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					Visualizing Diffusion Tensor Imaging
    Data with Merging Ellipsoids

        Wei Chen, Zhejiang University
    Song Zhang, Mississippi State University
      Stephen Correia, Brown University
        David Tate, Harvard University

           22 April 2009, Beijing
                  Background
• Diffusion Tensor Imaging (DTI)
  – Water diffusion in biological
     tissues.
  – Indirect information about the
     integrity of the underlying white
     matter.
                    Diffusion Tensors
     Dxx    Dxy    Dxz                            1         
                        
D   D yx   D yy   D yz   EE 1  (e1 e2   e3 ) 
                                                       2       (e e
                                                                 1 2    e3 ) 1
     Dzx    Dzy    Dzz                            
                                                            3 
                                                                
                        




 Primary diffusion direction
                                         3
      ei  eigenvector (i )  i  max(k )
                                        k 1
            Fractional anisotropy
• Degree of anisotropy
   -represents the deviation from
    isotropic diffusion

                1  2  3
    let :  
                     3
           (1   ) 2  (2   ) 2  (3   ) 2
                                                    0 1
         3
    FA 
         2               
                         2     2
                               1   2
                                      2
                                       3
Tensor at (155,155,30)
                Diffusion tensor:
                 10^(-3)*
                   0.5764 -0.3668 0.1105
                  -0.3668 0.8836 -0.1152
                   0.1105 -0.1152 0.8373

                Eigenvalue=
                  0.0003
                           0.0008
                                     0.0012
                Eigenvector:
                   0.8375 -0.1734 0.5182
                   0.5432 0.3669 -0.7552
                  -0.0592 0.9140 0.4015

                Primary diffusion direction:
                (0.5182 -0.7552 0.4015)
FA at (155,155,30)
              Diffusion tensor:
               10^(-3)*
                 0.5764 -0.3668 0.1105
                -0.3668 0.8836 -0.1152
                 0.1105 -0.1152 0.8373

              Eigenvalue=
                0.0003
                         0.0008
                                  0.0012

              FA = 0.5133
   Tensor Displayed as Ellipsoid
   isotropic                               anisotropic




Courtesy: G. Kindlmann

λ1 = λ2 = λ3                λ1 > λ2 > λ3                 λ1 > λ2 = λ3

                 Eigenvectors define alignment of axes
• Integral Curves               • Glyphs
   – Show topography              – Shows entire diffusion tensor
                                    information
   – Lost information because
     a tensor is reduced to a     – Topography information may
     vector                         be lost or difficult to interpret
   – Error accumulates over       – Too many glyphs  visual
     curves                         clutter; too few  poor
                                    representation
            Our contributions
• A merging ellipsoid method for DTI visualization.
  – Place ellipsoids on the paths of DTI integral curves.
  – Merge them to get a smooth representation


• Allows users to grasp both white matter
  topography/connectivity AND local tensor
  information.
  – Also allows the removal of ellipsoids by using the
    same method used to cull redundant fibers.
                                 Methods
1) Compute diffusion tensors:




2) Compute integral curves:




      p(0) = the initial point

      e1 = major vector field

      p(t) = generated curve
                                Methods
3) Sampling an integral curve, and place an elliptical function at each si :




      Streamball method [Hagen1995] employs spherical functions
                         λ1 = λ2 = λ3, e1 = e2 = e3

4) Construct a metaball function:




   R = truncation radius,
   si is the center of the ith ellipitical function.
   a = −4:0/9:0; b = 17:0/9:0; c = −22:0/9:0.
                           Methods
5) Define a scalar influence field:




6) The merging ellipsoids representation denotes an isosurface
extracted from a scalar influence field F(S; x)
                             Methods




Visualizing eight diffusion tensors along an integral curve with (a) glyphs, (b)
standard spherical streamballs [Hagen1995], and (c) merging ellipsoids
               Parameters
• The degree of merging or separation
  depends on three factors.
• 1st: the iso-value C adjusted interactively
  – Shows merging or un-merging
• 2nd: the truncation radius R

• 3rd: the placement of the ellipsoids.
  – Currently, uniform sampling
                   Parameters




Visualizing eight diffusion tensors with different iso-values:
(a) 0.01, (b) 0.25, (c) 0.51, (d) 0.75, (e) 0.85, (f) 0.95.
 The truncation radius R is 1.0.
              Parameters




