ODE-free survival modeling with full pharmacokinetic profiles in the IMPRES-M framework
Lorenzo Cifelli, Csaba Katai, Jeroen Elassaiss-Schaap
PD-value, Utrecht, the Netherlands
Introduction:
Time to event analysis (TTE) characterizes the probability that an event, such as a disease progression, occurs, and in pharmacometrics TTE is applied to understand the effect of the drug on a disease. A pharmacometric time-to-event model typically consists of a parametric distribution, which characterizes the hazard function using a baseline hazard and covariate effects [1]. In particular, it is reasonable to assume that a drug has a continuous effect on the disease over time. It is therefore preferred to include exposure as a continuous time-varying covariate. Estimation of event probability requires integration of the hazard function. When exposure is defined using an ordinal differential equation (ODE) system, this operation becomes challenging due to numerical instability.
A solution to avoid numerical integration of ODEs would be to use explicit solutions. Even if a standard compartmental model characterizes the exposure as a sum of negative exponential functions, employing an ODE solver is preferred due to the complexity of derivation of explicit form of the integral and the length of its solution; in other cases an explicit solution may not be available. Hence, it is common practice to compute summary statistics (e.g., minimum, maximum, average) of the exposure over time intervals and include them as time-invariant covariates. However, this approach, though often convenient, may result in suboptimal models, certainly in the case of missed doses, changing schedules or drug holidays.
Objectives:
The objective therefore is to derive an ODE-free methodology based on the IMPRES-M framework, a non-parametric regression approach, to estimate a time-to-event model where the exposure is treated as continuous function in an efficient and accurate manner.
Methods:
Following common approaches, we considered a straightforward time-to-event model, where the hazard function is characterized by a constant baseline hazard and a linear effect of exposure, where exposure is treated as a continuous function of time. Employing the IMPRES-M framework [2], the exposure function is estimated from data by non-parametric regression, and baseline hazard and the linear effect of exposure is computed by maximum likelihood estimation.
To assess the goodness of fit of the proposed methodology, we simulated time-to-event data with continuous exposure effect for a limited period of treatment. The baseline hazard was kept constant and exponential right censoring was applied. The survival curve yielded by our non-linear regression method was compared with the Kaplan-Meier (KM) estimator. We also compared it with an alternative model where the hazard was derived using average concentration, a summary statistic. This comparison required the addition of a placebo curve.
Results:
The IMPRES-M framework (patent pending) allows to estimate the exposure function as a linear combination of B-spline functions. The integral of a B-spline function has a closed form, and this feature was exploited to derive an explicit formula for the integration of the hazard function.
This offers a computational advantage compared to solving non-linear ODEs , as the integral involves linear algebraic operations, resulting in improved numerical efficiency and stability. The derived model was able to describe the simulated data adequately, i.e. within the 95-% CI of the KM estimate. In addition, the proposed approach outperforms the alternative approach based on summary statistic since the latter fails to follow the continuous effect of exposure on the probability function, especially when exposure changes rapidly.
Conclusions:
We propose a new methodology based on impulse-response regression of the exposure function to introduce it as a continuous time-varying covariate on the hazard function in TTE modeling.
While a straightforward model for the hazard function was selected as a starting point, the methodology can be extended to different baseline hazard and non-linear exposure-effect relationships. It would be additionally useful to evaluate the possibility of including PK and other covariates in the context of joint modeling, as has been done in other contexts [3].
References:
[1] Holford, Nick. "A time to event tutorial for pharmacometricians." CPT: pharmacometrics & systems pharmacology 2.5 (2013): 1-8.
[2] Elassaiss-Schaap J et al., PAGE 2024, Abstract title: “Construction of IMPRES-M, a non-parametric impulse-response modeling method, in the context of varying pharmacokinetic profiles”
[3] Andrinopoulou, E. R., Eilers, P. H., Takkenberg, J. J., & Rizopoulos, D. (2018). Improved dynamic predictions from joint models of longitudinal and survival data with time-varying effects using P-splines. Biometrics, 74(2), 685-693.