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Resolution of complex tissue microarchitecture using the diffusion orientation transform (DOT)
Abstract
This article describes an accurate and fast method for fiber orientation mapping using multidirectional diffusion-weighted magnetic resonance (MR) data. This novel approach utilizes the Fourier transform relationship between the water displacement probabilities and diffusion-attenuated MR signal expressed in spherical coordinates. The radial part of the Fourier integral is evaluated analytically under the assumption that MR signal attenuates exponentially. The values of the resulting functions are evaluated at a fixed distance away from the origin. The spherical harmonic transform of these functions yields the Laplace series coefficients of the probabilities on a sphere of fixed radius. Alternatively, probability values can be computed nonparametrically using Legendre polynomials. Orientation maps calculated from excised rat nervous tissue data demonstrate this technique's ability to accurately resolve crossing fibers in anatomical regions such as the optic chiasm. This proposed methodology has a trivial extension to multiexponential diffusion-weighted signal decay. The developed methods will improve the reliability of tractography schemes and may make it possible to correctly identify the neural connections between functionally connected regions of the nervous system.
... Even at the early stages of DT-MRI, it was clear that the Gaussian assumption had important limitations. It provided a useful tool allowing clinical studies, but the underlying diffusion processes were not accurately described because of the over-simplified fitting, so that more evolved techniques with more degrees-of-freedom naturally arose, such as High Angular Resolution Diffusion Imaging [6][7][8] HARDI] or Diffusion Kurtosis Imaging [9, DKI]. It seems obvious that more degrees-of-freedom require more diffusion images to be acquired, but the requirement of an accurate angular resolution also implies the need for a finer angular contrast, which translates in the need for stronger gradients to probe diffusion, i.e., higher b-values [10]. ...
... The most common techniques rely on the assumption of a Gaussian diffusion profile and a steady state regime of the diffusion process that yields to the well-known Diffusion Tensor (DT) approach. Alternatively, a more general expression for E(q) can be used [8]: ...
... Although the mono-exponential assumption in Eq (6) may seem restrictive, it has been successfully adopted before for single-shell, HARDI models to accurately describe several predominant diffusion directions within the imaged voxel [7,8,35,36]. Moreover, it allows to get rid of the dense sampling required by the original formulations of RTOP, RTPP, and RTAP, as long as the volumetric integrals over the whole q-space are replaced by surface integrals over one single shell. ...
In diffusion MRI, the Ensemble Average diffusion Propagator (EAP) provides relevant micro-structural information and meaningful descriptive maps of the white matter previously obscured by traditional techniques like Diffusion Tensor Imaging (DTI). The direct estimation of the EAP, however, requires a dense sampling of the Cartesian q-space involving a huge amount of samples (diffusion gradients) for proper reconstruction. A collection of more efficient techniques have been proposed in the last decade based on parametric representations of the EAP, but they still imply acquiring a large number of diffusion gradients with different b-values (shells). Paradoxically, this has come together with an effort to find scalar measures gathering all the q-space micro-structural information probed in one single index or set of indices. Among them, the return-to-origin (RTOP), return-to-plane (RTPP), and return-to-axis (RTAP) probabilities have rapidly gained popularity.
In this work, we propose the so-called “Apparent Measures Using Reduced Acquisitions” (AMURA) aimed at computing scalar indices that can mimic the sensitivity of state of the art EAP-based measures to micro-structural changes. AMURA drastically reduces both the number of samples needed and the computational complexity of the estimation of diffusion properties by assuming the diffusion anisotropy is roughly independent from the radial direction. This simplification allows us to compute closed-form expressions from single-shell information, so that AMURA remains compatible with standard acquisition protocols commonly used even in clinical practice. Additionally, the analytical form of AMURA-based measures, as opposed to the iterative, non-linear reconstruction ubiquitous to full EAP techniques, turns the newly introduced apparent RTOP, RTPP, and RTAP both robust and efficient to compute.
... However, in heterogeneous fiber populations with crossing, kissing, fanning or bending fiber configurations, the unimodal Orientation Distribution Function (ODF) assumed by DTI is no longer capable of resolving the orientation of distinct fiber populations within a voxel. Numerous methods have been introduced [4][5][6][7][8][9] to compute the underlying ODF with less stringent assumptions than those employed in DTI. However, the assumption that the ODF profiles are antipodally symmetric has prevailed through much of the literature. ...
... Our method relies on well-established symmetric ODF profiles where the reconstruction of the profiles is performed by following any of the classical HARDI techniques. Here, we utilized Diffusion Orientation Transform (DOT) [7] and Spherical Deconvolution (SD) [5,9] using HARDI tools [31]. After the estimation of a multi-directional representation of the local fiber orientations (ODFs), we develop a method that exploits voxel-based ODFs in a conic spatial neighborhood to capture underlying asymmetry of the fiber populations in a voxel. ...
... One such sample simulated experiment is shown in Fig. 7. In~(a), the reconstructed ODFs for a set of circularly bending fibers using the just described DSI method is shown. In (b) the symmetric ODFs using one of the classical methods, more specifically, the DOT approach [7], are depicted. In (c) the AODFs at each voxel constructed by our method in Section 2.1.3 ...
Characterization of anisotropy via diffusion MRI reveals fiber crossings in a substantial portion of voxels within the white-matter (WM) regions of the human brain. A considerable number of such voxels could exhibit asymmetric features such as bends and junctions. However, widely employed reconstruction methods yield symmetric Orientation Distribution Functions (ODFs) even when the underlying geometry is asymmetric. In this paper, we employ inter-voxel directional filtering approaches through a cone model to reveal more information regarding the cytoarchitectural organization within the voxel. The cone model facilitates a sharpening of the ODFs in some directions while suppressing peaks in other directions, thus yielding an Asymmetric ODF (AODF) field. We also show that a scalar measure of AODF asymmetry can be employed to obtain new contrast within the human brain. The feasibility of the technique is demonstrated on in vivo data obtained from the MGH-USC Human Connectome Project (HCP) and Parkinson's Progression Markers Initiative (PPMI) Project database. Characterizing asymmetry in neural tissue cytoarchitecture could be important for localizing and quantitatively assessing specific neuronal pathways.
... Accordingly, techniques with more degrees-of-freedom naturally arose, such as diffusion kurtosis imaging (DKI) 2 or methods based on high angular resolution diffusion imaging (HARDI). 3,4 The trend over the last decade has been to acquire a large number of diffusion-weighted images distributed over several shells (ie, with several gradient strengths) and with moderate-to-high b-values to estimate more advanced diffusion descriptors, such as the ensemble average diffusion propagator (EAP). 5 The estimation relies on model-free, nonparametric approaches that can accurately describe most of the relevant diffusion phenomena. ...
... Note that, although D(q) is independent of q, the signal attenuation, E(q), still has q-dependence. This assumption, although restrictive, is used to define certain diffusion representations in HARDI, 4,23 where only one data shell (ie, b-value) is usually acquired. ...
Purpose
The apparent propagator anisotropy (APA) is a new diffusion MRI metric that, while drawing on the benefits of the ensemble averaged propagator anisotropy (PA) compared to the fractional anisotropy (FA), can be estimated from single‐shell data.
Theory and Methods
Computation of the full PA requires acquisition of large datasets with many diffusion directions and different b‐values, and results in extremely long processing times. This has hindered adoption of the PA by the community, despite evidence that it provides meaningful information beyond the FA. Calculation of the complete propagator can be avoided under the hypothesis that a similar sensitivity/specificity may be achieved from apparent measurements at a given shell. Assuming that diffusion anisotropy (DiA) is nondependent on the b‐value, a closed‐form expression using information from one single shell (ie, b‐value) is reported.
Results
Publicly available databases with healthy and diseased subjects are used to compare the APA against other anisotropy measures. The structural information provided by the APA correlates with that provided by the PA for healthy subjects, while it also reveals statistically relevant differences in white matter regions for two pathologies, with a higher reliability than the FA. Additionally, APA has a computational complexity similar to the FA, with processing‐times several orders of magnitude below the PA.
Conclusions
The APA can extract more relevant white matter information than the FA, without any additional demands on data acquisition. This makes APA an attractive option for adoption into existing diffusion MRI analysis pipelines.
... Nevertheless, the diffusion mechanisms cannot be accurately described by DT-MRI because of the oversimplified Gaussian fitting. Accordingly, more evolved techniques with more degrees-of-freedom naturally arose, such as High Angular Resolution Diffusion Imaging ( Tuch et al., 2003;¨ Ozarslan et al., 2006, HARDI) or Diffusion Kurtosis Imaging (Hansen and Jespersen, 2016, DKI). The trend over the last decade has consisted in acquiring a large number of diffusion-weighted images distributed over several shells (i.e. with several gradient strengths) and with moderate-to-high b-values to estimate more advanced diffusion descriptors, as the Ensemble Average Diffusion Propagator ( ¨ Ozarslan et al., 2013, EAP). ...
... Note that, although D(q) no longer depends on q 0 , E(q) does. This assumption, although restrictive, is used to define certain diffusion modalities in HARDI ( Descoteaux et al., 2006;¨ Ozarslan et al., 2006), where only one shell is usually acquired. This simplification was initially intended to overcome the limitations of the DT by allowing the diffusion to be evaluated across many orientations, as opposed to the single orientation described by the DT. ...
The Propagator Anisotropy (PA) is a measurement of the orientational variability inside a tissue estimated from diffusion MRI using the Ensemble Average diffusion Propagator (EAP). It is based on the quantification of the angular difference between the propagator in a specific voxel and its isotropic counterpart. The PA has shown the ability to reveal microstructural information of interest and meaningful descriptive maps inside the white matter. However, the use of PA is not generalized among the clinical community, due to the great amount of data needed for its calculation, together with the associated long processing times. In order to calculate the PA, the EAP must also be properly estimated. This task would require a dense sampling of the Cartesian q-space. Alternatively, more efficient techniques have been proposed in the last decade. Even so, all of them imply acquiring a large number of diffusion gradients with different b-values and long processing times.
In this work, we propose an alternative implementation to drastically reduce the number of samples needed, as well as boosting the estimation procedure. We avoid the calculation of the whole EAP by assuming that the diffusion anisotropy is roughly independent from the radial direction. With such an assumption, we achieve a closed-form expression for a measure similar to the PA but using information from one single shell: the Apparent Propagator Anisotropy (APA). The new measure remains compatible with standard acquisition protocols commonly used for HARDI (based on just one b-value). The intention of the APA is not to exactly replicate the PA but inferring microstructural information with comparable discrimination power as the PA but using a reduced amount of data.
We report extensive results showing that the proposed measures present a robust behavior in clinical studies and they are computationally efficient and robust when compared with PA and other anisotropy measures.
... Using dMRI data for understanding intravoxel white matter structure is a mathematically challenging problem. Several strategies have been proposed in literature [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Additionally, the difficulty is increased given that there is no "goldstandard" for evaluating the process. ...
... Several papers propose methods to determine white matter geometry given dMRI data [6,11,12,14,[18][19][20]. These methods utilize classical least-squares regression, which does not allow variable selection or shrinkage to be performed. ...
The human brain is a complex system of neural tissue that varies significantly between individuals. Although the technology that delineates these neural pathways does not currently exist, medical imaging modalities, such as diffusion magnetic resonance imaging (dMRI), can be leveraged for mathematical identification. The purpose of this work is to develop a novel method employing machine learning techniques to determine intravoxel nerve number and direction from dMRI data. The method was tested on multiple synthetic datasets and showed promising estimation accuracy and robustness for multi-nerve systems under a variety of conditions, including highly noisy data and imprecision in parameter assumptions.
... Two different types of HARDI datasets are used to evaluate our approach: from the physical diffusion phantom of tractometer and from an in vivo human brain. For each dataset, we firstly constructed ODF fields using DOTmethod [24] and then applied the proposed algorithms to estimate fiber paths. ...
... Tuch presented a simple linear matrix formulation that was provided to construct ODFs using radial basic function (RBF) [26]. Diffusion orientation transform (DOT) converts water diffusivity profiles into probability profiles under the monoexponential signal decay assumption through computing PDF at a fixed distance from the origin [24,29,30]. Spherical deconvolution (SD) estimates fiber orientations by assuming that a single response function can adequately describe HARDI signals measured from any fiber bundle [31]. ...
The study of neural connectivity has grown rapidly in the past decade. Revealing brain anatomical connection improves not only clinical measures but also cognition understanding. In order to achieve this goal, we have to track neural fiber pathways first. Aiming to estimate 3D fiber pathways more accurately from orientation distribution function (ODF) fields, we presented a novel tracking method based on nonuniform rational B-splines (NURBS) curve fitting. First, we constructed ODF fields from high angular resolution diffusion imaging (HARDI) datasets using diffusion orientation transform (DOT) method. Second, under the angular and length constraints, the consecutive diffusion directions were extracted along each fiber pathway starting from a seed voxel. Finally, after the coordinates of the control points and their corresponding weights were determined, NURBS curve fitting was employed to track fiber pathways. The performance of the proposal has been evaluated on the tractometer phantom and real brain datasets. Based on several measure metrics, the resulting fiber pathways show promising anatomic consistency.
