MPH5016 Computed Tomography Study Notes

Principles of Computerized Tomography (CT)

• Conceptual Overview: The human body is visualized as a finite number of discrete slices with find thickness. Each slice is a collection of discrete, contiguous cubic volume elements known as voxels.
• Scanning Orientations:
  • Transverse (Axial): Standard imaging plane, representing the $x$-$y$ plane.
  • Z-axis: Perpendicular to the scanning/image plane, typically parallel to the body's longitudinal axis.
  • Sagittal: Approximated by the $y$-$z$ plane.
  • Coronal: Approximated by the $x$-$z$ plane.
• Voxel Dimensions: The edge length of a voxel is determined by the matrix size, field of view (FOV), and slice thickness.
• Contrast in CT: CT is a digital modality with higher contrast than plain radiographs. In plain radiography, voxels along a line are summed (integrated), leading to low local contrast due to the superposition of structures (e.g., bone over soft tissue). In CT, contrast is given directly by the attenuation values of neighboring voxels, not line integrals through the entire object.
  • Contrast Example: A CT image might show soft tissue structures with $50\%$ contrast ($\Delta \text{CT} = 28\,\text{HU}$), whereas an X-ray might show the same at $0.23\%$ contrast.

Mathematical Basis of CT Measurement

• Intensity Calculation: For monochromatic rays in a homogeneous medium:
  • $I = I_0 \cdot e^{-\mu \cdot d}$
  • $P = \ln\left(\frac{I_0}{I}\right) = \mu \cdot d$
  • $\mu = \frac{1}{d} \cdot \ln\left(\frac{I_0}{I}\right)$
  • Where $I$ is attenuated intensity, $I_0$ is primary intensity, $\mu$ is the linear attenuation coefficient, and $d$ is absorber thickness.
• Inhomogeneous Objects: Attenuation depends on the local value $\mu_i$ for increments $d_i$, requiring line integrals:
  • $I = I_0 \cdot e^{-\int_0^{d} \mu \, ds}$
  • $P = \ln\left(\frac{I_0}{I}\right) = \sum \mu_i \cdot d_i$
• Polychromatic Rays: The linear attenuation coefficient depends strongly on X-ray energy ($E$). Measurement requires integration over energy intervals:
  • $I = \int_0^{E_{\text{max}}} I_0(E) \cdot e^{-\int_0^{d} \mu(E)ds} \, dE$
• Temporal and Physiological Factors: $\mu$ may also depend on time ($t$), particularly with contrast media or lung volume changes (inspiration/expiration). The final measured quantity is given as $\mu(x, y, z, E, t)$.

Hounsfield Unit (HU) Scale and Display

• HU Definition: CT numbers quantify the spatial distribution of the linear attenuation coefficient relative to water. They are defined as:
  • $\text{CT Value} = \frac{\mu_T - \mu_{\text{water}}}{\mu_{\text{water}}} \times 1000\,\text{HU}$
• Fixed Points: Water is defined as $0\,\text{HU}$, and Air is $1000\,\text{HU}$ at all X-ray energies.
• HU Calculation Example: If $\mu_{\text{tissue}} = 0.05\,\text{cm}^{-1}$ and $\mu_{\text{water}} = 0.10\,\text{cm}^{-1}$, the HU is $\frac{0.05 - 0.10}{0.10} \times 1000 = -500\,\text{HU}$.
• Typical HU Values:
  • Compact Bone: $+1000$ ($+300$ to $+2500$)
  • Liver: $+60$
  • Blood: $+55$
  • Grey Matter: $+35$
  • White Matter: $+25$
  • Muscle: $+25$
  • Water: $0$
  • Fat: $-90$ (due to lower density $0.96\,\text{g/cm}^3$ and $Z_{\text{eff}} \approx 5.88$)
  • Lung: $-750$
  • Air: $-1000$
• HU Constraints: The scale typically ranges from $-1024$ to $+3071\,\text{HU}$ (4096 steps, requiring 12-bit depth).
• Windowing and Levelling:
  • Windowing: Adjusts the range of CT numbers assigned to grayscale.
  • Levelling: Changes the center of that range.
  • Formulas:
    • $\text{Upper Grey Level} = \text{Window Level (WL)} + \frac{\text{Window Width (WW)}}{2}$
    • $\text{Lower Grey Level} = \text{WL} - \frac{\text{WW}}{2}$
  • Clinical Window Settings:
    • Brain: WW 80, WL 40
    • Lung: WW 2000, WL -200
    • Liver: WW 150, WL 30
    • Bone (Spine): WW 1800, WL 400

