Why Dose Increments Need a Schedule
An escalation study needs a planned set of dose levels, but the size of a step matters. A fixed increase in milligrams can be a large relative change at a low dose and a small one at a high dose. Conversely, repeating the same percentage increase keeps the relative jump constant, even as the absolute amount added grows. A Fibonacci-style schedule is one way to make relative increments generally smaller as doses rise.
The schedule specifies candidate dose levels, not which level participants should receive next. Escalation decisions still depend on the protocol’s toxicity rules, the observed dose-limiting toxicities (DLTs), and any applicable stopping limits. In particular, a progressively smaller planned increment is not a guarantee that the next dose is safe.
Under this indexing, the first two increases are each 100%, followed by increases of about 66.7%, 50.0%, 40.0%, 33.3%, and 28.6%. The percentage is calculated relative to the dose immediately before the increase. The term “Fibonacci-style” is used for this modified percentage schedule; it does not mean that the doses are the ordinary Fibonacci numbers. For example, the ordinary sequence \(1,2,3,5,8\) has relative increases of 100%, 50%, 66.7%, and 60%, which do not shrink at every step.
Constructing the Ladder
The recurrence is multiplicative: calculate the next dose by multiplying the current dose by one plus the planned relative increase. For example, a 40% increase means multiplying the preceding dose by \(1.40\), not adding 40 units and not taking 40% of the starting dose. The index \(j\) identifies the level from which the increase is made.
Worked Example: Calculating Levels From a 10 mg/m² Starting Dose
Suppose a hypothetical protocol starts at \(D_1=10\) mg/m² and uses the modified schedule as written, before considering formulation constraints or rounding.
The first two increases are 100%: \(D_2=10(1+1)=20\) mg/m² and \(D_3=20(1+1)=40\) mg/m². The next increase is \(2/3\), so \(D_4=40(1+2/3)=66.67\) mg/m². Then \(D_5=66.67(1+2/4)=100.0\) mg/m², and \(D_6=100.0(1+2/5)=140.0\) mg/m².
Continuing, \(D_7=140.0(1+2/6)=186.7\) mg/m² and \(D_8=186.7(1+2/7)=240.0\) mg/m². The relative increases are therefore 100%, 100%, 66.7%, 50.0%, 40.0%, 33.3%, and 28.6%. Each percentage is based on the immediately preceding level.
The absolute increments are not the same as the percentages: for example, the increase from 10 to 20 mg/m² is 10 mg/m², while the increase from 140.0 to 186.7 mg/m² is 46.7 mg/m². The schedule reduces relative increments, not necessarily absolute increments.
Why the Relative Increments Shrink
For levels after the first two, the increase fraction is \(r_j=2/j\). As \(j\) gets larger, the denominator grows, so \(r_j\) gets smaller. This directly establishes the diminishing relative step: \(2/(j+1)<2/j\) for positive \(j\). It does not establish that the resulting dose is safe, or that toxicity risk rises smoothly with dose.
The recurrence also gives a useful check on calculations. For \(m\geq 3\), multiply the factors from level 3 to level \(m\):
In the final line, the numerator factors \(5,6,\ldots,m+1\) and denominator factors \(3,4,\ldots,m-1\) cancel except for \(m(m+1)/(3\cdot4)\). Since \(D_3=4D_1\), this convention implies \(D_m=D_1m(m+1)/3\) for \(m\geq3\). Substituting \(m=4\) gives \(D_4=10D_1(2/3)=6.667D_1\), as in the worked calculation. This formula is a check, not a substitute for stating the protocol’s dose levels and units.
Worked Example: Checking a Planned Dose Against the Recurrence
A hypothetical ladder starts at 6 mg and proposes 24 mg at level 3 and 40 mg at level 4. Check whether the level-4 dose follows the stated schedule.
The first two 100% increases give \(D_2=6(2)=12\) mg and \(D_3=12(2)=24\) mg. The increase from level 3 uses \(2/3\), so the prescribed next dose is \(D_4=24(1+2/3)=24(5/3)=40\) mg.
The proposed 40 mg level is consistent. Its absolute increase is 16 mg, which is \(16/24=0.667\), or 66.7% of the preceding dose. The check uses the relative increase from 24 mg, not the percentage of the 6 mg starting dose.
Rounding, Formulations, and Dose Limits
A mathematically generated level may not be a dose that can be administered accurately. Available capsule strengths, infusion preparation, body-surface-area calculations, or protocol-specific dose limits can require rounding or a change to the nominal ladder. Rounding is not a purely statistical detail: it can alter the realized percentage increase, create duplicate levels, or make a step larger than intended.
A sound protocol specifies how to handle such constraints in advance. It should state the dose units, the dose basis (for example, per square metre or per kilogram), the rounding rule, permitted dose levels, and any maximum dose. When an adjusted dose no longer follows the nominal recurrence exactly, the actual levels and their realized increments should be recorded explicitly. Do not silently round each step and then describe the resulting ladder as if its original percentages were preserved.
Worked Example: Rounding to Feasible 10 mg Levels
Return to the hypothetical 10 mg/m² starting dose. Suppose, for illustration, that the protocol permits only multiples of 10 mg/m² and rounds each nominal level to the nearest 10 mg/m². The nominal levels through level 8 are 10, 20, 40, 66.67, 100, 140, 186.7, and 240 mg/m². The rounded levels are 10, 20, 40, 70, 100, 140, 190, and 240 mg/m².