... where Y lm are the SH basis functions of order l and degree m with coefficients given by C lm . As the signal at order 8 is 500 times smaller than the one at order zero, we can capture 99.9% of diffusion-weighted signals using SH order of up to 8 in SH fusion methods (Mirzaalian et al. 2018;Descoteaux et al. 2007;Ozarslan et al. 2006;Tournier et al. 2007). Lastly, fused dMRI signals on b = 1000 and b = 2000 s/mm 2 shells can be re-estimated with inverse SH transformation, respectively. ...
While 7T diffusion magnetic resonance imaging (dMRI) has high spatial resolution, its diffusion imaging quality is usually affected by signal loss due to B1 inhomogeneity, T2 decay, susceptibility, and chemical shift. In contrast, 3T dMRI has relative higher diffusion angular resolution, but lower spatial resolution. Combination of 3T and 7T dMRI, thus, may provide more detailed and accurate information about the voxel-wise fiber orientations to better understand the structural brain connectivity. However, this topic has not yet been thoroughly explored until now. In this study, we explored the feasibility of fusing 3T and 7T dMRI data to extract voxel-wise quantitative parameters at higher spatial resolution. After 3T and 7T dMRI data was preprocessed, respectively, 3T dMRI volumes were coregistered into 7T dMRI space. Then, 7T dMRI data was harmonized to the coregistered 3T dMRI B0 (b = 0) images. Last, harmonized 7T dMRI data was fused with 3T dMRI data according to four fusion rules proposed in this study. We employed high-quality 3T and 7T dMRI datasets (N = 24) from the Human Connectome Project to test our algorithms. The diffusion tensors (DTs) and orientation distribution functions (ODFs) estimated from the 3T-7T fused dMRI volumes were statistically analyzed. More voxels containing multiple fiber populations were found from the fused dMRI data than from 7T dMRI data set. Moreover, extra fiber directions were extracted in temporal brain regions from the fused dMRI data at Otsu’s thresholds of quantitative anisotropy, but could not be extracted from 7T dMRI dataset. This study provides novel algorithms to fuse intra-subject 3T and 7T dMRI data for extracting more detailed information of voxel-wise quantitative parameters, and a new perspective to build more accurate structural brain networks.
... This parameter, which is proportional to the inverse of diffusion sensitizing gradient directions, is a crucial design element in dMRI (de Figueiredo et al., 2011;Basser and Pierpaoli, 1996). The number of gradient directions required to fit the basis can vary depending on the method used to represent the dMRI signal, ranging from a minimum of six gradients in DTI to several dozens or hundreds in High Angular Resolution Diffusion Imaging (HARDI) techniques (Tuch et al., 2003;Tristán-Vega et al., 2009;Ö zarslan et al., 2006). In clinical settings, it is necessary to optimize the number of gradient directions to minimize the examination duration and ensure patient comfort. ...
The objective of this study is to evaluate the efficacy of deep learning (DL) techniques in improving the quality of diffusion MRI (dMRI) data in clinical applications. The study aims to determine whether the use of artificial intelligence (AI) methods in medical images may result in the loss of critical clinical information and/or the appearance of false information. To assess this, the focus was on the angular resolution of dMRI and a clinical trial was conducted on migraine, specifically between episodic and chronic migraine patients. The number of gradient directions had an impact on white matter analysis results, with statistically significant differences between groups being drastically reduced when using 21 gradient directions instead of the original 61. Fourteen teams from different institutions were tasked to use DL to enhance three diffusion metrics (FA, AD and MD) calculated from data acquired with 21 gradient directions and a b-value of 1000 s/mm2. The goal was to produce results that were comparable to those calculated from 61 gradient directions. The results were evaluated using both standard image quality metrics and Tract-Based Spatial Statistics (TBSS) to compare episodic and chronic migraine patients. The study results suggest that while most DL techniques improved the ability to detect statistical differences between groups, they also led to an increase in false positive. The results showed that there was a constant growth rate of false positives linearly proportional to the new true positives, which highlights the risk of generalization of AI-based tasks when assessing diverse clinical cohorts and training using data from a single group. The methods also showed divergent performance when replicating the original distribution of the data and some exhibited significant bias. In conclusion, extreme caution should be exercised when using AI methods for harmonization or synthesis in clinical studies when processing heterogeneous data in clinical studies, as important information may be altered, even when global metrics such as structural similarity or peak signal-to-noise ratio appear to suggest otherwise.
... It has proven its effectiveness in modeling white matter brain structures along with the fiber intersection regions. There are numerous methods to reconstruct the architecture of brain tissue using HARDI data Tournier, Calamante, Gadian and Connelly (2004); Özarslan, Shepherd, Vemuri, Blackband and Mareci (2006) ;Tournier, Yeh, Calamante, Cho, Connelly and Lin (2008). Each approach provides the orientations of several white-matter tracts inside a voxel. ...
Background and objective:
We focus on three-dimensional higher-order tensorial (HOT) images using Finsler geometry. In biomedical image analysis, these images are widely used, and they are based on the diffusion profiles inside the voxels. The diffusion information is stored in the so-called diffusion tensor D. Our objective is to present new methods revealing the architecture of neural fibers in presence of crossings and high curvatures. After tracking the fibers, we achieve direct 3D image segmentation to analyse the brain's white matter structures.
Methods:
To deal with the construction of the underlying fibers, the inverse of the second-order diffusion tensor D, understood as the metric tensor D-1, is commonly used in DTI modality. For crossing and highly curved fibers, higher order tensors are more relevant, but it is challenging to find an analogue of such an inverse in the HOT case. We employ an innovative approach to metrics based on higher order tensors to track the fibers properly. We propose to feed the tracked fibers as the internal initial contours in an efficient version of 3D segmentation.
Results:
We propose a brand-new approach to the inversion of a diffusion HOT, and an effective way of fiber tracking in the Finsler setting, based on innovative classification of the individual voxels. Thus, we can handle complex structures with high curvatures and crossings, even in the presence of noise. Based on our novel tractography approach, we also introduce a new segmentation method. We feed the detected fibers as the initial position of the contour surfaces to segment the image using a relevant active contour method (i.e., initiating the segmentation from inside the structures).
Conclusions:
This is a pilot work, enhancing methods for fiber tracking and segmentation. The implemented algorithms were successfully tested on both synthetic and real data. The new features make our algorithms robust and fast, and they allow distinguishing individual objects in complex structures, even under noise.
... При использовании тензорной модели для работы с диффузионными данными невозможно различить на уровне вокселей такие конфигурации волокон, как пересечение, соприкосновение (kissing), изгиб и «веер» [39]. В таком случае может потребоваться использование моделей диффузии «высокого порядка» [40][41][42][43][44][45][46][47]. ...
... In general, data at larger b-values offer not only a better description of the signal but also more parameters related to the tissue microstructure. Unsurprisingly, acquisitions at higher b-values have led to significant advances, e.g., in resolving more than one major fiber orientations within a voxel (Tournier et al., 2004;Tuch, 2004;Özarslan et al., 2006;Wedeen et al., 2012). ...
Diffusion MR is sensitive to the microstructural features of a sample. Fine-scale characteristics can be probed by employing strong diffusion gradients while the low b -value regime is determined by the cumulants of the distribution of particle displacements. A signal representation based on the cumulants, however, suffers from a finite convergence radius and cannot represent the ‘localization regime' characterized by a stretched exponential decay that emerges at large gradient strengths. Here, we propose a new representation for the diffusion MR signal. Our method provides not only a robust estimate of the first three cumulants but also a meaningful extrapolation of the entire signal decay.
... Other methods assume a 'single-shell' spherical sampling of the qspace, which relies on the use of orthonormal basis functions on the sphere. Examples of these methods include the diffusion orientation transform, which relies on a pointwise convergent expansion of the plane wave (Ozarslan et al 2006, Canales-Rodríguez et al 2010; the numerical Q-ball model, which relies on the Funk Radon transform (Tuch 2004); the analytical Q-ball model, which relies on the use of spherical harmonics with the Funk-Hecke transform (Descoteaux et al 2007). Later, the sharpening deconvolution transform that performs the deconvolution in the ODF space was introduced to reconstruct fODF from dODF, and a further Laplace-Beltrami regularisation to increase its robustness to noise (Descoteaux et al 2009). ...
Diffusion MRI (dMRI) tractography is currently the only imaging technique that allows for non-invasive delineation and visualisation of white matter (WM) tracts in vivo prompting rapid advances in related fields of brain MRI research in recent years. One of its major clinical applications is for pre-surgery planning and intraoperative image guidance in neurosurgery, where knowledge about the location of WM tracts nearby the surgical target can be helpful to guide surgical resection and optimise surgical outcomes. Surgical injuries to these WM tracts can lead to permanent neurological and functional deficits, making the accuracy of tractography reconstructions paramount. The quality of dMRI tractography is influenced by many modifiable factors, ranging from MRI data acquisition through to the post-processing of the tractography output, with the potential of error propagation based on decisions made at each and subsequent processing steps. Research over the last 25 years has significantly improved the anatomical accuracy of tractography. An updated review about tractography methodology in the context of neurosurgery is now timely given the thriving research activities in dMRI, to ensure more appropriate applications in the clinical neurosurgical realm. This article aims to review the dMRI physics, and tractography methodologies, highlighting recent advances to provide the key concepts of tractography-informed neurosurgery, with a focus on the general considerations, the current state of practice, technical challenges, potential advances, and future demands to this field.
... High angular resolution diffusion imaging (HARDI) technique is another one. It further enables us to create ber orientation distribution function (fODF) models [32][33][34][35][36] , which offer accurate tractography and apparent ber density (AFD) for each streamline 47,48 . Here, we aimed to see if changes in microstructure measures of the medial regions are associated with cognitive function in PD patients, using DTI and HARDI techniques. ...
Functional neuroimaging studies of patients with Parkinson’s disease (PD) have repeatedly identified over-activations in midline structures (medial prefrontal cortex, anterior cingulate cortex, posterior cingulate cortex, and precuneus), especially in those without comorbid dementia. Here, we investigated whether the different cognitive profiles in PD were linked to measures of diffusion microstructure in medial regions of the brain. Using magnetic resonance based diffusion weighted imaging (DWI) in healthy volunteers (HV) and PD patients with and without mild cognitive impairment (PD-nonMCI and PD-MCI), applying diffusion tensor imaging (DTI) and high angular resolution diffusion imaging (HARDI) techniques, we observed: 1) increased fractional anisotropy (FA) in the precuneus and the anterior cingulate in the PD-nonMCI participants compared with the HV; 2) an association between precuneus FA and executive and memory function, respectively, in PD and HV; 3) a negative correlation between age and midline structure FA in PD but not HV; and 4) a differential association between cognitive scores and apparent fiber density (AFD) of the posterior cingulate-precuneus bundle in HV vs. PD. Together, these findings suggest that white matter reorganization of the posterior medial microstructures might serve a compensatory role for damaged basal ganglia function in PD-nonMCI.
... These components can potentially be used to track the fibers at crossings. The other observation is about fuzziness in finding maxima using the CT-ODF method, which is potentially better than DOT, QBI, etc [41] [42]. These maxima provide the direction of underlying fibers. ...
The geodesic ray-tracing method has shown its effectiveness for the reconstruction of fibers in white matter structure. It provides multiple solutions and is robust to noise and curvatures of fibers. The choice of the metric tensor has a significant impact on the outcome of this method. The existing metrics are not sufficient in the construction of white matter tracts as geodesics. We propose a way to choose the appropriate conformal class of metrics where the metric gets scaled according to tensor anisotropy. We used the logistic functions, which are commonly used in statistics as cumulative distribution functions. To prevent deviation of geodesics from the actual paths, we propose a hybrid ray-tracing approach. Furthermore, we employ diagonal projection of 4th order tensor to perform fiber tracking in crossing regions. Results from synthetic and real data experiments elucidate the method.
... These components can potentially be used to track the fibers at crossings. The other observation is about fuzziness in finding maxima using the CT-ODF method, which is potentially better than DOT, QBI, etc [41] [42]. These maxima provide the direction of underlying fibers. ...
... A major drawback of the diffusion tensor model is its inability to represent complex intravoxel topologies such as crossing fibers [24][25][26]. As illustrated in Fig. 20.3, the diffusion tensor is only capable of representing one principal fiber direction; when two or more fiber populations of approximately equal density are present in a single voxel, the leading eigenvector of the diffusion tensor is no longer well defined. ...
