More MR Fingerprinting

1. More MR Fingerprinting

2. Key Concepts • Traditional parameter mapping has revolved around fitting signal equations to data with tractable analytical forms • mcDESPOT is perhaps among the more complicated but still uses a multi-component matrix exponential model of the SPGR and SSFP signals • Generate unique signal time courses for each set of T1/T2/M0/B0 parameters • Vary the free variables in a bSSFP sequence (TR and flip angle) with inversions every 200 TRs • The resulting signal can be numerically found via Bloch simulation

3. Key Concepts • Use a dictionary and good lookup scheme to fit the acquired data • The reconstruction cost of using complex signal models is often that the fitting becomes very expensive • In mcDESPOT, the matrix exponential equation is calculated thousands of times for every voxel • In MRF, the Bloch simulation would similarly have to be run many times to find a good fit • Instead, if the parameter space is explored beforehand, we can store a dictionary of signal evolutions • This frontend loads all the computation time • Any change in the model or pulse sequence would require a recomputation of the entire dictionary

4. Interpretation • To me, MRF is the generalization of parameter mapping from analytical equations to numerical simulations • This poses two new problems: • Excitation problem: what is the best choice of signal parameters to optimize parameter estimation? • “optimize” is obviously a loaded word here, but we want a sequence that is robust to system imperfections and fast • Reconstruction problem: how do we efficiently find the parameters from acquired data?

5. Excitation Problem • This is not well addressed by the MRF abstract and they default to a random choice of sequence variables • Most likely sub-optimal, resulting in long acquisition: 500 frames, 10min per slice • Success depends on whether TR and flip angle choice of bSSFP produce enough incoherence between different tissues (i.e. T1/T2/M0/B0 sets) • May have to generalize even further, with complete freedom in RF excitation and gradients to achieve reasonable times with a random approach • Especially if they expand the model to include diffusion or multi-component behavior

6. Excitation Problem • Could be seen as an optimization problem of the form: , • where T is the total time and Σ is the correlation matrix of the signal evolutions for various tissues, e.g. WM, GM, CSF, fat, lesion • The Frobenius norm is the root sum of squares of all the elements in a matrix • In other words, find the sequence that best reduces total scan time and correlation between the signal evolutions of different tissues • This is the general form, could of course constrain it to only choose variables within a bSSFP framework

7. Reconstruction Problem • Orthogonal Matching Pursuit is their dictionary lookup method of choice • Previously used by Donevaet al. in T1/T2 mapping from T1 Look-Locker and T2 spin echo data • OMP solves the following problem: • where D is the dictionary, s is a set of weights for the over-complete basis functions in D, and K is the sparsity

8. Orthogonal Matching Pursuit • Essentially a CS reconstruction: the signal evolution is sparse in the dictionary space (ideally it’s only one entry) • Find a K-sparse representation that best matches the data • May be strange to think about but T1/T2/M0/B0 maps are a way to compress the acquired data set by using knowledge of the signal behavior • Matching Pursuit works by successively adding the most correlated entries from the dictionary with each iteration • Given a fixed dictionary, first find the one entry that has the biggest inner product with the signal • Then subtract the contribution due to that, and repeat the process until the signal is satisfactorily decomposed. • OMP is a refinement of this process that gives it additional useful properties

9. Orthogonal Matching Pursuit • Properties: • For random linear measurements, requires O(K ln N) samples – not sure how this applies to MRF • For any finite size dictionary N, converges to projection onto span of D within N iterations • After any n iterations, gives the optimal approximation for that subset of the dictionary • Fast convergence, within K iterations • Applicable to any dictionary scheme • mcDESPOT could benefit from this

10. Spatial Acceleration • Random spatial encoding like in CS can also be utilized in this framework • Doneva et al. achieve such acceleration by taking advantage of the robustness of OMP • Sampling incoherently in the spatial domain produces noise-like interference in the images, OMP can still fit well through this noise • Of course sampling pattern must change between frames • Can think of it like the inherent denoising in CS reconstruction • Results would be improved by including a spatial constraint on sparsity in a transform domain • Is the wavelet domain sparse for their sorts of images? Probably. • Parallel imaging also provides information

11. Temporal Acceleration • Typical temporal acceleration is achieved by exploiting smoothness in the temporal direction • This is the case for both dynamic imaging and parameter mapping • This seems tricky with a random pulse sequence • If the signal evolution appears random, then this is inherently very hard to compress • May be an advantage of a better solution to the Excitation problem • Alternatively, perhaps random time course can achieve good results in shorter time than a solution that enforces smoothness and the net speed ends up being similar (seems likely)