What We Mean When We Say “Dose”: Units, BED, EQD2, and a Calculator You Can Use

When a community program starts talking seriously about theranostics, one question comes up almost immediately, usually from a radiation oncologist or a medical oncologist sitting across the table: how does the dose from this therapy compare to what the patient already got from external beam?

It is a fair question. It is also a question that cannot be answered with a single number, because the word “dose” is doing at least four different jobs in that sentence. Some of those jobs belong to radiation protection. Some belong to radiobiology. Only one of them is the quantity we actually calculate when we do patient dosimetry, and it is not the one most people mean when they say “how much radiation did they get.”

A terminology post would normally open with a glossary. The confusion in practice comes from which quantity answers which question rather than from the definitions themselves, so this post builds the quantities in the order they actually stack. At the end, we are publicly releasing the calculator we built to do the last part of this math.

01 Three Quantities That All Get Called “Dose”

One place the protection quantities do belong is patient release. The release calculation under 10 CFR 35.75 is built on dose to other individuals, expressed in millisievert or millirem, and that lives squarely in the equivalent dose family. Different question, different quantity, and appropriately so.

02 Why Gray Alone Does Not Settle the Question

Answering that question in gray produces a comparison that looks valid and is not.

External beam delivers 2 Gy in a couple of minutes, then stops for 24 hours, then does it again. Radiopharmaceutical therapy delivers its dose continuously over days, at a dose rate that is highest in the first hours and decays exponentially from there. The physics is the same. The biology is not.

During a slow delivery, cells are repairing sublethal damage while the dose is still being deposited. Damage that would have combined into a lethal lesion during a fast fraction gets repaired before its partner arrives. The result is that a gray delivered slowly does less biological damage than a gray delivered quickly, and the gap is not small.

This is why you cannot put a kidney absorbed dose from PRRT next to a kidney constraint from external beam and draw a conclusion. The numbers are in the same units and they are not comparable.

03 BED: Accounting for Repair During Delivery

The linear quadratic model gives us a way across. For conventional fractionated radiotherapy, biologically effective dose is:

where D is total dose, d is dose per fraction, and α/β is the tissue-specific ratio that describes how sensitive that tissue is to fraction size. Low α/β tissues like kidney (around 2.6 Gy) and prostate (around 1.5 Gy) care a great deal about fraction size. High α/β tissues like bone marrow (around 10 Gy) care much less.

For an exponentially decaying dose rate, the fraction size term is replaced by a protraction factor, usually written G:

where λ _eff = ln(2) / T _eff is the effective decay constant in the organ, and μ _repair = ln(2) / T _repair is the sublethal damage repair constant. G is the fraction of the quadratic damage term that survives repair during delivery, and it falls between 0 and 1. This formulation follows the extension of the linear quadratic model to continuously decaying dose rates described by Millar and by Dale and Jones.

If the radiopharmaceutical clears fast relative to repair, G approaches 1 and the therapy behaves like an acute exposure. If it clears slowly, which is the usual case for ¹⁷⁷ Lu in kidney, G collapses toward a few percent and most of the quadratic damage never materializes. Two parameters drive this: the effective half-life in that specific organ in that specific patient, and the repair half-time of that tissue.

Representative published values. The full parameter set, with citations and evidence levels, is built into the calculator linked below.

04 EQD2: Saying It in a Language Radiation Oncology Already Speaks

BED is the right physical quantity, but it is not a number most clinicians have intuition for. EQD2 fixes that by converting BED into the equivalent dose delivered in conventional 2 Gy fractions:

Now we have something usable. A radiation oncologist who has spent twenty years thinking in 2 Gy fractions can look at an EQD2 number and know immediately whether it is a lot or a little, and can add it to prior external beam dose in a way that at least approximately makes sense. Although it should not be fully assumed that the doses are truly additive.

05 A Worked Example

Take a kidney receiving 5 Gy absorbed dose from one cycle, with an effective half-life of 50 hours, α/β of 2.6 Gy, and a repair half-time of 2.5 hours.

λ _eff is 0.0139 per hour and μ _repair is 0.277 per hour, yielding G = 0.048. BED is 5.46 Gy and EQD2 is 3.08 Gy, equivalent to 1.5 conventional 2 Gy fractions.

Delivering the same 5 Gy as a single external beam fraction yields a BED of 14.6 Gy and an EQD2 of 8.3 Gy.

That 5 Gy is also not delivered in a single afternoon. With a 50 hour effective half-life, 28 percent of the dose is deposited in the first 24 hours, and roughly 14 days are required for 99 percent of it to arrive. The patient walked out of the department on day one with most of their dose still ahead of them.

