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The accuracy of conventional superposition or convolution methods for scatter correction in kV-CBCT is usually compromised by the spatial variation of pencil-beam scatter kernel (PBSK) due to finite size, irregular external contour and heterogeneity of the imaged object. This study aims to propose an analytical method to quantify the Compton single scatter (CSS) component of the PBSK, which dominates the spatial distribution of total scatter assuming that multiple scatter can be estimated as a constant background and Rayleigh scatter is the secondary source of scatter. The CSS component of PBSK is the line integration of scatter production by incident primary photons along the beam line followed by the post-scattering attenuation as the scattered photons traverse the object. We propose to separate the object-specific attenuation term from the line integration and equivalently replace it with an average value such that the line integration of scatter production is object independent but only beam specific. We derived a quartic function formula as an approximate solution to the spatial distribution of the unattenuated CSS component of PBSK. The “effective scattering center” is introduced to calculate the average attenuation. The proposed analytical framework to calculate the CSS was evaluated using parameter settings of the On-Board Imager kV-CBCT system and was found to be in high agreement with the reference results. The proposed method shows highly increased computational efficiency compared to conventional analytical calculation methods based on point scattering model. It is also potentially useful for correcting the spatial variant PBSK in adaptive superposition method.

In kV cone beam CT, the 2D projection on the planar detector is contaminated by significantly increased scatter due to the large irradiated object volume compared to conventional CT modalities with fan-beam geometry. The increased scatter can be comparable to the primary X-ray transmission in magnitude and thus severely degrades image quality by introducing quantification errors and image artifacts [

The X-ray scatter consists of any secondary photons other than the incident primary photons and is mainly produced by direct photon interactions with medium, including Compton (incoherent) and Rayleigh (coherent) scattering. In Compton scattering, an incident photon transfers part of its energy to an outer shell atomic electron and is deflected from its incident direction. In Rayleigh scattering, the incident photon only changes its direction with no energy transferred. Analytic formulas have been discovered to calculate the distribution of the first-order Compton scatter using the Klein-Nishina formula and Rayleigh scatter using the Thomson formula [

Many methods have been proposed and developed for determining the spatial distribution of scatter in CBCT including: measurements using beam-stopper-array [

In this study, we propose an analytical model to calculate the spatial distribution dominant CSS component of the PBSK aiming to improve the computational accuracy of the PBSK based superposition method for scatter correction. The CSS component of the PBSK can be analytically formulated as line integration over the pencil-beam range. In the proposed method, the attenuation term is separated from the line integration and approximated by using an average attenuation path length and an effective attenuation coefficient. For the line integration of the unattenuated CSS, we derived a compact formula for easy implementation (Section 2.1). We also introduced the effective scattering center on the PBSK beam line as the point source of the unattenuated CSS so that the average attenuation could be efficiently calculated (Section 2.2). Exact calculations of the CSS of the PBSK with varying parameter settings and the integrated CSS from point scattering targets in a slab phantom were benchmarked to evaluate the performance of the proposed model (Section 2.3).

_{−} to x_{+} with the origin defined at the isocenter. The pencil-beam has a finite size of cross section area A_{pb} when projected at the detector plane. Denoting d Φ S ( x , r ) as the differential fluence of the CSS produced from a beam segment at x to point P with off-axis distance r on the detector plane, the spatial distribution of the CSS can be written as a line integration as Equation (1).

Φ S ( r ) = ∫ x − x + d Φ S ( x , r ) ⋅ e − μ S ( x , r ) ⋅ l ( x , r ) (1)

The attenuation path length l ( x , r ) and attenuation coefficient μ S ( x , r ) of the scattered photons when traversing the object are dependent of the external contour and heterogeneity of the object, which makes the scatter kernel object-dependent and spatially variant.

We assume that for a particular PBSK, the variations of the attenuation path length and attenuation coefficient of the CSS are smooth with respect to the beam segment x and the differential attenuation term can be equivalently replaced by an average value that is independent of the integration variable x. Mathematically, the above line integration is approximately calculated using Equation (2).

∫ x − x + d Φ S ( x , r ) ⋅ e − μ S ( x , r ) ⋅ l ( x , r ) ≈ ( ∫ x − x + d Φ S ( x , r ) ) ⋅ e − μ ¯ S ( r ) ⋅ l ¯ ( r ) (2)

Hereinafter, we denote the first part in the bracket, on the right side of Equation (2), as the “unattenuated CSS component of PBSK” and the second term as the “post-scattering attenuation of CSS”. In the following, we will derive analytical formulas as approximate solutions for the two parts.

