Photon Fluence and Dose Estimation in Computed Tomography using a Discrete Ordinates Boltzmann Solver Edward T. Norris, and Xin Liu

Abstract—In this study, cone-beam single projection and axial CT scans are modeled with a software package – DOCTORS, which solves the linear Boltzmann equation using the discrete ordinates method. Phantoms include a uniform 35 cm diameter water cylinder and a non-uniform abdomen phantom. Series simulations were performed with different simulation parameters, including the number of quadrature angles, the order of Legendre polynomial expansions, and coarse and fine mesh grid. Monte Carlo simulations were also performed to benchmark DOCTORS simulations. A quantitative comparison was made between the simulation results obtained using DOCTORS and Monte Carlo methods. The deterministic simulation was in good agreement with the Monte Carlo simulation on dose estimation, with a rootmean-square-deviation (RMSD) difference of around 2.87%. It was found that the contribution of uncollided photon fluence directly from the source dominates the local absorbed dose in the diagnostic X-ray energy range. The uncollided photon fluence can be calculated accurately using a ‘ray-tracing’ algorithm. The accuracy of collided photon fluence estimation is largely affected by the pre-calculated multigroup cross-sections. The primary benefit of DOCTORS lies in its rapid computation speed. Using DOCTORS, parallel computing using GPU enables the cone-beam CT dose estimation nearly in real-time. Index Terms—Computed tomography, discrete ordinates, dose, photon fluence, GPU

I

I. INTRODUCTION

X-ray attenuation-based CT imaging, the mechanisms responsible for a material’s attenuation are primarily the photoelectric effect and Compton scattering [1-3]. These interactions determine the energy transfer between photons and material as well as the photon distribution throughout the CT system. A complete description of the photon distribution and energy transfer is essential for estimating patient dose and for designing an optimized CT system. Stochastic methods (e.g., Monte Carlo simulation) have been used extensively in the past [4-20], and are generally considered to be the gold standard for estimating photon distributions and CT doses. However, they require a large number of particle histories and, therefore, a lengthy computation time is needed to reduce statistical N

This work was supported in part by the U.S. Nuclear Regulatory Commission under Grant NRC-HQ-13-G-38-0026. Edward T. Norris was with Missouri University of Science and Technology, Rolla, MO 65401 USA. He is now with the Department of Energy, Washington DC (e-mail: [email protected] mst.edu).

uncertainty to an acceptable level. There is, however, no statistical error associated with deterministic methods, so they can be comparatively efficient in large regions where the highly resolved spatial fluence must be known to within a tight uncertainty bound. While hybrid stochastic-deterministic methods advantageously combine Monte Carlo and deterministic techniques and are more computationally efficient than a pure Monte Carlo simulation, they result in a cumbersome computational framework, due to the combination of two different methodologies, and can also have lengthy computation times [21-22]. We have explored three methodologies including Monte Carlo, hybrid Monte Carlo, and deterministic methods to solve photon transport problems [23-27], and have found that the deterministic method provides accurate results that are comparable to a Monte Carlo simulation. In addition, the deterministic method has the highest computational efficiency among the three methodologies. Although deterministic photon dose estimation has been widely used in the field of radiation therapy [28-32], its use has not been fully investigated for CT imaging. Because CT imaging is fundamentally different from radiation therapy, in terms of X-ray photon energy, interaction mechanisms, beam shape, and source trajectory, a CT-specific approach to deterministic photon dose calculation is required. To date, we have developed several deterministic simulations of a CT system and its subcomponents [25-27]. In the course of these efforts, we discovered that the deterministic solution of the linear Boltzmann equation, based on the discrete ordinates method (i.e., SN method), is the most promising method, due to its scalability and parallelizability. Computer codes that are based on the discrete ordinates method have been extensively used in radiation shielding calculations and nuclear reactor analyses and, recently, have been utilized in clinical radiation therapy calculations. They have not, however, been applied to diagnostic imaging. Recently, we have developed a software application called DOCTORS (Discrete Ordinate Computed TOmography and Radiography Simulator) [33]. In this paper, we examined the accuracy and runtime of DOCTORS to compute energy-resolved photon fluence and dose distribution Xin Liu is with Missouri University of Science and Technology, Rolla, MO 65401 USA. (e-mail: [email protected]).

