{"entity":{"id":"linear-quadratic-model","kind":"technology","name":"The linear-quadratic model and fractionation","aka":[],"tldr":"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. Relative biological effectiveness rescales the gray before the linear-quadratic formula is applied; a proton plan at RBE 1.1 and the same plan at a variable, LET-weighted RBE are different EQD2 maps.","status":"established","asOf":"2026-09-17","links":[{"label":"Wikipedia","url":"https://en.wikipedia.org/wiki/Linear-quadratic_model"}],"tags":["radiation-wave1"],"related":[],"cancers":[],"sections":["radiation"],"technologies":["hypofractionated-radiotherapy","sbrt","imrt-igrt","proton-therapy","carbon-ion"],"targets":[],"drugs":[],"companies":[],"institutions":[],"pathways":[],"terms":["alpha-beta-ratio","biologically-effective-dose","hypofractionation","gray-unit","linear-energy-transfer","relative-biological-effectiveness"],"trials":[],"people":[],"bottlenecks":[],"keyPapers":[],"journals":[],"dependsOn":[],"notes":[],"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.","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"],"since":1980},"route":"/technologies/linear-quadratic-model/","neighbours":{"section":[{"id":"radiation","kind":"section","name":"Radiation Therapy","route":"/fronts/radiation/"}],"technology":[{"id":"carbon-ion","kind":"technology","name":"Carbon-ion therapy","route":"/technologies/carbon-ion/"},{"id":"hypofractionated-radiotherapy","kind":"technology","name":"Hypofractionated radiotherapy","route":"/technologies/hypofractionated-radiotherapy/"},{"id":"imrt-igrt","kind":"technology","name":"IMRT / IGRT (modern external beam)","route":"/technologies/imrt-igrt/"},{"id":"proton-therapy","kind":"technology","name":"Proton therapy","route":"/technologies/proton-therapy/"},{"id":"sbrt","kind":"technology","name":"SBRT / SABR (stereotactic radiotherapy)","route":"/technologies/sbrt/"},{"id":"fractionation-repopulation-models","kind":"technology","name":"The four Rs and accelerated repopulation","route":"/technologies/fractionation-repopulation-models/"},{"id":"tumour-control-probability-models","kind":"technology","name":"Tumour control and normal tissue complication probability (TCP and NTCP)","route":"/technologies/tumour-control-probability-models/"}],"term":[{"id":"alpha-beta-ratio","kind":"term","name":"Alpha/beta ratio","route":"/terms/alpha-beta-ratio/"},{"id":"biologically-effective-dose","kind":"term","name":"Biologically effective dose (BED) and EQD2","route":"/terms/biologically-effective-dose/"},{"id":"gray-unit","kind":"term","name":"Gray unit (Gy)","route":"/terms/gray-unit/"},{"id":"hypofractionation","kind":"term","name":"Hypofractionation (fewer, larger radiotherapy doses)","route":"/terms/hypofractionation/"},{"id":"linear-energy-transfer","kind":"term","name":"Linear energy transfer (LET)","route":"/terms/linear-energy-transfer/"},{"id":"relative-biological-effectiveness","kind":"term","name":"Relative biological effectiveness (RBE)","route":"/terms/relative-biological-effectiveness/"}],"pairing":[{"id":"proton-vs-imrt","kind":"pairing","name":"Caution: proton therapy vs IMRT","route":"/pairings/proton-vs-imrt/"}],"idea":[{"id":"idea-bio2-let-rbe-ab-selects-protons","kind":"idea","name":"Use LET, RBE and alpha/beta to decide when protons beat IMRT","route":"/ideas/idea-bio2-let-rbe-ab-selects-protons/"}]}}