Incorporating reversible and irreversible transverse relaxation effects into Steady State Free Precession (SSFP) signal intensity expressions for fMRI considerations

Magnetic Resonance Imaging xxx (2012) xxx–xxx

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Incorporating reversible and irreversible transverse relaxation effects into Steady State Free Precession (SSFP) signal intensity expressions for fMRI considerations Robert V. Mulkern a, Mukund Balasubramanian a,⁎, Darren B. Orbach a, Dimitrios Mitsouras b, Steven J. Haker b a b

Department of Radiology, Children's Hospital Boston, Boston, MA, USA Department of Radiology, Brigham and Women's Hospital, Boston, MA, USA

a r t i c l e

i n f o

Article history: Received 21 March 2012 Revised 10 October 2012 Accepted 14 October 2012 Available online xxxx Keywords: Transverse relaxation Irreversible Reversible Steady state free precession fMRI

a b s t r a c t Among the multiple sequences available for functional magnetic resonance imaging (fMRI), the Steady State Free Precession (SSFP) sequence offers the highest signal-to-noise ratio (SNR) per unit time as well as distortion free images not feasible with the more commonly employed single-shot echo planar imaging (EPI) approaches. Signal changes occurring with activation in SSFP sequences reflect underlying changes in both irreversible and reversible transverse relaxation processes. The latter are characterized by changes in the central frequencies and widths of the inherent frequency distribution present within a voxel. In this work, the well-known frequency response of the SSFP signal intensity is generalized to include the widths and central frequencies of some common frequency distributions on SSFP signal intensities. The approach, using a previously unnoted series expansion, allows for a separation of reversible from irreversible transverse relaxation effects on SSFP signal intensity changes. The formalism described here should prove useful for identifying and modeling mechanisms associated with SSFP signal changes accompanying neural activation. © 2012 Elsevier Inc. All rights reserved.

1. Introduction and background The most common MR imaging approach for studying neural activation is single-shot echo planar imaging (SS-EPI) with an echo ⁎ time on the order of the transverse relaxation time T ⁎ 2 = 1/R2 [1,2]. Here, R⁎ 2 is the transverse relaxation rate associated with combined effects of both the reversible relaxation rate R′2, and the irreversible relaxation rate R 2, where one often encounters the relation R⁎ 2= R2 + R′2. Despite its popularity, well-known drawbacks to SS-EPI include spatial distortions due to susceptibility, particularly near tissue-air interfaces in the pre-frontal lobes, low signal-to-noise ratio (SNR) and, perhaps, an oversensitivity to changes in and around larger vessels somewhat remote from activated parenchyma. For over a decade, steady state free precession (SSFP) sequences have been suggested as an intriguing alternative to SS-EPI [3–10] since they provide high spatial resolution, high SNR images with minimal spatial distortion and may be acquired quite rapidly, particularly with the use of short repetition time (TR) periods. The complicated frequency response of the magnitude SSFP signal, characterized by “transitions zones” and “passbands” separated by 1/(2TR), implies that the usual mechanisms affecting signal intensity changes will ⁎ Corresponding author. 300 Longwood Avenue, Boston, MA 02115, USA. Tel.: +1 857 218 4990; fax: +1 617 730 0550. E-mail address: [email protected] (M. Balasubramanian).

exhibit different sensitivities in these two regimes. Furthermore, tissue contrast in SSFP is a complicated function of pulse sequence parameters and tissue relaxation properties and it is not readily apparent from the standard SSFP signal intensity equations [11–17] how changes in the widths of intrinsic or applied frequency distributions will affect SSFP signal intensity. Here we derive analytic expressions for the SSFP signal which explicitly incorporate the longitudinal and irreversible transverse relaxation rates, R1 and R2 respectively, as well as the central frequency ω0, and width parameters R′2, etc., associated with frequency distributions responsible for reversible relaxation [18]. The expressions derived allow for separate evaluation of the effects of reversible versus irreversible transverse relaxation processes on the fMRI related signal changes observed in SSFP images. This may prove useful in optimizing sequence parameters and for modeling and/or interpreting the physical mechanisms associated with the SSFP signal intensity changes accompanying neuronal activation. 2. Experimental Methods Motor activation studies of a healthy male volunteer, aged 45 years, were performed using a 3 T scanner (Siemens Trio system, Erlangen, Germany) with a 32 channel head coil. Studies were performed according to the guidelines of the local institutional review board and written informed consent was obtained. A block design

