NeuroImage 89 (2014) 92–109
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NeuroImage journal homepage: www.elsevier.com/locate/ynimg
Whole brain mapping of water pools and molecular dynamics with rotating frame MR relaxation using gradient modulated low-power adiabatic pulses Ovidiu C. Andronesi a,⁎,1, Himanshu Bhat b, Martin Reuter c, Shreya Mukherjee a, Peter Caravan a, Bruce R. Rosen a a b c
Athinoula A. Martinos Center for Biomedical Imaging, Department of Radiology, Massachusetts General Hospital, Harvard Medical School, Boston, MA 02114, USA Siemens Medical Solutions USA Inc, Charlestown, MA 02129, USA Athinoula A. Martinos Center for Biomedical Imaging, Department of Neurology, Massachusetts General Hospital, Harvard Medical School, Boston, MA 02114, USA
a r t i c l e
i n f o
Article history: Accepted 7 December 2013 Available online 15 December 2013 Keywords: Rotating frame relaxation (T1ρ, T2ρ) Macromolecular content Correlation time Exchange time Bound water fraction Neurodegeneration
a b s t r a c t Nuclear magnetic resonance (NMR) relaxation in the rotating frame is sensitive to molecular dynamics on the time scale of water molecules interacting with macromolecules or supramolecular complexes, such as proteins, myelin and cell membranes. Hence, longitudinal (T1ρ) and transverse (T2ρ) relaxation in the rotating frame may have a great potential to probe the macromolecular fraction of tissues. This stimulated a large interest in using this MR contrast to image brain under healthy and disease conditions. However, experimental challenges related to the use of intense radiofrequency irradiation have limited the widespread use of T1ρ and T2ρ imaging. Here, we present methodological development to acquire 3D high-resolution or 2D (multi-)slice selective T1ρ and T2ρ maps of the entire human brain within short acquisition times. These improvements are based on a class of gradient modulated adiabatic pulses that reduce the power deposition, provide slice selection, and mitigate artifacts resulting from inhomogeneities of B1 and B0 magnetic fields. Based on an analytical model of the T1ρ and T2ρ relaxation we compute the maps of macromolecular bound water fraction, correlation and exchange time constants as quantitative biomarkers informative of tissue macromolecular content. Results obtained from simulations, phantoms and five healthy subjects are included. © 2013 Elsevier Inc. All rights reserved.
Introduction Longitudinal (T1ρ) and transverse (T2ρ) relaxation in the rotating frame of nuclear magnetic resonance can probe water molecules loosely bound (hydration layer) to macromolecules. The macromolecular fraction is tissue specific and changes with disease, hence T1ρ and T2ρ have been proposed (Sepponen et al., 1985) as a candidate for noninvasive imaging solid tissue matrix, albeit indirectly through the relaxation of water interacting with macromolecules. Several studies have investigated the value of T1ρ and T2ρ imaging in the context of neurological diseases, including Alzheimer's Disease (Borthakur et al., 2008; Haris et al., 2011), Parkinson Disease (Haris et al., 2011; Michaeli et al., 2007), stroke (Jokivarsi et al., 2009, 2010a), and brain cancer (Aronen et al., 1999; Hakumaki et al., 1999). These studies revealed that T1ρ and T2ρ relaxation times have a broader range of values compared to
⁎ Corresponding author at: Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, 149 Thirteenth Street, Suite 2301, Charlestown, MA 02129, USA. Fax: +1 617 726 7422. E-mail address: [email protected] (O.C. Andronesi). 1 Presented in part at the 21st Annual Meeting of ISMRM, Salt Lake City, UT, 20–26 Apr, 2013. 1053-8119/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.neuroimage.2013.12.007
T1 and T2 across disease spectrum, and better separate patients from controls. Also, T1ρ and T2ρ relaxation times seem to correlate better and earlier with therapy response or disease progression when compared to other MRI markers such as laboratory frame T1 and T2 relaxation, or ADC values (Duvvuri et al., 2001; Hakumaki et al., 2002; Jokivarsi et al., 2010a). Experimental challenges related to high specific absorption rate (SAR), non-selective spatial excitation, long acquisition times, and artifacts from B0 and B1 inhomogeneity currently prevent a wide spread adoption and usage of T1ρ and T2ρ in the neuroimaging community. Here, we present methodological development for clinically feasible and robust T1ρ and T2ρ mapping of the whole human brain based on a recent class of gradient modulated adiabatic pulses (Andronesi et al., 2010). Further, using an analytical model and fitting of relaxation curves we extract tissue specific maps for parameters of molecular dynamics such as correlation times, exchange times, and water pools. Nuclear spin relaxation in the rotating frame is obtained when a radiofrequency field (B1) is applied either along or perpendicular to the nuclear magnetization for longitudinal T1ρ relaxation or transverse T2ρ relaxation, respectively. Typically, T1ρ and T2ρ imaging of the human brain is done with B1 amplitudes in the range of 0.5–1 kHz and durations between 50 and 100 ms. The combination of high B1
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amplitude and long duration leads to increased SAR, which is often circumvented by increasing repetition times (TR) leading to long acquisition times (TA). Moreover, in all imaging studies so far T1ρ and T2ρ relaxation have been achieved via spatially non-selective radiofrequency (RF) irradiation, which further imposes limits on the TR in multislice experiments due to relaxation and saturation of magnetization in neighboring slices. As a result, rotating frame relaxation imaging is usually performed as a single slice or very few (1–4) slices at a time. Three main approaches are currently used to achieve T1ρ relaxation: i) the on-resonance continuous wave (CW) method where the magnetization is first flipped in the transverse plan and a B1 field with a constant amplitude is applied along magnetization for spin-lock (Aronen et al., 1999), ii) the off-resonance continuous wave (CW) method where an offset is used to create an effective field and magnetization is flipped along the direction of the effective field (Ramadan et al., 1998), and iii) a train of adiabatic inversion pulses producing repeated passages of the longitudinal magnetization between the + Z and − Z orientations via amplitude and frequency modulation of B1 field (Michaeli et al., 2008). In the case of T2ρ relaxation the initial magnetization is first flipped perpendicular to the direction of the spin-lock or the effective field for precession around it. The spin-lock method is easier to implement and model analytically, but it is more sensitive