Locally Linearized Approximation of RF Pulse Propagators for Faster Simulation of Magnetization Dynamics With Slice Profile Effects in 2D MR Fingerprinting

ABSTRACT

Purpose

The objective for this study is to decrease the computation time of magnetic resonance fingerprinting dictionary computations that include slice profile effects.

Theory and Methods

A locally linearized approximation to RF pulse propagators across RF scaling factors was derived and used to decrease the computational burden of MRF dictionary computation. MR fingerprinting phantom experiments were carried out at 3T to assess the practical feasibility of using dictionaries approximated with the proposed approach.

Results

Using linearizations at 10 ${\mathrm{B}}_1^{+}$ scaling factors produced root mean square differences in estimated $T_1$ and $T_2$ of 1.17% and 1.98%, respectively, between dictionaries generated using a full Bloch simulation and the locally linear approximation while decreasing computation time by a factor of 10.

Conclusion

The proposed locally linear approximation significantly accelerates slice‐profile‐aware MRF simulations while maintaining high accuracy, enabling practical incorporation of realistic RF pulse effects into large‐scale dictionary computations.

1. Introduction

Magnetic resonance fingerprinting (MRF) leverages transient flip‐angle steady‐state free precession acquisitions to encode tissue‐specific signal evolutions, enabling rapid and robust quantitative MRI (qMRI [1]). By probing a high‐dimensional parameter space through dynamic acquisition, MRF achieves efficient and precise estimation of relaxation parameters. However, its accuracy can be significantly degraded by confounding factors such as transmit field (${\mathrm{B}}_1^{+}$) inhomogeneity and imperfect slice profiles [2]. Prior work has shown that incorporating ${\mathrm{B}}_1^{+}$ variation and Bloch‐simulated responses to individual RF pulses into the MRF dictionary can substantially reduce bias in two‐dimensional acquisitions [3]. Despite these benefits, such comprehensive modeling dramatically increases the computational burden, requiring prohibitively long simulation times or access to high‐performance computing resources. While in many applications dictionary computation is a one‐time process, it may be the case that the time between repetitions of the MRF train can change at runtime (e.g., in cardiac triggered acquisitions), and the dictionary becomes scan‐specific [4]. There is therefore a pressing need to reduce the computational demand of fingerprinting dictionary calculations.

In this work, we propose a locally linear approximation (LLA) of RF pulse propagators with respect to RF scaling factors to reduce the computational burden associated with slice‐profile‐corrected 2D MRF dictionary generation. Instead of explicitly simulating the effect of every RF pulse across hundreds to thousands of time points in an MRF sequence, we compute the RF pulse propagator and its derivative with respect to the RF scaling factor $\beta$ at a small number of $\beta$ values (e.g., $\le 10$). RF propagators at arbitrary $\beta$ are then efficiently interpolated using a cubic spline basis. This approach is motivated by the nonlinear dependence of the magnetization response on RF scaling (Figure 1), which precludes simple linear scaling of precomputed signals. We demonstrate (1) the theoretical formulation of the LLA, (2) its accuracy in both simulation and phantom experiments, and (3) the feasibility of rapidly generating slice profile‐corrected MRF dictionaries using Bloch simulations of RF waveforms extracted directly from MRF Pulseq .seq files at the RF raster time resolution.

FIGURE 1.

Non‐linearity in RF pulse scaling and its locally linear approximation. Different scaling factors ($\beta =[1.0,1.5]$) for the same pulse shape result in different transverse magnetization excitation profiles. Components of the RF pulse propagator matrix as a function of $\beta$ are shown along with their derivatives evaluated at $\beta =0$ and $\beta =1.5$. The proposed $N_{\beta }=2$ cubic spline‐based locally linear approximation is shown.

2. Theory

Let the RF waveform in the MR fingerprinting frame $m$ be modeled as a scaled and phase‐rotated version of a discretized complex RF pulse,

$$b_n^{(m)}={\beta }_m{e}^{i{\varphi }_m}{\tilde{b}}_n,n=1,\dots ,N,$$ (1)

where ${\tilde{b}}_n={\tilde{b}}_{x,n}+i{\tilde{b}}_{y,n}$ denotes a non‐scaled and non‐phase‐rotated RF waveform sample, ${\beta }_m$ is the RF scaling factor for frame $m$, and ${\varphi }_m$ is a frame‐dependent RF phase.