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Please cite this article as: Mulkern RV, et al, Incorporating reversible and irreversible transverse relaxation effects into Steady State Free Precession (SSFP) signal intensity expressions for fMRI considerations, Magn Reson Imaging (2012), http://dx.doi.org/10.1016/j.mri.2012.10.002

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paradigm was used, with each trial consisting of four repetitions of 30s of rest (OFF) followed by 30s of bilateral finger tapping (ON), resulting in a trial time of 4 minutes. During each trial, either a BOLD EPI dataset or a balanced SSFP (TrueFISP) dataset was acquired. The following imaging parameters were used for the BOLD EPI acquisition: flip angle = 90°, pixel bandwidth = 2298 Hz/pixel, TE = 40 ms and TR = 1 s, with 240 timepoints acquired for each trial. The (in-plane) field-of-view was 256 mm with a 64 × 64 matrix, and 17 slices with a thickness of 5 mm each were positioned axially, resulting in a voxel size of 4 × 4 × 5mm. For the TrueFISP acquisition, the corresponding imaging parameters were flip angle = 70°, pixel bandwidth = 355 Hz/pixel, TE = 2.5 ms and TR = 5 ms, with 87 timepoints acquired for each trial. The (in-plane) field-ofview was 256 mm with a 128 × 128 matrix, and 4 slices with a thickness of 5 mm each were positioned axially, resulting in a voxel size of 2 × 2 × 5mm. Thus the TrueFISP acquisition was performed with higher in-plane resolution than the BOLD EPI acquisition, albeit at a cost to the temporal sampling rate and the spatial coverage in the slice direction. Data processing for the BOLD EPI and TrueFISP datasets was carried out using FEAT (fMRI Expert Analysis Tool) Version 5.98, part of FSL (FMRIB's Software Library, www.fmrib.ox.ac.uk/fsl). The following pre-statistics processing was applied: spatial smoothing using a Gaussian kernel of FWHM 8 mm; grand-mean intensity normalization of the entire 4-dimensional dataset by a single multiplicative factor; highpass temporal filtering (Gaussian-weighted least-squares straight line fitting, with σ = 60.0 s). Time-series statistical analysis was carried out using FILM with local autocorrelation correction [19]. Z (Gaussianized T/F) statistic images were thresholded using clusters determined by ZN2.3 and a (corrected) cluster significance threshold of P = 0.05 [20]. 3. Theory Our starting point is the standard set of equations developed in the past [11–17] to model the steady state response of a spin system to a θx-τ-θx-τ… sequence, where θx is the flip angle and τ is the repetition time (TR). It is recognized that modern, fully balanced, SSFP sequences [21] with vendor defined acronyms like FIESTA (General Electric Medical Systems), TrueFISP (Siemens Medical Systems), or BFFE (Philips Medical Systems) utilize a sequence of alternating phase θx-τ-θ-x-τ… RF pulses. As Hinshaw [13] noted, however, the alternating pulse version used in practice may be considered equivalent to the non-alternating pulse version when the primary resonance offset ω0 is set such that ω0τ = π (or any odd multiple of π), a step we shall take as warranted below. The steady state transverse magnetization from a string of θx pulses, received as the complex SSFP signal S as a function of time t, from t = 0, taken as immediately after a pulse, through to t = τ, immediately before the next pulse, is extracted from the literature [17] as follows:

S ðt Þ ¼

n o
c  e−iωt e−R2 t −eiωðτ−t Þ e−R2 ðτþt Þ 1−ðb=a Þcosωτ

a

;

ð1Þ

with a ≡ 1−e

−ðR1 þ2R2 Þτ

b ≡ ð1 þ cosθÞe

h i −R τ −2R2 τ −cosθ e 1 −e ;

