Biomedical Imaging Computed Tomography Patrícia Figueiredo IST 2013-2014
Overview Basic principles X ray attenuation projection Slice selection and line projections Projection reconstruction Instrumentation Beam collimation Gas ionization chambers From the 1 st to the 4 th generation Spiral / Helical CT Multi-slice CT Image reconstruction The Radon transform: filtered backprojection Iterative reconstruction methods
Basic principles Image orientation: z x y
Basic principles X ray attenuation projection: Coronal image z x y
Basic principles X ray attenuation projection: Coronal image Computed tomography: (Trans)axial image x z y x z y
Basic principles X ray attenuation projection: { } ( z) I ( z) exp ( x z) I =, dx 0 µ { } ( y, z) I0 ( y, z) exp µ ( x, y z) I =, dx -The object consists of a distribution of attenuation coefficients. -The intensity of the detected X ray beam reflects the projection of the attenuation coefficients across the beam direction z µ 11 µ 12 µ 13 µ 14 µ 15 µ 16 I 0 (y,z) y µ 21 µ 22 µ 23 µ 24 µ 25 µ 26 µ 31 µ 32 µ 33 µ 34 µ 35 µ 36 µ 41 µ 42 µ 43 µ 44 µ 45 µ 46 µ 51 µ 52 µ 53 µ 54 µ 55 µ 56 µ 61 µ 62 µ 63 µ 64 µ 65 µ 66 I(y,z) x
Basic principles Line projections L 1 L{ θ 1 } L 2 L{ θ 2 } y p L { f ( x, y) } f x( l), y( l) L ( )dl z x
Basic principles Radon transform: Object space Projection space
Basic principles { } π Projection reconstruction: fˆ 1 ( r, θ ) = p ( xʹ ) h( xʹ ) dθ = R p ( xʹ ) 0 φ φ Object Image 1 ang. 2 ang. 4 ang. 8 ang. 16 ang. 32 ang.
Basic principles Radon transform:, p ( xʹ ) { f ( x, y) } R { f ( x y) } f ( x( l), y( l) )dl L φ R φ The sinogram Symmetry at π: p ( xʹ ) = p ( xʹ ) φ ± π φ Object space Projection space Periodicity at 2π: p ( xʹ ) = p ( xʹ ) φ + 2π φ f(x,y) R φ p φ (x ) φ φ 4 φ 3 θ φ 2 φ 0 φ 1 -r r x xʹ = rcos ( φ θ)
Image reconstruction Projection reconstruction 2D 3D Filtered backprojection (FB) Backprojection filtering (BF) True Three-Dimensional Reconstruction (TTR) Generalized TTR (GTTR) Parallel beam mode Fan beam mode Parallel beam mode Fan beam mode Parallel beam mode Cone beam mode Planar-Integral Projection Reconstruction (PPR) Iterative reconstruction Fourier reconstruction Algebraic Reconstruction Technique (ART) Maximum Likelihood (ML) or Expectation Maximization (EM) Direct Fourier Reconstruction (DFR) Direct Fourier Imaging (in MRI)
Image reconstruction Backprojection: effect of a finite number of projections Object Image 1 ang. 2 ang. 4 ang. 8 ang. 16 ang. 32 ang.
Image reconstruction Backprojection: effect of a finite number of projections streak artifacts
Image reconstruction Backprojection: coverage π ( r, φ) p ( xʹ ) h( x ) dφ fˆ ʹ = 0 φ y y x Symmetry at π: p ( xʹ ) p ( xʹ φ ± π = φ ) p ( xʹ ) = p ( xʹ ) Periodicity at 2π: y φ + 2π φ p θ (x ) L φ x r φ θ q x x f(x,y) Sampling requirements for φ: Coverage of a total scan angle of 180º, usually 360º to reduce partial volume effects.
