程序代写 BMEN90021, Lecture set 4: CT

X-ray Computed Tomography (CT)
• X-rayimagesproduceplanar(2-d) projections of objects
• Difficult (if not impossible) to determine the depth information within the object from the x-ray.
• eg. Spinal column & jaw are superimposed

• In contrast to x-rays:
• CT reconstructs the object information, with depth, from collections of cross-sectional projections.
1 BMEN90021, Lecture set 4: CT

Imaging directions
Axial (transverse) Coronal Sagittal
BMEN90021, Lecture set 4: CT

CT: the main idea
• Thinx-raybeamsare transmitted covering a field-of-view.
• Process is repeated at range of angles with respect to the object.
• Basedonthesetof
cross-sectional measurements, reconstruct the actual attenuation at each point in the scanned slice.
3 BMEN90021, Lecture set 4: CT

Tomography
• TomographyfromtheGreekwords: • “Slice”
• “To write”
• MRI,PET,SPECTarealso“tomographic” techniques, but:
• The term “Computed Tomography” is reserved for x-ray computed tomography.
4 BMEN90021, Lecture set 4: CT

First CT image
• 1 October 1971
• Frontal lobe tumour,
including bleeding
BMEN90021, Lecture set 4: CT

X-rays vs CT: cancer
BMEN90021, Lecture set 4: CT

X-Ray vs CT: pneumothorax
Normal x-ray
Pneumothorax X-ray CT
BMEN90021, Lecture set 4: CT

CT applications: tumours
BMEN90021, Lecture set 4: CT

CT applications: bleeding
Subdural haemotoma due to traumatic brain injury
BMEN90021, Lecture set 4: CT

CT: Completely RELATED pic
BMEN90021, Lecture set 4: CT

History of CT

Starts at EMI works in radar then computing
discovers x-rays
The Beatles
sign to EMI

reconstruction of function from projections
EMI sells computing division, keeps Hounsfield on, funded by Beatles money.
The Beatles
make squillions

Builds first CT machine (brain)
“EMI scanner”
mathematics of CT
First helical scanner
Cormack & Hounsfield
win in Medicine
BMEN90021, Lecture set 4: CT
The Beatles
Multi-Slice CT

• https://www.youtube.com/watch?v=cjtHNxf01tQ
• C:\Users\kstok\OneDrive-TheUniversityof Melbourne\Documents\Work\Abstracts & Presentations\Presentations\uCT40 animation
BMEN90021, Lecture set 4: CT

CT output image information
• Basedonthesetofcross-sectionalmeasurements, reconstruct the actual attenuation at each point in the scanned slice.
• That is, construct a matrix of image intensities, where each pixel has an intensity = “CT number” in Hounsfield units (HU):
• μ: Linear attenuation coefficient
• CT number of air = -1000
• CT number of water = 0
• CT number of bone = wide range, 100’s to > 1000 HU.
13 BMEN90021, Lecture set 4: CT

Parallel beam coordinate system
• X-ray beams make angle θ with y-axis
• Define new coordinate system (r,s):
14 BMEN90021, Lecture set 4: CT

The intensity profile
Measured intensities along r
15 BMEN90021, Lecture set 4: CT

More accurate intensity profile
• Rememberfromx-raysthatthereisneverjustone x-ray energy in the beam.
• Should integrate over energy distribution, σ(E): Practically, common assumption is that the beam is
“monochromatic” = one energy:
Q: What if we divide by I0 and take the logarithm?
16 BMEN90021, Lecture set 4: CT

The attenuation profile
17 BMEN90021, Lecture set 4: CT

Sinogram & Radon transform • Attenuationprofiles,pθ(r),canbemeasuredfor
angles ranging from 0 to 2π. • Actually only need 0 to π.
Each pθ(r) is a row of the sinogram. • Stackallpθ(r)’sintoa2-ddataset=sinogram.
• Mathematically,thetransformationofafunction f(x,y) into its sinogram p(r,θ) is called the…
• Radon transform:
18 BMEN90021, Lecture set 4: CT

