Fast MR signal simulations of microvascular and diffusion contributions using histogram‐based approximation and recurrent neural networks

Abstract

Purpose

Accurate MR signal simulation, including microvascular structures and water diffusion, is crucial for MRI techniques like fMRI BOLD modeling and MR vascular Fingerprinting (MRF), which use susceptibility effects on MR signals for tissue characterization. However, integrating microvascular features and diffusion remains computationally challenging, limiting the accuracy of the estimates. Using advanced modeling and deep neural networks, we propose a novel simulation tool that efficiently accounts for susceptibility and diffusion effects.

Methods

We used dimension reduction of magnetic field inhomogeneity matrices combined with deep learning methodology to accelerate the simulations while maintaining their accuracy. We validated our results through an in silico study against a reference method and in vivo MRF experiments.

Results

This approach accelerates MR signal generation by a factor of almost 13 000 compared to previously used simulation methods while preserving accuracy.

Conclusion

The MR‐WAVES method allows fast generation of MR signals accounting for microvascular structures and water‐diffusion contribution.

1. INTRODUCTION

The diffusion of water molecules in biological tissues is a fundamental process influencing MRI signals. In homogeneous space, diffusion follows an isotropic Brownian motion described by Fick's law, with a characteristic diffusion coefficient specific to the medium. ¹ , ² However, in biological tissues, cell membranes, microvascular structures, and other barriers lead to restricted and anisotropic diffusion, altering the signal properties in diffusion‐sensitive MRI sequences. ³

In diffusion‐weighted MRI, the effect of diffusion is quantified by signal attenuation in the presence of diffusion‐sensitizing gradients, allowing the measurement of apparent diffusion coefficients (ADC) that reflect tissue microstructure. Similarly, in susceptibility‐based MRI techniques such as blood oxygenation level‐dependent (BOLD) functional MRI ⁴ and dynamic susceptibility contrast (DSC) ⁵ imaging, diffusion plays a key role in modulating the phase and magnitude of MR signals in the presence of microscopic magnetic field inhomogeneities. The interaction between water diffusion and local field variations around microvascular structures influences contrast mechanisms, making diffusion an essential factor in accurately modeling and interpreting MR signals in advanced imaging methods.

Accurately simulating MR signals in the presence of microvascular structures and water‐diffusion effects plays an important role in the development of new imaging biomarkers, helping to understand brain physiology and improve diagnostic precision. This is the case for MR vascular Fingerprinting (MRvF ⁶ , ⁷ ) where microvascular simulations are directly compared to in vivo MR signal time courses. However, the computational burden of such simulations remains a critical bottleneck for the generation of large‐scale dictionaries. Initial MRvF studies ⁸ , ⁹ utilize a GESFIDSE sequence (Gradient‐Echo Sampling of Free Induction Decay and Spin Echo) ¹⁰ pre‐ and post‐USPIO injection, comparing concatenated acquired signals with a “dictionary” of simulated signals. This approach enables the quantification of Cerebral Blood Volume (CBV), mean vessel radius (R), relaxation time T2, and oxygenation (SO). Dictionary generation relies on a previously published tool ¹¹ that accounts for susceptibility ¹² and diffusion. Unlike previous diffusion models for which effects were simulated using random walk (Monte Carlo) methods, ¹³ , ¹⁴ , ¹⁵ , ¹⁶ here a deterministic convolution process is performed using a Gaussian model of motion, where the random displacements of water molecules due to diffusion are modeled as a Gaussian distribution of displacements over time. ¹⁷ A Gaussian kernel representing the probability density function of displacements is convolved with the MR signal at each small time step (<0.5 ms) to capture this process. This convolution allows for the precise accumulation of phase shifts induced by diffusion, ensuring the accurate evolution of the MR signal. The small time step is critical to capture the cumulative effect of microscopic diffusion events accurately, but it also significantly increases the computational burden. Consequently, this approach requires intensive computations, leading to long simulation times (often days for thousands of signals), which constrained prior studies to small‐scale dictionaries (less than 30 000 signals). Given the vast diversity of microvascular structures in the human brain and the critical role of water‐diffusion effects in ensuring simulation realism, there is a pressing need for more efficient simulation tools.

In this work, we introduce a fast method for Water‐diffusion and Vascular Effects Simulations (MR‐WAVES). Our contribution is two‐fold: First, as already introduced in a previous study, ¹⁸ we accelerate intra‐voxel microvascular susceptibility calculations by moving from a three‐dimensional (3D) to a one‐dimensional (1D) representation. This time, we also investigate the accuracy of the 1D histogram approximation of 3D magnetic field inhomogeneity matrices, depending on the sampling parameters, such as the bins and the edges of the distribution. Secondly, we introduce a direct approach to diffusion using a Recurrent Neural Network (RNN), which transforms a zero‐diffusion simulated signal based on specified microvascular properties and a fixed ADC value. The combined approach removes the need for constant time steps and, similar to Bloch simulations, computes the magnetization only when events (RF, gradient, etc.) occur.

To validate our method, we compare MR‐WAVES against the described brute‐force method, highlighting the accuracy and computational efficiency trade‐offs. Finally, we will demonstrate the applicability of our method on MR vascular Fingerprinting pre‐clinical data, showcasing its potential to enable larger, more comprehensive analyses of vascular properties in the brain.