Evolution and Generations of CT Systems

• 1st Generation (Translate-Rotate): Pencil beam with a single detector. Translations produced intensity profiles; $180$ projections at $1^\circ$ increments with $160$ data points each. Scan time: 5 minutes.
• 2nd Generation (Translate-Rotate): Small fan beam with multiple detectors. Faster than 1st generation.
• 3rd Generation (Rotate-Rotate): Wide fan beam and large detector arc. Eliminates translation. Scan times reduced to 20 seconds (1976) and later to sub-second.
• 4th Generation (Stationary-Rotate): Stationary ring of detectors fully encircling the patient with a rotating X-ray tube. Less common than 3rd generation due to cost/limitations.
• Continuous Rotation and Slip-Ring Technology: Replaced cables with concentric metal bands and gliding contacts, enabling helical (spiral) scanning and reducing scan times to $1$ second or less.
• Multi-Detector CT (MDCT): Multiple detector rows (e.g., 4, 16, 64, 128, 256, 320 slices) allowing for fast volume scanning. Introduction of 4-slice scanners in 1998 reduced rotation time to $500\,\text{ms}$.
• Performance Metrics Over Time (1972 vs 2010):
  • Rotation time: $300\,\text{s}$ to $0.27-0.35\,\text{s}$.
  • Spatial Resolution: $3\,\text{lp/cm}$ to $12-25\,\text{lp/cm}$.
  • Data per $360^\circ$ scan: $57.6\,\text{kB}$ to $0.1-1.0\,\text{GB}$.
  • Power: $2\,\text{kW}$ to $80-120\,\text{kW}$.

Scanner Hardware and Component Geometry

• Isocenter: The center of gantry rotation and reconstructed images (e.g., pixel 256,256 on a 512x512 matrix). FOV size is typically $50\,\text{cm}$.
• Magnification ($M$): $M = \frac{\text{Source to Detector (B)}}{\text{Source to Isocenter (A)}}$. If $B=95\,\text{cm}$ and $A=50\,\text{cm}$, $M=1.9$. This affects detector width (e.g., $1.9 \times 0.5\,\text{mm} = 0.95\,\text{mm}$).
• X-Ray Tube Alignment: Mounted with the anode disk parallel to the rotation plane to reduce gyroscopic effects. The anode-cathode axis is parallel to the $z$-axis, eliminating heel-effect spectral changes along the fan angle.
• Anti-Scatter Grid: Focused on the X-ray source. 2D grids are required for multi-slice CT.
• Collimators:
  1. Fixed housing collimator (rough definition).
  2. Fixed collimator (fine precision).
  3. Adjustable pre-patient collimator (variable slice width).
  4. Adjustable detector-side collimator (reduces penumbra for thin slices).
  5. Anti-scatter grid (minimizes scatter).
• Bow-tie Filters: Beam-shaping filters that reduce intensity at the periphery where the patient is thinner. They equalize dose and reduce noise variance across the detector. Using a head bow-tie for a body scan causes over-concentration of dose towards the center.