of a cone-beam CT with uniform and non-uniform phantoms. II. METHODS The photon transport process can be described by the steadystate linear Boltzmann transport equation. The steady-state linear Boltzmann transport equation is given by the following: [34] #∙∇ &⃑ + 𝜎* (𝑟⃑, 𝐸)0𝜑2𝑟⃑, 𝐸, Ω # 3 = ∬ 𝜎6 2𝑟⃑, 𝐸 7 → 𝐸, Ω #∙Ω # 73 ∙ !Ω 9: # 7 )𝑑Ω # 7 𝑑𝐸 7 + 𝑆2𝑟⃑, 𝐸, Ω #3 𝜑(𝑟⃑, 𝐸 7 , Ω (1) # 7 3 is the angular fluence (photons/cm2) at where, 𝜑2𝑟⃑, 𝐸 7 , Ω # ; 𝜎* (𝑟⃑, 𝐸) is the total position 𝑟⃑, with energy 𝐸, and direction Ω macroscopic interaction cross section, including scattering and # 3 is the external source absorption cross sections; 𝑆2𝑟⃑, 𝐸, Ω #∙ which simplifies the actual physical source; 𝜎6 2𝑟⃑, 𝐸 7 → 𝐸, Ω 73 # Ω is the macroscopic differential scattering cross section which represents the probability that photons at position 𝑟⃑ are # to energy 𝐸 and scattered from energy 𝐸′ and direction Ω′ 7 # # # direction Ω; Ω ∙ Ω is the cosine of the scattering angle. The left side of the Equation (1) represents photon loss in a differential volume at position 𝑟⃑, and the right side represents the gain of photons in the same differential volume and position. Thus, the linear Boltzmann transport equation actually describes the photon conservation of a fixed volume in a steady state. A. Discrete Ordinates Method Analytic solutions of the Boltzmann transport equation can only be obtained for the simplest problems. Realistic, multidimensional, and energy-dependent problems must be solved numerically. The discrete ordinates method discretizes # , onto a discrete phase the continuous variables, 𝑟⃑, 𝐸, and Ω space so that Equation (1) can be solved numerically [34]. Detailed derivation of discretized Boltzmann equation can be found in our previous publication [26]. Here, only the final discretized linear Boltzmann transport equation is given C C CC7 C7 FGH # ? ∙ &∇⃑ + 𝜎@,A,B !Ω 0𝜙@,A,B,? = ∑L ?7JH ∑CIJC 𝜎6,??7 𝜙@,A,B,?7 𝜔? + C 𝑆@,A,B,? (2)

where, 𝑛 represents the discrete direction; 𝑖, 𝑗, and 𝑘 represent the 3D spatial discrete mesh grid; g represents the energy group; CC7 𝜎6,??7 is energy-group wised scattering cross section which is often expressed as a function of Legendre expansion [26]. The C angular fluence 𝜙@,A,B,? in equation (2) can be solved numerically with proper boundary conditions. In our study, vacuum boundary conditions are applied. The scalar fluence at each voxel (𝑖, 𝑗, 𝑘) is obtained by integration over all of the discrete angles and approximated by the quadrature formula, C

L(LRS)

𝜙@,A,B = ∑?JH

C

𝑤? 𝜙@,A,B,?

