Physics Model-based Autoencoding for Magnetic Resonance Fingerprinting
Authors: Juyeon Heo
Pingfan Song
Weiyang Liu
Adrian Weller
Venue: Submitted to ICLR 2023 (2023)
Keywords: Magnetic Resonance Fingerprinting (MRF)
Physics based representation learning
Medical imaging
Anti-causal representation learning
Quantitative Magnetic Resonance Imaging (QMRI)

Abstract: Magnetic Resonance Fingerprinting (MRF) is a promising paradigm to achieve fast quantitative Magnetic Resonance Imaging (QMRI). However, current MRF methods suffer from slow imaging speeds and poor generalization performance on radio frequency pulse sequences generated with varied settings. To address this challenging task, we propose a novel model-based MRF method that learns better representations by integrating a fast and differentiable MRI physics model as causal regularization. The proposed approach adopts a supervised auto-encoder framework consisting of an encoder and a decoder, where the encoder predicts the target tissue properties (anti-causal task) and the decoder reconstructs the inputs (causal task). Specifically, the encoder embeds high-dimensional MRF time sequences to a low-dimensional tissue property space, while the decoder exploits an MRI physics model to reconstruct the input signals using the estimated tissue properties and associated MRI settings. The causal regularization induced by the decoder improves the generalization performance and uniform stability of the approach, leading to the best performance on tissue property estimation, outperforming state-of-the-art competing methods.

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PHYSICS-BASED DECODING IMPROVES MAGNETIC RESONANCE FINGERPRINTING
Anonymous authors Paper under double-blind review
ABSTRACT
Magnetic Resonance Fingerprinting (MRF) is a promising paradigm to perform fast quantitative Magnetic Resonance Imaging (QMRI). However, existing MRF methods suffer from slow imaging speeds and poor generalization performance on radio frequency pulse sequences generated in various scenarios. To address these issues, we propose a novel MRI physics-informed learning approach for MRE. The proposed approach adopts a supervised encoder-decoder framework, where the encoder predicts the target tissue properties and the decoder reconstructs the inputs using the MRF physics. Specifically, the encoder embeds high bigh-dimensional MRF time sequences into low-dimensional tissue property space, while the de- coder exploits an MRI physics model to reconstruct the input signals using the predicted tissue properties and associated MRI settings. It allows to learn better representations by integrating a fast and differentiable MRI physics model as the physics-infon -nnfois- regularization. The thysics-based decoder improves the gener- alization performance and uniform stability by a considerable margin in practical out-of-distribution settings. Extensive experiments verified the effectiveness of the proposed physics-based decoding and achieved state-of-the-art performance tissue property estimation.
INTRODUCTION
Quantitative Magnetic Resonance Imaging (QMRI) is used to identify tissue's intrinsic properties, such the spin-lattice magnetic relaxation time (T1), the spin-spin magnetic relaxation time (T2), and other physical properties. Compared to conventional weighted (qualitative) MRI that focuses on tissue's contrast brightness and darkness, QMRI reveals tissue's 's intrinsic properties with quan- titative values and associated physical interpretations. Since different tissues are characterized by their distinct properties values, QMRI shows great potential to reduce subjectivity. with advan- tages in many areas including diagnosis, tissue characterization, investigation of disease pathologies, etc. [3, 30, 61].
Magnetic Resonance Fingerprinting (MRF) provides alternative QMRI framework achieve multi-property quantification simultaneously [43]. Given a pseudo-random radio frequency (RF) pulse sequence, distinct magnetic response a.k.a. fingerprint, signature, signal evolution from each specific tissue is observed and then used to predict the target tissue properties. Therefore, multi-property quantification boils down an inverse problem that aims to infer underlying tissue properties from the magnetic responses.
Various approaches have been developed to solve the MRF problem, using model-based techniques, e.g. dictionary matching (DM), compressive sensing, as well as learning-based / data-driven tech- niques [6, 9, 11, 15, 16, 18, 25, 38, 43, 46, 47, 50, 55, 57, 58]. spite of good performance in particular situations, they rarely take the MRI dynamics into consideration. This can cause reduced robustness and generalizability to potential data shifts occurred in practical scenarios with serious negative consequences. For example, the TI and T2 value range and distribution are patient-specific and subject to pathological tissue types, development phase and other factors, which may cause label shift. In addition, specific RF settings often be applied different situations, hospitals, and MRI instruments, MRF models are naturally expected to be able to handle such varied cases and be generalized different RF settings. Motivated by these issues, we aim develop new MRF

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Embedding loss Reconstruction loss
| 8.1| J|x X.1
Update
Bloch Decader B
Encodoe E
magnetic responses
prodiction
propeity
Quantitative T1, é x
Response
Reconstruced magnetic responses
X F ON
E(X,)
B(O,.. (9)
e cx
Figure I: Diagram of the proposed BlochNet, physics- -based encoder-decoder for MRF. BlochNet adopts supervised encoder-decoder framework where the encoder solves an inverse problem that predicts tissue properties from input magnetic responses while the decoder leverages Bloch equa- tion based MRI physics model to reconstructs the input responses from the estimated tissue prop- erties. Such design helps the encoder capture generalizable mapping effectively with the aid of physics-based feedback from the Bloch decoder.
approach that combines the benefits of both model-based and learning- baaning-based techniques achieve superior robustness and generalizability.