−R2 τ

h

1−e

−R1 τ

h i −R τ c ≡ i m00 sinθ 1−e 1 ;

i

; and

ð2aÞ ð2bÞ ð2cÞ

where ω is the angular frequency of a single isochromat in the rotating frame, m00 the equilibrium magnetization (taken as unity in all

further calculations), and R1 and R2 are the longitudinal and irreversible transverse relaxation rates empirically inserted into the Bloch equations. The two terms in the numerator (curly brackets) of Eq. (1) may be loosely identified as the FID and Echo components respectively whose difference, multiplied by the other factors, yields the signal. Following the essence of Ma and Wehrli's treatment of the reversible and irreversible transverse relaxation process within a spin echo sequence [18], “reversible” transverse relaxation mechanisms are now incorporated into this formalism by considering a distribution of frequencies ρ(ω), multiplying Eq. (1) by ρ(ω) and then integrating the resulting expression over all real ω. The three specific distributions we consider are Lorentzian, Gaussian and uniform (i.e., rectangular), each centered at ω0 and normalized such that integration over all real ω leads to unity. They are defined through: Lorentzian : ρðωÞ ¼

′

R2 π

!

1      ; ω− ω0 þ iR′2 ω− ω0 −iR′2



2 2 1 −ðω−ω0 Þ =ð2σ Þ ; and Gaussian : ρðωÞ pffiffiffiffiffiffi e σ 2π  Uniform : ρðωÞ ¼

1=ð2ΔωÞ for ω0 −Δω < ω < ω0 þ Δω 0 otherwise:

ð3Þ

ð4Þ

ð5Þ

The parameters characterizing the widths of these three probability distributions are R′2, σ and Δω, respectively, with R′2 related to the full width at half maximum (FWHM) via 2R′2, σ related to the width of the Gaussian via FWHM ≈2.35σ and Δω reflecting the width of the uniform distribution with FWHM being simply 2Δω. These frequency distributions are meant to represent the inherent, or background frequency distribution present within any imaged voxel and are separate from frequency distributions imposed during the application of imaging gradients, a condition justified by the use of balanced gradients in SSFP. Other distributions are of course possible but these three are physically reasonable distributions commonly encountered in the field. In addition, they share the convenient property of allowing for analytic integration where required, as seen below. The purely mathematical step of integrating the product of Eq. (1) with any of the three distributions leads to a time domain signal which includes all the effects of both reversible and irreversible relaxation processes, resonance offsets, and T1 relaxation. The width parameters R′2, σ and Δω characterize the reversible relaxation processes for the Lorentzian, Gaussian and uniform distributions, respectively. In performing the final mathematical step for each of the three distributions, we take advantage of a series expansion hitherto unused to our knowledge in previous literature on this topic, resulting in a convenient closed form solution for the Lorentzian distribution and single sum series solutions for the Gaussian and uniform distributions which converge more rapidly than the double sum series solution suggested by Gyngell [17]. Specifically, defining h = b/a and recognizing h as a real number between 0 and 1 for pulse sequence parameters and tissue relaxation times encountered in practice, the following expansion may be incorporated into Eq. (1): ∞ X 1 jn j inωτ ðð1−gÞ=h Þ e ; ð6Þ ¼ ð1=gÞ 1−h cos ωτ n¼−∞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi where g≡ 1−h 2 . This identity is proven in Appendix A. Using Eq. (6), we find that the integrations required for the Lorentzian distribution are most conveniently carried out using contour integration, evaluating the residue from the pole in either the upper or lower half of the complex plane as appropriate [22].

Please cite this article as: Mulkern RV, et al, Incorporating reversible and irreversible transverse relaxation effects into Steady State Free Precession (SSFP) signal intensity expressions for fMRI considerations, Magn Reson Imaging (2012), http://dx.doi.org/10.1016/j.mri.2012.10.002

R.V. Mulkern et al. / Magnetic Resonance Imaging xxx (2012) xxx–xxx

Requisite integrals involving the Gaussian distribution require using the relation [23] ∫e

2 2

−p x qx

dx ¼


 pffiffiffi  q2 =ð4p2 Þ 2 π=p e for R p > 0:

ð7Þ

For the uniform distribution, the integrals required are trivially performed leading to sinc type functions (vide infra). As Ganter [24] previously found, based on entirely different considerations, a closed form solution is obtained for the Lorentzian distribution time domain signal SLor(t): S Lor ðt Þ ¼ FIDLor ðt Þ−ECHOLor ðt Þ; where

