More MR Fingerprinting

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

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

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?

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

Excitation Problem Could be seen as an optimization problem of the form: find 𝜌 𝑡 ,𝑔 𝑥 [𝑡], 𝑔 𝑦 [𝑡], 𝑔 𝑧 [𝑡] min. Σ 𝐹 +𝜆𝑇 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

Reconstruction Problem Orthogonal Matching Pursuit is their dictionary lookup method of choice Previously used by Doneva et al. in T1/T2 mapping from T1 Look-Locker and T2 spin echo data OMP solves the following problem: find 𝑥−𝐷𝑠 2 s.t. 𝑠 0 ≤𝐾 where D is the dictionary, s is a set of weights for the over-complete basis functions in D, and K is the sparsity

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

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

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

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)