2. METHODS

2.1. Computations of magnetic field inhomogeneities

3D microvasculatures were segmented from microscopy‐imaged batches of a healthy mouse brain. The microscopy dataset ¹⁹ of mice brain voxels was processed to obtain a set of MRI‐sized microvascular voxels (i.e., 248 × 248 × 744 $\mu m^3$) and segmented using the VesselExpress software ²⁰ to derived binary masks representing the vascular network (see Figure 1 left). The 3D microvascular volumes were characterized in terms of total blood volume in the voxel (CBV) and mean vessel radius (R), using the same software and a computation resolution of 2 × 2 × 2$\mu m^3$, accounting for vessel radii that are at least 2$\mu m$. To achieve a large diversity of voxels, representing healthy and pathological vascular networks, two types of data augmentation were performed as illustrated in Figure 2: Dilation, using two different structuring elements: A $3\times 3\times 3$ cube (Figure 2A middle) and a sphere with a radius size of 2 (Figure 2A right); and merging (Figure 2B), where two or more segmented MR‐size voxels were superimposed and the final merged segmentation(right panel) was obtained by voxel‐wise logical OR of the two binary 3D. From 20 900 voxels, around 3500 merged voxels were added, and around 20 600 dilated voxels. In the end, this resulted in 45 000 segmented voxels.

FIGURE 1.

Illustrated example of frequency inhomogeneities distribution computation. A microscopy voxel is binary segmented, and susceptibility values associated with the vascular and extravascular components are then used to compute magnetic field inhomogeneity maps. The equivalent frequency inhomogeneity maps are saved as a 3D frequency distribution in the voxel.

FIGURE 2.

Microvascular voxel data augmentation. Examples of augmentation processes: (A) dilation with a cubic kernel (top middle), dilation with a spherical kernel (top right), and (B) merging of two voxels (bottom line). For each case, a 3D view of the microstructure, a 2D segmentation example, and a 1D frequency distribution are shown.

Magnetic susceptibility maps were computed with (${\chi }_{USPIO}$ = 3.5 ppm) and without the presence of a USPIO (Ultrasmall superparamagnetic iron oxide) contrast agent (CA) (see Figure 1). The SO and hematocrit values are fixed for each voxel. From this, 3D magnetic field maps were computed in Python 3.8.10 for each voxel using a Fourier method. ¹² The simulations were performed for a static magnetic field strength of 4.7 T.

2.2. MR Sequence

MR simulations were performed for a GESFIDSE sequence with 32 gradient echoes (repetition time TR = 4000 ms, 32 gradient echoes, $\mathrm{\Delta }$ TE = 3.3 ms, Spin‐Echo = 60 ms), before and after USPIO contrast agent injection. All the following numerical simulations of MR signals are made separately for pre‐ and post‐CA signals.

For the MRF in vivo study (see 2.6), signals acquired before and after USPIO injection were concatenated along the time dimension, resulting in a single 64‐timepoint fingerprint per voxel, composed of 32 pre‐CA and 32 post‐CA echoes. Among diverse possible combinations of pre‐ and post‐CA signals into one fingerprint, it has been shown that the concatenation led to better estimates. ²¹

2.3. Reference: 3D simulations

As a reference in our study, 3D matrices of magnetic field inhomogeneities were computed using a first‐order perturbation approach to Maxwell's magnetostatic equation, combined with the Fourier transformation technique. ¹² Using the fully 3D matrices of size $248\times 248\times 744$, the MR simulations were carried out in MATLAB R2024a, ²² using parallel computation of 64 CPU cores and intensive RAM allocations ⁹ without and with water‐diffusion effect. In the latter case, an apparent diffusion coefficient (ADC) of 850 $\mu {\mathrm{m}}^2/\mathrm{s}$ was chosen, considering it as a mean value in healthy brain tissues. ²³ Assuming that the motion of water molecules follows a normal distribution in space, ¹⁷ diffusion effects were modeled by the convolution of a Gaussian kernel. To ensure the accuracy of the motion simulation, computations were made using a 0.5ms time step. Note that no water exchange between vascular and extravascular compartments is modeled, and no constraint is applied to the diffusion direction. For computational efficiency, signals were first computed for a fixed T₂ value of 100ms. Varying T₂ effects were then added a posteriori by multiplying each GESFIDSE signal by the appropriate exponential decay functions. ⁹

To maintain reasonable computation time, 20 000 microvascular structure voxels among the 45 000 ones, combined with Sobol ²⁴ distributed values of SO and T₂, were used to generate two 20 000‐signal sets, with and without diffusion effect using the brute force approach. We respectively called these methods Ref and Ref in the following of this article (see Figure 4).

FIGURE 4.

Schematic illustration of the proposed method compared to the reference method. The reference method relies on computed 3D frequency inhomogeneity maps with $248\times 248\times 744$ intra‐voxel contribution, to compute signals with a 0.5 ms time‐step to account for water‐diffusion Gaussian motion. The proposed method undersamples the 3D inhomogeneity map to simplify computation and perform microvascular simulation at a time‐step equal to the repetition time (TR), and the water‐diffusion is a posteriori modeled by the RNN.

2.4. MR‐WAVES approach

The source code for our implementation is available here. The proposed MR‐WAVES approach, as introduced, combines two modules that are separately detailed in the two following subsections.

2.4.1. 1D simulations without diffusion effect

Using the Larmor equation (1), the 3D magnetic field ($\delta B$) maps derived from the microvascular voxel can be seen as 3D frequency inhomogeneities ($\delta f$) maps.

$$\delta f=\frac{\gamma }{2\pi }·\delta B$$ (1)

where $\gamma$ is the gyromagnetic ratio (e.g., $\gamma /2\pi =42.58$ MHz/T for protons).

