X-Ray Radiography: Beer-Lambert Attenuation Imaging

Exposure Technique

Tube voltage (kVp) 80 kVp

Tube current-time (mAs) 100 mAs

Object-to-detector distance 5 cm

Projection angle (AP → lateral) 0°

Detector Readout

Avg. transmission

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Bone/lung contrast

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Geometric blur U_g

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Entrance skin dose

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How it works

An X-ray beam loses intensity exponentially as it crosses tissue — the Beer-Lambert attenuation law:

I = I₀ · exp(−Σ μᵢ·xᵢ)
μ = linear attenuation coefficient (cm⁻¹)
xᵢ = path length through tissue i (cm)

Each detector pixel sums the attenuation along its own ray from the point source, through every tissue the ray crosses (soft tissue, lung air, bone), to that pixel — exactly how a real projection radiograph forms. Bone (high effective Z) attenuates mostly by the photoelectric effect, whose cross-section falls steeply as ∝E⁻³, so raising kVp washes out bone/soft-tissue contrast fast. Soft tissue attenuates mostly by Compton scattering, which is nearly energy-independent, so its contrast fades far more slowly — the reason this simulator's bone-to-lung contrast collapses much faster than its overall brightness as you raise kVp.

Two more real effects are modelled:

Quantum mottle — detected photon count scales with mAs, and shot noise scales as 1/√N; low-mAs images look visibly grainier.
Geometric unsharpness — a real focal spot isn't a point, so edges blur by U_g = f · OID/SOD (focal-spot size × object-to-detector distance ÷ source-to-object distance). Pulling the detector away from the body (raising OID) blurs and magnifies the image, exactly as in real magnification radiography.

Entrance skin dose scales roughly with mAs linearly and with kVp² (steeper tube output at higher voltage) — the same trade-off radiographers balance every exposure: enough kVp/mAs for a diagnostic image, as little dose as achievable (ALARA).

X-ray source Torso phantom Detector / radiograph

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