In this work, we propose physics- model, called BlochNet, to learn generalizable representa- tions for MRF, as shown in Fig. 1. BlochNet adopts a supervised encoder-decoder framework where the encoder solves the inverse problem that predicts tissue properties from input magnetic responses, while the decoder leverages Bloch equation based MRI physics model to reconstructs the input re- sponses from the estimated tissue properties. The Bloch equations, a.k.a. equations of motion of nuclear magnetization, are a set ordinary differential equations that model the magnetization dy- function of relaxation times T1 and T2 [4].
namies and calculate the nuclear magnetization They enable the formulation of the magnetic response tissue with specific intrinsic properties T1, T2 under varied magnetic field (we provide more details in Section 3.3 and Appendix). The physics-basee decoder acts as causal regularization to perform supervised reconstruction to the input magnetic responses from estimated tissue properties as well as the associated pscudo-random excitation pulse sequence, such repetition time (TR), time of echo (TE), and radio frequency flip angle (FA) over time. The rationale underlying the design is that domain knowledge such as physics principles can help to reduce the solution space of inverse problem by applying addi- tional constraints. This contributes to finding a better solution for an (ill-posed) inverse problem [31, 35, 41, 45]. Our results verify that the proposed model exhibits improved generalization perfor- mance from synthetic to real MRF data in two different out-of-distribun the test data is generated from seen and unseen RF pulses in the first and second setting, respectively.
(OOD) settings, where
Our major contributions include:
Our proposed method BlochNet is the first approach incorporate physics-informed learning to solve MRF problems, where the Bloch equation- based MRI physics model serves as the decoder and guides the encoder to learn underlying robust representations by providing useful feedback for searching possible tissue properties.
Compared to earlier methods, BlochNet shows consistently better generalization performance across synthetic, phantom and real MRF data, and across different types of RF pulse sequences. This demonstrates well the benefits of physics-based decoding in MRF in practical scenarios.
We provide fast and end-to-end solvable MRI physics model that can be used directly as a differentiable module in neural networks (e.g. g., acts decoder in BlochNet).
RELATED WORKS AND PRELIMINARIES
2.1 MODEL-BASED AND LEARNING-BASEE MRF APPROACHES
As tissue properties result in magnetic responses through the MRI dynamics, quantifying tissue's 's properties via QMRI/MR is typical anti-causal task. The core idea of MRF is based fact that
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for each specific tissue, a pseudo-random pulse sequence leads to an unique magnetic response (i.e. magnetizition along the temporal dimension) which can serve as an identifiable signal signature, analogue to a "fingerprint", for the corresponding tissue. Once the magnetic responses are obtained, estimation of tissue properties from responses reduces to a pattern recognition problem. In the orig- inal MRF work [43], this is addressed via dictionary matching (DM) which finds the best matching entry in pre-computed dictionary for each inquiry magnetic response. Accordingly, the best match- ing dictionary entry leads to multiple tissue properties directly via look-up-table (LUT) operation. More specifically, the pre-computed dictionary is composed of a number of magnetic responses for a variety of tissues characterized by the values of their intrinsic properties, such as TI, T2 relaxation times, etc. In this way, each dictionary entry is associated with specific tissue and its proper- ties. Thus, once the best matching entry (i.e. most correlated element with respect to the enquiry in terms of their inner product, a.k.a. f distance) is found, it directly leads to multiple properties simultancously through a LUT. However, high computation and storage burden makes DM-based approaches prohibitively time-consuming and values of tissue properties increases, because the size of the dictionary and lookup table increases exponentially accordingly. To alleviate such drawbacks, other model-based MRF approaches were proposed to improve the speed, incorporate additional useful priors, reduce the computational com- plexity [6. 11, 38, 46, 47, 58].
when the number of types and
Model-based MRF methods tend to suffer from burdens of storing the (enormous) dictionary and computational overhead. To address these shortcomings, learning-based approaches have been proposed for fast MRF by climinating/replacing the dictionary by a compact neural net- work [8. 9, 15, 16, 18, 20, 25, 50, 55, 57]. In particular, motivated by the success of deep learning [19, 36] in a number of tasks [13, 17, 21, 22, 24, 32, 34), there is an emerging trend [8, 9, 15, 16, 18, 20, 25, 50, 55, 57] that suggests to use a trained neural network as an al- ternative substitute for the MRF dictionary and LUT so that the time-consuming dictionary match- ing operation can be eliminated and replaced by an efficient inference through a trained network. In fact, well designed and tuned neural network is capable of approximating arbitrarily complex functions, therefore should also be able to approximate the response- to to-property mapping function. Although these learning-bass -baee MRF approaches demonstrated better performance in terms of speed and accuracy in comparison with model-based variants, they also exhibit some limitations, such as degraded robustness and generalizability various data shifts and duttof-distiibution (OOD) data samples. Moreover, most of them are designed empirically without taking into account the MRI physics underlying the imaging process.
