Sodium MRI: Methods and Applications ⇑ Guillaume Madelin A, Jae-Seung Lee A,B, Ravinder R

Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47

Contents lists available at ScienceDirect

Progress in Nuclear Magnetic Resonance Spectroscopy

journal homepage: www.elsevier.com/locate/pnmrs

Review Sodium MRI: Methods and applications ⇑ Guillaume Madelin a, Jae-Seung Lee a,b, Ravinder R. Regatte a, Alexej Jerschow b, a New York University Langone Medical Center, Department of Radiology, Center for Biomedical Imaging, New York, NY 10016, USA b Chemistry Department, New York University, 100 Washington Square East, New York, NY 10003, USA

Edited by David Neuhaus and David Gadian article info abstract

Article history: Sodium NMR spectroscopy and MRI have become popular in recent years through the increased availabil- Received 12 February 2014 ity of high-ﬁeld MRI scanners, advanced scanner hardware and improved methodology. Sodium MRI is Accepted 12 February 2014 being evaluated for stroke and tumor detection, for breast cancer studies, and for the assessment of Available online 7 March 2014 osteoarthritis and muscle and kidney functions, to name just a few. In this article, we aim to present an up-to-date review of the theoretical background, the methodology, the challenges, limitations, and Keywords: current and potential new applications of sodium MRI. Sodium MRI Ó 2014 Elsevier B.V. All rights reserved. Quadrupolar coupling Relaxation In vivo MRI Naþ–Kþ-ATPase Spherical tensors

Contents

1. Background...... 15 1.1. Introduction...... 15 1.2. Sodium in biological tissues ...... 16 1.3. Beginnings of biological sodium NMR ...... 16 1.4. Sodium NMR properties ...... 16 2. Theoretical formalism and pulse sequences ...... 17 2.1. Tensors and hamiltonians...... 17 2.1.1. Irreducible spherical tensor operators (ISTOs) ...... 17 2.1.2. The Zeeman term and the equilibrium density matrix ...... 18

2.1.3. The quadrupolar Hamiltonian ðHQÞ...... 18 2.1.4. The effect of a residual quadrupolar interaction ...... 18 2.2. Evolution and conversion between tensors ...... 19 2.3. Full solution of the evolution equations ...... 20 2.4. Biexponential decay of the free induction decay (FID) ...... 21 2.5. TQF buildup behavior ...... 21 2.6. Relaxation behavior of populations, third-rank tensor buildup after inversion pulse – IR-TQF experiment ...... 21 2.7. Analysis of biexponential decays and optimal settings...... 21 2.8. Notes on phase cycling and pulse flip angles ...... 21 2.9. Triexponential relaxation ...... 22 3. Summary of motional regimes and their effects on spectra ...... 22

3.1. Isotropic motion with motional narrowing ðx0sc 1Þ ...... 23 3.2. Isotropic motion in the intermediate to slow motion regime ðx0sc 1Þ; ðx0sc > 1Þ ...... 23

3.3. Partial alignment and slow motion ðx0sc > 1Þ ...... 23

⇑ Corresponding author. E-mail address: [email protected] (A. Jerschow). http://dx.doi.org/10.1016/j.pnmrs.2014.02.001 0079-6565/Ó 2014 Elsevier B.V. All rights reserved. G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 15

3.4. Compartment models ...... 23 3.5. Discrete Exchange Model (DEM) ...... 23 3.6. Berendsen-Edzes Model (BEM)...... 23 3.7. Other models ...... 24 4. Pulse sequences for compartment and fluid suppression ...... 24 4.1. Techniques based on shift reagents ...... 24 4.2. Diffusion ...... 24 4.3. Inversion recovery (IR) ...... 25 4.4. Multiple quantum filters...... 26 4.5. Optimal control pulse shape design...... 26 5. MRI implementation ...... 27 5.1. Image readout sequences ...... 27 5.2. Sodium quantification...... 27 6. Biomedical applications ...... 27 6.1. Brain...... 27 6.1.1. Stroke ...... 28 6.1.2. Tumors ...... 29 6.1.3. Multiple sclerosis ...... 30 6.1.4. Alzheimer’s disease (AD) ...... 32 6.1.5. Huntington’s disease ...... 32 6.2. Breast ...... 32 6.3. Heart ...... 32 6.4. Skeletal muscle ...... 33 6.4.1. Diabetes...... 33 6.4.2. Muscular channelopathy ...... 33 6.4.3. Myotonic dystrophy ...... 33 6.4.4. Hypertension...... 33 6.5. Cartilage...... 34 6.5.1. Osteoarthritis ...... 34 6.5.2. Cartilage repair ...... 35 6.5.3. Intervertebral disk ...... 35 6.6. Abdomen ...... 35 6.6.1. Kidney ...... 36 6.6.2. Prostate ...... 36 6.6.3. Uterus ...... 36 6.7. Whole body ...... 37 6.8. Cancer and chemotherapeutic response ...... 37 6.8.1. Chemotherapy assessment...... 37 6.8.2. Tumor resistance to therapy ...... 37 7. Conclusion ...... 38 Acknowledgements ...... 38 Appendix A.A.1. Subspace independent of quadrupolar coupling ...... 38 A.1.1. m ¼3 subspaces ...... 38 A.1.2. m ¼ 0 subspace...... 39 A.1.3. Example: inversion recovery ...... 39 A.2. Subspace depending on quadrupolar coupling ...... 40 A.2.1. m ¼1 subspaces ...... 40 A.3. Example: FID when fQ ¼ 0 ...... 41 A.4. m ¼2 subspaces ...... 41 A.5. Summary of decay constants ...... 41 A.6. Triple quantum filter coherence pathways, flip angle, and phase dependence...... 41 A.7. Double quantum filter coherence pathways, flip angle, and phase dependence...... 42 A.8. Summary of pulse phase choices in MQF sequences ...... 43 References ...... 43

1. Background example, in cartilage and in intervertebral disk tissue, sodium ions balance the negative charges of glycosaminoglycans, which 1.1. Introduction leads to an uptake of water into the tissue. The measurement of the sodium concentration directly in the tissues of interest Sodium is the most abundant cation present in the human or in vivo hence is of great interest for providing additional bio- body, and performs a number of vital body functions. The balance chemical information about both normal and abnormal body between intra- and extra-cellular sodium concentration is partic- functions. ularly crucial for insuring the proper functioning of a cell, and Although proton MRI has been highly successful, very often the disturbances of this balance are often signs of disorders (e.g. in information from standard MRI cannot provide direct biochemical stroke or in tumors). In addition, sodium takes part in such pro- markers for cell integrity and tissue viability, or for following cesses as nerve signal transmission and muscle action. A further changes in tissue viability upon treatment. Sodium MRI could pro- frequent function is the regulation of osmotic pressure. For vide some of this complementary information in a quantitative and 16 G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 non-invasive manner. A fact often overlooked is that 23Na also yields the second strongest nuclear magnetic resonance (NMR) signal among all nuclei present in biological tissues, after proton 1H spins. Due to the increase of the available magnetic fields for MRI scanners (1.5 T, 3 T, 7 T, 9.4 T), improved hardware capabilities such as strong gradient strengths with high slew rates, and new double-tuned radiofrequency (RF) coils, sodium MRI is now possible within reasonable measurement times (10–15 min) with a resolution of a few millimeters and has already been applied in vivo in many human organs such as brain, cartilage, kidneys, heart, as well as in muscle and breast. The majority of current sodium MRI applications can be understood without much sophisticated theory. On the other hand, contrast and quantification can in many situations be significantly affected by the underlying spin dynamics which are governed either by a residual or a fluctuating quadrupolar interaction, lead- + + ing to line splittings in the former, and to biexponential relaxation Fig. 1. Schematics of the sodium–potassium pump (Na /K -ATPase). in the latter case. In tissues, a complex mix of the two cases often persists, including the confounding factors of inhomogeneities and The Na+/K+-ATPase (also called sodium–potassium pump, or exchange. Taking full stock of all underlying phenomena and the just sodium pump) is present within the membrane of every ani- information content available from advanced methods both mal cell [19,20]. It is a plasma membrane-associated protein com- ex vivo and in vivo can allow one to extract further tissue parame- plex that is expressed in most eukaryotic cells, whose main ters and provide opportunities for new imaging contrast. function is to maintain the sodium and potassium gradients across In this review article we focus on regimes ranging from liquids the membrane. It therefore participates in the resting potential of to semi-solids, thereby specifically excluding applications in the the cell, by pumping three sodium ions out of the cell while pump- solid state. For applications in solids, we wish to refer the reader ing two potassium ions into the cell (see Fig. 1). to other review articles that cover those fields, their methodology The ion transport is performed against the electrochemical Na+ and applications [1–6]. + and K gradients existing across the cell membrane and therefore 23 Also, we wish to emphasize that Na NMR and MRI applica- requires energy provided by adenosine triphosphate (ATP) hydro- tions in semi-solids are certainly not limited to health-related lysis. This large electrochemical gradient is essential to protect fields, and studies have appeared in fields related to materials the cell from bursting as a result of osmotic swelling and also cre- science as well [7,8]. In this review article, however, we put an ates a potential that is used for transmitting nerve impulses and for emphasis on health-related and physiologically relevant pumping ions (such as protons, calcium, chloride, and phosphate) applications. and metabolites and nutrients (such as glucose and amino acids) A number of excellent review articles focusing on 23Na NMR in or neurotransmitters (such as glutamate) across the cell mem- semi-solids and 23Na MRI should be highlighted here as well brane. ATPase activity and ion transport are intimately linked [9–16]. + + and are two aspects of the same function. Regulation of Na /K - ATPase therefore plays a key role in the etiology of some pathological processes. For example, when the demand for ATP exceeds its 1.2. Sodium in biological tissues + + production, the ATP supply for the Na /K -ATPase will be insuffi- cient to maintain the low intra-cellular sodium concentration Sodium is a vital component of the human body. It is an impor- and thus an increase of intra-cellular sodium concentration can tant electrolyte that helps maintain the homeostasis of the organ- be observed [18,19]. ism through osmoregulation (maintaining blood and body fluid volume) and pH regulation [17]. It is also involved in cell physiology through the regulation of the transmembrane electrochemical 1.3. Beginnings of biological sodium NMR gradient, thus partaking in heart activity, in the transmission of nerve impulses, and in muscle contractions. Sodium concentra- Biological tissues were already investigated with sodium NMR tions are very sensitive to changes in the metabolic state of tissues spectroscopy in the early 1970s [21,22] and with sodium MRI in and to the integrity of cell membranes. the early 1980s [23–25], first on animals in vivo and then on hu- The intra-cellular fraction makes up approximately 80% of the man brain [26] and human heart and abdomen [27]. Sodium MRI tissue volume with a sodium concentration of 10–15 mM, and was thereafter applied to brain tumor and ischemia detection in the extra-cellular volume fraction (including the vascular compart- the late 1980s [28]. In the 1990s there was an increase of interest ment) accounts for the rest, with a sodium concentration of 140– in sodium MRI, due to the increase of the magnetic fields in scan- 150 mM. Cells in healthy tissues maintain this large sodium con- ners, improvements of electronics and RF coils, and new rapid se- centration gradient between the intra-cellular and extra-cellular quences that allowed one to acquire sodium images within a few compartments across the cell membrane, and any impairment of minutes with millimeter resolution [29]. New contrasts such as tri- the energy metabolism or disruption of the cell membrane integ- ple quantum filtering [30–32] provided further promise of tissue rity leads to an increase of the intra-cellular sodium concentration. discrimination and characterization. This trend continued and The sodium flux in and out of cells can occur by several mecha- intensified through the 2000s until today. nisms. Examples of these include voltage- and ligand-gated Na+ channels, Na+/Ca+ exchangers (NCX), Na+/H+ exchangers (NHE), 1.4. Sodium NMR properties + + + Na /bicarbonate (HCO3 ) cotransporters, Na /K /2Cl cotransporters, Na+/Mg+ exchangers and most importantly the Na+/K+-ATPase The NMR-active isotope of sodium is 23Na with close to 100% [18]. natural abundance. Its Larmor frequency is approximately 5% G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 17

Table 1 selectively detected with multiple quantum filtered (MQF) se- Ranges of sodium concentrations and relaxation times in some human tissues in vivo quences [35], thereby providing additional opportunities for the (T and the fast and slow components of T , i.e. T , and T , respectively). These 1 2 2;fast 2;slow identification of intra-cellular sodium. ranges are based on Refs. [11,36–42] and references therein. WM – white matter, GM – grey matter, CSF – cerebrospinal fluid.

+ 2. Theoretical formalism and pulse sequences Tissue [Na ] (mM) T1 (ms) T2;fast (ms) T2;slow (ms)

Brain 2.1. Tensors and hamiltonians WM 20–60 15–35 0.8–3 15–30 GM 30–70 15–35 0.8–3 15–30 CSF 140–150 50–55 – 55–65 2.1.1. Irreducible spherical tensor operators (ISTOs) Cartilage 250–350 15–25 0.5–2.5 10–30 The ensemble spin-states of a spin 3/2 system can be described Blood 140–150 20–40 2–3 12–20 by a 4 4 density matrix Muscle 15–30 12–25 1.5–2.5 15–30 0 1 q11 q12 q13 q14 B C X B q q q q C B 21 22 23 24 C q ¼ðqijÞ¼ qijjiihjj¼@ A; q31 q32 q33 q34 13 23 ij larger than the one of C, and 26% of the proton frequency. Na q q q q has a nuclear spin of 3/2, and hence exhibits a nuclear quadrupolar 41 42 43 44 moment. The NMR sensitivity of sodium is 9.2% of the proton sen- where the spin-states jii2fj1i; j2i; j3i; j4ig correspond to the eigen- sitivity and the concentration in vivo is approximately 2000 times states of the z-component of the spin angular momentum operator, 3 1 1 3 lower than the water proton concentration. As a consequence, so- Iz, with the magnetic quantum numbers 2 ; 2 ; 2 ; 2, respectively. 1 dium MRI has on average a signal-to-noise (SNR) ratio which is While in spin-2 NMR it is common to use the Cartesian tensor oper- 3000–20,000 times lower than the proton MRI SNR (depending ators or product operators [43–45], irreducible spherical tensor on its concentration and visibility in organs or tissues). operators (ISTOs) are more suitable for expanding the matrix into The nuclear quadrupolar moment Q interacts with the electric individual components for a spin P 1=2. The ISTOs for the spin

field gradients (EFG) generated by the electronic distribution 3=2 case are made up of a set of 16 operators Tlm with rank around the nucleus [3,33,34]. In liquids this interaction is often l ¼ 0; 1; 2; 3 and order m ¼l; ...; þl [46–48], allowing a complete averaged to zero. In the intermediate (semi-solid) regime, e.g. in and orthogonal expansion of the form biological tissues, the quadrupolar interaction results in a biexpo- X c T ; 1 nential relaxation behavior, and with anisotropic motion, line split- q ¼ lm lm ð Þ l;m tings may appear. The dominance of the quadrupolar relaxation mechanism for NMR signals can allow a sensitive characterization with clm being complex numbers. Several tensor normalizations are of the molecular environment of the sodium ions. A short T2 com- in use, which differ by a pre-factor which depends on l and the spin ponent T2;fast ¼ 0:5–5 ms generally contributes to 60% of the signal, number I. The most common normalization is the one listed in and a long component T 15–30 ms corresponds to 40% of the 2;slow ¼ [46,47]p,ffiffiffi which provides the spin-independent relationship signal. In order to detect both T2 components, imaging techniques I ¼ 2T11, and also allows one to couple tensors using Cle- with ultrashort echo times (UTE) of less than 0.5 ms are required. bsch–Gordan factors without additional conversion factors. With Typical ranges of sodium concentrations and relaxation times in this convention, the ISTOs have the matrix representations as listed some human tissues in vivo are given in Table 1. It has been shown in Table 2. The ISTOs can also be written in the form of linear that in the slow-motion regime, where biexponential relaxation is combinations of Cartesian or raising and lowering operators (Iz observed, multiple quantum coherences (MQC) can be created and and I ¼ Ix iIy) as shown in Table 3.

Table 2 m y Matrix expressions of the irreducible spherical tensors Tl;m in the jii basis. The expressions for Tl;m are found by Tl;m ¼ð1Þ Tl;m, where y means hermitian conjugation (i.e. simple transposition in the case of real entries of these matrices). 0 1 1000 B C B 0100C T ¼ B C 0;0 @ 0010A 0001 0 qffiffi 1 0 1 3 00 0 0 3 00 2 B 2 ffiffiffi C B 1 C B p C B 0 2 00C B 00 2 0 C T ¼ B C T ¼ B qffiffi C 1;0 @ 1 A 1;1 B C 00 2 0 @ 3 A 00 0 2 00 0 3 2 00 0 0 0 qffiffi 1 3 000 B 2 C 0 pffiffiffi 1 0 pffiffiffi 1 B qffiffi C 0 3 00 00 3 0 B 3 C B C B pffiffiffi C B 0 2 00C B C B C B qffiffi C B 0000ffiffiffi C B 00 0 3 C T2;0 ¼ B C T2;1 ¼ @ p A T2;2 ¼ @ A B 3 C 0003 00 0 0 B 00 2 0 C @ qffiffi A 0000 00 0 0 3 00 0 2 0 1 0 1 3ffiffiffiffi 3ffiffiffiffi 0 1 0 1 p 000 0 p 00 3 3ffiffi 2 10 10 000 p B C B qffiffiffiffi C 00 2 0 2 B 9ffiffiffiffi C B C B C B C B 0 p 00C B 0033 0 C B 000 3 C B 000 0C T ¼ B 2 10 C T ¼ B 10 C T ¼ B 2 C T ¼ B C 3;0 B 9ffiffiffiffi C 3;1 B C 3;2 @ A 3;3 B C 00p 0 3 000 0 @ 000 0A @ 2 10 A @ 00 0 pffiffiffiffi A 10 000 p3ffiffiffiffi 000 0 000 0 2 10 00 0 0 18 G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47

Table 3 jV zzj P jV yyj P jjV xx , and to define V zz ¼ eq (where e is the unit Relationship between irreducible spherical tensor (ISTO) and Cartesian operators electric charge, and q the field gradient per unit charge) and to de- (spin-independent). fine the anisotropy parameter ISTO Cartesian decomposition Description V xx V yy 6 T00 1 Identity g ¼ ; 0 g 1: ð6Þ V zz T10 Iz Longitudinal magnetization 1ffiffi T11 p I Rank 1 SQC 2 The quadrupolar interaction is then represented by the quadru- T20 p1ffiffi 3I2 IIðÞþ 1 Quadrupolar order polar Hamiltonian 6 z T 1 Rank 2 SQC 21 2 ½Iz; I þ X2 2 M T22 1 I Rank 2 DQC H 2 Q ¼ xQ ðÞ1 F2;mT2m; ð7Þ 3 T30 p1ffiffiffiffi Octupolar order m¼2 5Iz ðÞ3IIðÞþ 1 1 Iz 10 qffiffiffiffihi T31 1 3 3 1 Rank 3 SQC 4 10 5Iz IIðÞþ 1 2 ; I where the quadrupolar coupling angular frequency is qffiffihi þ T32 1 3 2 Rank 3 DQC 2 2 4 Iz; I e qQ þ xQ ¼ ; ð8Þ 3 T33 p1ffiffi I Rank 3 TQC 6h 2 2

SQC, DQC and TQC stand for single, double, and triple quantum coherences. with Q the nuclear quadrupole moment. The spatial tensors Flm in The anticommutator for the operators A and B is defined as ½A; B ¼ AB þ BA. the principal axis frame (PAS) of Vab take the form þ rffiffiffi 2.1.2. The Zeeman term and the equilibrium density matrix PAS 3 F2;0 ¼ ; The Zeeman Hamiltonian in angular frequency units (or units of 2 PAS ð9Þ J/h) is given by F2;1 ¼ 0; H PAS g z ¼x0Iz ¼x0T10; ð2Þ F ¼ : 2;2 2 with x ¼ c B the Larmor angular frequency, c ¼ 70:808493 0 Na 0 Na The dependence of the interaction on orientation of the PAS 106 rad T1 s1 the sodium gyromagnetic ratio, and B the magnetic 0 with respect to the magnetic field can be described via field. The chemical shift range of 23Na is very small, and its contri- X bution is therefore ignored in this equation and for much of the rest l PAS Flm ¼ Dm0;mða; b; cÞFlm0 ð10Þ of this article (a notable exception is the discussion of dynamic fre- m0;m quency shifts and paramagnetic shifts which will be mentioned l using the Wigner rotation matrices ðD 0 Þ for the Euler angles later). m ;m a; b; c [49]. The Zeeman interaction is by far the most dominant interaction in most 23Na NMR applications, In this case, one can write the equi- 2.1.4. The effect of a residual quadrupolar interaction librium density matrix via the Boltzmann distribution as We may consider two cases here, with very similar outcomes:

expðhx0Iz=kTÞ 1 (1) HZ HQ and no motion or (2) partial averaging and downscal- qeq ¼ 1 þ hx0Iz=4kT; ð3Þ Z 4 ing of the quadrupolar interaction such that HZ hHQ i. In both cases one can truncate the quadrupolar Hamiltonian to the secular where Z ¼ Trfexpðhx0Iz=kTÞg; 1 is the identity matrix, and the last component, step is made by use of the high-temperature approximation. Since rffiffiffi 1 does not lead to any observables, and does not participate in hi H 3 xQ 2 the evolution, one typically omits this term and writes Q ¼ xQ T20 ¼ 3I IðI þ 1Þ : ð11Þ 2 2 z qeq ¼ aIz; ð4Þ This interaction then predicts the appearance of three reso-

nance lines with distance 3xQ =2p between them as shown sche- where the Boltzmann factor a ¼ hcB0=4kT is likewise frequently matically in Fig. 2 type-a. The relative intensity of the lines is omitted. 3:4:3, which originates from the observable signal as determined by TrfIy qg. The relative ratio then amounts to the relative values 2.1.3. The quadrupolar Hamiltonian ðH Þ Q of the squares of the off-diagonal elements of the T tensor. The quadrupolar interaction arises from the electrostatic energy 1;1 The outer transitions are typically called the satellite transitions, term involving the charge distribution of the nucleus and the elec- and the inner one is called the central transition. tric field gradient around the nucleus [3,33,34]. The quadrupolar In the second case, where the quadrupolar coupling is the result term is the first non-vanishing term in a series expansion, with of some downscaled coupling via reorientation motion, x would higher order terms (hexadecapole) being negligibly small. Q represent the temporally and spatially averaged coupling constant Since the nuclear quadrupole moment can be assumed to be hx F i in Eq. (11). Strictly speaking, x would also contain the constant, the nuclear quadrupole interaction is determined by Q 2;0 qffiffiffiffiffiffiffiffiffiffiffiffiffiQ g2 the orientation, magnitude and temporal average of the electric term from the anisotropy factor, 1 þ 3 , but since this factor field gradient (EFG) generated by the electronic configuration of can rarely be measured independently in motionally averaged sys- the molecular environment surrounding the nucleus. If Vðx; y; zÞ tems, we will not carry it along here. is the electrostatic potential produced by the electrons at the point In solids, situations may arise when the secular approximation ðx; y; zÞ, the EFG can be described by the tensor Vab with the is no longer justified. It is common to include the second order per- components: turbation term in the overall Hamiltonian. We neglect this compo- 2 nent here, as this case is not relevant to tissues or semi-solids as @ V V ¼ ; ðwith a; b ¼ x; y; or zÞ: ð5Þ considered here. ab @a@b Fig. 2 shows different motional regimes that can be observed,

If we choose the principal axes of the symmetric tensor Vab as and their ensuing spectra. The effect described here corresponds coordinate axes, the cross-terms vanish, i.e. V xy ¼ V xz ¼ V yz ¼ 0. It to the ‘type-a’ category as classiﬁed by Rooney and Springer [12– is common to label the remaining components such that 14]. ‘Type-b’ spectra arise when the sample is inhomogeneous G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 19 Wiley-Blackwell 1991 Ó

Fig. 2. Typical energy levels and corresponding NMR spectra of sodium nuclei in different environments. Four types of motionally narrowed SQ spectra are possible (a, b, c, and d). The type-a (crystal-like) and type-b (powder-like) spectra are shown of Na+ in aqueous suspensions of oriented and unoriented dodecyl sulfate micelles, respectively. The type-c (homogeneous, biexponential, super-lorentzian-like) spectrum is that of Na+ in an aqueous solution that has a high concentration of micelle-solubilized gramicidin channels. The type-d (liquid-like) spectrum is from NaCl in H2O. We can make the correspondence xQ ¼ 3xQ in type-a. Reproduced with permission from Ref. [12]. and contains regions with a distribution of quadrupolar couplings, 2. The quadrupolar coupling Hamiltonian can only change the leading to a distribution of level shifts as indicated in Fig. 2. The ef- rank l. fect of a fluctuating quadrupolar interaction is illustrated with the 3. Quadrupolar relaxation can only change the rank l. ‘type-c’ regime and leads to biexponential relaxation as discussed later. Finally, if the fluctuations are very fast, or in the absence of After a 90y pulse acting on equilibrium magnetization, one ob- a quadrupolar interaction one recovers the spectrum of ‘type-d’ tains I p1ffiffi T T . A constant residual quadrupolar cou- q ¼ x ¼ 2 ð 1;1 1;1Þ where all transitions are equivalent. In such a case, the spin transi- pling (type-a) would lead to evolution within the following tions do not differ from those of a spin 1/2 nucleus. subspaces

2.2. Evolution and conversion between tensors fT1;1; T2;1; T3;1g and fT1;1; T2;1; T3;1gð13Þ

due to the commutation relationships In this section we first provide a general overview of the effects ffiffiffi and conversions that typically occur in experiments, and provide a p ½T2;0; T1;1¼ 3T2;1; full solution later. The evolution of a 23Na nuclear spin can be de- pffiffiffi scribed by the master equation given as 6 3 4ffiffiffi ½T2;0; T2;1¼ T1;1 þ p T3;1; 5 5 d b H b th qðtÞ¼i½ ; qðtÞ CfqðtÞq g; ð12Þ 3ffiffiffi dt ½T2;0; T3;1¼p T2;1: ð14Þ 5 where th is the density operator in the thermal equilibrium state, q b H the spin Hamiltonian, and Cb the relaxation superoperator. On the other hand, if only quadrupolar relaxation were active, The total Hamiltonian H in the rotating frame can for most the double-commutators ½T2;0; ½T2;0; q , ½T2;1; ½T2;1; q , and 23 ½T2;2; ½T2;2; q in the Redfield relaxation expressions that make practical purposes in Na NMR of tissues and liquids be assumed b up C can only connect operators within the following subspaces to be equal to HQ (one may wish to allow for inhomogeneous broadening, in which case a distribution of HZ terms would be fT1;1; T3;1g; fT1;1; T3;1g; fT1;0; T3;0g: ð15Þ added).

