Precise Inference and Characterization of Structural Organization (PICASO) of tissue from molecular diffusion

Inverse resolution of spatially varying diffusion coefficient using Physics-Informed neural networks

Resolving the diffusion coefficient is a key element in many biological and engineering systems, including pharmacological drug transport and fluid mechanics analyses. Additionally, these systems often have spatial variation in the diffusion coefficient which must be determined, such as for injectable drug-eluting implants into heterogeneous tissues. Unfortunately, obtaining the diffusion coefficient from images in such cases is an inverse problem with only discrete data points. The development of a robust method that can work with such noisy and ill-posed datasets to accurately determine spatially-varying diffusion coefficients is of great value across a large range of disciplines. Here, we developed an inverse solver that uses physics informed neural networks (PINNs) to calculate spatially-varying diffusion coefficients from numerical and experimental image data in varying biological and engineering applications. The residual of the transient diffusion equation for a concentration field is minimized to find the diffusion coefficient. The robustness of the method as an inverse solver was tested using both numerical and experimental datasets. The predictions show good agreement with both the numerical and experimental benchmarks; an error of less than 6.31% was obtained against all numerical benchmarks, while the diffusion coefficient calculated in experimental datasets matches the appropriate ranges of other reported literature values. Our work demonstrates the potential of using PINNs to resolve spatially-varying diffusion coefficients, which may aid a wide-range of applications, such as enabling better-designed drug-eluting implants for regenerative medicine or oncology fields.

Practical computation of the diffusion MRI signal based on Laplace eigenfunctions: permeable interfaces

The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch-Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold-standard reference model for the diffusion MRI signal arising from different tissue micro-structures of interest. A closed form representation of this reference diffusion MRI signal, called matrix formalism, which makes explicit the link between the Laplace eigenvalues and eigenfunctions of the tissue geometry and its diffusion MRI signal, was derived 20 years ago. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion-encoding sequences and b-values at negligible additional cost. In a previous publication, we presented a simulation framework that we implemented inside the MATLAB-based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for biological cells subject to impermeable membrane boundary conditions. In this work, we extend our simulation framework to include geometries that contain permeable cell membranes. We describe the new computational techniques that allowed this generalization and we analyze the effects of the magnitude of the permeability coefficient on the eigendecomposition of the diffusion and Bloch-Torrey operators. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI.

Accelerated diffusion and relaxation-diffusion MRI using time-division multiplexing EPI

PURPOSE

To develop a time-division multiplexing echo-planar imaging (TDM-EPI) sequence for approximately two- to threefold acceleration when acquiring joint relaxation-diffusion MRI data with multiple TEs.

METHODS

The proposed TDM-EPI sequence interleaves excitation and data collection for up to 3 separate slices at different TEs and uses echo-shifting gradients to disentangle the overlapping echo signals during the readout period. By properly arranging the sequence event blocks for each slice and adjusting the echo-shifting gradients, diffusion-weighted images from separate slices can be acquired. Therefore, we present 2 variants of the sequence. A single-TE TDM-EPI is presented to demonstrate the concept. Next, a multi-TE TDM-EPI is presented to highlight the advantages of the TDM approach for relaxation-diffusion imaging. These sequences were evaluated on a 3 Tesla scanner with a water phantom and in vivo human brain data.

RESULTS

The single-TE TDM-EPI sequence can simultaneously acquire 2 slices with a maximum b value of 3000 s/mm² and 2.5 mm isotropic resolution using interleaved readout windows with TE ≈ 78 ms. With the same b value and resolution, the multi-TE TDM-EPI sequence can simultaneously acquire 2 or 3 separate slices using interleaved readout sections with shortest TE ≈ 70 ms and ΔTE ≈ 30 ms. Phantom and in vivo experiments have shown that the proposed TDM-EPI sequences can provide similar image quality and diffusion measures as conventional EPI readouts with multiple echoes but can reduce the overall relaxation-diffusion protocol scan time by approximately two- to threefold.

CONCLUSION

TDM-EPI is a novel approach to acquire diffusion imaging data at multiple TEs. This enables a significant reduction in acquisition time for relaxation-diffusion MRI experiments but without compromising image quality and diffusion measurements, thus removing a significant barrier to the adoption of relaxation-diffusion MRI in clinical research studies of neurological and mental disorders.

