NMR spectroscopy using Rabi modulated continuous wave excitation

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NMR spectroscopy using Rabi modulated continuous wave excitation James C. Korte a,∗ , Kelvin J. Layton b , Bahman Tahayori c , Peter M. Farrell a , Stephen M. Moore d , Leigh A. Johnston a a

Department of Electrical & Electronic Engineering, University of Melbourne, Parkville, Victoria 3010, Australia Institute for Telecommunications Research, University of South Australia, Mawson Lakes, South Australia 5095, Australia Monash Institute of Medical Engineering, Department of Electrical and Computer Systems Engineering, Monash University, Clayton, Victoria 3166, Australia d IBM Research Australia, Carlton, Victoria 3053, Australia b c

a r t i c l e

i n f o

Article history: Received 12 November 2015 Received in revised form 6 July 2016 Accepted 18 October 2016 Available online xxx Keywords: NMR Continuous wave excitation Amplitude modulation Bloch equations Spectroscopy Inverse problem Nonlinear system modelling Optimal experimental design

a b s t r a c t We present a proof of principle method to reconstruct nuclear magnetic resonance spectra after perturbing the spin system with a series of Rabi modulated continuous wave excitations. This continuous wave method provides an exciting alternative to pulsed Fourier transform methods which dominate magnetic resonance techniques. Applications include the measurement of ultra-short relaxation samples, which would be beneficial for sodium imaging or the assessment of bone and connective tissues. It is known that under Rabi modulated continuous wave excitation the spin system will reach a substantial periodic orbit. In a first experiment, we confirm that these periodic orbits are affected by off-resonance effects, and therefore encode chemical shift information. Spectroscopy is posed as an inverse problem in a second experiment and we apply a nonlinear estimation model to obtain nuclear magnetic resonance spectra from experimental measurements. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction The first NMR spectra were obtained under continuous wave (CW) radio frequency (RF) excitation by sweeping either the strength of the main magnetic field or the carrier frequency of the excitation magnetic field, with the spectrum generated by plotting the magnitude of the resulting NMR signal against the sweep range [1,2]. These methods were overshadowed by the more efficient technique in which a sample is excited by a powerful, short duration, RF pulse [3]. The spectrum is obtained directly from a Fourier transform of the free induction decay (FID) signal which is a broadband combination of all precessing isochromats. A resurgence of interest in CW NMR spectroscopy methods [4,5] has influenced the development of new magnetic resonance imaging (MRI) techniques capable of imaging samples with ultrafast spin-spin relaxation [6,7]. The optimality of pulse excitation sequences is also challenged in [8,9] with the suggestion that CW excitation may improve signal intensity for lower energy

∗ Corresponding author.

excitation. Recent work [10], inspired by quantum optics [11], has experimentally demonstrated that a spin system excited by a Rabi modulated CW achieves substantial periodic steady-state magnetisation. The frequency components of this steady-state magnetisation are restricted to harmonics of the excitation modulation frequency [9] and a maximum harmonic magnitude is achieved when a secondary resonance condition is met [10]. The Rabi resonance condition is also being investigated in CW electron paramagnetic resonance (EPR) [12]. In this work, we report on two proof of concept experiments that demonstrate that the spin system response to Rabi modulated CW excitation contains chemical shift information. We extend the original experiment [10], which is limited to on-resonance excitation, to investigate the response of the spin system to off-resonance Rabi modulated CW excitation. We propose a CW method, by which to perform NMR spectroscopy, posed as an inverse problem. Chemical shift information is encoded in a series of Rabi modulated CW excitations. A forward model is constructed from a periodic solution of the Bloch equation, and is used to reconstruct a simple NMR spectrum of ethanol. An algorithm based on the A-optimality criteria [13] was used to select a theoretically optimal set of excitation parameter pairs.

http://dx.doi.org/10.1016/j.bspc.2016.10.006 1746-8094/© 2016 Elsevier Ltd. All rights reserved.

