NIH Public Access Author Manuscript J Magn Reson. Author manuscript; available in PMC 2012 April 1. NIH-PA Author Manuscript Published in final edited form as: J Magn Reson. 2011 April ; 209(2): 269–276. doi:10.1016/j.jmr.2011.01.022. Relaxation Dispersion in MRI Induced by Fictitious Magnetic Fields Timo Liimatainen1,2,*, Silvia Mangia2, Wen Ling2, Jutta Ellermann2, Dennis J. Sorce2, Michael Garwood2, and Shalom Michaeli2 1Department of Biotechnology and Molecular Medicine A.I.Virtanen Institute for Molecular Sciences, University of Eastern Finland, Kuopio, Finland 2Center for Magnetic Resonance Research and Department of Radiology, University of Minnesota, Minneapolis, USA Abstract NIH-PA Author Manuscript A new method entitled Relaxation Along a Fictitious Field (RAFF) was recently introduced for investigating relaxations in rotating frames of rank ≥ 3. RAFF generates a fictitious field (E) by applying frequency-swept pulses with sine and cosine amplitude and frequency modulation operating in a sub-adiabatic regime. In the present work, MRI contrast is created by varying the orientation of E, i.e. the angle between E and the z axis of the second rotating frame. When > 45°, the amplitude of the fictitious field E generated during RAFF is significantly larger than the RF field amplitude used for transmitting the sine/cosine pulses. Relaxation during RAFF was investigated using an invariant-trajectory approach and the Bloch-McConnell formalism. Dipoledipole interactions between identical (like) spins and anisochronous exchange (e.g., exchange between spins with different chemical shifts) in the fast exchange regime were considered. Experimental verifications were performed in vivo in human and mouse brain. Theoretical and experimental results demonstrated that changes in induced a dispersion of the relaxation rate constants. The fastest relaxation was achieved at ≈ 56°, where the averaged contributions from transverse components during the pulse are maximal and the contribution from longitudinal components are minimal. RAFF relaxation dispersion was compared with the relaxation dispersion achieved with off-resonance spin lock T1ρ experiments. As compared with the offresonance spin lock T1ρ method, a slower rotating frame relaxation rate was observed with RAFF, which under certain experimental conditions is desirable. NIH-PA Author Manuscript Keywords Relaxation; rotating frame; relaxation along fictitious field; relaxation dispersion; human brain; mouse brain * Corresponding author: Timo Liimatainen, Ph.D., Department of Biotechnology and Molecular Medicine, A.I.Virtanen Institute for Molecular Sciences, University of Eastern Finland, Yliopistonranta 1, 70210 Kuopio, Finland, Tel. +358 40 355 3903, FAX: +358 17 163 030, timo.liimatainen@uef.fi. Publisher's Disclaimer: This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Liimatainen et al. Page 2 Introduction NIH-PA Author Manuscript Rotating frame longitudinal and transverse relaxation experiments (with characteristic time constants T1ρ and T2ρ, respectively) are capable of probing slow molecular motions at high magnetic fields (B0) with the motional correlation times in the ns to ms time scale [1-5]. For some biomedical applications, both T1ρ and T2ρ have been shown to be more informative than laboratory frame T1 and T2 in assessing specific tissue pathologic processes, for example cartilage degeneration, gene therapy-induced apoptotic response in BT4C gliomas, cerebral ischemia, and brain iron accumulation and neuronal integrity in patients with Parkinson's disease [6-11]. Although T1ρ and T2ρ measurements are commonly pursued with time-invariant radiofrequency (RF) pulses, these experiments can also be conducted using a train of amplitude-(AM) and frequency-modulated (FM) pulses, each of which completes an adiabatic full-passage (AFP) [12-14]. NIH-PA Author Manuscript A potential limitation to the wide spread exploitation of rotating frame relaxation in living systems is the required RF power delivered to the sample (i.e., specific absorption rate (SAR) and concern about tissue heating). To satisfy the adiabatic condition in adiabatic T1ρ and T2ρ experiments, the RF power deposition can sometimes exceed allowable SAR levels for human studies at high magnetic fields. The RF power needed in the classical spin-lock experiments often limits its applicability for human studies, although RF power can sometimes be reduced by using off-resonance irradiation to create the locking field, Beff [15]. However, in both