## Introduction

Radiotherapy plays an important role in treating cancer and is often applied after surgical resection and current with chemotherapy, to achieve tumor control1. Recent radiation therapy uses modulated radiation intensity, so called intensity modulated radiation therapy (IMRT), which enables preserving organs at risk2,3. Also, stereotactic radiation therapy (SRT) gives high dose to solid tumors from many different angles4,5. Such advanced irradiation techniques require relatively longer dose-delivery times compared to previously used methods such as conformal therapy (3D-CRT)2,6, inducing cell recovery (radioresistance) by sub-lethal damage repair (SLDR)7,8,9. On the other hand, cancer stem-like cells (CSCs)10 with intrinsic or acquired radioresistance have higher DNA damage repair capability compared to non-CSCs11. Considering these biological characteristics, the possibility of recovery of CSCs between dose fractionations cannot be ignored. Therefore, the mechanisms underlying radioresistance in stem-like cells urgently need to be elucidated.

Radioresistant cell lines, such as oral squamous carcinoma SAS-R and HSC2-R, can be experimentally established after fractionated X-ray irradiations with 2 Gy/day for more than 30 days, which is often performed as a standard radiotherapy plan12. The CSC content can be evaluated using specific surface markers for CSCs and tumor formation ability13. Although fundamental biological data of radioresistant cell populations, including CSCs, have been experimentally accumulated14,15, the radioresistance mechanism remains uncertain because of the limited amount of experimental data. To solve this issue and to efficiently study the underlying mechanisms, we focused on a theoretical analysis using a biophysical model for predicting cell death. Among several biophysical models16,17,18,19,20,21, the integrated microdosimetric-kinetic (IMK) model has a unique feature that explicitly considers CSC content and DNA repair capability in heterogeneous cell populations22,23,24, enabling the theoretical evaluation of mechanisms underlying radioresistance and cell recovery of radioresistant cell lines.

## Materials and methods

### Cell culture

In this study, we chose the human oral squamous carcinoma cell lines SAS and HSC2, and their radioresistant counterparts SAS-R and HSC2-R as models12. These cell lines were obtained from the Cell Resource Center for Biomedical Research, Institute of Development, Aging and Cancer, Tohoku University. It should be noted that SAS-R and HSC2-R acquired radioresistance after exposed to daily 2 Gy of X-rays for more than 1 year12. These cell lines were maintained at 37 ℃ and 5% CO2 environment in Roswell Park Memorial Institute 1640 medium (Thermo Fisher Scientific, Inc. Tokyo, Japan) supplemented with 10% heat-inactivated fetal bovine serum (FBS) (Japan Bio Serum, Fukuyama, Japan) and 1% penicillin/streptomycin (Life Technologies, CA, USA).

In single dose experiment, the cultured cells were irradiated with kilo-voltage X-rays (150 kVp, 1.0 Gy/min) through an additional filter 0.5 mm aluminum and 0.3 mm copper using an X-ray generator (MBR-1520R-3; Hitachi Medical Co., Ltd., Tokyo, Japan). Split-dose experiment was performed with 4 Gy in two divided doses (i.e., 2 Gy irradiation twice) at various inter-fraction times. The same X-ray generator as the single dose experiment was used in case of split-dose experiment. To achieve various dose rates, 0.1 and 0.25 Gy/min, we performed multi fractionation experiment. For 0.1 Gy/min, 6 or 10 Gy irradiation was fractionated and irradiation time was 60 and 100 min, respectively. For 0.25 Gy/min, irradiation time was 24 and 40 min. The fractionation size was 1 Gy (1.0 Gy/min). The dose-averaged linear energy transfer (LETD) and the dose-mean lineal energy (yD) were estimated to be 1.53 kV/µm and 4.68 keV/μm, respectively, using the Particle and Heavy Ion Transport code System (PHITS) ver. 3.2125, and the in-house code of WLTrack for electrons26. The dose in air was monitored with a thimble ionization chamber placed next to the sample during irradiation. The uncertainty of the absorbed dose measured by the thimble ionization chamber was ± 1%.

