Abstract
The current radiosensitive studies are described with linear-quadratic (LQ) cell survival (S) model for one fraction with a dose d. As result of assuming all sublethally damaged cells (SLDCs) are completely repaired during the interfractions, that is, no presence of SLDCs, the survived cells are calculated for a n-fractionated regimen with the LQ S(n,D) model. A mathematically processed subpart of LQS(n,D) is the biologically effective dose (BED) that is used for evaluating a so-called “biological dose.” The interactions of ionizing radiation with a living tissue can produce partial death or sublethal damage from healthy or sublethally damaged cells. The proportions of the killed and sub-lethally damaged cells define the radiation biological effects (RBEfs). Computational simulations using RBEFs for fractionated regimens let calculating tumor control probability. While the derivation of the LQ S(n,D) considers a 100% cell repair, that is, 0% of sublethally damaged cells (SLDCs), the radiobiological simulators take into account the presence of SLDCs, as well as a cell repair <100% during the interfractions and interruption. Given “biological dose” does not exist, but RBEf, there was need for creating the BED. It is shown how some uses of BED, like the derivation of EQ2D expression, can be done directly with the LQ S(n,D).
Highlights
In 1989, an article published in [1] introduced the term BED, biologically effective dose, as a linear-quadratic (LQ)-based formula
The BED expression was a result of a mathematical derivation of the exponential part of the LQ S model for treatments with n fractions and dose per fraction d, the LQ S(n,D) where D = nd; and this model was obtained assuming that all sublethally damaged cells are wholly repaired during the interfraction period
While the derivation of the LQ S(n,D) model for fractionated regimen considers a 100% cell repair, that is, 0% of sublethally damaged cells (SLDCs), radiobiological simulator methodology takes into account the presence of SLDCs, as well as a cell repair
Summary
In 1989, an article published in [1] introduced the term BED, biologically effective dose, as a linear-quadratic (LQ)-based formula. The radiosensitivity studies function of the absorbed dose (d) are described with the cell survival (S), which is complement of cell kill (K), and probabilistically S = 1-K These studies are widely modeled with the LQ S(d) for one fraction as LQSðdÞ 1⁄4 exp ÀÀαd À βd2Á (1). The authors of [4] considered that: “BED represents the physical dose required for a given effect if the dose were to be delivered by infinitely small doses per fraction or, in the case of continuous radiation rates, at a very low dose rate.”. Except the probabilistic treatments of the tumor control/normal tissue complication probability (TCP/NTCP), many stochastic processes/effects in areas of the ionizing radiations interacting with living tissues have not been probabilistically treated nor modeled, which has led deficiencies, like replacement in the evaluations of cell survival (S)—a probabilistic metric—by BED, a non-probabilistic, a mathematical derivation from the LQ S(n,D) formalism
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