Learning-based Optimization of Acquisition Schedule for Magnetization Transfer Contrast MR Fingerprinting

Abstract

Magnetization transfer contrast MR fingerprinting (MTC-MRF) is a novel quantitative imaging method that simultaneously quantifies free bulk water and semisolid macromolecule parameters using pseudo-randomized scan parameters. To improve acquisition efficiency and reconstruction accuracy, the optimization of MRF sequence design has been of recent interest in the MRF field, but has been challenging due to a large number of degrees of freedom to be optimized in the sequence. Herein, we propose a framework for learning-based optimization of the acquisition schedule (LOAS), which optimizes RF saturation-encoded MRF acquisitions with a minimal number of scan parameters for tissue parameter determination. In a supervised-learning framework, scan parameters were subsequently updated to minimize a pre-defined loss function that can directly represent tissue quantification errors. We evaluated the performance of the proposed approach with a numerical phantom and in in vivo experiments. For validation, MRF images were synthesized using the tissue parameters estimated from a fully connected neural network (FCNN) framework and compared with references. Our results showed that LOAS outperformed existing indirect optimization methods with regard to quantification accuracy and acquisition efficiency. The proposed LOAS method could be a powerful optimization tool in the design of MRF pulse sequences.

1. Introduction

Magnetization transfer contrast (MTC) experiments measure the transfer (or exchange) of magnetization from semisolid macromolecular protons to surrounding free bulk water molecules^1–5. While the effect of low-concentration macromolecular protons on the water resonance is not detectable after a single exchange, the cumulative effect of repeated saturation and exchange allows indirect assessment of the semisolid macromolecular protons through the free bulk water signal. MTC imaging offers great potential for the characterization of comprehensive tissue composition (e.g., myelin and mobile protein concentrations), and hence, could act as a biomarker for the clinical diagnosis of tissue disorders^6–9. Typically, the MTC effect has been measured by calculating the MT ratio (MTR), which reflects the difference between two images obtained with and without RF saturation of macromolecular protons; however, the ratio is dependent on the choice of scan parameters and tissue relaxation effects. The most promising MTC quantification currently requires fitting of MTC-weighted signals acquired from repeated and serial image acquisitions with various saturation powers and frequency offsets using the analytical solution of the Bloch equations^2,10–12. Although the model-based quantification approach can largely remove the dependence of experimental settings and separate MTC contributions from relaxation effects, it typically requires a long scan time and the post-processing for MTC quantification is computationally complex. In addition, the quantification accuracy is sensitive to fitting parameters, thus limiting its clinical utility.

Recently, a strategy called “MR fingerprinting (MRF)” was introduced as a novel acquisition and reconstruction approach to overcome time constraints¹³. Imaging scan parameters are intentionally changed for every acquisition to generate transient-state signal evolutions. Then, tissue parameters are quantified by comparing experimentally measured signal profiles with a simulated library of signal variation patterns based on the sequence used^14–17. However, the size of the library is dramatically increased when a multiple-exchange model is taken into account, and thus, the construction of a huge library and an exhaustive search are unavoidable^18–21. Moreover, a conventional MRF library with discrete values could lead to quantization errors due to the large dictionary step size. Recent advances in deep neural networks have opened a new possibility to solve general inverse problems in MRF reconstruction in an efficient manner and to produce reliable estimates of tissue parameters^22–24. A deep-learning architecture can be designed to quantify multiple tissue parameters by learning the mapping relation between fingerprints and tissue parameter spaces, which bypasses the exhaustive dictionary search. Recent studies have proposed combining MTC- and chemical exchange saturation transfer (CEST)-MRF with deep-learning techniques to improve the quantification accuracy and accelerate the imaging time^25–29.

To further accelerate data acquisition and improve accuracy, an optimal design of the MRF acquisition schedule is critical for efficient and accurate tissue parameter-mapping^20,30,31. However, optimization of the MRF schedule has been challenging due to a large number of degrees of freedom available in an MRF sequence. Most of the current methods are based on indirect measurements, such as maximizing the signal discrimination between tissue types³⁰, or maximizing the signal-to-noise ratio (SNR) efficiency using the constrained Cramer-Rao lower bound (CRLB)³¹. However, MRF schedules optimized by the aforementioned indirect metrics cannot guarantee the low reconstruction errors for each tissue parameter. In this study, to optimize RF saturation-encoded MRF acquisitions with a minimal number of saturation parameters for tissue parameter determination, we proposed a learning-based optimization framework. Unlike the optimization methods based on indirect measurements, the proposed approach can optimize scan parameters by directly computing quantitative errors in tissue parameters.