Each discretized sample $n$ within the RF pulse is applied for a duration $\mathrm{\Delta }t$, and the magnetization is propagated according to

$${\mathbf{M}}_{n+1}={\mathbf{A}}_n^{(m)}{\mathbf{M}}_n,{\mathbf{M}}_n={[M_{x,n},M_{y,n},M_{z,n},1]}^T.$$

For each time step,

$${\mathbf{A}}_n^{(m)}=\mathbf{R}{\mathbf{Q}}_n^{(m)}\mathbf{R},$$

where $\mathbf{R}$ accounts for half‐step relaxation and recovery,

$$\mathbf{R}=\left[\begin{array}{llll}{E}_2& 0& 0& 0\\ 0& E_2& 0& 0\\ 0& 0& E_1& M_0(1-E_1)\\ 0& 0& 0& 1\end{array}\right],$$ (2)

with

$$E_1=e^{-\mathrm{\Delta }t/(2T_1)},E_2=e^{-\mathrm{\Delta }t/(2T_2)}.$$

For off‐resonance $\mathrm{\Delta }{B}_0$ in Hz and slice‐select gradient‐induced frequency offset $f_s$ in Hz, the effective field in frame $m$ during time step $n$ is

$${\mathit{\omega }}_n^{(m)}=2\pi \left[\begin{array}{l}\Re \left({\beta }_m{e}^{i{\varphi }_m}{\tilde{b}}_n\right)\\ \Im \left({\beta }_m{e}^{i{\varphi }_m}{\tilde{b}}_n\right)\\ \mathrm{\Delta }{B}_0+f_s\end{array}\right].$$ (3)

Its magnitude is

$${\mathrm{\Omega }}_n^{(m)}=\Vert {\mathit{\omega }}_n^{(m)}\Vert .$$

Rather than approximating RF and off‐resonance effects as sequential rotations, which introduces operator splitting error, they are treated as a single simultaneous rotation about the effective field over the interval $\mathrm{\Delta }t$. The corresponding affine propagator is

$${\mathbf{Q}}_n^{(m)}=\left[\begin{array}{ll}{\mathit{ℛ}}_n^{(m)}& 0\\ 000& 1\end{array}\right],$$ (4)

where the $3\times 3$ rotation matrix ${\mathit{ℛ}}_n^{(m)}$ is obtained from Rodrigues' formula, representing a rotation about the effective field ${\mathit{\omega }}_n^{(m)}$ over the $\mathrm{\Delta }t$ interval [5, 6],

$${\mathit{ℛ}}_n^{(m)}=\mathbf{I}+\frac{\sin ({\mathrm{\Omega }}_n^{(m)}\mathrm{\Delta }t)}{{\mathrm{\Omega }}_n^{(m)}}{\mathbf{K}}_n^{(m)}+\frac{1-\cos ({\mathrm{\Omega }}_n^{(m)}\mathrm{\Delta }t)}{{({\mathrm{\Omega }}_n^{(m)})}^2}{\left({\mathbf{K}}_n^{(m)}\right)}^2,$$ (5)

with

$${\mathbf{K}}_n^{(m)}=\left[\begin{array}{lll}0& -{\omega }_{z,n}^{(m)}& {\omega }_{y,n}^{(m)}\\ {\omega }_{z,n}^{(m)}& 0& -{\omega }_{x,n}^{(m)}\\ -{\omega }_{y,n}^{(m)}& {\omega }_{x,n}^{(m)}& 0\end{array}\right].$$ (6)

The full RF pulse propagator for frame $m$ is therefore

$${\mathbf{A}}^{(m)}({\beta }_m)={\mathbf{A}}_N^{(m)}({\beta }_m){\mathbf{A}}_{N-1}^{(m)}({\beta }_m)\cdots {\mathbf{A}}_1^{(m)}({\beta }_m).$$ (7)