Considering the beam segment as a point target, the number of Compton photons produced from segment B to pixel P, as shown in

d N S ( x , r ) = Φ p r i ( x ) ⋅ d σ d Ω ( θ x ) ⋅ λ e ( x ) ⋅ d x ⋅ d Ω x (3)

The parameter Φ p r i ( x ) is the fluence of incident primary photons at the segment x, d σ d Ω ( θ x ) is the Compton interaction cross section (given by the

Klein-Nishina formula [_{pxl}) with respect to segment B. They are given by the following equations.

Φ p r i ( x ) = Φ P ⋅ e μ ( x − x − ) ⋅ ( s + h s − x ) 2 (4)

λ e ( x ) = ρ e ⋅ A p b ⋅ ( s − x s + h ) 2 (5)

d Ω x = A p x l ⋅ cos ( θ x ) r 2 + ( h + x ) 2 (6)

d σ d Ω ( θ x ) = r 0 2 2 ( h ν ′ h ν ) 2 ( h ν h ν ′ + h ν ′ h ν − sin 2 ( θ x ) ) (7)

In above, Φ P is the projection fluence of the primary beam at the detector plane, µ is the linear attenuation coefficient of the primary photons, ρ_{e} is the volumetric electron density, r_{0} is the classic electron radius, hν and hv' are the energies of incident and scattered photons, respectively. The relation between hν and hv' is [

h ν ′ h ν = 1 1 + h ν m 0 c 2 ( 1 − cos ( θ x ) ) (8)

where m 0 c 2 is the electron rest energy (~511 keV).

We define short denotations for the relative photon energy and the cosine of scattering angle as following:

E = h ν m 0 c 2 , p = cos ( θ x ) (9)

Substituting Equations (4)-(9) into Equation (3), the differential fluence of CSS can be given by Equation (10).

d Φ S ( x , r ) = d N S ( x , r ) A p x l = ( r 0 2 2 ⋅ ρ e ⋅ A p b ) ⋅ Φ P ⋅ e μ ( x − x − ) r 2 + ( h + x ) 2 ⋅ g ( p ) ⋅ d x (10)

The term g(p) is a function of p and relative photon energy E, given by Equation (11).

g ( p ) = ( 1 1 + E − E p ) 2 ( p 3 + E p ( 1 − p ) + p 1 + E − E p ) (11)

We derived that under the conditions of E ( 1 − p ) < 1 and r h + x < 1 , the integration of d Φ S ( x , r ) as in Equation (10) over x ∈ ( x − , x + ) can be approximated to be a compact form (Equation (12)) as a quartic function of off-axis distance r (see the details in the Appendix).

Φ S ( r ) = ∫ x − x + d Φ S ( x , r ) ≈ ( r 0 2 2 ⋅ ρ e ⋅ A p b ) ⋅ Φ P ⋅ ( C 0 − C 2 ⋅ r 2 + C 4 ⋅ r 4 ) (12)

The coefficients C_{k} (k = 0, 2, 4) are independent of off-axis distance r and given by the following:

C 0 = 2 ⋅ ∫ x − x + e μ ( x − x − ) ( h + x ) 2 d x C 2 = 2 ( E + 2 ) ⋅ ∫ x − x + e μ ( x − x − ) ( h + x ) 4 d x

C 4 = ( 7 4 E 2 + 11 2 E + 25 4 ) ⋅ ∫ x − x + e μ ( x − x − ) ( h + x ) 6 d x (13)

Upon given incident photon energy and imaging system geometry, these coefficients are specified by the pencil-beam range and can be pre-calculated as a lookup table with varying pencil-beam lengths.

In the coordinate system of the detector plane, we define u → as the calculating point of a detector pixel and u → ′ as the projection position of a pencil-beam. The general formula to calculate the unattenuated CSS at u → contributed by pencil-beam at u → ′ is:

Φ S ( u → , u → ′ ) = ( r 0 2 2 ⋅ ρ e ⋅ A p b ) ⋅ Φ P ( u → ′ ) ⋅ ( C 0 ( u → ′ ) − C 2 ( u → ′ ) ⋅ ( u → − u → ′ ) 2 + C 4 ( u → ′ ) ⋅ ( u → − u → ′ ) 4 ) (14)