To accelerate simulations, we replaced the full 3D spatial distribution of magnetic field inhomogeneities with a 1D histogram of frequency offsets, neglecting spatial arrangements and assuming that the frequency offsets can be described statistically rather than spatially. ¹⁸ The 1D histogram was sampled using the numpy.histogram function in Python, providing a discrete approximation of the frequency distribution:

$$\mathscr{H}(\delta f)={\{({\delta f}_i,p_i)\}}_{i=1}^N$$ (2)

where ${\delta f}_i$ represents the center of the $i\text{th}$ frequency bin, and $p_i$ its corresponding normalized weight such that:

$$\sum _{i=1}^N{p}_i=1$$ (3)

Individual Bloch simulations ²⁵ were performed for each frequency bin $\delta f_i$, resulting in a complex MR signal $S_i$:

$$S_i=\text{Bloch}(\delta f_i;\theta )$$ (4)

where $\theta$ encompasses the relevant sequence and tissue parameters. The total simulated MR signal $S$ associated with the 3D volume was then obtained as a weighted sum over the histogram (see Figure S1):

$$S=\sum _{i=1}^N{p}_i·S_i$$ (5)

This allowed for fast computation of MR signals at time steps defined by sequence events while accounting for the contribution of microvascular properties (SO, CBV, R). We called this approach $MRWAVE{S}_{nodiff}$ because it doesn't include the water‐diffusion effect (see Figure 4).

Simulations were performed with a mix of Python 3.8.10 and MATLAB R2024a, using parallel computation on 64 CPU cores and intensive RAM allocations.

2.4.2. Fast simulation of the water‐diffusion effect on MR signals using neural networks

In the brute‐force approach, the effect of water diffusion was modeled by the convolution of a Gaussian kernel, updating the MR signal evolutions every 0.5ms time step, which is highly time‐consuming and limits the number of MR signals that can be generated. Yet, the effects of water diffusion that we have observed on the GESFIDSE signals are mainly a shape modification between the signal without and the signal with diffusion.

Recurrent Neural Networks (RNNs) are well‐suited for sequential data, ²⁶ therefore MRI time courses ²⁷ , ²⁸ , ²⁹ as they maintain a memory of previous states, allowing them to model temporal dependencies in time‐series data. While Long Short‐Term Memory (LSTM) networks have already been successfully applied in MR Fingerprinting for signal modeling or parameters estimation, ³⁰ , ³¹ , ³² we opted here to use Gated Recurrent Units (GRU) due to their simpler architecture and faster training times. GRUs retain the key capability of handling long‐term dependencies while being computationally more efficient, making them well‐suited for our application. We implemented and trained a GRU RNN to model the water diffusion effect on MR signals as a surrogate for time‐consuming deterministic methods. Training and testing of the GRU‐RNN were conducted using signals simulated from the brute‐force methods Ref and Ref .

Model parameters

The model followed a GRU‐RNN architecture (see Figure 3) with a single output dimension representing the magnetization vector. The network input was a signal without water diffusion‐effects, along with a set of (SO, CBV, R) tissue parameter values. The network processed batches of 5 signals, while the RNN layer contained 32 units. An Adam optimizer, a popular gradient‐based optimization algorithm, was used to optimize the model with a Mean Squared Error (MSE) loss. Learning rate adjustments were handled such that the learning rate was reduced when the validation loss plateaued, using a factor of 0.8 and patience of 20 epochs. Additionally, a minimum learning rate of $1e^{-6}$ was enforced to prevent it from diminishing to very small values. Model checkpoints were saved based on the training loss, with the best model retained in an HDF5 file named according to the experiment setup. Early stopping was employed with a patience of 15 epochs, halting the training process if the validation loss did not improve, to prevent overfitting. Together, these callbacks ensured efficient training while avoiding overfitting and dynamically adjusting the learning rate based on performance trends. Although dropout and learning rate scheduling were tested, they did not yield performance improvements in the final model and were therefore excluded for simplicity. Additionally, we note that the diversity of the training dataset naturally acts as a form of regularization. This variability ensures that the model does not memorize specific signal profiles, improving its robustness and generalization to unseen data.

FIGURE 3.

Schematic illustration of the GRU‐RNN layers architecture. The zero‐diffusion MR signal (pre‐ or post‐CA) of 32 time points is taken as input and concatenated with a triplet of microvascular parameters. The effect of water‐diffusion for an AD of 850 $\mu {\mathrm{m}}^2/\mathrm{s}$ is then applied through GRU layers, leading to an output MR signal of 32 time points.

Training set

As for the simulation process, the signals were split into pre‐ and post‐part, and the network was trained in two separate models. All signal magnitudes were normalized. The prediction is also made for pre‐ and post‐CA parts of the signals that are then concatenated. The training set was made of 85% of 20 000 signals from Ref and Ref , that is, 17 000 signals simulated using the brute‐force method for the GESFIDSE sequence described previously, once without diffusion effect, and then with diffusion effect to serve as ground truth. 5% of the dataset (i.e., 1000 signals) were used for validation.

Testing set

The testing of the model was made on the remaining signals not used for training or validation, that is, 2000 signals with and without diffusion as the ground truth for evaluating the model. Here we call the method GRU to specify that the vascular simulations are done using the 3D matrices brute‐force method, but the diffusion is added with the GRU‐RNN (see Figure 4).