Backprojection
Q. How can we reconstruct the 2-d image of attenuation coefficients from the sinogram?
A. Assign the profile values pθ(r) to all points x & y along
Backprojection of 4 views
Surface views
those beam lines.
Original object of attenuation coefficients
Complete backprojected image Note it is BLURRED
BMEN90021, Lecture set 4: CT

Same diagrams, different book…
Backprojection of a dot object
Intensity profile formation
(a) Dot object & FOV, with 4 views
(b) Backprojection using P1 and P3
(c) Backprojection using P1 – P4
(d) Backprojection using an infinite number of angles.
20 BMEN90021, Lecture set 4: CT

Backprojection
BMEN90021, Lecture set 4: CT

• Inreality,thereisalimitednumberofdetectors
number of projections (views of the object) number of detectors
M = number of different angles from which the object is imaged.
N = number of samples along r
22 BMEN90021, Lecture set 4: CT

Sampling: Discrete sinogram
• Thediscretesinogramisamatrixofintensityvalues = digital image.
23 BMEN90021, Lecture set 4: CT

Minimum number of detectors
• Rememberthatpθ(r)iszerooutside|r|>FOV/2.
• Ifx-raybeamshavewidthΔs,thencancalculate minimum number of detectors required.
24 BMEN90021, Lecture set 4: CT

Implication of x-ray beam width An attenuation profile Its Fourier transform
spatial frequency variable
Profile of x-ray beam Its Fourier transform
25 BMEN90021, Lecture set 4: CT

Implication of x-ray beam width
Smoothed projection resulting from convolution of attenuation profile with beam block.
Pulse train of samples along r.
Multiplication of (e) and (g) gives sampled signal.
Fourier transforms
BMEN90021, Lecture set 4: CT
Reduced high frequency content
Reciprocal pulse train
Convolution of (f) and (h): aliasing is present.

• Wantcopiesof(f)tobeatleastseparatedsuchthat the first zero-crossing coincide in (j):
i.e. Two samples per beam width are required = NYQUIST RATE
Coloured individually to see the aliasing.
Width of main lobe = 2/Δs
BMEN90021, Lecture set 4: CT

The Nyquist criterion • FromSuetensAppendixA:
Spatial frequency spectrum
If bandlimited
Simulated bowl of water and iron rod
Reconstruction after noise added to sinogram.
Aliasing artifact due to insufficient detectors.
BMEN90021, Lecture set 4: CT

Nyquist & aliasing (1)
29 BMEN90021, Lecture set 4: CT

Nyquist & aliasing (2)
Q. What is the spatial extent of a patient?
30 BMEN90021, Lecture set 4: CT

Bandlimited patient (object)
• Spatialextentofapatientisalwayslimited.
• Therefore aliasing unavoidable to a certain extent.
• Practicalsituation:
• Field-of-view = 50cm
• Beam width = 1mm
• Requires ~1000 detector channels.
• GE: 888 detectors, 984 views per 360 ̊.
• Siemens: 768 detectors, 1056 views per 360 ̊.
31 BMEN90021, Lecture set 4: CT

Back to backprojection…
• Remembertheprocessofbackprojection:
• Sampledversion:
32 BMEN90021, Lecture set 4: CT

The need for interpolation
• Ingeneral,thepoints are not the discrete grid
points rn.
• Thereforeinterpolation is required.
33 BMEN90021, Lecture set 4: CT

Goal of CT reconstruction
• Ourgoalistoanswerthequestion“Giventhe
sinogram, p(r,θ), what is the original function, f(x,y)?”
• We need a mathematical expression for the inverse Radon
transform,
The Projection Theorem = Central Slice Theorem answers this question.
34 BMEN90021, Lecture set 4: CT

Projection Theorem
• Considerthe2-dFourierTransformoff(x,y):
and consider the 1-d Fourier Transform of pθ(r): • For variable θ, Pθ(k) becomes P(k,θ):
35 BMEN90021, Lecture set 4: CT

Direct Fourier reconstruction • BasedontheProjectionTheorem:
1. Calculate the 1-d FT of all projections:
2. Plot all Pθ(k) on a polar grid, then interpolate to rectangular coordinates, Fθ(kx,ky).
3. Calculate the 2-d IFT of Fθ(kx,ky):
BMEN90021, Lecture set 4: CT
Interpolation causes artifacts, therefore use filtered backprojection.