2. Theory

Transient-state signal model

The MTC-MRF signal model is based on modified Bloch equations for a two-pool exchange system (w: free bulk water pool, m: semisolid macromolecule pool). In the presence of MT exchange, the time evolution for longitudinal magnetization in the two pools during an RF saturation can be described with tissue parameters and scan parameters, such as RF saturation power (B₁), frequency offset (Ω), saturation time (Ts), and delay time (Td)^2,32:

$$d{M}_x^w/dt=-M_x^w/T_2^w-2\pi \Omega M_y^w$$ [1]

$$d{M}_x^m/dt=-M_x^m/T_2^m-2\pi \Omega M_y^m$$ [2]

$$d{M}_y^w/dt=2\pi \Omega M_x^w-M_y^w/T_2^w-{\omega }_1{M}_z^w$$ [3]

$$d{M}_y^m/dt=2\pi \Omega M_x^m-M_y^m/T_2^m-{\omega }_1{M}_z^m$$ [4]

$$d{M}_z^w/dt={\omega }_1{M}_y^w-(1/T_1^w+k_{wm})M_z^w+k_{mw}{M}_z^m+M_0^w/T_1^w$$ [5]

$$d{M}_z^m/dt={\omega }_1{M}_y^m-(1/T_1^m+k_{mw})M_z^m+k_{wm}{M}_z^w+M_0^m/T_1^m$$ [6]

where T₁ⁱ and T₂ⁱ are the longitudinal and transverse relaxation times of pool i, respectively; k_ij is the proton exchange rate from pool i to pool j; M₀ⁱ is the equilibrium magnetization of pool i; ω₁ is the RF saturation amplitude (= 2πγB₁); and γ is the gyromagnetic ratio. The analytical solution of time-dependent longitudinal magnetization of the free bulk water pool from Eqs. [1] – [6] can be written as:

$$M_z^w\left(t\right)=\left(M_0^w-M_{ss}^w\right)e^{\lambda t}+M_{ss}^w$$ [7]

where

$$M_{ss}^w=M_0^w\frac{\frac{1}{T_1^m}(k_{mw}{M}_0^m{T}_1^w)+\alpha }{(k_{mw}{M}_0^m{T}_1^w)\left(R_{rfm}+\frac{1}{T_1^m}\right)+\alpha \left[1+{\left(\frac{{\omega }_1}{2\pi \Omega }\right)}^2\left(\frac{T_1^w}{T_2^w}\right)\right]}$$ [8]

$$\lambda =-\frac{1}{2}\left(\alpha +\beta -\sqrt{{(\alpha -\beta )}^2+4k_{mw}^2{M}_0^m}\right)$$ [9]

$$\alpha =1/T_1^m+k_{mw}+R_{rfm}$$ [10]

$$\beta =1/T_1^w+k_{wm}+R_{rfw}$$ [11]

and where M_ss^w is the longitudinal magnetization of the free bulk water pool under steady-state (dM/dt = 0), and the saturation of the magnetization for the free bulk water and semisolid pools is governed by the RF absorption rates as follows:

$$R_{rfw}=\frac{{\omega }_1^2\cdot T_2^w}{1+{(2\pi \Omega T_2^w)}^2},R_{rfm}=\frac{{\omega }_1^2\cdot T_2^m}{1+{(2\pi \Omega T_2^m)}^2}$$ [12]

During the relaxation delay time period, the magnetization vector was computed solely by the relaxation recovery process in the absence of B₁, and the magnetization vector at the endpoint of each dynamic scan served as an initial condition for the next dynamic scan. Finally, the transient-state MTC-MRF signal evolution (S_MTC-MRF) can be described by^25,26:

$$S_{\mathit{\text{MTC}}-\mathit{\text{MRF}}}(T_1^w,T_1^m,T_2^w,T_2^m,k_{mw},M_0^m;B_1,\Omega ,Ts,Td)=[M_0^w(1-e^{-Td/T_1^w})-M_{ss}^w]e^{\lambda Ts}+M_{ss}^w$$ [13]

The analytical solution for the transient-state signal model was incorporated with the deep-learning framework to optimize an MRF acquisition schedule with respect to the scan parameters (B₁, Ω, Ts, and Td).

3. Methods

3.1. MTC-MRF acquisition schedules

3.1.1. Linear acquisition schedule

A linear acquisition schedule was generated by linearly increasing values from the minimum to maximum points of each scan parameter range. Specifically, the parameter ranges were: 8 – 50 ppm for frequency offsets; 0.9 – 2 μT for RF saturation powers; 0.4 – 2 s for RF saturation times; and 3.5 – 4.5 s for relaxation delay times. The range of scan parameters was determined by specific absorption rate (SAR) constraints. In particular, the limited range of the RF saturation power was used to stay within the clinically permitted SAR, mainly due to the use of the SAR-intensive, time-interleaved parallel transmission (pTX)-based RF saturation and 3D TSE readout with multiple inversion RF pulses. The same range of scan parameters was applied to the other acquisition schedules.

