A Monte Carlo investigation of Swank noise for thick, segmented, crystalline scintillators for radiotherapy imaging

Abstract

Thick, segmented scintillating detectors, consisting of 2D matrices of scintillator crystals separated by optically opaque septal walls, hold considerable potential for significantly improving the performance of megavoltage (MV) active matrix, flat-panel imagers (AMFPIs). Initial simulation studies of the radiation transport properties of segmented detectors have indicated the possibility of significant improvement in DQE compared to conventional MV AMFPIs based on phosphor screen detectors. It is therefore interesting to investigate how the generation and transport of secondary optical photons affect the DQE performance of such segmented detectors. One effect that can degrade DQE performance is optical Swank noise (quantified by the optical Swank factor I_opt), which is induced by depth-dependent variations in optical gain. In this study, Monte Carlo simulations of radiation and optical transport have been used to examine I_opt and zero-frequency DQE for segmented CsI:Tl and BGO detectors at different thicknesses and element-to-element pitches. For these detectors, I_opt and DQE were studied as a function of various optical parameters, including absorption and scattering in the scintillator, absorption at the top reflector and septal walls, as well as scattering at the side surfaces of the scintillator crystals. The results indicate that I_opt and DQE are only weakly affected by absorption and scattering in the scintillator, as well as by absorption at the top reflector. However, in some cases, these metrics were found to be significantly degraded by absorption at the septal walls and scattering at the scintillator side surfaces. Moreover, such degradations are more significant for detectors with greater thickness or smaller element pitch. At 1.016 mm pitch and with optimized optical properties, 40 mm thick segmented CsI:Tl and BGO detectors are predicted to provide DQE values of ∼29% and 42%, corresponding to improvement by factors of ∼29 and 42, respectively, compared to that of conventional MV AMFPIs.

INTRODUCTION

Megavoltage (MV) active matrix, flat-panel imagers (AMFPIs) are routinely used for verifying patient positioning in radiation treatment rooms¹ and are being investigated for cone-beam computed tomography (CBCT).² However, the performance of conventional MV AMFPIs is severely limited by the very low (∼2%) quantum efficiency (QE) of their x-ray detector, which consists of a copper plate and a phosphor screen. As a result, the detective quantum efficiency (DQE) for conventional MV AMFPIs is only ∼1%,³ which is much lower than that for kilovoltage AMFPIs (∼40% to 80%).^{4, 5, 6} In order to significantly improve portal imaging performance,⁷ as well as reduce the dose requirement for MV CBCT imaging,^{7, 8, 9} it is necessary to substantially increase the DQE of MV AMFPIs.

Toward achieving significantly increased DQE, various forms of high-efficiency x-ray detectors have been theoretically and empirically examined by many research groups. These detectors are based on concepts involving high pressure gas (e.g., xenon) chambers employing tungsten walls,^{10, 11, 12} thick optical fibers detecting Cerenkov radiation,¹³ thick HgI₂ photoconductors,¹⁴ segmented phosphors based on photopolymer matrices,¹⁵ and segmented crystalline scintillators which consist of 2D matrices (or 1D arrays) of scintillator crystals separated by optically opaque septal walls.^{7, 16, 17, 18, 19} Among these approaches, segmented crystalline scintillators offer significantly improved QE, with only limited loss in spatial resolution, and possibly no substantial increase in noise compared to conventional AMFPIs. In order to examine this strategy, 1D segmented arrays incorporating zinc tungstate (ZnWO₄) and cadmium tungstate (CdWO₄) scintillators,^{17, 18, 19} as well as 2D segmented matrices employing bismuth germanate (BGO), thallium-doped cesium iodide (CsI:Tl) and CdWO₄ scintillators,^{7, 20, 21, 22} have been investigated by different research groups. Recently, a 40 mm thick segmented CsI:Tl detector, with an element-to-element pitch of 1.016 mm and an area of 16.25×16.25 cm², has been developed and evaluated by our group.¹⁶ A prototype MV AMFPI incorporating this detector exhibited a DQE of ∼22% for a 6 MV x-ray beam.¹⁶ However, the construction and performance of this initial prototype were far from optimal.

The use of computational tools, such as Monte Carlo simulation, to guide the design of these segmented detectors can be of assistance in optimizing their DQE performance, given the complexity of their structures. For example, previous Monte Carlo simulations of radiation transport have indicated that MV AMFPIs employing 10 to 40 mm thick, segmented CsI:Tl and BGO detectors could offer radiation DQE (DQE_rad) values ∼10 to 50 times higher than the DQE measured from conventional AMFPIs.⁷ Encouraged by these early findings, it is interesting to next investigate how the generation and transport of secondary optical photons could affect the DQE performance of AMFPIs employing such segmented detectors.