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We anticipate that the results we present here will find application in the acquisition of magnetic resonance information from ultra-fast relaxation samples. The proposed excitation method can theoretically maintain an observable steady-state magnetisation for such samples and presents an alternative to existing ultra-fast relaxation methods such as sweep imaging with Fourier transformation (SWIFT) [6], Ultra-short echo time imaging (UTE) [14] and zero-echo time imaging (ZTE) [15]. Such methods have beneficial clinical applications such as the assessment of bone and connective tissue [16], measurement of sodium concentration in brain tissue [17] and the detection of iron oxide nano-particles delivered to tumours [18].

T

where C(l) = [ Cx(l) Cy(l) Cz(l) ] are the Fourier coefficients. In matrix notation, the relationship becomes M = eT C

(6)

where



e = ...

e−2iωm t

C = ...

C(−2)



e−iωm t C(−1)



Cobs ˛, ωm , ırf , T1 , T2 ≈

We define Rabi modulated excitation as an amplitude modulated RF field with the envelope function ωe (t) = ω1 (1 + ˛ cos ωm t),

C(1)

C(2)

...

...

T

T

.



J K   

(j)

C ˛, ωm , ω1 , ıω0

(k)

 

(j)

, ırf , T1 , T2 p ω1

 

p ıω 0

(k)

 (7)

j=1 k=1

=

(1)

where ω1 = B1 is the average of the excitation envelope,  is the gyromagnetic ratio, B1 is the excitation field strength, ˛ is the modulation factor and ωm is the modulation frequency. It is known that under this excitation the spin system achieves a significant periodic steady-state magnetisation and that the magnitude of this steadystate is maximised when the Rabi resonance condition, ωm = ω1 , is met [10].

C(0)

e2iωm t

The bulk magnetisation in the frequency domain, C, is predicted by a periodic solution of the Bloch equations (B.4) as described in Appendix B. The observed bulk magnetisation (3) is transformed into the frequency domain by substitution of (6) and (B.4) into (3)

2. Theory 2.1. Rabi modulated excitation

1 eiωm t

K  

R ˛, ωm , ıω0

(k)

 

, ırf , T1 , T2 p ıω0

(k)



k=1

where



R ˛, ωm , ıω0 , ırf , T1 , T2 =

J  

  

(j)

C ˛, ωm , ω1 , ıω0 , ırf , T1 , T2

(j)



p ω1

.

(8)

j=1

2.2. Observed NMR signal 2.3. Spectroscopy as an inverse problem The observed noise-free NMR signal under Rabi modulated excitation is,



Mobs t, ˛, ωm , ırf , T1 , T2







=





M t, ˛, ωm , ω1 , ıω0 , ırf , T1 , T2 p (ω1 ) p ıω0



dω1 dıω0 (2)

where ıω0 is any deviation from the Larmor frequency, ω0 , due to main field inhomogeneities, chemical shift effects or applied gradients. The p(ıω0 ) distribution represents deviations from the Larmor frequency, p(ω1 ) represents inhomogeneities in the strength of the excitation field and ırf is an offset of the excitation carrier frequency from the Larmor frequency. The signal equation can be approximated by numerical integration over a regular grid of J excitation amplitudes and K off-resonances



Mobs t, ˛, ωm , ırf , T1 , T2

  J

≈



K

M

(j) (k) t, ˛, ωm , ω1 , ıω0 , ırf , T1 , T2

  p

(j) ω1

  p ıω0

(k)

 .

The steady-state magnetisation is described by a complex Fourier series, restricted to harmonics of the modulation frequency, ωm ,

∞ 

⎡

(9)



(1)

R ˛(1) , ωm , ıω0

(1)



 ⎢  (2) (2) ⎢ R ˛ , ωm , ıω (1) ⎢ =⎢ ⎢ .. ⎣.   (1) (N) R ˛(N) , ωm , ıω   (1)  

..





. . . p ıω0

(1) = Cobs ˛(1) , ωm



(K)



⎤



(2)

(K)





⎥ ⎥ ⎥ ⎥ ⎥ ⎦

.

. . .

(N)

· · · R ˛(N) , ωm , ıω0

0

= p ıω0

(1)

· · · R ˛(2) , ωm , ıω0

0

H



· · · R ˛(1) , ωm , ıω0

(K)



(K)



 T

(2)

Cobs ˛(2) , ωm





(N)

. . . Cobs ˛(N) , ωm

 T

.