3.1.2. Interior-Point (IP)-based acquisition schedule

An acquisition schedule was generated using an IP algorithm by maximizing signal discrimination between different tissue types³⁰. The optimization problem was solved by minimizing the correlation coefficient between MRF signals of the pre-defined dictionary, which could be formulated as follows:

$${\text{min}}_x{\Vert I-D_x^T{D}_x\Vert }_2^2$$ [14]

where $x\in {ℝ}^{4\times q}$ (d: length of the schedule) is the RF saturation and acquisition schedule, $I\in {ℝ}^{N\times M}$ is the identity matrix, and D is the dictionary of MTC-MRF signals for all possible tissue parameter values. A total of ten thousand tissue parameter sets (N = 10,000) were contained in the dictionary, which were randomly sampled from each tissue parameter range. The signals in the dictionary were normalized to power one for accurate calculation of the correlation coefficients between MTC-MRF profiles. The IP algorithm was initialized with a pseudo-random (PR) schedule (see 3.1.4).

3.1.3. Cramer-Rao Lower Bound (CRLB)-based acquisition schedule

An acquisition schedule was optimized by minimizing CRLB values for the MTC-MRF signal model (Eq. 13)³¹. The CRLB with respect to tissue parameters $p\in {ℝ}^q$, where q is the number of the parameters, can be defined as:

$$\mathit{\text{Var}}\left(p\right)\ge V\left(p\right)=I^{-1}\left(p\right)$$ [15]

where $V\left(p\right)\in {ℝ}^{q\times q}$ represents the CRLB matrix and I(p) is the Fischer Information Matrix (FIM) defined as:

$$I(p)=E\left[\left(\frac{\partial \ell (S_{\mathit{\text{MTC}}-\mathit{\text{MRF}}}(p,x);p)}{\partial p}\right){\left(\frac{\partial \ell (S_{\mathit{\text{MTC}}-\mathit{\text{MRF}}}(p,x);p)}{\partial p}\right)}^T\right]$$ [16]

where $\ell$ is the log-likelihood function, S_MTC-MRF is the MTC-MRF signal, and x is the RF saturation and acquisition schedule. With a white Gaussian noise, the Fisher information matrix can be expressed as:

$$I\left(p\right)=\frac{1}{{\sigma }^2}{\sum }_n{J}_n^T\left(p\right)J_n\left(p\right)$$ [17]

where $J_n\left(p\right)=\partial S_{\mathit{\text{MTC}}-\mathit{\text{MRF}}}/\partial p\in {ℝ}^{1\times q}$ is the Jacobian matrix of the MTC-MRF signal at n^th repetition time (TR), and σ is the standard deviation of the noise. The CRLB optimization focuses on minimizing the variance of tissue parameter estimates, thereby maximizing SNR efficiency. For any given tissue parameter dataset, the objective function of the CRLB optimization can be written as follows:

$${\text{min}}_x{\sum }_{i=1}^N\mathit{\text{Tr}}\left(WV\left(p\right)\right)$$ [18]

where x is the RF saturation schedule, N is the number of tissue parameter sets used to calculate the objective function, and W is the weighting function, which was heuristically determined. The tissue parameter dataset used for CRLB optimization was the same as that used for IP optimization.

3.1.4. Pseudo-random (PR) acquisition schedule

A pseudo-random acquisition schedule was generated by increasing spectral and temporal incoherence between dynamic scans, which could reduce information redundancy in multiple image acquisitions. The PR schedule was the same as that used in our previous studies^25,26.

3.1.5. Learning-based optimization of acquisition schedule (LOAS)

A fully-connected neural network (FCNN) was designed to optimize the RF saturation schedule for tissue parameter quantification. MTC-MRF signal profiles were generated using initial scan parameters and tissue parameters through a forward two-pool Bloch transform. The simulated MTC-MRF signals were fed to the FCNN architecture as an input, which outputs tissue parameter estimates (Fig. 1). The estimated tissue parameters were then compared to the input (ground-truth) for a loss calculation. The loss function was a mean square difference between the ground-truth and the estimated tissue parameters. The calculated loss was back-propagated to update scan parameters to minimize the loss via the adaptive moment estimation (ADAM) optimizer. Thus, the scan parameters were updated iteratively through epochs toward decreasing quantification errors. The objective function of the architecture was formulated as:

$$\widehat{p}=f\left(S_{\mathit{\text{MTC}}-\mathit{\text{MRF}}}\left(p,x;\eta \right),T_2^w;\theta \right)$$ [19]

where p represents the ground-truth tissue parameters (input), x is the RF saturation and acquisition schedule, S_MTC-MRF represents the MTC-MRF signal, η is the noise, T₂^w is the transverse relaxation time of the free bulk water, θ represents the parameters (weights) of the neural network, and $\widehat{p}$ represents the estimated tissue parameters (output). The FCNN architecture (f) consisted of FC (ds+1,256) – ReLU – FC (256,256) – ReLU – FC (256,4)– Sig, where FC(n,m) is a fully connected layer of input size n and output size m, ReLU is the rectified linear unit activation function, Sig is the sigmoid activation function, and ds is the number of dynamic scans. To normalize the MTC parameters, the sigmoid function was adopted at the last layer. The learning-based optimization architecture was trained by minimizing the mean square difference (loss function, L_θ) between p and $\widehat{p}$, as follows:

$$L_{\theta }=\mathit{\text{mean}}|p-\widehat{p}{|}_2^2$$ [20]

Figure 1.

