- Dedication
- Foreword
- Preface
- Contributors
- Diffusion MRI
- I What Are We Studying and Why?
- 1 The Functional Anatomy of White Matter: From Postmortem Dissections to In Vivo Virtual Tractography
- 2 Neurobiology of White Matter Disorders
- 3 Invasive Methods for Tracing White Matter Architecture
- II The Basics of Diffusion MRI
- 4 Physics of Diffusion
- 5 Magnetic Resonance Diffusion Imaging: Introduction and Concepts
- 6 Anisotropic Diffusion: From the Apparent Diffusion Coefficient to the Apparent Diffusion Tensor
- 7 What Makes Diffusion Anisotropic in the Nervous System?
- 8 Biophysics of Diffusion in Cells
- 9 Extracting Geometric Properties of White Matter with q-Space Diffusion MRI (QSI)
- 10 The Cumulant Expansion: An Overarching Mathematical Framework For Understanding Diffusion NMR
- III Experimental Aspects of Diffusion MRI
- 11 Hardware Considerations for Diffusion MRI
- 12 EPI-Based Pulse Sequences for Diffusion Tensor MRI
- 13 Non-EPI Pulse Sequences for Diffusion MRI
- 14 Phase Errors in Diffusion-Weighted Imaging
- 15 Optimal Approaches to Diffusion MRI Acquisition
- 16 Least Squares Approaches to Diffusion Tensor Estimation
- 17 Image Distortion and Its Correction in Diffusion MRI
- 18 Artifacts in Diffusion MRI
- 19 Quality Assurance for Diffusion MRI
- 20 Statistical Issues in Diffusion Tensor MRI
- 21 Visualization of Diffusion MRI Data
- IV Fiber Tracking
- 22 Deterministic White Matter Tractography
- 23 Probabilistic Fiber Tracking
- 24 Connectivity-Based Parcellation of Gray Matter
- 25 Diffusion Tensor Microimaging and Its Applications
- 26 Anatomical Validation of DTI and Tractography
- V Crossing Fibers
- 27 Mathematics of Crossing Fibers
- 28 The Biophysics of Crossing Fibers
- VI Diffusion MRI of Brain Disorders
- 29 Strategies for Patient–Control Comparison of Diffusion MR Data
- 30 Diffusion Tensor Imaging in Brain Development
- 31 Diffusion in Acute Stroke
- 32 Diffusion in Chronic Stroke and Small Vessel Disease
- 33 Diffusion Imaging in Brain Tumors
- 34 Diffusion Tensor MRI in Multiple Sclerosis
- 35 Diffusion MRI in Epilepsy
- 36 DTI and Tractography in Neurosurgical Planning
- 37 Diffusion MRI in Psychiatric Disorders
- 38 Diffusion Tensor Imaging in Aging and Age-Related Neurodegenerative Disorders
- VII Diffusion MRI Beyond White Matter
- 39 Diffusion Imaging in Gray Matter
- 40 Diffusion Imaging of the Optic Nerve, Spinal Cord, and Peripheral Nerve
- 41 Diffusion Imaging in Muscle
- 42 Insights into Diffusion Tensor Imaging from Animal Models of White Matter Pathology
- VIII How It All Started—Personal Perspectives from the Pioneers
- 43 Reminiscences about the Development of Pulsed Field Gradient Spin Echo NMR (PFGSE-NMR)
- 44 Diffusion, Diffraction, and Microimaging
- 45 Diffusion MRI: Conception, Birth, and Adolescence
- 46 DWI in Cerebral Ischemia: The Early Days
- 47 Invention and Development of Diffusion Tensor MRI (DT-MRI or DTI) at the NIH
- Author Index
- Subject Index
(p. 152) The Cumulant Expansion: An Overarching Mathematical Framework For Understanding Diffusion NMR
- Chapter:
- (p. 152) The Cumulant Expansion: An Overarching Mathematical Framework For Understanding Diffusion NMR
- Author(s):
Valerij G. Kiselev
- DOI:
- 10.1093/med/9780195369779.003.0010
This chapter discusses a general property of diffusion-weighted signals from inhomogeneous media such as porous samples or biological tissues. In simple terms, this property is similar to the Taylor expansion in mathematics that allows one to approximate any smooth function as a polynomial in a sufficiently small vicinity of a given point. In the present case, it is the logarithm of the signal that is expanded in powers of the diffusion-weighted gradient in the vicinity of the zero gradient value. This expansion is termed the cumulant expansion. Mathematically, it is valid for small gradients, but it turns out that the domain of small gradients is practically large enough to include the typical clinical measurements. Thus this approach is useful to account for this mathematical signal property when discussing experimental data. The coefficients of the cumulant expansion are expressed in terms of the correlation functions of the molecular velocity. These functions characterize the medium in a way that is independent from the measurement technique. The diffusion tensor is the simplest member of this. The overarching nature of this mathematical framework makes some overlap with other chapters of this book unavoidable.
