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Fund BioImag 2012 4-1 4: From x-ray to image – Computed Tomography 1.What factors influence contrast in x-ray imaging ? Beam hardening Sensitivity and resolution considerations 2.What is the fundamental basis for image reconstruction using x-ray absorption ? Radon Transform 3.How can x-ray images be reconstructed? Sinogram Backprojection vs. filtered backprojection Central Slice Theorem 4.Examples After this course you 1.Understand the consequences of the Bremsstrahlung continuum on image contrast 2.Understand how Compton scattering reduces image contrast and how its influence can be reduced 3.Are familiar with the Radon transform 4.Understand the principle of matrix reconstruction and backprojection 5.Understand the major mechanisms leading to CT contrast

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Fund BioImag 2012 4-2 Computed Tomography invented 1973 Physiology and Medicine engineerphysicist Primitive pancreatic lesion (neoplastic) Godfrey Hounsfield Allan Cormack Nobel prize 1979 First clinical CT- 1972

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Fund BioImag 2012 4-3 CT of animal models transgenic mice and cancer models microCT (~50 µm spatial resolution) Mouse: blood vessels (red) micro-CT scanner heart and lungs abdomen rat spine kidneys with IV contrast full body skeleton

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Fund BioImag 2012 4-4 4-1. Absorption in the real world Linear attenuation coefficient μ : linear attenuation coefficient Unit: [cm -1 ] n n-dn dx But, µ=f( E, Z, ) If µ is constant in x Two consequences: Beam hardening Depth dependent contrast Contrast is “well-defined” for monochromatic x-rays The measurement that is wanted : (x,y) What is measured : n(x) ( for a homogeneous object of thickness x)

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Fund BioImag 2012 4-5 Filtered Energy Spectrum: Bremsstrahlung and characteristic emission Minimal energy is zero, but: Soft x-rays (low energy) are filtered by instrument i(E ): complex function Define effective photon energy E eff E eff E eff is increased by instrument (filtering of soft x-rays) Hard x-rays (high energy) Maximal x-ray energy is = kinetic energy of e - (eV cathode ) Interaction with orbital electrons Soft x-rays (low energy) Bremsstrahlung continuum i(E )

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Fund BioImag 2012 4-6 Beam Hardening Effective energy depends on depth A similar consequence arises in tissue:Ideal: Monochromatic x-rays(E ( ) = ( )) Reality: Polychromatic, multienergetic i(E ) Absorption is not uniform with E Contrast changes with large objects and depth Excessive radiation dose to superficial tissue Absorption is not uniform with E Contrast changes with large objects and depth Excessive radiation dose to superficial tissue “Solution”: Reduce i(E ) for soft x-rays (e.g. 3mm Al eliminates 90% of 20keV photons)

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Object (lesion) Fund BioImag 2012 4-7 Scattering further reduces image contrast Scattering increases with FOV (field-of-view) of irradiation Further reduction of image contrast Solution: Anti-scatter grid (collimator) With E ↑: Compton scattering → increased masking of object. Photoelectric effect: complete absorption of photons → Objet (lesion) easily detected Collimation principle: establishes directionality of x-rays Detector Slice selection High Z, : stop x-rays

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Fund BioImag 2012 4-8 Signal- & Contrast-to-Noise Ratio Signal: I(d) (no. photons detected) I 0 (no. photons irradiated) Contrast: I(d) due to µ(d) differences i(E ), C produces reduced contrast Compton scattering: Antiscatter grid CT intensity can be measured in absolute terms (CT-number) Soft tissue: Typically has weak contrast (small Hounsfield units) HU : attenuation normalized to water ( 0) range from -1000 (air) to +3000 (bone and contrast agents) soft tissues: -300 to +100

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Fund BioImag 2012 4-9 4-2. Getting to the image: The Radon transform Given a certain beam intensity (no. of photons) I at a given position y, I(y 0 ), the beam intensity at y+ y is I(y) y0y0 yy y … lim y→0 I0I0 y x Considering a two-dimensional objet: Recursive application to derive I(y 0 +2 y) I 0 : intensity of incident x-ray beam Radon transform g(x) Definition Radon transform g(x) Definition Radon transform of a point-like homogeneous object rectangular object g(x)

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Fund BioImag 2012 4-10 Sinogram: Each point has a unique trajectory Detector: is moved in circular motion around object (indicated by angular position ) Each point in space is uniquely represented by Amplitude R and phase of sinusoidal trajectory in Sinogram (sic!): (x,y) → (R, ) Radon transform of a circular object Radon transform = projection of object Trajectory: Rsin( + 0)

