A mathematical modelling of Elkind recovery

Abstract

Elkind recovery is investigated from the viewpoint of mathematical modelling. We express the N-target/1-hit model in terms of simultaneous differential equations, where each step corresponds to the cells in which specific targets have been hit. The recovery effects are taken into account in the same manner as the Whack-A-Mole model, which makes dose rate an essential parameter of the model. Using the model, we reproduced the essential features of Elkind recovery, such as the reappearance of the shoulder in the survival curve, the increase of the survival rate with the number of splits, and the increase of the recovery rate with the length of the non-irradiation period.

1. INTRODUCTION

The biological effects of radiation differ depending on the dose rate even at the same total dose, which is called the dose-rate effects (Ruhm et al., 2016). The dose-rate effects are considered to be due to the recovery of damage that occurs during irradiation (Hall, 1994). Therefore, especially at low dose rates, the recovery effects are very important in considering the biological effects of radiation. Recovery distinguishes between physical/chemical and biological processes. In physical processes, once a wound occurs, it does not heal spontaneously; the number of wounds only increases over time. In contrast, biological processes result in various types of recovery: At the DNA level, DNA damage is repaired; at the cellular level, damaged cells die or are removed by competition with healthy cells and are replaced by new cells.

Sublethal damage (SLD) recovery or Elkind recovery (Elkind and Sutton, 1959, 1960) is a well-known phenomenon. When cultured cells are acutely irradiated with radiation, they lose their ability to proliferate in response to the dose. Elkind and Sutton showed that irradiation of cultured cells in two split doses at a sufficient interval resulted in a higher number of viable colonies than a single exposure at the same total dose. They postulated that this is the result of radiation-induced damage to cells (sublethal damage) that is restored during the period of non-irradiation. They then argued that this is understandable if a series of damage is required for cells to lose their ability to proliferate.

In this article, we describe the above phenomenon in terms of a mathematical model that takes account of the recovery effects. Previously, we proposed the Whack-A-Mole (WAM) model to describe DNA mutations of germ cells (Manabe et al., 2015) as a mathematical model that takes recovery effects into consideration. This model is given by a differential equation with time as a variable, consisting of two terms: one term that increases the number (fraction) of mutated cells and one term that decreases them. In the WAM model, the origin of the recovery is the removal of the mutated cells. In the case of Elkind recovery, we treat the recovery at the cellular level, and the origin of the recovery is the repair of the DNA damage.

2. MATERIALS AND METHODS

SLD can be understood by assuming that multiple factors (targets) are involved for the cell to die (loss of proliferative ability). We denote the number of targets as N and assume that cells do not die and continue to have the ability to form colonies while the number of damaged targets is N-1 or less (N-target model). If each target is damaged with a single hit, we refer to the model as the N-target/1-hit model. SLD recovery can be understood as the repair of DNA damage, especially double-strand break (DSB), during the non-irradiation period of fractionated irradiation (Utsumi et al., 2001).

For simplicity, we assume that all targets are destroyed stochastically at the same rate. According to the probability theory, the probability that k out of N targets are broken when irradiated with a dose D is given as follows:

$$P_{\mathrm{k}}={\displaystyle \frac{N!}{k!(N-k)!}({\mathrm{e}}^{-D/D_0}{)}^{\mathrm{N}-\mathrm{k}}(1-{\mathrm{e}}^{-\mathrm{D}/{\mathrm{D}}_0}{)}^{\mathrm{k}}}$$

(1)

where D₀ is the average dose required to destroy one target. Using this formula, the probability of survival is derived as $S=1-P_{\mathrm{N}}=1-(1-{\mathrm{e}}^{-\mathrm{D}/{\mathrm{D}}_0}{)}^{\mathrm{N}}$ .

In this study, Elkind recovery is investigated from the viewpoint of mathematical modelling. We formulate the N-target/1-hit model that takes account of the recovery effects in terms of simultaneous differential equations for P_k as follows:

$$\begin{array}{rl}& {\displaystyle \frac{\mathrm{d}{P}_0}{\mathrm{d}t}=-{\lambda }_0{P}_0+{\mu }_1{P}_1}\\ & {\displaystyle \frac{\mathrm{d}{P}_{\mathrm{k}}}{\mathrm{d}t}=-({\lambda }_{\mathrm{k}}+{\mu }_{\mathrm{k}})P_{\mathrm{k}}+{\lambda }_{\mathrm{k}-1}{P}_{\mathrm{k}-1}+{\mu }_{\mathrm{k}+1}{P}_{\mathrm{k}+1}(k=1,\dots ,N-2)}\\ & {\displaystyle \frac{\mathrm{d}{P}_{\mathrm{N}-1}}{\mathrm{d}t}=-({\lambda }_{\mathrm{N}-1}+{\mu }_{\mathrm{N}-1})P_{\mathrm{N}-1}+{\lambda }_{\mathrm{N}-2}{P}_{\mathrm{N}-2}}\\ & {\displaystyle \frac{\mathrm{d}{P}_{\mathrm{N}}}{\mathrm{d}t}={\lambda }_{\mathrm{N}-1}{P}_{\mathrm{N}-1}}\end{array}$$

(2)

where λ terms represent the creation of one damage and μ terms denote the recovery, which is given as λ_k = (N − k) d/D₀ with d being the (time-dependent) dose rate of radiation and μ_k = k/τ with τ being the average recovery time. The factor (N − k) in λ_k comes from the number of un-hit targets, and the factor k in μ_k comes from the number of targets that have been hit already. When all μ_k's vanish (i.e. no recovery), we obtain the same formula as Eq. (1) with $D={\int }_0^{\mathrm{t}}d(s)\mathrm{d}s$ . By writing the equations with time as a variable, dose rate is introduced as an essential parameter of the theory. The recovery effect is introduced in a most simple way; stochastic recovery with the same rate for all the damages. We did not take into account the effect of the cell cycle, which can be justified if each of the cultured cells is randomly distributed at various points in the cell cycle. By changing the time dependence of the dose rate d(t), the model can be applied to various cases, including not only single acute irradiation but also fractionated irradiation and chronic (continuous) irradiation.