One optical effect that degrades DQE involves the optical component of Swank noise. Swank noise arises from variations in the pulse height distribution (PHD) – the distribution of the number of optical photons detected for each interacting X ray.²³ This noise has both radiation and optical components. Radiation Swank noise represents the variation in the absorbed energy distribution (AED) – the distribution of the amount of energy absorbed in the scintillator for each interacting X ray. (In this representation, the AED includes any variations in the absorbed energy due to the use of a polyenergetic radiation beam.) Optical Swank noise represents the variation in the optical pulse distribution (OPD) – the distribution of the number of optical photons detected for each unit of energy absorbed in the scintillator. Swank noise and its components are quantified by Swank factors ranging from 0 to 1, with larger values representing lower noise. The total Swank factor (I) may be considered to consist of two main components, both of which degrade DQE: the radiation Swank factor (I_rad) and the optical Swank factor (I_opt).²³

In the absence of additive noise and noise power aliasing, the zero-spatial-frequency DQE of an AMFPI may be expressed as follows:^{23, 24}

$$\text{DQE}=\text{QE}\times I.$$ (1)

In the absence of optical effects, the zero-frequency value of DQE_rad is obtained from the product of QE and I_rad.²⁵ In principle, an increase in scintillator thickness will lead to a higher QE (due to greater radiation attenuation) and a larger I_rad (due to more complete absorption of the x-ray energy), resulting in a higher DQE_rad. However, when scintillator thickness increases, optical photons have to traverse longer pathways, on average, to reach the underlying flat-panel array, which leads to more light loss and thus a smaller I_opt – resulting in smaller I. It is anticipated that such decreases in I_opt will, at least partially, offset the potential gains in DQE obtained through increasing scintillator thickness.²²

In this article, Monte Carlo simulations of radiation and optical transport were performed to investigate Swank noise and DQE for MV AMFPIs based on thick, segmented scintillating detectors. This study was conducted as part of a program of research to support the development of such sophisticated, high-efficiency detectors.

METHODS

A glossary of the symbols and abbreviations used in this article appears in Table 1.

Table 1.

Glossary of the symbols and abbreviations used in this article.

Performance metrics		Detector optical properties
QE	Quantum efficiency	μA	Material absorption coefficient
I	Swank factor	μA-sci	Absorption coefficient: Scintillator
Irad	Radiation Swank factor	μS	Material scattering coefficient
Iopt	Optical Swank factor	μS-sci	Scattering coefficient: Scintillator
DQE	Detective quantum efficiency	α	Surface absorptivity
DQErad	Radiation DQE	αtop	Absorptivity: Top reflector
		αwall	Absorptivity: Septal walls
Distributions		β	User-selected scalar for roughness
AED	Absorbed energy distribution	${\vec{n}}_0$	Normal to a surface
OPD	Optical pulse distribution	${\vec{n}}_R$	Normal to a local microfacet
PHD	Pulse height distribution	$\vec{v}$	Unit vector with random direction
Mi	The ith moment of a distribution	θ	Angular tilt of ${\vec{n}}_R$ from ${\vec{n}}_0$
		θmax	Surface roughness
Detector geometric properties		θmax−0.9	Roughness resistance
Tsci	Thickness of scintillator crystals	G	Scintillator conversion gain
Wsci	Width of scintillator crystals	GCsI:Tl	Conversion gain of CsI:Tl
ϕsci	Aspect ratio of scintillator crystals	E	Energy deposition in scintillator
PE	Element-to-element pitch	N	No. of photons generated per E
Wwall	Width of septal walls	EA	Absorbed energy per X ray
		NG	No. of generated photons per X ray
Functions		ND	No. of detected photons per X ray
round	Rounding to nearest integers	NE	No. of detected photons per MeV
Poisson	Generating Poisson integers	η	Optical detection efficiency
norm	Normalizing vectors	γ	Critical angle: Total internal reflection

Segmented scintillating detectors

Figure 1 shows a schematic view of a portion of the general structure of the simulated AMFPIs, each of which consists of a segmented scintillating detector coupled to a photodiode array (represented in the simulation by a 0.001 mm thick a:Si-H layer). The detector consists of a 1 mm thick copper plate (serving as a radiation build-up layer and a top optical reflector) coupled to the incident x-ray side of the segmented scintillator. The scintillator consists of 81×81 scintillator crystals separated by polystyrene septal walls. The width of the septal walls W_wall is 0.05 mm throughout this study. Monte Carlo simulations were performed for a variety of segmented BGO and CsI:Tl detector designs with thicknesses T_sci of 10 to 40 mm and element-to-element pitches P_E (the distance between the center of two adjacent crystals) of 0.508 to 1.016 mm. Note that the pixel pitch of the indirect-detection flat-panel array presently being used to evaluate prototype segmented scintillating detectors by our group is 0.508 mm.²⁶ The aspect ratio of the scintillator crystals, ϕ_sci, is defined by

$${\varphi }_{\text{sci}}=\frac{T_{\text{sci}}}{W_{\text{sci}}}=\frac{T_{\text{sci}}}{P_E-W_{\text{wall}}},$$ (2)

where W_sci is the width of the scintillator crystal.

Figure 1.

Three-dimensional schematic drawing, not to scale, of a representative portion of the megavoltage AMFPIs simulated in this study. Each simulated detector consists of a 2D matrix of scintillator crystals, separated by septal walls, with an overlying top copper plate. The detector is coupled to a photodiode array, which is represented in the simulation by a thin layer of silicon. See main text for further details.