All model computations were made on an NVIDIA GeForce RTX 3060 GPU.

2.5. In silico validation study

MR simulations without diffusion

We first wanted to assess the accuracy of the dimension reduction technique of the 3D frequency inhomogeneity matrices. For this, we adjusted the sampling strategy in the histogram (i.e., its number of bins and its edges) to control the granularity of the frequency representation (i.e., the sampling step in Hz) and find a trade‐off between accuracy and computational efficiency. We made this with and without UPSIO contribution in the susceptibility computation.

The fully resolved 3D frequency matrices served as the reference for the accuracy evaluation. For both pre‐ and post‐CA simulations, we then compared the set of 20 000 MR signals computed from the brute‐force method (Ref , using the 3D matrices) and sets of 20 000 MR signals computed with the proposed MR‐WAVES method (MRWAVES ) using different 1D histogram samplings of the 3D matrices.

The accuracy was quantified using the signal‐wise Normalized Root Mean Squared Error (NRMSE) between the brute‐force signals and those obtained with different sampling parameters.

MR simulations with diffusion

Then we compared the proposed method, MR‐WAVES, against the brute‐force method, incorporating this time the diffusion effects by testing the trained RNN model on the 2000 signals. To further investigate the results, this testing was made on the 2000 signals computed from the brute‐force methods using full 3D matrices (Ref ), and with the 1D distribution histograms (MRWAVES ). We thus obtained 2000 output signals under a method we called MRWAVES . We used the Normalized Root Mean Squared Error (NRMSE) across dictionary entries to evaluate the accuracy of the result outputs.

To further investigate the validity of our approach, we also studied the dependency of the GESFIDSE signal on the voxel CBV. A total of 116 CBV values between 0.13% and 19.42% were selected among the 3D realistic voxels, for a mean vessel radius R constrained between 5.81 $\mu m$ and 5.91 $\mu m$. The oxygenation was set to SO = 81%, T2 = 100 ms and ADC fixed at 850 $\mu$ m. Specifically, we used the 6th echo (${\text{TE}}_{\text{GE}}\approx$ 20 ms) to estimate GE‐like and the 17th echo (${\text{TE}}_{\text{SE}}\approx$ 60 ms) for SE‐like relaxation rates. The relaxation rates were computed using Equations 6 and 7.

$$\mathrm{\Delta }{R}_2^{\ast }=-\frac{\ln (S_{\text{post}}^{\text{GE}}/S_{\text{pre}}^{\text{GE}})}{{\text{TE}}_{\text{GE}}}$$ (6)

$$\mathrm{\Delta }{R}_2=-\frac{\ln (S_{\text{post}}^{\text{SE}}/S_{\text{pre}}^{\text{SE}})}{{\text{TE}}_{\text{SE}}}$$ (7)

where $S_{\text{pre}}$ and $S_{\text{post}}$ are the normalized GESFIDSE signal magnitude values for the specific TE, before and after contrast injection.

2.6. In vivo validation study

We evaluated the accuracy of our simulation method against the brute‐force method in an MR Fingerprinting ³³ application, in healthy rats. The sets of MR signals computed for the in silico validation of our approach are used here as MRF dictionaries. The animal data are extracted from previous MRvF studies. ³⁴

2.6.1. Animal models

Wistar rats ($N=8,268\pm 23$ g) were imaged with the below‐described MR protocol. All procedures were reviewed and approved by the local ethics committee (Comité éthique du GIN n°004), were performed under permits 380 820 and A3851610008 (for experimental and animal care facilities) from the French Ministry of Agriculture (Articles R214–117 to R214–127 published on February 7, 2013), and reported in compliance with the ARRIVE guidelines (Animal Research: Reporting in Vivo Experiments).

2.6.2. MRI acquisition

MRI was conducted with a horizontal bore 4.7 T Biospec animal imager (Bruker Biospin, Ettlingen, Germany; IRMaGe facility) with an actively decoupled cross‐coil setup (body coil for radiofrequency transmission and quadrature surface coil for signal reception) and Paravision 5.0.1.

A GESFIDSE sequence (repetition time TR = 4000 ms, 32 gradient‐echoes, $\mathrm{\Delta }$ TE = 3.3 ms, Spin‐Echo = 60 ms; NEX = 1, 5 slices, FOV = 30 × 30 mm, matrix = 128 × 96 zero‐filled to $128\times 128$ and voxel size = 234 $\times$ 234 $\times$ 800 $\mu m^3$, acquisition duration 6 min 24 s) was performed before and after the manual injection of the ultrasmall superparamagnetic iron oxide nanoparticles (USPIO) P904 (200$\mu$ mol/kg body weight; Guerbet, Roissy, France). A three‐minute delay after the first injection was applied before starting the second GEFIDSE acquisition.

Anatomical T2‐weighted (T2w) images were acquired for ROI delineation, using a turbo spin‐echo MRI sequence (TR = 4000 ms, echo‐time TE = 33 ms, NEX = 2, 31 slices, FOV = 30 $\times$ 30 mm, matrix = 128 $\times$ 128 and voxel size = 234 $\times$ 234 $\times$ 800 $\mu m^3$, acquisition duration 4 min 17 s).

2.6.3. Image analysis

Quantitative parameter maps of CBV, R, SO, and T2 were computed from the GESFIDSE data acquired on healthy animals, relying on a standard dictionary‐matching process (inner‐product ³³ ), made on MP3, ³⁵ a MATLAB‐based image processing software, using an in‐house fingerprinting extension module.

