## Thursday, July 16, 2009

### Dose Escalation and Modified Fabonacci Series

Dose-escalation is a type of clinical trial design in which the amount of the drug is increased with each cohort that is added. Each cohort is called 'dose cohort' and the size of each cohort could be different depending on the nature of the study. Typically the cohort size is around 10 subjects. Dose-escalation study design is used to determine how a drug is tolerated in people and it is often used in first-in-men trial. In dose escalation study, a new cohort should not be initiated before safety data in the current or previous cohort has been fully assessed. Sometimes, it may be useful to pre-define a safety stopping rule to prevent the increase of the dose cohort if something bad happens.

One thing for a dose escalation study is how to determine the dose space (ie, how much increase in terms of the dose comparing with the previous cohort). One schema to determine the dose space is so called 'Modified Fabonacci Series'.

In the 12th century, Leonardo Fibonacci discovered a simple numerical series that is the foundation for an incredible mathematical relationship behind phi.

Starting with 0 and 1, each new number in the series is simply the sum of the two before it.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .

The ratios of the successive numbers in the Fibonacci series quickly converge on Phi. After the 40th number in the series, the ratio is accurate to 15 decimal places.

### 1.618033988749895 . . .

A “modified” Fibonacci series uses starting numbers other than 0 and 1. For example, the starting number is 1 and 3, we will have a modified Fibonacci series as following:

1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349...

The modified Fibonacci series has been used in Phase I dose escalation study to determine the dose space.

Assuming the d1 is the starting dose for the first cohort, according to the modified Fabonacci series, the next dose cohort will be
d2=2d1, and then d3=1.67d2, d4=1.5d3,... If the start dose is 5 mg and a study with 5 cohorts, the dose schema will be:

Cohort 1 (5 mg) -> Cohort 2 (10 mg) -> Cohort 3 (15 mg) -> Cohort 4 (25 mg) -> Cohort 5 (40 mg)

As we can see, with the dose increase, the ratio between two consecutive doses are getting smaller and smaller.

Further reading on the modified Fabonacci series and its application in dose escalation studies: