Additionally, this model is applied easily in clinical practice and relates the fundamental variables of a radi-otherapy treatment: dose, time and number of fractions. However, this model has some disadvantages, such as its short range of applicability on different tissues. The aforementioned models have been substituted by those deduced from the cell survival equations, including the Linear Quadratic model.
The interpretation of the effects of radiation on healthy and tumorous tissues has to be made according to the effect caused by the cells that compose these tissues. The survival cells are those that carry out their reproductive activities, along with their functions, after irradiation. Cell death takes place when the cell stops carrying out its function in the differentiated cells, and ceases its reproductive activities in un-differentiated cells. The methods used in the determination of the cell survival curves exposed to radiation are grouped into two categories: in-vitro and in-vivo.
In-vitro methods are based on the ability of some cells to survive and reproduce in a culture medium, whereas in-vivo methods are based on cell counts in certain organs after being irradiated . A survival curve is presented in semi-algorithmic form see Figure 3 , with the real dose value on the x-axis, and the natural logarithm of the survival fraction on the y-axis. In accordance with empirically-acquired data, interpolations and extrapolations have been generated in order to obtain cell survival curves.
These mathematical procedures, along with their theoretical assumptions, originated a number of survival curve models, such as:. In this model  each cell is assumed to contain a single target; the inactivation of this leads to cell death. In addition, a single hit inactivates that single target. Mathematically , the number of targets dN inactivated by a small dose dD is proportional to the number of targets N and to the dose dD.
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The value D0 characterizes the intrinsic sensitivity of a cell population. Integrating 5 , we obtain:. Where No is the initial number of cells and N is the number of cells surviving a dose of radiation. Considering the cell as a single target, the survival of a cell population is represented by the survival s, and it is expressed as:. Expression 7 represents the survival curve for a single target with a single hit. Observing Figure 4a , we can infer linear behavior in a semilogarithmic scale. This model allows the interpretation of survival curves in mammalian cells with very sensitive ratios that correspond to tumorous tissues.
It also allows the deduction of the cell response to low dose rates using high LET radiations.
In this model , the cell is considered to have multiple targets, and the inactivation of all of them is necessary in other to achieve cell death. Each target is inactivated by a single hit. According to the previous interpretation, and according to logical deduction, the survival equation for this model is:. Where S is the fraction of survival, D the total dose, Do is a constant that corresponds to the average dose of a hit per target, therefore maintaining the interpretation of the previous model, and n is the number of targets of the cell.
The survival curve obtained for this model has an initial slope equal to zero, which makes it impossible to correctly model the behavior of the majority of mammalian cells because, empirically, the survival curves generally show an initial negative slope. Nevertheless, as the dose increases, the behavior obtained corresponds to the empirical survival curves. The Linear Quadratic Model  is a mathematical formulation which uses a second-degree polynomial and assumes that each cell is killed because of the inactivation of two or more targets.
Therefore, when there is an impact, two possibilities are considered: a that the cell injury is irreparable, and b that the injury is reparable by the cell itself. The interpretation of this in relation to the cell survival curve is that the curve is determined through two components, one linear proportional to the radiation dose, related to the irreparable injury , and the other proportional to the square of the radiation dose related to the reparable region. It is important to consider that the inductive particles of the reparable lesion can act separately for a period of time.
After a critical period, the injury is repaired. Figure 5 analyses the effect that the LET has on cell survival. Thus, the survival curve is linear. At higher doses, cell survival is proportional to the square of the dose and the curve tends to be concave and downward. When, in the Linear Quadratic model , the linear term equals the quadratic, meaning the proportion of reparable lesions equals the irreparable, the model shall be expressed as:.
Hence, these characteristics show that the exhaustion in this type of cells produces acute clinical effects. This belongs to cells which are more differentiated, have a slow turnover capacity and better tolerate irradiation through low fractions, due to a notable capacity to repair sub-lethal lesions.
