Collapsed Cone Convolution Superposition (from ADAC)

The ADAC Pinnacle3 Collapsed Cone Convolution Superposition Dose Model
Todd McNutt, Ph.D. – Director of Product Development
ADAC’s Pinnacle3 3D treatment planning system uses a Collapsed Cone Convolution Superposition computation to determine the dose distribution in patients from external photon beams. The Pinnacle3 Convolution Superposition dose model is a true three-dimensional dose computation which intrinsically handles the effects of patient heterogeneities on both primary and secondary scattered radiation. This computation method is uniquely able to account for dose distributions in areas where the electronic equilibrium is perturbed, such as tissue-air interfaces and tissue-bone interfaces. While other convolution techniques account for the effects of patient heterogeneities on primary radiation, they neglect the effects of heterogeneities on scattered radiation in the final dose distribution. In addition, the nature of the Convolution Superposition dose model makes it ideally suited to handle optimization and intensity modulated radiation therapy planning.

The Convolution Superposition Dose Model
The Pinnacle3 Convolution Superposition dose algorithm is based on the work of Mackie, et al. Rather than correcting measured dose distributions, the Convolution Superposition algorithm computes dose distributions from first principles and, therefore, can account for the effects of beam modifiers, the external patient contour, and tissue heterogeneities on the dose distribution.
The Convolution Superposition dose model consists of four parts:
• Modeling the incident energy fluence as it exits the accelerator head. • Projection of this energy fluence through the density representation of a patient to compute a TERMA (Total Energy Released per unit Mass) volume. • A three-dimensional superposition of the TERMA with an energy deposition kernel using a ray-tracing technique to incorporate the effects of heterogeneities on lateral scatter. • Electron contamination is modeled with an exponential falloff which is added to the dose distribution after the photon dose is computed. The following sections describe each part of the model in more detail.

Modeling the Incident Energy Fluence as it exits the Accelerator
The incident energy fluence distribution is modeled as a two-dimensional array which describes the radiation exiting the head of the linear accelerator. The parameters defining this array are defined during physics data modeling.
The starting point for photon modeling is a uniform plane of energy fluence describing the intensity of the radiation exiting the accelerator head. The fluence model is then adjusted to account for the flattening filter, the accelerator head, and beam modifiers such as blocks, wedges and compensators.
• The “horns” in the beam produced by the flattening filter are modeled by removing an inverted cone from the distribution. • Off-focus scatter produced in the accelerator head is modeled by defining a 2D Gaussian function as a scatter source and adjusting the incident energy fluence based on the portion of the Gaussian distribution visible from each point in the incident energy fluence plane. • The geometric penumbra is modeled by convolving the array with a focal spot blurring function. • During planning, the shape of the field produced by blocks or multi-leaf collimators is cut out of the array leaving behind the corresponding transmission through the shape-defining entity. • Beam modifiers such as wedges and compensators are included in the array by attenuating the energy fluence by the corresponding thickness of the modifier. For static wedges and compensators, a radiological depth array is also stored which allows for proper modeling of the beam hardening due to the presence of the beam modifiers during the projection of the incident fluence array. Dynamic beam delivery with intensity-modulation or dynamic wedges is easily handled using the incident energy fluence array. For these beams, the radiological depth array is not needed to account for beam hardening.

Projection of Energy Fluence through a CT Patient Representation
The incident energy fluence plane is projected through the CT patient representation and attenuated using mass attenuation coefficients. These coefficients are stored in a three-dimensional lookup table as a function of density, radiological depth, and off-axis angle. Patient heterogeneities are taken into account with the density dependence. Beam hardening through the patient is accounted for with the radiological depth dependence, and the off-axis softening of the energy spectrum is produced with the off-axis angle dependence. To account for the changes in the photon energy spectrum at different locations in the beam, the mass attenuation coefficient lookup table is produced using a weighted sum of several mono-energetic tables.
The TERMA (Total Energy Released per unit Mass) volume is computed by projecting the incident energy fluence through the patient density volume using a ray-tracing technique. A given ray’s direction is determined based on the position of the radiation source and the particular location in the incident fluence plane. At each voxel in the ray path, the TERMA is computed using the attenuated energy fluence along the ray and the mass attenuation coefficient at the particular density, radiological depth, and off-axis angle.

3D Superposition of an Energy Deposition Kernel
The three-dimensional dose distribution in the patient is computed by superposition of the TERMA volume with the energy deposition kernel. The kernel represents the spread of energy from the primary photon interaction site throughout the associated volume. Poly-energetic kernels are produced by combining a series of Monte Carlo-generated mono-energetic energy deposition kernels. The superposition is carried out using a ray tracing technique similar to that used in the projection of the incident energy fluence. The kernel is inverted so that the dose can be computed in only a portion of the patient (TERMA) volume if desired. This allows for point dose computation and decreases computation time.
The rays from the dose deposition site are cast in three dimensions. At each voxel of the TERMA traversed along a ray, the contribution of dose to the dose deposition site is computed and accumulated using the TERMA and the kernel value at the current radiological distance. Using the radiological distance along the ray also allows the kernel to be scaled to account for the presence of heterogeneities with respect to scattered radiation in all directions.
The dose computation described above determines the dose from a single beam. Multiple beams are computed independently and the entire 3D dose distribution is created by adding the dose from each beam together according to the corresponding beam weight.

