Robust Monte-Carlo Simulations in Diffusion-MRI: Effect of the Substrate Complexity and Parameter Choice on the Reproducibility of Results

Molecular Imaging with Aquaporin-Based Reporter Genes: Quantitative Considerations from Monte Carlo Diffusion Simulations

Aquaporins provide a unique approach for imaging genetic activity in deep tissues by increasing the rate of cellular water diffusion, which generates a magnetic resonance contrast. However, distinguishing aquaporin signals from the tissue background is challenging because water diffusion is influenced by structural factors, such as cell size and packing density. Here, we developed a Monte Carlo model to analyze how cell radius and intracellular volume fraction quantitatively affect aquaporin signals. We demonstrated that a differential imaging approach based on subtracting signals at two diffusion times can improve specificity by unambiguously isolating aquaporin signals from the tissue background. We further used Monte Carlo simulations to analyze the connection between diffusivity and the percentage of cells engineered to express aquaporin and established a mapping that accurately determined the volume fraction of aquaporin-expressing cells in mixed populations. The quantitative framework developed in this study will enable a broad range of applications in biomedical synthetic biology, requiring the use of aquaporins to noninvasively monitor the location and function of genetically engineered devices in live animals.

Monte Carlo simulation of water diffusion through cardiac tissue models

Monte Carlo diffusion simulations are commonly used to establish a reliable ground truth of tissue microstructure, including for the validation of diffusion-weighted MRI. However, selecting simulation parameters is challenging and affects validity and reproducibility. We conducted experiments to investigate critical conditions in Monte Carlo simulations, such as tissue representation complexity, simulated molecules, update duration, and compartment size. Results show significant changes in microstructure characteristics when parameters are altered, emphasizing the importance of careful control for a reliable ground truth.

Tractography passes the test: Results from the diffusion-simulated connectivity (disco) challenge

Estimating structural connectivity from diffusion-weighted magnetic resonance imaging is a challenging task, partly due to the presence of false-positive connections and the misestimation of connection weights. Building on previous efforts, the MICCAI-CDMRI Diffusion-Simulated Connectivity (DiSCo) challenge was carried out to evaluate state-of-the-art connectivity methods using novel large-scale numerical phantoms. The diffusion signal for the phantoms was obtained from Monte Carlo simulations. The results of the challenge suggest that methods selected by the 14 teams participating in the challenge can provide high correlations between estimated and ground-truth connectivity weights, in complex numerical environments. Additionally, the methods used by the participating teams were able to accurately identify the binary connectivity of the numerical dataset. However, specific false positive and false negative connections were consistently estimated across all methods. Although the challenge dataset doesn't capture the complexity of a real brain, it provided unique data with known macrostructure and microstructure ground-truth properties to facilitate the development of connectivity estimation methods.

CACTUS: a computational framework for generating realistic white matter microstructure substrates

Monte-Carlo diffusion simulations are a powerful tool for validating tissue microstructure models by generating synthetic diffusion-weighted magnetic resonance images (DW-MRI) in controlled environments. This is fundamental for understanding the link between micrometre-scale tissue properties and DW-MRI signals measured at the millimetre-scale, optimizing acquisition protocols to target microstructure properties of interest, and exploring the robustness and accuracy of estimation methods. However, accurate simulations require substrates that reflect the main microstructural features of the studied tissue. To address this challenge, we introduce a novel computational workflow, CACTUS (Computational Axonal Configurator for Tailored and Ultradense Substrates), for generating synthetic white matter substrates. Our approach allows constructing substrates with higher packing density than existing methods, up to 95% intra-axonal volume fraction, and larger voxel sizes of up to 500μm3 with rich fibre complexity. CACTUS generates bundles with angular dispersion, bundle crossings, and variations along the fibres of their inner and outer radii and g-ratio. We achieve this by introducing a novel global cost function and a fibre radial growth approach that allows substrates to match predefined targeted characteristics and mirror those reported in histological studies. CACTUS improves the development of complex synthetic substrates, paving the way for future applications in microstructure imaging.

Molecular imaging with aquaporin-based reporter genes: quantitative considerations from Monte Carlo diffusion simulations

Aquaporins provide a new class of genetic tools for imaging molecular activity in deep tissues by increasing the rate of cellular water diffusion, which generates magnetic resonance contrast. However, distinguishing aquaporin contrast from the tissue background is challenging because water diffusion is also influenced by structural factors such as cell size and packing density. Here, we developed and experimentally validated a Monte Carlo model to analyze how cell radius and intracellular volume fraction quantitatively affect aquaporin signals. We demonstrated that a differential imaging approach based on time-dependent changes in diffusivity can improve specificity by unambiguously isolating aquaporin-driven contrast from the tissue background. Finally, we used Monte Carlo simulations to analyze the connection between diffusivity and the percentage of cells engineered to express aquaporin, and established a simple mapping that accurately determined the volume fraction of aquaporin-expressing cells in mixed populations. This study creates a framework for broad applications of aquaporins, particularly in biomedicine and in vivo synthetic biology, where quantitative methods to measure the location and performance of genetic devices in whole vertebrates are necessary.

Diffusion MRI signal cumulants and hepatocyte microstructure at fixed diffusion time: Insights from simulations, 9.4T imaging, and histology

Purpose

Relationships between diffusion‐weighted MRI signals and hepatocyte microstructure were investigated to inform liver diffusion MRI modeling, focusing on the following question: Can cell size and diffusivity be estimated at fixed diffusion time, realistic SNR, and negligible contribution from extracellular/extravascular water and exchange?

Methods

Monte Carlo simulations were performed within synthetic hepatocytes for varying cell size/diffusivity L/D0, and clinical protocols (single diffusion encoding; maximum b‐value: {1000, 1500, 2000} s/mm²; 5 unique gradient duration/separation pairs; SNR = {∞, 100, 80, 40, 20}), accounting for heterogeneity in (D0,L) and perfusion contamination. Diffusion (D) and kurtosis (K) coefficients were calculated, and relationships between (D0,L) and (D,K) were visualized. Functions mapping (D,K) to (D0,L) were computed to predict unseen (D0,L) values, tested for their ability to classify discrete cell‐size contrasts, and deployed on 9.4T ex vivo MRI‐histology data of fixed mouse livers

Results

Relationships between (D,K) and (D0,L) are complex and depend on the diffusion encoding. Functions mapping D,K to (D0,L) captures salient characteristics of D0(D,K) and L(D,K) dependencies. Mappings are not always accurate, but they enable just under 70% accuracy in a three‐class cell‐size classification task (for SNR = 20, bmax = 1500 s/mm², δ = 20 ms, and Δ = 75 ms). MRI detects cell‐size contrasts in the mouse livers that are confirmed by histology, but overestimates the largest cell sizes.

Conclusion

Salient information about liver cell size and diffusivity may be retrieved from minimal diffusion encodings at fixed diffusion time, in experimental conditions and pathological scenarios for which extracellular, extravascular water and exchange are negligible.

Does powder averaging remove dispersion bias in diffusion MRI diameter estimates within real 3D axonal architectures?

Noninvasive estimation of axon diameter with diffusion MRI holds the potential to investigate the dynamic properties of the brain network and pathology of neurodegenerative diseases. Recent studies use powder averaging to account for complex white matter architectures, but these have not been validated for real axonal geometries from regions that contain fibre crossings. Here, we present 120-304μm long segmented axons from X-ray nano-holotomography volumes of a splenium and crossing fibre region of a vervet monkey brain. We show that the axons in the complex crossing fibre region, which contains callosal, association, and corticospinal connections, are larger and exhibit a wider distribution than those of the splenium region. To accurately estimate the axon diameter in these regions, therefore, sensitivity to a wide range of diameters is required. We demonstrate how the q-value, b-value, signal-to-noise ratio and the assumed intra-axonal parallel diffusivity influence the range of measurable diameters with powder average approaches. Furthermore, we show how Gaussian distributed noise results in a wider range of measurable diameter at high b-values than Rician distributed noise, even at high signal-to-noise ratios of 100. The number of gradient directions is also shown to impose a lower bound on measurable diameter. Our results indicate that axon diameter estimation can be performed with only few b-shells, and that additional shells do not improve the accuracy of the estimate. For strong gradients available on human Connectom and preclinical scanners, Monte Carlo simulations of diffusion confirm that powder averaging techniques succeed in providing accurate estimates of axon diameter across a range of sequence parameters and diffusion times, even in complex white matter architectures. At relatively low b-values, the diameter estimate becomes sensitive to axonal microdispersion and the intra-axonal parallel diffusivity shows time dependency at both in vivo and ex vivo intrinsic diffusivities.

The diffusion-simulated connectivity (DiSCo) dataset

The methodological development in the mapping of the brain structural connectome from diffusion-weighted magnetic resonance imaging (DW-MRI) has raised many hopes in the neuroscientific community. Indeed, the knowledge of the connections between different brain regions is fundamental to study brain anatomy and function. The reliability of the structural connectome is therefore of paramount importance. In the search for accuracy, researchers have given particular attention to linking white matter tractography methods – used for estimating the connectome – with information about the microstructure of the nervous tissue. The creation and validation of methods in this context were hampered by a lack of practical numerical phantoms. To achieve this, we created a numerical phantom that mimics complex anatomical fibre pathway trajectories while also accounting for microstructural features such as axonal diameter distribution, myelin presence, and variable packing densities. The substrate has a micrometric resolution and an unprecedented size of 1 cubic millimetre to mimic an image acquisition matrix of 40×40×40 voxels. DW-MRI images were obtained from Monte Carlo simulations of spin dynamics to enable the validation of quantitative tractography. The phantom is composed of 12,196 synthetic tubular fibres with diameters ranging from 1.4 µm to 4.2 µm, interconnecting sixteen regions of interest. The simulated images capture the microscopic properties of the tissue (e.g. fibre diameter, water diffusing within and around fibres, free water compartment), while also having desirable macroscopic properties resembling the anatomy, such as the smoothness of the fibre trajectories. While previous phantoms were used to validate either tractography or microstructure, this phantom can enable a better assessment of the connectome estimation’s reliability on the one side, and its adherence to the actual microstructure of the nervous tissue on the other.

Comparative analysis of signal models for microscopic fractional anisotropy estimation using q-space trajectory encoding

Microscopic diffusion anisotropy imaging using diffusion-weighted MRI and multidimensional diffusion encoding is a promising method for quantifying clinically and scientifically relevant microstructural properties of neural tissue. Several methods for estimating microscopic fractional anisotropy (µFA), a normalized measure of microscopic diffusion anisotropy, have been introduced but the differences between the methods have received little attention thus far. In this study, the accuracy and precision of µFA estimation using q-space trajectory encoding and different signal models were assessed using imaging experiments and simulations. Three healthy volunteers and a microfibre phantom were imaged with five non-zero b-values and gradient waveforms encoding linear and spherical b-tensors. Since the ground-truth µFA was unknown in the imaging experiments, Monte Carlo random walk simulations were performed using axon-mimicking fibres for which the ground truth was known. Furthermore, parameter bias due to time-dependent diffusion was quantified by repeating the simulations with tuned waveforms, which have similar power spectra, and with triple diffusion encoding, which, unlike q-space trajectory encoding, is not based on the assumption of time-independent diffusion. The truncated cumulant expansion of the powder-averaged signal, gamma-distributed diffusivities assumption, and q-space trajectory imaging, a generalization of the truncated cumulant expansion to individual signals, were used to estimate µFA. The gamma-distributed diffusivities assumption consistently resulted in greater µFA values than the second order cumulant expansion, 0.1 greater when averaged over the whole brain. In the simulations, the generalized cumulant expansion provided the most accurate estimates. Importantly, although time-dependent diffusion caused significant overestimation of µFA using all the studied methods, the simulations suggest that the resulting bias in µFA is less than 0.1 in human white matter.

A machine learning-based method for estimating the number and orientations of major fascicles in diffusion-weighted magnetic resonance imaging

Accurate modeling of diffusion-weighted magnetic resonance imaging measurements is necessary for accurate brain connectivity analysis. Existing methods for estimating the number and orientations of fascicles in an imaging voxel either depend on non-convex optimization techniques that are sensitive to initialization and measurement noise, or are prone to predicting spurious fascicles. In this paper, we propose a machine learning-based technique that can accurately estimate the number and orientations of fascicles in a voxel. Our method can be trained with either simulated or real diffusion-weighted imaging data. Our method estimates the angle to the closest fascicle for each direction in a set of discrete directions uniformly spread on the unit sphere. This information is then processed to extract the number and orientations of fascicles in a voxel. On realistic simulated phantom data with known ground truth, our method predicts the number and orientations of crossing fascicles more accurately than several classical and machine learning methods. It also leads to more accurate tractography. On real data, our method is better than or compares favorably with other methods in terms of robustness to measurement down-sampling and also in terms of expert quality assessment of tractography results.

Bundle-Specific Axon Diameter Index as a New Contrast to Differentiate White Matter Tracts

In the central nervous system of primates, several pathways are characterized by different spectra of axon diameters. In vivo methods, based on diffusion-weighted magnetic resonance imaging, can provide axon diameter index estimates non-invasively. However, such methods report voxel-wise estimates, which vary from voxel-to-voxel for the same white matter bundle due to partial volume contributions from other pathways having different microstructure properties. Here, we propose a novel microstructure-informed tractography approach, COMMITAxSize, to resolve axon diameter index estimates at the streamline level, thus making the estimates invariant along trajectories. Compared to previously proposed voxel-wise methods, our formulation allows the estimation of a distinct axon diameter index value for each streamline, directly, furnishing a complementary measure to the existing calculation of the mean value along the bundle. We demonstrate the favourable performance of our approach comparing our estimates with existing histologically-derived measurements performed in the corpus callosum and the posterior limb of the internal capsule. Overall, our method provides a more robust estimation of the axon diameter index of pathways by jointly estimating the microstructure properties of the tissue and the macroscopic organisation of the white matter connectivity.

Resolving bundle-specific intra-axonal T2 values within a voxel using diffusion-relaxation tract-based estimation

At the typical spatial resolution of MRI in the human brain, approximately 60-90% of voxels contain multiple fiber populations. Quantifying microstructural properties of distinct fiber populations within a voxel is therefore challenging but necessary. While progress has been made for diffusion and T1-relaxation properties, how to resolve intra-voxel T2 heterogeneity remains an open question. Here a novel framework, named COMMIT-T2, is proposed that uses tractography-based spatial regularization with diffusion-relaxometry data to estimate multiple intra-axonal T2 values within a voxel. Unlike previously-proposed voxel-based T2 estimation methods, which (when applied in white matter) implicitly assume just one fiber bundle in the voxel or the same T2 for all bundles in the voxel, COMMIT-T2 can recover specific T2 values for each unique fiber population passing through the voxel. In this approach, the number of recovered unique T2 values is not determined by a number of model parameters set a priori, but rather by the number of tractography-reconstructed streamlines passing through the voxel. Proof-of-concept is provided in silico and in vivo, including a demonstration that distinct tract-specific T2 profiles can be recovered even in the three-way crossing of the corpus callosum, arcuate fasciculus, and corticospinal tract. We demonstrate the favourable performance of COMMIT-T2 compared to that of voxelwise approaches for mapping intra-axonal T2 exploiting diffusion, including a direction-averaged method and AMICO-T2, a new extension to the previously-proposed Accelerated Microstructure Imaging via Convex Optimization (AMICO) framework.

Axon morphology is modulated by the local environment and impacts the noninvasive investigation of its structure-function relationship

Axonal conduction velocity, which ensures efficient function of the brain network, is related to axon diameter. Noninvasive, in vivo axon diameter estimates can be made with diffusion magnetic resonance imaging, but the technique requires three-dimensional (3D) validation. Here, high-resolution, 3D synchrotron X-ray nano-holotomography images of white matter samples from the corpus callosum of a monkey brain reveal that blood vessels, cells, and vacuoles affect axonal diameter and trajectory. Within single axons, we find that the variation in diameter and conduction velocity correlates with the mean diameter, contesting the value of precise diameter determination in larger axons. These complex 3D axon morphologies drive previously reported 2D trends in axon diameter and g-ratio. Furthermore, we find that these morphologies bias the estimates of axon diameter with diffusion magnetic resonance imaging and, ultimately, impact the investigation and formulation of the axon structure-function relationship.

Diffusion MRI simulation of realistic neurons with SpinDoctor and the Neuron Module

The diffusion MRI signal arising from neurons can be numerically simulated by solving the Bloch-Torrey partial differential equation. In this paper we present the Neuron Module that we implemented within the Matlab-based diffusion MRI simulation toolbox SpinDoctor. SpinDoctor uses finite element discretization and adaptive time integration to solve the Bloch-Torrey partial differential equation for general diffusion-encoding sequences, at multiple b-values and in multiple diffusion directions. In order to facilitate the diffusion MRI simulation of realistic neurons by the research community, we constructed finite element meshes for a group of 36 pyramidal neurons and a group of 29 spindle neurons whose morphological descriptions were found in the publicly available neuron repository NeuroMorpho.Org. These finite elements meshes range from having 15,163 nodes to 622,553 nodes. We also broke the neurons into the soma and dendrite branches and created finite elements meshes for these cell components. Through the Neuron Module, these neuron and cell components finite element meshes can be seamlessly coupled with the functionalities of SpinDoctor to provide the diffusion MRI signal attributable to spins inside neurons. We make these meshes and the source code of the Neuron Module available to the public as an open-source package. To illustrate some potential uses of the Neuron Module, we show numerical examples of the simulated diffusion MRI signals in multiple diffusion directions from whole neurons as well as from the soma and dendrite branches, and include a comparison of the high b-value behavior between dendrite branches and whole neurons. In addition, we demonstrate that the neuron meshes can be used to perform Monte-Carlo diffusion MRI simulations as well. We show that at equivalent accuracy, if only one gradient direction needs to be simulated, SpinDoctor is faster than a GPU implementation of Monte-Carlo, but if many gradient directions need to be simulated, there is a break-even point when the GPU implementation of Monte-Carlo becomes faster than SpinDoctor. Furthermore, we numerically compute the eigenfunctions and the eigenvalues of the Bloch-Torrey and the Laplace operators on the neuron geometries using a finite elements discretization, in order to give guidance in the choice of the space and time discretization parameters for both finite elements and Monte-Carlo approaches. Finally, we perform a statistical study on the set of 65 neurons to test some candidate biomakers that can potentially indicate the soma size. This preliminary study exemplifies the possible research that can be conducted using the Neuron Module.

Challenges for biophysical modeling of microstructure

The biophysical modeling efforts in diffusion MRI have grown considerably over the past 25 years. In this review, we dwell on the various challenges along the journey of bringing a biophysical model from initial design to clinical implementation, identifying both hurdles that have been already overcome and outstanding issues. First, we describe the critical initial task of selecting which features of tissue microstructure can be estimated using a model and which acquisition protocol needs to be implemented to make the estimation possible. The model performance should necessarily be tested in realistic numerical simulations and in experimental data - adapting the fitting strategy accordingly, and parameter estimates should be validated against complementary techniques, when/if available. Secondly, the model performance and validity should be explored in pathological conditions, and, if appropriate, dedicated models for pathology should be developed. We build on examples from tumors, ischemia and demyelinating diseases. We then discuss the challenges associated with clinical translation and added value. Finally, we single out four major unresolved challenges that are related to: the availability of a microstructural ground truth, the validation of model parameters which cannot be accessed with complementary techniques, the development of a generalized standard model for any brain region and pathology, and the seamless communication between different parties involved in the development and application of biophysical models of diffusion.