The Cumulant Expansion: An Overarching Mathematical Framework For Understanding Diffusion NMR

Valerij G. Kiselev

in Diffusion MRI

Published on behalf of Oxford University Press

Published in print November 2010 | ISBN: 9780195369779
Published online September 2012 | e-ISBN: 9780199965144 | DOI:
The Cumulant Expansion: An Overarching Mathematical Framework For Understanding Diffusion NMR

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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.

Chapter.  14046 words.  Illustrated.

Subjects: Neuroscientific Techniques

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