06 Adding Up Cycles

The place I see this go wrong most often is cumulative dose across a treatment course.

The tempting move is to add up the absorbed doses from four cycles, get 20 Gy, and run that total through the BED equation. That is not correct. The quadratic term is quadratic, so pushing the summed dose through it inflates the damage estimate considerably.

Repair half-times are measured in hours. PRRT cycles are spaced six to eight weeks apart. Repair between cycles is essentially complete, which means each administration should be converted to BED on its own, using its own effective half-life, and the BED values summed:

The total-dose method overstates cumulative BED by 25 percent with these parameters.

For four cycles at 5 Gy each with the parameters above, summing per-cycle BED gives 21.8 Gy. Running the 20 Gy total through the equation gives 27.3 Gy, a 25 percent overestimate. If you are deciding whether a patient can safely receive a fifth cycle against a 40 Gy BED kidney limit, that difference matters.

On those limits: the thresholds most commonly cited in the PRRT literature come from Bodei and colleagues, at roughly 40 Gy BED for patients without renal risk factors and 28 Gy BED for patients with them. Risk factors here means the usual suspects, including hypertension, diabetes, prior nephrotoxic therapy, and existing renal impairment. These are reference points from a specific body of evidence, not regulatory limits, and the honest summary is that the field has a range of numbers rather than a settled one.

07 Where This Came From

We built this math out because we needed it. Our group has been running two-time-point and multi-time-point dosimetry on our ¹⁷⁷ Lu-DOTATATE and ¹⁷⁷ Lu-PSMA-617 patients, calculating absorbed dose with Torch, and then getting asked by radiation oncology colleagues how those numbers stack up against external beam.

We presented that work as Abstract #251971 at last year's Society of Nuclear Medicine and Molecular Imaging meeting, extracting daily integrated dose rates from patient time-activity curves at 24 hour intervals and converting them into several EQD2 scales rather than a single average: EQD2 over the time to 99 percent delivery, EQD2 based on the maximum 24 hour dose, and an EQD2 explicitly adjusted for protraction. Different scales answer different questions, and collapsing them into one average hides the thing you usually want to see.

The calculator started as the internal tool for that analysis. It has been useful enough in conversations with referring physicians that keeping it internal stopped making sense.

08 Releasing the Calculator

(If it is asleep to save energy, just wake it up)

Give it an absorbed dose and an organ effective half-life, and it returns BED, EQD2, the equivalent number of 2 Gy fractions, and the G factor it used to get there. It ships with α/β ratios and repair half-times for fourteen tissues, including kidneys, bone marrow, liver, spleen, salivary and lacrimal glands, lungs, heart, spinal cord, bladder, breast, prostate, and thyroid, plus a custom option if you want to supply your own parameters.

Beyond the single calculation, the tool does four other things. The first is temporal delivery analysis, reporting time to 99 percent delivery, the percent of dose deposited at 6, 12, 24, 48, and 72 hours and at one week, and plotting cumulative dose, BED, and EQD2 across the delivery period. The second is cycle-by-cycle treatment planning, where prior cycles are entered with their own absorbed doses and effective half-lives alongside a planned cycle, and per-administration BED values are summed as described above. The incorrect total-dose result is displayed next to it so the size of the error is visible rather than asserted.

The third is safety assessment against organ limits, with kidney limits switching between 40 Gy and 28 Gy BED by risk status and other organs using EQD2 tolerances converted to BED, reported as percent of limit used, remaining capacity, and a cumulative BED chart with the limit drawn on it. The fourth is a references tab listing every α/β value and repair half-time in the tool with its source citation and an evidence level, since several of these parameters are extrapolated from similar tissues rather than measured directly and a user should know which is which.

09 Why ARC Is Publishing This

A large part of the barrier to community theranostics is vocabulary. The conversation between nuclear medicine, radiation oncology, and referring physicians requires a shared set of terms that nobody was formally taught, because until recently very few people needed them.

A community program that can put an EQD2 number in front of a radiation oncologist, explain how it was derived, and defend the assumptions behind it is a program that gets taken seriously in tumor board.

We will walk through this material at the next ARC regional workshop and add the tool to the training materials library alongside the dosimetry content. If your site runs the calculator with parameters other than the defaults, particularly α/β values or repair half-times for the organs where the published evidence is extrapolated rather than measured, we would like to hear which values you used and why, since those are the parameters most likely to need revision as corridor data accumulates.