to artifacts induced by B0 and B1 inhomegeneities. These artifacts can be compensated over a reduced B0 and B1 range by composite pulses and phase alternation of the CW spin-lock (Witschey et al., 2007). On the other hand, due to its simultaneous frequency and amplitude sweep adiabatic pulses can cover a much larger bandwidth of B0 offsets and can tolerate several-folds of B1 inhomogeneity above the adiabatic threshold (Garwood and DelaBarre, 2001). In addition, adiabatic pulses have lower SAR compared to a CW spin-lock of the same amplitude. Our focus in this work was to address the major limitations of current T1ρ and T2ρ imaging techniques, namely high SAR and long acquisition times, while at the same time compensating for B0 and B1 inhomogeneities. Gradient modulated adiabatic pulses such as GOIAW(16,4) (Andronesi et al., 2010) simultaneously meet all these requirements. The Gradient Offset Independent Adiabaticity (GOIA) design (Tannus and Garwood, 1997) lowers the maximum B1 amplitude by 55%, resulting in 80% less SAR, compared to conventional adiabatic pulses of the same bandwidth and duration. Alternatively, GOIAW(16,4) can be made shorter and still have lower B1 amplitude than conventional adiabatic pulses, allowing more flexibility in choosing the duration of T1ρ and T2ρ preparation. Compared to a CW spin-lock of the same amplitude GOIA-W(16,4) pulses would have 20% less SAR. The use of gradient modulation in GOIA-W(16,4) pulses naturally provides slice-selectivity for T1ρ and T2ρ. In the case of 2D imaging, slice selectivity allows the use of a short TR to acquire a stack of slices without saturating magnetization of neighboring slices. Although strategies have been proposed for the CW spin-lock method to reduce SAR (Wheaton et al., 2004a) or acquire multiple slices with less magnetization saturation (Wheaton et al., 2004b), these have limitations in fully achieving those goals. In terms of the SAR deposition, GOIA pulses compare favorably also to relaxation along the fictitious field (RAFF) method (Liimatainen et al., 2010) which uses sub-adiabatic RF frequency and amplitude modulations to lower B1 requirements, at the cost of more sensitivity to B1 inhomogeneity and without slice selectivity. Here we demonstrate the use of GOIA-W(16,4) pulse trains in combination with 2D spin-echo EPI (EPI-SE) and 3D turbo FLASH (TFL) readouts as a new method for robust and feasible T1ρ and T2ρ imaging of the human brain. Theory Relaxation in the rotating frame of GOIA pulses Relaxation in the rotating frame has been an important research topic reach in applications since the beginning of nuclear magnetic resonance
93
(Redfield, 1957). The semi-classical theory of Bloch–Wangsness– Redfield (BWR), often used for practical reasons, treats the spins quantum mechanically and the lattice as random perturbation described by classical statistics (Abragam, 1961; Redfield, 1957; Wangsness and Bloch, 1953). Significant progress has been made over the last decade in laying out the analytical framework for relaxation in the rotating frame under adiabatic pulses (Mangia et al., 2009; Michaeli et al., 2008; Sorce et al., 2007). In this work we extend the theory of rotating frame relaxation for the case of gradient modulated adiabatic pulses. GOIA pulses (Tannus and Garwood, 1997) are obtained by simultaneously modulating the amplitude B1(t) and frequency Δω(t) of RF field, together with the slice selection gradient G(t) according to: Zt γ2 B2 t ′ 1 1 ′ dt Δωðt Þ ¼ Gðt Þ Q G t′
ð1Þ
0
where Q is the adiabatic factor, and γ is the nuclear gyromagnetic ratio. In particular, for GOIA-W(16,4) pulses (Andronesi et al., 2010) the modulation functions B1(t) and G(t) are based on the wideband uniform rate and smooth truncation WURST (Kupce and Freeman, 1995) function: h i16 B1 ðt Þ ¼ B1;max 1− sin π=2 2t=T p −1 h i4 Gðt Þ ¼ Gmax ð1−g m Þ þ gm sin π=2 2t=T p −1
ð2Þ
where Tp is the pulse duration, B1,max is the maximum RF amplitude, Gmax is the maximum gradient, and gm ∈ [0,1] is the gradient modulation factor. The larger the gm factor the more the gradient drops in the middle of the pulse and correspondingly lesser RF amplitude is required, an effect also known as the variable rate selective excitation VERSE (Conolly et al., 1988). Alternatively, GOIA-HS(8,4) (Andronesi et al., 2010) pulses are obtained if stretched hyperbolic-secant functions are used instead of WURST. Besides pulse duration and gradient modulation factor, the exponents of trigonometric functions can be varied too, providing great flexibility in obtaining new pulses. Modulations of GOIA-W(16,4) pulse of 4 ms duration and 5 kHz bandwidth are shown in Figs. 1A–C. In order to calculate the rates of rotating frame relaxation we need the effective field (ωeff) and the magnetization nutation angle (α): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωeff ðt; xÞ ¼ ðγB1 ðt ÞÞ2 þ ðΔωðt Þ þ γxGðt ÞÞ2 α ðt; xÞ ¼ atanðγB1 ðt Þ=ðΔωðt Þ þ γxGðt ÞÞÞ:
ð3Þ
The adiabatic condition states that the effective field should be much greater than the rate of the nutation angle ωeff ðt; xÞ≫α˙ ðt; xÞ at all times ˙ and positions. In practice an adiabatic factor Q ¼ ωeff =α≥5 and a timebandwidth product R ≥ 20 are needed to have an adiabatic pulse. Plots of the effective field modulation, nutation angle and inverse of adiabatic factor during the GOIA-W(16,4) pulse are shown in Figs. 1D–F. The adiabatic factor is greater than 5 for the most part of the pulse with the exception of short periods in the vicinity of 1.2 ms and 2.8 ms where it decreases to 3. By increasing the R product the pulse becomes more adiabatic. An important observation is necessary at this point. Due to gradient modulation the effective field and nutation angle in Eq. (3) become dependent on the spatial coordinate x in the slice direction. However, there is dependency only of the intra-slice isochromats, and not of the slice position relative to the isocenter. The effect of slice position is canceled by the additional frequency modulation that is used to excite slices off-isocenter. For simplicity this term was not included in Eq. (1), but more details are given in Andronesi et al. (2010). Thus, the slices at isocenter and off-isocenter positions have the same effective field and nutation angle, and consequently the relaxation rate does
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Fig. 1. Plots of GOIA-W(16,4) pulse: A) B1 amplitude modulation function; B) frequency sweep; C) gradient modulation; D) effective field; E) nutation angle; F) inverse of the adiabatic factor; G) slice profile at the isocenter; H) slice profile off-isocenter. A pulse duration of 4 ms and bandwidth of 5 kHz were assumed. In Fig. 1D the rate of the nutation angle is also shown with the dashed line.
