Fig. 5

The estimated critical time (\(T_{{{\text{critic}}}}\)) and dose rate (\(R_{{{\text{critic}}}}\)). a The four-dimensional (4D) theoretical \(T_{{{\text{critic}}}}\) spectrum and the estimated clinical \(T_{{{\text{critic}}}}\) (dot points) using the radiobiological parameters derived from patient data. b The theoretical \(R_{{{\text{critic}}}}\) spectrum and the estimated clinical \(R_{{{\text{critic}}}}\) and \(R_{{{\text{critic}}\left( k \right)}}\) (blue and green dot points, respectively) using the radiobiological parameters derived from patient data (see Table 1). The clinical \(R_{{{\text{critic}}}}\) and \(R_{{{\text{critic}}\;\left( k \right)}}\) were estimated using the derived repopulation \(T_{{\text{p}}}\) and for when \(T_{{\text{p}}} = T_{{\text{k}}}\), respectively. The \(T_{{{\text{critic}}}}\) and \(R_{{{\text{critic}}}}\) spectra are predicted by Eqs. 6 and 7 for a range of \(\alpha\), \(T_{{\text{p}}}\) and initial dose rate, \(R_{0}\) parameters