Purpose

Limited human intuition of the Bloch equations’ nonlinear dynamics, particularly over long periods of non-steady-state time evolution or in regimes such as off-resonance excitation, is an obstacle to fully exploiting the vast parameter space of potential MR pulse sequences [1]. Model-based approaches such as optimal control and magnetic resonance fingerprinting [2,3] have been exploring computer-generated, non-intuitive sequences, but with typically limited roles as part of a larger canonical imaging sequence. Our previous work [4] introduced a computational graph approach to modeling the Bloch equations for efficient optimal control, and demonstrated generation of original pulse sequences with non-intuitive gradient waveforms to perform Fourier spatial encoding. In this work, we show the same core AUTOSEQ framework extended with a multilayer fully-connected neural network to perform fast quantitative MR parameter measurement. By employing continuous off-resonant excitation with simultaneous continuous receive, we demonstrate in simulated experiments the ability to quantify T1 and T2 parameters in a single TR, and expect this system to be extensible to other MR parameters of interest.

METHOD AND EXPERIMENTS

We model the Bloch equations with a discrete-time state-space model in the rotating frame, incorporating off-resonant RF excitation. This dynamic model is represented as a directed acyclic graph (DAG) (Figure 1A) with recurring discrete-time cells (similar to a recurrent neural network) in order to be efficiently optimized upon with autodifferentiation [5]. In this work, we chose to perform experiments of duration 1 second, with discrete timesteps of 1ms. The network was trained with simulated NMR samples on a grid of T1 from 100 to 1000 ms in 10ms increments, and T2 from 40 to 500 ms in 10ms increments. T2* was set to 50ms, and was implemented by summation of 16 isochromats experiencing various field in homogeneities. RF flip angle, phase, and off-resonance frequency for each timepoint n=1..1000 was initialized randomly at the start of training. The receive signal at each sample time point is concatenated and the signal vector is input into a four-layer fully-connected network with 4000 nodes per hidden layer (Figure 1B). This stage of the network is trained to map the signal vector to MR parameters, and is similar to the MR fingerprinting Deep Reconstruction Network (DRONE) [6] in that the universal approximation theorem is employed to perform a regression mapping of MR signal data to parameters. An important point to note is that the entire system – the pulse sequence Bloch simulation components and the regression network - is connected and differentiable and therefore able to be jointly optimized with stochastic gradient descent with a mean squared error loss with respect to the target MR parameters. The pulse sequence is optimized to provide the most relevant and discriminating signal to the regression network, which is optimized to accurately decode the MR parameters. In our simultaneous Tx-Rx setup, we transmit continuously during the experiment with the ADC is open (Figure 2D), and isolation of transmit and receive channels can be obtained by constraining the transmit RF frequency to be off-resonant over a certain frequency threshold (in our experiment, 100 Hz) through addition of a penalty in the optimization loss function (Figure 2C). After training, we validate our network on a set of T1 and T2 parameter pairs unseen during training. We show that a generated pulse sequence (Figure 2A-C) is able to perform T1 (Fig. 4A) and T2 (Fig. 4B) estimation. The reconstructed T1 and T2 values and their linear regressions showed excellent agreement with the ground truth. The discriminative ability of our system is highlighted in Fig. 3, which demonstrates improved signal separation with AUTOSEQ-generated pulse sequences compared with random pulse sequences. Furthermore, we demonstrate excellent noise robustness with a noise analysis using additive white gaussian noise (AWGN), showing low root mean squared error (RMSE) of estimated parameters across a wide range of SNR.

DISCUSSION

In addition to isolation of frequencies, decoupling of transmit and receive can be achieved through Tx-Rx circuitry and coil geometry design [8]. Previous parameter estimation approaches have demonstrated the advantage of crafting pulse sequences for particular timescales of parameter measurement [7]. Our method would allow for the automatic generation of such sequences, for arbitrary timescales, simply by selecting the appropriate training set or by weighting the loss function appropriately. Our future work includes other uses of more sophisticated loss functions that optimize over experimental considerations such as RF power, extension into other MR parameters such as diffusion coefficients, and ultimately into higher-dimensional applications such as measuring other spectroscopic parameters and imaging and with the inclusion of gradients.

Acknowledgements

B.Z. was supported by National Institutes of Health / National Institute of Biomedical Imaging and Bioengineering F32 Fellowship (EB022390).