We assume the average post-scattering attenuation term (as in Equation (2)) can be equivalently calculated using the attenuation path length from a specific point on the beam line to the calculating point r and the corresponding attenuation coefficient determined by the scattering angle formed between them. We define this specific point as the “effective scattering center” of the integrated unattenuated CSS. The position of the effective scattering center on the beam line can be solved from Equation (15) as an approximate solution (see the details in the Appendix):

e μ ⋅ x ¯ ( r ) = ∫ x − x + w ( x , r ) ⋅ e μ ⋅ x ( h + x ) 2 d x ∫ x − x + w ( x , r ) ( h + x ) 2 d x , with w ( x , r ) = ( h + x ) 2 − r 2 ( h + x ) 2 + r 2 (15)

Similar to the quartic formula coefficients C_{k} (k = 0, 2, 4) (Equation (13)), the position of effective scattering center as a function of r can be pre-calculated as a lookup table with varying pencil-beam lengths for given incident photon energy and imaging system geometry.

To evaluate the performance of the derived analytical solutions as described in the preceding sections, we used the specifications of the Varian On-Board Imager (OBI) system (Varian Medical Systems, Palo Alto, CA) [^{2}.

The accuracy of using the quartic formula (Equation (12)) to approximate the exact line integration of the unattenuated CSS is dependent of three parameters including the primary photon energy (hν), the isocenter-to-detector height (h) and the size of the object (x_{−}, x_{+}). The profiles of the unattenuated CSS were calculated and compared between the quartic formula solutions and the exact line integrations with varying hν (40 keV, 60 keV, 100 keV and 130 keV), h (40 cm, 50 cm, 60 cm and 70 cm) and (x_{−}, x_{+}) ((−5, 5) cm, (−10, 10) cm, (−15, 15) cm and (−20, 20) cm).

As the object external contour plays a major role in determining the post-scattering attenuation of the CSS, we select two representative types of external contour―round and flat―for evaluating the performance of the effective scattering center method for calculating the average post-scattering attenuation. The profiles of CSS as a function of off-axis distance r along the radial and axial cross sections of a cylinder (corresponding to circular and flat contours, respectively) with three different sizes (10, 20 and 40 cm in diameter) are calculated using the exact differential attenuation term and the effective scattering center method, with comparisons made.

A preliminary evaluation of applying the proposed model as an analytical solution for the CSS distribution from a volumetric object was performed on a slab water phantom (30 cm in thickness) with an incident cone beam X-rays (26 × 20 cm^{2} at the isocenter plane and ~40 × 30 cm^{2} at the detector plane). The primary photon energy is 58keV, which is the mean energy of the 125 kVp X-ray tube voltage used in the Varian OBI system. The SAD is 100 cm and the isocenter-to-detector height is 50 cm.

The benchmarked CSS distribution was calculated by conventional analytical method using point scattering target, in which the irradiation volume is discretized as voxels and the CSS contribution from each voxel to each detector pixel is calculated analytically with exact implementation of the Klein-Nishina formula followed by exact calculation for the attenuation term [

The relative root-mean-square-error (RMSE) in the CSS distribution is used as the metric to assess the accuracy of the proposed model. The RMSE is calculated using Equation (16), with Φ S m o d e l , Φ S r e f the CSS distribution on the detector (pixel dimension is n i × n j ) calculated by the proposed model and the benchmarked method, respectively.

RMSE = 1 n i ⋅ n j ∑ i , j ( Φ S m o d e l ( i , j ) − Φ S r e f ( i , j ) Φ S r e f ( i , j ) ) 2 ⋅ 100 % (16)

The computations were performed in MATLAB (MathWorks, Inc., Natick, MA) with a single processor (Intel^{®} i5, 2.4 GHz, 4 GB RAM).

The profiles of CSS calculated by the quartic formula (Equation (12)) and the exact line integration for photon energies 40 keV, 60 keV, 100 keV and 130 keV are shown in _{−}, x_{+}) = (−10, 10) cm. The deviation of the quartic formula compared to the exact line integration increases with the off-axis radius and slightly increases with photon energy. The maximum differences at r = 20 cm are +1.3%, +1.6%, +1.9% and +2.2% for hν =40 keV, 60 keV, 100 keV and 130 keV, respectively.

(−15, 15) cm and (−20, 20) cm with the same hν = 58 keV and h = 50 cm. The ranges of pencil-beam correspond to nominal object diameters 10 cm, 20 cm, 30 cm and 40 cm, respectively. With larger pencil-beam range, the quartic formula increasingly matches the exact line integration. The maximum differences at r = 20 cm are +2.1%, +1.5%, +1.0% and +0.7% for beam length = 10 cm, 20 cm, 30 cm and 40 cm, respectively.

