Understanding Noise in Low-Dose CT

 This blog post is about low-dose CT imaging, focusing on how the reduction in radiation dosage can inadvertently introduce increased image noise and artifacts, ultimately impacting the overall image quality. Low-dose CT imaging aims to minimize patient radiation exposure while producing diagnostically valuable images. Reducing radiation exposure for patient safety often leads to a diminished signal-to-noise ratio, as there are fewer photons reaching each detector. Accordingly, reducing radiation exposure often leads to compromised image quality. 

The conventional Poisson noise model is inadequate in accurately depicting the complexities of low-dose CT imaging. In reality, the degradation of image quality goes beyond mere heightened image noise. It is also influenced by artifacts arising from sinogram inconsistency, which is caused by subject-dependent beam-hardening associated with the polychromatic X-ray beam.  Moreover, these sources of image degradation interact with each other  during the image reconstruction process, leading to further deterioration.  Therefore, understanding the multifaceted interplay of various factors contributing to image degradation extends beyond addressing solely quantum and electronic noise.

X-ray beam and Radiation exposure in low-dose CT

In CT imaging, X-ray beams are directed through a patient positioned between an X-ray source and a detector. These beams are emitted in various directions by rotating the gantry housing the X-ray source and detector assembly. Subsequently, the detector acquires projection data $P(\varphi, s)$, where $\Theta_{\varphi} =(\cos\varphi,\sin\varphi),~\varphi\in [0,2\pi)$ represents the projection angle, and $s$ denotes the detector position. Here, the projection data $P$ is defined as $P = \ln ( \frac{I_{out}}{I_{in}} )$, where $I_{in}$ denotes the intensity of X-ray photons in the incoming beam, and $I_{out}$, measured by the detector, represents the intensity of X-ray photons transmitted through the body.

  • Polychromaytic X-ray beam: The incident X-ray beams composed of a spectrum of photons with various energies. different energies ranging from around 60 kiloelectronvolts (keV) to 140 keV. Tube voltage settings (kVp) act as a control parameter that regulates the maximum energy of the X-ray photons produced.  Lower energy X-rays (closer to 60 keV) are more readily absorbed by soft tissues and are suitable for imaging structures with lower contrast, such as soft tissues. Higher energy X-rays (closer to 140 keV) can penetrate tissues more effectively and  higher kVp settings tend to produce images with higher contrast and reduced image noise. As a result, higher energy X-rays are better suited for imaging denser structures like bone or metal implants.  
  • Polychromatic X-ray and Energy-dependent attenuation coefficient $\mu$: The projection data $P$ for polychromatic X-rays can be described using the integral form: $ P(\varphi,s)=\int \eta(E) P_E(\varphi,s)dE$, where $P_E$ denotes the projection data at energy level $E$, and $\eta(E)$ represents the fractional energy distribution of photons at energy $E$. The support of $\eta(E)$ lies within the interval $[0,200]$, with $\int \eta(E) dE=1$. The X-ray attenuation coefficient ($\mu_E$) varies with energy $E$. An illustration below shows the relationship between the attenuation coefficient $\mu_E$ and the fractional energy distribution $\eta(E)$.  The figures below illustrate how the value of $\mu$ for each material varies with energy $E$.  For soft tissue, $\mu(\text{soft tissue}, 30 ~ \text{keV}) \approx 0.38 ~ \text{cm}^{-1}$ and $\mu(\text{soft tissue}, 60 ~ \text{keV}) \approx 0.21 ~ \text{cm}^{-1}$. For water, $\mu(\text{water}, 10 ~ \text{keV}) \approx 5 ~ \text{cm}^{-1}$ and $\mu(\text{water}, 100 ~ \text{keV}) \approx 0.17 ~ \text{cm}^{-1}$. For bone, $\mu(\text{bone}, 10 ~ \text{keV}) \approx 144 ~ \text{cm}^{-1}$ and $\mu(\text{bone}, 100 ~ \text{keV}) \approx 0.40 ~ \text{cm}^{-1}$.         

  • Lambert-Beer Law for Polychromatic X-rays. Notably, the attenuation coefficient $\mu$ of hard objects fluctuates significantly with energy below 60keV, while that of soft tissue remains relatively stable across different energy levels.  CT image reconstruction relies on the Lambert--Beer law, which is expressed as follows: $$P(\varphi,s)=-\ln(\int\eta(E) e^{-\mathcal R \mu_E(\varphi,s)}dE), $$ where $\mathcal R \mu(\varphi,s)=\iint \mu(x_1,x_2) \delta ((x_1,x_2) \cdot (\cos\varphi,\sin\varphi)-s) d x_1 d x_2$ is the Radon transform.  The CT model commonly reconstructs $\mu$ without accounting for its dependency on energy ($E$), following the ideal linear model $P = \mathcal R \mu$.  As a result, a discrepancy emerges between the nonlinear projection data $P$, influenced by the polychromatic nature of X-rays, and the linear mathematical model $P = \mathcal R \mu$ built upon the ideal monochromatic assumption.