not depend on the slice position. As we will show next, what is relevant for the measurement is the average relaxation rate over the pulse duration and slice thickness, which can be exactly calculated. A physical model that satisfactorily approximates the rotating frame relaxation of water in-vivo includes two main relaxation mechanisms: dipolar coupling and exchange interactions (Michaeli et al., 2004; Michaeli et al., 2005; Michaeli et al., 2008; Sorce et al., 2007). Random fluctuations due to reorientation of the direct proton–proton dipolar coupling and exchange of protons between sites with different chemical shifts or diffusion through microscopic field inhomogeneities lead to decoherence of the spin system. The stochastic nature of random perturbations is described by the spectral density functions of the type (n = 0,1,2): 2 J n ðτÞ τ= 1 þ ðnωτÞ
ð4Þ
where τ may represent the correlation time τc or exchange time τex, and ω may represent the Larmor frequency ω0 = γβ0 or the effective field ωeff from Eq. (3). The most commonly employed model (Jokivarsi et al., 2009; Mangia et al., 2009; Michaeli et al., 2008; Sierra et al., 2008) for deriving rotating frame relaxation in tissues assumes a two sites exchange model (2SX) at equilibrium between the free water
(slow relaxing component) and the water bound to macromolecules (fast relaxing component), as shown schematically in Fig. 2. Relaxation rates in the rotating frame (Blicharski, 1972) can be cast in a time dependent form for adiabatic pulses as instantaneous relaxation rates (Mangia et al., 2009; Michaeli et al., 2008). In the case of GOIA-W(16,4) pulses the instantaneous relaxation rate for dipolar coupling interaction becomes dependent on the intra-slice isochromat at position x 2
3 2 2 4 3 sin α ðt; xÞ cos α ðt; xÞ 3 sin α ðt; xÞ þ þ 2 2 7 7 1 þ ωeff ðt; xÞτc 1 þ 2ωeff ðt; xÞτc 7 7 7 2 2 2 þ 3 sin α ðt; xÞ 8−6 sin α ðt; xÞ 5 þ 2 2 1 þ ðω0 τc Þ 1 þ ð2ω0 τc Þ 2 3 2 2 3 3 cos α ðt; xÞ−1 þ 7 6 7 6 4 6 30 sin2 α ðt; xÞ cos2 α ðt; xÞ 7 3 sin α ðt; xÞ 6 þ Dc 2 2 þ 7 dd 6 7 τc 6 1 þ ω ðt; xÞτ R1ρ ðt; xÞ ¼ 7 1 þ 2ωeff ðt; xÞτ c 40 eff c 6 7 7 6 2 2 4 5 20−6 sin α ðt; xÞ 8 þ 12 sin α ðt; xÞ þ 2 2 1 þ ðω0 τc Þ 1 þ ð2ω0 τ c Þ ð5Þ 6 6 Dc dd 6 R1ρ ðt; xÞ ¼ τ 6 10 c 6 4
where the dipolar coupling constant Dc = 2I(I + 1)[μ0ℏγ2/(4πr3)]2, I = 1/2 is the spin number, and the physical constants are μ 0 = 4π 10− 7 H/m the vacuum magnetic permeability, ħ =1.055 · 10−34 Js the Planck's constant, γ = 2π 42.576·106 rad/(sT) the gyromagnetic ratio, and r = 1.58 Å is the proton–proton distance in water molecule. Simulations of dipolar relaxation rates for GOIA-W(16,4) pulses are shown in Figs. 3A,C. Similarly, the instantaneous relaxation rates for the anisochronous exchange interaction can be obtained in the form of
Fig. 2. Two site exchange (2SX) model depicting the pool of free water and pool of water bound to macromolecules. In each pool water molecules experience dipolar coupling relaxation governed by the correlation times τc of each pool. The two pools are considered in an equilibrium anisochronous exchange determined by the exchange time τex and the chemical shift difference δωex. The model is specified by four independent parameters: correlation time τc,b of bound water, τex, δωex and bound water fraction Pb (Pf = 1 − Pb and τc,f ≈ 5 ps). The fast exchange limit (FXL) is assumed in-vivo δωexτex ≪ 1.
ex
2
ex
2
R1ρ ðt; xÞ ¼ P b P f ðδωex Þ τex
sin2 α ðt; xÞ 2 1 þ ωeff ðt; xÞτex
ð6Þ
2
R2ρ ðt; xÞ ¼ P b P f ðδωex Þ τex cos α ðt; xÞ where Pb is the pool of bound water, Pf is the pool of free water (Pb + Pf = 1), and δωex is the chemical shift difference between the
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Fig. 3. Simulations of longitudinal and transverse rotating frame relaxation rates for dipolar coupling (Eq. (5)) and exchange (Eq. (6)) interactions during GOIA-W(16,4) pulses of 4 ms duration, 5 kHz bandwidth and 0.382 kHz peak amplitude. For exchange interaction (Figs. 3B,D) Pb was fixed at 0.15 and δωex was assumed 0.8 ppm (≈100 Hz at 2.89 T). The B0 field was considered 2.89 T.
two exchanging sites. Simulations for exchange relaxation rates during GOIA-W(16,4) pulses are presented in Figs. 3B,D. The intrinsic instantaneous relaxations rates Eqs. (5)–(6) are further used to compute the observed relaxation rates. First, the average dipolar and exchange relaxation rates over pulse duration need to be calculated for each slice isochromat:
Rdd;ex ðxÞ 1ρ;2ρ;b; f
1 ¼ Tp
ZT p
dd;ex R1ρ;2ρ;b; f dd;ex R1ρ;2ρ;b; f ðt; xÞ
dt:
ð7Þ
1 1 ¼ T p Δx
Δx=2 Z ZT p
dd;ex
R1ρ;2ρ;b; f ðt; xÞ dtdx −Δx=2 0
ð10Þ
dd dd ex Robs 1ρ;2ρ ¼ P b R1ρ;2ρ;b þ P f R1ρ;2ρ; f þ R1ρ;2ρ :
0
For each isochromat we can then calculate the observed (or apparent) relaxation rate averaged over the pulse duration. Under in-vivo conditions the fast exchange limit δωexτex ≪ 1 (FXL) is assumed (Jokivarsi et al., 2009; Sierra et al., 2008), and the observed relaxation rate is obtained as a linear combination between dipolar relaxation of free and bound water, and exchange relaxation: dd dd ex Robs 1ρ;2ρ ðxÞ ¼ P b ðxÞR1ρ;2ρ;b ðxÞ þ P f ðxÞR1ρ;2ρ; f ðxÞ þ R1ρ;2ρ ðxÞ:
ð8Þ
By integrating the exponential decays of all isochromats over the slice thickness we can obtain the time-spatial average of the observed relaxation rate: Robs 1ρ;2ρ ¼ −
Simulations for the average observed relaxation rates during GOIAW(16,4) pulses are presented in Fig. 4 as a function of model parameters. In the case when the change in the relaxation rates and initial signal is small among isochromats Eqs. (7)–(9) may be linearized and the average relaxations can be very well approximated by:
# " Δx=2 Z 1 1 dx ln S0 ðxÞexp −Robs ð x Þ T p 1ρ;2ρ Tp ΔxS0
ð9Þ
−Δx=2
where the mean initial magnetization (signal) over the slice is S0 ¼
Simulations performed in Figs. 5C–F show the results for the average observed relaxation rates obtained from Eqs. (7) to (9) or (10). The 2SX/FXL model depicted in Fig. 2 is described by a maximum number of seven independent parameters: two correlation times (for bound and free water), two proton–proton distances (for bound and free water), population of bound site (Pf = 1 − Pb), exchange time, and chemical shift difference. However, three parameters can be readily eliminated considering the same proton–proton distance (1.58 Å) for both bound and free water, and a standard value for the correlation time of free water (5 ps). In this case the observed relaxation rate is determined by the four remaining parameters: the correlation time of bound water, exchange time, chemical shift difference, and the population of bound water. In the ideal situation of no measurement noise a minimum number of four experimental time points are necessary to obtain the molecular parameters of interest, however experimentally extra time points are beneficial. Fig. 4 shows simulations of the observed rotating frame relaxation rate under GOIA-W(16,4) pulses for relevant in-vivo model parameters.