Monte Carlo code

A recently implemented Monte Carlo code, MANTIS (v2.0),²⁷ was used to perform simulations for the determination of QE, I_rad, and I_opt for MV AMFPIs employing segmented BGO and CsI:Tl detectors. DQE was calculated using Eq. 1 while DQE_rad was obtained from the product of QE and I_rad. MANTIS is a combination of a radiation transport code (PENELOPE-2005)²⁸ and an optical transport code (DETECT-II).²⁹PENELOPE is employed to simulate the energy deposition for each incident X ray. Once x-ray energy is deposited in a scintillator material, DETECT-II is triggered to generate optical photons which are emitted isotropically. For an energy E deposited in the scintillator crystal, N optical photons are generated using the following expression:

$$N=\text{Poisson}\left[\text{round}\left[E\times G\right]\right],$$ (3)

where Poisson is a function to generate integers with a Poisson distribution, round is a rounding function, and G is the average optical conversion gain of the scintillator material (i.e., 54000 and 8500 photons∕MeV for CsI:Tl and BGO, respectively).^{30, 31} The optical code simulates the transport of each of the N photons. The optical simulation of a photon is terminated once the optical photon is absorbed in the segmented detector or detected by the underlying photodiode array. Each material involved in the simulation was assigned a refractive index, an optical absorption coefficient μ_A, and an optical scattering coefficient μ_S. For simplicity, these and other optical parameters used in the study are assumed to be wavelength independent.

In DETECT-II simulations, optical surfaces can be defined as partially or totally absorptive, with a user-defined surface absorptivity α, which can range from 0 to 1. Optical photons that are not absorbed will be either reflected back to the original medium, or refracted into the next medium. For surfaces allowing light transmission, the code determines if a light photon is reflected or refracted at the surface using Fresnel’s law. For surfaces not allowing light transmission (i.e., reflectors), all photons that are not absorbed will be reflected back toward the original medium.

For a smooth surface, the direction of reflected photons is determined using the normal to the surface, ${\vec{n}}_0$, and the rule for specular reflection. The direction of refracted photons is calculated using ${\vec{n}}_0$ and Snell’s law. As illustrated in Fig. 2, for each optical reflection or refraction occurring at a rough surface S₀, the code calculates a normal ${\vec{n}}_R$ to a local microfacet S_R having an angle θ with respect to ${\vec{n}}_0$. The distribution of θ is governed by a user-selected scalar β, which can range from 0 to 1. For each optical interaction, the code generates a unit vector with random direction $\vec{v}$ and calculates ${\vec{n}}_R$ as follows:

$${\vec{n}}_R=\text{norm}\left[{\vec{n}}_0+\beta \vec{v}\right],$$ (4)

where norm is the normalization function for vectors. In this article, surface roughness is expressed in terms of the maximum value of θ, θ_max. The correspondence between the selected value of β and θ_max is given by

$${\theta }_{\text{max}}=\text{arcsin}\left(\beta \right).$$ (5)

In this representation, θ_max corresponds to the largest angle between the plane of the local microfacets and the plane determined by S₀. Roughness can be measured by means of extracting a profile of the surface topology from which a distribution of θ angles for local microfacets can be determined. θ_max is related to the maximum value of that distribution.

Figure 2.

Illustration of the surface roughness model used by DETECT-II. S₀ is an interface plane adjoining media 1 and 2, and S_R is a local microfacet that is tilted at an angle θ with respect to S₀. The definitions of ${\vec{n}}_0$, β, $\vec{v}$, and θ_max are given in the main text. Note that ${\vec{v}}_R$ is equal to the vector sum of ${\vec{n}}_0$ and $\beta \vec{v}$. Since $\vec{v}$ is a unit vector with random direction, the end points of ${\vec{v}}_R$ are located at points along a dotted circle of radius β, centered at point O. The dashed line, originating from the point of optical interaction, is tangent to the circle. Note that this model can generate nonphysical solutions for which the vector representing the reflected or refracted photon points toward the wrong medium. In such cases, the code will abandon the solution and repeat the calculation by generating a new $\vec{v}$ until a physical solution is obtained.

Monte Carlo simulations

In this article, the phrase “detector design” will refer to a collection of detector configurations that employ the same scintillator material (i.e., BGO or CsI:Tl) and have the same detector geometry (i.e., T_sci, P_E, and W_wall). For each detector design, QE and I_rad were obtained using a customized version of MANTIS in which optical transport is disabled. For all radiation simulations, the cut-off energies for electrons, positrons, and x-ray photons (the minimum energy below which tracking of a quantum is terminated) were set to a common value of 0.01 MeV. The radiation source was a parallel 6 MV photon treatment beam³² with an area of 10×10 mm². Depending on the element-to-element pitch, 1.016 or 0.508 mm, the beam covers ∼10×10 or 20×20 detector elements, respectively. The beam area is large enough to accurately simulate the effects occurring at the septal walls and small enough compared to detector area to ensure containment of lateral secondary radiation. The choice of a parallel beam over a pencil beam provides better accuracy in the determination of QE and Swank factor. For a given detector design, 10⁶ x-ray histories were used in the determination of QE and I_rad, whereas, for a given detector configuration, 10⁵ histories were used in the determination of I_opt. (Simulations involving optical interactions were more computationally intensive and thus necessitated fewer histories.) The simulations reported in this article were performed using a 64-bit Linux cluster with up to two hundred 1.8 GHz AMD Opteron processors and consumed a total of ∼350000 CPU hours.

The QE was determined directly from the fraction of the incident X rays that deposit energy in the scintillator crystals (which is a standard output from MANTIS). In addition, a customized output was used to record the energy deposited in the scintillator crystals by each interacting X ray. This information was used to generate the absorbed energy distribution from which I_rad is calculated. For selected detector designs, I_opt was determined as a function of the various detector optical properties, including absorption and scattering in the scintillator crystals, absorption at the top reflector and at the septal walls, as well as scattering at the side surfaces of the crystals. For the determination of I_opt, MANTIS was used to provide the number of optical photons detected as well as the amount of energy absorbed in the scintillator crystals for each interacting X ray. The ratio of these two quantities yielded the number of light photons detected per unit of absorbed energy for each interacting X ray. This number was used to generate the OPD from which I_opt was determined.

