*b*-values) and longer echo times are used.

*, denoted by the dimensionless parameter*

**Kurtosis***, is a long recognized statistical metric for quantifying the shape of a probability distribution. By definition, a Gaussian distribution has*

**K***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.

*) relative to baseline (*

**S***) during application of diffusion-sensitizing gradients can be expressed as*

**So****S** = **S**_{o}e^{−bD} *or* **S**/**S**_{o} = e^{−bD}

**S**=

**S**e

_{o}^{−bD}

**S**/

**S**= e

_{o}^{−bD}

*is the apparent (or measured) diffusion coefficient and*

**D***is a factor reflecting the strength and duration of the pulsed diffusion gradients. Under this model, a semilogarithmic plot of signal attenuation*

**b***ln(*vs

**S**/**So**)*should be a straight line with slope =*

**b***.*

**D***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.

**S** = **S**_{o}e^{−bD + b²D²K/6}

**S**=

**S**e

_{o}^{−bD + b²D²K/6}

*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

*and*

**q-Ball imaging***, are briefly described in the*

**diffusion spectrum imaging****Advanced Discussion**.

### Advanced Discussion (show/hide)»

Both DTI and DKI suffer from the "crossing fiber problem" that occurs in voxels where anatomic structures constraining diffusion in different directions coexist. Such a situation is common in the white matter tracts of the brainstem and cerebral hemispheres.

** Q-Ball Imaging** is a method to resolve multiple intravoxel fiber orientations that does not require specific assumptions on the shape of the diffusion probability curve. The method, still confined to research laboratories, requires probing the tissue with at least 60 gradient directions including ones with high amplitude (>4000 s/mm²). Postprocessing is performed using a complex filtered back projection method (the Funk-Radon transform). Image acquisition time is reasonable (10-20 min). The technique seems promising but is not yet fully validated.

Even more advanced is ** Diffusion Spectrum Imaging (DSI)**. DSI studies are very long (30-60 min), requiring over 200 gradient directions to be sampled using gradients of 8000-12000 s/mm² (far beyond the range of standard clinical systems). DSI is relatively hypothesis-free, producing a full 3D diffusion probability density function map, allowing accurate depiction of fiber crossings with high angular resolution.

**References**

Balanda KP, MacGillivray HL. Kurtosis: a critical review. Am Statistician 1988; 42:111-119. (A paper for you mathematicians out there. The actual definition of kurtosis and how to draw

*K*>0 versus

*K*<0 probability distributions is more complicated than my simple drawing suggests.)

Hagmann P, Jonasson L, Maeder P, et al. Understanding diffusion MR imaging techniques: from scalar diffusion-weighted imaging to diffusion tensor imaging and beyond. RadioGraphics 2006; 25:S205-223.

Jensen JH, Helpern JA, Ramani A, Lu H, Kaczynski K. Diffusional kurtosis imaging: the quantification of non-Gaussian water diffusion by means of magnetic resonance imaging. Magn Reson Med 2005; 53:1432-1440. (original paper where DKI was first proposed)

Jensen JH, Helpern JA. MRI quantification of non-gaussian water diffusion by kurtosis analysis. NMR Biomed 2010; 23:698-710. (follow-up paper to the original with a more comprehensive review and explanation)

Hori M, Aoki S, Fukunaga I, et al. A new diffusion metric, diffusion kurtosis imaging, used in the serial examination of a patient with stroke. Acta Radiologica Short Reports 2012;1:2 DOI: 10.1258/arsr.2011.110024

Tuch DS. Q-Ball imaging. Magn Reson Med 2004; 52:1358-1372.

Wedeen V, Hagmann P, Tseng WY, reese TG, Weisskoff RM. Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging. Magn Reson Med 2005; 54:1377-1386.

Yablonskiy DA, Sukstanskii AL. Theoretical models of the diffusion weighted MR signal. NMR Biomed 2010; 23:661-681.