2.2 PHYYSICS LEARNING
Another highly related line of research is physics-infrmmed machine learning, a.k.a. model-based deep learning [29, 49, 54, 56] where a physics-based prior is embedded in the learning process. One of the typical examples is physics-informed neural networks (PINNs) [51] which are usually trained with both a data fitting loss and a regularization loss induced by physical priors, such as PDE, sparsity, low-rank, etc. Combining physics prior in designing neural networks was shown to be beneficial to a broad range of applications [5. 14, 26, 44, 48]. PINNs are able to generalize well with limited number of training data. In a similar spirit, we aim to incorporate the physics model that describes the MRI process into our neural networks to improve its data efficiency and generalization.
Physics-informed/ model-based learning also appears in medical imaging domain [2. 7, 8, 27, 61, 62]. This provides a promising path to incorporate domain/expert knowledge with learned priors, and further combine the benefits of model-based methods with learning- methods to achieve improved interpretability, transparency, and robustness. [8] proposes incorporate the Bloch dy- namics in an encoder-decoded framework. However, the decoder in their work, is actually learned network rather than an MR physics model. Actually, the decoder just simulates and approximates the Bloch equations, instead of leveraging the exact physics model as in our case. Therefore, [8] is less physics-oriented than our proposed approach in essence. Reference [28, 37J focus on how to design better scan parameters to obtain RF pulse sequences with higher sensitivity and effectiveness. To this end, it is formulated as an optimization problem that involves automatic differentiation of the Bloch equation. We admit that RF sequence optimization is very important for MRF. However, designing more sensitive and effective RF sequences is beyond the scope of our manuscript, as our goal is to improve the performance, efficiency, and robustness of mapping magnetic response to
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tissue properties. [33] applies the analytic form of Bloch equations to generate synthetic qualitative MR images. Even though they include a physics model in the training procedure, they only consider one conventional RF pulse sequence setting (a fixed TE/TR/FA value) where simple analytic form of Bloch equations exists. In our work, various RF settings are applied where complex non-analytic forms of Bloch equations are exploited, and therefore the encoder could capture generalizable rep- resentations from diverse magnetic responses, for example, from different hospitals. In addition, the application scenario of their work is in the synthesis of weighted MR contrasts, i.e. qualitative MRI domain, whereas our work focuses on quantitative MRI, in particular MRF problem. [42] proposes supervised learning framework to generate MR images based on differentiable Bloch equation sim- ulations from tissue parameters to be close to the target image contrast, such as shorter scan times, or higher SNR. [52] implements the fast EPG code and optimizes tissue parameters by repeating back propagation several times, leading to a much longer inference time than learning-based MRF methods. Our method, however, has the same inference time as deep neural network- based MRF methods since the Bloch equations are used only in the training time to provide additional useful feedback for training the encoder.
2.3 MRF PROBLEM SETTING
Generation model (Bloch equations) The data generation model of producing magnetic responses from tissue properties is based on the MRI physics model formulated by the Bloch equations [4]- Given the RF pulse sequence whose parameter setting & {FA, TR, TE} consists of flip angles F E CL, repetition times T' E IR_ and echo times T'L E R² across L time points, the temporal signal evolution Xn E C for each individual voxel n is associated with tissue properties such as }. O, = (Tl,, T2,, through the Bloch differential equations B(O,): RP - C=,p=2.
X, B(O, ())
Vn 1.........
The Bloch equations B, shown as Equation equation are composed of of linear ordinary differential equations that represent a mapping from per-voxel intrinsic tissue properties to the cor- responding temporal signal evolution that records the magnetization response of proton dipoles to dynamic excitations induced by the RF sequence.
d.M. 7
May T2
M.
Mo T1
M
N7 x
M x
M,/T2 /T2
[M, - Mo)/T1
where M = [Ma, My, Mz]T is magnetization with May and M2 as the transverse and longitudinal components, respectively. Mo is the equilibrium magnetization; B is the magnetic field; 7 is the gyromagnetic ratio. Note that X,, May. meaning only the transverse component of magneti- zation is read out and used to infer tissue parameters. (see Appendix for more details about Bloch equations and their differentiabilityy)
Inverse model Given the magnetic response X,, E CL for the n-th voxel, the inverse process aims address an inverse problem that maps the response back the corresponding tissue properties O,,
+, =((m) Vn E L......,
where g denotes the inverse mapping function. Note that, estimation of tissue properties O from magnetic responses X requires long enough sequences L > to create unique signal evolutions that distinguish different tissues. Hence, the magnetic responses live low-dimensional (nonlinear) sub-maut monimoln of Ct.
