Simulation study to evaluate the identifiability of a non-linear multi-level joint model for the follow-up of patients with cancer

Maxime Beaulieu[1], Jérémie Guedj[1], Marion Kerioui[1]

[1] Université de Paris, IAME, INSERM F-75018 Paris, France

Introduction:

In oncology, the sum of longest diameters (SLD) is commonly used to follow the dynamics of patient tumor size over time. However, the SLD has several limitations, including neglecting the heterogeneity of tumor kinetics, especially in response to treatment, which can be explained by the location of the target lesion[1].  Accounting for the inter-lesion variability by following lesions at the individual level could help to better anticipate the treatment response.

For this purpose, using data from patients with metastatic cancer[2], a semi-mechanistic model was developed to follow the non-linear tumor kinetics and to estimate its impact on the patient's risk of death, adjusted by the lesion's host organ[3]. However, this model is complex including a non-linear mixed effects model (NLME) with two levels of random effects, at patient and lesion level to characterise both within and between patient variability. This raises the question of the amount of data needed, i.e. the number of patients or the number of target lesions, for accurate parameter estimation. A simulation study allows us to generate different populations with a variable amount of available information, allowing us to study the numerical identifiability and its estimation quality in these different situations.

Objectives: 

1. Validate the numerical identifiability of the model in a clinical trial context
2. Assess the amount of data needed to obtain an unbiased estimate of the endpoints
3. Test the limits of the model in more constraining contexts

Methods: 

The joint model consists of a longitudinal sub-model corresponding to the tumor dynamics of each target lesion present in each organ of each patient (described by a NLME) and a survival sub-model. The joint model admits a link parameter between these two sub-models. The joint modelling avoids bias in the estimation of the longitudinal and link parameters[4].

To achieve these objectives, we simulated four scenarios. i) A large population (N=300) inspired by a real clinical trial, ii) the same situation but with a reduced population (N=150), iii) a reduced population (N=150) but with twice as many individual target lesions followed, iv) a scenario to evaluate the capacity of the model to differentiate the inter-patient variability from the inter-lesion variability in a restricted population (N=150). In all scenarios, patients had up to five target lesions (up to ten in the third scenario) located in the lymph nodes, lungs, liver and bladder. For each scenario, we simulated fifty datasets.

Then the model was run on each dataset. Model inference was done in a Bayesian framework, using the HMCNUTS[5] algorithm developed on Stan[6]. The convergence of the Markov chains was assessed using the Rhat criterion[7], the estimation error of the parameters was quantified by the relative estimates errors (REE), and the estimation accuracy was evaluated using 95% coverage rates[8] for each parameter. Individual fits were also constructed to check the goodness of fit of the model to the simulated data in the different scenarios.

Results: 

Of the fifty inferences in the first three scenarios, more than 90% of the data sets validated our criterion for Markov chains to converge in a reasonable number of iterations. The fourth scenario had more difficulty converging with a rate of 70%. The REE associated with all parameters are negligible in the first scenario, we observed a slight deterioration of the parameter estimates when the population is halved but this is mitigated when the number of target lesions followed within each patient is increased. As for the coverage rates, we observed a good accuracy in the estimation of the parameters, and this, in all the scenarios. Finally, the individual fits show a good fit of the model to the observations in all scenarios.

Conclusions: 

In this simulation study, we did not encounter any major convergence problems, even when the population was small. The model does not have any numerical identifiability problems, there is no bias in the estimation of the parameters as well as a high accuracy. This work shows that the number of patients included in a clinical trial can be reduced by increasing the number of target lesions followed without losing. There would therefore be an advantage to question the RECIST1.1 criterion[9] limiting the number of target lesions followed to five because following more lesions is less expensive than adding more patients to the study.

References:

1. Kerioui M, Desmée S, Mercier F, Lin A, Wu B, Jin JY, et al. Assessing the impact of organ-specific lesion dynamics on survival in patients with recurrent urothelial carcinoma treated with atezolizumab or chemotherapy. ESMO Open. 2022;7: 100346. doi:10.1016/j.esmoop.2021.100346
2. Powles T, Durán I, van der Heijden MS, Loriot Y, Vogelzang NJ, De Giorgi U, et al. Atezolizumab versus chemotherapy in patients with platinum-treated locally advanced or metastatic urothelial carcinoma (IMvigor211): a multicentre, open-label, phase 3 randomised controlled trial. The Lancet. 2018;391: 748–757. doi:10.1016/S0140-6736(17)33297-X
3. Kerioui M, Desmée S, Bertrand J, Mercier F, Jin JY, Bruno R, et al. Nonlinear multilevel joint model for individual lesion kinetics and survival to characterize intra-individual heterogeneity in patients with advanced cancer. 2022. Available: https://hal.archives-ouvertes.fr/hal-03695061
4. Rizopoulos D. Joint Models for Longitudinal and Time-to-Event Data. 0 ed. Chapman and Hall/CRC; 2012. doi:10.1201/b12208
5. Hoffman MD, Gelman A. The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo. arXiv; 2014. Available: http://arxiv.org/abs/1111.4246
6. Carpenter B, Gelman A, Hoffman MD, Lee D, Goodrich B, Betancourt M, et al. Stan : A Probabilistic Programming Language. J Stat Softw. 2017;76. doi:10.18637/jss.v076.i01
7. Gelman A. Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper). Bayesian Anal. 2006;1: 515–534. doi:10.1214/06-BA117A
8. Morris TP, White IR, Crowther MJ. Using simulation studies to evaluate statistical methods. Stat Med. 2019;38: 2074–2102. doi:10.1002/sim.8086
9. Eisenhauer EA, Therasse P, Bogaerts J, Schwartz LH, Sargent D, Ford R, et al. New response evaluation criteria in solid tumours: Revised RECIST guideline (version 1.1). Eur J Cancer. 2009;45: 228–247. doi:10.1016/j.ejca.2008.10.026

Reference: PAGE 31 (2023) Abstr 10698 [www.page-meeting.org/?abstract=10698]

Poster: Drug/Disease Modelling - Oncology