Filtered Backprojection (FBP)
• UsepolarversionofinverseFouriertransform:
• Define: • Then:
Q: Where does the |k| come from?
37 BMEN90021, Lecture set 4: CT

FBP derivation
2D Fourier Transform w.r.t x and y
2D Inverse Fourier Transform w.r.t kx and ky
Projection Theorem
Polar coordinates defined as
BMEN90021, Lecture set 4: CT

k dk dθ becomes |k| dk dθ to ensure we are still multiplying by the distance to the origin.
FBP derivation
Polar coordinates defined as
Note: k is the distance from the origin. Because we are acquiring over
the limits of k must be redefined as so that we integrate over all space.
BMEN90021, Lecture set 4: CT

Steps in FBP
1. Filter the sinogram, p(r,θ):
2. Backproject the filtered sinogram:
(a) Ram-Lak filter (b) (c)
BMEN90021, Lecture set 4: CT
Q: How to create Ram- Lak from b) and c)?

Improvements to Ram-Lak
• Ram-Lakrampfilteramplifieshigh- frequency noise.
• Therefore best to use a low-pass filter to suppress highest frequencies. Multiply by a filter H(k):
BMEN90021, Lecture set 4: CT

Fan-beam FBP
• Alternativetoparallel-beamCT,usedinmore modern scanners.
β: Angle between y-axis and beam centre-line
θ: Angle between line through (x,y) and y-axis
Υ: Angle between line through (x,y) and beam
centre-line
Q: Why is an angle range of 0 ↔ (π+ fan-angle) required to capture all projection lines?
BMEN90021, Lecture set 4: CT

Fan-beam reconstruction • 2methods
1. Rebinning (reordering) data into parallel beams and solving as for parallel-beam FBP.
• Requires interpolation
2. Adapt the FBP equation to suit the fan-beam geometry. After some algebra…
43 BMEN90021, Lecture set 4: CT

Imaging in 3-d: Circular CT
• Usecirculartube-detectorrotations,andshiftthe
table position to discrete locations
• = AXIAL scanning
• Requires two slices per slice thickness to minimise aliasing.
44 BMEN90021, Lecture set 4: CT

Imaging in 3-d: Helical CT (early 1990s)
• X-raytuberotatescontinuouslyaroundpatientwhile
table shifts.
• Tube maps out a “screw” shape with respect to the patient = spiral or helix
• “Table feed” (TF) = axial distance table moves to map out a full rotation of the x- ray tube.
• To avoid aliasing, TF = 1/2 * slice thickness.
45 BMEN90021, Lecture set 4: CT

Imaging in 3-d: Multi-slice CT • Inmodernscanners,multiplerowsofdetectors
replace the single row.
• Can acquire multiple projections at a single view/angle.
• Still acquire with helical trajectory.
• For small number of rows (e.g. 4), assume beams are parallel in z-direction
• For large number of rows (e.g. 16), tilted slices are reconstructed, then axial slices are interpolated.
• For very large number of rows (e.g. 64), 3D reconstruction techniques required.
46 BMEN90021, Lecture set 4: CT

Imaging in 3-d: Multi-slice CT
• Pitch = TF / total width of stack of slices
• Not just single beam
• Can acquire full image in single rotation
• Reduced scan time
• Operatorcanspecifyslicesthickerthan single detector width
• Combine multiple detector rows via smoothing operation
• Raises SNR
47 BMEN90021, Lecture set 4: CT

Imaging in 3-d: Volumetric CT • Exploitsthemultipledetectors
• Can even acquire a full volume from a single view
• Circular cone-beam geometry
• Helical cone-beam geometry
• Each geometry has its own 3-d reconstruction algorithm
• 3-d version of the Projection theorem
• Variants of FBP
48 BMEN90021, Lecture set 4: CT

• Diagramofstandardmulti-sliceCT:
Q: What is the purpose of the bow-tie filter?
BMEN90021, Lecture set 4: CT

