Tumour Volume Doubling Time Calculator
Tumour Volume Doubling Time Calculator
Doubling time from two measurements and the interval between them. The trap is volume against diameter: volume goes as the cube of diameter, so a doubling of volume is a 26% increase in diameter, and reading a diameter ratio as a volume ratio makes a tumour look three times more indolent than it is.
Tumour volume doubling time
Two measurements and an intervalTwo diameters, 14 mm and 18 mm, 120 days apart
The formula, and the factor of three
From two diameters: doubling time = interval × ln2 ÷ [ 3 × ln(d₂ ÷ d₁) ]
because a sphere has V = (π/6)d³, so ln(V₂/V₁) = 3 × ln(d₂/d₁) exactly
and therefore one volume doubling is a diameter increase of 2^(1/3) − 1 = 26.0%, not 100%
- the cube relationship
- the whole content of the mode switch. Volume scales as the cube of diameter, so the log of a volume ratio is exactly three times the log of the diameter ratio. Feed diameters into the volume form and the answer is three times too long: 14 mm to 18 mm over 120 days is 110 days, and the same numbers read as volumes give 331
- a doubling of volume is 26% of diameter
- the cube root of two is 1.2599. A nodule going from 10 mm to 12.6 mm has doubled in volume. This is also why RECIST’s 20% diameter criterion for progression is a far larger change in volume than it sounds: 20% on diameter is 1.2³ = 1.73, a 73% increase in volume
- what the volume formula assumes
- a spherical lesion growing exponentially. Real tumours are neither, and growth decelerates as a lesion outgrows its blood supply. Published work on non-spherical lesions substitutes a modified volume, (π/6) × a × b², from the longest diameter and the largest perpendicular one
- two points, wide uncertainty
- a fit to two measurements and nothing more. One reproducibility study of stable nodules put the coefficient of variation of repeat volume measurement below 0.15 and the optimal follow-up interval at 81 days. Over a shorter interval, or a change under about 10%, measurement error can exceed the growth
Worked example
Two diameters, 14 mm and 18 mm, 120 days apart
Ratio of diameters: 18 ÷ 14 = 1.2857, and ln(1.2857) = 0.2513
Diameters, so multiply by three: 3 × 0.2513 = 0.7539
Doubling time = 120 × ln2 ÷ 0.7539 = 83.18 ÷ 0.7539 = 110 days
Now the error. Enter the same two numbers as volumes and the answer is 120 × ln2 ÷ 0.2513 = 331 days — three times longer, on identical measurements
A case where that matters more: 14 mm to 17 mm over 140 days is 167 days read as diameters and 500 days read as volumes, which lands on opposite sides of the 400-day convention
Check the cube relationship directly: a 26.0% increase in diameter is one volume doubling, so 10 mm to 12.6 mm over 90 days gives a doubling time of 90 days either way round
And the sanity check the formula must pass: a volume that doubles over 50 days has a doubling time of exactly 50 days
Diameter change against volume change, which is the whole problem
| Increase in diameter | Increase in volume | Volume doublings |
|---|---|---|
| 10% | 33% | 0.41 |
| 20% (the RECIST progression criterion) | 73% | 0.79 |
| 26.0% | 100% — one doubling | 1.00 |
| 50% | 238% | 1.75 |
| 100% — a doubled diameter | 700% | 3.00 |
Median volume doubling times in one cohort of resected primary lung cancers
| Group | Median volume doubling time |
|---|---|
| Squamous cell carcinoma | 70 days |
| Adenocarcinoma, all | 261 days |
| Adenocarcinoma without a ground-glass component | 177 days |
| Adenocarcinoma with a ground-glass component | 725 days |
| Other histology | 70 days |
An exponential fit to two points, and the geometry that ruins it
Doubling time turns two measurements and an interval into a growth rate. The arithmetic is the same one this project uses for PSA, thyroglobulin and hCG — the interval multiplied by the natural logarithm of two, divided by the natural logarithm of the ratio of the two values — and it rests on an assumption of exponential growth, which Schwartz set out in 1961 and everyone since has cited.