In order to follow the sequence of events in experiments at high Hence, in isotropic liquids only odd rank tensors T31 can be formed magnetic ﬁelds, one can follow these simple rules: [3,48]. under the inﬂuence of quadrupolar relaxation. In anisotropic media, where the residual quadrupolar interaction does not average to

1. A non-selective radiofrequency (RF) pulse changes the coher- zero, rank 2 tensors T21 can be also formed. ence order m within the limits of jjm 6 l. The amplitude induced A multiple-quantum ﬁltered (MQF) experiment can be used to by the rotation is given by the corresponding Wigner rotation distinguish between the two cases [35,50–52]. For example, the

element, so a rotation with ﬂip angle h and phase / of Tl;m to T2;1 term generated in the ﬁrst case can be converted to a 0 l Tl;m0 gives rise to a factor exp ðÞi/ðm mÞ dm;m0 ðhÞ, where double-quantum coherence (DQC) term T2;2 by another pulse, l dm;m0 ðhÞ is the reduced Wigner rotation matrix element. while this term is not available due to relaxation alone. The 20 G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47

2.3. Full solution of the evolution equations

In this section we provide the complete solution of the evolution equation, Eq. (12), based on residual quadrupolar coupling of the form of Eq. (11) and a fluctuating quadrupolar interaction of the form of Eq. (7) with the definitions in Eqs. (8) and (9) as the relaxation mechanism. Using the Redfield formalism, one can write the action of the relaxation superoperator as

ÈÉX2 ÂÃÂÃ b th m th C q q ¼ A ð1Þ T2;m; T2;m; q q ðÞJðmxÞiKðmxÞ ; m¼2 ð16Þ

where JðxÞ is the spectral density function, and iKðxÞ is the imag- inary portion arising in the Fourier Transform from the integration of the autocorrelation function from 0 to 1 (rather than from 1 to 1) [33], and A a constant used for conversion between different conventions as explained below. The K term gives rise to a small frequency shift, called the dynamic frequency shift. For the typical case of exponential autocorrelation functions, the relationship

KðxÞ¼xscJðxÞ holds. The shift is rarely observable [57], since it Fig. 3. Multiple quantum filter (MQF) pulse sequence. For the triple quantum filter is small (typically within the linewidth of the transitions [58]). (TQF), h1 ¼ h2 ¼ h3 ¼ 90 , and a typical phase cycle is /1 ¼ 30 ; 90 ; 150 ; 210 ; We therefore neglect it for the rest of this treatment. 270 ; 330 ; /2 ¼ /1; /3 ¼ 90 , /acq ¼ 0; 180 . For the double quantum filter (DQF), The convention used in our article is characterized by A ¼ 1 and h h h 90, and a typical phase cycle is / 0; 90; 180; 270, 1 ¼ 2 ¼ 3 ¼ 1 ¼ has the advantage that all quadrupolar-coupling-related expres- /2 ¼ /1 90 ; /3 ¼ 90 ; /acq ¼ 0; 180 . For DQF with magic angle pulses (DQF- sions are included in JðxÞ. As a result, the expressions as listed be- MA), h1 ¼ 90 ; h2 ¼ h3 ¼ 54:7 , and the phase cycling is the same as for DQF. For the IR-TQF sequence, h1 ¼ 180 ; h2 ¼ h3 ¼ 90 , and no additional 180 pulse is used. The low can be used directly for all fluctuation regimes, including phase cycle is /1 ¼ 30 ; 90 ; 150 ; 210 ; 270 ; 330 ; /2 ¼ /1 90 ; /3 ¼ 90 , direct fluctuations of electric field gradients, for example due to /acq ¼ 0; 180 . The delay d is kept as short as possible to avoid signal decay. These ion solvation or due to motion near a polyelectrolyte. For ease of minimal phase cycles could be augmented by an independent 2-step cycle on the comparison between different conventions, we list here the follow- first pulse (with an accompanying 0; 180 cycle on the receiver), and a 2-step cycle on the 180 pulse, giving a 12 or a 24-step cycle for the TQF/IR-TQF experiments, ing definitions of the spectral density functions and conversion fac- and a 8 or 16-step cycle for the DQF. tors A for an isotropic tumbling model: Convention I; ð17Þ A ¼ 1; ð18Þ corresponding pulse sequence is shown in Fig. 3, along with the 2 2 different phase cycles that would be used for the experiments. ð2pÞ C sc JðxÞ¼ Q : ð19Þ One can also separate the regimes to some degree using a triple- 2 60 1 þðxscÞ quantum filtered (TQF) experiment by a judicious choice of pulse flip angles, as discussed in more detail below. Here CQ is the ‘quadrupole coupling constant’ and has the usual def- 2 Since multiple quantum coherences (MQCs) cannot be detected inition qCQffiffiffiffiffiffiffiffiffiffiffiffiffi¼ e qQ=h (as noted before, the term from the anisotropy directly, the final RF pulse converts the MQCs into single quantum g2 factor, 1 þ 3 , would be contained in here as well, which we drop coherences (SQCs), T ; T ; T , which then evolve under the ac- 11 21 31 for simplicity). Convention I is the one used in this article, as well as, tion of relaxation and the residual quadrupolar interaction (RQI) for example in Refs. [10,59–61]. into detectable SQC (T ) during the acquisition time t .A 1;1 acq Other conventions are in use, of which two popular ones are 180 RF pulse can be applied in the middle of the preparation time s in order to refocus any chemical shifts, or more typically mag- Convention II; ð20Þ netic field inhomogeneities. Since MQFs are very sensitive to RF 1 A ¼ ; ð21Þ imperfections, it has been shown that filters without this refocus- 3 ing pulse generate a better signal and are more robust to RF 2 2 ð2pÞ C sc inhomogeneities for in vivo MQF MRI [53]. Recently, procedures Q JðxÞ¼ 2 ð22Þ 20 1 þðxscÞ have been published for avoiding B0 inhomogeneities by separately acquiring and storing the individual phase cycling steps and used for example in Refs. [48,62], and subsequent post-processing before co-adding the individual signals [54–56]. Specific absorption rate (SAR) considerations also fa- Convention III; ð23Þ vor the sequence without the 180° pulse. The formation of DQC 1 ð2pÞ2 A ¼ C2 ; ð24Þ and TQC is detected separately by choosing the appropriate RF 3 40 Q pulse flip angles and phase cycling. Typical phase cycles and 2sc choices of RF pulse flip angles are given in the caption of Fig. 3. JðxÞ¼ 2 ; ð25Þ 1 þðxscÞ DQF can detect the contribution of two tensors, T22; T32, due to the slow motion regime or/and RQI in anisotropic media. If which is used for example in [35,63]. Note that for C given in Refs. h1 ¼ h2 ¼ h3 ¼ 90 , contributions from both T21 and T31 can be [35,63], the relationship C ¼ 3A holds. detected. If h1 ¼ 90 and h2 ¼ h3 ¼ 54:7 (magic angle), only the Closed solutions can be obtained by using the tensor operator contribution of T22 is detected, which arises only from the aniso- basis or the single transition basis. There is no obvious advantage tropic RQI. Only T33 due to the slow motion regime can be of one vs. the other. Eventually, it is most useful to represent the detected by the TQF sequence. final evolution expressions in the form of spherical tensor G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 21

Table 4 2.6. Relaxation behavior of populations, third-rank tensor buildup Conversion amplitudes in the equation of motion, Eq. (12), from a tensor T to a l2 ;m after inversion pulse – IR-TQF experiment tensor Tl1 ;m. The expressions for the negative orders m are identical except for complex conjugation. For brevity, we use the notation Jm ¼ Jðmx0Þ here. One can observe the relaxation behavior of the Iz-term by fol- m l1 l2 Amplitude lowing the populations after a 180° inversion pulse as a function of 011 6 ðJ þ 4J Þ the delay . The Zeeman term also has a biexponential relaxation 5 q1 ﬃﬃ 2 s 013 18 2 behavior according to Eq. (A.11): 5 5ðJ1 J2Þ

0226ðJ1 þ J2Þ R s R s 6Jðx Þs 6Jð2x Þs qffiffi e 1;fast þ 4e 1;slow ¼ e 0 þ 4e 0 : ð28Þ 031 2 4 5ðJ1 J2Þ 033 6 In an IR-TQF (Fig. 3) experiment, one first generates the non- 5 ð4J1 þ J2Þ equilibrium state q ¼Iz ¼T10 by a 180° pulse and then the 111 3 ð3J þ 5J þ 2J Þ 5 pffiffiffi0 1 2 third-rank component builds up as a function of s, which can sub- 112 9 i 2x 5 qffiffi Q sequently be filtered out by a TQF element. The T term builds up 113 30 9 3ðJ J Þ 5 5 0 2 according to Eq. (A.11) with an amplitude proportional to 121 p3ffiffi ix 2 Q R s R s 6Jðx Þs 6Jð2x Þs e 1;fast e 1;slow e 0 e 0 : 29 1223ðqJ0 ffiffiffiffiþ J1 þ 2J2Þ ¼ ð Þ 123 3 3i xQ qffiffi10 The buildup of the third rank component can again be thought 131 2 3ðJ J Þ of as arising from the faster relaxation of the populations in the q5ffiffi 0 2 132 6 outer energy levels due to larger fluctuations. 2i 5xQ 133 3 5 ð2J0 þ 5J1 þ 3J2Þ 2.7. Analysis of biexponential decays and optimal settings 2223ðJ þ 2J þ J Þ p0 ffiffiffi 1 2 3 223i 3xQ 2pffiffiffi The buildup of the third-rank component as measured by the 2322i 3xQ TQF and IR-TQF experiments can be analyzed as illustrated in 2333ðJ0 þ J2Þ Fig. 4. Fig. 4b illustrates the optimal setting for the delay s based 3333ðJ1 þ J2Þ on the two relaxation rates R2 and R1 in the curve. Fig. 4c can be used to determine the maximum achievable signal at the optimal delay setting. It is clear that this signal goes to zero as the relaxa- transformations. The results are easy to interconvert from one tion rates become equal to each other, indicating that no third-rank basis to another. The right hand side of Eq. (12) can be evaluated components were built up. Fig. 4d shows how the maximum in the tensor basis as summarized in Table 4. Note, that these achievable signal depends on the ratios Jðx0Þ=Jð0Þ and results are completely equivalent to those given in other works Jð2x0Þ=Jð0Þ in the TQF experiment. The largest amplitudes are [48,62] when considering that there Convention II for scaling achieved in the cases where the ratios go to zero (for example in JðxÞ, and normalized tensors withÂÃ symmetric and antisymmetric the very slow motion regime, as discussed below). linear combinations (T s p1ffiffi T T , and T a p1ffiffi ½T lmð Þ¼ 2 lm þ l;m lmð Þ¼ 2 lm Tl; m) were used. From these expressions, one can then derive full 2.8. Notes on phase cycling and pulse flip angles analytical solutions to the evolution for a number of special cases. The complete solution is given in Appendix A, and we highlight One can distinguish between the TQF, IR-TQF, and DQF experi- some of the main results in the text below. ments on the basis of a suitable combination of flip angles with appropriate phase cycles [15,64,65]. At the core of the selection 2.4. Biexponential decay of the free induction decay (FID) process is a 6-step cycle (with 60° intervals) for the first two pulses in the TQF/IR-TQF experiments (selecting triple quantum coher- The free-induction decay after single pulse excitation in the ab- ences), and a 4-step cycle with 90 intervals for the DQF experi- sence of quadrupolar coupling (fQ ¼ 0) can be found from Eq. ment (selecting double quantum coherences), as illustrated in (A.29) to be biexponential Fig. 3. For the TQF experiments, the flip angles of the second and third 2 R t 3 R t 2 ½3Jðx Þþ3Jð2x Þt 3 ½3Jð0Þþ3Jðx Þt e 2;slow þ e 2;fast ¼ e 0 0 þ e 0 : ð26Þ pulses should be 90° to maximize the signal intensities (see a com- 5 5 5 5 plete list of the flip-angle dependencies of different pathways in The origin of the biexponential relaxation can be understood as Appendix A). For the DQF experiment, suppose that the coherence arising from the faster relaxation of the satellite transitions pathways involving the second-rank tensors T2;1 need to be exclu- compared to the central transition, because they experience larger sively selected, the existence of which will show whether there is a fluctuations from the instantaneous quadrupolar interactions. residual quadrupolar coupling or not. By setting the flip angle of either the second or the third pulse (h2 and/or h3) to the magic angle, the coherence pathways involving the third-rank tensors 2.5. TQF buildup behavior T3;2 will be suppressed, since they have a sin h2 sin h3ð1 2 2 3 cos h2Þð1 3 cos h3Þ dependence (see Appendix A). By changing According to Eq. (A.29), a free evolution delay s after a single both flip angles from 90° to the magic angle, the signal from the pulse gives rise to the appearance of a third-rank tensor with an coherence pathways involving the second-rank tensors T2; 1 is amplitude proportional to reduced by sin h2 sin h3. Such a sequence is also typically called

R s R s ½3Jðx Þþ3Jð2x Þs ½3Jð0Þþ3Jðx Þs DQF-MA (with MA short for magic angle). e 2;slow e 2;fast ¼ e 0 0 e 0 : ð27Þ The flip-angle-based selections mentioned here represent spe- The third-rank component, when converted into triple-quan- cial cases of rank-selection procedures, which are in principle tum coherences is filtered out using a TQF element (Fig. 3). Record- extendible to any spin value [66,67]. ing the buildup behavior of this component can be used to extract There is one important requirement on the phases of the second relaxation properties, or to filter out slow motion sodium. and third pulses in the TQF and IR-TQF sequences that are needed 22 G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47

A BC

R t R t R t Fig. 4. (A) Plots of a biexponential function e 1 e 2 with R2 ¼ 1:1R1 (red), 2R1 (blue), 5R1 (purple), and 10R1 (green). The dashed line is the plot of the function e 1 alone. R t R t R t R t (B) Plot of a biexponential function e 1 e 2 with R2 ¼ 10R1. As visual guides, the two functions, e 1 and 1 e 2 were plotted (red and blue dashed lines, respectively). R1 R R2 R1 R ln R ln R The biexponential function reaches its maximum 1 1 1 when t ¼ 2 1 . (C) Plot of the maximum of the biexponential function eR1 t eR2 t as a function of the R2 R2 R2 R1 ratio R1=R2 ð0 < R1 6 R2Þ. This ratio is equal to Jð2x0Þ=Jðx0Þ for longitudinal relaxation and ½Jðx0ÞþJð2x0Þ=½Jð0ÞþJðx0Þ for transverse relaxation. (D) Plot of the maximum of

R1 t R2 t the biexponential function e e for the transverse relaxation: R1 ¼ 3Jðx0Þþ3Jð2x0Þ and R2 ¼ 3Jð0Þþ3Jð2x0Þ with Jð0Þ P Jðx0Þ P Jð2x0Þ > 0. With a Jðx0Þ=Jð0Þ and aþb aþb 1b 1b 6 6 b Jð2x0Þ=Jðx0Þ, the maximum is given as 1þa 1þa with 0 < b a 1.

to choose the correct coherent pathways. The phase of the second (for example, for fluid or extra-cellular suppression, as discussed pulse in the TQF sequence should be same as the phase of the first below), which is guaranteed regardless. Therefore, a minimal 6- pulse in order to convert T3;1 to T3;3. If the phase of the second step cycle could work well for TQF/IR-TQF, and a 4-step cycle for (non-180°) pulse is shifted by 90 relative to the phase of the first DQF. Considerations regarding B1 inhomogeneity suppression pulse, no triple quantum coherences are generated from T3;1. One may require longer cycles, to suppress additional coherence path- can find an analogy with the spin-1/2 case, where x magnetization ways as mentioned above and described in Refs. [54–56]. can be converted to z magnetization by a y pulse, but not by an x pulse. Similar considerations apply for the conversion of T to 3;3 2.9. Triexponential relaxation T3;1, and from T30 to T3;3. The different pathways and the acquired phase factors are listed in Appendix A. As a result, in a min- Triexponential relaxation can be observed when cross-corre- imal 6-step phase cycle in the TQF and IR-TQF experiments, one lated relaxation mechanisms are included in the calculations. can in principle distinguish between these experiments by a sim- These could arise from an interference between paramagnetic ple 90 phase shift of the second (non-180) pulse. Details are given and quadrupolar interactions [68,69] (or between chemical shift in Appendix A, and the minimal 6-step cycle is listed in the caption and quadrupolar interactions), but are not common. Other mani- of Fig. 3. festations of these mechanisms are the buildup of a second rank In the presence of B inhomogeneities and other possible flip- 1 component, which complicates analyses which rely on the clean angle errors, however, a clean separation between TQF and IR- separation between a quadrupolar coupling and relaxation effects. TQF is not possible on this basis alone, and it is best to augment Paramagnetic-quadrupolar cross-correlation has been observed for the phase cycle by at least an additional two-step cycle on the first sodium in solutions with high concentrations of paramagnetic pulse (with accompanying 0; 180 shifts for the receiver, and an- molecules [68,69]. other on the 180 pulse (giving a minimum of 12 or 24 steps for TQF/IR-TQF or 8 and 16 for DQF). This precise separation between TQF and IR-TQF is required when the respective buildup curves are 3. Summary of motional regimes and their effects on spectra to be analyzed and fitted. In vivo, such a separation is, however, not always necessary, as one is most concerned with a separation be- Depending on the motional regime of the system, its energy lev- tween a normal single-quantum and a triple-quantum behavior els and relaxation rate change, giving one or many NMR peaks in a G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 23

spectrum, as shown on Fig. 2 [12]. We summarize below several of 2. If xQ > J2, the relaxation eigenvalues corresponding to the the possible motional regimes and their effects on the spectra outer transitionsqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi are complex and the satellite transitions are [12,14,15] for a single type of sodium, where the motion could 2 2 shifted by 3 ðxQ Þ J2 from the central transition. This case be characterized by a single correlation time sc. This case would either describe reorientation motion, or also the case where EFG corresponds to a powder-like type-b spectrum in Fig. 2, which fluctuations are characterized by such a correlation time. Following was acquired from unoriented micelles in aqueous solution this section, we provide an overview of exchange models for (unoriented liquid crystal). The DQF signal from T22 can also describing exchange between compartments, which can be super- be used to measure xQ . imposed on the motional classification as laid out here. 3. If xQ J2, the energy levels are all shifted by the RQI resulting in three distinct peaks in the spectrum. The central transition

3.1. Isotropic motion with motional narrowing ðx0sc 1Þ and the two satellite transitions are separated by 3xQ . This frequency separation between the lines provides therefore indirect In a system of rapid motion such as a ﬂuid, the quadrupolar information about the RQI and the magnitude of ordering in the interaction is averaged to zero on a time scale of 2p and the four en- system. This case can be described by a crystal-like type-a spec- x0 3 trum in Fig. 2 (the spectrum in the Figure was obtained from ergy levels of the spin 2 are equally spaced by the angular fre- 23 quency x0. The Na spectrum is then composed of a single sodium ions in aqueous solution with oriented micelles). The resonance line at x0, as shown in the type-d spectrum in Fig. 2 relative areas of the peaks are 3:4:3 as also discussed earlier. (a spectrum shown of NaCl in aqueous solution). Both transverse and longitudinal relaxation proceed as monoexponential decays. Anisotropic molecular tumbling may also require the incorpora- Such spectra can be obtained, for example, from synovial ﬂuids tion of one or two additional correlation times into the spectral in the body. In this regime, the three transitions behave exactly density expressions [70], to account for the principal axis motion. in the same way, and the spin 3/2 nucleus can be treated as a pseu- do 1/2 spin, simply with a fast T1 and T2 relaxation rate. 3.4. Compartment models

3.2. Isotropic motion in the intermediate to slow motion regime In tissues, Na+ aquo cations exist in compartmentalized spaces ðx0sc 1Þ; ðx0sc > 1Þ (intra- and extra-cellular compartments for example) and encoun- ter an abundance of charged macromolecules. The nature of tissue The satellite and central transitions have different longitudinal is such that most of the sodium spectra are likely to be of type c or and transverse relaxation rates as outlined earlier. type b, or intermediates or superpositions of these. Several models This situation can correspond to a biological-like type-c were therefore developed in order to interpret sodium spectra in spectrum in Fig. 2, which was obtained from solubilized micelles biological tissues and are described in more detail in Refs. [12– in an aqueous solution. It is a ‘‘homogeneous’’ (‘‘biexponential’’, 14] and references therein. The single-compartment model is also ‘‘super-Lorentzian’’) spectrum, in which the satellite peaks have known as the Debye Model (DM). Further models are summarized essentially coalesced into a single, broad, homogeneous peak. The below. relaxation rates are:

R1;fast ¼ 6Jð0Þ; ð30Þ 3.5. Discrete Exchange Model (DEM)

R1;slow ¼ 6Jðx0Þ; ð31Þ The DEM is a more complex model based on equilibrium chem- R2;fast ¼ 3Jð0Þþ3Jðx0Þ; ð32Þ ical exchange between distinct sites that have different quadrupo- R ¼ 3Jðx Þþ3Jð2x Þ; ð33Þ 1;slow 0 0 lar properties. These individual sites can each be characterized by a with the spectral density function given as Debye model (single sc). For example, the James–Noggle exchange occurs between two different type-d sites and can yield only a 2 C2 ð2pÞ Q sc resultant type-d spectrum. Bull exchange happens between a JðxÞ¼ 2 ð34Þ 60 1 þðxscÞ type-d site and a type-c site and can yield either type of resultant spectrum, depending on the lability of the exchange on the time- under Convention I (Eq. (19)). scale determined by the properties of the two sites. Chan exchange occurs between a type-d site and a type-a site and can produce all 3.3. Partial alignment and slow motion ðx0sc > 1Þ kinds of resultant spectra, depending on the lability of the ex- In this regime, the quadrupolar interaction is not completely change or the random orientation distribution of the type-a sites. averaged to zero, and a residual quadrupolar interaction (RQI) A two-site DEM model, however, has at least six (or even eight, if has to be considered. The single-quantum (SQ) relaxation rates are temperature is included) independent parameters to describe the resulting spectra, and a three-site exchange model requires at least k ¼ 3ðJ þ J Þ; 1;0 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13 independent parameters. Analyzing the tissue with a realistic ð35Þ 2 2 DEM is therefore very difficult. k1;1 ¼ 3ðJ0 þ J1 þ J2Þ3 J2 ðxQ Þ as shown in Eq. (A.23), where we have used Jm ¼ðmx0Þ as in 3.6. Berendsen-Edzes Model (BEM) Table 4.