Diffusion MRI can probe neural fiber orientations through an in vivo measurement of water diffusion on a subvoxel scale, ultimately enabling tracking structural connections in the brain. This chapter introduces the principles behind resolving fiber orientations in diffusion MRI and provides an overview of the models and techniques that have advanced this field over the years. We will first introduce the concept of q→-space as the fundamental descriptor of the diffusion propagator. Subsequently, we will discuss the main developments toward practical, scan-time-efficient fiber orientation modeling, starting with diffusion tensor imaging and moving on to higher-order representations of increasingly complex fiber topologies. We will particularly discuss spherical deconvolution as the most widely adopted approach to modeling fiber orientation distributions. Finally, we will point out the major differences and similarities between the various techniques, distinguishing discrete versus continuous representations and model-based versus data-driven methods.
... First, the diffusion orientation transform (DOT) shares similarities with q-ball imaging as both are based on the earlier q-space approach. In contrast, DOT assumes diffusion is Gaussian along the radial direction and uses this assumption to perform the Fourier transform in (4) using fewer diffusion weighted image samples [95]. The DOT diffusion ODF is then obtained by analytically integrating the resulting PDF along the radial direction. ...
... All these methods can be used to investigate tissue geometry, but they are not all equally applicable in all situations. Other methods such as the high angular resolution diffusion imaging (HARDI) [26,30] and diffusion kurtosis imaging (DKI) [17] were also proposed. To obtain the non-Gaussian property of the signal higher b-values are required [20]. ...
Diffusion magnetic resonance imaging (dMRI) is a relatively modern technique used to study tissue microstructure in a non-invasive way. Non-Gaussian diffusion representation is related to the restricted diffusion and can provide information about the underlying tissue properties. In this paper, we analytically derive $n$-th order statistics of the signal considering a stretched-exponential representation of the diffusion. Then, we retrieve the Q-space quantitative measures such as the Return-To-the-Origin Probability (RTOP), Q-space mean square displacement (QMSD), Q-space mean fourth-order displacement (QMFD). The stretched-exponential representation enables the handling of the diffusion contributions from a higher $b$-value regime under a non-Gaussian assumption, which can be useful in diagnosing or prognosis of neurodegenerative diseases in the early stages. Numerical implementation of the method is freely available at https://github.com/TPieciak/Stretched.
... For example, several studies utilized synthetic data to highlight the failure of DTI to characterize intra-voxel orientation when more than a single fiber population is present (Alexander, 2005), a challenge which came to be known as the crossing fiber problem and is known to affect a majority of voxels in the brain (Schilling et al., 2017;Jeurissen et al., 2013). A large number of models have since been developed to solve this problem, many of which are first validated through simulations (Descoteaux et al., 2007;Aganj et al., 2010;Ozarslan et al., 2006;Tournier et al., 2007;Tuch et al., 2002;Yeh et al., 2010), typically by optimizing selection of algorithmic and acquisition parameters. ...
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.
... Some groups have developed alternate, nonparametric models to replace the tensor model, which can better resolve crossing fibres, such as diffusion spectrum imaging, 104,109 q-ball imaging, 103,105 spherical deconvolution, 102 persistent angular structure MRI, 58 and diffusion orientation transform. 85 However, these are outside the scope of this review (see Ref. 3 for a comprehensive review). Some groups have tried to resolve crossing fibres within the tensor model. ...
The orofacial region is psychologically important, given that it serves fundamental and important biological purposes. Chronic orofacial pain disorders affect the head and neck region. Although some have clear peripheral etiologies, eg, classic trigeminal neuralgia, others do not have a clear etiology (eg, muscular temporomandibular disorders). However, these disorders provide a unique opportunity in terms of elucidating the neural mechanisms of these chronic pain conditions: both the peripheral and central nervous systems can be simultaneously imaged. Diffusion-weighted imaging and diffusion tensor imaging have provided a method to essentially perform in vivo white matter dissections in humans, and to elucidate abnormal structure related to clinical correlates in disorders, such as chronic orofacial pains. Notably, the trigeminal nerve anatomy and architecture can be captured using diffusion imaging. Here, we review the trigeminal somatosensory pathways, diffusion-weighted imaging methods, and how these have contributed to our understanding of the neural mechanisms of chronic pain disorders affecting the trigeminal system. We also discuss novel findings indicating the potential for trigeminal nerve diffusion imaging to develop diagnostic and precision medicine biomarkers for trigeminal neuralgia. In sum, diffusion imaging serves both an important basic science purpose in identifying pain mechanisms, but is also a clinically powerful tool that can be used to improve treatment outcomes.
... Moreover, DTI has the significant limitation of being unable to represent multiple fiber orientations in a single voxel. Several methods have been developed to overcome this limitation such as q-ball imaging (QBI) ( Tuch, 2004 ), diffusion spectrum imaging (DSI) ( Wedeen et al., 2005 ), diffusional kurtosis imaging (DKI) ( Jensen et al., 2005 ), diffusion orientation transform (DOT) ( Ozarslan et al., 2006 ), neurite orientation dispersion and density imaging (NODDI) ( Zhang et al., 2012 ), diffusion basis spectrum imaging (DBSI) ( Wang et al., 2011 ) and generalized q-sampling imaging (GQI) ( Yeh et al., 2010 ). These methods reconstruct orientation distribution functions (ODFs) of fibers at each voxel, allowing multiple fiber populations in a voxel to be represented. ...
... This technique provides an approximation to the dODF directly, without the need to estimate the spin propagator. Other techniques have been proposed over the years, including more efficient implementations of QBI [48,49], QBI with solid angle consideration [50], the diffusion orientation transform [51], persistent angular structure MRI [52], and many others. ...
Over the past two decades, diffusion MRI has become an essential tool in neuroimaging investigations. This is due to its sensitivity to the motion of water molecules as they diffuse through the microstructural environment, allowing diffusion MRI to be used as a ‘probe’ of tissue microstructure. Furthermore, this sensitivity is strongly direction-dependent, notably in brain white matter, due to the alignment of structures that restrict or hinder the motion of water molecules, notably axonal membranes. This provides a means of inferring the orientation of fibres in vivo, and by use of appropriate fibre-tracking algorithms, of delineating the path of white matter tracts in the brain. The ability to perform so-called tractography in humans in vivo non-invasively is unique to diffusion MRI, and is now used in applications such as neurosurgery planning and more broadly within investigations of brain connectomics. This review describes the theory and concepts of diffusion MRI and describes its most important areas of application in the brain, with a strong focus on tractography.
... First, the diffusion orientation transform (DOT) shares similarities with q-ball imaging as both are based on the earlier q-space approach. In contrast, DOT assumes diffusion is Gaussian along the radial direction and uses this assumption to perform the Fourier transform in (4) using fewer diffusion weighted image samples [95]. The DOT diffusion ODF is then obtained by analytically integrating the resulting PDF along the radial direction. ...
Diffusion MRI (dMRI) is a powerful imaging protocol that is used in the assessment of the organization and integrity of fibrous tissue. The imaging works by measuring the diffusion of water molecules within the body. This diffusion is restricted by cell membranes and as a result, rates of diffusion are far less across tissue fibers than parallel to them. With enough diffusion measurements along different directions in 3D, we can noninvasively obtain a model of the diffusion pattern at different points within an imaged subject.
... This assumption is implicitly done when considering the diffusion tensor model, and it is also common in HARDI models [39,40,46] where a marginalization along the radial component is necessary. The acquired signal in the scanner is not directly E(q), but a scaled version of it, S(g): ...
... A number of modifications have been proposed since, including the use of spherical harmonics to speed up and improve the reconstruction, 45,46 its extension to multi-shell data, 41 and solid-angle considerations to provide a dODF equivalent to that provided by DSI. 47 A number of other approaches based on q-space have also been proposed over the years, including persistent angular structure MRI (PAS-MRI), 20 diffusion orientation transform, 48 generalized diffusion tensor imaging, 49,50 and many others. However, these are less commonly used, and in some cases are not developed specifically to estimate the directions of the fibres themselves, but rather are used to characterize the diffusion signal in the presence of crossing fibres. ...
Since the realization that diffusion MRI can probe the microstructural organization and orientation of biological tissue in vivo and non‐invasively, a multitude of diffusion imaging methods have been developed and applied to study the living human brain. Diffusion tensor imaging was the first model to be widely adopted in clinical and neuroscience research, but it was also clear from the beginning that it suffered from limitations when mapping complex configurations, such as crossing fibres. In this review, we highlight the main steps that have led the field of diffusion imaging to move from the tensor model to the adoption of diffusion and fibre orientation density functions as a more effective way to describe the complexity of white matter organization within each brain voxel. Among several techniques, spherical deconvolution has emerged today as one of the main approaches to model multiple fibre orientations and for tractography applications. Here we illustrate the main concepts and the reasoning behind this technique, as well as the latest developments in the field. The final part of this review provides practical guidelines and recommendations on how to set up processing and acquisition protocols suitable for spherical deconvolution. Diffusion MRI can probe non‐invasively and in vivo the microstructural organization and the orientation of white matter in the human brain. Diffusion tensor imaging was the first model to be widely adopted in clinical and research applications, but it has also shown important limitations. Alternative methods have been developed to extract and better describe multiple populations of fibre orientations in the brain. Here we review these techniques, focusing particularly on spherical deconvolution, its principles and how this method can be practically applied in real studies.
... Aside from characterizing microstructural development, DTI and tractography can be used to identify the sequential appearance of fiber bundles. In order to better characterize structural maturation and connectivity during the past decade numerous diffusion models (Assaf and Basser, 2005;Basser et al., 1994;Özarslan et al., 2006;Tournier et al., 2004) and algorithms used for tract reconstruction, i.e. tractography (Chao et al., 2007;Mori et al., 1999;Tuch, 2002;Weinstein et al., 1999) have been developed. However, given the rapid changes in the fetal brain from week to week and altered cortical microstructural development in prematurely born infants, model-based reconstructions that require the assumption of a fixed diffusivity have limitations and care must be taken in the interpretation of the biological meaning of the parameters obtained at each stage of development. ...
Early brain development, from embryonic period to infancy, is characterized by rapid structural and functional changes. These changes can be studied using structural and physiological neuroimaging methods. In order to optimally acquire and accurately interpret this data, concepts from adult neuroimaging cannot be directly transferred. Instead, one must have a basic understanding of fetal and neonatal structural and physiological brain development, and understand important modulators of this process. Here, we first review the major developmental milestones of transient cerebral structures and structural connectivity (axonal connectivity) followed by a summary of the contributions from ex vivo and in vivo MRI. Next, we discuss the basic biology of neuronal circuitry development (synaptic connectivity, i.e. ensemble of direct chemical and electrical connections between neurons), physiology of neurovascular coupling, baseline metabolic needs of the fetus and the infant, and functional connectivity (defined as statistical dependence of low-frequency spontaneous fluctuations seen with functional magnetic resonance imaging (fMRI)). The complementary roles of magnetic resonance imaging (MRI), electroencephalography (EEG), magnetoencephalography (MEG), and near-infrared spectroscopy (NIRS) are discussed. We include a section on modulators of brain development where we focus on the placenta and emerging placental MRI approaches. In each section we discuss key technical limitations of the imaging modalities and some of the limitations arising due to the biology of the system. Although neuroimaging approaches have contributed significantly to our understanding of early brain development, there is much yet to be done and a dire need for technical innovations and scientific discoveries to realize the future potential of early fetal and infant interventions to avert long term disease.
... More recent methods which can be labelled collectively as high angular resolution diffusion imaging (HARDI) provide models capturing information on the existence of multiple fiber orientations per voxel. Yet, HARDI is not a singular technique; rather, numerous methods exist to model DW-MRI data with more complex intra-voxel models such as Q-ball, spherical deconvolution, PAS-MRI, diffusion orientation transform [2][3][4][5][6]. ...
High Angular Resolution Diffusion Imaging (HARDI) models are used to capture complex intra-voxel microarchitectures. The magnetic resonance imaging sequences that are sensitized to diffusion are often highly accelerated and prone to motion, physiologic, and imaging artifacts. In diffusion tensor imaging, robust statistical approaches have been shown to greatly reduce these adverse factors without human intervention. Similar approaches would be possible with HARDI methods, but robust versions of each distinct HARDI approach would be necessary. To avoid the computational and pragmatic burdens of creating individual robust HARDI analysis variants, we propose a robust outlier imputation model to mitigate outliers prior to traditional HARDI analysis. This model uses a weighted spherical harmonic fit of diffusion weighted magnetic resonance imaging scans to estimate the values which had been corrupted during acquisition to restore them. Briefly, spherical harmonics of 6th order were used to generate basis function which were weighted by diffusion signal for detection of outliers. For validation, a single healthy volunteer was scanned for a single session comprising of two scans one without head movement and the other with deliberate head movement at a b-value of 3000 s/mm2 with 64 diffusion weighted directions with a single b0 (5 averages) per scan. The deliberate motion from the volunteer created natural artifacts in the acquisition of one of the scans. The imputation model shows reduction in root mean squared error of the raw signal intensities and improvement for the HARDI method Q-ball in terms of the Angular Correlation Coefficient. The results reveal that there is quantitative and qualitative improvement. The proposed model can be used as general pre-processing model before implementing any HARDI model in general to restore the artifacts which are created because of the outlier diffusion signal in certain gradient volumes.