The results with different truncation radii: (a) 0.3,
(b) 0.5, (c) 1.0. In all cases, the iso-value is 0.5.
                       Properties
• The entire merging ellipsoid representation is smooth.
• A diffusion tensor produces one elliptical surface.
• When two diffusion tensors are close, their ellipsoids
  tend to merge smoothly. If they coincide, a larger
  ellipsoid is generated.
• Provide iso-value parameters for users to interactively
  change sizes of ellipsoids.
   – Larger: ellipsoids merge with neighbors and provide a sense of
     connectivity
   – Smaller: provide better sense of individual tensors but has limited
     connectivity information
                   Comparison
• If the three eigenvectors are set as identical, our method
  becomes the standard streamball approach.


• If a sequence of ellipsoids are continuously distributed
  along an integral curve, the hyperstreamline
  representation is yielded.


• An individual elliptical function can be extended into
  other superquadratic functions, yielding the glyph based
  DTI visualization representation.
                 Experiments
• Scalar field pre-computed
   – Running time dependent on the grid resolution and
     number of tensors
   – Construction costs 15 minutes to 150 minutes with
     the volume dimension of 2563.




• Visualization of ellipsoids done interactively
   – Reconstruction of isosurface takes 0.5 seconds using
     un-optimized software implementation.
                Experiments
• DTI data from adult healthy control participant
  (age > 55).
• DTI protocol:
  – b = 0, 1000 mm/s2
  – 12 directions
  – 1.5 Tesla Siemens
• Experimental results performed on laptop P4 2.2
  GHz CPU & 2G host memory.
• Box = 34mm3
• Minimum path
  distance = 1.7mm
• Anatomic
  structures and
  relationships
  between tensors

     sagittal                             axial


                     coronal



     axial




                               sagittal   coronal
•   Box = 17mm3
•   Min path distance = 3.4mm
•   b = streamtubes
•   c = ellipsoids
•   d = merging ellipsoids
•   Note greater detail in d


       sagittal



                       coronal



       axial
•   Same ROI
•   Different iso-values
     •   a = 0.90
     •   b = 0.80
     •   c = 0.60
     •   d = 0.40
•   Different emphases on
    local diffusion tensor
    info vs. connectivity info
•   Forceps major
     •   Box = 17mm3
     •   Min path distance =
         3.4mm
•   Renderings
     •   b = streamtubes
     •   c = ellipsoids
     •   d = merging ellipsoids
•   More isotropic tensors
    vs. corpus callosum
•   Change from high to
    low anisotropy on same
    fiber seen with merging
    ellipsoid method
                   axial
•   Differences between
    tensors on a single
    curve.
      • Blue = more
         anisotropic
      • Red = more
         isotropic
•   Improves ability to
    identify problematic
    fibers or problematic
    sections on a curve
                      Evaluation
• Identify regions within a fiber that has low anisotropy and
  thus might be problematic.
   – Normal anatomy (e.g., crossing fibers)?
   – Injured?
   – At risk?


• Adjunct to conventional quantitative tractography
  methods
                          Evaluation
                                        •   Adjunct to conventional
                                            quantitative tractography methods
                                        •   Activate merging ellipsoids after
                                            tract selection to visually evaluate
                                            and select fibers with low or high
                                            anisotropy, even if length is same




•   Group comparison and statistical
    correlation with cognitive and/or
    behavioral measures
•   May reveal effects otherwise
    masked by larger number of normal
    fibers in the tract-of-interest
               Conclusions
• A simple method for simultaneous
  visualization of connectivity and local
  tensor information in DTI data.
• Interactive adjustment to enhance
  information about local anisotropy.
  – Full spectrum from individual glyphs to
    continuous curves
            Future Directions
• Statistical tests
  – Cingulum bundle in vascular cognitive
    impairment
     • Association with apathy?
  – Circularity?
     • Select fibers at risk based on visual inspection and
       then enter into statistical models?
• Intra-individual variability
• Inter-individual variability
  – Interhemispheric differences
        Acknowledgements
• This work is partially supported by NSF of
  China (No.60873123), the Research
  Initiation Program at Mississippi State
  University.
Distance between integral curves




  s = The arc length of shorter curve
  s0, s1 = starting & end points of s
  dist(s) = shortest distance from location s on the
  shorter curve to the longer curve.
  Tt ensures two trajectories labeled different if they
  differ significantly over any portion of the arc length.

				
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