A simulation-driven supervised learning framework to estimate brain microstructure using diffusion MRI.

We propose a framework to train supervised learning models on synthetic data to estimate brain microstructure parameters using diffusion magnetic resonance imaging (dMRI). Although further validation is necessary, the proposed framework aims to seamlessly incorporate realistic simulations into dMRI microstructure estimation. Synthetic data were generated from over 1,000 neuron meshes converted from digital neuronal reconstructions and linked to their neuroanatomical parameters (such as soma volume and neurite length) using an optimized diffusion MRI simulator that produces intracellular dMRI signals from the solution of the Bloch-Torrey partial differential equation. By combining random subsets of simulated neuron signals with a free diffusion compartment signal, we constructed a synthetic dataset containing dMRI signals and 40 tissue microstructure parameters of 1.45 million artificial brain voxels. To implement supervised learning models we chose multilayer perceptrons (MLPs) and trained them on a subset of the synthetic dataset to estimate some microstructure parameters, namely, the volume fractions of soma, neurites, and the free diffusion compartment, as well as the area fractions of soma and neurites. The trained MLPs perform satisfactorily on the synthetic test sets and give promising in-vivo parameter maps on the MGH Connectome Diffusion Microstructure Dataset (CDMD). Most importantly, the estimated volume fractions showed low dependence on the diffusion time, the diffusion time independence of the estimated parameters being a desired property of quantitative microstructure imaging. The synthetic dataset we generated will be valuable for the validation of models that map between the dMRI signals and microstructure parameters. The surface meshes and microstructures parameters of the aforementioned neurons have been made publicly available.

Incorporating interface permeability into the diffusion MRI signal representation while using impermeable Laplace eigenfunctions.

Objective. The complex-valued transverse magnetization due to diffusion-encoding magnetic field gradients acting on a permeable medium can be modeled by the Bloch-Torrey partial differential equation. The diffusion magnetic resonance imaging (MRI) signal has a representation in the basis of the Laplace eigenfunctions of the medium. However, in order to estimate the permeability coefficient from diffusion MRI data, it is desirable that the forward solution can be calculated efficiently for many values of permeability.Approach. In this paper we propose a new formulation of the permeable diffusion MRI signal representation in the basis of the Laplace eigenfunctions of the same medium where the interfaces are made impermeable.Main results.We proved the theoretical equivalence between our new formulation and the original formulation in the case that the full eigendecomposition is used. We validated our method numerically and showed promising numerical results when a partial eigendecomposition is used. Two diffusion MRI sequences were used to illustrate the numerical validity of our new method.Significance.Our approach means that the same basis (the impermeable set) can be used for all permeability values, which reduces the computational time significantly, enabling the study of the effects of the permeability coefficient on the diffusion MRI signal in the future.

A fractional order-based mixture of central Wishart (FMoCW) model for reconstructing white matter fibers from diffusion MRI.

This paper introduces an algorithm for reconstructing the brain's white matter fibers (WMFs). In particular, a fractional order mixture of central Wishart (FMoCW) model is proposed to reconstruct the WMFs from diffusion MRI data. The pseudo super diffusive modality of anomalous diffusion is coupled with the mixture of central Wishart (MoCW) model to derive the proposed model. We have shown results on multiple synthetic simulations, including fibers orientations in 2 and 3 directions per voxel and experiments on real datasets of rat optic chiasm and a healthy human brain. In synthetic simulations, a varying Rician distributed noise levels, σ=0.01-0.09 is also considered. The proposed model can efficiently distinguish multiple fibers even when the angle of separation between fibers is very small. This model outperformed, giving the least angular error when compared to fractional mixture of Gaussian (MoG), MoCW and mixture of non-central Wishart (MoNCW) models.

A generalized order mixture model for tracing connectivity of white matter fascicles complexity in brain from diffusion MRI.

This paper focuses on tracing the connectivity of white matter fascicles in the brain. In particular, a generalized order algorithm based on mixture of non-central Wishart distribution model is proposed for this purpose. The proposed algorithm utilizes the generalization of integer order based approach with the mixture of non-central Wishart distribution model. Pseudo super anomalous behavior of water diffusion inside human brain is the prime motivation of the the present study. We have shown results on multiple synthetic simulations with fibers orientations in two and three directions in each voxel as well as experiments on real data. Synthetic simulations were performed with varying noise levels and diffusion weighting gradient i.e. $b-$values. The proposed model performed outstanding especially for distinguishing closely oriented fibers.

Generalisation of continuous time random walk to anomalous diffusion MRI models with an age-related evaluation of human corpus callosum.