... Moreover, DTI has the significant limitation of being unable to represent multiple fiber orientations in a single voxel. Several methods have been developed to overcome this limitation such as q-ball imaging (QBI) ( Tuch, 2004 ), diffusion spectrum imaging (DSI) ( Wedeen et al., 2005 ), diffusional kurtosis imaging (DKI) ( Jensen et al., 2005 ), diffusion orientation transform (DOT) ( Ozarslan et al., 2006 ), neurite orientation dispersion and density imaging (NODDI) ( Zhang et al., 2012 ), diffusion basis spectrum imaging (DBSI) ( Wang et al., 2011 ) and generalized q-sampling imaging (GQI) ( Yeh et al., 2010 ). These methods reconstruct orientation distribution functions (ODFs) of fibers at each voxel, allowing multiple fiber populations in a voxel to be represented. ...
Accurate digital representation of major white matter bundles in the brain is an important goal in neuroscience image computing since the representations can be used for surgical planning, intra-patient longitudinal analysis and inter-subject population connectivity studies. Reconstructing desired fiber bundles generally involves manual selection of regions of interest by an expert, which is subject to user bias and fatigue, hence an automation is desirable. To that end, we first present a novel anatomical representation based on Neighborhood Resolved Fiber Orientation Distributions (NRFOD) along the fibers. The resolved fiber orientations are obtained by generalized q-sampling imaging (GQI) and a subsequent diffusion decomposition method. A fiber-to-fiber distance measure between the proposed fiber representations is then used in a density-based clustering framework to select the clusters corresponding to the major pathways of interest. In addition, neuroanatomical priors are utilized to constrain the set of candidate fibers before density-based clustering. The proposed fiber clustering approach is exemplified on automation of the reconstruction of the major fiber pathways in the brainstem: corticospinal tract (CST); medial lemniscus (ML); middle cerebellar peduncle (MCP); inferior cerebellar peduncle (ICP); superior cerebellar peduncle (SCP). Experimental results on Human Connectome Project (HCP)'s publicly available "WU-Minn 500 Subjects + MEG2 dataset" and expert evaluations demonstrate the potential of the proposed fiber clustering method in brainstem white matter structure analysis.
Diffusion -weighted Magnetic Resonance Imaging (dMRI) has long proven to be a versatile tool for the in-vivo microstructural investigation of the human brain, the spinal cord, or even muscle tissue. In contrast to conventional weighted MRI or functional MRI discussed in the preceding Chap. 4, it is quantitative in the sense that it directly infers on physical quantities with physical units, specifically the diffusion constant. In this chapter, we will first elaborate on the physical background before presenting experimental dMRI data and describe its processing. This includes pre-processing steps, i.e., the removal of artifacts, and the actual modeling of the data to infer on interesting and relevant quantities. We also discuss a structural adaptive smoothing method for dMRI data before concluding the chapter with fiber tracking within the brain white matter and the construction of structural connectivity networks.
Neural tissue is a hierarchical multiscale system with intracellular and extracellular diffusion compartments at different length scales. The normal diffusion of bulk water in tissues is not able to detect the specific features of a complex system, providing nonlocal, diffusion measurement averaged on a 10-20 μm length scale. Being able to probe tissues with sub-micrometric diffusion length and quantify new local parameters, transient anomalous diffusion (tAD) would dramatically increase the diagnostic potential of diffusion MRI (DMRI) in detecting collective and sub-micro architectural changes of human tissues due to pathological damage. In DMRI, the use of tAD parameters quantified using specific DMRI acquisition protocols and their interpretation has often aroused skepticism. Although the derived formulas may accurately fit experimental diffusion-weighted data, the relationships between the postulated dynamical feature and the underlying geometrical structure remains elusive, or at most only suggestive. This work aimed to elucidate and validate the image contrast and information that can be obtained using the tAD model in white matter (WM) through a direct comparison between different diffusion metrics and histology. Towards this goal, we compared tAD metrics extracted from pure subdiffusion (α-imaging) and super-pseudodiffusion (γ-imaging) in excised mouse spinal cord WM, together with T2 and T2* relaxometry, conventional (normal diffusion-based) diffusion tensor imaging (DTI) and q-space imaging (QSI), with morphologic measures obtained by optical microscopy, to determine which structural and topological characteristics of myelinated axons influenced tAD contrast. Axon diameter (AxDiam), the standard deviation of diameters (SDax.diam), axonal density (AxDens) and effective local density (ELD) were extracted from optical images in several WM tracts. Among all the diffusion parameters obtained at 9.4 T, γ-metrics confirmed a strong dependence on magnetic in-homogeneities quantified by R2* = 1/T2* and showed the strongest associations with AxDiam and ELD. On the other hand, α-metrics showed strong associations with SDax.diam and was significantly related to AxDens, suggesting its ability to quantify local heterogeneity degree in neural tissue. These results elucidate the biophysical mechanism underpinning tAD parameters and show the clinical potential of tAD-imaging, considering that both physiologic and pathologic neurodegeneration translate into alterations of WM morphometry and topology.
AMURA (Apparent Measures Using Reduced Acquisitions) was originally proposed as a method to infer micro-structural information from single-shell acquisitions in diffusion MRI. It reduces the number of samples needed and the computational complexity of the estimation of diffusion properties of tissues by assuming the diffusion anisotropy is roughly independent on the b-value. This simplification allows the computation of simplified expressions and makes it compatible with standard acquisition protocols commonly used even in clinical practice. The present work proposes an extension of AMURA that allows the calculation of general moments of the diffusion signals that can be applied to describe the diffusion process with higher accuracy. We provide simplified expressions to analytically compute a set of scalar indices as moments of arbitrary orders over either the whole 3-D space, particular directions, or particular planes. The existing metrics previously proposed for AMURA (RTOP, RTPP and RTAP) are now special cases of this generalization. An extensive set of experiments is performed on public data and a clinical clase acquired with a standard type acquisition. The new metrics provide additional information about the diffusion processes inside the brain.
Mapping the structural connectome and probing the cytoarchitecture of the cortical areas involved in cerebral connections are two of the greatest challenges that face the neuroimaging community. Methodological and technological issues still stand in the way of producing high-resolution maps, namely below the millimeter scale, on living subjects. Ultra-high magnetic fields (>7T) and powerful gradient sets (>300mT/m) are potent tools related to MRI that can overcome these hurdles. This thesis has the purpose of mapping human and non-human brains postmortem based on the combined use of the cited tools. Since clinical MRI systems do not offer this combination yet, the chosen approach was to scan brain samples with two preclinical systems (7T and 11.7T) equipped with strong gradients (440mT/m and 780mT/m) that enabled to produce cartographies at the mesoscopic scale in the frame of three complementary studies. Anatomical MRI, diffusion MRI (dMRI) and quantitative MRI (qMRI) as well as the brain fixation process were instrumental in these studies to finely map cerebral structures and connections, and collect cytoarchitectural and myeloarchitectural information as well. Dedicated post-processing pipelines were developed to map and analyze the substantial amount of data. The structural connectivities related to the socio-emotional circuits of breeding and wildlife animals were firstly investigated using T₂-weighted MRI and dMRI protocols. The breeding species was the Japanese quail, a bird which emotionality trait can be characterized through its specific response facing fearful situations called tonic immobility. A cohort of 21 subjects was formed to validate if its propensity to express fearfulness is reflected in its connectome. The study, in collaboration with INRAE Val de Loire (Nouzilly, France), consisted in investigating two lines of Japanese quails: the short (STI) and the long tonic immobility (LTI) lines. The 200-micron dMRI dataset led to a first structural connectivity atlas of the Japanese quail which revealed the existence of structural differences between the connectivity patterns characterizing the two lines. Then, 8 wildlife animals that lived in the ZooParc de Beauval (Saint-Aignan, France) were scanned at a submillimeter resolution. Their structural connectivity maps were generated and the labeling of the white matter bundles involved in their socio-emotional circuits was done. These two studies were stepping stones in observing the structural intracortical connectivity of the human cerebral cortex. The investigation of the human cerebral cortex at the mesoscale remains challenging but promising to better understand psychiatric, neurodegenerative and neurodevelopmental pathologies associated with cortex damage. In this ex vivo study, an entire human brain was provided by the Bretonneau University Hospital (Tours, France) after its extraction and fixation. The MRI dataset stems from two acquisition campaigns performed at 7T for qMRI and at 11.7T for anatomical MRI and dMRI, which lasted 2 years. The human brain sample was cut into 13 blocks in order to fit in the tunnels of the preclinical MRI scanners. For this thesis, the 200-micron qMRI and dMRI dataset enabled an automatic segmentation of the cortical layers inferred from MRI-based information. This segmentation aimed at better understanding the cytoarchitecture and myeloarchitecture of the human cerebral cortex.
Diffusion magnetic resonance imaging (dMRI) can be used to probe the connectivity and microstructure of human brain tissue non-invasively in vivo. The diffusion-weighted MR signal has sensitivity to micrometer-scale tissue properties averaged over the imaging voxel size, for example, intra-cellular and extra-cellular volumes and the 3D orientations of axonal bundles. It derives its contrast from sensitivity to the bulk displacement of water through what is known as thermal motion, Brownian motion, or passive (self-)diffusion. In the brain, cell membranes, organelles, and myelin sheaths create barriers and form biological compartments that constrain the displacement of water molecules, modifying the statistical behavior of bulk diffusion over time. This review therefore focuses on the use of dMRI for estimates of the local orientations and estimates of microstructure of fiber tracts, and on an understanding of dMRI signal mechanisms and appropriate signal processing and modeling for this purpose. It first discusses basic diffusion MRI acquisition principles and diffusion contrast and the constraints the acquisition places on the modeling of diffusion. It then sets out the diffusion tensor model, the most used model in dMRI that underlies diffusion tensor imaging, in a way which prepares a discussion of its limitations. The next sections set out advances in dMRI beyond DTI as focusing on tractography and connectomics, with a need to accurately model spatially complex fiber configurations and on diffusion microstructure, with a need to accurately model restricted diffusion and compartmentalization. Throughout, the emphasis is on a thorough understanding of basic principles and assumptions underlying techniques, as well as their possibilities and limitations for inference of brain connectivity, with a minimum of technical detail and mathematics. This review ends with an outlook on future developments emanating from current trends.
Validation and interpretation of diffusion magnetic resonance imaging (dMRI) requires detailed understanding of the actual microstructure restricting the diffusion of water molecules. In this study, we used serial block-face scanning electron microscopy (SBEM), a three-dimensional electron microscopy (3D-EM) technique, to image seven white and grey matter volumes in the rat brain. SBEM shows excellent contrast of cellular membranes, which are the major components restricting the diffusion of water in tissue. Additionally, we performed 3D structure tensor (3D-ST) analysis on the SBEM volumes and parameterised the resulting orientation distributions using Watson and angular central Gaussian (ACG) probability distributions as well as spherical harmonic (SH) decomposition. We analysed how these parameterisations described the underlying orientation distributions and compared their orientation and dispersion with corresponding parameters from two dMRI methods, neurite orientation dispersion and density imaging (NODDI) and constrained spherical deconvolution (CSD). Watson and ACG parameterisations and SH decomposition captured well the 3D-ST orientation distributions, but ACG and SH better represented the distributions due to its ability to model asymmetric dispersion. The dMRI parameters corresponded well with the 3D-ST parameters in the white matter volumes, but the correspondence was less evident in the more complex grey matter. SBEM imaging and 3D-ST analysis also revealed that the orientation distributions were often not axially symmetric, a property neatly captured by the ACG distribution. Overall, the ability of SBEM to image diffusion barriers in intricate detail, combined with 3D-ST analysis and parameterisation, provides a step forward toward interpreting and validating the dMRI signals in complex brain tissue microstructure.
A novel diffusion anisotropy metric is presented. It is based on dissimilarity between the acquired diffusion signal and its isotropic equivalent. Using the inner product of signals, a closed form expression is obtained, which allows its computation using spherical harmonics from a reduced set of acquired data, compatible with most popular diffusion MRI acquisition protocols. Results show that the proposed metric (1) is able to discriminate among different microstructure scenarios; (2) shows a robust behaviour in clinical studies.
Background
Diffusion magnetic resonance imaging (dMRI) is a popular non-invasive imaging technique applied for the study of nerve fibers in vivo, with diffusion tensor imaging (DTI) and high angular resolution diffusion imaging (HARDI) as the commonly used dMRI methods. However, DTI cannot resolve complex fiber orientations in a local area and HARDI lacks a solid physical basis.
New Method
We introduce a diffusion coefficient orientation distribution function (DCODF). It has a clear physical meaning to represent the orientation distribution of diffusion coefficients for Gaussian and non-Gaussian diffusion. Based on DCODF, we then propose a new HARDI method, termed as diffusion coefficient orientation distribution transform (DCODT), to estimate the orientation distribution of nerve fibers in voxels.
