Understanding the Fibons Volume Doubling Ruler
The Volume Doubling Ruler (also known as the Volume and Area Doubling Ruler) is a sophisticated medical instrument designed to bridge the gap between simple linear measurements and the complex biological reality of tumor growth. Invented by Dr. Colin Paul Spears, a renowned hematologist-oncologist, this tool provides a tangible, rapid method for clinicians to assess treatment efficacy by calculating “doubling time” directly from medical imaging.
The Clinical Foundation
Tumor volume doubling was originally described by Collins et al in 1956, using average
diameters of cancers on chest X-rays. Collins paper by 1986 was the most highly cited
(nearly 400 times) publication in the history of the American Journal of Roentgenology
(Chew, 1988). By 1970, tumors were measured as the summed products of perpendicular
diameters (SPD), still in use today by the World Health Organization (WHO). A lymphoma
workshop report from a Swiss conference in 1999 established the Lugano Criteria using
(SPD), which continues in regular updates to the present (Cheson). However, the RECIST
(Response Criteria in Solid Tumors) was established in 2010 and rapidly has become the
most commonly used method. By RECIST, diameters of up to 6 parameter lesions are
summed, and that sum compared over time for percentage change for showing treatment
response or tumor progression. The practicality, (speed and reproducibility) have
contributed to the popularity of RECIST, although the waterfall plots describing percent
change in tumor diameters in given patients does not include the tumor volumes.
Refinement of RECIST for PET-scan based response criteria included tumor metabolic
activity in PERCIST criteria, when size alone is misleading, as in GIST malignancies.
Using edge arcs to measure the size of chest tumors has been championed by the Memorial
Sloan-Kettering group. Volume Doubling Time of chest tumors has been an important
aspect of evaluating the malignancy of cancers, coin lesions vs. benign pathologies. The
different response criteria all use concepts of Volume Doubling for setting thresholds for
calling shrinkage for attaining a partial response, and reaching growth qualifying for official
progression.
Beyond Collins’ Algebra: the Spears Approach (1979, 1984, 2022-2025)
The VDR uses the number of Volume Doublings since a single cell division to produce a
volume of 1 cc. At 30 volume doublings, working backwards, this gives an initial cell volume
of about 931 ꙡL or about 10 times that of a red cell (not counting interstitial volume).
Conveniently, 30 Volume Doublings = 20 Area Doublings = 10 Radius (Diameter) Doublings
(Spears, 1984). Thus, our patented VDR includes scales for converting diameters (radii) into
volumes, and areas and their Doubling Numbers, n, where n A is the Area Doubling Number.
Our coinage of the term “The 1-2-3-4 of Doubling” is that 1 doubling of radius = 2 doublings
of area = 3 doublings of volume. And 4 times the area (cross-sectional or projection) =
surface area. This unifies Collins’ average diameter, the WHO and Lugano (Cheson) response
criteria and RECIST.
1. The Mathematical Foundation
The ruler is based on models of asymmetric binary division. In nature, growth often follows predictable mathematical progressions, as observed in phyllotaxis (the arrangement of leaves on a stem). Dr. Spears applied these geometric patterns to cell populations.
When a tumor grows, it doesn’t just get “longer”; it expands in volume. Because volume increases as a cube of the radius ($V = \frac{4}{3}\pi r^3$), a small change in diameter represents a massive change in the number of cancer cells. The ruler uses logarithmic scales to allow a physician to convert a 2D diameter measurement into a 3D volume doubling factor instantly.
2. The Interconvertible Scales
The ruler features a Cursor Slider and a transparent overlay that includes:
- Linear Scales: For cm/mm, Radius or Diameter, inches/fractions
- Volume Scale: From 1 cc (cm 3 ) to over 4000 cc using the Cursor-Slider, n = 30 to 42
- (over 4000 cc).