What makes a tumour measurement different from a concentration is that it has geometry. A PSA has no shape; a lesion does, and it can be reported as a volume or as a diameter. Volume scales as the cube of diameter, so the logarithm of a volume ratio is exactly three times the logarithm of the corresponding diameter ratio, and the three has to go into the denominator when diameters are used. Get that wrong and the answer is out by a factor of three in the direction that reassures: a lesion going from 14 mm to 18 mm over 120 days has a doubling time of 110 days, and the same two numbers treated as volumes give 331. At 14 mm to 17 mm over 140 days the two readings are 167 and 500 days, which fall on opposite sides of the conventional 400-day line. The thing to carry in your head is that one doubling of volume is a 26% increase in diameter, not a doubling of it — and that a doubled diameter is eight times the volume, three doublings rather than one. The same geometry is worth noticing in the response criteria: RECIST’s threshold for progression is a 20% increase in the sum of diameters, which for a spherical lesion is a 73% increase in volume.
The honest limitation is that this is a two-point fit, and a two-point fit has very wide uncertainty. Measurement variability is roughly proportional and does not shrink with the interval, while real growth accumulates, so the shorter the gap the more of the apparent change is noise. One reproducibility study of stable pulmonary nodules put the coefficient of variation of repeat semi-automated volume measurement below 0.15 throughout the range examined and concluded that the optimal follow-up interval — the shortest giving a false-positive growth rate under 5% — was 81 days. A change of under about 10% over a short interval is not distinguishable from measurement error at all. The 400-day convention is itself a proposal from the lung nodule literature for separating benign from malignant lesions rather than a validated threshold, and it travels badly: in one cohort of 371 resected primary lung cancers, ground-glass adenocarcinomas had a median doubling time of 725 days and were all cancers. One published reference tool puts it bluntly — volume doubling time “is not a useful clinical tool in most cases”. A figure from a cohort is not this patient’s outcome and a response category is not a diagnosis: a stratum in which 42 per cent were alive at fifteen years tells you about that stratum, not which 42 per cent. This page computes a published quantity and states the criteria behind it. It renders no dose, no prescription and no treatment decision — that is the treating team’s. Every coefficient, conversion factor and threshold here is attributed to the source it was read in and, where it is a prognostic figure, to its derivation cohort; where the treating protocol differs, the protocol takes precedence.
Frequently asked questions
How do I calculate tumour volume doubling time?
From two volumes: the interval multiplied by the natural logarithm of two, divided by the natural logarithm of the ratio of the second volume to the first. From two diameters the denominator is three times that logarithm, because volume goes as the cube of diameter. Only the ratio matters, so any consistent unit works.
Is a doubling of diameter the same as a doubling of volume?
No, and this is the error the page exists to prevent. A doubled diameter is eight times the volume, which is three doublings. One doubling of volume is a 26.0% increase in diameter, the cube root of two. Treating a diameter ratio as a volume ratio makes a doubling time come out three times too long.
What does a doubling time under 400 days mean?
400 days is a long-standing convention from the lung nodule literature for separating benign from malignant lesions, not a validated threshold, and it travels poorly outside that setting. In one cohort of 371 resected primary lung cancers, median doubling time was 70 days for squamous cell carcinoma and 725 days for adenocarcinoma with a ground-glass component — the latter well on the reassuring side of 400 days and still cancer.
How reliable is a doubling time from two scans?
Not very, especially over a short interval. One study of stable nodules put the coefficient of variation of repeat semi-automated volume measurement below 0.15 and found that the shortest interval giving a false-positive growth rate under 5% was 81 days. A change of under about 10% over a short gap cannot be separated from measurement error.
Related calculators
References
- Schwartz M. A biomathematical approach to clinical tumor growth. Cancer. 1961;14:1272–94.
- Obayashi K, Shimizu K, Nakazawa S, et al. The impact of histology and ground-glass opacity component on volume doubling time in primary lung cancer. J Thorac Dis. 2018;10(9):5428–34.
- Reproducibility of volumetric computed tomography of stable small pulmonary nodules with implications on estimated growth rate and optimal scan interval. PLoS ONE. 2015;10(9):e0138144.
- Radiology Tutor. Tumour volume doubling time.
Not medical advice. For healthcare professionals and education. Reference intervals vary by laboratory and assay — always use your own laboratory's. Never base a dose or a treatment decision on this page alone. Full disclaimer at calcengines.com/disclaimer/