Each Swank factor (I, I_rad, and I_opt) was calculated using an expression of the form²³

$$I=\frac{M_1^2}{M_0\times M_2},$$ (6)

where M_i is the ith order moment of a distribution P(x) obtained from³³

$$M_i=\int x^i\times P\left(x\right)dx.$$ (7)

The refractive indices of BGO, CsI:Tl, and the a:Si-H photodiode were assumed to be 2.15, 1.79, and 1.70, respectively.^{30, 31} The absorption and scattering coefficients for both BGO and CsI:Tl (μ_A-sci and μ_S-sci) were assumed to be 0.02 and 0 cm⁻¹, respectively,³⁴ unless otherwise stated. The assumption of μ_S-sci=0 cm⁻¹ corresponds to a scintillator that is perfectly nonscattering. All surfaces were assumed to be smooth, except for the side surfaces of the scintillator crystals whose roughness was defined by the parameter θ_max (with θ_max ranging from 0° to 30°). The top reflector and septal walls were simulated as reflectors (prohibiting optical transmission) with absorption characterized by absorptivities of α_top and α_wall, respectively. The bottom surface of the scintillator crystals was assumed to allow light transmission with no optical absorption and all photons passing into the photodiode layer are counted as detected signal. This corresponds to the absence of any optical coupling medium.

The values for the optical parameters investigated in this article correspond to ranges that encompass the estimated values for existing and anticipated segmented scintillators. For example, a recently developed CsI:Tl prototype scintillator,¹⁶ with a thickness of 40 mm and a pitch of 1.016 mm, exhibits an estimated, combined scattering and absorption coefficient of less than 0.02 cm⁻¹. In addition, given a reflectivity of more than ∼94% for the septal walls and the fact that the estimated I_opt for this prototype is ∼0.8,¹⁶ it was estimated that the absorptivity α_wall is only a few percentage while θ_max is less than ∼10°. While these are rough estimates, they nevertheless provide a benchmark for future prototype development.

RESULTS

QE, I_rad, and DQE_rad

Figure 3 shows results for QE, I_rad, and DQE_rad as a function of scintillator thickness for AMFPIs employing 10 to 40 mm thick segmented BGO and CsI:Tl detectors with element pitches of 1.016 and 0.508 mm. Detectors with greater T_sci and higher scintillator density (i.e., BGO) exhibit higher QE due to increased x-ray attenuation and larger I_rad due to more efficient absorption of x-ray energy. Moreover, detectors with larger P_E show slightly higher QE and I_rad due to their larger scintillator fill factor (90% at 1.016 mm versus 81% at 0.508 mm). The trends for DQE_rad are similar to those for QE and I_rad.

Figure 3.

Simulation results for (a) QE (b) radiation Swank factor (I_rad) and (c) radiation DQE (DQE_rad). The results are plotted as a function of scintillator thickness T_sci for segmented BGO and CsI:Tl detectors at pitches of 1.016 and 0.508 mm. In this and the remaining figures, lines are drawn between the points to guide the eye, unless otherwise indicated.

Validation of the use of reduced conversion gain for CsI:Tl

The amount of computational time required for the intended systematic examination of I_opt for CsI:Tl detectors would exceed that available for this study if the nominal conversion gain for that material, G_CsI:Tl (54000 photons∕MeV), were used. In order to overcome this limitation, a reduced value for conversion gain, 0.1×G_CsI:Tl, was used. The conditions under which such a reduction will still lead to a correct determination of I_opt are described in the Appendix0.

In order to validate the use of this reduction, a comparison of I_opt for values determined through simulations of 10 to 40 mm thick CsI:Tl detectors at pitches of 1.016 and 0.508 mm was performed using both G_CsI:Tl and 0.1×G_CsI:Tl. In these simulations, α_top, α_wall, and θ_max were assumed to be 100%, 4%, and 0°, respectively. The results of the calculations, shown in Fig. 4, demonstrate that the use of reduced gain results in negligible underestimation of I_opt for the examined detector designs. Moreover, given these results, it is reasonable to expect that the use of reduced gain should also be valid for CsI:Tl detectors at pitches between 0.508 and 1.016 mm. Reduced gain was therefore used for all remaining CsI:Tl detector simulations. In the case of the BGO detector simulations, a reduction in conversion gain is not required due to the more modest magnitude of that parameter (8500 photons∕MeV).

Figure 4.

Simulation results for optical Swank factor (I_opt) plotted as a function of T_sci. The results were obtained using the nominal and a reduced conversion gain, G_CsI:Tl and 0.1×G_CsI:Tl, respectively. Results are shown for 10 to 40 mm thick segmented CsI:Tl detectors at pitches of 1.016 and 0.508 mm.

I_opt and DQE

In this section, simulation results for I_opt and DQE are reported as a function of various detector optical properties for selected segmented detector designs. A summary of the comparative behaviors of I_opt for the BGO and CsI:Tl detectors appears in Sec. 3C5.

