In comparison to the established clonogenic assay, the MTS (or MTT) assays are sensitive and accurate methods with several advantages such as requiring a short time to assess the viability, acquiring the results easily and accurately [25, 27–32]. It measures all viable cells thus representing cells from a true tumour population rather than just clonogenic cells (i.e. cells that can form colonies) [33, 34].

Previous studies have shown that the cells exposed to the higher radiation doses cannot regain their exponential growth in a single MTT or MTS assay, and multiple MTS assays are required to obtain measurements of radiation survival that is comparable to that of clonogenic assays [27, 29]. Although previous studies have shown that MTS and MTT assays are suitable to investigate the radiation dose-response analysis [29], the clonogenic assay has been preferred to measure survival after radiation exposure, as it measures the sum of all modes of cell death, encompassing both early and late events such as delayed growth arrest [35]. Therefore, the *α*/*β* ratios derived from a metabolic activity assay at a single time point should be used cautiously in further radiobiological modelling. Additionally, the cells sustaining radiation-induced damage may exhibit delayed effects and there could be a lag between irradiation and subsequent biological events (e.g. cell cycle arrest, caspase activation) leading to cell death [10, 36]. The acute DNA damage induced by radiation and the subsequent cellular responses are influenced by multiple factors including radiation characteristic, dose rate and cell cycle phase [10, 36]. In RNT, the dose rate is one of the most important factors influencing the cell damage, cell repair and also the effective treatment time. The *T*_{crit} for all three lines ranges approximately 3–5.5 days. In this study, the MTS assays were performed eight days after the initial addition of ^{90}YCl_{3}, on average ≈ 4 days beyond these critical times where the treatment is effectively being insignificant and the delayed effects could occur. Thus, performing the MTS assay after the actual treatment time could still include the cell death due to the delayed effects.

The cell survival results provide insight into different damage responses to EBRT and ^{90}Y RNT. With EBRT, the dose was delivered at a constant dose rate of 277 Gy/h in ≈ 1.7 min, whereas ^{90}Y was delivered at a lower dose and varying rate (i.e. 0.013–0.13 Gy/h and 0.077–0.77 Gy/h) over 8 days. The dose rate and nature of radiation clearly affects the nature of cell damage. There are two types of cell damages: (1) double-strand break (DSB), where two proximal DNA strands are broken simultaneously by a single radiation event causing lethal damage (LD) (i.e. cell death) and (2) single-strand break (SSB), where each DNA strand broken by independent radiation events. When only a single DNA strand is damaged, the cell is considered sublethally damaged (SLD) and has the ability to repair itself typically within in 0.5-3 hours [37]. Therefore, at lower dose rates, the probability of accruing two sequential hits to damage before DNA repair is less. Moreover for an exponentially decreasing dose rate, the probability of DSB is even lower. Therefore LDR–^{90}Y RNT is less effective at causing lethal damage than HDR–^{90}Y RNT and EBRT.

The rate of cell kill through the DSB process is measured from the response component of the survival curve parameterised by *α*. The rate of cell kill through the SSB process is measured from the quadratic part of the survival curve, parameterised by *β* which is highly dependent on the initial dose rate. The value of *α* depends on the the radiation type. For high LET radiation such as alpha particles, DSB dominates the cell kill process and therefore *α* has higher significance than *β*. Therefore, for a low and exponentially decreasing dose rate ^{90}Y source, there is less contribution to SF from the *β**D*^{2} component than from the *α* term. This is evident from Fig. 6a (i and ii), as SSB repair results in the formation of a shoulder for the HDR–^{90}Y survival curve. However, as the EBRT dose rate is significantly higher than HDR–^{90}Y RNT (by a factor 400), the rate of lethal SSB production is much higher. Consequently, this results in delivering more lethal damage to cells at an even lower EBRT dose range (compared to ^{90}Y doses). Moreover for LDR–^{90}Y (0.013-0.13 Gy/h), there is even less contribution from the *β**D*^{2} component since the rate of SSB repair occurs even more rapidly (see Fig. 6a (ii)) than the rate of damage. Simulation results further confirmed that for both of HDR– and LDR–^{90}Y, for smaller delivered activities and doses (i.e. ≲ 3 Gy) the average number of hits per cell nucleus is ≲2. Additionally for doses \(\gtrsim \) 3 Gy, although the average number of hits is \(\gtrsim \) 2, the number of beta particle hits per cell nucleus decreases and falls below 2 after ≈ 5 days of irradiation. This is also consistent with the calculated *T*_{crit} for each cell lines (i.e. 3–5.5 days). This suggests that for LDR– ^{90}*Y* only ≈ 1–2 half-lives of delivered dose can potentially cause direct cell nucleus DNA damage.

