Introduction

In Magnetic Resonance Fingerprinting (MRF),¹ parameter estimation accuracy is dominated by undersampling artifacts. While in classical relaxometry techniques, the omission of data always leads to a larger error, in fingerprinting, undersampling artifacts can lead to both an increase or a decrease in error. This “temporal encoding efficiency” can be analyzed by the change in matching error upon omission of a single time point (leave-one-out). To accelerate the analysis, we propose a first-order perturbation of the undersampling error approximation by Stolk and Sibrizzi.² The analysis is used to identify temporal sequence segments of primary parameter encoding and, based on the insight, to subsequently shorten an exemplary MRF sequence by truncation.

Methods

The leave-one-out (LOO) match$$$\;\mathbf{\hat{\theta}}_i^{LOO}\left(\mathbf{x}\right)\;$$$upon omission of time point$$$\;t_i\;$$$during matching is defined by$$\mathbf{\hat{\theta}}_i^{LOO}\left(\mathbf{x}\right)=\text{arg}\max_{\mathbf{\theta}}\sum_{n=1}^N\left|I_n\left(\mathbf{x}\right)-S\left(t_n;\mathbf{\theta}\right)\right|^2-\left|I_i\left(\mathbf{x}\right)-S\left(t_i;\mathbf{\theta}\right)\right|^2,\qquad\text{(Eq. 1)}$$where$$$\;N\;$$$is the number of time points in the MRF sequence,$$$\;I_n\left(\mathbf{x}\right)\;$$$the reconstructed images, and$$$\;S\left(t_i;\mathbf{\theta}\right)\;$$$the signal model for each acquisition time point$$$\;t_i\;$$$and parameter vector$$$\;\mathbf{\theta}\;$$$(here$$$\;\mathbf{\theta}=\left(\log{T1},\log{T2},\rho\right)$$$). As a measure for temporal encoding efficiency, we use the change in normalized root-mean-square error$$$\;\Delta{}nRMSE_i\;$$$between matching with the full sequence$$$\;\mathbf{\hat{\theta}}\left(\mathbf{x}\right)\;$$$and each LOO match$$$\;\mathbf{\hat{\theta}}_i^{LOO}\left(\mathbf{x}\right)\;$$$(Figure 1)$$\Delta{}nRMSE_i=\sqrt{\left\langle\left(\frac{\hat{\mathbf{\theta}}^{LOO}_i\left(\mathbf{x}\right)-\mathbf{\theta}^{GT}\left(\mathbf{x}\right)}{\mathbf{\theta}^{GT}\left(\mathbf{x}\right)}\right)^2\right\rangle_\mathbf{x}}-\sqrt{\left\langle\left(\frac{\hat{\mathbf{\theta}}\left(\mathbf{x}\right)-\mathbf{\theta}^{GT}\left(\mathbf{x}\right)}{\mathbf{\theta}^{GT}\left(\mathbf{x}\right)}\right)^2\right\rangle_\mathbf{x}},$$with ground truth parameters$$$\;\mathbf{\theta}^{GT}\left(\mathbf{x}\right)\;$$$and$$$\;\langle\cdots\rangle_\mathbf{x}\;$$$denoting the mean over all spatial positions$$$\;\mathbf{x}$$$.

Equation (1) can be solved by iterating over all points in the sequence and performing matching with an adapted dictionary (LOO-Match). Using the approach by Stolk & Sibrizzi,² the matching result$$$\;\hat{\mathbf{\theta}}\;$$$can be approximated to first order using the acquisition’s point-spread function (PSF), the signal$$$\;S\left(t;\mathbf{\theta}_0\right)$$$, and its derivatives$$$\;\mathbf{D}_\mathbf{\theta}S\left(t;\mathbf{\theta}_0\right)\;$$$at an expansion point$$$\;\mathbf{\theta}_0$$$. Equally, an approximate solution to (1) can be obtained for each omitted time point (LOO-Stolk).

