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Modeling/Forecasting Patient Recruitment in Multicenter Clinical Trials Using Time-dependent Models

Clinical trials in the modern era are characterized by their complexity and high costs and usually involve hundreds/thousands of patients to be recruited across multiple clinical centers in many countries, as typically it is required to achieve a rather large sample size to prove the efficiency of a particular drug.

As the imperative to enroll vast numbers of patients across multiple clinical centers has become a major challenge, an accurate forecasting of patient recruitment is one of the key factors for the operational success of clinical trials. A classic Poisson-gamma (PG) recruitment model assumes time-homogeneous recruitment rates. However, there can be potential time-trends in the recruitment driven by various factors, for example, seasonal changes and exhaustion of patients on particular treatments in some centers. Recently, a few authors have considered some extensions of the PG model to time-dependent rates under some particular assumptions. In this chapter, a natural generalization of the original PG model to a PG model with non-homogeneous time-dependent rates is introduced. Some tests on homogeneity of the rates (non-parametric using a Poisson model and parametric using a PG model) are considered. The techniques for modeling and simulation of the recruitment using time-dependent model are discussed; for re-projection of the remaining recruitment, it is proposed to use a moving window and re-estimating parameters at every interim time. The results are supported by simulation of some artificial data sets.

1.1. Introduction

Contemporary late-phase clinical trials involve hundreds or even thousands of patients that need to be recruited across multiple clinical centers in many countries, as typically it is required to achieve a rather large sample size to prove the efficiency of a particular drug. Citing Ken Getz, director, Tufts Center for Study of Drug Development, USA, patient recruitment and retention are among the greatest challenges that the clinical research enterprise faces today, and they are a major cause of drug development delays.

To address these challenges, novel approaches and predictive analytic techniques are required for efficient data analysis, monitoring and decision-making.

Let us consider typical multicenter clinical trials used in the pharmaceutical industry where the patients are recruited by many clinical centers in different countries and then randomized to different treatments.

Using Poisson models for modeling patient recruitment in clinical trials is a well-accepted approach. Several papers use Poisson processes in different clinical centers with fixed recruitment rates (Carter et al. (2005); Senn (1997, 1998)).

However, in real trials, different centers typically have different capacity and productivity, and the recruitment rates vary. To model this variation, Anisimov and Fedorov (2007) introduced a Poisson-gamma (PG) model, where the patient arrival is modeled using Poisson processes with some rates and the variation in rates in different centers is modeled using a gamma distribution.

This model can be described in the framework of the empirical Bayesian approach where the prior distribution of the rates is a gamma distribution with parameters that are evaluated either using historical data or data provided by study investigators. In Anisimov and Fedorov (2007), a maximum likelihood technique was proposed for estimating the parameters of the rates and the Bayesian technique for adjusting the posterior distribution of the rates at any interim time using recruitment data in the individual centers. Various applications to real trials are considered in Anisimov and Fedorov (2007), Anisimov et al. (2007) and Anisimov (2011a).

This technique was developed further to account for random delays and the closure of clinical centers (Anisimov (2011a, 2020)) and for modeling events in event-driven trials (oncology) (Anisimov (2011b, 2020); Anisimov et al. (2022)).

Note that independently, Gajewski et al. (2008) later considered a similar approach to modeling recruitment as a Poisson process with gamma distributed rate, however their model is applicable to only one clinical center.

The PG model was also used in Mijoule et al. (2012) to consider some extensions related to other distributions of the recruitment rates and sensitivity analysis, in Minois et al. (2017) to evaluate the duration of recruitment process when historical trials are available, and in Bakhshi et al. (2013) for the evaluation of parameters of a PG model using meta-analytic techniques for historic trials. A survey on using mixed Poisson models is provided in Anisimov (2016).

It is also worth mentioning the recent investigations devoted to centralized statistical monitoring and forecasting recruitment performance and also forecasting patient recruitment under various restrictions and creating an optimal cost-efficient recruitment design (Anisimov and Austin (2020, 2022, 2023); Anisimov (2023)).

There are also other approaches to recruitment modeling described in the literature; however, they mainly deal with the analysis of the global recruitment and therefore have some limitations. These approaches typically require rather large number of centes and patients and cannot be applied to predicting recruitment on center/country level (see survey papers by Barnard et al. (2010), Heitjan et al. (2015) and Gkioni et al. (2019)).

Currently, the originally developed PG model (Anisimov and Fedorov (2007); Anisimov (2011a)) has gained world-wide recognition and in some recent papers is now called "one of the most popular techniques" and an "industry-standard" model.

This model assumes that patient recruitment rates do not change over time. However, in real trials there can be some time-trends, seasonable changes, and recruitment can change in some countries over time due to different reasons, for example, a slowdown in recruitment near the end of a trial.

Therefore, to capture these situations, in some recent papers (Lan et al. (2019); Best et al. (2022); Perperoglou et al. (2023); Turchetta et al. (2023); Urbas et al. (2022)), the standard PG model was extended to the cases where the recruitment rates can be time dependent.

Lan et al. (2019) proposed a non-homogeneous Poisson process model that allows for staggered center activation and heterogeneity within centers modeled by a gamma distribution and assuming that after a period of steady recruitment, the center mean recruitment rate gradually declines as a negative exponential with some coefficient. They calculated the posterior distribution and considered some applications to real trials. Urbas et al. (2022) proposed a more general time-dependent PG model that allows for a wider range of recruitment rate functions. The model is fitted using a maximum likelihood approach and is used to select the best model among a set of candidate models. Perperoglou et al. (2023) compared the performance of the standard PG model to two time-dependent models: the models proposed in Lan et al. (2019) and Urbas et al. (2022). The study found that the time-dependent models outperformed the PG model in terms of prediction accuracy, especially in trials with time-varying recruitment rates. Best et al. (2022) considered a standard PG model as the starting point and developed a more flexible version that introduces variation in rates over time as a function of COVID-19-dependent covariates, which is implemented in a fully Bayesian probabilistic framework. Turchetta et al. (2023) proposed a time-dependent PG model allowing recruitment rates to vary over time through B-splines that is fitted using a Bayesian approach. The model was evaluated in a simulation study and found to perform well in a variety of scenarios.

All five papers above contribute to the advancement of recruitment modeling in multicenter clinical trials by extending the standard time-homogeneous PG model to accommodate time-dependent recruitment rates.

The flexibility of the original PG model allows a rather natural extension to model time-dependent rates on the center level. However, the main question here is: What type of time-dependence to use, especially, how to detect the change time-point of the behavior of the rates, and how to test this dependence in real trials? Thus, the choice of a suitable time-dependent model is trial specific.

In this chapter, the authors consider a natural generalization of a PG model to a time-dependent model using a general framework of a standard PG model, which is similar to the way considered in Urbas et al. (2022). The basic methodology of using a PG model with time-dependent rates is developed by the first author in Anisimov (2024). This chapter extends some results of Anisimov (2024); in particular, a new PG criterion is investigated for testing the recruitment rates for homogeneity. In Anisimov (2024), a novel analytic methodology is also proposed for modeling/forecasting patient recruitment using the recent results in Anisimov and Austin (2020) about a PG approximation of the sums of PG processes in different centers that can be applied to modeling recruitment even in not so large countries/regions, which is also described in this chapter for completeness of presentation.

Note that this...