The discrete NMR spectrum, x, was reconstructed from the known forward model, H, and an observation vector, z, by solving the linear system (9). In this formulation (9), the relaxation constants, T1 and T2 , and the RF offset, ırf , are assumed to be known constants and are omitted for notational simplicity. 3. Methods and materials

C(l) eilωm t ,

(4)

l=−∞

˙ = M

where:

z

(3)

∞ 

Hx = z

x

j=1 k=1

M=

It is possible to reconstruct a NMR spectrum under Rabi modulated excitation as chemical shift effects are encoded in the observed NMR signal. A linear system can be constructed from (7) the spin system witha set of N excitation parameter by exciting

(1) (N) pairs (˛(1) , ωm ) . . . (˛(N) , ωm ) and subsequent measurement of the transverse magnetisation,

ilωm C(l) eilωm t ,

l=−∞

(5)

Two experiments were performed on a 4.7T Bruker BioSpec small bore MRI scanner, using the experimental protocol developed in [10] and summarised in Fig. 1. Rabi modulated excitation (1) was applied for an initial duration of T = 1000 ms to allow the magnetisation to reach a steady-state, after which the free induction decay

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the second experiment, an ethanol spectra, p(ıω0 ), is reconstructed from a series of steady-state harmonics measured under a range of Rabi modulated excitation envelopes. The distribution of off-resonances, p(ıω0 ), was measured using the field-mapping sequence MAPSHIM (Bruker Biospin) [19]. The distribution of excitation field strengths, p(ω1 ), was measured with a B1 mapping sequence [20]. Two spherical phantoms were used in this study, one filled with tap water and the other with ethanol. Both phantoms were doped with Magnevist® to reduce the experimental time. 3.1. The response of off-resonance spins

Fig. 1. Experimental protocol for the incremental measurement of the steady-state transverse magnetisation waveform. The spin system is perturbed by an excitation ωe causing a response Mxy . By repeatedly perturbing the spin system with an increasing excitation duration it is possible to construct an observed transverse magnetisation Mobs .

We investigated the effects of off-resonance on the observed steady-state magnetisation waveform by offsetting the excitation carrier frequency from the Larmor frequency. A spherical phantom of Gd-doped water (T1 = 287 ms, T2 = 150 ms) was selected for its narrow off-resonance distribution. Off-resonance measurements were taken from ırf =−400 Hz to ırf = 400 Hz with a 20 Hz increment. The excitation envelope had parameters ˛ = 1 and ω1 = ωm = 100 Hz. A discrete Fourier transform was applied to the measured steady-state magnetisation waveform to extract its harmonic components. A theoretical curve was generated from the periodic solution of the Bloch equations, numerically integrated over the measured p(ıω0 ) and p(ω1 ) distributions using (7). 3.2. Rabi modulated spectroscopy A proof of concept experiment was undertaken to demonstrate the utility of chemical shift information encoded in the steady-state magnetisation. Our objective was to reconstruct the spectrum of a spherical phantom of Gd-doped ethanol (T1 = 120 ms, T2 = 43 ms) from a series of CW excitations. A reference spectrum from a single, 2048-point, FID was acquired with a dwell time of 100 ␮s. Spectra and variance are plotted on a ppm scale, ı=

ıω0 × 106 + ıtms , ω0

(10)

where ıtms = 1.25 ppm is an offset commonly observed when using tetramethylsilane (TMS) as a reference compound. No reference compound was added to the doped ethanol sample used in our experiments.

Fig. 2. Spin system response of the water phantom under Rabi modulated excitation. (a) Measured (red circles) and theoretical (black line) periodic steady-state magnetisation waveform for excitation parameters ˛ = 1, ω1 = ωm = 100 Hz and ırf = 180 Hz. (b) Measured (coloured circles) and theoretical (black cross) harmonics of steadystate magnetisation. DC component (blue), first (green), second (purple), third (red) fourth (orange) and fifth (grey) harmonics. (For interpretation of the references to colour in this legend, the reader is referred to the web version of the article.)

