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					           The International Workshop on
“Mathematical Methods in Imaging and Image Analysis”
    East China Normal University, Shanghai, China


       Biomedical Tomography
          --Some history and new problems

                      Yi Li
    Hunan Normal University & University of Iowa

                  June 21-22, 2006
Joint work with
• Ge Wang (PI), Department of Radiology,
University of Iowa
• Ming Jiang, School of Mathematical
Sciences, Peking University
                    Motivations

• Diagonistics
• Treatments
• Results Accessments
         Diffusion approximation




Inverse Problem: Retrieve useful information on q from boundary
measurements.
Spiral CT
               Definition
• Tomography is the study of the reconstruction of
  two- and three-dimensional objects from one-
  dimensional slices.
• Tomography is imaging by sections or sectioning. A
  device used in tomography is called a tomograph,
  while the image produced is a tomogram. The
  method is used in medicine, biology, geology,
  materials science and other sciences. It is based on
  the mathematical procedure called tomographic
  reconstruction.
• tomography [t/&/ 'm/o/ gr/&/ fi] 名 断层摄影术
Outline
• Spiral Cone-Beam CT (computed
tomography)
• Bioluminescence/Flourscence Tomography
•Magnetic resonance imaging (MRI)
•Medical ultrasonography
Computed Tomography (CT)

                      Measurement

         y        r              Sinogram     
                                  P( , r )   p

                      x


             Object
X-rays       f ( x, y )                           t


                      Reconstruction
          Tomographic reconstruction

• The mathematical basis for tomographic imaging
  was laid down by Johann Radon. It is applied in
  Computed Tomography to obtain cross-sectional
  images of patients.
• The projection of an object at a given angle, θ is
  made up of a set of line integrals. In X-ray CT, the line
  integral represents the total attenuation of the beam
  of x-rays as it travels in a straight line through the
  object. As mentioned above, the resulting image is a
  2D (or 3D) model of the attenuation coefficient. That
  is, we wish to find the image f(x,y).
• Attenuation occurs exponentially in tissue:



where μ(x) is the attenuation coefficient at position x along
 the ray path. Therefore generally the total attenuation of
 a ray at position r, on the projection at angle θ, is given
 by the line integral:
where f(x,y) represents μ(x,y). This function is known as
 the Radon transform (or sinogram) of the 2D object,
 which tells us that if we had an infinite number of one-
 dimensional projections of an object taken at an
 infinite number of angles, we could perfectly
 reconstruct the original object, f(x,y). So to get f(x,y)
 back, from the above equation means finding the
 inverse Radon transform.
The pixel itself is displayed according to the mean attenuation of
the tissue that it corresponds to on a scale from −1024 to +3071
on the Hounsfield scale.


Water has an attenuation of 0 Hounsfield units (HU)


air is −1000 HU,


bone is typically +400 HU or greater


metallic implants are usually +1000 HU.
1963&1964: Allan McLeod Cormack of Tufts
University develop the theoretical underpinnings of
CAT scanning.

1972: The CT system was invented by Godfrey
Newbold Hounsfield of EMI Central Research
Laboratories using X-rays. Computerized axial
tomography (CAT)

1979: They shared a Nobel Prize in medicine.
• The first scanner took several hours to acquire the raw
  data and several days to produce the images. Modern
  multi-detector CT systems can complete a scan of the
  chest in less time than it takes for a single breath and
  display the computed images in a few seconds.
                  Fan-Beam Geometry

           Integral on any line through an object


Detector
  Arc




                      Object




                          Source
              Cone-Beam Geometry




• Point source
• Planar detector                                 Source
• Parallel data acquisition
• Volumetric coverage                    Object


                              Detector
                               Plate
          Spiral/Helical Cone-Beam CT


                                           Object
Spiral/helical cone-beam
scanning mode and an
approximate reconstruction
algorithm were first proposed
by Wang et al. in 1991
                                                    Source
Wang et al.: Scanning cone-
beam reconstruction
algorithms for x-ray
microtomography. Proc. SPIE     Detector
v. 1556, p. 99-112, July 1991    Plate
                                              General Locus
                                  Wang Algorithm

