Sperduto_Khan's Treatment Planning in Radiation Oncology, 5e

First page Previous page 29 Next page Last page

CHAPTER 20 Treatment Planning Algorithms: Photon Dose Calculations 445

within the medium by an amount e − αvΔt . Here σ is the macroscopic cross section for electrons and represents 1/ λ , where λ is the mean free path. Although not strictly a crosssection, σ is analogous to the photon attenua tion coefficient and has units of 1/length. For very short times Δ t , the number of electrons from this packet which have reached r + v Δt is ≈ N e ( 1 − σ v Δ t ) d V dΩ d E . At the same time scattered electrons from elsewhere in the medium may reach the same position ( r + v Δ t ) . This quantity may be determined by integrating the angular density over phase space multiplied by the probability for these interactions: N e scatter ( r + v ⋅ Δ t , t + Δ t ) = ∬ N e ( r , t ) { d  2 σ _ d E d Ω ( Ω ', E ′ ; Ω , E ) } d Ω d E , (20.7) where d ( Ω ′, E ′ ; Ω , E ) represents the doubly differen tial crosssection for electron scatter from energy E ′ and direction Ω ′ to energy E and direction Ω . In addition, any additional sources of electrons pro duced during time Δ t may also reach the position r + v Δ t . In this case, the number of additional electrons at r + v Δ t becomes Q ( r , W , E , t ) Δ t , where Q ( r , Ω , E , t ) represents the rate of electron production from other sources. The total number of electrons at position r + v Δ t is now given by the following equation: N e ( r + v Δ t , Ω , E , t + Δ t ) = N e ( r , Ω , E , t ) ( 1 − σv Δ t ) + { ∬ N e ( r , Ω ′ , E ′ , t )   d  2 σ _ d Ω d E ( E ′ , Ω ′ ; E , Ω ) } Δ t (20.8) Dividing the equation by Δ t and taking the limit Δ t → 0, we obtain 2 σ _ d Ω d E + Q ( r , Ω , E , t ) Δ t

Target

Patient independent components

Primary colimator

Vacuum window Flattening filter

Ion chamber

Phase-space plane 1

Jaws

Patient dependent structures

MLC

Phase-space plane 2

DISCRETE ORDINATES METHOD More recently, several authors have reported on a direct numerical solution of the Boltzmann transport equa tions (BTEs). The approach has been commercialized in the Varian Eclipse Treatment Planning System, under the name Acuros. In particular, this methodology has been proposed as an alternative to MC calculations, in order to produce accurate dose distributions with a substantially reduced calculation time. Derivation of the Transport Equations The linear BTE can be derived by assuming particle conservation within a small volume element of phase space. 38–40 We define a quantity called the angular density of electrons, N e ( r , Ω , E,t ) , which represents the probable number of electrons at location r and direction Ω with energy E at time t per unit volume per unit solid angle per unit energy. Ω represents the unit director in the direction of motion, that is, parallel to v . Thus, N e ( r , Ω , E, t ) d V d Ω d E represents the number of electrons at time t in a volume element d V about r , in a narrow beam of solid angle d Ω about Ω , and energy range d E about E . After a time Δ t , these electrons have moved to posi tion r + v Δ t and have been reduced due to collisions FIGURE 20.6. Illustration of the components of a typical Varian lin ear accelerator treatment head in photon beam mode. Phase space planes for simulating patient-dependent and patient-independent structures are also represented. (Reprinted from Chetty IJ, Cur ran B, Cygler JE, et al. Report of the AAPM Task Group No. 105: Issues associated with clinical implementation of Monte Carlo based photon and electron external beam treatment planning. Med Phys. 2007;34:4818–4853, with permission.) MLC.

[ N e ( r + v Δ t , Ω , E , t + Δ t ) − N e ( r , Ω , E , t ) ______________________ Δ t ]

lim Δ t → 0

+ σv N e ( r , Ω , E , t ) = ∬ N e ( r , Ω ', E ′ , t )

d  2 σ _ d Ω d E ( E ′ , Ω '; E , Ω ) d Ω ' d E ′

+ Q ( r , Ω , E , t ) (20.9) The limit term represents the total time derivative of N e for an observer moving with the packet of electrons (i.e., from r to r + v Δ t ). It may be rewritten to simplify the equation:

[ N e ( r + v Δ t , t + Δ t ) − N e ( r , t ) ________________ Δ t ] = lim  Δ t → 0 [ [ N e ( r , t + Δ t ) − N e ( r , t ) _____________ Δ t ] + lim  Δ t → 0

lim  Δ t → 0

N e ( r + v Δ t , t + Δ t ) − N e ( r , t + Δ t ) ___________________ Δ t ]

∂ N e ( r , t ) _ ∂ t

= v . ∇ N e ( r , t ) +

(20.10)

Made with FlippingBook Digital Publishing Software