RECRUITMENT IN MULTICENTRE TRIALS: PREDICTION & ADJUSTMENT Vladimir Anisimov, Darryl Downing, Valerii Fedorov Statistical Quantitative Sciences GlaxoSmithKline MODA 8 Almagro, Spain 4–8 June 2007
INTRODUCTION Patient recruitment modeling is an essential part of drug development process: Existing techniques of recruitment planning are mainly deterministic and do not account for various uncertainties. A large proportion of trials ( > 50%) fails to complete the enrolment in time. Monte Carlo simulation cannot solve the basic problems: it requires a substantial time and realistically cannot provide the solutions of multivariate optimal problems, calculate critical probabilities, etc. Advanced statistical methodology is developed in Research Statistics Unit. Anisimov, Downing, Fedorov MODA8 Almagro, Spain 4–8 June 2007 1
RECRUITMENT MODELING Multicentre clinical trial: n patients to be recruited by N clinical centres. Several sources of stochasticity: • stochastic fluctuations of recruitment process in each centre; • variation in recruitment rates between centres; • delays in centre initiation. Model: patients arrive according to Poisson processes with rates λ i – common assumption (Senn; Anisimov & Fedorov). number of patients n i ( t ) = Π λ i ( t − u i ) , t > u i – rates λ i vary between centres and not certain, – delays u i can be random variables. Anisimov, Downing, Fedorov MODA8 Almagro, Spain 4–8 June 2007 2
Modeling recruitment rates between centres Study A: – a typical example. Data: ¯ ν = ( ν (1) , ν (2) , ... ) , ν ( j ) – number of centres with j patients. n=629 pts, N=91 centres, ¯ ν = (7 , 11 , 8 , 8 , 9 , 8 , 9 , 7 , 2 , 4 , 1 , 3 , 3 , 4 , 0 , 0 , 2 , 1 , 1 , 2 , 1 , 0 , 0 , 0 , ... ) Variation in the number of patients between centres is substantial 7 centres – 1 pt, 11 centres – 2 pts,.., 2 centres – 20 pts, 1 centre – 21 pts. Poisson-gamma recruitment model: Rates λ i are modeled as a sample from a gamma distribution Ga ( α, β ) with unknown parameters ( α - shape parameter, β - rate parameter) (Bayesian setting or hierarchic random effects model) Anisimov, Downing, Fedorov MODA8 Almagro, Spain 4–8 June 2007 3
Validation of recruitment model Empirical data and mean number of centres 14 12 10 8 6 4 2 0 5 10 15 20 j Mean + 2 SD − dashed line Graph of real data (vector ¯ ν ) – step-wise green line, the mean number of centres with j patients (theoretical curve) – solid blue line, the mean + 2sd – dashed red line. Theoretical curve is constructed using parameters estimated by real data. Poisson-gamma model adequately reflects real data: working model for modeling patient recruitment when the number of centres is large enough ( N ≥ 20 ). If a few centres initiated, estimate rates individually in each centre. Anisimov, Downing, Fedorov MODA8 Almagro, Spain 4–8 June 2007 4
Recruitment modeling n ( t, N ) = � N i =1 n i ( t ) - total number of recruited patients up to time t ; T ( n, N ) – recruitment time. If all centres are initiated at time t 0 = 0 , then t k β αN 1 P ( n ( t, N ) = k ) = ( t + β ) k + αN , k = 0 , 1 , .. (1) k B ( k, αN ) - negative binomial distribution, B ( a, b ) – beta function. T ( n, N ) has a Pearson type VI distribution with p.d.f. t n − 1 β αN 1 p ( t, n, N, α, β ) = ( t + β ) n + αN , t ≥ 0 . (2) B ( n, αN ) Anisimov, Downing, Fedorov MODA8 Almagro, Spain 4–8 June 2007 5
Recruitment modeling Solution of dual problems: 1. Given n and N find the least t ∗ such that P ( n ( t, N ) ≥ n ) ≥ p . (3) 2. Given T and n find the least N ∗ such that P ( T ( n, N ) ≤ T ) ≥ p , (4) where p is some prescribed probability ( p = 0 . 9 ). Solutions can be found numerically or using normal approximation. Anisimov, Downing, Fedorov MODA8 Almagro, Spain 4–8 June 2007 6
Normal approximation At large N and n , T ( n, N ) − M ( n, N ) ≈ N (0 , 1) , S ( n, N ) where at αN > 2 , βn M ( n, N ) = E [ T ( n, N )] = αN − 1 , β 2 n ( n + αN − 1) S 2 ( n, N ) = Var [ T ( n, N )] = ( αN − 1) 2 ( αN − 2) , and N ∗ can be found from: M ( n, N ) + z p S ( n, N ) = T, where z p is a p -quantile of a normal distribution. Functions M ( n, N ) and S ( n, N ) are monotonically decreasing in N and a unique solution N ∗ exists. Anisimov, Downing, Fedorov MODA8 Almagro, Spain 4–8 June 2007 7
