Medical physics

Piezo-electric effect

A piezo-electric 压电 crystal changes shape a little when a p.d. is put across it, and the reverse: it makes an electromotive force 电动势 (e.m.f.) across itself when its shape is changed. Quartz and PZT are common examples. Two linked effects:

Applying a p.d. deforms the crystal; squeezing it makes a p.d.

• apply a p.d. → the crystal changes shape (used to make vibrations).
• change the shape (a wave squeezes it) → an e.m.f. appears (used to detect vibrations).

Piezo-electric transducer

A transducer 换能器 uses this effect to both make and detect ultrasound 超声波.

• an alternating p.d. (a few MHz) makes the crystal vibrate at the same frequency, sending out longitudinal 纵波 waves above $20\ \text{kHz}$ ($1$–$10\ \text{MHz}$ for medical imaging).
• the same crystal then detects: returning ultrasound makes it vibrate and produce an e.m.f.

So one transducer is both emitter and detector, switching between sending and listening.

A piezo-electric transducer both sends and detects ultrasound using a vibrating crystal

Pulse-echo imaging

To see inside the body (pulse-echo 脉冲回波 imaging):

1. the transducer sends a short pulse into the body.
2. at each tissue boundary, part of the pulse is reflected and part goes on.
3. the transducer detects each reflected pulse.
4. the time delay gives the depth: $d = c t / 2$ (there and back). The echo's amplitude gives the strength of the reflection.

Worked example. An ultrasound pulse returns to the transducer $60\ \mu\text{s}$ after it was sent. The speed of sound in the tissue is $1500\ \text{m s}^{-1}$. Find the depth of the reflecting boundary.

The pulse travels there and back, so $d = \dfrac{ct}{2}$:

5. sweeping across the body builds a 2-D image.

A real ultrasound image — the fan shape comes from the transducer sweeping across the body. Every bright speck is an echo from a boundary between tissues, and its depth was worked out from the echo's time delay, exactly as in the worked example above

A coupling gel 耦合剂 is put between the transducer and the skin to push out the air; without it almost all the ultrasound would reflect at the skin–air boundary and never enter the body.

An A-scan shows the transmitted pulse and the echoes from each tissue boundary

Specific acoustic impedance

The specific acoustic impedance 声阻抗 of a medium is

where $\rho$ is the density 密度 and $c$ the speed of sound. Unit: $\text{kg m}^{-2}\ \text{s}^{-1}$. Bone has large $Z$; air has small $Z$; soft tissue is in between.

Worked example. Find the specific acoustic impedance of soft tissue. (Density $1060\ \text{kg m}^{-3}$, speed of sound $1540\ \text{m s}^{-1}$.)

Reflection at a boundary

At a boundary between media of impedance $Z_{1}$ and $Z_{2}$, the intensity reflection coefficient 强度反射系数 (fraction reflected) is

• very different impedances: almost all is reflected (skin/air — hence the gel).
• very similar impedances: almost nothing is reflected, so the boundary cannot be seen.
• best for imaging: different enough to give an echo, but not so different that nothing passes on.

At a boundary, part of the pulse reflects and part transmits — a bigger impedance difference gives a bigger echo

Attenuation

As ultrasound goes through tissue, its intensity falls with distance:

where $\mu$ is the attenuation coefficient 衰减系数 (unit $\text{m}^{-1}$). The same form applies to X-rays.

Intensity falls exponentially with thickness; a larger attenuation coefficient (bone) falls off faster

Production

X-rays come from an X-ray tube X射线管:

1. a heated cathode 阴极 emits electrons by thermionic emission 热电子发射.
2. a high p.d. (tens to hundreds of kV) accelerates the electrons across a vacuum 真空 to a metal target 靶 (the anode 阳极, often tungsten 钨).
3. the electrons hit the target and slow sharply. Most of their kinetic energy 动能 becomes heat; a small part is emitted as X-ray photons 光子 (Bremsstrahlung 轫致辐射, "braking radiation"). Some electrons knock out inner electrons of the metal atoms, and the refilling emits characteristic 特征 X-ray lines.