Results
The method is verified on the simulated data, ISMRM-2015-Tracto-challenge data, and HCP datasets. The results show the superior capability of DCODT in resolving the complex distribution of multiple fiber bundles effectively.
Comparison with Existing Method(s)
The method is compared to other common model-free HARDI estimators. In the numerical simulations, DCODT achieves a better trade-off between the resolution and accuracy than the counterparts for high b-values. In the comparisons based on the challenge data, the improvement of DCODT is significant in scoring. The results on the HCP datasets show that DCODT provides fewer spurious lobes in the glyphs, resulting in more coherent fiber orientations.
Conclusions
We conclude that DCODT may be a reliable method to extract accurate information about fiber orientations from dMRI data and promising for the study of neural architecture.
Diffusion-weighted magnetic resonance imaging (dMRI) is a popular tool for noninvasively assessing properties of white matter in the brain. Among other uses, dMRI data can be used to produce estimates of anatomical connectivity on the basis of tractography. However, direct comparisons of anatomical connectivity as estimated through invasive neural tract-tracing experiments and dMRI-derived connectivity have shown only a moderate relationship in nonhuman primate (particularly macaque) studies. Tractography is plagued by known problems associated with resolution, crossing fibers, and curving fibers, among others. These problems lead to deficits in both sensitivity and specificity, which trade off with each other in multiple datasets. Although not yet examined quantitatively, there is reason to believe that some large white matter bundles, those with more topographic organization, may produce more accurate results than others. Moving forward, sophisticated analytical approaches and anatomical constraints may improve tractography accuracy. However, broadly speaking, dMRI-derived estimates of brain connectivity should be approached with caution.
Fiber orientation distribution estimation with diffusion magnetic resonance imaging (dMRI) is critical in white matter fiber tractography which is most commonly based on symmetric crossing structures. Ambiguous spatial correspondence between estimated diffusion directions and fiber geometry, such as asymmetric crossing, bending, fanning, or kissing, makes tractography challenging. Consequently, numerous tracts suggest intertwined connections in unexpected regions of the white matter or actually stop prematurely in the white matter. In this work, we propose a novel asymmetric fiber trajectory distribution (FTD) function defined on neighboring voxels based on a streamline differential equation from fluid mechanics. The spatial consistency with intra- and inter-voxel constraints is derived for FTD estimation by introducing the concept of divergence. At a local level, the FTD is a series of curve flows that minimize the energy function, characterizes the relations between fibers and joint fiber fragments within the same fiber bundle. Experiments are performed on FiberCup phantom, ISMRM 2015 Tractography challenge data, and in vivo brain dMRI data for qualitative and quantitative evaluations. Results show that our approach can reveal continuous asymmetric FTD details that are potentially useful for robust tractography.
In the first study comparing high angular resolution diffusion MRI (dMRI) in the human brain to axonal orientation measurements from polarization-sensitive optical coherence tomography (PSOCT), we compare the accuracy of orientation estimates from various dMRI sampling schemes and reconstruction methods. We find that, if the reconstruction approach is chosen carefully, single-shell dMRI data can yield the same accuracy as multi-shell data, and only moderately lower accuracy than a full Cartesian-grid sampling scheme. Our results suggest that current dMRI reconstruction approaches do not benefit substantially from ultra-high b-values or from very large numbers of diffusion-encoding directions. We also show that accuracy remains stable across dMRI voxel sizes of 1 mm or smaller but degrades at 2 mm, particularly in areas of complex white-matter architecture. We also show that, as the spatial resolution is reduced, axonal configurations in a dMRI voxel can no longer be modeled as a small set of distinct axon populations, violating an assumption that is sometimes made by dMRI reconstruction techniques. Our findings have implications for in vivo studies and illustrate the value of PSOCT as a source of ground-truth measurements of white-matter organization that does not suffer from the distortions typical of histological techniques.
In diffusion MRI, the Ensemble Average diffusion Propagator (EAP) provides relevant microstructural information and meaningful descriptive maps of the white matter previously obscured by traditional techniques like the Diffusion Tensor. The direct estimation of the EAP, however, requires a dense sampling of the Cartesian q-space. Due to the huge amount of samples needed for an accurate reconstruction, more efficient alternative techniques have been proposed in the last decade. Even so, all of them imply acquiring a large number of diffusion gradients with different b-values. In order to use the EAP in practical studies, scalar measures must be directly derived, being the most common the return-to-origin probability (RTOP) and the return-to-plane and return-to-axis probabilities (RTPP, RTAP).
In this work, we propose the so-called “Apparent Measures Using Reduced Acquisitions” (AMURA) to drastically reduce the number of samples needed for the estimation of diffusion properties. AMURA avoids the calculation of the whole EAP by assuming the diffusion anisotropy is roughly independent from the radial direction. With such an assumption, and as opposed to common multi-shell procedures based on iterative optimization, we achieve closed-form expressions for the measures using information from one single shell. This way, the new methodology remains compatible with standard acquisition protocols commonly used for HARDI (based on just one b-value). We report extensive results showing the potential of AMURA to reveal microstructural properties of the tissues compared to state of the art EAP estimators, and is well above that of Diffusion Tensor techniques. At the same time, the closed forms provided for RTOP, RTPP, and RTAP-like magnitudes make AMURA both computationally efficient and robust.
An important task for neuroscience is to accurately construct structural connectivity network of human brain. Tractography constructed based on high angular resolution diffusion imaging (HARDI) provides valuable information of human brain structural connections. Existing algorithms, mainly categorized as deterministic or probabilistic, come with inherent limitations (e.g., fiber direction uncertainty induced by noise, or anatomically unreasonable connections and heavy computational cost). In this study, a novel integrated algorithm was proposed to construct brain structural connectivity network by incorporating the deterministic path planning and probabilistic connection strength estimation, based on ensemble average propagator (EAP). We first estimated EAPs from multi-shell samples using the spherical polar Fourier imaging (SPFI), and then extracted diffusion orientations coinciding with neural fiber tracts. Only under angular constraints, the deterministic path planning algorithm was subsequently used to find all reasonable pathways between pairwise white matter (WM) voxels in different regions of interest (ROIs). Consequently, a train of consecutive WM voxels along each of the identified pathways was determined, and the connection strength of these pathways was computed by integrating their EAP alignment over a solid angle. The connection strength of a pair of WM voxels was assigned as the connection strength with the largest connection possibility. Finally, the connection strength between two ROIs was calculated as the sum of all the connection probabilities of each pair of WM voxels in the ROIs. A comparison against voxel-graph based probabilistic tractography method was performed on Fibercup phantom dataset, and the results demonstrated that the proposed method can produce better structural connection and is more computationally economical. Lastly, three datasets from Human Connectome Project (HCP) S1200 group were tested and their structural connectivity networks were constructed for topological analysis. The results showed great consistency in network metrics with previous WM network studies in healthy adults.
Diffusion-weighted magnetic resonance imaging (dMRI) is a non-invasive technique to probe the complex micro-architecture of the tissue being imaged. The diffusional properties of the tissue at the imaged resolution are well captured by the ensemble average propagator (EAP), which is a probability density function characterizing the probability of water molecule diffusion. Many properties in the form of imaging 'stains' can then be computed from the EAP that can serve as bio-markers for a variety of diseases. This motivates the development of methods for the accurate estimation of the EAPs from dMRI, which is an actively researched area in dMRI analysis. To this end, in the recent past, dictionary learning (DL) techniques have been applied by many researchers for accurate reconstruction of the EAP fields from dMRI scans of the central nervous system (CNS). However, most of the DL-based methods did not exploit the geometry of the space of the EAPs, which are probability density functions. By exploiting the geometry of the space of probability density functions, it is possible to reconstruct EAPs that satisfy the mathematical properties of a density function and hence yield better accuracy in the EAP field reconstruction. Using a square root density parameterization, the EAPs can be mapped to a unit Hilbert sphere, which is a smooth manifold with well known geometry that we will exploit in our formulation of the DL problem. Thus, in this paper, we present a general formulation of the DL problem for data residing on smooth manifolds and in particular the manifold of EAPs, along with a numerical solution using an alternating minimization method. We then showcase the properties and the performance of our algorithm on the reconstruction of the EAP field in a patch-wise manner from the dMRI data. Through several synthetic, phantom and real data examples, we demonstrate that our non-linear DL-based approach produces accurate and spatially smooth estimates of the EAP field from dMRI in comparison to the state-of-the-art EAP reconstruction method called the MAPL method, as well as the linear DL-based EAP reconstruction approaches. To further demonstrate the accuracy and utility of our approach, we compute an entropic anisotropy measure (HA), that is a function of the well known Rényi entropy, from the EAP fields of control and injured rat spinal cords respectively. We demonstrate its utility as an imaging 'stain' via a quantitative comparison of HA maps computed from EAP fields estimated using our method and competing methods. The quantitative comparison is achieved using a two sample t-test and the results of significance are displayed for a visualization of regions of the spinal cord affected most by the injury.
The Ensemble Average diffusion Propagator (EAP) provides relevant microstructural information and meaningful descriptive maps of the white matter previously obscured by traditional techniques like the Diffusion Tensor. The direct estimation of the EAP requires a dense sampling of the \({\mathbf {q}}\)-space data. Although alternative techniques have been proposed, all of them require a high number of gradients and several b-values to be calculated. Once the EAP is calculated scalar measures must be directly derived. In this work, we propose a method to drastically reduce the number of points needed for the estimation of one of the measures, the return-to-axis probability (RTAP), efficiently estimating the \({\mathbf {q}}\)-space diffusion measure from a single shell acquisition. The proposal avoids the calculation of the EAP assuming that the diffusion does not depend on the radial direction. By applying this assumption locally, we achieve closed-form expressions of the measure using information from only one b-value, compatible with acquisitions protocols used for HARDI. Results have shown that the measures are highly correlated with the same measures calculated with state-of-the-art EAP estimators and highly accelerated execution times.
Diffusion-attenuated MR signal for heterogeneous media has been represented as a sum of signals from anisotropic Gaussian sub-domains to the extent that this approximation is permissible. Any effect of macroscopic (global or ensemble) anisotropy in the signal can be removed by averaging the signal values obtained by differently oriented experimental schemes. The resulting average signal is identical to what one would get if the micro-domains are isotropically (e.g., randomly) distributed with respect to orientation, which is the case for “powdered” specimens. We provide exact expressions for the orientationally-averaged signal obtained via general gradient waveforms when the microdomains are characterized by a general diffusion tensor possibly featuring three distinct eigenvalues. This extends earlier results which covered only axisymmetric diffusion as well as measurement tensors. Our results are expected to be useful in not only multidimensional diffusion MR but also solid-state NMR spectroscopy due to the mathematical similarities in the two fields.
Since its inception 30 years ago, diffusion-weighted magnetic resonance imaging (dMRI) has advanced to become a common component of routine clinical MRI examinations. Diffusion-weighted magnetic resonance offers a way to measure anisotropic diffusion in-vivo, which has led to the development of techniques capable of characterising the orientation of diffusion within living tissue. These modelling techniques can be used to investigate the microstructure and connectivity of white matter tracts within the human brain. Such techniques have been used to study many neural networks within the human body. There is, however, a notable paucity of research utilising dMRI techniques to investigate the white matter tracts of the auditory brainstem. In this review we provide a brief introduction to the basic principles of dMRI analysis and consider some of the difficulties associated with applying dMRI techniques to study the auditory pathways of the brainstem. We also consider aspects of current dMRI methodologies relevant to the auditory brainstem to inform future research in this area.
Diffusion spectrum MRI (DSI) provides model-free estimation of the diffusion ensemble average propagator (EAP) and orientation distribution function (ODF) but requires the diffusion data to be acquired on a Cartesian q-space grid. Multi-shell diffusion acquisitions are more flexible and more commonly acquired but have, thus far, only been compatible with model-based analysis methods. Here, we propose a generalized DSI (GDSI) framework to recover the EAP from multi-shell diffusion MRI data. The proposed GDSI approach corrects for q-space sampling density non-uniformity using a fast geometrical approach. The EAP is directly calculated in a preferable coordinate system by multiplying the sampling density corrected q-space signals by a discrete Fourier transform matrix, without any need for gridding. The EAP is demonstrated as a way to map diffusion patterns in brain regions such as the thalamus, cortex and brainstem where the tissue microstructure is not as well characterized as in white matter. Scalar metrics such as the zero displacement probability and displacement distances at different fractions of the zero displacement probability were computed from the recovered EAP to characterize the diffusion pattern within each voxel. The probability averaged across directions at a specific displacement distance provides a diffusion property based image contrast that clearly differentiates tissue types. The displacement distance at the first zero crossing of the EAP averaged across directions orthogonal to the primary fiber orientation in the corpus callosum is found to be larger in the body (5.65 ± 0.09 μm) than in the genu (5.55 ± 0.15 μm) and splenium (5.4 ± 0.15 μm) of the corpus callosum, which corresponds well to prior histological studies. The EAP also provides model-free representations of angular structure such as the diffusion ODF, which allows estimation and comparison of fiber orientations from both the model-free and model-based methods on the same multi-shell data. For the model-free methods, detection of crossing fibers is found to be strongly dependent on the maximum b-value and less sensitive compared to the model-based methods. In conclusion, our study provides a generalized DSI approach that allows flexible reconstruction of the diffusion EAP and ODF from multi-shell diffusion data and data acquired with other sampling patterns.