### Colony formation assay

Clonogenic potency was evaluated using a colony formation assay. The appropriate number of cells were seeded on φ60 or φ100 cell culture dishes and irradiated with X-rays after 2 h incubation to allow cells to adhere to the bottom of the dish. The cells were fixed with methanol (Wako Pure Chemical Industries, Ltd., Osaka, Japan) 7–10 days after irradiation, and stained with Giemsa staining solution (Wako Pure Chemical Industries). Colonies with more than 50 cells were counted. The surviving fraction for each cell line was calculated from the ratio of the plating efficiency of irradiated cells to that of non-irradiated cells.

### Flow cytometric analysis for detecting the ALDH positive fractions

To input the reference value of CSC fraction into the theoretical model, we selected ALDH which is reported as CSC marker for head and neck cancer27, and measured the CSC fraction for each cell line. The ALDEFLUOR Kit (Catalog no. 01700) was purchased from STEMCELL Technologies, Inc. (Vancouver, Canada). Trypsinized cells were adjusted to a density of 1 × 106 cells/mL and washed twice with phosphate-buffered saline. Next, the cells were incubated for 45 min at 37 ℃ in the dark after the addition of ALDEFLUOR reagent (5 μL/106 cells). For the negative control, 1.5 mM diethylaminobenzaldehyde (DEAB) was added at a final concentration of 7.5 μM. Subsequently, the cells were centrifuged, resuspended in ALDEFLUOR assay buffer, and analyzed by direct immunofluorescence flowcytometry using a FACS Aria (BD Biosciences, Tokyo, Japan). The percentage of ALDH (+) was determined by subtract from the positive percentage of DEAB administration group with about 0.5% false positives. Noted that the ALDH (+) cell populations was assumed to correspond to a radioresistant population. In SAS-R and HSC2-R cells, the ALDH activity within six months (i.e., passage number of 20–50) after the last irradiation.

The experiment was repeated three times. The significances of differences between non-irradiated group and irradiated group were evaluated using Student’s t-test. P < 0.01 was considered to indicate a statistically significant difference, which is noted as ** in this study.

### A theoretical cell-killing model for analyzing measured survival data

To theoretically analyze the cell survival data measured by clonogenic survival assay, we employed a theoretical cell-killing model, the “IMK model”, which considers microdosimetric quantities along ionizing radiation tracks, cell recovery during irradiation, and the existence of stem-like cells22,23,24. In this section, we provide an overview of the IMK model used in this study.

### Surviving fraction after single-dose irradiation for a certain single cell population

In the IMK model, it is assumed that the nucleus is radiation sensitive target divided into hundreds of small sites (called domains) with a diameter of 1–2 μm. By considering the energy deposited in every domain and repair kinetics of induced DNA lesions in a domain, the model enables the evaluation of the impact of microdosimetry and SLDR on cellular damage after irradiation23,24. To consider the repair kinetics of DNA lesions during and after irradiation, the model incorporates potentially lethal lesions (PLLs) that can undergo one of three transformations: (i) a PLL can transform into a lethal lesion at a constant rate a (h−1); (ii) two PLLs can transform into a LL at a constant rate bd (h−1); (iii) a PLL can be repaired by DNA damage repair pathway (mainly non-homologous end joining (NHEJ)) at a constant rate c (h−1)28. By solving the kinetic equations of PLLs, the surviving fraction for various irradiation conditions (i.e., single-dose irradiation, split dose irradiation, and multi-fractionated irradiations) can be expressed based on the linear-quadratic relation.

First, we assume that a single dose is delivered to a single-cell population with the dose-delivery time T (h). This modeling was discussed in our previous report on the IMK model23. Thus, the surviving fraction of a certain single cell population S after single-dose irradiation can be expressed as:

$$- \ln S = \left( {\alpha_{0} + \frac{{y_{D} }}{{\rho \pi r_{d}^{2} }}\beta_{0} } \right)D + F\beta_{0} D^{2} ,$$
(1)

where D is the absorbed dose (= $$\dot{D}$$ T); $$\dot{D}$$ is the absorbed dose rate (Gy/h); yD is dose-mean lineal energy (keV/μm) representing the radiation track structure29; ρ and rd represent the density of liquid water (1.0 g/cm3) and radius of a domain (set as 0.5 μm in this study), respectively; F is the Lea-Catcheside time factor, which is expressed as

$$F = \frac{2}{{(a + c)^{2} T^{2} }}\left[ {(a + c)T + e^{ - (a + c)T} - 1} \right].$$
(2)

Note that (a + c) can be approximated as c, representing the SLDR rate (h−1)23. $$\alpha_{0}$$ and $$\beta_{0}$$ are the coefficients for the dose (Gy−1) and dose squared (Gy−2), respectively, which are in inversely proportional to (a + c)23 as

$$\alpha_{0} \propto \frac{1}{(a + c)} \cong \frac{1}{c}\;{\text{and}}\;\beta_{0} \propto \frac{1}{(a + c)} \cong \frac{1}{c}.$$
(3)

Using Eqs. (1)–(3), we can estimate tumor cell survival for various dose rates based on the cell-specific parameters of a certain single cell population [$$\alpha_{0} , \beta_{0} ,$$ (a + c)].

