen level in that areal is influenced, which consequently appears as intensity fluctuation in the fMRI time series. This is called the blood oxygen level dependent (BOLD) effect, and was first shown in vivo in 1990.24 ∗ The concept bases upon the fact that T2 -changes are an intrinsic marker for changes of blood oxygenation. While the oxygen-enriched oxyhemoglobin in blood is diamagnetic, the oxygen-depleted desoxyhemoglobin is paramagnetic due to its 4 unpaired electrons. Moreover, the dipole-dipole interactions of both molecules show different strength. The latter leads to a change in T2 depending on the (des-)oxyhemoglobin concentration of the blood in the imaged ∗ tissue, and both effects combined lead to an even stronger change in T2 as the change of magnetic property of the hemoglobin molecule promotes changes in local field inhomogeneity with positive addition of both effects. ∗ While the hemodynamic response of T2 values is confined to smaller vessels than that of T2 values,55 this effect is dependent on field strength and shows especially beyond 3 T.56 This ∗ relationship justifies the generally more time-efficient T2 -weighted sequences being the most common choice for fMRI at fields up to 3 T.

3.9.1 Hemodynamic Response Function

Right after neuronal activation and the onset of increased oxygen consumption of a group of neurons, the body reacts with blood vessel dilatation in the activated area in order to supply the activated region with oxygen and glucose. The hemodynamic response function (HRF), shown on the left side of Figure 3.4, models the linear time-invariant response of the fMRI signal to neuronal activation. The vessel dilatation leads to a supercompensation of the actual consumption of oxygen, which then leads to a temporally dispersed rise in signal. It is important to note that the real HRF is dependent on the imaged area and the subject.57 Functional MRI 47

∗ =

Figure 3.4: The shape of the hemodynamic response to a binary activation model is given by the convolution of the HRF with the activation model. The ﬁgure shows a schematic representation of the process with mathematical operations in red.

Figure 3.4 shows the modeling process of the hemodynamic response to an activation pattern which is assumed to be known and binary. This model can be used for the statistical detection of previously unknown locations with known activation periods.

3.9.2 Functional Connectivity

The brain is a neuronal network of physically and functionally connected regions. The physical network is highly complex, and made up of several billion neurons, which form the network nodes at the lowest level. Its function is mostly modeled by a hierarchical approach, in which a small number, i.e., hundreds to thousands, of nodes form a functional entity, which then again forms a node in the next higher layer of the model. At the macroscopic level, the brain consists of a few hundred regions58–60 that can be related to integral functions, e.g. sensory processing, motoric control, emotion, memory, behavior, etc. One important application of fMRI is researching the functional connections of the macroscopic brain network. The so-called resting state networks of the brain, i.e., the spontaneously shown functional connectivity at rest, has aroused a great deal of research interest61 since its discovery.62 One common approach to identify functional connectivity from fMRI time series is the calculation of Pearson correlation coefficients between the resting state time series from specifically chosen seeds, i.e., voxels or contiguous regions, and the rest of the brain. In other words, the correlations of spontaneous BOLD fluctuations which are apparent in the time series of the brain are assumed to contain relevant information about the functional connectivity. 48 Magnetic Resonance Imaging

3.9.3 Advanced fMRI Techniques

Segments of this subsection originate from the ﬁrst-author-publications "Trading oﬀ spatio- temporal properties in 3D high-speed fMRI using interleaved stack-of-spirals trajectories"63 and "Targeted Partial Reconstruction for Real-Time fMRI with Arbitrary Trajectories".64

High-speed fMRI

The method that was used in this thesis is a GRE acquisition, which is the most common choice for fMRI in neuroscience at field strengths up to 3 T. The trade-off between spatial and ∗ temporal resolution is inherent to any time-series acquisition in MR, and T2 -weighed GRE- fMRI is used due to its efficient data sampling and high BOLD-signal contrast. Although

T2-weighed SE- or bSSFP-fMRI show more localized BOLD-signal response and pronounced micro-vascular specificity,55 this effect is mainly observed at field strengths higher than 3 T.56 In fMRI, the selection of the spatial resolution, typically ranging from less than 1 mm up to 5 mm, depends on the focus of the performed study. The investigation of cortical columns, e.g., requires the smallest possible voxel size, realized in localized portions of the cortical ribbon at a high field strength in combination with reduced field-of-view techniques and tailored gradient and receive coils.65 For the detection and classification of macroscopic functional network structure in the brain, a resolution of several millimeters in each dimension is widely accepted. Although the cortical ribbon has a mean thickness of around 3 mm,66 the number of functionally specialized brain regions are not more than a few hundred parcels,58–60 where each parcel has a size of the order of a centimeter or more. Macroscopic whole brain fMRI is a method of great interest in the field of neuroscience, used for the analysis of resting-state networks, and an important tool in clinical research. It has been proven useful for investigation of functional architecture of the healthy brain67 and is being used in various clinical applications, especially in neurological and psychiatric disorders such as Alzheimer’s disease,68 schizophrenia,69 depression70, 71 and bipolar disorders.72, 73 While the intrinsic temporal resolution of fMRI is limited by the width and delay (both around 5-6 s) of the hemodynamic response to neuronal activation, a high temporal resolution can enable un-aliasing and correction of physiological fluctuations, allow for the characterization of subject- and ROI-dependent HRF waveforms, and leads to increased sensitivity for mapping task-based activation and functional connectivity as well as for detecting dynamic changes in connectivity over time.74–77 The larger number of data points also holds the potential for faster and/or more accurate detection of resting-state networks.78, 79 Moreover, BOLD signal changes have been shown to contain measureable fluctuations at frequencies up to 5 Hz, which are not accessible by conventional whole brain EPI.80–84 A reduction of sampling time without penalty in spatial resolution is achieved by incorporating parallel imaging techniques and efficient sampling patterns, leading to temporal resolutions from below 100 to several hundred milliseconds. In the range of high-speed methods that aim to not compromise the point spread function, the simultaneous multi-slice (SMS) EPI approach has led to multiple times the temporal resolution at EPI-comparable image quality,74 introducing increased but manageable g-factor noise.85 Acceleration in SMS-EPI Functional MRI 49 is limited by slice leakage artifacts and subsequent false-positive activation,86 where the use of split-slice GRAPPA87 has helped to reduce the former. A segmented multi-shot approach to 3-dimensional Cartesian readouts with high flexibility in the acquisition and acceleration schemes at the cost of increased physiological noise is 3D EPI.88, 89 3-dimensional Cartesian single-shot readouts of one or multiple slabs per brain have been realized through echo volumar imaging,75, 90, 91 which was shown to enhance BOLD sensitivity compared to EPI but offers less spatial resolution than SMS-EPI. High-speed methods achieving sub-100 ms resolution incorporate strong variable density sampling in non-Cartesian read-outs, i.e., MREG, as described above, or omitting Fourier encoding in one or two dimensions entirely and letting the point spread function be determined only by the receiver coil configuration in Inverse Imaging (InI).92, 93

Real-Time fMRI

The ultimate purpose of the image reconstruction method presented in Chapter 7 is the application of MREG trajectories in real-time feedback fMRI (rtfMRI). Examples of potential applications of rtfMRI include neurofeedback, providing direct feedback information from the activity of targeted brain regions, networks, or other physiological measures such as connectivity,94–98 and include the alteration of brain function and behavior via voluntary self-regulation for treatment of stroke,99 depression,100 pain,101 and addiction102, 103 amongst others. "The potential of MR-Encephalography for BCI/Neurofeedback applications with high temporal resolution" was investigated in a co-authored publication by Lührs et al.,104 using the concept of the presented reconstruction method applied to single-shot MREG in comparison with SMS-EPI. The SENSE formalism and its derivatives are well-established methods,37, 105 albeit computationally expensive. Amongst previously existing reconstruction methods in high-speed fMRI, Geometric-decomposition Coil Compression for Cartesian sampling106 in combination with SMS-EPI has been shown to reduce the reconstruction time of fMRI datasets to marginally longer than the scan time without a signiﬁcant SNR penalty.107 However, no frame-by-frame real-time feedback capability was shown. A simpliﬁcation of the signal equation to a low number of arbitrarily shaped voxels representing the regions of interest,108–110 enables the direct inversion of the forward operator and fast evaluation, but can introduce leakage artifacts.111

4 Own Contributions

This thesis extends the line of research around the 3D fMRI method MREG, and aims at enabling and demonstrating the ﬂexibility of the method beyond the capabilities of previously published fMRI methods. The ﬁrst part, consisting of Chapters 5 and 6, focuses on understanding and enhancing the acquisition scheme of MREG, while the second part, Chapter 7, aims at the application of MREG trajectories in real-time feedback fMRI (rtfMRI).

While the latest MREG single-shot patterns were exploring minimum feasible TR at common fMRI resolution, i.e., nominal 3-4 mm isotropic, and opened up new fields of investigation,80, 112 the method’s variable density sampling artifacts - most notably blurriness and regionally strong signal drop-out in the presence of strong off-resonance gradients - can be an application-limiting factor. The stack-of-spirals trajectory already alleviates the signal drop-out when compared to the concentric-shells trajectory,29 and translates off-resonance into signal shift rather than signal attenuation due to its linear read-out along the slowest encoding direction (see content of the last chapter, Section 3.5, for explanation). However, Chapter 5 investigates the consequences of variable density sampling in long read-outs, which, as shown, can be the source of additional artifacts, which add up to the effects that are described in Section 3.5. Chapter 5 represents groundwork for Chapter 6, in which sampling density parameter sets are identified that let the user trade-off between ultra-high temporal resolution and spatial signal quality in 3D high-speed fMRI using a stack-of-spirals trajectory. Chapter 6 represents the results of the first-author publication "Trading off spatio-temporal properties in 3D high-speed fMRI using interleaved stack-of-spirals trajectories".63 It is aimed at broadening the bandwidth of applications of 3-dimensional spherical read-outs, identifying the best possible compromise of image quality and tSNR per unit time, rather than pushing the limits of temporal and spatial resolution. This was done by choosing an approach that closes the gap of used parameters in between conventional high-speed fMRI and the previously published ultra-fast methods operating at sub-100 ms repetition times. It is shown that the investigated and in fMRI previously unexplored parameter region of mild variable density sampling leads to excellent tSNR maps in conjunction with no visible decay of image quality. The necessary flexibility to achieve this is gained by multi-shot acquisition in an interleaved manner that minimizes intra-frame signal model inconsistencies. The method proposed in Chapter 6 maintains the synergies of a temporal resolution that enables direct filtering of physiological artifacts for the highest statistical power, and three-dimensional read-outs with optimal use of encoding capabilities of multi-coil arrays for efficient sampling and high SNR. 52 Own Contributions

Chapter 7 represents the results of the first-author publication "Targeted Partial Recon- struction for Real-Time fMRI with Arbitrary Trajectories".64 It presents the Targeted Partial Reconstruction applied as a partial SENSE method with MREG acquisition. The trajectories presented in Chapter 6 necessitate a SENSE-type image reconstruction, where a conventional SENSE reconstruction renders the extraction of localized information from the raw data in real-time applications, i.e., in less than TR, unfeasible. For whole-brain fMRI, the computational complexity of the inverse problem calls for computation clusters, as even the reconstruction of one of several thousand time frames requires in the order of minutes on a single core of a contemporary CPU. The Targeted Partial Reconstruction generates minimal reconstructions, i.e., mean or reduced-FOV reconstructions, over one or several freely selected target regions. A similar idea was previously presented for back-projection reconstructions in computed tomography.113 This provides a tool for reconstructing exactly the information that is necessary, and is useful during experiments using sampling patterns that require more time for full image reconstruction than the experiment allows for the generation of feedback. Target regions could be defined, e.g., on the basis of the subject’s anatomy or previous functional scans, or chosen from a pre-defined brain atlas. One or more reconstruction vectors are computed per target prior to the experiment, allowing the reconstruction of the signal from the target region as one or more scalar products, i.e., a straightforward matrix operation. Computation time is then sufficiently fast for the real-time reconstruction of several regions even at very short repetition times (below 100 ms), enabling a kind of ultra-fast fMRI feedback experiments that have not been conducted like this before. While the aim of the development presented in Chapter 7 is the use in fast real-time feedback fMRI, it is generally applicable to any scenario requiring a fast real-time target reconstruction or fast reduced-FOV reconstruction, independent of the type of trajectory. It should also be noted here that only the reconstruction framework is modified while the data acquisition can remain unchanged. As such, all recorded information still remains available for full post-experimental analysis based on offline reconstruction, if desired. 5 Investigation of Off-Resonance-Gradient Vulnerability due to Variable-Density Sampling

Contents

5.1 Motivation ...... 54 5.2 Theory ...... 54 5.2.1 Oﬀ-Resonance ...... 54 5.2.2 Variable-Density Sampling ...... 55 5.3 Methods ...... 57 5.3.1 Oﬀ-resonance Model ...... 57 5.3.2 Trajectories ...... 58 5.3.3 Simulation Model and Image Reconstruction ...... 59 5.4 Results ...... 60 5.5 Discussion ...... 63 5.6 Conclusion ...... 64 54 Investigation of Off-Resonance-Gradient Vulnerability

5.1 Motivation

Previous trajectories for 3D whole-brain fMRI developed by our group have provided ultra- fast acquisition at the expense of image quality using variable-density sampling (VDS) in combination with highly undersampled sensitivity-encoded read-outs. While achieving sampling rates below 100 ms per frame, those single-shot trajectories lead to signal drop-out, blurring, and undersampling artifacts in areas with strong magnetic susceptibility effects. Major artifact sources of those trajectories are related to non-equidistant sampling in VDS, off-resonance vulnerability due to very long read-out times, and - as shown in this chapter - the combination of these. The stack-of-spirals trajectory has been shown to be a highly efficient 3D read-out pattern with favorable properties in the presence of off-resonances,29 as it combines the gradient- encoding efficiency of spirals with the robustness of the point spread function of Cartesian imaging. In multi-dimensional trajectories, the point spread function in the presence of main field inhomogeneity is most heavily influenced by the slowest component of the read-out, which is a linear progression along the Cartesian kz-direction in the stack of spirals. In contrast, radially slow in-out or out-in schemes like spirals in 2D or concentric shells in 3D lead to blurring and signal loss. Consequently, this thesis aims at further optimizing high-speed 3D fMRI using the stack-of- spirals trajectory scheme. For a better understanding of the multitude of complex imaging properties that are affected by deviations from the idealized k-space imaging concept, this chapter is restricted to simple 1D simulations that demonstrate an effect that has not been well-described previously. The simulations emulate the 2 encoding schemes on different time- scales within the stack-of-spirals trajectory, namely the sampling along the stack dimension and the spiral-sampling scheme in the kx-ky-plane. The simulations are implemented by linear and in-out k-space traversal in 1D, respectively.

5.2 Theory

5.2.1 Oﬀ-Resonance

The consequences of field inhomogeneities ωor(r) on the global and local signal encoding were formalized in Chapter 3.2.2. To summarize the effects of Equation (3.15): field inhomogeneities affect the global encoding terms via e−iωor(rn)t, the local gradients of the field inhomogeneities induce a locally dependent k-space, and the transverse decay is influenced by macroscopic field structure. The values of the macroscopic field structure, in the following referred to as absolute off-resonance, and the off-resonance gradients are the source of different effects, and need to be carefully discriminated. Off-resonance effects are highly dependent on the used read-out trajectory, and the effects of absolute off-resonance have been described extensively in the literature,114 and are formally Theory 55 described in Section 3.5. Briefly, for the trajectory patterns used in this work: (1) In linear k-space traversal, such as used in echo planar imaging or in the stack-dimension of the stack-of-spirals trajectory, the off-resonance term adds an additional phase to the data at each line or element of the stack, and thus induces a shift in image space that can be understood with the Fourier shift theorem. While the shift theorem is applicable in situations of monotonous read-out speed/phase accrual in k-space, a varying read-out speed can lead to the same effect in approximation. (2) In radial in-out or out-in k-space traversal, which is in 2D most efficiently realized by spirals, the additional phase accrual along the radial direction and the resulting phase inconsistencies in k-space and lead to signal attenuation and blurring. The effects of signal decay are also dependent on the read-out pattern, but in the used trajectories mainly result in blurring in addition to the decay-related signal attenuation in an idealized measurement. In the presence of off-resonances, the trajectory distortion leads to altered effective echo times.114 Lastly, the effects of local k-space (see Section 3.2.2),

kn(t) ≡ k(t) + ∇ωor(rn) · t, (5.1) will be investigated in this chapter. It is important to note that local k-space is not apparent in point spread function simulations with an oﬀ-resonance model, as the term e−ikn(t)·(r−rn) in Equation (3.15) vanishes when ρ⊥(r) = δ(r − rn) is assumed.