A scheme of the learning-based optimization of the acquisition schedule (LOAS). MRF signals are synthesized using initialized scan parameters, noise, and tissue parameters (Input) and fed to the fully connected neural network (FCNN). The FCNN outputs tissue parameter estimates (Output). A loss function is a mean square error between the ground-truths and estimated tissue parameters. The calculated loss was back-propagated with an ADAM optimizer to update scan parameters.

For the training dataset, forty million sets of tissue parameters were randomly sampled from the pre-defined range of each parameter. Specifically, the ranges of tissue parameters were: 5 – 100 Hz for K_mw; 2 – 17 % for M₀^m; 1 – 100 μs for T₂^m; and 0.2 – 3.0 s for T₁^w. In addition, T₁ of macromolecular pool was set to a constant value of 1 s because it was not well determined^2,33. The networks were trained using Tensorflow with ADAM as the optimizer³⁴. The initial learning rate was set to 10⁻³ with a batch size of 200. Training was implemented on an NVIDIA TITAN RTX GPU (Santa Clara, CA).

3.2. MTC-MRF Bloch simulations

Two Bloch-McConnell equation-based simulation studies were performed to evaluate the reconstruction accuracy and efficiency of MTC-MRF schedules. For the first simulation study, four digital phantoms were constructed to evaluate the accuracy of tissue parameters (k_mw, M₀^m, T₂^m, and T₁^w) with various schedules (Linear, IP, CRLB, PR, and LOAS). For each phantom with a matrix size of 100 × 500, one tissue parameter had five uniformly sampled values (5, 25, 50, 75, and 100 Hz for K_mw, 2, 6, 10, 14, and 17 % for M₀^m, 1, 25, 50, 75, and 100 μs for T₂^m, and 0.2, 0.9, 1.6, 2.3, and 3.0 s for T₁^w), while other three tissue parameter values were randomly selected. The parameter ranges were given as 5 to 100 Hz for k_mw, 2 to 17% for M₀^m, 1 to 100 μs for T₂^m, and 0.2 to 3.0 s for T₁^w. Longitudinal magnetization evolutions of the free bulk water protons were simulated with the tissue parameters and scan parameters, and fed to the newly trained 6-layer FCNN architecture for tissue quantifications. Quantification results of the FCNN method were compared with those of the Bloch-fitting approach.

In the second digital phantom study, the efficiency of the MRF schedule optimized by the LOAS method was evaluated with various numbers of dynamic scans (#5, #10, #20, #30, and #40). The optimization was performed for each dynamic scan number schedule. The quantification accuracy was evaluated by calculating normalized root mean square errors (nRMSE) between ground-truths and tissue parameters estimated by the FCNN-based quantification method. The Bloch simulations were performed on a 64-bit Windows operating system (12-CORE, 3.8-GHz AMD processor and 32 GB of memory) using MATLAB (The MathWorks, Natick, MA).

3.3. In vivo MRI experiments

MRI experiments were performed on a 3T MRI system (Achieva dStream, Philips Healthcare, Best, The Netherlands), using a body coil for transmission and a 32-channel head coil for reception. Six subjects were included in the in vivo experiments, two females and four males (mean age: 36 years; range: 27 to 41 years of age). All subjects were examined with the approval of the institutional ethics committee, and written informed consent was obtained prior to the study. 3D MTC-MRF images were obtained from a fat-suppressed (spectral pre-saturation with inversion recovery, SPIR) multi-shot TSE pulse sequence with a four-fold (2 × 2) compressed sensing (CS) acceleration in the K_y-K_z plane^35,36. The imaging parameters of MTC-MRF were: TE = 6 ms; FOV = 212 × 186 × 60 mm³; spatial resolution = 1.8 × 1.8 × 4 mm³; slice-selective 120° refocusing pulses; turbo factor = 104; and slice oversampling factor = 1.4. The pTX technique was used to achieve continuous RF saturation (100% duty-cycle saturation over 2 s, BW = 24.5 Hz), which is capable of reducing amplifier limitations while detecting highly sensitive saturation effects on clinical scanners^35,37. In addition, the pTX-based pseudo-continuous-wave RF irradiation allowed for a larger degree of freedom for the RF saturation schedule and a simple analytical solution of the two-pool exchange model. Forty dynamic scans were acquired with varied frequency offsets, RF saturation powers, RF saturation times, and relaxation delay times. For amide proton transfer (APT) and nuclear Overhauser enhancement (NOE) imaging, another series of saturated images was obtained with a saturation power of 1.2 μT, a saturation time of 2 s, a delay time of 4 s, and six frequency offsets (±3, ±3.5, and ±4 ppm), following the MTC-MRF scans. For normalization, an unsaturated image (S₀) was acquired by turning off the saturation pulse. In addition, water saturation shift-referencing (WASSR) images (26 frequency offsets range of ±1.2 ppm at intervals of 0.125 ppm, B₁ = 0.5 μT)³⁸ and multi-echo gradient-spin-echo (GRASE)^39,40 images (TR = 3 s, echoes = 20, 40, 60, 80, and 100 ms, TSE factor = 5) were acquired for B₀ and T₂ maps, respectively. Note that water T₂ maps were separately measured due to lack of water spin information from MTC-MRF images with far off-resonance frequency offsets (Ω >8ppm).