                 2p                
                1        2 ( )                            (  )t            ( )
g ( x, y , z )                   2 
                                       R ( p,  ,  ) f [             p]                  dpd
                2 0 [  (  )  s ]                      ( )  s       ( )  p  
                                                                           2         2   2



   t  x cos   y sin 
   s   x sin   y cos          Wang, Lin, Cheng, Shinozaki, Kim: Scanning cone-beam
                                   reconstruction algorithms for x-ray microtomography. Proc.
    ( )  t 2  s 2              SPIE v. 1556, p. 99-112, July 1991
         (  )[s  z  h(  )]
                                 Wang, Lin, Cheng, Shinozaki: A general cone-beam
                ( )  s          reconstruction algorithm. IEEE Trans. on Medical Imaging
                                   12:486-496, 1993
                                   Wang, Liu, Lin, Cheng, Shinozaki: Half-scan cone-beam X-ray
                                   microtomography formula. Journal of Scanning Microscopy
                                   16:216-220, 1994
        Implication for Medical CT




Wang, G, Lin, TH, Cheng PC, Shinozaki DM, Kim, HG: Scanning cone-beam
reconstruction algorithms for x-ray microtomography. Proc. SPIE Vol. 1556, p. 99-112,
July 1991 (Scanning Microscopy Instrumentation, Gordon S. Kino; Ed.)
          Danielsson Conjecture



               • Data collected from a PI-segment is
                        sufficient for exact reconstruction

               • Data collected outside
                        the PI-segment is irrelevant!
Pi-Line
                Danielsson et al.: Towards exact reconstruction
                for helical cone-beam scanning of long objects – A
                new detector arrangement and a new
                completeness condition. Proc. 1997 Meeting on
                Fully 3D Image Reconstruction in Radiology and
                Nuclear Medicine, pp. 141-144, 1997
           Katsevich Theorem (2002)

                                                    y ( s2 )
                                                                                     e
                                                                      y ( s1 )
                                                                                            u(s, x)
                                  
                                                                                 y ( s0 )
                                   Object   f (x)
                                                      ( s0 , x )
                                                                                 Source


                                                Pi-Line


                                                       2p
                              1                 1                                                1
Detector
 Plate
                  f ( x)  
                             2p      ( x) | x  y(s) |  q
                                  I PI                  0
                                                             D f ( y (q), ( s, x,  )) | q  s
                                                                                                sin 
                                                                                                      dds

                         e(s, x)   (s, x)  u(s, x)               ( s, x,  )  cos ( s, x)  sin e(s, x)

           Katsevich A: A general scheme for constructing inversion algorithms for cone beam
           CT. Int'l J. of Math. and Math. Sci. 21:1305-1321, 2003
            Axioms for Resolution

1.   R p( z ) is continuous                                   (Continuity)

2.   R p( z )  0 if            p( z )   ( z )              (Regularity)

3.   R  p ( z  c )   R p ( z )                            (Translation)


                  R p( z )
4.   R p(cz )             Rcp( z )  R p( z )            (Scaling)
                     c
5.   R p1 ( z )  p2 ( z )  F R p1 ( z ), R p2 ( z )   (Combination)


                R[ p ( z )]     z 2 p ( z )dz

                         Wang, Li: IEEE Sig. Proc. Letters 6:257-258, 1999
   Motivation for Optical Imaging
• Physics — Non-ionizing radiation: related to
  molecular conformation (the shape, outline, or
  form of something, determined by the way in
  which its parts are arranged; any of the
  arrangements of a molecule that result from
  atoms being rotated about a single bond) of
  tissue
• Provide functional as opposed to anatomical
  information.
                Motivation cont…
• Optics — High intrinsic contrast:
   – Optical absorption:
      » Angiogenesis (the formation and differentiation of
        blood vessels)
      » Apoptosis (programmed cell death)
      » Necrosis (localized death of living tissue)
      » hyper-metabolism (Increased Speed)
      » exogenous contrast agents
   – Optical scattering: Size of cell nuclei
   – Optical polarization: Collagen
             Motivation cont…

• Physiology — Functional imaging:
   – Oxygen saturation of hemoglobin (血色素)
   – Total hemoglobin
   – Enlargement of cell nuclei
   – Orientation of collagen
   – Denaturation of collagen
   – Blood flow (Doppler)
             Motivation cont…