Now define a non‐phase‐rotated propagator ${\tilde{\mathbf{A}}}_n(\beta )$ using the nominal RF waveform ${\tilde{b}}_n$. Introduce the $z$‐rotation matrix

$$\mathbf{Z}(\varphi )=\left[\begin{array}{llll}\cos \varphi & -\sin \varphi & 0& 0\\ \sin \varphi & \cos \varphi & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right].$$ (8)

Because the Bloch dynamics are equivariant under rotations about the $z$‐axis, the frame‐$m$ propagator can be written exactly as

$${\mathbf{A}}_n^{(m)}(\beta )=\mathbf{Z}({\varphi }_m){\tilde{\mathbf{A}}}_n(\beta )\mathbf{Z}(-{\varphi }_m),$$ (9)

and hence

$${\mathbf{A}}^{(m)}(\beta )=\mathbf{Z}({\varphi }_m)\tilde{\mathbf{A}}(\beta )\mathbf{Z}(-{\varphi }_m),$$ (10)

where

$$\tilde{\mathbf{A}}(\beta )={\tilde{\mathbf{A}}}_N(\beta ){\tilde{\mathbf{A}}}_{N-1}(\beta )\cdots {\tilde{\mathbf{A}}}_1(\beta ).$$ (11)

To construct a locally linear approximation (LLA), only the non‐phase‐rotated propagator $\tilde{\mathbf{A}}(\beta )$ needs to be tabulated as a function of $\beta$. Its derivative with respect to RF scaling is

$$\frac{d\tilde{\mathbf{A}}}{d\beta }=\sum _{k=1}^N{\tilde{\mathbf{A}}}_N{\tilde{\mathbf{A}}}_{N-1}\cdots {\tilde{\mathbf{A}}}_{k+1}\frac{d{\tilde{\mathbf{A}}}_k}{d\beta }{\tilde{\mathbf{A}}}_{k-1}\cdots {\tilde{\mathbf{A}}}_2{\tilde{\mathbf{A}}}_1,$$ (12)

with

$$\frac{d{\tilde{\mathbf{A}}}_k}{d\beta }=\mathbf{R}\frac{d{\tilde{\mathbf{Q}}}_k}{d\beta }\mathbf{R}.$$ (13)

Since $\mathbf{R}$ does not depend on $\beta$, only ${\tilde{\mathbf{Q}}}_k$ contributes to the derivative. A derivation of $d{\tilde{\mathbf{Q}}}_k/d\beta$ is provided in the Supporting Information.

Suppose that $\tilde{\mathbf{A}}({\beta }_1)$, $\frac{d\tilde{\mathbf{A}}}{d\beta }({\beta }_1)$, $\tilde{\mathbf{A}}({\beta }_2)$, and $\frac{d\tilde{\mathbf{A}}}{d\beta }({\beta }_2)$ have been precomputed at neighboring RF scaling values ${\beta }_1$ and ${\beta }_2$. Then for any $\beta \in [{\beta }_1,{\beta }_2]$, the propagator is approximated by cubic Hermite spline interpolation [7], which preserves derivatives at ${\beta }_1$ and ${\beta }_2$.

$$\begin{array}{ll}\hfill & \tilde{\mathbf{A}}(\beta )\approx H_{00}(t)\tilde{\mathbf{A}}({\beta }_1)+h{H}_{10}(t)\frac{d\tilde{\mathbf{A}}}{d\beta }({\beta }_1)\\ \hfill & +H_{01}(t)\tilde{\mathbf{A}}({\beta }_2)+h{H}_{11}(t)\frac{d\tilde{\mathbf{A}}}{d\beta }({\beta }_2),\end{array}$$ (14)

where

$$h={\beta }_2-{\beta }_1,t=\frac{\beta -{\beta }_1}{h},$$ (15)

and

$$\begin{array}{llll}\hfill H_{00}(t)& =2t^3-3t^2+1,& \hfill H_{10}(t)& =t^3-2t^2+t,\end{array}$$ (16)

$$\begin{array}{llll}\hfill H_{01}(t)& =-2t^3+3t^2,& \hfill H_{11}(t)& =t^3-t^2.\end{array}$$ (17)