Δx=2
1=Δx ∫ S0 ðxÞ dx. Note that under the most general conditions the cor-
Diffusion weighting during rotating frame relaxation under GOIA pulses
−Δx=2
relation times, exchange times, exchange rate, water pools and initial signal can vary spatially across isochromats. In certain conditions, such as activation, the model parameters could vary also on the time scale of the pulse. This variation is captured implicitly in Eqs. (6)–(9).
The use of the gradient modulation during relaxation in the rotating frame of GOIA pulses raises the question of what is the effect of diffusion on the signal decay. The effect of diffusion may be expected particularly for T2ρ where magnetization processes around the
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effective field and diffusion may produce signal decay similarly like in spin echo experiments. In the case of T1ρ magnetization spends most of the time longitudinally and there is no evolution of the spin echo type. A few papers have investigated the effects of diffusion during adiabatic pulses in spin echo experiments (Sun and Bartha, 2007; Valette et al., 2010). In deriving an expression for the b-value produced by the T2ρ preparation with GOIA-W(16,4) pulses we will follow the approach proposed in Valette et al. (2010). Most generally, the b-value associated with a gradient G(t) is obtained as the integral of the square of the gradient moment k(t) Zt bðt Þ ¼
k
2
′ ′ t dt
0
ð11Þ
′ k t ¼γ
′
″ ″ G t dt :
To calculate the gradient moment for the GOIA-W(16,4) pulse we can use the commonly assumed approximation (Park and Garwood, 2009; Pipe, 1995) that for a given isochromat located at position x within the slice (−Δx/2 ≤ x ≤ Δx/2) the magnetization is flipped instantaneously when Δω(tx) = γxG(tx). Under this assumption, it has been shown (Valette et al., 2010) that during adiabatic frequency-swept pulses the only contribution to diffusion weighting comes from the phase scrambling due to gradient, while the scrambling effect due to the quadratic phase induced by the frequency modulation of the B1 field cancels out. The 180° flip of magnetization will change the sign of the gradient moment at the instant time tx
′ k t ;x ¼
′
Zt γ
″ ″ G t dt ;
′
0 ≤t ≤t x
0
′ > Zt Ztx > > > ″ ″ ″ ″ > > dt γ G t −2 γ G t dt ; > :
0
ð12Þ ′
t x bt ≤T p
0
Zt x
2
B1 ðt Þ=Gðt Þdt ¼ Qx
ð13Þ
0
with B1(t) and G(t) as defined by Eq. (2). While solving Eq. (11) by integration of B1(t) and G(t) modulations given in Eq. (2) leads to a very complicated analytical solution, it can be shown that the inversion time for each isochromat can be simply calculated from: tx ¼
Tp 2x Gðt Þ 1þ ; Δx Gmax 2
Δx Δx ; : x∈ − 2 2
ð14Þ
This can be verified also numerically by simulations of the slice profile during the pulse as shown in Fig. 5A. Plots of the inversion times and gradient moment are shown in Figs. 5B and G, respectively. From Eqs. (12)–(14) it can be seen that the gradient moment and bvalue become dependent on the isochromat position. Considering a pair of two GOIA-W(16,4) pulses as the diffusion weighting element the spatial dependent b-value can be calculated: Z
2T p
bðxÞ ¼
2
k ðt; xÞ dt: 0
1 6 1 ln6 D 4ΔxS0
#
Δx=2 Z
ð16Þ
S0 ðxÞexpð−bðxÞ DðxÞÞ dx −Δx=2
where D(x) is the diffusion coefficient, which most generally may vary Δx=2
across slice, and the mean coefficient D ¼ 1=Δx
∫ DðxÞdx, and S0 is
defined similarly like in the case of relaxation. In the case there is small change in b-value b(x), diffusion coefficient D(x) and initial signal S0(x) across the slice Eq. (16) can be linearized 1 Δx
Δx=2 Z
bðxÞdx
ð15Þ
ð17Þ
−Δx=2
Assuming that rotating frame relaxation and diffusion weighting are acting independently their combined effect on the signal decay during the train of GOIA-W(16,4) pulses: h i Sðt m Þ ¼ Sð0Þ exp − R2ρ t m þ b þ br D t m =2T p ;
ð18Þ
t m ¼ 4m T p ; m ¼ 0; 5 where tm is the T2ρ preparation time (multiple of four GOIA-W(16,4) pulses) and br is the b-value associated with the ramping of the gradient at the beginning and the end of the preparation time. The calculated bvalues are 0.294, 0.589, 0.872, 1.178 and 1.473 s⋅mm−2 for preparation times of 16, 32, 48, 64 and 80 ms, respectively (GOIA-W(16,4) of 4 ms duration and 5 kHz bandwidth). We found experimentally (Fig. 6) that an effect of diffusion weighting can be observed on the signal decay in the case of T2ρ preparation for 2D imaging sequences. However, the experimental effect is much higher than predicted by the above theory: h i 3 Sðt m Þ ¼ Sð0Þ exp − R2ρ t m þ β D t m
where the flipping time tx ∈ [0,Tp] is given by the solution of
γ
b¼−
b¼
0
8 > > > > > > >
Contents lists available at ScienceDirect
NeuroImage journal homepage: www.elsevier.com/locate/ynimg
Whole brain mapping of water pools and molecular dynamics with rotating frame MR relaxation using gradient modulated low-power adiabatic pulses Ovidiu C. Andronesi a,⁎,1, Himanshu Bhat b, Martin Reuter c, Shreya Mukherjee a, Peter Caravan a, Bruce R. Rosen a a b c
Athinoula A. Martinos Center for Biomedical Imaging, Department of Radiology, Massachusetts General Hospital, Harvard Medical School, Boston, MA 02114, USA Siemens Medical Solutions USA Inc, Charlestown, MA 02129, USA Athinoula A. Martinos Center for Biomedical Imaging, Department of Neurology, Massachusetts General Hospital, Harvard Medical School, Boston, MA 02114, USA
a r t i c l e
i n f o
Article history: Accepted 7 December 2013 Available online 15 December 2013 Keywords: Rotating frame relaxation (T1ρ, T2ρ) Macromolecular content Correlation time Exchange time Bound water fraction Neurodegeneration