ð8aÞ

( )

′ c h −iω t − R þR t e 0 e ð 2 2Þ −ðiω0 þR′2 Þτ ag h− ( ð1−g Þe )

c −iω0 t −ðR2 −R′2 Þt 1−g ; and þ e e ′ ag he−ðiω0 −R2 Þτ −1 þ g

FIDLor ðt Þ ¼



ECHOLor ðt Þ ¼



ð8bÞ

(

R2 −R′2

ð8cÞ Time domain signals for the Gaussian distribution SGauss(t) and the uniform distribution SUniform(t) are also obtained as the difference between their respective FID and Echo components, given by the following relations: Gaussian distribution: S Gauss ðt Þ ¼ FIDGauss ðt Þ−ECHOGauss ðt Þ; where

ð9aÞ



FIDGauss ðt Þ ¼

c −R2 t −ðt 2 σ 2 Þ=2 −iω0 t e e e ag

∞ X c −R2 t 1−g n −ððt−nτÞ2 σ 2 Þ=2 −iω0 ðt−nτÞ þ e e e ag h n¼1



∞ c −R2 t X 1−g n −ððtþnτÞ2 σ 2 Þ=2 −iω0 ðtþnτÞ þ e e ; and e ag h n¼1 ð9bÞ



ECHOGauss ðt Þ ¼

c −R2 ðτþt Þ −ððτ−t Þ2 σ 2 Þ=2 iω0 ðτ−t Þ e e e ag



∞ c −R2 ðτþt Þ X 1−g n −ðððnþ1Þτ−t Þ2 σ 2 Þ=2 þ e e ag h n¼1

∞ c −R2 ðτþt Þ X 1−g n iω ððnþ1Þτ−t Þ ×e 0 þ e ag h n¼1 2

2

−ðððn−1ÞτþtÞ σ Þ=2 −iω0 ððn−1Þτþt Þ

:

ð9cÞ ð9cÞ

S Uniform ðt Þ ¼ FIDUniform ðt Þ−ECHOUniform ðt Þ; where

ð10aÞ

×e

e



FIDUniform ðt Þ ¼

c −R t −iω t sin ðΔωt Þ e 2e 0 ag ðΔωt Þ

∞ c −R2 t X 1−g n iω0 ðnτ−t Þ sin ðΔωðnτ−t ÞÞ e þ e ag h ðΔωðnτ−t ÞÞ n¼1



∞ n c −R2 t X 1−g −iω0 ðnτþt Þ sin ðΔωðnτþt ÞÞ e e ; and þ ag h ðΔωðnτ þ t ÞÞ n¼1 ð10bÞ



ECHOUniform ðt Þ ¼

c −R2 ðτþt Þ iω0 ðτ−t Þ sin ðΔωðτ−t ÞÞ e e ag ðΔωðτ−t ÞÞ

∞ c −R2 ðτþt Þ X 1−g n iω0 ððnþ1Þτ−t Þ e e þ ag h n¼1

∞ sin ðΔωððn þ 1Þτ−t ÞÞ c −R2 ðτþt Þ X 1−g n  þ e ðΔωððn þ 1Þτ−t ÞÞ ag h n¼1 e

−iω0 ððn−1Þτþt Þ

Uniform distribution:

sin ðΔωððn−1Þτ þ t ÞÞ : ðΔωððn−1Þτ þ t ÞÞ ð10cÞ

)

c iω0 ðτ−t Þ −ð h Þτ e−ð Þt e e ðiω0 −R′2 Þτ ag h− ð 1−g Þe ( )

′ ′ c iω0 ðτ−t Þ −ðR2 −R2 Þτ −ðR2 þR2 Þt 1−g þ : e e e ′ ag heðiω0 þR2 Þτ −1þg R2 þR′2

