Energy-Threshold Bias Calculator: A Physics-Model Based Adaptive Correction Scheme for Photon-Counting CT

TL;DR

The paper addresses spectral inconsistency in photon-counting CT, caused by inter-pixel energy-threshold bias and spectral skew, which manifest as ring and band artifacts in reconstructions. It introduces ETB-Cal, a physics-based two-term model that separates a fixed spectral skew term $g_i(E)$ from a pixelwise energy-threshold bias $\Delta E_i^k$, enabling robust, computationally efficient, pixel-wise spectral corrections with minimal calibration data and no XRF materials. The method is validated through numerical simulations and physical experiments on phantoms, showing substantial reductions in non-uniformity (e.g., from 29.3 HU to 5.8 HU in MEPT and 27.9 HU to 3.2 HU in the Kyoto head) and improved material-decomposition accuracy. ETB-Cal also demonstrates favorable comparisons against two existing model-based methods and remains compatible with pileup correction, highlighting its practical potential for real-world PCCT spectral correction and quantitative imaging.

Abstract

Photon-counting detector based computed tomography (PCCT) has greatly advanced in recent years. However, spectral inconsistency, referring to inter-pixel variations in detected counts per energy bin, can easily leads to ring or band artifacts and inaccuracies in CT reconstructed images. This work proposes a novel physics-model based method to correct for spectral inconsistency by modeling it through two terms: (1) a fixed spectral skew term (energy threshold-independent filtration function) determined at a given energy threshold, and (2) a variable energy-threshold bias term that can be directly calculated by using our spectral model as the threshold changes. After the two terms being computed out in the calibration stage, they will be incorporated into our spectral model to adaptively generate the spectral correction vectors as well as the material decomposition vectors if needed, pixel-by-pixel for PCCT projection data. Using a minimum set of parameters with explicit physics meaning, such an energy-threshold bias calculator (ETB-Cal) has advantages of computational efficiency, robustness in implementation, and convenience with no need of X-ray fluorescence materials in calibration. To validate our method, both numerical simulations and physical experiments using multiple phantoms were carried out on a tabletop PCCT system, with preliminary results showing a significant reduction in non-uniformity, from 29.3 to 5.8 HU for Gammex multi-energy phantom versus no correction (comparatively, 8.3 HU was achieved by a polynomial-involving model-based approach with no explicit modeling and calculating of energy threshold bias but more calibration data required), and from 27.9 to 3.2 HU for the Kyoto head phantom.

Figures (14)

• Figure 1: Schematic diagram of PCCT data acquisition mechanism, which illustrates the sources of spectral inconsistency in the proposed physics-based two-term ($\Delta E^{k}_{i}$ and $g_{i}(E)$) factorized spectral model and the specific forms of the model parameters. $E^{k}_{T}$ is the $k$-th nominal energy threshold with bias $\Delta E^{k}_{i}$ varying across different detector pixels, $g_{i}(E)$ is the spectral skew term determined by thickness deviations ($\Delta \textit{L}_\textit{b,i}$) of different material combinations.
• Figure 2: The overall framework of the spectral inconsistency correction. Based on the proposed spectral model, $\alpha_{i}$, $\Delta D_{i}$, $\Delta T_{i}$, $\Delta E^{k}_{i}$ can be determined through a series of attenuation measurements with known $\mu _{n}$ and $L _{n}$. By applying these parameters to beam hardening correction process and material decomposition process, we can correct the ring and band artifacts resulting from the spectral inconsistency in the reconstruction images.
• Figure 3: Energy response functions $\bar{R}(E,E^\prime)$, when E = 40, 80, and 120 keV. Dashed line: schematic of the energy response function, which incorporates three principal components: (1) the primary photopeak resulting from incident photon energy deposition, (2) the characteristic fluorescence escape peak, and (3) a modeled constant Compton continuum. Solid line: The energy response function, generated by our Monte Carlo simulations, used in our experiments. Based on the decoupling of the energy threshold bias (ETB) term and spectral skew term $[\Delta E^{k}_{i}$ and $g_{i}(E)]$, a reasonably appropriate base energy threshold $E^{0}_{T}$ can be selected in the low-energy range.
• Figure 4: Experiment setup of photon-counting CT: (a) Phantom experiment; (b) XRF-based calibration experiment.
• Figure 5: Parameter-solving accuracy of ETB-Cal algorithm with a single-material spectral skew term. The $\Delta D_{i}$ (selected as CdTe) distributed from -0.075 to 0.1 mm, which accounted for both inter-module and intra-module random variations, and the $\Delta E_{i}$ from -10 keV to 10 keV following a truncated gaussian distribution were incorporated. Both the absolute and relative errors in the solutions of $\Delta D_{i}$ and $\Delta E_{i}$ were minimal, and excellent results were achieved in correcting inconsistencies in the reconstructed images of a virtual water phantom containing four slightly denser (1.05 times) regions, without the addition of noise.