Sperduto_Khan's Treatment Planning in Radiation Oncology, 5e

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440 SECTION III Treatment Planning: Physics and Dosimetric Principles

dose at a point is proportional to the energy fluence of photons at the same point. The main criterion for CPE is that the energy fluence of photons must be constant out to the range of electrons set in motion in all directions. This does not occur in general in heterogeneous media, near the beam boundary, or for intensity-modulated beams. Electrons produced in the head of the accelerator and in air between the accelerator and the patient are called contamination electrons . The interaction of these electrons in and just beyond the buildup region contributes signifi cantly to the dose, especially if the field is large. Perturbation in electron transport can be exaggerated near heterogeneities. For example, the range of electrons is three to five times as long in lung as in water, and so beam boundaries passing through lung have much larger penumbral regions. Bone is the only tissue with an atomic composition significantly different from that of water. This can lead to perturbations in dose of only a few percent, 8 and so perturbations in electron scattering or stopping power are rarely taken into account. Bone can therefore be treated as “high-density water.” SUPERPOSITION/CONVOLUTION ALGORITHM The most common photon dose calculation in use for radiotherapy planning today is the superposition/convo lution algorithm. 8–19 This method incorporates a model based approach in describing the underlying physics of the interactions, while still being able to calculate dose in a reasonable time. The convolution/superposition method begins by modeling the indirect nature of dose deposition from photon beams. Primary photon interactions are dealt with separately from the transport of scattered photons and electrons set in motion. Dose Calculation under Conditions of CPE To begin, we consider the special case of dose determina tion under conditions of CPE. In this case, the total energy absorbed by charged particles at position r is the same as the total energy that escapes due to photon interactions at the same location. Thus, the primary dose D p and the first-scattered dose from a parallel beam of monoener getic photons can be computed as 9 D p ( r ) = ( K c( r )) P = ( μ en _ ρ ) P Ψ  P ( r ) = ( μ en __ ρ ) P ϕ P ( r = 0) h v p e − μr (20.1) where Ψ P ( r ) and ( K c ( r ) ) P are the primary energy fluence and collision kerma, respectively, at point r , ( µ en / ρ ) p is the mass energy absorption coefficient, ϕ P ( r = 0 ) is the pri mary photon fluence at the surface of the phantom, hν P is the primary photon energy, and μ is the attenuation

coefficient of primary photons. The total dose is the sum of the primary and scatter components D tot ( r ) = D P ( r ) + ∫ D P ( r ′ ) ( μ en ) scat _ ( μ en ) P   ( hv ) scat _ ( hv ) P   d P scat ( θ , r ′ ) _ d V e  − μ scat ( r ′ - r ) d V (20.2) where d P scat ( θ , r ′ ) / d V is the probability per unit volume of a primary photon being scattered into a solid angle cen tered about angle θ . These equations are complicated enough, but they do not take into account any secondary or higher-order pho ton scatter. They also neglect beam divergence and do not take into account tissue heterogeneities. They are valid only for CPE situations, so that the dose computation is not valid in the buildup region or near the field boundar ies, and the scatter dose is perturbed by heterogeneities lying between the scatter site at r ′ and the point r , where the total dose is being computed. Convolution/Superposition Method Unfortunately, Equation (20.1) is simplistic because it does not take into account the finite range of charged particles. In other words, the energy fluence that was present at the point the charged particles were set in motion upstream should replace the energy fluence in Equation (20.1). We may think of this energy fluence as that originating upstream (i.e., assuming that the charged particles all moved linearly downstream), but in reality, the particles may originate from any location around the calculation point, as long as it is within the particles’ range. Thus, rather than a single effective photon interac tion site, this expression for dose becomes a convolution integral about r : D ( r ) = ∫ K c ( r ′ ) A c ( r − r ′ ) d r ′ (20.3) where A c ( r − r ′ ) describes the contribution of charged particle energy that gets absorbed per unit volume at r ′ from interactions at r ′ and the integration is over all values of r ′ that make up volume d r ′ . The charged particle energy absorption kernel has a finite extent because the range of charged particles set in motion is finite. Equation (20.3) requires knowledge of the energy fluence due to both primary and scattered photons at all points. Time-consuming transport methods, such as the method of discrete ordinates or the MC method, would be needed to compute the scattered component accurately. A simpler solution is to utilize a scatter ker nel that includes the scattered photon component along with the contribution from charged particles. The ker nel is no longer finite because photon scatter (which has no range) is included. Now, only primary photons are explicitly transported. A convolution equation that separates primary photon transport and a kernel that accounts for the scattered photon and electrons set in

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