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- Dedication
- Foreword
- Preface
- Contributors
- Diffusion MRI
- I What Are We Studying and Why?
- 1 The Functional Anatomy of White Matter: From Postmortem Dissections to In Vivo Virtual Tractography
- 2 Neurobiology of White Matter Disorders
- 3 Invasive Methods for Tracing White Matter Architecture
- II The Basics of Diffusion MRI
- 4 Physics of Diffusion
- 5 Magnetic Resonance Diffusion Imaging: Introduction and Concepts
- 6 Anisotropic Diffusion: From the Apparent Diffusion Coefficient to the Apparent Diffusion Tensor
- 7 What Makes Diffusion Anisotropic in the Nervous System?
- 8 Biophysics of Diffusion in Cells
- 9 Extracting Geometric Properties of White Matter with q-Space Diffusion MRI (QSI)
- 10 The Cumulant Expansion: An Overarching Mathematical Framework For Understanding Diffusion NMR
- III Experimental Aspects of Diffusion MRI
- 11 Hardware Considerations for Diffusion MRI
- 12 EPI-Based Pulse Sequences for Diffusion Tensor MRI
- 13 Non-EPI Pulse Sequences for Diffusion MRI
- 14 Phase Errors in Diffusion-Weighted Imaging
- 15 Optimal Approaches to Diffusion MRI Acquisition
- 16 Least Squares Approaches to Diffusion Tensor Estimation
- 17 Image Distortion and Its Correction in Diffusion MRI
- 18 Artifacts in Diffusion MRI
- 19 Quality Assurance for Diffusion MRI
- 20 Statistical Issues in Diffusion Tensor MRI
- 21 Visualization of Diffusion MRI Data
- IV Fiber Tracking
- 22 Deterministic White Matter Tractography
- 23 Probabilistic Fiber Tracking
- 24 Connectivity-Based Parcellation of Gray Matter
- 25 Diffusion Tensor Microimaging and Its Applications
- 26 Anatomical Validation of DTI and Tractography
- V Crossing Fibers
- 27 Mathematics of Crossing Fibers
- 28 The Biophysics of Crossing Fibers
- VI Diffusion MRI of Brain Disorders
- 29 Strategies for Patient–Control Comparison of Diffusion MR Data
- 30 Diffusion Tensor Imaging in Brain Development
- 31 Diffusion in Acute Stroke
- 32 Diffusion in Chronic Stroke and Small Vessel Disease
- 33 Diffusion Imaging in Brain Tumors
- 34 Diffusion Tensor MRI in Multiple Sclerosis
- 35 Diffusion MRI in Epilepsy
- 36 DTI and Tractography in Neurosurgical Planning
- 37 Diffusion MRI in Psychiatric Disorders
- 38 Diffusion Tensor Imaging in Aging and Age-Related Neurodegenerative Disorders
- VII Diffusion MRI Beyond White Matter
- 39 Diffusion Imaging in Gray Matter
- 40 Diffusion Imaging of the Optic Nerve, Spinal Cord, and Peripheral Nerve
- 41 Diffusion Imaging in Muscle
- 42 Insights into Diffusion Tensor Imaging from Animal Models of White Matter Pathology
- VIII How It All Started—Personal Perspectives from the Pioneers
- 43 Reminiscences about the Development of Pulsed Field Gradient Spin Echo NMR (PFGSE-NMR)
- 44 Diffusion, Diffraction, and Microimaging
- 45 Diffusion MRI: Conception, Birth, and Adolescence
- 46 DWI in Cerebral Ischemia: The Early Days
- 47 Invention and Development of Diffusion Tensor MRI (DT-MRI or DTI) at the NIH
- Author Index
- Subject Index