3 PROPOSED METHOD
3.1 REPRESENTATION LEARNING FOR MRF
Here, present representation learning method to achieve fast and robust MRF, The proposed approach adopts a supervised encoder-decoder framework where the encoder predicts tissue proper- ties from input signatures, while the decoder reconstructs the inputs signatures from the estimated
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tissue properties based on MRI physics model. We highlight that sophisticated MRI physics model is tailored and exploited as the decoder which plays the role of causal regularization and help the encoder learn generalizable representations effectively. The rationale is based on the fact that representations that reflect the underlying data generation mechanism tend to exhibit stronger gen- eralization to out-domain distribution, and the domain-specifhe physics based knowledge help to reduce the solution space, thereby contributing to better solution [31, 35, 45].
Encoder
Encoder
O
Decader
P
Encoder
Bloch Equationn
(a) Previsous method
(b) Standard encoder-decoder
(c) BlochNet: Physics- encoder-decoder
Figure 2:
Two baseline methods and our BlochNet. BlochNet exploits physics-based decoder for helping the encoder learn generalizable representation.
3.2 ENCODER AND DECODER
In the proposed approach, the encoder uses three-layers fully connected neural network to predict TI, T2 parameters from fixed time-length input signatures. In contrast, the decoder leverages the Bloch equation based MRF physics model, acting as a physics-informed regularization, to perform supervised reconstruction to the input signatures from estimated tissue properties and the given RF pulse +.
Given an enquiry signature X, E CL for the n-th voxel, the encoder solves an inverse problem and outputs predicted tissue properties é, = (T'n, T2, This operation nonlinearly maps the input signature from high dimensional manifold low dimensional manifold.
θ = E(X,) Vn I....,N
Given the RF sequence settings t and the estimated tissue properties Ô.. the decoder reconstructs the input signature via solving Bloch equations using extended phase graph (EPG) formalism [23, 59].
X, = B()
Vn l....,.,
where B denotes the Bloch equation based decoder (see Appendix for more details).
3.3 FAST BLOCH DECODER BASED ON EFFICIENT EPG
Since there is general analytic solution the Bloch equations, numerical solutions such as EPG formalization are often adopted (see Appendix for details about EPG). However, a significant limi- tation of the released EPG code [59] is its slow computation speed in solving the Bloch equations. To circumvent this, recurrent neural network [40] and generative adversarial networks [60] have been applied as surrogates for the Bloch equation. However, these require lot of training data and still may generate inaccurate sequences on unseen tissue parameters and RF pulse settings due to overfitting. Instead, we adapt the EPG code [59] achieve a much more efficient implementation, making it practical to use the exact MRI physics model as decoder in the training procedure. A key change involves incorporating the torch jit package, and using batch- macch-wieeae computation for the Bloch stages, leading to 500 times faster generation of magnetic responses for 1,000 sequences on CPU. More details can be found in the appendix.
In experiment setting, properties including the longitudinal Tl and transverse T2 relaxation times, simultaneously encoded for each voxel. This setting could be further extended to include other properties, e.g. proton density P., off-resonance frequencies, T2*, diffusion and perfusion.
"Orr code, including the fast and end-to-end solvable EPG-Bloch code, is released on our GitHub repository [text left blank during peer review].
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3.4 LOSS FUNCTION
The loss function consists of two parts: the mean squared error (MSE) between the ground truth and the predicted tissue properties, referred embedding loss, and the MSE between the input and the reconstructed signatures, referred to as reconstruction loss,
Σ G1 - 0n11 + 111x, xnl1
N
EXPERIMENT RESULTS
In this section, perform evaluation on the proposed method and conduct comparison with other state-of-the-ar MRF methods. We evaluate the generalization performance of all models across different data distributions and different RF pulse sequences.
4.1
DATA SETTINGS
4.1.1
SYNTHETIC DATA
The synthetic signature data X is generated by solving the Bloch equations using the adapted fast EPG formulation for set of ground-truth tissue properties θ = (T1, given the fixed RF pulse sequences with setting +- [F A, TR, T } Specifically, the tissue properties θ are combinatorial pairs of 7'1 values that range from to 5000, and 72 values that range from to 2500, following the settings used in [47, 55]. We also perform normalization for each signal evolution X,, remove the impact the proton density which plays the role of a scale factor front the Bloch equations. This prevents the encoder from predicting tissue parameters based solely on magnitude sequence instead of intrinsic properties of sequence. Every flip angle FA was constrained to be in the interval o to 120 degree, and TE, TR in second-scale, Note that predicted tissue properties θ,, are in logarithmic scale and they are reversed back original scale before getting into the Bloch decoder.
