Attenuation of X-rays (linear attenuation coefficient and half value layer)

Attenuation is fundamental to diagnostic imaging: it is what allows different tissues to produce varying signal intensities on the detector, creating the image.

Exponential Attenuation Law

In a uniform medium, the decrease in X-ray intensity with thickness follows an exponential relationship:

I = I₀e^−μx

Where:

I₀: incident intensity
I: transmitted intensity
μ: linear attenuation coefficient (cm⁻¹) – Specific to each tissue! More on this next.
x: tissue thickness (cm)

Interpretation in plain terms:
Each additional centimetre of material removes the same fraction (not the same number) of photons from the beam. This gives the characteristic exponential decay curve.

This equation assumes a monoenergetic beam and a homogeneous material.
In reality, diagnostic X-rays are polyenergetic, and low-energy photons are preferentially absorbed (beam hardening), so attenuation is not perfectly exponential. This equation provides a close approximation.

Linear and Mass Attenuation Coefficients

Linear Attenuation Coefficient (μ) (LAC)

The linear attenuation coefficient defines the fractional decrease in beam intensity per unit thickness of material.

Represents the probability per unit path length that a photon will interact with matter.
Units: cm⁻¹
Depends on:
- Photon energy (E): μ decreases as energy increases (beam more penetrating).
- Atomic number (Z): μ increases strongly with Z (more electrons, higher probability of photoelectric absorption).
- Density (ρ) of the material: μ increases with density (more atoms per unit volume).

This is a key point. If energy, density or atomic number changes, the LAC changes! The LAC is specific to each tissue/material at a specific energy.

Higher μ = greater attenuation.

Mass Attenuation Coefficient (μ/ρ)

To compare materials of different densities, μ is divided by ρ:

Normalises μ to material density, allowing comparison between materials.
This expresses attenuation per gram rather than per centimetre.
Units: cm²/g
Useful when comparing tissues or materials of different densities (e.g. fat vs bone).

μ / ρ = μ (cm⁻¹) / density (g/cm³)

Factors affecting attenuation

Factor	Effect	Explanation
Photon energy (kVp)	↑ Energy → ↓ Attenuation	Higher-energy photons more likely to pass through tissue without interacting.
Atomic number (Z)	↑ Z → ↑ Attenuation	More electrons and stronger nuclear fields increase interaction probability.
Density (ρ)	↑ Density → ↑ Attenuation	More atoms per unit volume → higher chance of interaction.
Thickness (x)	↑ Thickness → ↑ Attenuation	Longer path length through material → more interactions.

Makes sense right?!

Half-Value Layer (HVL)

We’ve mentioned the HVL previously. It’s the thickness of material required to reduce beam intensity by 50%.

HVL = ln⁡(2) / μ

Higher HVL = more penetrating (harder) beam.
HVL increases with increasing photon energy.

HVL provides a practical way to measure beam quality and filtration adequacy in quality control testing.

Meaning

It describes the penetrating power (beam hardness) of the X-ray beam.
A higher HVL means the beam is harder (more penetrating).
It provides a practical alternative to quoting μ, which depends on units and material.

Relationship Between μ and HVL

Because attenuation is exponential, each additional HVL removes half of the remaining photons:

Number of HVLs	Fraction Transmitted
1	50 %
2	25 %
3	12.5 %
4	6.25 %

Why is attenuation not perfectly explained by the exponential attenuation law we looked at earlier (I = I₀e^−μx)?

A couple reasons…

1. Narrow vs Broad Beam Geometry

You may hear the terms narrow vs broad beam geometry come up in exams, so I thought I’d briefly mention it here. In a nutshell:

The exponential law assumes narrow beam geometry (no scattered photons reach the detector).
In reality, diagnostic imaging uses broad beam geometry, where scattered photons contribute to detector signal.
This makes the measured attenuation less than predicted, because scatter adds back some intensity.

Practical takeaway: measured attenuation in patients is slightly less than theoretical attenuation due to scatter contribution.

2. Beam Hardening

In a polyenergetic beam, low-energy photons are absorbed more readily, leaving behind higher-energy photons, the beam becomes harder (more penetrating). The above equation assumes a monoenergetic beam.

This causes:

Non-linear attenuation through thick objects.
Increased mean photon energy with depth.
Artefacts in CT (cupping or streaks).

Key Takeaways and Exam Tips

Attenuation = reduction in intensity due to absorption + scatter.
Exponential law: I = I₀e^−μx
Linear attenuation coefficient (μ): probability of interaction per unit thickness (cm⁻¹).
μ (linear attenuation coefficient) depends on energy, Z, density, and thickness.
Mass attenuation coefficient (μ/ρ): normalised for density (cm²/g).
HVL = ln(2)/μ → quantifies beam hardness.
Higher energy → lower attenuation (more penetration).
Differential attenuation forms the basis of image contrast.
Beam hardening = preferential removal of low-energy photons → non-linear attenuation.
Each HVL halves intensity: after n HVLs, transmitted intensity = (½)ⁿ.
Common exam question: “Define attenuation and describe the factors that influence it.” “Define the half-value layer and explain how it relates to beam quality and filtration.”

Up next:

Next, a detailed look at the Photoelectric Effect. Let’s discuss this key interaction in detail, including its mechanism, energy dependence, and role in producing image contrast.