Radiotherapy Fractionation 2D: The Linear-Quadratic Model

Fraction Prescription

Dose per fraction 2.0 Gy

Course length 30 fx

Tumor α/β ratio 10 Gy

Normal tissue α/β 3 Gy

Course Progress

Fraction 0 / 300%

Live Readouts

Tumor cells alive

100.0%

Normal tissue alive

100.0%

Tumor BED

0.0 Gy

Normal BED

0.0 Gy

Therapeutic ratio

1.00×

Total dose

0.0 Gy

How it works

Cell killing by a single radiation dose d (in Gy) follows the linear-quadratic (LQ) model, the standard equation used to plan real fractionated radiotherapy:

Surviving fraction:  SF = exp(-(αd + βd²))
Biological effect:   BED = n·d·(1 + d / (α/β))

Every tissue has its own α/β ratio: high for fast-dividing tumors (≈10 Gy here), low for slow, late-responding normal tissue (≈3 Gy). A low α/β makes the βd² term dominate, so normal tissue is disproportionately punished by a large dose per fraction — but barely notices when the same total dose is split into many small fractions with time to repair between them. A high-α/β tumor is far less sensitive to fraction size, so it keeps taking damage either way.

That gap is the whole reason radiotherapy is fractionated: splitting a course into 20–30 small daily doses kills nearly as much tumor as one huge dose would, while sparing far more of the surrounding healthy tissue — exactly what the therapeutic ratio readout (tumor kill ÷ normal-tissue kill) tracks below. The lower chart plots the per-fraction survival curve SF(d) for both tissues over 0–8 Gy so you can see the crossover directly, with a marker at the currently dialed-in dose.

Deliver Next Fraction — applies one dose of the current slider value to every still-alive cell in both clusters, independently, with kill probability 1 − SF for that tissue's α/β.
Auto-Deliver Course — plays through the full prescribed number of fractions automatically.
Each surviving cell shrinks away individually and stochastically — the aggregate curve you see is the Monte-Carlo average of the same exponential the LQ formula predicts.
Drag the cross-section to pan, scroll/pinch to zoom — the view does not affect the physics.

Tumor cells Healthy tissue

drag to pan · scroll to zoom