All three cell lines showed higher sensitivity to EBRT than ^{90}Y. The HT29 and HCT116 cell lines were the most radioresistant and radiosensitive cell lines, respectively. This is also consistent with previous studies [38, 39]. Similar sensitivity for both cell lines was observed for ^{90}Y irradiation results. The *α*/*β* ratios for both EBRT and ^{90}Y were derived from Eq. 5. However, several models have been developed to incorporate the effect of DNA repair and the exponentially changing dose rate of RNT [10, 37]. The extended LQ model includes the kinetics of DSB creation, repair, and misrepair to estimate the true fraction of surviving cells in an irradiated cell population (Eqs. 9a and 9b).

$$\begin{array}{*{20}l} \text{-ln(SF)} &= \alpha\,D+\beta\,G\,D^{2} \end{array} $$

(9a)

$$\begin{array}{*{20}l} G &= \frac{2}{D^{2}}\,\int_{-\infty}^{\infty}\,\dot{D}(t)\,dt\,\int_{-\infty}^{t}\,\dot{D'}(t')\,\,e^{- \mu\,(t-t')}\,dt' \end{array} $$

(9b)

where *G* is the Lea–Catcheside factor and *μ* is the DNA repair time constant [40]. The first integral represents the physical absorbed dose. The integrand of the second integral over *t*^{′} refers to the first SSB of two DSBs required to cause lethal damage. Also, the integral over *t* refers to the second SSB of remaining of two DSBs to cause the unrepairable lethal damage. The exponential term describes the repair and reduction in 2 SSB → DSB process due to decreasing dose rate. For a constant EBRT dose rate where the dose is delivered acutely and the irradiation time is very short, the *G*–factor approaches unity and Eq. 9a simplifies to its general form (Eq. 5). This means the rate of SSB → DSB process production is higher than the DNA repair rate, and therefore, the kinetics of DSB creation are negligible.

In comparison to EBRT, the ^{90}Y RNT dose rate is much lower and exponentially decreasing (Eq. 1), so the difference in the kinetics of DSB creation has more impact on cell survival. The *G* factor can be derived for ^{90}Y radiation using Eqs. 10a, 10b.

$$\begin{array}{@{}rcl@{}} G\,=2\,\bigg(\frac{\dot{D_{0}}}{D}\bigg)^{2}\,\int_{0}^{T} e^{-\lambda t}\,dt\,\int_{-\infty}^{t} e^{-\lambda t'}\,e^{- \mu\,\left(t-t'\right)}\,dt' \end{array} $$

(10a)

$$\begin{array}{@{}rcl@{}} = \frac{2}{\lambda-\mu}\,\bigg(\frac{\lambda}{1-e^{-\lambda T}}\bigg)^{2}\,\bigg(\frac{1-e^{-(\lambda+\mu) T}}{\lambda+\mu}\,-\frac{1-e^{-2 \lambda T}}{2\lambda}\bigg) \end{array} $$

(10b)

For long irradiation times (i.e. *T* → *∞*) the *G* factor reduces to:

$$\begin{array}{@{}rcl@{}} G_{\infty}\,=\frac{\lambda}{\lambda+\mu}\,=\frac{\tau_{1/2}}{T_{1/2}+\tau_{1/2}} \end{array} $$