(FID) was measured. A steady-state magnetisation waveform, Mobs , was incrementally acquired by selecting the first FID point, and repeating the process with an increase, , to the excitation duration on each repetition. Whilst this method is inefficient, it allows the measurement of the steady-state magnetisation waveform, such as Fig. 2a, without hardware modification. In the first experiment, the steady-state harmonics, Cobs , are measured over a range of ırf and compared to the predicted steadystate harmonics by integrating the periodic solution of the Bloch equations over the measured p(ıω0 ) and p(ω1 ) distributions. In

3.2.1. Excitation parameter selection Experimental parameters were selected from a set of candidate parameter pairs,  = {(˛, ωm )}, defined by a dense grid adhering to the power limitation of the excitation coil, as illustrated in Fig. 3. Two reduced sets of 500 parameter pairs were then selected for experiments. The first set, grid , was selected as a coarse grid of the candidate parameter pairs. The second set, opti , was constructed iteratively to minimise the theoretical variance of the reconstructed spectrum, as detailed in Algorithm 1. At each measurement selection step, k, in Algorithm 1, the covariance, Pk , for every candidate parameter pair in the candidate set was calculated using,



Pk = Pk−1 − Pk−1 HTk Hk Pk−1 HTk + W

−1

Hk Pk−1

(11)

where W is the covariance of observation noise. The optimal parameter pair was selected from the candidate set using the A-optimality criteria [21], minimise Tr (Pk ) (˛,ωm ) ∈ 

(12)

where Tr denotes the matrix trace.

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The initial covariance was set to P0 =  2 I where the variance,  2 = 10,000, and I is an identity matrix, providing an initial condition with negligible regularisation. The observation noise 2 I, with a standard deviation covariance matrix was set to W = obs −3  obs = 7.2 × 10 , which is approximately 5% of the mean signal.

Fig. 3. Potential (light grey) excitation parameter pairs , set of gridded (dark grey) excitation parameter pairs grid , set of optimal (black) excitation parameter pairs opti for a 101 point reconstruction from −600 Hz to 600 Hz and a homogeneous B1 field.

Algorithm 1. Select the optimal set, opti , of N excitation parameter pairs from a candidate set, . function OptimalSet(, N) P0 ←  2 I opti ← {} an empty set k ←1 measurement counter while k ≤ N do  (˛ , ωm ) ← argmin Tr (Pk (˛, ωm )) (˛,ωm ) ∈ 

 ) to opti add (˛ , ωm  remove (˛ , ωm ) from    ) update covariance Pk ← Pk (˛ , ωm k ←k+1 end while return opti end function

The selected excitation parameter pair was then added to opti and used to update Pk . The selection cycle was repeated until the required number of excitation parameter pairs were selected. The variance of the optimal set is shown in Fig. 4.

3.2.2. Measurement Steady-state magnetisation waveforms were acquired for every CW excitation parameter pair in grid and opti . The excitation carrier frequency was offset from the system frequency by ırf = 400 Hz to centre the spectrum and reduce the required bandwidth. The DC component and first five harmonics were extracted from each steady-state magnetisation and recorded in a measurement vector z. Relaxation constants T1 and T2 were measured before each experiment, using a rapid acquisition with relaxation enhancement with variable repetition time (RARE-VTR) scan, to ensure accuracy of the encoding matrix H. Scans were taken with six TR values (200, 400, 800, 1500, 3000 and 4500 ms) and five TE values (11, 33, 55, 77, and 99 ms) over a single 1 mm slice, FOV 4 cm, 64 × 64 matrix. The measured relaxation maps were spatially averaged to produce a single T1 and T2 constant value per phantom. All measurements were duplicated to verify the reproducibility of results. 3.2.3. Reconstruction The spectrum, x, was reconstructed by least squares optimisation with a non-negative constraint: minimise Hx − z 2 .