Filtered back-projection π fˆ ʹ Filtering: simple backprojection (no filter) ( r, φ) pφ ( x ) dφ = 0
Filtered back-projection π fˆ ʹ ʹ Filtering: filtered backprojection ( r, φ) p ( x ) h( x ) dφ = 0 φ
Filtered back-projection Filtering: simple vs filtered backprojection π π fˆ ( r, φ) p ( xʹ ) dφ fˆ ( r, φ) p ( xʹ ) h( xʹ ) dφ = 0 φ = 0 φ
Filtered back-projection Filtering: effect of noise
Filtered back-projection CONTINUAR AQUI - MEBiom Filtering: effect of different filter functions
Image reconstruction Iterative reconstruction The ray-by-ray method: p m N = n = 1 W mn f n To estimate the value of the image cell f n from the projection data p m = to solve the inverse problem of M linear equations with N unknowns Iterative method: Mean square error (MSE) Expectation maximization (EM) Maximum likelihood (ML)
Image reconstruction Iterative reconstruction Jean-Baptiste Thibault et al., GE Medical Systems
Instrumentation
Instrumentation Beam collimators: 1 st collimator: width ~45º 2 nd collimator, perpendicular to 1 st : thickness ~1-5 mm Slice profile: -Δz/2 +Δz/2
Instrumentation 1 st generation systems Parallel beam 1 pencil beam source 1 detector translating together rotating together scanning time ~4-5 min
Instrumentation 2 nd generation systems Fan beam: equilinear geometry 1 thin fan beam source multiple detectors translating together rotating together But fewer rotations required: scanning time ~20 s
Instrumentation 3 rd generation systems Fan beam: equiangular geometry 1 wide (~30-45 ) fan beam source ~512-1000 detectors covering object no translations rotating together scanning time ~1-3 s Use pulsed X ray sources to take advantage of significant dead time
Instrumentation 4 th generation systems Fan beam: equiangular geometry 1 wide fan beam source complete ring of detectors source rotating detector ring stationary ~ scanning time No cumulative detector drift, But very expensive (BGO-PMT detectors)
Instrumentation 3 rd generation CT T = X ray tube D = Detectors X = X ray fan beam R = Rotation direction Typical parameters: kvp 140 kv E eff ma Pulse f 70-80 kev 70-320 ma 2-4 ms 0.6 1.6 mm Thickness 1-5 mm Matrix 512 1024 (Resolution~0.35 mm) Nb detectors~1000
Instrumentation X ray detectors: X rays must be converted into radiation accessible to human vision Type of X ray detectors: - itensifying screen + photographic emulsion - cassette of photostimulable phosphor + laser scanner - scintillation detectors - crystals: NaI (Tl), CsI(Tl), BGO coupled to a photo-multiplier tube (PMT) or a photodiode array (e.g. TFT) - gas ionizing detectors: - ionizing chamber, proportional counter, Geiger-Muller counter Main characteristics of X ray detectors : - Sensitivity - Efficiency - Linearity - Energy resolution - Dead time
Instrumentation Ionization chamber: X rays gas ionization electron-ion pairs electron/ions attracted to cathode/anode electric current amplifier Array of interlinked Xenon-filled (~1000) ionization chambers (Z Xe = 66, P = 20 atm) ~1mm X X - + Xe+ e- 10cm - spatial resolution ADC - simplicity - more compact X - efficiency - Role of antiscatter grid X
Instrumentation Scintillation detectors: X rays crystal excitation electron-hole pairs electron-hole pairs collected at p-n junctions electric current pre-amplifier X rays crystal excitation optical photons photocathode ionization photoelectrons electron multiplication electric current
Instrumentation Conventional CT configurations One slice at a time: time inefficient susceptible to artifacts due to motion between slices
Instrumentation Spiral / Helical CT Data are acquired as the patient table moves continuously along z, simultaneously with the source/detectors rotation, tracing out a spiral/helix for the X ray trajectory. Continuous scanning requirements: - source: high heat capacity and efficient cooling - detectors: high efficiency Only one projection is acquired exactly in the image plane. All other projections have to be interpolated.
Instrumentation Spiral / Helical CT Spiral pitch: p = d / S p d = table feed per rotation S = collimated slice thickness A. p<1: slice overlap higher dose B. p>2: slice gaps lower resolution blurring 1<p<2: typical values
Instrumentation Multislice CT An array of detectors is incorporated along z: Spiral pitch: p ms = d / S single d = table feed per rotation S single = single slice collimated beam width 4-slice: p ms <8 8-slice: p ms <16 16-slice : p ms <32
Instrumentation Multislice CT - Multislice helical scans produce a set of interleaved helices interpolation is (even) more difficult to visualize - Images are reconstructed at optimized oblique planes and are then filtered to produce axial images.
Instrumentation Multislice CT - Multislice helical scans produce a set of interleaved helices interpolation is (even) more difficult to visualize - Images are reconstructed at optimized oblique planes and are then filtered to produce axial images.
Instrumentation Multislice vs single-slice CT Advantages: - Same acquisition in shorter time - Larger volumes in same time - Thinner slices: better spatial resolution - Can get isotropic volumes Disadvantages: -Larger beam width (relative to slice) -Higher dose for same quality -Cone beam artefacts
Instrumentation
Image characteristics Estimated object function CT numbers (Hounsfield units, HU): CT ij µ ij µ H = µ H 2 O 2 O 1000
Image characteristics Dosimetric quantities CT Dose Index: D z is absorbed dose at position z T is slice thickness CTDI = 1 T + 7T 7T D z dz Effective dose:
Image characteristics Spatial resolution: X ray tube effective focal spotsize f (~0.6-1.6mm) Scanner (translation and) rotation steps (~512x512-1024x1024: ~0.35x0.35mm 2 ) Collimated single-slice thickness (~0.5-5mm) and table feed Signal to noise ratio (SNR): - X ray tube voltage (~140kV): kvp SNR - X ray tube current and exposure time (~2-4ms): ma s SNR - X ray filtration (effective energy ~70-80 kev): filtration SNR Contrast to noise ratio (CNR): - X ray energy: E I scatt /I primary CNR - Object size (thickness): thickness I scatt /I primary CNR - Field-of-view: FOV I scatt CNR - Artefacts!
Image characteristics Artefacts: - Streak artefacts: undersampling due to finite number of projections (interaction owith motion, beam hardening or scatter) increase nb rotation steps / decrease rotation step. - Ring artefacts: imbalances in detector sensitivity calibration is performed using spatially uniform test objects. - Beam hardening: results in more attenuation in the center of the object than around the edge, but algorithms assume monochromatic X ray beams and hence uniform attenuation coefficient X ray beam filtering; calibration; reconstruction corrections. - Partial volume effects: thick slices can include, and mix up, different tissue types decrease slice thickness (using multislice systems).
Image characteristics Streak artefacts: Barrett J F, and Keat N Radiographics 2004;24:1679-1691
Image characteristics Ring artefacts: when a detector is out of calibration: Barrett J F, and Keat N Radiographics 2004;24:1679-1691
Image characteristics Beam hardening effects: Barrett J F, and Keat N Radiographics 2004;24:1679-1691
Image characteristics Beam hardening effects Monoenergetic X rays: Polichromatic X rays: Beam hardening: I i = I i 0 exp µ ij j I i0 = Ii0 i i0 µ ij 0 j Ema x ( E) I = I ( E) exp de µ ij = µ ij i 0i exp µ ij 0 j Ema x ( E) I = I ( E) ( E) de Non-linear relation between p and µ artefacts p i = ln I I i i0 I i = ln 0 E max I 0i ( E) exp µ ( E) 0 E max I 0i j ( ) E de ij de
Image characteristics Beam hardening effects - minimized by: Filtration: a flat piece of attenuating material is used to pre-harden the beam before it passes through the patient (so that it becomes closer to monochromatic). Monochromatic X ray Polychromatic X ray Barrett J F, and Keat N Radiographics 2004;24:1679-1691
Image characteristics Beam hardening effects - minimized by: Calibration correction: using phantoms in a range of sizes. Uncalibrated Calibrated Barrett J F, and Keat N Radiographics 2004;24:1679-1691
Image characteristics Beam hardening effects - minimized by: Reconstruction: an iterative correction algorithm may be applied when images of bony regions are being reconstructed. Uncorrected Corrected Uncorrected Corrected Barrett J F, and Keat N Radiographics 2004;24:1679-1691
Image characteristics Partial volume effects Thick slice Thin slice Barrett J F, and Keat N Radiographics 2004;24:1679-1691
Image characteristics Partial volume effects: multi-slice vs single-slice CT
Image characteristics Scout image for CT Head CT
References Webb, Introduction to Biomedical Imaging, Wiley 2003. Cho, Foundations of Medical Imaging, Wiley 1993. Hendee, Medical Imaging Physics, Wiley 2002.