The most important biological damage of cells after exposure of ionizing radiation is believed to be DNA damage. Many kinds of DNA damages will generate in a cell [3]. Damages or losses of nucleotide bases and DNA single-strand breaks (SSBs) are potentially to be repaired without errors of genetic information because of the existing right genetic information in the complementary DNA strand. Numbers of double-strand breaks (DSBs) are also rejoined to reconstruct DNA strand by the nonhomologous end joining (NHEJ) and homologous recombination (HR) systems. Damaged site is reconstructed by using the genetic information in complementary strand of sister chromatid by the HR system and shows error-free repair. Some nucleotide bases near the damage site are removed to clean up the damaged end, rejoin the strands without correct genetic information in the NHEJ process; thus it will be an error-prone rejoining. In addition, productions of complex lesion of DNA breaks are expected for heavy-ion exposure, because the density of ionization must be very high at around the traversal of the ion.

When cells are exposed to ionizing radiation, DNA damages in the form of SSBs, DSBs, base damage, or their combinations are frequent events. It is known that the complexity and severity of DNA damage depend on the quality of radiation and the microscopic dose deposited in small segments of DNA, which is often related to the linear transfer energy (LET) of the radiation. Experimental studies have suggested that under the same dose, high-LET radiation induces more small DNA fragments than low-LET radiation, which affects Ku binding with DNA end efficiently and might be a main reason for high-LET radiation-induced RBE since DNA DSB is a major cause for radiation-induced cell death. In this work, we proposed a mathematical model of DNA fragments rejoining according to NHEJ mechanism.

One set of chromosomes in a human individual consists of 22 pairs of autosomes (any of the chromosomes in a cell other than the sex chromosomes) and two sex chromosomes, all of which are inherited from one’s parents. In the DNA synthetic phase of the cell cycle, one copy of the genetic information is created and associated with the original one as a pair of sister chromatids, bound at the centromere, in order that the same genetic information may be distributed to two daughter cells in the coming mitotic phase. In the metaphase of the mitotic phase, DNA, distributed homogeneously within a cell nucleus in other phases, condenses to form 46 structures that can be microscopically observed as chromosomes. In the anaphase, 46 pairs of sister chromatids are pulled apart to transmit the genetic information to the two daughter cells.

In the cells of which DNA has been damaged by radiation exposure, abnormality can be observed as a chromosomal aberration which is effective as an indicator of radiation exposure as it is easy to quantify and correlates well with radiation-caused cell death. Various types of chromosomal aberration (Fig. 4.5) are caused by radiations, depending on the phase of the cell when its DNA is damaged. If it occurs during the period from the first pause (G1 phase) to the early DNA synthetic phase (early S phase), the so-called chromosomal aberration can be observed, which appears to be caused by simultaneous cleavage of both of the sister chromatids. If a cleaved chromosome is not reunited, “terminal deletion” and “fragment formation” are observed. After a chromosome is cleaved at the two sites, the center section turns over and is then reunited with the remaining fragments. This phenomenon is called “inversion.” If the termini of this center section rejoin with one another, a “circular chromosome” is formed, with the fragments left over. After two chromosomes are cleaved, they can rejoin with each other to form a dicentric chromosome, which has two kinetochores (structures of protein associated with DNA located at the centromere region). This phenomenon is called “mutual translocation.” On the other hand, if DNA damage occurs late in the DNA synthetic phase (late S phase) or the second pause (G2 phase), chromatid aberration is observed, with damage showing on either of the sister chromatids (Fig. 4.5).

Many of the multiplied human cells undergo cell cycle about once a day, with one parent cell dividing into two daughter cells; the number of the cells doubles again and again. A tens-of-micron single cell proliferates into a macroscopic colony with a diameter of several millimeters after a dozen days or so. When these cells are irradiated, some of them die owing to the resulting damage, while others survive. The surviving cells can form a colony, but the colonies show different morphology from those made by unirradiated cells, and the number of colonies they form is usually smaller. There are several cell death modes, but they are generally classified into reproductive death and interphase death.

In order to quantitatively describe the properties of radiobiological effects using survival curves, models to explain the shapes of these curves have been proposed. The target theory is the most classical model. This model assumes “parts of a cell are sensitive to ionizing radiations and regarded as targets for radiations”; “the targets are much smaller than the entire cell, but essential for survival of the cell”; and “when these targets are hit by radiations, the cell can lose its functions to become inactivated (killed).” This is the idea of “all or nothing”; cells can survive, unless they are hit. The linear quadratic model takes the place of this theory now especially in the field of radiotherapy.