The effective scattering center position on the pencil-beam as a function of off-axis distance was determined using Equation (15) for three different pencil-beam ranges (−5, 5) cm, (−10, 10) cm and (−20, 20) cm, with the corresponding cylinder radius (R) of 5 cm, 10 cm and 20 cm, respectively.

In the benchmarked calculations, the phantom volume within the X-ray field consists of [26, 20, 30] voxels with voxel size 1 × 1 × 1 cm^{3} and in the model calculations, the irradiated phantom volume consists of [26, 20] pencil-beams with beam size 1 × 1 cm^{2}. The CSS distribution on the detector plane consists of [80, 60] pixels with pixel size 0.5 × 0.5 cm^{2}.

The general form of PBSK applied by superposition or convolution methods for scatter calculation is usually obtained by measurements or Monte Carlo simulations using large slabs or disks and has symmetric, spatially invariant formulation [

The quartic formula as an approximation of the line integration of unattenuated CSS was derived under the conditions of E ( 1 − p ) < 1 and r h + x < 1 . In

kV-CBCT, the maximum photon energy is 130keV and the maximum value of E is 130/511 = 0.25. As shown in

r h + x are truncated in the approximation, the error of the quartic formula increases with the off-axis distance r. Assuming h = 50 cm and the object size less than 40 cm in diameter, the condition of r h + x < 1 may be violated at off-axis

distance r beyond 30 cm for beam segment (x) at the lower end of the pencil-beam and increased error can be caused by using the quartic formula. However, as the detector active area has a maximum size of 40 × 30 cm^{2}, such increased error only happens at the periphery of the detection area. Using larger isocenter-to-detector height (h) can increase the accuracy of the quartic formula, as shown in

length, the value of r h + x is reduced for beam segments at the upper end

which have more contribution to scatter due to larger incident primary photon fluence, and thus the truncation error is reduced as well.

We introduced the concept of effective scattering center as the equivalent scatter point source of the PBSK for calculating the average post-scattering attenuation. An analytical equation (Equation (15)) was obtained to determine the position of the effective scattering center with a first-order approximation made.

The scope of the current study has not considered medium heterogeneity in analytical calculation of the CSS component of PBSK. However, as the object-specific variance of the PBSK is determined by the post-scattering attenuation term, the impact of medium heterogeneity may be accounted for by ray tracing the attenuation path length from the effective scattering center. In addition, the electron binding effect in Compton interaction was considered as a secondary effect to the distribution of CSS and a correction factor may be included in future work.

The method proposed in this study shows highly increased computational efficiency compared to the conventional analytical calculation method based on point scattering model. It is also potentially useful for correcting the spatial variant PBSK in adaptive superposition calculation for the purpose of scatter correction in kV-CBCT.

This work awarded US patent US 9,615,807 B2, April 2017 [

Liu, J. and Bourland, J.D. (2018) Analytical Calculation of the Compton Single Scatter Component of Pencil Beam Scatter Kernel for Scatter Correction in kV Cone Beam CT (kV-CBCT). International Journal of Medical Physics, Clinical Engineering and Radiation Oncology, 7, 214-230. https://doi.org/10.4236/ijmpcero.2018.72019

From Equation (11), g(p) can be rewritten as:

g ( p ) = f 1 ( p − 1 ) + f 2 + f 3 1 p − 1 − 1 E + f 4 1 ( p − 1 − 1 E ) 2 + f 5 1 ( p − 1 − 1 E ) 3 (A1)

The coefficients f_{j} (j = 1, 2, 3, 4, 5) are

f 1 = 1 E 2 , f 2 = 2 E 3 + 3 E 2 − 1 E , f 3 = 3 E 4 + 6 E 3 + 1 E 2 − 1 E , f 4 = 1 E 5 + 3 E 4 + 1 E 3 , f 5 = − 1 E 4 − 1 E 3 (A2)

Applying Taylor series expansion for the last three terms in Equation (A1):

1 p − 1 − 1 E = − E ⋅ ∑ n = 0 ∞ ( − 1 ) n ( E ( 1 − p ) ) n 1 ( p − 1 − 1 E ) 2 = E 2 ⋅ ∑ n = 0 ∞ ( − 1 ) n ( n + 1 ) ( E ( 1 − p ) ) n 1 ( p − 1 − 1 E ) 3 = − E 3 ⋅ ∑ n = 0 ∞ ( − 1 ) n ( n + 1 ) ( n + 2 ) 2 ( E ( 1 − p ) ) n (A3)