- Area Scale: From 1.2 cm 2 to over 300 cm 2 , n A = 20 to 28
- Subcm Scale: Field of small circles with Tabled Data, n = 24 (n A = 16) to 30 (n A = 20)
- from 0.016 cc to 1 cc
- Doubling Time: (Time 2 – Time 1 [days, weeks, years]) ÷ change in n
- Response Velocity: when n is negative, getting smaller over time.
Clinical Application: Measuring Object Response
In the medical field, specifically in oncology and radiology, the ruler is used to evaluate Object Response—how a tumor reacts to chemotherapy, radiation, or surgery.
Step-by-Step Usage:
- Baseline Measurement: A clinician uses the transparent ruler to measure the diameter of a spherical or near-spherical mass on a CT scan, X-ray, or digital screen.
- Follow-up Comparison: After a treatment cycle, a second measurement is taken.
- Applying the Slider: By aligning the initial measurement with the follow-up on the ruler’s specialized scales, the clinician can immediately see if the volume has doubled, tripled, or—ideally—decreased by specific doubling factors.
- Determining Velocity: By factoring in the time elapsed between scans, the Tumor Doubling Time (TDT) is calculated. A fast doubling time indicates an aggressive malignancy, while a slow or negative doubling time indicates a successful therapeutic intervention.
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Why It Matters in Modern Oncology
Precision Beyond RECIST
Current medical standards often rely on RECIST (Response Evaluation Criteria in Solid Tumors), which uses simple linear changes. However, linear changes can be misleading. The Fibons ruler provides a more “volumetric” perspective, which is a truer reflection of the total tumor burden and the number of viable malignant cells.
Immediate Chair-Side Assessment
In a world of complex software, the Volume Doubling Ruler offers a “low-tech, high-intelligence” solution. It allows doctors to provide immediate feedback to patients during a consultation without needing to wait for complex volumetric rendering from a radiology lab.
Educational & Research Utility
The ruler serves as a powerful educational tool for medical students and researchers to visualize the exponential nature of biological growth. It transforms abstract mathematical concepts into a physical, manipulatable interface.
Response Criteria:
RECIST:
Response = ≥ 30% decrease in diameter
= ≥ Δn < –1.54
Progression = ≥ 20% increase in diameter
= ≥ Δn ≥ +0.79
WHO:
Response = ≥ 50% decrease in Sum of Perpendicular Diameters (SPD)
= ≥ Δn < –1.5
Progression = ≥ 25% increase in SPD
= ≥ Δn ≥ +0.48
Cheson (Lugano):
Response = ≥ 50% decrease in SPD
= ≥ Δn ≤ –1.5
Progression = > 50% increase SPD
= ≥ Δn ≥ +0.88
The VDR is Based on the importance of Tumor (Volume) Doubling Time in Response Criteria.
Partial Response is a 50% decrease in the SPD (Δn A = -1) or 1.5 Volume Doublings
decrease (Δn = -1.5), universally agreed upon by WHO, Lugano (Cheson) and RECIST Criteria.
But they differ in criteria of Progression. RECIST is > 20% increase diameter(s) or
Δn = 0.79, Lugano is > 50% increase in the SPD (Δn > 0.88) and WHO, an increase of just
a 25% in SPD (Δn = 0.48).
Mathematical Relationship between Doubling and Fibonacci:
Pascal’s Triangle as a left-adjusted array of running sums: Doubling
1
11
121
1331
14641
Sum (∑)
1
2
4
8
16
Fibonacci Numbers as a left adjusted G n-1 + G n-c = G n c = 2, c-adjusted Pascal Triangle:
1
1
11
12
131
143
1561
G n Sum (∑)
1
1
2
3
5
8
13
Fibonacci Phyllotaxis is the appearance of sequential Fibonacci numbers in opposite spirals
in plants and in the numbers of branches (leaves) around trunks (stems) such as 8 branches in 3
360 degree turns, and 21/34 and 34/55 spirals in the seedheads of compound flowers such
sunflowers, Gerbera, and Cotula species of the Asteraceae (Daisie) family.
“Start where you are. Use what you have. Do what you can.” — The guiding philosophy behind the practical design of the Fibons Ruler.