Absorption and scattering in scintillator crystals

Figures 5a, 5b show results for I_opt as a function of μ_A-sci (assuming μ_S-sci is 0 cm⁻¹) and μ_S-sci (assuming μ_A-sci is 0 cm⁻¹), respectively, for segmented CsI:Tl detectors at various values of T_sci and P_E. In these simulations, α_top, α_wall, and θ_max were assumed to be 100%, 2%, and 0°, respectively. The results indicate that I_opt decreases with increasing μ_A-sci and μ_S-sci. Moreover, the rate of decrease is seen to be steeper for detectors with greater thickness but not obviously different for detectors of different pitches. Furthermore, the decline in I_opt with increasing μ_A-sci is found to be approximately linear. [Note that, when both values of μ_A-sci and μ_S-sci are concurrently nonzero, the resulting decrease in I_opt approximately corresponds to the combined decline observed from the individual dependencies illustrated in Figs. 5a, 5b]. In addition, for the detector configurations illustrated in Fig. 5, the effects of μ_A-sci and μ_S-sci on DQE (which are not shown) are minimal for detectors thinner than 20 mm (i.e., DQE changes by less than 0.008) and are still relatively modest for detectors between 20 and 40 mm thick (i.e., DQE changes by less than 0.03).

Figure 5.

Simulation results for I_opt as a function of (a) scintillator absorption coefficient μ_A-sci and (b) scintillator scattering coefficient μ_S-sci. Results are shown for 10 to 40 mm thick segmented CsI:Tl detectors at pitches of 1.016 and 0.508 mm.

In the simulations reported in the remainder of this article, μ_A-sci and μ_S-sci are assumed to be fixed at 0.02 and 0 cm⁻¹, respectively, for both BGO and CsI:Tl. These values represent optical properties that are close to those of the CsI:Tl material used in our present detector development efforts.³⁴ Although BGO is a more transparent material with negligible levels of optical absorption and scattering, the use of the same μ_A-sci and μ_S-sci values for both scintillators allows direct comparison of the behavior of the corresponding I_opt as a function of other optical properties. Slightly overestimating the value of μ_A-sci for BGO leads to a small underestimate of I_opt and DQE.

Absorption at the top reflector

In Fig. 6, simulation results for I_opt are shown as a function of α_top for 20 and 40 mm thick segmented BGO and CsI:Tl detectors at pitches of 1.016 and 0.508 mm. In these simulations, α_wall and θ_max were assumed to be 2% and 0°, respectively. For all examined detector designs, I_opt decreases approximately linearly with increasing α_top. This decline is explained in the discussion of the results for Fig. 8. Moreover, I_opt for each BGO configuration is higher than that for its CsI:Tl counterpart. The same trends were also found for the 10 and 30 mm thick detectors, the results for which are not shown. In addition, for the configurations illustrated in Fig. 6, the effect of α_top on DQE (which is not shown) is relatively small (i.e., less than ∼0.017).

Figure 6.

Simulation results for I_opt plotted as a function of top reflector absorptivity α_top. Results are shown for 20 and 40 mm thick segmented BGO and CsI:Tl detectors at pitches of 1.016 and 0.508 mm.

Figure 8.

Simulation results for I_opt plotted as a function of the product of the aspect ratio of the scintillator crystals, ϕ_sci, and α_wall. Results are shown for AMFPIs employing segmented BGO or CsI:Tl detectors configured with a reflective (α_top equal to 0%) or an absorptive (α_top equal to 100%) top reflector. The solid and dashed lines correspond to fits to the simulation results using fourth order polynomial functions.

Absorption at septal walls

Figure 7 illustrates simulation results for I_opt and DQE as a function of α_wall for 10 to 40 mm thick segmented BGO and CsI:Tl detectors at pitches of 1.016 and 0.508 mm. In these simulations, α_top and θ_max were chosen to be 100% and 0°, respectively. The results shown in Figs. 7a, 7c indicate that, for all examined detector designs, I_opt decreases as α_wall increases. Moreover, this decline is more significant for detectors with greater T_sci and smaller P_E, the reason for which is explained in the discussion of the Fig. 8 results below. In addition, I_opt for each BGO configuration is higher than that for its CsI:Tl counterpart. Also, as shown in Figs. 7b, 7d, DQE generally decreases with increasing α_wall. Moreover, this decrease is more pronounced for detectors with greater T_sci and smaller P_E, diminishing the advantage of increasing scintillator thickness. In addition, at pitches of 1.016 and 0.508 mm, the DQE is not strongly affected by increasing α_wall for detectors thinner than ∼20 and ∼10 mm, respectively (at least up to an α_wall value of 6%). Finally, at both pitches and at a given value of α_wall, the 20 mm thick BGO detector generally provides higher DQE than CsI:Tl detectors up to 40 mm thick.

Figure 7.

Simulation results for I_opt and DQE plotted as a function of septal wall absorptivity α_wall for 10 to 40 mm thick segmented BGO and CsI:Tl detectors. Results for I_opt and DQE for detectors with 1.016 mm pitch are shown in (a) and (b), while results for detectors with 0.508 mm pitch are shown in (c) and (d), respectively.