The frame‐specific RF pulse propagator is then recovered as

$${\mathbf{A}}^{(m)}(\beta )=\mathbf{Z}({\varphi }_m)\tilde{\mathbf{A}}(\beta )\mathbf{Z}(-{\varphi }_m).$$ (18)

Thus, a single LLA table over $N_{\beta }$ values of $\beta$ can be reused across all frames in an RF pulse family, while frame‐to‐frame global RF phase variations are handled by $z$‐rotations rather than by constructing separate propagator tables.

Components of $\tilde{\mathbf{A}}$ at row $r$ and column $c$ ($A_{rc}$) are plotted as a function of $\beta$ in Figure 1 for an on‐resonance isochromat, and the depicted $N_{\beta }=2$ LLA demonstrates how the cubic spline interpolation is performed on these components using derivatives calculated at the $\beta$ control points.

3. Methods

3.1. Implementation Details

From a single‐slice, single‐repetition inversion‐recovery FISP MRF [8] pulse sequence generated using the Pulseq framework [9, 10], RF and slice‐select gradient objects were extracted and used to simulate magnetization dynamics throughout each TR directly on the RF raster time. The number of slice positions (i.e., Bloch simulation isochromats) and the number of simulated tissues are specified by the user. A Python interface to a custom CUDA kernel was written to parallelize simulations over intraslice positions and tissues. The RF pulse propagator and its locally linearized approximation were implemented as described in the Theory and the Supporting Information. Code and data to reproduce the results of this study are provided at https://gitlab.com/mickevicius‐lab/papers/locally‐linearized‐rf‐scaling‐approximation. All computations were performed on a server equipped with a 48 GB GPU (RTX A6000, NVIDIA, Santa Clara, CA) to leverage the high degree of parallelization possible with graphics processing hardware for MRF dictionary simulation [11, 12].

3.2. Approximation Error and Runtime

The original adiabatic inversion‐prepared FISP MRF flip angle pattern was used to simulate tissue fingerprints from a Hanning‐windowed sinc pulse with a duration of 4 ms and bandwidth of 2 kHz (the pulse shape shown in Figure 1). Tissues were simulated using 401 isochromats and 1000 combinations of relaxation times with $T_1\in [100,2500]$ ms, $T_2\in [30,500]$ ms, with $T_2<T_1$. Relaxation was considered during the adiabatic inversion pulse. Simulations were performed for the reference of calculating a new RF pulse propagator for every pulse in the MRF pattern along with LLA approximations using $N_{\beta }\in [2,40]$. This was repeated for quadratic RF phase spoiling increments of $\mathrm{\Delta }\psi =0^{\circ }$ (IR FISP MRF) and $\mathrm{\Delta }\psi =117^{\circ }$ (IR FLASH MRF). The root mean square error and runtime, including the time needed for the calculation of the $\mathbf{A}$ and $d\mathbf{A}$ tables at each $N_{\beta }$ value, were calculated as a function of $N_{\beta }$.

3.3. Dictionary Computation

Dictionaries for IR‐FISP MRF with constant TR of 11 ms and $\mathrm{\Delta }\psi =0^{\circ }$ was generated for 1 923 064 combinations of $T_1\in [20,2500]$ ms, $T_2\in [5,1500]$ ms and ${\mathrm{B}}_1^{+}\in [0.5,1.5]$ with timing extracted from a corresponding Pulseq sequence file. Simulated responses were simulated in batches of 4096 tissues using 401 spin isochromats (a relatively large number similar to that used by Zhao et al. [13]) along the slice dimension. The dictionaries were calculated without the LLA approximation and using $N_{\beta }\in [2,3,\dots ,10]$. The $N_{\beta }$ values were chosen to surround and include the apparent “elbow” at $N_{\beta }=7$ in the RMSE vs. $N_{\beta }$ curves generated in Section 3.2. The runtime to generate each dictionary was recorded.