(11)

where *T* and *τ* are the physical half-life of ^{90}Y and DNA repair half-life, respectively. Since *G*_{∞} only depends on the physical half-life of ^{90}Y and the DNA repair half-life, its value for both HDR– and LDR–^{90}Y RNT can be determined using Eq. 11. However, as the DNA repair half-lives for all the three cell lines in our study were not experimentally derived, *G*_{∞} for HDR– and LDR–^{90}Y was calculated using Eq. 11 and assuming an average value of *τ*_{1/2}=1.5 h [11], giving *G*_{∞}=0.022. Such a relatively small *G*_{∞} value for ^{90}Y suggests less contribution from 2 SSB → DSB production (e.g. *β* *D*^{2}) as the dose rate is relatively lower compared to EBRT. In comparison, *G*_{∞}≈1 for EBRT, using an irradiation time ≈ 1.7 min. This is more evident for the LDR–^{90}Y survival curve since for a given dose, the initial dose rate is ≈ 0.013–0.13 Gy/h. This is also consistent with the simulation results. For example, for LDR–^{90}Y, for smaller delivered activities and doses (i.e. ≲ 3 Gy), the average number of hits per cell nucleus is ≲ 2 during the 8 days irradiation time (see Fig. 7a). Therefore, there will be very low probability for 2 SSB → DSB process at this dose rate levels. These results highlight the importance of initial dose rate in radionuclide therapy.

In RNT, dose rate is inversely proportional to the radionuclide half-life for a given total dose. Therefore, a longer half-life radionuclide such as ^{90}Y delivers dose at a relatively lower rate for a fixed total dose. Equation 11 implies that for a similar absorbed dose, radionuclides with a shorter half-life could potentially result in better dose response for RNT. Since radionuclides with shorter half-lives deliver dose at much higher initial rate, the DNA damage process can compete with DNA repair rate and thus result in better dose response.

The *G*_{∞} factor was calculated and plotted (see Fig. 9 in Appendix) for a few commonly used isotopes in RNT and emission tomography for comparison. These calculations assume *μ* to be 0.42 h ^{−1} (ln(0.5)/15 hours). The *G*_{∞} factor also plays an important role for high LET radiation. For example ^{213}Bi with a 46 min half-life which decays ≈2*%* as high LET alpha and ≈98*%* low energy beta particles, delivers dose at a reasonably high dose rate. Currently ^{213}Bi is the only radionuclide that can deliver dose at a dose rate close to the EBRT dose rate (*G*_{EBRT} ≈ 1, *G*_{Bi} ≈ 0.7). The ability to deliver a very localised dose at very high dose rate has a significant impact on the RNT [41]. For both EBRT and ^{90}Y, the HT29 and HCT116 cell lines were the most radioresistance and radiosensitive cell lines, respectively. HT-29 cells have the TP53 mutation which may increase the cell resistance to radiation [38]. The *α*/*β* derived from the HDR–^{90}Y cell survival curve was larger than that derived from EBRT by a factor of ≈ 8 for all three cell lines. This can be attributed to a relatively lower dose rate of ^{90}Y. Also for each given dose, the total HDR–^{90}Y absorbed dose (6.2–55.5 Gy) was up to six times greater than the EBRT dose (1–9 Gy).

Additionally, different *α* and *β* values derived for HDR–^{90}Y RNT and EBRT reflect on the different relative radiobiological effectiveness of ^{90}Y RNT compared to EBRT.

These results are subject to several assumptions. Firstly, for all the calculations and formalisms, it was assumed that there is no significant cell repopulation during 8 days of incubation. A more complex formulation is required to offset the repopulation effect [10, 42]. Secondly, it was assumed that ^{90}YCl_{3} was uniformly distributed in the well. This assumption was used to calculate the absorbed dose to cells for each ^{90}YCl_{3} activity.