By substituting Equations (A2) and (A3) into Equation (A1), g(p) then becomes:

g ( p ) = β 0 + β 1 ⋅ ( p − 1 ) + ∑ n = 2 ∞ ( − 1 ) n β n ( E ( 1 − p ) ) n (A4)

The coefficients are given as following:

β 0 = 2 , β 1 = 4 E + 4 , β 2 = 3 E 2 + 8 E + 7 , β n = ( 1 2 E + 1 2 ) n 2 + ( 1 E 3 + 3 E 2 + 5 2 E + 3 2 ) n + ( − 2 E 3 − 3 E 2 + 1 E + 2 ) ( for n ≥ 2 ) . (A5)

Because of lim n → ∞ β n + 1 / β n = 1 , the series in Equation (A4) is convergent when E ( 1 − p ) < 1 and the high order terms (n > 2) are truncated hereinafter.

The Taylor series expansion for p (the cosine of the scattering angle, as in Equation (9)) is:

p = h + x r 2 + ( h + x ) 2 = 1 − 1 2 ( r h + x ) 2 + 3 8 ( r h + x ) 4 + o ( ( r h + x ) 6 ) (A6)

Then g(p) can be written in the form of r / ( h + x ) as following:

g ( p ) = 2 − ( 2 E + 2 ) ( r h + x ) 2 + ( 7 4 E 2 + 7 2 E + 9 4 ) ( r h + x ) 4 + o ( ( r h + x ) 6 ) (A7)

Also,

1 r 2 + ( h + x ) 2 = 1 ( h + x ) 2 [ 1 − ( r h + x ) 2 + ( r h + x ) 4 + o ( ( r h + x ) 6 ) ] (A8)

Thus,

g ( p ) r 2 + ( h + x ) 2 = 1 ( h + x ) 2 [ 2 − ( 2 E + 4 ) ( r h + x ) 2 + ( 7 4 E 2 + 11 2 E + 25 4 ) ( r h + x ) 4 + o ( ( r h + x ) 6 ) ] (A9)

Under the condition of r / ( h + x ) < 1 and neglecting the high order terms (≥6) in Equation (A9), the differential form of Compton single scatter d Φ S ( x , r ) as in Equation (10) is given as a quartic function of r as following:

d Φ S ( x , r ) ≈ ( r 0 2 2 ⋅ ρ e ⋅ A p b ) ⋅ Φ P ⋅ [ c 0 ( x ) − c 2 ( x ) ⋅ r 2 + c 4 ( x ) ⋅ r 4 ] d x (A10)

The coefficients c_{k}(x) with k = 0, 2, 4 are given as following:

c 0 ( x ) = 2 ⋅ e μ ( x − x − ) ( h + x ) 2 c 2 ( x ) = 2 ( E + 2 ) ⋅ e μ ( x − x − ) ( h + x ) 4 c 4 ( x ) = ( 7 4 E 2 + 11 2 E + 25 4 ) ⋅ e μ ( x − x − ) ( h + x ) 6 (A11)

First consider the scatter reaching the detector point on the pencil-beam’s axis (i.e. r = 0). The scatter produced from segment x is

d Φ S ( x , r = 0 ) ∝ e μ x ( h + x ) 2 d x . The attenuation path length is given by l ( x , r = 0 ) = x − x − and the scattering angle is 0 such that the attenuation coefficient is µ. The effective scattering center position x ¯ can be accurately determined by:

e μ ⋅ x ¯ = ∫ x − x + e μ ⋅ x ( h + x ) 2 d x ∫ x − x + 1 ( h + x ) 2 d x (A12)

For r ≠ 0, the general form of d Φ S ( x , r ) is given by Equation (10) and the attenuation path length is determined by intersection of the ray with the object contour. It becomes complicated to accurately calculate the effective scattering center position using Equation (2). However, we can approximate the effective scattering center position for r ≠ 0 by introducing a correction factor based on the above equation for r = 0. Assume the attenuation path length is linear with x and change in the attenuation coefficient is negligible (for small scattering angle, the scatter photon energy by Equation (8) is very close to the primary photon energy). Meanwhile, approximate the term g(p) to the first order. Then the effective scattering center position can be approximately determined by Equation (15).