### Surviving fraction after split-dose irradiation for a certain single cell population

Next, we assume that a certain single cell population is exposed to two irradiation doses with inter-fraction time $$\tau$$ (h). Based on previous reports23,30, the surviving fraction for split-dose irradiation can be expressed as:

$$- \ln S(\tau ) = \mathop \sum \limits_{i = 1}^{2} \left[ {\left( {\alpha_{0} + \frac{{y_{D} }}{{\rho \pi r_{d}^{2} }}\beta_{0} } \right)D_{i} + \beta_{0} D_{i}^{2} } \right] + 2\beta_{0} e^{ - (a + c)\tau } D_{1} D_{2} .$$
(4)

Note that we assume that two doses (D1 and D2) are acutely delivered to the cell population. By using the surviving fractions taking the limits of the fractionation time ($$\tau \to 0, \tau \to \infty$$), S(0), and S($$\infty$$), the SLDR rate can be calculated from the experimental split-dose cell recovery curve. The SLDR rate can be expressed based on the previous modeling23 as

$$(a + c) = \frac{{\mathop {\lim }\limits_{\tau \to 0} \frac{1}{S}\frac{{{\text{d}}S}}{{{\text{d}}\tau }}}}{{\ln \frac{S(\infty )}{{S(0)}}}}.$$
(5)

As perprevious reports, the mean value of (a + c) for cancer cell lines ranges from 1.506 to 2.218 (h−1)23.

### Surviving fraction considering progeny cells and cancer stem-like cells

Cells that survive fractionated irradiation can acquire a greater radioresistant ability than the non-irradiated (non-radioresistant) parental cells, as shown in Fig. 1A. Experiments suggest that the radioresistant cell line exhibits enhanced DNA repair capability and a higher CSC fraction compared to non-radioresistant cell lines14,31. To account for these cellular characteristics, we introduced a two-cell population model22 and the enhancement factor of SLDR, wSLDR. Using the model for a single cell population (i.e., Eqs. (1) and (2)), the surviving fractions for progeny cells and CSCs can be expressed as

$$- \ln S_{{\text{p}}} = \left( {\alpha_{{0{\text{p}}*}} + \frac{{y_{D} }}{{\rho \pi r_{{\text{d}}}^{2} }}\beta_{{0{\text{p}}*}} } \right)D + \frac{{2\beta_{{0{\text{p}}*}} }}{{(a + c)_{{{\text{p}}*}}^{2} T^{2} }}\left[ {(a + c)_{{{\text{p}}*}} T + e^{{ - (a + c)_{{{\text{p}}*}} T}} - 1} \right]D^{2}$$
(6)
$$- \ln S_{{\text{s}}} = \left( {\alpha_{{0{\text{S}}}} + \frac{{y_{D} }}{{\rho \pi r_{{\text{d}}}^{2} }}\beta_{{0{\text{s}}}} } \right)D + \frac{{2\beta_{{0{\text{s}}}} }}{{(a + c)_{{\text{H}}}^{2} T^{2} }}\left[ {(a + c)_{{\text{H}}} T + e^{{ - (a + c)_{{\text{H}}} T}} - 1} \right]D^{2}$$
(7)

where Sp and Ss are the surviving fractions for progeny cells and CSCs, respectively; [$$\alpha_{{0{\text{p}}*}} , \beta_{{0{\text{p}}*}} ,$$ (a + c)p*], and [$$\alpha_{{0{\text{s}}}} ,\beta_{{0{\text{s}}}} ,$$ (a + c)H] are model parameters for progeny cells and CSCs, respectively; (a + c)p* is the SLDR rate of progeny cells, and (a + c)H is the SLDR rate of CSCs. Based on the experimental report15, we assumed that the DNA repair efficiency (SLDR rate) of progeny cells in the radioresistant cell populations was enhanced compared to that of the non-radioresistant parental cells. With this assumption, we can express that the set of parameters for progeny cells can be modulated by the (a + c) value. The parameter set of [$$\alpha_{{0{\text{p}}*}} , \beta_{{0{\text{p}}*}} ,$$ (a + c)p*] can be expressed as follows

