Fibonacci scheme

One way to solve the problem of unphysically short atomic distances is to project onto the Rpm subspace only those grid points included in a selected strip (gray area), with width of a (cos a + sin a) in the A per subspace. The slope of RPai shown in Fig. 1 is 0.618..., an irrational number related to the golden mean [( /5 + l)/2 = 1.618...]. As a result, the projected ID structure contains two segments (denoted as L and S), and their distribution follows a ID quasiperiodic Fibonacci sequence [2] (c.f. Table 1). From another viewpoint, the ID quasiperiodic structure on the par subspace can be conversely decomposed into periodic components (square lattice) in a (higher) 2D space. The same strip/projection scheme holds for icosahedral QCs, which are truly 3D objects but apparently need a more complex and abstract 6D... 

Some version of the modified Fibonacci escalation scheme is probably the most frequently used escalation scheme, particularly in oncologic Phase I studies. However, its pre-eminence is fading. The sequence... 

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). 

The main differences between the diverse designs encountered in drug development are related to the dose-escalation scheme, the number of patients per level, and the stopping rule definition. The oldest and most frequently used dose-escalation method for the last 20 years is the well known standard method based on Fibonacci series. Because of the limitations of this method, more sophisticated approaches have been developed, namely ... 

Cycle Parameterization. Another scheme makes use of the fact that some PRNGs have more than one cycle. If we choose the seeds carefully, then we can ensure that each random sequence starts out in a different cycle, and so two sequences will not overlap. Thus the seeds are parameterized (i.e., sequence i gets a seed from cycle i, the sequence number being the parameter that determines its cycle). This is the case for the lagged fibonacci generator described in the next section. 

The goal of dose escalation is to determine the maximum tolerated dose both efficiently and conservatively. Optimally, any scheme should not produce long and expensive Phase I studies, and at the same time should avoid the risks of overdosing and serious adverse events. One approach is to double doses with each escalation until a pharmacological response is observed, and proceed more conservatively with subsequent escalations, for example, calculating increases based upon a modified Fibonacci series (Table 4.1). 

When Archimedes computed % by his approximation of the circle by a sequences of polygons, or when the Sumerians approximated V2 by an incredible numerical scheme, which was much later rediscovered by Newton, they all were well aware of the fact that they are dealing with the unusual numbers [484]. As early as in the year 1202, the population growth was evaluated by the number of immature pairs, i.e., A +i = A + A ./ with Ao=0,Aj = 1, continuing 1,2,3,5,8,13,21,34,55,89,144,... (meaning that the state at time n+1 requires information from the both previous states, n and n-1, known as two-step loops) called the Fibonacci sequence. The ratio of A +i/A is steadily approaching some particular number, i.e., 1,618033988..., which can be equaled to (l+V5)/2) and which is well-known as the famous golden mean ( proportio... 

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