5.2.2 Variable-Density Sampling

The concept of variable-density sampling (VDS) describes k-space location-dependent sampling density. For this concept it is assumed that in MR images the power spectral density is roughly radially symmetric and concentrated near the k-space origin,115 so that high sampling density around the k-space origin is assumed to have the most decisive impact on image quality. VDS thus enables to flexibly allocate scan time based on the signal characteristics in time critical applications. For example, previous MREG methods made use of pronounced VDS27–29 to achieve whole brain fMRI at a nominal resolution of 3 mm with TR below 100 ms. The following simulations in this subsection recapitulate the properties of the previously published method of VDS.115 Figure 5.1 depicts point spread functions in 1D with different degrees of linearly varying undersampling factors, and identical extent of k-space sampled. The point spread functions are calculated over 3 times the encoded FOV in order to depict the aliasing behavior of signals close to the image edges.1 This was done by increasing the dimension of the iDFT and applying only every third line of the iDFT matrix to the simulated signal of a point source. The sampling densities in this example range from just fulfilling the Nyquist criterion at the k-space center, corresponding to an undersampling

1Only 2 times the FOV is necessary to depict the aliasing behavior of signals close to the image edges. The shown simulations are consistent with the representation of a PSF in the literature. 56 Investigation of Off-Resonance-Gradient Vulnerability

Figure 5.1: Point spread functions of full sampling (subfigure (a)) and VDS with undersampling factors linearly increasing from 1 at the k-space center to 1.25 (subfigure (b)) and 2 (subfigure (c)) at the periphery. The reconstruction spans 3 times the encoded FOV to depict the aliasing behavior of signals close to the image edges. The signal was simulated on a grid 5 times finer than the reconstruction grid, with the simulated Kronecker-delta lying in the middle of two reconstruction points. The resulting leakage effect accounts for the fact that spin ensembles are not confined to the reconstruction grid. This approach shows essentially the envelope of the continuous PSF. VDS leads to a varying FOV, yielding an aliasing pattern that is incoherently spread out in the range of the nominal FOVs corresponding to the used sampling densities.

factor of R = 1, to mild undersampling and strong undersampling in the periphery, i.e., R = 1.25 and R = 2 respectively. The case of strongly varying sampling density is similar to the parameter range, i.e., a factor of 2, used in the in-plane dimension of the MREG single-shot stack-of-spirals trajectory. The point spread functions show an unchanged width of the virtually imaged point due to the identical extent of k-space sampled. However, the nominal FOV, which is defined by the sampling density and manifests itself in the distance between aliased points, is only apparent in the fully sampled case. VDS leads to a variable FOV, yielding an aliasing pattern that is incoherently spread out in the range of the nominal FOVs corresponding to the used sampling densities. The simulation in Figure 5.2 gives a more comprehensive view of the consequences of VDS in an idealized environment without off-resonance. As the point spread functions suggest, small objects whose extent falls short of the minimal FOV are depicted with no to only negligible degradation, i.e., Gibbs ringing-like artifacts for the strongly variable case. The reconstruction of a large object that covers the FOV, however, suffers from high frequency artifacts that increase in intensity with increasing distance from the image center, as high-frequency portions from one side of the image fold into the other side. This behavior attests that - in an idealized measurement - mild VDS induces small noise-like artifacts at the edges of the image, due to the sampling density being locally reduced beyond the Nyquist criterion in regions of k-space containing high-frequency information. The intensity and spatial extent of these artifacts increase with stronger sampling variation, and the degree of VDS needs to be adjusted according to the accepted level of noise-like artifacts and the relevance of low-intensity high-frequency details. The artifact appearance and the resulting image degradation can thus be traded off against a decrease in acquisition time, according to the requirements of the application of interest. Methods 57

Figure 5.2: Examples of test objects/ground truths imaged with full sampling, and mild and strong VDS with parameters as in Figure 5.1. The reconstructions of the small rectangular feature in the left column span 3 times the encoded FOV, similar to PSFs, to show the aliasing behavior of features close to the image edges. When applied to the FOV-covering object in the right column, VDS leads to image degradation in areas close to the image edges. The image degradation increases with the degree of VDS and the distance from the image center.

5.3 Methods

5.3.1 Oﬀ-resonance Model

Imaging simulations were performed using the same ground truths as in Figure 5.2, which are shown together with the modeled off-resonance maps in Figure 5.3. These objects were chosen to: (1) Reveal isolated effects of off-resonance gradients on small portions of an image using 58 Investigation of Off-Resonance-Gradient Vulnerability

a rectangle shape as ground truth, as shown on the left of Figures 5.2 and 5.3. The modeled off-resonance consists of a constant gradient in the region containing the rectangle. (2) Illustrate the effects of a more complex off-resonance map with varying gradients over most of the FOV, using the instance on the right of Figure 5.3. The off-resonance map exhibits two features: a broad dip over the FOV, which is also sometimes seen in human heads depending on the shim; and a strong rise on one side of the FOV which distorts the trajectory enough to lead to signal drop-out, similar to orbitofrontal areas around the sinuses in human heads.

Figure 5.3: Each column depicts a test object/ground truth for the simulation of oﬀ-resonance- induced artifacts in 1D, with corresponding oﬀ-resonance and -gradient maps.

The off-resonance (gradient) intensity was chosen to provoke the effects that are observed in the later used trajectory, i.e., according to the simulated image resolution. The validity of the off-resonance values in conjunction with the trajectory is discussed in Section 5.5

5.3.2 Trajectories

With stack-of-spirals trajectories in mind, the 1D simulation was performed with 2 types of read-out patterns: linear k-space traversal from −|kmax| to |kmax|, simulating the in-stack direction, and in-out traversal from 0 to ±|kmax|, simulating a 1D spiral. Read-out times were chosen according to stack-of-spirals analogy: 75 ms for the linear traversal in the kz-direction, and 5 ms for the in-out pattern, with TE = 35 ms like the innermost spiral in the single-shot Methods 59 whole-brain trajectory.29 As already shown in Section 5.2.2, this chapter compares 3 sampling density schemes: constant Nyquist-conform sampling, and VDS with linearly increasing undersampling factors R ∈ [1, 1.25], and R ∈ [1, 2]. VDS was calculated in the following way: starting at the k-space origin, the distance from the sampling point kn to the next sampling point kn+1 in units of Nyquist-conform spacing was calculated by |kn|/|kmax| · Rmax. To isolate the effect of VDS in the presence of off-resonance, the timing of the trajectory samples is allocated k-space location-dependent in identical manner for all trajectories, i.e., all trajectories have the same duration, and VDS trajectories exhibit varying sampling speed along k-space. Effects of varying sampling speed along the trajectory, like apparent in spherical stacks of spirals along the kz-direction, are not investigated in this chapter. The trajectories were adjusted for a resolution of 256 image points, i.e., 0.75 mm at a hypothetical FOV of 192 mm. This results in a resolution that is 4 times higher than in one dimension of the later presented 3D fMRI trajectories. Mind that, at identical read-out time of 75 ms, the simulated k-space traversal speed is consequently 4 times higher than in the single-shot stack of spirals, and off-resonance gradients of 4 times the magnitude result in the same effects.

5.3.3 Simulation Model and Image Reconstruction

To mimic the behavior of an MRI reconstruction, i.e., appearance of leakage artifacts, Gibbs ringing, and intra-voxel dephasing, the reconstruction is performed on a coarser grid than the simulated measurement and the corresponding ground truth and oﬀ-resonance maps. In this chapter, the grid sizes diﬀer by a factor of 5. Analoguous to Section 3.4.1, the measurement operator E of a simulated measurement

s = Eρgt, (5.2) with ground truth ρgt and simulated signal s, was an explicit matrix created in MatLab, reading 2π E = exp(−i k˜ m + iω t ), (5.3) nm N n m n with ground truth index m ∈ [1,N], ground truth dimension N, trajectory index n, and kñ = kn · FOV . Due to the 1D reduction, an explicit matrix formation was computationally sufficiently efficient. Image reconstruction was performed by solving

Er∗s = Er∗Erρr, (5.4) for the reconstructed image ρr using a conjugate gradient algorithm. The matrix Er was formed by extracting every 5th column of E, representing the grid size reduction of the reconstruction. The reconstructions were Tikhonov-regularized with a regularization parameter of 20, to ensure convergence of the solution. The above process is called the 60 Investigation of Off-Resonance-Gradient Vulnerability

"off-resonance corrected" reconstruction, while the reconstruction without correction is formed from the matrix without an off-resonance term: 2π E0 = exp(−i k˜ m). (5.5) nm N n This reconstruction is used to show the effects of non-corrected off-resonances, which represents a more realistic scenario in the context of MR imaging. Signal decay during read-out was not considered in this chapter, since the effects resulting from decay, i.e., blurring and signal attenuation, have been extensively described in the literature116 and no new insight is expected from the analysis of signal decay in the VDS context.

5.4 Results

Figure 5.4 shows the alteration of trajectories due to the presence of an off-resonance gradient, specifically the off-resonance gradient shown in the left column of Figure 5.3. From the trajectory deviation described by Equation (5.1) it follows that, in a linear sampling scheme, the k-space coverage of the trajectory is compressed/stretched by the same factor by which the sampling densities are in-/decreased. The in-out sampling scheme can be understood as two simultaneous linear schemes in both k-space directions, starting from the same point, which leads to compression of one trajectory arm and stretching of theother. As both shown trajectories have similar TE, their idealized k-space origin is shifted by a similar amount.

Figure 5.4: Subfigure (a) shows the timing scheme of both used trajectory types. Subfigures (b) and (c) show the actual undersampling parameters without and with an off-resonance gradient, respectively. The off-resonance gradient used for the calculation of (c) is identical to the one shown in the left column of Figure 5.3, i.e., 31.4 mT/m/ms. In (c), the colors correspond to the same trajectories as in (a), and the line styles correspond to the same undersampling factors as in (b). In (b) and (c), the lines only extend up to the actual sampled area of k-space, which, together with the shown undersampling factors, defines the whole trajectory. While the shapes of the distorted trajectories depend on the sampling scheme, the similar TEs, i.e., 37.5 ms vs. 35 ms, lead to an almost identical shift of the ideal k-space center. In (c), the ratios of the sampling density at the true k-space center relative to full sampling are: 0.92 and 0.72 for mild and strong VDS, respectively, in the linear scheme; and 0.94 and 0.79 for the in-out scheme. Results 61

Figure 5.5: Corresponding to the modeled ground truth shown in Figure 5.3, the left column shows reconstructions of a rectangular shape with constant off-resonance in the measurement model, and the right column shows a FOV-covering object with a more complex off-resonance map from Figure 5.3. The gray lines show the reconstruction with full sampling and without modeled off-resonance. Color and linestyle coding correspond to Figure 5.4. The green vertical lines mark multiples of the effective FOV, calculated by multiplying the sampling density ratios determined in Figure 5.4c) with the FOV. Rows 1 and 3 show reconstructions without off-resonance in the reconstruction model, rows 2 and 4 with off-resonance in the reconstruction model. In the presence of off-resonance gradients, the shift of the real k-space center into less densely sampled areas of the trajectory in conjunction with VDS leads to a shift from high-frequency to low frequency aliasing. While off-resonance correction in the reconstruction model corrects for signal shifts, it cannot undo increased signal drop-out and low-frequency aliasing resulting from undersampling. For a full description of effects see text body. 62 Investigation of Off-Resonance-Gradient Vulnerability

Figure 5.5 illustrates how trajectory deviations in the presence of off-resonance gradients (left column) and more complex off-resonance maps (right column) translate into image artifacts, with and without incorporation of off-resonance into the reconstruction model. As pointed out in Section 5.2.1, the effects of off-resonance gradients are not apparent in the point spread function. However, the rectangular object with underlying off-resonance is used in this simulation to represent the effects of a small feature, e.g., voxel-sized, in a continuous object. Reconstructing a small feature with underlying off-resonance gradient leads to 3 observable effects: (1) In the case of the linear sampling scheme, the object width is smaller in the non-corrected reconstruction. This can be understood in 2 manners: (i) The off-resonances induce a location-dependent signal shift leading to - in this case - compression of the object; (ii) The compressed trajectory leads to denser k-space samples and therewith a larger FOV with less resolution. For the in-out scheme, no signal shift is observed. According to Section 5.2.1, blurring is induced by off-resonance in this scheme. This effect is undone by the off-resonance correction. (2) The object shapes change due to the partial k-space sampling, which excludes some object information. In theory, this effect is always present. However, due to the spectral properties of the imaged object and no extreme off-resonance gradient, only the strong VDS case results in a visibly altered shape. Non-acquired information cannot be restored by the reconstruction model, thus this effect is not undone by the off-resonance correction. (3) The aliasing behavior is dependent on the degree of VDS. The change of sampling density at the real k-space center relative to the fully sampled trajectory is shown in Figure 5.4. This leads to a transition from high-frequency to low-frequency VDS artifacts, and thus an effectively reduced FOV. As indicated by green markers in Figure 5.5, the FOV reduction is proportional to the change of sampling density at the k-space center in Figure 5.4. Like the previously described effect, this behavior results from insufficient information and cannot be undone by off-resonance correction. The reconstruction of a large object with complex underlying off-resonance map on the right column of Figure 5.5 leads to 2 observable effects of importance that are not immediately obvious from the left column: (1) Signal drop-out in regions of very high off-resonance gradients. When the trajectory is strongly distorted, the real k-space center is locally not or very sparsely sampled, which leads to signal drop-out. This effect was previously described.29 Additional sampling sparsity due to strong VDS leads to enhancement of this effect, as can be seen in the left portion of the reconstructions. In this particular in-out pattern only strong VDS leads to noticeable signal drop-out. Off-resonance correction undoes signal shifts, but cannot recover non-acquired signal. (2) The VDS-induced reduced FOV of signal that originates from the high off-resonance- gradient-region in the left part of the image folds into the object reconstruction in Discussion 63

the right part of the image. This leads to noticeable image degradation in the case of strong VDS. Mild VDS with a reduction of this effect leads to folding artifacts only at the most outer right part of the image. As described above, this effect cannot be undone by off-resonance correction.

5.5 Discussion

The results demonstrate that the main assumption of VDS, which is the concentration of power spectral density around the k-space center, is corrupted by the trajectory distortions that appear in the presence of off-resonance gradients. Consequently, high-frequency artifacts that were deemed acceptable for a given application can be transformed into low-frequency artifacts, i.e., reduction of effective FOV, which are not acceptable. Also, an increased susceptibility to signal drop-out due to strong VDS was shown. Moreover, the simulations shown in this chapter reveal that the effects of VDS-induced signal degradation cannot be corrected even when an off-resonance map is available. This stands in contrast to the reversibility of effects of absolute off-resonance on the point spread function. While a thorough analysis was made only for one off-resonance gradient value, that value was chosen such that all effects except for signal drop-out are apparent. In general, the occurring effects are highly dependent on the trajectory, off-resonance gradient value, target resolution, and the object itself. The resulting artifacts in the image reconstruction are a consequence of completely omitting parts of k-space, which is equivalent to zero-filling, and regional over- or undersampling of local k-space. As such the general behavior cannot be generally foreseen. A distinction of cases as described before,29 i.e., (i) off-resonance gradient exceeds the mean phase-encoding gradient with reversed polarity, (ii) off-resonance gradient falls below the mean phase-encoding gradient with reversed polarity, and (iii) off-resonance gradient boosts the mean phase-encoding gradient due to same polarity, has not been presented in this chapter; these cases will result in distinct artifact behavior. The image resolution in this chapter was increased by a factor of 4 in comparison to the next chapter, while TE was left the same. The idealized k-space as well as the resulting sampling speed along k-space were consequently increased by the same factor. When compared to the effects in the next chapter using MREG, the parameters used in this chapter lead to mitigation of the trajectory deviations due to off-resonance gradients proportional to the resolution increase. This means that the examples shown in this chapter are equivalently occurring at roughly2 1/4th of the off-resonance gradient value in the next chapter, when the resolution is reduced by a factor of 4. The trajectory timing was set solely according to k-space location in this chapter, in the same manner for all 3 sampling density types in order to isolate the sampling density effects. This results in constant k-space traversal speed for all sampling times. On the other hand, the stack-of-spirals trajectories for fMRI that will be presented in the next chapter exhibit faster

2Not considering varying read-out speed in the next chapter. 64 Investigation of Off-Resonance-Gradient Vulnerability

kz travel speed in the periphery than in the center, which theoretically leads to additional artifacts in the presence of absolute oﬀ-resonance.