In the first three validation studies, MRF parameter maps were reconstructed

1. with a 20 000‐entry MRF dictionary simulated without diffusion effect using brute‐force method (Ref ) versus MR‐WAVES (MRWAVES )

2. with a 2000‐entry MRF dictionary with diffusion computed using the brute‐force method (Ref ) versus the proposed GRU‐RNN for diffusion only (GRU )

3. with a 2000‐entry MRF dictionary with diffusion computed using the brute‐force method (Ref ) versus the whole MR‐WAVES (MRWAVES )

Results using a dictionary computed with the proposed and the brute‐force methods were compared using relative difference and Bland‐Altman analysis.

Finally, as the speed of MR‐WAVES simulations allows large‐scale dictionary generation, we used it to produce an MRF dictionary of 135 000 MR signals computed using 45 000 voxels with varying [35%–95%] SO and [45–150 ms] T₂ values. Mean parameter values were computed in manually drawn ROIs: Skull‐extracted brain, cortex, and striatum areas. Examples of ROIs can be seen in Figure 9. Due to the long computation time of the brute‐force method, no reference dictionary of 135 000 MR signals was computed.

FIGURE 9.

(A) One‐slice MRvF parametric maps for each animal of the study, reconstructed using a 135 000‐entry dictionary computed with $MRWAVE{S}_{diff}$. (B) Mean and standard deviation of parameter estimates in brain ROIs are displayed on the anatomical T2w scan on the left.

For diffusion‐related studies, the dictionaries used in the in vivo MRF reconstruction were not drawn from the RNN training or test datasets but independently generated.

3. RESULTS

3.1. Brute force 3D simulations

Excluding the computation of the 3D magnetic field inhomogeneities matrices, the brute force 3D simulations Ref and Ref compute respectively at a speed of 3 and 0.2 signals/s (i.e., 1 h 51 min and 25 h 03 min for 20 000‐entry datasets). For the reference method, the computation time increased linearly with the number of signals simulated (data not shown).

3.2. Fast 1D simulations without diffusion effect

Figure 2 highlights the effect of the microvascular characteristics of the voxel on the 1D frequency histogram. Changes in the CBV and R values (here related to data‐augmented voxels) have a strong effect on the frequency distributions. The sensitivity to this effect is of high importance to maintain the accuracy of the simulation compared to fully resolved 3D matrices.

After conducting numerous experiments adjusting the sampling of the 3D matrices using 1D distribution histograms, the optimal strategy that balances computational efficiency and accuracy was identified. The results, shown in Figure 5, highlight the final approach based on a specific number of bins and bin edges that effectively represent the optimal frequency “step” in Hz, which is the focus of our analysis. Only the most relevant tests are presented, computed for fixed histogram edges (−200; 200 Hz) and the adapted number of bins to vary the sampling step. For example, on a −200; 200 Hz interval, achieving a 2 Hz sampling step means that the histogram is made of 200 bins.

FIGURE 5.

NRMSE computed between proposed ($MRWAVE{S}_{nodiff}$ with several sampling) and reference ($Re{f}_{nodiff}$) dictionaries for pre (blue) and post‐CA (red) simulated MR signals. NRMSE is computed for different sampling steps (from 10 to 0.5 Hz for post‐CA and 0.25 Hz for pre‐CA) of the histogram frequency distribution into a microvascular voxel. As the number of bins in the histogram increases, the sampling step decreases, and so does the NRMSE between the two methods.

As could be expected, Figure 5 shows that the NRMSE between the brute‐force method (Ref ) and our proposed approach (MRWAVES ) decreases with smaller sampling step sizes, reflecting a more accurate description of the 3D matrices using less discretized 1D histograms.

As described previously (2.1), the magnetic susceptibility of the vascular component is adjusted to account for the contrast‐agent effect, which broadens the frequency distribution, as can be seen in Figure 1. Given that the Full‐Width at Half Maximum (FWHM) of the post‐CA distributions is larger than that of the pre‐CA distributions, a larger frequency interval (i.e., histogram edges) is necessary for post‐CA signals to ensure a realistic description of the voxel's frequency inhomogeneity (see Figure 5 where NRMSE stabilizes at larger step sizes for post‐CA signals).

Based on these observations, we set the 1D distribution parameters as follows: For pre‐CA, a range from −100 to 100 Hz with a 0.5 Hz step; for post‐CA, a range from −200 to 200 Hz with a 1 Hz step. This approach balances computational efficiency and simulation accuracy, yielding an NRMSE of 0.07% for pre‐CA and 0.16% for post‐CA compared to the fully resolved 3D matrix method. On the signal concatenating pre‐ and post‐CA parts, the normalization slightly increased the NRMSE to 0.26%.

Using this description, the $MRWAVE{S}_{nodiff}$ method computes 20 000 MR signals with varying microvascular properties (SO, CBV, R) and T2 in 32 s. This corresponds to a simulation speed of 625 signals per second, achieving a 200‐fold acceleration compared to the Ref approach (3.1).

Using the GESFIDSE data acquired on the 8 animals of our study, MRF parameters maps of (CBV, R, SO, and T₂) are reconstructed in 20 s of matching time per animal, using dictionaries computed with both methods. In Figure S2, examples of reconstructed MRF maps are shown for one animal of our study, comparing the reconstruction made with the brute force and our proposed method using frequency histogram distributions. Relative difference maps are shown on the bottom line. Density histograms of the relative difference across the voxels of the slices are also shown, exhibiting a strong similarity between both methods.

