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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

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