Modelling prognostic capabilities of tumor size: application to colorectal cancer

Transcription

1 Session 3: Epidemiology and public health Modelling prognostic capabilities of tumor size: application to colorectal cancer Virginie Rondeau, INSERM

2 Modelling prognostic capabilities of tumor size : application to colorectal cancer Agnieszka KROL 1 Loïc FERRER 1 Jean-Pierre PIGNON 2 Cécile PROUST-LIMA 1 Michel DUCREUX 3 Olivier BOUCHÉ 4 Stefan MICHIELS 2 Virginie RONDEAU 1 1 Biostatistics Team, INSERM U897, Bordeaux, 2 INSERM U1018 CESP, Gustave Roussy, U. Paris-Sud, Villejuif, 3 Institut Gustave Roussy, Villejuif, 4 Centre Hospitalier Universitaire, Reims

3 Tumor evaluations in clinical trials Virginie Rondeau BRIO 10 Nov / 19

4 Categorical criteria - RECIST and WHO WHO Bidimensional size, target lesions determined before treatment Progression : >25% increase of one or more lesions and/or appearance of new lesions RECIST (v1.1) Unidimensional size, max 2 lesions per organ and up to 5 total Progression : >20% increase over smallest sum observed (> 5 mm absolute increase) and/or appearance of new lesions 4 categories (Complete Response, Partial Response, Progressive Disease, Stable Disease) dichotomization : response or no response / progression or no progression Virginie Rondeau BRIO 10 Nov / 19

5 Objective Does the continuous tumor size and/or appearance of new lesions enable better prediction of the OS than times of progression? Virginie Rondeau BRIO 10 Nov / 19

6 Observed data For individual i (i = 1,..., N) we observe : Longitudinal biomarker (left-censored) : Y i (t ik ) that represent a true observed value Yi o (t ik ) or a censored value Yi c (t ik ) Recurrences : T ij = min(t ij, C i, T i ) and δ ij = 1 {T ij =T ij } Death : T i = min(c i, T i ) and δ i = 1 {T i =T i } Virginie Rondeau BRIO 10 Nov / 19

7 Joint model for longitudinal data, recurrent events and a terminal event System of linear mixed-effects model and two hazard functions : Yi (t ik ) = m i (t ik ) + ɛ i (t ik ) = X i,l (t ik ) β l + Z i (t ik ) b i + ɛ i (t ik ) r ij (t v i, b i ) = r 0 (t) exp ( ) v i + X ij,r β r + g(b i, t) η r λ i (t v i, b i ) = λ 0 (t) exp ( ) αv i + X i,tβ t + h(b i, t) η t (Biomarker) (Recurrences) (Death) ( ) u i = (b T i, v i ) T B1 0 N (0, B) with B = 0 σv 2 measurement errors iid, ɛ i (t ik ) N (0, σɛ) 2 g(b i, t) and h(b i, t) - subject-specific link function r 0 (t), λ 0 (t) - baseline hazard functions Virginie Rondeau BRIO 10 Nov / 19

8 Estimation Joint marginal likelihood L i (θ) = n i f Y ui (Y i (t ik ) u i ; θ) r i u i k=1 j=1 f T r u i (T ij, δ ij u i ; θ) f T t u i (T i, δ i u i ; θ)f ui (u i ; θ)du i ni - number of biomarker measurement of individual i, r i - number of recurrent events of individual i Parameters to estimate θ = (β l, β r, β t, η r, η t, α, r 0 ( ), λ 0 ( ), B, σ ɛ) Penalized maximum likelihood estimation using Marquardt algorithm Baseline hazard functions approximation using splines : smooth estimation Integrals approximated using Gauss-Hermite quadrature : approach of iterated integrals and Genz algorithm (HRMSYM Fortran subroutine) Virginie Rondeau BRIO 10 Nov / 19

9 Dynamic predictions H i (t) - history of recurrences of individual i until t Y i (t) - history of the biomarker of individual i until t Predicted probability of terminal event T i in a horizon [t, t + w] P(T i t + w T i > t, F i (t), X i ; θ) F i (t) = H i (t), F i (t) = Y i (t) or F i (t) = {H i (t), Y i (t)} Virginie Rondeau BRIO 10 Nov / 19

10 Use two measures of predictive abilities for an internal validation Expected Prognostic Observed Cross-Entropy (EPOCE) Commenges et al., 2012 in a time window, "the lower the better" Brier score (with cross validation) The inverse probability of censoring weighted error estimator (data-based Brier score) Gerds and Schumacher, 2006 comparison of predictions and actual observed events Virginie Rondeau BRIO 10 Nov / 19

11 Clinical trial FFCD Follow-up : Phase III randomized multi-center clinical trial (53 centers in France), 407 patients Tumor evaluation every 8 weeks, max 4 target lesions in 2 dimensions Progression defined with the WHO criteria : more than 25% increase of one or more lesions observed and/or appearance of new lesions (on the best response obtained) Change of line : progression, unacceptable toxicity, decision of investigator Virginie Rondeau BRIO 10 Nov / 19

12 Clinical trial FFCD Objectives : Which of longitudinal biomarker, times of appearance of new lesions or times of progression provide the most accurate prediction the overall survival? To identify the prognostic factors on the outcomes of interest To evaluate the treatment effect Virginie Rondeau BRIO 10 Nov / 19

13 Data Biomarker definition : sum of the longest diameters n ij SLD ij = d ijk, j = 0, 1,..., n i, i = 1,..., 407 k=1 n i {0, 1,..., 17} - number of visits of individual i, n ij {1, 2, 3, 4} - number of target lesions measured during visit j, d ijk - max diameter of lesion k measured during visit j of individual i Left-censoring Virginie Rondeau BRIO 10 Nov / 19

14 Data : FFCD N=402 patients analyzed (53 centers in France) Observed : 6.18 tumor size measurements per patient 1.05 appearance of new lesions per patient 1.82 progression per patient 321 deaths Virginie Rondeau BRIO 10 Nov / 19

15 Results of the trivariate model Biomarker : SLD New lesions Death Covariate Est. (SE) p-value HR (95% CI) HR (95% CI) Intercept 2.81 (0.28) < Time 0.29 (0.12) Treatement (C/S) 0.17 (0.14) ( ) 1.12 ( ) Treatement (C/S) Time 0.40 (0.15) Age (60-69/<60 years) 0.22 (0.17) ( ) 1.10 ( ) Age ( 70/<60 years) 0.02 (0.16) ( ) 1.58 ( ) Sex (Women/Men) 0.27 (0.14) ( ) 1.06 ( ) Baseline WHO PS (1/0) 0.11 (0.15) ( ) 1.87 ( ) Baseline WHO PS (2/0) 0.47 (0.21) ( ) ( ) significant decreasing value of SLD with time (-0.29), and decreasing with time more pronounced for the combination arm (-0.40) strong effect of WHO performance status 2 on the risk of death, new lesions and on repeated tumor size (larger tumor size) no significant associations with gender and age smaller centers had an increase risk of death Virginie Rondeau BRIO 10 Nov / 19

16 Results of the trivariate model Association parameters (with the random effects) strong positive association between tumor size and deaths strong positive association between tumor size and new lesions strong positive association between appearances of new lesions and death Virginie Rondeau BRIO 10 Nov / 19

17 Comparison with the alternative models - predictive ability Comparison of the models in terms of the predictive ability of the overall survival Joint modelling of times of progression and time of death (M1) Joint modelling of times of appearance of new lesions and time of death (M2) Joint modelling of tumor size (SLD) and time of death (M3) Joint modelling of tumor size (SLD), times of appearance of new lesions and time of death (M4) Measures of predictive ability using internal validation Brier score (10-fold cross-validation) EPOCE (CVPOL - approximated cross-validation) Virginie Rondeau BRIO 10 Nov / 19

18 Results - EPOCE Virginie Rondeau BRIO 10 Nov / 19

19 Results - Brier score Virginie Rondeau BRIO 10 Nov / 19

20 Conclusion Advantages of using joint models for simultaneous analysis of prognostic factors Comparison of joint models of different types in terms of predictive accuracy Proposition of a new trivariate joint model FFCD : Improvement of predictive abilities using tumor size and appearance of new lesions Perspectives : implementation of the proposed model into the R package frailtypack extension of the model - adjustment for toxicity, changes of treatment lines application to other clinical trials : external validation of the model, incorporation of information on progression of non-target disease Virginie Rondeau BRIO 10 Nov / 19