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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
dds
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 ) Rcp( 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