on- and off-resonance T1ρ experiments, a separate RF pulses are needed to place the magnetization vector initially at the locking angle, which can be challenging to achieve accurately in the presence of non-uniform RF field (B1), especially at high magnetic fields. To satisfy the adiabatic condition, a sufficiently slow frequency sweep must take place to minimize the fictitious field component, γ-1dα/dt ŷ, that exists in a second rotating frame, (the ωeff –frame) where γ is the gyromagnetic ratio and α is the time-dependent angle between the effective RF field Beff and the axis of quantization in the first rotating frame, z (collinear with B0). By convention B1 is assumed to be along the x -axis of the first rotating frame. The vector sum of γ-1dα/dt ŷ and Beff leads to a fictitious effective field E in the ωeff– frame. NIH-PA Author Manuscript Recently we introduced a rotating frame relaxation experiment that exploits E called Relaxation Along a Fictitious Field (RAFF). RAFF does not require an initial rotation of the magnetization to a specific locking angle. As compared to continuous wave (CW) T1ρ and adiabatic T1ρ and T2ρ methods, RAFF experiments can be performed with reduced RF power because the stationary spin-locking field E is produced by AM and FM functions (sine and cosine, respectively) operating in a sub-adiabatic condition [16]. In the initial RAFF method, we set the angle between Beff and E to = 45° when designing the sine/ cosine pulse, by using parameters satisfying the condition Beff = |γ-1dα/dt|. In the present work, we further explore the effect of sub-adiabatic rotation with FM pulses, and use different amplitudes of the fictitious component (γ-1dα/dt) to produce different locking angles and amplitudes of E. With this goal, we first extend the experimental capabilities of the RAFF method, and then use an invariant trajectory method to model the evolution of the transverse and longitudinal components during these sine/cosine pulses. Finally, we present our initial developments of relaxation theory during RAFF. This theory covers dipolar interactions between like spins in the Redfield limit (fast rotational motion) [17] and anisochronous exchange (e.g., exchange between spins with different chemical shifts) in the fast exchange regime described here using both the invariant trajectory approach and the Bloch-McConnell formalism. These theoretical efforts are meant to predict general features of the dispersion of relaxation rate constants induced by varying in RAFF, while J Magn Reson. Author manuscript; available in PMC 2012 April 1. Liimatainen et al. Page 3 NIH-PA Author Manuscript recognizing that they would be inadequate for simulation purposes of intrinsic relaxation parameters of tissue in vivo. Based on theoretical predictions and on experimental verifications that we obtained in vivo from human and mouse brain, we demonstrate that dispersion of relaxation rates are indeed induced by altering the orientation of E in the RAFF method, in analogy to how altering the orientation of Beff induces dispersion of relaxation rate constants in the CW spin-lock experiment. This property can be utilized to expand the possible MRI contrasts that can be obtained with the RAFF method, thus offering a novel tool to investigate tissues in vivo with the sensitivity of rotating frame methods and with acceptable RF power levels. Finally, our results demonstrate that relaxation rate constants are slower with RAFF as compared to conventional off-resonance spin-lock T1ρ. This property is especially beneficial at high magnetic fields where relaxation pathways such as anisochronous exchange are significantly accelerated. Theory Description of the RAFF method NIH-PA Author Manuscript The ωRF–frame rotates around the laboratory z axis with the time-dependent pulse frequency ωRF(t). By convention, the axes of the ωRF-frame are labeled x , y , z . In this reference frame, the net effective field Beff(t) is the vector sum of B1(t) x̂′ and the fictitious field γ-1Δω(t) ẑ′ = γ-1(ω0 - ωRF(t)) ẑ′, where Δω is the offset frequency (i.e., the difference between the Larmor frequency (ω0 = γB0) and the pulse frequency (ωRF)). The amplitude of Beff(t) is given by (1) and the angle between Beff and the first rotating frame z axis is (2) NIH-PA Author Manuscript where ω1(t) = γB1(t). When the adiabatic condition, |γ-1dα/dt| ≪ Beff, is well satisfied, the trajectory of the net magnetization (M) can be approximately described as a simple nutation about Beff(t) with the angular velocity