Welcome to Medical Physics (9702 A Level)

Welcome to one of the most exciting chapters in A Level Physics! Medical Physics shows us how all the concepts we&rsquove learned—waves, quantum mechanics, and nuclear physics—are applied directly to diagnose and treat the human body.
This chapter is highly visual and conceptual. Don’t worry if the formulas seem complicated; focus first on understanding the physical processes involved. Let&rsquos dive into how we use physics to see inside the body!

24.1 Production and Use of Ultrasound

The Piezoelectric Effect: The Heart of Ultrasound

Ultrasound imaging relies on special materials called piezoelectric crystals. These crystals are amazing because they link mechanical stress (changes in shape) directly to electrical energy (potential difference).

1. Generation of Ultrasound (Mechanical from Electrical):
When an alternating potential difference (p.d.) is applied across the crystal, the crystal “pings” by changing its shape rapidly. If the frequency of the p.d. is in the megahertz range, the crystal produces ultrasound waves (high-frequency sound waves, beyond the range of human hearing).

2. Detection of Ultrasound (Electrical from Mechanical):
When an incoming ultrasound wave hits the crystal, it causes the crystal to vibrate and change shape slightly. This mechanical deformation generates an electromotive force (e.m.f.) across the crystal. This electrical signal is then processed to create an image.

Think of it like a microphone and speaker combined! The same crystal can transmit and receive.

Diagnostic Imaging using Reflection

Ultrasound is used for diagnostic information, like checking fetal development or examining soft tissues. It works by sending out short pulses and measuring the “echoes” that return.

The Process:

  1. The piezoelectric transducer sends an ultrasound pulse into the body.
  2. The pulse travels until it hits a boundary between two different tissues (e.g., muscle and fat).
  3. A portion of the wave is reflected back (the echo) to the transducer, while the rest continues.
  4. The transducer detects the echo and the time delay is measured.

The depth of the boundary is calculated using the time delay and the speed of sound \(c\) in the medium: \(Distance = c \times \Delta t / 2\). We divide by two because the pulse traveled to the boundary and back.

Specific Acoustic Impedance (Z)

The amount of reflection at a boundary depends on how different the two materials are. This difference is quantified by Specific Acoustic Impedance (Z).

Definition: The specific acoustic impedance of a medium is the product of the density of the medium \(\rho\) and the speed of sound \(c\) in that medium.

\[Z = \rho c\]

Unit: \(kg \, m^{-2} \, s^{-1}\) (Since \(\rho\) is \(kg\, m^{-3}\) and \(c\) is \(m \, s^{-1}\)).

Intensity Reflection Coefficient (\(I_R/I_0\))

The ratio of the reflected intensity (\(I_R\)) to the incident intensity (\(I_0\)) is called the intensity reflection coefficient. This coefficient determines the “brightness” of the echo, hence the contrast in the image.

For a boundary between medium 1 (impedance \(Z_1\)) and medium 2 (impedance \(Z_2\)), the intensity reflection coefficient is:

\[\frac{I_R}{I_0} = \frac{(Z_1 - Z_2)^2}{(Z_1 + Z_2)^2}\]

Key Takeaway for Contrast:

  • If \(Z_1 = Z_2\) (impedance matched), then \(I_R/I_0 = 0\). There is no reflection, and the wave passes straight through. (This is good for waves passing through skin, but bad if we want to see the boundary!)
  • If \(Z_1\) is very different from \(Z_2\) (e.g., tissue to air or tissue to bone), \(I_R/I_0\) is large, resulting in a strong echo.

Did you know? Before using an ultrasound probe, doctors apply a gel. This gel acts as an impedance matching layer between the transducer and the skin, ensuring minimal reflection at the skin surface, allowing more ultrasound energy to enter the body!

Attenuation of Ultrasound

As ultrasound travels through matter, its intensity decreases because the energy is absorbed by the tissue. This loss of intensity is called attenuation.

The decrease in intensity follows an exponential decay relationship:

\[I = I_0 e^{-\mu x}\]

Where:

  • \(I\) is the intensity after traveling distance \(x\).
  • \(I_0\) is the initial intensity.
  • \(\mu\) is the attenuation coefficient (in units of \(m^{-1}\)). This coefficient depends on the medium and the frequency of the ultrasound.
Higher frequency ultrasound attenuates more quickly but provides better resolution.

Quick Review: Ultrasound

Purpose: Soft tissue imaging.
Mechanism: Piezoelectric effect generates and detects echoes.
Contrast determined by: Difference in Specific Acoustic Impedance (\(Z = \rho c\)).

24.2 Production and Use of X-rays

Production of X-rays

X-rays are high-energy electromagnetic waves. They are produced in an evacuated X-ray tube by accelerating electrons into a heavy metal target (usually Tungsten).

Step-by-Step Production:

  1. Electrons are emitted (often thermionically) from a heated filament (cathode).
  2. A very large potential difference (typically 50 kV to 150 kV) is applied between the cathode and the anode (metal target).
  3. The electrons are accelerated to very high kinetic energy.
  4. These high-speed electrons strike the metal target.
  5. When the electrons decelerate rapidly upon impact, they release energy in the form of X-rays. (This is related to the process called Bremsstrahlung, or "braking radiation").