3.4. Image processing for APT and NOE imaging

MTC images at ±3.5 ppm, synthesized using scan parameters, estimated tissue parameters from the FCNN framework, and acquired T₂ relaxation time, were used as reference baseline images for APT and NOE imaging. APT and NOE images were calculated based on the subtraction method⁴¹:

$$\mathit{\text{CEST}}=Z_{\mathit{\text{ref}}}\left(\Delta \Omega \right)-Z_{\mathit{\text{lab}}}\left(\Delta \Omega \right)$$ [21]

$$Z=S_{\mathit{\text{sat}}}/S_0$$ [22]

where Z_lab is the normalized label signal intensity (from free water + semisolid macromolecule + multiple CEST pools), Z_ref is the normalized reference signal intensity containing only MTC and direct saturation effects (from free water + semisolid macromolecules), ΔΩ is the frequency offset of a specific labile group, S_sat is the signal intensity measured with RF saturation, and S₀ is the signal intensity measured without RF saturation. Here, ΔΩ of +3.5 ppm and −3.5 ppm were selected for APT and NOE images, respectively^42,43. Z_ref (±3.5ppm) images were synthesized whereas Z_lab (±3.5ppm) images were experimentally acquired.

3.5. Validation of MTC-MRF with synthetic MRI

Due to the lack of in-vivo ground truth information, a synthetic MRI technique was adopted to validate the proposed LOAS method using in-vivo images. The in-vivo tissue parameter maps reconstructed from the FCNN framework using 40 MTC-MRF images acquired with the PR schedule were defined as the reference (namely, Reference). Then, synthetic 3D MTC-MRF images were generated using the reference tissue parameters and the MRF schedules (LOAS, PR, CRLB, IP, and Linear) via a two-pool Bloch transform, and fed to 6-layer FCNN architectures newly trained with the MRF schedules. The reference tissue parameters were used to validate water and MTC quantification, comparing tissue parameters estimated with the various MRF schedules to the explicitly known ground-truth, and to gauge how tissue parameter estimation errors depend on the particular choice of scan parameters (e.g., MRF schedule and the number of scans). To assess the efficiency of the LOAS schedule, the quantification accuracy was evaluated by calculating nRMSE between the reference and tissue parameters estimated with various numbers of scans from 5 to 40 in an increment of 5. Eight FCNN architectures that corresponded to the different dynamic scan numbers were separately trained and used to estimate tissue parameters. In total, 176 different FCNN architectures were separately trained and tested with the various MRF schedules: 78 for CLRB; 58 for IP; and 40 for LOAS, PR, CRLB, IP, and Linear, with eight different dynamic scan numbers. Moreover, the accuracy of estimated MTC at 3.5 ppm, APT, and NOE signals were evaluated. A Bland-Altman analysis was performed to assess the agreement of tissue parameters estimated from the different dynamic scan numbers.

4. Results

4.1. Reconstruction error vs. optimization cost

Reconstruction errors of LOAS and optimization costs of CRLB and IP were monitored through epochs or iterations, as shown in Fig. 2. Overall training loss of LOAS (Fig. 2A), CRLB (Fig. 2C), and correlation (Fig. 2E) values decreased as the iteration and epoch proceeded. In the test process, reconstruction errors (converted to nRMSE) of the LOAS for each tissue parameter decreased with the epoch, as shown in Fig. 2B. However, the reconstruction errors of CRLB and IP were invariant even the optimization costs decreased with iterations, as shown in Figs. 2D and 2F. Note that the CRLB algorithm ran for 58 iterations to minimize the objective function, and thus, the corresponding 58 MRF schedules were used to train each FCNN architecture for water and MTC quantification. The reconstruction errors for each parameter were calculated from the test dataset, with 58 respectively trained FCNN models (Fig. 2D). The IP algorithm ran for 78 iterations and the reconstruction errors for each parameter (Fig. 2F) were evaluated in the same way as with the CRLB. The RF saturation schedules optimized by LOAS, CRLB, and IP strategies, and PR and Linear schedules are shown in Fig. 3 (also see Supporting Information Table S1).