Depending on the magnitude of J2, three situations can occur: The BEM seems is a more realistic model which focuses on the EFG tensor projection ﬂuctuations caused by motions of the inner + 1. If xQ < J2, the relaxation rates are real and there is no line split- hydration shell of the ion Na (aq), that are very powerful and very ting despite the presence of the RQI. The single-pulse spectrum rapid. Slower modulations also occur, usually of lower amplitudes, is the sum of three Lorentzians. The RQI inﬂuences the line- and are generated as the aquo ion diffuses and encounters macro- widths and amplitudes of the components. This case can be still molecules. The BEM mechanism is based on the concept of a sam- described by a type-c spectrum in Fig. 2. In this case, the DQF ple domain, as experienced by diffusing aquo ions, which must be

signal due to even rank coherence T22 can be used to detect at least as large as the average volume sampled by the diffusional and measure xQ . excursions during the lifetime of an NMR coherence. The BEM can 24 G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47

A -x (y, -y) (y, -y)x (y, -y) Acquisition

23Na

Gfreq

Gphas B C 5 1.0 5 1.0

4 0.8 4 0.8

3 0.6 3 0.6 mm mm 2 0.4 2 0.4

1 0.2 1 0.2

0 0.0 0 0.0 012345 012345 mm mm

Fig. 5. 2D 23Na images obtained with the pulse sequences shown in (A) – the quadrupolar jump-and-return (QJR) sequence. (B) The signal is excited by a hard 90 pulse. (C) The QJR sequence with a delay of 2.5 ms was used to excite the signal. The phantom consists of two concentric tubes (3 mm and 5 mm od, respectively), the inner tube ﬁlled with Pf1 bacteriophage solution (fQ ¼ 205 Hz) and an outer with 50 mM NaCl solution. Reproduced with permission from Ref. [109].

therefore be useful for describing spectra by modeling domains (homeostasis, energetic state and sodium pump function) [74].In with the same (type-a) or random (type-b) orientations. A type-c other cases, it is of interest to separate out the immobile sodium spectrum can also be described using BEM for Na+ in blood plasma, fraction in extra-cellular space (such as in osteoarthritis). These for example, where rapid modulations of the inner hydration techniques can be broadly classified as techniques for compart- sphere domain happen together with isotropic slow fluctuations ment and fluid separation, and we summarize them here. of the EFG by diffusion. 4.1. Techniques based on shift reagents 3.7. Other models These techniques are based on the use of a 23Na chemical shift Some other more complex models such as continuous diffusion reagent based on lanthanide chelates, such as Tm (DOTP)5, 7 3 models, models with distributions of sc values, or Debye models DyðPPPÞ2 or Dy (TTHA) [75–77]. These compounds are thought with a distribution of xQ values were also developed in order to to not penetrate the cell membrane and therefore create a fre- describe more accurately sodium spectra in biological tissues. quency offset for the sodium nuclei in the extra-cellular space. The Augustine model [71–73], for example, also describes the Depending on concentration, the shift can be as large as 20– important case of sodium ions in the vicinity of polyelectrolytes. 40 ppm (the largest ones are typically seen for the Tm and Dy com- In this case, relaxation is induced by the diffusion motion of the pounds). These compounds do not cross the blood–brain barrier. ions within the EFG generated by the polyelectrolytes. Furthermore, they are moderately toxic and thus cannot be used in humans. 4. Pulse sequences for compartment and fluid suppression 4.2. Diffusion The most popular sequences currently in use for in vivo sodium MRI are the single-pulse and the inversion recovery (IR) sequence. Diffusion-based techniques can separate the sodium signal Others, such as TQF have, with notable exceptions, mostly been from the intra-cellular and extra-cellular compartments based on used in animal imaging, due to their generally large SAR require- the differences between the motional properties of the ions in ments, especially at high fields when sodium MRI is more sensitive. these two compartments [78–80]. The fast relaxation rates and In many pathological states, the sodium concentration increase is the low gyromagnetic ratio of sodium require the use of very large detected, which can be caused by either an increase of intra-cellu- magnetic field gradients and fast switching, which is challenging lar sodium concentration, the increase of extra-cellular volume but not impossible in clinical MRI scanners. Other ex vivo studies with a constant concentration (140 mM), or an increase of vascu- have used a TQF diffusion filter experiment to separate compart- larization. It is widely believed that the most accurate approach ments. For TQF, the action of the gradient is larger by a factor 3 to study the health of tissues in vivo should be taken by isolating than for SQ coherences. The diffusion attenuation is hence larger the sodium NMR signal from the intra-cellular compartment. by a factor 9. This technique, in combination with 1D spectroscopy, Intra-cellular sodium concentration and relaxation properties allowed one to distinguish several compartments in the optic should give access to additional useful information on cell viability nerve ex vivo [81]. G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 25

AB1 0.5 PBS 0.4

0.3 Agarose 0.5 0.2

0.1

0 0

C 0.5 D 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0 0 E F

23 Fig. 6. Na images of a phantom consisting of an inner 3 mm tube filled with 20% agarose gel (T1;slow ¼ 31:5 ms, T1;fast ¼ 30:3 ms, T2;slow ¼ 28:5 ms, and T2;fast ¼ 3:1 ms) and an outer 5 mm tube filled with PBS (T1 ¼ T2 ¼ 56:4 ms). The images were recorded by using (A) a hard 90 pulse, (B) inversion recovery with a delay of 35.5 ms, (C) spin-echo TQF, and (D) an optimal pulse. The intensities were scaled with respect to the pixel of the highest intensity from the image (A). (E) Comparison of averaged intensities of PBS and agarose gel in the images (A–D). (F) The waveform of the optimal pulse used to get the image (D). Parts of the figures are reproduced with permission from Ref. [60].

Fig. 7. k-Space trajectories for different UTE sequences. (A) 3D radial sequence. (B) Twisted projection imaging (TPI) or 3D cones types of sequences.

4.3. Inversion recovery (IR) sodium or free sodium in ﬂuids can be signiﬁcantly longer than

the T1 of the intra-cellular sodium, IR can be used to eliminate The IR technique can be employed simply based on the differ- the signal contribution from either environment [82–89]. We dis- ence in the T1 relaxation of the sodium nuclei in different cuss sodium quantiﬁcation issues with this technique in the next compartments. As the T1 relaxation time of the extra-cellular section. 26 G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47

Fig. 8. Example of a tissue sodium map (TSC) calculation in articular cartilage. (1) ROIs are drawn over the noise background (which serve as the 0 mM sodium concentration reference) and over reference phantoms (here, 150, 200, 250 and 300 mM). (2) The same number of voxels with higher signal intensity is selected from each ROI on each phantom, and all voxels from the noise background are selected. (3) The mean signal intensity is calculated for the noise and for each phantom, and plotted versus their known sodium concentrations. Simple linear regression is then performed for ﬁtting the points with a function I ¼ a TSC þ b, with I the image intensity, and TSC the tissue sodium concentration of the phantoms (in mM), a and b the unknown constants to be found by regression. (4) The TSC map calculated by TSC ¼ðI bÞ=a.

4.4. Multiple quantum ﬁlters

Since only slow-motion sodium leads to the generation of triple-quantum coherences, a TQF or IR-TQF experiment allows one in principle to separate out the slow-moving sodium [12,14,30,35,57,90–97]. Extra-cellular fluids typically provide for a fast motion regime, where no significant TQF buildup can be observed. As a result, a TQF experiment can therefore be used for the selective detection of intra-cellular sodium. One drawback is the relatively low sensitivity of this method (10% of the single-pulse intensity is a typical in vivo value), and of course another is the need for a relatively long phase cycle. Nonetheless, the technique provides for a very clean separation of the signals. Several reports estimate that the TQF NMR signal in biological tissues comes primarily from the intra-cellular sodium [95,98– 102], but this statement is still controversial and studies on physiological samples have shown a significant contribution of the extra-cellular sodium in MQ-filtered spectra [74,103,104]. DQF experiments can also be used to select signals from ordered environments (such as cartilage) [105–107], by exploiting the appearance of a residual quadrupolar interaction.

4.5. Optimal control pulse shape design

Pulse optimization strategies based on optimal control theory (OCT) have been used to design rf pulses with enhanced performance in spin 1/2 systems [108]. Recently, this approach has also been implemented for 23Na NMR and MRI. In particular, it has been applied to the problem of optimal excitation of the central transi- Fig. 9. Examples of sodium brain images. (A) Twisted projection imaging (TPI) at tion under quadrupolar coupling and relaxation [61], the selective 3 T of a healthy brain. (B) Triple-quantum filtered (TQF)-TPI at 3 T of a healthy brain. Courtesy of Professor F. Boada, New York University Medical Center/University of detection of ordered sodium ions [109], and for the differentiation Pittsburgh Medical Center. between fast and slow sodium ions [60]. The results demonstrated G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 27 that OCT can be quite successful in improving the performance of 23Na NMR and MRI in terms of sensitivity and contrast. The first application of OCT to 23Na NMR was the optimal excitation of the central transition under quadrupolar coupling [110].It led to the discovery of a simple excitation sequence with the central transition reaching the theoretical maximum in the absence of relaxation [109]. The sequence resembles a pulse sequence consisting of two pulses with opposite phases and separated by a delay. By setting the flip angles of the two pulses to the same value, the signals from the sodium ions with no quadrupolar couplings are automatically suppressed. Furthermore, if the flip angle for both pulses is set to the magic angle 54, the satellite signals disap- pear, while the signal of the central transition is proportional to Wiley-Blackwell 2011

2 Ó sin ðpfQ sÞ. Here fQ ¼ 3xQ =2p is the quadrupolar splitting (in Hz) between the resonance lines and s is the duration of the delay between two magic-angle pulses. When the quadrupolar coupling constant and the delay are matched, the signal from the central Fig. 10. Representative perfusion weighted imaging (PWI), diffusion weighted transition can reach 94% of the theoretical maximum. Therefore, imaging (DWI) at 1.5 T and sodium TPI at 4.7 T of the brain of a patient with acute ischemic stroke. These images show the hypoperfused (Tmax + 4s) perfusion maps, the pulse sequence, named the quadrupolar jump-and-return se- the DWI hyperintense region (core) in dotted outline and the PWI-DWI mismatch quence [109], can be used to selectively excite the sodium ions tissue (penumbra) in solid outline, and corresponding sodium images, acquired 4 h in ordered environments (Fig. 5). 180 pulses during the delay and 25.5 h after symptom onset. Reproduced with permission from From Ref. can be used to remove unwanted effects due to static field [141]. inhomogeneities. For situations where quadrupolar relaxation is the dominant evolution mechanism (without a residual line splitting due to The data acquired with all these non-Cartesian sequences are quadrupolar coupling), IR and the TQF can be used to selectively reconstructed using different methods, such as regridding [123– detect such slowly-tumbling sodium ions by exploiting the differ- 125], non-uniform fast Fourier transform (NUFFT) algorithms ence in the relaxation times or the accessible states. IR is a more [126,127], or other iterative methods [128,129]. Undersampled sensitive technique, while TQF is a cleaner one. An optimal shape data can further be processed by compressed sensing [130]. was derived via OCT, which was more efficient than both of these for generating contrast (Fig. 6), thus showing that there is still con- 5.2. Sodium quantification siderable potential in improving sodium excitation sequences. The optimal shapes were further optimized for robustness in the pres- Sodium concentration quantification is generally performed by ence of variations of static and rf magnetic fields [60]. placing phantoms of known sodium concentration and known relaxation times within the field-of-view of the images. It is best to use phantoms with relaxation times that match approximately 5. MRI implementation the relaxation times of the investigated tissue in order to reduce uncertainties in the quantification. A linear regression from the 5.1. Image readout sequences phantom signals, corrected for relaxation, is then used to produce the tissue sodium concentration (TSC) map. For example, reference Due to the short (biexponential) T2 relaxation of sodium in tis- phantoms for brain or muscle imaging are typically composed of sues and in vivo, ultrashort TE (UTE) sequences are recommended 2–3% agar gel with sodium concentrations within the range of to acquire the images. The first sodium images were acquired with TSC usually found in these tissues (10 mM up to 150 mM). For car- 3D gradient echo sequences [111] where the echo time (TE) was tilage, 4–6% agar gels can be used, with sodium concentrations be- minimized by using a non-selective hard pulse, thus avoiding the tween 100 and 350 mM. Fig. 8 shows an example of the TSC use of a slice refocusing gradient, and by applying read and phase quantification procedure in cartilage. gradients with the maximum slew rates and magnitudes. TE as short as 2–3 ms can be achieved with this method, although longer delays are more typical. The shortest TEs (<1 ms) can generally be 6. Biomedical applications obtained by acquiring the data from the center of k-space in a radial or spiral fashion. The most common type of this kind of se- We present here an overview of recent biomedical applications quence is the 3D radial sequence which has been used widely for of sodium MRI for assessing diseases and therapies non-invasively sodium MRI [112]. This method can be further improved with re- and quantitatively in vivo, sorted by the targeted organs or dis- spect to signal-to-noise ratio (SNR) and time efficiency by modify- eases. In this overview, we also focus mostly on human in vivo ing the density of acquisition points along the projections and also MRI, which best illustrate the breadth of applications that newer by twisting the radial projection in the outer k-space in an optimal methodology has enabled. manner in order to fill k-space more homogeneously and thus to improve the point spread function of the acquisition method. This 6.1. Brain is done in sequences such as the density adapted radial sequence [113], the twisted projection imaging (TPI) and its variations Since the first experiments on sodium MRI, many studies have [29,114,115], the 3D cones [116] and in the FLORET sequences been performed on brain, first to show the feasibility of quantita- [117]. Examples of k-space trajectories from 3D radial and TPI se- tive brain sodium MRI, and then to evaluate its possible use for quences are shown in Fig. 7. These latter sequences now represent assessing diseases such as tumors, stroke, multiple sclerosis (MS), the de facto standard for sodium MRI. Other UTE sequences, which Alzheimer’s disease (AD) or Huntington’s disease (HD). include SPRITE [118,119], SWIFT [120], ZTE [121], and PETRA [122] The strategies employed in these areas are to use sodium MRI as are still under investigation for their use for in vivo sodium MRI. a probe for the changes in cellular integrity and viability through 28 G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 changes in intra-cellular sodium concentration and/or extra-cellular volume due to these pathologies [11,26,28,131–133]. Examples of single pulse and TQF brain sodium images are shown in Fig. 9A and B. Other examples of sodium images with new TQF schemes can be found in Refs. [37,54,134] and with IR in Refs. [82,135]. Thulborn et al. [136] proposed to measure tissue sodium concentration (TSC) via 23Na MRI (along with proton MRI) and cell volume fraction (CVF) for assessing diseased tissues in the brain (tumor, stroke). TSC is the volume fraction weighted mean of the intra-cellular sodium concentration ð½Nain ¼ 10—15 mMÞ and the extra-cellular sodium concentration ([Na]ex = 140–150 mM). Therefore TSC = CVF [Na]in +(1 CVF) [Na]ex. CVF is around 0.8 in normal brain tissues. After correction of the fractional average water content in normal brain (0.8), the average TSC in brain is calculated as 45–55 mM, which is the range of values usually measured experimentally. An increase of TSC indicates a loss of tissue viability. TSC maps can be obtained directly from quantitative Fig. 11. Proton and sodium images from a patient with glioblastoma (WHO grade sodium MRI where reference phantoms with known sodium con- IV) of the left medial frontal lobe. (A) T1 weighted MRI, (B) T1 weighted MRI with centrations and relaxation times are placed within the FOV. CVF contrast medium (rim enhancement), (C) T2 weighted MRI showing cystic and solid can be estimated from TSC through knowledge of [Na] and [Na] , portions of the lesion and perifocal brain edema, (D) DWI showing elevated ADC in ex values in the center of the tumor, (E) sodium MRI showing increased sodium signal obtained from TQF sodium MRI, for example [137]. in the tumor, (F) sodium MRI with fluid suppression by inversion recovery (IR), also Brain sodium images have been acquired at fields of up to 9.4 T showing increased sodium signal mainly at the center of the tumor. Proton images in vivo [138,139]. (A–D) were acquired at 3 T while sodium images (E and F) were acquired at 7 T. Reproduced with permission from Ref. [135]. Copyright 2011, Wolters Kluwer Health. 6.1.1. Stroke Stroke is the third largest cause of death in the USA and a leading cause of long-term disability. Stroke can be classified into two subtypes: ischemic and hemorrhagic. Most of the strokes are ischemic (around 87%). It is very important to intervene as early as possible after symptom onset in order to reperfuse viable tissues and minimize tissue loss in order to improve recovery. The usual treatment is thrombolysis by intravenous injection of recombinant tissue plasmogen (tPA) within a 3-h window after onset [140], which induces recanalization of blocked arteries and potentially reperfusion of ischemic tissues. There is a risk, however, also of hemorrhagic transformation or edema formation due to reperfusion in non-viable tissues which can reverse the expected outcome of thrombolysis. It is therefore important to assess the viability of tissues and their likelihood of recovery before the treatment is applied. A combination of proton diffusion weighted imaging (DWI) and perfusion weighted imaging (PWI) [142] has been proposed to help Oxford University Press 2010 identify patients with more probable improvement in outcome Ó after thrombolysis: apparent diffusion coefficient (ADC) maps from DWI can identify regions of cerebral ischemia (water restriction) within minutes after ischemic onset, reflecting cytotoxic edema, and PWI can detect regions with perfusion deficits (regions at risk) within seconds of ischemic onset. It has been proposed that the mismatch area between a larger abnormal PWI region and a smal- ler lesion in the ADC map can represent the ischemic penumbra, where the tissue is at risk for infarction but still viable, and could Fig. 12. Example of brain sodium image on a patient with multiple sclerosis (MS). therefore be saved [143–145]. It has been found, however, that (A) Proton density MRI, (B) T1 weighted MRI, (C) sodium MRI, and (D) correspond- DWI does not necessarily represent ischemic tissue damage, that ing TSC map. Reproduced with permission from Ref. [158]. the limit of salvageable tissue is not limited to the volume of the DWI–PWI mismatch, and that PWI and DWI do not provide information on the duration of the acute period of ischemia, and cannot membrane. An increase in TSC can be associated with an increase establish the time of symptom onset. As a result, patients would be of intra-cellular sodium due to the loss of integrity of the cell, excluded from thrombolysis treatment based on this analysis and also with an increase of extra-cellular volume when cells are [141,146]. dying. Studies in animal models and humans have shown that so- A more direct method, such as quantitative sodium MRI, that dium MRI can measure increases in TSC over the first few hours could give accurate spatial information on the tissue viability and and days after induction of cerebral ischemia, and that sodium also time information about stroke onset could prove to be very MRI could have a potential utility for stroke management useful for stoke management. TSC is very sensitive to cell homeo- [141,146–149]. These studies have further shown that an elevation stasis and any loss of cellular energy production will impair the of 50% in TSC above the TSC value in the homologous region in the Na+/K+-ATPase and induce loss of ion balance across the cell contralateral brain hemisphere was consistent with completed G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 29 Springer Science and Business Media 2007 Ó Fig. 13. Examples of breast cancer images from a patient with 5.5 cm infiltrating poorly differentiated ductal carcinoma (T3) at the 12 o’clock position in the left breast. (A)

Fat suppressed T2 weighted MRI showing a mass with T2 intermediate signal and edematous T2 bright retroareolar glandular tissue (arrow). (B) Fat suppressed T1 weighted MRI post-Gd injection with contour levels in yellow, showing enhancement of the mass at 12 o’clock but not in the retroareolar glandular tissue. (C) Sodium MRI with contours from B superimposed in blue. The region with edema is indicated by the green arrow. Reproduced with permission from Ref. [168].

infarction (corresponding to 70 mM in humans). This value could DWI and PWI. T2-weighted and T1-weighted images with and therefore serve as a threshold for tissue viability and help decision without gadolinium enhancement are used for detecting the loca- making about which treatment should be more suitable for the pa- tion and dimension of the tumor. DWI gives information on the ex- tient. It is also suggested that the different rates of loss of tissue tent of vasogenic edema while excluding cytotoxic edema, and PWI viability are reﬂected in the different rates of change in TSC values can detect the regions of the tumor with high vascularity, which between cortex and basal ganglia. Therefore, clinical decisions to are consistent with high-grade tumors [136]. All these changes, use thrombolytic agents may use different time windows depend- however, are generally late events in tumor development. Adding ing on the location of stroke. These studies showed that sodium sodium MRI to the protocol would therefore provide direct and MRI can be used as a useful complement to DWI and PWI for man- more rapid biochemical information on the tumor metabolism, aging patients with acute and subacute stroke and that TSC evolu- and also help monitor the immediate effects of therapy. Further tion can help guide the thrombolysis protocol outside the 3-h time applications could include a study of the effects of cancer therapies window currently used for treatment decision making. over time. Some examples of sodium images in human stroke are shown in Ouwerkerk et al. [38] combined proton and sodium MRI at 1.5 T Fig. 10C. In this study, Tsang et al. [141] showed that sodium signal to measure the TSC in brain and to determine how TSC is altered in intensity cannot be predicted by PWI and that it was not altered within the PWI-DWI mismatch tissue, irrespective of the interval between symptom onset and image acquisition, indicating pre- served viable tissue in this region. A combination of DWI, PWI and sodium MRI could therefore provide useful information on tissue viability in patients with stroke, despite an unknown symptom onset time. Further studies could include optimized sodium acquisitions, perhaps based on TQF or IR preparation for increasing the weighting of the images towards the intra-cellular sodium content, and thus providing help for stroke management in a quantitative and non-invasive manner.

6.1.2. Tumors

Tumor malignancy can be characterized by angiogenesis and Wiley-Blackwell 2005 cell proliferation [150], among other effects. Unregulated cell divi- Ó sion, leading to tumor growth, can be initiated by changes in Na+/ H+ exchange kinetics and therefore changes in intra-cellular and extra-cellular pH [151]. This mechanism, associated with reduced Na+/K+-ATPase activity [152] leads to increased intra-cellular sodium concentration that can therefore also be associated with tumor malignancy [38,153]. Most likely, the increase in total sodium concentration in malignant tumors depends on both changes Fig. 14. Comparison between 1H and 23Na cardiac images at 1.5 T of healthy 1 in extra-cellular volume fraction and in intra-cellular sodium volunteers. Registered axial H MRI FSE images (A and C), cardiac-gated at 60– content. Similarly, tumor neovascularization and increase in inter- 70 bpm, echo train length = 8, effective TE = 17 ms. Coil sensitivity adjusted axial slices from three-dimensional 23Na TPI data sets of the heart (B and D) with TR/ stitial space both lead to an increased extra-cellular volume frac- TE = 100/0.17 ms, and adiabatic half passage (AHP) excitation from healthy human tion and are also associated with the potential for tumor volunteers. The 23Na image intensity was adjusted for the coil receiving sensitivity 1 proliferation [154]. Overall, total sodium concentration levels in using a computer-generated B1 field estimate of the coil. The H image data were malignant tumors are likely to be elevated, and therefore could interpolated in three dimensions so that images (A and C) match the spatial positions of the images (B and D). The resulting in-plane resolution was 128 128, perhaps be measured by quantitative single pulse sodium MRI at the same slice spacing as the 23Na data. The three-dimensional 23Na image data non-invasively. Implementing TQF or IR in the sodium acquisition were interpolated to 128 128 pixels in-plane to facilitate copying of contours and could also provide information more specific to changes in the in- ROI. Yellow contours are drawn in the 1H images (A and C) and copied to the 23Na tra-cellular sodium content by reducing the weight of fluids (from images (B and D) (shown as gray lines) to guide placement of the ROIs (black lines 23 edema) and/or extra-cellular sodium in image contrast. in the Na images). ROIs were placed in the lateral LV wall, anterior LV wall, and septum (ROI numbers 1–3 or 1–4 in all images). Additional ROIs were placed in the The conventional MRI protocol for brain tumor scanning is left ventricle for LV blood (ROI number 5) and in the right ventricle for RV blood 1 based on T2-, and T1-weighted H images in combination with (ROI number 6). Reproduced with permission from Ref. [174]. 30 G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 Springer Science and Business Media 2010 Ó

Fig. 15. Sodium images in the muscle at rest (A) and post-exercise (B). 3D FLASH proton MRI (C) was performed after sodium MRI to delineate muscle anatomy: TA is tibialis anterior, S is soleus and G is gastrocnemius. In healthy subjects (D), the sodium signal intensity increases significantly in S and G just after exercise, but not in the control muscle TA. The sodium intensity then decreases to near baseline in both S and G with a half-time of around 22 min. In diabetic subjects (E), sodium signal intensity increases significantly in S and G just after exercise, but not in the control muscle TA. The sodium intensity then decreases to near baseline with half-times of around 37 min in S and 27 min in G. Reproduced with permission from Ref. [179]. malignant tumors. Sodium images were acquired with a UTE TPI sequence. Mean TSC (in mmol/kg wet weight) was measured as 60 for grey matter (GM), 70 for white matter (WM), 135 for cerebrospinal fluid (CSF), 115 for vitreous humor, 100 for tumor, 70 for unaffected contralateral tissue, and 100 for regions surrounding the tumors (detected with FLAIR hyperintense proton image). Sig- nificant differences in TSC were demonstrated for both tumors and surrounding FLAIR hyperintense tissues versus GM, WM, CSF, and contralateral brain tissue. This work shows that UTE sodium MRI can be used to quantify absolute TSC in patients with brain tumors and shows increased sodium concentration of 50–60% in tumors relative to that in normal brain regions. These measurements cannot, however, define whether TSC increases are due to changes in extra-cellular volume, intra-cellular sodium content or neovascularization. Nagel et al. [135] recently applied sodium MRI with and without IR to 16 patients with brain tumors of different grades (WHO grades I–IV) at 7 T. Some examples of proton and sodium images are presented in Fig. 11. The authors found that TSC increased in 15 of 16 brain tumors before therapy and that IR imaging enabled Lippincott Williams and Wilkins/Wolters Kluwer Health 2013 Ó further differentiation of these lesions by suppressing CSF and edema signal. Further, all glioblastomas (grade IV) demonstrated higher IR sodium signal intensities as compared with WHO grade I–III Fig. 16. Examples of sodium images (row A) and proton images (row B) of the lower tumors. It was also noted that contrary to total TSC signal, the IR leg of a young healthy volunteer (left column) and of an older volunteer with sodium signal correlated with the histologic MIB-1 proliferation hypertension (right column). Tubes with solutions containing 10, 20, 30, and rate of tumor cells. This study shows that a combination of sodium 40 mmol/L of NaCl are arranged below the extremity, and used as calibration standards for quantification of the tissue sodium concentration. Tissue Na+ content MRI with and without T1 relaxation weighting through IR can reveal important physiological tissue characteristics and help char- increased in the old subject compared to the young subject. No significant difference in muscle water content is visible (B). Reproduced with permission from acterize and grade tumors. Ref. [187]. Very recently Fiege et al. [134] combined single pulse and TQF sodium imaging in a single acquisition scheme and applied this new sequence to 6 healthy brains and 3 brains with tumors at 4 T. In this very preliminary work, however, the authors detected 6.1.3. Multiple sclerosis a decrease of signal in tumor regions on the TQF images compared Another interesting application of sodium MRI in the brain is to normal tissue, probably due to suppression of edema around the the assessment of multiple sclerosis (MS). MS is an inflammatory tumors. Due to the very low resolution of the images (10 mm iso- neurological disease characterized by focal and diffuse inflamma- tropic) and low SNR of the TQF images, it is difficult to reach defin- tion in white matter (WM) and grey matter (GM), by demyelina- itive conclusions yet. tion of the axons and by neuro-axonal injury and loss, but the G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 31

sodium channels can promote the reverse action of the Na+/Ca2+ exchanger, which leads to a metabolic cascade, and results in an overload with intra-axonal calcium and axon degeneration [156,157]. The ﬁrst application of sodium MRI to MS at 3 T was demonstrated recently [158] in 17 patients with relapsing–remitting MS (rrMS) and 13 normal subjects (Fig. 12). Images were acquired at 3 T with a 3D radial sequence and a birdcage coil. The absolute TSC was measured in lesions and in several areas of normal- appearing white and grey matter in patients (NAWM and NAGM in proton MRI), and corresponding areas of white and grey matter in controls. Mean sodium concentrations were 20 and 30 mM in WM and GM in controls, and 27 and 36 mM in corresponding NAWM and NAGM in MS patients. In this preliminary study, TSC in MS patients was therefore elevated in acute and chronic lesions compared to areas of NAWM. TSC averaged over lesions and over regions of NAWM and NAGM matter was positively associated

with T2-weighted and T1-weighted lesion volumes from proton MRI. The expanded disability status scale score showed a mild, positive association with the mean TSC in chronic lesions, in regions Fig. 17. Cartilage sodium concentration maps from a healthy volunteer (control) of NAWM and NAGM. More studies need to be performed to under- and a patient with osteoarthritis (OA). Images were acquired with a 3D radial stand the pathophysiological mechanisms involved in tissue injury sequence (R3D) and IR WURST (IRW). IR WURST was used to suppress ﬂuids through inversion recovery (IR) with an adiabatic WURST pulse in order to increase in MS and their link to sodium images. Separation of intra-cellular the sensitivity of the method to a change of sodium concentration within the and extra-cellular sodium by TQF [137] or IR could prove to be use- cartilage only. MED = femoro-tibial medial and LAT = femoro-tibial lateral cartilage. ful for this purpose. This work shows that abnormal values of TSC Reproduced with permission from [86]. measured non-invasively with sodium MRI in patients with rrMS might reﬂect changes in the composition of the lesions and/or cellular and molecular mechanisms contributing to neurodegener- changes in metabolic integrity. ation are still poorly understood [155]. Studies have shown that Another recent study by Zaaraoui et al. [159] expanded the the accumulation of sodium in axons through non-inactivating method to patients with early and advanced rrMS. It was found Springer Science and Business Media 2012 Ó