Calculating Minimum Wavelength (\(\lambda_{min}\)):
The maximum energy an X-ray photon can have occurs when the entire kinetic energy of the accelerated electron is converted into a single photon. If \(V\) is the accelerating potential difference, the electron kinetic energy is \(eV\).

\[E_{max} = eV = hf_{max} = \frac{hc}{\lambda_{min}}\]

Thus, the minimum wavelength of the X-rays produced is:

\[\lambda_{min} = \frac{hc}{eV}\]

X-ray Imaging and Contrast

X-rays are primarily used to image dense structures, most commonly bones, because the absorption of X-rays by a material depends heavily on its density and proton number (Z).

Contrast in X-ray Imaging:

  • High Z materials (like bone/calcium): Absorb X-rays strongly. They appear bright white on the film (or detector).
  • Low Z materials (like soft tissue/air): Absorb X-rays weakly. They appear dark (or grey).
The contrast is the difference in X-ray intensity transmitted through different tissues, allowing us to distinguish them.

Attenuation of X-rays

Like ultrasound, X-ray intensity is reduced (attenuated) as it passes through matter. The key absorption processes here are the photoelectric effect and Compton scattering.

The intensity \(I\) decreases exponentially with the thickness \(x\) of the material:

\[I = I_0 e^{-\mu x}\]

Where:

  • \(I\) is the intensity after thickness \(x\).
  • \(I_0\) is the initial intensity.
  • \(\mu\) is the attenuation coefficient for X-rays.
Don't confuse the attenuation coefficients for ultrasound and X-rays! They use the same symbol \(\mu\) and the same exponential decay law, but their values and the underlying physical mechanisms are different.

Computed Tomography (CT) Scanning

CT scanning provides a much more detailed 3D image of internal structures compared to a standard 2D X-ray image.

The CT Process (Building a 3D Image):

  1. Multiple Projections (2D Slice): An X-ray beam is passed through a specific section (slice) of the body. Detectors measure the transmitted intensity. The X-ray source and detectors then rotate around the patient, taking hundreds of X-ray images of that same section from different angles.
  2. Computer Processing: A powerful computer combines these multiple 2D images. Using complex mathematical algorithms, the computer calculates the density and attenuation of every tiny volume element (voxel) within that slice, creating a detailed 2D cross-section image.
  3. 3D Reconstruction: This whole process (steps 1 & 2) is repeated for adjacent sections along the length of the patient's body (the axis of the structure). The stacked 2D slices are then combined digitally to form a comprehensive 3D image of the internal structure.

24.3 PET Scanning (Positron Emission Tomography)

The Tracer and Decay

PET scanning is a functional imaging technique, meaning it looks at how the body works (e.g., metabolic activity) rather than just structure.

1. The Tracer:
A tracer is a substance containing radioactive nuclei that is introduced into the body (often injected). This substance is chosen because it is naturally absorbed by the tissue being studied (e.g., highly active tissues like cancer cells or brain regions absorb more sugar).

2. Beta-Plus (\(\beta^+\)) Decay:
The radioactive nuclei used in PET tracers decay via \(\beta^+\) decay (positron emission).
In \(\beta^+\) decay, a proton turns into a neutron, emitting a positron (\(e^+\) or \(\beta^+\)) and an electron neutrino (\(\nu_e\)).
Remember: A positron is the antiparticle of an electron. It has the same mass but opposite charge (\(+e\)).

Annihilation and Gamma Ray Production

The key physical mechanism in PET scanning is annihilation.

1. Annihilation:
The emitted positron travels a short distance (a few millimeters) in the tissue before encountering an electron (\(e^-\)). When a particle interacts with its antiparticle, annihilation occurs.
The total mass of the electron and positron is converted entirely into energy, following Einstein’s mass-energy equivalence principle, \(E = mc^2\).
Crucially, mass-energy and momentum are conserved during this process.

2. Gamma Photon Emission:
To conserve momentum, the annihilation event produces a pair of gamma-ray photons (often called annihilation radiation). These two photons travel in exactly opposite directions (180° apart).

Calculating Photon Energy (24.3.5):
The total mass annihilated is \(m = m_e + m_{e^+} = 2m_e\).
The total energy released is \(E_{total} = (2m_e)c^2\).
Since this energy is shared equally between the two photons:
$$E_{photon} = m_e c^2$$
The rest mass energy of a single electron (or positron) is approximately \(0.511 \text{ MeV}\). Therefore, each gamma photon has an energy of about 0.51 MeV.

Detection and Image Creation

PET relies on detecting this pair of opposing gamma rays.

1. Detection:
The patient is surrounded by a ring of detectors. The gamma-ray photons travel outside the body and strike these detectors.

2. Localization (Timing):
Since the two photons arrive simultaneously at detectors 180° apart, the system knows that the annihilation event occurred somewhere along the line connecting those two detectors. By analyzing the arrival times (and the time difference, if any) of many pairs of photons, the computer can precisely localize the annihilation event.

3. Image Formation:
Areas that absorbed more tracer (e.g., metabolically active regions) will produce more annihilation events. By mapping the concentration of these annihilation events, an image is created that highlights the areas of high metabolic activity in the body.

Key Takeaway: PET vs. X-ray/Ultrasound

Ultrasound & X-rays: Used for structural imaging (bones, tumors, organs).
PET Scan: Used for functional imaging (metabolism, blood flow). It shows *where* the body is working hardest.