3.3. GRU‐RNN simulations of the water‐diffusion effect on MR signals

The training on the GRU RNN lasts approximately 300 s. On the 2000 testing signals, the prediction time is just 3 s. Considering that input signals are here computed using Ref , it means a total simulation time of 670 s for 2000 entries, meaning a speed of 3 signals per second compared to 0,2 signals per second for the Ref brute force method. Figure 6A illustrates examples of predicted signals, comparing them with both the input signals (without diffusion) and the expected ground truth for the pre‐ and post‐CA sections. It can be seen that the prediction closely matched the ground truth signals from Ref . Figure 6B presents the normalized root mean square error (NRMSE) calculated between the 2000 predicted signals (GRU ) and their corresponding ground truth (Ref ) across the parameter space. The overall error is low (<4%) and uniformly distributed across different parameter values. However, notable exceptions are observed for signals corresponding to large vessels (R>$15\mu$ m and CBV>5%), where the error is highest. These cases can be considered outliers, as such values are uncommon in healthy rat brains.

FIGURE 6.

2000 signals without diffusion computed with the brute force method (from $Re{f}_{nodiff}$) are used as input in the GRU‐RNN. Results are compared with brute force signals computed with diffusion (from $Re{f}_{diff}$). (A) GRU‐RNN prediction examples ($GR{U}_{diff}$) for two voxels with fixed CBV, R, and SO2 values, where the signal is computed without and with CA contribution. Ground truth signal as well as zero‐diffusion signal used for the prediction are shown. (B) NRMSE between GRUdiff prediction and Refdiff ground truth, across the whole parameter space.

In Figure S3, we show an example of the results on one slice from one of the 8 healthy animals. The relative difference maps highlight the similarity between the results using both simulation methods.

3.4. Complete MR‐WAVES method

3.4.1. MRWAVES versus Ref for small MRF dictionary generation

The mean NRMSE between 2000 signals (pre and post‐CA concatenation) simulated with the brute‐force method (Ref ) on one side and MR‐WAVES (MRWAVES , combining 1D histograms and GRU‐RNN) on the other side is 2.15% for pre and 1.16% for post‐CA signals, reflecting that the main error originates from the GRU‐RNN, for which the error was also of this order. Using MR‐WAVES, simulation time is less than 5 s, including fast microvascular simulations and GRU‐RNN inference time, that is, a simulation speed of around 400 signals per second. For a small dictionary, loading the external data and the inference model, and saving the signals used a significant part of the simulation time, artificially decreasing the simulation speed.

In Figure 7A, MRF maps of CBV, R, SO and T₂ computed with a dictionary with diffusion from the Ref method and the proposed MR‐WAVES dictionary combining fast simulations and fast diffusion representation (MRWAVES ) are shown for single‐slice examples of one animal, supplemented with relative difference maps.

FIGURE 7.

(A) One slice example of MRvF parametric maps reconstructed from a 2000‐entry dictionary computed for an ADC of 850 $\mu {\mathrm{m}}^2/\mathrm{s}$ using the brute force method ($Re{f}_{diff}$, top) and the proposed MRWAVES ($MRWAVE{S}_{diff}$, bottom). Relative difference maps are also shown for each estimated parameter. (B) Mean and standard deviation of parameter estimates, in brain ROIs drawn from anatomical T2w images, across all the animals of the study group.

Visually, the resulting maps computed from the MR‐WAVES model are of the same quality as the reference maps. This observation is confirmed by the mean and standard deviation values for the estimated parameters across the whole animal group (Figure 7B), and the Bland‐Altman plots in the whole brain for all animals in Figure 8. For the CBV, the plot shows a small bias of around −0.15, indicating that MR‐WAVES tends to slightly overestimate CBV compared to the brute‐force method. The limits of agreement (±1.96 SD) range is narrow, suggesting a good level of agreement between the methods. For the R estimates, the healthy population plot reveals a larger bias of approximately −0.4, with all data points within the limits of agreement, though some variance is observed. For the SO parameter, analysis shows a minimal bias and excellent agreement between methods, as the differences hover around zero. For T₂, the healthy population shows good agreement between methods, with a bias close to zero and minimal spread, indicating consistency. Overall, the Bland‐Altman analysis highlights good agreement between the brute‐force method and MR‐WAVES.

FIGURE 8.

Bland‐Altman analyses across all animals of the study in three ROIs of the brain. Bias is indicated by the red line. Limits of agreements (±1.96 SD) are indicated by the blue lines. Each point represents one animal.

3.4.2. MRWAVES for large MRF dictionary generation

Finally, a large MRF dictionary of 135 000 entries is studied. Using MR‐WAVES and considering that intra‐voxel frequency distributions are precomputed for a large database of tissue parameters, the generation of the dictionary took 74 s for pre‐ and post‐CA signals, including the inference of the RNN in 22 s. Based on our observation of the simulation time using the brute‐force method linearly increasing with respect to the number of simulated signals, the equivalent simulation of an MRF dictionary of 135 000 entries would take 278 h. Thus, we observe an acceleration factor of 13 000 with the proposed MR‐WAVES method.

