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Gauthier ARP, Stocek N, Newling B. Diffusion tensor imaging of anisotropic inhomogeneous turbulent flow. Phys Rev E 2022; 106:015108. [PMID: 35974538 DOI: 10.1103/physreve.106.015108] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/05/2021] [Accepted: 07/08/2022] [Indexed: 06/15/2023]
Abstract
Inhomogeneous anisotropic turbulent flow is difficult to measure, and yet it commonly occurs in nature and in many engineering applications. This work aims to introduce a technique based on magnetic resonance imaging which can spatially map the degree of turbulence as well as the degree of anisotropy. Our interpretation relies on the eddy diffusion model of turbulence, and combines this with the technique of diffusion tensor imaging. The result is an eddy diffusion tensor, which is represented by a symmetric three-by-three matrix. This tensor contains a wealth of information about the magnitude and directions of the turbulent fluctuations; however, the correlation time must be considered before interpreting this information. In the constricted pipe flow used in this study, the turbulence is greatest in magnitude in the space surrounding the core of the turbulent jet, and the turbulence is highly anisotropic.
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Affiliation(s)
- Amy-Rae P Gauthier
- UNB MRI Centre, University of New Brunswick, 8 Bailey Drive, Fredericton, New Brunswick E3B 5A3, Canada
| | - Noah Stocek
- UNB MRI Centre, University of New Brunswick, 8 Bailey Drive, Fredericton, New Brunswick E3B 5A3, Canada
| | - Benedict Newling
- UNB MRI Centre, University of New Brunswick, 8 Bailey Drive, Fredericton, New Brunswick E3B 5A3, Canada
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Koay CG, Yeh PH, Ollinger JM, İrfanoğlu MO, Pierpaoli C, Basser PJ, Oakes TR, Riedy G. Tract Orientation and Angular Dispersion Deviation Indicator (TOADDI): A framework for single-subject analysis in diffusion tensor imaging. Neuroimage 2015; 126:151-63. [PMID: 26638985 DOI: 10.1016/j.neuroimage.2015.11.046] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/22/2015] [Revised: 11/05/2015] [Accepted: 11/18/2015] [Indexed: 11/19/2022] Open
Abstract
The purpose of this work is to develop a framework for single-subject analysis of diffusion tensor imaging (DTI) data. This framework is termed Tract Orientation and Angular Dispersion Deviation Indicator (TOADDI) because it is capable of testing whether an individual tract as represented by the major eigenvector of the diffusion tensor and its corresponding angular dispersion are significantly different from a group of tracts on a voxel-by-voxel basis. This work develops two complementary statistical tests based on the elliptical cone of uncertainty, which is a model of uncertainty or dispersion of the major eigenvector of the diffusion tensor. The orientation deviation test examines whether the major eigenvector from a single subject is within the average elliptical cone of uncertainty formed by a collection of elliptical cones of uncertainty. The shape deviation test is based on the two-tailed Wilcoxon-Mann-Whitney two-sample test between the normalized shape measures (area and circumference) of the elliptical cones of uncertainty of the single subject against a group of controls. The False Discovery Rate (FDR) and False Non-discovery Rate (FNR) were incorporated in the orientation deviation test. The shape deviation test uses FDR only. TOADDI was found to be numerically accurate and statistically effective. Clinical data from two Traumatic Brain Injury (TBI) patients and one non-TBI subject were tested against the data obtained from a group of 45 non-TBI controls to illustrate the application of the proposed framework in single-subject analysis. The frontal portion of the superior longitudinal fasciculus seemed to be implicated in both tests (orientation and shape) as significantly different from that of the control group. The TBI patients and the single non-TBI subject were well separated under the shape deviation test at the chosen FDR level of 0.0005. TOADDI is a simple but novel geometrically based statistical framework for analyzing DTI data. TOADDI may be found useful in single-subject, graph-theoretic and group analyses of DTI data or DTI-based tractography techniques.
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Affiliation(s)
- Cheng Guan Koay
- National Intrepid Center of Excellence (NICoE), Bethesda, MD, USA; Section on Tissue Biophysics and Biomimetics, NICHD, National Institutes of Health, Bethesda, MD, USA; NorthTide Group, LLC, USA.