1 Fig. 18. A. Sagittal T2-weighted H MR image (left) of the right knee with an osteochondral allograft transplantation (arrowhead) at the weight-bearing aspect of the medial femoral condyle. There is synovial fluid at the articular surface (arrows) in the conventional 23Na TSC map (middle). Hyperintense signal is seen from synovial fluid at the articular surface (arrows) and to a lesser extent in a subchondral location at the repair site (arrowhead). On the TSC map from 23Na-IR MRI (right), there is a visible 1 suppression of signal from free sodium within synovial fluid (arrows) and also in the subchondral location (arrowhead). B. Axial T2-weighted H MR image (left) of the right knee, with a juvenile cartilage cell implantation (arrowhead) at the medial facet of the patella. Structures containing free sodium, such as synovial fluid and popliteal vessels, are indicated by arrows. On the conventional 23Na TSC map (middle), hyperintense signal is seen from synovial fluid at the articular surface and within the popliteal vessels (arrows). On the 23Na-IR TSC map (right panel), a suppression of signal from free sodium within synovial fluid and within popliteal vessels is visible (arrows). The calibration phantoms are seen at top left of the sodium images. Reproduced with permission from Ref. [87]. 32 G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 that TSC increased inside demyelinated lesions in both groups of to disruption of the sodium–potassium pump in cell membranes, patients. TSC was also increased in NAMW and NAGM of advanced quantitative sodium MRI would be a good candidate to detect tu- rrMS patients, but not in early rrMS. mors in the breast and also assess the degree of malignancy and follow-up chemotherapy. To test this hypothesis, Ouwerkerk 6.1.4. Alzheimer’s disease (AD) et al. [168] applied quantitative sodium MRI at 1.5 T and TPI acqui- Finding biomarkers for detecting early signs of AD and tracking sition to patients with benign and malignant breast tumors before response to treatments is the subject of intense research. Methods biopsy. T2 and T1 weighted contrast-enhanced proton MRI were such as sampling the cerebro-spinal fluid (CSF) for biochemical also acquired. Sodium and proton images were co-registered to al- analysis of biomarkers, positron emission tomography (PET), and low quantification of TSC in normal and suspicious tissues based MR imaging (through regional volumetric analysis) or spectros- on 1H MRI contrast enhancement, with histology confirmed by copy (N-acetylaspartate with 1H NMR or glutamate with 13C biopsy. The measured TSC were on average higher by 60% in histo- NMR) have been proposed. logically proven malignant lesions compared to glandular tissue. A first study on the applicability of sodium MRI to Alzheimer’s TSC in benign tumors was significantly higher than in adipose tis- disease (AD) was performed by Mellon et al. [160]. The hypothesis sue but at the same level as in glandular tissue (34 mM). An exam- here is that alterations of the sodium levels in brain due to cell ple of these results is shown in Fig. 13. death or loss of viability characteristics of AD could be measured Increased TSC can arise from an increase of the extra-cellular with sodium MRI non-invasively and could give useful comple- volume fraction (EVF) due to changes in cellular organization, from mentary information for the assessment of early AD. In these stud- an increase of the vascular volume, or from increases in the intra- ies, a small increase (7.5%) of the relative sodium signal intensity cellular sodium concentration due to impaired energy metabolism was found in the brains of patients with AD compared to controls. and Na+/K+-ATPase activity, or a combination of all these factors. Further, it was found that this signal intensity enhancement was Further studies based on fluid suppressed 23Na MRI, combined moderately inversely correlated with hippocampal volume mea- with sensitive proton MRI techniques such as DWI or T1 contrast sured from T1 weighted inversion recovery proton images. No con- enhancement, could further help in assessing biochemical infor- clusive explanation on a physiological basis can be provided at the mation in breast cancer. moment for this sodium content increase. Possible explanations A multiparametric multinuclear combination of MRI techniques are that the extra-cellular volume increases due to cell death and could prove to be useful for grading benign and malignant tumors, fluid invasion, the intra-cellular sodium concentration increases and also for monitoring the response to chemotherapy [169].It due to an impairment of the Na+/K+-ATPase due to amyloid beta was even shown by Jacobs et al. [170] that a multimodal combina- channels in the membrane, or perhaps that a combination of some tion of 1H+23Na MRI with computed tomography (CT) and proton of these effects is at play. More studies that allow fluid suppression emission tomography (PET) was feasible and could help to evaluate and/or intra-cellular sodium isolation (IR, TQF) need to be per- the complex tumor micro-environment by examining the changes formed in order to selectively study these possible aspects of AD in morphology, sodium concentrations and glucose metabolism in progression. response to therapy.

6.1.5. Huntington’s disease 6.3. Heart A very recent preliminary study [161] showed that patients with Huntington’s disease (n ¼ 13) also present increased TSC in Acute myocardial infarction (MI) can lead to an increase in the the whole brain compared to healthy controls (n ¼ 13), in structur- intra-cellular sodium concentration due to loss of ionic homeosta- ally affected regions of the brain, but also in some non-affected re- sis and to an enlarged extra-cellular volume as a consequence of gions. Similar to the AD case, no satisfying explanation of these TSC myocardial edema formation or scar formation [171,172]. Quanti- increases could be proposed due to the limited data, low resolution tative sodium MRI appears therefore to be a good candidate to try and lack of differentiation between intra-cellular and extra-cellular to detect cardiac infarction by measuring localized increased so- sodium content. Further studies may help explain these observed dium content in cardiac tissues and thereby help to differentiate TSC variations which are generally linked to changes in cellular viable from non-viable tissue. Some preliminary studies using sur- and metabolic integrity that lead up to structural degeneration. face coils were performed over the years to test the feasibility of cardiac sodium MRI for assessing infarction [111,173–176]. 6.2. Breast Ouwerkerk et al. [174] measured the myocardial TSC of healthy volunteers. Mean TSC was approximately 43 mM in the left ven- Treatments for breast cancer such as prophylactic mastectomy tricular (LV) free wall, 53 mM in the septum and 17 mM in adipose or chemoprevention are more effective when the disease is de- tissue. The same team then applied the method to twenty patients tected at an early stage. Mammography is the standard method with nonacute MI (Fig. 14). Mean TSC for MI was measured to be for breast cancer screening [162] with high specificity (>90%) but higher by 30% than for noninfarcted tissues in LV regions (signifi- has a very variable sensitivity (30–60%) due to difficult lesion cant difference). The mean TSC in regions adjacent to MI regions detection in dense breast tissue [163–165]. Ultrasound (US) is a was intermediate between MI and normal tissue sodium content. supplemental method for women with high-risk and dense breast The study also concludes that TSC increase was not related to in- tissue that can increase screening sensitivity, but can also lead to farct age, size or global ventricular function. The measured sodium higher false-positive rates [166]. Standard proton MRI has a higher TSC or pixel intensity changes may be attributable to loss of cellu- sensitivity (75–90%) and is less affected by the breast tissue den- lar integrity, inhibition of the Na+/K+-ATPase function due to en- sity, but has lower specificity (70–90%) [163,166]. False-positive ergy depletion and changes in the sodium concentration gradient diagnoses from these screening methods result in increased pa- between intra-cellular and extra-cellular volumes, and possibly tient anxiety, overdiagnosis and unnecessary biopsy, or overtreat- also changes in the molecular environment of the sodium ions. ment [167]. Future studies might include imaging at higher fields for Sodium MRI could improve this situation by adding more infor- increasing the SNR and spatial resolution for better identification mation about the physiology and metabolism of suspicious lesions, of infarcted tissues and adjacent areas, and the use of sequences such as cellular integrity and energy metabolism. As proliferating that could separate intra-cellular from extra-cellular sodium and/ tumors may cause increases in the sodium content of tissues due or relaxation time weighted (TPI associated with TQF or IR), for a G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 33 better estimation of intra-cellular TSC, which is likely to be more relaxation measurements, could prove to be useful to assess the sensitive to cell energy impairment and viability. origin of the sodium intensity decrease observed in this study.

6.4.2. Muscular channelopathy 6.4. Skeletal muscle Patients with hypokalemic periodic paralysis (hypoPP) or para- myotonia congenita (PC), two different kinds of muscular channel- Sodium MRI has the potential to provide insights into muscle opathies, were scanned with sodium MRI in a study by Nagel et al. physiology and disorders. In muscle tissues, the electrochemical [40]. These rare diseases are considered to be caused by genetic gradient across the cell membrane is maintained by the Na+/K+- mutations of the voltage-gated sodium channels in muscular cells, ATPase mechanism, but when an action potential is generated, which can also be characterized by elevated myoplasmic sodium at there is a rapid influx of sodium ions and efflux of potassium ions rest and after cooling (for provoking PC pathology effects). via the sodium and potassium channels, leading to muscle contrac- Three sodium techniques were used to assess the disease: total tion. During intense contractile activity, the persistent influx and tissue sodium concentration (23Na-TSC), T -weighted sodium efflux of ions, which degrades the transmembrane Na+ and K+ gra- 1 imaging (23Na-T ) and inversion recovery (23Na-IR). The latter dients, can lead to a loss of membrane excitability and muscle con- 1 was used to suppress most of the extra-cellular and vascular so- tractility [177], which is believed to represent one of the main dium and vasogenic edema. All 23Na sequences showed signifi- mechanisms of muscle fatigue [178]. cantly higher signal intensities in hypoPP compared with PC It has already been shown that many disease states, such as dia- patients and healthy subjects, and provocation in PC induced a sig- betes, starvation and hypothyroidism, can be linked to a decrease nificant increase (> 20%) in the 23Na-IR signal and a corresponding in Na+/K+-ATPase activity in skeletal muscle [177]. Therefore so- decrease of muscle strength. Signal intensities from 23Na-T and dium MRI has the potential to play a role in imaging of these dis- 1 23Na-TSC also have a non-significant tendency to be higher after orders. For example, Bansal et al. [39] studied the change of provocation in PC. This study indicates that 23Na-IR could provide sodium concentration and relaxation in muscle after voluntary a stronger weighting toward intra-cellular sodium than 23Na-T muscle contractions. The authors showed that the sodium inten- 1 or 23Na-TSC, and therefore may allow a better visualization of sity in the images increased by 34% in the exercised muscle changes in the intra-cellular sodium content. A combined applica- and then diminished with a half-life time of 30 min. At the same tion of 23Na-TSC and 23Na-IR could therefore improve the analysis time, the calculated sodium concentration did not change signifi- of pathophysiological changes in muscles of patients with muscu- cantly, while the long T component of the sodium relaxation 2 lar channelopathies by measuring the changes in intra-cellular increased. This work therefore suggests that the change in inten- concentration and T relaxation time. sity in the sodium images is mainly due to a change in the so- 1 The same kind of study was also performed on patients with dium-macromolecule interaction rather than a change in TSC. We hyperkalemic periodic paralysis (hyperPP) by Amarteifio et al. describe below a few examples of sodium MRI studies of different [181]. They showed that sodium MRI can detect increased myo- muscular diseases. plasmic sodium content in HyperPP patients with permanent weakness, as they are affected by an incomplete inactivation of 6.4.1. Diabetes muscular sodium channels [182]. In this case too, 23Na-IR is more 23 23 Sodium MRI has been studied recently in diabetic patients. sensitive to intra-cellular changes than Na-TSC and Na-T1.In Some results from the study by Chang et al. [179] are presented conclusion, sodium overload may cause muscle degeneration in Fig. 15. Pre- and post-exercise sodium intensity was measured developing with age and sodium MRI could therefore help moni- in sodium images in healthy volunteers and in patients with diabe- toring medical treatments that reduce this overload. tes, in the tibialis anterior (TA) as control muscle, in the soleus (S) and the gastrocnemius (G) muscles. It was found that the muscle 6.4.3. Myotonic dystrophy sodium signal intensities (in S and G) increase significantly imme- Myotonic dystrophy has been linked to alterations in sodium diately after exercise and that afterwards this sodium signal recov- channel conductance regulation, which can cause elevated muscle ers down to baseline more slowly in diabetics than in healthy fiber concentrations that correlate with disease severity [183]. subjects. Constantinides et al. [184] have found in their study that the The increase in sodium signal intensity could be due to two mean TSC measured with sodium MRI after exercise was elevated main factors: (1) an increase in total muscle sodium content due by 16% and 22% in two healthy volunteers, and 47% and 70% in two to an increase in the volumes of both the intra-cellular and ex- dystrophic muscles compared with those at normal resting levels. tra-cellular compartments after exercise as well as an increase of These results in patients with myotonic dystrophy are consistent intra-cellular concentration by muscle cell depolarization and (2) with the known imbalance in sodium homeostasis in dystrophic alterations in the macromolecular environment and therefore muscle fibers [185]. Quantitative sodium imaging could be a valu- changes in T2 relaxation times or in the proportions of the long able tool for characterizing the early onset, pathogenesis, and mon- + + and short T2 components. Na /K -ATPase preserves the muscle itoring of pharmacologic treatment of dystrophic muscle, but more membrane excitability by maintaining the Na+ and K+ concentra- data need to be acquired to confirm these findings. tion gradients across the cell membrane and therefore helps protect muscles against fatigue [177]. Diabetics, however, have a 6.4.4. Hypertension decreased Na+/K+-ATPase activity and decreased numbers of Na+ Another possible application of quantitative sodium MRI in and K+ pumps on the muscle cell membrane, which has been muscle is to measure the increase of body sodium content due to attributed to insulin resistance [180]. This effect would result in hypertension, which is linked to a disturbed total body sodium a decreased ability to extrude intra-cellular sodium ions into the regulation. Kopp et al. [186] measured the sodium content in the extra-cellular space and leads to elevated intra-cellular sodium triceps surea of healthy volunteers and of patients with primary content in muscle with a slower recovery to baseline. aldosteronism, before and after treatment. They found a 29% This preliminary study shows that sodium MRI could therefore increase in muscle Na+ content in patients with aldosteronism be applied to patients with diabetes in order to evaluate those who compared with normal women and men. This tissue Na+ content are at risk of diabetic muscle infarction. Moreover, fluid and/or was then reduced to normal levels (20–25 mM) after successful extra-cellular sodium suppression by TQF or IR, along with T2 treatment without accompanying weight loss. The authors suggest 34 G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 Wiley-Blackwell 2006 Ó

Fig. 19. Examples of kidney images. (A) Scheme of a human kidney. Central coronal slices of the 3D sodium images of a human kidney under normal conditions (B) and after 12-h water deprivation (C). The sodium gradient increases signiﬁcantly by 25% after water deprivation. Reproduced with permission from Ref. [214]. that sites such as muscle (and also skin) could serve to store Na+ aggregation of the PG. These changes lead to an alteration of the nonosmotically by binding of the Na+ ions to proteoglycans within mechanical properties of cartilage, which can therefore lose its the extra-cellular compartment without apparent accompanying load- and shear-bearing functions. At present, there does not exist ﬂuid retention or changes in serum Na+ concentration in patients either a known cure or a preventive treatment for OA, and present with primary aldosteronism. treatments focus mainly on pain management with analgesics and In a subsequent study [187], Kopp et al. measured the sodium in quality of life (e.g., exercise and weight loss). If these methods are muscle and skin of 57 patients with hypertension and 56 control ineffective, joint replacement surgery may be considered. Early subjects at 3 T. Representative images are shown in Fig. 16. The detection of OA, prior to irreversible morphological changes, and authors noticed an increase of Na+ deposition in skin in both an accurate method for quantifying the effects of potential treat- men and women with increasing age, and an age-dependent in- ments are therefore of fundamental importance. crease of sodium content in muscle in men, but not women. A Radiography is the standard method used to detect gross loss of slight increase of sodium concentration in both muscle and skin cartilage by measuring joint space narrowing, but does not image was detected in men and women with refractory hypertension, cartilage directly. Proton MRI, such as T1-weighted, T2-weighted compared to age-matched controls. These studies show promise for hypertension assessment by sodium MRI. These techniques could also help testing the role of Na+ for assessing long-term car- diovascular risk in populations.

6.5. Cartilage

Cartilage is a dense connective tissue that can be found in many parts of the body such as articular joints between bones (hyaline cartilage), in the ear and nose (elastic cartilage) or in intervertebral disk (fibrocartilage). Here, we will mainly focus on articular hyaline cartilage, which consists of a small population of chondrocytes (5% of volume) within a large extra-cellular matrix (ECM) made of type II collagen fibers (15–20% of the volume), proteoglycans (PG; 3–10%) and water (65–80%) and does not contain blood vessels. PGs further consist of a protein core and negatively charged glycosaminoglycan (GAG) side chains, which endow cartilage with a negative fixed charge density (FCD). This FCD attracts free-floating positive coun- ter-ions, such as Na+, which in turn attract water molecules within

the cartilage through osmotic pressure. The negative charge of the IOP Publishing Ltd 2012

GAG side chains also provides a strong electrostatic repulsive force Ó between the PG molecules and is responsible for the compressive stiffness of cartilage. The collagen ﬁbers serve to immobilize the PG and provide a tensile force opposing the tendency of the PG to expand the cartilage. Due to these properties, articular cartilage can provide synovial joints with lubrication and can also serve to absorb mechanical shocks and to distribute load over the underlying bone [36,188].

6.5.1. Osteoarthritis Osteoarthritis (OA) is the most common form of arthritis in synovial joints and a leading cause of chronic disability, mainly in the elderly population. From the biochemical point of view, OA is a degenerative disease of the articular cartilage, characterized by a reduction of FCD (or GAG) concentration, possible Fig. 20. Whole body sodium MRI of a volunteer. Reproduced with permission from changes of size and organization of the collagen fibers, and Ref. [219]. G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 35 and proton density imaging, can provide morphological informa- cartilage did not result in better clinical outcome of the MACT pa- tion on damage of cartilage, such as fissuring and partial- or full- tients, which could then corroborate the hypothesis that the in- thickness cartilage defects, but does not give any information on crease of signal might not originate only from GAG content but GAG content within the ECM or the structure of the collagen fiber also from fluids. The absence of fluid suppression and of TSC quan- network. New methods for functional proton MRI of cartilage are tification make the results of this study difficult to interpret and under development, such as T1q mapping [189], T2 mapping reproduce in other research labs. As a short TR of 10 ms was used, [190], GAG chemical exchange saturation transfer (gagCEST) the contribution from fluid signal is expected to be strongly atten- [191–193], delayed gadolinium enhanced MRI of cartilage (dGEM- uated, but still present. The long TE (3.77 ms) for sodium acquisi-

RIC) [194] and diffusion weighted imaging (DWI) [195]. T2 map- tion might also lead to a significant loss of signal in native and ping and DWI can give information about collagen fiber and repair cartilage (which might have different T2 relaxation behavior water mobility, while the other methods can provide information — generally biexponential and very short in cartilage). Histological on the FCD or GAG content, indirectly through the use of contrast measurements from biopsies from repair cartilage have shown that agents (dGEMRIC), or directly (T1q mapping, gagCEST). All these more fibrocartilage (with less GAG) is present in BMS repair regions methods are still under investigation for assessing their specifici- [208,209], and more hyaline-like cartilage is present in MACT re- ties and sensitivities for detecting early OA. gions [210], which could be correlated with the results of this study. It has been shown that sodium concentration has a strong cor- Chang et al. [87] evaluated the feasibility of fluid-suppressed relation with FCD and GAG content in cartilage [196–198], there- sodium imaging with inversion recovery (IR) at 7 T for detecting fore quantitative sodium MRI could also be useful for detecting cartilage repair in the knee of eleven patients. These patients had directly GAG loss in early OA [199,41,200]. Several studies have al- undergone different reparative and restorative cartilage repair pro- ready been performed on healthy and OA cartilage, and show that cedures. Images were acquired during follow-up after several in general TSC in healthy cartilage is in the range of 250–350 mM, weeks (range: 12–151 weeks) after surgery. Selected T2-weighted while it is typically lower than 250 mM in OA cartilage [200]. Be- proton and TSC maps from sodium images are shown in cause of the low resolution of the sodium images (>3 mm), due Figs. 18A and B, with and without fluid suppression, respectively. to the presence of synovial fluid or joint effusion and also possible These images were obtained from a person who had undergone cartilage thickening, the sensitivity of the method to measure osteochondral allograft transplantation, and one with juvenile car- small changes of TSC within the cartilage should include fluid sup- tilage cell implantation. In all cases, the area of cartilage repair was pression by either TQF [31,201] or IR [83] to suppress signals from more easily detectable on fluid-suppressed TSC maps, visible as a surrounding fluids. ‘hole’, thus indicating that GAG is not present in those areas. This An example of quantitative sodium MRI of cartilage at 7 T in pilot study shows the feasibility of fluid-suppressed sodium MRI presented in Fig. 17, where fluid suppression was obtained by IR for cartilage repair imaging and indicates that sodium MRI could with an adiabatic inversion pulse [83]. The images were acquired prove useful for detecting the evolution of GAG content during fol- with a 3D radial sequence. It is shown that fluid suppression allows low-up of the repair procedure. More data from more subjects and a better differentiation between control and OA patient. This work over more time points need to be acquired for evaluating the utility is still preliminary but it is reproducible and repeatable [88]. Fur- of this imaging technique to assess cartilage repair in the knee ther information could be obtained from sodium T1 and T2 rates, joint. For best quantification results, further studies could also take which are expected to be very sensitive to any changes of their into account the changes in T1 and T2 sodium relaxation times environment, such as GAG depletion or collagen fiber rupture. within the repair and adjacent regions in cartilage, as they may Some reviews on the potential of sodium MRI and OA are pre- also vary due to changes in the cartilage matrix structure and sented in Refs. [36,89]. GAG concentrations. The quadrupolar interaction itself has been shown to increase Two other preliminary studies have compared sodium MRI with with degradation in cartilage ex vivo [16,197,202], and this effect dGEMRIC and gagCEST for evaluating cartilage in patients after re- has also been shown to occur in osteoarthritic and osteoporotic pair procedures. Schmitt et al. [207] scanned 12 MACT patients cartilage [202]. These interactions could, in principle be used for with both sodium MRI at 7 T and dGEMRIC at 3 T. A significant cor- producing alternative image contrast (quadrupolar contrast relation (R2 ¼ 0:7, p = 0.001) was found between the results from [61,109,110,203,204]), but its implementation in imaging has not these two methods after MACT procedure. Trattnig et al. [206] been explored yet. scanned 12 patients (5 MFX and 7 MACT) at 7 T with sodium MRI and gagCEST. A moderate correlation (R2 ¼ 0:49, p not given) 6.5.2. Cartilage repair was found between the results from these two methods after re- Sodium MRI could also prove to be useful for assessing cartilage pair procedures. repair [87,205–207]. For example, Zby` nˇ et al. [205] compared sodium and proton images at 7 T from 9 subjects who underwent 6.5.3. Intervertebral disk reparative bone marrow stimulation (BMS) procedures (micro frac- The processes governing GAG and sodium concentration are ture and subchondral drilling) and 9 matched subjects who under- very similar in the intervertebral disk. Ex vivo studies have demon- went restorative cell-based procedures (MACT). No sodium strated that sodium concentrations could also be used to report on quantification was performed, but a sodium normalized mean sig- GAG conditions in the disk [198]. Insko et al. [211] demonstrated nal intensity (NMSI) was calculated. For comparison between dif- the feasibility of sodium MRI of the intervertebral disk (IVD) ferent subjects, a repair-to-reference ratio was also calculated for in vivo at 4 T using a surface coil and a 3D gradient echo sequence. each subject. Significantly lower average sodium NMSI were found This technique could be used to measure the proteoglycan content in repair tissue compared to native healthy cartilage, from both in fibrocartilage and help detect early degenerative changes in IVD. MACT and BMS repair procedures. Higher sodium NMSI were mea- In a subsequent recent study, Wang et al. [212], showed that so- sured in MACT subjects compared to BMS subjects, which were dium MRI can correlate with loss of proteoglycans in the spine. interpreted as resulting from a better quality of repair from MACT due to higher GAG content. As no fluid suppression was performed 6.6. Abdomen in the sodium acquisition, this increase of sodium signal within the repair region could also arise from partial fluid invasion of the Quantitative sodium MRI could also prove to be useful for scaffold in MACT. It is noticed that higher sodium NMSI in repair assessing cell viability in abdominal organs for the purposes of 36 G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 Wiley-Blackwell 2005 Ó