Figure 2.

(A) A training loss with respect to epoch numbers from the LOAS architecture. (B) Test losses (converted to nRMSE values) of each tissue parameter. (C) Optimization cost as a function of iteration numbers from CRLB. Fifty-eight FCNN architectures were trained with corresponding MRF schedules obtained at each iteration. (D) nRMSE values for the estimated tissue parameters with respect to iteration numbers. (E) Optimization cost with respect to iteration numbers from IP. Seventy-eight FCNN architectures were trained with corresponding MRF schedules obtained at each iteration. (F) nRMSE values for the estimated tissue parameters with respect to iteration numbers.

Figure 3.

Optimized MRF schedules (B₁, Ω, Ts, and Td) with 40 dynamic scans from the LOAS, CRLB, IP, PR, and Linear methods.

4.2. Digital phantom studies

Optimized MRF schedules were evaluated using digital phantoms encoded by the two-pool transient-state MTC model. Water and MTC parameters estimated from the deep-learning FCNN and Bloch fitting-based quantification methods were compared to the ground-truths, as shown in Fig. 4. Overall, the nRMSE values of the FCNN method were lower than those of the Bloch fitting method except for T₁^w estimated using LOAS, PR, and CRLB schedules. Nevertheless, both quantification methods still showed a high degree of accuracy for T₁^w estimation. The FCNN-based quantification approach with the LOAS schedule showed the highest accuracy in the quantification of tissue parameters. The average nRMSE values of k_mw, M₀^m, T₂^m, and T₁^w between the FCNN and the ground-truth were 15.2%, 6.6%, 3.5%, and 1.3% for LOAS, 15.9%, 7.4%, 3.8%, and 1.6% for PR, 15.3%, 6.6%, 4.0%, and 1.3% for CRLB, 21.1%, 9.8%, 7.4%, and 2.0% for IP, and 32.4%, 16.1%, 14.1%, and 8.8% for Linear, respectively. Tissue parameters estimated from the FCNN with different numbers of dynamic scans (e.g., #40, #30, #20, #10, and #5 dynamic scans) were compared to the ground-truth and nRMSE values were calculated to evaluate the scan efficiency of the MTC-MRF (Fig. 5). A reduction of the number of dynamic scans resulted in increased nRMSE values. However, the parameters estimated from the LOAS schedule with 10 dynamic scans were not significantly different from the ground-truths. Moreover, significant correlations were observed between all estimated parameters and the ground-truth in all dynamic scan numbers (p< 0.05 for all parameters) and the correlation coefficients were all over 0.8, even for #5 dynamic scans (see Supporting Information Table S2). The nRMSE values with the dynamic scan numbers of #40, #30, #20, #10, and #5 were 15.2%, 15.8%, 16.9%, 18.8%, and 20.7% for k_mw, 6.6%, 6.9%, 7.7%, 8.8%, and 11.0% for M₀^m, 3.5%, 3.9%, 4.5%, 6.0%, and 8.7% for T₂^m, and 1.3%, 1.4%, 1.7%, 2.2%, and 2.9% for T₁^w, respectively.

Figure 4.

Bloch equation-based digital phantom experiments with the optimized MRF schedules. (A) Four ground-truth (GT1, GT2, GT3, and GT4) digital phantoms. (B) Water and MTC parameter maps estimated with various MRF schedules. (C) The resulting parameter maps from the Bloch-fitting and FCNN methods were compared to the ground-truths by calculating nRMSE.

Figure 5.

Bloch equation-based digital phantom experiments with different numbers of dynamic scans. (A) Parameter maps reconstructed from FCNN with respect to number of dynamic scans with LOAS schedules. (B) Reconstruction accuracy was evaluated for each tissue parameter with various dynamic scan numbers.

4.3. In vivo MRI studies

To evaluate the LOAS method with in-vivo data, synthetic MTC-MRF images were generated using the five MRF schedules and reference tissue parameters obtained from the PR schedule with 40 dynamic scans (Fig. 6). Then, synthetic images, as a test dataset, were fed into the newly-trained FCNN with each MRF schedule. The estimation accuracy was evaluated by calculating the mean absolute error (MAE) between the reference and tissue parameter estimates with the different MRF schedules. The lowest MAE values (0.95 for k_mw, 0.36 for M₀^m, 0.97 for T₂^m, and 0.02 for T₁^w) were obtained with the LOAS method. In addition, good agreements were still observed for the PR and CRLB-based schedules. However, the IP and Linear schedules showed poor quantification quality.

