Table 1.

Overview of the examined dose-response models together with a summary of their inherent parameters.

Statistics	Model	Dose-response, P(D)	Dose-volume response, P(D,V)	Inherent model parameters	D50 / Gy	γ50
Binomial	Critial volume	(1−e−D/D0)N0	$\text{∑}_{\mathit{\text{t}}=\mathit{\text{M}}+1}^{\mathit{\text{N}}}\left(\begin{array}{l}\mathit{\text{N}}\\ \mathit{\text{t}}\end{array}\right){\mathit{\text{p}}}^{\mathit{\text{t}}}{\left(1-\mathit{\text{P}}\right)}^{\mathit{\text{N}}-\mathit{\text{t}}}\S$	D0, N0, M, N	−D0 ln(1−(M/N)1/N0)	$ln{\mathit{\text{N}}}_0{\left(1-\frac{1}{{\mathit{\text{N}}}_0}\right)}^{{\mathit{\text{N}}}_0-1}$ Implicit
	Relative seriality	${\mathit{\text{e}}}^{-{\mathit{\text{N}}}_0{\mathit{\text{e}}}^{-\left(\mathit{\text{D}}/{\mathit{\text{D}}}_0\right)}}$	${\left[1-{\left(1-\mathit{\text{P}}{(\mathit{\text{D}})}^{\mathit{\text{s}}}\right)}^{\frac{\mathit{\text{V}}}{{\mathit{\text{V}}}_{\text{ref}}}}\right]}^{1/\mathit{\text{s}}}$	D0, N0, s	D0(ln N0−ln ln 2)	$\frac{ln2}{2}ln\left(\frac{{\mathit{\text{N}}}_0}{ln2}\right)$
Poisson	Inverse tumor	${\mathit{\text{e}}}^{-N_0{e}^{-\left(\mathit{\text{D}}/{\mathit{\text{D}}}_0\right)}}$	${\mathit{\text{e}}}^{{-\mathit{\text{N}}}_0{\mathit{\text{e}}}^{-\left({\mathit{\text{D}}/\mathit{\text{D}}}_0\right)}}+\mathit{\text{k}}ln\left(\mathit{\text{V}}/{\mathit{\text{V}}}_{\text{ref}}\right)$	D0, N0, k	D0(ln N0−ln ln 2)	$\frac{ln2}{2}ln\left(\frac{N_0}{ln2}\right)$
	Critical element	$e^{-N_0{e}^{-\left(\mathit{\text{D}}/{\mathit{\text{D}}}_0\right)}}$	$1-{\left(1-\mathit{\text{P}}(\mathit{\text{D}})\right)}^{\frac{\mathit{\text{V}}}{{\mathit{\text{V}}}_{\text{ref}}}}$	D0, N0	D0(ln N0−ln ln 2)	$\frac{ln2}{2}ln\left(\frac{{\mathit{\text{N}}}_0}{ln2}\right)$
Probit	Lyman	$\frac{{\gamma }_{50}}{{\mathit{\text{D}}}_{50}}{\int }_0^D{e}^{-\frac{1}{\pi }{\left(\gamma \frac{x-{\mathit{\text{D}}}_{50}}{{\mathit{\text{D}}}_{50}}\right)}^2}\mathit{\text{dx}}$	$\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\mathit{\text{t}}}{e}^{-\frac{t^2}{2}}\mathit{\text{dt}}\Vert$	D50, m, n	D50	$\frac{1}{\mathit{\text{m}}\sqrt{\pi }}$
Logit	Parallel architecture	$\frac{1}{1+{\left(\frac{{\mathit{\text{D}}}_{50}^{\text{FSU}}}{\mathit{\text{D}}}\right)}^{\mathit{\text{k}}}}$	$\frac{1}{\sqrt{2\pi {\sigma }^2}}{\int }_0^{\mathit{\text{f}}}{\mathit{\text{e}}}^{-\frac{{\mathit{\text{v}}-\mathit{\text{v}}}_{50}}{2{\sigma }^2}}\mathit{\text{dv}}$	D50, k, v50, σ	$\frac{{\mathit{\text{D}}}_{50}^{\text{FSU}}}{{\left(\frac{1}{{\mathit{\text{v}}}_{50}}-1\right)}^{1/\mathit{\text{k}}}}$	k/4
Weibull	Weibull distribution	$1-{\mathit{\text{e}}}^{-{\left(\mathit{\text{D}}/{\mathit{\text{A}}}_1\right)}^{{\mathit{\text{A}}}_2}}$	$1-exp\left[-{\left({\frac{\mathit{\text{D}}\left(\frac{\mathit{\text{V}}}{{\mathit{\text{V}}}_{\text{ref}}}\right)}{{\mathit{\text{A}}}_1}}^b\right)}^{{\mathit{\text{A}}}_2}\right]$	A1, b, A2	$\frac{{\mathit{\text{A}}}_1}{{\left(ln2\right)}^{1/{\mathit{\text{A}}}_2}}$	$\frac{{\mathit{\text{A}}}_{2}ln2}{2}$
