A new model of shouldered survival curves.

Recently, the linear-quadratic equation has been used to construct the dose-response relationships of ionizing radiation. The radiobiological theory on which this relationship is based indicates that at low doses, the risk of a biological lesion being formed should depend linearly on dose if a single event is required or quadratically on dose if two events are required. The same approach has also been used to construct the shouldered survival curves, which indicate a lower response of cell killing at low doses of low linear energy transfer (LET) radiation than at high doses because of repair. However, a different approach is possible, derived from the concept of generating the hybrid lognormal distribution, in which the hybrid form of linear and logarithmic components of a random variable is used. The hybrid form is a formulation of the phenomenon in which there is a feedback mechanism against the large change in the random variable. This paper presents a new model of shouldered survival curves, called a hybrid scale model, which has two parameters: the inactivation constant and the protective factor. In the model, the surviving fraction, normalized by a protective factor plotted in a hybrid scale, is assumed to be linear against the dose. This simple model provides an implication of the shoulder of survival curve and the effect of recovery time of radiation damage, as well as giving a good to the well-known data of split-dose experiments with mammalian cells.


Introduction
Typical survival curves ofexposed mammalian cells may have a steep slope ofa semilogarithmic plot for densely ionizing radiation, but for sparsely ionizing radiation they usually have a small slope at low doses, followed by a curved shoulder leading to a substantially steeper slope at higher doses. A shouldered dose response indicates a lower effectiveness of cell killing at low doses of low linear energy transfer (LET) radiation than at high doses because some ofthe radiation damage has been repaired. This repair was demonstrated by Elkind and Sutton (1) in experiments with Chinese hamster cells irradated with two or more doses of X-rays separated by intervals of time.
The shouldered survival curves, S(D), are usually described as a function of dose, D, by various models as follows: a) the single-target plus multitarget single-hit type (1) where AD0 is the inverse of the slope of the initial slope of the curve and ,,Do is the inverse of the sensitivity of each of the n targets, and b) The linear-quadratic form S(D) = exp{-(aD + OD2)} (2) Department of Health Physics, Japan Atomic Energy Research Institute, Tokai-Mura, Ibaraki-Ken, 319-11, Japan. This paper was presented at the International Biostatistics Conference on the Study of Toxicology that was held May 13-25, 1991, in Tokyo, Japan.
where a and (3 are, respectively, the linear and the quadratic coefficient. Equation 2 may be generalized as a polynomial form of D. The report ofthe United Nations Scientific Committee on the Effects ofAtomic Radiation [UNSCEAR (2)] states that the initial slope, exp(-D/iDo) in Equation 1 and at in Equation 2 is still a matter of debate, but this question is immaterial in the dose-response models ofradiation-induced cancer, as both functions satisfactorily describe the experimental data for surviving fractions between 1.0 and 0.1.
In contrast to models based on the target theory, Hug and Kellerer (3) derived a different form ofsurviving fraction, S(D), based on a concept of reactivity, R(D), for the slope of the survival curve and compensational capability, K(D), for reducing the reactivity as follows: where R' is the final value of R(D) when the compensational capability diminishes, Ko is the initial compensational capability before irradation, and K = Koexp(-'yD) and R(D) = R'-Koexp(--yD). This model was reported to fit the data of Elkind and Sutton (1) well. This paper presents another possibility ofthe model building, extended from the concept of generating the hybrid lognormal distribution, which is defined as lnpX+pX-N(A,a2), 0 <xv< X,p >0 (4). The hybrid form of linear and logarithmic terms ofthe random variable is a formulation ofthe phenomenon in which a feedback mechanism constrains the larger variation in the range of large values. Then we can expect that there might be a feedbck mechanism in biological systems that mitigates the large decrease of surviving fraction incurred by the given dose because of repair.
The same equation also results from differentiating Equation 4 with respect to D. Equation 6 may be interpreted as the slope, dS/dD, ofthe survival curve on linear-linear coordinates, reduced by decreasing the absolute value of b in the reciprocal of (1 + pS) via the negative feedback mechanism of dS/dD with tie feedback parameter of p because ofrepair. Putting the simultaneous equations of the feedback mechanism, dS/dD=b'S and b'=b-pdS/dD, and removing b' we get Equation 6.
If the slope of survival curve on linear-linear coordinates is dS/dD=bS, the surviving fraction is S(D)=exp(bD) or In S(D)=bD, b< 0, where b is the inactivation constant. Then the Defining the hybrid scale as y=hyb (t)=ln(t)+t in much the same manner as the log scale defined by y=ln(t), we can write the differentiation ofEquation 6 as dhyb (pS)/dD=b, which means that the slope ofthe survival curve on the semihybrid (a hybrid scale of surviving fraction) plot is constant. Thus the shoulder ofsurvival curve disappears ifwe plot the data on semihybrid paper by introducing a protective factor, p.

