Derivation of a comprehensive semi-empirical proton RBE model from published experimental cell survival data collected in the PIDE database

We aimed to develop a comprehensive proton relative biological effectiveness (RBE) model based on accumulated cell survival data in the literature. Our approach includes four major components: (1) Eligible cell survival data with various linear energy transfers (LETs) in the Particle Irradiation Data Ensemble (PIDE) database (72 datasets in four cell lines); (2) a cell survival model based on Poisson equation, with <i>α</i> and <i>β</i> defined as the ability to generate and repair damage, respectively, to replace the classic linear-quadratic model for fitting the cell survival data; (3) hypothetical linear relations of <i>α</i> and <i>β</i> on LET, or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfrac><mml:mrow><mml:mi>α</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow><mml:mi>L</mml:mi> <mml:mi>E</mml:mi> <mml:mi>T</mml:mi></mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> <mml:mrow><mml:msub><mml:mi>α</mml:mi> <mml:mi>x</mml:mi></mml:msub> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:msub><mml:mtext>α</mml:mtext> <mml:mi>α</mml:mi></mml:msub> <mml:mo>+</mml:mo> <mml:msub><mml:mi>b</mml:mi> <mml:mi>α</mml:mi></mml:msub> <mml:mtext> </mml:mtext> <mml:mo>∗</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>L</mml:mi> <mml:mi>E</mml:mi> <mml:mi>T</mml:mi></mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfrac><mml:mrow><mml:mi>β</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow><mml:mi>L</mml:mi> <mml:mi>E</mml:mi> <mml:mi>T</mml:mi></mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> <mml:mrow><mml:msub><mml:mi>β</mml:mi> <mml:mi>x</mml:mi></mml:msub> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:msub><mml:mtext>α</mml:mtext> <mml:mi>β</mml:mi></mml:msub> <mml:mo>-</mml:mo> <mml:msub><mml:mi>b</mml:mi> <mml:mi>β</mml:mi></mml:msub> <mml:mtext> </mml:mtext> <mml:mo>∗</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>L</mml:mi> <mml:mi>E</mml:mi> <mml:mi>T</mml:mi></mml:mrow> </mml:math> ; and (4) a multi-curve fitting (MCF) approach to fit all cell survival data into the survival model and derive the <i>a_α</i> , <i>b_α</i> , <i>a_β</i> , and <i>b_β</i> values for each cell line. Dependences of these parameters on cell type were thus determined and finally a comprehensive RBE model was derived. MCF showed that (<i>a_α</i> , <i>b_α</i> , <i>a_β</i> , <i>b_β</i> ) = (1.09, 0.0010, 0.96, 0.033), (1.10, 0.0015, 1.03, 0.023), (1.12, 0.0025, 0.99, 0.0085), and (1.17, 0.0025, 0.99, 0.013) for the four cell lines, respectively. Thus, <i>a_α</i> = 1.12 ± 0.04, <i>b_α</i> = 0.0019 ± 0.0008, <i>a_β</i> = 0.99 ± 0.03, and <i>b_β</i> = 0.013 ∗ <i>α_x</i> , and approximately <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>α</mml:mi> <mml:mo>∼</mml:mo> <mml:mn>1.12</mml:mn> <mml:mo>∗</mml:mo> <mml:msub><mml:mi>α</mml:mi> <mml:mi>x</mml:mi></mml:msub> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow><mml:mn>0.99</mml:mn> <mml:mo>-</mml:mo> <mml:mn>0.013</mml:mn> <mml:mo>∗</mml:mo> <mml:msub><mml:mi>α</mml:mi> <mml:mi>x</mml:mi></mml:msub> <mml:mo>∗</mml:mo> <mml:mi>L</mml:mi> <mml:mi>E</mml:mi> <mml:mi>T</mml:mi></mml:mrow> <mml:mo>)</mml:mo></mml:mrow> <mml:mo>∗</mml:mo> <mml:msub><mml:mi>β</mml:mi> <mml:mi>x</mml:mi></mml:msub> </mml:mrow> </mml:math> . Consequently, a relatively reliable and comprehensive RBE model with dependence on LET, <i>α_x</i> , <i>β_x</i> , and dose per fraction was finally derived for potential clinical application.