Diffusion MRI measures of the human brain provide key insight into microstructural variations across individuals and into the impact of central nervous system diseases and disorders. One approach to extract information from diffusion signals has been to use biologically relevant analytical models to link millimetre scale diffusion MRI measures with microscale influences. The other approach has been to represent diffusion as an anomalous transport process and infer microstructural information from the different anomalous diffusion equation parameters. In this study, we investigated how parameters of various anomalous diffusion models vary with age in the human brain white matter, particularly focusing on the corpus callosum. We first unified several established anomalous diffusion models (the super-diffusion, sub-diffusion, quasi-diffusion and fractional Bloch-Torrey models) under the continuous time random walk modelling framework. This unification allows a consistent parameter fitting strategy to be applied from which meaningful model parameter comparisons can be made. We then provided a novel way to derive the diffusional kurtosis imaging (DKI) model, which is shown to be a degree two approximation of the sub-diffusion model. This link between the DKI and sub-diffusion models led to a new robust technique for generating maps of kurtosis and diffusivity using the sub-diffusion parameters ßSUB and DSUB. Superior tissue contrast is achieved in kurtosis maps based on the sub-diffusion model. 7T diffusion weighted MRI data for 65 healthy participants in the age range 19-78 years was used in this study. Results revealed that anomalous diffusion model parameters α and ß have shown consistent positive correlation with age in the corpus callosum, indicating α and ß are sensitive to tissue microstructural changes in ageing.

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.

The localization regime in a nutshell.

High diffusion-sensitizing magnetic field gradients have been more and more often applied nowadays to achieve a better characterization of the microstructure. As the resulting spin-echo signal significantly deviates from the conventional Gaussian form, various models have been employed to interpret these deviations and to relate them with the microstructural properties of a sample. In this paper, we argue that the non-Gaussian behavior of the signal is a generic universal feature of the Bloch-Torrey equation. We provide a simple yet rigorous description of the localization regime emerging at high extended gradients and identify its origin as a symmetry breaking at the reflecting boundary. We compare the consequent non-Gaussian signal decay to other diffusion NMR regimes such as slow-diffusion, motional-narrowing and diffusion-diffraction regimes. We emphasize limitations of conventional perturbative techniques and advocate for non-perturbative approaches which may pave a way to new imaging modalities in this field.

Apparent diffusion coefficient measured by diffusion MRI of moving and deforming domains.

The modeling of the diffusion MRI signal from moving and deforming organs such as the heart is challenging due to significant motion and deformation of the imaged medium during the signal acquisition. Recently, a mathematical formulation of the Bloch-Torrey equation, describing the complex transverse magnetization due to diffusion-encoding magnetic field gradients, was developed to account for the motion and deformation. In that work, the motivation was to cancel the effect of the motion and deformation in the MRI image and the space scale of interest spans multiple voxels. In the present work, we adapt the mathematical equation to study the diffusion MRI signal at the much smaller scale of biological cells. We start with the Bloch-Torrey equation defined on a cell that is moving and deforming and linearize the equation around the magnitude of the diffusion-encoding gradient. The result is a second order signal model in which the linear term gives the imaginary part of the diffusion MRI signal and the quadratic term gives the apparent diffusion coefficient (ADC) attributable to the biological cell. We numerically validate this model for a variety of motions and deformations.

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.

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.

Global sensitivity analysis of skeletal muscle dMRI metrics: Effects of microstructural and pulse parameters.

PURPOSE: Estimating microstructural parameters of skeletal muscle from diffusion MRI (dMRI) signal requires understanding the relative importance of both microstructural and dMRI sequence parameters on the signal. This study seeks to determine the sensitivity of dMRI signal to variations in microstructural and dMRI sequence parameters, as well as assess the effect of noise on sensitivity. METHODS: Using a cylindrical myocyte model of skeletal muscle, numerical solutions of the Bloch-Torrey equation were used to calculate global sensitivity indices of dMRI metrics (FA, RD, MD, λ1 , λ2 , λ3 ) for wide ranges of microstructural and dMRI sequence parameters. The microstructural parameters were: myocyte diameter, volume fraction, membrane permeability, intra- and extracellular diffusion coefficients, and intra- and extracellular T2 times. Two separate pulse sequences were examined, a PGSE and a generalized diffusion-weighted sequence that accommodates a larger range of diffusion times. The effect of noise and signal averaging on the sensitivity of the dMRI metrics was examined by adding synthetic noise to the simulated signal. RESULTS: Among the examined parameters, the intracellular diffusion coefficient has the strongest effect, and myocyte diameter is more influential than permeability for FA and RD. The sensitivity indices do not vary significantly between the two pulse sequences. Also, noise strongly affects the sensitivity of the dMRI signal to microstructural variations. CONCLUSIONS: With the identification of key microstructural features that affect dMRI measurements, the reported sensitivity results can help interpret dMRI measurements of skeletal muscle in terms of the underlying microstructure and further develop parsimonious dMRI models of skeletal muscle.

