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