X-Ray CT, Endoscopy and MRI

1Slicing without a knife

On 1 October 1971, at Atkinson Morley's Hospital in London, a woman with a suspected brain tumour lay with her head in a new machine built by the engineer Godfrey Hounsfield. An X-ray beam scanned round her head for several minutes, a computer then spent hours processing the readings, and the image showed a dark, round cyst in the brain itself. No ordinary radiograph could have shown it: a radiograph flattens the whole thickness of the body into one shadow, and soft tissues absorb X-rays so similarly that they all look the same grey. Hounsfield shared the 1979 Nobel Prize in Physiology or Medicine with Allan Cormack, who had worked out the mathematics independently. This lesson shows how CT, MRI and the endoscope each get round the limits of the plain radiograph that you met in Unit 7.

2Computed tomography (CT)

In a CT scanner an X-ray tube and an arc of several hundred detectors face each other across a ring, the gantry, and rotate together around the patient (Figure 4.1). Each detector measures the intensity $II$ that crosses the body along one ray. From Lesson 1, for a ray crossing small elements of thickness $\mathrm{\Delta }x\backslash Delta\; x$:

$$I=I_0{e}^{-({\mu }_1+{\mu }_2+\cdots +{\mu }_n)\mathrm{\Delta }x}⟹\ln \frac{I_0}{I}=\sum _i{\mu }_i\mathrm{\Delta }xI\; =\; I\_0\backslash ,e^\{-(\backslash mu\_1\; +\; \backslash mu\_2\; +\; \backslash dots\; +\; \backslash mu\_n)\backslash Delta\; x\}\; \backslash quad\backslash Longrightarrow\backslash quad\; \backslash ln\backslash frac\{I\_0\}\{I\}\; =\; \backslash sum\_i\; \backslash mu\_i\backslash ,\backslash Delta\; x$$

So each reading gives the sum of the attenuation coefficients along one line. One sum cannot say where along the line the absorber is, but sums measured along thousands of lines at hundreds of angles can. A computer solves for the value of $\mu \backslash mu$ in every small square (pixel) of the slice, using a method called filtered back-projection or newer iterative methods. Modern scanners move the patient steadily through the rotating gantry (a helical or spiral scan), collecting many slices in a few seconds, which the computer can also stack into a three-dimensional image.

Figure 4.1: A CT scanner viewed along the patient's body (not to scale).

Figure 4.2: The smallest possible CT problem: four unknown pixels and five measured ray sums.

Calculated by Ganzaa from NIST mass attenuation coefficients; real scanners use a spread of photon energies, so measured CT numbers differ somewhat.

Muscle and fat differ by only about 13% in $\mu \backslash mu$, far too little to separate on a radiograph where many tissues overlap. In CT each pixel stands alone, so the 130 HU difference shows plainly; the display can even be set to stretch a narrow "window" of CT numbers over the whole grey scale. CT is fast and widely available, and it is the first test after a head injury or a stroke. Its cost is dose: from Lesson 2, a CT of the abdomen gives about 77 times the effective dose of a chest radiograph, so each CT scan must be justified.

3Optical imaging: the endoscope

Some parts of the body can be looked at directly with light, if the light can be taken round corners. An endoscope is a flexible tube passed through the mouth (to view the oesophagus, stomach and duodenum) or through a small cut (keyhole surgery). It relies on the optical fibre: a glass core of refractive index $n_{1}n\_1$ surrounded by a cladding of lower index $n_{2}n\_2$. Light striking the core–cladding boundary at more than the critical angle $CC$, where $\sin C=\frac{n_2}{n_1}\backslash sin\; C\; =\; \backslash frac\{n\_2\}\{n\_1\}$, is totally internally reflected, and bounces along the fibre with very little loss, even when the fibre bends.

Figure 4.3: Light guided along an optical fibre by total internal reflection.

Light entering the end of the fibre is guided only if it arrives within the acceptance angle ${\theta }_a\backslash theta\_a$. Applying Snell's law at the end face (from air) and the critical-angle condition at the side gives

$$\sin {\theta }_a=\sqrt{n_1^2-n_2^2}\backslash sin\backslash theta\_a\; =\; \backslash sqrt\{n\_1^2\; -\; n\_2^2\}$$

An endoscope contains two bundles. An incoherent bundle, with fibres arranged at random, carries light down to illuminate the tissue. A coherent bundle, with tens of thousands of fibres in exactly the same arrangement at both ends, carries the image back: each fibre carries one dot of the picture. Many modern endoscopes instead put a tiny camera chip at the tip and use the fibres only for lighting. Extra channels let the doctor blow in air, wash the lens, take a small tissue sample (biopsy), or pass a laser fibre to seal a bleeding vessel. Endoscopy uses visible light only, so it gives no ionising dose.

4Magnetic resonance imaging (MRI)

MRI maps the hydrogen nuclei (protons) in the body's water and fat. A proton has spin, which makes it a tiny magnet. In the strong, steady field $B_{0}B\_0$ of the scanner (usually 1.5 T or 3.0 T, made by a superconducting coil cooled by liquid helium), each proton's magnetic axis precesses around the field direction, like a spinning top leaning in gravity, at the Larmor frequency:

$$f=\bar{\gamma }{B}_0,\bar{\gamma }=42.58\mathrm{M}\mathrm{H}\mathrm{z}{\mathrm{T}}^{-1}\text{ for hydrogen}f\; =\; \backslash bar\{\backslash gamma\}\backslash ,B\_0,\; \backslash qquad\; \backslash bar\{\backslash gamma\}\; =\; 42.58\backslash ,\backslash mathrm\{MHz\backslash ,T^\{-1\}\}\backslash \; \backslash text\{for\; hydrogen\}$$

Slightly more protons line up with the field than against it, so the tissue has a small net magnetisation $\mathit{M}\backslash boldsymbol\{M\}$ along $B_{0}B\_0$.

Figure 4.4: Precession, resonance and relaxation.

1. Resonance. A pulse of radio waves at exactly the Larmor frequency, from a coil around the patient, is absorbed by the protons and tips $\mathit{M}\backslash boldsymbol\{M\}$ away from $B_{0}B\_0$, for example through 90°.
2. Signal. The tipped magnetisation keeps precessing; as it sweeps round, it induces an alternating e.m.f. in the receiver coil (Faraday's law, Unit 3).
3. Relaxation. The signal dies away as $\mathit{M}\backslash boldsymbol\{M\}$ returns to equilibrium. The recovery along $B_{0}B\_0$ has a time constant $T_{1}T\_1$; the loss of the rotating part, as the protons get out of step, has a time constant $T_{2}T\_2$. Both differ between fat, water, grey matter, white matter and diseased tissue, which is where MRI's excellent soft-tissue contrast comes from.
4. Position. Gradient coils add small fields that vary with position, so protons at different places precess at slightly different frequencies; the computer decodes the frequencies into positions. Switching the gradients on and off makes the loud knocking during a scan.

MRI safety. There is no ionising radiation, but the magnet is always on. Iron and steel objects (scissors, oxygen cylinders, keys) are pulled violently into the bore, and some pacemakers and metal implants are unsafe, so every patient and visitor is screened before entering the room. The RF pulses warm the body slightly and the gradient noise is loud enough to require ear protection. Scans take longer than CT, and the narrow tunnel distresses some patients.