In an X-ray tube electrons from the heated cathode are accelerated onto a metal target anode

Minimum wavelength

The most energy one X-ray photon can have is the full kinetic energy of one accelerated electron, lost in a single event. For accelerating p.d. $V$, the KE is $eV$, so

This is the short-wavelength cut-off. The continuous Bremsstrahlung spectrum tails off above $\lambda_{\text{min}}$, with sharp characteristic peaks set by the target metal.

A typical X-ray spectrum — a continuous Bremsstrahlung curve cut off at $\lambda_0$, with sharp characteristic peaks

Imaging with X-rays

X-rays pass through the patient onto a detector. Tissues that attenuate 衰减 more (bone, high $Z$) cast a stronger shadow and look lighter; tissues that attenuate less (soft tissue, lung) look darker.

The contrast 对比度 is the difference in attenuation between tissues. A contrast medium 造影剂 (e.g. a barium meal) can be given to make soft tissues stand out.

A real chest X-ray: dense bone absorbs more X-rays and looks white; the air-filled lungs let X-rays through and look dark

Attenuation law

Higher-energy X-rays penetrate further (smaller $\mu$); bone has a much larger $\mu$ than soft tissue. To find the thickness for a given fraction, take logs: $x = \dfrac{1}{\mu} \ln\dfrac{I_{0}}{I}$. The half-value thickness 半值厚度 $x_{1/2} = \ln 2 / \mu$ halves the intensity (like half-life in decay).

Worked example. X-rays pass through $3.0\ \text{cm}$ of tissue with attenuation coefficient $\mu = 40\ \text{m}^{-1}$. Find the fraction of the intensity that gets through.

Computed tomography (CT)

A computed tomography 计算机断层扫描 (CT) scan builds a 3-D image:

1. the tube and detectors rotate around the patient, taking many images of one thin slice from different angles.
2. a computer combines these into a 2-D cross-section of the slice.
3. the patient is moved along, and the next slice is imaged.
4. the slices are stacked into a 3-D image.

CT shows far more than a single X-ray, because overlapping soft tissues are separated by the many-angle reconstruction.

In a CT scan the X-ray tube and detectors rotate around the patient to image a slice from many angles

Tracer

A tracer 示踪剂 is a substance with radioactive nuclei put into the body. It is taken up more by the tissue being studied (e.g. a tumour takes up more glucose-tagged tracer due to its high metabolism 代谢). Its decay is detected from outside.

In positron emission tomography 正电子发射断层扫描 (PET), the tracer is a $\beta^{+}$ emitter — it gives out a positron 正电子. A common one is fluorine-18 on a glucose analogue (FDG).

Annihilation

When a particle meets its antiparticle 反粒子 they annihilate 湮灭: their mass turns into electromagnetic energy. In PET:

• a positron travels a few mm before meeting an electron 电子.
• they annihilate. Energy and momentum 动量 are conserved.
• since the total momentum is about zero, two photons are produced going in opposite directions, each $511\ \text{keV}$ ($= m_{e} c^{2}$).

PET: annihilation gives two 511 keV photons in opposite directions; a coincidence fixes the line

Energy of the annihilation photons

By energy conservation, the total photon energy equals the pair's rest energy 能量:

Each photon has $h f = m_{e} c^{2} \approx 8.2 \times 10^{-14}\ \text{J} \approx 0.51\ \text{MeV}$, with $\lambda \approx 2.4 \times 10^{-12}\ \text{m}$.

Reconstructing the image

The two photons leave the body in opposite directions and hit detector rings around the patient. Recording the two simultaneous arrivals (a "coincidence") fixes the line the annihilation happened on. Many coincidences from many angles let the computer build a 3-D map of the tracer — showing tissues with high metabolic activity. Comparing the two arrival times can refine the position along that line (time-of-flight PET).

A finished PET image of the brain: warm colours (red, yellow) mark where the tracer collected -- the most active tissue

• Ultrasound reflects at boundaries (acoustic impedance); a coupling gel reduces reflection at the skin.
• X-rays are attenuated as $I = I_0 e^{-\mu x}$; contrast media (barium, iodine) absorb strongly.
• PET uses a positron emitter; annihilation gives two $\gamma$-rays in opposite directions, located by the detector ring.