Detector Technologies

• Xenon Detectors: High-pressure xenon chambers with tungsten plates. Charges produced by ionizing xenon are collected at $500\,\text{V}$. Advantages: stable, fast decay, low afterglow. Disadvantage: Quantum Detection Efficiency (QDE) is only $60-70\%$; difficult to construct for MDCT.
• Scintillation Detectors: Ceramic phosphors (e.g., $\text{Gd}_2\text{O}_2\text{S}$) coupled with photodiodes. Energy-integrating detectors (EID). GDE is approximately $80\%$ due to collimation plates (e.g., $0.2\,\text{mm}$ thick Tungsten plates between $1\,\text{mm}$ cells).
• Photon Counting Detectors (PCD): Direct-conversion semiconductor (e.g., Cadmium Telluride). Bins photons by energy thresholds. Eliminates electronic noise.
  • Challenges:
    • Cross talk: Fluorescent X-rays or Compton scatter hitting neighboring cells.
    • Charge Sharing: Clouds of charge extending across cell borders, leading to incorrect energy registration and double-counting.
    • Pulse Pile-up: Occurs at high count rates (hundreds of millions per $\text{mm}^2$); pulses overlap, appearing as a single high-energy pulse.

Image Reconstruction Algorithms

• Sinogram: The raw data map where the $x$-axis is the attenuation profile and the $y$-axis is angulation.
• Back Projection: "Smearing" projections back onto the matrix. Simple back projection causes $1/r$ blurring.
• Filtered Back Projection (FBP): Applies a convolution kernel ($h(x)$) to projections before back projection to sharpen the image. Standard for decades.
• Iterative Reconstruction (IR): Numerically intensive process using update loops. An initial "guess" image is forward-projected and compared to measured data to create an error matrix for correction. IR yields higher Signal-to-Noise Ratio (SNR) and allows dose reduction.
  • Vendors: GE (ASiR, Veo), Siemens (SAFIRE, ADMIRE), Philips (iDose), Canon (AIDR).
• Reconstruction Kernels: Soft (B10-B20 for noise reduction), Medium (B30-B50 for routine), Sharp (B60-B70 for bone/lung), Ultra-Sharp (B80-B90 for inner ear).

Scan Acquisition Modes

• Scout/Survey Radiograph: Tube is stationary while the table moves. Used for planning and Automatic Exposure Control (AEC/Tube Current Modulation).
• Axial (Step-and-shoot): Table is stationary during rotation. Discontinuous table motion increases scan time. Potential for beam overlap and increased dose at slice edges.
• Helical (Spiral): Table moves at constant speed during continuous rotation.
  • Pitch: $\text{Pitch} = \frac{\text{Table feed per rotation}}{nT}$, where $nT$ is nominal beam width.
    • Pitch $< 1$: Overscanning; higher dose (e.g., Cardiac).
    • Pitch $= 1$: Contiguous scanning.
    • Pitch $> 1$: Underscanning; lower dose.
• Cone Beam CT (CBCT): Required for MDCT with many rows where beam divergence is non-negligible ($\text{Delta } z > T$). Requires 3D reconstruction.

Radiation Dosimetry in CT

• MSAD (Multiple Scan Average Dose): Estimate for multiple slice examinations.
• CTDI (Computed Tomography Dose Index):
  • $\text{CTDI}_{100} = \frac{1}{NT} \int_{-50\,\text{mm}}^{+50\,\text{mm}} D(z) \, dz$ (Measured with a $100\,\text{mm}$ ionization chamber).
  • Weighted CTDI ($\text{CTDI}_w$): $\frac{1}{3} \text{CTDI}_{100,c} + \frac{2}{3} \text{CTDI}_{100,p}$ (c=center, p=peripheral).
  • Volume CTDI ($\text{CTDI}_{\text{vol}}$): $\frac{\text{CTDI}_w}{\text{pitch}}$ (units: mGy).
• DLP (Dose Length Product): $\text{DLP} = \text{CTDI}_{\text{vol}} \times L$ (units: mGy\u00b7cm), where $L$ is scan length.
• Size-Specific Dose Estimate (SSDE): Modifies $\text{CTDI}_{\text{vol}}$ using conversion factors ($k$) based on patient dimensions (AP, Lateral) or Water Equivalent Diameter ($D_w$).
  • Water Equivalent Diameter ($D_w$): Accounts for both geometry and density: $D_w = 2 \sqrt{\frac{A_{\text{ROI}}}{\pi} \times \left[ \frac{\text{MPV}_{\text{ROI}}}{1000} + 1 \right]}$.
• Tube Current Modulation (TCM): AEC adjusting mA along the $z$-axis (longitudinal) or $x$-$y$ plane (angular) to maintain consistent image noise.