3

Although we focus here on the magnitude signals commonly employed in fMRI studies, note that the equations derived above for the SSFP signal of each distribution are complex in form, and therefore magnitude, phase, real and imaginary components can all be extracted and analyzed through this formalism. 4. Results and discussion SSFP fMRI studies have been carried out with both short and long repetition time periods as well as with small and large flip angles. We first deploy the expressions derived above to simulate brain parenchymal signal in the long τ (TR) regime at high and low flip angles. Fig. 1 presents, for all three distributions, the frequency response, magnitude signal versus ω0/(2π), for a high flip angle of 70° (top) and for a low flip angle of 10° (bottom) for a τ value of 0.05 s and sampled at mid-repetition time t = τ/2. The simulations were performed with generic values for brain parenchyma relaxation rates R1 = 1 s −1, R2 = 10s −1 and reversible relaxation rates (distribution widths) of 5 s −1. Under these conditions, the frequency responses of the Gaussian and uniform distributions are practically identical while the maximum of the Lorentzian distribution is smaller than the other two in the passband, a difference which is larger at small versus high flip angles. Clearly, however, the differences between the distributions are relatively minor. Despite the fact that the Lorentzian distribution has no well-defined variance, we do not agree that it leads to “unphysical results”, as previously stated by Ganter [24]. All three frequency responses shift by 1/(2τ) in transitioning from high to low flip angles, as is also the case for single isochromat frequency responses generated with Eq. (1) under these conditions. Time domain magnitude signal simulations throughout the long repetition time interval of 0.05 s under passband conditions (ω0τ = (2n + 1)π for the high flip angle of 70° and ω0τ = 2nπ for the low flip angle of 10°, where n is any integer) are provided in Fig. 2 for generic brain parenchymal relaxation parameters for all three distributions. Gaussian and uniform distributions show similar decays of signal throughout the τ interval under both flip angle conditions while the Lorentzian distribution yields less signal overall throughout the interval and has a faster decay with t, particularly under low flip angle conditions. Again, however, the Lorentzian does not yield “unphysical results” [24], but rather, small (but real) quantitative differences with the other distributions that we attribute to contributions from the larger frequency content in the “wings” of the Lorentzian frequency distribution. In addition, the decays appearing in Fig. 2 appear very

Please cite this article as: Mulkern RV, et al, Incorporating reversible and irreversible transverse relaxation effects into Steady State Free Precession (SSFP) signal intensity expressions for fMRI considerations, Magn Reson Imaging (2012), http://dx.doi.org/10.1016/j.mri.2012.10.002

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Fig. 1. Simulated SSFP magnitude signals as functions of central frequency ω0/(2π) for a high flip angle of 70° (top plot) and a low flip angle of 10° (bottom plot) for Lorentzian (blue +), uniform (red ∘) and Gaussian (green *) distributions. The following parameters were employed: R1 = 1 s−1, R2 = 10s−1, R′2 = σ= Δω = 5 s−1, τ = 0.05 s and with t = τ/2. (For interpretation of the references to color in the figure legends, please refer to the web version of this article.)

similar to those reported experimentally by Zhong et al. [9] who used multi-echo sampling throughout 70 ms τ intervals and demonstrated activation-induced changes of SSFP signal intensity similar in magnitude to gradient echo EPI imaging within the visual cortex. Since the original SSFP fMRI work by Scheffler et al. [3] emphasized changes occurring within transition bands (primarily from frequency induced changes), we also examined the effects of R′2 on transition band signals. Fig. 3 utilized the Lorentzian distribution equations to

simulate the signal magnitude as a function of frequency through transition bands for typical brain parenchymal relaxation rates and three different values of R′2 (0 s −1, 5 s −1 and 10 s −1). The top plot in Fig. 3 was generated using the same sequence parameters employed by Scheffler et al. [3], with a relatively large flip angle (50°) and TR (τ = 0.044 s) while the lower plot was generated using a small flip angle (10°) and TR (τ = 0.01 s), where the signal readout was at time t = τ/2 in both cases. The general effect of increasing R′2 is to lower and

Fig. 2. Time domain signals in the long τ regime (τ = 0.05 s) for a high flip angle of 70° (top plot) and for a low flip angle of 10° (bottom plot). Passband conditions were applied by using ω0τ = π and ω0 = 0 for high and low flip angles, respectively, as demonstrated necessary by Fig. 1. The same brain parenchymal parameters were employed as in Fig. 1 and with the same symbols employed for each of the three distributions: Lorentzian (blue +), uniform (red ∘) and Gaussian (green *).