3.4. Phantom Experiments

The same Pulseq file used to generate the dictionaries in Section 3.3 was executed on a 3T system (Signa Premier, GE HealthCare, Waukesha, WI) in a quantitative imaging phantom (Model 137 Mini Hybrid Phantom, CaliberMRI, Boulder, CO) using a 32‐channel head coil (Nova Medical, Inc. Wilmington, MA). The slice thickness was 5 mm, the matrix size was $192\times 192$, and the field‐of‐view was 200 mm $\times$ 200 mm. The sequence used a radial readout that rotated by a tiny golden angle of $23.628^{\circ }$ [14] between MRF frames with two‐fold oversampling and a sampling dwell time of $8\mu \mathrm{s}$. The MRF train was repeated to acquire 21 spokes in each frame with golden angle $111.246^{\circ }$ spacing [15, 16]. An undersampled (i.e., 21 spokes per frame) gridding reconstruction was performed using the BART toolbox [17].

A gradient echo Bloch‐Siegert shift‐based ${\mathrm{B}}_1^{+}$ map [18] was acquired at the same resolution as the MRF data. An 8 ms Fermi pulse with a nominal amplitude of $24\mu \mathrm{T}$ and offset frequencies of $\pm 4\text{kHz}$ was used to induce the Bloch‐Siegert phase shift. A seventh‐order Lagrange polynomial was fit to the ${\mathrm{B}}_1^{+}$ map. The acquired ${\mathrm{B}}_1^{+}$ scaling factor map was quantized to the ${\mathrm{B}}_1^{+}$ resolution of the dictionaries.

Dictionary matching was performed on the magnitude of the gridded image time series by finding the dictionary entry with the highest inner product using the acquired ${\mathrm{B}}_1^{+}$ value. The RMSE in $T_1$ and $T_2$ was calculated within each voxel in the phantom at each of the tested $N_{\beta }$ values.

4. Results

Example IR‐FISP MRF fingerprints are shown with and without quadratic RF phase increments in Figure 2. With $N_{\beta }=2$, there is a discrepancy between the reference and approximated signals that visually gets less severe as $N_{\beta }$ increases to a value of four. This trend holds both with and without the quadratic RF spoiling increment of $117^{\circ }$.

FIGURE 2.

Results from the MR fingerprinting simulation experiment. The flip angle pattern on the top was used in conjunction with a Hanning‐windowed sinc pulse with a time‐bandwidth product of eight. Reference and $N_{\beta }=2$ fingerprints are shown for IR FISP and IR FLASH MRF sequences. The absolute error relative to the reference is shown for $N_{\beta }=2$ and $N_{\beta }=4$.

Quantitative results from the simulations can be seen in Figure 3. Averaged over the 1000 tissues and MRF frames, the RMSE decreased rapidly as a function of $N_{\beta }$ with an elbow appearing near $N_{\beta }=7$. The full reference Bloch simulation took 10 s. The LLA approximations increased linearly as a function of $N_{\beta }$ with $N_{\beta }=10$ achieving an order of magnitude reduction in computation time compared with the full simulation. LLA runtime calculations included the time necessary to compute the $\mathbf{A}$ and $d\mathbf{A}/d\beta$ grid.

FIGURE 3.

MR fingerprinting (MRF) root mean square error (RMSE) and runtime of the locally linear RF scaling approximation as a function of the number of calculated RF propagators $N_{\beta }$. These simulations were performed for 1000 tissues.

The results for the MRF dictionary matching experiment are shown in Figures 4 and 5. Both the $T_1$ and $T_2$ maps start with relatively high error near 50% with $N_{\beta }=2$, and values rapidly approaching the full Bloch simulation reference by $N_{\beta }=10$. The RMSE as a percentage of the reference, averaged over all voxels in the phantom, is plotted as a function of $N_{\beta }$ in Figure 5. The error as a function of $T_1$ flattens just over 1% at $N_{\beta }=6$, while $T_2$ does not begin to flatten at just under 2% RMSE by $N_{\beta }=9$. Similarly to the 1000‐tissue simulations above, the calculation of the full dictionary is still an order of magnitude faster with $N_{\beta }=10$ vs. the full Bloch simulation. The full Bloch simulation took 5 h and 20 min while the $N_{\beta }=10$ dictionary took 32 min.

