Modified Fibonacci method

Since many patients in phase I clinical trials are treated at doses of chemotherapeutic agents that are below the biologically active, they have a reduced the chance to get therapeutic benefit. Therefore we decided to adopt an accelerated dose escalation design followed by a modified Fibonacci method to reduce the number of such patients . 

The optimum seeking methods which have been found to be particularly useful are the modified Fibonacci search (search by golden section) for one-dimensional searches and the Hooke-Jeeves search for multi-dimensional searches. Beveridge and Schechter (8) give a complete description of these searches. 

Buoen et al. (9) reported that the dose-escalation schemes used in FTIH studies could be categorized as linear, logarithmic, modified Fibonacci, or miscellaneous. The latter included dose-escalation regimens in which the three standardized methods are combined. The authors reported that in 12 out of the 105 studies they reviewed a linear escalation method with fixed dose increment was used. A logarithmic dose-escalation scheme in which the relative dose increment was the same (e.g., 100%) was used in 22 studies. Four of the studies used a modified version of the Fibonacci escalation scheme, which is frequently used in cancer Phase 1 trials (6, 12-14). For most of the studies reviewed (i.e., 63.8%, or 67 studies) the dose-escalation schemes used did not seem to follow one particular scheme. In some cases two of the escalation schemes described above were combined (e.g., starting with a logarithmic escalation to convert later into a modified Fibonacci sequence), while for other studies, no escalation scheme was apparent. The doses appeared to have been chosen arbitrarily (11). 

It is also possible to modify the Fibonacci method to obtain a series that minimizes the maximum final interval of uncertainty when a point is already positioned in the starting interval. In this case, the final interval is not univocally determined by the number of points, but different widths can be obtained depending on the position of the first point and the function to be minimized. 

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