The results shown in Figs. 7a, 7c demonstrate that I_opt decreases with higher α_wall, greater T_sci, and smaller P_E. As indicated by Eq. 2, greater T_sci and smaller P_E both result in greater ϕ_sci. In order to examine the combined influence of ϕ_sci and α_wall on I_opt, results for segmented BGO and CsI:Tl detectors configured with an absorptive and a reflective top reflector are plotted as a function of ϕ_sci×α_wall in Fig. 8. The four sets of simulation results include some of the results illustrated in Figs. 67. In addition, the two sets of results for the reflective top reflector also include results obtained from simulations performed at a value of α_wall equal to 4%. In all simulations, θ_max was assumed to be 0°. For a given set of results, the small difference in I_opt observed for simulation results that correspond to the same value of ϕ_sci×α_wall is a consequence of the competing, nonlinear effects of ϕ_sci and α_wall on I_opt when there is optical absorption in the scintillator crystals (i.e., when μ_A-sci is not equal to 0 cm⁻¹). For example, at a value of 1.75 for ϕ_sci×α_wall, the detector configuration with a greater T_sci, the same P_E, but smaller α_wall shows slightly lower I_opt. For the smallest examined value of ϕ_sci×α_wall, ∼0.1, I_opt is greater than 0.99 for all four sets of simulations. As ϕ_sci×α_wall increases, I_opt declines in all cases mainly due to increased optical absorption at the septal walls. Specifically, greater ϕ_sci increases the number of optical interactions occurring at the walls, whereas higher α_wall increases the chance of absorption for each interaction event. For a given top reflector absorptivity, BGO offers higher I_opt compared to CsI:Tl at the same value of ϕ_sci×α_wall. This difference initially increases as ϕ_sci×α_wall increases but remains approximately constant beyond a ϕ_sci×α_wall value of ∼1.9. For a given scintillator material, the I_opt obtained with a reflective top reflector is higher than that obtained with an absorptive top reflector. Initially, this difference increases with increasing ϕ_sci×α_wall but does not further increase beyond ϕ_sci×α_wall values of ∼1.9 and 1.1 for BGO and CsI:Tl, respectively. For both BGO and CsI:Tl, this difference remains less than ∼0.06 for ϕ_sci×α_wall values up to ∼3.5.

Scattering at the side surfaces of the scintillator crystals

Figure 9 shows simulation results for I_opt and DQE as a function of θ_max for 10 to 40 mm thick segmented BGO and CsI:Tl detectors at pitches of 1.016 and 0.508 mm. In these simulations, α_top and α_wall were assumed to be 100% and 2%, respectively. As shown in Figs. 9a, 9c, I_opt decreases with increasing θ_max, except between 0° and 5°, where a slight increase is observed for some of the 10 and 20 mm thick detectors. Moreover, those detectors with greater T_sci and smaller P_E are more affected by increasing θ_max. In addition, although each BGO detector offers higher I_opt than its CsI:Tl counterpart at θ_max equal to 0°, this difference is reduced and, for thicker detectors, reversed as θ_max increases. In particular, the reversal of this difference occurs at progressively smaller θ_max values for detectors with greater T_sci and smaller P_E. As shown in Figs. 9b, 9d, DQE generally decreases with increasing θ_max and this decrease is more significant for detectors with greater T_sci and smaller P_E, diminishing the DQE advantage of increasing scintillator thickness. In addition, at pitches of 1.016 and 0.508 mm, the DQE for detectors thinner than ∼20 and ∼10 mm, respectively, is not significantly affected by increasing θ_max (at least up to θ_max values of 20°). Finally, at both pitches and for a given value of θ_max, the 20 mm thick BGO detectors provide higher DQE than CsI:Tl detectors up to 40 mm thick.

Figure 9.

Simulation results for I_opt and DQE as a function of the roughness of the side surfaces scintillator crystals, θ_max, for 10 to 40 mm thick segmented BGO and CsI:Tl detectors. Results for I_opt and DQE for detectors with 1.016 mm pitch are shown in (a) and (b), while results for detectors with 0.508 mm pitch are shown in (c) and (d), respectively.

The results shown in Figs. 9b, 9d demonstrate that, for all the detector designs examined, the decline of DQE with increasing θ_max is not steep at small values of θ_max. For purposes of this study, the θ_max value at which DQE drops to 90% of its value at θ_max equal to 0° is defined as the roughness resistance of the scintillator crystal side surfaces, θ_max−0.9. In Fig. 10, values for θ_max−0.9 are plotted as a function of T_sci for BGO and CsI:Tl detectors at pitches of 1.016 and 0.508 mm. In the figure, θ_max−0.9 is seen to decrease in an asymptotic-like manner as T_sci increases. Moreover, at a given T_sci, θ_max−0.9 is significantly higher for those detectors with the larger pitch. In addition, at the larger pitch, each BGO detector offers a higher value of θ_max−0.9 than its CsI:Tl counterpart. By comparison, at the smaller pitch, BGO offers higher θ_max−0.9 only when T_sci is less than ∼30 mm.

Figure 10.

Results for the roughness resistance, θ_max−0.9, of the scintillator crystal side surfaces as a function of T_sci. Results, shown for 10 to 40 mm thick segmented BGO and CsI:Tl detectors at pitches of 1.016 and 0.508 mm, were obtained from the I_opt results appearing in Fig. 9.

Figure 11 shows simulation results for I_opt as a function of ϕ_sci for BGO and CsI:Tl detectors at θ_max values of 10° and 20°. The four sets of results shown in this figure were obtained using detector thicknesses varying from 10 to 40 mm and pitches of 0.508, 0.65, 0.8, and 1.016 mm, and include some of the results shown in Fig. 9. In all simulations, α_top and α_wall were assumed to be 100% and 2%, respectively. For all four sets of simulations, I_opt is observed to decrease as ϕ_sci increases. At a θ_max value of 10°, BGO consistently offers a higher I_opt than CsI:Tl. However, at a θ_max value of 20°, CsI:Tl provides higher I_opt for values of ϕ_sci greater than ∼40. Note that, no general trends were found when the I_opt results for the BGO and CsI:Tl detectors shown in Figs. 911 were plotted as a function of ϕ_sci×θ_max.