5.6 Conclusion

The effect of increased image drop-out due to sampling of a smaller extent of k-space in the presence of off-resonance and the alteration of the effective TE in GRE-fMRI due to off-resonance gradients are well-described phenomena in MRI. However, the influence of VDS in combination with off-resonances has not been analyzed thoroughly to date. The transition from assumed high-frequency to actual low-frequency artifacts in VDS reconstructions and their image-deteriorating consequences suggest that mild VDS schemes should be preferred in the context of long GRE read-outs and strong off-resonance gradients. While the presented VDS scheme with a sampling variation by a factor of 2 seems high, previous MREG sampling schemes used this or higher factors in all sampling dimensions. However, it has to be noted that the properties of a spherically shaped stack-of-spirals trajectory are not separable into 1D-problems. Nevertheless, this chapter aims at a conceptional isolation of effects, and the 3D-implementation follows in the next chapter. 6 Trading Off Spatio-Temporal Properties in 3D High-Speed fMRI Using Interleaved Stack-of-Spirals Trajectories

Segments of this chapter originate from the ﬁrst-author-publication "Trading oﬀ spatio- temporal properties in 3D high-speed fMRI using interleaved stack-of-spirals trajectories".63

Contents

6.1 Motivation ...... 66 6.2 Material and Methods ...... 67 6.2.1 Trajectory Design ...... 67 6.2.2 Data Acquisition ...... 68 6.2.3 Signal Simulation and Image Reconstruction ...... 70 6.2.4 Time Series Processing ...... 70 6.2.5 Thermal Noise Propagation ...... 71 6.2.6 tSNR ...... 71 6.2.7 Functional Characterization ...... 71 6.3 Results ...... 72 6.3.1 Image Quality ...... 72 6.3.2 Thermal Noise Propagation ...... 73 6.3.3 tSNR ...... 76 6.3.4 Functional Characterization ...... 77 6.4 Discussion ...... 80 6.4.1 Image Quality ...... 80 6.4.2 SNR ...... 82 6.4.3 Functional Characterization ...... 83 6.5 Conclusion ...... 83 66 Trading Off Spatio-Temporal Properties in 3D High-Speed fMRI

6.1 Motivation

In fMRI, the defining properties of an imaging sequence are functional sensitivity and specificity (determined by the sequence type, i.e., GRE), spatial fidelity (determined by the trajectory), and tSNR. The analysis methods for the suggested fMRI read-out method in this chapter were chosen accordingly. As outlined in Chapter 5, this chapter addresses the need to improve the method of MREG by enhancement of image quality in the presence of off-resonance gradients. Conceptionally shown in Chapter 5, some aspects of the image degradation are caused by the strong degree of variable-density sampling (VDS) employed in single-shot MREG and cannot be corrected by off-resonance modeling. The advantageous properties of the stack-of-spirals trajectory for the application in long GRE read-outs were already analyzed,32 thus the basic properties of this trajectory are not further discussed. An increase of sampling density, as discussed in Chapter 5, at constant resolution and near-constant fMRI-relevant TE values, is realized by interleaving the stack of spirals in the spiral-plane. Interleaving of spirals is a well-established method, as identical segments of read-out are applied with only a rotation along the z-axis, where the response of the x- and y-gradients is almost identical in contemporary scanners. This leads to approximately identical signal behavior in every shot with spirals in the x-y-plane, providing robust means of increasing sampling density with unchanged imaging behavior. SNR in MRI, in its definition of mean signal to thermal noise ratio, is proportional to voxel volume - in addition to field strength, echo time, flip angle, and sampling density. In fMRI, however, time series fluctuations are primarily influenced by physiological effects,117 especially at larger voxel sizes and high field strength, as physiological signals scale linearly with the voxel signal and consequently dominate over the thermal noise portion. Together with possible partial volume effects118 of the functional signal that has a mean FWHM of 3-4 mm at 3 T,55 tSNR and functional contrast-to-noise ratio relations become more complex. At constant repetition times, it was found that 1.5 mm isotropic voxel size leads to the largest activated volume for a motor task at 3 T.118 However, spatial smoothing to larger voxel sizes attenuates the uncorrelated portions of physiological fluctuations if those are a relevant factor and high resolution is not the main aim of the experiment.119 When the increase of statistical power due to more samples per unit time by decrease of TR using lower spatial resolution is taken into account, though, the acquisitions with lower spatial resolution lead to slightly higher detection power of task-activation. However, the same study observes no significant differences in functional connectivity with lower spatial resolution.120 This chapter focuses on the effect on spatial and temporal properties of the resulting time series when varying the sampling densities in a spherical acquisition at a constant nominal resolution of 3 mm. It should be noted that the point spread function of a spherical 3 mm acquisition has approximately the same FWHM of a 4 mm Cartesian acquisition with less leakage, as previously noted,121 due to the shape of the Fourier transformed k-space sampling window (see Section 3.3.3). Material and Methods 67

6.2 Material and Methods

6.2.1 Trajectory Design

122, 123 The trajectories were generated by smoothly connecting 2D spirals, stacked along the kz- direction, incorporating a slew rate optimization for reduced peripheral nerve stimulation.32 For the sake of eﬃcient connections, inward- and outward-directed spirals were alternated.

The spiral at kz = 0 starts in the center in order to minimize TE. Care was taken that the k-space center was sampled in every shot despite the smooth connections. The spirals have varying diameters such that the stack’s envelope is a sphere with radius |kmax| = 1/(2 · nr), with the nominal resolution nr = 0.003 m. The multi-shot trajectories are interleaved stacks of spirals, where each stack is rotated along the kz-axis by 2π/N in every shot in the N-fold segmented case, as shown schematically in Figure 6.1.

Figure 6.1: Schematic example of an interleaved 2-shot stack-of-spirals trajectory. The number and extent of the spirals in the stack was reduced for visual clarity. This ﬁgure was used in a ﬁrst-author publication.63

Table 6.1 shows the sampling parameters of one interleave for all used trajectories, adjusted to a ﬁeld of view of 192 × 192 × 150 mm3. Calculating a single mean undersampling factor from the number of data points was not considered a meaningful measure for k-space reduction: slew-rate optimized gradients and constant dwell time lead to trajectory-dependent and non-constant oversampling along the spirals’ constantly varying read-out directions.

The spacing in kz-direction of the stack of spirals is increased beyond Nyquist spacing. It is variable in some cases, and varies linearly from Rzsp,min to Rzsp,max. The undersampling factors within a spiral depend on the spiral’s kz position and are calculated from the globally deﬁned spiral undersampling parameters Rr,min and Rr,max: the undersampling factor of the center of a spiral linearly progresses from Rr,min to Rr,max as a function of the kz-position of the spiral on the interval [0, |kmax|]; the undersampling factor in the periphery of a spiral 68 Trading Off Spatio-Temporal Properties in 3D High-Speed fMRI

1-shot 2-shot 3-shot 4-shot Rr,min 3 4 7 8 Rr,max 6 8 7 8 Rzsp,min 1.56 1.7 1.7 1.6 Rzsp,max 3.9 2 2 1.6 Spirals 23 29 29 33 Samples (total / per shot) 14014 / 14014 25828 / 12914 37350 / 12450 49480 / 12370 TE [ms] 34.9 32.7 31.8 31.8 TR (total / per shot) [ms] 96 / 96 180 / 90 264 / 88 352 / 88 Flip angle [◦] 23 22 22 22

Table 6.1: Parameters for one shot of all 4 versions of the stack-of-spirals trajectory with diﬀerent degree of segmentation. All trajectories cover the same extent of k-space, i.e., a sphere with radius 1/(2 · 0.003 m). The outermost spirals of each trajectory cover only a small fraction of a spiral.

is always Rr,max. The concept of varying sampling densities in the parameter plane of resolution over sampling

time is outlined in Figure 6.2. It shows the parameter plane of nominal resolution vs. TR and the parameters of all trajectories that are the subject of this work. Extrapolations of the

sampling parameters of the single-shot and 4-shot stack of spirals are shown, where TR is assumed to be directly proportional to sampling time, i.e., acquired data points, for simplicity. These lines are meant to represent lower and upper bounds of sampling densities that make sense in the context of high-speed fMRI. The lower bound yields image quality established by the single-shot publication,29 and the upper bound leads to theoretical image quality comparable to full sampling. This plot is supposed to illustrate the concept of the trade-oﬀ zone, and approximate this zone in the shown parameter plane based on a proportionality relation between voxel volume and sampling time. It is not based on measured and quantiﬁed data.

6.2.2 Data Acquisition

All experiments of this and the next chapter were performed on a 3 T system (Magnetom Prisma ﬁt, Siemens, Erlangen) with a 64 channel head/neck coil array, using the upper 52 brain-relevant coil elements covering the brain. Written informed consent was obtained in accordance with a protocol approved by the Ethical board of the University Medical Center Freiburg. With a focus on whole-brain fMRI, a 150 mm slab was chosen in all experiments. Flip angles

always correspond to the Ernst angle calculated for a T1 of 1200 ms, with TR speciﬁed in Table 6.1. Dwell time was kept at 5 µs, and care was taken that no trajectory exceeded 23.5 mT/m during read-out, which lies below the maximally resolvable gradient amplitudes according to the Nyquist criterion of data acquisition (see Section 3.3.3). Slew rates were held below 170 mT/m/ms at all times, which is slightly under the maximum possible 200 mT/m/ms, in order to be on the safe side regarding peripheral nerve stimulation. Gradient spoilers were applied on all gradient axes before each excitation pulse to minimize Material and Methods 69

Figure 6.2: The parameter plane of nominal resolution vs. required sampling time for a whole brain 3D GRE fMRI image. The sampling parameters used in this chapter are depicted by yellow points. The underlaid shading is a schematically interpolated representation of image artifact intensity, idealized without physiological artifacts, from strongest (black) to minimal (white). Areas identified with "I", "II", and "III" represent parameter sets with minimal static image artifacts ("I"), trade-off between sampling time and variable density artifacts ("II"), and severe artifacts ("III"). The red dashed lines represent an interpolation of minimal and maximal sampling densities used in this work. Interpolations depict dependencies inversely proportional to the cube root of sampling time. Note that this plot is supposed to illustrate resolution flexibility, and shows only a conceptional idea. No scalar measure for artifact strength can be deduced, the dependence of TR on sampling density exhibits discontinuities due to sequence overhead, and it is not unambiguous in the case of VDS.

52 B0-fluctuation-induced time series variation due to steady-state effects. The spoilers were played out at maximally 21.6 mT/m. Fat saturation pulses were implemented in between gradient spoilers and excitation, in the way provided by the vendor, i.e., engulfed by gradient lobes with reversed polarity at maximally 8 mT/m. A prescan was implemented for anatomical localization, receiver coil sensitivity, off-resonance, and rough signal decay 1 estimation . It consisted of a double-echo gradient-echo sequence with TE = 4.92/7.38 ms, ◦ TR = 529 ms, flip angle 60 , and total scan time of 70 s. Time series to measure tSNR were measured for all trajectories in 3 volunteers, which were advised to hold as still as possible. All trajectories were measured consecutively for 2 to 4 minutes per session per volunteer. For the comparison of 1- vs. 3-shot resting state results, two separate stack-of-spirals datasets were acquired from 15 health volunteers each, each scan lasting 5 minutes. Subjects were instructed to fixate their eyes to a presented fixation cross and not to sleep. Cardiac and respiratory signals were recorded.

1 ∗ ∗ 2 echoes and the coarse resolution are not enough for highly reliable estimation of a T2 map. The T2 map was, however, only used for one simulation shown in Figures 6.6 and 6.7, which is used to assess eﬀect on image quality. Quantitative imprecision does not alter the quality of occurring eﬀects. 70 Trading Off Spatio-Temporal Properties in 3D High-Speed fMRI

6.2.3 Signal Simulation and Image Reconstruction

Like in the last chapter, Section 5.3.3, the signal equation,

s = Eρ, (6.1) with MR signal s, encoded object ρ, and measurement-modeling operator E, was used for (i) signal simulation from an image representing the ground truth ρ = ρgt, and/or for (ii) image reconstruction of simulated/measured signal s by pseudoinversion. The forward operator E was implemented as described in Section 3.4.1, on a 64 × 64 × 50 grid resulting in 3 mm isotropic grid spacing. The non-uniform FFT utilized the Michigan Image Reconstruction Toolbox.39 For reconstruction of acquired data, a gradient delay correction was applied.124 In the case of simulation and reconstruction incorporating previously acquired ∗ B0 and/or T2 maps into the signal model, a time-segmented approach (see Section 3.4.1) with 10 segments was used. For details on image reconstruction, see Section 3.4.2. The reconstruction of thermal noise is sensitive to both regularization parameters and truncation of the conjugate gradient algorithm, as the noise ampliﬁcation of the problem inversion is determined by the smallest eigenvalues of the system matrix E, which are manipulated by the Tikhonov regularization. Consequently, λ was chosen as low as possible, but high enough such that the necessary number of iterations for complete convergence of the thermal noise was not prohibitive regarding accessible computational resources. All time series of the thermal noise simulation and the tSNR measurements were reconstructed at λ = 0.05 with 100 iterations. The reconstructions of the resting-state datasets were reconstructed with oﬀ-resonance correction and thus with a higher regularization parameter due to computational constraints, i.e., at λ = 0.2 with 40 iterations.

6.2.4 Time Series Processing

All the time series in this work were motion corrected post-reconstruction with MCFLIRT from the FSL toolbox.125 A dynamic global field shift correction126 was applied to the raw data of every shot individually in the reconstruction step, where every interleaf from a reference frame served as an individual reference for the interleaves of subsequent shots. All time series were subject to linear detrending of the absolute values. For the resting-state data sets, cardiac and respiratory signals that were recorded during acquisitions were removed retrospectively as a confound.127 Time series were band-pass filtered with cut-off frequencies of 0.01 to 0.5 Hz. The resting-state datasets were spatially smoothed with a Gaussian kernel of full-width at half-maximum of 5 mm. No spatial smoothing was applied to the reconstructed images for the tSNR experiments. Material and Methods 71

6.2.5 Thermal Noise Propagation

Noise propagation maps were simulated for each trajectory by the Monte Carlo pseudo-replica method.51 The sensitivity maps for this step were measured in a spherical water phantom. 2000 noise realizations were simulated for every trajectory, and the standard deviations of their real part were used for the maps. The reference noise for the noise propagation maps was calculated with a fully sampled spherical stack-of-spirals trajectory, using parameters (Rr,min,Rr,max,Rzsp,min,Rzsp,max) = (1, 1, 1, 1) in the algorithm for the trajectory calculation. Aiming at a comparison with "full sampling", the sampling density along the spirals of the reference trajectory was adjusted such that its number of data points approximately equals that of a Cartesian grid inscribed by a sphere of radius 1/(2 · 0.003 m), i.e., 110102 vs. 107256 data points, respectively.

6.2.6 tSNR

For the calculation of tSNR, the standard deviations of linearly detrended time series were formed, and their mean values divided by the standard deviations. For histograms and mean values, the brain was masked with the Brain Extraction Tool from the FSL toolbox.125 The static signal intensities and slow signal fluctuations in the image time series, e.g., typical BOLD fluctuations, can be assumed to remain unaffected by the number of shots, when static image artifacts are not taken into account. However, fluctuations that lead to significant signal change within the acquisition window of one image can lead to artifacts with an intensity that depends on the window size, i.e., number of shots. As the motion and field variation induced by breathing lies in that time scale, the additional information of band-filtered time series’ tSNR in the breathing band was formed by dividing the standard deviations of band-filtered signals by the mean of the unfiltered signals. Band-filtering was accomplished using 4th order Type II Chebyshev filters with stopband attenuation of 40 dB. Breathing band frequencies were determined by plotting the absolute values of Fourier spectra of various time series, and visually identifying the respiratory peak and, if apparent, its first harmonic. For the analysis of time series spectra of all 4 time series, the voxel-wise Fourier transforms in the time domain were normalized by the square root of time points, and the mean of absolute values was formed. With this method, all spectral components that play a role in numerous voxels appear.

6.2.7 Functional Characterization

A rigid transformation matrix was calculated for functional images by ﬁrst registering them to T1 weighted anatomical images and then by registering those to the MNI-152 template. By using the inverse of the already calculated transformation matrix, the Stanford FIND 72 Trading Off Spatio-Temporal Properties in 3D High-Speed fMRI atlas59 and AAL128 atlas in MNI space were registered back to individual functional spaces. To assess diﬀerences between the two datasets in regions prone to susceptibility artifacts, left and right hippocampus parcels of AAL, posterior cingulate cortex (PCC) and medial prefrontal cortex (mPFC) parcels of the FIND atlas were used as seed regions for resting state network (RSN) analysis. By using average time courses of the seed regions, seed correlation maps were calculated for each individual. A group analysis was then performed using group-level t-tests.