ωeff(t) = γBeff(t). In the sub-adiabatic condition, however, the rapid sweep of Beff(t) results in a non-negligible fictitious field vector in ωeff– frame (double-primed axis labels, x , y , z ). This frame is sometimes referred to as “doubly rotating” because it rotates with frequency ω0 − ωRF around the z-axis of the laboratory frame and simultaneously with frequency dα/dt around the y -axis of the ωRF-frame. The magnitude of the fictitious field component that arises from the rotation around the y -axis is equal to γ-1dα/dt and is directed along the y axis. Note, y and y axes are the same, and Beff is directed along the z axis by convention [16]. The net effective field in the ωeff–frame is therefore the vector sum of Beff(t) ẑ″ and γ-1dα/dt ŷ . Accordingly, the time-dependent amplitude of this net effective field is (3) J Magn Reson. Author manuscript; available in PMC 2012 April 1. Liimatainen et al. Page 4 and the angle between E and Beff is NIH-PA Author Manuscript (4) When Beff(t) and dα/dt are time invariant throughout the pulse, the amplitude of E remains constant. From Eq. (4), it can be seen that a change of γ-1dα/dt relative to Beff(t) creates the possibility to vary the locking angle . It can also be seen that the magnitude of E is greater than the peak RF amplitude, B1max = γ-1 ω1max (Fig. 1, and Eq. (3)). In our prior work, E was kept stationary with = 45° by choosing AM and FM shapes based on sine and cosine functions with equal amplitudes [16]. However, the angular velocity of the sweep (dα/dt) depends on the arguments of the sine and cosine functions. It is straightforward to verify that a stationary angle of any arbitrary value can be produced. To accomplish this, let us modulate the pulse amplitude and offset frequency according to sine and cosine functions having maximum amplitude of and frequency ω, NIH-PA Author Manuscript (5) (6) In this case, the amplitude of Beff(t) can be simplified based on Eq. (1) to (7) This result, when combined with Eq. (4), leads to (8) NIH-PA Author Manuscript It can be verified using Eqs. 2, 5, and 6) that, to produce a given angle , ω must equal dα/dt. That is, (9) Therefore, the amplitude and frequency modulation functions based on Eqs. (5, 6, 8 and 9) are J Magn Reson. Author manuscript; available in PMC 2012 April 1. Liimatainen et al. Page 5 (10) NIH-PA Author Manuscript (11) The analytical form for the phase modulation function of the pulse, which is typically applied in the MR scanners, can be obtained by (12) Substitution of the frequency offset function given by Eq. (11) into Eq. (12) yields the sine/ cosine phase modulation function of the pulse, NIH-PA Author Manuscript (13) In this work, the RAFF pulse was calculated using Eqs. (5,6), and the length of each RAFF pulse element was chosen as (14) for all angles of used. In experiments and simulations, the basic RAFF pulse element to produce a specified was concatenated to create a set of windowless RAFF trains. Examples of the basic RAFF pulse element ( = 5°, 45°, and 85°) are shown in Fig. 1. In previous work NIH-PA Author Manuscript was added to the middle section of the RAFF element to achieve [16], a phase of self-refocusing, as used in the adiabatic plane rotation pulse, BIR-4 [18]. For an arbitrary choice of the time-invariant parameter , the RAFF modulation functions, including selfrefocusing, are: (15) J Magn Reson. Author manuscript; available in PMC 2012 April 1. Liimatainen et al. Page 6 NIH-PA Author Manuscript An additional phase of π must be added to the phase function during the times when the amplitude modulation function is negative (i.e., a π phase shift is the equivalent of a change in sign of ω1(t)). Notably, the period of the sine and cosine functions is longer than the RAFF pulse element when (ω1 (t)tan ( ))-1 ≥ Tp/4; in this case, the amplitude of the RAFF with < 24° (Fig. 1). pulse element does not reach If M is initially aligned with Beff, during the RAFF pulse it precesses on a cone having an axis defined by the vector E. Wigner rotation matrices can be used to transform from the ωRF–frame (x , y , z ) to the ωeff–frame (x , y , z ), followed by a final rotation around x by the angle to align the axis of quantization (z ) along E. Invariant Trajectory When the spin rotation is characterized by rotational correlation times on the order of ps/ns, the Redfield relaxation theory (or fast motion limit) with the perturbation treatment of the spin dynamics applies because the corresponding relaxation rate constants are larger than the NMR