\begin{aligned} \left( {a + c} \right)_{{{\text{p}}*}} & = \left( {a + c} \right)_{{\text{p}}} \left[ {{\text{for}}\;{\text{non-radioresistant}}\;{\text{cells}}} \right] \\ & = (a + c)_{{\text{H}}} \left[ {{\text{for}}\;{\text{radioresistant}}\;{\text{cells}}} \right] \\ \end{aligned}
(8)
$$w_{{{\text{SLDR}}}} = \frac{{(a + c)_{{\text{H}}} }}{{(a + c)_{{\text{p}}} }}$$
(9)
$$\alpha_{{0{\text{p}}*}} = \frac{{\alpha_{{0{\text{p}}}} }}{{w_{{{\text{SLDR}}}} }}$$
(10)
$$\beta_{{0{\text{p}}*}} = \frac{{\beta_{{0{\text{p}}}} }}{{w_{{{\text{SLDR}}}} }},$$
(11)

where (a + c)p is the inherent SLDR rate in non-radioresistant parent (progeny) cells. The model parameters for progeny cells, progeny cells with increased SLDR rates, and CSCs are summarized in Fig. 1B.

Similar to single-dose irradiation, the surviving fractions of progeny cells and CSCs in case of split-dose irradiation can be expressed as

$$- \ln S_{{\text{p}}} (\tau ) = \mathop \sum \limits_{i = 1}^{2} \left[ {\left( {\alpha_{{0{\text{p}}*}} + \frac{{y_{D} }}{{\rho \pi r_{{\text{d}}}^{2} }}\beta_{{0{\text{p}}*}} } \right)D_{i} + \beta_{{0{\text{p}}*}} D_{i}^{2} } \right] + 2\beta_{{0{\text{p}}*}} e^{{ - (a + c)_{{{\text{p}}*}} \tau }} D_{1} D_{2}$$
(12)
$$- \ln S_{{\text{s}}} (\tau ) = \mathop \sum \limits_{i = 1}^{2} \left[ {\left( {\alpha_{{0{\text{s}}}} + \frac{{y_{D} }}{{\rho \pi r_{{\text{d}}}^{2} }}\beta_{{0{\text{s}}}} } \right)D_{i} + \beta_{{0{\text{s}}}} D_{i}^{2} } \right] + 2\beta_{{0{\text{s}}}} e^{{ - (a + c)_{{\text{H}}} \tau }} D_{1} D_{2}$$
(13)

As shown in Fig. 1, we considered that the cancer cell line is composed of two cell populations: progeny cells and CSCs. By considering the surviving fraction for the cell population containing the progeny cells and CSCs, the overall surviving fraction S can be expressed as

$$S = S_{{\text{p}}} f_{{\text{p}}} + S_{{\text{s}}} f_{{\text{s}}}$$
(14)

where $$f_{{\text{p}}}$$ is the fraction of progeny cells, and $$f_{{\text{s}}}$$ is the fraction of CSCs. Note that fp + fs = 1. As shown in Fig. 1B, the differences in cell-specific parameters between non-radioresistant parental cell lines and radioresistant cell lines are (i) an increase in the SLDR rate in progeny cells and (ii) fraction of CSCs. As shown in Fig. 1C, using Eqs. (6)–(12) and (15), we analyzed the surviving fractions of non-radioresistant cell lines (SAS and HSC2) and radioresistant cell lines (SAS-R and HSC2-R) measured in this study.

### Determination of the model parameters

The model parameters were determined by applying the present model to the experimental survival data after single-dose and split-dose irradiation. The steps to obtain the model parameters are: (i) determination of the (a + c) values using the experimental split-dose cell recovery curve and Eq. (5), and (ii) determination of the cell-specific parameters [α0p, β0p, (a + c)p, α0S, β0S, wSLDR] by applying Eqs. (6)–(12) and (15) to the experimental surviving fraction after single-dose irradiation of both non-radioresistant and radioresistant cell lines. For the latter determination, we used the Markov chain Monte Carlo (MCMC) simulation32,33. The algorithms of MCMC are summarized in previous reports23,34, which enable the evaluation of the uncertainties of model parameters.