Please cite this article as: Mulkern RV, et al, Incorporating reversible and irreversible transverse relaxation effects into Steady State Free Precession (SSFP) signal intensity expressions for fMRI considerations, Magn Reson Imaging (2012), http://dx.doi.org/10.1016/j.mri.2012.10.002

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Fig. 3. Frequency response simulations through the transition band and halfway into adjacent passbands for the Lorentzian distribution and three different values of R′2, 0 s−1 (blue +), 5 s−1 (red ∘) and 10 s−1 (green *), for a flip angle of 50° and τ = 0.044 s (top plot) and a flip angle of 10° and τ = 0.01 s (bottom plot), with t = τ/2. Brain parenchymal parameters of R1 = 1 s−1 and R2 = 10 s−1 were used for both plots.

broaden the transition band somewhat, although clearly for small changes in R′2 the largest contributor to signal changes within the transition band result from frequency offsets, which can be attributed to blood susceptibility changes. The expressions derived in Sec. 3 also allow us to examine in detail the sensitivity of SSFP signals to changes in both the reversible and

irreversible relaxation rates in both short and long τ regimes, as demonstrated in Fig. 4, in which the Lorentzian distribution was assumed. The longitudinal relaxation rate R1 was fixed at 1 s −1 and a high flip angle of 70° was employed with passband conditions assumed (ω0τ = π). The signal as a function of R′2 for a fixed value of R2 of 10 s −1 is shown in the top plot while the lower plot depicts signal

Fig. 4. SSFP magnitude signal changes as R⁎2 changes from 12 s−1 to 18 s−1 by either altering R′2 from 2 to 8 s−1 with R2 fixed at 10 s−1 (top plots) or by altering R2 from 7 to 13 s−1 with R′2 fixed at 5 s−1 (bottom plots). Plots were generated using the Lorentzian distribution formulae under passband conditions (ω0τ = π) with R1 = 1 s−1, flip angle of 70°, t = τ/2 and for repetition times τ = 0.05 s (blue +), τ = 0.025 s (red ∘) and τ = 0.005 s (green *). General findings are that increasing the repetition time increases the sensitivity to R′2 effects (top plots) but that generally R2 changes will play a larger role than R′2 changes (lower plots) for all three repetition times examined under these conditions.

Please cite this article as: Mulkern RV, et al, Incorporating reversible and irreversible transverse relaxation effects into Steady State Free Precession (SSFP) signal intensity expressions for fMRI considerations, Magn Reson Imaging (2012), http://dx.doi.org/10.1016/j.mri.2012.10.002

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R.V. Mulkern et al. / Magnetic Resonance Imaging xxx (2012) xxx–xxx

as a function of R2 for a fixed value of R′2 of 5 s −1. In each case, simulations were performed for short, intermediate and long repetition times τ of 0.005, 0.025 and 0.05 s. Of note is the relative lack of sensitivity to R′2 at the shortest repetition time of 0.005 s, increasing somewhat at the longer τ values. In contrast, there is a marked and similar sensitivity of signal to the irreversible relaxation rate R2 at short, intermediate and long τ values. Fig. 4 helps us interpret the motor activation studies performed on the healthy volunteer with both a standard EPI BOLD acquisition and a short (τ = 0.005 s) repetition time SSFP acquisition under passband conditions. The experimental results from this study are shown in Fig. 5, where the magnitude of the activation changes in the motor cortex (crosshairs) for both acquisitions is observed to be similar and on the order of 1–2%. The activation in the EPI BOLD study is due to changes in R⁎2 = R2 +R′2, which must be on the order of 0.5–1 s −1 to achieve the observed 1–2% signal intensity changes that accompany this activation. Fig. 4 indicates that for the conditions of our SSFP experiment (τ = 0.005 s, θ = 70°), R′2 changes of this magnitude, and even much larger, would not alter the SSFP signal intensity, while R2 changes of this order would lead to signal changes consistent with those observed in the SSFP fMRI experiment. Thus the signal intensity changes accompanying motor activation from the finger tapping task can be attributed, in both cases, to changes in the irreversible transverse relaxation rate R2 as opposed to changes in the reversible relaxation rate R′2, an interpretation not evident from the EPI BOLD results alone. Dharmakumar et al. [8] and Lee et al. [10] have suggested that the random diffusion or motion of spins within local field inhomogeneities around vessels—inhomogeneities which are sensitive to the oxygenation state of the blood— may be the primary mechanism for passband SSFP functional contrast. This mechanism would, of course, be expected to contribute to irreversible as opposed to reversible transverse relaxation and so is consistent with our results. To conclude, the standard SSFP signal intensity expressions previously developed for a single isochromat ω have been generalized

