A parametric fitting technique for rapid determination of a skin-dose correction factor for angle of beam incidence during image-guided endovascular procedures

Abstract

Skin dose is dependent on the incident beam angle and corrections are needed for accurate estimation of the risk of deterministic effects of the skin. To obtain the angular correction factors (ACF’s), EGSnrc Monte Carlo (MC) software was used to calculate the skin dose as a function of incident x-ray beam angle at the center of the field for beam energies from 60 to 120 kVp, field sizes from 5 to 15 cm, and thicknesses of Cu beam filters from 0.2 to 0.5 mm. All MC simulations used 3×10¹⁰ incident photons. The dose was averaged over a 1 mm depth on the entrance surface of a 40×40 cm by 20 cm thick water phantom and was then normalized to the incident primary dose which was calculated using NIST mass energy absorption coefficients and by integrating over the beam energy spectrum. The Matlab tool, ‘cftool’, was used to fit these normalized dose values to power law equations as a function of incident beam angle, with coefficients that were fit to polynomials as a function of kVp. Separate fitting was done for different beam sizes and beam filters. The skin dose values calculated using the ACF determined from the fitted functional formulas agreed with that calculated by MC with a mean absolute percentage error (MAPE) less than 3% over the entire range of incident angles and kVp values. This fitting technique allows an ACF to be quickly determined for accurate skin dose calculation.

Graphical Abstract

1. INTRODUCTION

Accurate estimation of the skin dose is important to evaluate the risk of deterministic cutaneous effects^1–2. The incident angle of the x-ray beam on the patient’s skin varies during fluoroscopically-guided procedures as shown in Fig. 1 and dose changes with the angle of beam incidence relative to the skin surface due to changes in primary attenuation and backscatter as illustrated in Fig. 2. The dose dependence on incident angle varies with field size and beam energy including kVp and beam filter. A parameter-dependent angular correction is thus needed for accurate estimation of dose to the cutaneous layer and a parametric fitting technique was developed to quickly determine the proper angular correction factor (ACF) to use for real-time dose estimation using our Dose Tracking System (DTS) during fluoroscopically-guided interventional (FGI) procedures^3–4.

Fig. 1.

Fluoroscopic procedures utilize various angulations of the c-arm gantry that, along with the body contours, can result in the incident beam being non-normal to the skin surface. The incident angle (θ) is defined as the angle of the beam central axis with the skin surface and normal incident is defined as the angle of 90 degrees.

Fig. 2.

a) Cross-section through the cutaneous layer showing the increased path length [t/sin(θ)] due to the x-ray going through the skin of thickness t at an angle (θ). b) Figure of the backscatter point spread function (BSPSF) at 60 kVp for different incident angles calculated by EGSnrc Monte Carlo (MC) software using a 1 mm² beam averaged over a 1 mm depth on the entrance surface of a 40 × 40 cm by 20 cm thick water phantom; the value was normalized to the incident primary dose. This BSPSF demonstrates the change in direction and magnitude of backscatter with angle of incidence.

2. METHODS

EGSnrc Monte-Carlo (MC) software was used to calculate the skin dose as a function of incident x-ray beam angle at the center of the field for beam energies from 60 to 120 kVp, field sizes from 5 to 15 cm, and thicknesses of Cu beam filters from 0.2 to 0.5 mm. All MC simulations used 3×10¹⁰ incident photons. The dose was averaged over a 1 mm depth on the entrance surface of a 40 × 40 cm by 20 cm thick water phantom (Figure 3) and was then normalized to the incident primary dose which was calculated using NIST mass energy absorption coefficients⁵ and by integrating over the beam energy spectrum. The Matlab tool, ‘cftool’, was used to fit these normalized dose values to power law equations as a function of incident beam angle, with coefficients that were fit to polynomials as a function of kVp. Separate fitting was done for different beam sizes and beam filters. This fitting technique allows an angular correction factor (ACF) to be quickly determined for accurate skin dose calculation (see Figure 4 and Figure 5).

Fig. 3.

Geometry of the Monte Carlo (MC) calculation of skin dose. The incident angle (θ) is the angle of the central axis with the skin surface. Field size is defined in a plane perpendicular to the central axis at its intersection with the skin. Results are for a 0.1 cm thickness on the entrance surface of a 40×40 cm, 20 cm thick water-equivalent phantom.