anisotropies (in frequency units) modulated by the rotational motion [2,17,19]. Here, the condition applicable to non-viscous liquids should be satisfied, ωeff τc ≪1. The invariant trajectory approach introduced by Griesinger and Ernst [20] can be used to calculate the relaxation rate constants in the presence of RF irradiation. For dipolar auto-relaxations of like spins, the effective relaxation rate constant Reff is given by: NIH-PA Author Manuscript (16) where R1 and R2 are free precession longitudinal and transverse relaxation rate constants respectively, while Clong and Ctr represent their relative weightings and the integration is over the duration of the pulse of Tp. The weighted average depends on the trajectory of the normalized magnetization vector M(t) = [Mx(t),My(t),Mz(t)], with the condition (17) being satisfied. Similar equations have been found for evaluating exchange-induced relaxation rate constants when setting R2 and R1 in Eq. (16) to NIH-PA Author Manuscript (18) As will be shown below, this approach agrees well with the Bloch-McConnell treatment. Evaluating exchange induced relaxations using invariant trajectory approach In order to use the invariant trajectory approach to calculate relaxation rate constants in RAFF, the magnetization trajectory during RAFF irradiation needs to be calculated. The generalized matrix describing rotation around an arbitrary axis in three-dimensional space is given by normalized unit vectors [u, v, w] which has the form J Magn Reson. Author manuscript; available in PMC 2012 April 1. Liimatainen et al. Page 7 NIH-PA Author Manuscript (19) for time invariant angle ϕ. In the ωeff–frame, the amplitude of E in rad/s is ωE, so the angle of rotation of M at time t is ωEt and the direction of M is defined by the normalized unit vectors [u=0, v= sin( ), w=cos( )]. It is noticed here that is time invariant. When viewing from the ωeff-frame, M initially aligned along z will evolve according to: (20) NIH-PA Author Manuscript Transformation from the ωeff–frame to the ωRF–frame is performed around y axis with the rotation matrix given by: (21) Thus, the magnetization trajectory in the ωRF–frame is described as: NIH-PA Author Manuscript (22) Now, the Mx(t) and My(t) components of M(t) are given by: (23) J Magn Reson. Author manuscript; available in PMC 2012 April 1. Liimatainen et al. Page 8 Here, . Finally, substituting Eq. (18) in Eq. (16) the exchange induced NIH-PA Author Manuscript relaxation rate constant ( fast exchange regime is ) in the case of two-site anisochronous exchange (2SX) in the (24) with Mx(t) and My(t) given by Eq. (23). Here, PA and PB are populations of exchanging sites A and B with chemical shift difference ω in rad/s and exchange correlation time τex. It should be noted that the consideration above is valid for like spin at one specific site A or B. Bloch-McConnell Formulation of the Relaxations during RAFF Relaxations during RF irradiation due to dipolar interactions (like spins) and induced by anisochronous exchange between two pools A and B can be described using BlochMcConnell equations written in the phase-modulated rotating frame [21]: NIH-PA Author Manuscript (25) where ΔA,B are the chemical shifts relative to on-resonance in rad/s of exchanging groups A and B, respectively ( ω = |ΔA + ΔB|), PA and PB are the populations of the exchanging sites, NIH-PA Author Manuscript and are the exchange rate constants, and T1,2,A,B=1/R1,2,A,B are the relaxation time constants at sites A and B, respectively. Here, the sine/cosine modulation and functions (Eq. (15)) were used. The RF pulse amplitude was set to the pulse duration (Tp) was set according to Eq. (14). For the simulations, fifteen linearly spaced angles between 5 and 85° were generated. The pulse train was extended by increasing the number of basic RAFF elements up to a total length of 144.82 ms. The R1 and R2 were calculated considering dipolar interactions between isolated identical spins (at specific sites A and B), following: (26) and J Magn Reson. Author manuscript; available in PMC 2012 April 1. Liimatainen et al. Page 9 NIH-PA Author Manuscript (27) where τc is the rotational correlation time and ω0 is the Larmor precession frequency, b= −μ0ħγ2/(4πr3), μ0=4π · 10-7 H/m is the vacuum permeability, ħ = 1.055 · 10-34 Js is Planck's constant, and r is the internuclear distance in meters. The dipolar interaction theory used in this work is an over-simplification for biological tissues, where a variety of dipolar mechanisms contribute to relaxation. However, the