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CT number and unit
Example water, air, bone
- CT# = (u-u_water)/u_water*1000
- u_water = 0.1928 cm^-1
- CT#(water) = 0
- CT#(air) = -1000
- CT#(bone) = +250...+1000
- Hounsfield Units
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Attenuation
Lambert-Beers Law
- I = I_0 * exp[-(tau+sigma+sigma_gamma)]*L for monochromatic incident x-ray beam
- L: thickness
- Scattering attenuation coefficients:
- tau: photoelectric
- sigma: compton
- sigma_gamma: coherent
- Lambert-Beers Law
- :I = I_0 * exp[-mu*L], mu = mu(E)
- E: photon energy
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Sources of CT measurement error
JH, ch.7
- X-ray beam not monochromatic, mu=mu(E)
- Scattered radiation, non-primary photons
- Non-Linearity of detector/data acquisition system
- Scanned object movement during scanning
- Partial volume effect
- Focal spot drift
- Mechanical vibration
- Off-focal radiation
- CT gantry misalignment
- Metal objects in scanning plane
- X-ray tube arcing
- X-ray photon starvation
- Projection sampling deficiency
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Scattered Radiation
Related Artifacts
JH41
- Not all detected photons are primary
- Adds "low frequency bias" to true attenuation
- This is non-linear when the logarithm is applied to the signal to obtain lin integrals
- p = -ln(I/I_0) = Line integral [mu(x) dx]
- Leads to shading and streaking artifacts
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Beam Hardening
Reasons and Consequences
- Back-projection assumes energy-independent attenuation coefficient mu or monoenergetic x-ray beam
- Both conditions typically not fullfilled
- Resulting artifacts in recontructed images:
- Cupping, shading, streaking
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Total Intensity of Bremstrahlung Radiation
Why x-ray pruduction with electrons?
- X-ray intensity proportional to (Z2*z4*e6)/m2
- Charged particle beam, mass m, charge z*e
- Target charge Z*e
- Electrons 3*106 times more efficient than protons
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Energy-wavelength conversion
- E = h*nu = h * c / lambda
- E = 1240 / lambda[nm] eV
- lambda = 1240 / E_eV nm
- 1 keV ~ 1.2 nm
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Low contrast differentiation of tissues
Photoelectric effect
HS32
- Probability of photoelectric interaction, P
- P ~ 1/E3
- P ~ Z3
- Need lower energy x-ray photons for differentiation
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X-ray energy transfer
Compton vs Photoelectric
HS34
- Typically more energy is transferred in a photoelectric interaction than in a Compton interaction
- Graph here
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CT Inventors
Year and Nobel Prize
- Allan M. Cormack, work 1955-1963, reconstruct attenuation coefficients, images disk w/ different materials, Tufts University
- Godfrey N. Hounsfield, work 1967-1971, first clinically available CT scan, EMI Labs, England
- Nobel prize 1979 (both), Physiology and Medicine
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5 CT scanner generations
- 1. Translate source and detector while scanning object at one given projection angle, rotate for new angle, repeat. One source. One detector cell.
- 2. Translate less, rotate. Possible due to larger detector, several cells. Povides more coverage.
- 3. Rotate both source and detector while scanning. Possible due to object fully covered in radiation "cone" and detector large enough to capture radiation.
- 4. Stationary detector, rotate source
- 5. Stationary detector, stationary source (Imatron)
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Mathematical formulation for reconstruction
HS7
- J. Radon, 1917, Austrian mathematician
- "Object could be replicated from an infinite set of its projections"
- Tomographic reconstruction problem = inverse Radon transform
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