# The linear-quadratic model and fractionation

Source: https://onco.cc/technologies/linear-quadratic-model/  
OnCo record `linear-quadratic-model` (Technology). Data CC BY-NC 4.0, attribute "Data from OnCo (onco.cc)"; commercial use needs a licence.

## TL;DR

The equation radiotherapy uses to compare schedules: cell kill has a part proportional to dose and a part proportional to dose squared, and the ratio between them (alpha over beta) tells you how much a tissue cares about the size of each fraction.

## Summary

The linear-quadratic model describes the surviving fraction of cells after a dose d as exp(minus alpha d minus beta d squared). Tissues and tumours with a high alpha/beta ratio (around 10 Gy, most tumours and acutely reacting tissues) are little affected by fraction size; those with a low ratio (around 3 Gy for late-reacting normal tissue, and perhaps 1.5 Gy for prostate cancer) are very sensitive to it. Jack Fowler and others turned this into the biologically effective dose, which lets clinicians compare a 25-fraction course with a five-fraction one and underpins hypofractionation, stereotactic dosing and re-irradiation sums. The model breaks down at very large fractions and ignores repopulation unless extended.

## Fields

- Kind: Technology
- Status: established
- Last checked: 2026-09-16
- Tags: radiation-wave1
- Principle: Radiation cell kill has a linear component from single lethal events and a quadratic component from pairs of sublethal events that combine; their ratio sets each tissue's sensitivity to fraction size.
- Since: 1980
- Strengths: Simple and predictive within the clinical range; Explains why hypofractionation works in breast and prostate cancer; Basis for dose summation
- Limitations: Overestimates kill at very large fractions; Ignores time, hypoxia and immune effects unless extended; Alpha/beta values carry wide uncertainty

## Sources

- Wikipedia: https://en.wikipedia.org/wiki/Linear-quadratic_model

## Connected records

- fronts: [Radiation Therapy](https://onco.cc/fronts/radiation/)
- technologies: [Hypofractionated radiotherapy](https://onco.cc/technologies/hypofractionated-radiotherapy/), [IMRT / IGRT (modern external beam)](https://onco.cc/technologies/imrt-igrt/), [SBRT / SABR (stereotactic radiotherapy)](https://onco.cc/technologies/sbrt/)
- terms: [Alpha/beta ratio](https://onco.cc/terms/alpha-beta-ratio/), [Biologically effective dose (BED) and EQD2](https://onco.cc/terms/biologically-effective-dose/), [Gray unit (Gy)](https://onco.cc/terms/gray-unit/), [Hypofractionation (fewer, larger radiotherapy doses)](https://onco.cc/terms/hypofractionation/)

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