Binomial	Critial volume	(1−e−D/D0)N0	$\sum _{\mathit{\text{t}}=\mathit{\text{M}}+1}^{\mathit{\text{N}}}\left(\begin{array}{l}\mathit{\text{N}}\\ \mathit{\text{t}}\end{array}\right){\mathit{\text{P}}}^{\mathit{\text{t}}}{\left(1-\mathit{\text{P}}\right)}^{\mathit{\text{N}}-\mathit{\text{t}}}\S$	D0, N0, M, N	−D0 ln(1−(M/N)1/N0)	$ln{\mathit{\text{N}}}_0{\left(1\text{−}\frac{1}{{\mathit{\text{N}}}_0}\right)}^{{\mathit{\text{N}}}_0-1}$ Implicit
	Relative seriality	$e^{-{\mathit{\text{N}}}_0{e}^{-\left(\mathit{\text{D}}/{\mathit{\text{D}}}_0\right)}}$	${\left[1-{\left(1-\mathit{\text{P}}{(\mathit{\text{D}})}^{\mathit{\text{s}}}\right)}^{\frac{\mathit{\text{V}}}{{\mathit{\text{V}}}_{\text{ref}}}}\right]}^{1/\mathit{\text{s}}}$	D0, N0, s	D0(ln N0−ln ln 2)	$\frac{ln2}{2}ln\left(\frac{{\mathit{\text{N}}}_0}{ln2}\right)$
Poisson	Inverse tumor	$e^{-{\mathit{\text{N}}}_0{e}^{-\left(\mathit{\text{D}}/{\mathit{\text{D}}}_0\right)}}$	${\mathit{\text{e}}}^{-{\mathit{\text{N}}}_0{e}^{-\left(\mathit{\text{D}}/{\mathit{\text{D}}}_0\right)}}+\mathit{\text{K ln}}\left(\mathit{\text{V}}/{\mathit{\text{V}}}_{\text{ref}}\right)$	D0, N0, k	D0(ln N0−ln ln 2)	$\frac{ln2}{2}ln\left(\frac{{\mathit{\text{N}}}_0}{ln2}\right)$
	Critical element	$e^{-{\mathit{\text{N}}}_0{e}^{-\left(\mathit{\text{D}}/{\mathit{\text{D}}}_0\right)}}$	$1-{\left(1-\mathit{\text{P}}(\mathit{\text{D}})\right)}^{\frac{\mathit{\text{V}}}{{\mathit{\text{V}}}_{\text{ref}}}}$	D0, N0	D0(ln N0−ln ln 2)	$\frac{ln2}{2}ln\left(\frac{{\mathit{\text{N}}}_0}{ln2}\right)$
Probit	Lyman	$\frac{{\gamma }_{50}}{{\mathit{\text{D}}}_{50}}{\int }_0^{\mathit{\text{D}}}{\mathit{\text{e}}}^{\frac{1}{\pi }{\left(\gamma \frac{x-{\mathit{\text{D}}}_{50}}{{\mathit{\text{D}}}_{50}}\right)}^2}\mathit{\text{dx}}$	${\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\mathit{\text{t}}}{\mathit{\text{e}}}^{-\frac{{\mathit{\text{t}}}^2}{2}}\mathit{\text{dt}}}_{\Vert }$	D50, m, n	D50	$\frac{1}{\mathit{\text{m}}\sqrt{\pi }}$
Logit	Parallel architecture	$\frac{1}{1+{\left(\frac{{\mathit{\text{D}}}_{50}^{\text{FSU}}}{\mathit{\text{D}}}\right)}^{\mathit{\text{k}}}}$	$\frac{1}{\sqrt{2\pi {\sigma }^2}}{\int }_0^{\mathit{\text{f}}}{\mathit{\text{e}}}^{-\frac{\mathit{\text{v}}-{\mathit{\text{v}}}_{50}}{2{\sigma }^2}}\mathit{\text{dv}}$	D50, k, v50, σ	$\frac{{\mathit{\text{D}}}_{50}^{\text{FSU}}}{{\left(\frac{1}{{\mathit{\text{v}}}_{50}}-1\right)}^{1/\mathit{\text{k}}}}$	k/4
Weibull	Weibull distribution	$1-{\mathit{\text{e}}}^{-{(\mathit{\text{D}}/{\mathit{\text{A}}}_1)}^{{\mathit{\text{A}}}_2}}$	$1-exp\left[-{\left(\frac{\mathit{\text{D}}{\left(\frac{\mathit{\text{V}}}{{\mathit{\text{V}}}_{\text{ref}}}\right)}^{\mathit{\text{b}}}}{{\mathit{\text{A}}}_1}\right)}^{{\mathit{\text{A}}}_2}\right]$	A1, b, A2	$\frac{{\mathit{\text{A}}}_1}{{\left(ln2\right)}^{1/{\mathit{\text{A}}}_2}}$	$\frac{{\mathit{\text{A}}}_{2}ln2}{2}$

$\S P=P(D)\left(\frac{V}{V_{\text{ref}}}\right)$; ‖ t is given by equation (19)