Fig. 4.

Example fitted curves. The Matlab tool, ‘cftool’, was used to fit a) normalized dose values to power law formulas as a function of incident beam angle: f(Angle) = (a1)*(Angle)^(a2) + (a3) and b) the coefficients of equations at each kVp were fit to polynomials as a function of kVp: a_n = p1*(kVp)^3 + p2*(kVp)^2 + p3*(kVp) + p4.

Fig. 5.

Example of coefficients determined for fitting equations for the 15 × 15 cm field size and beam filter thicknesses of 0.5 mm Cu.

The incident primary dose without backscatter was calculated as

$${\text{P}}_0\left(\text{E}\right)=\text{N}\left(\text{E}\right)\times \text{E}\times \frac{{\pi }_{\text{en}}}{\rho }\left(\text{E}\right)$$ (1)

integrated over energy E for the beam spectrum, where N(E) is the photon fluence and $\frac{{\pi }_{\text{en}}}{\rho }\left(\text{E}\right)$ is the mass energy absorption coefficient for water published by NIST.⁵

The angular correction factor (ACF) at the angle of incidence θ was calculated as

$$\text{Angular correciton factor }\left(\theta \right)=\frac{\text{Monte Carlo calculated skin dose }\left(\theta \right)}{\text{Incident primary dose}}$$ (2)

The ACF is the MC calculated skin dose normalized to the incident primary as a function of incident angle.

The ACF can then be used to determine the skin dose.

$$\text{Skin dose }\left(\theta \right)=\text{Incident primary dose x Angular correciton factor }\left(\theta \right)$$ (3)

3. RESULTS AND DISCUSSION

3.1. Fitted ACF vs incident beam angle

The curves in Fig. 6 show that the fitted ACF decreases with decreasing incident angle due to the longer path length of the primary x-rays through the skin thickness, increases with increasing kVp due to more Compton scattering, increases with increasing field size due to more surrounding scattering, increases with increasing thickness of Cu filter due to beam hardening effect.

Fig. 6.

Fitted curves of the angular correction factor (ACF) for different kVps and field sizes, and thickness of a) 0.2, b) 0.3, c) 0.4, and d) 0.5 mm Cu beam filter, at the field center as a function of incident angle to the surface of a 40 × 40 cm by 20 cm thick water phantom.

3.2. Fitted ACF vs thickness of Cu filter

The curves in Fig. 7 show that the fitted ACF increases with increasing thickness of Cu filter due to beam hardening effect.

Fig. 7.

Fitted curves of the angular correction factor (ACF) for a) 60, b) 80, c) 100, and d) 120 kVp, and for a 10 cm field, at different incident angle, at the field center as a function of thickness of Cu filter to the surface of a 40 × 40 cm by 20 cm thick water phantom.

3.3. Coefficients for fitting equations

The results of Tables 1, 2, 3, and 4 show the coefficients for fitting equations as a function of kVp for different field sizes and thickness of Cu filters. The coefficients of equations at each kVp were fit to polynomials as a function of kVp: a_n = p1*(kVp)^3 + p2*(kVp)^2 + p3*(kVp) + p4 and the angular correction factor is calculated as ACF = (a1)*(Angle)^(a2) + (a3).

Table 1.

Coefficients for fitting equations for 0.2 mm thickness of Cu filter for 5 × 5, 10 × 10, and 15 × 15 cm field sizes.

Field size : 5 cm × 5 cm
	p1	p2	p3	p4
a1	1.598E-06	−0.0004815	0.04654	−2.305
a2	3.146E-07	−1.02E-04	0.008689	−0.7587
a3	8.333E-08	−2.63E-05	0.002392	1.264
Field size : 10 cm × 10 cm
	p1	p2	p3	p4
a1	−5.123E-06	0.001307	−0.1086	1.948
a2	1.458E-08	−8.63E-05	0.01068	−0.6237
a3	1.604E-06	−0.0004675	0.04146	0.461
Field size : 15 cm × 15 cm
	p1	p2	p3	p4
a1	−5.917E-06	0.00152	−0.1247	2.162
a2	2.000E-06	−0.0006229	0.05786	−1.893
a3	5.979E-06	−0.00169	0.1502	−2.433

Table 2.

Coefficients for fitting equations for 0.3 mm thickness of Cu filter for 5 × 5, 10 × 10, and 15 × 15 cm field sizes.