| | - Ping-Hong Yeh
- National Intrepid Center of Excellence (NICoE), Bethesda, MD, USA; The Henry M. Jackson Foundation for the Advancement of Military Medicine, Bethesda, MD, USA
| | - John M Ollinger
- National Intrepid Center of Excellence (NICoE), Bethesda, MD, USA
| | - M Okan İrfanoğlu
- The Henry M. Jackson Foundation for the Advancement of Military Medicine, Bethesda, MD, USA; Section on Tissue Biophysics and Biomimetics, NICHD, National Institutes of Health, Bethesda, MD, USA
| | - Carlo Pierpaoli
- Section on Tissue Biophysics and Biomimetics, NICHD, National Institutes of Health, Bethesda, MD, USA
| | - Peter J Basser
- Section on Tissue Biophysics and Biomimetics, NICHD, National Institutes of Health, Bethesda, MD, USA
| | - Terrence R Oakes
- National Intrepid Center of Excellence (NICoE), Bethesda, MD, USA
| | - Gerard Riedy
- National Intrepid Center of Excellence (NICoE), Bethesda, MD, USA; National Capital Neuroimaging Consortium, Bethesda, MD, USA
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Özarslan E, Westin CF, Mareci TH. Characterizing magnetic resonance signal decay due to Gaussian diffusion: the path integral approach and a convenient computational method. CONCEPTS IN MAGNETIC RESONANCE. PART A, BRIDGING EDUCATION AND RESEARCH 2015; 44:203-213. [PMID: 27182208 PMCID: PMC4864615 DOI: 10.1002/cmr.a.21354] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/05/2023]
Abstract
The influence of Gaussian diffusion on the magnetic resonance signal is determined by the apparent diffusion coefficient (ADC) and tensor (ADT) of the diffusing fluid as well as the gradient waveform applied to sensitize the signal to diffusion. Estimations of ADC and ADT from diffusion-weighted acquisitions necessitate computations of, respectively, the b-value and b-matrix associated with the employed pulse sequence. We establish the relationship between these quantities and the gradient waveform by expressing the problem as a path integral and explicitly evaluating it. Further, we show that these important quantities can be conveniently computed for any gradient waveform using a simple algorithm that requires a few lines of code. With this representation, our technique complements the multiple correlation function (MCF) method commonly used to compute the effects of restricted diffusion, and provides a consistent and convenient framework for studies that aim to infer the microstructural features of the specimen.
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Affiliation(s)
- Evren Özarslan
- Department of Physics, Bođaziçi University, Bebek, Ýstanbul, Turkey
- Department of Radiology, Brigham and Women’s Hospital, Harvard Medical School, Boston, MA, USA
- Corresponding author.
| | - Carl-Fredrik Westin
- Department of Radiology, Brigham and Women’s Hospital, Harvard Medical School, Boston, MA, USA
| | - Thomas H. Mareci
- Department of Biochemistry and Molecular Biology, University of Florida, Gainesville, FL, USA
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Shetty AN, Chiang S, Maletic-Savatic M, Kasprian G, Vannucci M, Lee W. Spatial Mapping of Translational Diffusion Coefficients Using Diffusion Tensor Imaging: A Mathematical Description. CONCEPTS IN MAGNETIC RESONANCE. PART A, BRIDGING EDUCATION AND RESEARCH 2014; 43:1-27. [PMID: 27441031 PMCID: PMC4948124 DOI: 10.1002/cmr.a.21288] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
In this article, we discuss the theoretical background for diffusion weighted imaging and diffusion tensor imaging. Molecular diffusion is a random process involving thermal Brownian motion. In biological tissues, the underlying microstructures restrict the diffusion of water molecules, making diffusion directionally dependent. Water diffusion in tissue is mathematically characterized by the diffusion tensor, the elements of which contain information about the magnitude and direction of diffusion and is a function of the coordinate system. Thus, it is possible to generate contrast in tissue based primarily on diffusion effects. Expressing diffusion in terms of the measured diffusion coefficient (eigenvalue) in any one direction can lead to errors. Nowhere is this more evident than in white matter, due to the preferential orientation of myelin fibers. The directional dependency is removed by diagonalization of the diffusion tensor, which then yields a set of three eigenvalues and eigenvectors, representing the magnitude and direction of the three orthogonal axes of the diffusion ellipsoid, respectively. For example, the eigenvalue corresponding to the eigenvector along the long axis of the fiber corresponds qualitatively to diffusion with least restriction. Determination of the principal values of the diffusion tensor and various anisotropic indices provides structural information. We review the use of diffusion measurements using the modified Stejskal-Tanner diffusion equation. The anisotropy is analyzed by decomposing the diffusion tensor based on symmetrical properties describing the geometry of diffusion tensor. We further describe diffusion tensor properties in visualizing fiber tract organization of the human brain.
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Affiliation(s)
- Anil N Shetty
- Texas Children's Pavilion for Women, Department of Obstetrics and Gynecology, Baylor College of Medicine, Houston 77030, TX
| | - Sharon Chiang
- Department of Statistics, Rice University, Houston, TX
| | - Mirjana Maletic-Savatic
- Departments of Pediatrics and Neuroscience, Program in Developmental Biology Jan and Dan Duncan Neurological Research Institute, Texas Children's Hospital, Houston, TX
| | - Gregor Kasprian
- Texas Children's Pavilion for Women, Department of Obstetrics and Gynecology, Baylor College of Medicine, Houston 77030, TX
| | | | - Wesley Lee
- Texas Children's Pavilion for Women, Department of Obstetrics and Gynecology, Baylor College of Medicine, Houston 77030, TX
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