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4: From x-ray to image – Computed Tomography. What factors influence contrast in x-ray imaging ? Beam hardening Sensitivity and resolution considerations What is the fundamental basis for image reconstruction using x-ray absorption ? Radon Transform - PowerPoint PPT Presentation
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Introduction au cours
Fund BioImag 20134-14: From x-ray to image Computed TomographyWhat factors influence contrast in x-ray imaging ? Beam hardeningSensitivity and resolution considerationsWhat is the fundamental basis for image reconstruction using x-ray absorption ?Radon TransformHow can x-ray images be reconstructed? Sinogram Backprojection vs. filtered backprojection Central Slice TheoremExamples
After this course youUnderstand the consequences of the Bremsstrahlung continuum on image contrastUnderstand how Compton scattering reduces image contrast and how its influence can be reduced Are familiar with the Radon transform Understand the principle of matrix reconstruction and backprojection Understand the major mechanisms leading to CT contrast Fund BioImag 20134-2Computed Tomographyinvented 1971
Physiology and Medicine engineerphysicist
Primitive pancreatic lesion (neoplastic) Godfrey Hounsfield Allan Cormack Nobel prize 1979
First clinical CT- 1972Fund BioImag 20134-3CT of animal models transgenic mice and cancer models
microCT (~50 m spatial resolution)
Mouse: blood vessels (red) micro-CT scannerheart and lungsabdomenrat spinekidneys with IV contrastfull body skeletonFund BioImag 20134-44-1. What does absorption in the real world imply ?Linear attenuation coefficient m : linear attenuation coefficient Unit: [cm-1]nn-dndxBut, =f(En,Z, r)
If is constant in x Two consequences:Beam hardeningDepth dependent contrastContrast is well-defined for monochromatic x-raysThe measurement that is wanted: m(x,y)What is measured: n(x)
(m for a homogeneous object of thickness x)Fund BioImag 20134-5What does the Energy Spectrum of an x-ray tube really look like ?filtered Bremsstrahlung and characteristic emission
Minimal energy is zero, but:Soft x-rays (low energy)are filtered by instrumenti(En): complex functionDefine effective photon energy EeffEeffEeff is increased by instrument (filtering of soft x-rays) Hard x-rays (high energy)Maximal x-ray energy is = kinetic energy of e- (eVcathode)Interaction with orbital electronsSoft x-rays (low energy)Bremsstrahlung continuumi(En)Fund BioImag 20134-6What is the consequence of energy-dependent absorption ?Beam Hardening - Effective energy depends on depthA similar consequence arises in tissue:
Ideal: Monochromatic x-rays (En(l) = d(l0))Reality: Polychromatic, multienergetic i(En)
Absorption is not uniform with EnContrast changes with large objects and depthExcessive radiation dose to superficial tissueSolution: Reduce i(En) for soft x-rays (e.g. 3mm Al eliminates 90% of 20keV photons)
Object (lesion)Fund BioImag 20134-7
How does x-ray scattering impact CNR ?Scattering increases with FOV (field-of-view) of irradiation Further reduction of image contrast
Solution: Anti-scatter grid (collimator)
With En : Compton scattering increased masking of object. Photoelectric effect: complete absorption of photons Object (lesion) easily detectedCollimation principle: establishes directionality of x-raysDetectorSlice selectionHigh Z, r: stop x-rays
Fund BioImag 20134-8How is CNR quantified ?
Signal: I(d) (no. photons detected) I0 (no. photons irradiated)
Contrast: DI(d) due to (d) differencesi(En), mC produces reduced contrastCompton scattering: Antiscatter gridCT 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 Fund BioImag 20134-94-2. What is the basis of image reconstruction ? The Radon transformGiven a certain beam intensity (no. of photons) I at a given position y, I(y0), the beam intensity at y+Dy is
I(y)y0DyDy lim Dy0I0yxConsidering a two-dimensional objet:Recursive application to derive I(y0+2Dy)
I0: intensity of incident x-ray beamRadon transform g(x)Definition
Radon transform of a point-like homogeneous objectrectangular objectg(x)Fund BioImag 20134-10
Does each pixel have a unique trajectory ?SinogramDetector: is moved in circular motion around object (indicated by angular position f)Each point in space is uniquely represented by Amplitude R and phase f0 of sinusoidal trajectory in Sinogram (sic!): (x,y) (R,f0)f
Radon transform of a circular objectRadon transform = projection of objectTrajectory:Rsin(f+f0)Fund BioImag 20134-11Can a CT image be constructed by Matrix inversion ?
In principle such an n2 matrix can be inverted. Too complex, computationally intensive and unstable ln(I1/I0) = -m1Dx-m2 Dx ln(I2/I0) = -m3Dx-m4 Dx ln(I3/I0) = -m1Dy-m3 Dy ln(I4/I0) = -m2Dy-m4 Dy Decomposing an object into a 2x2 matrix requires a minimum of 4 measurements:
Setting Dx=Dy yields a linear 2x2 inversion problem linking mk to Ik
CT was introduced in 1970 simple reconstruction algorithm!Fund BioImag 20134-124-3. What algorithm is adapted to 1970s computing power ?Backprojection reconstructionBasic 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):11111111 1 1 1 1 1 122 Projections4 Projections8 Projections16 Projections
xy
180 backprojections: reconstruction is complete.the next 601st 60 ProjectionsFund BioImag 20134-13Why does simple Backprojection have poor spatial resolution ?
dxdy=dA=rsin(df)drrdfdrdrdfrrdfThe 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/rWHY?Solution ?dfdA rNumber of rays (projections): constant with dfBut: pixel size = dxdy dA = const No. of rays 1/rFund BioImag 20134-14
How can good image resolution be maintained ? Filtered Backprojection
Example: Reconstruction of a slice from projections
Fund BioImag 20134-15How is Backprojection linked to Fourier transform ? Central Slice TheoremFTIFTk-space (Fourier space)kxkyMRI: Acquired DataImage spacexyFinal Image (CT: acquired data)
See also: Signals and Systems (SV)
FT of projection 1 line in k-space Reconstruction using FT :artifactskx=0Fund BioImag 20134-16
Bone (calcification) : bright (high absorption)Air is darkDifferent densities of tissue give inter-mediate results
Imaging of mummified bodiesDislodged arrow head
X-ray CT : Examples (Human)Fund BioImag 20134-17CT: 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)
13m micro CT of mouse placenta vasculature
Micro-CT of mouse femor boneFund BioImag 20134-18
CT: SummaryMain contrast is bone vs. soft tissue (or air) (calcium content i.e. e- density r)Contrast agents (increase Zeff) allow depiction of vessel architecture and lesions
Insect with iodine contrast agentSNR and CNR:Intensity can be increased by cathode currentHigh spatial resolution possible(limited only by radiation dose in humans)Fish larvae
18Fund BioImag 20134-19How have CT scanners evolved ?Generations of CT scannersFirst Generation Parallel beam designOne/two detectorsTranslation/rotation
2nd GenerationSmall fan beamTranslation/rotationLarger no. of detectors
3rd GenerationMultiple detectorsLarge fan beam
4th GenerationDetector ringLarge fan beam
EQ I0 = \I(0,,i(En)dEn)