complete description of dipolar relaxation is outside of the scope of this work. The simulations of the two-site exchange were carried out using Eq. (24) with T1 and T2 calculated using τc = 2·10-12 s for both sites A and B. The decay of M (M0=[0 0 1]) during the pulse was estimated by solving Eq. (25) using the Runge-Kutta numerical method. Simulations were performed with magnetization initially oriented along z M0=[0 0 1] and then for inverted M0=[0 0 -1] to include the steady state in the analysis. The approach of placing magnetization initially along the +z or −z axis was previously described in detail [16], and was shown to facilitate the analysis of the relaxations during sine / cosine pulses. Materials and Methods NIH-PA Author Manuscript All human experiments were performed according to procedures approved by the Institutional Review Board of the University of Minnesota Medical School. After obtaining informed consent, MRI measurements on human brain (of 5 healthy volunteers) were performed with a 4 T magnet (Oxford Instruments) interfaced to a Varian UNITYINOVA console. A volume coil based on the transverse electromagnetic design was utilized for brain imaging [22]. Images were acquired using fast spin echo readout, TR = 4.5 s, 15 ms echo spacing, number of echoes 8, matrix size 256 × 128, FOV = 256 × 256 mm2, and slicethickness = 4 mm. Maps of the RAFF relaxation rate constant (RRAFF) were generated with 0, 36, 72, 108, 144 ms preparation pulse train lengths. Sine/cosine pulse power calibration was performed using LASER localization [23] in voxels placed in the area of interest and using 400 μs hard pulse to be calibrated as 90o excitation. NIH-PA Author Manuscript A total of three c57bl mice from the University of Eastern Finland, National Laboratory Animal Center, Finland were imaged in this study. The mice were housed in a controlled environment with free access to food and water. Animal experiments were reviewed and approved by the national Animal Experiment Board and conducted in accordance with the guidelines set by the European Community Council Directives 86/609/EEC. Mice were scanned in a 9.4 T vertical magnet (Oxford Instruments, Plc., Witney, UK) interfaced to a Varian DirectDrive console. A quadrature volume transceiver coil (diameter 23 mm) (Rapid Biomedical GmbH, Rimpar, Germany) was used for mouse brain imaging. In all experiments, anesthesia was induced using 4.5 % and maintained with 1.5 % isoflurane during the experiment in oxygen/N2O with fractions 0.21:0.79. Temperatures of animals were maintained using a circulating water heater. For off-resonance CW T1ρ, spin lock durations between 20 and 120 ms (4 values linearly spaced) were selected with hard pulse excitation and 180° phase flip in the middle of the CW pulse. Pulse durations were selected similarly for RAFF but linearly spaced up to 144 ms. For both setups, γB1/(2π) = 625 Hz (also for hard pulse excitation). For RAFF, a standard hyperbolic secant pulse (HS1), with Tp = 4 ms, time-bandwidth product (R) = 10, and peak amplitude = 2.5 kHz, was used for initial inversion to allow steady state fitting. A fast spin-echo sequence was used for readout with TR = 4 s, effective TE = 6.85 ms, echo spacing 3.3 ms, number of echoes in the pulse train = 4, field-of-view = 19.2 × 19.2 mm2, and matrix size = 128 × 128 for readout. J Magn Reson. Author manuscript; available in PMC 2012 April 1. Liimatainen et al. Page 10 NIH-PA Author Manuscript Experimental details of the acquisition strategy used for obtaining RAFF maps were described in detail in [16]. Briefly, signal intensity decay and rise during sine/cosine pulses for the different angles of RAFF were collected consequently with and without initial inversion of magnetization. The results with and without initial inversion of magnetization were simultaneously fitted using model that takes into account the formation of a steady state [16] (28) in which S0,±Z are the initial amplitudes of magnetization before the application of sine/ cosine pulse (with and without inversion), and SSS is the amplitude of magnetization when the pulse train length approaches infinity. Relaxation maps were generated pixel-wise using Matlab 7.1 (Mathworks Inc., Natick, MA, USA). Results and Discussion NIH-PA Author Manuscript During the sine/cosine pulse, M undergoes precession around E in the second rotating frame (the ωeff–frame) (Figs. 1 and 2). Using Bloch simulations we demonstrated previously that E behaves as a spin-locking field [16]. A unique