Relapse Period (REL)

Relapse Period (REL) in Q-TWiST Analysis

The relapse period is the state of a Q-TWiST analysis that holds the time a patient lives after the disease has come back or progressed. Its mean length in each trial arm is multiplied by a weight, $u_{REL}$, so the relapse period is where failure to control the cancer enters quality-adjusted survival. For a health economist the state matters because a treatment that delays relapse moves time out of REL and into better-valued states, while a treatment that only extends life after progression adds REL time whose value rests entirely on the weight. Goldhirsch and colleagues, in a reanalysis of an adjuvant breast cancer trial, defined the state as time after systemic relapse, while later analyses in advanced cancer start it at progression.

Where the relapse period sits among the Q-TWiST states

Goldhirsch and colleagues, reanalysing Ludwig Trial III in 463 evaluable postmenopausal women with node-positive breast cancer, divided overall survival into time with toxicity (TOX), time without symptoms and toxicity (TWiST) and time after systemic relapse (REL). They weighted the TOX and REL periods by utility coefficients relative to TWiST. The relapse period therefore adds $u_{REL}$ multiplied by its restricted mean duration to the Q-TWiST total, and its weight is set apart from the weight on the toxicity period.

When the relapse period starts and ends

The start of REL depends on the setting. In adjuvant breast cancer, where patients have no detectable disease after surgery, the state begins at relapse. Goldhirsch and colleagues used systemic relapse, and Cole, Gelber and Goldhirsch, in an IBCSG meta-analysis of eight randomised trials, described the third state as the period following disease recurrence. In advanced cancer there is no disease-free period, so the state begins at progression. Lee and colleagues, analysing lenvatinib plus everolimus in renal cell carcinoma, defined REL as time after disease progression.

The relapse period ends at death or at the limit of the analysis, whichever comes first. Glasziou, Simes and Gelber estimated each state's mean duration from the areas between partitioned survival curves restricted to a chosen time, and Gelber and colleagues described mean durations restricted to the follow-up limits of the trial. A patient still alive after relapse at the limit contributes REL time only up to that limit, so REL in a Q-TWiST analysis is a restricted quantity and not lifetime post-progression survival.

$$\text{REL}_L = \text{OS}_L-\text{PFS}_L$$

where $\text{REL}_L$ is the mean relapse period restricted to the limit $L$, $\text{OS}_L$ is restricted mean overall survival, the area under the survival curve up to $L$ as in restricted mean survival time, and $\text{PFS}_L$ is restricted mean progression-free survival, or relapse-free survival in the adjuvant setting. The relapse period is the area between the two curves. The subtraction holds only when both curves are measured from the same time origin, normally randomisation. Because the states are mutually exclusive, toxicity that happens after progression stays in REL and does not count as TOX.

The relapse utility weight

The weight $u_{REL}$ runs from 0, where a month after relapse counts for nothing, to 1, where it counts as fully as a month without symptoms or toxicity. Goldhirsch and colleagues used 0.5 for both TOX and REL as their illustrative case, and Lee and colleagues used the same values in their base case. In Lee and colleagues' sensitivity analysis only the TWiST weight was varied, between 0.55 and 0.9, with TOX and REL held at 0.5, so that study did not test the relapse weight itself.

The relapse weight acts against the benefit of delaying relapse. A treatment that postpones relapse shortens REL and lengthens TWiST, and the lower the REL weight, the more that shift is worth. A treatment that adds survival only after progression lengthens REL, and that extra time adds little when $u_{REL}$ is low. Assigned weights such as 0.5 are utility-like coefficients set by the analysts and need not come from a valuation of life after relapse.

Threshold utility analysis with the relapse weight

Glasziou and colleagues proposed showing results as a threshold analysis across the utility weights, and Gelber and colleagues later made threshold utility analysis the emphasis of the method. The difference in Q-TWiST between two arms is a straight-line function of the toxicity and relapse weights, and the line where the difference is zero divides the square of possible weight pairs into a region favouring each arm.

When the more effective arm spends less time after relapse, a high relapse weight erodes its advantage, so the region favouring that arm lies towards low $u_{REL}$. A patient who regards life after relapse as nearly as good as life without symptoms can then prefer the arm with shorter survival. This is a two-weight form of threshold analysis.

Worked example with an adjuvant regimen that delays relapse (illustrative)

All figures are illustrative, in months, restricted to 84 months of follow-up and measured from randomisation. Without adjuvant therapy, restricted mean overall survival is 62 months and restricted mean relapse-free survival is 50 months, with no toxicity. With adjuvant chemotherapy, restricted mean overall survival is 66 months and restricted mean relapse-free survival is 58 months, of which 6 months are spent with toxicity.

1. Relapse periods from the survival curves. The relapse period is restricted overall survival minus restricted relapse-free survival in each arm, which gives 12 months without treatment (62 minus 50) and 8 months with chemotherapy (66 minus 58). TWiST with chemotherapy is 58 minus 6, or 52 months, and without treatment TWiST is the full 50 months of relapse-free time.

