Free BED and EQD2 calculator: LQ or LQL?

LQL-Equiv is a free, open-source (MIT) calculator of BED and EQD2 under the LQ and LQL models. It runs in the browser, and nothing entered leaves the page. Its authors are Cyril Voyant and Daniel Julian. It is a research and teaching tool, not a medical device: not for choosing or changing a prescription.

Cyril Voyant · Mines Paris – PSL, Centre O.I.E. · 2 November 2026

Reading the quantities. BED is biologically effective dose; EQD2 is the equivalent dose expressed in 2 Gy fractions. LQ means linear-quadratic and LQL linear-quadratic-linear. n counts fractions, d is dose per fraction and Gy is the gray (energy deposited per unit mass). The tissue parameter α/β and transition dose dT are expressed in Gy. NTCP and TCP denote normal-tissue complication and tumour-control probabilities within the stated model.

LQ or LQL?

For n fractions of dose d, the linear-quadratic model gives BED = n · d · (1 + d / (α/β)), and EQD2 = BED / (1 + 2/(α/β)). The LQ curve keeps bending upward as the dose per fraction grows. The LQL model keeps it up to a transition dose dT, then continues in a straight line.

Below the transition dose, the two models agree. Above it, they part. For one fraction of 15 Gy to the rectum (α/β = 3.9 Gy, dT = 7.8 Gy):

  • LQ: 15 × (1 + 15/3.9) = 72.7 Gy;
  • LQL: 7.8 × (1 + 7.8/3.9) + 5 × (15 − 7.8) = 59.4 Gy, or 18 % lower.

The model must therefore be stated whenever hypofractionated and conventional schedules are compared.

What the calculator covers

  • Documented parameters for 34 organs at risk and 20 tumour sites.
  • Gaps, two fractions a day and tumour repopulation.
  • NTCP and TCP as model illustrations, never validated against clinical outcomes.
  • The 2026 version reproduces the 2014 software on 4,438 schedules: 69,125 of 69,131 values agree.

Other calculators

Other tools exist, such as the eviQ BED and EQD2 calculators. Check which model and which α/β each one uses before comparing results.

Limits

The result is only as good as the model and its parameters (α/β, dT, slope of the linear tail), which remain uncertain.

To read and to use

References

  1. Cyril Voyant, Daniel Julian. LQL-Equiv. SoftwareX 36, 103085 (2026).
  2. Cyril Voyant, Daniel Julian, Stéphane Muraro, Véronique Bodez, Morgane Pinpin, Delphine Leschi, Rudy Oozeer, Guiddi Wided, Marie-Aimée Acquaviva, Sylvain Prapant, Omar Gahbiche, Noureddine Bouaouina. Improving clinical decision-making in radiotherapy: a comparative analysis of linear-quadratic (LQ) and linear-quadratic-linear (LQL) dose models. Clinical Oncology 45, 103893 (2025).
  3. Cyril Voyant et al.. LQL-Equiv, original software. Reports of Practical Oncology and Radiotherapy (2014).