title: Anomalous diffusion measured by a twice-refocused spin echo pulse sequence: analysis using fractional order calculus.
abstract: OBJECTIVE
  To theoretically develop and experimentally validate a formulism based on a fractional order calculus (FC) diffusion model to characterize anomalous diffusion in brain tissues measured with a twice-refocused spin-echo (TRSE) pulse sequence.
  
  
  METHODS
  The FC diffusion model is the fractional order generalization of the Bloch-Torrey equation. Using this model, an analytical expression was derived to describe the diffusion-induced signal attenuation in a TRSE pulse sequence. To experimentally validate this expression, a set of diffusion-weighted (DW) images was acquired at 3 Tesla from healthy human brains using a TRSE sequence with twelve b-values ranging from 0 to 2600 s/mm(2). For comparison, DW images were also acquired using a Stejskal-Tanner diffusion gradient in a single-shot spin-echo echo planar sequence. For both datasets, a Levenberg-Marquardt fitting algorithm was used to extract three parameters: diffusion coefficient D, fractional order derivative in space β, and a spatial parameter μ (in units of μm). Using adjusted R-squared values and standard deviations, D, β, and μ values and the goodness-of-fit in three specific regions of interest (ROIs) in white matter, gray matter, and cerebrospinal fluid, respectively, were evaluated for each of the two datasets. In addition, spatially resolved parametric maps were assessed qualitatively.
  
  
  RESULTS
  The analytical expression for the TRSE sequence, derived from the FC diffusion model, accurately characterized the diffusion-induced signal loss in brain tissues at high b-values. In the selected ROIs, the goodness-of-fit and standard deviations for the TRSE dataset were comparable with the results obtained from the Stejskal-Tanner dataset, demonstrating the robustness of the FC model across multiple data acquisition strategies. Qualitatively, the D, β, and μ maps from the TRSE dataset exhibited fewer artifacts, reflecting the improved immunity to eddy currents.
  
  
  CONCLUSIONS
  The diffusion-induced signal attenuation in a TRSE pulse sequence can be described by an FC diffusion model at high b-values. This model performs equally well for data acquired from the human brain tissues with a TRSE pulse sequence or a conventional Stejskal-Tanner sequence.
authors.affiliation: School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, China; Center for Magnetic Resonance Research, University of Illinois at Chicago, Chicago, Illinois, USA; Department of Bioengineering, University of Illinois at Chicago, Chicago, Illinois, USA.
authors.forename: Qing
authors.initials: Q
authors.lastname: Gao
authors.forename: Girish
authors.initials: G
authors.lastname: Srinivasan
authors.forename: Richard L
authors.initials: RL
authors.lastname: Magin
authors.forename: Xiaohong Joe
authors.initials: XJ
authors.lastname: Zhou
mesh_terms: D000465:Algorithms; D016477:Artifacts; D001921:Brain / Q000473:pathology*; D001931:Brain Mapping / Q000379:methods; D002555:Cerebrospinal Fluid / Q000378:metabolism; D004058:Diffusion; D038524:Diffusion Magnetic Resonance Imaging / Q000379:methods*; D006801:Humans; D007091:Image Processing, Computer-Assisted / Q000379:methods; D015233:Models, Statistical