(3)

where 𝑁 is the discrete ordinates order that are referred as SN quadratures which results in a total of 𝑁(𝑁 + 2) directions; 𝑤?

is the weight associate with each discrete direction. Once the C group photon fluence 𝜙@,A,B is solved, the absorbed dose or kerma can be obtained by applying the conversion factors from ICRP publications [35]. The collision kerma is usually considered equal to the absorbed dose due to the low X-ray energy used in CT scanning [14]. The accuracy of the discrete ordinates method depends on the number of discrete angles, energy groups, and the size of the mesh grid. However, a large number of discrete angles, energy groups, and a mesh grid would inevitably slow down the computation. Thus, a trade-off between accuracy and computation speed has to be evaluated when performing a simulation. In addition, discrete ordinates methods suffer from ‘rayeffect’ in a weakly scattered media [34]. Ray-effect arises due to the restriction of particle transport to a set of discrete directions. To mitigate ‘ray-effect’, the first-collision source method was employed [36]. The fluence in each voxel is composed of uncollided photons directly from the source and collided photons scattered from other voxels, energies, and directions. Thus, equation (1) can be rewritten as two equations solving for the uncollided fluence 𝜙V , and collided fluence 𝜙W , independently. Both 𝜙V , and 𝜙W obey the linear Boltzmann transport equation and the sum of 𝜙V and 𝜙W is the total fluence 𝜙. The uncollided photon transport equation has no scatter term since scattered particles are not considered in the uncollided fluence. The lack of a scatter term makes the uncollided fluence computable with a raytracing algorithm. The uncollided fluence is then used to compute the first-collision source. The firstcollision source is used in the same way as the external source as shown in equations (1) and (2) to solve for the collided fluence in each voxel. B. The Discrete Ordinates Boltzmann Solver We have recently developed dedicated discrete ordinates software, named DOCTORS, which computes photon fluence distribution and equivalent dose in a patient using the discrete ordinates method. The graphical user interface (GUI) of DOCTORS is shown in Figure 1. The first tab is the geometry input where a user provides a 3-D reconstructed image, such as DICOM (digital imaging and communications in medicine) images or raw CT numbers. As soon as the data is read in, it is converted into a series of dosimetrically equivalent materials representative of a human patient. Although there is no direct relation between CT numbers and tissue types, CT numbers can be converted to tissue types quite accurately based on a stoichiometric calibration [37-39]. The next three tabs identify the cross-section dataset, quadrature, anisotropy treatment, and whether GPU is used for the photon transport computation. The final tab, as shown in Figure 1, defines the X-ray source. A user can select a source type from a number of built-in options available for analysis, including point sources, fan beams, and cone beams. Multi-fan beams and multi-cone beams can be arranged about an object to mimic the source rotation in a CT scan. Each source type is described by its position and energy distribution. Two additional parameters, 𝜑 and 𝜃, describe the azimuthal and polar angels subtended by the fan or cone beam, respectively.

Fig. 1. The graphical user interface of DOCTORS.

Once all necessary data is loaded, the “Launch Solver” button becomes active and will remain so until the user creates a conflicting set of input that would prevent the solver from being able to run. When the user clicks on the “Launch Solver” button, the computation process begins and the photon fluence information will be displayed in the output dialog. Currently, DOCTORS cannot automatically identify specific organs, thus it is only able to compute the equivalent dose or whole-body effective dose via dose deposition. Organ-specific effective dose can be converted from the equivalent dose manually if specific organs could be identified. C. Verification against Monte Carlo Simulation The Monte Carlo simulation is considered the gold standard for patient dose estimation. DOCTORS has the capability to automatically generate an input file for the Monte Carlo simulation, using CT data and source specification provided by the user. The user can easily verify the results obtained from DOCTORS by comparison to a Monte Carlo simulation with the exact same input parameters. This feature greatly reduces the amount of work required to manually generate a Monte Carlo simulation input file. Currently, the only supported file format for a Monte Carlo simulation is the MCNP [40] input file format. MCNP, a widely-used Monte Carlo simulation package, has more than 50 years of history. The version of MCNP used in this study is MCNP6. Only analog Monte Carlo (i.e. no variance reduction techniques applied) simulation were performed in order to achieve a perfect analog to the photon transport in the patient. To achieve good statistics (i.e.

