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Summary

A repair/misrepair cell kinetics model is superimposed onto the track structure model of Katz to provide for a repair mechanism. The model is tested on the repair-dependent data of Yang et al. and provides an adequate description of that data. The misrepair rate determines the maximum relative biological effectiveness (RBE), but similar results could arise from indirect X-ray lethality not included in the present model.

Introduction

It was Schaefer's original suggestion that track structure would be an important parameter in controlling biochemical reactions in cell response to ionizing radiation (ref. 1). Further insight was gained through Katz's introduction of a geometric view to sensitive biological sites (ref. 2), resulting in a phenomenological model that has been successful in interpreting and extrapolating cell response data for high-energy heavy ion (HZE) particles (ref. 3). The main criticism of the Katz model has been the lack of a clear mechanism for repair processes (ref. 4). We propose herein a step toward a simple repair model in which kinetic coefficients are related to the Katz parameters.

Repair/Misrepair Track Model

We consider a three-level system which contains single- and double-level ionizing radiation transitions and a repair/misrepair process for single-level cells. The appropriate equations are given as

_n0 = ?kn0 + rn1 (1)

_n1 = ksn0 ? ksn1 ? ffn1 (2)

_n2 = kDn0 + ksn1 + mn1 (3)

where ni is the number of cells in level i, is the total repair/misrepair rate coefficient, r is the repair rate coefficient, m is the misrepair rate coefficient, ks is the single-step radiation transition coefficient, kD is the double-step radiation transition coefficient, and k = ks + kD. The uppermost level is assumed not to repair and that misrepair appears as a transition to the upper level independent of the radiation source (that is, an inherent biochemical rate). If we consider very high exposure over a limited time period, then repair processes may be neglected and the solutions to equations (1) to (3) are approximately

n0(tr) = n0(0)e?ktr (4)

n1(tr) ss n0(0)kstre?ktr (5)

n2(tr) = n0(0)kD
k

?
1 ? e?ktr ?

+ n0 k2s
k2

h
1 ? (1 + ktr) e?ktr i
(6)

where ni(tr) is the i-level population after exposure period tr and n0(0) is the initial population. The symbol represents reaction rates related to radiationinduced lesions within the nucleus (presumably chromosomes) and the rates are assumed proportional to particle flux (primary ions or secondary charged products). The lesions are chemically active species neutralized by enzyme activity at rate constants i. After the exposure, the repair processes proceed at rates dependent on the experimental protocol (ref. 5). The repair proceeds as

_n0 = rn1 (7)

_n1 = ?ffn1 (8)

_n2 = mn1 (9)

The solutions are found, assuming a constant rate of repair throughout the repair period, as

n0(t) = n0(tr) + r n1(tr)(1 ? e?fft) (10)

n1(t) = n1(tr)e?fft (11)

n2(t) = n2(tr) + mn1(tr)(1 ? e?fft) (12)

So, the final state of the system is given in terms of repair/misrepair ratios as

n0(1) = n0(0)
?
e?ktr + r kstre?ktr ?
(13)

n2(1) = n0(0) ? n0(1) (14)

In the limit of low exposure we find

n0(1)
n0 ss 1?ktr

?

1 ? r ks
k

?

? 1
2

?

2 r ks
k ? 1

?

k2t2r (15)

which displays typical two-target response in the limit as r ! 1 and kD ! 0. We now seek an understanding of the kinetic coefficients in terms of the Katz model.

Katz Model

The cellular track model of Katz has been described extensively (refs. 2 and 3). The track model attributes biological damage from energetic ions to the secondary electrons (delta rays) produced along