Figure 11.

Simulation results for I_opt as a function of ϕ_sci for BGO and CsI:Tl detectors at θ_max values of 10° and 20°. The solid and dashed lines correspond to fits to the simulation results using fourth order polynomial functions.

Comparison of I_opt for BGO and CsI:Tl detectors

The simulation results reported in Secs. 3C2 through 3C4 indicate that, at a θ_max value of 0°, BGO detectors offer higher I_opt than their CsI:Tl counterparts. However, as θ_max increases, the difference in I_opt between the BGO and CsI:Tl detectors is reduced and, for thicker detectors, is eventually reversed. It is believed that this complex behavior is the result of the higher refractive index of BGO, 2.15, compared to that of CsI:Tl, 1.79. Since the refractive index of the photodiode array, 1.70, is lower than that of both scintillators, total internal reflection occurs at the bottom surface of the scintillator crystals starting at critical angles γ of 52° and 72° for the BGO and CsI:Tl detectors, respectively. As a result of these different γ values, light photons incident at angles between 52° and 72° can be detected for the CsI:Tl detectors but not for the BGO detectors. For the case of θ_max equal to 0°, these photons, compared to those incident at angles smaller than 52°, have traveled through pathways that are less perpendicular to the photodiode array. On average, they have experienced more interactions at the partially absorptive septal walls. The detection of these additional photons results in widening of the optical pulse distribution, reducing I_opt for the CsI:Tl detectors. However, as θ_max increases, the aforementioned correlation between a photon’s incident angle on the bottom surface of the scintillator crystals and the number of interactions it incurs is weakened, which reduces the difference in I_opt between the BGO and CsI:Tl detectors. In the case of larger θ_max values, CsI:Tl has the advantage of a higher mean value for the optical pulse distribution, resulting in slightly higher I_opt as observed in Figs. 911.

DISCUSSION AND CONCLUSIONS

The theoretical investigation of optical Swank noise and DQE reported in this article illustrates how Monte Carlo techniques can be used to explore the numerous parameter choices involved in the design of segmented scintillating detectors so as to achieve optimized prototypes that provide greatly improved DQE compared to conventional MV AMFPIs. Such techniques allow examination of Swank noise contributions originating from both optical and radiation transport. In the present study, the Swank noise due to the 6 MV poly-energetic x-ray spectrum and the Swank noise resulting from the absorbed energy distributions are combined in the AEDs, resulting in the combined radiation Swank factor I_rad. As shown in Fig. 3b, I_rad is on the order of 0.5, causing the DQE to decrease by approximately a factor of 2. Additional degradation can occur if the optical properties of the scintillator crystals and the septal walls are not carefully chosen. The effect of these properties on DQE is quantified by the optical Swank factor I_opt. The simulation results reported in Sec. 3 demonstrate that I_opt is larger for detectors with smaller ϕ_sci (i.e., smaller T_sci and larger P_E). Although the use of larger P_E leads to higher DQE at zero spatial frequency, it is also necessary to ensure that the element pitch is sufficiently small to avoid significant loss of spatial resolution and DQE at high spatial frequencies. At a given value of ϕ_sci, I_opt generally decreases with increasing μ_A-sci, μ_S-sci, α_top, α_wall, or θ_max. Among these five optical properties, only α_wall and θ_max significantly degrade DQE at ϕ_sci values greater than ∼20 – such as for 10 and 20 mm thick detectors at P_E values of 0.508 and 1.016 mm, respectively. In addition, the results shown in Figs. 79 can be used to determine the value of α_wall and θ_max beyond which a thicker detector does not provide a higher DQE. For example, for BGO detectors at a pitch of 0.508 mm, DQE does not increase as T_sci increases from 30 to 40 mm when α_wall is greater than ∼3% (when θ_max is equal to 0°) or when θ_max is larger than ∼8° (when α_wall is equal to 2%).

At θ_max equal to 0°, the results shown in Fig. 8 (and the corresponding fits) can be used to estimate I_opt for detectors having values of T_sci, P_E, and α_wall that are not specifically examined in this study. Note that such estimates may not be valid beyond the parameter ranges examined in this study. Moreover, since the results in Fig. 6 indicate that I_opt decreases approximately linearly with increasing α_top, the Fig. 8 results can also be used to estimate I_opt for α_top values between 0% and 100%. In addition, at θ_max of 10° and 20°, the results illustrated in Fig. 11 (and the corresponding fits) can be used to estimate I_opt for detectors having values of T_sci and P_E that are within the examined range of parameters but not specifically investigated in this study. Finally, in the development of future prototype detectors, the θ_max−0.9 values shown in Fig. 10 could provide guidance for avoiding significant loss of DQE due to scattering at the side surfaces of the scintillator crystals.

The examination of hypothetical detector designs reported in Sec. 3 assumes the use of polystyrene septal walls that have radiation properties generally consistent with the type of polymer walls used in recent prototype detectors.^{16, 35} Alternatively, septal walls could conceivably be made of metal (e.g., silver foil or tungsten powder with reflective coating). In radiation simulations carried out for BGO and CsI:Tl detectors involving 0.05 mm thick tungsten septal walls, the replacement of polymer walls with tungsten walls was found to result in higher DQE_rad – by up to ∼0.04 and 0.08 at P_E values of 1.016 and 0.508 mm, respectively. However, metal walls are typically more absorptive (e.g., α_wall is greater than 10%) than polymer walls, resulting in lower I_opt that may mitigate the higher levels of DQE_rad. Were it possible to produce metal walls with low absorptivity, the I_opt behaviors reported in Sec. 3 should also be applicable to detectors employing such walls.

