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© 2003 American Society for Clinical Oncology
Modified Fibonacci SearchUniversity of Alabama at Birmingham, Birmingham, AL To the Editor: Although alternative phase I dose-escalation schemes have emerged recently,1 the most frequently used scheme for more than two decades has been said to use the modified Fibonacci search. The number sequence, wherein the next number equals the sum of the two previous numbers (1, 1, 2, 3, 5, 8, 13, 21. . .) is familiar to many people, but the modification used in phase I trials to give decreasing increases (2n, 3.3n, 5n, 7n, 9n, 12n, 16n as multiples of the initial dose, or 100%, 65%, 52%, 40%, 29%, 33%, 33% increase over the previous dose) is not so straightforward. Fibonacci himself is seldom identified, and the source of this scheme is almost never cited. In fact, the original proposal for this type of dose escalation2 seems to have been referenced only once in the Journal of Clinical Oncology3 and rarely elsewhere. Who was Fibonacci? How was the scheme derived, and by whom? The eminent mathematician known as Leonardo Fibonacci (about 1170 to 1240) was born in Pisa to the Bonacci family; Fibonacci is apparently a nickname applied to him several centuries later. He was educated in what is now Algeria and learned the Hindu-Arabic numerals, which he introduced to Europe 800 years ago in his book, "Liber abaci" (book of the abacus). Among other problems presented in that book is one about how many pairs of rabbits can be produced from a single pair under specified conditions; the solution given is 1, 2, 3, 5, 8, 13, 21, and so on. Fibonacci numbers have many ramifications in nature, mathematics, and science.4 At the beginning of the modern chemotherapy era after World War II, the starting dose and rate of dose escalation for clinical trials of new agents posed major problems, as they still do. Once the initial dose was chosen, 50% to 100% escalations were common, but this risked abruptly moving from a nontoxic to a substantially toxic dose. Marvin Schneiderman, a statistician at the National Cancer Institute, seems to have been the first to propose in writing a scheme of decreasing increases for drug testing. His article,2 from a 1965 symposium, says, in part, "A decreasing step suggestion also has been made. This is due to Bellman (reference 32, page 342) in another context, and I have not seen it in any published account of preliminary dose finding. The Bellman suggestion is a form of Fibonacci search. Three decisions have to be made here: the initial dose d, the maximum possible dose d', and N, the number of steps allowable in moving upward from dose d to dose d'. By taking a Fibonacci series of length N + 1, inverting the order, and spacing the doses in proportion to the N intervals in the series, one would take smaller and smaller steps in moving from d to d'. This cautious approach has considerable appeal." Unfortunately, Bellman’s book,5 mentioned above, has only 337 pages, and Fibonacci is not listed either in the table of contents or in the index. There is, beginning on page 34 of Bellman’s book, a section entitled "Locating the Unique Maximum of a Concave Function" that includes a discussion of Fibonacci numbers, but not the matter of inverting the order, and neither Bellman nor Schneiderman provides the derivation, for a nonmathematician, of the multiples that subsequently emerged in the cancer literature. Investigators at the National Cancer Institute seem to have been the first to report the use of this scheme in cancer chemotherapy trials. Hansen et al6 gave increasing doses of lomustine to subsequent sets of patients, saying that they used "escalation in decreasing steps" and that "the dose was escalated with a modification of the Fibonacci search scheme" citing Schneiderman’s article, but they don’t elaborate. Goldsmith et al7 do indicate the multiples n, 2n, 3.3n, 5n, 7n, 9n, and so on without showing their derivation. A review of articles in the Journal of Clinical Oncology that use a modified Fibonacci search indicates that not everyone means the same thing. For example, 20%3 or 30%8 increments have been mentioned, although most authors mean an escalation involving decreasing steps, even if the details vary. It might be best, for the sake of clarity, to label decreasing increment schemes as such, specifying the increments, without invoking Fibonacci. REFERENCES
1. Eisenhauer EA, O’Dwyer PJ, Christian M, et al: Phase I clinical trial design in cancer drug development. J Clin Oncol 18:684–692, 2000 2. Schneiderman MA: Mouse to man: Statistical problems in bringing a drug to clinical trial—Proceedings of the fifth Berkeley symposium on mathematical statistics and probability (vol IV). Berkeley, CA, University of California Press, 1967, pp 855–866 3. Wolff SN, Fer MF, McKay CM, et al: High dose VP-16-213 and autologous bone marrow transplantation for refractory malignancies: A phase I study. J Clin Oncol 1:701–705, 1983[Abstract] 4. Livio M: The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number. New York, NY, Broadway Books, 2002 5. Bellman RE: Dynamic Programming. Princeton, NJ, Princeton University Press, 1957
6. Hansen HH, Selawry OS, Muggia FM, et al: Clinical studies with 1-(2-chloroethyl)-3-cyclohexyl-1-nitrosourea (NSC 79037). Cancer Res 31:223–227, 1971
7. Goldsmith MA, Slavik M, Carter SK: Quantitative prediction of drug toxicity in humans from toxicology in small and large animals. Cancer Res 35:1354–1364, 1975
8. Propper DJ, de Bono J, Saleem A, et al: Use of positron emission tomography in pharmacokinetic studies to investigate therapeutic advantage in a phase I study of 120-hour intravenous infusion XR5000. J Clin Oncol 21:203–210, 2003 This article has been cited by other articles:
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Copyright © 2003 by the American Society of Clinical Oncology, Online ISSN: 1527-7755. Print ISSN: 0732-183X
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