Tabl decunice
(a) Three different Flip angles
(b) Magnetic responses
Figure 3: Flip angles of three RF pulse sequences, including FISP, Spline5, Spline Noisy, and corresponding magnetic responses generated using the Bloch equations for 16,384 data points(in different color).
4.1.2 PHANTOM MRI DATA
We exploit the fuzzy version of the brain phantom of the BrainWel Brain Database [10] construct a set of realistic, high-resolution Ti maps and T2 maps as ground truth, which faithfully exhibit spatial distribution for different tissue compositions, including CSF, Grey matter, White matter, etc. Then, we select a few slices from each pair of T1/T2 phantom volume across 10 subjects. After removing non-brain parts, these slices are then vectorized column vectors and stacked 2D matrix, leading to the ground truth tissue properties & E IR where N 85, 645 denotes the total number of voxels from all slices. Given O, signatures X NxL are generated using the Bloch equation- based MRI physics model. (More details are provided in the appendix.)
4.1.3 ANATOMICAL. MRI DATA
The anatomical data is from [47, 55]. Specifically, brain MRI scans were first acquired from healthy subject using GE Signa 3T HDXT scanner with Fast Imaging Employing Steady-state

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Acquisition (FIESTA) and Spoiled Gradient Recalled Acquisition in Steady State (SPGR) at four different flip angles (3", 5", 12", 20%). Then corrections [39] are implemented, followed by using DESPOTI and DESPOT2 algorithms [12] obtain T1, T2 maps of size of 128 128 for reference, leading to θ E NR with N 7, 499 voxels after removing non-brain parts. Then the signatures X are generated from the Bloch equations with FISP RF pulse sequence.
4.2 BASELINE METHODS
We compare approach with 6 representative state-of-the-arr MRF methods, including dictionary matching (DM) [43], Fully-connected deep neural network (FC) [9]. Hybrid deep learning (HY- DRA) [55] well two auto-encoder methods with RNN encoder and RNN decoder(RNN-RNNN) and FC encoder and FC decoder(FC-FCC) respectively. The first three models are trained with only embedding loss, whereas auto-encoder models have an additional reconstruction loss for the de- coder, apart from the embedding loss for the encoder. In constrast to the two auto-encoder models where the decoder is learned together with the encoder during training, our physics- baase model exploits the Bloch equation based decoder and keeps the decoder fixed, while only the encoder is updated during training. The idea use the decoder as causal regularization which gives recon- struction loss guide the encoder to find more generalizable representations than baseline models.
Dictionary matching
Autoencoder (FC-FC)
Autoencoder (RNN-RNN)
BlochNet (FC-Bloch)
FC
RNN
HYDRA
Phantom data Anatomical data
18.6652
0.0554
0.0486
0.1597
0.0519 0.0801
0.0443 0.0889
0.0409
19.4883
0.0812
0.092
0.3088
0.0748
Table Generalization performance across different data distributions: synthetic data for training while phantom (top row) and anatomical data (bottom row) for testing.
5000
PC
5000
Hydra
5000
Autbencoder(FC..E
5000
BlachNet(FCC
Ext mellon Reforance
4000
4000
4000
2000
2000
2000
2000
1000
2006 Neferrnce (ms)
Balerence (mm)
Reference (ma)
Relerence 11 (mml
(a) T1 gold standard and predicted values for four models.
500
FC
500
Hydra
500
Autoencoder1FCFF
500
BlochNee//FC)
300
300
200
200
200
200
200
100
Extimation
Delerence
100
500
200
300
450
1.0
200
500
300
400
Setereenc 12 (m))
Deleeence (m)))
Refference
Reeeeencce T2 (mm)
(b) T2 gold standard and predicted values for four models.
Figure 4: Comparison the generalization performance across different RF pulse sequences for 4 models. Blue line: gold-standard. Red dots: predicted values for tissue properties TI and T2.
Dictionary matching
Autoencoder (FC-FC)
Autoencoder (RNN-RNN)
BlochNet
FC
RNN
HYDRA
(FC-Bloch)
Phantom data Anatomical data
27.1955
0.8415
0.7973
0.3593 0.695
0.7018
0.6148
0.1574
16.6151
0.7305
1.06603
0.4081
0.9385
0.2667
Table Generalization performance across different RF pulse sequences: SplineNoisy!1 in training. FISP in testing.
Spline5 and
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4.3 EXPERIMENTS OF EVALUATING GENERALIZATION PERFORMANCE
We evaluate the generalization performance of various models on two types of experiment settings: 1) across different data distributions, including synthetic, phantom and anatomical MRF data; 2) across different RF pulse sequences with different flip angles.