Portable simulation framework for diffusion MRI.

The numerical simulation of the diffusion MRI signal arising from complex tissue micro-structures is helpful for understanding and interpreting imaging data as well as for designing and optimizing MRI sequences. The discretization of the Bloch-Torrey equation by finite elements is a more recently developed approach for this purpose, in contrast to random walk simulations, which has a longer history. While finite element discretization is more difficult to implement than random walk simulations, the approach benefits from a long history of theoretical and numerical developments by the mathematical and engineering communities. In particular, software packages for the automated solutions of partial differential equations using finite element discretization, such as FEniCS, are undergoing active support and development. However, because diffusion MRI simulation is a relatively new application area, there is still a gap between the simulation needs of the MRI community and the available tools provided by finite element software packages. In this paper, we address two potential difficulties in using FEniCS for diffusion MRI simulation. First, we simplified software installation by the use of FEniCS containers that are completely portable across multiple platforms. Second, we provide a portable simulation framework based on Python and whose code is open source. This simulation framework can be seamlessly integrated with cloud computing resources such as Google Colaboratory notebooks working on a web browser or with Google Cloud Platform with MPI parallelization. We show examples illustrating the accuracy, the computational times, and parallel computing capabilities. The framework contributes to reproducible science and open-source software in computational diffusion MRI with the hope that it will help to speed up method developments and stimulate research collaborations.

SpinDoctor: A MATLAB toolbox for diffusion MRI simulation.

The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium can be modeled by the multiple compartment Bloch-Torrey partial differential equation. Under the assumption of negligible water exchange between compartments, the time-dependent apparent diffusion coefficient can be directly computed from the solution of a diffusion equation subject to a time-dependent Neumann boundary condition. This paper describes a publicly available MATLAB toolbox called SpinDoctor that can be used 1) to solve the Bloch-Torrey partial differential equation in order to simulate the diffusion magnetic resonance imaging signal; 2) to solve a diffusion partial differential equation to obtain directly the apparent diffusion coefficient; 3) to compare the simulated apparent diffusion coefficient with a short-time approximation formula. The partial differential equations are solved by P1 finite elements combined with built-in MATLAB routines for solving ordinary differential equations. The finite element mesh generation is performed using an external package called Tetgen. SpinDoctor provides built-in options of including 1) spherical cells with a nucleus; 2) cylindrical cells with a myelin layer; 3) an extra-cellular space enclosed either a) in a box or b) in a tight wrapping around the cells; 4) deformation of canonical cells by bending and twisting; 5) permeable membranes; Built-in diffusion-encoding pulse sequences include the Pulsed Gradient Spin Echo and the Oscillating Gradient Spin Echo. We describe in detail how to use the SpinDoctor toolbox. We validate SpinDoctor simulations using reference signals computed by the Matrix Formalism method. We compare the accuracy and computational time of SpinDoctor simulations with Monte-Carlo simulations and show significant speed-up of SpinDoctor over Monte-Carlo simulations in complex geometries. We also illustrate several extensions of SpinDoctor functionalities, including the incorporation of T2 relaxation, the simulation of non-standard diffusion-encoding sequences, as well as the use of externally generated geometrical meshes.

Diffusion MRI simulation in thin-layer and thin-tube media using a discretization on manifolds.

The Bloch-Torrey partial differential equation can be used to describe the evolution of the transverse magnetization of the imaged sample under the influence of diffusion-encoding magnetic field gradients inside the MRI scanner. The integral of the magnetization inside a voxel gives the simulated diffusion MRI signal. This paper proposes a finite element discretization on manifolds in order to efficiently simulate the diffusion MRI signal in domains that have a thin layer or a thin tube geometrical structure. The variable thickness of the three-dimensional domains is included in the weak formulation established on the manifolds. We conducted a numerical study of the proposed approach by simulating the diffusion MRI signals from the extracellular space (a thin layer medium) and from neurons (a thin tube medium), comparing the results with the reference signals obtained using a standard three-dimensional finite element discretization. We show good agreements between the simulated signals using our proposed method and the reference signals for a wide range of diffusion MRI parameters. The approximation becomes better as the diffusion time increases. The method helps to significantly reduce the required simulation time, computational memory, and difficulties associated with mesh generation, thus opening the possibilities to simulating complicated structures at low cost for a better understanding of diffusion MRI in the brain.

