Proof that gmT₂ is the reciprocal of gmR₂

Abstract

When performing multiexponential analysis of relaxation times in the slow exchange regime, it is often convenient to simplify the resulting distribution to one or more relaxation time constants. In doing so, what averaging method should be used and should the rate constant or time constant be reported? This note outlines why the geometric mean is most appropriate and provides a proof that the geometric mean rate and geometric mean time constants are reciprocals, which is counter-intuitive unless one considers that averaging values in a distribution is different than physically averaging relaxation species.

Multi‐gradient‐echo myelin water fraction imaging: Comparison to the multi‐echo‐spin‐echo technique

“…The gm is the mean time on a logarithmic scale and is the reciprocal of the geometric mean . This does not hold true for the arithmetic means of and , making the geometric mean an appropriate choice for distributions of exponential components .…”

MRI‐based myelin water imaging: A technical review

“…Additional metrics that can be extracted from ME T 2 relaxation mapping include the geometric mean T 2 ( ) and the width of the IE water peak (which makes up more than 80% of the water in the normal brain). The is the mean T 2 time on a logarithmic scale and is the reciprocal of the geometric mean R 2 [ ] . This does not hold true for the arithmetic means of T 2 and R 2 , making the geometric mean an appropriate choice for distributions of exponential components 24,38).…”