It has also been suggested that metal walls having higher than desired absorptivity could be used if these walls were fabricated so as to exhibit graded absorptivity.¹⁶ Such walls would be made to provide less optical absorption near the top of the detector (i.e., on the side near the x-ray source) compared to the bottom (i.e., on the side next to the array) – which could potentially improve I_opt. As an initial test of this hypothesis, a 40 mm thick CsI:Tl detector was simulated with dual-grade septal walls – such that the absorptivity of the top half of the walls, α_wall-1, was fixed at 10%, while that of the bottom half, α_wall-2, was fixed at a value ranging from 10% to 14%. The result of these simulations indicates that I_opt decreases as α_wall-2 increases, showing no gain in I_opt for septal walls graded in this manner. However, it is conceivable that, in order to achieve a gain in I_opt, the walls need to be divided into many more grades, with optimized absorptivity applied to each grade, or varied in a continuous manner. Simulations to guide the optimization of such graded walls are of interest but are beyond the scope of the present study.

In summary, results from this theoretical investigation suggest that, with optical properties optimized with the help of computer simulations, AMFPIs based on thick, segmented CsI:Tl and BGO detectors offering significantly improved DQE [up to ∼29% and 42%, corresponding to ∼29 and 42 times higher than that of conventional MV AMFPIs, respectively, as shown in Fig. 7b] can be achieved. It is anticipated that such greatly increased DQE will result in significant enhancement of performance for electronic portal imaging and facilitate acquisition of MV CBCT images with soft-tissue visualization at clinically acceptable doses.⁹

APPENDIX: JUSTIFICATION FOR REDUCED GAIN IN CsI:Tl SIMULATIONS

This appendix describes the conditions under which, in the present study, a 90% reduction in the conversion gain for CsI:Tl detectors will still lead to a correct determination of I_opt. In the MANTIS simulations, an X ray deposits its energy in the scintillator in multiple (k) steps. Each energy deposition, E_k, generates N_k optical photons, as determined by the conversion gain and Eq. 3. For each interacting X ray, the total energy absorbed in the scintillator, E_A, is given by

$$E_A=\sum _k{E}_k,$$ (A1)

while the total number of optical photons generated, N_G, is given by

$$N_G=\sum _k{N}_k.$$ (A2)

Due to light loss in the x-ray detector, only a fraction η of the N_G optical photons are detected in the photodiode array. (η is referred to as the optical detection efficiency.) Therefore, for each interacting X ray, the number of optical photons detected, N_D, is given by

$$N_D=\eta \times N_G,$$ (A3)

so that the number of optical photons detected per unit of energy absorbed in the scintillator, N_E, may be determined from

$$N_E=N_D∕E_A.$$ (A4)

As indicated in Sec. 2, the distribution of N_E, P(N_E), referred to as the optical pulse distribution, is used to determine I_opt by means of Eqs. 6, 7. By substituting Eqs. 3 and A1 to A3 into Eq. A4, N_E may be expressed in terms of η, E_k, and G as follows:

$$N_E=\frac{\eta \times \sum _k\left(\text{Poisson}\left[\text{round}\left[E_k\times G\right]\right]\right)}{\sum _k{E}_k}.$$ (A5)

Since MANTIS rounds the product E_k×G to the nearest integer before generating light photons, the use of reduced conversion gain for CsI:Tl (i.e., G=0.1×G_CsI:Tl) may result in a reduction in N_G greater than the intended 90%. This additional loss in N_G occurs when 0.5≤(E_k×G)<5. For this range of E_k×G values, while the nominal gain will result in generation of a few optical photons (i.e., 1 to 5), the reduced gain will lead to the generation of no light photons (since E_k×0.1×G is rounded to zero). Therefore, in order to achieve the intended 90% reduction in N_G, the probability of 0.5≤(E_k×G)<5 must be negligible.

It is of interest to note that, when the nominal gain for the CsI:Tl detectors is used (i.e., when G=G_CsI:Tl), N_G is found to be very large (∼40 000 to 110 000), resulting in a very small statistical uncertainty for η (∼0.3% to 0.5%). This uncertainty remains small (∼1.0% to 1.6%) even when N_G is reduced by 90%. Therefore, if N_G is reduced by the intended 90% when reduced gain is used, N_E will also be reduced by ∼90% as intended (given that the value of η will remain relatively unchanged). As a result of the reduction in N_E, the first and second order moments of P(N_E), M₁ and M₂, will be reduced to 10% and 1% of their original values, respectively. As can be seen from Eq. 6, since the value of M₀ remains unchanged, the effect of such reductions in M₁ and M₂ will cancel out, leaving the value of I_opt unchanged.

In summary, in order to demonstrate the validity of using reduced gain, it is necessary to show that the probability of 0.5≤(E_k×G)<5 is not high enough to affect the determination of I_opt. For each CsI:Tl detector design (i.e., for a collection of CsI:Tl detector configurations offering the same radiation transport properties and the same value of G), such validation can be demonstrated by comparing I_opt values determined from simulations using the nominal and reduced gains. This comparison only needs to be performed for one configuration of each design, since the value of E_k×G is the same for all configurations of that design.