Spectral calibration of photon-counting detectors at high photon flux

Abstract

Background: Calibration of photon-counting detectors (PCDs) is necessary for quantitatively accurate spectral computed tomography (CT), but the calibration process can be complicated by nonlinear flux-dependent physical factors such as pulse pile-up.

Purpose: This work develops a method for spectral sensitivity calibration of a PCD-based spectral CT system that incorporates nonlinear flux dependence and can thus be employed at high photon flux.

Methods: A calibration model for the spectral response and polynomial flux dependence is proposed, which incorporates prior x-ray source spectrum and PCD models and that has a small set of parameters for adjusting to the spectral CT system of interest. The model parameters are determined by fitting transmission data from a known object of known composition: a step-wedge phantom composed of different thicknesses of aluminum, a bone equivalent, and polymethyl methacrylate (PMMA), a soft-tissue equivalent. This fitting employs Tikhonov regularization, and the regularization strength and the polynomial order for the intensity modeling are determined by bias and variance analysis. The spectral calibration and nonlinear intensity correction is validated on transmission measurements through a third material, Teflon, at different x-ray photon flux levels.

Results: The nonlinear intensity dependence is determined to be accurately accounted for with a third-order polynomial. The calibrated spectral CT model accurately predicts Teflon transmission to within 1% for flux levels up to 50% of the detector maximum.

Conclusions: The proposed PCD calibration method enables accurate physical modeling necessary for quantitative imaging in spectral CT. Furthermore, the model applies to high flux settings so that acquisition times will not be limited by restricting the spectral CT system to low flux levels.

Keywords: calibration; photon-counting detectors; spectrum estimation.

Figure 1:

Normalized models for the X-ray source spectrum and DxRay PCD detector response. the top and middle graphs show the functions R(E) and D_w(E), respectively, from Eq. (4). The bottom graph shows the product R(E)D_w(E).

Figure 2:

Model error as measured by ΔT_w, defined in Eq. (12), averaged over all 64 detector pixels. The energy window w increases along the column index, and the X-ray flux level increases with the row index. The different curves in each panel indicate the N_α value (linear corresponds to N_α = 1, and quartic corresponds to N_α = 4). the x-axis of each plot indicates log(γ).

Figure 3:

The grid of plots shows the standard deviation of the normalized spectrum model $s_w\left(E,{\beta }_w^{*}\right)$, averaged over E, and the NLF function $F_w\left(T,{\alpha }_w^{*}\right)$, averaged over T. Recall that the transmission fraction T varies from 0 to 1.

Figure 4:

The grid of plots shows the relative standard deviation (the standard deviation divided by the mean) of X-ray transmission through a 1% Gadolinium solution of different thicknesses.

Figure 5:

The calibrated transmission model, averaged over all 64 detector pixels, for all energy windows and different flux levels. The left column shows the normalized spectral sensitivities. The middle column shows the non-linear fluence function, and the right column shows the non-linear flux function based on the assumption that it is proportional to the fluence function and the slope near zero flux is unity.

Figure 6:

Top row: predicted X-ray transmission fractions through a 3.81 cm slab of Teflon for energy window w = 1 at the various flux levels. The measured transmission fraction (“+” symbols) is shown together with the calibration model without non-linear fluence fitting (red “ * “ symbols) and with the proposed non-linear fluence fitting (black curve). Bottom row: prediction of the Teflon slab thickness obtained by inverting the calibrated transmission model using the measured data as input. The Teflon slab thickness prediction is displayed as a difference from the true value of 3.81 cm.

Figure 7:

Same as Fig. 6 except the displayed results are for energy window w = 2.

Figure 8:

Same as Fig. 6 except the displayed results are for energy window w = 3.

Figure 9:

Same as Fig. 6 except the displayed results are for energy window w = 4.