a b s t r a c t Nuclear magnetic resonance (NMR) relaxation in the rotating frame is sensitive to molecular dynamics on the time scale of water molecules interacting with macromolecules or supramolecular complexes, such as proteins, myelin and cell membranes. Hence, longitudinal (T1ρ) and transverse (T2ρ) relaxation in the rotating frame may have a great potential to probe the macromolecular fraction of tissues. This stimulated a large interest in using this MR contrast to image brain under healthy and disease conditions. However, experimental challenges related to the use of intense radiofrequency irradiation have limited the widespread use of T1ρ and T2ρ imaging. Here, we present methodological development to acquire 3D high-resolution or 2D (multi-)slice selective T1ρ and T2ρ maps of the entire human brain within short acquisition times. These improvements are based on a class of gradient modulated adiabatic pulses that reduce the power deposition, provide slice selection, and mitigate artifacts resulting from inhomogeneities of B1 and B0 magnetic fields. Based on an analytical model of the T1ρ and T2ρ relaxation we compute the maps of macromolecular bound water fraction, correlation and exchange time constants as quantitative biomarkers informative of tissue macromolecular content. Results obtained from simulations, phantoms and five healthy subjects are included. © 2013 Elsevier Inc. All rights reserved.
Introduction Longitudinal (T1ρ) and transverse (T2ρ) relaxation in the rotating frame of nuclear magnetic resonance can probe water molecules loosely bound (hydration layer) to macromolecules. The macromolecular fraction is tissue specific and changes with disease, hence T1ρ and T2ρ have been proposed (Sepponen et al., 1985) as a candidate for noninvasive imaging solid tissue matrix, albeit indirectly through the relaxation of water interacting with macromolecules. Several studies have investigated the value of T1ρ and T2ρ imaging in the context of neurological diseases, including Alzheimer's Disease (Borthakur et al., 2008; Haris et al., 2011), Parkinson Disease (Haris et al., 2011; Michaeli et al., 2007), stroke (Jokivarsi et al., 2009, 2010a), and brain cancer (Aronen et al., 1999; Hakumaki et al., 1999). These studies revealed that T1ρ and T2ρ relaxation times have a broader range of values compared to
⁎ Corresponding author at: Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, 149 Thirteenth Street, Suite 2301, Charlestown, MA 02129, USA. Fax: +1 617 726 7422. E-mail address: [email protected] (O.C. Andronesi). 1 Presented in part at the 21st Annual Meeting of ISMRM, Salt Lake City, UT, 20–26 Apr, 2013. 1053-8119/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.neuroimage.2013.12.007
T1 and T2 across disease spectrum, and better separate patients from controls. Also, T1ρ and T2ρ relaxation times seem to correlate better and earlier with therapy response or disease progression when compared to other MRI markers such as laboratory frame T1 and T2 relaxation, or ADC values (Duvvuri et al., 2001; Hakumaki et al., 2002; Jokivarsi et al., 2010a). Experimental challenges related to high specific absorption rate (SAR), non-selective spatial excitation, long acquisition times, and artifacts from B0 and B1 inhomogeneity currently prevent a wide spread adoption and usage of T1ρ and T2ρ in the neuroimaging community. Here, we present methodological development for clinically feasible and robust T1ρ and T2ρ mapping of the whole human brain based on a recent class of gradient modulated adiabatic pulses (Andronesi et al., 2010). Further, using an analytical model and fitting of relaxation curves we extract tissue specific maps for parameters of molecular dynamics such as correlation times, exchange times, and water pools. Nuclear spin relaxation in the rotating frame is obtained when a radiofrequency field (B1) is applied either along or perpendicular to the nuclear magnetization for longitudinal T1ρ relaxation or transverse T2ρ relaxation, respectively. Typically, T1ρ and T2ρ imaging of the human brain is done with B1 amplitudes in the range of 0.5–1 kHz and durations between 50 and 100 ms. The combination of high B1
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amplitude and long duration leads to increased SAR, which is often circumvented by increasing repetition times (TR) leading to long acquisition times (TA). Moreover, in all imaging studies so far T1ρ and T2ρ relaxation have been achieved via spatially non-selective radiofrequency (RF) irradiation, which further imposes limits on the TR in multislice experiments due to relaxation and saturation of magnetization in neighboring slices. As a result, rotating frame relaxation imaging is usually performed as a single slice or very few (1–4) slices at a time. Three main approaches are currently used to achieve T1ρ relaxation: i) the on-resonance continuous wave (CW) method where the magnetization is first flipped in the transverse plan and a B1 field with a constant amplitude is applied along magnetization for spin-lock (Aronen et al., 1999), ii) the off-resonance continuous wave (CW) method where an offset is used to create an effective field and magnetization is flipped along the direction of the effective field (Ramadan et al., 1998), and iii) a train of adiabatic inversion pulses producing repeated passages of the longitudinal magnetization between the + Z and − Z orientations via amplitude and frequency modulation of B1 field (Michaeli et al., 2008). In the case of T2ρ relaxation the initial magnetization is first flipped perpendicular to the direction of the spin-lock or the effective field for precession around it. The spin-lock method is easier to implement and model analytically, but it is more sensitive to artifacts induced by B0 and B1 inhomegeneities. These artifacts can be compensated over a reduced B0 and B1 range by composite pulses and phase alternation of the CW spin-lock (Witschey et al., 2007). On the other hand, due to its simultaneous frequency and amplitude sweep adiabatic pulses can cover a much larger bandwidth of B0 offsets and can tolerate several-folds of B1 inhomogeneity above the adiabatic threshold (Garwood and DelaBarre, 2001). In addition, adiabatic pulses have lower SAR compared to a CW spin-lock of the same amplitude. Our focus in this work was to address the major limitations of current T1ρ and T2ρ imaging techniques, namely high SAR and long acquisition times, while at the same time compensating for