Duarte Coello, Roberto De Jesus

Repetti, Audrey

Wiaux, Yves

English

In the context of quantitative Magnetic Resonance Imaging (qMRI), the recently proposed Magnetic Resonance Fingerprinting (MRF) technique has significantly reduced the acquisition time. MRF reconstructions are based on the correlation of multiple aggressively undersampled time acquisitions. Coil sensitivity calibration is a crucial step in the reconstruction process to obtain accurate results. Traditionally, MRF calibration is performed upstream of the imaging process, using a different acquisition. This technique makes the overall acquisition time consuming and may introduce artefacts in the image estimate. Usual MRI self-calibration methods, reconstructing independently the time acquisitions, are not suitable for highly undersampled MRF data. In this work, leveraging recent developments in non-convex optimisation, we propose the first self-calibration method for MRF, exploiting the correlation in the time acquisitions

Heriot Watt Pure

Self-calibration for Magnetic Resonance FingerprintingRoberto Duarte∗, Audrey Repetti∗†, and Yves Wiaux∗∗ Institute of Sensors, Signals and Systems, Heriot-Watt University, Edinburgh, EH14 4AS.† Department of Actuarial Mathematics & Statistics, Heriot-Watt University, Edinburgh, EH14 4AS.Abstract—In the context of quantitative Magnetic Resonance Imag-ing (qMRI), the recently proposed Magnetic Resonance Fingerprinting(MRF) technique has significantly reduced the acquisition time. MRFreconstructions are based on the correlation of multiple aggressivelyundersampled time acquisitions. Coil sensitivity calibration is a crucialstep in the reconstruction process to obtain accurate results. Traditionally,MRF calibration is performed upstream of the imaging process, using adifferent acquisition. This technique makes the overall acquisition timeconsuming and may introduce artefacts in the image estimate. UsualMRI self-calibration methods, reconstructing independently the timeacquisitions, are not suitable for highly undersampled MRF data. Inthis work, leveraging recent developments in non-convex optimisation,we propose the first self-calibration method for MRF, exploiting thecorrelation in the time acquisitions.I. INTRODUCTIONIn MRF, the objective is to estimate the parameters associatedwith each of the N voxels of an imaged volume, from degraded un-dersampled measurements Y ∈ CQ×L×C [1]. These measurementsare acquired from C coils, with an excitation sequence of lengthL. For each coil, and each excitation, Q degraded measurementsare acquired. Let M ∈ CN×L be the response of the imagedvolume of interest. For every excitation l ∈ {1, . . . , L} and coil c ∈{1, . . . , C}, the corresponding observation Y:,l,c ∈ CQ is given byY:,l,c = Ω:,:,lFDiag(S:,c)M:,l + η:,l,c, where Ω ∈ {1, 0}Q×N×Lis the concatenation of L selection matrices, F ∈ CN×N modelsthe 2D discrete Fourier transform, S ∈ CN×C is the concatena-tion of the C sensitivity maps, and Diag is a function that takesa vector and returns the corresponding diagonal matrix. Finally,η ∈ CQ×L×C is a realisation of a random i.i.d. Gaussian noise.Let h : CN×C ×CN×L → CQ×L×C be the linear mapping definingthe complete acquisition process, i.e. Y = h (S,M)+η. The imagevolume response M belongs to a non-convex set B that models theconstraints from the physics of the magnetic resonance as well asother relevant phenomena such as the partial volume effect [1], [2].The magnetisation sequence can be obtained by defining M as aminimiser of F (S,M) = 1/2‖Y−h(S,M)‖22, under the constraintM ∈ B, when S is fixed and has been pre-calibrated.Traditional MRF methods estimate S beforehand, using differentacquisitions, making the data acquisition process more expensive. Inaddition, since calibration is performed by a single acquisition, Sis calibrated up to a voxel-wise global phase. Whilst estimating Mhas been the focus of works such as [1], [2], the lack of advancedtechniques for the coil sensitivity estimation can significantly affectthe image reconstruction. Moreover, due to the highly undersampledMRF acquisitions, MRI self-calibration methods are not suitable.Hence the need for developing joint calibration and imaging methodsfor MRF. In the next section, we propose an optimisation approach toestimate simultaneously the sensitivity maps S and the image volumeresponse M.II. JOINT CALIBRATION AND IMAGING FOR MRFWe propose to minimiseM∈B,S∈RN×CF (S,M). To solve this problem,we use an alternating forward-backward approach [3], described inAlgorithm 1, where ∇MF (resp. ∇SF ) denotes the partial gradientof F with respect toM (resp. S). Under technical conditions [3], thesequence (M(k),S(k))k∈N generated by Algorithm 1 is guaranteedto converge to a critical point of the objective function of interest.Algorithm 1 Joint Calibration and Imaging for MRF1: Input: S(0) ∈ CN×C , M(0) ∈ B, (I, J) ∈ N2∗, (µ, ν) ∈ (R+∗ )22: for k = 0, 1, . . . do3: M˜(0) = M(k) and S˜(0) = S(k)4: for i = 0, 1, . . . , I − 1 do5: M(i) = PB(M˜(i) − µ∇MF(S(k),M˜(i)))6: end for7: M(k+1) = M˜(I)8: for j = 0, 1, . . . , J − 1 do9: S(j) = S˜(j) − ν∇SF(S˜(j),M(k+1))10: end for11: S(k+1) = S˜(J)12: end forIII. RESULTSWe test our algorithm with a partial volume phantom using themodel defined in [2]. As shown in Fig. 1, the estimated tissueproportion maps are close to the ground truth. While the algorithmis able to accurately estimate the parameters and the proportions ofeach tissue, the obtained proton densities are not accurate. This isdue to the ambiguity in the operator h, i.e. for any voxel n, the normof Mn,: can be absorbed in Sn,: and vice-versa. In future work, weaim to address the ambiguity by adding more prior information inthe model such as, spatial regularisation on the proton density andsmoothness in the sensitivity maps.0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1Fig. 1: Tissue proportion maps. The first row correspond to the Ground Truth andthe second row to our reconstructions. The columns correspond, from left to right, toAdipose Tissue, White Matter, Muscle, Gray Matter and Cerebrospinal fluid.REFERENCES[1] M. Davies, G. Puy, P. Vandergheynst, and Y. Wiaux, “A CompressedSensing Framework for Magnetic Resonance Fingerprinting,” SIAM J.Imaging Sci., vol. 7, no. 4, p. 2623, Nov. 2014.[2] R. Duarte, A. Repetti, P. A. Go´mez, M. Davies, and Y. Wiaux, “Greedyapproximate projection for magnetic resonance fingerprinting with partialvolumes,” Tech. Rep., 2018, https://arxiv.org/abs/1807.06912.[3] E. Chouzenoux, J.-C. Pesquet, and A. Repetti, “A block coordinatevariable metric forward-backward algorithm,” Journal of Global Opti-mization, pp. 1–29, Feb. 2016.

Self-calibration for Magnetic Resonance Fingerprinting

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Self-calibration for Magnetic Resonance Fingerprinting

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