(13)

x ∈ [0,∞)

The forward model, H, was constructed from the Fourier series approximation of the Bloch equation, numerically integrated over the measured p(ω1 ) distribution using (9). The optimisation algorithm was initialised with the spectrum, x, set as a vector of zeros. 3.2.4. Spectrum analysis Let x be the area normalised version of the reconstructed spectra, x. Lorentzian peaks were fitted to x, to assess the accuracy of peak locations and the ratio of peak areas. A Lorentzian peak is defined as





L ı, , , a =

a







(14)

2

(ı − ) + 2

where ı is a discrete vector of off-resonance,  is the peak centre,  is the half width at half maximum (HWHM) and a is the peak area. As ethanol has three distinct peaks, the objective function is minimise x − ,,a

where

⎡ ⎢

1

3  



L ı, i , i , ai 2 + g 1 −

i=1

⎤ ⎥

 = ⎣ 2 ⎦ , 3

3 

ai 2

(15)

i=1

⎡ ⎢

1

⎤ ⎥

 = ⎣ 2 ⎦ , 3

⎡ ⎢

a1

⎤ ⎥

a = ⎣ a2 ⎦ a3

and g is a weighting factor for the unity area term. The constrained minimisation function, fmincon, from MATLAB was used to solve optimisation problems (13) and (15). 4. Results 4.1. The response of off-resonance spins Fig. 4. Trace of covariance matrix of reconstructed spectra (12) for the gridded (grey triangles) set of excitation parameter pairs grid and the optimal (black circles) set of excitation parameter pairs opti . Mean variance for the optimal set is 1.4 × 10−4 and 2.7 × 10−4 for the gridded set.

The measured and theoretical steady-state magnetisation waveform, for a single measurement, ırf = 180 Hz, is shown in Fig. 2. The spectrum of this steady-state magnetisation (Fig. 2b) shows

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Fig. 6. Harmonic curves of the water phantom under Rabi modulated CW excitation with parameters ˛ = 1 and ω1 = ωm = 100 Hz. Measured (circles) and theoretical (solid line) frequency coefficients of steady state magnetisation. DC component (blue), first (green), second (purple), third (red) fourth (orange) and fifth (grey) harmonics. (For interpretation of the references to colour in this legend, the reader is referred to the web version of the article.)

Fig. 5. Measured distributions of the water phantom. (a) Off-resonance distribution p(ıω0 ) extracted from the histogram of a B0 field map. (b) RF power distribution p(ω1 ) extracted from the histogram of a B1 field map.

Fig. 7. NMR spectrum of doped ethanol phantom from reference FID (grey) and reconstruction from Rabi modulated CW excitation (black). (a) Reconstruction from the grid measurement set. (b) Reconstruction from the opti measurement set.

that information is restricted to harmonics of the Rabi frequency, ω1 = 100 Hz. The measured p(ıω0 ) and p(ω1 ) distributions, used in the generation of the predicted harmonic curves, are shown in Fig. 5a and b, respectively. The variation in the harmonics is shown over a range of off-resonances (Fig. 6). The experimental measurements agree with the theoretical curves and show that the relative strengths of the harmonic components are influenced by off-resonance effects.

reference FID are comparable. The measured and reconstructed spectra differ slightly from those observed in [22] with larger relative peak shifts, enlarged CH3 peak areas and reduced OH peak areas. Spectra reconstructed from the optimal parameter set, opti , are slightly more accurate than those reconstructed from the gridded parameter set, grid , which suffer from an artifact near 0.25 and 6.25 ppm. 5. Discussion

4.2. Rabi modulated spectroscopy Reconstructed ethanol spectra from the reference FID, gridded parameter set grid and optimal parameter set opti are shown in Fig. 7. These reconstructions demonstrate that under Rabi modulated CW excitation, chemical shift information can be encoded in the steady-state magnetisation. The results of the Lorentzian fitting are shown in Table 1 where the reference entries for relative peak shifts (ı1 , ı2 ) and area ratios (aratio ) are experimental results from [22]. The results for both parameter sets, for both measurements, and the result from the

