- Introduction
Radiations such as - rays, UV rays, etc. have ionizing effects on tissues and have the ability kill cells and inactivate viruses. A detailed analysis of the survival of such cells and viruses are imperative in many fields, including radiation oncology, public health and environmental photobiology. However, the mathematical and statistical tools based on target theory in radiobiology are mentioned in literature (Alpen, 1988, Smith, 1989, Hall, 2000) and are wide in use in the analysis of experimental data involving biophysical mechanisms of the repair of radiation damage. This work demonstrates the use of probabilistic approaches in descriptions of radiation survival giving a detailed study of radiation survival that combine elementary concepts of cellular repair with the ease on which the assumptions the target theory has been based on.
1.1 Target Theory
Critical analysis of survival functions were initially based on the target theory, which hypothesized that one or more precisely selected targets within a cell, had to be hit to cause inactivation (Lea, 1956). There are a few target based theories that cover the survival functions which are currently in use, described mathematically in (1), (2) and (3), provide a comprehensive mathematical account of the range of observed survival data. The single target single hit (STSH) model (1) assumes that a cell is deactivated by a single hit and results in single exponential survival. This is the zeroth term in a statistical Poisson distribution, where P represents the probability of survival, D is dose and D0 is the dose that results in an average of one lethal hit per cell. The multi-target single hit (MTSH) model (2) describes the method in which cells resistant to incident radiations survive low doses (Sutherland, John C., 2006, Physics in Medicine and Biology, p 4883). It presumes that each of m critical targets must be hit at least once for inactivation, where Dm is the dose that produces an average of one hit in each of the m targets. The intermediate region can be fit by a multi-target and single hit combined (MT+) model shown in (3). A cell is said to be inactivated either if a single marked out sub-target is hit at least once, or if all m marked out targets are hit together at least once. When Dm and D0 approach infinity, equations (1) and (2) puts a tab on equation (3). Thus, the single hit target theory can be generalized with MT+.
SMTSH[D, Dm, m] = 1 - ( 1 – e ( -D / D0 ) )m (2)
1.2 Effective damages
Many types of lacerations are formed in the basic DNA strands by ionizing and UV radiations (Friedberg et al., 1995, Smith, 1989, Hall, 2000), but the number of such lacerations measured per unit dose greatly exceeds the number of lethal events. Thus, it becomes difficult to extract the precise molecular nature of such exposure related effects. However, it has been proposed that there could be a certain repair process involved which enabled tissues to survive such radiations. (Elkind and Sutton, 1959). When radiation is incident on a cell, Coulomb interaction processes can alter the charge and energy levels of the atoms in it resulting in a series of physical or chemical changes that eventually results in cellular damage. Direct interaction predominates in the interaction of densely ionizing particles with biological samples.
However, in indirect interactions, the radiation is incident on atoms and molecules in the tissues to create free radicals. Extremely reactive and short lived free radicals such as H2O+ and OH• (hydroxyl radical) can be produced when ionizing radiation interacts with water present in the cell that can damage the target in the cell.
Free radicals are extremely reactive molecules due to their unpaired electron and have the ability to break the chemical bonds and cause changes in the composition of the sample that can lead to biological damage
- Theory
- Survival probability
Assume that a system of 0 identical cells or viruses is irradiated uniformly so that for a given set of experimental conditions only survive. The probability of survival for a cell in this population is thus S = /N0(Sutherland, John C., 2006, Physics in Medicine and Biology, p 4883) . Only those irradiations are considered where the duration of the exposure is short compared to biological processes such as DNA repair or the cell cycle. Let represent the average number of lethal damages induced initially per target in the sample. The “lethal damages” includes all damages that can contribute to inactivation whether or not they can be repaired. Assuming that the damages are randomly distributed throughout the targets in the sample, the probability that a particular target will experience exactly i such damages is given by the Poisson probability function
(Bevington and Robinson, 1992). The probability that a target will receive no lethal damages and thus be certain to survive, is PP[ ,0] e- , which is the result obtained in classical single-target- single-hit-theory. In this stochastic model only cells that receive lesser doses of those radiations causing a fixed number of lethal damages are said to survive.
Introducing a survival function in terms on the average damage frequency, is helpful since its focus is on the actual damage as well as being a dimensionless quantity. While measuring different levels of damage is possible, it becomes tedious to ascertain which are lethal and to ensure that all lethal damages are accounted for. It is much simpler to measure the dose precisely. Therefore, experimental survival functions usually are expressed as functions of dose. If damage levels are a linear function of dose D, then the average frequency of lethal damages is D/ D1, where D1 is the dose that corresponds to the initial consideration of an average of one lethal damage per member of the population.
The elements of the cells surviving these ionizing effects can be determined through direct or indirect methods. Some typical examples of such survival curves are shown in Fig. 1. The targets in the Fig 1 are subjected to both densely and sparsely ionizing radiations. As can be seen, the type of radiation affects the curve profile. Highly ionizing radiations reveal a profile of the cell survival that is approximately an exponential function of dose, as can be seen from the log plot as roughly a straight line. But in the case of lightly ionizing radiation, there is an initial slope after which there is a linear region that becomes almost straight at elevated dose values. A few reasons for cells being less radiosensitive could be: insufficiency of oxygen to create a hypoxic state, the addition of chemical radical scavengers, slowing down the dose rate or multi-fractionated irradiation, and cells coordinated in the late S phase of the cell cycle. As the complexity of different mathematical approaches varies, different shapes of cell survival curves can be defined.
