understand that computed tomography (CT) scanning produces a 3D image of an internal structure by first combining multiple X-ray images taken in the same section from different angles to obtain a 2D image of the section, then repeating this process a

Production of X‑rays (Section 24.2)

• Bremsstrahlung (braking radiation) – a continuous spectrum produced when high‑energy electrons are decelerated in the electric field of atomic nuclei in the target.
• Characteristic radiation – discrete spectral lines emitted when an inner‑shell electron is removed and an outer‑shell electron fills the vacancy. The line energy is specific to the target material (e.g. tungsten K‑α ≈ 59 keV).

Minimum (cut‑off) wavelength

When an electron is accelerated through a potential difference V, its kinetic energy is eV. If the entire kinetic energy is converted into a single photon, the shortest possible wavelength is

\$\lambda_{\min}= \frac{hc}{eV}\$

Example – 100 kV X‑ray tube

\$\$\lambda_{\min}= \frac{6.626\times10^{-34}\,\text{J·s}\; \times 3.00\times10^{8}\,\text{m s}^{-1}}

{1.60\times10^{-19}\,\text{C}\; \times 1.0\times10^{5}\,\text{V}}

\approx 1.24\times10^{-11}\,\text{m}=0.0124\;\text{nm}\$\$

X‑ray tube components

Interaction of X‑rays with Matter

The intensity I of a mono‑energetic X‑ray beam after passing through a material of thickness x obeys the exponential attenuation law

\$I = I_{0}\,e^{-\mu x}\$

• Linear attenuation coefficient μ (cm⁻¹) depends on photon energy and the atomic composition of the material.
• Differences in μ between tissues generate contrast on a radiograph.
• Contrast agents (e.g. iodine, barium) contain high‑Z elements that increase local μ, improving visibility of vessels or the gastrointestinal tract.

Radiation Protection – ALARA principle

Production of Ultrasound (Section 24.1)

Piezo‑electric transducer

• Electrical pulse → crystal vibrates → emits a short burst of sound (1–20 MHz).
• Returning echoes cause the crystal to vibrate again → electrical signal.
• Bandwidth: Determines axial resolution; a broader bandwidth gives a shorter pulse and finer detail.
• Focusing: Either acoustic lenses or electronic phasing of an array concentrates the beam at a chosen depth, improving lateral resolution.

Speed of sound and acoustic impedance

In soft tissue

\$c \approx 1540\ \text{m s}^{-1}\$

Acoustic impedance

\$Z = \rho\,c\$

where ρ is the density (kg m⁻³).

Reflection at an interface

For two media with impedances Z₁ and Z₂, the intensity reflection coefficient is

\$\frac{I{r}}{I{i}} = \left(\frac{Z{1}-Z{2}}{Z{1}+Z{2}}\right)^{2}\$

Depth calculation (time‑of‑flight)

Echoes travel to the reflector and back, so the depth d is

\$d = \frac{c\,t}{2}\$

Example: An echo returns after 130 µs.

\$\$d = \frac{1540\ \text{m s}^{-1}\times130\times10^{-6}\ \text{s}}{2}

\approx 0.10\ \text{m}=10\ \text{cm}\$\$

Imaging modes

Clinical use & safety

• Real‑time imaging of moving structures – e.g. fetal scanning, cardiac echocardiography.
• High‑resolution superficial imaging – breast, musculoskeletal, eye.
• No ionising radiation → safe for repeated or paediatric examinations.

Case study: A 28‑week pregnancy scan uses B‑mode to visualise fetal anatomy and colour Doppler to map the umbilical artery flow, providing information on growth and placental health without any radiation risk.

Use of X‑rays in Imaging

• Conventional radiography – a single projection; contrast arises from differences in attenuation.
• Computed tomography (CT) – multiple projections are combined to produce cross‑sectional images, giving true anatomical localisation.

Computed Tomography (CT) Scanning (Section 24.2)

Conceptual overview

1. Acquire many X‑ray projections of a thin slice while the X‑ray source and detector rotate around the patient (typically 0°–360°).
2. Reconstruct a two‑dimensional attenuation map of that slice using a mathematical algorithm.
3. Advance the patient table (or move the gantry) to image the next adjacent slice.
4. Stack the series of slices to form a three‑dimensional data set that can be displayed in any plane or rendered volumetrically.

Acquisition of projections

• Fan‑beam geometry – a narrow X‑ray beam and a linear detector array; most common in clinical scanners.
• Cone‑beam geometry – a wide beam and a two‑dimensional detector; used for volume‑CT (spiral/ helical scanning).
• During one rotation (≈ 0.5–1 s) the detector records intensity I(θ,s) for each angle θ and detector position s.
• Key parameters:

  • Slice thickness – usually 0.5–5 mm; thinner slices give higher spatial resolution but increase dose.
  • Pitch = table travel per rotation ÷ slice thickness; pitch > 1 reduces dose but may introduce gaps.
  • Collimation – defines the width of the fan/cone and limits scatter.

Image reconstruction methods

For filtered back‑projection the reconstruction formula can be written as

\$\$\mu(x,y)=\int{0}^{\pi}\!\!\left[\int{-\infty}^{\infty}

R(\theta,s)\,| \omega |\,e^{i\omega\,(s-x\cos\theta-y\sin\theta)}\,d\omega\right]d\theta\$\$

where R(θ,s) is the Radon transform of the measured data and |ω| is the frequency‑domain filter.

CT numbers – Hounsfield Units (HU)

Reconstructed attenuation values are scaled relative to water:

\$\text{CT}=1000\;\frac{\mu-\mu{\text{water}}}{\mu{\text{water}}}\$

• Air ≈ –1000 HU
• Water = 0 HU (reference)
• Fat ≈ –100 HU
• Soft tissue ≈ 20–80 HU
• Bone ≈ +700 to +3000 HU

Formation of the 3‑D image

• Each reconstructed slice is stored as a matrix of pixels (2‑D). Adding the slice index creates a three‑dimensional array of voxels.
• Stacking voxels yields a volumetric data set that can be visualised in:

  • Axial, coronal and sagittal planes.
  • Multiplanar reconstructions (MPR) – arbitrary oblique slices.
  • Volume rendering – realistic 3‑D images for surgical planning.

• Window‑level adjustments change the displayed HU range to emphasise bone, soft tissue or lung.

Key Points to Remember

• High‑energy electrons striking a high‑Z target produce X‑rays via bremsstrahlung and characteristic radiation; the cut‑off wavelength follows λ_min=hc/eV.
• Attenuation of X‑rays obeys I = I₀e⁻ᵐᵘˣ; contrast arises from differences in μ and can be enhanced with iodine or barium agents.
• Radiation protection follows ALARA – minimise time, maximise distance, use shielding, and monitor dose.
• Ultrasound is generated by a piezo‑electric transducer; bandwidth and electronic focusing determine resolution. Modes include B‑mode (2‑D imaging), M‑mode (motion), and Doppler (flow).
• Depth of a reflector is calculated from echo time: d = ct/2. Example: a 130 µs echo corresponds to 10 cm depth.
• CT acquires many angular X‑ray projections of a thin slice, reconstructs a 2‑D attenuation map (filtered back‑projection or iterative reconstruction), assigns Hounsfield Units, and stacks successive slices to produce a 3‑D voxel dataset.
• Key CT parameters – slice thickness, pitch, collimation – balance spatial resolution, coverage and radiation dose.