6.3 Results

6.3.1 Image Quality

Point spread functions from the center of the field of view and reconstructions of simulated signals from the GRE reference scan images without modeled off-resonances or signal decay are shown in Figure 6.3, as well as the difference images of the simulation and the GRE reference. While the fully sampled sphere with a nominal resolution of 3 mm mainly leads to loss of edge information, i.e., Gibbs ringing, over the full 3 mm Cartesian sampling, the highly variably undersampled single-shot trajectory introduces clear spiral artifacts and aliasing with intensities up to 8% of the maximum image intensity. 2- to 4-shot trajectories lead to minimal z-aliasing in the 1-2% range. The strongest artifact intensity of full image reconstruction is located at the center of the field-of-view, where the encoding power of the receiver coil array is lowest, and in areas where aliasing from the center occurs. 2- to 4-shot trajectories show a clear dependence of incoherent artifacts and loss of edge information with decreasing sampling densities, however, with artifact intensity below Gibbs ringing intensity of the fully sampled sphere. The cross sections of Figures 6.4 to 6.8 are identical, and all simulations and reconstructions shown in Figures 6.3 to 6.8 originate from the same dataset for the sake of comparability. Figure 6.4 shows the ground truth used for the simulations in Figures 6.5 to 6.7. It is obvious ∗ that T2 estimation in highly off-resonant areas is biased to lower values, as previously shown in Equation (3.15) . Reconstructions of simulated signals including off-resonance in the simulation and recon- ∗ struction model are shown in Figure 6.5, without effects of T2 . The single-shot trajectory with its longest TE and most pronounced variable density sampling clearly suffers from the strongest signal drop-out and general image degradation. Both image degradation and drop out in presence of strong off-resonance gradients become less with higher sampling densities and lower TE. 3- and 4-shot trajectories, which have the same TE and only differ in sampling densities, show no obvious different off-resonance susceptibilities. However, the lower sampling density of the 3-shot trajectory leads to slightly pronounced ring-like artifacts. ∗ Figures 6.6 and 6.7 show the same simulation with simulated T2 decay. The reconstruction model incorporates the decay ground truth in Figure 6.6 to assess theoretical information Results 73

Figure 6.3: The first line shows point spread functions from the center of the field of view for 1-to 4-shot stack-of-spirals trajectories and a fully sampled sphere. Corresponding reconstructions of simulated signals from the GRE reference scan images without modeled off-resonances or signal decay are shown in the second line. The difference images of the simulation and the GRE reference with Cartesian sampling, capped at ±10%, are shown in the last line. The incoherent artifacts of the 2-shot trajectory show an intensity below the Gibbs ringing of the fully sampled sphere, and are becoming less with the denser sampling of 3- and 4-shot trajectories. This figure was used in a first-author publication.63 loss. The reconstruction in Figure 6.7 does not model the decay, as this is the common case in MRI. Both Figures 6.6 and 6.7 have different image contrast than Figure 6.5, and both images are blurred. While Figure 6.6 appears as a blurred version of Figure 6.5 with different contrast, the blurring in Figure 6.7 is even stronger and an additional artifact in the form of spurious signals distributed over the image can be observed. Figure 6.8 shows reconstructions of all trajectories from a single scan session. The imaging quality that was previously observed in Figure 6.7 is confirmed, considering the different weighting of the MREG and reference scans.

6.3.2 Thermal Noise Propagation

Noise Propagation maps of 1- to 4-shot stacks of spirals at 3 mm nominal resolution are depicted in Figure 6.9a). Intersections through the central slices of the maps are shown in Figure 6.9b). While the intersection proﬁles of the maps show a clear dependence on the undersampling, i.e., number of shots in this case, both 3- and 4-shot trajectories lead to greatly reduced noise propagation over 1- and 2-shot trajectories, with the 3-shot trajectory’s values lying 5% above the 4-shot’s on average over the whole FOV. 74 Trading Off Spatio-Temporal Properties in 3D High-Speed fMRI

Figure 6.4: Ground truth of the simulations shown in Figures 6.5-6.7, with identical cross sections. The red lines mark the cross sections of the shown slices.

Figure 6.5: Reconstructions of simulated signals including off-resonances from the GRE reference ∗ scan of the scan session shown in Figure 6.8. No T2 decay was incorporated in the model to isolate the off-resonance effects as in Chapter 5 .The red lines mark the cross sections of the shown slices. The TE values from 1- to 4-shot trajectories in ms are: 34.9, 32.7, 31.8, and 31.8. Artifacts that originate from low sampling density are pronounced in the presence of off-resonance artifacts that distort the acquired k-space trajectory shape, and are the strongest near the sinuses in the shown slices. Off-resonance-artifact intensity decreases with number of shots, where only minimal changes can be observed from the 3- to 4-shot case. Results 75

∗ Figure 6.6: Reconstructions of simulated signals, identical to Figure 6.5 except for additional T2 decay in the forward model and reconstruction. Blurring is induced and signal is partly attenuated in regions of strong decay. Oﬀ-resonance artifacts remain unchanged.

∗ Figure 6.7: Reconstructions of simulated signals, identical to Figure 6.6 except for uncorrected T2 decay, as commonly done in MR reconstructions. Blurring is pronounced in all cases compared ∗ to Figure 6.6, and T2 signal weighting can be observed. Some additional artifacts in the form of spurious signals distributed over the image can be observed. 76 Trading Off Spatio-Temporal Properties in 3D High-Speed fMRI

Figure 6.8: First shot of time series subsequently acquired with all 4 trajectories in a single scan session. The red lines mark the intersection positions of shown slices. Compared to the effects that are observed in simulations in Figure 6.7, the non-approximated and continuous object leads to slightly different results with the same quality as in Figure 6.7. The first shot of the MREG sequence exhibits different signal weighting than the reference scan. Some signal from the cerebellum is missing due to the excitation profile. This figure was used in a first-author publication.63

Figure 6.9: Every 5th slice of the noise propagation maps obtained by the pseudo-replica method for all trajectories in subfigure (a). (b) shows the section at (y = 32, z = 25) marked by the red line in (a). The mean values of the maps are 1.71, 1.39, 1.20, and 1.14 for 1- to 4-shot trajectories, respectively, over the whole field of view. This figure was used in a first-author publication.63

6.3.3 tSNR

The tSNR histograms of the linearly detrended time series of all 4 trajectories have the same properties in all 3 subjects, as shown in Figure 6.10. The 2-, 3-, and 4-shot histograms are of very similar shape, and lie above 1-shot histograms at values larger than ca. 60. At values of over ca. 70, 3-shot histograms lie marginally over the 4-shot ones, both lying over 2-shot, and more pronounced over 1-shot histograms. Below the intersections the relations invert. One reason for the lack of improvement in tSNR of the 4-shot vs. 3-shot trajectory can Results 77 be found in the consistently decreased tSNR of the 4-shot scans in the respiratory band, which can be seen in the 3rd column of Figure 6.10. While the shortest read-out leads to the highest values in the respiratory band, the 2- and 3-shot trajectories do not lead to clearly distinguishable results. The breathing bands [Hz] for subjects 1 to 3 were: [0.16, 0.58], [0.21, 0.68], and [0.18, 0.59]. While low-frequency tSNR is similar for all trajectories, some non-reproducible variability is caused by spontaneous BOLD ﬂuctuations. The high-frequency band reﬂects the results of the noise propagation simulations shown in Figure 6.9, meaning that more shots and therewith more datapoints lead to higher SNR when only thermal noise is considered. An exception is the 4-shot scan in subject 3, where the tSNR histogram is identical to the 3-shot scan, for an unknown reason. The mean spectral components of all time series are shown in in Figure 6.11. The 1-shot trajectory enables to clearly distinguish the higher harmonics of the cardiac pulsation up to the 3rd harmonic at a heart rates of 75/min and 50/min. At 3 and 4 shots, the higher harmonics that were unaliased by the high sampling rate of the 1-shot trajectory partly fold into the low-frequency and respiratory frequency spectra. The tSNR maps of the detrended time series of subject 2, which correspond to the middle plot in the left column of Figure 6.10, are shown in Figure 6.12. While 3- and 4-shot trajectories lead to the most detailed structures in the maps, the 3-shot case leads to approximately the same mean values over the whole brain, i.e., 49.6 vs. 48.1 in this subject. 2 shots lead to slight blurring and lower tSNR values at a mean of 43.8. The single-shot trajectory exhibits pronounced blurring and decreased tSNR of 40.2 on average. The mean values for subject 1 and 3, from 1- to 4-shot trajectories, are: 47.5, 55, 58.3, 58.6, and 46.6, 53.1, 54.9, 55.4.

6.3.4 Functional Characterization

Group t-maps are shown in Figure 6.13, a threshold of t>4 is applied for both datasets. A default mode network (DMN) pattern is detected when the seed is placed at the posterior cingulate cortex (PCC), as shown in subfigure (a). A main difference between 1- and 3-shot trajectories is a larger frontal component in the 3-shot sequence, particularly in the ventral part of the frontal cortex which is closer to the frontal sinus. Similarly, when the seed is placed to the medial prefrontal cortex (mPFC), almost no significant long-distance connections were detected in 1-shot trajectories, as shown in subfigure (d). The other two seeds placed in left and right hippocampus are shown in radiological convention in subfigures (c) and (d), respectively. In the 1-shot dataset, significant connectivity was limited to the contralateral side of the hippocampal network. On the other hand, in the 3-shot dataset, wide-spread cortical connections are revealed in addition to the hippocampal network, notably involving DMN areas for a seed location in the left hippocampus. 78 Trading Off Spatio-Temporal Properties in 3D High-Speed fMRI

Figure 6.10: The left column shows tSNR histograms of linearly detrended time series of consecutive scans with all 4 trajectories, for all 3 subjects. 3- and 4-shot trajectories exhibit very similar histograms. Columns 2-4 show the corresponding partial tSNR of band-filtered time series, with bands I, II, and III being low-frequency-, respiratory-, and high-frequency-bands, respectively, as shown in Figure 6.11. Low-frequency tSNR is similar for all trajectories, but BOLD fluctuations are scan dependent and lead to variability. The 4-shot trajectory leads to a clear shift to lower partial tSNR values in the respiratory band. The high-frequency band reflects the results of the thermal noise propagation simulations well, with exception of the 4-shot scan in subject 3. Results 79

Figure 6.11: Mean of the absolute values of time series spectra over the whole brains of all subjects, normalized by the square root of time points N. All subplots have the same scaling. The green shading indicates the values that were identified as respiratory band, i.e., band II. At a heart rate of 75/min in subjects 1 and 3, cardiac pulsation leads to signal fluctuations in the frequency band around 1.25 Hz. Higher harmonics thereof, marked with arrows with coloring according to the order of the harmonic, can be clearly identified in the 1-shot scan. In the 2-shot scan, a foldover of the 2nd harmonic can be seen at around 1.8 Hz, and ripples at around 0.6 Hz indicate a foldover of the 3rd harmonic. In the 3-shot scan, the 1st harmonic folds right into the base frequency, and the 2nd harmonic into the lowest frequencies. In the 4-shot case, the 1st harmonic folds into the respiratory band, and the 2nd harmonic foldover can be seen at around 0.9 Hz. Subject 2 had a heart rate of around 50/min, which leads to resolution of 2 harmonics in the 2-shot scan, and fold-over of only the 2nd harmonic in the 3-shot scan. The middle part of this figure was used in a first-author publication.63 80 Trading Off Spatio-Temporal Properties in 3D High-Speed fMRI

Figure 6.12: Every 5th slice of the tSNR maps from subject 2 is shown. The mean values of the maps over the whole brain were 40.2, 43.8, 49.6, and 48.1 for 1- to 4-shot trajectories, respectively. Structural detail is most apparent in 3- and 4-shot cases. 2-shot and especially 1-shot trajectories exhibit blurred maps. This ﬁgure was used in a ﬁrst-author publication.63

6.4 Discussion

6.4.1 Image Quality

While point spread functions in sensitivity-encoded imaging are location-, off-resonance-, and signal decay-dependent, and therewith do not give a complete model of the imaging properties of a trajectory, they can reveal obvious aliasing, which is pronounced in the single-shot trajectory. It has to be noted that the presented point spread functions were calculated for a point originating from the least sensitivity-encoded area in the center of the field-of-view. They consequently represent the worst case, and voxels in the brain cortex exhibit ameliorated properties. For a more intuitive picture of the imaging properties of the trajectories, the middle and lower rows of Figure 6.3 show that 3- and 4-shot trajectories lead to differences in the image that are barely noticeable visually, compared to the fully sampled sphere. Furthermore, the 2-shot case leads to an artifact level below Gibbs ringing amplitude, however showing increasingly structured artifacts. The minimal z-aliasing of the 2- to 4-shot trajectories that is revealed by the point spread functions does not lead to any pronounced artifacts in the fully reconstructed image. The simulation that incorporates off-resonances in the signal model as well as the recon- Discussion 81

Figure 6.13: Seed based analysis results of 1-shot vs. 3-shot trajectories: four different seeds are used, as indicated in the subfigure titles; (a) posterior cingulate cortex (PCC), (b) left hippocampus, (c) right hippocampus,(d) medial prefrontal cortex (mPFC). 1-shot and 3-shot results are shown in upper and lower rows of each subfigure, respectively. The frontal component of the DMN is less significant in the single-shot case (a), and PCC to mPFC connection is not observable when the seed is placed at mPFC (d). Cortico-hippocampal connections are visible when the 3-shot trajectory is used in (b) and (c). Red to yellow color scale is used with a threshold of t>4 and identical range in each subfigure. This figure was used in a first-author publication,63 and was produced using code written by Burak Akin.

struction reveals another quality of artifacts. Off-resonance artifacts mainly depend on 29 the sampling speed along the slowest encoding direction, as well as TE, which both are disadvantageous in the single-shot case. These parameters are more favorable and chosen similar (traveling speed through k-space center) or equal (TE) for 3-and 4-shot trajectories, and slightly less favorable in the 2-shot’s. An additional impact on off-resonance artifacts is produced in conjunction with variable density sampling: off-resonance gradients constitute an additional constant encoding gradient in local k-space,29, 129 and deform the locally acquired trajectory, as described in Section 3.2.2. k-space is then acquired only partially or less densely in reference to the k-space frame that is targeted, and furthermore the acquisition of the k-space center is displaced to lower density portions of a variable density acquisition. The resulting lower local k-space densities or losses in parts of the imaged subject lead to increased artifacts, which are especially apparent in the single-shot trajectory, but can also be seen above the sinuses29 to a much smaller degree when comparing the 2-shot vs. 3/4-shot acquisitions due to less sampling density in kz-direction. 82 Trading Off Spatio-Temporal Properties in 3D High-Speed fMRI

∗ T2 decay adds pronounced blurring to the images due to the length of the read-out. While impracticable in reconstructions of real data due to the challenges associated with measure- ∗ ment of a sufficiently resolved and correct T2 map, the simulations show that blurring is in principle reducible to a fair amount. Considering the linear relation of necessary k-space points for image reconstruction of a certain grid size, it can be extrapolated that a nominal resolution of 2 mm would be feasible at around 500 ms with the sampling parameters of the 3 mm 2-shot variant in 5 to 6 shots. Off-resonance effects due to variable density sampling would presumably reduce in their severity due to the larger extent of k-space being covered with each shot, if TE was held constant. However, we see the strengths of the segmented 3D read-out at a TR of below 300 ms due to the accumulation of physiological artifacts. It also has to be noted that despite its Cartesian structure in the kz-dimension, the spherical stack of spirals leads to more blurring in addition to signal shifting in presence of strong off-resonances and long read-outs. This effect is not apparent in EPI read-outs, which translate off-resonances purely in signal shifting. A detailed comparison of stack-of-spirals vs. EPI trajectories is however out of scope of this article.