Therefore, it can be inferred that the chronic effects appear when there is an exhaustion of slow turnover cells, and hence their alterations are shown after a prolonged period of tolerance. In order to apply the linear quadratic model to dose fractionation in time intervals, three conditions must be satisfied:. Under these assumptions, the mathematical form of the linear quadratic model is established for a fractionated administration of the dose:. Where D is the overall dose administrated in N fractions, dk is the dose of the k th fraction. Figure 6 shows the behavior of the fractionated dose in equal time intervals.
Conversely, when the size of the dose per fraction decreases, the total dose required to achieve a determined isoeffect increases more for slow response tissues than for acute response tissues. This technique is the basis of radiotherapeutic techniques, such as hyperfractionation, which seeks to control tumors with responses similar to the acute response tissues without damaging the adjacent differentiated tissues. It is necessary to clarify that between each small dose, a period of time that allows for sub-lethal repair is necessary .
This paper sets the conceptual bases for the study of Physico-Mathematical models, which have been used in the development of radiobiology. In addition, it is only the beginning of deeper studies on the current models, such as the quadratic model of cell survival curves. This model is still being researched, with the purpose of consolidating it as a solid theoretically-based model, using mechanistic ideas and statistical arguments. In recent years, the linear quadratic model has been widely used in radiotherapy, using both single and fractionated doses.
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Its simplicity allows comfortable application in clinical contexts since, endowed with biological parameters with easy interpretations, it allows a satisfactory description of the cell survival curves in homogenous cell populations. Along with the technical processes and technological developments, one of the main tools for improving clinical results in radiotherapy with minimal conditions in healthy tissues is the integral development of radiobiology.
This provides the conceptual basis for cellular behavior under the effects of ionizing radiation and highlights new courses of action in the face of oncological disease. Joiner y A. Hillar, Radiation Safety Training Guide.
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Fowler y B. Orton y F. Saunders Company, Hall, Radiobiology for the Radiologist. Second Edition. Publishers Philadelphia, Ciudad, D. Guiraldo, A. Sanchez, W. Sanjuanbenito y S. Services on Demand Article. English pdf Article in xml format Article references How to cite this article Automatic translation Send this article by e-mail. Plazas 2 , E. Introduction A variety of radiobiological studies and models were developed during the first decades on the 20th century.
Radiobiology, along with technological development, will continue to find solutions to improve the techniques and protocols of treatments, with the aim of maintaining radiotherapy as one of the most effective cancer treatment methods  2. Overview The irradiation of any biological system produces a succession of physical, chemical and biological processes that can be approximated in a wide timeline See Figure 1. In fact, this is one of the fundamental principles in radiotherapy - cancer cells are highly proliferative, thus they are in constant mitotic activity, which means they are highly vulnerable to radiation   Tumor growth is characterized by the disproportionate and unorganized growth of the tissue structure; it is, in general, an abnormal growth, with a deficit of blood vessels.
This is why the temporal influence turns out to be a key factor, since by fractioning the therapeutic radiation dose 1 in time intervals, the healthy tissues and organs will have more capacity to recover in time, which is not the case with cancerous tumors   In therapies using ionizing radiation for the eradication of cancerous tumors there are other important physical factors. This is shown in the cell survival curves See Figure 5 where cell survival is analyzed according to the dose for low or high LET radiation  3.
Physico-Mathematical Models The development of biological concepts and physical parameters that describe the effects of ionizing radiation on various tissues are described below. Models Based on Empirical Isoeffect In Strandqvist  developed a new scientific approach to the effects of ionizing radiation, introducing the isoeffect curve dose-isoeffect , which represents the way the total dose of a fractionated treatment changes, depending on the total treatment time to produce an equivalent isoeffect see Figure 2.
He thus obtained a family of curves Figure 2 , where an optimal region can be found which favors the curing of the tumor with minimal complications  This was expressed as: Where D is the total dose, T is treatment time, p is a potency less than 1 to be determined , and k is the constant of proportionality an interpretation of which will be presented later in this paper. Expression 2 then becomes: NSD is used to calculate the tolerance of normal tissue in the tumor region and represents the tolerance of normal connective tissue, which is given at ret  The NSD concept was well received, because it simplified the comparison of treatments.