The sensitivity of diffusion MRI to microstructural properties and experimental factors

Diffusion MRI is a non-invasive technique to study brain microstructure. Differences in the microstructural properties of tissue, including size and anisotropy, can be represented in the signal if the appropriate method of acquisition is used. However, to depict the underlying properties, special care must be taken when designing the acquisition protocol as any changes in the procedure might impact on quantitative measurements. This work reviews state-of-the-art methods for studying brain microstructure using diffusion MRI and their sensitivity to microstructural differences and various experimental factors. Microstructural properties of the tissue at a micrometer scale can be linked to the diffusion signal at a millimeter-scale using modeling. In this paper, we first give an introduction to diffusion MRI and different encoding schemes. Then, signal representation-based methods and multi-compartment models are explained briefly. The sensitivity of the diffusion MRI signal to the microstructural components and the effects of curvedness of axonal trajectories on the diffusion signal are reviewed. Factors that impact on the quality (accuracy and precision) of derived metrics are then reviewed, including the impact of random noise, and variations in the acquisition parameters (i.e., number of sampled signals, b-value and number of acquisition shells). Finally, yet importantly, typical approaches to deal with experimental factors are depicted, including unbiased measures and harmonization. We conclude the review with some future directions and recommendations on this topic.

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.

Practical computation of the diffusion MRI signal of realistic neurons based on Laplace eigenfunctions

The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch-Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold-standard reference model for the diffusion MRI signal arising from different tissue micro-structures of interest. A closed form representation of this reference diffusion MRI signal called matrix formalism, which makes explicit the link between the Laplace eigenvalues and eigenfunctions of the biological cell and its diffusion MRI signal, was derived 20 years ago. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion-encoding sequences and b-values at negligible additional cost. Up to now, this representation, though mathematically elegant, has not been often used as a practical model of the diffusion MRI signal, due to the difficulties of calculating the Laplace eigendecomposition in complicated geometries. In this paper, we present a simulation framework that we have implemented inside the MATLAB-based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for realistic neurons using the finite element method. We show that the matrix formalism representation requires a few hundred eigenmodes to match the reference signal computed by solving the Bloch-Torrey equation when the cell geometry originates from realistic neurons. As expected, the number of eigenmodes required to match the reference signal increases with smaller diffusion time and higher b-values. We also convert the eigenvalues to a length scale and illustrate the link between the length scale and the oscillation frequency of the eigenmode in the cell geometry. We give the transformation that links the Laplace eigenfunctions to the eigenfunctions of the Bloch-Torrey operator and compute the Bloch-Torrey eigenfunctions and eigenvalues. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI.

Retrospective harmonization of multi-site diffusion MRI data acquired with different acquisition parameters

A joint and integrated analysis of multi-site diffusion MRI (dMRI) datasets can dramatically increase the statistical power of neuroimaging studies and enable comparative studies pertaining to several brain disorders. However, dMRI data sets acquired on multiple scanners cannot be naively pooled for joint analysis due to scanner specific nonlinear effects as well as differences in acquisition parameters. Consequently, for joint analysis, the dMRI data has to be harmonized, which involves removing scanner-specific differences from the raw dMRI signal. In this work, we propose a dMRI harmonization method that is capable of removing scanner-specific effects, while accounting for minor differences in acquisition parameters such as b-value, spatial resolution and number of gradient directions. We validate our algorithm on dMRI data acquired from two sites: Philadelphia Neurodevelopmental Cohort (PNC) with 800 healthy adolescents (ages 8-22 years) and Brigham and Women's Hospital (BWH) with 70 healthy subjects (ages 14-54 years). In particular, we show that gender and age-related maturation differences in different age groups are preserved after harmonization, as measured using effect sizes (small, medium and large), irrespective of the test sample size. Since we use matched control subjects from different scanners to estimate scanner-specific effects, our goal in this work is also to determine the minimum number of well-matched subjects needed from each site to achieve best harmonization results. Our results indicate that at-least 16 to 18 well-matched healthy controls from each site are needed to reliably capture scanner related differences. The proposed method can thus be used for retrospective harmonization of raw dMRI data across sites despite differences in acquisition parameters, while preserving inter-subject anatomical variability.

