Chapter 1: X-rays and Attenuation

As X-rays travel through matter, photoelectric absorption and Compton scattering gradually remove photons from the beam, so the intensity decays exponentially with the distance travelled. For a uniform slab of thickness $L$, the incident intensity $I_0$ and the transmitted intensity $I$ obey the relation below. The proportionality constant $\mu$ is called the linear attenuation coefficient; it is determined by the material, its density, and the X-ray energy.

$$I = I_0\, e^{-\mu L}$$

In an inhomogeneous object such as the human body, $\mu$ varies from place to place. Splitting the path $L$ into small segments $dl$ and multiplying the transmittances $\exp(-\mu\, dl)$ segment by segment puts a line integral of $\mu$ in the exponent. This is the general form of the Beer–Lambert law.

$$I = I_0 \exp\!\left(-\int_{L} \mu(x,y)\, dl\right)$$

A detector can only measure the intensity ratio $I/I_0$. Taking its negative logarithm, however, yields exactly the line integral $p$ of $\mu$ along the path. This quantity $p$ is called a projection. CT is the technique of recovering the spatial distribution $\mu(x,y)$ from projections measured along many directions.

$$p \;=\; -\ln\frac{I}{I_0} \;=\; \int_{L} \mu(x,y)\, dl$$

How a scanner makes the measurement

An X-ray tube accelerates electrons into a metal target, producing a spectrum of bremsstrahlung and characteristic X-rays. Tube voltage (kVp) mainly changes the maximum photon energy and spectrum, while tube-current–time product (mAs) mainly changes the photon count. A bow-tie filter reduces exposure through the thinner periphery of the patient and evens out the intensity reaching the detector.

Conventional CT detectors convert X-rays to visible light in a scintillator and integrate the signal with photodiodes. Before scanning, dark and air measurements correct detector offsets and gains. The ring artifacts in Chapter 11 illustrate what can happen when this calibration is imperfect. In the equations above, $I_0$ corresponds to the air measurement and $I$ to the measurement through the object.

Simulation: line integral along a single ray

A single ray is drawn over the selected phantom, which represents the distribution of $\mu$. Changing the angle $\theta$ or detector position $s$ updates the $\mu$ profile, line integral $p$, and transmittance $I/I_0$. When the ray crosses a highly attenuating region, such as the outer shell of the Shepp–Logan phantom, $p$ rises and the transmittance drops. The free-draw tab accepts a custom $\mu$ distribution.

From projections to a sinogram

A projection is the line integral of the attenuation coefficient along a ray: $p = -\ln(I/I_0)$. Because one ray supplies only one value, CT measures a row of rays at many angles. The next chapter organizes those measurements into a two-dimensional dataset called a sinogram.

References

• Hounsfield GN. Computerized transverse axial scanning (tomography): Part 1. Description of system. British Journal of Radiology 46, 1016–1022 (1973).
• Bushberg JT et al. The Essential Physics of Medical Imaging, 4th ed. Wolters Kluwer (2020).