Fiber tractography using diffusion weighted MRI (DWI) is a primary tool for mapping structural connectivity in the human brain in vivo. However, this method suffers from a number of inherent limitations that have a significant impact on its capability in faithfully constructing fiber bundles for specific function. In this paper, a novel tractography algorithm combining DWI and functional MRI (fMRI) was proposed. Specifically, a spatio-temporal correlation tensor that characterizes the anisotropy of fMRI signals in white matter was introduced to complement the estimation of fiber orientation density function from DWI. The proposed method has been demonstrated to identify functional pathways implicated in fMRI task. It can effectively follow tracts in the genu of the corpus callosum that connects to the frontal lobe cortex, obtain connections between the thalamus and the anterior insula under sensory simulation, and reconstruct optic radiations in the visual circuit under visual stimulation. Taken together, the method we proposed in this work may benefit our understanding of structure-function relations in the human brain.
The diffusion tensor model is nonspecific in regions where micrometer structural patterns are inconsistent at the millimeter scale (i.e., brain regions with pathways that cross, bend, branch, fan, etc.). Numerous models have been proposed to represent crossing fibers and complex intravoxel structure fromin vivodiffusion weighted magnetic resonance imaging (e.g., high angular resolution diffusion imaging-HARDI). Here, we present an empirical comparison of two HARDI approaches-persistent angular structure MRI (PAS-MRI) and Q-ball-using a newly acquired reproducibility dataset. Briefly, a single subject was scanned 11 times with 96 diffusion weighted directions and 10 reference volumes for each of two [Formula: see text] values (1000 and [Formula: see text] for a total of 2144 volumes). Empirical reproducibility of intravoxel fiber fractions (number/strength of peaks), angular orientation, and fractional anisotropy was compared with metrics from a traditional tensor analysis approach, focusing on [Formula: see text] values of 1000 and [Formula: see text]. PAS-MRI is shown to be more reproducible than Q-ball and offers advantages at low [Formula: see text] values. However, there are substantial and biologically meaningful differences between the intravoxel structures estimated both in terms of analysis method as well as by [Formula: see text] value. The two methods suggest a fundamentally different microarchitecture of the human brain; therefore, it is premature to perform meta-analysis or combine results across HARDI studies using a different analysis model or acquisition sequences.
INTRODUCTION Application of diffusion sensitizing gradients along different directions in a pulsed gradient spin echo experiment has provided a way to observe anisotropic diffusion that occurs within fibrous tissues like muscle and white-matter. In addition to quantification of anisotropy, a natural and very important application of anisotropic diffusion imaging has been fiber orientation mapping that enables one to visualize the anatomical and functional connections in the brain. The simplest method in modeling anisotropic diffusion has employed a rank-2 diffusion tensor (1), which assumes a Gaussian displacement profile. Although this model is successful in extracting fiber orientations in relatively simple systems in which the fibers are orientationally coherent, it fails when there is orientational heterogeneity in the voxel of interest. To overcome this difficulty, diffusion tensor imaging has been generalized so that the signal attenuation observed by the application of diffusion gradients along many directions is better quantified (2). The generalized (higher-rank) diffusion tensor is a totally symmetric tensor and has (r+1)(r+2)/2 distinct elements, where r is the rank of the tensor. The choice of r is typically done based on the number of orientations along which the signal is measured. In this work, we present the application of generalized diffusion tensor imaging to an excised rat brain and demonstrate that this technique is capable of producing orientation maps even in complex fiber structures. METHODS A series of diffusion weighted images of an excised rat brain in phosphate-buffered saline was acquired at 17.6T using a Bruker Avance imaging system. The imaging parameters were TR=2500ms, TE=28ms, ∆=17.8ms, δ=2.2ms, resolution=150µmX150µmX300µm. Diffusion-weighted images were acquired along 81 directions specified by the tessellations of an icosahedron on the hemisphere, with a b-value of 1500 s/mm 2 , along with a single image acquired at b≈0.
The relationship between brain structure and complex behavior is governed by large-scale neurocognitive networks. The availability of a noninvasive technique that can visualize the neuronal projections connecting the functional centers should therefore provide new keys to the understanding of brain function. By using high-resolution three-dimensional diffusion magnetic resonance imaging and a newly designed tracking approach, we show that neuronal pathways in the rat brain can be probed in situ. The results are validated through comparison with known anatomical locations of such fibers.
Functional imaging with positron emission tomography and functional MRI has revolutionized studies of the human brain. Understanding the organization of brain systems, especially those used for cognition, remains limited, however, because no methods currently exist for noninvasive tracking of neuronal connections between functional regions [Crick, F. & Jones, E. (1993) Nature (London) 361, 109-110]. Detailed connectivities have been studied in animals through invasive tracer techniques, but these invasive studies cannot be done in humans, and animal results cannot always be extrapolated to human systems. We have developed noninvasive neuronal fiber tracking for use in living humans, utilizing the unique ability of MRI to characterize water diffusion. We reconstructed fiber trajectories throughout the brain by tracking the direction of fastest diffusion (the fiber direction) from a grid of seed points, and then selected tracks that join anatomically or functionally (functional MRI) defined regions. We demonstrate diffusion tracking of fiber bundles in a variety of white matter classes with examples in the corpus callosum, geniculo-calcarine, and subcortical association pathways. Tracks covered long distances, navigated through divergences and tight curves, and manifested topological separations in the geniculo-calcarine tract consistent with tracer studies in animals and retinotopy studies in humans. Additionally, previously undescribed topologies were revealed in the other pathways. This approach enhances the power of modern imaging by enabling study of fiber connections among anatomically and functionally defined brain regions in individual human subjects.
A methodology is presented for estimation of a probability density function of cerebral fibre orientations when one or two fibres are present in a voxel. All data are acquired on a clinical MR scanner, using widely available acquisition techniques. The method models measurements of water diffusion in a single fibre by a Gaussian density function and in multiple fibres by a mixture of Gaussian densities. The effects of noise on complex MR diffusion weighted data are explicitly simluated and parameterised. This information is used for standard and Monte Carlo streamline methods. Deterministic and probabilistic maps of anatomical voxel scale connectivity between brain regions are generated.
A methodology is presented for estimation of a probability density function of cerebral fibre orientations when one or two fibres are present in a voxel. All data are acquired on a clinical MR scanner, using widely available acquisition techniques. The method models measurements of water diffusion in a single fibre by a Gaussian density function and in multiple fibres by a mixture of Gaussian densities. The effects of noise on complex MR diffusion weighted data are explicitly simluated and parameterised. This information is used for standard and Monte Carlo streamline methods. Deterministic and probabilistic maps of anatomical voxel scale connectivity between brain regions are generated.
The relationship between brain structure and complex behavior is governed by large-scale neurocognitive networks. The availability of a noninvasive technique that can visualize the neuronal projections connecting the functional centers should therefore provide new keys to the understanding of brain function. By using high-resolution three-dimensional diffusion magnetic resonance imaging and a newly designed tracking approach, we show that neuronal pathways in the rat brain can be probed in situ. The results are validated through comparison with known anatomical locations of such fibers. Ann Neurol 1999;45:265–269
A procedure that rapidly generates an approximate parametric representation of macromolecular surface shapes is described. The parametrization is expressed as an expansion of real spherical harmonic basis functions. The advantage of using a parametric representation is that a pair of surfaces can be matched by using a quasi-Newton algorithm to minimize a suitably chosen objective function. Spherical harmonics are a natural and convenient choice of basis function when the task is one of search in a rotational search space. In particular, rotations of a molecular surface can be simulated by rotating only the harmonic expansion coefficients. This rotational property is applied for the first time to the 3-dimensional molecular similarity problem in which a pair of similar macromolecular surfaces are to be maximally superposed. The method is demonstrated with the superposition of antibody heavy chain variable domains. Special attention is given to computational efficiency. The spherical harmonic expansion coefficients are determined using fast surface sampling and integration schemes based on the tessellation of a regular icosahedron. Low resolution surfaces can be generated and displayed in under 10 s and a pair of surfaces can be maximally superposed in under 3 s on a contemporary workstation.
THE transport of fluids in porous media is of importance in a wide range of areas, such as oil recovery, heterogeneous catalysis and biological perfusion. The pulsed gradient spin-echo (PGSE) NMR technique has been used for many years to characterize diffusion and flow in such systems1–3. The analogy between NMR measurements in a field gradient and diffraction has been pointed out in the context of NMR imaging4 and, more recently, diffraction-like effects in the PGSE experiment have been discussed for diffusion in both impermeable5 and connected6 structures. The gradient pulse area plays the role of a wavevector, q, which can probe the structure in which the fluid diffuses. Here we report experimental confirmation of these predicted effects from proton NMR studies of a water-saturated, orientationally disordered, loosely packed array of monodisperse polystyrene spheres. The PGSE-NMR experiments may thus be used to provide an indirect, averaged image of the internal structure of porous solids at a resolution higher than that achievable with conventional NMR imaging. This is particularly advantageous for measurements on large samples, as the resolution available with the PGSE method depends only on the available range of gradient pulse amplitude and duration and is unconstrained by the factors determining resolution in conventional NMR imaging.
The Bloch—Torrey equations are modified to include the case of anisotropic, restricted diffusion and flow. The problem of solving these modified equations for the amplitude of a spin echo in a time-dependent magnetic-field gradient subject to restricting boundary conditions is discussed. This problem is solved for a number of selected cases. In particular, it is found that a magnetic-field gradient applied in short, intense pulses is effective in defining the time during which nuclear displacements take place. A simplified equation, suitable for the pulsed-gradient experiment, is presented and solved for two different examples of systems showing restricted diffusion. A procedure for analyzing the data from pulsed-gradient measurements is suggested, and its merits are discussed. Suggestions are made of systems which may well be expected to show restricted, anisotropic diffusion or interesting flow properties.
A derivation is given of the effect of a time-dependent magnetic field gradient on the spin-echo experiment, particularly in the presence of spin diffusion. There are several reasons for preferring certain kinds of time-dependent magnetic field gradients to the more usual steady gradient. If the gradient is reduced during the rf pulses, H1 need not be particularly large; if the gradient is small at the time of the echo, the echo will be broad and its amplitude easy to measure. Both of these relaxations of restrictions on the measurement of diffusion coefficients by the spin-echo technique serve to extend its range of applicability. Furthermore, a pulsed gradient can be recommended when it is critical to define the precise time period over which diffusion is being measured.
The theoretical expression derived has been verified experimentally for several choices of time dependent magnetic field gradient. An apparatus is described suitable for the production of pulsed gradients with amplitudes as large as 100 G cm−1. The diffusion coefficient of dry glycerol at 26°±1°C has been found to be (2.5±0.2)×10−8 cm2 sec−1, a value smaller than can ordinarily be measured by the steady gradient method.
For non-Gaussian distributed random displacement, which is common in restricted diffusion, a second-order diffusion tensor is incapable of fully characterizing the diffusion process. The insufficiency of a second-order tensor is evident in the limited capability of diffusion tensor imaging (DTI) in resolving multiple fiber orientations within one voxel of human white matter. A generalized diffusion tensor imaging (GDTI) method was recently proposed to solve this problem by generalizing Fick's law to a higher-order partial differential equation (PDE). The relationship between the higher-order tensor coefficients of the PDE and the higher-order cumulants of the random displacement can be derived. The statistical property of the diffusion process was fully characterized via the higher-order tensor coefficients by reconstructing the probability density function (PDF) of the molecular random displacement. Those higher-order tensor coefficients can be measured using conventional diffusion-weighted imaging or spectroscopy techniques. Simulations demonstrated that this method was capable of quantitatively characterizing non-Gaussian diffusion and accurately resolving multiple fiber orientations. It can be shown that this method is consistent with the q-space approach. The second-order approximation of GDTI was shown to be DTI.
Diffusion-weighted single voxel experiments conducted at b-values up to 1 × 104 smm−2 yielded biexponential signal attenuation curves for both normal and ischemic brain. The relative fractions of the rapidly and slowly decaying components (f1, f2)are f1 = 0.80 ± 0.02, f2 = 0.17 ± 0.02 in healthy adult rat brain and f1 = 0.90 ± 0.02, f2 = 0.11 ± 0.01 in normal neonatal rat brain, whereas the corresponding values for the postmortem situation are f1 = 0.69 ± 0.02, f2 = 0.33 ± 0.02. It is demonstrated that the changes in f1 and f2 occur simultaneously to those in the extracellular and intracellular space fractions (fex, fin) during: (i) cell swelling after total circulatory arrest, and (ii) the recovery from N-methyl-D-aspartate induced excitotoxic brain edema evoked by MK-801, as measured by changes in the electrical impedance. Possible reasons for the discrepancy between the estimated magnitude components and the physiological values are presented and evaluated. Implications of the biexponential signal attenuation curves for diffusion-weighted imaging experiments are discussed.
