Electron Beam Dose Planning Using Discrete Gaussian Beams: Mathematical background

Brahme, Anders; Lax, Ingmar; Andreo, Pedro

doi:10.3109/02841868109130436

Cited by 97 publications

(58 citation statements)

References 32 publications

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“…In particular, Hogstrom et al, 1 Brahme, 2 Perry and Holt, 3 Werner et al, 4 Mandour et al, 5 Lax et al, 6 and others cited therein, have extended the application of Fermi 7 and Eyges 8 electron scattering theory, including energy loss, to clinically relevant use.…”

Section: Introductionmentioning

confidence: 99%

Application of measured pencil beam parameters for electron beam model evaluation

2003

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Most current electron beam models, as are used in commercial treatment planning systems, combine measured broad beam central axis depth dose data with measured or modeled functions to approximate radial scatter and heterogeneity effects. In this paper, we extend a recently developed pencil beam model to calculate doses outside the field edge and doses in heterogeneous media. We have also explored use of this model as a tool for evaluating commercial electron planning programs. The algorithm we have developed, based on the concept of the lateral buildup ratio (LBR), enables calculation of dose at any point in an irregular electron field, and is capable of generating both on- and off-axis depth dose curves and isodose profiles. This model includes the effects of density and mass-angular scattering power in measured broad beam central axis depth dose data, which when combined with small field reference data, can be used to generate LBR ratios. From these ratios one can infer the depth dependent, effective pencil beam radial spread parameter a in water or other materials, which can be used to model any arbitrary field. We have used this approach to calculate fractional depth doses for small fields incident on aluminum and cork, which we have then compared against measurements and the calculations of several commercial planning systems.

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Section: Introductionmentioning

confidence: 99%

Application of measured pencil beam parameters for electron beam model evaluation

2003

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“…In this paper, however, we shall adopt Eq. (9) as our definition of atr so as to conform with earlier n~t a t i o n~~'~* '~ for purely scattering problems. The analysis of this section applies to either form of H and hence we use the convenient compact notation in Eq.…”

Section: Introductionmentioning

confidence: 97%

One dimensional beam transport

Prinja

Pomraning

1996

Transport Theory and Statistical Physics

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Approximations to the transverse integrated scalar flux for a particle beam, whose corresponding angular flux satisfies the one dimensional transport equation, are developed under the assumption that the angular distribution remains peaked about the original beam direction. Asymptotic expansions, in the form of a power series in z q + , are obtained by scaling appropriately small terms in the transport equation. Similar results are also generated by developing a truncated hierarchy of (1 -p)n-moments of the angular flux. A low order truncation is shown to be sufficient to obtain the scalar flux with a high order error. Although formally different, all methods are seen to yield the same asymptotic expansion of the scalar flux. This is not unreasonable as all methods exploit the same feature of the solution, namely, that highly forward peaked scattering results in an initially peaked distribution remaining peaked.

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“…Based on the simple Fokker-Plank approximation of the multiple scattering collision term of the Boltzmann equation, the first analytical solution for pencil beam transport was derived by Fermi in 1941 [3] and Eyges in 1948 [4] taking energy losses within the continuous slowing down approximation (CSDA) into account. This pencil beam solution has further been generalized to describe the lateral spread of arbitrary Gaussian beams of known initial mean square radial " r 2 ð0Þ, angular spread, " 2 ð0Þ, and covariance " r ð0Þ [5,6], of clinical electron beams in a patient during radiation therapy of malignant tumors [5][6][7][8], as well as to describe the transport of light ions in different media [9][10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Solution of the Boltzmann equation for primary light ions and the transport of their fragments

Kempe

Brahme

2010

Phys. Rev. ST Accel. Beams

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The Boltzmann equation for the transport of pencil beams of light ions in semi-infinite uniform media has been calculated. The equation is solved for the practically important generalized 3D case of Gaussian incident primary light ion beams of arbitrary mean square radius, mean square angular spread, and covariance. The transport of the associated fragments in three dimensions is derived based on the known transport of the primary particles, taking the mean square angular spread of their production processes, as well as their energy loss and multiple scattering, into account. The analytical pencil and broad beam depth fluence and absorbed dose distributions are accurately expressed using recently derived analytical energy and range formulas. The contributions from low and high linear energy transfer (LET) dose components were separately identified using analytical expressions. The analytical results are compared with SHIELD-HIT Monte Carlo (MC) calculations and found to be in very good agreement. The pencil beam fluence and absorbed dose distributions of the primary particles are mainly influenced by an exponential loss of the primary ions combined with an increasing lateral spread due to multiple scattering and energy loss with increasing penetration depth. The associated fluence of heavy fragments is concentrated at small radii and so is the LET and absorbed dose distribution. Their transport is also characterized by the buildup of a slowing down spectrum which is quite similar to that of the primaries but with a wider energy and angular spread at increasing penetration depths. The range of the fragments is shorter or longer depending on their nuclear mass to charge ratio relative to that of the primary ions. The absorbed dose of the heavier fragments is fairly similar to that of the primary ions and also influenced by a rapidly increasing energy loss towards the end of their ranges. The present analytical solution of the Boltzmann equation accurately accounts for the loss of primary particles as well as their energy losses and multiple scattering. At the same time these quantities for the fragments are also accurately derived as based on the generalized Gaussian solution of the primaries and compared both with Monte Carlo and experimental data. The results are useful for fast transport calculations and biologically optimized therapy planning with light ion beams.

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Electron Beam Dose Planning Using Discrete Gaussian Beams: Mathematical background

Cited by 97 publications

References 32 publications

Application of measured pencil beam parameters for electron beam model evaluation

Application of measured pencil beam parameters for electron beam model evaluation

One dimensional beam transport

Solution of the Boltzmann equation for primary light ions and the transport of their fragments

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