feature of the RAFF method is that the amplitude of E can be larger than the input RF amplitude used for the sine/cosine pulses (Eqs. (1-3)), and the amplitude of E approaches its maximal value with → 90°. Thus, RAFF might be exploited in rotating frame relaxation experiments to achieve a large spinlock field without increasing the RF amplitude. NIH-PA Author Manuscript In Fig. 2, the trajectories of M in the first rotating frame (the ωRF–frame) are presented. The simulations were performed using the Runge-Kutta algorithm to solve the Bloch equations. The trajectories of M for a given between 5 and 85° demonstrate that M nutates only slightly from z axis when is small and large. With small , M undergoes slow nutation to a small angle. On the other hand, with large values of (when → 90°), fast oscillations of both frequency and amplitude modulations occur (see Eqs. (5 and 6)). For large angles, E is close to y , and in the second rotating frame, M precesses in the plane almost perpendicular to E with increasing number of the trajectory cycles as increases. The latter phenomenon becomes evident especially with high values of . The number of these cycles depends on how close E is to y , or equivalently, how fast Beff rotates relative to z . Despite the high amplitude of the effective field, the effective tip angle of M may remain small due to the rapid change in the direction of E in the ωRF–frame. For intermediate values of , M nutates with larger angles. Results of calculations of the relaxation rate constant due to dipolar interactions between like spins (RRAFF,di) are presented in Fig. 3. This analysis of RAFF was performed using the Bloch equations. It can be seen that when T1 ≈ T2, which corresponds to rotational correlation times shorter than 5·10-10 s, RRAFF,di is independent of the angle . With longer correlation times, when T1 ≠ T2, a significant dependence of RRAFF on was found. The greatest RRAFF values were observed for angles ranging between 50 and 60°. The simulations of the exchange-induced relaxation rate constant (RRAFF,ex) were performed using Bloch-McConnell formalism for two-site equilibrium exchange Eq. (24). In Fig. 4, it is shown that, for small τex < 10-5 s and large τex > 10-2 s, RRAFF,ex values are independent of . This property of RRAFF,ex is similar to the RRAFF,di dependence on for the correlation times τc ≥ 10-9 s (Fig. 4). In the intermediate region of correlation times, RRAFF,ex attains its maximum at = 56°. J Magn Reson. Author manuscript; available in PMC 2012 April 1. Liimatainen et al. Page 11 NIH-PA Author Manuscript The relative weighting contributions of the magnetization components during RAFF were investigated using invariant trajectory approach Eqs. (16-24). In Fig. 5a, the integrated magnetization weighting, i.e., normalized coefficients of transverse, Ctr, and longitudinal, Clong, components Eq. (18) are shown. It can be seen that the maximal weighting of transverse magnetization occurs with = 56° where the contribution of the longitudinal magnetization weighting is minimal. On the other hand, with the smaller and larger the contribution of Mz becomes greater and the contribution of Mx and My reduce. This suggests that largest RRAFF should be detected when = 56°. The calculation of the RRAFF rate constants were performed using Eqs. (19-24) for invariant trajectory and for BlochMcConnell approaches Eq. (25). Evidently, when M spends more time close to the xy-plane the values of RRAFF are greater (Fig. 5a). It can be seen in Fig. 4 and in Fig. 5B that these two methods correspond well and closely describe exchange-induced relaxation rate constants dependencies on the angle . NIH-PA Author Manuscript In Fig. 6A–C, a representative example of the human brain relaxation mapping with RAFF from one healthy subject at 4T is presented along with the anatomical image in Fig. 6D. The multi-subject (n=5) averaged rate constants from the ROIs indicated in Fig. 6D from substantia nigra (SN) and CSF areas are shown in Fig. 6E. The rate constants measured with RAFF (RRAFF) are significantly dependent on as can be seen in Fig. 6E. Here, a superior difference in RRAFF, greater than double, between the measurements with RAFF at 56° and 74° was detected. The differences in tissue RRAFF values obtained with small and large s are significantly bigger than those in CSF (almost no difference), suggesting good sensitivity of RAFF to slow dynamics in tissues. The results of the RAFF measurements in mouse cortex are shown in Fig. 7. Here, the comparison