6.4.2 SNR

The noise propagation of 1- and 2-shot trajectories show over-proportional thermal noise enhancement, when compared to 3- and 4-shot maps. This is presumably evoked by the distinct variable density sampling beyond the encoding capabilities of the receiver array. However, as shown in the histograms of Figure 6.10, the impact on the temporal SNR is not prohibitive to the usefulness of the 1- and 2-shot trajectories, as the 3D read-outs yield sufficient signal for the method to remain in a regime dominated by physiological noise. It has to be noted that, while regularization is necessary for the pseudo-inversion in order to insure image convergence and avoid noise divergence, it has implications on the reconstruction, especially on the intensity of thermal noise. This means that noise propagation values may be affected by the different effects of regularization in each trajectory’s reconstruction. Tikhonov regularization of the inverse problem with regularization strength λ leads to an inversion 2 2 of modified eigenvalues (i + λ )/i instead of i. While the system matrices that describe the measurement processes with different trajectories were too large for an Eigencomponent analysis due to computational limits, it was ensured that λ was small enough not to visibly affect the components involved in the simulated image reconstruction, and that the noise amplitude dependence on λ was roughly trajectory-independent in the chosen parameter range. As a concequence of the physiologically dominated noise a prolongation of the acquisition window beyond 3-shot does not lead to an increase in tSNR. This is at least partly due to respiration, one of the main physiological artifact sources, inducing time series fluctuations on the time scale of the used multi-shot sampling rates. Considering the near-identical imaging properties of 3- and 4-shot trajectories, the advantage in tSNR per unit time of roughly 4/3 yields an excellent compromise in favor of 3 shots. While tSNR per unit time increases with decreasing length of acquisition window in this noise regime, it comes at increasing cost of Conclusion 83 spatial fidelity below 3 shots. The Fourier spectra of the time series show a strength of the single-shot acquisition, which is complete temporal unfolding of the cardiac pulsation with its higher harmonics. It is noteworthy that a part of the partial tSNR degradation in the respiratory band of the multi-shot acquisitions possibly derives from the higher harmonics of cardiac pulsation.

6.4.3 Functional Characterization

We investigated brain regions which are dominantly aﬀected by susceptibility artefacts reported in single-shot MREG.29 Those regions are above the frontal sinus, covering the medial and ventral prefrontal cortex of the brain, which constitute the frontal part of the DMN,130 as well as the basal areas of the right and left temporal lobes. When the seed is located close to these artifact regions, areas of signiﬁcant functional connectivity are limited to surrounding voxels in 1-shot acquisitions. On the other hand, in 3-shot acquisitions long-distance connections are detected successfully, including for example cortical connections to the hippocampus, as reported in previous fMRI studies,131 in relation with associative memory.132 The analysis was restricted to the two trajectories in which we see most potential: the ultra-fast single-shot variant which has already seen wide application, and the 3-shot variant, which we found to represent the sweet spot of acceleration vs. image quality with regard to tSNR maps.

6.5 Conclusion

This chapter demonstrates the flexibility in the parameter combination of sampling time and resolution when using different and partly variable sampling densities in sensitivity encoded functional MRI. With the use of segmented 3D read-outs for whole brain acquisition, the aim was to find the best possible compromise of image quality and tSNR per unit time, rather than pushing the limits of temporal and spatial resolution with the highest variable density sampling that was used in previous MREG methods. It was shown that highly variable density sampling leads to increased off-resonance artifact susceptibility on top of its point spread function artifacts, while in the encountered noise regime it has the benefit of higher tSNR per unit time. While long read-out windows beyond 300 ms can lead to increased physiological artifacts in the breathing range, all used parameters lead to useful tSNR values. Accounting for the generally high SNR of the method, the 3-shot trajectory yields an excellent balance of tSNR per unit time and imaging quality that is visually indistinguishable from higher sampling densities. It also delivers a temporal resolution that is sufficient for direct filtering of physiological noise. This method could prove favorable for high-speed whole brain acquisitions in fMRI at macroscopic resolutions due to its synergies of a temporal resolution that enables direct 84 Trading Off Spatio-Temporal Properties in 3D High-Speed fMRI

ﬁltering of physiological artifacts for highest statistical power, and 3D read-outs with optimal use of encoding capabilities of multi-coil arrays for eﬃcient sampling, including variable density capability, and high SNR. 7 Targeted Partial Reconstruction for High-Speed fMRI Neurofeedback

Segments of this chapter originate from the ﬁrst-author-publication "Targeted Partial Recon- struction for Real-Time fMRI with Arbitrary Trajectories".64

Contents

7.1 Motivation ...... 86 7.2 Theory ...... 86 7.2.1 Voxel-Wise Reconstruction ...... 87 7.2.2 Volume-wise Reconstruction ...... 88 7.2.3 Efficiently Approximated Reduced-FOV Reconstruction ...... 91 7.3 Methods ...... 94 7.3.1 Data Acquisition ...... 94 7.3.2 General Reconstruction Details ...... 94 7.3.3 Voxel-wise Reconstruction ...... 95 7.3.4 Volume-wise Reconstruction ...... 95 7.3.5 Efficiently Approximated Reduced-FOV Reconstruction ...... 96 7.3.6 Feedback Scan Workflow ...... 97 7.4 Results ...... 98 7.4.1 Voxel-wise Reconstruction ...... 98 7.4.2 Volume-wise Reconstruction ...... 98 7.4.3 Efficiently Approximated Reduced-FOV Reconstruction ...... 102 7.4.4 Feedback Scan Implementation Performance ...... 103 7.5 Discussion ...... 105 7.5.1 Workflow and Implementation ...... 105 7.5.2 Reconstruction quality ...... 106 7.5.3 Reconstruction phase correction ...... 106 7.6 Conclusion ...... 107 86 Targeted Partial Reconstruction for High-Speed fMRI Neurofeedback

7.1 Motivation

As described in Chapter 4, the aim of this chapter is to find a reconstruction framework that makes the use of highly accelerated non-Cartesian trajectories in rtfMRI possible. The non-Cartesian trajectories for high and ultra-high speed fMRI methods that were presented in the last chapter necessitate a computationally expensive SENSE-type image reconstruction. This reconstruction is most conveniently formulated as an inverse problem (see Section 3.4.1) solved by inversion algorithms whose duration renders the extraction of localized information from the raw data in real-time applications, i.e., in less than TR, unfeasible. This chapter presents a Targeted Partial Reconstruction, applied as a partial SENSE method with MREG acquisition, which generates minimal reconstructions over one or several freely selected target regions. For example, such regions could be defined on the basis of the subject’s anatomy or previous functional scans, or chosen from a pre-defined brain atlas. Only one or - if feasible and necessary - more reconstruction vectors are computed per target prior to the experiment, allowing the reconstruction of the signal from the target region as one or more scalar products, i.e., a straightforward matrix operation. Computation time is then sufficiently fast for the real-time reconstruction of several regions even at very short repetition times (below 100 ms). The ultimate purpose of this development is the use in fast real-time feedback fMRI, yet it is generally applicable to any scenario requiring a fast real-time target reconstruction or fast reduced-FOV reconstruction, independent of the type of trajectory. It should also be noted here that only the reconstruction framework is modified while the data acquisition remains unchanged. As such, all recorded information still remains available for full post-experimental analysis based on offline reconstruction, if desired.

7.2 Theory

As outlined in the introduction, the primary aim of this chapter is to develop an ROI-speciﬁc reconstruction that is applicable to non-Cartesian trajectories in less than TR, i.e., faster than the order of 100 ms in the case of MREG. This will be realized by replacing the computationally expensive pseudo-inversion of every time-frame by a two-step method, which consists of: (1) Explicitly determine the required lines of the reconstruction matrix, or linear combinations thereof, which correspond to the individual voxels, or linear combinations thereof, that are to be reconstructed. This step is as expensive as a full-frame reconstruction for every aimed-for reconstructed value, however it is signal-independent and consequently feasible as a precalculation step prior to the actual feedback scan. This step’s computational cost rises linearly with the desired number of reconstructed values. It will result in a partial reconstruction matrix that is small enough to ﬁt in random access memory (RAM). (2) Application of the explicitly precalculated reconstruction matrix to the signal data in Theory 87

real-time. This step’s expense can be controlled by the number of reconstructed values per time-frame. As such, the full-frame reconstruction is decomposed into a highly expensive precalculation and a minimal and controllable real-time reconstruction. Note that Step 1) has a major constraint regarding the precalculations for the case of sensitivity encoded imaging. These require coil sensitivity maps for the formulation of the forward operator of the inverse problem, and these maps are object-dependent. This means that the precalculations have to be performed while the subject to be scanned rests in the scanner, waiting for the feedback- scan. Another constraint is the size of the reconstruction matrix, which need to be stored in RAM. For the contemplated application, though, the computational eﬀort1 of step 1) is the bottleneck. Step 1) is derived for individual voxels in Section 7.2.1, and for linear combinations of voxels, e.g., volume-sums, in Section 7.2.2, together with a previously published method for volume-wise reconstruction. An eﬃcient approximation of step 1) for adjacent voxels or volumes is presented in Section 7.2.3.

7.2.1 Voxel-Wise Reconstruction

Shortly recapitulated (see Section 3.4.1), the discretized MR signal equation is

s = Eρ, (7.1)

(C·N) with complex MR signal s ∈ C representing N data points from each of the C acquisition M channels, complex image vector ρ ∈ C , and measurement-modeling forward operator (C·N)×M E ∈ C . Formally, the solution in the least-squares sense of the signal equation by pseudo-inversion of the forward operator is (E∗E)†E∗s = ρ, (7.2)

E∗ denoting the conjugate transpose of E, where the invertibility of E∗E is ensured by an appropriate encoding pattern.

A voxel value can be expressed by a scalar product in image space, or in signal space by making use of Equation (7.2) and the properties of the dot product respective to linear operations: hρ, δi = h(E∗E)†E∗s, δi = hs,E(E∗E)†δi = hs, vi, (7.3) where δ is Kronecker delta representing the single voxel whose value is to be expressed, and

v ≡ E(E∗E)†δ (7.4)

1Speed tests were done with: Berkeley Advanced Reconstruction Toolbox, IMPATIENT MRI, gpuNUFFT, self-implementation of NUFFT with MIRT gridding operator, self-implementation of Toeplitz method.39, 133–136 The tested GPU was a NVIDIA Tesla C1060. The self-implemented nuFFT with MIRT’s gridding operator proved to be the most eﬃcient on a 64 × 50 × 50 grid. 88 Targeted Partial Reconstruction for High-Speed fMRI Neurofeedback corresponds to one complex conjugated line of the reconstruction matrix (E∗E)†E∗. The complex conjugation results from the sesquilinearity of the scalar product. Consequently, a computationally intensive full reconstruction, i.e., evaluation of (E∗E)†E∗ or its conjugate transpose in Equation (7.4), is independent of the data s and can be performed prior to the time-critical application. The reconstruction of the signal within the targeted voxel can then be performed by the scalar product on the right side of Equation (7.3). This approach yields voxel values that are fully equivalent to the results of the full frame- wise reconstruction for any linear reconstruction.2 This can be seen in Equation (7.3), where ρ represents the reconstructed image in Equations (7.2) and (7.3). The term (E∗E)† can be replaced by any linear reconstruction, such as the Tikhonov-regularized variant ∗ ∗ −1 M×M ∗ (E E + Γ Γ) , Γ = diag(γ1, ..., γM ) ∈ R , as the property hx, Ayi = hA x, yi holds for every matrix A of adequate dimensions. In theory, it is now possible to create a reduced-FOV reconstruction

Rs = ρpartial, (7.5) where ρpartial is a partial image and the partial reconstruction matrix R is composed of P explicitly precalculated lines of the complete reconstruction matrix for the voxels represented by δ1, ..., δP : ∗ ∗ ∗ † ∗ † R ≡ (v1, ..., vP ) = E(E E) δ1, ..., E(E E) δP . (7.6)

In the following, reconstruction via R will be referred to as target-wise reconstruction, as opposed to the standard full frame-wise reconstruction (E∗E)†E∗ in Equation (7.2). With the introduced formalism it is possible to compute and explicitly store the simpliﬁed reconstruction matrix R before the experiment, given that R is chosen small enough to be stored explicitly and Rs can be evaluated faster than or equally fast as TR.

7.2.2 Volume-wise Reconstruction

While it is theoretically possible to ﬁnd a reduced-FOV reconstruction for any number of ROIs that need high-speed reconstruction, it might be computationally too demanding to reconstruct those ROIs at full resolution in real-time. Also, the (weighted) mean values of the targeted ROIs might suﬃce for the application. In such cases, Equations (7.3) to (7.6) can analogously be derived:

hρ, V i = h(E∗E)†E∗s,V i = hs,E(E∗E)†V i = hs, vi, (7.7)

2Note that when computing Equations (7.2) and (7.4) using algorithms that do not fully converge, Equation (7.3) is not exactly fulfilled anymore, and residual differences may remain between frame-wise and volume- wise reconstruction. This is also the case for the used truncated conjugate gradient algorithm, as the different known components on both sides of the linear systems that arise when computing Equations (7.2) and (7.4) determine which subspaces the algorithm visits until termination within a certain tolerance. The parameters chosen in this work ensured that resulting residuals were at least an order of magnitude below thermal noise level. Theory 89 where V is a complex vector of length M containing arbitrary voxel weights representing a ROI, and v ≡ E(E∗E)†V (7.8) is a linear combination of lines of the complete reconstruction matrix, weighted by V .

When considering P volumes expressed by the weights V1, .., VP , such as those shown in Figure 7.1, their reconstruction becomes

Rs = ρROI, (7.9) with the partial reconstruction matrix

∗ ∗ ∗ † ∗ † R ≡ (v1, ..., vP ) = E(E E) V1, ..., E(E E) VP . (7.10)

Each line of R is a linear combination of lines of the complete reconstruction matrix E(E∗E)†, and ρROI is a vector containing the corresponding sums over the image m weighted by V1, ..., VP . This can be understood as an incomplete basis change in spatial coordinates. Again, as for the case of single voxels, this approach yields results that are fully equivalent to the results of the full frame-wise reconstruction for any linear reconstruction.3

Figure 7.1: Example of one slice of an image ρ and a number of target volumes Vn within this slice taken from a functional atlas. This ﬁgure was used in a ﬁrst-author publication.64

The equivalency with sums over fully reconstructed voxels distinguishes the proposed reconstruction from previously applied simplified volume-wise approaches,108–110 in which the target volumes are considered arbitrarily shaped voxels. These methods incorporate a strongly simplified signal model, which assumes pre-defined intensity and phase distributions from the ROIs: s = EsimplifiedρROI, (7.11) where the terms are analogous to Equation (7.1). Omitting relaxation and off-resonance effects, Esimplified, as described in previous publications, Z simplified −ikar Eab = Vb(r)c(r)e dr (7.12) Vb can be deduced from a spatial discretization of E,

−ikarb Eab = c(rb)e , (7.13)

3See footnote 2. 90 Targeted Partial Reconstruction for High-Speed fMRI Neurofeedback by adhering to the same grid size:

simpliﬁed X Eab ≈ Vb(rm)Eam. (7.14) m

The index m iterates over all grid positions rm, c(r) denotes a receiver coil sensitivity map, and Vb are the same volume weights as employed above. Multi-channel operators can be defined likewise, concatenating the channel-wise operators along the a-index. After this simplification, the forward operator’s second dimension reduces from the total number of voxels to the number of considered ROIs, and, depending on the number of ROIs, it becomes computationally feasible to explicitly calculate and save the pseudo-inverse (Esimplified∗Esimplified)†Esimplified∗. The time series of ρROI can then be obtained by application of the pseudo-inverse to s - ROI-wise, if desired:

∗ ∗ (Esimplified Esimplified)†Esimplified s = ρROI. (7.15)

The simplification of the signal model expressed by Equation (7.12), however, does not consider variable signal fluctuations within the ROIs. Resulting deviations in time series from calculating Equation (7.15) as opposed to the introduced Equations (7.5)/(7.9) are case-dependent and cannot be generally quantified - an application example is shown later in Section 7.4. Note that, while the pseudo-inversion of Esimplified becomes explicitly feasible in the shown application without the method of conjugate gradients on large vectors, this approach does - even for small numbers of ROIs - not necessarily compute faster than Equation (7.10). The reason is the computation of Equation (7.14), which becomes an additional, potentially expensive step, as will also be shown later.

Phase Correction for rtfMRI

In the sought field of application, i.e., fMRI, sums over voxels are usually performed using magnitude data. Thus, inter-voxel phase differences do not contribute to the evaluation of the sums. The proposed method, however, implicitly incorporates linear combinations in the complex domain, which can induce artifacts due to intra-volume dephasing. Due to physiological changes the dephasing artifacts in MRI time series are generally dynamic, but can be divided into static and dynamic parts. After global navigator correction the static part is evoked by rigid anatomy and shimmed fields, and the dynamic part mainly by breathing and cardiac pulsatility. For correction, the static phases φ can then be formally −iφ −iφ included in the complex weights V−φ by applying e to V , i.e., V−φ = e V , φ being a diagonal matrix. In analogy to Equations (7.3)-(7.6), neglecting the dynamic parts, the magnitude sum can then be expressed as

∗ † h|ρ|,V i = hρ, Vφi = hs,E(E E) Vφi = hs, vφi, (7.16) resulting in a weighted magnitude sum reconstruction matrix Rφ. In a time series, Rφ reconstructs the approximate magnitude sum when all phases vary around a stable mean.