Figure 6.

A schematic of the validation process using a synthetic MRI technique. Synthetic MTC-MRF images were generated using the reference maps and MRF schedules optimized with the LOAS, PR, CRLB, IP, and Linear methods and fed to newly-trained FCNN frameworks corresponding to the optimized schedules. Difference images between the reference and the estimated maps are shown. The mean difference value of each map is shown in the insert (white). Note that the MTC proton concentration was converted to absolute units using the water proton concentration (110 M).

Figure 7 shows the accuracy of tissue parameters, MTC at 3.5 ppm, APT, and NOE signals estimated using LOAS with different numbers of data sampling points. When the number of dynamic scans decreased, nRMSE values increased for water and MTC parameters. However, MTC, APT, and NOE contrast-weighted signals were relatively insensitive to the number of dynamic scans. Compared to the reference values obtained from 40 dynamic scans, parameter quantification with 10 dynamic scans was accurate, where nRMSE values were 3.44%, 3.23%, 1.84%, 2.03%, 0.49%, 0.49%, and 0.49% for k_mw, M₀^m, T₂^m, T₁^w, Z_ref(3.5ppm), APT, and NOE, respectively. The estimated tissue parameter maps and contrast-weighted images (Z_ref(3.5ppm), APT, and NOE) are shown in Fig. 8.

Figure 7.

Quantification accuracy of LOAS with respect to the number of dynamic scans (#40, #35, #30, #25, #20, #15, #10, and #5) for water and MTC parameters, Z_ref (3.5ppm), APT, and NOE signal intensities from six healthy volunteers.

Figure 8.

Quantitative MTC parameter maps, water T₁ map, Z_ref (3.5ppm), APT, and NOE images of a representative human brain of a healthy volunteer. Difference images of the estimated maps with #40 dynamic scan vs. #30, #20, #15, #10, and #5 dynamic scans. The mean difference value of each map is shown in the insert (white).

A Bland Altman analysis was performed to evaluate the agreement among tissue parameter estimates for different numbers of dynamic scans, as shown in Fig. 9. No significant bias for the measurement of k_mw, M₀^m, T₂^m, and T₁^w was observed over all dynamic scan numbers, except the dynamic scan number of #5 in k_mw. The analysis revealed that the 95% confidence interval (CI) of the agreement became wider and more outliers were presented at a smaller number of dynamic scans. Particularly, when reducing the dynamic scan number from 10 to 5, a huge expansion of the 95% CI was observed for all tissue parameters. The Pearson correlation coefficient values between the dynamic scan number of #40 and the dynamic scan numbers of #30, #20, #10, and #5, respectively were 0.93, 0.88, 0.85, 0.75 for k_mw, 0.99, 0.99, 0.99, and 0.98 for M₀^m, 0.98, 0.97, 0.97, and 0.94 for T₂^m, and 0.99, 0.99, 0.98, and 0.97 for T₁^w (p < 0.01 for all). The structural similarity index measure (SSIM) values between the dynamic scan number of #40 and the dynamic scan numbers of #30, #20, #10, and #5, respectively, were 0.97, 0.96, 0.94, 0.90 for k_mw, 0.98, 0.98, 0.97, and 0.95 for M₀^m, 0.99, 0.98, 0.98, and 0.97 for T₂^m, and 0.99, 0.99, 0.98, and 0.97 for T₁^w (p < 0.01 for all).

Figure 9.

Bland-Altman plots comparing MTC (A) exchange rate, (B) concentration, (C) T₂ relaxation time, and (D) water T₁ relaxation time between #40 dynamic scans and #30, #20, and #10, respectively. The central solid lines show the mean bias, and the upper and lower dashed lines show the variation limits (±1.96 standard deviation).

5. Discussion

In this study, the MRF optimization problem was solved using a deep-learning model trained with a large amount of simulated MTC-MRF signals. In the supervised-learning framework, RF saturation parameters were updated toward minimizing a loss function that could directly calculate tissue quantification errors. Therefore, the LOAS method improved parametric reconstruction quality compared to existing indirect optimization methods^19,31, enabling a further four-fold temporal acceleration by reducing the number of dynamic scans, in addition to the CS acceleration in two phase encoding directions (eight-fold acceleration in total).