    • Simple understanding of the mismatch between $P$ and the linear model $P = \mathcal R \mu$: The figure below showcases a toy example of reconstructed CT images, illustrating the Beer-Lambert law for bichromatic energy. Despite using homogeneous triangular subjects (water on top, iron on bottom), the reconstructed image displays variation along the vertical direction (depending on the thickness) when employing the standard reconstruction method.

    The figure below illustrates streaking artifacts resulting from a model mismatch caused by beam hardening effects: the average energy of the transmitted X-ray beam is higher in the horizontal direction compared to the vertical direction.

    • Tube current represents the number of X-ray photons emitted per unit time. Increasing the tube current results in a higher number of X-ray photons being emitted, which leads to a higher signal-to-noise ratio and improved image quality. However, higher tube currents also result in increased radiation exposure to the patient.
    • Patient's radiation dose: Radiation interacts with human tissues through penetration, scattering, or absorption, with key interactions like Compton scattering and photoelectric absorption playing crucial roles in diagnostic imaging. Compton scattering occurs when a photon interacts with an atom's outer electron, causing both scattering and ionization, thus contributing to the patient's radiation dose. Radiation exposure in CT refers to the amount of ionizing radiation patients receive during the procedure. The radiation dose delivered to one slice of the body is affected by tube voltage, tube current, exposure time, source-to-image distance, and others. The radiation dose in microsieverts  ($\mu$Sv) for low-dose CT scans can vary based on factors such as the imaging protocol, patient size, anatomical region scanned, and the specific CT scanner utilized. Typically, radiation doses for low-dose CT scans range from approximately 1 to 10 microsieverts per scan. 
    • Collimation restricts the X-ray beam's cross-sectional area to match the image receptor's size, minimizing unnecessary radiation exposure to surrounding tissues.
    • Sparse data sampling, utilized in low-dose CT imaging, involves acquiring fewer projections during the scan. 
    • Local CT, also known as limited-area CT or focused CT, scans only a specific region of interest, resulting in only that area being exposed to X-ray radiation. A prominent example of local CT application is seen in dental cone-beam computed tomography (CBCT) scans.

    Noise and Artifacts in Low-Dose CT

    In the context of low-dose CT studies, where sinogram data may be noisy, undersampled, or both, efforts have predominantly concentrated on enhancing image quality by addressing noise and artifacts within the image space, rather than directly manipulating the sinogram data. The standard filtered backprojection (FBP) method commonly used to reconstruct  CT images from imperfect sinograms. Subsequently, denoising or image restoration from these reconstructed images can be achieved through various iterative methods that utilize image priors as a form of regularization. This preference for working within the image space is mainly due to the significant challenges associated with directly denoising sinogram data without referencing the reconstructed images. Maintaining sinogram consistency presents a complex challenge, as deviating from the range space of the forward model during denoising can severely compromise the quality of the resulting reconstructed image, potentially leading to further degradation rather than improvement.

    As previously discussed, reducing the tube voltage (kVp) in low-dose CT scans is an effective means of minimizing radiation exposure to patients. Lower kVp settings generate X-ray photons with diminished energy level. Noting that lower-energy photons exhibit increased absorption by tissues and that the attenuation coefficient varies more at lower energy levels, this can increase the sinogram inconsitency induced from beam-hardening effects (that are depending on the imaging subject). This inconsistency, combined with  noise from low SNR, can synergistically contribute to image degradation and streaking artifacts in reconstructed images through FBP. 

    Similarly, when noise combines with undersampled data, such as sparse data sensing and data truncation in local CT, their interaction can further degrade image quality. Therefore, it is desirable for practical studies in low-dose CT and its simulation to consider these compounded factors.

    Reference

    [1]  Deep Learning for for Dental Cone-Beam Computed Tomography, CM Hyun, T Bayaraa, SM Lee, H Jung, JK Seo, Deep Learning and Medical Applications, Springer Nature, 2023, https://link.springer.com/book/10.1007/978-981-99-1839-3

    [2] Characterization of metal artifacts in X‐ray computed tomography, HS Park, JK Choi, JK Seo, Communications on Pure and Applied Mathematics, 2017

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