To establish a general methodology for quantifying streamline-based diffusion fiber tracking methods in terms of probability of connection between points and/or regions.
The commonly used streamline approach is adapted to exploit the uncertainty in the orientation of the principal direction of diffusion defined for each image voxel. Running the streamline process repeatedly using Monte Carlo methods to exploit this inherent uncertainty generates maps of connection probability. Uncertainty is defined by interpreting the shape of the diffusion orientation profile provided by the diffusion tensor in terms of the underlying microstructure.
Two candidates for describing the uncertainty in the diffusion tensor are proposed and maps of probability of connection to chosen start points or regions are generated in a number of major tracts.
The methods presented provide a generic framework for utilizing streamline methods to generate probabilistic maps of connectivity. J. Magn. Reson. Imaging 2003;18:242–254.
Generalized diffusion tensor imaging uses tensors of arbitrary ranks to model the angular variations in the diffusivities measured by magnetic resonance imaging (MRI) methods. However, a diffusivity profile alone is not readily capable of producing distinct fiber orientations. In this work, we show how it is possible to get the displacement probability profile for water molecules from the higher rank diffusion tensors and validate the technique via simulations of one, two and three fiber systems. Finally, we present fiber orientation results for an image from an excised rat brain.
We determine a statistic called the (radially) persistent angular structure (PAS) from samples of the Fourier transform of a three-dimensional function. The method has applications in diffusion magnetic resonance imaging (MRI), which samples the Fourier transform of the probability density function of particle displacements. The PAS is then a representation of the relative mobility of particles in each direction. In PAS-MRI, we compute the PAS in each voxel of an image. This technique has biomedical applications, where it reveals the orientations of microstructural fibres, such as white-matter fibres in the brain.
Scanner time is a significant factor in determining the amount of data available in clinical brain scans. Here, we use measurements acquired for diffusion-tensor MRI, which is a routine diffusion imaging technique, but extract richer information. In particular, PAS-MRI can resolve the orientations of crossing fibres.
We test PAS-MRI on human brain data and on synthetic data. The human brain data set comes from a standard acquisition scheme for diffusion-tensor MRI in which the samples in each voxel lie on a sphere in Fourier space.
Histological and histochemical investigations in albino rats showed that the neuroglial cells in the chiasm are fewer and smaller than those in the intraorbital portion of the optic nerve. The subpial astrocytic layer and pia mater were thin on the ventral surface of the chiasm and the pial vessels readily detached. The optic nerve had a thicker subpial layer of astrocytes. These structural differences implied concomitant differences in function, which may explain why certain toxins produce lesions in limited zones of the optic nerves or chiasm.
The anisotropy of the spin-diffusion coefficient Ds of water protons in skeletal muscle has been studied by pulsed NMR methods. The mid-portion of the tibialis anterior muscle of mature male rats was placed in a special sample holder by means of which the muscle fiber orientation theta relative to the diffusion direction could be varied over the range 0 degrees less than or equal to theta less than or equal to 90 degrees. The value of Ds(theta) was determined for theta = 0 degrees, 45 degrees, and 90 degrees. The measured anisotropy Ds(0)/Ds(90) was 1.39, and the value of Ds(0) was 1.39 X 10(-5) cm2/s. These results are interpreted within the framework of a model calculation in which the diffusion equation is solved for a regular hexagonal network similar to the actin-myosin filament network. The large anisotropy, and the large reduction in the value of Ds measured parallel to the filament axes lead to two major conclusions: (a) interpretations in which the reduction in Ds is ascribed to the effect of geometrical obstructions on the diffusion of "free" water are ruled out; and, (b) there is a large fraction of the cellular water associated with the proteins in such a way that its diffusion coefficient is substantially reduced.
Quantitative measurements of perfusion and molecular diffusion were made in human white matter in two orientations of the motion-sensitization gradient to document anisotropy of these parameters. Measurements were localized to a 10 X 10-mm tissue column oriented in an anterior-to-posterior direction in the left cerebral hemisphere just above the body of the left ventricle. This region was selected because of the relatively high directionality of white matter fibers. In this study of five healthy volunteers, strong diffusion anisotropy was observed in all cases. Twofold or greater anisotropy was commonly observed, with the higher diffusion value associated with motion sensitivity along the fiber directions. By combining data from both gradient orientations in all cases, diffusion values of solid tissue ranged from 0.38 X 10(-3) mm2/sec to 1.12 X 10(-3) mm2/sec, and measured perfusion fractions were in the range of 2%-5% (excluding areas highly contaminated by cerebrospinal fluid). Little or no perfusion-fraction anisotropy was observed; however, perfusion measurements were limited by noise. Data were collected without cardiac gating by using a technique that offers good immunity to bulk tissue motion artifacts.
The diffusion behavior of intracranial water in the cat brain and spine was examined with the use of diffusion-weighted magnetic resonance (MR) imaging, in which the direction of the diffusion-sensitizing gradient was varied between the x, y, and z axes of the magnet. At very high diffusion-sensitizing gradient strengths, no clear evidence of anisotropic water diffusion was found in either cortical or subcortical (basal ganglia) gray matter. Signal intensities clearly dependent on orientation were observed in the cortical and deep white matter of the brain and in the white matter of the spinal cord. Greater signal attenuation (faster diffusion) was observed when the relative orientation of white matter tracts to the diffusion-sensitizing gradient was parallel as compared to that obtained with a perpendicular alignment. These effects were seen on both premortem and immediate postmortem images obtained in all axial, sagittal, and coronal views. Potential applications of this MR imaging technique included the stereospecific evaluation of white matter in the brain and spinal cord and in the characterization of demyelinating and dysmyelinating diseases.
The diagonal and off-diagonal elements of the effective self-diffusion tensor, Deff, are related to the echo intensity in an NMR spin-echo experiment. This relationship is used to design experiments from which Deff is estimated. This estimate is validated using isotropic and anisotropic media, i.e., water and skeletal muscle. It is shown that significant errors are made in diffusion NMR spectroscopy and imaging of anisotropic skeletal muscle when off-diagonal elements of Deff are ignored, most notably the loss of information needed to determine fiber orientation. Estimation of Deff provides the theoretical basis for a new MRI modality, diffusion tensor imaging, which provides information about tissue microstructure and its physiologic state not contained in scalar quantities such as T1, T2, proton density, or the scalar apparent diffusion constant.
This paper describes a new NMR imaging modality--MR diffusion tensor imaging. It consists of estimating an effective diffusion tensor, Deff, within a voxel, and then displaying useful quantities derived from it. We show how the phenomenon of anisotropic diffusion of water (or metabolites) in anisotropic tissues, measured noninvasively by these NMR methods, is exploited to determine fiber tract orientation and mean particle displacements. Once Deff is estimated from a series of NMR pulsed-gradient, spin-echo experiments, a tissue's three orthotropic axes can be determined. They coincide with the eigenvectors of Deff, while the effective diffusivities along these orthotropic directions are the eigenvalues of Deff. Diffusion ellipsoids, constructed in each voxel from Deff, depict both these orthotropic axes and the mean diffusion distances in these directions. Moreover, the three scalar invariants of Deff, which are independent of the tissue's orientation in the laboratory frame of reference, reveal useful information about molecular mobility reflective of local microstructure and anatomy. Inherently tensors (like Deff) describing transport processes in anisotropic media contain new information within a macroscopic voxel that scalars (such as the apparent diffusivity, proton density, T1, and T2) do not.
We review several methods that have been developed to infer microstructural and physiological information about isotropic and anisotropic tissues from diffusion weighted images (DWIs). These include Diffusion Imaging (DI), Diffusion Tensor Imaging (DTI), isotropically weighted imaging, and q-space imaging. Just as DI provides useful information about molecular displacements in one dimension with which to characterize diffusion in isotropic tissues, DTI provides information about molecular displacements in three dimensions needed to characterize diffusion is anisotropic tissues. DTI also furnishes scalar parameters that behave like quantitative histological or physiological 'stains' for different features of diffusion. These include Trace(D), which is related to the mean diffusivity, and a family of parameters derived from the diffusion tensor, D, which characterize different features of anisotropic diffusion. Simple thought experiments and geometrical constructs, such as the diffusion ellipsoid, can be used to understand water diffusion in isotropic and anisotropic media, and the NMR experiments used to characterize it.
This paper investigates the use of color to represent the directional information contained in the diffusion tensor. Ideally, one wants to take into account both the properties of human color vision and of the given display hardware to produce a representation in which differences in the orientation of anisotropic structures are proportional to the perceived differences in color. It is argued here that such a goal cannot be achieved in general and therefore, empirical or heuristic schemes, which avoid some of the common artifacts of previously proposed approaches, are implemented. Directionally encoded color (DEC) maps of the human brain obtained using these schemes clearly show the main association, projection, and commissural white matter pathways. In the brainstem, motor and sensory pathways are easily identified and can be differentiated from the transverse pontine fibers and the cerebellar peduncles. DEC maps obtained from diffusion tensor imaging data provide a simple and effective way to visualize fiber direction, useful for investigating the structural anatomy of different organs. Magn Reson Med 42:526-540, 1999.
This paper investigates the use of color to represent the directional information contained in the diffusion tensor. Ideally, one wants to take into account both the properties of human color vision and of the given display hardware to produce a representation in which differences in the orientation of anisotropic structures are proportional to the perceived differences in color. It is argued here that such a goal cannot be achieved in general and therefore, empirical or heuristic schemes, which avoid some of the common artifacts of previously proposed approaches, are implemented. Directionally encoded color (DEC) maps of the human brain obtained using these schemes clearly show the main association, projection, and commissural white matter pathways. In the brainstem, motor and sensory pathways are easily identified and can be differentiated from the transverse pontine fibers and the cerebellar peduncles. DEC maps obtained from diffusion tensor imaging data provide a simple and effective way to visualize fiber direction, useful for investigating the structural anatomy of different organs. Magn Reson Med 42:526–540, 1999. © 1999 Wiley-Liss, Inc.
Fiber tract trajectories in coherently organized brain white matter pathways were computed from in vivo diffusion tensor magnetic resonance imaging (DT-MRI) data. First, a continuous diffusion tensor field is constructed from this discrete, noisy, measured DT-MRI data. Then a Frenet equation, describing the evolution of a fiber tract, was solved. This approach was validated using synthesized, noisy DT-MRI data. Corpus callosum and pyramidal tract trajectories were constructed and found to be consistent with known anatomy. The method's reliability, however, degrades where the distribution of fiber tract directions is nonuniform. Moreover, background noise in diffusion-weighted MRIs can cause a computed trajectory to hop from tract to tract. Still, this method can provide quantitative information with which to visualize and study connectivity and continuity of neural pathways in the central and peripheral nervous systems in vivo, and holds promise for elucidating architectural features in other fibrous tissues and ordered media.
The apparent diffusion tensor (ADT) imaging method was extended to account for multiple diffusion components. A biexponential ADT imaging experiment was used to obtain separate images of rapidly and slowly diffusing water fractions in excised rat spinal cord. The fast and slow component tensors were compared and found to exhibit similar gross features, such as fractional anisotropy, in both white and gray matter. However, there were also some important differences, which are consistent with the different structures occupying intracellular and extracellular spaces. Evidence supporting the assignment of the two tensor components to extracellular and intracellular water fractions is provided by an NMR spectroscopic investigation of homogeneous samples of brain tissue. Magn Reson Med 45:580-587, 2001.
Fundamental relationships between diffusion tensor (DT) and 3D q-space MRI are derived which establish conditions when these two complementary MR methods are equivalent. It is shown that the displacement distribution measured by q-space MRI in both the large displacement (i.e., large r) and the long-wavelength (i.e., small q) limits is the same 3D Gaussian displacement distribution assumed in DT-MRI. In these limiting cases, q-space MR yields a dispersion tensor that is identical to the effective DT, D, measured in DT-MRI. An experiment is then proposed to measure D using q-space methods. These findings establish that the effective DT, measured in DT-MRI, characterizes molecule motions on a coarse length-scale. Finally, the feasibility of and requirements for performing 3D q-space MRI on a clinical scanner are considered.
The methods of group theory are applied to the problem of characterizing the diffusion measured in high angular resolution MR experiments. This leads to a natural representation of the local diffusion in terms of spherical harmonics. In this representation, it is shown that isotropic diffusion, anisotropic diffusion from a single fiber, and anisotropic diffusion from multiple fiber directions fall into distinct and separable channels. This decomposition can be determined for any voxel without any prior information by a spherical harmonic transform, and for special cases the magnitude and orientation of the local diffusion may be determined. Moreover, non-diffusion-related asymmetries produced by experimental artifacts fall into channels distinct from the fiber channels, thereby allowing their separation and a subsequent reduction in noise from the reconstructed fibers. In the case of a single fiber, the method reduces identically to the standard diffusion tensor method. The method is applied to normal volunteer brain data collected with a stimulated echo spiral high angular resolution diffusion-weighted (HARD) acquisition.