4.3.1 GENERALIZATION ACROSS DIFFERENT DATA DISTRIBUTIONS
In practice, acquiring anatomical MRF data with ground truth TI, T2 values is time-consuming and expensive. Due to limited labeled anatomical data, it is common practice to use a large amount of synthetic data to train models avoid potential overfitting, and then perform validation on anatom- ical data [6. 11, 18, 38, 46, 47, 55, 58J. Following the same routine, we perform model training on synthetic MRF data, followed by model testing on phantom and anatomical data, in order to evaluate the generalization performance of trained models across different data distributions. In this experiment, we perform model training on synthesized MRF data introduced in Section 4.1.1 for our model and baseline models, and then compare their performance on unseen phantom and anatomical data described in Section 4. 1.2 and 4.1.3.
Table 1 includes the mean squared error (MSE) between the ground truth and predicted tissue prop- erties for seven approaches on phantom and anatomical data. Note that these errors are computed in log-scale. As shown in the table, the dictionary-matchin because the pre-computed dictionary and LUT did not cover the OOD data samples that could be quite different from the already contained dictionary entries. This caused poor inner product based matching results, and restricted the generalization.
approach gave the worst performance,
Interestingly, the results show that the reconstruction loss provides benefits to between-data gen- eralization for autoencoder models(FC-FC or RNN-RNN), in comparison with non-autoencoder models(FC or RNN), respectively, on both phantom and anatomical MRF data. Furthermore, our BlochNet outperforms all other models, indicating that reconstruction loss from the physics-based decoder has the best regularization effect that contributes to improved encoder training. In particular, when the physics- based decoder reconstructs input signatures from estimated tissue properties using Bloch equations, the corresponding reconstruction loss serves as additional guidance and constraint for the encoder to effectively capture the underlying anti-causal mapping. This in turn leads to better generalization performance.
Figure 5 shows the predicted tissue properties using various models anatomical MRF data. All models perform well in the middle range of tissue properties and lead to small errors. However, each individual models show different prediction characteristics. Specifically, HYDRA, shown in the third column of 5, suffers from a higher loss at the rim region of the brain, and leads to larger errors than other models. Autoencoder(FC-FC) model, shown in the forth column, demonstrates a better prediction of T2 values than the non-autoencoder(FC) model in the second column. The proposed BlochNet outperforms other comparison models with the least prediction error and most stable performance across the whole range of both TI and T2 values, as shown in the fifth column.
4.3.2 GENERALIZATION ACROSS DIFFERENT RF PULSE SEQUENCES
In this experiment, we perform model training on one RF pulse sequence and evaluate the trained models on another different RF pulse sequence. Specifically, we adopted 3 different RF pulse se- quences, including FISP [43], Spline5 [40], Spline Figure 3a. (More details are provided in Appendix.) FISP is used exclusively in the testing stage, while Spline5 and Splinel Noisy are used exclusively in the training stage. Under such settings, the performance of our BlochNet and other six models is compared in Table 2.
[40] with their flip angles shown in
Both Table 2 show that the embedding loss for all models are much higher than Table regard- less of phantom and anatomical MRF data. This is because the signatures generated with Spline5 and Spline I INoisy RF pulse sequences have notable difference from signatures for FISP, as shown in Figure 3b, which are particularly challenging cases. In spite of degraded performance for all models, the results clearly show the advantage of autoencoder (FC-FC or RNN-RNN) models over non-autoencoder MFCels(FC or RNN), which confirms the benefits of incorporating decoder to derive the reconstruction loss as additional regularization. Furthermore, the proposed BlochNet demonstrates significant gains over the competing methods in such challenging cases on both phan-
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T1 gold-standard
FC
Hydra
300
Autoencader(FC-F[]
BlochNee
4000
200
200
200
200
2002
150
150
159
190
2090
1000
(a) hoaddstandard T1 and errors between gold-stand aalddstandard and predicted T1 values of four models.
Hydra
Autoencoder(FC-FF)
BlochNeetF
2000
200
200
200
200
150
350
150
(b) stand-tandarr T2 errors between gold-standard and predicted values four models.
Figure 5: Generalization performance across different data distributions: data in training is syn- thetic while testing is anatomical MRI for four different models.
tom and anatomical MRF data. For example, the reconstruction loss baseline models is larger than 0.3 while our BlochNet gives 0.1574 phantom MRI data. In all these settings, approach leads to at least 50% smaller loss than the best competing approach.
These differences in MSE losses shown in Figure 4. FC model (left) makes poor predictions both TI and T2 values with high variance, predicting different values even when the sequence is gen- erated from the same tissue properties. This happens when the model cannot infer tissue properties from input signatures generated from different combinations of Tl and T2 values. Autoencoder (FC- FC) model (third column) shows more aligned and better inferences with lower variance, but still has high deviation between predicted and gold-standard values. In comparison, our BlochNet outputs predictions that are closest to the gold-standard with the lowest error. This confirms the benefits of physies-based decoder that guides the encoder to learn the underlying anti-causal mechanism effectively.
