Chapter 9: Radiation Dose and Image Quality

In Chapters 4 and 8 we added noise to the projections and watched the reconstruction turn grainy, but noise was just "a setting", and we never asked where it comes from. In a real scanner, what sets the noise level is the amount of X-rays the patient absorbs: the radiation dose. More dose buys a cleaner image at the cost of higher radiation risk. The optimization at the heart of CT is achieving diagnostic image quality with as little dose as possible. This chapter derives the noise level from photon statistics, then sets up the rulers on both sides of the trade: dose metrics (CTDI, DLP) and image-quality metrics (MTF, NPS).

Photon statistics and noise

X-ray detection is inherently stochastic: even at constant beam intensity the number of detected photons fluctuates with a Poisson distribution, whose variance equals its mean ($\mathrm{Var}[I] = \bar I$). For a transmission measurement with expectation $\bar I = I_0 e^{-p}$, first-order error propagation through the log transform $p' = -\ln(I/I_0)$ gives

$$\sigma_p^2 \simeq \frac{\mathrm{Var}[I]}{\bar I^2} = \frac{1}{\bar I} = \frac{e^{p}}{I_0}$$

Two important consequences hide in this short formula. First, $\sigma_p \propto 1/\sqrt{I_0}$: quadrupling the photon count (≈ tube current-time product mAs ≈ dose) only halves the noise; the exchange rate between dose and noise is a square root. Since FBP is linear, the same scaling carries over to the image noise $\sigma$. Second, the $e^p$ in the numerator: rays with strong attenuation get exponentially noisier. This is why large patients, bone, and metal blow up the noise, and the photon starvation of Chapter 11 is this effect taken to its limit.

On a real scanner the main dose knobs are mAs and tube voltage (kVp). mAs scales $I_0$ directly; kVp changes the spectrum itself, moving both $\mu$ and contrast along with dose (this leads to the energy story of Chapter 13). The simulations in this chapter simplify to varying $I_0$ only.

Patient thickness changes with table position and projection angle. Automatic exposure control (AEC) estimates attenuation from the scout image or recent projections and modulates tube current longitudinally and angularly. Together with the bow-tie filter introduced in Chapter 1, it reduces unnecessary exposure while maintaining the target image quality. The rule that halving mAs halves dose is therefore an approximation for otherwise fixed conditions.

Simulation: dose, noise, and low-contrast detectability

A low-contrast phantom places inserts of contrast $\Delta A = 0.08 / 0.04 / 0.02 / 0.01$ in a uniform disk, reconstructed by FBP while you vary $I_0$. The plot on the right shows the measured noise $\sigma$ at all dose levels against $1/\sqrt{\text{relative dose}}$: the points fall on a straight line through the origin ($\sigma \propto 1/\sqrt{\text{dose}}$). As the dose drops, the lowest-contrast inserts sink into the noise first. The rule of thumb that a feature becomes undetectable when its CNR (contrast-to-noise ratio) falls below roughly 3–5 is known as the Rose criterion.

Measuring dose: from CTDI to effective dose

To manage dose you must first measure it. The starting point of CT dosimetry is the CT Dose Index. A single axial rotation deposits a dose profile $D(z)$ along the patient axis; integrating it over ±50 mm inside a standard acrylic cylinder (16 cm head / 32 cm body) and dividing by the nominal beam width $nT$ gives

$$\mathrm{CTDI}_{100} = \frac{1}{nT} \int_{-50\,\mathrm{mm}}^{+50\,\mathrm{mm}} D(z)\, dz$$

Because of scatter, the dose profile has tails wider than the beam itself, and CTDI folds those tails back into an "effective dose per rotation". A weighted average of center and periphery measurements gives $\mathrm{CTDI}_w = \frac{1}{3}C + \frac{2}{3}P$, and correcting for helical pitch yields the $\mathrm{CTDI}_{vol}$ [mGy] displayed on every scanner console. Multiplying by scan length gives $\mathrm{DLP} = \mathrm{CTDI}_{vol} \times L$ [mGy·cm], a surrogate for the whole examination, and an anatomy-specific coefficient $k$ converts that into an estimate of effective dose, $E \approx k \cdot \mathrm{DLP}$ [mSv].