Towards microstructure fingerprinting: Estimation of tissue properties from a dictionary of Monte Carlo diffusion MRI simulations

Many closed-form analytical models have been proposed to relate the diffusion-weighted magnetic resonance imaging (DW-MRI) signal to microstructural features of white matter tissues. These models generally make assumptions about the tissue and the diffusion processes which often depart from the biophysical reality, limiting their reliability and interpretability in practice. Monte Carlo simulations of the random walk of water molecules are widely recognized to provide near groundtruth for DW-MRI signals. However, they have mostly been limited to the validation of simpler models rather than used for the estimation of microstructural properties. This work proposes a general framework which leverages Monte Carlo simulations for the estimation of physically interpretable microstructural parameters, both in single and in crossing fascicles of axons. Monte Carlo simulations of DW-MRI signals, or fingerprints, are pre-computed for a large collection of microstructural configurations. At every voxel, the microstructural parameters are estimated by optimizing a sparse combination of these fingerprints. Extensive synthetic experiments showed that our approach achieves accurate and robust estimates in the presence of noise and uncertainties over fixed or input parameters. In an in vivo rat model of spinal cord injury, our approach provided microstructural parameters that showed better correspondence with histology than five closed-form models of the diffusion signal: MMWMD, NODDI, DIAMOND, WMTI and MAPL. On whole-brain in vivo data from the human connectome project (HCP), our method exhibited spatial distributions of apparent axonal radius and axonal density indices in keeping with ex vivo studies. This work paves the way for microstructure fingerprinting with Monte Carlo simulations used directly at the modeling stage and not only as a validation tool.

Cumulant expansions for measuring water exchange using diffusion MRI

The rate of water exchange across cell membranes is a parameter of biological interest and can be measured by diffusion magnetic resonance imaging (dMRI). In this work, we investigate a stochastic model for the diffusion-and-exchange of water molecules. This model provides a general solution for the temporal evolution of dMRI signal using any type of gradient waveform, thereby generalizing the signal expressions for the Kärger model. Moreover, we also derive a general nth order cumulant expansion of the dMRI signal accounting for water exchange, which has not been explored in earlier studies. Based on this analytical expression, we compute the cumulant expansion for dMRI signals for the special case of single diffusion encoding (SDE) and double diffusion encoding (DDE) sequences. Our results provide a theoretical guideline on optimizing experimental parameters for SDE and DDE sequences, respectively. Moreover, we show that DDE signals are more sensitive to water exchange at short-time scale but provide less attenuation at long-time scale than SDE signals. Our theoretical analysis is also validated using Monte Carlo simulations on synthetic structures.

Diffusion imaging of reversible and irreversible microstructural changes within the corticospinal tract in idiopathic normal pressure hydrocephalus

The symptoms of idiopathic normal pressure hydrocephalus (iNPH) can be improved by shunt surgery, but prediction of treatment outcome is not established. We investigated changes of the corticospinal tract (CST) in iNPH before and after shunt surgery by using diffusion microstructural imaging, which infers more specific tissue properties than conventional diffusion tensor imaging. Two biophysical models were used: neurite orientation dispersion and density imaging (NODDI) and white matter tract integrity (WMTI). In both methods, the orientational coherence within the CSTs was higher in patients than in controls, and some normalization occurred after the surgery in patients, indicating axon stretching and recovery. The estimated axon density was lower in patients than in controls but remained unchanged after the surgery, suggesting its potential as a marker for irreversible neuronal damage. In a Monte-Carlo simulation that represented model axons as undulating cylinders, both NODDI and WMTI separated the effects of axon density and undulation. Thus, diffusion MRI may distinguish between reversible and irreversible microstructural changes in iNPH. Our findings constitute a step towards a quantitative image biomarker that reflects pathological process and treatment outcomes of iNPH.

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NODDI and WMTI provide markers of reversible and irreversible changes in iNPH.

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Measures of axon orientation indicated recovery from stretching after surgery.

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Axon density remained low after surgery, suggesting chronic neuronal damage.

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Axon stretching in simulations differentially affected diffusion metrics.

Effective potential for magnetic resonance measurements of restricted diffusion

The signature of diffusive motion on the NMR signal has been exploited to characterize the mesoscopic structure of specimens in numerous applications. For compartmentalized specimens comprising isolated subdomains, a representation of individual pores is necessary for describing restricted diffusion within them. When gradient waveforms with long pulse durations are employed, a quadratic potential profile is identified as an effective energy landscape for restricted diffusion. The dependence of the stochastic effective force on the center-of-mass position is indeed found to be approximately linear (Hookean) for restricted diffusion even when the walls are sticky. We outline the theoretical basis and practical advantages of our picture involving effective potentials.