CONCLUSION
We propose BlochNet, novel physics-informed learning model, to perform quantification of mul- tiple tissue properties from their magnetic responses, a challenging ill-posed inverse problem in quantitative MRI. By integrating the encoder-decoder framework with Bloch equation based MRI physics, an additional reconstruction error resulting from the Bloch decoder can be exploited to regu- larize the training of the encoder, and accordingly allows the well-train encoder to achieve improved multi-property quantification performance. Experiments demonstrate that method consistently outperforms competing methods with better robustness and generalizability.
In future work, will consider the k-space subsampling cireumstance and spatial information that may lead to faster and more efficient QMRI/MRF. We will also explore more varied RF settings, for example, varying TR, TE and FA, has been reported [3] that RF pulse sequence with multiple pseudo-random parameters may improve the identifiability of resulted magnetic responses.

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13


Reviews:
[{"review_id": "jk3o2D1CqO", "reviewer": "Reviewer_29qt", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The authors propose a \"auto-encoder\" for MR Fingerprinting, which is an imaging method in magnetic resonance that captures multiple tissue properties at once (opposed to the \"standard\" T1 or T2 imaging, which captures a single tissue property, or their weighted variants T1w and T2w which only capture relative weightings of that property). The standard physical model taking these properties to the observed signal are the Bloch Equations, a system of ODEs.\n\nThe authors propose to learn an encoder, which (optimally) reconstructs the tissue properties from the timeseries observed measurements, and use the extended phase graph (EPG) method to approximate the forward model (Bloch Eq.s).\n\nThey then present results of various baselines (the standard dictionary method, and several deep architectures) on both a phantom and anatomical data (human brain).", "summary_of_the_review": "I think this work is significant and novel, and important for the deep-learning/MRI sub-field, but I think it requires a lot more description of the exact construction of the deep learning method to be publication quality and/or reproducible. It is a good idea, and has the pieces of a good manuscript, but missing details in section 3 and inconsistency with section 4 are strong negatives.\n\nGiven a revision fixing these problems (or an explanation of what I have missed/misunderstood), I am willing to change my recommendation.", "strength_and_weaknesses": "Strengths:\n* This method demonstrates an excellent example of physical models being incorporated into deep architectures, and, conversely, deep architectures informing the estimation of real physical properties.  \n* The experimental results are promising, especially in the case of cross-sequence learning.  \n* This, in my opinion, is an excellent use case for deep learning in MRI, since the standard fingerprinting already uses a dictionary method (i.e., is already as a field conditioned on some form of learning), and since the inverse problem here is relatively stiff for some parameters/playbacks.  \n\nWeaknesses (extended comments in Clarity section):\n* The authors do not describe the EPG method of reconstruction or its use. Even considering appendix A.1/A.2, usage is opaque. Since this portion is describing a critical portion of the method, care should taken so that this entire section is as explicit as possible.  \n* The same comment as above, but for the RNN(?) or FC network(?) used as an encoder. This section is completely opaque, except for the fact that some form of learning on the sequence of measurements has occurred. Moreover, section 3.0.2. implies that an RNN is used on the readout timeseries (of fixed length L), while Section 4 refers to BlochNet as (FC-Bloch).  ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "cazorRdb81-", "reviewer": "Reviewer_qrEB", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The authors propose BlochNet, an approach to reconstructing magnetic resonance fingerprinting (MRF) data. The approach takes the input images, solves for the parameter maps through an inverse problem, and then re-encodes the parameters into multi-contrast images. The encoder is implemented using a neural network and the decoder is implemented using the extended phase graph algorithm. Comparisons are made to other MRF reconstruction techniques across different RF pulse schedule. \n", "summary_of_the_review": "The authors propose an auto encoder for MRF based on NN (encoder) + Bloch equations (decoder). Though powerful, the approach is not new and the paper does not provide any novelty.\n", "strength_and_weaknesses": "Strengths:\n- The paper defined the problem well and makes the benefit of QMRI clear.\n- The paper proposes a solution to directly incorporating the Bloch equations into the reconstruction\n\nWeaknesses:\n- The paper incorrectly claims to be the first to incorporate the Bloch equations into the decoder. This is simply not true. See for example, [1-4]. In addition, the paper is not the first to accelerate EPG to make it usable as a decoder. See for example, [5]. It is unclear how the paper differs from these works, especially with respect to [1] and [5], except that these works can also be used to optimize the pulse sequence. The authors will need to explain how their approach is fundamentally different than these existing works.