Adaptive Convolution Superposition
An Adaptive Convolution Superposition approach has also been implemented in Pinnacle3. This uses the calculation technique described above with some slight modifications. The speed of the computation is increased by adaptively varying the resolution of the dose computation grid depending on the curvature of the TERMA and dose distribution. First, the dose in a coarse 3D grid is computed and then the curvature in the TERMA distribution is assessed. In regions where the curvature is high, the dose is computed at intermediate points to provide higher resolution. The system adaptively increases the resolution in regions of high curvature until an acceptable resolution is used. In regions of low curvature, the dose is interpolated from the coarse dose grid. This technique decreases the computation time by a factor of 2-3 without compromising the accuracy of the Convolution Superposition calculation in the presence of heterogeneities.

Other Model-Based Algorithms
Other model-based algorithms, including 3D Fast Fourier Transform (FFT) techniques and differential pencil beam models which use FFTs on two-dimensional planes, use a projection of the incident energy fluence similar to that used in Pinnacle3 to determine the TERMA volume. They differ in that they do not use the superposition technique in the convolution process.
The FFT techniques require the assumption of an invariant kernel, which inherently assumes a homogeneous density representation during the convolution process. This technique reduces the accuracy of the computation because it ignores the effects of heterogeneities on laterally scattered radiation. Some post-computation corrections may be performed to help alleviate the error. In contrast, by scaling the rays from the primary dose deposition site, the Convolution Superposition method accurately and intrinsically models the effects of lateral scatter from tissue heterogeneities, a requirement for calculating dose from conformal and intensity modulated fields.
Although the FFT algorithms have a fast computation speed per computation point, they require full computation of dose over the entire TERMA volume. For irregular field calculations, point doses, plane doses, or other situations where the planner is only interested in the dose to smaller regions, the FFT algorithms still require the dose to be computed over the entire volume. The Convolution Superposition method can accurately compute the dose to a single point with the inverted energy deposition kernel. Therefore, depending on the planning situation, the desired dose calculation may be faster using the Convolution Superposition calculation than when using the FFT calculation.
The ability to define a smaller calculation matrix also results in a calculation speed advantage with the Convolution Superposition model for inverse planning and optimization of intensity modulated beams where computation of dose need only be performed in limited regions during the optimization process.
Further Reading
This paper provides an overview of the Convolution Superposition dose computation method used in the ADAC Pinnacle3 3D treatment planning system. For further information on this method and other dose computation techniques, please refer to the publications listed below:
R. Mohan, C. Chui, L. Lidofsky, “Energy and angular distributions of photons from medical linear accelerators,” Med. Phys. 12, 592-597 (1985). T.R. Mackie, J.W. Scrimger, J.J. Battista, “A convolution method of calculating dose for 15-MV x-rays,” Med. Phys. 12, 188-196 (1985). T.R. Mackie, A. Ahnesjo, P. Dickof, A. Snider, “Development of a convolution/superposition method for photon beams,” Use of Comp. In Rad. Ther., 107-110 (1987). A. Ahnesjo, P. Andreo, A. Brahme, “Calculation and application of point spread functions for treatment planning with high energy photon beams,” Acta. Oncol., 26, 49-56, (1987). T.R. Mackie, A.F. Bielajew, D.W.O. Rogers, J.J. Battista, “Generation of photon energy deposition kernels using the EGS Monte Carlo code,” Phys. Med. Biol. 33, 1-20 (1988). T.R. Mackie, P.J. Reckwerdt, T.W. Holmes, S.S. Kubsad, “Review of convolution/superposition methods for photon beam dose computation,” Proceedings of the Xth ICCR, 20-23, (1990). T.R. Mackie, P.J. Reckwerdt, M.A. Gehring, T.W. Holmes, S.S. Kubsad, B.R. Thomadsen, C.A. Sanders, B.R. Paliwal, T.J. Kinsella, “Clinical implementation of the convolution/superposition method,” Proceedings of the Xth ICCR, 322-325, (1990). N. Papanikolaou, T.R. Mackie, C. Meger-Wells, M. Gehring, P. Reckwerdt, “Investigation of the convolution method for polyenergetic spectra,” Med. Phys. 20, 1327-1336 (1993). M.B. Sharpe, J.J. Battista, “Dose calculations using convolution and superposition principles: The orientation of the dose spread kernels in divergent x-ray beams,” Med. Phys., 20, 1685-1694 (1993). T.R. McNutt, T.R. Mackie, P. Reckwerdt, N. Papanikolaou, B.R. Paliwal, “Calculation of portal dose images using the convolution/superposition method,” Med. Phys. 23(4) (1996). T.R. Mackie, P.J. Reckwerdt, T.R. McNutt, M. Gehring, C. Sanders, “Photon dose computations,” Teletherapy: Proceedings of the 1996 AAPM Summer School, Ed. J. Palta, T. R. Mackie., AAPM-College Park, MD, (1996). ADAC Laboratories 540 Alder Dr. Milpitas, CA 95035 Tel: (408)321-9100 (800)538-8531 Fax: (408)577-0907 www.adaclabs.com
MBA-X0010, Rev. B