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Joint T1 and T2 Mapping with Tiny Dictionaries and Subspace-Constrained Reconstruction1Biomed NMR, Max Planck Institute for Biophysical Chemistry, Goettingen, Germany, 2Institute for Diagnostic and Interventional Radiology, University Medical Center Göttingen, Goettingen, Germany, 3partner site Göttingen, German Centre for Cardiovascular Research (DZHK), Goettingen, Germany
Synopsis
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.
INTRODUCTION
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 patterns1–5. 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.
THEORY
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.
METHODS
Sensitization of the response signal to T1 and T2 relaxation was realized by an Inversion Recovery Hybrid-State Free Precession (HSFP) experiment5 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 mm3 in TACQ=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 decomposition6–8 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 ESPIRIT9 and spatial correlations across subspace coefficients were exploited by a locally low rank regularizer8,10 $$$R$$$. 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.
RESULTS
The subspace-constrained reconstruction yields a set of coefficient maps that represent the full temporal signal dynamics in the chosen basis (Figure 2). Quantitative maps (Figure 3) are generated from this subspace representation a) by projection onto the Bloch-response manifold and b) by following the matching approach1. A quantitative comparison reveals similar overall accuracy and precision of the two methods. Figure 4 shows T1, T2 and proton density maps of a transverse section of a human brain. The in vivo acquisition protocol and the reconstruction parameters are identical to the presented phantom study.DISCUSSION & CONCLUSION
We demonstrated efficient multi-parametric mapping with tiny dictionaries and subspace-constrained reconstruction. Approximating the Bloch-response manifold by piece-wise constant functions has the potential to overcome problems associated with large dictionaries in quantitative parametric mapping. The presented method preserves accuracy and precision and allows to efficiently map higher-dimensional signal representations to the parametric domain.Acknowledgements
The authors would like to thank Dr. Jakob Assländer for providing the optimized HSFP excitation pattern.References
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