B0 and B1 inhomogeneities. Gradient modulated adiabatic pulses such as GOIAW(16,4) (Andronesi et al., 2010) simultaneously meet all these requirements. The Gradient Offset Independent Adiabaticity (GOIA) design (Tannus and Garwood, 1997) lowers the maximum B1 amplitude by 55%, resulting in 80% less SAR, compared to conventional adiabatic pulses of the same bandwidth and duration. Alternatively, GOIAW(16,4) can be made shorter and still have lower B1 amplitude than conventional adiabatic pulses, allowing more flexibility in choosing the duration of T1ρ and T2ρ preparation. Compared to a CW spin-lock of the same amplitude GOIA-W(16,4) pulses would have 20% less SAR. The use of gradient modulation in GOIA-W(16,4) pulses naturally provides slice-selectivity for T1ρ and T2ρ. In the case of 2D imaging, slice selectivity allows the use of a short TR to acquire a stack of slices without saturating magnetization of neighboring slices. Although strategies have been proposed for the CW spin-lock method to reduce SAR (Wheaton et al., 2004a) or acquire multiple slices with less magnetization saturation (Wheaton et al., 2004b), these have limitations in fully achieving those goals. In terms of the SAR deposition, GOIA pulses compare favorably also to relaxation along the fictitious field (RAFF) method (Liimatainen et al., 2010) which uses sub-adiabatic RF frequency and amplitude modulations to lower B1 requirements, at the cost of more sensitivity to B1 inhomogeneity and without slice selectivity. Here we demonstrate the use of GOIA-W(16,4) pulse trains in combination with 2D spin-echo EPI (EPI-SE) and 3D turbo FLASH (TFL) readouts as a new method for robust and feasible T1ρ and T2ρ imaging of the human brain. Theory Relaxation in the rotating frame of GOIA pulses Relaxation in the rotating frame has been an important research topic reach in applications since the beginning of nuclear magnetic resonance
93
(Redfield, 1957). The semi-classical theory of Bloch–Wangsness– Redfield (BWR), often used for practical reasons, treats the spins quantum mechanically and the lattice as random perturbation described by classical statistics (Abragam, 1961; Redfield, 1957; Wangsness and Bloch, 1953). Significant progress has been made over the last decade in laying out the analytical framework for relaxation in the rotating frame under adiabatic pulses (Mangia et al., 2009; Michaeli et al., 2008; Sorce et al., 2007). In this work we extend the theory of rotating frame relaxation for the case of gradient modulated adiabatic pulses. GOIA pulses (Tannus and Garwood, 1997) are obtained by simultaneously modulating the amplitude B1(t) and frequency Δω(t) of RF field, together with the slice selection gradient G(t) according to: Zt γ2 B2 t ′ 1 1 ′ dt Δωðt Þ ¼ Gðt Þ Q G t′
ð1Þ
0
where Q is the adiabatic factor, and γ is the nuclear gyromagnetic ratio. In particular, for GOIA-W(16,4) pulses (Andronesi et al., 2010) the modulation functions B1(t) and G(t) are based on the wideband uniform rate and smooth truncation WURST (Kupce and Freeman, 1995) function: h i16 B1 ðt Þ ¼ B1;max 1− sin π=2 2t=T p −1 h i4 Gðt Þ ¼ Gmax ð1−g m Þ þ gm sin π=2 2t=T p −1
ð2Þ
where Tp is the pulse duration, B1,max is the maximum RF amplitude, Gmax is the maximum gradient, and gm ∈ [0,1] is the gradient modulation factor. The larger the gm factor the more the gradient drops in the middle of the pulse and correspondingly lesser RF amplitude is required, an effect also known as the variable rate selective excitation VERSE (Conolly et al., 1988). Alternatively, GOIA-HS(8,4) (Andronesi et al., 2010) pulses are obtained if stretched hyperbolic-secant functions are used instead of WURST. Besides pulse duration and gradient modulation factor, the exponents of trigonometric functions can be varied too, providing great flexibility in obtaining new pulses. Modulations of GOIA-W(16,4) pulse of 4 ms duration and 5 kHz bandwidth are shown in Figs. 1A–C. In order to calculate the rates of rotating frame relaxation we need the effective field (ωeff) and the magnetization nutation angle (α): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωeff ðt; xÞ ¼ ðγB1 ðt ÞÞ2 þ ðΔωðt Þ þ γxGðt ÞÞ2 α ðt; xÞ ¼ atanðγB1 ðt Þ=ðΔωðt Þ þ γxGðt ÞÞÞ:
ð3Þ
The adiabatic condition states that the effective field should be much greater than the rate of the nutation angle ωeff ðt; xÞ≫α˙ ðt; xÞ at all times ˙ and positions. In practice an adiabatic factor Q ¼ ωeff =α≥5 and a timebandwidth product R ≥ 20 are needed to have an adiabatic pulse. Plots of the effective field modulation, nutation angle and inverse of adiabatic factor during the GOIA-W(16,4) pulse are shown in Figs. 1D–F. The adiabatic factor is greater than 5 for the most part of the pulse with the exception of short periods in the vicinity of 1.2 ms and 2.8 ms where it decreases to 3. By increasing the R product the pulse becomes more adiabatic. An important observation is necessary at this point. Due to gradient modulation the effective field and nutation angle in Eq. (3) become dependent on the spatial coordinate x in the slice direction. However, there is dependency only of the intra-slice isochromats, and not of the slice position relative to the isocenter. The effect of slice position is canceled by the additional frequency modulation that is used to excite slices off-isocenter. For simplicity this term was not included in Eq. (1), but more details are given in Andronesi et al. (2010). Thus, the slices at isocenter and off-isocenter positions have the same effective field and nutation angle, and consequently the relaxation rate does
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Fig. 1. Plots of GOIA-W(16,4) pulse: A) B1 amplitude modulation function; B) frequency sweep; C) gradient modulation; D) effective field; E) nutation angle; F) inverse of the adiabatic factor; G) slice profile at the isocenter; H) slice profile off-isocenter. A pulse duration of 4 ms and bandwidth of 5 kHz were assumed. In Fig. 1D the rate of the nutation angle is also shown with the dashed line.