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Fund BioImag 2012 4-11 Algebraic reconstruction: Matrix inversion In principle such an n 2 matrix can be inverted. Too complex, computationally intensive and unstable ln(I 1 /I 0 ) = - 1 x- 2 x ln(I 2 /I 0 ) = - 3 x- 4 x ln(I 3 /I 0 ) = - 1 y- 3 y ln(I 4 /I 0 ) = - 2 y- 4 y Decomposing an object into a 2x2 matrix requires a minimum of 4 measurements: Setting x= y yields a linear 2x2 inversion problem linking k to I k CT was introduced in 1970 simple reconstruction algorithm!

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Fund BioImag 2012 4-12 4-3. Backprojection reconstruction Basic reconstruction principle: Along the measured projection direction fill in each pixel constant numbers corresponding to the Radon transform (projection intensity). Repeat for next orientation of the projection, sum the values in overlapping pixels. Illustration with gray shades (point-like object): 11111111111111 1 1 1 1 1 1 1 2 2 Projections 4 Projections 8 Projections16 Projections x’ y’ 180 backprojections: reconstruction is complete. the next 60 1 st 60 Projections

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Fund BioImag 2012 4-13 Simple Backprojection has poor spatial resolution dxdy=dA= sin(d )dr d d dd dd dd The reconstruction falls off with 1/r (in analogy to the decrease of light intensity in 2D) Backprojection has poor spatial resolution: Reconstruction of a point- object falls of with 1/r WHY? Solution ? dd dA Number of rays (projections): constant with d But: pixel size = dxdy dA = const No. of rays 1/

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Fund BioImag 2012 4-14 Filtered Backprojection is used to maintain good image resolution Example: Reconstruction of a slice from projections

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Fund BioImag 2012 4-15 Central Slice Theorem: Two Spaces linked by Fourier Transform FT IFT k-space (Fourier space) kxkx kyky MRI: Acquired Data Image space x y Final Image (CT: acquired data) See also: Signals and Systems (SV) FT of projection → 1 line in k-space Reconstruction using FT : artifacts k x =0

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Fund BioImag 2012 4-16 Bone (calcification) : bright (high absorption) Air is dark Different densities of tissue give inter-mediate results Imaging of mummified bodies Dislodged arrow head X-ray CT : Examples (Human)

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Fund BioImag 2012 4-17 CT: Examples (mouse) Pre- Post-contrast 3D CT scan of rodent spine treated with human mesenchymal stem cells (transduced with the human BMP-9 gene via an adenoviral vector) significant bone formation at the treatment sites (arrows) 13µm micro CT of mouse placenta vasculature Micro-CT of mouse femor bone

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Fund BioImag 2012 4-18 CT: Conclusion Main contrast is bone vs. soft tissue (or air) (calcium content i.e. e - density ) Contrast agents (increase Z eff ) allow depiction of vessel architecture and lesions Insect with iodine contrast agent SNR and CNR: 1.Intensity can be increased by cathode current 2.High spatial resolution possible (limited only by radiation dose in humans) Fish larvae

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Fund BioImag 2012 4-19 Generations of CT scanners First Generation Parallel beam design One/two detectors Translation/rotation 2nd Generation Small fan beam Translation/rotation Larger no. of detectors 3rd Generation Multiple detectors Large fan beam 4th Generation Detector ring Large fan beam

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20 Fund BioImag 2012 X Basic principle of CT -Reconstruction of 2 dimensional image- Simple Backprojection Projection Data Blur x y x y curvilinear integral of absorption coefficient regarding Y object X-ray tube X-ray detector array Data Acquisition field Reconstruction field X X Y

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21 Fund BioImag 2012 Basic principle of CT -Reconstruction of 2 dimensional image- Filtered Backprojection Projection Data x y ＊ x Multidirectional Backprojection Reconstruction Filter ω or x Filtered Projection data x X

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Fund BioImag 2012 4-22 Projection reconstruction – by example courtesy of Dr. K. Kouris 1st step in tomography = recording projections

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Fund BioImag 2012 4-23 Medial temporal lobe O’Brien et al, Dement Geriatr Cogn Disord 2000

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If F {f(x,y)} = F(u,v) or F( , ) then F( , ) = F { g (R) } The Fourier Transform of a projection at angle is a line in the Fourier transform.

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