In the MCMC simulation, the prior distributions of α0p, β0p, α0S, β0S, and wSLDR were set to be uniform. We also assumed that the parameters of stem-like cells [$$\alpha_{{0{\text{s}}}} ,\beta_{{0{\text{s}}}}$$] are smaller than those of progeny cells [$$\alpha_{{0{\text{p}}}} ,\beta_{{0{\text{p}}}}$$] based on our previous study22. The prior distribution of (a + c)p was obtained from the model analysis using the experimental split-dose cell recovery curve of the non-radioresistant cell lines (SAS, HSC2), while that of fs was the experimental flowcytometric data itself. The set of model parameters θ[α0p, β0p, (a + c)p, α0S, β0S, wSLDR, fp, fs] was sampled following the likelihood P(di|θ) and the transition probability αP as follows:

$$P(d|\theta ) = \mathop \prod \limits_{i = 1}^{N} [P(d_{i} |\theta )] = \mathop \prod \limits_{i = 1}^{N} \left\{ {\frac{1}{{\sqrt {2\pi \sigma } }}\exp \left[ { - \frac{{\left( { - \ln S_{\exp i } + \ln S_{{{\text{cal}}i}} } \right)^{2} }}{{2\sigma^{2} }}} \right]} \right\}$$
(15)
$$\alpha_{{\text{P}}} = \frac{{P\left( {\theta^{{{\text{candidate}}}} | d} \right)}}{{P\left( {\theta^{(t)} |d} \right)}}$$
(16)

where di (i = 1 ~ N) is the experimental survival data [di = (Di, − ln Sexpi)], Sexp is the experimental surviving fraction measured by the colony formation assay, Scal is the surviving fraction calculated by the model, and P(θ|d) and P(θcandidate|d) are the posterior likelihood for the candidate (t + 1)-th and the previous (t)-th conditions, respectively, as reported previously24.

After sampling the set of parameters via MCMC, we calculated the mean values and standard deviations of the model parameters. Note that the parameters fS and (a + c)p were updated using the MCMC simulation. Using the mean values, we calculated the surviving fraction to analyze the radiosensitivity of non-radioresistant and radioresistant cell lines. To check whether the present IMK model is in agreement with the experimental data, we calculated the coefficient of determination (R2) value.

## Results

### Determination of SLDR rate from split-dose irradiation experiment

To obtain the prior information on the SLDR rate (i.e., c in h−1), we measured the surviving fraction after split-dose irradiation and applied Eq. (5) to the measured survival data of non-radioresistant SAS and HSC2 cell lines. Figure 2 shows the split-dose cell recovery curve, where (A) is the curve of SAS and (B) is that of HSC2. The open circles represent the experimental results, while the two dotted lines indicate the initial slopes of dS/dτ and S(∞) expressed in Eq. (5), respectively. As shown in Fig. 2, the cell surviving fraction of both cell lines increased monotonically in the interval range of 0–3 h, and the recovery was saturated in the interval range of 6–24 h. Cell recovery between dose fractionation reflects the SLDR of cell survival23.

By using the values of dS/dτ, S(∞) and S(0) described in Fig. 2 and Eq. (5), the SLDR rates ((a + c) value) of SAS and HSC2 were estimated to be 1.31 ± 0.69 (h−1) and 1.45 ± 0.93 (h−1), respectively. Even in the case of the same oral squamous carcinoma cell line, the SLDR rate of HSC2 was slightly higher than that of SAS. The previous MK model analysis suggested that PLLs correspond to DNA double-strand breaks35. It is well known that two major repair pathways, NHEJ and homologous recombination (HR), can repair radiation-induced DNA double strand-breaks (DSBs)36, but the SLDR rate mainly comprises the faster component of NHEJ37. The difference between SAS and HSC might be attributed to the cell-cycle distributions. However, judging from their uncertainties, the results suggest that the (a + c) values of SAS and HSC2 are similar.