not depend on the slice position. As we will show next, what is relevant for the measurement is the average relaxation rate over the pulse duration and slice thickness, which can be exactly calculated. A physical model that satisfactorily approximates the rotating frame relaxation of water in-vivo includes two main relaxation mechanisms: dipolar coupling and exchange interactions (Michaeli et al., 2004; Michaeli et al., 2005; Michaeli et al., 2008; Sorce et al., 2007). Random fluctuations due to reorientation of the direct proton–proton dipolar coupling and exchange of protons between sites with different chemical shifts or diffusion through microscopic field inhomogeneities lead to decoherence of the spin system. The stochastic nature of random perturbations is described by the spectral density functions of the type (n = 0,1,2): 2 J n ðτÞ τ= 1 þ ðnωτÞ
ð4Þ
where τ may represent the correlation time τc or exchange time τex, and ω may represent the Larmor frequency ω0 = γβ0 or the effective field ωeff from Eq. (3). The most commonly employed model (Jokivarsi et al., 2009; Mangia et al., 2009; Michaeli et al., 2008; Sierra et al., 2008) for deriving rotating frame relaxation in tissues assumes a two sites exchange model (2SX) at equilibrium between the free water
(slow relaxing component) and the water bound to macromolecules (fast relaxing component), as shown schematically in Fig. 2. Relaxation rates in the rotating frame (Blicharski, 1972) can be cast in a time dependent form for adiabatic pulses as instantaneous relaxation rates (Mangia et al., 2009; Michaeli et al., 2008). In the case of GOIA-W(16,4) pulses the instantaneous relaxation rate for dipolar coupling interaction becomes dependent on the intra-slice isochromat at position x 2
3 2 2 4 3 sin α ðt; xÞ cos α ðt; xÞ 3 sin α ðt; xÞ þ þ 2 2 7 7 1 þ ωeff ðt; xÞτc 1 þ 2ωeff ðt; xÞτc 7 7 7 2 2 2 þ 3 sin α ðt; xÞ 8−6 sin α ðt; xÞ 5 þ 2 2 1 þ ðω0 τc Þ 1 þ ð2ω0 τc Þ 2 3 2 2 3 3 cos α ðt; xÞ−1 þ 7 6 7 6 4 6 30 sin2 α ðt; xÞ cos2 α ðt; xÞ 7 3 sin α ðt; xÞ 6 þ Dc 2 2 þ 7 dd 6 7 τc 6 1 þ ω ðt; xÞτ R1ρ ðt; xÞ ¼ 7 1 þ 2ωeff ðt; xÞτ c 40 eff c 6 7 7 6 2 2 4 5 20−6 sin α ðt; xÞ 8 þ 12 sin α ðt; xÞ þ 2 2 1 þ ðω0 τc Þ 1 þ ð2ω0 τ c Þ ð5Þ 6 6 Dc dd 6 R1ρ ðt; xÞ ¼ τ 6 10 c 6 4
where the dipolar coupling constant Dc = 2I(I + 1)[μ0ℏγ2/(4πr3)]2, I = 1/2 is the spin number, and the physical constants are μ 0 = 4π 10− 7 H/m the vacuum magnetic permeability, ħ =1.055 · 10−34 Js the Planck's constant, γ = 2π 42.576·106 rad/(sT) the gyromagnetic ratio, and r = 1.58 Å is the proton–proton distance in water molecule. Simulations of dipolar relaxation rates for GOIA-W(16,4) pulses are shown in Figs. 3A,C. Similarly, the instantaneous relaxation rates for the anisochronous exchange interaction can be obtained in the form of
Fig. 2. Two site exchange (2SX) model depicting the pool of free water and pool of water bound to macromolecules. In each pool water molecules experience dipolar coupling relaxation governed by the correlation times τc of each pool. The two pools are considered in an equilibrium anisochronous exchange determined by the exchange time τex and the chemical shift difference δωex. The model is specified by four independent parameters: correlation time τc,b of bound water, τex, δωex and bound water fraction Pb (Pf = 1 − Pb and τc,f ≈ 5 ps). The fast exchange limit (FXL) is assumed in-vivo δωexτex ≪ 1.
ex
2
ex
2
R1ρ ðt; xÞ ¼ P b P f ðδωex Þ τex
sin2 α ðt; xÞ 2 1 þ ωeff ðt; xÞτex
ð6Þ
2
R2ρ ðt; xÞ ¼ P b P f ðδωex Þ τex cos α ðt; xÞ where Pb is the pool of bound water, Pf is the pool of free water (Pb + Pf = 1), and δωex is the chemical shift difference between the
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Fig. 3. Simulations of longitudinal and transverse rotating frame relaxation rates for dipolar coupling (Eq. (5)) and exchange (Eq. (6)) interactions during GOIA-W(16,4) pulses of 4 ms duration, 5 kHz bandwidth and 0.382 kHz peak amplitude. For exchange interaction (Figs. 3B,D) Pb was fixed at 0.15 and δωex was assumed 0.8 ppm (≈100 Hz at 2.89 T). The B0 field was considered 2.89 T.
two exchanging sites. Simulations for exchange relaxation rates during GOIA-W(16,4) pulses are presented in Figs. 3B,D. The intrinsic instantaneous relaxations rates Eqs. (5)–(6) are further used to compute the observed relaxation rates. First, the average dipolar and exchange relaxation rates over pulse duration need to be calculated for each slice isochromat:
Rdd;ex ðxÞ 1ρ;2ρ;b; f
1 ¼ Tp
ZT p
dd;ex R1ρ;2ρ;b; f dd;ex R1ρ;2ρ;b; f ðt; xÞ
dt:
ð7Þ
1 1 ¼ T p Δx
Δx=2 Z ZT p
dd;ex
R1ρ;2ρ;b; f ðt; xÞ dtdx −Δx=2 0
ð10Þ
dd dd ex Robs 1ρ;2ρ ¼ P b R1ρ;2ρ;b þ P f R1ρ;2ρ; f þ R1ρ;2ρ :
0
For each isochromat we can then calculate the observed (or apparent) relaxation rate averaged over the pulse duration. Under in-vivo conditions the fast exchange limit δωexτex ≪ 1 (FXL) is assumed (Jokivarsi et al., 2009; Sierra et al., 2008), and the observed relaxation rate is obtained as a linear combination between dipolar relaxation of free and bound water, and exchange relaxation: dd dd ex Robs 1ρ;2ρ ðxÞ ¼ P b ðxÞR1ρ;2ρ;b ðxÞ þ P f ðxÞR1ρ;2ρ; f ðxÞ þ R1ρ;2ρ ðxÞ:
ð8Þ
By integrating the exponential decays of all isochromats over the slice thickness we can obtain the time-spatial average of the observed relaxation rate: Robs 1ρ;2ρ ¼ −
Simulations for the average observed relaxation rates during GOIAW(16,4) pulses are presented in Fig. 4 as a function of model parameters. In the case when the change in the relaxation rates and initial signal is small among isochromats Eqs. (7)–(9) may be linearized and the average relaxations can be very well approximated by:
# " Δx=2 Z 1 1 dx ln S0 ðxÞexp −Robs ð x Þ T p 1ρ;2ρ Tp ΔxS0
ð9Þ
−Δx=2
where the mean initial magnetization (signal) over the slice is S0 ¼
Simulations performed in Figs. 5C–F show the results for the average observed relaxation rates obtained from Eqs. (7) to (9) or (10). The 2SX/FXL model depicted in Fig. 2 is described by a maximum number of seven independent parameters: two correlation times (for bound and free water), two proton–proton distances (for bound and free water), population of bound site (Pf = 1 − Pb), exchange time, and chemical shift difference. However, three parameters can be readily eliminated considering the same proton–proton distance (1.58 Å) for both bound and free water, and a standard value for the correlation time of free water (5 ps). In this case the observed relaxation rate is determined by the four remaining parameters: the correlation time of bound water, exchange time, chemical shift difference, and the population of bound water. In the ideal situation of no measurement noise a minimum number of four experimental time points are necessary to obtain the molecular parameters of interest, however experimentally extra time points are beneficial. Fig. 4 shows simulations of the observed rotating frame relaxation rate under GOIA-W(16,4) pulses for relevant in-vivo model parameters.