Field size : 5 cm × 5 cm
	p1	p2	p3	p4
a1	9.708E-07	−0.0002219	0.01711	−1.382
a2	−2.854E-07	0.0001031	−0.0124	−0.1491
a3	−8.333E-08	2.75E-05	−0.003067	1.444
Field size : 10 cm × 10 cm
	p1	p2	p3	p4
a1	1.583E-06	−0.0003763	0.02904	−1.809
a2	2.021E-07	3.24E-05	−0.01287	0.2466
a3	−2.708E-07	0.0001238	−0.01652	2.191
Field size : 15 cm × 15 cm
	p1	p2	p3	p4
a1	2.521E-06	−0.0007813	0.08117	−3.892
a2	8.958E-07	−2.15E-04	0.01418	−0.4634
a3	−2.083E-07	1.53E-04	−0.02617	3.052

Table 3.

Coefficients for fitting equations for 0.4 mm thickness of Cu filter for 5 × 5, 10 × 10, and 15 × 15 cm field sizes.

Field size : 5 cm × 5 cm
	p1	p2	p3	p4
a1	9.083E-07	−0.0002721	0.02604	−1.728
a2	−7.083E-08	1.99E-05	−0.003179	−0.4791
a3	−6.250E-08	2.00E-05	−0.002225	1.411
Field size : 10 cm × 10 cm
	p1	p2	p	p4
a1	3.700E-06	−0.001047	0.0962	−3.833
a2	9.646E-07	−0.0002406	0.01653	−0.667
a3	−1.042E-07	5.75E-05	−0.009308	1.997
Field size : 15 cm × 15 cm
	p1	p2	p3	p4
a1	−1.208E-06	0.0004237	−0.04354	0.149
a2	1.931E-06	−0.0005546	0.05074	−1.686
a3	5.375E-06	−0.001555	0.1437	−2.287

Table 4.

Coefficients for fitting equations for 0.5 mm thickness of Cu filter for 5 × 5, 10 × 10, and 15 × 15 cm field sizes.

Field size : 5 cm × 5 cm
	p1	p2	p3	p4
a1	2.356E-06	−0.0007278	0.07385	−3.329
a2	−6.750E-07	0.0001306	−0.006233	−0.5891
a3	−5.417E-07	0.0001363	−0.01076	1.611
Field size : 10 cm × 10 cm
	p1	p2	p3	p4
a1	7.229E-06	−0.001934	0.1666	−5.688
a2	4.283E-06	−0.001088	0.08618	−2.588
a3	1.271E-06	−0.000305	0.02214	1.071
Field size : 15 cm × 15 cm
	p1	p2	p3	p4
a1	1.000E-06	−0.0002913	0.02963	−2.171
a2	−3.463E-06	0.0009655	−0.08729	2.284
a3	−5.521E-06	0.001538	−0.139	5.909

3.4. Fitted ACF vs MC calculated ACF

Fig. 8–9 shows the fitted curves for ACF using the coefficient of Table 1–4 and the MC calculated data points. The ACF determined from the fitted functional formulas agreed with that calculated by MC with a mean absolute percentage error (MAPE) of 2.2% over the entire range of incident angles, thickness of Cu filters, and kVp values.

Fig. 8.

MC data points and fitted curves of the angular correction factor (ACF) as a function of incident angle for different beam filters for a a) 5 × 5, b) 10 × 10, and c) 15 × 15 cm field size at 120 kVp.

Fig. 9.

MC data points and fitted curves of the angular correction factor (ACF) as a function of incident angle for different kVps. Results are shown for a) 5 × 5, b) 10 × 10, and c) 15 × 15 cm field sizes and at 60, 80, 100, and 120 kVp, for a beam filter thickness of 0.3 mm Cu.

4. CONCLUSIONS

Skin dose is dependent on the angle of beam incidence, with the value normalized to the primary entrance dose decreasing with decreasing angle of incidence and increasing with increasing kVp and thickness of filter. The angular correction factor was able to be parameterized as a function of kVp for a range of beam filters for rapid input into the algorithm of a real-time dose mapping software. The skin dose values calculated using the ACF determined from the fitted functional formulas agreed with those calculated by MC with a mean absolute percentage error (MAPE) just over 2% over the entire range of incident angles and kVp values.