Application of Model
There is much survival data concerning established cells exposed to ionizing radiation. Data suitable to test the model come from split-dose experiments with V79-1 cells after2.5 and 23 hr ofincubation at 730C following a first dose of5.05 Gy (1). Hug and Kellerer (3) also used the same data to test their model of Equation 3 shown above. However, the data must be obtained by reading the plots of Figure 11 of Elkind and Sutton (1). Table 1 shows these data. The given data are a set of (DiS,), (i=I to n), where Di is the ith dose, and Si is the surviving fraction of cells where e-is an error term. Equation 8 was used for both the hybrid scale model and the linear-quadratic model given by Equation 2, where for the latter model xi=Di, y =lnSi ID,, bo=a and bi=-(3. The model given by Equations 1 and 3 were not used here because ofthe complexities ofthe calculation. Results and Discussion Figure 1 shows the results offitting the proposed model to the data ofthe surviving fraction ofmammalian cells irradiated with doses separated by three incubation periods of0.0, 2.5, 23.0 hr. Figure la-c shows the given data points and the survival curves estimated by the proposed model. Each set ofdata points lies on each fitted curve of surviving fraction for its incubation period. This means that the proposed model is likely to be applicable to these data. Figure Id shows the linearity of all sets of surviving fraction on a semihybrid plot (a hybrid scale of surviving fraction, H=lnpS+pS; see Table 2 for p), although these survival curves on a semilog plot have shoulders. The theory of hybrid scale (S) predicts that the survival curve on a semihybrid plot locates higher for strong protective systems than for weak protective systems, where the degree ofprotection ofa system is defined by the protective factor p. Therefore, the lower location of survival curve for 2.5 hr suggests that it is less protective than that for 0 hr. The proximity of survival curves for 0 and 23 hr reflects that both give similar protective conditions, that is, the complete recovery of cells irradiated with 23-hr split-doses. Table 2 gives estimated parameters of the proposed model to the data for each incubation period, including the correlation coefficients between D/(1-S) and linS/(1-S). All the absolute values of each of the correlation coefficients are close to 1 because ofthe goodness offit ofthe proposed model to the data. The protective factor p for 2.5 hr is the smallest in all three cases because of degraded protective conditions. The inactivation constants b are similar among three cases. If the model given by Equation 4 is used, that is, a lnpt p, the protective factor is about 2 for 0 and 23 hr and about 1 for 2.5 hr; but the inactivation constants are not so different from those shown in Table 2. Thus the method of estimating parameters needs to be studied further.
The linear-quadratic model of the survival curve, applied to the data in the form given by Equation 8, gives a = 0.2551, (3 = 0.0281, and r = -0.9675 for 0 hr; a = 0.2850, (3 = 0.0373, and r = -0.9300 for 2.5 hr; and ai = 0.1917, (3 = 0.0329, r = -0.9778 for 23 hr. where r is the correlation coefficient. Therefore, the data did not fit the linear-quadratic model as well as the hybrid scale model. The hybrid scale model can also be applied to the doseresponse curve, which is concave upward on semilog paper, for low LET radiation. Then the response plotted in a logarithmic scale is linear against the given dose plotted in a hybrid scale. This application has been given in another papper (6).

Conclusion
The new concept of a hybrid scale model, extended from the hybrid lognormal distribution, was applied to data of shouldered survival curves. The hybrid scale model has two parameters: the protective factor p and the inactivation constant b. The model gave a good fit to the data of split-dose experiments with mammalian cells (1). The model also provides an explanation ofthe shoulder of survival curve and the effect of recovery time. This model is as simple as the linear-quadratic model of S(D)=exp(-aD-(3D2) but is applicable to the data both in the low-and the high-dose ranges. However, the method of estimating parameters needs to be studied further.