ellipsoid. This is a little different from the ADC since that was a projection on the axis, while λ is an actual measurement of the ellipsoid we have calculated. The diffusivity along the principal axis, λ1 is also called the longitudinal diffusivity or the axial diffusivity or even the parallel diffusivity λ∥. Historically, this is closest to what Richards originally measured with the vector length in 1991. The diffusivities in the two minor axes are often averaged to produce a measure of radial diffusivity This quantity is an assessment of the degree of restriction due to membranes and other effects and proves to be a sensitive measure of degenerative pathology in some neurological conditions. It can also be called the perpendicular diffusivity (). Another commonly used measure that summarizes the total diffusivity is the Trace—which is the sum of the three eigenvalues, where is a diagonal matrix with eigenvalues , and on its diagonal. If we divide this sum by three we have the mean diffusivity, which equals ADCi since where is the matrix of eigenvectors and is the diffusion tensor. Aside from describing the amount of diffusion, it is often important to describe the relative degree of anisotropy in a voxel. At one extreme would be the sphere of isotropic diffusion and at the other extreme would be a cigar or pencil shaped very thin prolate spheroid. The simplest measure is obtained by dividing the longest axis of the ellipsoid by the shortest = (λ1/λ3). However, this proves to be very susceptible to measurement noise, so increasingly complex measures were developed to capture the measure while minimizing the noise. An important element of these calculations is the sum of squares of the diffusivity differences = (λ1 − λ2)2 + (λ1 − λ3)2 + (λ2 − λ3)2. We use the square root of the sum of squares to obtain a sort of weighted average—dominated by the largest component. One objective is to keep the number near 0 if the voxel is spherical but near 1 if it is elongate. This leads to the fractional anisotropy or FA which is the square root of the sum of squares (SRSS) of the diffusivity differences, divided by the SRSS of the diffusivities. When the second and third axes are small relative to the principal axis, the number in the numerator is almost equal the number in the denominator. We also multiply by so