### Measurement of the ALDH-positive fractions by flowcytometric analysis

The IMK model includes several model parameters. To efficiently determine the parameter sets, we experimentally determined the ALDH (+) fractions (assuming fs value) in the cell lines used in this study. In general, the abundance of ALDH (+) in the cell population can be measured by flowcytometric analysis. In this study, ALDH was used to determine the radioresistant population. Figure 3 shows the experimental results of the ALDH (+) fraction of each cell line: (A) SAS and SAS-R, and (B) HSC2 and HSC2-R. In Fig. 3, the percentages of ALDH (+) cells were 0.97 ± 0.68% for SAS, 9.65 ± 3.65% for SAS-R, 1.36 ± 0.32% for HSC2, and 12.61 ± 6.11% for HSC2-R. From these measured ALDH (+) fractions, it was confirmed that the SAS-R and the HSC2-R included more stem-like cells than SAS and HSC2, which suggests that the radioresistance acquired after fractionated irradiation is intrinsically related to the increment of ALDH (+) fractions.

### Dose–response curve fitting by theoretical model

Using the experimental values of (a + c) and fs as prior information, we analyzed the dose–response curve of the cell surviving fraction. We performed the MCMC simulation to determine the parameter sets (α0p*, β0p*, (a + c)p, α0s, β0s, wSLDR) for non-radioresistant SAS and HSC2 cell lines. Figure 4 shows the distribution of the model parameters (α0p*,β0p*, α0s,β0s) after the MCMC simulation; (A) SAS cell line, and (B) HSC2 cell line. As expected, the parameters of the progeny cells tended to be lower than those of CSCs. The mean values and standard deviations of the model parameter sets are summarized in Table 1. As shown in Table 1, the weighting factor of the SLDR rate, wSLDR is significantly higher than 1.0, indicating that the cell recovery effects of the progeny cells in both SAS-R and HSC2-R cell lines can be enhanced.

Figure 5 shows the comparison of cell survival predicted by the IMK model and the experiments. The accuracy of the model was evaluated by the R2 value. The results indicated that the IMK model, which considered changes in the CSC fraction and SLDR rate, complied well with the experimental data. Notably, as shown in Fig. 5, the cell survival curves exhibited a sigmoid nature in the relationship between dose and logarithmic survival. The radioresistance exhibited by CSCs against the high dose range (i.e., 10–15 Gy in Fig. 5) showed a similar tendency as our previous report on CSCs22. In addition, as shown in Fig. 5, the tendency of the dose response was more pronounced in radioresistant cell lines (SAS-R and HSC2-R) than in non-radioresistant cell lines (SAS and HSC2). These comparisons in Fig. 5 suggest that the changes in both the CSC fractions and DNA repair efficiency (i.e., SLDR) are essential to reproduce the experimental dose–response curves for cell survival.

### Prediction of repair capability during irradiation by the IMK model

To determine the repair capability of CSCs during fractionated radiotherapy, the cell recovery of the four cell lines in the intervals between irradiation was evaluated. Figure 6 shows the relative radiosensitivity at various time intervals between the irradiations. The relative sensitivity was calculated from the ratio of the surviving fraction after fractionated radiation to that after acute irradiation. As shown in Fig. 5, the relative radiosensitivity estimated by the IMK model (Eqs. (13)–(15)) showed good agreement with the experimental values. As per these comparison results, both the cell survival of non-resistant and resistant cells were saturated at intervals of approximately 3 h, suggesting that the cell recovery during dose fractionation is dominant until a 3 h interval.

In addition to the acute irradiation (Fig. 5) and split-dose irradiation (Fig. 6), we further evaluated the dose-rate effects of non-resistant and resistant cell lines. Figure 7 shows the relationship between the absorbed dose rate and surviving fractions of the four cell lines. Dose-rates of 0.1 and 0.25 Gy/min were achieved by multi-fractionated irradiations. As shown in Fig. 7, while the survival of SAS and SAS-R estimated by the IMK model seemed to be slightly lower than the experimental survival, that of the four cell lines was in good agreement with the experimental results, as indicated by the R2 values. Consistent with the general theory of dose-rate effects that cellular damage leading to cell death is reduced at dose-rates 1.0 Gy/min to 1.0 Gy/h38 cell-killing effects were saturated at a dose rate higher than 1.0 Gy/min. In the same manner as the general tendency of dose-rate effects (Fig. 7), the survival results at dose rates below 1.0 Gy/min exhibited significant recovery. The increase in cell survival was also reproduced by the IMK model with high accuracy, suggesting that both the CSC content and the SLDR rate play an important role in predicting the dose-rate effects of non-resistant and resistant cell lines.