Analytic derivation of central axis percent depth dose calculations in transition zones with loss of electronic equilibrium

John M Pacyniak


Purpose: The study of megavoltage photon dose distribution behind and near small areas of low and high density material is best understood with Monte Carlo (MC) dose calculation or direct measurements which may not be always be possible. This is especially true for air-tissue area where the replacement of soft tissue scattering material by air results in the loss of electronic equilibrium and changes in the lateral spread of the beam as well. Monte Carlo calculations are the standards to correctly evaluate in homogeneities in transition zones. If one could develop a model with sufficient accuracy to obtain similar results, this would be very helpful clinically.

Methods: To this end, we have developed an exponential model and derive an explicit expression that accounts for the under dosage. The model is an extension of a much earlier work done with electrons and photons. Our analytic model is based on the experience of the underlying physics assuming exponential attenuation of photons in matter.

Results: It differs from a similar work by solving the problem correctly and introducing parameters that can be traced to direct measurements without the need of extensive statistical data analysis. It combines the generation of free electrons through ionization and their attenuation to a simple differential equation for the central axis depth dose. It involves two parameters, which can be obtained from 1) direct beam measurements, 2) primary photon attenuation coefficients from physics tables and 3) iteration techniques.

Conclusion: The simplicity of the model allows us to extend our derivation to situations such as transitions zones of different densities in areas such as head and neck and lung. A clinical example is illustrated to demonstrate the problems encountered in treating cancer of the larynx.


Monte Carlo, Electronic Disequilibrium, Heterogeneity

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