Considering the LOO problem as a small perturbation to the overall least-squares problem, we can perform a formal perturbation expansion of$$$\;\mathbf{\hat{\theta}}_i^{LOO}\left(\mathbf{x}\right)\;$$$to first-order in$$$\;\epsilon\;$$$(LOO-Perturbation, LOOP) by making use of Stolk’s analytical approximation$$\mathbf{\hat{\theta}}_i^{LOO}\left(\mathbf{x}\right)=\text{arg}\max_{\mathbf{\theta}}\sum_{n=1}^N\left|I_n\left(\mathbf{x}\right)-S\left(t_n;\mathbf{\theta}\right)\right|^2-\epsilon\left|I_i\left(\mathbf{x}\right)-S\left(t_i;\mathbf{\theta}\right)\right|^2$$ $$\approx\hat{\mathbf{\theta}}^{Stolk}\left(\mathbf{x}\right)+\epsilon\Delta\hat{\mathbf{\theta}}^{LOOP}_i\left(\mathbf{x}\right)+O\left(\epsilon^2\right),$$defining$$\hat{\mathbf{\theta}}^{LOOP}_i\left(\mathbf{x}\right)=\hat{\mathbf{\theta}}^{Stolk}\left(\mathbf{x}\right)+\Delta\hat{\mathbf{\theta}}_i^{LOOP}\left(\mathbf{x}\right).$$

In the classical encoding regime, the Cramér-Rao lower bound is an estimator for the noise-dependent parameter encoding error. For comparison, we report its relative change alongside ΔnRMSE.

As an example, a FISP-MRF^3,4 sequence with constant repetition time of 15 ms was simulated using the extended phase graph formalism.⁵ For acquisition, two interleaves of a 40-fold undersampled Archimedean spiral (6 ms duration) were reconstructed on a 128x128 grid. The numerical phantoms were generated with the same resolution deliberately committing the “inverse crime”⁶ such that matching errors were not biased by downsampling errors. To evaluate the exact LOO-Match, a MRF dictionary was generated on a logarithmic T1/T2 scale with 0.5% T1 and 1% T2 resolution and 63 220 atoms. In each iteration, one dictionary element was omitted and the dictionary re-normalized.

To analyze the sequence encoding efficiency and its dependency on the sampled object, 20 brains from the Brain Web Database (www.bic.mni.mcgill.ca/brainweb/) were randomly sliced to form 128 slices with randomized T1/T2 values with a maximal parameter variation of 25% to remain within the linearizable signal regime.

Results & Discussion

Figure 2 shows the leave-one-out comparison of ΔnRMSE for the exact dictionary matching, Stolk’s approximation, and our perturbation approach using a checkerboard phantom (see Ref. 2). Calculation of the LOO analysis took 60 minutes using dictionary matching, 10 minutes using Stolk’s approximation and 5 seconds employing our perturbation expansion. Our perturbation approach is in very good agreement with Stolk’s approximation and follows closely the matching approach.

Figure 3 shows an animation of the statistical analysis that is generating the full LOO results shown in Figure 4a. Integrating Figure 4a over time (Figure 4b), allows to study cancellation effects between subsequent time points. Constant regions in Figure 4b can be interpreted as intervals of no or little parameter encoding. Colored areas denote changing regions and coincide with strong parameter encoding.

Compared to the CRB, ΔnRMSE can be both positive and negative, thus omission of time points during the matching process can be beneficial for removing undersampling artifacts from the final parameter maps. This is contrary to classical encoding regimes, here represented by the CRB, where omission of data always leads to an increase in matching error.

Using the LOO findings of Figure 4 it becomes apparent that encoding is sufficient for T1 after approximately 4 seconds, T2 after 5 seconds, and proton density after ~6 seconds. Hence, the sequence can be considerably shortened if small matching errors are tolerated. In Figure 5, dictionary matching results obtained after truncation of the sequence are shown. Truncation of the sequence from 7.5s per interleave to 5s results in an increase of 10% in nRMSE for T2 and proton density, while T1 remains unchanged. Further truncation of the sequence results in increasing reconstruction error, which is in qualitative agreement with our LOO-Perturbation analysis.

Conclusion

In this work, we have introduced a method to visualize parameter encoding in MR fingerprinting on a temporal basis and proposed a fast way to evaluate it using perturbation theory. The findings can be directly used to compare sequences, identify key encoding time points, and perform truncation in cases where encoding is prematurely finished.

Acknowledgements

No acknowledgement found.