Image Quality and Quality Control (QC)

• Spatial Resolution: Ability to resolve small features.
  • In Plane: Affected by focal spot, detector width, and kernel. Typically $0.5\,\text{mm}$.
  • Z-axis: Determined by slice width.
  • Measurement: Evaluated via Point Spread Function (PSF), Full Width at Half Maximum (FWHM), and Modulation Transfer Function (MTF).
• Low Contrast Detectability: Ability to distinguish objects with similar $\mu$ values. Limited by noise ($\sigma \propto \sqrt{N}$ where $N$ is photon count).
• Routine QC Measures:
  • Tube Warmup: Daily to prevent target cracking.
  • Air Calibration: Daily to correct detector sensitivity differences (prevents ring artefacts).
  • Water CT Number: Tolerance $0 \pm 4\,\text{HU}$.
  • Uniformity: Difference between center and edge ROIs in water phantom ($0 \pm 4\,\text{HU}$).
  • Linearity: Checking HU for various substances (Teflon $\approx 1016$, Acrylic $\approx 140$, Polyethylene $\approx -75$).
• Medical Physics QA: Table increment assessment, laser alignment, and slice thickness measurement using ramps.

CT Artefacts

• Beam Hardening: Effective energy increases as it passes through the object. Causes "cupping" (center appears darker). Minimized by filtration and calibration corrections.
• Partial Volume Averaging: Occurs when a voxel contains multiple materials (e.g., bone and soft tissue). Reduced by narrower slices.
• Ring Artefacts: Caused by dead/uncalibrated detector cells.
• Aliasing: Caused by undersampling (increments between projections are too large).
• Streak/Metal Artefacts: Due to photon starvation and extreme beam hardening by high-$Z$ objects. Exacerbated by motion.
• Motion: Causes mis-registration and blurring.

Dual Energy CT (DECT)

• Principle: Exploits energy dependence of the mass attenuation coefficient to differentiate materials (e.g., Calcium vs. Iodine).
• Acquisition Methods:
  • Dual Source: Two X-ray tube/detector pairs offset by $90^\circ$.
  • Fast kV Switching: Rapidly changing kV during a single rotation.
  • Dual Layer Detector: Superficial layer catches low-energy photons; deeper layer catches high-energy photons.
  • Split Beam Filtration: Different filters (Tin/Gold) on halves of the beam.
• Applications:
  • Renal Calculi: Identifying Uric Acid ($Z \text{ elements: H, C, N, O}$) vs. non-uric acid stones.
  • Virtual Non-Contrast (VNC): Subtracting iodine signal from contrast images to simulate pre-contrast scans.
  • Virtual Monochromatic Images: Synthesizing images at specific energies (e.g., $70\,\text{keV}$) to increase conspicuity or reduce artefacts.

Questions & Discussion

• Q: Ratio of linear attenuation coefficient for ROI (-30 HU) to water?
  • A: $\text{HU} = ((\mu / \mu_{\text{water}}) - 1) \times 1000$. $\mu / \mu_{\text{water}} = (-30 / 1000) + 1 = 0.97$.
• Q: Filtered Back Projection process?
  • A: It is the standard reconstruction method using a convolution kernel to adjust voxel values from projections.
• Q: Scout views?
  • A: Produced line-by-line at a fixed projection angle.
• Q: CT spatial resolution comparison?
  • A: CT has better contrast resolution but lower spatial resolution than plain film radiography.
• Q: Pitch and Dose?
  • A: Pitch is inversely proportional to dose.
• Q: HU for air?
  • A: It is $-1000\,\text{HU}$, not $0$.
• Q: Effective dose values?
  • A: Head CT $\approx 2\,\text{mSv}$; Chest CT $\approx 8\,\text{mSv}$; Abdomen/Pelvis $\approx 10\,\text{mSv}$.
• Q: Ring Artefacts?
  • A: Can occur if a single detector is faulty.