Keep in mind that $\mathrm{CTDI}_{vol}$ characterizes the scanner's output into a standard phantom, not the dose to an individual patient: at equal $\mathrm{CTDI}_{vol}$, a smaller patient absorbs more. The Size-Specific Dose Estimate (SSDE) corrects for patient size. Operationally, dose management rests on the ALARA principle (as low as reasonably achievable) and on diagnostic reference levels (DRLs) compiled per country and examination type.

Measuring image quality: MTF and NPS

Against the noise level $\sigma$ stands the other half of image quality: sharpness. As Chapter 4 showed, switching the reconstruction filter from Ram-Lak to Hann reduces noise but blurs the image. MTF and NPS quantify both sides on a common axis of spatial frequency.

The MTF (modulation transfer function) tells you what fraction of the contrast of a sinusoidal pattern at frequency $f$ survives the imaging chain. Imaging a point-like object yields the point spread function (PSF); integrating it along one direction gives the line spread function (LSF), whose 1D Fourier magnitude is the MTF. The frequencies where the MTF drops to 50% / 10% (MTF50 / MTF10) are standard single-number summaries of resolution.

The NPS (noise power spectrum) decomposes the noise variance $\sigma^2$ over spatial frequency. It is measured by scanning a uniform phantom repeatedly and averaging the 2D periodograms of the noise-only difference images. Integrated over all frequencies it returns $\sigma^2$, so the NPS tells you whether a given σ is made of coarse blobs or fine grain. FBP noise is far from white: the ramp filter tilts the spectrum up toward high frequencies, giving CT noise its characteristic texture.

The efficiency of the whole chain including the detector is measured by the DQE (detective quantum efficiency), the ratio of output to input SNR²; here we focus on comparing reconstruction filters.

Iterative reconstruction and DLR are nonlinear, so their resolution and noise can depend on dose, contrast, and the object. Conventional MTF and NPS measurements alone are then insufficient. Modern task-based assessment combines a contrast-specific task transfer function (TTF), NPS, and a detectability index $d'$ for a defined observation task.

Simulation: measuring MTF and NPS

The MTF is measured from the reconstructed PSF of a tiny disk, and the NPS from 6 noise difference images of a uniform water phantom, simultaneously for three filters. Ram-Lak holds the highest MTF but pays with strong high-frequency NPS (grainy images); Hann is the reverse. Note that changing $I_0$ scales the NPS up and down without changing its shape, and that the texture of the noise difference image changes with the filter. Since the measurement uses a disk of finite size, the curves are effective MTFs including the disk's shape factor, which does not affect the comparison between filters.

Image quality is defined by the task

σ, MTF, and NPS are machine-side rulers, but the clinical question is one of detectability: whether a given lesion can be seen. The Rose criterion is its simplest estimate from signal area, contrast, and noise; more rigorous task-based assessment replaces the human reader with statistical model observers (NPW, CHO). This connects directly to the evaluation problem of deep-learning reconstruction discussed in Chapter 12: methods that improve RMSE or PSNR can nevertheless reduce low-contrast detectability, so "numerically cleaner" and "diagnostically better" are not the same thing.

Research on dose reduction follows three broad paths: better detectors (photon-counting CT, Chapter 13), acquiring less data (sparse-view reconstruction, next chapter), and compensating with prior knowledge (the regularization of Chapter 8 and the learned priors of Chapter 12). The next chapter takes the second path: can we still reconstruct an image after throwing away most of the projections?

References

• Shope TB, Gagne RM, Johnson GC. A method for describing the doses delivered by transmission x-ray computed tomography. Medical Physics 8, 488–495 (1981) — the original CTDI paper.
• ICRP. Publication 103: The 2007 Recommendations of the International Commission on Radiological Protection (2007).
• AAPM. Report 204: Size-Specific Dose Estimates (SSDE) in Pediatric and Adult Body CT Examinations (2011).
• AAPM. Report 220: Use of Water Equivalent Diameter for Calculating Patient Size and SSDE in CT (2014).
• ICRU. Report 87: Radiation Dose and Image-Quality Assessment in Computed Tomography. Journal of the ICRU 12(1) (2012) — the standard reference for MTF/NPS/detectability measurement.
• Samei E et al. Performance evaluation of computed tomography systems: Summary of AAPM Task Group 233. Medical Physics 46, e735–e756 (2019).
• Rose A. The Sensitivity Performance of the Human Eye on an Absolute Scale. JOSA 38, 196–208 (1948) — the Rose criterion.