05 Biology
Survival fraction — LQ model
Cell survival after a single dose or n fractions.
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Simulation
Survival fraction — LQ model — Change the numbers; the scene follows.
Where it works
Linear accelerator
Isocenter
In the treated volume — tumour and OARs at isocenter, after the dose has been delivered.
Open this machineFormula
Typical values
Variables
Results
S
Survival fraction
4.1614e-10
−ln S
Log cell kill
21.6
α/β
Ratio
10Gy
Curve
Explanation
What it means
The linear-quadratic model writes cell kill as S = exp[−n(αd + βd²)]. The linear term αd dominates at low dose per fraction; the quadratic βd² (double-strand breaks from two tracks) grows with fraction size. α/β is low for late-reacting tissues (~3 Gy) and higher for most tumours (~10 Gy). This is a working relation in Radiobiology.
Where it is used
Clinically it sits on the Linear accelerator — Isocenter. In the treated volume — tumour and OARs at isocenter, after the dose has been delivered. Radiobiology sits between the prescription and the organ-at-risk: LQ, BED, EQD2, and why 2 Gy is not 2 Gy if the fraction size changed. Use it to compare regimens, not to invent one.
Linear accelerator · Open this machineHow to use it
Presets: early/tumour α=0.3, β=0.03; late α=0.15, β=0.05; prostate often α/β≈1.5. The curve vs fraction size (at fixed n) shows why hypofractionation hits low-α/β targets harder. Change one input and watch the curve and the simulation follow.
Symbols
- αLinear coefficient0.3 Gy⁻¹
- βQuadratic coefficient0.03 Gy⁻²
- dDose per fraction2 Gy
- nNumber of fractions30
Worked example
A typical case from the default values: α = 0.3 Gy⁻¹ (Linear coefficient); β = 0.03 Gy⁻² (Quadratic coefficient); d = 2 Gy (Dose per fraction); n = 30 (Number of fractions). Substituting into the relation gives S = 4.1614e-10; −ln S = 21.6; α/β = 10 Gy. These are teaching numbers — align them with your machine.
Typical values give
- S = 4.1614e-10
- −ln S = 21.6
- α/β = 10Gy
Where it comes from
The displayed formula is the working relation. Cell survival after a single dose or n fractions. Usual reference: Hall & Giaccia / Fowler. Derive it in the specialty lesson, then return here to pin the numbers.
Reference: Hall & Giaccia / Fowler
Assumptions & limits
Equal-effect per fraction, full repair between fractions, no repopulation or hypoxia. Not a TCP model unless you add clonogen number (see TCP).
Pitfalls
α/β is a model parameter, not a measured organ. BED from incomplete repair or a changed overall time is not the simple n·d·(1+d/(α/β)). Never EQD2 a stereotactic dose with an α/β you did not state. Equal-effect per fraction, full repair between fractions, no repopulation or hypoxia. Not a TCP model unless you add clonogen number (see TCP).
Keep this
Always write α/β and the fraction size next to a BED or EQD2. Equal-effect per fraction, full repair between fractions, no repopulation or hypoxia. Not a TCP model unless you add clonogen number (see TCP).
In this specialty