A high degree of freedom and flexibility of MRF scan parameters pose challenges for sequence optimization^13,44,45. Current MRF optimization methods are based on indirect optimization metrics. The CRLB-based MRF optimization technique minimizes the Cramer-Rao lower bound of tissue parameter estimates, which is equivalent to maximizing SNR efficiency³¹. The IP algorithm-based optimization method minimizes the signal correlation, or maximizes the discrimination between MRF signals of different tissue properties^20,30. The indirect optimization metrics, however, cannot guarantee an optimal solution for tissue parameter quantification. In this study, the uncertainty of the objective functions in the CRLB and IP methods were evaluated by calculating direct tissue parameter errors for each iteration. Interestingly, the CRLB and correlation metrics were inconsistent with the tissue parameter errors, although the optimization cost values decreased with the number of iterations. In contrast, the LOAS updated the MRF scan parameters based on the gradient of the estimated tissue parameter error, and thus, the training loss of the LOAS directly reflected the tissue parameter error. It is interesting to note that Bang-Bang behavior was observed in the CRLB and IP optimization results, in which optimal values were often found at either the lower bound or the upper bound of the pre-defined scan parameter ranges (Fig. 3). Such a structural behavior was also observed in a previous study³¹. We speculate that this could be attributed, in part, to the limited ranges of scan parameters, thus limiting the maximum encoding efficiency. In our case, particularly, RF saturation powers and durations, as well as relaxation delay times, were highly constrained by SAR.

The proposed method was validated using simulated digital phantoms, and in vivo experiments against other sub-optimal MTC-MRF schedules. The performances of the various optimization methods were retrospectively evaluated by solving Bloch equations since it would not be feasible to prospectively acquire many images with different MRF schedules and dynamic scan numbers in a limited scan time. Our results showed that the LOAS outperformed other state-of-the-art methods^30,31 in terms of quantification accuracy and acquisition efficiency. Moreover, the LOAS takes advantage of compatibility with various types of noise distributions, while the CRLB optimization requires the inherent assumption of a Gaussian noise distribution. In addition, the LOAS method could be integrated with the deep-learning-based tissue quantification or Bloch-fitting approaches. Importantly, the LOAS framework can be extended to include any other analytical signal models, tissue properties, sequences, and target scan parameters, which makes it extremely versatile and suitable for a wide range of multi-parameter MRF applications. A super-Lorentzian lineshape has been known to provide a good approximation for RF absorption rate by a semisolid MTC pool^2,46,47. However, the calculation of numerical integration in the super-Lorentzian lineshape is significantly time-consuming. Instead, the Lorentzian lineshape was used in this study to reduce the computational burden on generating massive amounts of training dataset for 176 different FCNN architectures.

For water and MTC quantification, a deep-learning neural network was used to learn a mapping relation between a high-dimensional MR fingerprint space and a low-dimensional tissue parameter space. Although the FCNN required a long training time to optimize the weights of the network (20 hours for 40 million datasets), the neural network had low computational complexity in the test phase. A conventional Bloch equation-based fitting approach enabled tissue parameter estimation at a high computational cost (two hours for an image matrix of 256 × 256 × 9 × 40). In comparison, the FCNN framework enabled almost instantaneous estimation of tissue parameters (four seconds for an image matrix of 256 × 256 × 9 × 40). Furthermore, we exploited flexibility and degrees of freedom in MRF sequence parameters to improve quantification accuracy and efficiency. With the LOAS method, ten dynamic scans provided the best trade-off between quantification accuracy and acquisition efficiency, possibly by four-fold reduction in scan time. Corresponding tissue parameter maps are shown in Supporting Information Figure S1. Notably, the optimized MTC-MRF framework can provide baseline MTC reference signals for APT and NOE imaging and potentially be extended to quantitative CEST and NOE parameter mappings (e.g., proton exchange rate and concentration) within a clinically feasible scan time.

6. Conclusion

We proposed a learning-based optimization framework to accelerate data acquisition and improve quantification accuracy for magnetization transfer contrast MR fingerprinting. Unlike optimization methods based on indirect measurements, the proposed approach optimized scan parameters by directly calculating quantitative errors of the tissue parameters and significantly improved the accuracy and also reduced data acquisition time. The flexible LOAS framework could be a powerful optimization tool for MRF pulse sequence design.

Supplementary Material

Supporting Table1

Supporting Information Table S1. LOAS-based MTC-MRF schedules for dynamic scan numbers 40 and 10.

Supporting figure1

Supporting Information Figure S1. Quantitative water and MTC parameter maps from the LOAS, PR, CRLB, IP, and Linear schedules, and their corresponding difference images. The reference images (Ref) were obtained from deep-learning reconstruction with 40 MTC-MRF images acquired from the PR schedule. The mean difference value of each map is shown in the insert (white).

Supporting Table2

Supporting Information Table S2. The correlation coefficients between ground-truth values of digital phantoms and estimated parameters for free bulk water and semisolid MTC parameters for five different dynamic scan numbers.

Abbreviations:

MRF

MR Fingerprinting

MTC

Magnetization Transfer Contrast

APT

Amide Proton Transfer

CEST

Chemical Exchange Saturation Transfer

LOAS

Learning-based Optimization of Acquisition Schedule

FCNN

Fully Connected Neural Network