Sperduto_Khan's Treatment Planning in Radiation Oncology, 5e

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CHAPTER 20 Treatment Planning Algorithms: Photon Dose Calculations 443

curves can be obtained by scaling the contamination elec tron depth–dose curve with the surface dose and add ing this component to the convolution-computed dose distribution.

Water

Kernel Spatial Variance and Phantom Heterogeneities

1.0

Air

The convolution equation assumes that the kernel is spa tially invariant in that the kernel value depends only on the relative geometrical relationship between the interac tion and dose deposition sites and not on their absolute position in the phantom. When this is true, the convo lution calculation can be done in Fourier space, saving much time. Unfortunately, this is not the case as the ker nel varies with position. The effects of hardening and divergence of the beam are small and can be calculated in a number of ways. A multiplicative correction to the terma in the patient can be used to correct for hardening of the kernel. 17,26 Alter natively, several kernels valid for different depths in the phantom can be used as a basis for interpolation to a spe cific depth. 17,19 Liu et al. showed that the correction as a function of depth is nearly linear, and not employing any correction results in ∼ 4% discrepancy at 30 cm depth. Tilting the kernel to match the beam divergence results in only a minor improvement in accuracy for the worst-case examples. 19 Phantom heterogeneities are a more serious problem. Modeling the transport of electrons and scattered pho tons through a heterogeneous phantom would require a unique kernel at each location. Each kernel would be superimposed on the dose grid and weighted with respect to the primary terma. What is required to make the calculation tractable is to modify a kernel, computed in a homogeneous medium, to be reasonably representa tive in a heterogeneous situation. If most of the energy between the primary interaction site and the dose depo sition site is transported on the direct path between these sites, it is possible to have a relatively simple correction to the convolution equation based on ray-tracing between the interaction and dose deposition sites, and on scal ing the path length by density to get the radiologic path length between these sites. The convolution equation modified for radiologic path length is called the super position equation: D ( r ) = ∫ T P ( ρ r ′ ⋅ r ′ ) A ( ρ r − r ′ ⋅ ( r − r ′ ) ) d r ′ (20.5) where ρ r - r ′ ⋅ ( r − r ′ ) is the radiologic distance from the dose deposition site to the primary photon interaction site and ρ r ⋅ r ′ is the radiologic distance from the source to the photon interaction site. Woo and Cunningham 15 compared the modified kernel using range scaling for a complex heterogeneous phantom with a kernel computed de novo for a particu lar interaction site inside the phantom. The results shown

2.0

Depth (cm)

Water

3.0

1.0

2.0

Radial distance (cm)

in Figure 20.5 indicate that the agreement is not perfect, but the computational trends are clearly in evidence in that isovalue lines contract in high-density regions and expand in low-density regions. MONTE CARLO The MC technique of radiation transport consists of using well-established probability distributions governing the individual interactions of electrons and photons to simu late their transport through matter. MC methods are used to perform calculations in all areas of physics and math for any problems that involve a probabilistic nature. Several excellent reviews of MC calculations in radiation therapy exist, 27–31 as well as an American Association of Physicists in Medicine (AAPM) Task Group Report which discusses its clinical implementation. 32 Although the MC method had been proposed for some time, it was not capable of being fully utilized until the development of the digital computer in the 1940s. Radia tion transport was one of the first uses for this methodol ogy at that time, and public codes, such as Monte Carlo N-Particle Transport code (MCNP), began appearing as early as the 1950s. In photon transport calculations, the Electron Transport (ETRAN) code, developed by the FIGURE 20.5. Comparison of Monte Carlo-generated 6-MeV pri mary photon kernel in a water phantom containing a ring of air. The dotted line is a kernel modified for the heterogeneous situation using range scaling from the one derived in a homogeneous phan tom. The continuous line is a kernel computed expressly for the het erogeneous situation. It is impractical to compute kernels for every possible heterogeneous situation, and there is sufficient similarity to warrant the range scaling approximation. (Reprinted from Woo MK, Cunningham JR. The validity of the density scaling method in pri mary electron transport for photon and electron beams. Med Phys. 1990;17:187–194, with permission.)

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