From this simulated large dictionary, MRF maps of CBV, R, SO, and T₂ on animal models are computed and shown for one slice of each animal in Figure 9A. In Figure 9B, detailed estimation values are plotted for all animals with the corresponding standard deviation. A detailed description of individual parameter estimated values is provided in Table 1. MRF estimates of microvascular and relaxometry parameters are consistent across animals and in line with expected literature values. ⁹

TABLE 1.

ROI values for whole brain, cortex, and striatum for each animal of the study (computed with the 135 000 entry MR‐WAVES dictionary).

Region	CBV (%)	R (μm)	SO (%)	T (ms)
Brain	4.1590	5.2948	73.2511	60.6642
	4.8557	5.4027	66.7667	61.1601
	3.7131	4.8959	72.2098	57.9048
	3.5615	4.9131	76.0095	56.0804
	4.4469	5.1905	73.5398	58.3167
	4.0613	4.9498	73.9492	57.1638
	5.0453	5.3207	69.9437	58.7428
	4.1632	5.1035	73.4670	58.2198
Cortex	3.3207	4.5049	77.8120	57.2103
	2.8993	4.3204	72.7278	56.5981
	2.7885	4.2270	79.3297	53.9291
	2.3842	3.8678	78.8777	53.0398
	3.4252	4.2866	78.2382	54.5659
	3.4364	4.3209	77.6453	55.9036
	4.9560	4.9892	76.4490	57.5645
	3.0758	4.2030	78.9589	54.7975
Striatum	2.8166	4.4798	72.0547	55.9499
	3.4512	4.8342	76.1648	54.3757
	2.3605	4.0911	75.2158	54.7879
	2.2498	4.0942	85.2227	51.4786
	3.1726	4.5476	77.0392	53.3658
	3.1312	4.5599	78.3934	54.1147
	3.4182	4.8302	76.3880	57.6723
	2.7379	4.2582	79.0244	54.2772

It can be observed that the results are different from those obtained with smaller dictionaries (Figure 7A), especially for estimated values of R and SO that seem to be increased whereas the CBV and T₂ values are consistent with previous observations but smoother because the sampling of the parameters in the dictionary is thinner. We propose a highlighted comparison in Figure S4, where representative animal slices and ROIs across the whole group are shown for MRvF maps derived from a 2000‐entry and a 135 000‐entry dictionary.

3.4.3. Relaxation changes versus CBV

Figure S5 shows the dependence of $\mathrm{\Delta }{R}_2^{\ast }$ and $\mathrm{\Delta }{R}_2$ on cerebral blood volume (CBV), computed using the MR‐WAVES framework. In our simulations, each CBV value corresponds to a distinct microvascular geometry extracted from microscopy acquisitions, as described in Section 2.1. Because a single simulation is run per 3D vascular network, some variability is observed across the CBV range. Nevertheless, a linear relationship between $\mathrm{\Delta }{R}_2^{\ast }$ or $\mathrm{\Delta }{R}_2$ and CBV is observed in the low‐CBV range (4%–10%), consistent with the trend described by Boxerman and colleagues, ³⁶ with a steeper slope for $\mathrm{\Delta }{R}_2^{\ast }$. However, for higher CBV values, the relationship becomes nonlinear due to structural coupling effects: Larger CBV values often arise from more or larger vessels, which disproportionately affect transverse relaxation because the vascular volume fraction scales with radius squared. Although the mean radius remains nearly constant across samples, the effective vessel size distribution shifts with CBV, explaining the deviation from linearity observed at high CBV. This structural coupling leads to $\mathrm{\Delta }{R}_2$ and $\mathrm{\Delta }{R}_2^{\ast }$ values that evolve with CBV and that could mimic the Boxerman $\mathrm{\Delta }{R}_2$(R) and $\mathrm{\Delta }{R}_2^{\ast }$(R) curves. ¹¹ , ³⁶ Notably, $\mathrm{\Delta }{R}_2^{\ast }$ is systematically greater than $\mathrm{\Delta }{R}_2$ and shows a plateau at higher CBV values. Meanwhile, $\mathrm{\Delta }{R}_2$ exhibits a peak near 10% CBV before declining. CBV variations were selected as the primary parameter of interest here because realistic 3D microvascular structures inherently include diverse vessel size distributions, such that different geometries may share a similar average radius despite structural heterogeneity.

4. DISCUSSION AND CONCLUSIONS

In this paper, we introduced MR‐WAVES, a novel simulation method for incorporating water‐diffusion and vascular effects into MR signal simulations. By combining a simplified representation of intra‐voxel magnetic field inhomogeneities with the GRU‐RNN, MR‐WAVES achieves over 13 000 times the computational speed of previously proposed brute‐force methods while maintaining the accuracy of the signal simulations.

In silico studies show that concerning the first fast simulation tools utilizing pre‐computed intra‐voxel frequency distributions, our results align closely with brute‐force simulation outputs, as confirmed by statistical analyses on simulated MR signals.

For the RNN implementation, two models were trained on separate datasets for pre‐CA and post‐CA signals, following the two‐stage acquisition protocol. Since the recurrence in the RNN model takes into account time‐dependent data points in the processed inputs, this approach reduces errors by decoupling the last time point of the pre‐CA signal from the first time point of the post‐CA signal. The predicted outputs showed an NRMSE of up 2% compared to deterministic simulations of water diffusion.