with CW spin lock T1ρ was also performed at 9.4 T, and the dependencies on angles (for RAFF) and α (for CW T1ρ) were investigated. It can be seen that with RAFF the maximum rate constant is observed at = 56°. As expected with CW T1ρ, the maximum rate constant is measured at α= 90° since T1ρ → T2 in the Redfield limit (fast motional regime) [17]. This experiment demonstrates an advantage of RAFF as compared to conventional CW T1ρ for rotating frame relaxation measurements at high magnetic fields (3T and above), where the relaxations induced by exchange and diffusion in local magnetic gradients (collectively referred to here as dynamic averaging) are significantly accelerated. This is particularly important for probing slow motional correlation times when the relaxation rates could be so large that the signal escapes detection. NIH-PA Author Manuscript Our theoretical description, although performed using the most parsimonious relaxation models, well described our in vivo results. At this stage of investigation we did not intend to describe the details of specific relaxation pathways during RAFF in the human brain, for example, relaxations due to dipolar interactions, dynamic averaging (e.g., anisochronous/ isochronous mechanisms), residual dipolar interactions, or cross relaxations. Detailed theoretical treatments are necessary to completely understand the relaxation during RAFF in vivo. The developed RAFF method provides a possibility to generate novel contrast in MR which is based on the relaxations in the presence of the fictitious field E. Based on the theoretical and experimental results presented in this work, the RAFF method provides excellent sensitivity to spin dynamics since it comprises characteristics of both rotating frame relaxation methods, T1ρ and T2ρ. In addition, due to its highly efficient refocusing properties, the RAFF method has similarities to the Carr-Purcell-Meiboom-Gill (CPMG) method [24], while simultaneously having characteristics of the classical rotating frame rotary echo technique [5]. Due to its sensitivity to different relaxation channels in vivo particularly in the slow motional regime, MRI contrast based on RAFF has potential utility for a number of in vivo applications. J Magn Reson. Author manuscript; available in PMC 2012 April 1. Liimatainen et al. Page 12 Conclusions NIH-PA Author Manuscript Here, we presented a method to create relaxation dispersion using a (partially) fictitious field E and altering its magnitude and orientation, in a doubly rotating frame of reference. The amplitude of E can be greater than the RF amplitude used in transmitting RAFF pulses and thus provides an alternative means to increase the magnitude of the spin-lock field without increasing the RF power. In the presence of dipolar interactions and exchange-induced relaxation between spins with different chemical shifts, the relaxation rate during RAFF has a large dependence on the tilt angle of the fictitious field E. Maximum values of RRAFF were shown to occur at = 56°, which is consistent with the weighted contributions to the relaxations from transverse and longitudinal magnetization components using an invariant trajectory analysis and Bloch-McConnell formalism. The RAFF method comprises properties of the second rotating frame relaxation method with sensitivity to slow motion and efficient refocusing of the rotating frame rotary echo, and thus holds potential to generate unique contrast for MRI. With RAFF, relaxation rates are slower than with conventional off resonance spin-lock T1ρ. This suggest its applicability for the rotating frame relaxation measurements at high magnetic fields (3T and above) where the relaxations induced by dynamic averaging are accelerated. Acknowledgments NIH-PA Author Manuscript The authors the following agencies for financial support: Instrumentarium Science Foundation (TL), Orion Corporation Research Foundation (TL), Finnish Cultural Foundation Northern Savo (TL), and NIH grants P30 NS057091, P41 RR008079, R01 NS061866, and R21 NS059813. References NIH-PA Author Manuscript 1. Abergel D, Palmer A III. On the use of the stochastic Liouville equation in nuclear magnetic resonance: application to R1ρ relaxation in the presence of exchange. Concepts Magn Reson A. 2003; 19:134–148. 2. Abragam, A. Principles of nuclear magnetism. Oxford University Press Inc.; Oxford: 1963. p. 517-522. 3. Fischer M, Majumdar A, Zuiderweg E. Protein NMR relaxation: theory, applications and outlook. Prog NMR Spectrosc. 1998; 33:207–272. 