Table 1.

Summary of phenomenological RBE proton models based on a similar formalism of the LQ model (equation 1)^{12 13}

Model	Cell lines (size)	RBEmax	RBEmin	Parameter values
				p0	p1	p2	p3
Carabe14	V79 (44)	${\displaystyle p_0+p_1\frac{LE{T}_d}{{\left(\alpha /\beta \right)}_x}}$	${\displaystyle p_2+p_3\frac{LE{T}_d}{{\left(\alpha /\beta \right)}_x}}$	0.843	0.413644	1.09	0.01612
Chen15	V79 (14)	${\displaystyle \frac{p_0}{{\alpha }_x}+\frac{1-e^{-p_{1}LE{T}_d^2}}{{\alpha }_x{p}_{2}LE{T}_d}}$	1.0	0.1	0.0013	0.045	--
McNamara16	Multiple (285)	${\displaystyle p_0+p_1\frac{LE{T}_d}{{\left(\alpha /\beta \right)}_x}}$	${\displaystyle p_2+p_3\sqrt{{\left(\alpha /\beta \right)}_x}LE{T}_d}$	0.99064	0.35605	1.1012	−0.0039
Peeler17	Multiple (48)	${\displaystyle p_0+p_1\frac{LE{T}_d^3}{{\left(\alpha /\beta \right)}_x}}$	${\displaystyle p_2+p_3{\left(\alpha /\beta \right)}_{x}LE{T}_d^3}$	0.75	0.00143	1.24	0.00074
Tilly18	Multiple (11)	${\displaystyle 1+p_1\frac{LE{T}_d}{{\left(\alpha /\beta \right)}_x}}$	1.0	0.309* or 0.550964	--	--	--
Rørvik12	Multiple (85)	${\displaystyle p_0+p_1\frac{LE{T}_d}{{\left(\alpha /\beta \right)}_x}}$	1.0	1.0	0.645	--	--
Wilkens19	V79 (19)	${\displaystyle \frac{p_0+p_{1}LE{T}_d}{{\alpha }_x}}$	1.0	0.1	0.02	--	--
Wedenberg20	V79 (19)	${\displaystyle p_0+p_1\frac{LE{T}_d}{{\left(\alpha /\beta \right)}_x}}$	1.0	1.0	0.434	--	--
Model	Cell lines (size)		RBEmin	Other
Belli21	V79 (6)	${\displaystyle \frac{1}{0.13}{\int }_0^{\mathrm{\infty }}\alpha \left(E\right)d\left(E\right)dE}$	${\displaystyle \frac{1}{0.214}\sqrt{{\int }_0^{\mathrm{\infty }}\beta \left(E\right)d\left(E\right)dE}}$	$\alpha \left(E\right)$and $\beta \left(E\right)$ are biological weighting functions
Jones22	Multiple (28)	${\displaystyle 1+\frac{LE{T}_d-0.22}{30.28}\times }$ ${\displaystyle \left(\frac{{\alpha }_u}{{\alpha }_x}-1\right)}$	${\displaystyle \sqrt{1+\frac{LE{T}_d-0.22}{30.28}\left(\frac{{\beta }_u}{{\beta }_x}-1\right)}}$	${\alpha }_u=2.696\left(1-e^{-3.92{\alpha }_x}\right)$ ${\beta }_u=0.06\left(1-e^{-50{\beta }_x}\right)$

RBE, relative biological effectiveness.

The definition of the RBE_min and RBE_max functions for each model differs as shown, and the fit parameters (p₀, p₁, p₂ and p₃) are summarized were applicable. The type of in vitro data and size of each dataset used in the fitting is indicated. The last two models have been separated from the table since the formulations differ. Only the linear fit of the Rørvik et al¹³ is shown here for simplicity. *For low and high (α/β)_x, respectively.