Volkert Roeloffs^{1}, Martin Uecker^{2,3}, and Jens Frahm^{1,3}

Dictionaries as used in multi-parametric mapping are typically very large in size, take long to compute, and scale exponentially with the number of parameters. Here, we break the bond between dictionary size and representation accuracy by two modifications: First, we approximate the Bloch-response manifold by piece-wise linear functions, and second, we allow the sampling grid to be refined adaptively depending on the precision needed. Phantom and in vivo studies demonstrate efficient multi-parametric mapping with tiny dictionaries and subspace-constrained reconstruction. The presented method preserves accuracy and precision with dictionaries reduced in size by a factor of 10 and beyond.

Multi-parametric mapping has the potential to detect subtle disease effects earlier than conventional imaging. As traditional mapping methods are typically very time consuming, more efficient methods have recently been presented that break with simple signal models and employ more sophisticated excitation patterns^{1–5}. One way to deal with more complex signal responses is to generate lookup tables with signal prototypes (i.e. dictionaries). However, these dictionaries are typically very large in size, take long to compute, and scale exponentially with the number of parameters. Here, we present a new mapping approach based on piece-wise constant approximations with adaptive dictionary sampling that allows to reduce dictionary sizes by a factor of 10 and beyond.

We propose to break the bond between dictionary size and representation accuracy by two modifications: First, we approximate the Bloch-response manifold by piece-wise linear functions and consider the dictionary as a set of support points. As a consequence, mapping to the parametric domain becomes continuous rather than discretized by a chosen sampling grid. Second, we allow the sampling grid to be refined adaptively during the generation of the dictionary depending on the precision needed. To this end, an initial grid is iteratively refined in regions where the locally linear approximation is not accurate enough. More specifically, in the vicinity of reference position $$$x=(T_1,T_2)^{\top}$$$ the local linear approximation $$$Y - y \approx A(X-x)$$$ holds, where $$$A$$$ is the Jacobi matrix, $$$y$$$ the subspace representation and $$$X$$$ and $$$Y$$$ neighborhoods in parameter domain and subspace domain, respectively. The approximation error

$$ E(X)=\frac{||A(X-x)-(Y-y)||}{||Y||} $$

can be reduced by adding new sampling points $$$X_{\text{new}}$$$ at positions closer to the reference position $$$x$$$. These points can be generated by simply shrinking the old neighborhood $$$X$$$ according to

$$X_{\text{new}}-x=\frac{1}{2}(X-x).$$

New neighbors are added until all approximation errors $$$E(X)$$$ are smaller than a defined threshold. illustrates the differences between the adaptively sampled and the heuristically sampled dictionary as proposed in Ma et al.^{1}. The number of entries can be reduced by more than a magnitude (tiny dictionaries). However, for the proposed manifold projection, the Jacobian matrix for each entry is stored additionally.

Sensitization of the response signal to T1 and T2 relaxation was realized by an Inversion Recovery Hybrid-State Free Precession (HSFP) experiment^{5} with a flip angle pattern optimized for maximal mapping efficiency. This flip angle pattern was implemented on a Siemens Magnetom Prisma with 2D Golden-Angle radial sampling. Imaging was performed with a spatial resolution of 1×1×5 mm^{3} in T_{ACQ}=4.3s. Signal time courses and corresponding gradients were computed using the analytic expression for HSFP and slice profile effects were taken into account explicitly. To further reduce dictionary size and to minimize noise amplification, we formulate the reconstruction as a subspace-constrained linear inverse problem. The subspace basis was determined by performing a singular value decomposition^{6–8} on the full adaptive dictionary and a subspace size of $$$K=4$$$ was chosen heuristically. Then, the following minimization problem was solved:

$$ x^* = \arg\min_x {\left\lVert y-\mathcal{P}_{\vec{k}}\mathcal{F}S\Phi_K x\right\rVert}_2^2 + \lambda R(x) $$

