## 2006 How to calculate the doubling time of a solitary pulmonary nodule?

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**Dr. Wai-Kei Lam, Department of Medicine, North District Hospital**

Solitary pulmonary nodule (SPN) is defined as a single spherical lesion of 3 cm or less in diameter completely surrounded by lung without any associated atelectasis or lymphadenopathy.1 It is usually an incidental finding and can be seen on 0.09-0.2 % of chest x-rays (CXR) in mass screening studies.2,3

Radiologically the spatial (size, margin, internal characteristics, density, calcifications) and temporal (post-contrast enhancement and growth rate) characteristics of an SPN can suggest benign or malignant aetiology.4 Resection is advisable for nodules with high probability of malignancy. Difficulty arises from nodules having low to intermediate probability of malignancy, technically difficult for fine-needle aspiration or transbronchial biopsy, size being too small for a reliable positron-emission tomography evaluation in patients who are operable but with a relatively high operative risk. Repeat imaging for any nodule growth, or more precisely observing volume doubling time of the nodule is an often-employed strategy when a single imaging study is inconclusive.

Doubling time calculation is based on an exponential growth model.5 Suppose in the beginning there is one tumor cell and each tumor cell divides into two cells every day:

There are three “doublings” in the number of tumor cells from the end of day 3 to end of day 6. The number of doublings can be counted on the graph, or calculated as log2(64) - log2(8) = 6 - 3 = 3. Assuming tumor volume is proportional to the number of tumor cells (say, by a factor of α), the “number of doublings” a tumor has undergone from an initial volume of V1 to a volume of V2 can therefore be expressed as:

Number of doublings = log2(αV2) – log2(αV1)

= log2(V2/V1)

Doubling time is the time required per doubling. In the above example the doubling time is 3 days / 3 doublings = 1 day. With a known time interval between V1 and V2,

Doubling time = time interval (Ti) / number of doublings

= Ti / [log2(V2/V1)]

Changing logarithm base 2 to logarithm base 10 (for the ease of calculation using conventional calculators) (log2X = log X / log 2),

= Ti ․log 2 / log (V2/V1),

or = Ti ․log 2 / [3 log (d2/d1)]

by assuming a spherical volume V = 4/3․ ․(d/2)3 with d as the diameter of the sphere, and as log (xy) = y log (x).

For example on CXR a mass is 0.8 cm in diameter and 200 days later its diameter increases to 1.0 cm, the doubling time of the mass is 200 days x log 2 / [3 x log (1.0/0.8)] = 207 days. There is a margin of measurement error. The 95 % limits of intra-observer agreement of diameter measured was found to be 1.32 mm for SPN less than 2 cm in diameter on computerized tomography (CT).6 Put this into our example it would result in an error of about ± 40 days for the estimated doubling time.

Bronchogenic carcinoma usually has a doubling time from 20 to 400 days, though in adenocarcinoma and bronchioloalveolar carcinoma the doubling time can be as long as 1346 days. 7-11 Infection, infarction, lymphoma or fast-growing metastasis from choriocarcinoma and osteosarcoma have doubling time less than 20 days.7 Measurement of SPN diameter can be performed on the screen on any picture archiving and communication system (PACS) station. With the advent of multi-detector CT volume of the nodule can be measured directly. This does not assume the shape of the nodule being a sphere and is particularly useful for nodules with asymmetrical growth.12 Computer-assisted algorithm can minimize inter-observer variability on volumetric measurement.13

Practically one can follow-up SPN 5 mm or smaller with CT at 3, 6, 12, 24 months, or 6-monthly for less than 4 mm nodules and 3-monthly for 4-8 mm nodules. Some has advocate it is sufficient to repeat CT at one year for nodule ≤ 4 mm and at 6 months for nodules 4-7 mm in size. Nevertheless non-invasive or invasive diagnostic methods should be considered for nodules greater than 8 mm.4,14 No change in SPN size on CXR for 2 years only has a 65 % positive predictive value for a benign outcome.15 A CT should be performed to further characterize the nodule, and a one more-year CXR follow-up is recommended.16

References:

1. O’Donovan PB. The radiologic appearance of lung cancer. Oncology (Williston Park) 1997; 11:1387-1402.

2. Comstock GW, Vaughan RH, Montgomery G. Outcome of solitary pulmonary nodules discovered in an x-ray screening program. N Engl J Med 1956; 254:1018-1022.

3. Good CA, Wilson TW. The solitary circumscribed pulmonary nodule: study of seven hundred five cases encountered roentgenographically in a period of three and one-half years. J Am Med Assoc 1958; 166:210-215.

4. Moses DA, Ko JP. Multidetector CT of the solitary pulmonary nodule. Semin Roentgenol 2005; 40:109-125.

5. Gurney JW. Doubling time. Available at: http://www.chestx-ray.com/SPN/DoublingTime.html. Accessed December 30, 2006.

6. Revel MP, Bissery A, Bienvenu M, et al. Are two-dimensional CT measurements of small non-calcified pulmonary nodules reliable? Radiology 2004; 231:453-458.

7. Nathan MH, Collins VP, Adams RA. Differentiation of benign and malignant pulmonary nodules by growth rate. Radiology 1962; 79:221

8. Weiss W. Peripheral measurable bronchogenic carcinoma. Growth rate and period of risk after therapy. Am Rev Resp Dis 1971; 103:198

9. Yankelevitz DF, Gupta R, Zhao B et al. Small pulmonary nodules: evaluation with repeat CT-preliminary experience. Radiology 1999; 212:561-566.

10. Yankelevitz DF, Reeves AP, Kostis WJ et al. Small pulmonary nodules: volumetrically determined growth rate based on CT evaluation. Radiology 2000; 217:251-256.

11. Aoki T, Nakata H, Watanabe H et al. Evolution of peripheral lung adenocarcinomas: CT findings correlated with histology and tumor doubling time. AJR Am J Roentgenol 2000; 174:763-768.

12. Zerhouni EA, Stitik FP, Siegelman SS et al. CT of the pulmonary nodule: a cooperative study. Radiology 1986; 160:319-327.

13. Schwartz LH, Ginsberg MS, DeCorato D et al. Evaluation of tumor measurements in oncology: use of film-based and electronic techniques. J Clin Oncol 2000; 18:2179-2184.

14. Jett JR, Midthun DE. Update on CT screening and nodule evaluation. Pulmonary Perspectives 2005; 22:1-3.

15. Hasegawa M, Sone S, Takashima S et al. Growth rate of small lung cancers detected on mass CT screening. Br J Radiol 2000; 73:1252-1259.

16. GC Ooi, Khong PL, Yau YY. Advances in imaging of the solitary pulmonary nodule. Hong Kong Med J 2004; 10:107-116.