A

here to incorporate a distribution of isochromats for three separate cases: Lorentzian, Gaussian and uniform distributions. The widths of these distributions are responsible for so-called reversible relaxation effects and thus the expressions derived herein allow for the separation of reversible from irreversible transverse relaxation processes when applying SSFP approaches to functional magnetic resonance imaging. Acknowledgments This work was supported by a grant from the National Institutes of Health (NIH R21 NS076859) and a CHB-MIT Collaborative Research Fellowship Grant. Appendix A The goal is to prove the identityq inffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eq. (6). Let Q = (1-h cos ωτ) −1  ffi 2 and define p≡(1-g)/h, where g≡ 1−h : Eq. (6) may then be written as ( ) ∞ ∞ X X n inωτ n −inωτ p e þ p e Q ¼ ð1=gÞ 1 þ n¼1 n¼1 ( ) ∞
∞
  X X iωτ n −iωτ n : ¼ ð1=g Þ 1 þ pe þ pe n¼1

ðA1Þ

n¼1

Since h is a real number between 0 and 1 for pulse sequence parameters and tissue relaxation times encountered in practice, |pe iωτ| = |pe -iωτ| = |p| = |(1-g)/h| is also a real number between 0 and 1; both sums in Eq. (A1) are therefore recognized as convergent geometric series leading to  Q ¼ ð1=gÞ 1 þ

B

p e−iωτ −p

þ

p : eiωτ −p

ðA2Þ

1.5

Signal change (%)

1 0.5 0 -0.5 -1 -1.5

C

D

0

50

100 150 Time (s)

200

0

50

100 150 Time (s)

200

1.5

Signal change (%)

1 0.5 0 -0.5 -1 -1.5

Fig. 5. Experimental results from motor activation (bilateral finger tapping: four repetitions of 30s OFF; 30s ON) studies using (A) standard EPI BOLD and (C) passband SSFP with a short repetition time (τ = 0.005 s). The time course for the voxel in motor cortex indicated by the crosshair in (A) and (C) is shown in (B) and (D), respectively, with time plotted against signal change. For both EPI BOLD and passband SSFP, significant activation (on the order of 1–2% signal change) is observed.

Please cite this article as: Mulkern RV, et al, Incorporating reversible and irreversible transverse relaxation effects into Steady State Free Precession (SSFP) signal intensity expressions for fMRI considerations, Magn Reson Imaging (2012), http://dx.doi.org/10.1016/j.mri.2012.10.002

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Combining the three terms in braces in Eq. (A2) with one common denominator and using Euler identities as appropriate leads to (

) 1−p2 : Q ¼ ð1=gÞ 1 þ p2 −2p cos ωτ

ðA3Þ

Substituting back p = (1-g)/h into Eq. (A3) and performing some further algebraic manipulation leads to (

) h 2 −1 þ g Q ¼ ð1=gÞ : 1−g−h ð1−gÞcos ωτ

ðA4Þ

Factoring (1-g) out of the denominator in Eq. (A4) and noting that 1-h 2 = g 2 leads to (

) −g 2 þ g Q ¼ ð1=g Þ ð1−gÞð1−h cos ωτÞ 1 : ¼ 1−h cos ωτ

ðA5Þ

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Please cite this article as: Mulkern RV, et al, Incorporating reversible and irreversible transverse relaxation effects into Steady State Free Precession (SSFP) signal intensity expressions for fMRI considerations, Magn Reson Imaging (2012), http://dx.doi.org/10.1016/j.mri.2012.10.002