GlaxoSmithKline MODA 8 Almagro, Spain 48 June 2007 INTRODUCTION - - PowerPoint PPT Presentation

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 λi – common assumption (Senn; Anisimov & Fedorov). number of patients ni(t) = Πλi(t − ui), t > ui – rates λi vary between centres and not certain, – delays ui can be random variables.

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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 λi 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)

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Validation of recruitment model

5 10 15 20 2 4 6 8 10 12 14

Empirical data and mean number of centres

Mean + 2 SD − dashed line j

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.

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Recruitment modeling n(t, N) = N

i=1 ni(t) - total number of recruited patients up to time t;

T(n, N) – recruitment time. If all centres are initiated at time t0 = 0, then

P(n(t, N) = k) =

1 kB(k, αN) tkβαN (t + β)k+αN , k = 0, 1, .. (1)

• negative binomial distribution, B(a, b) – beta function.

T(n, N) has a Pearson type VI distribution with p.d.f. p(t, n, N, α, β) = 1

B(n, αN)

tn−1βαN (t + β)n+αN , t ≥ 0. (2)

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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.

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Normal approximation At large N and n, T(n, N) − M(n, N) S(n, N) ≈ N (0, 1), where at αN > 2, M(n, N) = E[T(n, N)] = βn αN − 1, S2(n, N) = Var[T(n, N)] = β2n(n + αN − 1) (αN − 1)2(αN − 2), and N∗ can be found from: M(n, N) + zpS(n, N) = T, where zp 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.

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Delays in centre initiation General case – different times of centres initiation {ui}: n(t, N) is a non-homogeneous Poisson process, cumulative rate in the interval [0, t] Σ(t) =

N

• i=1

λi · [t − ui]+, Combined analytic and simulation tools are created. CDF of T(n, N):

P(T(n, N) ≤ t) = P(ΠΣ(T) ≥ n) = P(Ga(n, 1) ≤ Σ(t)).

{ui} 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.

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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 [ai, bi] 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 t1 are given. Task: using data adjust predictions and # of centers if required.

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Initial stage of prediction

• Delays ui in centre initiations are generated in given intervals [ai, bi]

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.

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Initial prediction – mean and confidence boundaries

2 4 6 8 20 40 60 80 100 120

Predicted number of patients

Mean and confidence boundaries Time 1 3 5 7 9

2 4 6 8 10 20 40 60 80 100 120 140

Mean and predicted 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).

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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

• n + S2(T, N)

= zp. N∗ can be found numerically.

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Prediction of ongoing study – intermediate stage Advanced statistical technique Recruitment data in each centre at interim time t1: – date of centre initiation ui, – # of recruited patients ki. Several steps:

• 1. Estimation:

τi = t1 − ui – duration of active recruitment, ki – negative binomial distribution with parameters (α, τi/β). Log-likelihood function L(α, β):

N

• i=1

ln Γ(ki + α) − N ln Γ(α) − K1 ln β −

N

• i=1

(ki + α) ln(1 + τi/β), where K1 = N

i=1 ki.

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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:

• λi = Ga(α + ki, β + τi),

E

λi = mi = (α + ki)/(β + τi), Var λi = s2

i = (α + ki)/(β + τi)2,

(α, β) – estimated values. Predicted over time Mean[n(t, N)] and V ar[n(t, N)] are constructed using re-estimated mi and s2

i .

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Prediction of ongoing study – intermediate stage

• 3. Remaining recruitment time.

Re-estimated overall rate (random variable)

• Λ =

N

• i=1
• λi,

predicted remaining time

• T(K2, N) = Ga(K2, 1)/

Λ. If τi ≡ τ,

• Λ = Ga(αN + K1, β + τ), and
• T(K2, N) =

Ga(K2, 1) Ga(αN + K1, 1)(β + τ). – Pearson type VI distribution. If τi are different, use approximation or simulation.

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Adaptive adjustment Find optimal number of centres needed to complete recruitment before deadline with probability p (Problem 2). If relation:

P(

T(K2, N) ≤ T − t1) ≥ p is true – OK.

• 4. Adaptive adjustment.

If it’s not true: calculate the number M of additional centres to achieve:

P(

T (K2, N + M) ≤ T − t1) ≥ p.

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Adaptive adjustment Let τi ≡ τ, and all new centres initiated after time t1 with the same delay d. The number of additional centres M is: M ≥ A + Bz2

p/2 + zp

• AB + Q + B2z2

p/4

αB , A = K2 − (αN + K1)(T − t1)/(β + τ), B = (T − t1 − d)/β, Q = K2 + (αN + K1)(T − t1)2/(β + τ)2.

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STUDY DESIGN 5: Find optimal # of centres and patients accounting for study costs and risk per delay: (n∗, N∗) = arg min R(n, N) given P(T(n, N) ≤ T) ≥ p. Example: R(n, N) = C1n + C2N + C3E[(T(n, N) − T)]+, C1 and C2 – patient and centre costs, C3 - cost per excess the recruitment deadline. For particular recruitment scenarios R(n, N) can be calculated numerically

• r using Monte Carlo simulation.

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CASE STUDY – validation of predictive technique Real completed study: initial plan: 366 patients should be recruited by 75 centres, finally, 109 centres were initiated. Retrospective analysis: study was divided on 4 periods, each 64 days: 1st period – 15 recruited patients, after 2nd period – 118 pts, after 3d period – 254 pts. Prediction was constructed after each period (using previous data) and compared with real course of recruitment.