Bow tie filter
• Ensureconsistentenergyspectrumreachingthe detectors
• PeripheryofFOV:
• Patient less thick = less attenuation
• Therefore do not require as many low energy x-rays
• Reducespatientdosage.
BMEN90021, Lecture set 4: CT

Equipment: Scanner generations First generation
Fan beam Multiple detectors Reduced scanning time
Increased number of detectors (~1000)
Wider fan beam
Intense pulses (2-4ms) per view Spiral / helical possible
Detectors in 360 deg ring (~5000) Only x-ray tube rotates
No real decrease in scan time compared to 3rd generation.
BMEN90021, Lecture set 4: CT

Cone-beam CT
• higherspatialresolution
• lowradiationdose
• fastacquisition
• largefieldofview,highversatility
Currently limited to qualitative analyses
• Lowimagecontrast
• Notcalibratedtobonedensity
52 BMEN90021, Lecture set 4: CT

High resolution peripheral quantitative CT
• ClinicalCTforhigher-resolutionimagingofperipheral structures (hands, wrists, feet)
• Maximum sample diameter 140 mm
• resolution 60 – 80 μm
• Calibrated & quantitative
Klingberg, E, et al, Arthritis research & therapy, 2013 BMEN90021, Lecture set 4: CT

• SimilarprinciplesasclinicalCT,but
• smaller scale objects
• greatly increased resolution (0.5 nm – 50 μm)
• Cabinet CT: sample rotates • Quantitative
Mouse kidney,
Zagorchev et al, J Angiog Res, 2010
54 BMEN90021, Lecture set 4: CT

CT imaging
HRpQCT-1 HRpQCT-2
cone-beam CT; cone-beam CT; manufacturer’s software in-house software
Mys et al, Bone, 2018 Mys et al, JBMR, 2019

Cone-beam CT imaging
Manufacturer
Adapted reconstruction Adapted reconstruction and beam hardening correction
Mys et al, Bone, 2018

Weight-bearing CT
• Cone-beamCT
• Imaginginaweight-bearing position
• Accurate position of bones & joints relative to each other
• Alignment
57 BMEN90021, Lecture set 4: CT

Photon-counting CT
• Photon-countingdetectors(directradiography)directly transform X‐ray photons into electrical signals, rather than solid-state scintillation (indirect).
• Improved spatial resolution
• No increase in dose
• Improves contrast
• Multi-energy information
https://www.siemens-healthineers.com/computed- tomography/technologies-and-innovations/photon-counting-ct
58 BMEN90021, Lecture set 4: CT

Image quality: Resolution
• Resolutiondependson
• x-ray focal spot (size where electrons hit anode)
• size of detector channels
• rotation of tube-detectors, producing blur
• reconstruction kernel in FBP
• any interpolation processes
59 BMEN90021, Lecture set 4: CT

Image quality: Noise
• Noiseisdominatedbyquantumnoise=statistical nature of x-ray production.
• Totalnoisedependson:
• Total exposure (up the dose, increase SNR)
• Reconstruction algorithm
• Turns measured signal noise into structured image noise.
• In metal objects: dark & bright streaks radiating out
Simulated water bowl and metal rod Quantum noise added to measured signal
60 BMEN90021, Lecture set 4: CT

Image quality: Artifacts • Undersampling
• Too few detectors causes a sharp edge in the projection to be blurred, which backprojects artifact into reconstructed image.
• Too few views causes alternating dark and bright streaks in peripheral part of image where sampling density smallest
61 BMEN90021, Lecture set 4: CT

Image quality: Artifacts
• Beamhardening
• Low-energy x-rays are absorbed as beam goes through tissue.
• The harder the beam, the less it is further attenuated.
• Beam hardening causes streaks connecting highly attenuated areas.
Artifact-free reconstruction of plexiglass plate & 3 amalgam fillings
BMEN90021, Lecture set 4: CT

Image quality: Artifacts
• Up to 30% of detected x-rays
due to scatter.
• Presence of scatter causes underestimation of attenuation.
• Scatter causes streaks artifacts:
• The larger the attenuation, the smaller the measured intensity, therefore the larger the effect due to scatter.
Artifact-free reconstruction of plexiglass plate & 3 amalgam fillings
BMEN90021, Lecture set 4: CT