2. Q-TWiST with weights of 0.5 for TOX and REL.

$$\text{Q-TWiST}_{chemo} = 0.5 \times 6 + 52 + 0.5 \times 8 = 59$$

where 6, 52 and 8 are the chemotherapy arm's months in TOX, TWiST and REL and 0.5 is the weight on TOX and on REL.

$$\text{Q-TWiST}_{none} = 0.5 \times 0 + 50 + 0.5 \times 12 = 56$$

where 0, 50 and 12 are the untreated arm's months in the three states. Chemotherapy gains 3 months of quality-adjusted time.

3. Threshold line. The differences are 6 months of TOX, 2 months of TWiST and minus 4 months of REL.

$$\Delta\text{Q} = 6u_{TOX}+2-4u_{REL}$$

where $\Delta\text{Q}$ is the Q-TWiST difference, chemotherapy minus no treatment, and $u_{TOX}$ and $u_{REL}$ are the toxicity and relapse weights. The arms are equal when $u_{REL} = 0.5 + 1.5u_{TOX}$. With toxicity weighted at 0.2 and relapse at 0.9, chemotherapy scores 0.2 × 6 + 52 + 0.9 × 8 = 60.4 months and no treatment scores 50 + 0.9 × 12 = 60.8 months, so no treatment is ahead by 0.4 months. A patient who values life after relapse highly and toxicity poorly would favour no adjuvant chemotherapy even though it gives shorter survival. Once $u_{TOX}$ exceeds one third, chemotherapy is preferred at every relapse weight between 0 and 1.

Ludwig Trial III and later applications

Goldhirsch and colleagues reported that, with TOX and REL each weighted 0.5 against 1.0 for TWiST, mean Q-TWiST by 7 years in Ludwig Trial III was 6.7 months longer with chemoendocrine therapy than with no adjuvant therapy (P = .05) and 4.1 months longer than with endocrine therapy alone (P = .20). They concluded that quality-adjusted survival analysis should be used in assessing the costs and benefits of toxic adjuvant therapy. At those weights the added toxicity of chemoendocrine therapy did not cancel its survival advantage.

Cole, Gelber and Goldhirsch extended the approach to a meta-analysis of 1,229 patients aged 49 years or younger, randomised in eight trials of chemotherapy against no adjuvant systemic therapy, with a Q-TWiST analysis for each trial. In advanced renal cell carcinoma, Lee and colleagues reported mean Q-TWiST of 14.7 months with lenvatinib plus everolimus and 11.0 months with everolimus alone, a gain of 3.7 months (95% confidence interval 1.3 to 6.3).

Relapse time in partitioned survival models and appraisal

The relapse period has the same structure as the progressed state of a partitioned survival model, which also takes the area between the overall survival and progression-free survival curves. A partitioned survival model built for appraisal usually extrapolates the curves over a lifetime and applies preference-based utility values to the progressed state to produce QALYs, while a Q-TWiST analysis stays within trial follow-up and often uses assigned weights tested by threshold analysis.

Boundaries with nearby states and measures

The Time with Symptoms entry defines a single phase with symptoms of progression or treatment toxicity, so the relapse period is one part of that phase. In the Q-TWiST analyses of Goldhirsch, Cole and their colleagues, relapse and toxicity are separate states with separate weights.

REL also differs from relapse-free survival. Relapse-free survival measured from randomisation corresponds to TOX plus TWiST, and REL is the remainder of restricted survival, while relapse-free survival measured from the end of treatment does not fit the partition directly. Post-progression survival reported as a separate outcome carries no weight, while REL is restricted to the analysis limit and always used with a weight.

Common misreadings of the relapse period

A longer REL in one arm is sometimes read as a worse result, but longer REL can come from longer overall survival with the same relapse timing. The sign of its effect on Q-TWiST depends on $u_{REL}$, since any REL time with a positive weight adds to quality-adjusted survival.

Comparing REL across studies without checking whether the state starts at relapse or at progression, and where the restriction limit falls, can mislead. A short follow-up truncates REL most, because relapse comes late in the patient's course, so a Q-TWiST result with a short limit captures little REL time whatever value $u_{REL}$ takes.

Sources

• Cole BF, Gelber RD, Goldhirsch A. A quality-adjusted survival meta-analysis of adjuvant chemotherapy for premenopausal breast cancer. Statistics in Medicine. 1995;14(16):1771-1784. https://doi.org/10.1002/sim.4780141606
• Gelber RD, Cole BF, Gelber S, Goldhirsch A. Comparing treatments using quality-adjusted survival: the Q-TWiST method. The American Statistician. 1995;49(2):161-169. https://doi.org/10.1080/00031305.1995.10476135
• Glasziou PP, Simes RJ, Gelber RD. Quality adjusted survival analysis. Statistics in Medicine. 1990;9(11):1259-1276. https://doi.org/10.1002/sim.4780091106
• Goldhirsch A, Gelber RD, Simes RJ, Glasziou P, Coates AS. Costs and benefits of adjuvant therapy in breast cancer: a quality-adjusted survival analysis. Journal of Clinical Oncology. 1989;7(1):36-44. https://doi.org/10.1200/JCO.1989.7.1.36
• Lee CH, Wan Y, Smith A, Xie R, Motzer RJ. Quality-adjusted time without symptoms or toxicity (Q-TWiST) for lenvatinib plus everolimus versus everolimus monotherapy in patients with advanced renal cell carcinoma. European Urology Open Science. 2021;31:1-9. https://pmc.ncbi.nlm.nih.gov/articles/PMC8385286