generate a formula that allows us to convert the signal attenuation of an mri voxel into a numerical measure of diffusion—the diffusion coefficient d. when various barriers and restricting factors such as cell membranes and microtubules interfere with the free diffusion, we are measuring an "apparent diffusion coefficient", or adc, because the measurement misses all the local effects and treats the attenuation as if all the movement rates were solely due to brownian motion. the adc in anisotropic tissue varies depending on the direction in which it is measured. diffusion is fast along the length of (parallel to) an axon, and slower perpendicularly across it. once we have measured the voxel from six or more directions and corrected for attenuations due to t2 and t1 effects, we can use information from our calculated ellipsoid tensor to describe what is happening in the voxel. if you consider an ellipsoid sitting at an angle in a cartesian grid then you can consider the projection of that ellipse onto the three axes. the three projections can give you the adc along each of the three axes adcx, adcy, adcz. this leads to the idea of describing the average diffusivity in the voxel which will simply be ( a d c x + a d c y + a d c z ) / 3 = a d c i +adc_+adc_)/3=adc_ we use the i subscript to signify that this is what the isotropic diffusion coefficient would be with the effects of anisotropy averaged out. the ellipsoid itself has a principal long axis and then two more small axes that describe its width and depth. all three of these are perpendicular to each other and cross at the center point of the ellipsoid. we call the axes in this setting eigenvectors and the measures of their lengths eigenvalues. the lengths are symbolized by the greek letter λ. the long one pointing along the axon direction will be λ1 and the two small axes will have lengths λ2 and λ3. in the setting of the dti tensor ellipsoid, we can consider each of these as a measure of the diffusivity along each of the three primary axes of the 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 λ ⊥ = ( λ 2 + λ 3 ) / 2. =(\lambda _+\lambda _)/2.} 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 ( λ ⊥ ). another commonly used measure that summarizes the total diffusivity is the trace—which is the sum of the three eigenvalues, t r ( λ ) = λ 1 + λ 2 + λ 3 (\lambda )=\lambda _+\lambda _+\lambda _ where λ is a diagonal matrix with eigenvalues λ 1 , λ 2 and λ 3 on its diagonal. if we divide this sum by three we have the mean diffusivity, m d = ( λ 1 + λ 2 + λ 3 ) / 3 =(\lambda _+\lambda _+\lambda _)/3} which equals adci since t r ( λ ) / 3 = t r ( v − 1 v λ ) / 3 = t r ( v λ v − 1 ) / 3 = t r ( d ) / 3 = a d c i \mathrm (\lambda )/3&=\mathrm (v^v\lambda )/3\\&=\mathrm (v\lambda v^)/3\\&=\mathrm (d)/3\\&=adc_\end where v is the matrix of eigenvectors and d 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 1 / 2 so that fa has a maximum value of 1. the whole formula for fa looks like this: f a = 3 ( ( λ 1 − e ⁡ [ λ ] ) 2 + ( λ 2 − e ⁡ [ λ ] ) 2 + ( λ 3 − e ⁡ [ λ ] ) 2 ) 2 ( λ 1 2 + λ 2 2 + λ 3 2 ) =-\operatorname )^+(\lambda _-\operatorname )^+(\lambda _-\operatorname )^)}}^+\lambda _^+\lambda _^)}}}} the fractional anisotropy can also be separated into linear, planar, and spherical measures depending on the "shape" of the diffusion ellipsoid. for example, a "cigar" shaped prolate ellipsoid indicates a strongly linear anisotropy, a "flying saucer" or oblate spheroid represents diffusion in a plane, and a sphere is indicative of isotropic diffusion, equal in all directions. if the eigenvalues of the diffusion vector are sorted such that λ 1 ≥ λ 2 ≥ λ 3 ≥ 0 \geq \lambda _\geq \lambda _\geq 0} , then the measures can be calculated as follows: for the linear case, where λ 1 ≫ λ 2 ≃ λ 3 \gg \lambda _\simeq \lambda _ , c l = λ 1 − λ 2 λ 1 + λ 2 + λ 3 =-\lambda _+\lambda _+\lambda _ for the planar case, where λ 1 ≃ λ 2 ≫ λ 3 \simeq \lambda _\gg \lambda _ , c p = 2 ( λ 2 − λ 3 ) λ 1 + λ 2 + λ 3 =-\lambda _)}+\lambda _+\lambda _ for the spherical case, where λ 1 ≃ λ 2 ≃ λ 3 \simeq \lambda _\simeq \lambda _ , c s = 3 λ 3 λ 1 + λ 2 + λ 3 =+\lambda _+\lambda _ each measure lies between 0 and 1 and they sum to unity. an additional anisotropy measure can used to describe the deviation from the spherical case: c a = c l + c p = 1 − c s = λ 1 + λ 2 − 2 λ 3 λ 1 + λ 2 + λ 3 =c_+c_=1-c_=+\lambda _-2\lambda _+\lambda _+\lambda _ there are other metrics of anisotropy used, including the relative anisotropy (ra): r a = ( λ 1 − e ⁡ [ λ ] ) 2 + ( λ 2 − e ⁡ [ λ ] ) 2 + ( λ 3 − e ⁡ [ λ ] ) 2 3 e ⁡ [ λ ] =-\operatorname )^+(\lambda _-\operatorname )^+(\lambda _-\operatorname )^\operatorname }}} and the volume ratio (vr): v r = λ 1 λ 2 λ 3 e ⁡ [ λ ] 3 =\lambda _\lambda _ ^ applications the most common application of conventional dwi (without dti) is in acute brain ischemia. dwi directly visualizes the ischemic necrosis in cerebral infarction in the form of a cytotoxic edema, appearing as a high dwi signal within minutes of arterial occlusion. with perfusion mri detecting both the infarcted core and the salvageable penumbra, the latter can be quantified by dwi and perfusion mri. dwi showing necrosis (shown as brighter) in a cerebral infarction dwi showing restricted diffusion in the medial dorsal thalami consistent with wernicke encephalopathy dwi showing cortical ribbon-like high signal consistent with diffusion restriction in a patient with known melas syndrome another application area of dwi is in oncology. tumors are in many instances highly cellular, giving restricted diffusion of water, and therefore appear with a relatively high signal intensity in dwi. dwi is commonly used to detect and stage tumors, and also to monitor tumor response to treatment over time. dwi can also be collected to visualize the whole body using a technique called 'diffusion-weighted whole-body imaging with background body signal suppression' (dwibs). some more specialized diffusion mri techniques such as diffusion kurtosis imaging (dki) have also been shown to predict the response of cancer patients to chemotherapy treatment. the principal application is in the imaging of white matter where the location, orientation, and anisotropy of the tracts can be measured. the architecture of the axons in parallel bundles, and their myelin sheaths, facilitate the diffusion of the water molecules preferentially along their main direction. such preferentially oriented diffusion is called anisotropic diffusion. tractographic reconstruction of neural connections via dti dti of a healthy human brachial plexus. taken from wade et al., 2020. the imaging of this property is an extension of diffusion mri. if a series of diffusion gradients (i.e. magnetic field variations in the mri magnet) are applied that can determine at least 3 directional vectors (use of 6 different gradients is the minimum and additional gradients improve the accuracy for "off-diagonal" information), it is possible to calculate, for each voxel, a tensor (i.e. a symmetric positive definite 3×3 matrix) that describes the 3-dimensional shape of diffusion. the fiber direction is indicated by the tensor's main eigenvector. this vector can be color-coded, yielding a cartography of the tracts' position and direction (red for left-right, blue for superior-inferior, and green for anterior-posterior). the brightness is weighted by the fractional anisotropy which is a scalar measure of the degree of anisotropy in a given voxel. mean diffusivity (md) or trace is a scalar measure of the total diffusion within a voxel. these measures are commonly used clinically to localize white matter lesions that do not show up on other forms of clinical mri. applications in the brain: tract-specific localization of white matter lesions such as trauma and in defining the severity of diffuse traumatic brain injury. the localization of tumors in relation to the white matter tracts (infiltration, deflection), has been one of the most important initial applications. in surgical planning for some types of brain tumors, surgery is aided by knowing the proximity and relative position of the corticospinal tract and a tumor. diffusion tensor imaging data can be used to perform tractography within white matter. fiber tracking algorithms can be used to track a fiber along its whole length (e.g. the corticospinal tract, through which the motor information transit from the motor cortex to the spinal cord and the peripheral nerves). tractography is a useful tool for measuring deficits in white matter, such as in aging. its estimation of fiber orientation and strength is increasingly accurate, and it has widespread potential implications in the fields of cognitive neuroscience and neurobiology. the use of dti for the assessment of white matter in development, pathology and degeneration has been the focus of over 2,500 research publications since 2005. it promises to be very helpful in distinguishing alzheimer's disease from other types of dementia. applications in brain research include the investigation of neural networks in vivo, as well as in connectomics. applications for peripheral nerves: brachial plexus: dti can differentiate normal nerves (as shown in the tractogram of the spinal cord and brachial plexus and 3d 4k reconstruction here) from traumatically injured nerve roots. cubital tunnel syndrome: metrics derived from dti (fa and rd) can differentiate asymptomatic adults from those with compression of the ulnar nerve at the elbow carpal tunnel syndrome: metrics derived