The phases in Rφ can be approximated by drawing a sample from a short time series acquired as a premeasurement with subsequent full reconstruction of one frame. Theory 91

7.2.3 Eﬃciently Approximated Reduced-FOV Reconstruction

A signiﬁcant constraint of the Targeted Partial Reconstruction are the available computational resources for both the computation of step 1), i.e., Equation (7.6), and step 2), i.e., Equation

(7.5), as mentioned at the beginning of this chapter. TR together with the execution time of R in step 2) give an upper bound for the number of reconstructed volumes, while step 1) needs to be efficient enough to precalculate R. While the computational efficiency of step 2) can be enhanced by specific linear algebra libraries such as BLAS, and computational effort can be decreased by reduction of the dimension of the multi-channel reconstruction problem by Principal Component Analysis, an efficient approximation for step 1) is addressed in this subchapter. It consists of exploiting the smooth and slow varying structure of receiver coil sensitivity maps and the Fourier Shift Theorem to approximate the reconstruction of a voxel from neighboring voxels with minimal effort, and thus reducing computational effort to precalculate R for a full-resolution reconstruction of a contiguous ROI. Two approximations are presented: while the first is trivial and is an extension to methods that have been used in the context of motion correction, the second is original work and refines the approximation for larger voxel-reconstruction shifts. For the sake of simplicity, formulations are restricted to one dimension, as they are trivially generalized to more dimensions.

Fourier Shift Theorem Applied on Single Reconstruction Vectors

With knowledge of the reconstruction vm of a speciﬁc voxel value ρm, in a MRI measurement with linear gradients and without sensitivity encoding, the neighboring voxel value ρm+1 can be accessed by exploiting the Fourier shift theorem:

ρm = hρ, δmi = hs, vmi (7.17) ⇒ ρm+1 = hρ, δm+1i = hs, vm+1i = hssh+1, vmi = hs, vm,sh+1i, where the subscript sh + 1 stands for a shift of spatial index +1, and implies the left- n o multiplication by diag eik1 , ..., eikN for the signal of every channel or by its conjugate for the reconstruction, N being the length of the used k-space trajectory with coordinates kn. In sensitivity encoded MRI, however, shifts of the object - either by motion in the scanner or by application of the Fourier shift theorem to the data - lead to errors in the reconstruction model. Unlike for motion-induced shifts, in the case of signal data shifts arising from phase oﬀsets the error is reduced: the physical coil load remains unchanged and the reconstruction becomes identical to one with unshifted data and inversely shifted coil sensitivity maps. Small shifts of the object due to motion are unavoidable in any MRI scan, however, and the small resulting artifacts are deemed acceptable (not to be confused with intra-acquisition motion). Neglecting the induced coil map shift for the reconstruction of directly neighboring voxels will be investigated in this context, as it would then be possible to use Equation (7.17) to generate an approximated reconstruction of a small contiguous ROI with full resolution by only precalculating the reconstruction of the most central voxel and applying phases to it. 92 Targeted Partial Reconstruction for High-Speed fMRI Neurofeedback

Interpolation Between Multiple Reconstruction Vectors

A more reﬁned approximation for larger shifts, which incorporates the interpolation of the 4 reconstruction vm by vm+i and vm−j for small positive integers i and j, is derived in the following. From Equations (7.4)/(7.8), it follows that

E∗v ≡ E∗E(E∗E)†V = V 0, (7.18) where V 0 on the right side of the equation represents the reconstruction result, which might diﬀer from V if E∗E is not invertible, e.g., when the reconstruction is achieved by pseudoinversion or with Tikhonov regularization: E∗E(E∗E + Γ∗Γ)−1V = V 0. This does not interfere with the following derivation. While this method can be applied to any volume V , the aim of a full-resolution reconstruction implies the application to single voxels. V 0 then becomes the point spread function PSF 5. Expanding the forward model E in Equation (7.18) for the reconstruction of a voxel at position m leads to X l∗ ∗ ∗ ∗ l C S F G vm = PSFm (7.19) l where Cl is a diagonal matrix containing the receiver coil sensitivity map of receiver channel l, S is a diagonal matrix containing the nuFFT scaling factors, F is the Fourier matrix of l adequate dimension, G is the nuFFT gridding matrix, and vm is the part of the reconstruction vector that is applied to the signal of channel l. For nearby voxels the point spread functions are assumed6 to be shift-invariant function s, so that shifted and neighboring reconstructions can be set to be equal:

X l ∗ ∗ ∗ ∗ l (Csh+1) S F G vm,sh+1 = PSFm,sh+1 l X l∗ ∗ ∗ ∗ l C S F G vm+1 = PSFm+1 (7.20) l X l ∗ ∗ ∗ ∗ l X l∗ ∗ ∗ ∗ l ⇒ (Csh+1) S F G vm,sh+1 = C S F G vm+1. l l

In this case, the subscript sh + 1 stands for a circular shift of magnitude +1 along the diagonal of Cl, and for a phase multiplication like introduced in Equation (7.17) for the reconstruction vector vm. Inserting 1 = Cl∗S∗F ∗FS∗−1Cl∗−1 in every term of the left side of Equation (7.20) leads to X l∗ ∗ ∗ l∗−1 l ∗ ∗ ∗ l X l∗ ∗ ∗ ∗ l C S F FC (Csh+1) F G vm,sh+1 = C S F G vm+1. (7.21) l l

4This method is not restricted to integers, however neighboring voxels are represented by integers. 5The point spread function of a Tikhonov-regularized reconstruction is (E∗E + Γ∗Γ)−1E∗Eδ, which is identical to E∗E(E∗E + Γ∗Γ)−1δ. With E = E∗E, G = Γ∗Γ, and [(E + G)−1, G] = 0 (G is diagonal): E(E + G)−1 = (E + G)(E + G)−1 − G(E + G)−1 = (E + G)(E + G)−1 − (E + G)−1G = (E + G)−1E. [E, E †] = 0, on the other hand, follows directly from the pseudoinverse properties. 6In highly undersampled SENSE imaging, the point spread function varies over space, as the encoding properties of conventional receiver arrays vary over space.27 However, the encoding properties are changing smoothly and slowly over space, such that for neighboring voxels it can be assumed to be identical. Theory 93

The left side of Equation (7.21) follows from the fact that Cl and S are diagonal matrices and therewith trivially permute. It follows that if

l∗−1 l ∗ ∗ ∗ l ∗ l FC (Csh+1) F G vm,sh+1 = G vm+1, ∀l (7.22) is fulfilled, Equation (7.21) is fulfilled. Further assuming G∗ to have linear independent 7 ∗ 8 columns , i.e., GG to be invertible , Equation (7.22) can be used to approximate vm when vm+i and vm−j are known, for small i and j, as subsequently shown. l∗−1 l ∗ ∗ By first left-inverting the convolution FC (Csh+1) F on both sides of Equation (7.22) and re-indexing (m → m − 1), it follows for the case i = j = 1:

l ∗ −1 l ∗−1 l∗ ∗ ∗ l vm±1,sh∓1 = (GG ) GF (Csh∓1) C F G vm

1 ⇒ (vl + vl ) = (7.23) 2 m−1,sh+1 m+1,sh−1 1 ∗ (GG∗)−1GF ((Cl )∗−1 + (Cl )∗−1) Cl F ∗G∗vl ≈ vl 2 sh+1 sh−1 m m

1 l ∗−1 l ∗−1 l∗−1 l∗−1 if 2 ((Csh+1) + (Csh−1) ) ≈ C , which is formally the case if the diagonal of C , i.e., the conjugated reciprocal of the coil sensitivity map of channel l, is an approximately linear function over the distance over which it is interpolated. As the coil sensitivity maps are smooth and thus low varying functions, this assumption holds for distances up to a few mm or even cm relatively well, e.g., for a head coil array. Generalized to unequal distances of the interpolated vector from the known vectors in all channels, Equation (7.23) results in

1 (j v + i v ) ≈ v . (7.24) i + j m−i,sh+i m+j,sh−j m

Equation (7.24) is easily generalized to multiple dimensions and enables to relatively in- expensively interpolate, e.g., the reconstruction at full resolution of a small cube if the reconstructions of the corner voxels of that cube are known. This result will later enable to precalculate a reconstruction of a volume of 5 × 5 × 5 voxels in step 1) of the Targeted Partial Reconstruction for an rtfMRI scan in a reasonable amount of time, whereas a full precalculation would be prohibitive.

7This is trajectory dependent. The assumption bases on the fact that undersampled trajectories’ G∗ have more lines than columns, and efficient, non-redundant sampling adds information with every data point. 8 l In this case vm+1 can formally deduced from vm channel-wise: vm+1 = ∗ −1 l∗−1 i∗ ∗ ∗ l ∗ −1 (GG ) GF C Csh+1F G vn,sh+1. However, the explicit computation of (GG ) is not feasible due to memory requirements in realistic applications and the channel-wise computation with a conjugate gradient algorithm is not significantly more efficient then just calculating both vm and vm+1 directly from Equation (7.4). In (undersampled) Cartesian imaging, on the other hand, G is a line selection operation and G∗G becomes an identity matrix. The left side of Equation (7.22) then becomes just a convolution (in the formulation of the Fourier convolution theorem) of the reconstruction vector. 94 Targeted Partial Reconstruction for High-Speed fMRI Neurofeedback

7.3 Methods

All reconstruction methods mentioned in Section 7.2 were tested for validity in reconstructions on MREG datasets in MATLAB. The real-time evaluation of previously targeted ROIs was carried out as a proof-of-concept experiment with MREG, which requires a computationally expensive iterative reconstruction of the images at full resolution. It was implemented in C++ as a so-called functor in the SIEMENS Image Reconstruction Environment (ICE), and the explicit matrix R was loaded and applied by the functor for an exemplary case in a test subject for performance tests and to identify constraints of feasibility.

7.3.1 Data Acquisition

FMRI data acquisition and prescan were performed analogous to the last chapter, Section 6.2.2. Written informed consent was obtained in accordance with a protocol approved by the Ethical board of the University Medical Center Freiburg. The single-shot stack-of-spirals trajectory29 with the parameters from Chapter 6 was used for the ROI localizer, the scan for validation and comparison of the reconstruction, as well as the feedback scan. The data for validation and comparison of the reconstruction, as well as the functional localizer for the feedback scan, originates from an fMRI scan of a normal volunteer with block paradigm. During four periods of 15 s with resting periods of 20 s a flickering checkerboard was displayed via a mirror in the scanner, and the volunteer was advised to execute finger tapping with his right hand during the task period. During the feedback test, the volunteer was advised to repeat the finger tapping to generate signal fluctuations in the previously localized feedback ROI.

7.3.2 General Reconstruction Details

The signal equation was inverted as described in Section 3.4. To speed up convergence, the regularization parameter was chosen such that fast convergence was guaranteed, i.e., λ = 0.2. All reconstructions in this chapter were stopped at 50 iterations. Equation (7.2) then leads to ρ = (E∗E + Γ∗Γ)−1E∗s, and thus v = (E(E∗E + Γ∗Γ))−1V . Note that the pre-computation of each ROI is algorithmically identical to the full reconstruction of one time frame, as expressed in Equations (7.2)-(7.4), and thus leads to identical computational load. During the feedback session, the matrix R was pre-computed in MATLAB on an external computer connected to the scanner console via LAN, and subsequently transferred to the SIEMENS Measurement and Reconstruction System (MARS), where it was loaded into memory by the ICE functor. The MARS was equipped with an Intel Xeon E5-2690 (@2.90 GHz), while the external computer was equipped with an Intel Xeon X7560 (@2.27 GHz) running MATLAB in single thread mode; computation times were calculated by averaging Methods 95 over 5 reconstructions of one ROI. For use during the feedback scan, a straightforward, non-optimized matrix evaluation was implemented in the SIEMENS Image Calculation Environment (ICE, version VD13d). According to Equation (7.3), one dot product for each ROI was implemented in C++, by a for-loop without specific parallelization. To reduce artifacts due to global field fluctuations and drift caused by heating, a frame-by-frame global dynamic off-resonance correction126 was implemented in the reconstruction ICE functor. The correction was applied by multiplying a linear phase to the raw data on the MARS before any further processing. The phase was acquired twice during each read-out: right before applying the read-out gradients and at the data point at TE after gradients have slowed down.

7.3.3 Voxel-wise Reconstruction

For validation of the target-wise reconstruction, ﬁrst a full frame-wise reconstruction was performed, from which the time series of one voxel as target was extracted and compared with the time series extracted using the Targeted Partial Reconstruction. The voxel selected for demonstrational purpose represented an activated area located in the motor cortex.

7.3.4 Volume-wise Reconstruction

As introduced in Section 7.2.2, the ROI-wise Targeted Partial Reconstruction incorporates summation over the target ROIs with complex summands. For investigation of usefulness of the reconstructions’ results, the 2 types of summation that arise in Section 7.2.2, i.e., complex summation and its statically phase-corrected version, were compared to the conventional method of magnitude summation. A second investigation of this subchapter deals with the results of reconstructing a compartment- wise signal model as introduced in existing literature. These results are compared with complex summation over smaller voxels from the full-resolution signal equation, as larger voxels approximate a linear accumulation of signal from smaller voxels.

Comparison of Time Series of Target-Summation Methods

The time series of 3 possible summations over the voxels in the volume of interest are investigated: from the magnitude sums over a volume for each frame, Smag, as used in conventional fMRI methods, from the complex sums, Scom, and from the statically phase- corrected complex sums, Scorr, as feasible with the presented method. For demonstration of the effects occurring in complex and phase-corrected sums compared to the magnitude sums, two activated regions located in the visual and motor cortices were identified by thresholding an activation map obtained from a general linear model analysis, and subsequently evaluated using the three types of sums. All three time series evaluated by the three types of sums were normalized to the first frame of the magnitude sum.

To examine the behavior of Smag, Scom, and Scorr within a greater range of volumes across 96 Targeted Partial Reconstruction for High-Speed fMRI Neurofeedback the sample, a number of target ROIs were identiﬁed by coregistration of the Stanford FIND functional atlas59 with a reference image. The three time series per ROI were scaled by the mean of the magnitude sum to account for target size and signal strength in the subsequent investigation. For the analysis of the time series dependence on intra-volume dephasing, the R dephasing of a ROI was characterized by Dφ, using the complex sum ρdV divided by the magnitude sum R |ρ|dV over the ﬁrst frame of that ROI: Z Z Dφ = 1 − ( ρdV )/( |ρ|dV ). (7.25)

Addressing the potential use of the presented method for feedback generation from changes in connectivity from pre-deﬁned ROIs, the time series of all registered functional ROIs were cross-correlated after applying the magnitude and phase-corrected sums to obtain Smag and Scorr. The correlation matrices of these time series are subsequently called Cmag and Ccorr.

Comparison with Existing Methods

The previously published method108–110 for compartment-wise reconstruction, i.e., by modeling arbitrarily shaped voxels and inverting a simplified signal equation, was tested for conformity to the full reconstruction in the investigated scenario and compared to the presented Targeted Partial Reconstruction. The sum over the full reconstruction is regarded as the standard in this investigation. The time series of the above mentioned activated ROI from the visual cortex, as well as of the full coregistered atlas, were therefore calculated by a signal model which contained voxels with the shape of the ROIs. To account for signal from outside the registered ROIs, the model of the measured field of view was completed by two additional large voxels representing the rest of the brain and the region outside of the brain (model 1), or by 320 large cuboid voxels (24 × 24 × 15 mm3), where the arbitrarily shaped ROIs were trimmed from these cuboids such that no overlap between to voxels/ROIs exists (model 2). The explicit calculation and inversion of this signal model 2 was a complementary investigation; it should be noted, however, that it was computationally too expensive for real-time applications with execution in between reference scan and feedback scan. An even coarser signal model was considered too simplified for an approximate depiction of spatially non- homogenous brain signals. Computations were performed on the same external computer as the partial reconstruction. The discrepancy of results between the simplified signal equation and the sum over voxel values from a full reconstruction was measured by forming the standard deviation of the differences between the time series’ magnitudes along the temporal dimension.

7.3.5 Eﬃciently Approximated Reduced-FOV Reconstruction

The two full-resolution reduced-FOV approximation methods, later referred to as "shifting method" (Fourier Shift Theorem Applied on Single Reconstruction Vectors) and "interpolation method" (Interpolation in Between Multiple Reconstruction Vectors), were compared with Methods 97 the full reconstruction. All voxels in 3 concentric cubes of edge lengths 5, 7, and 9 were reconstructed with both methods. The cubes are located in the frontal lobe, roughly at the supplementary motor area, which is a neurofeedback-relevant area.137 For time series comparison, each voxel’s time series was linearly detrended, demeaned, and the standard deviation from the diﬀerence9 to the full reconstruction calculated for every voxel in the cubes.

