# Radiation Biology for Physical Scientists/Mathematics of Radiation Biology

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## Metrics
### Ionization Density: Linear Energy Transfer

Ionization from radiation in biological material leads to a random and uneven distribution of deposited energy in cells. The spatial distribution of the energy imparted by the charged particle is quantified by the Linear Energy Transfer (LET) metric. It is the quotient of the average energy imparted and the distance traversed by the radiation with units of keV/μm.
Radiation can be reclassified into low LET or high LET radiation based on their LET value. The demarcation value between low and high LET is about 10 keV/μm.
| Low LET Radiation | High LET Radiation |
| X ray | Alpha Particle |
| Gamma ray | Neutron |
| | Proton |

### Relative Biological Effectiveness

As the LET of radiation increases, the ability of the radiation to produce biological damage also increases. The relative biological effectiveness (RBE) compares the dose of test radiation to the dose of standard radiation to produce the same biological effect. The standard radiation has been taken as 250 kVp X rays for historical reasons, but is now recommended to be Cobalt 60 gamma rays.
Mathematically, the RBE is defined by the following ratio:
    $RBE={\frac {Dose_{s}}{Dose_{t}}}$
where $Dose_{s}$ is the dose from standard radiation to produce an effect and $Dose_{t}$ is the dose from test radiation to produce the same effect.
The RBE peaks when the separation between ionizing events coincides with the diameter of the DNA double helix (~ 2 nanometers).

### Oxygen Enhancement Ratio

In the presence of molecular oxygen (as little as a few hundred ppm) damage to DNA caused by free radicals can become "fixed" (i.e. permanent). This oxygen effect is considered since two-thirds of DNA damage is caused by free radicals.Tumor cells that are oxygen depleted (i.e. hypoxic) are thus more highly resistant to ionizing radiation.
The oxygen enhancement ratio determined by calculating the ratio of doses in hypoxic and normaxic conditions for a given isoeffect. Mathematically, it is expressed as:
    $OER={\frac {Dose_{h}}{Dose_{n}}}$
where $Dose_{h}$ is the dose to produce an effect in hypoxic conditions and $Dose_{n}$ is the dose to produce the same effect in normoxic conditions.
The oxygen enhancement ratio (OER) is typically lower for high LET radiation than for low LET radiation. The OER for electrons produced by x-rays may be as high as 3 while that for alpha particles is close to unity.

### Biologically Effective Dose
## Cell Survival Curves

The biological effects of radiation have historically been measured with cell survival curves. These curves model the relationship between a given dose of radiation and the fraction of cells surviving in cell cultures. Examples of cell-survival curves are shown on the left.
Several mathematical methods have been developed to define the shape with the Linear Quadratic Model being most used.

### Linear Quadratic Model

This model assumes there are two main ways to generate double strand breaks and subsequently cell death. The first way is caused by a single particle breaking both strands and is proportional to dose. This is the linear component of cell killing. The second way involves two independent breaks in opposite strand and is proportional to dose squared. This is the quadratic component of cell killing.
The combined effect of the linear and quadratic components of cell kill on the cell surviving fraction is given by
where alpha is a constant describing the linear component while the small constant beta describes the quadratic component. Alpha has units of Gray$^{-1}$ while beta has units of Gray$^{-2}$.
The ratio of alpha/beta gives the dose at which the linear and quadratic components of cell killing are equal. The typical values for tumours are ~3Gray and ~10Gray for normal tissues.