Affine Transforms and Covariance Matrices in Neuro-Diffusion Images

Diffusion-weighted imaging, a cornerstone in neurology, holds significant importance. The DWI protocols, meticulously designed for brain evaluation, with a diffusion filter applied to the brain's surface, yield invaluable insights into molecular-based properties. This aspect of neuro-MRI, particularly in diffusion imaging, presents neuroradiologists with a wealth of crucial information, enlightening them about neurologic spatial resolution, temporal resolution, and molecular diffusion filters. The focus here is on the diffusion filter applied to the neurologic-based MR field of view, underscoring its pivotal role in brain tissue evaluation.?

Here, the value information embedded within DWI-neurologic sequences is the evaluation of dynamic flow within the capillary-microtubule spaces within white matter brain tissue. The importance of this type of evaluation concerning the white tissue of the brain is that because neurological demyelinating diseases project themselves specifically upon the brain tissue of white matter, diffusion-oriented imaging is the prime choice for evaluating such neurologic pathology; diffusion-weighted imaging is only beneficial when both MR quality assurance and MR quality control are maintained; it is beneficial to illustrate the distinction between QA and QC within the context of MR physics when it comes to diffusion-based evaluations for neurological imaging. Here, the American College of Radiology standardized QA and QC for MRI. Using the "MR Phantom," this "phantom" is a cylindrical container filled with a specific fluid; there are phantoms whose contents are water, but many more types of fluids can fill the volumetric space.

Given the importance of applying Brownian motion to MRI phantoms for MRI QA and QC-ACR testing, it is best to detail the role Brownian motion plays in ACR testing. Brownian motion is the property that allows scientists and engineers to model a given stochastic process, empowering them with the knowledge to understand and interpret the outcomes. A "stochastic" process is a characteristic of a given outcome that follows some random probability. An "outcome" under the gauze of Brownian motion is a "sample path." Therein, each sample path is a discrete symmetric "random walk." In as much, each node within the sample path is independent of the other nodes within the given set of nodes that allocate themselves to the sample path where this set of nodes making up the sample path is convoluted with some normal probability distribution.?

Transitioning back to the MR Phantom, the phantom is a translucent ellipsoid, a clear and distinct tool used in neuro-MRI for quality assurance and quality control testing, detailed shown below:?

As can be seen from Figure 1, there is an emphasis on gray scaling the image to see the embedded contrast within the ellipsoid. Moreover, as depicted by the green-asterisk graphics, there is a reason there are two graphics of that detail. Expressly, Brownian motion under the context of being a discrete "random walk," such a property of Brownian motion retains the characteristic of quadratic variation; more formally, this property is the quadratic variation of Brownian motion; this quadratic variation converges the sample path between the two graphics within Figure 1 to simulate the neurological fiber pathway concerning human neurologic white matter. Notably, the trait of labeling a "white matter tract" in the neurology framework is that since a neurologic fiber pathway is a continuous deformation of white tissue, a "tract" is a segment of that fiber pathway. Therefore, by performing a sample pathway on a metaphorical neurologic white matter pathway, sampling a segment of that pathway (i.e., a "tract"), because the diffusion probability density of a fiber pathway, as previously illustrated, is a continuous deformation, the properties of neurologic diffusion of a given tract is representative of the entire fiber pathway.?

Here, the quadratic variance of two nodes of a given sample path is a segment (i.e., tract); the computed value of the quadratic variance will provide a number between 0 and 1. With this, because the output of quadratic variance of brain white matter is of the embodiment of a continuous deformation, this output maps to a probability distribution-density of the given liquid molecules within the ACR phantom, where given the molecular properties of the given phantom fluid, each given unique fluid occupying that volumetric space of the ACR MR-phantom will have its unique normal distribution. Thus, the computation of quadratic variance is the probability distribution of molecular diffusion underlying a standardized normal distribution by the given liquid that occupies the volume of the ACR-MR phantom. In detail, each of those embedded graphics within Figure 1 respectively, embodies an effective diffusion tensor where this tensor is:?

Albeit both effective diffusion tensors, representative of the two graphics embedded within Figure 1, taking the matrix product of these two given effective diffusion tensors, the trace of the output matrix of that matrix product provides a probability density of molecular diffusion representing a neurologic white matter tract under the projection of a pre-defined normal distribution that is encoded within the liquid occupying the volume of the ACR-MR Phantom.?