Method for evaluating bow tie filter angle-dependent attenuation in CT: theory and simulation results

Abstract

Purpose: Dosimetry in computed tomography (CT) is increasingly based on Monte Carlo studies that define the dose in the patient (in mGy) as a function of air kerma (free in air) at isocenter (mGy). The accuracy of Monte Carlo studies depends in part on the accuracy of the characterization of the bow tie filter for a given CT scanner model. A simple method for characterizing the bow tie filter attenuation profile in CT scanners would therefore be very useful. The theory behind such a method is proposed.

Methods: A measurement protocol is discussed mathematically and demonstrated using computer simulation. The proposed method requires the placement of a radiation monitor at the periphery of the CT field, and the time domain signal (kerma rate versus time) is measured with good temporal resolution (-200 Hz or better) and with all other objects (e.g., patient couch) retracted from the field of view. Knowledge of the source to isocenter distance (or alternately, the isocenter to probe distance) is required. The stationary detector records the kerma rate versus time signal as the gantry rotates through several revolutions. From this temporal data, signal processing techniques are used to extract in-phase peaks, as well as out-of-phase kerma rate levels. From these data, the distance from isocenter to the probe can be determined (or, alternatively, the source to isocenter distance), and the angle-dependent bow tie filter attenuation can be computed. By measuring the angle-dependent bow tie filter attenuation at several kVp settings, the bow tie composition versus fan angle can be computed using basis decomposition techniques.

Results: The simulations illustrated that with 2% added noise in the kerma rate versus time signal, the attenuation properties of a hypothetical two component (aluminum and polymethyl methacrylate) bow tie filter could be determined (r2 > 0.99). Although the computed basis material thicknesses were not exactly equal to the actual thicknesses, their combined attenuation factors matched that of the actual filter across kVp's to within an average of 0.057%.

Conclusions: It is concluded that the proposed method may provide a simple noninvasive approach to characterizing the performance of bow tie filters in CT systems; however, experimental validation is necessary.

Figures

Figure 1

The geometrical basis of the theoretical development of the proposed technique is illustrated. (a) The gantry angle α and the fan angle θ are illustrated. The gantry angle is defined as α=0 when the isocenter, probe, and source are coaligned and when the source is on the probe side of isocenter. For the fan angle, θ=0 in the center of the field of view (at isocenter). The bow tie filter function is symmetric about θ=0. (b) The distances s, r, and g are defined, along with the coordinates of the x-ray tube, x-ray probe, and isocenter. The source to isocenter distance is given by s, the radius from the isocenter to the x-ray probe is r, and the distance from the x-ray source to the x-ray probe is given by g. The values of s and r remain constant through gantry rotation as do the probe coordinates, while g changes along with the x-ray tube coordinates.

Figure 2

To demonstrate the proposed method, a simple hypothetical bow tie filter was designed. The thicknesses of the polymethyl methacrylate (PMMA) and aluminum (Al) components of the filter, as a function of fan angle, are shown. The coefficients for generating the PMMA and Al thicknesses are provided in Table 1.

Figure 3

(a) The raw waveform X(t) is illustrated for a 5.5 s acquisition at 200 Hz. The gantry rotation time (and hence the period of the waveform) was 1.0 s. The peaks in the waveform correspond to points where the x-ray tube is closest to the x-ray source. (b) The first step in defining the bow tie filtration properties requires that the raw waveform X(t) be analyzed to identify the locations in the waveform corresponding to the peaks. The results of the automated detection scheme illustrate the location of the detected even-π peaks (tall vertical lines) and the odd-π phase locations (short vertical lines). (c) Using the extracted locations identified in (b), the Y(t) waveform corresponding to the effect of the inverse square law only was computed and is shown along with the original X(t) waveform (solid line), which includes the influences of the inverse square law and the bow tie filter F(θ). The difference between X(t) and Y(t) is solely dependent on the bow tie filter characteristics, and this is illustrated as the blackened region at the fifth peak in the figure. (d) The ratio of Y(t) to X(t) in (c) is shown in this figure as a solid line, as is the value of the fan angle θ (in radians) as the dotted line. Plotting these two curves with respect to each other is how the bow tie filter function, F(θ), is computed.

Figure 4

(a) The bow tie function F(θ) is shown versus θ, computed from the two waveforms illustrated in Fig. 3d. The gray data points are the raw data, and the solid circles are the average values binned in 1° intervals. This curve is for the 120 kVp spectrum. (b) A computer fitting routine was used to further smooth the data shown in (a). Here, the smoothed attenuation versus angle profiles for 80, 100, 120, and 140 kVp are illustrated.

Figure 5

(a) The actual filter thickness for PMMA and Al is shown as the solid line, and the computed values are shown as circles. These data are for the standard basis decomposition approach and clearly demonstrate that the thickness of Al is overestimated while the thickness of PMMA is underestimated. (b) The computed attenuation factors for the standard basis decomposition procedure are compared against the actual attenuation factors for 80 kVp (open circles), 100 kVp (squares), 120 kVp (diamonds), and 140 kVp (gray circles). For low attenuation factors corresponding to thicker objects, a pronounced difference between actual and computed attenuation factors is seen. The error at higher attenuation levels is thought to be due to beam hardening.

Figure 6

(a) The actual thickness of PMMA and Al is shown (solid lines), and the PMMA (black circles) and Al (open circles) thicknesses computed using the proposed iterative method are shown as symbols. Although more accurate than the simpler method whose results are shown in Fig. 5a, this approach does show a compensatory effect where overestimates of one material are balanced by underestimates of the other. (b) The computed attenuation is shown plotted versus the actual attenuation, and for the iterative method, the results are quite accurate—as indicated by the ∼unit slope and ∼zero intercept. The key to the symbols is defined in Fig. 5b.