Image quality: Artifacts • Motion
• A short, sharp movement of object while being imaged causes two streaks: one from object to position of tube when movement started, the other from object to position of movement when object stopped.
• Cardiac motion etc result in blurring of moving parts.
• Metalstreakartifact
Presence of metal implants causes disruption to x-ray transmission:
• Beam hardening
• Partial voluming • Noise
64 BMEN90021, Lecture set 4: CT

Image quality: Artifacts
• Stairstepartefact
• In helical CT
• Pitch is too large (i.e. not enough sampling in z)
• Windmillartefact
• Helical cone-beam CT
• z-aliasing due to interpolation between detector rows from cone-beam projections.
Boas & Fleischmann
BMEN90021, Lecture set 4: CT

Biological effects & safety
• RadiationdosesrelativelyhighinCT
• 10-100 times higher than for x-ray
• eg. Head CT: 1-2 mSv each
• eg. Chest CT: 5-8 mSv each
• Mustbeextracarefultolimitdose
Correct use of equipment
Low tube current
• Modulated tube current : lower current for smaller FOVs
Limited scan range
Maintaining equipment in optimal condition
• Use phantoms to calibrate daily
66 BMEN90021, Lecture set 4: CT

Biological effects & safety
• CTdoseindex(CTDI)
• Dose absorbed by standard acrylic phantom for 360° rotation.
• Varies across the image plane, higher in periphery:
• For helical scanning:
CTDI’s are independent of scan length. Therefore introduce “Dose length product” (DLP)
67 BMEN90021, Lecture set 4: CT

Dose examples
NOTE: Values all refer to standard phantoms! They do not distinguish between patients.
68 BMEN90021, Lecture set 4: CT

Example: Cardiac CT
• Howtoimageabeatingheartin3-d?
• Select phase of heart with ECG signal
• Collect sufficient data for each z-value (Table feed or “pitch”)
• eg. 128 row system with 0.5mm detectors, 0.33 second rotation time, heart rate of 60bpm, 20cm can be imaged in 4 seconds.
69 BMEN90021, Lecture set 4: CT

CT: foreign body diagnosis
Patient presented 20 days post accident (nail gun) with double vision, but no neurological effects. Was sure nail fell down after hitting nose.
X-Ray confirmed presence of foreign body.
CT provided detailed positional information required for surgical planning.
A, Pohchi A, K, Athar Y, Shiekh RA. An intraorbital metallic foreign body. Indian J Ophthalmol [serial online] 2014 [cited 2016 Mar 20];62:1098-100. Available from: http://www.ijo.in/text.asp?2014/62/11/109 8/146756
BMEN90021, Lecture set 4: CT

CT: foreign body
BMEN90021, Lecture set 4: CT

Bone structure – femoral head
• YoungNormal Osteoporotic
BMEN90021, Lecture set 4: CT

Bone structure – lumbar spine
• YoungNormal Osteoporotic
BMEN90021, Lecture set 4: CT

CT: image processing & analysis
• Bone(structure)analysesandvisualizationareofgreat importance in
• Diagnosis • Treatment • Prevention
2947 N / 282 mm2
6212 N / 397 mm2
Von Mises Stress [MPa]
6358 N / 493 mm2
BMEN90021, Lecture set 4: CT
1% Applied Strain

CT: image processing & analysis • Imageprocessingcanquantify
Trabecular thickness, number, separation Cortical thickness
Bone volume fraction
Tb.Th Trabecular thickness
Tb.Sp Trabecular Separation
Tb.N Trabecular Number
BV/TV Bone volume ratio
BMEN90021, Lecture set 4: CT

Distance Transform
BMEN90021, Lecture set 4: CT

Direct measures
DT Object DT Background
BMEN90021, Lecture set 4: CT

CT: image processing & analysis
• Imageprocessingcanquantify • Joint alterations
BMEN90021, Lecture set 4: CT

CT: image processing & analysis
• Imageprocessingcanquantify
• Joint alterations
• Cartilage composition
Contrast agent CA4+
Negatively charged GAGs
Cationic contrast agent
BMEN90021, Lecture set 4: CT

BMEN90021, Lecture set 4: CT

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