Abstract—In this study, cone-beam single projection and axial CT scans are modeled with a software package – DOCTORS, which solves the linear Boltzmann equation using the discrete ordinates method. Phantoms include a uniform 35 cm diameter water cylinder and a non-uniform abdomen phantom. Series simulations were performed with different simulation parameters, including the number of quadrature angles, the order of Legendre polynomial expansions, and coarse and fine mesh grid. Monte Carlo simulations were also performed to benchmark DOCTORS simulations. A quantitative comparison was made between the simulation results obtained using DOCTORS and Monte Carlo methods. The deterministic simulation was in good agreement with the Monte Carlo simulation on dose estimation, with a rootmean-square-deviation (RMSD) difference of around 2.87%. It was found that the contribution of uncollided photon fluence directly from the source dominates the local absorbed dose in the diagnostic X-ray energy range. The uncollided photon fluence can be calculated accurately using a ‘ray-tracing’ algorithm. The accuracy of collided photon fluence estimation is largely affected by the pre-calculated multigroup cross-sections. The primary benefit of DOCTORS lies in its rapid computation speed. Using DOCTORS, parallel computing using GPU enables the cone-beam CT dose estimation nearly in real-time. Index Terms—Computed tomography, discrete ordinates, dose, photon fluence, GPU

I

I. INTRODUCTION

X-ray attenuation-based CT imaging, the mechanisms responsible for a material’s attenuation are primarily the photoelectric effect and Compton scattering [1-3]. These interactions determine the energy transfer between photons and material as well as the photon distribution throughout the CT system. A complete description of the photon distribution and energy transfer is essential for estimating patient dose and for designing an optimized CT system. Stochastic methods (e.g., Monte Carlo simulation) have been used extensively in the past [4-20], and are generally considered to be the gold standard for estimating photon distributions and CT doses. However, they require a large number of particle histories and, therefore, a lengthy computation time is needed to reduce statistical N

This work was supported in part by the U.S. Nuclear Regulatory Commission under Grant NRC-HQ-13-G-38-0026. Edward T. Norris was with Missouri University of Science and Technology, Rolla, MO 65401 USA. He is now with the Department of Energy, Washington DC (e-mail: [email protected] mst.edu).

uncertainty to an acceptable level. There is, however, no statistical error associated with deterministic methods, so they can be comparatively efficient in large regions where the highly resolved spatial fluence must be known to within a tight uncertainty bound. While hybrid stochastic-deterministic methods advantageously combine Monte Carlo and deterministic techniques and are more computationally efficient than a pure Monte Carlo simulation, they result in a cumbersome computational framework, due to the combination of two different methodologies, and can also have lengthy computation times [21-22]. We have explored three methodologies including Monte Carlo, hybrid Monte Carlo, and deterministic methods to solve photon transport problems [23-27], and have found that the deterministic method provides accurate results that are comparable to a Monte Carlo simulation. In addition, the deterministic method has the highest computational efficiency among the three methodologies. Although deterministic photon dose estimation has been widely used in the field of radiation therapy [28-32], its use has not been fully investigated for CT imaging. Because CT imaging is fundamentally different from radiation therapy, in terms of X-ray photon energy, interaction mechanisms, beam shape, and source trajectory, a CT-specific approach to deterministic photon dose calculation is required. To date, we have developed several deterministic simulations of a CT system and its subcomponents [25-27]. In the course of these efforts, we discovered that the deterministic solution of the linear Boltzmann equation, based on the discrete ordinates method (i.e., SN method), is the most promising method, due to its scalability and parallelizability. Computer codes that are based on the discrete ordinates method have been extensively used in radiation shielding calculations and nuclear reactor analyses and, recently, have been utilized in clinical radiation therapy calculations. They have not, however, been applied to diagnostic imaging. Recently, we have developed a software application called DOCTORS (Discrete Ordinate Computed TOmography and Radiography Simulator) [33]. In this paper, we examined the accuracy and runtime of DOCTORS to compute energy-resolved photon fluence and dose distribution Xin Liu is with Missouri University of Science and Technology, Rolla, MO 65401 USA. (e-mail: [email protected]).

of a cone-beam CT with uniform and non-uniform phantoms. II. METHODS The photon transport process can be described by the steadystate linear Boltzmann transport equation. The steady-state linear Boltzmann transport equation is given by the following: [34] #∙∇ &⃑ + 𝜎* (𝑟⃑, 𝐸)0𝜑2𝑟⃑, 𝐸, Ω # 3 = ∬ 𝜎6 2𝑟⃑, 𝐸 7 → 𝐸, Ω #∙Ω # 73 ∙ !Ω 9: # 7 )𝑑Ω # 7 𝑑𝐸 7 + 𝑆2𝑟⃑, 𝐸, Ω #3 𝜑(𝑟⃑, 𝐸 7 , Ω (1) # 7 3 is the angular fluence (photons/cm2) at where, 𝜑2𝑟⃑, 𝐸 7 , Ω # ; 𝜎* (𝑟⃑, 𝐸) is the total position 𝑟⃑, with energy 𝐸, and direction Ω macroscopic interaction cross section, including scattering and # 3 is the external source absorption cross sections; 𝑆2𝑟⃑, 𝐸, Ω #∙ which simplifies the actual physical source; 𝜎6 2𝑟⃑, 𝐸 7 → 𝐸, Ω 73 # Ω is the macroscopic differential scattering cross section which represents the probability that photons at position 𝑟⃑ are # to energy 𝐸 and scattered from energy 𝐸′ and direction Ω′ 7 # # # direction Ω; Ω ∙ Ω is the cosine of the scattering angle. The left side of the Equation (1) represents photon loss in a differential volume at position 𝑟⃑, and the right side represents the gain of photons in the same differential volume and position. Thus, the linear Boltzmann transport equation actually describes the photon conservation of a fixed volume in a steady state. A. Discrete Ordinates Method Analytic solutions of the Boltzmann transport equation can only be obtained for the simplest problems. Realistic, multidimensional, and energy-dependent problems must be solved numerically. The discrete ordinates method discretizes # , onto a discrete phase the continuous variables, 𝑟⃑, 𝐸, and Ω space so that Equation (1) can be solved numerically [34]. Detailed derivation of discretized Boltzmann equation can be found in our previous publication [26]. Here, only the final discretized linear Boltzmann transport equation is given C C CC7 C7 FGH # ? ∙ &∇⃑ + 𝜎@,A,B !Ω 0𝜙@,A,B,? = ∑L ?7JH ∑CIJC 𝜎6,??7 𝜙@,A,B,?7 𝜔? + C 𝑆@,A,B,? (2)

where, 𝑛 represents the discrete direction; 𝑖, 𝑗, and 𝑘 represent the 3D spatial discrete mesh grid; g represents the energy group; CC7 𝜎6,??7 is energy-group wised scattering cross section which is often expressed as a function of Legendre expansion [26]. The C angular fluence 𝜙@,A,B,? in equation (2) can be solved numerically with proper boundary conditions. In our study, vacuum boundary conditions are applied. The scalar fluence at each voxel (𝑖, 𝑗, 𝑘) is obtained by integration over all of the discrete angles and approximated by the quadrature formula, C

L(LRS)

𝜙@,A,B = ∑?JH

C

𝑤? 𝜙@,A,B,?

(3)

where 𝑁 is the discrete ordinates order that are referred as SN quadratures which results in a total of 𝑁(𝑁 + 2) directions; 𝑤?

is the weight associate with each discrete direction. Once the C group photon fluence 𝜙@,A,B is solved, the absorbed dose or kerma can be obtained by applying the conversion factors from ICRP publications [35]. The collision kerma is usually considered equal to the absorbed dose due to the low X-ray energy used in CT scanning [14]. The accuracy of the discrete ordinates method depends on the number of discrete angles, energy groups, and the size of the mesh grid. However, a large number of discrete angles, energy groups, and a mesh grid would inevitably slow down the computation. Thus, a trade-off between accuracy and computation speed has to be evaluated when performing a simulation. In addition, discrete ordinates methods suffer from ‘rayeffect’ in a weakly scattered media [34]. Ray-effect arises due to the restriction of particle transport to a set of discrete directions. To mitigate ‘ray-effect’, the first-collision source method was employed [36]. The fluence in each voxel is composed of uncollided photons directly from the source and collided photons scattered from other voxels, energies, and directions. Thus, equation (1) can be rewritten as two equations solving for the uncollided fluence 𝜙V , and collided fluence 𝜙W , independently. Both 𝜙V , and 𝜙W obey the linear Boltzmann transport equation and the sum of 𝜙V and 𝜙W is the total fluence 𝜙. The uncollided photon transport equation has no scatter term since scattered particles are not considered in the uncollided fluence. The lack of a scatter term makes the uncollided fluence computable with a raytracing algorithm. The uncollided fluence is then used to compute the first-collision source. The firstcollision source is used in the same way as the external source as shown in equations (1) and (2) to solve for the collided fluence in each voxel. B. The Discrete Ordinates Boltzmann Solver We have recently developed dedicated discrete ordinates software, named DOCTORS, which computes photon fluence distribution and equivalent dose in a patient using the discrete ordinates method. The graphical user interface (GUI) of DOCTORS is shown in Figure 1. The first tab is the geometry input where a user provides a 3-D reconstructed image, such as DICOM (digital imaging and communications in medicine) images or raw CT numbers. As soon as the data is read in, it is converted into a series of dosimetrically equivalent materials representative of a human patient. Although there is no direct relation between CT numbers and tissue types, CT numbers can be converted to tissue types quite accurately based on a stoichiometric calibration [37-39]. The next three tabs identify the cross-section dataset, quadrature, anisotropy treatment, and whether GPU is used for the photon transport computation. The final tab, as shown in Figure 1, defines the X-ray source. A user can select a source type from a number of built-in options available for analysis, including point sources, fan beams, and cone beams. Multi-fan beams and multi-cone beams can be arranged about an object to mimic the source rotation in a CT scan. Each source type is described by its position and energy distribution. Two additional parameters, 𝜑 and 𝜃, describe the azimuthal and polar angels subtended by the fan or cone beam, respectively.

Fig. 1. The graphical user interface of DOCTORS.

Once all necessary data is loaded, the “Launch Solver” button becomes active and will remain so until the user creates a conflicting set of input that would prevent the solver from being able to run. When the user clicks on the “Launch Solver” button, the computation process begins and the photon fluence information will be displayed in the output dialog. Currently, DOCTORS cannot automatically identify specific organs, thus it is only able to compute the equivalent dose or whole-body effective dose via dose deposition. Organ-specific effective dose can be converted from the equivalent dose manually if specific organs could be identified. C. Verification against Monte Carlo Simulation The Monte Carlo simulation is considered the gold standard for patient dose estimation. DOCTORS has the capability to automatically generate an input file for the Monte Carlo simulation, using CT data and source specification provided by the user. The user can easily verify the results obtained from DOCTORS by comparison to a Monte Carlo simulation with the exact same input parameters. This feature greatly reduces the amount of work required to manually generate a Monte Carlo simulation input file. Currently, the only supported file format for a Monte Carlo simulation is the MCNP [40] input file format. MCNP, a widely-used Monte Carlo simulation package, has more than 50 years of history. The version of MCNP used in this study is MCNP6. Only analog Monte Carlo (i.e. no variance reduction techniques applied) simulation were performed in order to achieve a perfect analog to the photon transport in the patient. To achieve good statistics (i.e.