The MRF in vivo validation study that we conducted used the proposed simulation methods for the computation of parametric maps for CBV, R, SO, and T₂ while accounting for the effects of water diffusion. With the proposed GRU‐RNN, the reconstruction was effective in generating accurate quantitative maps for CBV, R, SO, and T₂. The slight prediction errors appear to have a limited impact on matching, especially given the noise in acquired signals due to undersampling. However, during RNN training, we observed minor overfitting, which could contribute to prediction errors for data outside the training parameter space. To address this, we propose to further employ data augmentation techniques or add dropout layers to the RNN architecture. Additionally, for longer sequences like MRF‐bSSFP, more robust networks like LSTMs ³² may be considered for longer sequences to improve the diffusion effect simulations.

When the full protocol MR‐WAVES was tested against a brute‐force method with diffusion on a dictionary of 2000 entries, differences in the reconstructed parametric maps were a little bit more pronounced, as expected. The approximation of intra‐voxel inhomogeneities in the first part of MR‐WAVES could contribute to error propagation through the RNN. Because relaxation was modeled using a simple exponential form, introducing bias from prior simulation errors, we believe that incorporating T₂ from the beginning of the Bloch simulations could further reduce the estimate errors, although this would extend simulation time.

While our network is trained on 3D simulations and applied to 1D‐simulated signals for efficiency, we acknowledge that direct 1D diffusion modeling using the Bloch‐Torrey equation is also a valid alternative. However, we believe it may miss some of the spatial interactions captured by our training using 3D‐simulated signals.

In the in vivo MRF application involving a larger dictionary, R and SO values were higher than those observed in the 2000‐entry experiment. This could result from differences in parameter distributions. With the 135 000‐entry dictionary, our results are in line with previous experiments ⁹ for which the method has been validated against histological and analytical estimations. Our SO estimates specifically align in cases with a simplified cylindrical vascular model, though they exceed those derived from more realistic microscopy‐based structures. While the 1D approximation limits spatial fidelity and might oversimplify the microvasculature representation, increasing the density of frequency bins improves the accuracy of the simulated signal by reducing discretization artefacts.

Similarly to previous MRvF studies, ⁹ , ¹⁸ while our dataset was augmented using dilation and merging strategies to produce variability in CBV and R, all data ultimately originate from the same mouse microscopy dataset. A follow‐up study will aim to focus on the impact of new microvasculatures, such as those derived from real human data ³⁷ or synthetically generated ones, to extend the parameter space coverage and potentially improve the estimates.

Future work will also focus on adapting MR‐WAVES to incorporate multiple ADCs in the MR simulations framework. Preliminary tests suggest that certain neural network architectures can effectively learn water diffusion representations across varying ADC values. This extension has the potential to improve microvascular measurements in pathological tissues such as stroke or tumor, where the fixed ADC value of 850 $\mu$ m may no longer be appropriate ³⁸ within the MRF dictionary. While the sensitivity of the GESFIDSE sequence to variations in ADC is relatively limited, appropriate sequence design could even enable MRF‐based estimation of ADC maps, which is particularly relevant for stroke diagnosis. Future developments of MR‐WAVES could also incorporate diffusion anisotropy or non‐Gaussian propagators, especially for applications in white matter or tumor tissues. This would require either modifying the convolution kernel in the brute‐force method or retraining the GRU‐RNN on anisotropic datasets. Such extensions would likely increase model complexity but could improve physiological specificity in complex tissues.

Finally, our proposed MR‐WAVES method appears to be generalizable to a broad range of MR sequences. Notably, in a previous study, ¹⁸ fast microvascular simulations based on 1D frequency distributions were successfully implemented with balanced steady‐state free precession (bSSFP) sequences. This approach enabled non‐contrast perfusion estimates in human volunteers, leveraging simplified frequency distributions for microvascular modeling. Our initial observations suggest that the water‐diffusion effect model can be rapidly trained on signals from sequences other than GESFIDSE and produces similarly accurate results. This also opens the possibility of developing more robust neural network architectures capable of learning signal representations from multiple sequences simultaneously. Such an approach could provide a generalizable simulation tool, however, leading to longer training times and possibly reduced estimation accuracy.

The proposed MR‐WAVES also aligns with the ongoing efforts in microstructure MRI to refine model‐based tissue characterization. ³⁹ , ⁴⁰ , ⁴¹ , ⁴² By efficiently simulating diffusion processes at different scales, it could contribute to resolving key challenges outlined in recent literature, such as the identification of intra‐ versus extra‐cellular diffusion components and the MRI characterization of microstructures.

In summary, the proposed tool provides a powerful framework for fast MR simulations in microvascular structures considering the water‐diffusion effect. Fast numerical computations of such tissue properties effects serve to improve both our understanding of the BOLD effect and the quantification of microvascular tissue parameters using MRF methods. By enabling more flexible and computationally efficient simulations, it has the potential to broaden the application of MRF‐based techniques and microstructure diffusion‐MRI.

FUNDING INFORMATION

The MRI facility IRMaGe is partly funded by the French program “Investissement d'avenir” run by the Agence Nationale de la Recherche, grant “Infrastructure d'avenir en Biologie et Santé” [ANR‐11‐INBS‐0006]. The project is supported by the Agence Nationale de la Recherche [ANR‐20‐CE19‐0030 MRFUSE].

Supporting information

Data S1. Supporting Information.

Coudert T, Silva Martins Marçal M, Delphin A, et al. Fast MR signal simulations of microvascular and diffusion contributions using histogram‐based approximation and recurrent neural networks. Magn Reson Med. 2025;94(5):2234‐2248. doi: 10.1002/mrm.30629