The derivation of homogenized diffusion kurtosis models for diffusion MRI.

We use homogenization theory to establish a new macroscopic model for the complex transverse water proton magnetization in a voxel due to diffusion-encoding magnetic field gradient pulses in the case of biological tissue with impermeable membranes. In this model, new higher-order diffusion tensors emerge and offer more information about the structure of the biological tissues. We explicitly solve the macroscopic model to obtain an ordinary differential equation for the diffusion MRI signal that has similar structure as diffusional kurtosis imaging models. We finally present some validating numerical results on synthetic examples showing the accuracy of the model with respect to signals obtained by solving the Bloch-Torrey equation.

Pseudo-diffusion effects in lung MRI.

Magnetic resonance imaging of lung tissue is strongly influenced by susceptibility effects between spin-bearing water molecules and air-filled alveoli. The measured lineshape, however, also depends on the interplay between susceptibility effects and blood-flow around alveoli that can be approximated as pseudo-diffusion. Both effects are quantitatively described by the Bloch-Torrey-equation, which was so far only solved for dephasing on the alveolar surface. In this work, we extend this model to the whole range of physiological relevant air volume fractions. The results agree very well with in vivo measurements in human lung tissue.

Spin dephasing around randomly distributed vessels.

We analyze the gradient echo signal in the presence of blood vessel networks. Both, diffusion and susceptibility effects are analytically emphasized within the Bloch-Torrey equation. Solving this equation, we present the first exact description of the local magnetization around a single vessel. This allows us to deduce the gradient echo signal of parallel vessels randomly distributed in a plane, which is valid for arbitrary mean vessel diameters in the range of physiological relevant blood volume fractions. Thus, the results are potentially relevant for gradient echo measurements of blood vessel networks with arbitrary vessel size.

Rapid solution of the Bloch-Torrey equation in anisotropic tissue: Application to dynamic susceptibility contrast MRI of cerebral white matter.

Blood vessel related magnetic resonance imaging (MRI) contrast provides a window into the brain's metabolism and function. Here, we show that the spin echo dynamic susceptibility contrast (DSC) MRI signal of the brain's white matter (WM) strongly depends on the angle between WM tracts and the main magnetic field. The apparent cerebral blood flow and volume are 20% larger in fibres perpendicular to the main magnetic field compared to parallel fibres. We present a rapid numerical framework for the solution of the Bloch-Torrey equation that allows us to explore the isotropic and anisotropic components of the vascular tree. By fitting the simulated spin echo DSC signal to the measured data, we show that half of the WM vascular volume is comprised of vessels running in parallel with WM fibre tracts. The WM blood volume corresponding to the best fit to the experimental data was 2.82%, which is close to the PET gold standard of 2.6%.

Microscopic interpretation and generalization of the Bloch-Torrey equation for diffusion magnetic resonance.

In order to bridge microscopic molecular motion with macroscopic diffusion MR signal in complex structures, we propose a general stochastic model for molecular motion in a magnetic field. The Fokker-Planck equation of this model governs the probability density function describing the diffusion-magnetization propagator. From the propagator we derive a generalized version of the Bloch-Torrey equation and the relation to the random phase approach. This derivation does not require assumptions such as a spatially constant diffusion coefficient, or ad hoc selection of a propagator. In particular, the boundary conditions that implicitly incorporate the microstructure into the diffusion MR signal can now be included explicitly through a spatially varying diffusion coefficient. While our generalization is reduced to the conventional Bloch-Torrey equation for piecewise constant diffusion coefficients, it also predicts scenarios in which an additional term to the equation is required to fully describe the MR signal.

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

Inferring the microstructure of complex media from the diffusive motion of molecules is a challenging problem in diffusion physics. In this paper, we introduce a novel representation of diffusion MRI (dMRI) signal from tissue with spatially-varying diffusivity using a diffusion disturbance function. This disturbance function contains information about the (intra-voxel) spatial fluctuations in diffusivity due to restrictions, hindrances and tissue heterogeneity of the underlying tissue substrate. We derive the short- and long-range disturbance coefficients from this disturbance function to characterize the tissue structure and organization. Moreover, we provide an exact relation between the disturbance coefficients and the time-varying moments of the diffusion propagator, as well as their relation to specific tissue microstructural information such as the intra-axonal volume fraction and the apparent axon radius. The proposed approach is quite general and can model dMRI signal for any type of gradient sequence (rectangular, oscillating, etc.) without using the Gaussian phase approximation. The relevance of the proposed PICASO model is explored using Monte-Carlo simulations and in-vivo dMRI data. The results show that the estimated disturbance coefficients can distinguish different types of microstructural organization of axons.