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CASE STUDY – predictions

100 150 200 250 300 350 400 50 100 150 200 250 300 350

Prediction after 1st period

Time # of patients 100 150 200 250 300 350 50 100 150 200 250 300 350

Adjusted prediction after 2nd period

Time # of patients

Predicted number of patients over time after 1st and 2nd periods: mean and 95% confidence boundaries. Target - 366 patients.

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CASE STUDY – adaptive adjustment

100 150 200 250 300 350 50 100 150 200 250 300 350

Prediction and adjustment after 1st period

Time # of patients

Real recruitment was going slower. Recommendations on the adaptive adjustment after 1st period to reach the planned date: to reach target on average – add 50 new centres (can be added only with delay ∼ 2 months). In reality 32 centres were added, study was completed ∼ 1.5 months later.

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CASE STUDY – all predictions

50 100 150 200 250 300 350 50 100 150 200 250 300 350

All predictions after 1st, 2nd and 3d periods

Time # of patients

Combined plot of all predictions and real data: 95% prediction regions constructed by 3 intermediate time points. Target 366 pts.

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CONCLUSIONS Advanced statistical methodology for modeling & predicting patient recruitment is developed:

• stochastic recruitment model adequately reflects real data
• patient recruitment is predicted with confidence boundaries
• the optimal number of centres can be calculated
• Optimal study design problem combining study duration, costs

and risks can be solved Methodology was validated on many real trials.

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References

• 1. V. Anisimov, V. Fedorov, Modeling of enrolment and estimation of

parameters in multicentre trials, GSK BDS TR 2005-01, 2005

http://www.biometrics.com/7E5C600B533062FD80257146004333DC.html

• 2. V. Anisimov, V. Fedorov, Design of multicentre clinical trials with random

enrolment, in book ”Advances in Statistical Methods for the Health Sci- ences”, Balakrishnan, N.; Auget, J.-L.; Mesbah, M.; Molenberghs, G. (Eds.) Birkhˆ auser, 2006, Ch. 25, pp. 387-400.

• 3. V. Anisimov, V. Fedorov, Modeling, prediction and adaptive adjustment
• f recruitment in multicentre trials, Statistics in Medicine (early view).

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BACK UP SLIDES

Centre occupation properties Predicting # of centres recruited particular # of patients Completed trial: n patients recruited by N centres, ni – # of patients recruited by centre i. Characterization of vector (n1, .., nN): ν(n, N, j) – # of centres recruited exactly j patients, j = 0, 1, .., n. A measure of centre occupation: m(n, N, α, j) = E[ν(n, N, j)], j = 0, 1, .., n. m(n, N, α, j) = N

n

j

B(α + j, α(N − 1) + n − j)

B(α, α(N − 1)) , j = 0, 1, 2, ..n. Anisimov, Fedorov (2005, 2006)

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Predicting mean number of “empty” centres If j = 0, m(n, N, α, 0) is the mean number of empty centres. Case n = 400, different N, mean[λ] = 1, different values of standard deviation sd=sd[λ]. sd = 0 – case of deterministic and equal rates.

N \sd 0.5 0.7 0.85 1 1.1 1.2 20 0.01 0.15 0.47 0.9 1.41 1.94 40 0.25 1.07 2.25 3.55 4.88 6.18 60 0.07 1.16 3.13 5.41 7.71 9.91 11.98 80 0.53 3.07 6.44 9.9 13.19 16.24 19.03 100 1.81 6.17 11 15.61 19.84 23.67 27.13

• Table. Mean number of empty centres.

The variation in rates increases the number of empty centres. Clear evidence why in trials with large number of centres the number of “empty” centres is often substantial.

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Mean number of centres recruited a particular number of patients

10 20 30 40 1 2 3 4 5 6 7

Mean number of centres with j patients

j

n = 400, N = 40.

10 20 30 40 1 2 3 4 5 6 7

Mean number of centres with j patients

j

n = 800, N = 60. Dotted brown line – λi are deterministic and equal; dash-dotted green line – λi are random, sd[λ] = 0.5; solid blue line – sd[λ] = 0.7; dashed red line – sd[λ] = 1; where mean[λ] = 1 for all cases.

Uncertainty in rates increases the variation in the number of patients between centres.

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Patient recruitment prediction Mean and sd of n(t, N):

E[n(t, N)] = M(t, N), Var[n(t, N)] = M(t, N) + S2(t, N),

where M(t, N) =

N

• i=1

M(t, ai, bi, mi), S2(t, N) =

N

• i=1

S2(t, ai, bi, m, s2

i ).

Functions M(·) and S2(·): M(t, a, b, m) = mt − m(a + b)/2, S2(t, a, b, m, s2) = (m2 + s2)(b − a)2/12 + s2(t − (a + b)/2)2, for t > b. Similar formulae are derived for t < b. Approximate prediction confidence boundaries over time calculated using normal approximation.

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Parameter estimation Properties of Maximum likelihood, Least squares and Moment methods estimators are investigated using an exhaustive Monte Carlo simulation.

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.2 0.4 0.6 0.8

Comparison of empirical densities of estimators for MLM, MLS, MM

n=720, N=60, alpha=2, 5000 runs, solid line − MLS, dashed − MLM, dotted − MM alpha density 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.2 0.4 0.6 0.8 1.0

Comparison of empirical densities of estimators for MLM, MLS, MM

n=1200, N=100, alpha=2, 15000 runs, solid line − MLS, dashed − MLM, dotted − MM alpha density

Empirical density function of the estimator α. Scenarios: n = 720, N = 60 (left); n = 1200, N = 100 (right). Real parameter α = 2. All three types of estimators show a similar behaviour, asymptotically unbiased and normally

• distributed. Proposed technique works well for trials with N > 20.

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