Comparison of the beam mixing models proposed by Lam and Zaider & Rossi and a derived Dt extension for the Zaider & Rossi model

In the radiotherapy of cancer with heavy ions, treatment planning systems like TRiP [1] use externally calculated tables for the biological effect of monoenergetic ion beams and derive the effect of a therapeutic mixed beam from these data by means of a beam-mixing model. In TRiP, the effects of monoenergetic beams are predicted by the Local Effect Model (LEM) [2]. The full simulation approach of this model [3] predicts ion and energy dependent threshold values, Dt, for the transitions between linearquadratic and pure linear part of dose-effect curves (LinearQuadratic-Linear (LQL) model), but in the previously used version, the single particle approximation, such individual Dts could not be estimated. Instead, as an approximation, an independence of the D t value from the beam quality was assumed. This could be exploited by TRiP by using the very efficient beam-mixing model proposed by Zaider & Rossi [4]. In principle this model does not include a D t and an extension is not straight-forward [5]. Nevertheless, for a constant Dt, the model could be extended by applying the Dt threshold after the actual beam-mixing (“constantDt extension”). As this approach could not be used for the varying Dts of the full-simulation method, the much more flexible beam-mixing method proposed by Lam [6] has to be introduced. However, this model is conceptually different from the Zaider & Rossi approach and the predicted RBE-weighted doses (Relative Biological Effectiveness) deviate by a few percent. A theoretical understanding of the differences is therefore highly interesting, especially for the comparison with previous TRiP/LEM results. In the LQ model (LQL without D t threshold) dose-effect curves, (D), are described by = αD + βD. Formally, this could be separated in a linear effect α = αD and a quadratic effect β = βD. As the Lam method can handle any dose-effect curve, the method could, formally, be applied to both effect-curves separately, resulting in α and β , and providing a new mixed effect = α + β . This is not the mixed-beam prediction of the original Lam model but, interestingly, it could be proven, that is identically to the result predicted by the Zaider & Rossi model. In addition to a theoretical description of the model differences, this immediately leads to a Dt extension of the Zaider & Rossi method: For monoenergetic including a D t, this threshold can be moved to the β part by using α = αD and β = − α. In this extension, individual ion and energy dependent Dt thresholds can be used, but for constant D t this model does not exactly lead to the previously used constantDt method. This is shown in the figure, where relative differences between RBE-weighted doses calculated by