FIGURE 4.

Dictionary‐matched $T_1$ and $T_2$ maps for the reference full Bloch simulations and locally linearized approximations (LLA) with $N_{\beta }$ of 2 through 10. Difference maps are shown for each LLA approximation relative to the reference dictionary as a percentage of the reference value.

FIGURE 5.

(Left) Root mean square error in $T_1$ and $T_2$ of the LLA approach and full Bloch simulations as a function of $N_{\beta }$. (Right) Dictionary computation time as a function of $N_{\beta }$. The full‐scale dictionary computation time is shown for reference.

5. Discussion

A locally linear approximation of the RF propagator across $\beta$ in dynamic‐flip angle pulse sequences was derived and applied to magnetic resonance fingerprinting dictionary calculations, including slice profile effects. Using this framework for a 2D IR‐FISP MRF acquisition resulted in less than a 2% difference in $T_1$ and $T_2$ values compared with a full‐scale Bloch simulation with an order of magnitude less computation time. The method works for complex‐valued pulses and was shown to also work for the simulation of quadratic RF phase spoiled MRF patterns.

Exact pulse sequence timing was used in the generation of the MRF dictionaries by simulating tissue fingerprint responses directly at the RF raster time resolution with timings extracted from a Pulseq file. While this ensures high‐fidelity magnetization dynamics, it also contributes substantially to computational cost. In practice, simulation time could be further reduced by adopting coarser temporal discretizations. However, this would introduce additional approximation error, particularly for high‐amplitude RF pulses. The proposed locally linear approximation provides a complementary strategy that preserves temporal fidelity while reducing the computational burden associated with RF scaling. While $N_{\beta }=10$ resulted in less than 2% RMSE in $T_1$ and $T_2$ in this work, appropriate $N_{\beta }$ values will likely depend on RF pulse shape, pulse sequence type, and application. One practical strategy for selecting $N_{\beta }$ is to use the minimum value for which the resulting RMSE is comparable to the uncertainty or noise in the estimated $T_1$ and $T_2$ values.

A limitation of this study is the use of a single‐pool magnetization model. Multi‐pool exchanging systems, including magnetization transfer, were not considered, despite their relevance to many quantitative MRI and MRF applications. Extending the proposed locally linear approximation to RF pulse propagators derived from Bloch‐McConnell dynamics [19, 20, 21] may enable similar computational savings in MT‐sensitive and other exchange‐sensitive fingerprinting frameworks.

Future work may include the incorporation of the LLA framework into OpenMRF [22] and MRZero [23] packages for broader use within the open source MR community. Also, investigating the potential use of LLA in MRF sequences optimized for joint ${\mathrm{B}}_1^{+}$ mapping [24] is an interesting avenue of future research. Finally, this LLA framework may also be useful for accelerating simulations involving adiabatic pulses, whose magnetization response exhibits a strong nonlinear dependence on the RF scaling factor $\beta$.

In summary, a locally linear approximation to RF pulse propagators with respect to RF scaling factors was developed to accelerate slice‐profile‐corrected MRF dictionary generation. By decoupling RF phase from scaling and interpolating propagators across a small number of scaling factors, the proposed approach achieves substantial reductions in computation time while maintaining high accuracy in estimated relaxation parameters. These results demonstrate that accurate modeling of RF pulse effects in 2D MRF can be made computationally practical, potentially enabling broader adoption of slice‐profile‐aware and ${\mathrm{B}}_1^{+}$‐corrected quantitative MRI methods in both research and clinical settings.

Supporting information

Data S1. Supporting Information.

Data Availability Statement

The data that support the findings of this study are openly available in GitLab at: https://gitlab.com/mickevicius‐lab/papers/locally‐linearized‐rf‐scaling‐approximation.