This work details the observation of non-Gaussian apparent diffusion coefficient (ADC) profiles in multi-direction, diffusion-weighted MR data acquired with easily achievable imaging parameters (b approximately 1000 s/mm(2)). A technique is described for modeling the profile of the ADC over the sphere, which can capture non-Gaussian effects that can occur at, for example, intersections of different tissue types or white matter fiber tracts. When these effects are significant, the common diffusion tensor model is inappropriate, since it is based on the assumption of a simple underlying diffusion process, which can be described by a Gaussian probability density function. A sequence of models of increasing complexity is obtained by truncating the spherical harmonic (SH) expansion of the ADC measurements at several orders. Further, a method is described for selection of the most appropriate of these models, in order to describe the data adequately but without overfitting. The combined procedure is used to classify the profile at each voxel as isotropic, anisotropic Gaussian, or non-Gaussian, each with reference to the underlying probability density function of displacement of water molecules. We use it to show that non-Gaussian profiles arise consistently in various regions of the human brain where complex tissue structure is known to exist, and can be observed in data typical of clinical scanners. The performance of the procedure developed is characterized using synthetic data in order to demonstrate that the observed effects are genuine. This characterization validates the use of our method as an indicator of pathology that affects tissue structure, which will tend to reduce the complexity of the selected model.
Several new MR techniques have been introduced to infer direction through diffusion in multiple nerve fiber bundles within a voxel. To date, however, there has been no physical model reported to evaluate these methodologies and their ability to determine fiber orientation. In this article a model of diffusion analogous to nerve fibers is presented. Diffusion measurements at multiple closely spaced angles of 15 degrees in samples with different fiber orientations are compared with theoretical calculations for restricted diffusion in cylindrical geometry. Orientational diffusion measurements are shown to reflect fiber geometry and theoretical predictions to within 10%. Simulations of fiber crossings within a voxel suggest fiber orientation does not correspond to the direction of the largest measured diffusion coefficient, but theoretical knowledge of signal decay curves can predict the shape of these diffusion coefficient contours for given fiber orientation probabilities.
Magnetic resonance (MR) diffusion tensor imaging (DTI) can resolve the white matter fiber orientation within a voxel provided that the fibers are strongly aligned. However, a given voxel may contain a distribution of fiber orientations due to, for example, intravoxel fiber crossing. The present study sought to test whether a geodesic, high b-value diffusion gradient sampling scheme could resolve multiple fiber orientations within a single voxel. In regions of fiber crossing the diffusion signal exhibited multiple local maxima/minima as a function of diffusion gradient orientation, indicating the presence of multiple intravoxel fiber orientations. The multimodality of the observed diffusion signal precluded the standard tensor reconstruction, so instead the diffusion signal was modeled as arising from a discrete mixture of Gaussian diffusion processes in slow exchange, and the underlying mixture of tensors was solved for using a gradient descent scheme. The multitensor reconstruction resolved multiple intravoxel fiber populations corresponding to known fiber anatomy. Ma
A new method for mapping diffusivity profiles in tissue is presented. The Bloch-Torrey equation is modified to include a diffusion term with an arbitrary rank Cartesian tensor. This equation is solved to give the expression for the generalized Stejskal-Tanner formula quantifying diffusive attenuation in complicated geometries. This makes it possible to calculate the components of higher-rank tensors without using the computationally-difficult spherical harmonic transform. General theoretical relations between the diffusion tensor (DT) components measured by traditional (rank-2) DT imaging (DTI) and 3D distribution of diffusivities, as measured by high angular resolution diffusion imaging (HARDI) methods, are derived. Also, the spherical tensor components from HARDI are related to the rank-2 DT. The relationships between higher- and lower-rank Cartesian DTs are also presented. The inadequacy of the traditional rank-2 tensor model is demonstrated with simulations, and the method is applied to excised rat brain data collected in a spin-echo HARDI experiment.
While functional brain imaging methods can locate the cortical regions subserving particular cognitive functions, the connectivity between the functional areas of the human brain remains poorly understood. Recently, investigators have proposed a method to image neural connectivity noninvasively using a magnetic resonance imaging method called diffusion tensor imaging (DTI). DTI measures the molecular diffusion of water along neural pathways. Accurate reconstruction of neural connectivity patterns from DTI has been hindered, however, by the inability of DTI to resolve more than a single axon direction within each imaging voxel. Here, we present a novel magnetic resonance imaging technique that can resolve multiple axon directions within a single voxel. The technique, called q-ball imaging, can resolve intravoxel white matter fiber crossing as well as white matter insertions into cortex. The ability of q-ball imaging to resolve complex intravoxel fiber architecture eliminates a key obstacle to mapping neural connectivity in the human brain noninvasively.
Diffusion NMR may provide, under certain experimental conditions, micro-structural information about confined compartments totally non-invasively. The influence of the rotational angle, the pulse gradient length and the diffusion time on the diffusion diffraction patterns and q-space displacement distribution profiles was evaluated for ensembles of long cylinders having a diameter of 9 and 20 microm. It was found that the diffraction patterns are sensitive to the rotational angle (alpha) and are observed only when diffusion is measured nearly perpendicular to the long axis of the cylinders i.e., when alpha= 90 degrees +/- 5 degrees under our experimental conditions. More importantly, we also found that the structural information extracted from the displacement distribution profiles and from the diffraction patterns are very similar and in good agreement with the experimental values for cylinders of 20 microm or even 9 microm, when data is acquired with parameters that satisfy the short gradient pulse (SGP) approximation (i.e., delta -->0) and the long diffusion time limit. Since these experimental conditions are hardly met in in vitro diffusion MRI of excised organs, and cannot be met in clinical MRI scanners, we evaluated the effect of the pulse gradient duration and the diffusion time on the structural information extracted from q-space diffusion MR experiments. Indeed it was found that, as expected, accurate structural information, and diffraction patterns are observed when Delta is large enough so that the spins reach the cylinders' boundaries. In addition, it was found that large delta results in extraction of a compartment size, which is smaller than the real one. The relevance of these results to q-space MRI of neuronal tissues and fiber tracking is discussed.
To characterize anisotropic water diffusion in brain white matter, a theoretical framework is proposed that combines hindered and restricted models of water diffusion (CHARMED) and an experimental methodology that embodies features of diffusion tensor and q-space MRI. This model contains a hindered extra-axonal compartment, whose diffusion properties are characterized by an effective diffusion tensor, and an intra-axonal compartment, whose diffusion properties are characterized by a restricted model of diffusion within cylinders. The hindered model primarily explains the Gaussian signal attenuation observed at low b values; the restricted non-Gaussian model does so at high b. Both high and low b data obtained along different directions are required to estimate various microstructural parameters of the composite model, such as the nerve fiber orientation(s), the T2-weighted extra- and intra-axonal volume fractions, and principal diffusivities. The proposed model provides a description of restricted diffusion in 3D given by a 3D probability distribution (average propagator), which is obtained by 3D Fourier transformation of the estimated signal attenuation profile. The new model is tested using synthetic phantoms and validated on excised spinal cord tissue. This framework shows promise in determining the orientations of two or more fiber compartments more precisely and accurately than with diffusion tensor imaging.
Diffusion-weighted magnetic resonance imaging can provide information related to the arrangement of white matter fibers. The diffusion tensor is the model most commonly used to derive the orientation of the fibers within a voxel. However, this model has been shown to fail in regions containing several fiber populations with distinct orientations. A number of alternative models have been suggested, such as multiple tensor fitting, q-space, and Q-ball imaging. However, each of these has inherent limitations. In this study, we propose a novel method for estimating the fiber orientation distribution directly from high angular resolution diffusion-weighted MR data without the need for prior assumptions regarding the number of fiber populations present. We assume that all white matter fiber bundles in the brain share identical diffusion characteristics, thus implicitly assigning any differences in diffusion anisotropy to partial volume effects. The diffusion-weighted signal attenuation measured over the surface of a sphere can then be expressed as the convolution over the sphere of a response function (the diffusion-weighted attenuation profile for a typical fiber bundle) with the fiber orientation density function (ODF). The fiber ODF (the distribution of fiber orientations within the voxel) can therefore be obtained using spherical deconvolution. The properties of the technique are demonstrated using simulations and on data acquired from a volunteer using a standard 1.5-T clinical scanner. The technique can recover the fiber ODF in regions of multiple fiber crossing and holds promise for applications such as tractography.
Magnetic resonance diffusion tensor imaging (DTI) provides a powerful tool for mapping neural histoarchitecture in vivo. However, DTI can only resolve a single fiber orientation within each imaging voxel due to the constraints of the tensor model. For example, DTI cannot resolve fibers crossing, bending, or twisting within an individual voxel. Intravoxel fiber crossing can be resolved using q-space diffusion imaging, but q-space imaging requires large pulsed field gradients and time-intensive sampling. It is also possible to resolve intravoxel fiber crossing using mixture model decomposition of the high angular resolution diffusion imaging (HARDI) signal, but mixture modeling requires a model of the underlying diffusion process.
Recently, it has been shown that the HARDI signal can be reconstructed model-independently using a spherical tomographic inversion called the Funk–Radon transform, also known as the spherical Radon transform. The resulting imaging method, termed q-ball imaging, can resolve multiple intravoxel fiber orientations and does not require any assumptions on the diffusion process such as Gaussianity or multi-Gaussianity. The present paper reviews the theory of q-ball imaging and describes a simple linear matrix formulation for the q-ball reconstruction based on spherical radial basis function interpolation. Open aspects of the q-ball reconstruction algorithm are discussed. Magn Reson Med 52:1358–1372, 2004.
This paper details the derivation of rotationally invariant scalar measures from higher-rank diffusion tensors (DTs) and functions defined on a unit sphere. This was accomplished with the use of an expression that generalizes the evaluation of the trace operator to tensors of arbitrary rank, and even to functions whose domains are the unit sphere. It is shown that the mean diffusivity is invariant to the selection of tensor rank for the model used. However, this rank invariance does not apply to the anisotropy measures. Therefore, a variance-based, general anisotropy measure is proposed. Also an information theoretical parametrization of anisotropy is introduced that is frequently more consistent with the meaning attributed to anisotropy. We accomplished this by associating anisotropy with the amount of orientational information present in the data, regardless of the imaging technique used. Using a simplified model of fibrous tissue, we simulated anisotropy values with varying orientational complexity and tensor models. Simulations suggested that a lower-rank tensor model may produce artificially low anisotropy values in voxels with complex structure. This was confirmed with a spin-echo experiment performed on an excised rat brain.
Several studies have shown that in tissues over an extended range of b-factors, the signal decay deviates significantly from the basic monoexponential model. The true nature of this departure has to date not been identified. For the current study, line scan diffusion images of brain suitable for biexponential diffusion tensor analysis were acquired in normal subjects on a clinical MR system. For each of six noncollinear directions, 32 images with b-factors ranging from 5 to 5000 s/mm2 were collected. Biexponential fits yielded parameter maps for a fast and a slow diffusion component. A subset of the diffusion data, consisting of the images obtained at the conventional range of b-factors between 5 and 972 s/mm2, was used for monoexponential diffusion tensor analysis. Fractional anisotropy (FA) of the fast-diffusion component and the monoexponential fit exhibited no significant difference. FA of the slow-diffusion biexponential component was significantly higher, particularly in areas of lower fiber density. The principal diffusion directions for the two biexponential components and the monoexponential solution were largely the same and in agreement with known fiber tracts. The second and third diffusion eigenvector directions also appeared to be aligned, but they exhibited significant deviations in localized areas.
A procedure that rapidly generates an approximate parametric representation of macromolecular surface shapes is described. The parametrisation is expressed as an expansion of real spherical harmonic basis functions. The advantage of using a parametric representation is that a pair of surfaces can be matched by using a quasi-Newton algorithm to minimise a suitably chosen objective function. Spherical harmonics are a natural and convenient choice of basis function when the task is one of search in a rotational search space. In particular, rotations of a molecular surface can be simulated by rotating only the harmonic expansion coecients. For the rst time, this rotational property is applied to the 3D molecular similarity problem in which a pair of similar macromolecular surfaces are to be maximally superposed. The method is demonstrated with the superposition of antibody VH domains. Special attention is given to computational eciency. The spherical harmonic expansion coecients are deter...
Quantum Mechanics Restricted diffusion in cylindrical geometry
- F Schwabl
Schwabl, F., 1989. Quantum Mechanics. Springer-Verlag, Berlin. Sö derman, O., Jö nsson, B., 1995. Restricted diffusion in cylindrical geometry. J. Magn. Reson., A (117), 94 – 97.