\n\n[1] Philip K Lee, Lauren E Watkins, Timothy I Anderson, Guido Buonincontri, Brian A Hargreaves, Flexible and efficient optimization of quantitative sequences using automatic differentiation of Bloch simulations, Magnetic resonance in medicine 2019\n[2] Beomgu Kang, Byungjai Kim, HyunWook Park, Hye-Young Heo, Learning-based optimization of acquisition schedule for magnetization transfer contrast MR fingerprinting, NMR in Biomedicine 2021\n[3] A. Loktyushin, K. Herz, N. Dang, F. Glang, A. Deshmane, S. Weinmüller, A. Doerfler, B. Schölkopf, K. Scheffler, M. Zaiss, MRzero - Automated discovery of MRI sequences using supervised learning, Magnetic resonance in medicine 2021\n[4] Evan Scope Crafts, Hengfa Lu, Huihui Ye, Lawrence L. Wald, Bo Zhao, An efficient approach to optimal experimental design for magnetic resonance fingerprinting with B-splines, Magnetic resonance in medicine 2022\n[5] Somnath Rakshit, Ke Wang, and Jonathan I Tamir, A GPU-accelerated Extended Phase Graph Algorithm for differentiable optimization and learning, ISMRM 2021\n\n- The paper throws around a lot of jargon surrounding causal and anti-causal mapping, but this makes it harder to read and interpret.\n- The paper is heavy in exposition but not heavy in description. This makes it difficult to discern the contribution vs. background material.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}, {"review_id": "b51-H8Zfe9U", "reviewer": "Reviewer_F4Mm", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper introduces a physics-based approach, named BlochNet, for magnetic resonance fingerprinting. Unlike existent works BlochNet directly incorporates Bloch equations in the decoder for improved generalizability. The experiments show improvement compared to conventional and learning-based approaches.", "summary_of_the_review": "Overall, the novelty is quite limited. motivation and experimental result does not align well. The novelty and experiments need to be improved for ICLR.", "strength_and_weaknesses": "Strengths:\n-  Including physics of the problem is  important for the medical imaging applications.\n- The paper experimental results generally align with earlier reports on the importance of physics-based models (Monga et al, SPM 2021)\n\nWeaknesses:\n\n1) Physics-based models are a well-studied field of work across different domains. The physics-based models are known to be more robust and generalizable compared to data-driven/learning-based approaches (Monga et al, SPM 2021). In fact, for MRI reconstruction tasks such as fastMRI challenge (Muckley et al, TMI 2021), physics-based models have been consistently preferred over data-driven/learning-based approaches. Thus, including the physics of the problem in the network architecture is not a novel contribution, but a straightforward extension of MRF.\n \n2) The generalization performance is questionable. While physics-based model improves the performance over learning-based models, they do not necessarily address the out-of-distribution challenges\n\n    a) Based on the findings in Table 1, auto-encoder models such as RNN-RNN do not suffer from the out-of-distribution challenge.  This somehow contradicts the motivation of the study which introduces physics-based modeling to improve the generalization performance. More concrete experiments must be done to show a clear advantage when facing different data distributions. \n\n   b) Figure 4 results show that BlochNet improves compared to other methods in presence of different RF-Pulse sequences, however, compared to reference it is still far off. Thus, it does not really solve the OOD challenge.\n\n3) Including the physics of the problem tends to have a counter impact on the inference time. A comparative study comparing the inference time is important as inference time plays an important role in medical imaging applications.\n\n4) Please revise the writing as there are repetitive statements (see the last paragraph of the section 2.2)  \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "HJKi5oJbEy", "reviewer": "Reviewer_oTWC", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper presents a deep learning based reconstruction for Magnetic Resonance Fingerprinting (MRF). The proposed approach based on a AutoEncoder scheme, where the decoder block is inspired by the physics. The so-called BlochNet allows to integrated priori knowledge on the physics driving the generation of MRI, and ensuring that the latent space respects the Bloch equations.", "summary_of_the_review": "The proposed approach seems interesting with the use of prior physics to drive the training of an autoencoder. However, the authors should discuss how their proposed approach is different form missing references, and compare their technique with this state of the art.", "strength_and_weaknesses": "Strengths:\n1.\tThe authors contribute to the field of MR fingerprinting by presenting a network that builds on MR physics models and demonstrates a promising generalization performance.\n2.\tProviding a fast implementation for solving Bloch equations appears to be an impactful contribution to the research community. Sharing the GitHub code improves the reproducibility of their work.\n3.\tA thorough evaluation of the BlochNet and multiple baseline comparisons was performed on three datasets.\n4.\tThey present appealing visualizations\n\nWeaknesses:\n1.\tThe literature review appears incomplete. An example is the work of Kim et al. who proposed a Bloch equation-based decoder in ‘Fat-saturated image generation from multi-contrast MRIs using generative adversarial networks with Bloch equation-based autoencoder regularization’ published in Medical Image Analysis, 2021. The authors should also check the recent contributions by Jesse Hamilton, especially “Deep learning reconstruction for cardiac magnetic resonance fingerprinting T1 and T2 mapping”, MRM, 2021. This brings questions to the stated novelty of the proposed work.\n2.\tThe discussion of causal and anti-causal mapping appears contextless\n3.\tThere are major grammar and format errors that need to be addressed (e.g. wordy sentences, misplaced citations, inconsistencies in the use of capital and small initial letters as well as blank spaces)\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject