Four-state occupancy with a time to treatment discontinuation curve

Extends the three-state partition with a third curve, time to treatment discontinuation (TTD), which splits progression-free time into on-treatment and off-treatment states. It applies the general N-state rule of TSD 19: the first state is its own curve, each later alive state is its curve minus the curve of the state before it, and the dead state is 1 minus OS. Five of the appraisals reviewed in TSD 19 used this structure.

Signature

P_ON = S_TTD; P_OFF = S_PFS - S_TTD; P_PD = S_OS - S_PFS; P_D = 1 - S_OS
Inputs
InputsDefinitionUnit
S_TTDProbability of still being on initial treatment at time t since model entry, where stopping treatment or death ends the time on treatment, as TSD 19 defines TTDprobability from 0 to 1
S_PFSProgression-free survival at time t since model entryprobability from 0 to 1
S_OSOverall survival at time t since model entryprobability from 0 to 1
Output
P_ONProportion of the starting cohort progression-free and still on initial treatment at time tproportion from 0 to 1
P_OFFProportion of the starting cohort progression-free after stopping initial treatment at time tproportion from 0 to 1
P_PDProportion of the starting cohort alive after progression at time tproportion from 0 to 1
P_DProportion of the starting cohort dead at time tproportion from 0 to 1

Function

Partitioned survival state occupancy and area-under-the-curve QALY function

Maps a set of survival curves that are not mutually exclusive, usually progression-free survival (PFS) and overall survival (OS) for each treatment arm, to the share of the cohort in each health state over time, and then to mean time in each state and QALYs as utility-weighted areas under the curves. State membership is read from the curves rather than built from transition probabilities, which is what separates a partitioned survival model from a state transition model. Linked records cover the parts this package does not repeat: the restricted mean from a Kaplan-Meier curve (HE-FM-ADMC-003), QALYs summed over periods (HE-FM-QALY-001), discounted totals (HE-FM-DR-002), a treatment curve from a baseline curve and a hazard ratio (HE-FM-HR-003) and the continuous discount rate (HE-FM-CONT-003).

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Implementations

  • Excel

    Four partitioned survival state shares from three curve columns

    With TTD in column B, PFS in column C and OS in column D from row 2, the four formulas filled down return the on-treatment, off-treatment, progressed and dead shares.

    =B2; =C2-B2; =D2-C2; =1-D2

Assumptions

  • Nested TTD, PFS and OS curves at every time

    At every time S_TTD is no higher than S_PFS and S_PFS no higher than S_OS. This holds when initial treatment stops at or before progression or death. People cannot move back to a lower-indexed state, so no one restarts initial treatment once it has stopped.

  • Same arm and time origin for the three partitioning curves

    TTD, PFS and OS describe the same treatment arm and are measured from model entry, so the differences between them are shares of one cohort at the same time.

Worked examples

  • Four-state shares at one year with an illustrative TTD of 0.30

    Taking the article's standard-care PFS and OS at one year and an illustrative 30% still on initial treatment, 6.8% are progression-free off treatment, while the progressed and dead shares are unchanged from the three-state example.

    S_TTD = 0.30; S_PFS = 0.36788; S_OS = 0.60653; P_ON = 0.30; P_OFF = 0.06788; P_PD = 0.23865; P_D = 0.39347

Common errors

  • Using a TTD curve that includes treatment beyond progression

    If some people stay on initial treatment after progression, the TTD curve can lie above PFS. With 40% on treatment at one year against PFS of 0.368, the off-treatment share is about minus 0.032, and the on-treatment state then holds progressed people valued at progression-free costs and utility.

Sources

  • N-state partition and TTD as a third curve in DSU TSD 19

    Woods B, Sideris E, Palmer S, Latimer N, Soares M. NICE DSU Technical Support Document 19: partitioned survival analysis for decision modelling in health care: a critical review. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2017. Section 2.2 and Table 1: state membership in an N-state model, the first state from its own survival function, each later alive state from its function minus the prior one, and the dead state as 1 minus the final function; section 3.2.4: five appraisals used TTD, PFS and OS to partition patients into on-treatment and progression-free, off-treatment and progression-free, progressed and dead.

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Canonical Identity

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