Cure Survival Data in Oncology Study Shih-Yuan (Connie) Lee, Ph.D. Takeda JSM 2018 meeting

Generate the Random Variable Outline Introduction Generate the Random Variable Parametric Method Non-Parametric Method Application Sample Size Calculation Event Projection Summary

Introduction For some caners, the cure of the disease is a possibility especially in the earlier line of treatment (i.e. Hodgkin Lymphoma) Viviani S, Zinzani PL, Rambaldi A, Brusamolino E, Levis A, Bonfante V, Vitolo U, Pulsoni A, Liberati AM, Specchia G, Valagussa P, Rossi A, Zaja F, Pogliani EM, Pregno P, Gotti M, Gallamini A, Rota Scalabrini D, Bonadonna G, Gianni AM; Michelangelo Foundation; Gruppo Italiano di Terapie Innovative nei Linfomi; Intergruppo Italiano Linfomi. ABVD versus BEACOPP for Hodgkin's lymphoma when high-dose salvage is planned. N Engl J Med. 2011 Jul 21;365(3):203-12. doi: 10.1056/NEJMoa1100340. Events of first progression were defined as progression while receiving treatment, a lack of complete remission at the end of treatment, relapse, or death from any cause. The probability of event-free survival is shown in Panel B. Events were defined as progression while receiving treatment, a lack of complete remission at the end of treatment, relapse, death from any cause, discontinuation of treatment owing to life-threatening toxic effects, and secondary leukemia.

Proper vs. Improper Survival Curve S(t): Survival Function H(t): Cumulative Hazard Function Proper Random Variable H(∞)= ∞, S(∞)=exp(-H(∞))=0 Improper Random Variable H(∞)< ∞ (bounded), S(∞)=exp(-H(∞))>0

Generate the Random Variable Parametric Method

Formulation for Cure Survival Function H(∞) is bounded (<∞) in cure setting and EXP(-H(∞)) is the cure rate An arbitrary proper CDF (F(t)=1-G(t)) G(t) arbitrary proper survival → ∞

Generate Random Variable Inverse transform Generate Random Variable U ~ Uniform(0, 1) Find T such that CDF of T, F(T) = U (i.e. T=F-1(U)) Since 1-U also ~Uniform (0,1), this implies the Survival of T, S(T)=U, (i.e. T=S-1(U)) Since we can write the Cure Survival Function as Where G(T) is the proper survival function (i.e. Weibull Survival Function) -H(∞) will be bounded (i.e. a constant) with exp(-H(∞)) is the Cure Rate

Example: Weibull Cure Random Variable Using the Inverse Transformation Need to be >0 If <=0, then T is ∞(i.e. return a large number)

Adding Treatment Effect Under the Proportional Hazard (PH) Assumption Under the proportional hazard (PH) assumption, the treatment effect θ can be applied to the survival function

HR=0.75 (Active vs. Control) G(t): Weibull Shape=1 2 year survival=0.8 for the control arm Cure rate=50% for the control arm

Generate the Random Variable Non-parametric Method

Non-parametric Method Based on the Discrete Hazard Function The probability of event occurred between t1 and t2 given the subject is event free at t1 is Time   Survival Number Failed Number Left 0.0000 1.0000 20 0.9500 1 19 2.0000 0.9000 2 18 3.0000 0.8500 3 17 5.0000 * . 16 6.0000 0.7969 4 15 7.0000 0.7438 5 14 8.0000 13 9.0000 12 10.0000 11 11.0000 0.6761 6 10 For example: The subject is had event at time 1 is: (1-0.95)/1=0.05 The subject is event free at time 1. The chance of getting an event at time 2 is: (0.95-0.9)/0.95=0.053

Graphical Illustration Porb. of getting an event at time 1 Event free at time 1. Porb. of getting an event at time 2 Event at time1 (Stop) Bernoulli (p=0.05) 1 Bernoulli (p=0.053) Event at time 2 (Stop) 1 Censored at time 1 (Continue) Censored at time 2 … Continue till event or the last F/U time This could be computational intensive depending on the time interval and sample size evaluated

Method Illustrated Using the Simulation

Sample Size Calculation Application Sample Size Calculation

Number of Events Required Does not depend on the distribution assumption

Sample Size Estimation Example 2 year PFS (Active: 81% vs. Control: 73%) HR=0.67 (assuming an emergent plateau in the PFS event rate after 2 year) Alpha: 1-sided 0.025 Power: 90% Study duration: 60 months for the primary analysis. 36 months of accrual, a 5% annual dropout rate and 24 months of F/U after last patient in

Sample Size Calculation Plug into the Software Typically on the exponential distribution assumption Number of event=262 Sample size=790 Simulation Approach Generate R.V. based on the protocol assumption (Assuming cure after 2 years) Number of event~260 ~ 1240 patients were required

Event Projection for Cure Survival Curve Non-Parametric Method Application Event Projection for Cure Survival Curve Non-Parametric Method

Graphic Illustration of Event Projection Simulation Observed Event Time Future Survival Time Lost to follow up Ongoing at the data cut off 0 (FPI) Current Data Cutoff Calendar Time

Event Projection for Cure Survival Data Typical exponential distribution with memory-less property will not hold Parametric method such as Piecewise Exponential Distribution with very low rate at the tail is an option Non-parametric method using the discrete hazard function based on the observed K-M curve

Non-parametric Method Based on the Discrete Hazard Function The probability of event occurred between t1 and t2 given the subject is event free at t2 is Time   Survival Number Failed Number Left 0.0000 1.0000 20 0.9500 1 19 2.0000 0.9000 2 18 3.0000 0.8500 3 17 5.0000 * . 16 6.0000 0.7969 4 15 7.0000 0.7438 5 14 8.0000 13 9.0000 12 10.0000 11 11.0000 0.6761 6 10 For example: The subject is event free at time 6. The chance of getting an event at time7 is: (0.7969-0.7438)/0.7969=0.0666

Non-parametric Method Based on the Discrete Hazard Function Time 12 24 36 48 At Risk 1334 1227 1137 584 110

Summary Introduce two methods to simulate cure survival data which can easily incorporate the treatment effect under PH or non-PH assumption For a study with potential curable disease, we should not use the traditional software (under exponential distribution assumption) for sample size estimation. Project future event number using traditional exponential distribution is not appropriate. A pure non-parametric approach could be an option based on the observed K-M survival curve

Acknowledgement Dr. Alex Tsodikov and Dr. Jeremy Taylor, Department of Biostatistics, University of Michigan Takeda Cancer Immunotherapy Working Group

Estimate Sample Size using East Estimated Sample Size using EAST: N=790

Sample Size Calculation Naïve Approach Plug into the formula: # of event required: ~262 N is the total sample size (with 1:1 Randomization Ratio , 0.5*N is sample size in each arm) Assuming 5% annual dropout rate, the cumulative dropout by the end of year 2 is ~10% (1-0.95*0.95) (1-0.1)*(0.5*N*(1-0.81)+0.5*N*(1-0.73))=262 → N=1266 Simulation Approach Generate R.V. based on the protocol assumption ~ 1240 patients were required in about 60 months