Using Stochastic Analytics in Subject Recruitment in Clinical Trials
One of the biggest challenges in planning and conducting clinical trials is the modeling and predicting of subject accrual. About 20% of all trials are terminated or closed due to inability to recruit subjects. Additionally, about 80% of completed trials fail to meet recruitment goals.1 Predicting subject accrual for clinical trials has long been a topic of significant interest. It affects resource allocation and budgeting not only during the planning of a clinical trial, but also during the operation of an ongoing trial. There are several factors that contribute to inadequately predicting subject enrollment, including: 1) not appropriately assessing recruitment rates for potential sites; 2) underestimating the effect of eligibility criteria on recruitment; 3) length of study; 4) drug supply variations; and 5) visit schedules. Any improvement in the planning and monitoring of a trial may reduce costs. Moreover, a trial's inability to recruit the planned sample size undermines the ability to draw scientifically sound inferences based on the study results. Employing statistical methods that can 1) render early warning signs that a trial will not reach its recruitment goal, based on observed accrual, and 2) give the flexibility to test certain situations for improving recruitment before implementation (such as adding sites), can prove invaluable for a Sponsor.2
Traditionally, for predicting recruitment, constant or piecewise constant assumptions are used (Figure 1). The conventional practice for enrollment monitoring assumes that accrual is constant; therefore, it uses a linear extrapolation to project future accrual. This practice assumes the accrual process to be free of random variations. Real trial accrual, however, rarely has an enrollment pattern that is constant or the shape of a step-function. The observed number of subjects can vary greatly from day to day or month to month (depending on the unit of time that is being modeled), as well as from site to site, during the recruitment phase of a trial (Figure 2). Typically, total enrollment increases in the beginning of a trial as the sites start to enroll, at each site and overall (Figure 3). Even after all sites are actively recruiting and the overall accrual rate for each site reaches capacity, the day to day enrollment rate still varies. The average enrollment rate, however, can remain constant or oscillate. These nuances in recruitment patterns across time and between sites should be considered when deriving a study's initial recruitment prediction, as well as when using observed recruitment counts to improve recruitment predictions during a study.
There are two major classes of statistical models that are used for the prediction of accrual. One uses Brownian motion and the other one is based on the Poisson process. Browning motion was derived from the observation of pollen grains moving randomly in water by the Scottish botanist Robert Brown, and it is now used in many fields of study to describe a variety of phenomena. In a Brownian motion, variation of the recruitment process is modeled by assuming independent increment (i.e., what occurs in one increment of time does not influence what occurs in another increment of time) structure. It models variance (i.e., measures how far the data is spread out from its mean) independent of expected or predicted arrival. Several variations of the Brownian motion model have also been proposed,3 such as an extension of the Brownian motions, called the fractional Brownian motion model, in which the time increments are no longer assumed to be independent, and the correlation between the time increments are assumed to be negative or positive (i.e., increasing or decreasing over time intervals). If the parameter is 0.5, then the model reduces to simple Brownian motion.4 The fractional Brownian motion model may be more appropriate if one is interested in cumulative accrual because it can serve as a continuous approximation of the underlying enrollment process, which is discrete, or countable.
A Poisson process models count data. One of the main assumptions of a Poisson process is that the number of events (subjects enrolling in a study) in nonoverlapping intervals are independent for all intervals. Since the number of subjects enrolled in a trial is an integer that increases over time with random daily increments, its characteristics can be naturally captured by a Poisson process.3
As with the Brownian motion models, several applications of the Poisson process have been proposed to model accrual.5-9 These models incorporate the Bayesian approach, which allows for the use of prior knowledge (knowledge of the accrual rates per site by way of historical data or pure guessing). A particular Poisson process model, the Poisson-Gamma model, introduces a multisite accrual model, which assumes the underlying accrual rate to be constant at each site while allowing for staggered site start-up times. In the special case of all sites starting at the same time, the overall enrollment across sites reduces to a homogenous Poisson process, where the overall accrual rate is the sum of the rates across all sites. However, in a more general setting, when all sites do not start at the same time, the underlying overall enrollment rate assumes the shape of a step-function. This model becomes a double stochastic model, varying accrual both over time and between sites. Notably, this approach addresses the issues with accrual that are present in many multisite clinical trials (e.g., possible staggered site start-up time and differential enrollment capacity across sites). This model can be further modified to accommodate for non-constant accrual rate over time using a nonhomogeneous Poisson-Gamma process.
The Poisson-Gamma model is very flexible and can incorporate multi-regional study designs, seasonal trends, added sites over time, as well as pauses in recruitment. Poisson-Gamma models can aid in determining initial recruitment rates and monitor recruitment during a trial. Different scenarios for study design can be used to determine how overall recruitment rates and duration of study are affected. Currently, the homogenous and nonhomogeneous Poisson processes are more widely used for modeling accrual. This is because these models better utilize all available accrual information and directly address the multiregional and multisite nature of clinical trials, as well as other factors that can affect recruitment over the course of the trial.
TRI recognizes that random fluctuation is inherent in the accrual process. TRI statisticians have experience in using and applying stochastic approaches in modeling subject accrual in clinical trials. Our knowledge in these techniques enables us to accumulate efficiencies in recruitment predictions and monitoring, greatly increasing the chances of saving money and meeting trials goals and objectives.
- Anonymous, Subject recruitment and retention: barriers to success. Editorial, Applied Clinical Trials Website (April 1, 2004) http://www.appliedclinicaltrialsonline.com/subject-recruitment-and-retention-barriers-success
- Modeling and prediction of subject accrual and event times in clinical trials: a systematic review; Xiaoxi Zhang, Qi Long; Clinical Trials 2012; 9:681-688.
- Ross SM. Stochastic Processes (2nd edn). Wiley, New York, 1995.
- Zhang Q, Lai D. Fractional Brownian motion and long term clinical trial recruitment. J Stat Plan Infer 2010; 141: 1783-88. doi: 10.1016/j.jspi.2010.11.028.
- Gajewski BJ, Simon SD, Carlson SE. Predicting accrual in clinical trials with Bayesian posterior predictive distributions. Stat Med 2008; 27: 2328–40. doi: 10.1002/sim.3128
- Anisimov VV, Fedorov VV. Modelling, prediction and adaptive adjustment of recruitment in multicentre trials. Stat Med 2007; 26: 4958-75. doi: 10.1002/sim.2956.
- Anisimov, V.V.: Recruitment modeling and predicting in clinical trials. Pharmaceutical Outsourcing 10, 44-48 (2009)
- Zhang X, Long Q. Stochastic modeling and prediction for accrual in clinical trials. Stat Med 2010; 29(6): 649-58. doi: 10.1002/sim.3847.
- Zhang X, Long Q. Bayesian modeling and prediction of patient accrual in multi-regional clinical trials. 2012 Technical Report 12-01, Department of Biostatistics and Bioinformatics, Emory University.
About the Author
Chandra S. Thames is the Director of Biostatistics and Clinical Data Management at TRI. She has over 21 years of statistical analysis experience including 17 years in clinical research. She has conducted collaborative research for a pharmaceutical company with an academic research university on using Stochastic processes for predicting subject recruitment in clinical trials. She has also developed standards and improved statistical methodologies for assisting in predicting/forecasting subject recruitment.