Fig. 21. Examples of sodium MRI and proton DWI studies on tumor and chemotherapeutic response in a rat glioma. (A) Sodium images (top) and proton ADC maps (bottom) of a BCNU-treated 9L rat glioma acquired at days 0, 7, and 23 (left to right). BCNU injection is performed at day 17 after tumor implantation. The central sodium image at day 7 shows a dramatic increase of sodium concentration throughout the entire tumor area. Image at day 23 shows tumor regrowth after tumor shrinking started at day 9 and its maximum regression at day 16. (B) Correlation of tumor sodium concentration and ADC in rat glioma obtained at various points following a single dose of BCNU. (C) Time course after tumor implantation of sodium (left) and ADC (right) variations for a naive type 9L glioma. Sodium and ADC data are given in percent relative to normal contralateral brain. All data are presented as mean ± standard deviation. Sodium concentration steadily increased in the naive tumor at a rate of 2.4% per day while ADC was practically unchanged (1.4% per day). (D) Time course after tumor implantation of sodium (left) and ADC (right) variations for a glioma created from a resistant 9L cell line. Sodium concentration steadily increased in the resistant tumor at a rate of 5.8% per day while ADC was practically unchanged (1.2% per day). Sodium values were corrected for partial volume effects. Figures A and B were reproduced with permission from Ref. [221]. Figures C and D were reproduced with permission from Ref. [224].

detecting and diagnosing diseases in liver, gallbladder, pancreas, (intra-cellular, extra-cellular, vascular compartments). Differences kidney, spleen, prostate, uterus and others [213]. Very few studies in the relaxation times of sodium within the tissues also contribute have been performed for the moment on the abdomen, mainly due to the overall measurements. to lack of suitable body sodium coils and the necessity for the A following study by Rosen et al. [215] showed sodium MRI at acquisition sequences to take into account the cardiac and respira- 3 T on a patient with a transplanted kidney, who was previously tory movements of the body, which can seriously perturb the qual- diagnosed with end-stage hypertensive nephropathy, to measure ity of the images. We survey here some preliminary work on the the corticomedullary sodium gradient and to assess the quality kidney, prostate and uterus. of the new kidney. The kidney was imaged 4 months after transplantation. The measured mean medulla/cortex SNR ratio was 6.6.1. Kidney 1.8, and the gradient slope was 1.1 (arbitrary units/mm), which The kidneys are essential in regulating homeostatic functions in was lower than the gradient observed in the healthy kidney in the body such as extra-cellular fluid volume, acid–base equilibrium the previous study. This difference was interpreted as being due (pH), electrolyte concentrations, and blood pressure (via maintain- to the effects of the recent transplantation. Although resolution ing salt and water balance). This role depends tightly on the regu- and SNR were low and the physiology/biochemistry involved is lation of extra-cellular sodium in the kidney, which builds up a not fully known, such findings are encouraging for possible appli- concentration gradient from the cortex to the medulla. Thus, renal cations of quantitative sodium MRI for assessing renal functions in function is tightly dependent on this corticomedullary gradient different diseases such as nephropathy, renal failure as well as in and mapping this gradient with sodium MRI could help assess kid- kidney transplantation. ney impairments. A first study in humans by Maril et al. [214] presented sodium 6.6.2. Prostate MRI of kidneys before and after water deprivation, as shown in Sodium MRI of the human prostate was recently tested in vivo Fig. 19. This work was performed at 3 T with a surface coil and data at 3 T [216,217] and compared with diffusion MRI. The prostate were acquired with a 3D gradient echo sequence. The results and its different compartments were identifiable, with measured showed that the sodium signal intensity increased linearly from TSC of around 60 mM in the central zone and 70 mM in the periph- the cortex to each of the medullae with a mean slope of 1.6 (in eral zone. This method could be a potential radiological biomarker arbitrary units per mm) and then decreased towards the renal pel- of prostate cancer and of treatment response. vis. After 12 h water deprivation, this gradient increased significantly by 25%. The sodium gradient change in the kidney may 6.6.3. Uterus reflect changes in the concentration in each kidney compartment Uterine leiomyomata (fibroids) were investigated by diffusion (cortex, medulla), but also within micro-compartments therein weighted imaging (DWI) and quantitative sodium MRI by Jacobs G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 37 et al. [218] at 1.5 T. Uterine leiomyomata are solid masses arising 6.8.1. Chemotherapy assessment from the muscle of the uterus (myometrium) and can be associated It has already been demonstrated that DWI is highly sensitive to with menstrual pain and loss of reproductive function. The goal microscopic changes in cellular structure and can be useful for the was to monitor the treatment of fibroids on patients that were quantitative assessment of tumor response to therapeutic insult by treated using MRI guided high-intensity focused ultrasound sur- observing the microscopic changes in water diffusion due to cellu- gery (MRg-HIFUS). Regions where the tissue was treated were lar destruction [222]. These microscopic changes generally precede clearly identified on both DWI and sodium images. The TSC in nor- macroscopic changes in tumor volume, which occur at much later mal myometrium tissue was approximately 36 mM with an appar- stages after therapeutic insult. On the other hand, quantitative ent diffusion coefficient (ADC) of 2.2 mm2/s. The TSC was 28 mM in sodium MRI can give direct information on the intra-cellular and untreated fibroids and increased to 42 mM in treated tissues while extra-cellular sodium concentrations, which depend on the intact- ADC was 1.75 mm2/s in untreated fibroids and decreased to ness of the sodium channels and the Na+/K+-ATPase mechanism in 1.3 mm2/s in treated tissues. the cell membrane as well as on the cellular energy consumption. The mechanisms involved in the changes in ADC and TSC in Therefore, any disruption in cellular integrity can lead to abnormal treated uterine tissue are still unknown but these changes may imbalances that are detectable using sodium MRI. provide an empirical measure of the efficacy of the treatment. Schepkin et al. [223] studied the possibility of using DWI and DWI is sensitive to translational motion and changes in the intra- sodium MRI for assessing non-invasively the effect of chemothera- and extra-cellular compartments available for diffusion of water peutic treatment of tumors in vivo. The study was performed on 9L molecules and can reveal disruptions or restrictions to the move- gliosarcomas implanted in rats and then treated with varying ment of these molecules within a tissue. Sodium MRI is more doses of BCNU. MRI images were acquired at 21.1 T. The main re- sensitive to the gradient of ion concentration between the sults are shown in Fig. 21A and B. This study demonstrated a very intra- and extra-cellular compartments, which depends on the good correlation between DWI and sodium MRI, and thus indicated Na+/K+-ATPase activity within the cell membrane and its energy that sodium MRI may help to reveal the efficacy of tumor therapy consumption. Thermal ablation disrupts the cell membrane and within a few days following treatment in a dose-dependent alters the cellular integrity and perfusion of the tissue. The ADC manner. and sodium changes within the diseased uterine tissue before Babsky et al. [220] implanted fibrosarcoma (RIF-1) tumors in and after treatment may therefore reflect a combination of mice and monitored the effects of 5-fluorouracil (5FU) treatment complex changes in tissues such as modification of the water with proton DWI, single pulse and TQF sodium MRI and also PET environment, disruption of Na+/K+-ATPase function, decreased for measuring fluorodeoxyglucose (FDG) uptake in the tumor. Tu- vascularization or cytotoxic edema. This preliminary study sug- mor volumes significantly decreased in treated animals and gests that DWI (ADC map) coupled with quantitative sodium slightly increased in control tumors one, two, and three days concentration images may provide non-invasive biomarkers for post-treatment. Single-pulse sodium signal intensity (SI) and water the uterine leiomyomata response to therapy or for the investiga- ADC increased in treated tumors but not in control tumors during tion of malignant tumors by probing biochemical changes that the same period. TQF sodium SI and FDG uptake were significantly can give information beyond the usual anatomical imaging lower in treated tumors compared with control tumors 3 days after parameters. 5FU treatment. The authors concluded that the correlated increases in single pulse sodium SI and water ADC following chemotherapy might reflect an increase in extra-cellular space, while the 6.7. Whole body lower TQF sodium signal and FDG uptake in treated tumors compared with control tumors suggested a shift in tumor metabolism Most human body parts have been scanned with sodium MRI from glycolysis to oxidation and/or a decrease in cell density. A separately for the moment, and recently a study by Wetterling multiparametric approach using 1H+23Na MRI, PET and histology et al. [219] demonstrated whole body sodium MRI in vivo at 3 T. could therefore provide a clearer understanding of the relationship A new asymmetrical birdcage coil was used to acquire the data between metabolic and ionic changes in tumors caused by in 5 segments of 10 min acquisitions each with a 3D radial se- chemotherapy. quence, and a nominal isotropic resolution of 6 mm. The resulting Kline et al. [84] used sodium MRI with IR to monitor the re- composite image of the 5 acquisitions is shown in Fig. 20. Despite sponse to chemotherapy of mouse xenograft tumors propagated the low resolution, one can easily detect many different organs from human prostate cancer cell lines, in order to increase the such as the brain, the left and right ventricles of the heart, the liver, weighting of the images toward intra-cellular sodium content. A the bladder, the spine or articular cartilage. Possible applications of 37% increase in sodium signal intensity was observed between whole body sodium imaging could be for example the detection images before and 24 h after administration of antineoplastics, and malignancy assessment of cancer with metastases, and moni- which may be due to an increase of intra-cellular sodium in the toring the response to chemotherapy. treated tumors. These findings can be matched with experiments with these same drugs and cells treated in culture, where a significant intra-cellular sodium elevation (10–20 mM) was detected 6.8. Cancer and chemotherapeutic response using a ratiometric fluorescent dye. Flow cytometry also showed that this intra-cellular sodium increase preceded cell death by Although the following studies were not performed on humans apoptosis. Sodium MRI with IR seems to increase the weight of but on rats or mice in vivo, it is useful to mention these here as they the images to intra-cellular sodium and could help monitoring represent important preliminary work in preparation for potential apoptosis of tumor cells induced by chemotherapy in a non-inva- human applications on cancer and chemotherapeutic response sive manner. assessment [84,220,221]. The goal of cancer therapy is to eliminate neoplastic cells, but the assessment of the therapeutic efficiency 6.8.2. Tumor resistance to therapy in vivo may be difficult. Different functional imaging modalities Schepkin et al. [224] also studied the capability of sodium MRI such as PET (rate of glycolysis), DWI MRI (extra-cellular volume) and DWI for monitoring tumor resistance to chemotherapy in or sodium MRI (intra-cellular sodium) could prove to be useful intracranial rat 9L gliomas, in order to see if these methods could for this purpose. be used as biomarkers for drug resistance. It was first determined 38 G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 that implanted resistant 9L cells created tumors with significantly Acknowledgements reduced TSC (57 mM) compared with nonresistant (naive) 9L cells (78 mM) and that corresponding differences in ADC were less pro- A.J. acknowledges funding from National Science Foundation nounced but still statistically significant. CHE 0957586, and NIH/NIBIB/NIAMS 1 R01 EB016045-01. R. R. R. The main results are presented in Fig. 21C–D which shows the acknowledges funding from National Institutes of Health, time course evolution of sodium content and ADC in naive and 1R01AR053133, 1R01AR056260, and 1R01AR060238, and J.S.L. resistant tumors after chemotherapeutic injection (BCNU). Both so- acknowledges funding from National Institutes of Health dium and ADC can differentiate resistant from naive tumors. It is K25AR060269. We thank Dr. Uzi Eliav, Dr. Philip Kuchel, Dr. Ivan shown that sodium content and ADC vary at very different rates Maximov, Dr. Charles S. Springer Jr., and Dr. Keith Thulborn for in the two kinds of tumors after treatment and thus must depend commenting on our manuscript and providing helpful suggestions. on different mechanisms. Many parameters are involved, such as tumor volume change, increased extra-cellular volume, increased Appendix A intra-cellular sodium concentration due to deficit in ATP production, increase of blood supply, and lactate overproduction (indica- The full solution for the evolution in the presence of both relax- tor of increased glycolysis). As lactate production tends to correlate ation and residual quadrupolar interaction is developed here in the with tumor aggressiveness, the increasing intra-cellular sodium single transition basis. The connection to the tensor conversions is content could be an indicator of growing tumor malignancy. The given as well. increase in intra-cellular sodium is likely the main mechanism in- In the single transition basis, we can write the Liouville–von volved in the detected increase of the sodium signal. Neumann (LvN) Equation (Eq. (12)) in terms of the elements of This study shows that TSC measured with sodium MRI could the density operator q as give important information about the level of drug resistance to rs chemotherapy and that it is more sensitive than DWI for detecting d X X q ¼ı ðH q q H Þ C fq q ð0Þg; ðA:1Þ small changes in tumor resistance. dt rs rk ks rk ks rstu tu tu Xk X tu H ¼ı rstuqtu Crstufqtu qtuð0Þg; ðA:2Þ 7. Conclusion tu tu

where Hrstu ¼ Hrtdus drtHus. 23 We have reviewed here the theoretical basis of Na NMR and If one arranges the elements of the density operator q according MRI, different pulse sequences, as well as in vivo MRI applications. to The detection of sodium signals in vivo is challenging, and sodium images typically have low SNR (around 20–40), low resolution (2– ðq11; q22; q33; q44; q12; q23; q34; q21; q32; q43; q13; q24; q31;q42;q14;q41Þ; 10 mm) and long acquisition times (10–30 min). Such constraints, however, are often outweighed by the improved diagnostic infor- 5 5 where qkl ¼ 2 k 2 l and jai is an eigenstate of the z compo- mation that can be obtained from measuring sodium concentra- nent of the spin angular momentum with the eigenvalue a, i.e. tions, such as is the case in osteoarthritis, stroke, and muscle b I jai¼ajai, the relaxation superoperator C becomes block-diago- disorders, for example. Pioneering work in these areas has already z nal, and each block conserves the magnetic quantum number demonstrated unique contrast, in complement to existing 1H m ¼ l k. In the same representation, the chemical shift and the techniques. quadrupolar interaction are diagonal. Therefore, one can solve Due to the low gyromagnetic ratio of the sodium nuclei (26% of the free evolution by examining the individual m subspaces the proton gyromagnetic ratio), strong magnetic field gradients separately. need to be used for acquiring images with higher resolution, which All spectral density functions listed here follow Convention I can therefore be limited by the hardware capabilities of clinical (Eq. (19)). scanners (currently around 40 mT/m) and physiological constraints on gradient slew rates (200 T/(m s)). There are definite advantages to sodium MRI at high static fields (P3 T), although A.1. Subspace independent of quadrupolar coupling important data have been acquired even at 1.5 T. New methodological advances, such as multichannel capabili- There are 3 subspaces in which the quadrupolar interaction ties [225,226] with parallel data acquisition using array coils and does not contribute to the spin evolution: m ¼ 0; 3. parallel reconstruction algorithms [227], and also new reconstruction schemes such as compressed sensing (CS), that allow A.1.1. m ¼3 subspaces rapid undersampled data acquisitions [228,130] have provided These are the simplest subspaces with single dimensionality. significant speed up and sensitivity boosts. An optimized combi- The relevant equations are: nation of parallel-CS-NUFFT reconstruction associated with optimized sequences such as TPI at high field could therefore allow d q ¼3½Jðx ÞþJð2x Þq and one to significantly reduce the acquisition time to a few minutes dt 14 0 0 14 and still generate images of good resolution (1–2 mm) and main- d q ¼3½Jðx ÞþJð2x Þq : ðA:3Þ tain reasonable SNR. Such a procedure would position sodium dt 41 0 0 41 MRI well within the realm of practical clinical imaging tech- Therefore, the solution are niques. Assessing the intra-cellular sodium content is more challenging, as the intra-cellular sodium concentration is significantly 3½Jðx0ÞþJð2x0Þt lower. Recent pulse excitation techniques based on optimal con- q14ðtÞ¼e q14ð0Þ and trol theory [60,61] could also provide additional sensitivity and 3½Jðx0ÞþJð2x0Þt q41ðtÞ¼e q41ð0Þ: ðA:4Þ contrast enhancements. Given the many promising pilot studies that have already been performed, one can expect many new The triple quantum coherences decay exponentially with a decay 23 applications and studies of Na MRI to emerge in the near constant 3½Jðx0ÞþJð2x0Þ, which is equal to R2;slow as will be seen future. below. G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 39

0 1 0 1 20 1 0 13 th q11 3½Jðx0ÞþJð2x0Þ 3Jðx0Þ3Jð2x0Þ 0 q11 q11 B C B C 6B C B th C7 d B q C B 3Jðx0Þ 3½Jðx0ÞþJð2x0Þ 0 3Jð2x0Þ C 6B q C B q C7 B 22 C ¼B C 6B 22 C B 22 C7; ðA:5Þ @ A @ A 4@ A @ th A5 dt q33 3Jð2x0Þ 03½Jðx0ÞþJð2x0Þ 3Jðx0Þ q33 q33 th q44 0 3Jð2x0Þ3Jðx0Þ 3½Jðx0ÞþJð2x0Þ q44 q44

A.1.2. m ¼ 0 subspace The dimensionality of the m ¼ 0 subspace is 4, and the corre-

0 1 0 1 3½Jðx0ÞþJð2x0Þ 3Jðx0Þ3Jð2x0Þ 0 1 1 11 B C B C B 3Jðx Þ 3½Jðx ÞþJð2x Þ 0 3Jð2x Þ C 1 B 111 1 C B 0 0 0 0 C B C @ A ¼ @ A 3Jð2x0Þ 03½Jðx0ÞþJð2x0Þ 3Jðx0Þ 2 1 111 0 3Jð2x Þ3Jðx Þ 3½Jðx ÞþJð2x Þ 1111 0 0 0 0 0 1 0 1 ðA:6Þ 00 0 0 1111 B C B C B 0 6Jðx Þ 00C 1 B 1111C B 0 C B C @ A @ A; 006Jð2x0Þ 0 2 1 111

00 0 6½Jðx0ÞþJð2x0Þ 1 1 11

spondingÀÁ part of the LvN equation is given by For the m ¼ 0 subspace, one can set up the following states corre- th th th th where q11; q22; q33; q44 ¼ð3=2; 1=2; 1=2; 3=2Þ. Since the ma- sponding to the zeroth-order spherical tensors (indicated in the trix is real and symmetric, one can easily diagonalize it: subscript), and the matrix exponentiation gives the evolution operator: 0 1 0 1 0 1 0 1 1 3 rffiffiffi 1 1 þ a þ b þ ab 1 a þ b ab 1 þ a b ab 1 a b þ ab B C B C B C B 1 C 1 B 1 C 3B 1 C B C ^ B C ^ B C ^ B C L 1 B 1 a þ b ab 1 þ a þ b þ ab 1 a b þ ab 1 þ a b ab C uT0;0 ¼ ; uT1;0 ¼ ; uT2;0 ¼ ; and e 0 t B C; @ 1 A 2 @ 1 A 2@ 1 A 4 @ 1 þ a b ab 1 a b þ ab 1 þ a þ b þ ab 1 a þ b ab A 1 3 1 1 a b þ ab 1 þ a b ab 1 a þ b ab 1 þ a þ b þ ab 0 1 1 ðA:7Þ B C 3 B 3 C where a e6Jðx0Þt and b e6Jð2x0Þt. Note that Eq. (A.5) is inhomo- u^ ¼ pffiffiffiffiffiffi B C: ðA:10Þ T3;0 @ A L t 2 10 3 geneous, so e 0 should be applied as follows: 0 1 0 1 20 1 0 13 1 th th q11ðtÞ q11 q11ð0Þ q11 B C B th C 6B C B th C7 In terms of these vectors, the result in Eq. (A.9) is given as B q22ðtÞ C B q C L 6B q22ð0Þ C B q C7 B C B 22 C ¼ e 0t6B C B 22 C7: ðA:8Þ 0 1 @ A @ th A 4@ A @ th A5 q33ðtÞ q33 q33ð0Þ q33 q11 th th B C ÂÃ q44ðtÞ q44 q44ð0Þ q44 B q22 C 2 B C ¼ u^ e6Jðx0Þt þ 4e6Jð2x0Þt u^ @ q A T1;0 5 T1;0 33 ðA:11Þ A.1.3. Example: inversion recovery q 44 pffiffiffiffiffiffi After applying an inversion pulse to the thermal equilib- 2 2 10 ½e6Jðx0Þt e6Jð2x0Þt u^ ; rium state, the diagonal elements of the density operator can be 5 3 T3;0 obtained by setting ðq11ð0Þ; q22ð0Þ; q33ð0Þ; q44ð0ÞÞ ¼ ð3=2; 1=2; 1=2; 3=2Þ in Eq. (A.8): which describes the recovery of the longitudinal magnetization 0 1 0 1 20 1 0 13 back to the thermal equilibrium and the buildup of octupolar order q ðtÞ th 3=2 th 11 q11 q11 (T30). The recovery and buildup are governed by two exponentially B C B th C 6B C B th C7 B q22ðtÞ C B q C L 6B 1=2 C B q C7 decaying functions with the decay constants 6Jðx0Þ and 6Jð2x0Þ. B C B 22 C ¼ e 0t6B C B 22 C7 @ A @ th A 4@ A @ th A5 After a reading pulse applied at time t, the first point of the FID or q33ðtÞ q33 1=2 q33 the integration of the Fourier-transformed spectrum gives the size q ðtÞ qth 3=2 qth 44 44 0 1 0 144 ðA:9Þ of the longitudinal magnetization, which is proportional to 1 1 1 2 e6Jðx0Þt 4e6Jð2x0Þt . To measure the buildup of the octupolar B C B C 5 ½ þ B 1 C B 1 C order, a multiple-quantum filtering technique can be applied, which ¼ aB C þ 2bB C: @ 1 A @ 1 A usually consists of two p=2 pulses with a phase cycle. For example, 1 1 the first p=2 pulse excites the triple-quantum coherences T3;3 from octupolar order, the phase cycle selects only the triple-quantum 40 G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 coherence, and the second p=2 pulse converts the triple-quantum A.2.1. m ¼1 subspaces coherences into the single-quantum coherence T3;1, which is The m ¼1 subspaces are each three-dimensional. The LvN subsequently converted into observable magnetization through equation for the m ¼ 1 subspace is given as quadrupolar coupling or quadrupolar relaxation. The intensity at where fQ is the distance in Hz from the central peaks to the satellite the resonance frequency should be proportional to a function peaks. This three-dimensional subspace breaks down to one single e6Jðx0Þt e6Jð2x0Þt. and one two-dimensional subspaces. The eigenvalues of the matrix are A.2. Subspace depending on quadrupolar coupling k ¼3½JðxÞþJð2x Þ; and 1;0 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA:23Þ The remaining four subspaces with m ¼1; 2 are affected by 2 2 2 k1; ¼3½Jð0ÞþJðx0ÞþJð2x0Þ 9Jð2x0Þ 4p f ; the residual quadrupolar interaction. The following two-dimen- Q sional expressions will be useful for calculating the free evolution and the corresponding eigenvectors are in these subspaces. 0 1 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 A two-dimensional coupled ordinary linear differential equa- 0 2pıf 9Jð2x Þ2 4p2f 2 tion is given by B C B Q 0 Q C @ A B C 1 and @ 0 A: ðA:24Þ d y1 L11 L12 y1 0 ¼ Ly ¼ : ðA:12Þ 3Jð2x0Þ dt y2 L21 L22 y2 The general solution can be written as Therefore, by using Eq. (A.21) for the two-dimensional subspace, the free evolution in the m ¼ 1 subspace becomes y ðtÞ a1 b 0 1 0 10 1 1 ¼ Aek1t þ Bek2t 1 ; ðA:13Þ q12ðtÞ R11 0 R13 q12ð0Þ y2ðtÞ a2 b2 B C B CB C @ q ðtÞ A ¼ @ 0 R22 0 A@ q ð0Þ A; ðA:25Þ and the initial condition should satisfy 23 23 q34ðtÞ R31 0 R33 q34ð0Þ y1ð0Þ a1 b1 ¼ A þ B : ðA:14Þ where y2ð0Þ a2 b2

The general solution can also be expressed as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 R11 ¼ 2 2 2 y1ðtÞ y1ð0Þ R11 R12 y1ð0Þ 2 9Jð2x0Þ 4p fQ ¼ R ¼ : ðA:15Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y2ðtÞ y2ð0Þ R21 R22 y2ð0Þ k1;þt 2 2 2 e 2pıfQ þ 9Jð2x0Þ 4p fQ By equating Eqs. (A.13) and (A.15) and using the expression of Eq. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

(A.14), we get k1;t 2 2 2 þe 2pıfQ þ 9Jð2x0Þ 4p fQ ; 1 ÀÁ k1t k2t R11 ¼ e a1b2 e a2b1 ; ðA:16Þ a1b2 a2b1 ÀÁ 3Jð2x0Þ a b ÀÁ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1;þt k1;t 1 1 k1t k2t R13 ¼ e e ; R12 ¼ e e ; ðA:17Þ 2 2 2 a1b2 a2b1 2 9Jð2x0Þ 4p fQ a b ÀÁ 2 2 k1t k2t R21 ¼ e e ; ðA:18Þ a b a b k1;0t 1 2 2 1 R22 ¼ e ; 1 ÀÁ R ¼ ek1ta b ek2ta b ; ðA:19Þ 22 b b 2 1 1 2 ÀÁ a1 2 a2 1 3Jð2x0Þ k1;þt k1;t R31 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e e ; or in a simpler form, 2 9Jð2x Þ2 4p2f 2 0 Q k1t k2t k1t k2t 1 e a1b e a2b a1b ðe e Þ R ¼ 2 1 1 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k t k t k t k t a1b2 a2b1 a b ðe 1 e 2 Þðe 1 a b e 2 a b Þ 1 2 2 2 2 1 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1;þt 2 2 R33 ¼ e 2pıfQ þ 9Jð2x0Þ 4p fQ ðA:20Þ 2 2 2 2 9Jð2x0Þ 4p fQ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k t a1 k t b1 1 2 2 ¼ e ðÞþb2 b1 e ðÞa2 a2 : k1;t 2 2 þe 2pıfQ þ 9Jð2x0Þ 4p f : a1b2 a2b1 a2 b2 Q ðA:21Þ The expression for the m ¼1 subspace can be obtained by replac-

ing fQ with fQ .