- Results and Discussions
Eight overhead transparencies were taken and were labeled ‘Target’, ‘Dose 1’, ‘Dose 2’, ‘Dose 3’,.., ‘Dose 7’ to simulate test conditions. On the sheet labeled ‘Target’, a snaking line was drawn from the top to bottom as shown in fig 2.
On this target sheet fifty circles of about 1cm in diameter were drawn along the snaking line to represent the ‘target’ centers. On the sheets labeled ‘Dose 1 – 7’, about fifty dots were drawn spread over the entire transparency. These dots represented the random transfer of energy between a dose of radioactive particles/photons and the environment it was travelling through.
Using the ‘Target’ sheet to navigate through the sheet labeled ‘Dose 1’, a note was made on the number of dots that had survived the first dose of radiation. A target with two or more dots superimposed on them was presumed to have been ‘destroyed’. This was repeated for all seven sheets labeled ‘Dose’. The results of the experiment are listed in Table 1.
The graphical representation of Table one is shown in Fig 3.
per dose
When the data was plotted in terms of % number of survivers following each hit, the graph looked similar to that of Fig 3. The pattern obtained is as expected since it is a mere normalization of the graph shown in Fig 3. Subsequently, when the natural log of the % number of survivors were take, a similar curve was obtained. The corresponding plots are shown in Fig 4 and in Fig 5.
As can be seen, these curves match the curves shown in Fig 1. We will now discuss the possibilities of using other models that could possibly fit the data better.
- Bounded constant fraction repair: a stochastic model
Suppose that each of the values of j in the repair probability product is replaced by , where j represent the probability that damage j can be repaired, the average probability that a lethal damage is repairable. Thus, the probability that all i damages are repairable becomes i (Sutherland, John C., 2006, Physics in Medicine and Biology, p 4883). When the molecular damage type distribution is independent of the radiation dose, each class of molecular damage will appear periodically and a fixed fraction of all lethal damages will be repairable. However, as the function is an average value, accounting for all the information, it is not imperative to know how many such classes of molecular damage exist. The average probability that a lethal damage is not repairable and thus inherently lethal is 1 - . In formulating the effect of survival of cells that have received i lethal damages on the complete survival of a population that has received an average of lethal damages per cell, the repair probability produce must be multiplied by the Poisson probability that a cell will receive exactly i damages and such products summed over all values of i from 0 to (n 1), as indicated on the left hand side of (3). Note that e- has been removed from the summation because it does not depend on i and the i 0 term is included in the summation because both 0 and 0! equal unity. Summations related to the form shown in Equation 3 (usually with 1 ) have appeared previously in discussions of radiation survival.
(3)
A contribution of this work is to note that the summation in Equation 3 is the exponential sum
function, which has the closed form solution where [n] is Euler gamma function and [n,] is the upper incomplete Euler gamma function (Weisstein, 2005, Weisstein, 2003). The ratio [n,] / [n] is given a special name, the regularized Euler gamma function, denoted by
Q [n,], hence leading to the expression shown on the right hand ide of Equation 5. Note that the regularized gamma function takes two arguments and that in this case the second argument is the product of and either or D/ D1 .
While the summation in equation 3 assumes that n is an integer, it appears in the gamma function as a continuous variable, i.e., a real number. The quantity n in the last expression in equation (3) represents a value obtained by fitting that equation to survival data obtained experimentally. It is written with a bar to highlight that it is a stochastic parameter that characterizes the effective upper limit of survivable damages in the population.
While μ (or D1) reflects the extent of induction of damages by the radiation, and n show the contribution of repair in determining the probability of survival. Plots of survival functions generated from equation (3) for selected values of and n are shown in Fig 6
- Oxygen Effect
Various factors that can affect the number of target cells getting hit are the amount of oxygen, the ionizing extent of the radiation etc. Different mathematical tools have been formulated to specify the extent of biological damage. The amount of oxygen within a cell affects the biological effect of ionizing radiation, ie if the cell oxygenation is greater than anoxia, the greater is the biological effect of the ionizing radiation (Suntharalingam, N. et al. Review of Radiation Oncology Physics: A Handbook for Teachers and Students, Ch. 14). This is more so in the case of low ionizing radiations where the more the cell oxygenation above anoxia, the more is the biological effect until the oxygen saturates. As shown in Fig. 7, the effect is quite remarkable for lightly ionizing radiations, while for highly ionizing radiations it is much less so. This ratio of doses without and with oxygen to produce the same biological effect is called the oxygen enhancement ratio (OER).
OER= Dose to produce a given effect without oxygenDose to produce the same effect with oxygen
- Relative Biological Effectiveness
As the Linear Energy Transfer or ionization strength of the radiation changes, so does the ability of the radiation to impart biological damage. The relative biological effectiveness (RBE) compares the dose of test radiation to the dose of standard radiation to produce the same biological effect. Even though the standard radiation has been taken as 250 kVp X rays, 60Co lrays is now recommended. The RBE is defined by the following ratio
RBE= Dose from standard radiation to produce a given biological effectDose from test radiation to produce the same biological effect
The RBE changes not only with the type of radiation but also with the type of cell or tissue, biologic effect under study, the dose strength, the rate of dosage and fractionation (Suntharalingam, N. et al. Review of Radiation Oncology Physics: A Handbook for Teachers and Students, Ch. 14) In general, the RBE increases with the LET to reach a maximum RBE of 3–8 (depending on the level of cell kill) at LET ≈ 200 keV/m and then decreases because of excess energy, as shown in Fig. 8.
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