Magnetism

Diffusion Kurtosis Imaging

What is diffusion kurtosis and how does it differ from regular "diffusion"?

Standard diffusion weighted imaging (DWI) methods have incorporated Einstein's original concept that the diffusion water molecules follows a Gaussian (normal) distribution. While this assumption may be true for pure liquids and gels, it is most certainly incorrect for complex biological tissues with cell membranes that create compartments and barriers to diffusion. Non-Gaussian behavior becomes more noticeable when stronger gradients (higher b-values) and longer echo times are used.

Probability distributions reflecting the distance a
water molecule would likely diffuse in a fixed
time interval. Diffusion in pure liquids follows
Gaussian statistics, but tissue diffusion is
characterized by positive kurtosis.

Kurtosis, denoted by the dimensionless parameter K, is a long recognized statistical metric for quantifying the shape of a probability distribution. By definition, a Gaussian distribution has K = 0. Distributions that are more "peaked" and with less weight on their "shoulders" typically have a positive kurtosis (K>0). Diffusion in pure fluids is Gaussian, but biological tissues are characterized by a positive diffusion kurtosis. This reflects the heterogeneous diffusion environments experienced by water molecules as they encounter barriers, move between compartments, and undergo chemical exchange. A water molecule diffusing according to a K>0 distribution would typically not travel as far in a given time interval as one that followed Gaussian (K=0) statistics.

With Gaussian diffusion the predicted MR signal (S) relative to baseline (So) during application of diffusion-sensitizing gradients can be expressed as

S = S_oe^−bD or S/S_o = e^−bD

where D is the apparent (or measured) diffusion coefficient and b is a factor reflecting the strength and duration of the pulsed diffusion gradients. Under this model, a semilogarithmic plot of signal attenuation ln(S/So) vs b should be a straight line with slope = D.

Deviation of the expected MR signal in DWI from a mono-exponential model. At low b-values IVIM effects due to microscopic perfusion must be considered (red), while at high b-values kurtosis effects must be considered (green).

In the prior Q&A we showed how at low b-values intravoxel incoherent motion (IVIM) perfusion in capillaries caused a deviation of measured data from this expected line. The effect of diffusion kurtosis (green region) is best appreciated at high b-values (i.e., ≥ 1500 s/mm²). Kurtosis produces a deviation of the graph in a direction opposite to IVIM perfusion, resulting in lower than expected apparent diffusion coefficient.

A common method to account for kurtosis is to perform a one-step Taylor series expansion of the exponent in the standard diffusion signal equation above, yielding

S = S_oe^{−bD + b²D²K/6}

Acute cerebral infarction shown by trace DWI, ADC map, and mean kurtosis image. Note decreased kurtosis in center of DK image which is not appreciated on conventional DWI or ADC maps. This area corresponds to the more cystic area of infarction seen on the T2-FLAIR image 6 months later. (From Hori et al under CC BY)

By acquiring data in several directions and different b-values (the largest in the 2000−3000 s/mm² range), variables in this equation can be estimated. The value of D obtained can be thought of as the apparent diffusion coefficient corrected for kurtosis, while K can be considered the apparent or mean kurtosis averaged over three cardinal directions. In cranial imaging typical calculated kurtosis values might range from K=0 for CSF to K=0.7 for gray matter to K=1.0 for white matter.

Analogous to DTI it is possible to create diffusion kurtosis tensors and, for example, estimate axial and radial components of kurtosis. At least 15 different diffusion directions and two non-zero b-values must be probed to create such a tensor. Several even more complex methods, including q-Ball imaging and diffusion spectrum imaging, are briefly described in the Advanced Discussion.