0 1 0 10 1 q 3½Jð0ÞþJðx0ÞþJð2x0Þ 2pıfQ 03Jð2x0Þ q d B 12 C B CB 12 C @ q A ¼ @ 0 3½JðxÞþJð2xÞ 0 A@ q A; ðA:22Þ dt 23 23 q34 3Jð2x0Þ 0 3½Jð0ÞþJðx0ÞþJð2x0Þ þ 2pıfQ q34 G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 41 A.3. Example: FID when f ¼ 0 q ðtÞ R11 R12 q ð0Þ Q 13 ¼ 13 ; ðA:33Þ q24ðtÞ R21 R22 q24ð0Þ After applying a 90 pulse to the thermal equilibrium state, the where single-quantum coherences T1;1 and T1;1 are excited. The free evolution for T is given as 1;1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 pffiffiffiffiffiffiffiffi R11 ¼ 0 1 0 10 1 2 1 k1;þt k1;t 1 k1;þt k1;t 2 2 q ðtÞ ðÞe þ e 0 ðe e Þ 3=2 2 9Jðx0Þ 4p fQ B 21 C B 2 2 CB pffiffiffi C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ A @ k1;0t A@ A q32ðtÞ ¼ 0 e 0 2 ; 2 pffiffiffiffiffiffiffiffi k2;þt 2 2 e 2pıfQ þ 9Jðx0Þ 4p fQ q ðtÞ 1 ðÞek1;þt ek1;t 0 1 ðÞek1;þt þ ek1;t 43 2 2 3=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðA:26Þ k2;t 2 2 2 0 1 0 1 e 2pıfQ 9Jðx0Þ 4p f ; rffiffiffi Q 1 ffiffiffi 0 3 B C p B C ÀÁ k1;þt@ A k1;0t@ A 3Jðx0Þ ¼ e 0 þ 2e 1 : ðA:27Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2;þt k2;t 2 R12 ¼ e e ; 2 2 2 1 0 2 9Jðx0Þ 4p fQ ÀÁ These three vectors can be decomposed into corresponding vectors 3Jðx0Þ R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ek2;þt ek2;t ; for the first-order spherical tensors, 21 ¼ 2 9Jðx Þ2 4p2f 2 0 pffiffiffi 1 0 1 0 Q 3 ffiffiffi 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 B C p B C 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2;þt 2 2 u^T ¼ pffiffiffi @ 2 A; u^T ¼ 3@ 0 A; and R22 ¼ e 2pıfQ þ 9Jðx0Þ 4p f 1;1 ffiffiffi 2;1 Q 2 p 2 9J 2 4 2f 2 1 ðx0Þ p Q 0 3 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k t 2 ffiffiffi 2; 2 2 3 B p C þe 2pıfQ þ 9Jðx0Þ 4p fQ : u^T ¼ pffiffiffiffiffiffi @ 3 A: ðA:28Þ 3;1 10 1 The propagator for a m ¼2 subspace can be obtained by substitut-

ing fQ with fQ . The solution can then be written as 0 1 q ðtÞ A.5. Summary of decay constants B 21 C 2 3 2 2 @ A k1;0 t k1;þt ^ ffiffiffiffiffiffi k1;0t ffiffiffiffiffiffi k1;þt ^ q32ðtÞ ¼ e þ e uT1;1 þp e þ p e uT3;1 ; 5 5 15 15 Some of the results of this section are summarized in the table q43ðtÞ below. ðA:29Þ which describes the monotonic decay of the transverse magnetiza- Observables Experiment Decay constant tion and the buildup of the single-quantum coherence of the third- T Inversion 6Jð2x Þ1=T rank component, governed by two exponentially decaying functions 3;0 0 1;slow recovery with with the time constants k1;0 ¼ 3Jðx0Þþ3Jð2x0Þ and k1;þ ¼ TQF 6Jðx0Þ1=T1;fast 3Jð0Þþ3Jðx0Þ. Only the transverse magnetization is directly obser- ^ vable, so the FID should be proportional to the coefficient of uT1;1 in T3;1 Spin echo with 3½Jðx0ÞþJð2x0Þ 1=T2;slow Eq. (A.29) or 2e½3Jðx0Þþ3Jð2x0Þt þ 3e½3Jð0Þþ3Jðx0Þt . The buildup of the TQF, when single-quantum coherence of the octupolar order may be observed f ¼ 0 3½Jð0ÞþJðx0Þ 1=T through the triple-quantum filter technique, giving a signal propor- Q 2;fast tional to e½3Jðx0Þþ3Jð2x0Þt e½3Jð0Þþ3Jðx0Þt. T3;2 Delay after 3½Jð0ÞþJð2x0Þ DQF, when A.4. m ¼2 subspaces fQ ¼ 0 Each of the m ¼2 subspaces is two-dimensional. The LvN T Delay after TQF 3½Jðx ÞþJð2x Þ ¼ 1=T equation for the m ¼ 2 subspace is given as 3;3 0 0 2;slow d q 13 Central transition Spin echo, 3½Jðx0ÞþJð2x0Þ ¼ 1=T2;slow dt q 24 when 3½Jð0ÞþJðx0ÞþJð2x0Þ 2pıfQ 3Jðx0Þ ¼ 2pfQ > 3Jð2x0Þ 3Jðx Þ3½Jð0ÞþJðx ÞþJð2x Þ þ 2pıf 0 0 0 Q q 13 : Satellite transition Spin echo, 3½Jð0ÞþJðx0ÞþJð2x0Þ q24 when

ðA:30Þ 2pfQ > 3Jð2x0Þ The eigenvalues of the matrix are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T2;2 and T3;2 Delay after 3½Jð0ÞþJðx0ÞþJð2x0Þ 2 2 2 DQF, when k2; ¼3½Jð0ÞþJðx0ÞþJð2x0Þ 9Jðx0Þ 4p fQ ; ðA:31Þ 2pfQ > 3Jðx0Þ and the corresponding eigenvectors are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2pıf 9Jðx Þ2 4p2f 2 A.6. Triple quantum filter coherence pathways, flip angle, and phase Q 0 Q : ðA:32Þ dependence 3Jðx0Þ

Therefore, the free evolution in the m ¼ 2 subspace is described The coherence pathways of interest are T3;0 ! T3;3 ! T3;1 and by T3;1 ! T3;3 ! T3;1. Using the Wigner coefﬁcients, the transition 42 G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47

matrix elements of two consecutive pulses with the flip angles h2 A.7. Double quantum filter coherence pathways, flip angle, and phase and h3 and their corresponding phases /2 and /3 can be calculated dependence

(note, h2;3, and /2;3 refer to the deﬁnitions in the pulse sequence in Fig. 3). First, for the case starting with T3;0, the transition matrix The coherence pathways of interest are elements are given in the following table.

T2;1 ! T2;2 ! T2;1 and Coherence pathway 3 3 T3;1 ! T3;2 ! T3;1. D1;3ð/3; h3; /3ÞD3;0ð/2; h2; /2Þ pffiffi h 2 2 T3;0 ! T3;3 ! T3;1 5 3 ıð2/33/2Þ 2 3 16 e cos 2 sin h3 sin h2 For the second-order spherical tensors T2;1, the transition ma- pffiffi 2 h 2 2 T3;0 ! T3;3 ! T3;1 5 3 ıð4/3þ3/2Þ 3 trix elements are 16 e sin 2 sin h3 sin h2 pffiffi 2 h 2 2 T3;0 ! T3;3 ! T3;1 5 3 ıð4/33/2Þ 3 16 e sin 2 sin h3 sin h2 pffiffi Coherence pathway 2 2 h 2 2 D1;2ð/3; h3; /3ÞD2;1ð/2; h2; /2Þ T3;0 ! T3;3 ! T3;1 5 3 ıð2/3þ3/2Þ 2 3 16 e cos 2 sin h3 sin h2 ı / / 3 h h 3 h h T ; ! T ; ! T ; ð 3 2Þ 3 3 2 2 2 1 2 2 2 1 4e cos 2 sin 2 cos 2 sin 2 h 3 h h 3 h T2;1 ! T2;2 ! T2;1 ıð3/3þ3/2Þ 3 3 2 2 4e cos 2 sin 2 cos 2 sin 2 h h h 3 h For the combination of 0 and p=2 phases of the two pulses one T2;1 ! T2;2 ! T2;1 ıð/33/2Þ 3 3 3 2 2 4e cos 2 sin 2 cos 2 sin 2 has for the amplitudes: h 3 h h h T2;1 ! T2;2 ! T2;1 ıð3/3þ/2Þ 3 3 3 2 2 4e cos 2 sin 2 cos 2 sin 2 h 3 h h h T2;1 ! T2;2 ! T2;1 ıð3/3/2Þ 3 3 3 2 2 4e cos 2 sin 2 cos 2 sin 2 h h h 3 h /2 /3 Final state T2;1 ! T2;2 ! T2;1 ıð/3þ3/2Þ 3 3 3 2 2 4e cos 2 sin 2 cos 2 sin 2 pffiffi h 3 h h 3 h 0 0 5 3 2 2 T2;1 ! T2;2 ! T2;1 4eıð3/33/2Þ cos 3 sin 3 cos 2 sin 2 4 sin h3 sin h2 cos h3ðT3;1 T3;1Þ 2 2 2 2 pffiffi p ıð/ þ/ Þ 3 h3 h3 3 h2 h2 0 5 3 2 2 T2;1 ! T2;2 ! T2;1 4e 3 2 cos sin cos sin 2 4 sin h3 sin h2ðT3;1 T3;1Þ 2 2 2 2 pffiffi p 0 5 3 2 2 2 ı 4 sin h3 sin h2ðT3;1 þ T3;1Þ pffiffi p p 5 3 2 2 2 2 ı 4 sin h3 sin h2 cos h3ðT3;1 þ T3;1Þ For the initial states T2;1 T2;1 and ıðT2;1 þ T2;1Þ, which are

Hermitian, the ﬁnal states are given for the cases with /2 and /3 being either 0 or p=2as From this table, one can see that the coherence pathway can be suppressed when ð/2; /3Þ¼ð0; 0Þ and ðp=2; p=2Þ with h3 ¼ p=2. On the other hand, one can maximize the signal intensity by choosing /2 /3 T2;1 T2;1 ıðT2;1 þ T2;1Þ h2 ¼ h3 ¼ p=2 with ð/2; /3Þ¼ð0; p=2Þ or ð0; p=2Þ. The phase of the 0 0 ðT2;1 T2;1Þ cos h3 cos h2 ıðT2;1 þ T2;1Þ acquisition should be p=2 when ð/ ; / Þ¼ð0; p=2Þ and 0 when p 2 3 0 2 ıðT2;1 þ T2;1Þ cos h3 cos h2 ðT2;1 T2;1Þ ð/2; /3Þ¼ðp=2; 0Þ to obtain the absorption-like signal. (This is be- p 2 0 ıðT2;1 þ T2;1Þ ðT2;1 T2;1Þ cos h3 cos h2 cause T3;1 T31 and ıðT3;1 þ T31Þ are Hermitian.) p p 2 2 ðT2;1 T2;1Þ ıðT2;1 þ T2;1Þ cos h3 cos h2 For the single quantum coherences T3;1,

Coherence pathway 3 3 D1;3ð/3; h3; /3ÞD3;1ð/2; h2; /2Þ after taking out the common factor sin h3 sin h2. When h 2 h h 2 h T3;1 ! T3;3 ! T3;1 ıð2/32/2Þ 4 3 3 4 2 2 h ¼ h ¼ p=2, / is important to observe the signal, depending on 15e cos 2 sin 2 cos 2 sin 2 2 3 2 h 2 h h 4 h the initial state. T3;1 ! T3;3 ! T3;1 ıð2/34/2Þ 4 3 3 2 2 2 15e cos 2 sin 2 cos 2 sin 2 h 4 h h 4 h For the third-order spherical tensors T3;1, the transition matrix T3;1 ! T3;3 ! T3;1 15eıð4/3þ4/2Þ cos2 3 sin 3 cos2 2 sin 2 2 2 2 2 elements are h 4 h h 2 h T3;1 ! T3;3 ! T3;1 ıð4/3þ2/2Þ 2 3 3 4 2 2 15e cos 2 sin 2 cos 2 sin 2 h 4 h h 2 h T3;1 ! T3;3 ! T3;1 15eıð4/32/2Þ cos2 3 sin 3 cos4 2 sin 2 Coherence pathway 3 3 2 2 2 2 D1;2ð/3; h3; /3ÞD2;1ð/2; h2; /2Þ h 4 h h 4 h T3;1 ! T3;3 ! T3;1 15eıð4/34/2Þ cos2 3 sin 3 cos2 2 sin 2 2 2 2 2 5 ı / / 3 h T3;1 ! T3;2 ! T3;1 e ð 3 2Þ cos 3 1 3 cos h h 2 h h 4 h 2 2 ð þ 3Þ T3;1 ! T3;3 ! T3;1 15eıð2/3þ4/2Þ cos4 3 sin 3 cos2 2 sin 2 2 2 2 2 sin h3 cos3 h2 ð1 þ 3 cos h Þ sin h2 h 2 h h 2 h 2 2 2 2 T3; 1 ! T3; 3 ! T3; 1 ıð2/3þ2/2Þ 4 3 3 4 2 2 15e cos sin cos sin 5 ı 3/ / h 2 2 2 2 T ; ! T ; ! T ; ð 3 2Þ 3 3 1 3 2 3 1 2 e cos 2 ð1 þ 3 cos h3Þ 3 h3 3 h2 h2 sin 2 cos 2 ð1 þ 3 cos h2Þ sin 2 5 ıð3/ þ3/ Þ h3 T3;1 ! T3;2 ! T3;1 e 3 2 cos ð1 þ 3 cos h3Þ For the initial states T3;1 T3;1 and ıðT3;1 þ T3;1Þ, which are 2 2 3 h3 h2 3 h2 Hermitian, the ﬁnal states are given for the cases with /2 and /3 sin 2 cos 2 ð1 þ 3 cos h2Þ sin 2 h being 0 and p=2as T3;1 ! T3; 2 ! T3; 1 5 ıð/3þ3/2Þ 3 3 2 e cos 2 ð1 þ 3 cos h3Þ h3 h2 3 h2 sin 2 cos 2 ð1 þ 3 cos h2Þ sin 2 /2 /3 T3;1 T3;1 ıðT3;1 þ T3;1Þ 5 ı / 3/ 3 h T ; ! T ; ! T ; ð 3 2Þ 3 3 1 3 2 3 1 2 e cos 2 ð1 þ 3 cos h3Þ 0 0 ðT T Þ cos h cos h ıðT þ T Þ h h 3 h 3;1 3;1 3 2 3;1 3;1 sin 3 cos 2 ð1 þ 3 cos h Þ sin 2 p 2 2 2 2 0 2 ðT3;1 T3;1Þ cos h2 ıðT3;1 þ T3;1Þ cos h3 5 ı 3/ 3/ h T ; ! T ; ! T ; ð 3 2Þ 3 p 3 1 3 2 3 1 2 e cos 2 ð1 þ 3 cos h3Þ 2 0 ðT3;1 T3;1Þ cos h3 ıðT3;1 þ T3;1Þ cos h2 3 h3 h2 3 h2 p p sin 2 cos 2 ð1 þ 3 cos h2Þ sin 2 2 2 ðT3;1 T3;1Þ ıðT3;1 þ T3;1Þ cos h3 cos h2 h T3; 1 ! T3; 2 ! T3;1 5 ıð3/3þ/2Þ 3 2 e cos 2 ð1 þ 3 cos h3Þ 3 h3 3 h2 h2 sin 2 cos 2 ð1 þ 3 cos h2Þ sin 2 15 2 2 5 ı / / 3 h after taking out the common factor sin h sin h . When T ; ! T ; ! T ; ð 3þ 2Þ 3 16 3 2 3 1 3 2 3 1 2 e cos 2 ð1 þ 3 cos h3Þ h2 ¼ h3 ¼ p=2, there is only one combination of /2 and /3 depend- h3 3 h2 h2 sin 2 cos 2 ð1 þ 3 cos h2Þ sin 2 ing on the initial state, which can produce a signal. G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 43

As before, for the initial states T3;1 T3;1 and ıðT3;1 þ T3;1Þ, the ﬁnal states are given as

/2 /3 T3;1 T3;1 ıðT3;1 þ T3;1Þ

2 2 00ðT3;1 T3;1Þð1 3 cos h3Þð1 3 cos h2Þ ıðT3;1 þ T3;1Þ4 cos h3 cos h2 p 2 2 0 2 ıðT3;1 þ T3;1Þð1 3 cos h3Þð1 3 cos h2Þ ðT3;1 T3;1Þ4 cos h3 cos h2 p 2 2 2 0 ıðT3;1 þ T3;1Þ4 cos h3 cos h2 ðT3;1 T3;1Þð1 3 cos h3Þð1 3 cos h2Þ p p 2 2 2 2 ðT3;1 T3;1Þ4 cos h3 cos h2 ıðT3;1 þ T3;1Þð1 3 cos h3Þð1 3 cos h2Þ

5 after taking out the common factor 8 sin h3 sin h2. The phase /2 of References the first pulse controls the visibility of the signal depending on the [1] T. Bräuniger, M. Jansen, Solid-state NMR spectroscopy of quadrupolar nuclei initial state when h2 ¼ h3 ¼ p=2 or when either of h2 and h3 is set to in inorganic chemistry, Z. Anorg. Allg. Chem. 639 (2013) 857–879. the magic angle. [2] C. Fernandez, M. Pruski, Probing quadrupolar nuclei by solid-state nmr Normally it is not necessary to consider the evolution of T2;0, but spectroscopy: recent advances, in: Solid St. NMR, Springer, Berlin, 2012, pp. for completeness, it is: 119–188. [3] A. Jerschow, From nuclear structure to the quadrupolar NMR interaction and high-resolution spectroscopy, Prog. NMR Spectrosc. 46 (2005) 63–78. Coherence pathway 2 2 [4] S.E. Ashbrook, M.J. Duer, Structural information from quadrupolar nuclei in D1;2ð/3; h3; /3ÞD2;0ð/2; h2; /2Þ qffiffi solid state NMR, Concepts Magn. Reson., Part A 28 (2006) 183–248. T ! T ! T h h 3 h [5] M.J. Duer, Introduction to Solid-state NMR Spectroscopy, Blackwell, Oxford, 2;0 2;2 2;1 3eıð/32/2Þ cos3 3 sin 3 sin 2 qffiffi2 2 2 2 2004. 3 3 [6] S.E. Ashbrook, S. Wimperis, Nuclear quadrupole coupling: an introduction and T2;0 ! T2;2 ! T2;1 3 ıð3/32/2Þ h3 h3 h2 2e cos 2 sin 2 sin 2 crystallographic aspects, eMagRes (Encyclopedia of Magnetic Resonance), qffiffi 2009. 3 3 T2;0 ! T2;2 ! T2;1 3 ıð3/3þ2/2Þ h3 h3 h2 2e cos 2 sin 2 sin 2 [7] L.A. Rijniers, P.C.M.M. Magusin, H.P. Huinink, L. Pel, K. Kopinga, Sodium NMR qffiffi relaxation in porous materials, J. Magn. Reson. 167 (2004) 25–30, http:// T T T h h 3 h 2;0 ! 2;2 ! 2;1 3 ıð/3þ2/2Þ 3 3 3 2 dx.doi.org/10.1016/j.jmr.2003.11.008. 2e cos 2 sin 2 sin 2 [8] K.E. Washburn, G. Madelin, Imaging of multiphase fluid saturation within a porous material via sodium nmr, J. Magn. Reson. 202 (2010) 122–126, http:// dx.doi.org/10.1016/j.jmr.2009.10.001. [9] G. Madelin, R.R. Regatte, Biomedical applications of sodium MRI in vivo, J. The final state can be found when / and / are either 0 or =2: Magn. Reson. Imaging 38 (2013) 511–529. 2 3 p [10] S.C. Shekar, J.A. Tang, A. Jerschow, Dynamics of I ¼ 3=2 nuclei in isotropic slow motion, anisotropic and partially ordered phases, Concepts Magn. Reson., Part A 36 (2010) 362–387. /2 /3 Final state [11] R. Ouwerkerk, Sodium MRI, Methods Mol. Biol. 711 (2011) 175–201. 0 0 T2;1 T2;1 [12] W.D. Rooney, C.S. Springer, A comprehensive approach to the analysis and 0 p ıðT þ T Þ cos h interpretation of the resonances of spins 3/2 from living systems, NMR 2 2;1 2;1 3 Biomed. 4 (1991) 209–226. p 2 0 ðT2;1 T2;1Þ [13] C. Springer, Biological Systems: Spin-3/2 Nuclei, eMagRes (Encyclopedia of p p Magnetic Resonance), 2007. ıðT2;1 þ T2;1Þ cos h3 2 2 [14] W.D. Rooney, C.S. Springer, The molecular environment of intracellular sodium – Na-23 NMR relaxation, NMR Biomed. 4 (1991) 227–245. [15] R. Kemp-Harper, S.P. Brown, C.E. Hughes, P. Styles, S. Wimperis, Na-23 NMR qffiffi methods for selective observation of sodium ions in ordered environments, 3 Prog. NMR Spectrosc. 31 (1997) 157–181. after taking out the common factor 1 3 sin h sin h2. Hence, the sig- 2 2 3 2 [16] H. Shinar, G. Navon, Multinuclear NMR and microscopic MRI studies of the articular cartilage nanostructure, NMR Biomed. 19 (2006) 877–893. nal can be suppressed when /3 ¼ p=2 and h3 ¼ p=2. [17] M. Burnier, Sodium in Health and Disease, Informa Healthcare USA, Inc., New York, 2008. A.8. Summary of pulse phase choices in MQF sequences [18] E. Murphy, D.A. Eisner, Regulation of intracellular and mitochondrial sodium in health and disease, Circ. Res. 104 (2009) 292–303. [19] A.M. Rose, R. Valdes, Understanding the sodium-pump and its relevance to The table below summarizes the choice of the phases of the disease, Clin. Chem. 40 (1994) 1674–1685. pulses to obtain an absorption-like signal without changing the [20] J.C. Skou, M. Esmann, The Na, K-Atpase, J. Bioenerg. Biomembr. 24 (1992) phase of the acquisition. 249–261. [21] H.J. Berendsen, H.T. Edzes, The observation and general interpretation of sodium magnetic resonance in biological material, Ann. NY Acad. Sci. 204 Coherence pathway ð/2; /3Þ (1973) 459–485. [22] J.A. Magnuson, N.S. Magnuson, NMR studies of sodium and potassium in T3;0 ! T3;3 ! T3;1 ð0; p=2Þ various biological tissues, Ann. NY Acad. Sci. 204 (1973) 297–309. [23] D.A. Feinberg, L.A. Crooks, L. Kaufman, M. Brantzawadzki, J.P. Posin, M. T3;1 ! T3;3 ! T3;1 ð0; 0Þ Arakawa, J.C. Watts, J. Hoenninger, Magnetic-resonance imaging T2;1 ! T2;2 ! T2;1 ð0; 0Þ performance – a comparison of sodium and hydrogen, Radiology 156 T3;1 ! T3;2 ! T2;1 ðp=2; p=2Þ (1985) 133–138. [24] A.A. Maudsley, S.K. Hilal, Biological aspects of Na-23 imaging, British Med. Bull. 40 (1984) 165–166. [25] M.E. Moseley, W.M. Chew, M.C. Nishimura, T.L. Richards, J. Murphy-Boesch, G.B. Young, T.M. Marschner, L.H. Pitts, T.L. James, In vivo sodium-23 magnetic The spectral densities Jð0Þ, Jðx0Þ, and Jð2x0Þ can be estimated resonance surface coil imaging: observing experimental cerebral ischemia in by observing the build-up of T and T during inversion recovery the rat, Magn. Reson. Imaging 3 (1985) 383–387. 3;0 3;1 [26] S.K. Hilal, A.A. Maudsley, J.B. Ra, H.E. Simon, P. Roschmann, S. Wittekoek, Z.H. and spin echo sequences, respectively. Measuring the decays of the Cho, S.K. Mun, In vivo NMR imaging of sodium-23 in the human head, J. double-quantum and triple-quantum coherences can be useful to Comput. Assist. Tomogr. 9 (1985) 1–7. [27] J.B. Ra, S.K. Hilal, C.H. Oh, I.K. Mun, In vivo magnetic resonance imaging of validate the biexponential fitting curves on the build-up of T ; 3 0 sodium in the human body, Magn. Reson. Med. 7 (1988) 11–22. and T3;1, or to increase accuracy. See the main text for a discussion [28] W. Grodd, U. Klose, Sodium-MR-imaging of the brain – initial clinical-results, of the main results of the Appendix. Neuroradiology 30 (1988) 399–407. 44 G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47

