Designs of Dose Selection Studies in Phase I Oncology Trialsims.nus.edu.sg/events/2017/quan/files/ying.pdf · dose-finding studies Zohar et al (SIM 2011). – Looking for the MTD

July 6, 2017 Workshop on Design of Healthcare Studies 1

Designs of Dose Selection Studies in Phase I Oncology Trials

Ying Lu, Ph.D.Department of Biomedical Data Science

Center for Innovative Study Designs (CISD)Stanford Cancer Institute (SCI)

Stanford University School of MedicineStanford, CA, USA

ACKNOWLEDGEMENT• This is a joint work of

Bee Leng Lee, San Jose State University, San Jose, CA, USA

Shenghua Kelly Fan, California State University at East Bay, Hayward, CA, USA

Hua Jin, South China Normal University, Guangzhou, China

• The YL work is partially supported by P30 CA124435.


OUTLINE1. Introduction

2. Curve-free Bayesian decision-theoretic designs (CBDDs)

3. CBDD for drug combinations

4. Parametric stochastic model approaches

5. Discussion


1. INTRODUCTION• Cancer treatment types:

cytotoxic (chemotherapies)

cytostatic (molecularly targeted therapies)

immunotherapies (cellular, antibody, cytokine, etc)

• Increasing combination trials

• Challenges: overlapping toxicity prevents patients completing treatments (Kelley and Venook, ASCO, 2012)


1. INTRODUCTION

• All treatments should not harm patients

July 6, 2017 Workshop on Design of Healthcare Studies

Zohar et al. Statistics in MedicineVolume 30, Issue 17, pages 2109-2116, 23 FEB 2011

Dose …

DLT T

maximumtolerabledose (MTD)

5

1|

…

1. INTRODUCTION• Early phase (I and II) trials:

– phase I focuses on safe doses and pharmacology– phase I recommends dose for phase II trial (RP2D)– phase II focuses on proof of concept for efficacy

• RP2D– MTD: maximum tolerated dose– MED: maximum effective dose– OPD: optimal biologic dose

• Rule or model based determination of MTD• Frequentist versus Bayesian approaches• Curve-free versus model based Bayesian approaches


2. CBDD

• Components of a Bayesian decision-theoretic design– working model

– starting cohort

– utility function

– dose selection and restriction rule

– range for sample size

– stopping rules

• Fan, Wang, and Lu (SIM, 2012) presented a curve-free BDD.


2. CBDD – Working ModelWorking Model – for dose with an estimated toxicity

– toxicity response ~– toxicity rate ~ ,– mean toxicity rate ⁄– for observed toxic response , among patients, the

posterior distribution ~ ,– selection of prior distribution: , 1 ,

such that , 1 >1

– for … , ⋯ – alternative selection based on interquartile values at each

doseJuly 6, 2017 Workshop on Design of Healthcare

Studies 8

2. CBDD – Working Model

Working data and updated prior Beta(a,b)– Want to use the monotonicity property to extrapolate

information beyond current dose



Working data and updated prior Beta(a,b) at diff doses– monotonic dose-toxicity relationship

July 6, 2017

Toxicity at xi x1 x2 … xi xi+1 … xk

Yes Working data ti = 1 NA NA NA 1 1 1 1

Beta parameter a: a1 a2 … ai+1 ai+1+1 … ak+1

Beta parameter b: b1 b2 … bi bi+1 … bk

No Working data ti = 0 0 0 0 0 NA NA NA

Beta parameter a: a1 a2 … ai ai+1 … ak

Beta parameter b: b1+1 b2+1 … bi+1 bi+1 … bk

Workshop on Design of Healthcare Studies 10


Working data and updated prior Beta(a,b) at diff doses– For patients tested in dose , with experienced DLT,

update the posterior parameters as following:

∗ ; ∗

For , ∗ ; ∗

For , ∗ ; ∗

– ⋯ – Increased effective sample size


2. CBDD – STARTING COHORT


1. Start from lowest dose in most cases

2. Determine the size of dose cohort

3. Escalate the dose levels until the first toxicity is observed

4. Use the Bayesian utility function to determine the next dose selection based on posterior distributions derived using working model and data updates

2. CBDD – DOSE ALLOCATION• Utility functions:

– , ∑ , – ,

• Expected utility:

, ∗, ∗

∗, ∗∗

∗ ∗∗ 1, ∗

∗

∗ ∗

where , is regularized incomplete beta function.

• We can select dose to maximize the utility or the higher one in adjacent dose levels or .July 6, 2017 Workshop on Design of Healthcare

Studies 13

2. CBDD – SAMPLE SIZE AND STOPPING RULES

Range of sample size:– don’t stop before n=nmin

– must stop at n=nmax

Stopping rules:– Evidence that the lowest dose is too toxic

|– Current recommended dose is very likely the MTD

min |

All parameters , r1 and r2 are pre-determined.