Delays in centre initiation General case – different times of centres initiation { u i } : n ( t, N ) is a non-homogeneous Poisson process, cumulative rate in the interval [0 , t ] N � λ i · [ t − u i ] + , Σ( t ) = i =1 Combined analytic and simulation tools are created. CDF of T ( n, N ) : P ( T ( n, N ) ≤ t ) = P (Π Σ( T ) ≥ n ) = P ( Ga ( n, 1) ≤ Σ( t )) . { u i } can be viewed as a sample from a random population – in each centre two mixing levels: randomness in rate and in centre initiation date. CDF can be calculated very quickly using Monte Carlo simulation. Anisimov, Downing, Fedorov MODA8 Almagro, Spain 4–8 June 2007 8
RECRUITMENT PREDICTION Two basic stages of prediction: 1. initial stage (before study start): rates are given by site managers or estimated using historical data, time-intervals [ a i , b i ] for centre initiation are set. Task: predict # of recruited patients n ( t ) over time and recruitment time. 2. intermediate (or ongoing) stage: recruitment data at some interim point t 1 are given. Task: using data adjust predictions and # of centers if required. Anisimov, Downing, Fedorov MODA8 Almagro, Spain 4–8 June 2007 9
Initial stage of prediction • Delays u i in centre initiations are generated in given intervals [ a i , b i ] using a uniform distribution; • mean [ n ( t )] and sd [ n ( t )] are calculated in a closed form. • Confidence bands for n ( t ) and for recruitment time are created using normal approximation. • The number of centres required to complete the study by a certain time with a pre-specified probability can be calculated. • Optimal design problem: minimizing utility function accounting for risk per study delay and costs per centres can be solved. Anisimov, Downing, Fedorov MODA8 Almagro, Spain 4–8 June 2007 10
Initial prediction – mean and confidence boundaries Predicted number of patients Mean and predicted boundaries 140 120 120 100 100 80 80 60 60 40 40 20 20 0 0 0 1 2 3 4 5 6 7 8 9 0 2 4 6 8 10 Time Mean and confidence boundaries K=20, rate=1, a=1,b=5 Target 100 pts, 20 centres, mean[rate]=1, intervals [0 , 5] and [1 , 5] . Predicted mean and 90% confidence band for patient recruitment and recruitment time (left), 5 simulated trajectories of the realistic recruitment process and 95% boundaries (right). Anisimov, Downing, Fedorov MODA8 Almagro, Spain 4–8 June 2007 11
Optimal number of centres Find minimal number of centres N ∗ required to complete recruitment before time T with probability p (Problem 2): P ( T ( n, N ) ≤ T ) ≥ p . At large N , it is equivalent to M ( T, N ) − n = z p . � n + S 2 ( T, N ) N ∗ can be found numerically. Anisimov, Downing, Fedorov MODA8 Almagro, Spain 4–8 June 2007 12
Prediction of ongoing study – intermediate stage Advanced statistical technique Recruitment data in each centre at interim time t 1 : – date of centre initiation u i , – # of recruited patients k i . Several steps: 1. Estimation: τ i = t 1 − u i – duration of active recruitment, k i – negative binomial distribution with parameters ( α, τ i /β ) . Log-likelihood function L ( α, β ) : N N � � ln Γ( k i + α ) − N ln Γ( α ) − K 1 ln β − ( k i + α ) ln(1 + τ i /β ) , i =1 i =1 where K 1 = � N i =1 k i . Anisimov, Downing, Fedorov MODA8 Almagro, Spain 4–8 June 2007 13
Prediction of ongoing study – intermediate stage 2. Re-estimation of recruitment rates (empirical Bayesian approach): – for active centres - using real data, – for centres to be added in future - using parameters of the model. Active centres – re-estimated posterior rates: � λ i = Ga ( α + k i , β + τ i ) , λ i = s 2 i = ( α + k i ) / ( β + τ i ) 2 , E � λ i = m i = ( α + k i ) / ( β + τ i ) , Var � ( α, β ) – estimated values. Predicted over time Mean [ n ( t, N )] and V ar [ n ( t, N )] are constructed using re-estimated m i and s 2 i . Anisimov, Downing, Fedorov MODA8 Almagro, Spain 4–8 June 2007 14
Prediction of ongoing study – intermediate stage 3. Remaining recruitment time. Re-estimated overall rate (random variable) N � � � Λ = λ i , i =1 predicted remaining time T ( K 2 , N ) = Ga ( K 2 , 1) / � � Λ . � If τ i ≡ τ , Λ = Ga ( αN + K 1 , β + τ ) , and Ga ( K 2 , 1) � T ( K 2 , N ) = Ga ( αN + K 1 , 1)( β + τ ) . – Pearson type VI distribution. If τ i are different, use approximation or simulation. Anisimov, Downing, Fedorov MODA8 Almagro, Spain 4–8 June 2007 15
Adaptive adjustment Find optimal number of centres needed to complete recruitment before deadline with probability p (Problem 2). If relation: P ( � T ( K 2 , N ) ≤ T − t 1 ) ≥ p is true – OK. 4. Adaptive adjustment. If it’s not true: calculate the number M of additional centres to achieve: P ( � T ( K 2 , N + M ) ≤ T − t 1 ) ≥ p. Anisimov, Downing, Fedorov MODA8 Almagro, Spain 4–8 June 2007 16
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