[29] F.E. Boada, J.S. Gillen, G.X. Shen, S.Y. Chang, K.R. Thulborn, Fast three [63] T. Dinesen, B. Sanctuary, Relaxation of anisotropically oriented I = 3/2 nuclei dimensional sodium imaging, Magn. Reson. Med. 37 (1997) 706–715. in the multipole basis: evolution of the second rank tensor in the double [30] S. Wimperis, B. Wood, Triple-quantum sodium imaging, J. Magn. Reson. 95 quantum filtered nuclear magnetic resonance experiment, J. Chem. Phys. 101 (1991) 428–436. (1994) 7372–7380. [31] A. Borthakur, I. Hancu, F.E. Boada, G.X. Shen, E.M. Shapiro, R. Reddy, In vivo [64] G. Bodenhausen, H. Kogler, R. Ernst, Selection of coherence-transfer pathways triple quantum filtered twisted projection sodium MRI of human articular in NMR pulse experiments, J. Magn. Reson. 58 (1984) 370–388. cartilage, J. Magn. Reson. 141 (1999) 286–290. [65] M. Munowitz, A. Pines, Principles and applications of multiple-quantum [32] I. Hancu, F.E. Boada, G.X. Shen, Three-dimensional triple-quantum-filtered NMR, Adv. Chem. Phys. 66 (1987) 1–152. 23Na imaging of in vivo human brain, Magn. Reson. Med. 42 (1999) 1146– [66] J.D. van Beek, M. Carravetta, G.C. Antonioli, M.H. Levitt, Spherical tensor 1154. analysis of nuclear magnetic resonance signals, J. Chem. Phys. 122 (2005) [33] A. Abragam, The Principles of Nuclear Magnetism: The International Series of 244510. Monographs on Physics, Oxford University Press, Oxford, 1961. [67] S. Chandra Shekar, P. Rong, A. Jerschow, Irreducible spherical tensor analysis [34] C. Slichter, Principles of Magnetic Resonance, Springer, Berlin, 1990. of quadrupolar nuclei, Chem. Phys. Lett. 464 (2008) 235–239. [35] G. Jaccard, S. Wimperis, G. Bodenhausen, Multiple quantum NMR- [68] W. Ling, A. Jerschow, Relaxation-allowed nuclear magnetic resonance spectroscopy of S=3/2 spins in isotropic phase: a new probe for transitions by interference between the quadrupolar coupling and the multiexponential relaxation, J. Chem. Phys. 85 (1986) 6282–6293. paramagnetic interaction, J. Chem. Phys. 126 (2007) 064502. [36] A. Borthakur, E. Mellon, S. Niyogi, W. Witschey, J.B. Kneeland, R. Reddy, [69] U. Eliav, W. Ling, G. Navon, A. Jerschow, Magnetic alignment and Sodium and T1rho MRI for molecular and diagnostic imaging of articular quadrupolar/paramagnetic cross-correlation in complexes of Na with cartilage, NMR Biomed. 19 (2006) 781–821. LnDOTP5-, J. Magn. Reson. 216 (2012) 114–120. [37] A. Tsang, R.W. Stobbe, C. Beaulieu, Triple-quantum-filtered sodium imaging [70] D. Woessner, Nuclear spin relaxation in ellipsoids undergoing rotational of the human brain at 4.7 T, Magn. Reson. Med. 67 (2012) 1633–1643. brownian motion, J. Chem. Phys. 37 (1962) 647–654. [38] R. Ouwerkerk, K.B. Bleich, J.S. Gillen, M.G. Pomper, P.A. Bottomley, Tissue [71] S. Vyas, A. Weekley, B. Tenn, J. Flinders, T. Dieckmann, M. Augustine, Using sodium concentration in human brain tumors as measured with 23Na MR sodium cation organization to study the phase behavior of bicelle solutions, J. imaging, Radiology 227 (2003) 529–537. Phys. Chem. B 107 (2003) 10956–10961. [39] N. Bansal, L. Szczepaniak, D. Ternullo, J.L. Fleckenstein, C.R. Malloy, Effect of [72] D. Sobieski, N. Krueger, S. Vyas, M. Augustine, Nuclear spin relaxation of exercise on Na-23 MRI and relaxation characteristics of the human calf sodium cations in bacteriophage Pf1 solutions, J. Chem. Phys. 125 (2006) muscle, J. Magn. Reson. Imaging 11 (2000) 532–538. 244509. [40] A.M. Nagel, E. Amarteifio, F. Lehmann-Horn, K. Jurkat-Rott, W. Semmler, L.R. [73] S. Vyas, C. Hernandez, M. Augustine, Ordering of alkali halide salts dissolved Schad, M.A. Weber, 3 Tesla sodium inversion recovery magnetic resonance in bacteriophage Pf1 solutions: a nuclear magnetic resonance study, J. Chem. imaging allows for improved visualization of intracellular sodium content Phys. 116 (2002) 7109–7115. changes in muscular channelopathies, Invest. Radiol. 46 (2011) 759–766. [74] R.B. Hutchison, J.I. Shapiro, Measurement of intracellular sodium with NMR [41] E.M. Shapiro, A. Borthakur, A. Gougoutas, R. Reddy, 23Na MRI accurately methods, Concepts Magn. Reson. 3 (1991) 215–263. measures fixed charge density in articular cartilage, Magn. Reson. Med. 47 [75] R.K. Gupta, P. Gupta, R.D. Moore, Nmr-studies of intracellular metal-ions in (2002) 284–291. intact-cells and tissues, Annu. Rev. Biophys. Bioeng. 13 (1984) 221–246. [42] G. Madelin, A. Jerschow, R.R. Regatte, Sodium relaxation times in the knee [76] H. Naritomi, M. Kanashiro, M. Sasaki, Y. Kuribayashi, T. Sawada, Invivo joint in vivo at 7 T, NMR Biomed. 25 (2012) 530–537, http://dx.doi.org/ measurements of intracellular and extracellular Na+ and water in the brain 10.1002/nbm.1768. and muscle by nuclear-magnetic-resonance spectroscopy with shift-reagent, [43] R.R. Ernst, G. Bodenhausen, A. Wokaun, et al., Principles of Nuclear Magnetic Biophys. J. 52 (1987) 611–616. Resonance in One and Two Dimensions, vol. 14, Clarendon Press, Oxford, 1987. [77] P.M. Winter, N. Bansal, TmDOTP5- as a Na-23 shift reagent for the [44] P. Guntert, N. Schaefer, G. Otting, K. Wuthrich, Poma: a complete subcutaneously implanted 9L gliosarcoma in rats, Magn. Reson. Med. 45 mathematica implementation of the NMR product–operator formalism, J. (2001) 436–442. Magn. Reson., Ser. A 101 (1993) 103–105. [78] J.W. van der Veen, P. van Gelderen, J.H. Creyghton, W.M. Bovee, Diffusion in [45] Y. Zhang, F. Han, A. Jerschow, Product operator formalism, eMagRes red blood cell suspensions: separation of the intracellular and extracellular (Encyclopedia of Magnetic Resonance), 2013. NMR sodium signal, Magn. Reson. Med. 29 (1993) 571–574. [46] G.J. Bowden, W.D. Hutchison, Tensor operator-formalism for multiple- [79] A.R. Waldeck, A.J. Lennon, B.E. Chapman, P.W. Kuchel, 7Li and 23Na nuclear quantum NMR. 1. Spin-1 nuclei, J. Magn. Reson. 67 (1986) 403–414. magnetic resonance studies of transport and diffusion in liposomes. [47] G.J. Bowden, W.D. Hutchison, J. Khachan, Tensor operator-formalism for Comparison of transport rate constants estimated using pulsedfield multiple-quantum NMR. 2. Spins-3/2, spin-2, and spin-5/2 and general-i, J. gradient and magnetization-transfer procedures, Faraday Trans. 89 (1993) Magn. Reson. 67 (1986) 415–437. 2807–2814. [48] J.R.C. Van der Maarel, Thermal relaxation and coherence dynamics of spin 3/2. [80] P. Lundberg, P.W. Kuchel, Diffusion of solutes in agarose and alginate gels: 1H I. Static and fluctuating quadrupolar interactions in the multipole basis, and 23Na PFGSE and 23Na TQF NMR studies, Magn. Reson. Med. 37 (1997) Concepts Magn. Reson., Part A 19A (2003) 97–116. 44–52. [49] D. Varshalovich, A. Moskalev, V. Kersonskii, Quantum Theory of Angular [81] U. Eliav, X. Xu, A. Jerschow, G. Navon, Optic nerve: separating compartments Momentum, World Scientific, Teaneck, NJ, 1987. based on 23Na TQF spectra and TQF-diffusion anisotropy, J. Magn. Reson. 23 [50] G. Navon, H. Shinar, U. Eliav, Y. Seo, Multiquantum filters and order in tissues, (2013) 61–65. NMR Biomed. 14 (2001) 112–132. [82] R. Stobbe, C. Beaulieu, In vivo sodium magnetic resonance imaging of the [51] C.W. Chung, S. Wimperis, Optimum detection of spin-3/2 biexponential human brain using soft inversion recovery fluid attenuation, Magn. Reson. relaxation using multiple-quantum filtration techniques, J. Magn. Reson. 88 Med. 54 (2005) 1305–1310. (1990) 440–447. [83] G. Madelin, J.S. Lee, S. Inati, A. Jerschow, R.R. Regatte, Sodium inversion [52] G. Bodenhausen, Multiple-quantum NMR, Prog. NMR Spectrosc. 14 (1980) recovery MRI of the knee joint in vivo at 7 T, J. Magn. Reson. 207 (2010) 42– 137–173. 52. [53] R. Reddy, M. Shinnar, Z. Wang, J.S. Leigh, Multiple-quantum filters of spin-3/2 [84] R.P. Kline, E.X. Wu, D.P. Petrylak, M. Szabolcs, P.O. Alderson, M.L. Weisfeldt, P. with pulses of arbitrary flip angle, J. Magn. Reson., Ser. B 104 (1994) 148–152. Cannon, J. Katz, Rapid in vivo monitoring of chemotherapeutic response using [54] L. Fleysher, N. Oesingmann, M. Inglese, B0 inhomogeneity-insensitive triple- weighted sodium magnetic resonance imaging, Clin. Cancer Res. 6 (2000) quantum-filtered sodium imaging using a 12-step phase-cycling scheme, 2146–2156. NMR Biomed. 23 (2010) 1191–1198. [85] P. Rong, R.R. Regatte, A. Jerschow, Clean demarcation of cartilage tissue 23Na [55] C. Tanase, F.E. Boada, Triple-quantum-filtered imaging of sodium in presence by inversion recovery, J. Magn. Reson. 193 (2008) 207–209. of sodium in the presence of B0 inhomogeneities, J. Magn. Reson. 174 (2005) [86] G. Madelin, J. Babb, D. Xia, G. Chang, S. Krasnokutsky, S.B. Abramson, A. 270–278. Jerschow, R.R. Regatte, Articular cartilage: evaluation with fluid-suppressed [56] D.P. Fiege, S. Romanzetti, D.H. Y Tse, D. Brenner, A. Celik, J. Felder, N.J. Shah, 7.0-T sodium MR imaging in subjects with and subjects without B0 insensitive multiple-quantum filtered resolved sodium imaging using a osteoarthritis, Radiology 268 (2013) 481–491. phase-rotation scheme, J. Magn. Reson. 228 (2013) 32–36. [87] G. Chang, G. Madelin, O.H. Sherman, E.J. Strauss, D. Xia, M.P. Recht, A. [57] U. Eliav, H. Shinar, G. Navon, An observation of 23Na NMR triple-quantum Jerschow, R.R. Regatte, Improved assessment of cartilage repair tissue using dynamic shift in solution, J. Magn. Reson. 94 (1991) 439–444. fluid-suppressed 23Na inversion recovery MRI at 7 Tesla: preliminary results, [58] U. Eliav, G. Navon, Nuclear magnetic resonance line shapes of double and Eur. Radiol. 22 (2012) 1341–1349. triple quantum coherences of spin 3/2 nuclei, J. Chem. Phys. 95 (1991) 7114. [88] G. Madelin, J.S. Babb, D. Xia, G. Chang, A. Jerschow, R.R. Regatte, [59] A. Jerschow, Mathnmr: spin and spatial tensor manipulations in Reproducibility and repeatability of quantitative sodium magnetic mathematica, J. Magn. Reson. 176 (2005) 7–14. resonance imaging in vivo in articular cartilage at 3 T and 7 T, Magn. [60] J.S. Lee, R.R. Regatte, A. Jerschow, Optimal control NMR differentiation Reson. Med. 68 (2011) 841–849. between fast and slow sodium, Chem. Phys. Lett. 494 (2010) 331–336. [89] G. Madelin, A. Jerschow, R. Regatte, Sodium MRI with fluid suppression: will [61] J.S. Lee, R.R. Regatte, A. Jerschow, Optimal excitation of Na-23 nuclear spins in it improve early detection of osteoarthritis?, Imaging Med 3 (2011) 1–4. the presence of residual quadrupolar coupling and quadrupolar relaxation, J. [90] J.L. Allis, A.-M.L. Seymour, G.K. Radda, Absolute quantification of intracellular Chem. Phys. 131 (2009) 174501. Na+ using triple-quantum-filtered sodium-23 NMR, J. Magn. Reson. 93 (1991) [62] I. Hancu, J.R. van der Maarel, F.E. Boada, A model for the dynamics of spins 3/2 71–76. in biological media: signal loss during radiofrequency excitation in triple- [91] G. Navon, J.G. Werrmann, R. Maron, S.M. Cohen, 31P NMR and triple quantum quantum-filtered sodium mri, J. Magn. Reson. 147 (2000) 179–191. filtered 23Na NMR studies of the effects of inhibition of Na+/H+ exchange on G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 45

intracellular sodium and pH in working and ischemic hearts, Magn. Reson. [119] B.J. Balcom, R.P. MacGregor, S.D. Beyea, D.P. Green, R.L. Armstrong, T.W. Med. 32 (1994) 556–564. Bremner, Single-point ramped imaging with T-1 enhancement (SPRITE), J. [92] V.D. Schepkin, I.O. Choy, T.F. Budinger, D.Y. Obayashi, S.E. Taylor, W.M. Magn. Reson. A 123 (1996) 131–134. Decampli, S.C. Amartur, J.N. Young, Sodium TQF NMR and intracellular [120] D. Idiyatullin, C. Corum, J.Y. Park, M. Garwood, Fast and quiet MRI using a sodium in isolated crystalloid perfused rat heart, Magn. Reson. Med. 39 swept radiofrequency, J. Magn. Reson. 181 (2006) 342–349. (1998) 557–563. [121] M. Weiger, K.P. Pruessmann, F. Hennel, MRI with zero echo time: hard versus [93] J.M. Dizon, J.S. Tauskela, D. Wise, D. Burkhoff, P.J. Cannon, J. Katz, Evaluation sweep pulse excitation, Magn. Reson. Med. 66 (2011) 379–389. of triple-quantum-filtered 23Na NMR in monitoring of intracellular na [122] D.M. Grodzki, P.M. Jakob, B. Heismann, Ultrashort echo time imaging using content in the perfused rat heart: comparison of intra- and extracellular pointwise encoding time reduction with radial acquisition (PETRA), Magn. transverse relaxation and spectral amplitudes, Magn. Reson. Med. 35 (1996) Reson. Med. 67 (2012) 510–518. 336–345. [123] J.D. Osullivan, A fast sinc function gridding algorithm for fourier inversion in [94] J.S. Tauskela, J.M. Dizon, J. Whang, J. Katz, Evaluation of multiple-quantum- computer-tomography, IEEE Trans. Med. Imaging 4 (1985) 200–207. filtered 23Na NMR in monitoring intracellular Na content in the isolated [124] H. Sedarat, D.G. Nishimura, On the optimality of the gridding reconstruction perfused rat heart in the absence of a chemical-shift reagent, J. Magn. Reson. algorithm, IEEE Trans. Med. Imaging 19 (2000) 306–317. 127 (1997) 115–127. [125] H. Schomberg, J. Timmer, The gridding method for image reconstruction by [95] Y. Zhang, M. Poirer-Quinot, C.S. Springer Jr, J.A. Balschi, Discrimination of Fourier transformation, IEEE Trans. Med. Imaging 14 (1995) 596–607. intra-and extracellular 23Na+ signals in yeast cell suspensions using [126] J.A. Fessler, On NUFFT-based gridding for non-Cartesian MRI, J. Magn. Reson. longitudinal magnetic resonance relaxography, J. Magn. Reson. 205 (2010) 188 (2007) 191–195. 28–37. [127] L. Greengard, J. Lee, Accelerating the nonuniform fast Fourier transform, SIAM [96] A.M. Torres, D.J. Philp, R. Kemp-Harper, C. Garvey, P.W. Kuchel, Determination Rev. 46 (2004) 443–454. of Na+ binding parameters by relaxation analysis of selected 23Na NMR [128] T. Knopp, S. Kunis, D. Potts, A note on the iterative MRI reconstruction from coherences: RNA, BSA and SDS, Magn. Reson. Chem. 43 (2005) 217–224. nonuniform k-space data, Int. J. Biomed. Imaging 24727 (2007) 1–9. [97] S.M. Grieve, B. Wickstead, A.M. Torres, P. Styles, S. Wimperis, P.W. Kuchel, [129] K. Block, M. Uecker, J. Frahm, Undersampled radial MRI with multiple coils. Multiple-quantum filtered 17O and 23Na NMR analysis of mitochondrial Iterative image reconstruction using a total variation constraint, Magn. suspensions, Biophys. Chem. 73 (1998) 137–143. Reson. Med. 57 (2007) 1086–1098. [98] G. Navon, Complete elimination of the extracellular 23Na NMR signal in triple [130] G. Madelin, G. Chang, R. Otazo, A. Jerschow, R.R. Regatte, Compressed sensing quantum filtered spectra of rat hearts in the presence of shift reagents, Magn. sodium MRI of cartilage at 7 T: preliminary study, J. Magn. Reson. 214 (2012) Reson. Med. 30 (1993) 503–506. 360–365. [99] U. Eliav, H. Shinar, G. Navon, The formation of a 2nd-rank tensor in Na-23 [131] F.E. Boada, G. LaVerde, C. Jungreis, E. Nemoto, C. Tanase, I. Hancu, Loss of cell double-quantum-filtered nmr as an indicator for order in a biological tissue, J. ion homeostasis and cell viability in the brain: what sodium MRI can tell us, Magn. Reson. 98 (1992) 223–229. Curr. Top. Dev. Biol. 70 (2005) 77–101. [100] J. Pekar, P.F. Renshaw, J.S. Leigh, Selective detection of intracellular sodium by [132] T. Hashimoto, H. Ikehira, H. Fukuda, A. Yamaura, O. Watanabe, Y. Tateno, R. coherence-transfer NMR, J. Magn. Reson. 72 (1987) 159–161. Tanaka, H.E. Simon, In vivo sodium-23 MRI in brain tumors: evaluation of [101] T. Knubovets, H. Shinar, G. Navon, Quantification of the contribution of preliminary clinical experience, Am. J. Physiol. Imaging 6 (1991) 74–80. extracellular sodium to 23Na multiple-quantum-filtered NMR spectra of [133] S.S. Winkler, Na-23 magnetic-resonance brain imaging, Neuroradiology 32 suspensions of human red blood cells, J. Magn. Reson. 131 (1998) 92–96. (1990) 416–420. [102] T. Knubovets, H. Shinar, U. Eliav, G. Navon, A 23Na multiple-quantum-filtered [134] D.P. Fiege, S. Romanzetti, C.C. Mirkes, D. Brenner, N.J. Shah, Simultaneous NMR study of the effect of the cytoskeleton conformation on the anisotropic single-quantum and triple-quantum-filtered MRI of 23Na (SISTINA), Magn. motion of sodium ions in red blood cells, J. Magn. Reson., Ser. B 110 (1996) Reson. Med. 69 (2012) 1691–-1696. 16–25. [135] A.M. Nagel, M. Bock, C. Hartmann, L. Gerigk, J.O. Neumann, M.A. Weber, M. [103] L.A. Jelicks, R.K. Gupta, On the extracellular contribution to multiple quantum Bendszus, A. Radbruch, W. Wick, H.P. Schlemmer, W. Semmler, A. Biller, The filtered 23Na NMR of perfused rat heart, Magn. Reson. Med. 29 (1993) 130– potential of relaxation-weighted sodium magnetic resonance imaging as 133. demonstrated on brain tumors, Invest. Radiol. 46 (2011) 539–547. [104] L.A. Jelicks, R.K. Gupta, Double-quantum NMR of sodium-ions in cells and [136] K.R. Thulborn, A.M. Lu, I.C. Atkinson, F. Damen, J.L. Villano, Quantitative tissues – paramagnetic quenching of extracellular coherence, J. Magn. Reson. sodium MR imaging and sodium bioscales for the management of brain 81 (1989) 586–592. tumors, Neuroimaging Clin. N. Am. 19 (2009) 615–624. [105] J. Pekar, J.S. Leigh, Detection of biexponential relaxation in Na-23 facilitated [137] L. Fleysher, N. Oesingmann, R. Brown, D.K. Sodickson, G.C. Wiggins, M. by double-quantum filtering, J. Magn. Reson. 69 (1986) 582–584. Inglese, Noninvasive quantification of intracellular sodium in human brain [106] H. Shinar, T. Knubovets, U. Eliav, G. Navon, Sodium interaction with ordered using ultrahigh-field MRI, NMR Biomed. 26 (2013) 9–19. structures in mammalian red blood cells detected by Na-23 double quantum [138] I.C. Atkinson, L. Renteria, H. Burd, N.H. Pliskin, K.R. Thulborn, Safety of human NMR, Biophys. J. 64 (1993) 1273–1279. MRI at static fields above the FDA 8T guideline: sodium imaging at 9.4 T does [107] U. Eliav, G. Navon, Analysis of double-quantum-filtered NMR spectra of 23Na not affect vital signs or cognitive ability, J. Magn. Reson. Imaging 26 (2007) in biological tissues, J. Magn. Reson., Ser. B 103 (1994) 19–29. 1222–1227. [108] N.C. Nielsen, C. Kehlet, S.J. Glaser, N. Khaneja, Optimal control methods in [139] C.C. Mirkes, J. Hoffmann, G. Shajan, R. Pohmann, K. Scheffler, High-resolution NMR spectroscopy, eMagRes (Encyclopedia of Magnetic Resonance), 2010. quantitative sodium imaging at 9.4 Tesla, Magn. Reson. Med. (2014), http:// [109] J.S. Lee, R.R. Regatte, A. Jerschow, Selective detection of ordered sodium dx.doi.org/10.1002/mrm.25096. signals by a jump-and-return pulse sequence, J. Magn. Reson. 200 (2009) [140] The National Institute of Neurological Disorders and Stroke rt-PA Stroke 126–129. Study Group. Tissue plasminogen activator for acute ischemic stroke, N. Engl. [110] J.S. Lee, R.R. Regatte, A. Jerschow, Optimal nuclear magnetic resonance J. Med. 333 (1995) 1581–1587. excitation schemes for the central transition of a spin 3/2 in the presence of [141] A. Tsang, R.W. Stobbe, N. Asdaghi, M.S. Hussain, Y.A. Bhagat, C. Beaulieu, D. residual quadrupolar coupling, J. Chem. Phys. 129 (2008) 224510. Emery, K.S. Butcher, Relationship between sodium intensity and perfusion [111] T.B. Parrish, D.S. Fieno, S.W. Fitzgerald, R.M. Judd, Theoretical basis for deficits in acute ischemic stroke, J. Magn. Reson. Imaging 33 (2011) 41–47. sodium and potassium MRI of the human heart at 1.5 T, Magn. Reson. Med. [142] R. Luypaert, S. Boujraf, S. Sourbron, M. Osteaux, Diffusion and perfusion MRI: 38 (1997) 653–661. basic physics, Eur. J. Radiol. 38 (2001) 19–27. [112] S. Nielles-Vallespin, M.A. Weber, M. Bock, A. Bongers, P. Speier, S.E. Combs, J. [143] G. Schlaug, A. Benfield, A.E. Baird, B. Siewert, K.O. Lovblad, R.A. Parker, R.R. Wohrle, F. Lehmann-Horn, M. Essig, L.R. Schad, 3D radial projection Edelman, S. Warach, The ischemic penumbra: operationally defined by technique with ultrashort echo times for sodium MRI: clinical applications diffusion and perfusion MRI, Neurology 53 (1999) 1528–1537. in human brain and skeletal muscle, Magn. Reson. Med. 57 (2007) 74–81. [144] T. Neumann-Haefelin, H.J. Wittsack, F. Wenserski, M. Siebler, R.J. Seitz, U. [113] A.M. Nagel, F.B. Laun, M.A. Weber, C. Matthies, W. Semmler, L.R. Schad, Modder, H.J. Freund, Diffusion- and perfusion-weighted MRI. The DWI/PWI Sodium MRI using a density-adapted 3D radial acquisition technique, Magn. mismatch region in acute stroke, Stroke 30 (1999) 1591–1597. Reson. Med. 62 (2009) 1565–1573. [145] T. Ueda, W.T. Yuh, T. Taoka, Clinical application of perfusion and diffusion MR [114] F.E. Boada, G.X. Shen, S.Y. Chang, K.R. Thulborn, Spectrally weighted twisted imaging in acute ischemic stroke, J. Magn. Reson. Imaging 10 (1999) 305–309. projection imaging: reducing T2 signal attenuation effects in fast three- [146] F.E. Boada, Y.X. Qian, E. Nemoto, T. Jovin, C. Jungreis, S.C. Jones, J. Weimer, V. dimensional sodium imaging, Magn. Reson. Med. 38 (1997) 1022–1028. Lee, Sodium MRI and the assessment of irreversible tissue damage during [115] A.M. Lu, I.C. Atkinson, T.C. Claiborne, F.C. Damen, K.R. Thulborn, Quantitative hyper-acute stroke, Transl. Stroke Res. 3 (2012) 236–245. sodium imaging with a flexible twisted projection pulse sequence, Magn. [147] M.S. Hussain, R.W. Stobbe, Y.A. Bhagat, D. Emery, K.S. Butcher, D. Manawadu, Reson. Med. 63 (2010) 1583–1593. N. Rizvi, P. Maheshwari, J. Scozzafava, A. Shuaib, C. Beaulieu, Sodium imaging [116] P.T. Gurney, B.A. Hargreaves, D.G. Nishimura, Design and analysis of a intensity increases with time after human ischemic stroke, Ann. Neurol. 66 practical 3D cones trajectory, Magn. Reson. Med. 55 (2006) 575–582. (2009) 55–62. [117] J. Pipe, N. Zwart, E. Aboussouan, R. Robison, A. Devaraj, K. Johnson, A new [148] K.R. Thulborn, T.S. Gindin, D. Davis, P. Erb, Comprehensive MR imaging design and rationale for 3D orthogonally oversampled k-space trajectories, protocol for stroke management: tissue sodium concentration as a measure Magn. Reson. Med. 66 (2011) 1303–1311. of tissue viability in nonhuman primate studies and in clinical studies, [118] S. Romanzetti, M. Halse, J. Kaffanke, K. Zilles, B.J. Balcom, N.J. Shah, A Radiology 213 (1999) 156–166. comparison of three SPRITE techniques for the quantitative 3D imaging of the [149] K.R. Thulborn, D. Davis, J. Snyder, H. Yonas, A. Kassam, Sodium MR imaging of Na-23 spin density on a 4 T whole-body machine, J. Magn. Reson. 179 (2006) acute and subacute stroke for assessment of tissue viability, Neuroimaging 64–72. Clin. N. Am. 15 (2005) 639–653. 46 G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47

[150] D. Hanahan, R. Weinberg, The hallmarks of cancer, Cell 100 (2000) 57–70. [175] R. Ouwerkerk, P.A. Bottomley, M. Solaiyappan, A.E. Spooner, G.F. Tomaselli, [151] D. Rotin, D. Steelenorwood, S. Grinstein, I. Tannock, Requirement of the Na+/ K.C. Wu, R.G. Weiss, Tissue sodium concentration in myocardial infarction in H+ exchanger for tumor-growth, Cancer Res. 49 (1989) 205–211. humans: a quantitative Na-23 MR imaging study, Radiology 248 (2008) 88– [152] M. Spector, S. O’Neal, E. Racker, Reconstitution of the Na+K+ pump of Ehrlich 96. ascites tumor and enhancement of efficiency by quercetin, J. Biol. Chem. 255 [176] T. Pabst, J. Sandstede, M. Beer, W. Kenn, A. Greiser, M. von Kienlin, S. (1980) 5504–5507. Neubauer, D. Hahn, Optimization of ECG-triggered 3D Na-23 MRI of the [153] I.L. Cameron, N.K.R. Smith, T.B. Pool, R.L. Sparks, Intracellular concentration of human heart, Magn. Reson. Med. 45 (2001) 164–166. sodium and other elements as related to mitogenesis and oncogenesis [177] T. Clausen, Na+-K+ pump regulation and skeletal muscle contractility, Physiol. in vivo, Cancer Res. 40 (1980) 1493–1500. Rev. 83 (2003) 1269–1324. [154] N. Weidner, Tumor angiogenesis: review of current applications in tumor [178] M.J. McKenna, J. Bangsbo, J.M. Renaud, Muscle K+,Na+, and Cl+ disturbances prognostication, Semin. Diagn. Pathol. 10 (1993) 302–313. and Na+-K+ pump inactivation: implications for fatigue, J. Appl. Physiol. 104 [155] C. Bjartmar, B. Trapp, Axonal and neuronal degeneration in multiple sclerosis: (2008) 288–295. mechanisms and functional consequences, Curr. Opin. Neurol. 14 (2001) [179] G. Chang, L. Wang, M.E. Schweitzer, R.R. Regatte, 3D 23Na MRI of human 271–278. skeletal muscle at 7 Tesla: initial experience, Eur. Radiol. 20 (2010) 2039– [156] S.G. Waxman, Mechanisms of disease: sodium channels and neuroprotection 2046. in multiple sclerosis – current status, Nat. Clin. Pract. Neurol. 4 (2008) 159– [180] G. Sweeney, A. Klip, Mechanisms and consequences of Na+,K+-pump 169. regulation by insulin and leptin, Cell. Mol. Biol. 47 (2001) 363–372. [157] S.G. Waxman, M.J. Craner, J.A. Black, Na+ channel expression along axons in [181] E. Amarteifio, A.M. Nagel, M.A. Weber, K. Jurkat-Rott, F. Lehmann-Horn, multiple sclerosis and its models, Trends Pharmacolog. Sci. 25 (2004) 584– Hyperkalemic periodic paralysis and permanent weakness: 3-T MR imaging 591. depicts intracellular 23Na overload – initial results, Radiology 264 (2012) [158] M. Inglese, G. Madelin, N. Oesingmann, J. Babb, W. Wu, B. Stoeckel, J. Herbert, 154–163. G. Johnson, Brain tissue sodium concentration in multiple sclerosis: a sodium [182] Y.Y. Vilin, P.C. Ruben, Slow inactivation in voltage-gated sodium channels: imaging study at 3 tesla, Brain 133 (2010) 847–857. molecular substrates and contributions to channelopathies, Cell Biochem. [159] W. Zaaraoui, S. Konstandin, B. Audoin, A.M. Nagel, A. Rico, I. Malikova, E. Biophys. 35 (2001) 171–190. Soulier, P. Viout, S. Confort-Gouny, P.J. Cozzone, J. Pelletier, L.R. Schad, J.P. [183] T. Kushnir, T. Knubovets, Y. Itzchak, U. Eliav, M. Sadeh, L. Rapoport, E. Kott, G. Ranjeva, Distribution of brain sodium accumulation correlates with disability Navon, In vivo 23Na NMR studies of myotonic dystrophy, Magn. Reson. Med. in multiple sclerosis: a cross-sectional 23Na MR imaging study, Radiology 37 (1997) 192–196. 264 (2012) 859–867. [184] C.D. Constantinides, J.S. Gillen, F.E. Boada, M.G. Pomper, P.A. Bottomley, [160] E.A. Mellon, D.T. Pilkinton, C.M. Clark, M.A. Elliott, W.R. Witschey, A. Human skeletal muscle: sodium MR imaging and quantification-potential Borthakur, R. Reddy, Sodium MR imaging detection of mild Alzheimer applications in exercise and disease, Radiology 216 (2000) 559–568. disease: preliminary study, Am. J. Neuroradiol. 30 (2009) 978–984. [185] W.M. Hofmann, G.L. Denardo, Sodium flux in myotonic muscular dystrophy, [161] K. Reetz, S. Romanzetti, I. Dogan, C. Sass, C.J. Werner, J. Schiefer, J.B. Schulz, Am. J. Physiol. 214 (1968) 330–336. N.J. Shah, Increased brain tissue sodium concentration in Huntington’s [186] C. Kopp, P. Linz, L. Wachsmuth, A. Dahlmann, T. Horbach, C. Schofl, W. Renz, disease – a sodium imaging study at 4 T, Neuroimage 63 (2012) 517–524. D. Santoro, T. Niendorf, D.N. Muller, M. Neininger, A. Cavallaro, K.U. Eckardt, [162] C.H. Lee, D.D. Dershaw, D. Kopans, P. Evans, B. Monsees, D. Monticciolo, R.J. R.E. Schmieder, F.C. Luft, M. Uder, J. Titze, 23Na magnetic resonance imaging Brenner, L. Bassett, W. Berg, S. Feig, et al., Breast cancer screening with of tissue sodium, Hypertension 59 (2012) 167–172. imaging: recommendations from the society of breast imaging and the ACR [187] C. Kopp, P. Linz, A. Dahlmann, M. Hammon, J. Jantsch, D.N. Müller, R.E. on the use of mammography, breast MRI, breast ultrasound, and other Schmieder, A. Cavallaro, K.-U. Eckardt, M. Uder, et al., 23Na magnetic technologies for the detection of clinically occult breast cancer, J. Am. Coll. resonance imaging-determined tissue sodium in healthy subjects and Radiol. 7 (2010) 18–27. hypertensive patientsnovelty and significance, Hypertension 61 (2013) [163] E. Warner, H. Messersmith, P. Causer, A. Eisen, R. Shumak, D. Plewes, 635–640. Systematic review: using magnetic resonance imaging to screen women at [188] H. Mankin, V. Mow, J. Buckwalter, J. Iannotti, A. Ratcliffe, Form and function high risk for breast cancer, Ann. Int. Med. 148 (2008) 671–679. of articular cartilage, Orthop. Basic Sci. (1994) 1–44. [164] H.D. Nelson, K. Tyne, A. Naik, C. Bougatsos, B.K. Chan, L. Humphrey, Screening [189] R.R. Regatte, S.V. Akella, A. Borthakur, J.B. Kneeland, R. Reddy, In vivo proton for breast cancer: an update for the US Preventive Services Task Force, Ann. MR three-dimensional T1rho mapping of human articular cartilage: initial Int. Med. 151 (2009) 727–737. experience, Radiology 229 (2003) 269–274. [165] N. Calonge, D.B. Petitti, T.G. DeWitt, A.J. Dietrich, K.D. Gregory, D. Grossman, [190] T.J. Mosher, B.J. Dardzinski, Cartilage MRI T2 relaxation time mapping: G. Isham, M.L. LeFevre, R.M. Leipzig, L.N. Marion, et al., Screening for breast overview and applications, Semin. Musculoskelet. Radiol. 8 (2004) 355–368. cancer: US Preventive Services Task Force recommendation statement, Ann. [191] W. Ling, R.R. Regatte, G. Navon, A. Jerschow, Assessment of Int. Med. 151 (2009) 716–736. glycosaminoglycan concentration in vivo by chemical exchange-dependent [166] W. Berg, J. Blume, J. Cormack, E. Mendelson, D. Lehrer, M. Böhm-Vélez, E. saturation transfer (gagCEST), PNAS 105 (2008) 2266–2270. Pisano, R. Jong, W. Evans, M. Morton, et al., Combined screening with [192] J.-S. Lee, R.R. Regatte, A. Jerschow, Isolating chemical exchange saturation ultrasound and mammography vs mammography alone in women at transfer contrast from magnetization transfer asymmetry under two- elevated risk of breast cancer, JAMA 299 (2008) 2151–2163. frequency rf irradiation, J. Magn. Reson. 215 (2012) 56–63. [167] L.J. Esserman, I.M. Thompson, B. Reid, Overdiagnosis and overtreatment in [193] J.-S. Lee, P. Parasoglou, D. Xia, A. Jerschow, R.R. Regatte, Uniform cancer: an opportunity for improvement. Emerging overdiagnosis and magnetization transfer in chemical exchange saturation transfer magnetic overtreatment viewpoint, JAMA 310 (2013) 797–798. resonance imaging, Sci. Rep. 3 (2013). 1707: 1–5. [168] R. Ouwerkerk, M.A. Jacobs, K.J. Macura, A.C. Wolff, V. Stearns, S.D. Mezban, [194] A. Bashir, M.L. Gray, D. Burstein, Gd-DTPA2 – as a measure of cartilage N.F. Khouri, D.A. Bluemke, P.A. Bottomley, Elevated tissue sodium degradation, Magn. Reson. Med. 36 (1996) 665–673. concentration in malignant breast lesions detected with non-invasive 23Na [195] L. Filidoro, O. Dietrich, J. Weber, E. Rauch, T. Oerther, M. Wick, M.F. Reiser, C. MRI, Breast Cancer Res. Treat. 106 (2007) 151–160. Glaser, High-resolution diffusion tensor imaging of human patellar cartilage: [169] M.A. Jacobs, V. Stearns, A.C. Wolff, K. Macura, P. Argani, N. Khouri, T. feasibility and preliminary findings, Magn. Reson. Med. 53 (2005) 993–998. Tsangaris, P.B. Barker, N.E. Davidson, Z.M. Bhujwalla, D.A. Bluemke, R. [196] L.M. Lesperance, M.L. Gray, D. Burstein, Determination of fixed charge density Ouwerkerk, Multiparametric magnetic resonance imaging, spectroscopy and in cartilage using nuclear magnetic resonance, J. Orthop. Res. 10 (1992) 1–13. multinuclear (Na-23) imaging monitoring of preoperative chemotherapy for [197] W. Ling, R.R. Regatte, M.E. Schweitzer, A. Jerschow, Behavior of ordered locally advanced breast cancer, Acad. Radiol. 17 (2010) 1477–1485. sodium in enzymatically depleted cartilage tissue, Magn. Reson. Med. 56 [170] M.A. Jacobs, R. Ouwerkerk, A.C. Wolff, E. Gabrielson, H. Warzecha, S. Jeter, (2006) 1151–1155. D.A. Bluemke, R. Wahl, V. Stearns, Monitoring of neoadjuvant chemotherapy [198] G. Saar, B. Zhang, W. Ling, R.R. Regatte, G. Navon, A. Jerschow, Assessment of using multiparametric, 23Na sodium MR, and multimodality (PET/CT/MRI) glycosaminoglycan concentration changes in the intervertebral disc via imaging in locally advanced breast cancer, Breast Cancer Res. Treat. 128 chemical exchange saturation transfer, NMR Biomed. 25 (2012) 255–261. (2011) 119–126. [199] R. Reddy, E.K. Insko, E.A. Noyszewski, R. Dandora, J.B. Kneeland, J.S. Leigh, [171] M.A. Jansen, J.G. Van Emous, M.G. Nederhoff, C.J. Van Echteld, Assessment of Sodium MRI of human articular cartilage in vivo, Magn. Reson. Med. 39 myocardial viability by intracellular 23Na magnetic resonance imaging, (1998) 697–701. Circulation 110 (2004) 3457–3464. [200] A.J. Wheaton, A. Borthakur, E.M. Shapiro, R.R. Regatte, S.V. Akella, J.B. [172] M. Horn, C. Weidensteiner, H. Scheffer, M. Meininger, M. de Groot, H. Remkes, Kneeland, R. Reddy, Proteoglycan loss in human knee cartilage: quantitation C. Dienesch, K. Przyklenk, M. von Kienlin, S. Neubauer, Detection of with sodium MR imaging – feasibility study, Radiology 231 (2004) 900–905. myocardial viability based on measurement of sodium content: a 23Na- [201] R. Reddy, E.K. Insko, J.S. Leigh, Triple quantum sodium imaging of articular NMR study, Magn. Reson. Med. 45 (2001) 756–764. cartilage, Magn. Reson. Med. 38 (1997) 279–284. [173] R. Jerecic, M. Bock, S. Nielles-Vallespin, C. Wacker, W. Bauer, L.R. Schad, ECG- [202] K. Keinan-Adamsky, H. Shinar, S. Shabat, Y.S. Brin, M. Nyska, G. Navon, 23Na gated 23Na-MRI of the human heart using a 3D-radial projection technique and 2H magnetic resonance studies of osteoarthritic and osteoporotic with ultra-short echo times, MAGMA 16 (2004) 297–302. articular cartilage, Magn. Reson. Med. 64 (2010) 653–661. [174] R. Ouwerkerk, R.G. Weiss, P.A. Bottomley, Measuring human cardiac tissue [203] J. Choy, W. Ling, A. Jerschow, Selective detection of ordered sodium signals sodium concentrations using surface coils, adiabatic excitation, and twisted via the central transition, J. Magn. Reson. 180 (2006) 105–109. projection imaging with minimal T-2 losses, J. Magn. Reson. Imaging 21 [204] W. Ling, A. Jerschow, Frequency-selective quadrupolar MRI contrast, Solid St. (2005) 546–555. Nucl. Magn. Reson. 29 (2006) 227–231. G. Madelin et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 79 (2014) 14–47 47

[205] Š. Zby` nˇ, D. Stelzeneder, G. Welsch, L. Negrin, V. Juras, M. Mayerhoefer, P. emerging tumor chemotherapeutic resistance, NMR Biomed. 19 (2006) Szomolanyi, W. Bogner, S. Domayer, M. Weber, et al., Evaluation of native 1035–1042. hyaline cartilage and repair tissue after two cartilage repair surgery [224] V.D. Schepkin, F.C. Bejarano, T. Morgan, S. Gower-Winter, M. Ozambela Jr., techniques with 23Na MR imaging at 7 T: initial experience, Osteoarthr. C.W. Levenson, In vivo magnetic resonance imaging of sodium and diffusion Cartilage 20 (2012) 837–845. in rat glioma at 21.1 T, Magn. Reson. Med. 67 (2012) 1159–1166. [206] S. Trattnig, G.H. Welsch, V. Juras, M.E. Szomolanyi, P. Mayerhoefer, D. [225] Y. Qian, T. Zhao, G.C. Wiggins, L.L. Wald, H. Zheng, J. Weimer, F.E. Boada, Stelzeneder, T.C. Mamisch, O. Bieri, K. Scheffler, S. Zbyn, Na-23 MR imaging at Sodium imaging of human brain at 7 t with 15-channel array coil, Magn. 7 T after knee matrix-associated autologous chondrocyte transplantation: Reson. Med. 68 (2012) 1807–1814. preliminary results, Radiology 257 (2010) 175–184. [226] R. Brown, G. Madelin, R. Lattanzi, G. Chang, R.R. Regatte, D.K. Sodickson, G.C. [207] B. Schmitt, Š. Zby` nˇ, D. Stelzeneder, V. Jellus, D. Paul, L. Lauer, P. Bachert, S. Wiggins, Design of a nested eight-channel sodium and four-channel proton Trattnig, Cartilage quality assessment by using glycosaminoglycan chemical coil for 7 T knee imaging, Magn. Reson. Med. 70 (2013) 259–268. exchange saturation transfer and 23Na MR imaging at 7 T, Radiology 260 [227] L. Ying, Z.P. Liang, Parallel MRI using phased array coils, IEEE Sign. Proc. 27 (2011) 257–264. (2010) 90–98. [208] G. Knutsen, L. Engebretsen, T.C. Ludvigsen, J.O. Drogset, T. Grøntvedt, E. [228] M. Lustig, D. Donoho, J.M. Pauly, Sparse MRI: the application of compressed Solheim, T. Strand, S. Roberts, V. Isaksen, O. Johansen, Autologous sensing for rapid MR imaging, Magn. Reson. Med. 58 (2007) 1182–1195. chondrocyte implantation compared with microfracture in the knee. A randomized trial, J. Bone Joint Surg. 86 (2004) 455–464. Glossary [209] R. Gudas, E. Stankevicˇius, E. Monastyreckiene˙ , D. Pranys, R.J. Kalesinskas, Osteochondral autologous transplantation versus microfracture for the treatment of articular cartilage defects in the knee joint in athletes, Knee AD: Altzheimer’s disease Surg. Sports Traumatol. Arthrosc. 14 (2006) 834–842. ADC: apparent diffusion coefficient [210] W. Bartlett, J. Skinner, C. Gooding, R. Carrington, A. Flanagan, T. Briggs, G. BMS: bone marrow stimulation Bentley, Autologous chondrocyte implantation versus matrix-induced CEST: chemical exchange saturation transfer autologous chondrocyte implantation for osteochondral defects of the knee, CSF: cerebrospinal fluid a prospective randomized study, J. Bone Joint Surg., Br. 87 (2005) 640–645. CVF: cell volume fraction [211] E.K. Insko, D.B. Clayton, M.A. Elliott, In vivo sodium MR imaging of the DWI: diffusion weighted imaging intervertebral disk at 4 T, Acad. Radiol. 9 (2002) 800–804. DQC: double quantum coherence [212] C.Y. Wang, E. McArdle, M. Fenty, W. Witschey, M. Elliott, M. Sochor, R. Reddy, DQF: double quantum filter A. Borthakur, Validation of sodium magnetic resonance imaging of ECM: extra-cellular matrix intervertebral disc, Spine 35 (2010) 505–510. [213] J. James, C. Lin, H. Stark, B. Dale, N. Bansal, Optimization and characterization EFG: electric field gradient of sodium MRI using 8-channel 23Na and 2-channel 1H RX/TX coil, in: 13th FCD: fixed charge density International Conference on Biomedical Engineering, Springer, 2009, pp. FID: free induction decay 138–141. FOV: field of view [214] N. Maril, Y. Rosen, G.H. Reynolds, A. Ivanishev, L. Ngo, R.E. Lenkinski, Sodium GAG: glycosaminoglycan MRI of the human kidney at 3 Tesla, Magn. Reson. Med. 56 (2006) 1229– GM: grey matter 1234. IR: inversion recovery [215] Y. Rosen, R.E. Lenkinski, Sodium MRI of a human transplanted kidney, Acad. ISTO: irreducible spherical tensor operator Radiol. 16 (2009) 886–889. LvN: Liouville–von Neumann [216] D. Hausmann, S. Konstandin, F. Wetterling, S. Haneder, A. Nagel, D. Dinter, S. MA: magic angle Schönberg, F. Zöllner, L. Schad, Apparent diffusion coefficient and sodium MACT: matrix-associated autologous chondrocyte transplantation concentration measurements in human prostate tissue via hydrogen-1 and MI: myocardial infarction sodium-23 magnetic resonance imaging in a clinical setting at 3 T, Invest. MQC: multiple quantum coherence Radiol. 47 (2012) 1–6. MQF: multiple quantum filter [217] D. Hausmann, S. Konstandin, F.G. Zollner, S. Haneder, F. Wetterling, A.M. Nagel, D.J. Dinter, S.O. Schonber, L. Schad, Sodium imaging of the prostate at 3 MS: multiple sclerosis T, Proc. ISMRM (2012) 546. NMSI: normalized mean signal intensity [218] M.A. Jacobs, R. Ouwerkerk, I. Kamel, P.A. Bottomley, D.A. Bluemke, H.S. Kim, OA: osteoarthritis Proton, diffusion-weighted imaging, and sodium (23Na) MRI of uterine OCT: optimal control theory leiomyomata after MR-guided high-intensity focused ultrasound: a PBS: phosphate buffered saline preliminary study, J. Magn. Reson. Imaging 29 (2009) 649–656. PG: proteoglycan [219] F. Wetterling, D.M. Corteville, R. Kalayciyan, A. Rennings, S. Konstandin, A.M. PWI: perfusion weighted imaging Nagel, H. Stark, L.R. Schad, Whole body sodium MRI at 3 T using an RF: radio frequency asymmetric birdcage resonator and short echo time sequence: first images ROI: region of interest of a male volunteer, Phys. Med. Biol. 57 (2012) 4555–4567. RQI: residual quadrupolar interaction [220] A.M. Babsky, H. Zhang, S.K. Hekinatyar, G.D. Hutchins, N. Bansal, Monitoring SAR: specific absorption rate chemotherapeutic response in RIF-1 tumors by single-quantum and triple- SNR: signal-to-noise ratio quantum-filtered Na-23 MRI, H-1 diffusion-weighted MRI and PET imaging, SQC: single quantum coherence Magn. Reson. Imaging 25 (2007) 1015–1023. [221] V.D. Schepkin, B.D. Ross, T.L. Chenevert, A. Rehemtulla, S. Sharma, M. Kumar, TE: echo time J. Stojanovska, Sodium magnetic resonance imaging of chemotherapeutic TPI: twisted projection imaging response in a rat glioma, Magn. Reson. Med. 53 (2005) 85–92. TQC: triple quantum coherence [222] T.L. Chenevert, C.R. Meyer, B.A. Moffat, A. Rehemtulla, S.K. Mukherji, S.S. TR: repetition time Gebarski, D.J. Quint, P.L. Robertson, T.S. Lawrence, L. Junck, J.M. Taylor, T.D. TQF: triple quantum filter Johnson, Q. Dong, K.M. Muraszko, J.A. Brunberg, B.D. Ross, Diffusion MRI: a TSC: tissue sodium concentration new strategy for assessment of cancer therapeutic efficacy, Mol. Imaging 1 UTE: ultrashort TE (echo time) (2002) 336–343. WM: white matter [223] V.D. Schepkin, K.C. Lee, K. Kuszpit, M. Muthuswami, T.D. Johnson, T.L. Chenevert, A. Rehemtulla, B.D. Ross, Proton and sodium MRI assessment of