Δx=2
1=Δx ∫ S0 ðxÞ dx. Note that under the most general conditions the cor-
Diffusion weighting during rotating frame relaxation under GOIA pulses
−Δx=2
relation times, exchange times, exchange rate, water pools and initial signal can vary spatially across isochromats. In certain conditions, such as activation, the model parameters could vary also on the time scale of the pulse. This variation is captured implicitly in Eqs. (6)–(9).
The use of the gradient modulation during relaxation in the rotating frame of GOIA pulses raises the question of what is the effect of diffusion on the signal decay. The effect of diffusion may be expected particularly for T2ρ where magnetization processes around the
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effective field and diffusion may produce signal decay similarly like in spin echo experiments. In the case of T1ρ magnetization spends most of the time longitudinally and there is no evolution of the spin echo type. A few papers have investigated the effects of diffusion during adiabatic pulses in spin echo experiments (Sun and Bartha, 2007; Valette et al., 2010). In deriving an expression for the b-value produced by the T2ρ preparation with GOIA-W(16,4) pulses we will follow the approach proposed in Valette et al. (2010). Most generally, the b-value associated with a gradient G(t) is obtained as the integral of the square of the gradient moment k(t) Zt bðt Þ ¼
k
2
′ ′ t dt
0
ð11Þ
′ k t ¼γ
′
″ ″ G t dt :
To calculate the gradient moment for the GOIA-W(16,4) pulse we can use the commonly assumed approximation (Park and Garwood, 2009; Pipe, 1995) that for a given isochromat located at position x within the slice (−Δx/2 ≤ x ≤ Δx/2) the magnetization is flipped instantaneously when Δω(tx) = γxG(tx). Under this assumption, it has been shown (Valette et al., 2010) that during adiabatic frequency-swept pulses the only contribution to diffusion weighting comes from the phase scrambling due to gradient, while the scrambling effect due to the quadratic phase induced by the frequency modulation of the B1 field cancels out. The 180° flip of magnetization will change the sign of the gradient moment at the instant time tx
′ k t ;x ¼
′
Zt γ
″ ″ G t dt ;
′
0 ≤t ≤t x
0
′ > Zt Ztx > > > ″ ″ ″ ″ > > dt γ G t −2 γ G t dt ; > :
0
ð12Þ ′
t x bt ≤T p
0
Zt x
2
B1 ðt Þ=Gðt Þdt ¼ Qx
ð13Þ
0
with B1(t) and G(t) as defined by Eq. (2). While solving Eq. (11) by integration of B1(t) and G(t) modulations given in Eq. (2) leads to a very complicated analytical solution, it can be shown that the inversion time for each isochromat can be simply calculated from: tx ¼
Tp 2x Gðt Þ 1þ ; Δx Gmax 2
Δx Δx ; : x∈ − 2 2
ð14Þ
This can be verified also numerically by simulations of the slice profile during the pulse as shown in Fig. 5A. Plots of the inversion times and gradient moment are shown in Figs. 5B and G, respectively. From Eqs. (12)–(14) it can be seen that the gradient moment and bvalue become dependent on the isochromat position. Considering a pair of two GOIA-W(16,4) pulses as the diffusion weighting element the spatial dependent b-value can be calculated: Z
2T p
bðxÞ ¼
2
k ðt; xÞ dt: 0
1 6 1 ln6 D 4ΔxS0
#
Δx=2 Z
ð16Þ
S0 ðxÞexpð−bðxÞ DðxÞÞ dx −Δx=2
where D(x) is the diffusion coefficient, which most generally may vary Δx=2
across slice, and the mean coefficient D ¼ 1=Δx
∫ DðxÞdx, and S0 is
defined similarly like in the case of relaxation. In the case there is small change in b-value b(x), diffusion coefficient D(x) and initial signal S0(x) across the slice Eq. (16) can be linearized 1 Δx
Δx=2 Z
bðxÞdx
ð15Þ
ð17Þ
−Δx=2
Assuming that rotating frame relaxation and diffusion weighting are acting independently their combined effect on the signal decay during the train of GOIA-W(16,4) pulses: h i Sðt m Þ ¼ Sð0Þ exp − R2ρ t m þ b þ br D t m =2T p ;
ð18Þ
t m ¼ 4m T p ; m ¼ 0; 5 where tm is the T2ρ preparation time (multiple of four GOIA-W(16,4) pulses) and br is the b-value associated with the ramping of the gradient at the beginning and the end of the preparation time. The calculated bvalues are 0.294, 0.589, 0.872, 1.178 and 1.473 s⋅mm−2 for preparation times of 16, 32, 48, 64 and 80 ms, respectively (GOIA-W(16,4) of 4 ms duration and 5 kHz bandwidth). We found experimentally (Fig. 6) that an effect of diffusion weighting can be observed on the signal decay in the case of T2ρ preparation for 2D imaging sequences. However, the experimental effect is much higher than predicted by the above theory: h i 3 Sðt m Þ ¼ Sð0Þ exp − R2ρ t m þ β D t m
where the flipping time tx ∈ [0,Tp] is given by the solution of
γ
b¼−
b¼
0
8 > > > > > > >