where $$$y$$$ denotes the radial raw data, $$$\mathcal{P}_{\vec{k}}$$$ the projection onto the sampled k-space trajectory, $$$\mathcal{F}$$$ the Fourier transform, $$$S$$$ multiplication with the (predetermined) coil sensitivity profiles, $$$\Phi_K$$$ the temporal basis, and $$$x$$$ the unknown subspace coefficients. Coil sensitivity profiles $$$S$$$ were predetermined using ESPIRIT^{9} and spatial correlations across subspace coefficients were exploited by a locally low rank regularizer^{8,10} $$$R$$$. As a last step, pixel-wise projection of the subspace coefficients $$$x$$$ onto the Bloch-response manifold was realized by first identifying the best matching linear patch by nearest neighbor search in the dictionary and subsequent projection to the plane spanned by the Jacobians.

1. Ma D, Gulani V, Seiberlich N, Liu K, Sunshine JL, Duerk JL, Griswold MA. Magnetic resonance fingerprinting. Nature 2013;495:187.

2. Buonincontri G, Sawiak SJ. MR fingerprinting with simultaneous b1 estimation. Magnetic resonance in medicine 2016;76:1127–1135. doi: 10.1002/mrm.26009

3. Assländer J, Lattanzi R, Sodickson DK, Cloos MA. Relaxation in spherical coordinates: Analysis and optimization of pseudo-ssfp based mr-fingerprinting. 2017.

4. Assländer J, Glaser SJ, Hennig J. Pseudo steady-state free precession for mr-fingerprinting. Magnetic resonance in medicine 2017;77:1151–1161. doi: 10.1002/mrm.26202

5. Assländer J, Novikov DS, Lattanzi R, Sodickson DK, Cloos MA. Hybrid-state free precession in nuclear magnetic resonance. arXiv preprint 2018:1–12.

6. Petzschner FH, Ponce IP, Blaimer M, Jakob PM, Breuer FA. Fast MR parameter mapping using k-t principal component analysis. Magnetic resonance in medicine 2011;66:706–716. doi: 10.1002/mrm.22826

7. McGivney DF, Pierre E, Ma D, Jiang Y, Saybasili H, Gulani V, Griswold MA. SVD Compression for Magnetic Resonance Fingerprinting in the Time Domain. IEEE Trans. Med. Img. 2014;33:2311–2322.

8. Tamir JI, Uecker M, Chen W, Lai P, Alley MT, Vasanawala SS, Lustig M. T2 shuffling: Sharp, multicontrast, volumetric fast spin-echo imaging. Magnetic Resonance in Medicine 2016;00:n/a–n/a. doi: 10.1002/mrm.26102

9. Uecker M, Lai P, Murphy MJ, Virtue P, Elad M, Pauly JM, Vasanawala SS, Lustig M. ESPIRiT — An Eigenvalue Approach to Autocalibrating Parallel MRI : Where SENSE Meets GRAPPA. 2014;1001:990–1001. doi: 10.1002/mrm.24751

10. Roeloffs V, Rosenzweig S, Holme HCM, Uecker M, Frahm J. Frequency-modulated SSFP with radial sampling and subspace reconstruction: A time-efficient alternative to phase-cycled bSSFP. Magnetic Resonance in Medicine 2018.

The proposed adaptive dictionary generation (left) leads to an automatically adjusted sampling density in T1-T2-landscape. The 209 dictionary entries are much less than the 3331 entries of the heuristically sampled dictionary (right).

Reconstructed subspace coefficients (magnitude, individually windowed)

T1, T2 (in seconds) and proton density maps reconstructed with the proposed adaptive dictionary + manifold projection (top) and the conventional heuristically sampled dictionary + nearest neighbor matching (bottom). Rightmost column shows obtained quantitative values (red) in comparison to Gold Standard measurements (black).

T1, T2 (in seconds) and proton density maps of human brain generated from the proposed adaptive dictionary with manifold projection. Resolution: 1×1×5 mm^{3}, T_{ACQ}: 4.3 s