Yael Balbastre^{1}, Julio Acosta-Cabronero^{1}, Nadège Corbin^{1}, Oliver Josephs^{1}, John Ashburner^{1}, and Martina F Callaghan^{1}

We present an algorithm for inferring sensitivities from low-resolution data, acquired either as an external calibration scan or as an autocalibration region integrated into an under-sampled acquisition. The sensitivity profiles of each coil, together with an unmodulated image common to all coils, are defined as penalised-maximum-likelihood parameters of a generative model of the calibration data. The model incorporates a smoothness constraint for the sensitivities and is efficiently inverted using Gauss-Newton optimization. Using both simulated and acquired data, we demonstrate that this approach can successfully estimate complex coil sensitivities over the full (FOV), and subsequently be used to unfold aliased images.

Let us assume that each acquired image $$$\mathbf{x}_n\in\mathbb{C}^K$$$ is obtained from a true magnetization $$$\mathbf{r}\in\mathbb{C}^K$$$ multiplied by a sensitivity field $$$\mathbf{s}_n\in\mathbb{C}^K$$$ plus some Gaussian noise. Each sensitivity is assumed non-null and is therefore written as the exponential of another complex field: $$$\mathbf{s}_n=\exp\left(\mathbf{c}_n\right)$$$. The real part of $$$\mathbf{c}_n$$$ is the sensitivity’s log-magnitude ($$$\mathrm{Re}\left\{\mathbf{c}_n\right\}=\ln\left|\mathbf{s}_n\right|$$$); the imaginary part is the sensitivity’s phase ($$$\mathrm{Im}\left\{\mathbf{c}_n\right\}=\arg\left(\mathbf{s}_n\right)$$$). Smoothness is introduced by assuming that log-fields stem from a multivariate Gaussian with precision $$$\alpha_n\mathbf{L}$$$. This matrix is chosen to penalize sensitivities with high bending energy, under Neumann’s boundary conditions^{4}. Finally, we incorporate noise correlation between coils with precision $$$\boldsymbol{\Lambda}$$$. Voxels are indexed by $$$k$$$, while coil-wise images and sensitivities at voxel $$$k$$$ are given by $$$\mathbf{x}_k\in\mathbb{C}^N$$$ and $$$\mathbf{s}_k\in\mathbb{C}^N$$$ respectively. The negative log-likelihood of the model is:

$$\mathcal{L}=\frac{1}{2}\sum_{k=1}^K\left(\mathbf{s}_kr_k-\mathbf{x}_k\right)^{H}\boldsymbol{\Lambda}\left(\mathbf{s}_kr_k-\mathbf{x}_k\right)+\frac{1}{2}\sum_{n=1}^N\alpha_n\mathbf{c}_n^{H}\mathbf{L}\mathbf{c}_n+\mathrm{constant}.$$

Differentiating with respect to the magnetization distribution yields a closed-form update:

$$r_k=\frac{\mathbf{b}_k^H\boldsymbol{\Lambda}\mathbf{x}_k}{\mathbf{b}_k^H\boldsymbol{\Lambda}\mathbf{b}_k}.$$

No closed-form solution exists for log-sensitivities, leading us to use a Gauss-Newton (GN) update scheme. The complex Hessian and gradient of the log-likelihood data term with respect to $$$\mathbf{c}_n$$$ have as many elements as voxels, with values:

$$g_{nk}=\frac{\partial\mathcal{L}}{{\partial}c_{nk}}=\frac{1}{2}\left({r_k}{r_k^\star}\right)s_{nk}\boldsymbol{\Lambda}_n\mathbf{s}_k^\star-\frac{1}{2}{r_k}{s_{nk}}\boldsymbol{\Lambda}_n\mathbf{x}_k^\star,$$

$$h_{nk}=\frac{\partial^2\mathcal{L}}{{\partial}c_{nk}^\star{\partial}c_{nk}}=\frac{1}{2}\Lambda_{nn}\left({r_k}{r_k^\star}\right)\left(s_{nk}s_{nk}^\star\right).$$

Gradient and Hessian with respect to the real and imaginary parts can be obtained from their complex equivalent^{5}. We solve the GN inversion step, of the form $$$\left(\mathbf{H}+\alpha_n\mathbf{L}\right)^{-1}\left(\mathbf{g}+\alpha_n\mathbf{L}\mathbf{c}_n\right)$$$, by full-multigrid^{4}. Since $$$\mathbf{r}$$$ is an exponentiated barycentre of $$$\mathbf{x}_{1{\dots}N}$$$, it can be shown that $$$\sum_n{\alpha_n}\mathbf{c}_n=\mathbf{0}$$$, which we enforce after each GN iteration.

To validate our method, we applied it to a phantom image acquired with a 16-element-array-coil^{6}. We compared our sensitivity estimates with fields obtained by smoothing the coil images and dividing them with their sum-of-squares. The noise covariance was assumed diagonal and its coil-specific elements were estimated by fitting a Rician mixture to each magnitude image.To evaluate the algorithm’s accuracy, we estimated sensitivity fields from the central autocalibration region of a 4-fold accelerated dataset acquired with a 64-element-array-coil. The sensitivities were then used to unfold the aliased images.

- Pruessmann, K. P., Weiger, M., Scheidegger, M. B., & Boesiger, P. (1999). SENSE: sensitivity encoding for fast MRI. Magnetic resonance in medicine, 42(5), 952-962.ISO 690
- Griswold, M. A., Jakob, P. M., Heidemann, R. M., Nittka, M., Jellus, V., Wang, J., ... & Haase, A. (2002). Generalized autocalibrating partially parallel acquisitions (GRAPPA). Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine, 47(6), 1202-1210.ISO 690
- Uecker, M., Lai, P., Murphy, M. J., Virtue, P., Elad, M., Pauly, J. M., ... & Lustig, M. (2014). ESPIRiT—an eigenvalue approach to autocalibrating parallel MRI: where SENSE meets GRAPPA. Magnetic resonance in medicine, 71(3), 990-1001.
- Ashburner, J. (2007). A fast diffeomorphic image registration algorithm. Neuroimage, 38(1), 95-113.ISO 690
- Sorber, L., Barel, M. V., & Lathauwer, L. D. (2012). Unconstrained optimization of real functions in complex variables. SIAM Journal on Optimization, 22(3), 879-898.ISO 690
- Haldar, J. P.. Real MRI Dataset Samples. Retrieved from https://mr.usc.edu/download/data/.