• Physiology
   – Molecular imaging
      » Fluorescent (e.g. gene expression of enhanced
        (Idocyanine) green flourescent protein)
      » Bioluminescent (uptake of quantum dots)
Bioluminescence Imaging Work


Highest cited paper
Contag, Bachmann: Advances In Vivo Bioluminescence Imaging of Gene
Expression. Annu. Rev. Biomed. Eng. 4:235-260, 2003

Xenogen IVIS System
Most popular bioluminescent imaging systems on the market

Source depth determination
Coguoz, Troy, Jekic-MsMullen, Rice: Determination of depth of in-vivo
bioluminescent signals using spectral imaging techniques. Proc. SPIE
4967:37-45, 2003
   Bioluminescence Tomography
              (BLT)


First initiative

Pioneered by
Wang, Hoffman &
McLennan (2002)

Funded by NIBIB
      (R21/R33)
Optical Biopsy
                     BLT Prototype




Bioluminescence Tomography Device               In Vivo Micro-CT Scanner
   Designed by Wang, Hoffman, McLennan              Designed by Wang, Hoffman, BIR
     Built by Meinel, Suter, UI Med. Inst. Facility   Built by BIR
                BLT Models
• The Radiative Transfer Equation, or Linearized
  Boltzmann Equation:
BLT Models cont…
BLT Models cont…
BLT Models cont…
Physical Measurables
Diffusion approximation
     Inverse Problem for BLT




Or
Electron-Beam Micro-CT

               Source




           Mouse




                        Spiral locus


           Detector
            Plate
                        BLT Models cont…

 Diffusion Equation with Cauchy Data
                                                                       Measurement
   ( D u 0 )   a u 0  q 0
                                                                                          u0
                   u0 ( x )                                           g ( x)   D( x)       ( x).
u0 ( x) | x ,             | x                                                       v
                    


                                                                                      Diffused
                                       a ( x)  0 ,   D( x)  0                      Photons


                                                                          v
                                              Source?              
                                             q0 ( x )  0
                                                                       
Literature Review
       Uniqueness Theorem


                                 
        i
 r0i              1 : 1 , D1
          qi
r1i
                         j

                j :  j , Dj
First Journal Paper on BLT




 Wang G, Li Y, Jiang M: Med. Phys. 31:2289-2299, 2004
Uniqueness Theorem
Heterogeneous Mouse Phantom



                 T



            S        H   L




                     B
                  Validation with a Phantom


                                             10-10 W/mm3
          T                                   300
                                              250
                                              200
                                              150

S             H   L                           100
                                              50
                                              10
                                              5
                                              1
              B



    True                  With                              Without
    source distribution   modality fusion                   modality fusion
   Validation with a Mouse

                                      Reconstructed source

                                                pW/mm3


           True source




CT slice                 BLT reconstruction
                    First In Vivo Mouse Study
                                                                                                        7.0  10-15 W/mm2




                                                                                                                    0
        Anterior-posterior                  Right lateral           Posterior-anterior   Left lateral




 14
                                                                                                           10-12 W/mm3
                B
                                                            B

                        L                                       L
  0                                                                 T
                             T
                    H                                           H


-14
  -13            0               -13 (mm)
Multi-Spectral BLT/FLT
Multi-Spectral BLT/FLT
Multi-Spectral BLT/FLT cont…
       Multi-Spectral BLT/FLT cont…




Inverse Problem: Retrieve as much information as possible about the
sources.
Computational Optical Biopsy (COB)
                   Questions

Question 1: if certain priori knowledge is known about
 the sources, what can be done to narrow the range of
 possible solutions?
                  Questions cont…

Question 2: In practice no measurements are absolute,
 i.e. there are always noises/errors effecting the
 accuracy of the attempted measurements. It is
 therefore vitally important for us to understand the
 "stability" of the source distributions.

Again due to the general "non-uniqueness" feature of
 the inverse problem here, the stability is referred to
 such case module the "non-radiating" terms.
  Bolus-Chasing CTA

                                              Bolus Propagation
           Control        Predictive Filter         Model




                                                Comparison


                                     Table

 On-Line
             CTF Image Reconstruction          Image Analysis


             CT Volume Reconstruction          CT Angiography
Off-Line
      Acknowledgment


• Supported in part by NIH
• Supported in part by Xiao-Xiang Professorship
  Fund

				
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