Assessment of Diffusion Anisotropy of White Matter in Areas of the Brain with crossing fibers: A simulation study

Introduction: We present a model of simulation of diffusion in white matter. This model has been used in Diffusion Weighted Imaging researches as a tool for employing cylindrically constrained two-tensor models to identify two independent directions within a voxel and assessing the orientation angle as a parameter that influence on fractional anisotropy.
 
Materials and Methods:  We generate a random number generator with Maxwell-Boltzmann distribution. This distribution is chosen from quantum mechanics theory and is a physical assumption for the initial velocity of molecules before they strike the cylindrical geometry. First position of a molecule generates randomly and then this molecule can transmit to every point on the cylinder. So by determine a random θ and either a random Φ, we determining a random orientation for this molecule and until the molecule haven’t stroked a point on the cylinder this orientation not changed since the collision between molecules is not important because of their tiny dimensions. After tracing the molecules and recording all of the collision point’s characteristics, we can calculate Mean Free Path (MFP) along all directions, we can calculate diffusivity (D) along all directions. So we can calculate eigenvalues (λ1, λ2, λ3) and eigenvectors (V1, V2, V3). Therefore we have collision points data on a cylinder rotated by arbitrary angle so mean free paths and index vector associated to this state can be calculated.
Results: FA values for all states that involve only one fiber along a determined direction have calculated from three values of index vector. An overall FA value is calculated by situated two fiber crossing at an arbitrary angle in a voxel. Also variation of FA value with variation of angle between two fibers in a voxel have been calculated and assessed how this angle can change FA values.
Conclusion: In this paper an evaluation of FA value variation with variation of angle presented and results shown with increasing in angle between two bundle, FA value decreases. .FA has its must value when two bundle are oriented at the same direction and least value when they are crossing at 180.