4. Palmer A, Kroenke C, Loria J. Nuclear magnetic resonance methods for quantifying microsecondto-millisecond motions in biological macromolecules. Meth Enzymol. 2001; 339:204–238. [PubMed: 11462813] 5. 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NIH-PA Author Manuscript J Magn Reson. Author manuscript; available in PMC 2012 April 1. Liimatainen et al. Page 14 NIH-PA Author Manuscript NIH-PA Author Manuscript Fig. 1. RF pulse amplitude (ω1(t)) and phase (ϕ(t)) modulation functions corresponding to = 5° (A), = 45° (B), = 85° (C) with the refocusing scheme during RAFF; and planes of magnetization rotation for initial magnetization along z (M(t=0)=[0 0 1]) (D–F) are shown. Note, the scales of ω1(t) and ϕ(t) are equally scaled for separate . NIH-PA Author Manuscript J Magn Reson. Author manuscript; available in PMC 2012 April 1. Liimatainen et al. Page 15 NIH-PA Author Manuscript NIH-PA Author Manuscript Figure 2. Magnetization trajectories during the first segments of sine/cosine pulses with different angles are shown. Fifteen values were evenly distributed between 5° and 85°, and the RF pulse amplitude and frequency modulation functions generated with ω1max/(2π) = 625 Hz were used. The Runge-Kutta algorithm was used for simulating the Bloch equations. NIH-PA Author Manuscript J Magn Reson. Author manuscript; available in PMC 2012 April 1. Liimatainen et al. Page 16 NIH-PA Author Manuscript NIH-PA Author Manuscript Figure 3. Simulated relaxation rate constants due to dipolar interactions (RRAFF,di) between two identical spins as a function of angle and correlation time τc. For the simulations, Eq. (28) was used under the condition kABex=0, along with Eqs. (29 and 30). τc = 2·10-12 s was assumed for both sites A and B. ω1max/(2π)=625 Hz. NIH-PA Author Manuscript J Magn Reson. Author manuscript; available in PMC 2012 April 1. Liimatainen et al. Page 17 NIH-PA Author Manuscript NIH-PA Author Manuscript Fig. 4. Simulated exchange-induced relaxation rate constants (RRAFF,ex) between spins with different chemical shifts as a functions of and exchange correlation times (τex). For the simulations, Eqs. (28, 29 and 30) were used. For both sites A and B, τc = 2·10-12 s was assumed. Other assumptions: difference in chemical shifts, ω=0.85 ppm; PA = PB = 0.5; ω1max/(2π) = 625 Hz. NIH-PA Author Manuscript J Magn Reson. Author manuscript; available in PMC 2012 April 1. Liimatainen et al. Page 18 NIH-PA Author Manuscript Fig 5. NIH-PA Author Manuscript A) Magnetization weighting coefficients Ctr and Clong Eqs. (20, 21) averaged over time during the RAFF pulse, plotted as a function of . B) Theoretical simulations of exchange induced relaxations with RAFF using Bloch-McConnell formalism Eqs. (25 - 27) (dashed line) and invariant trajectory method Eq. (24) (solid line), as a function of for different exchange-correlation times. For both sites A and B, τc = 2·10-12 s was assumed. Other parameters: difference in chemical shifts, ω= 0.85 ppm; PA = PB = 0.5; ω1max/(2π) = 625 Hz. Exchange correlation times are shown in the inset of (B). NIH-PA Author Manuscript J Magn Reson. Author manuscript; available in PMC 2012 April 1. Liimatainen et al. Page 19 NIH-PA Author Manuscript NIH-PA Author Manuscript Fig. 6. Maps of the relaxation rate constant RRAFF obtained from the human brain of one representative volunteer at 4 T when using sine/cosine pulses with different angles: (A) 28°, (B) 56° and (C) 74°. T2 weighted anatomical reference image (D) and multi-subject averaged relaxation rate constants (E) from regions of interest (red and green in D represent grey matter and CSF, respectively), plotted as a function of . Data are presented as mean ± SD (n=5 for all angles except = 45° where number of studies n=4). NIH-PA Author Manuscript J Magn Reson. Author manuscript; available in PMC 2012 April 1. Liimatainen et al. Page 20 NIH-PA Author Manuscript Fig. 7. NIH-PA Author Manuscript (A) Maps of the relaxation rate constant (RRAFF) measured in the intact mouse brain using several settings of the locking angle . Also shown is a T2-weighted anatomical image of the same slice with the ROIs used for the analysis. The differences in the RAFF 45° rate constants between right and left hemispheres of the brain was found on average to be 6.3% (n=3) and the displayed are the averaged rate constants between both sides of the brain. (B) A comparison between RRAFF and CW off-resonance R1ρ. The mean ± SD values are shown (n = 3). NIH-PA Author Manuscript J Magn Reson. Author manuscript; available in PMC 2012 April 1.