3. RESULTS

Now, we show some calculation results obtained with the model. We show in Fig. 1 the calculation results for single acute irradiation and two split irradiations. The horizontal axis shows the irradiated dose, and the vertical axis shows the survival rate of irradiated cells on a logarithmic scale. The parameters used are as follows: D₀ is 1.4 Gy, the average recovery time τ is 2 h, and the number of targets N is 4. These values remain the same for the following calculations. In the split irradiation, irradiation stops once at 10Gy and is resumed after 20 h of non-irradiation time. The dose rate during irradiation is assumed to be 1 Gy min⁻¹. In single irradiation, a gradual decrease in survival (presence of a shoulder) is observed immediately after the start of irradiation, which is characteristic of the N-target model. Thereafter, a linear decrease is observed. The slope is mainly determined by D₀ as D₀⁻¹. In split irradiation, the shoulders reappear at the start of re-irradiation because the SLDs that had been generated during the first irradiation recovered during the non-irradiation period. The slope of the linear part in the second irradiation is the same as in the single irradiation. For the present parameters, the survival rate is 3.8 times greater for the fractionated irradiation than for the single irradiation at 10 Gy. This enhancement factor is essentially determined by the number of targets N, with a small modification due to the recovery effects. Our model can reproduce the reappearance of the shoulder which is one important feature of Elkind recovery.

Fig. 1.

Survival rate of irradiated cells. The dotted line is for acute single irradiation, and the solid line is for two split irradiations.

Next, the results for different non-irradiation durations are shown in Fig. 2. After the first irradiation up to 10 Gy at a dose rate of 1 Gy min⁻¹, we changed the non-irradiation period to 0.5, 1, 2, 4, 8, and 12 h and then restarted the irradiation. The curve at the bottom of Fig. 2 shows the ratio of the survival rate at 10-Gy irradiation with different interval times to that of single irradiation, with the right-hand scale used for the vertical axis and the horizontal axis representing the interval time. As the non-irradiation period increases from 0, the survival rate increases at first. It then gradually saturates and reaches the 3.8-fold increase mentioned earlier. This saturation behaviour is mainly determined by the recovery time τ.

Fig. 2.

Dependence of the cell survival rate on the duration of the non-irradiated period. Single exposure (thick solid line), 0.5-h interval (dashed line), 1-h interval (dot-dashed line), 2-h interval (dot–dot-dashed line), 4-h interval (short-dotted line), and 8-h interval (thin solid line). The thin solid line at the bottom shows the enhancement factor with respect to the single exposure at 10 Gy. The horizontal axis should be read as the interval time, and one should refer to the right axis.

Finally, Fig. 3 shows the results of varying the number of splits with the total dose fixed at D_tot and the total non-irradiation time fixed as T_NR. When the number of divisions is n, the dose per irradiation is given as D_tot/n, and one non-irradiation period is given as T_NR/(n-1). Therefore, as n increases, the irradiation period per exposure becomes shorter, and the survival rate becomes larger because of the appearance of shoulders. At the same time, the non-irradiation period per exposure also becomes shorter, and the recovery during each interval becomes smaller. It can be confirmed that at the limit of increasing n, the survival curve becomes almost linear, which is consistent with continuous irradiation at the average dose rate d_av; $S=e^{-{\mathrm{d}}_{\mathrm{av}}\tau /{\mathrm{D}}_0}$ . This is the rationale for considering slow irradiation as the limit of fractionated irradiation. In the present calculation, the total dose D_tot = 20 Gy, d = 1 Gy min⁻¹, the total irradiation time is 20 min, the total non-irradiation time is T_NR = 19 h and 40 min, and the time from the start to the end of irradiation is 20 h, and hence the average dose rate d_av = 1 Gy h⁻¹.

Fig. 3.

Dependence of the survival rate on the number of splits for a given total time for dose delivery. Single acute exposure (thick solid line), 2 splits (dashed line), 4 splits (dot-dashed line), 8 splits (dot–dot-dashed line), 16 splits (short-dotted line), and chronic exposure (thin solid line).

4. SUMMARY

We have demonstrated that the main features of Elkind recovery can be reproduced with a simple mathematical model that takes account of the recovery effects. By formulating the model in terms of simultaneous differential equations with time as a variable, dose rate is introduced as an essential parameter of the theory. We applied the model with various types of dose delivery, such as fractionated irradiations with various numbers of splits and various lengths of the interval period, and also continuous low–dose-rate irradiation. By solving the differential equations numerically, we showed that this model can reproduce the essential features of Elkind recovery, such as the reappearance of the shoulder in the survival curve, increase of the survival rate with the number of splits, and increase of the recovery rate with the length of the non-irradiation period. It has been shown that the linear decrease of the survival rate is described with the parameter D₀, while the saturation of the recovery with respect to the length of the interval is described with the parameter τ. It is also shown that the linear decrease of the survival rate in the case of chronic irradiation is given by the combination of the parameters as $\frac{1}{D_0}-\frac{1}{d_{\mathrm{av}}\tau }$ .