Calibration of the PLB forward-model and accuracy in real PCD applications

The basic procedure for calibration of the surrogate energies and intensities in the PLB model has been outlined previously in section 6.2. This section will now focus on its practical implementation and study the quantitative accuracy of the forward-model in real experimental settings. In the current set-up configuration we employ two staircase-shaped calibration phantoms of different materials with accurately known thicknesses to enable spectral decomposition into a two-material basis. The calibration phantoms are positioned between the x-ray tube and the sample holder using two additional linear-stages and can be moved separately enabling to cover the complete grid of calibration points (cf. figure 7.1 B).

A convenient choice of materials is poly-vinyl chloride (PVC) and poly-methyl methacrylate (PMMA) which are then often converted into the contributions of photoelectric effect and Compton scattering basis.

Figure 7.2 shows a schematic representation of the calibration process. Due to the placement of the calibration phantoms in a cone-beam geometry, effects of the cone angleαacross the detector lines need to be taken into account. In general, this leads to different calibration line-integrals on a pixel-by-pixel basis, since the path length of the x-ray beam through the object varies with the cone angle seen be each detector row. From the magnified view in figure 7.2 we obtain that this path length|~x|is linked to the nominal thickness|d~|of the calibration step via

|~x|= |d~|

cos(α/2) = |d~| cos

arctan

D/2 SID

, (7.2)

90

7.2 Calibration of the PLB forward-model and accuracy in real PCD applications SID

Detector

PVC PMMA

30o α

Collimator

D Source-spot

Cone-beam α/2

Figure 7.2:Overview of the PLB model calibration process in the experimental set-up. The schematic drawing summarizes the calibration process using PVC and PMMA staircase blocks and visualizes the geometric considerations required when determining pixel-by-pixel corrections for the cone-beam effects. The SID and other distances are not up to scale in this sketch.

whereDis the height of the detector measured in the same units as the SID. An analog calculation is valid for the perpendicular direction, i. e. the detector columns along the fan angleβ.

The currently available calibration phantoms offer up to eight steps per material. The maximum range for the calibration of the line-integrals is80mm for PMMA and48mm for PVC. Due to the benefits when working with a photoelectric effect and Compton scattering attenuation basis (cf. section 6.2), this grid of calibration points is typically converted into the associated line-integralsA_CandA_phaccording to equation 6.6 which is visualized by figure 7.3. Since the attenuation of x-rays follows an exponential curve when the magnitude of line-integrals increases, the gradient of the expected number of photon counts is large for small line-integral values and decreases towards larger thickness. Therefore, the density of calibration points is higher at low line-integral values to make the parameter estimation process more robust in regions with large gradients. Furthermore, the number of frames acquired at

7 Practical implementation of photon-counting based material decomposition

Figure 7.3:Calibration range and distribution of points in the experimental set-up. The current phantoms allow calibrations using up to eight steps for each material where the maximum thicknesses are80mm PMMA and48mm PVC (A). The corresponding conversion into coefficients of a photoelectric effect / Compton scattering basis are shown in plot B.

each combination of calibration steps is adjusted to keep the recorded photon statistic at a constant level for all calibration points. Therefore, a constant mean numberN of photons is registered during the calibration measurement process for each energy bin and line-integral. Prior to the fitting of the parameters, the images for each calibration step were normalized using their respective frame numbers.

In this way the effects of Poisson noise on the resulting PLB parameters were minimized, especially at larger line-integral values.

The experimental settings of the tube source and the detector for this study are summarized in table 7.1.

Peak voltage Filtration N THLL THLH Frame rate 120kVp 0.2mm Cu 5×10⁴ 27keV 52keV 10fps

Table 7.1:Acquisition settings for the investigation of the forward-model accuracy.The numberN refers to the total number of photon counts seen be the low PCD threshold.

In order to allow the experimental set-up to reach a steady state of operation, the source and detector were given approximately one hour of settling time before the calibration process was started. In this way, temperature-induced drifts of the x-ray source emission or the detector thresholds were minimized.

92

7.2 Calibration of the PLB forward-model and accuracy in real PCD applications

Calibration points Relative error / %

Relative error / %

Figure 7.4:Accuracy of the PLB forward-model in an experimental set-up. The surface plot in (A) shows the global performance of the forward-model after fitting to the calibration data corresponding to the grid in figure 7.3. The relative bias of the predicted counts barely exceeds1.5% (B). An examination of the bias histogram for all pixels and calibration points (C) yields that nearly all values lie within a3σinterval as determined by the Poisson statistic of the calibration measurement. The values in this figure are for the low PCD energy bin and the mean number of photons wasN_L = 2.8×10⁴andσ=√

N_L.

A quantitative examination of the forward-model accuracy in the low energy bin is shown in figure 7.4. For an arbitrary detector pixel, the number of photon counts determined by the forward-model corresponds well to the experimental data. In part A of the figure, this is represented by a 2D surface dependent on the basis material line-integrals. Panel B shows the relative bias of the predicted counts for the same pixel over all 64 points in the calibration grid. Since all calibration points were effectively acquired under the same photon statistic, a global relative noise level can be attributed to all points. For the present study, the number of photons in the low energy bin wasN_L= 2.8×10⁴. Usingσ =√

N_L the relative amount of noise contained within a3σinterval of the Poisson distribution was1.8%. The experimentally observed bias values lie all within this interval. A statistic evaluation for all detector

7 Practical implementation of photon-counting based material decomposition

pixels and all calibration points yields the bias histogram in figure 7.4 C. The behavior of the bias is characterized by a Gaussian distribution with mean value of zero and a FWHM corresponding to the uncertainty of count numbers (FWHM≈2.36σ).

From this it becomes evident that the accuracy of the forward-model over the complete detector and range of line-integrals is governed by the photon statistics and can hence be treated as unbiased. The performance is seemingly not affected by effects present in real PCD system. The following sections will therefore focus on investigations of the quantitative accuracy and quality of material-decomposed images.