However, it presented various problems: It does not represent a single equivalent dose because the isoeffect time calculated by Ellis used data from 4 to 30 fractions. NSD cannot be calculated when there is no variation in the volume treated or in therapy interruptions. NSD tables are useful for subcutaneous and skin tolerance only, and cannot be applied to tumors or other normal tissues  3.
Its mathematical expression is the following: Where n is the total number of fractions, d is the dose per fraction, and x is the coefficient of proportionality which relates the days of the week with the number of sessions per week. In-vitro methods are based on the ability of some cells to survive and reproduce in a culture medium, whereas in-vivo methods are based on cell counts in certain organs after being irradiated  A survival curve is presented in semi-algorithmic form see Figure 3 , with the real dose value on the x-axis, and the natural logarithm of the survival fraction on the y-axis.
These mathematical procedures, along with their theoretical assumptions, originated a number of survival curve models, such as: 1.
Single target - Single Hit Model, 2. The base pairs adenine, thymine guanine and cytosine are held together by weak hydrogen bonds. Adenine always pairs with thymine except in RNA where thymine is substituted by uracil and guanine always pairs with cytosine. The bonding of these base pairs can also be affected by the direct action of ionizing radiation. However, heavy charged particles such as alpha particles have a greater probability of causing direct damage compared to low charged particles such as X-rays which causes most of its damage by indirect effects.
The DNA base pairs form sequences called nucleotides which in turn form genes. Genes tell the cell to make proteins which determine cell type and regulate cell function. When such breaks occur, DNA usually repairs itself through a process called excision. The excision process has three steps:.
These repair processes are highly efficient since we have evolved as a species in a sea of radiation. DNA repair takes place continuously, involving every cell in our bodies several times per year. Occasionally, however, damage to the base pair can occur when the DNA is incorrectly repaired and the wrong nucleotide is inserted which can lead to cell death or a mutation. Remember your DNA is the code which determines the type and function of the cell.
There are two basic types of mutations:. To illustrate the effects of these mutations, consider the following phrase, read as a triplet code groups of three letters :. The fat cat are the hot hog. A deletion would have the same effect:. Two types of breaks in the sugar phosphate backbone can also be caused by ionizing radiation. A single strand break occurs when only one of the sugar phosphate backbones is broken. Single strand breaks are readily repaired using the opposite strand as a template. However, base pair substitutions and frameshift mutations can still occur.
Double-strand breaks are believed to be the most detrimental lesions produced in chromosomes by ionizing radiation. Because such breaks are difficult to repair, they can cause mutations and cell death. Unrejoined double strand breaks are cytotoxic they kill cells. Double strand breaks can also result in the loss of DNA fragments which, during the repair process, can cause the joining of non-homologous chromosomes chromosomes not of the same pair leading to the loss or amplification of chromosomal material.
These events can lead to tumorigenesis creation of tumour cells if, for example, the deleted chromosomal region encodes a tumour suppressor or if an amplified region encodes a protein with oncogenic potential cancer potential. If the genetic code is damaged and the cell does not undergo apoptosis cell suicide , the mutation may be passed on during cell division, perhaps leading to a cancer or other mutation.
In some cases a mutation may remain dormant for years and perhaps forever. Ionizing radiation can also impair or damage cells indirectly by creating free radicals. Free radicals are molecules that are highly reactive due to the presence of unpaired electrons on the molecule. Free radicals may form compounds, such as hydrogen peroxide, which could initiate harmful chemical reactions within the cells. As a result of these chemical changes, cells may undergo a variety of structural changes which lead to altered function or cell death.
Spatio-temporal radiation biology: new insights and biomedical perspectives
Eric J. Hall, Amato J. Stephen P. NDT Resource Center, www. Main menu Skip to primary content. Skip to secondary content. When ionizing radiation comes in contact with a cell any or all of the following may happen: It may pass directly through the cell without causing any damage.