The realized increase from 40 to 70 mg/m² is \(30/40=75.0\%\), rather than the nominal 66.7%. From 70 to 100 mg/m² it is \(30/70=42.9\%\), rather than 50.0%. From 140 to 190 mg/m² it is \(50/140=35.7\%\), rather than 33.3%. Rounding changes the relative increments, so the realized sequence must be checked rather than assumed to retain the nominal schedule.
These values illustrate arithmetic only; a clinical protocol would need to justify its formulation and rounding rule. If a maximum dose were 200 mg/m², the nominal next level of 240 mg/m² would exceed that cap. It should not be included just because the recurrence produces it; the protocol’s cap takes precedence.
Assumptions and Practical Checks
The schedule is meaningful only when its inputs and implementation are clear. At a minimum, check that the starting dose is positive and expressed in specified units; every planned level uses the same dose basis; the recurrence is applied to the immediately preceding level; and the resulting doses are feasible under the protocol. If the dose ladder is intended to ascend, its planned levels should be strictly increasing. A duplicate dose introduced by rounding is a signal that the nominal formula and practical dose set need reconciliation.
The underlying premise is that a ladder of smaller relative steps at higher levels is a useful way to organize dose escalation. It is not a statistical model of the dose-toxicity curve. In particular, the schedule alone does not require DLT probability to increase monotonically, quantify the probability of a DLT at a dose, or select a maximum tolerated dose. The earlier material on nonmonotone DLT patterns is relevant when observed toxicity does not follow a simple increasing pattern; the planned dose percentages cannot resolve that evidence.
Write the starting dose, units, index definition, and exact relative-increment rule. Do not rely on the label “Fibonacci” alone.
Apply each percentage to the immediately preceding dose, keeping enough precision to avoid avoidable arithmetic errors.
Use the stated rounding, formulation, and maximum-dose rules, then list the actual levels that can be administered.
For each adjacent pair, calculate \((D_{j+1}-D_j)/D_j\). Check for duplicates, unexpected large jumps, and violations of dose limits.
How This Schedule Compares With Alternatives
A fixed-amount ladder adds the same number of dose units at each step. Its relative increase is large at low doses and becomes smaller as the dose grows, but the absolute increase never adapts to the current dose. A fixed-percentage ladder instead multiplies every dose by the same factor; this preserves the relative increase but makes absolute increments grow with dose. The modified Fibonacci-style schedule varies the percentage by level, reducing the relative increase as the level index rises.
| Approach | What stays fixed | Implication for successive steps |
|---|---|---|
| Fixed absolute increment | Amount added in dose units | Relative increase generally shrinks as the dose grows |
| Fixed percentage | Relative increase | Absolute increment grows with the dose |
| Modified Fibonacci-style percentage schedule | A level-indexed rule for the relative increase | Relative increases shrink under the specified convention; absolute increments can still grow |
| Ordinary Fibonacci-number ladder | Additive sequence of dose values | Relative increases need not shrink at every step |
These are ways to construct planned levels, not substitutes for the protocol’s cohort-level escalation, de-escalation, or stopping rules. The rule-based design determines what action follows the observed outcomes; the ladder determines which dose levels are available for those decisions. A preplanned ladder also has limitations: it may not fit a drug with unusual pharmacology, narrow formulation options, or a safety concern that warrants smaller steps earlier. Clinical and pharmacological justification remains necessary.
Common Mistakes and Diagnostics
- Confusing the label with the exact sequence. “Fibonacci” does not uniquely specify a recurrence. State the percentage rule and index convention.
- Applying the percentage to the starting dose. Each increase is calculated from the immediately preceding level, not repeatedly from \(D_1\).
- Assuming a smaller relative increase means a smaller absolute increase. For instance, a 28.6% rise from 186.7 mg/m² is about 53.3 mg/m², larger than a 66.7% rise from 40 mg/m², which is about 26.7 mg/m².
- Treating nominal percentages as realized after rounding. Recalculate each adjacent percentage using the administered levels.
- Treating the schedule as a safety rule. The arithmetic does not predict toxicity or authorize escalation; the protocol’s DLT rules and dose limits govern those decisions.
A quick diagnostic is to tabulate each level alongside the absolute increment and realized relative increment. Under the unrounded convention above, the relative increments after the first two should decrease. If they do not, check the indexing, multiplication, and rounding. If they do decrease, that confirms the arithmetic only; it does not validate the clinical suitability of the ladder.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A ladder has \(D_1=8\) mg and follows the stated convention. Calculate \(D_2\), \(D_3\), and \(D_4\).
- From 40 mg to 60 mg, what is the relative increase? Which dose belongs in the denominator?
- Why is the ordinary Fibonacci sequence \(1,2,3,5,8\) not the same as a schedule with progressively smaller relative increments?
- A protocol rounds two nominal adjacent doses to the same administered dose. What should be checked before calling the ladder valid?
- Does a smaller relative increase at a higher dose establish that escalation is safe? Explain.
Solutions: (1) \(D_2=8(2)=16\) mg, \(D_3=16(2)=32\) mg, and \(D_4=32(1+2/3)=53.33\) mg. (2) The increase is \((60-40)/40=0.50\), or 50.0%; use the preceding dose, 40 mg, in the denominator. (3) Its relative increases are 100%, 50%, 66.7%, and 60%, so they do not decrease at every step. (4) Check the stated rounding rule, feasible dose strengths, and whether the duplicate levels need to be revised; a strictly ascending ladder cannot contain the same administered dose at adjacent levels. (5) No. A smaller relative increment describes the dose spacing, not the probability of toxicity or the safety of escalation; protocol rules and clinical evidence still govern the decision.