The results from the investigation of off-resonance spins verify that our periodic solution of the Bloch equations (7) can predict off-resonance behaviour of the spin system under Rabi continuous wave excitation. Furthermore, the hypothesis that spectral information is restricted to harmonics of the Rabi frequency [10] for the resonant case, = 0 Hz, has now been extended to off-resonance cases. Variations between prediction and measurement can be attributed to a number of factors such as measurement error of relaxation time constants, measurement error of field distributions

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Table 1 Results from the spectrum analysis, with a g = 0.01, of the reference and reconstructed ethanol spectra. The parameters of three fitted Lorentzian peaks are centre , half width half max  and area a. The chemical shift ı1 is between CH3 and CH2 and the chemical shift ı2 is between CH2 and OH. Peak centre, peak width and chemical shifts are listed in parts per million (ppm). CH3 peak

CH2 peak

OH peak

Summary





a





a





a

ı1

ı2

aratio

Reference Literature [22] FID

1.250 1.305

0.07

0.500 0.524

3.709 3.737

0.07

0.333 0.341

5.365 5.525

0.08

0.167 0.156

2.459 2.432

1.656 1.788

3.0: 2.0: 1.0 3.1: 2.0: 0.9

Measurement 1 Optimal Grid

1.272 1.270

0.05 0.05

0.533 0.527

3.751 3.753

0.04 0.04

0.332 0.334

5.589 5.567

0.07 0.06

0.135 0.138

2.479 2.483

1.838 1.815

3.2: 2.0: 0.8 3.2: 2.0: 0.8

Measurement 2 Optimal Grid

1.266 1.268

0.05 0.05

0.534 0.521

3.746 3.744

0.05 0.04

0.336 0.333

5.575 5.552

0.07 0.04

0.130 0.146

2.480 2.476

1.829 1.808

3.2: 2.0: 0.8 3.1: 2.0: 0.9

or phase error between the prediction model and measurements. Whilst the error introduced by each of these factors should be minimal, the combined effect of these errors on the prediction model, H, is the subject of ongoing investigation. Results from the Rabi modulated spectroscopy experiment demonstrate that it is possible to encode chemical shift in a series of CW excitations and reconstruct a spectrum. This method requires the measurement of relaxation constants to ensure an accurate forward model, H, this requirement may be removed in future experiments by joint estimation [23]. The reconstructions from the optimal and gridded parameter sets are comparable and the lack of an artifact around 0.25 and 6.25 ppm in the optimal reconstruction may be attributed to the lower theoretical variance in these regions as shown in Fig. 4. The reconstruction routine (13) also converges in less iterations when reconstructing with measurements from the optimal parameter set. Imperfections in the spectra reconstructed from measurements under Rabi modulated excitation can be attributed to error in the forward model, H. The receiver attenuation was fixed during each experiment which may have introduced error when measuring magnetisation waveforms with a lower relative power, where experimental imperfections begin to dominate. In this work spectroscopy is posed as an inverse problem thus it is important to consider the stability of the reconstruction. The low theoretical variance of the estimated spectra, shown in Fig. 4, provides an exact quantification of the predicted accuracy and demonstrates that the inverse problem is well conditioned. The reconstruction algorithm was tested with multiple starting points and converged to the same solution as presented. Currently, this method has the potential to produce non-physical spectra, such as the small spurious peaks around 4.5 ppm in Fig. 7. This is likely due to minor imperfections in experimental setup and with an improved acquisition strategy this should not be an issue. The proposed method is not time efficient and our current work involves improving the protocol efficiency by using gapped excitation [6] and in the future will consider hardware modification [24,25] to allow simultaneous transmit and receive. In [24,25], sideband modulation was used to spectrally isolate the excitation and induced NMR signal. Similar filtering could be used here but no sideband modulation would be required as the Rabi modulated response contains harmonic information outside the excitation bandwidth. With a gradient applied during the Rabi spectroscopy experiment, the reconstruction vector, x, is a projection of proton density along the gradient direction rather than a spectrum. Imaging will thus result from the acquisition of a sufficient number of projections, to be reconstructed using a filtered backprojection algorithm or an iterative algorithm, such as the conjugate gradient method.

The ability to maintain an observable steady-state magnetisation under Rabi modulated excitation may provide better SNR than methods that measure a single fast decaying FID signal. Measurement of ultra-short samples such as sodium, bone or connective tissue will be explored in future work. 6. Conclusion Rabi modulated CW excitation causes an observable periodic steady-state magnetisation with spectral information at harmonics of the Rabi frequency. This steady-state magnetisation is influenced by off-resonance effects, in a manner that can be accurately predicted by a periodic solution of the Bloch equations. In this paper, we have verified the existence of the off-resonance response, by experimental measurement of the spin-system under Rabi modulated excitation. We have further demonstrated the utility of the off-resonance harmonic response by encoding chemical shift information in a series of measurements and the reconstruction of a doped ethanol spectrum. We anticipate that with further investigation, and the development of a more efficient measurement protocol, it will be possible to reconstruct images from samples with ultra-fast spin–spin relaxation. Acknowledgements This work was supported by the Australian Research Council, the Elizabeth & Vernon Puzey Scholarship and National ICT Australia. Appendix A. The Bloch equation The behaviour of bulk magnetisation, predicted by the Bloch equation [26], in the standard rotating frame of reference is ˙ = (t)M + M where

⎡

1 M0 T1

(A.1)

⎤

1 T2



(t) = ⎢ −

−

⎢ ⎢ ⎣ 

−

0

0

1 T2

−ωe (t)

M

= Mx

M0

= [0 0 M0 ]T , = ıω0 + ırf .

My

Mz

⎥ ⎥ ⎦

ωe (t) ⎥ ,

T

−

1 T1

,

Here T1 and T2 are phenomenological relaxation constants, M0 is the bulk magnetisation at thermal equilibrium and represents deviation from the Larmor frequency. The deviation from the Larmor

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frequency has been split into two groups; the first, ıω0 , accounting for main field inhomogeneities, chemical shift effects or applied gradients, the second, ırf , for an offset to the excitation carrier frequency. Appendix B. Periodic solution of the Bloch equation In order to pose NMR spectroscopy as an inverse problem, we extend the periodic solution described in [9] to include off-resonance effects, . The steady-state magnetisation can be described by a complex Fourier series, restricted to harmonics of the modulation frequency, ωm , The transformation matrix, (t), in (A.1) can be decomposed as (t) = A + Deiωm t + De−iωm t where

⎡

−

(B.1)

⎤

1 T2



⎢ ⎢ A = ⎢ −

⎣

−

0

1 T2

ω1

−ω1

0

−

1 T1

⎡ ⎤ 0 0 0 ⎥ ω1 ˛ ⎥ ⎥ ⎢ ⎥ D = ⎣0 0 2 ⎦. ⎦ ω1 ˛ 0 −

0

2

Substituting (4) and (B.1) into (A.1) and rearranging terms yields ∞  



AC(l) − ilωm C(l) + DC(l−1) + DC(l+1) eilωm t

(B.2)

l=−∞

=−

1 M0 . T1

(B.2)

Written as a linear system





P ˛, ωm , ω1 , ıω0 , ırf , T1 , T2 C = Q (T1 , M0 )

(B.3)

the solution can then be found by





C ˛, ωm , ω1 , ıω0 , ırf , T1 , T2 = P−1 Q

(B.4)

where:

⎡

⎤

..

. ⎢ ⎢ A + 2iωm I ⎢D ⎢

P=⎢ ⎢

⎢ ⎢ ⎣

D A + iωm I D

⎥ ⎥ ⎥ D ⎥ ⎥ A D ⎥ ⎥ D A − iωm I D ⎥ D A − 2iωm I ⎦ ..

.

and, Q = [...

0

0

−

1 M0 T1

T

0

0

...] .

It has been shown that the series coefficients can be approximated by CN ≈ ((˛/2)N (ω1 /ωm )N )/N !, where N is the number of harmonics considered in the solution [27]. Therefore, the series solution coefficients always converge to zero. When ˛ω1 < 2ωm the

7

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Please cite this article in press as: J.C. Korte, et al., NMR spectroscopy using Rabi modulated continuous wave excitation, Biomed. Signal Process. Control (2016), http://dx.doi.org/10.1016/j.bspc.2016.10.006