Interpretation threshold utility analysis for quality adjusted survival time I am having trouble with the Interpretation of this figure. In this study Survival time of patients was split up in 3 Intervalls: TOX, TWIST and REL. TOX is time with Treatment side effects. REL is time after Progression until death. I understand that it shows correlation between the weights given for These Intervalls by patients and a score (Q-TWIST). However, I do not understand how the black contour lines for months gained are derived. Could somebody please help?

 A: This study was based on a re-analysis of a randomized clinical trial that examined whether, following surgery that appeared to remove non-small-cell lung cancer (NSCLC) successfully, patients got a benefit from subsequent (adjuvant) chemotherapy. The original study found that overall survival was longer for those receiving adjuvant chemotherapy. This paper asked whether that longer overall survival was worth it in terms of quality of life, as patients receiving chemotherapy face several months of toxic therapy-associated side effects that are avoided by those who omit adjuvant chemotherapy.
This study examined restricted mean survival in different states, in this case survival up through 6 years after study entry, for those with surgery alone versus those also receiving adjuvant chemotherapy. The states were defined as TOX for times with chemotherapy-related toxicities, TWiST for "Time Without Symptoms or Toxicity" (i.e, the good times), and REL for times between relapse of disease and death. The argument is that a Quality-adjusted life time, called Q-TWiST, can be calculated based on the quality of life during TOX or REL, expressed relative to the quality of life during actual TWiST.
Taking the quality of life during actual TWiST as 1, and calling the relative qualities of life during the other periods to be $u_{TOX}$ and $u_{REL}$ respectively, each in the range of [0,1], Q-TWiST is then the sum of TWiST plus the quality-weighted times in the other states:
$$\textsf{Q-TWiST} = \textsf{TWiST} + u_{TOX} \textsf{TOX} + u_{REL} \textsf{REL}.$$
Adjuvant chemotherapy versus surgery alone led to 6.8 months extra restricted mean survival overall. This included 6.3 extra months in the TOX state (only in the adjuvant chemotherapy group), but 2.8 fewer months in the REL state, for 3.4 extra months in actual TWiST. If you specify values of $u_{TOX}$ and $u_{REL}$ then the above formula gives you the corresponding difference in Q-TWiST between adjuvant chemotherapy and surgery alone. If you plug those values for differences in restricted mean survival among the states and solve for $u_{TOX}$ you get:
$$ u_{TOX} = \frac{\textsf{Q-TWiST}-3.4}{6.3} + \frac{2.8}{6.3} u_{REL} $$
expressed as an intercept that depends on Q-TWiST and a positive slope with respect to $u_{REL}$. So the contour lines represent the $u_{TOX}$ versus $u_{REL}$ tradeoffs that provide a particular Q-TWiST value, based on the results of the clinical trial. A quick check shows that this makes sense. If quality of life is 0 during TOX and REL, then the Q-TWiST difference at the bottom left of the plot equals the actual TWiST difference, 3.4 months. If quality of life during TOX and REL equals that of actual TWiST ($u_{TOX}=u_{REL}=1$), then the Q-TWiST difference at the top right of the plot is the total overall survival difference, 6.8 months.