7.3.6 Feedback Scan Workﬂow

The extraction of feedback from a target ROI V in the presented manner requires knowledge of the location of the ROI within the subject, as well as the location of the subject. Consequently, as shown schematically in Figure 7.2, before the main feedback scan session, a ROI localizer session with full reconstruction and analysis needs to be performed. In the case of a split into two diﬀerent scan sessions, an anatomical localizer is played out prior to both sessions - the ROI localizer and the feedback session. For sensitivity encoded imaging the anatomical localizers prior to either or both sessions can be extended to sensitivity estimation and potentially oﬀ-resonance estimation, which are needed for the forward model E. The latter together with the ROIs Vb can then be used for the precalculation of the partial reconstruction R right before the feedback scan.

Figure 7.2: Diagram of the workflow of a feedback scan with separate localizer protocol. This figure was used in a first-author publication.64

9As in the rest of the chapter, the diﬀerence is measured at 50 iterations. While fewer reconstruction vectors need to be calculated for the shifting and interpolation methods, adequate convergence of these fewer vectors is crucial. While convergence of a single-time-frame or single-voxel reconstruction is well approximated after 20 iterations instead of the used 50, if the reconstruction is inter-/extrapolated, premature interruption of the conjugate gradient lead to pronounced propagation of reconstruction residuals. 98 Targeted Partial Reconstruction for High-Speed fMRI Neurofeedback

7.4 Results

7.4.1 Voxel-wise Reconstruction

Figure 7.3 depicts the time series of one voxel extracted via target-wise reconstruction and full frame-wise reconstruction of the time series in (a), and their diﬀerence in (b). The time series were normalized to the ﬁrst value of each series, and for purpose of visual clarity only a time window of 32 s is shown. The standard deviation of the discrepancy in magnitude between both reconstructions is 0.0015% at 50 iterations. In over 400 equidistantly spaced voxels across the brain, the mean value of this is 0.0032% and does not exceed 0.034%. This shows that the implemented trajectory and reconstruction lead to accurate solutions at a precision several orders of magnitude below MRI noise level.

Figure 7.3: Subplot (a) shows the frame-wise full reconstruction (black) as well as the target reconstruction (red) of one voxel. (b) depicts the difference between both at 50 iterations. Note the stretched vertical scale, demonstrating that both signal reconstructions are identical considering the given SNR. This figure was used in a first-author publication.64

7.4.2 Volume-wise Reconstruction

Comparison of Time Series of Target-Summation Methods

Figure 7.4 compares the time series Smag, Scom, and Scorr of the two activated representative volumes from the visual and motor cortices. In the shown examples, the statically phase- corrected complex sums qualitatively coincide with the magnitude sums, whereas the signal of the complex sums is attenuated and has decreased BOLD fluctuation amplitude due to the structure of the intra-volume dephasing with a dephasing measure (Equation (7.25)) of Dφ = 0.43 in the visual cortex, and increased respiratory artifacts due to the structure of the intra-volume dephasing with Dφ = 0.84 in the motor cortex. Note that the signal deviations evoked by different types of summation are nonlinear and strongly depend on the static field structure, as well as on dynamical influences including spatial irregularities due Results 99

Figure 7.4: Time series obtained by magnitude sum (Smag), complex sum (Scom), and complex sum with static phase correction (Scorr) over two ROIs from the motor (a) and visual (b) cortices that have been identified by thresholding an activation map. The scales are offset, but identical. The intra-volume dephasing of the ROIs was measured as Dφ = 0.84 (a) and Dφ = 0.43 (b) (Equation [12]). The time window of stimulus is indicated by the shaded area. The static phase correction restores the signal of the magnitude sum almost perfectly. This figure was used in a first-author publication.64

Figure 7.5: Coregistered volumes of the used atlas, color coded according to their degree of dephasing in the investigated subject. The colors range from dark red (strong dephasing) to light yellow (marginal dephasing). Subplot (a) shows the values of the conventional nuFFT-reconstruction, (b) shows the values of a time-segmented off-resonance corrected version. The off-resonance correction significantly decreases the strongest dephasing values introduced by conventional reconstruction following a more basic signal model. This figure was used in a first-author publication.64 100 Targeted Partial Reconstruction for High-Speed fMRI Neurofeedback to physiological fluctuations. The static intra-volume dephasing in an ROI is naturally dependent on anatomy and shimming due to the gradient along occurring off-resonances, in combination with the spatial extent of the ROIs, and therefore subject-dependent. For the investigated case, Figure 7.5 spatially resolves all coregistered volumes of the used atlas and their dephasing, after conventional nuFFT-reconstruction and a time-segmented off-resonance corrected version in subplot (b). It is demonstrated that individual ROIs in certain areas of field variation suffer from strong dephasing which is greatly reduced by the off-resonance correction, while dephasing values in most ROIs remain approximately unchanged. Figure 7.6 shows the analysis of the time series in volumes obtained by coregistration with a functional atlas, with and without off-resonance correction. Following the previous notation, time series Scom and Scorr are compared to time series Smag in each ROI. As expected, the mean signal attenuation over the course of a whole time series shows a strong correlation to the dephasing (considering Equation (7.25)) over each ROI, with minor deviations from the linear relationship due to dynamical fluctuations of the time series. In contrast, the statically phase-corrected sums show restored signal strength at a level of at least 98.5%.

Standard deviations of the time series discrepancies to the magnitude sums Scom − Smag and Scorr − Smag show pronounced scattering with increasing values of ROI dephasing, where the corrected sums’ differ significantly less from the magnitude sums in all ROIs. Off-resonance correction by time-segmented reconstruction reduces all dephasing values and therewith time series discrepancies. However, it does not completely eliminate the effect, due to the spatial extent of the volumes in combination with model imperfections. The Targeted Partial Reconstruction not only leads to high fidelity of activation signals, but can also be used in the context of functional connectivity analysis. The connectivities Ccorr differ from Cmag by 0 ± 0.03 over the whole matrix, as shown component-wise in Figure 7.7.

Comparison with Existing Methods

The exemplary time series obtained by the complex sum over the full-resolution signal model, the Targeted Partial Reconstruction, and the compartment-wise models from the same ROI used for Figure 7.3a are shown in Figure 7.8. The magnitude diﬀerences in the whole time series partly shown in Figure 7.8a had standard deviations of 0.011 (Targeted Partial Reconstruction), 0.051 (compartment-wise model 1), 0.076 (compartment-wise model 2). The calculation and inversion of the matrix of the compartment-wise models took 185 s (137 s for the forward model and 48 s for the inversion) for model 1, and 1205 s (650 s for the forward model and 555 s for inversion) for model 2 on average on one core. The complexity of this calculation increases linearly with the number of voxels/ROIs in the the forward model, and is superlinear for the inversion depending on the used algorithm. Results 101

Figure 7.6: Subplot (a) depicts the ratios of mean signal strengths (µ) in the coregistered ROIs of the complex and phase-corrected sums (Scom and Scorr, respectively) to the magnitude sums (Smag) as a function of the dephasing in each ROI (considering Equation (7.25)) as filled circles. Diamonds mark the standard deviations (σ) of the demeaned and detrended differences to the magnitude sums normalized to their standard deviations. Uncorrected complex sums lead to a nearly linear dependency of signal decay, whereas the phase-corrected sums have nearly fully restored signals. The latter are more conform to the magnitude sums in the dynamical portion of the signal, as can be seen from the standard deviations. Subplot (b) contains similar data from the off-resonance-corrected reconstruction. Dephasing values and associated signal attenuation are reduced by the off-resonance correction. This figure was used in a first-author publication.64

Figure 7.7: The correlation matrices of all ROIs’ time series Smag (a) and Scorr (b) are shown. The matrix in (c) is obtained by forming the component-wise differences between the correlation matrices from (a) and (b). The correlation matrix of the phase-corrected sums agrees extremely well with that of the magnitude sums. This figure was used in a first-author publication.64 102 Targeted Partial Reconstruction for High-Speed fMRI Neurofeedback

Figure 7.8: Subfigure (a) shows the first 32 seconds of detrended and normalized time series from an activated ROI, obtained by the full reconstruction with subsequent sum (black), regarded as reference, by the Targeted Partial Reconstruction (red), and by reconstruction with compartment- wise signal models containing ROI-shaped voxels from a functional atlas (green and cyan). Model 1 (green) provides additionally one rest-brain and one outer-brain voxel, and model 2 (cyan) provides 320 large cubic voxels covering all non-ROI parts of the FOV to account for signal from outside the ROIs in the signal model. The difference between full reconstruction and these time series is shown in (b) and ranges over approximately 2% of the signal intensity for the case of the compartment-wise signal models, whereas it remains below 0.25% for the Targeted Partial Reconstruction. This figure was used in a first-author publication.64

7.4.3 Eﬃciently Approximated Reduced-FOV Reconstruction

The 9 × 9 × 9 voxel volume of the full reconstruction of the first time frame is shown in the first row of Figure 7.9 as a reference. The voxel-wise standard deviation of the demeaned difference of the approximated reconstructions to the full reconstruction is shown: in the second row for Fourier Theorem shifting from the center voxel’s reconstruction; in the third to fifth row for interpolation from the corner voxel’s reconstructions of the concentric 9×9×9, 7×7×7, and 5×5×5 volumes, respectively. While the standard deviations in the second row remain under 0.2% in the direct neighborhood of the center voxel, they exceed 0.5% starting at distances of three voxels. The standard deviations of the interpolations’ time series do not exceed 0.38%, 0.23%, and 0.18% in the cubes of edge length 9, 7, and 5, respectively. Figure 7.10 shows time series examples to illustrate the range of standard deviations in the 5 × 5 × 5 volume for both shifting and interpolation method. The minimal standard deviations are found in the voxels from which the reconstruction is inter-/extrapolated, i.e., the center voxel of the second row and the corner voxels of the interpolated cubes. Accordingly, subfigure (a) shows time series with 0.0015% (red line in the upper part of the subfigure) and 0.00085% (green line in the lower part) standard deviation. The subfigure also shows time series with approximately maximal standard deviations in the 5 × 5 × 5 volume, i.e., 0.17% for the case of the 5 × 5 × 5 interpolated reconstruction (green line in the upper part), and 0.49% for the case of the shifted reconstruction (red line in the lower part). While the maximal deviations in the interpolated volume do not exceed 0.5% in the peaks of the time series, the shifted reconstruction leads to deviations of over 1.4% in the peaks. Results 103

Figure 7.9: Exemplary reduced-FOV of a full reconstruction (first row) and standard deviations of the difference between time series of approximated reconstructions and the full reconstruction (second row: shifting method from the center voxel, third to fifth row: interpolation method from corner voxels of cubes with edge length 9, 7, and 5, respectively). The shifting method leads to standard deviations of up to 0.2% and 0.5% in the inner cubes of edge length 3 and 5, respectively. The interpolation method leads to standard deviations of maximally 0.38%, 0.23%, and 0.18%.

The execution times of the interpolation after calculation of the corner voxels’ reconstruction were: 58, 166, and 313 seconds for edge lengths of 5, 7, and 9, respectively. The execution times of the extrapolation after the calculation of the center voxel’s reconstruction were: 7, 22, and 45 seconds, respectively. These calculations were made in MATLAB on a single core. The precomputation of a single-voxel reconstruction took 71 s per voxel on average on a single core.

7.4.4 Feedback Scan Implementation Performance

In a test experiment with a 52 channel acquisition and 15000 data points per channel per frame the SIEMENS MARS evaluated the dot product for 10 volumes in 2.8±0.1 ms per ROI. The reconstruction matrix R is complex and was stored explicitly in a ﬂoating point data structure on the heap. Consequently, each line of R (one per ROI) consumed 52 · 15000 · 2 · 4 bytes, i.e., less than 6 MB. The pre-computation took 71 s per ROI on average on a single core. The complexity of these computations increase linearly with the number of ROIs. Note that this problem is easily parallelized and the computation time depends linearly on the number of channels. MARS preparation time was 618 ± 1 ms per ROI on average. A feedback example that was shown to the volunteer performing right hand ﬁnger tapping via a monitor 104 Targeted Partial Reconstruction for High-Speed fMRI Neurofeedback

Figure 7.10: Time series from two exemplary voxels marked by red and cyan squares in Figure 7.9 are shown in subfigure (a). The voxels were chosen because their time series reflect the range of deviations for both shifting and interpolation method in the inner 5 × 5 × 5 volume. The deviations are shown in subfigure (b). The minimal standard deviations are found in the voxels from which the reconstruction is inter-/extrapolated, i.e., the center voxel of the shifting method and the corner voxels of the interpolated cubes. They amount to 0.0015% (red line in the upper parts of the subfigures) and 0.00085% (green line in the lower parts) standard deviation. The examples for approximately maximal standard deviations amount to 0.17% (interpolation method, green line in the upper parts), and 0.49% (shifting method, red line in the lower parts). The maximal deviations in the interpolated volume do not exceed 0.5% in the peaks of the time series, the shifted reconstruction leads to deviations of over 1.4% in the peaks.

in the scanner room is shown in Figure 7.11.

Figure 7.11: Extracted time series example from a feedback test, in which the sum over the primary visual cortex was extracted and shown in the scanner room in real-time. This ﬁgure was used in a ﬁrst-author publication.64 Discussion 105

7.5 Discussion

7.5.1 Workﬂow and Implementation

The general constraint for real-time applications is that the reconstruction time cannot exceed TR, which limits the possible number of evaluated target volumes. In the online-tested application with 52 channels and 15000 data points per channel, the achievable number of volumes would easily suffice, e.g., for real-time evaluation of direct feedback from or connectivity between different brain regions. The reconstruction of a larger number of volumes could be achievable using a more efficient implementation, such as using optimized libraries for matrix operations (BLAS) and/or PCA coil compression. Preliminary offline tests in MATLAB suggest that at least 125 volumes with 22 virtual channels can be achievable in less than 100 ms on a current standard PC, enabling the on-the-fly evaluation of an entire functional atlas, or, e.g., a fully resolved 5 × 5 × 5 volume. In the case of many target ROIs, the bottleneck of our implementation was the pre-calculation time of over one minute per ROI per core during the feedback session, which, in a sensitivity- encoded reconstruction method, has to be executed between estimation of sensitivity maps and the feedback scan. Nevertheless, parallelization on multiple cores would be straight-forward across multiple ROIs or channels. Generally, the pre-calculations could be performed parallel to any protocol scans that are desired in the session, e.g., an MPRAGE anatomical scan. Using the presented efficient approximation of fully resolved reduced-FOV reconstruction it becomes feasible to precalculate a reduced-FOV reconstruction of, e.g., a 5 × 5 × 5 voxel volume, on a standard contemporary desktop PC with 8 threads in less than 2 minutes. Our implementation of the forward operator using the Michigan Image Reconstruction Toolbox for the nuFFT interpolation was extensively compared with other existing MRI reconstruction and nuFFT toolboxes implemented in MATLAB or C++, partly incorporating use of a GPU (NVIDIA Tesla C1060), and was found to be the fastest implementation in multi-thread mode with the used parameters and hardware. Even though tested GPU implementations have shown benefit for higher reconstruction matrix sizes, and other implementations would be possible, we decided to settle for this highly efficient CPU implementation with a large number of cores. When a field map is acquired, an off-resonance correction is possible via time segmentation of the forward operator.40 This, however, lengthens the precalculations roughly by a factor of the number of time segments. An alternative for single-voxel reconstructions is to consider only the off-resonance value of the reconstructed voxel and apply it as a global off-resonance shift. Every single-voxel reconstruction vector can then be premultiplied with the negative phases that would otherwise be multiplied to the raw data with a global off-resonance shift. This is an approximation; however, it comes at almost no cost. 106 Targeted Partial Reconstruction for High-Speed fMRI Neurofeedback

7.5.2 Reconstruction quality

In accordance with the derivation in Section 7.2, the target-wise reconstruction was exper- imentally shown to yield the same results as the standard reconstruction. The residual difference between both approaches of the iterative reconstruction problem was shown to be negligible under SNR considerations at a manageable number of iterations. This distinguishes the Targeted Partial Reconstruction from previously applied ROI-wise reconstructions that are based upon simplification of the fine voxel-wise to a coarse compartment-wise signal model that is fully inverted. A refinement of the compartment-wise model in terms of accounting for all regions of the FOV with large voxels, which was computationally already too expensive to be implemented in our experiment due to long precalculation of the inverse matrix, did not lead to any improvement. Most likely, these discrepancies are due to the violation of pre-defined intensity distributions in the signal model of the ROIs over time. When using Targeted Partial Reconstruction, discrepancies are significantly lower and can be attributed to voxel-wise accumulation of residuals from the conjugate gradient algorithm at the chosen number of iterations, as mathematically there is no other difference to the sum over the full reconstruction. For the approximated reduced-FOV reconstruction, it was shown that the interpolation method produces time series of a 5 × 5 × 5 volume with acceptable deviations from the full reconstruction, at an order of magnitude below SNR. While the reconstruction quality slowly degrades for 7 × 7 × 7 and 9 × 9 × 9 volumes, the time series’ standard deviations are still significantly below 0.5%. In the direct neighborhood of the center voxel, i.e., the inner 3 × 3 × 3 volume, the shifting method leads to deviations that are comparable to those reached in the interpolated 5 × 5 × 5 volume. Outside the direct neighborhood, it degrades increasingly and deviations reach noise level.

7.5.3 Reconstruction phase correction

Intra-volume-dephasing that occurs due to complex summation is inherent to the presented method. As the magnitude of a time series that results from complex sums depends on the individual phase and magnitude fluctuations of sub-components of the sums, it is generally not feasible to estimate the dynamic artifacts of an individual ROI. On the other hand, conformity to the magnitude sum is desirable. Results of our investigation showed that the statically phase-corrected complex sums are in very good agreement to the magnitude sums, even though a static phase correction cannot fully restore all dynamical properties of the magnitude sums. The restored mean signal strengths and the overall low signal deviations lead to a restored tSNR . The tendency of the corrected sums to exhibit lower signal deviations agrees with the behavior of a complex sum over an ensemble of components with weakly oscillating phases that results in a less pronounced oscillation when the phases are not spread out. It is important to note that, even though an off-resonance correction did reduce the overall intra-volume dephasing, it did not lead to the same time series restoration that the static phase correction could offer. Conclusion 107

The behavior of the connectivity matrices follows from the time series results, i.e., differences in the time series alter the correlation values. Analytical predictions are hardly feasible, but it can be stated that statically phase-corrected sums showed good agreement to the generally used magnitude sums’ correlations. Due to the cooperative volunteer, no detrimental motion effects were seen even in small ROIs. However, strong motion could potentially cause artifacts in the time series if the displacement is significantly larger than the target ROI. This could be addressed by prospective motion correction,138 which would be applicable to the reconstruction assuming that the change in sensitivity maps induced by the motion is not destructive for the image.139

7.6 Conclusion

A formalism was established for the targeted extraction of real-time feedback from arbitrary trajectories that allow a linear reconstruction, where the feedback from the target region is either equivalent to the sum over fully reconstructed voxels, or even fully resolved if the target region is small. This provides a tool for reconstructing exactly the necessary information, and is useful for experiments using sampling patterns that prohibit full image reconstruction in real-time. This was achieved by explicitly calculating and storing linear combinations of lines of the reconstruction matrix by an incomplete basis change in spatial coordinates, and therewith translating the iterative reconstruction from a frame-wise application to a ROI-wise application that is independent from signal data and can be executed before the real-time experiment. The Targeted Partial Reconstruction of a highly accelerated fMRI acquisition generates near-instantaneous signals as an enabling prerequisite for real-time fMRI applications. The results suggest that feedback signals could be generated within a time which is much shorter than hemodynamic delays. The actual feedback delay which can be realized in practice will, however, depend on the exact nature of data processing used in the feedback loop. It should be noted that the Targeted Partial Reconstruction is not limited to the special case of our MREG-sequence, i.e., sensitivity encoding or non-Cartesian trajectories, but can be very generally applied to any application based on linear reconstruction.

8 Conclusion and Outlook

8.1 Conclusion

The presented thesis extends the line of research around the 3D fMRI method MREG, and aims at enabling and demonstrating the flexibility of the method beyond the capabilities of previously published fMRI methods. As of yet, MREG was primarily focused on acquisition speed, and was limited to single- shot trajectories. As shown by simulation in Chapter 5, however, the k-space sampling density scheme used to achieve the ultra-fast image acquisition induces non-reversible image artifacts in the presence of off-resonance gradients. The chapter also identifies variable- density schemes that alleviate sampling-density induced artifacts but still offer a sped-up measurement compared to constant-density sampling. This effect was then translated comprehensively into MREG measurements using the stack-of-spirals trajectory in Chapter 6. The successful implementation of MREG as a multi-shot method as presented in Chapter 6 leads to substantial flexibility in the acquisition scheme. That flexibility enables the sequence user to trade-off between ultra-high temporal resolution and spatial signal quality. An evaluation of the parameter trade-off identifies excellent spatio-temporal data quality for 3-shot MREG, which is suggested as a viable alternative to single-shot MREG unless the ultra-short TR of single-shot MREG is necessary for the application. In the last part of the thesis, Chapter 7, the image reconstruction of MREG was revised for real- time applicability. The presented framework introduces a novel algorithm for reconstruction in the context of MRI, and it can be employed for any linear image reconstruction with arbitrary trajectory. The derived targeted partial reconstruction does not aim at reconstructing every image volume wholly, it can be tuned to extract just the image voxels or voxel sums that are necessary, e.g., in real-time neurofeedback scans, in order to allow its high-speed application. In conclusion, the methods of accelerated constant-density sampling fMRI and ultra-high- speed non-Cartesian fMRI were conceptionally connected, and a new parameter region of interest was located in the process. Moreover, the identified parameter region as well as the ultra-high-speed parameters were made accessible for real-time fMRI. 110 Conclusion and Outlook

8.2 Outlook

Within the course of this thesis, a number of segmented k-space trajectories with a GRE read-out were explored. That work, however, was preliminary and not written down due to the temporal constraints of a thesis. A multi-shot approach gives the flexibility to combine different read-out directions or different k-space segments in the slowest encoding direction in different trajectory segments. This approach, however, is highly vulnerable to unavoidable imperfections in the reconstruction model, especially in conjunction with long read-outs as in fMRI. The objective of any sampling scheme incorporating long read-outs has to be the image stability under imperfect and discretized signal models. One trajectory example that might lead to benign errors under imperfect signal/reconstruction model and also less off-resonance- induced signal loss than the presented trajectory is the read-out of half stack-of-spirals per shot, in opposite directions. Another example would be to use a cylindrical stack-of-spirals and shift it in the kz-direction. Variable-density-specific aliasing artifacts induced by off-resonance, which were described in this thesis, are hypothesized to be remedied by a spin echo acquisition, as the trajectory deviations can be chosen to vanish in the k-space center.

The reduction of the computational cost of MREG time series reconstruction is a prerequisite for bringing the method to wide application. This can be achieved, e.g., by a truncated singular value decomposition of the time series matrix, but also the method of image reconstruction by machine learning could provide acceleration.

Promising applications of MREG are the investigation of epileptic networks, dynamic evolution and high-frequency components of resting-state networks, simultaneous fMRI and EEG acquisition, and the study of CSF pulsations for the investigation of the glymphatic system of the brain. For the latter, the high temporal resolution enables the detection of CSF pulsations that play a key role in the metabolite waste clearance of the brain. However, at the present time, the contrast mechanism for the depiction of CSF pulsations has not yet been investigated or optimized properly, and the necessary temporal resolution for clinically relevant results is unknown. A Appendix

A.1 Symbols and Deﬁnitions

Physical Constants

γ Gyromagnetic ratio h ≡ 2π~ Planck constant kB Boltzmann constant µ0 magnetic constant

Quantum Mechanics

cms Complex weights of the spin quantum state Ei Energy eigenvalue of quantum state i Hˆ Hamiltonian operator

HˆB Hamiltonian in the rotating frame for a spin in the main magnetic ﬁeld HˆRF Hamiltonian in the rotating frame during the RF pulse ms Spin projection quantum number µˆ Magnetic moment operator

µˆS Spin magnetic moment operator ρˆ Density operator (matrix) ρˆeq Density operator (matrix) at equlibrium s Spin quantum number Sˆ Spin angular momentum vector-operator

Sˆi Spin angular momentum operator of component i σi Pauli matrix of component i |s, msi Spin quantum state in Dirac notation |ψi General quantum state in Dirac notation

Quantities, Vectors, and Matrices

3 A ∈ R Magnetic vector potential α Flip angle

αE Ernst angle 3 B ∈ R Magnetic field vector Bi Magnetic field component i B Magnetic field strength

BRF RF ﬁeld vector 112 Appendix

BRF RF field strength B1 Complex representation of transverse RF field B Receive field of receive coil

B⊥ Transverse magnitude of receive ﬁeld β ∈ C Complex signal decay time constant, including oﬀ-resonance c Receiver coil map cl Receiver coil map of channel l

cm Receiver coil map at position rm C Diagonal matrix with diagonal c

Cmag, Ccom, Ccorr Correlation matrices of time series Smag, Scom, Scorr from functional atlas Dφ ∈ [0, 1] Measure of intra-volume dephasing 3 E ∈ R Electric ﬁeld from the receive coil E Forward operator of the measurement

Enm Matrix entries of E F Fourier matrix 3 G ∈ R Linear encoding gradient G Interpolation matrix of the gridding operation Γ Tikhonov matrix 3 J ∈ R Source current density 3 k ∈ R k-space encoding vector K k-space ﬁlter function due to relaxation (and oﬀ-resonance) λ Regularization parameter M Magnetization density vector

Mi Magnetization density component i M0 Magnitude of the magnetization density at thermal equilibrium M⊥ Magnitude of the transverse magnetization density M˜ ⊥ Complex representation of M⊥ n Spin density PSF Point spread function

φRF Phase of RF field φframe Phase of rotating frame φM Phase of transverse magnetization density in laboratory frame φB Receive field phase Φ Magnetic flux 3 r ∈ R Image space vector R k-space reduction factor

Rr,min, Rr,max Minimal and maximal reduction factors of the spirals Rzsp,min, Rzsp,max Minimal and maximal spiral spacing in z-direction R Partial reconstruction matrix

ρ⊥ Imaged quantity proportional to transverse magnetization density −1 ρ¯ Fw -convolved discretized ρ⊥ ρ¯m ρ¯ at position rm ρ¯rec ρ¯ approximated by reconstruction

Smag, Scom, Scorr Time series of volumes that were obtained by magnitude, complex, or phase-corrected sum S Diagonal matrix with scaling factors of the nuFFT Symbols and Definitions 113

S Demodulated complex MR signal Sl Demodulated complex MR signal of channel l s Demodulated complex MR signal as vector

Sn Demodulated complex MR signal, received at time tn Scoil MR signal induced in a coil Sunﬁltered Coil signal multiplied with complex rotating unit vector σ2 Vector of noise variance in image space 2 σm Noise variance in image space, at position rm 2 σk Noise variance in k-space t Time

TE Echo time of the read-out TR Repetition time of the sequence T1 Longitudinal relaxation time constant T2 Transverse relaxation time constant 0 T2 Field-induced transverse decay time constant (microscopic) + T2 Field-induced transverse decay time constant (macroscopic) ∗ T2 Field- and relaxation-induced transverse decay time constant T Temperature (j) th th Ti Taylor term of the oﬀ-resonance map, i (summed to j ) v(l) Single reconstruction vector (or component of channel l), corresponding to one voxel or a linear combination of voxels

vi One of multiple v - if voxel-wise reconstruction: corresponding to voxel i V , Vi Vector of weights that specify inclusion to an ROI (with index i) w k-space sampling window χ magnetic susceptibility

ω0 Larmor frequency ωRF Frequency of the RF pulse ωnut On-resonance RF nutation frequency 3 ωeff ∈ R Effective RF nutation frequency vector ω0B Nominal main field Larmor frequency δω0 Deviation of Larmor frequency from nominal main field frequency enc δω0 Deviation of Larmor frequency induced by encoding gradients inhom δω0 Deviation of Larmor frequency induced by field inhomogeneities ωor Deviation of Larmor frequency induced by field inhomogeneities

Mathematical Symbols and Operations

≡ Deﬁned as 1 Identity matrix i Imaginary unit, unless used as index variable < Aˆ > Expectation value of Aˆ [A, B] Commutator of A and B ha, bi Scalar product of a and b diag {A} Diagonal of matrix A or diagonal matrix with diagonal A exp Exponential function Tr {A} Trace of operator A

Ri(φ) Rotation by angle φ about axis i 114 Appendix

AT Transpose of A A∗ Conjugate transpose of A A−1 Inverse of A A† Pseudoinverse of A a ∗ b Convolution of a and b a × b Vector product of a and b

am Value of a at position rm an Value of a, received at time tn ash+i Fourier shift theorem applied on vector a in the k-domain, corresponds to shift of i voxels d dt a Total derivative of a with respect to t ∇ Nabla operator

Ff Fourier transform of f −1 Ff Inverse Fourier transform of f S Ff Fourier series of f

δ(x) Dirac delta distribution

δ, δi Kronecker delta corresponding to further speciﬁed voxel (with index i) ei Basis vector of the 3-dimensional Euclidian space, axis i ijk Levi-Civita symbol µ(a) Mean of vector a σ(a) Standard deviation of vector a

XL Sha function/Dirac comb, with spacing L Abbreviations 115

A.2 Abbreviations

BOLD Blood oxygen-level dependent CG Conjugate gradient CPU Central processing unit CSF Cerebrospinal ﬂuid DMN Default mode network (i)DFT (Inverse) Discrete Fourier transform DTFT Discrete-time Fourier transform EEG Electroencephalography EPI Echo planar imaging FFT Fast Fourier transform fMRI Functional MRI FOV Field of view FT Fourier transform FWHM Full width at half maximum GPU Graphics processing unit GRE Gradient echo HRF Hemodynamic response function ICE Image Reconstruction Environment (from Siemens) LAN Local area network MARS Measurement and Reconstruction System (from Siemens) mPFC Medial prefrontal cortex MR Magnetic resonance MREG MR Encephalography MRI MR imaging NMR Nuclear magnetic resonance nuFFT Non-uniform FFT PCA Principal Component Analysis PCC Posterior cingulate cortex PSF Point spread function RAM Random access memory RF Radiofrequency ROI Region of interest RSN Resting state network rtfMRI Real-time fMRI SENSE Sensitivity encoded (parallel imaging) SMS Simultaneous multi-slice SNR Signal-to-noise ratio tSNR Temporal SNR VDS Variable-density sampling

Journal Publications

Bruno Riemenschneider, Burak Akin, Pierre LeVan, and Jürgen Hennig. Trading Oﬀ Spatio-Temporal Properties in 3d High-Speed fMRI Using Interleaved Stack-of-Spirals Trajectories. Magnetic Resonance in Medicine. submitted in 12/2019.

Bruno Riemenschneider conceived the presented idea, carried out a part of the experiments, all simulations, and all analysis except for the resting state analysis, and wrote the manuscript. Burak Akin helped with the resting state analysis, and the corresponding paragraphs in the manuscript, and helped carrying out a part of the experiments. Pierre LeVan and Jürgen Hennig supervised the project, discussed the results, and contributed to the ﬁnal manuscript.

Bruno Riemenschneider, Pierre LeVan, and Jürgen Hennig. Targeted partial reconstruction for real-time fMRI with arbitrary trajectories. Magnetic Resonance in Medicine, 81(2):1118–1129, 2019.

Bruno Riemenschneider conceived the presented reconstruction framework, carried out the experiments, all simulations, and all analysis, implemented the real-time pipeline on the used MRI scanner, and wrote the manuscript. Pierre LeVan and Jürgen Hennig supervised the project, discussed the results, and contributed to the ﬁnal manuscript.

Michael Lührs, Bruno Riemenschneider, Judith Eck, Amaia Benitez Andonegui, Benedikt A. Poser, Armin Heinecke, Florian Krause, Fabrizio Esposito, Bettina Sorger, Jürgen Hennig, and Rainer Goebel. The potential of MR-Encephalography for BCI/Neurofeedback applications with high temporal resolution. NeuroImage, 194:228–243, July 2019.

Bruno Riemenschneider installed the imaging sequence at the cooperation site, gave technical support for its use, discussed results, and helped writing the manuscript.

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Acknowledgments

Besondere Danksagung geht an

Jürgen Hennig für die lange Unterstützung während der Doktorarbeit, sowie die wissenschaftliche Freiheit, eigene Ideen entwickeln und durchführen zu können.

Pierre LeVan für sieben Jahre Schreibtischnachbarschaft, und die vielen beantworteten Fragen.

Maxim Zaitsev für die oﬀene Tür, und die Bereitschaft stets mit großem Know-how zu helfen.

meine Bürokollegen für die gegenseitige mentale Unterstützung, insbesondere an Burak für Gesellschaft in den vielen arbeitsreichen Nächten.

die gesamte Abteilung Medizinphysik für das entspannte Klima.

meine Familie für die fortwährende Unterstützung.