2. CBDD - AN EXAMPLEThe S-1 treatment based on the meta-analysis of three dose-finding studies Zohar et al (SIM 2011).

– Looking for the MTD at = 30% toxicity rate

July 6, 2017

Dose Level (mg/m2)

25 30 35 40 45

Toxicity rate (pi)

0.001 0.143 0.364 0.439 0.600


2. CBDD - AN EXAMPLEThe S-1 treatment based on the meta-analysis of three dose-finding studies Zohar et al (SIM 2011).

– Looking for the MTD at r = 30% toxicity rate

July 6, 2017

Dose Level (mg/m2)

25 30 35 40 45

Toxicity rate (pi)

0.001 0.143 0.364 0.439 0.600

c=4, ai=4pi , bi=4(1-pi)

(0.004,3.996) (0.57,3.43) (1.46,2.54) (1.76,2.24) (2.4,1.6)


2. CBDD - AN EXAMPLE

July 6, 2017

Prior distributions

Dose 25 30 35 40 45ai 0.004 0.572 1.456 1.756 2.4bi 3.996 3.428 2.544 2.244 1.6



July 6, 2017

Dose 25 30 35 40 45ai 0.004 0.572 1.456 1.756 2.4bi 4.996 3.428 2.544 2.244 1.6

Start from the lowest dose

Patient 1Dose 25Obs T 0

Posterior Beta



July 6, 2017

Dose 25 30 35 40 45ai 0.004 0.572 1.456 1.756 3.4bi 7.996 6.428 4.544 3.244 1.6

Continue to the first toxicity event

Patient 1 2 3 4 5Dose 25 30 35 40 45Obs T 0 0 0 0 1

Posterior Beta



July 6, 2017

Dose 25 30 35 40 45ai 0.004 0.572 1.456 1.756 3.4bi 8.996 7.428 5.544 3.244 1.6

Calculate OSLA and select the next dose

Patient 1 2 3 4 5 6Dose 25 30 35 40 45 35Obs T 0 0 0 0 1 0

Posterior Beta



July 6, 2017

Dose 25 30 35 40 45ai 0.004 0.572 4.456 5.756 7.4bi 14.99613.428 11.544 3.244 1.6

Continue until a stop rule applies

Patient 1 2 3 4 5 6 7 8Dose 25 30 35 40 45 35 40 35Obs T 0 0 0 0 1 0 1 1Patient 9 10 11 12 13 14 15 16Dose 35 35 35 35 35 35 35 35Obs T 1 0 0 1 0 0 0 0

Posterior Beta


2. CBDD - SIMULATIONS• Extensive simulation studies based on data

generated by power and logistic dose response models– with stopping rules

– without stopping rules

– no dose jump allowed

– prior misspecification

– uniform prior

• Repeated 10,000 times


2. CBDD - SIMULATIONS• The new design out performs CRM and CRML

in– dose allocation

– dose recommendation

– mean number of patients

– mean number of toxicity


2. CBDD - SIMULATIONS• Simulation studies investigated effect of prior

misspecification.– works well for over estimate toxicity rates

– Does not work well for underestimated rates

• Simulation studies investigated effect of 100% non-informative prior – Only 35% coverage of the MTD

– Recommended dose is often too low


2. CBDD - Extension to OBD


• OBD is based on a more rational and scientific endpoint to determine– drug hits the target

– the target is altered by drug

– the tumor is altered by hitting the target

– giving a dose fails to improve outcomes further

• Algorithms that placing more patients on MTDs are not ideal for cytostatic drugs.

• Instead of a single OBD, we are looking for a dose range that has biomarker activities (or efficacy if available) beyond a prespecified clinical significant level.



• Assuming the BOD endpoint is a binary variable

• For dose , the effective rate

• The goal of the current design is to find dose that and

• Admissible doses:

• Acceptable doses:

• Assume doses within should be connected, we denote be all possible connected subsets in .

• If or is empty set, there is no viable dose choice



• Two-Step processes1. CBDD to find the initial boundary of MTDs and estimated

admissible set

2. Let and be the number of patients tested and observing activities, respectively, in dose ∈ .

3. Two ways to find :- MLE: Define ∑ ∈ ; ∑ ∈ and

∑ ∉ ; ∑ ∉ ;

⁄ 1 ⁄ ⁄ 1 ⁄

Find ∗: ∗ max,…,

⁄



• Two-Step processes1. CBDD to find the boundary of MTD and estimated

admissible set

2. Let and be the number of patients tested and observing activities, respectively, in dose , ∈ .

3. Two ways to find :- Max Deviance: let ( ) be the clinically insignificant level of

activities:

⁄2 ⁄

2Find ∗: ∗ max

,…, ⁄



• Two-Step processes (Continue)4. If is empty, add a group of patients randomly in all

admissible doses.

5. Update the admissible set and

6. If is not empty, check | , 0.9, stop the search.

7. Otherwise, continue to sample around the boundary of , repeat 5-6 until all patients being used.



• Zang and Lee (SIM 2016) proposed a Phase I/II design– Unimode efficacy

– Using the PAVA regression for efficacy data

• We performed simulation only for one drug trial– Repeated 5,000

– Total sample size limited to 50 patients

– Parameters: 0.3, 0.03, 0.3, 0.2• Simulations for MD and combinations of two drugs are

on-going.



• Simulation scenarios (Zang and Lee, SIM 2016)

Scenario Endpoint Prob at Dose Level (1, 2, 3, 4, 5)

1 DLT (0.1, 0.2, 0.3, 0.4, 0.5)

Efficacy (0.05, 0.1, 0.18, 0.25, 0.3)

2 DLT (0.01, 0.05, 0.09, 0.15, 0.2)

Efficacy (0.1, 0.3, 0.4, 0.2, 0.05)

3 DLT (0.02, 0.06, 0.12, 0.3, 0.5)

Efficacy (0.3, 0.4, 0.2, 0.1, 0.05)

4 DLT (0.02, 0.06, 0.12, 0.3, 0.5)

Efficacy (0.1, 0.3, 0.3, 0.3, 0.3)

5 DLT (0.1, 0.2, 0.25, 0.4, 0.5)

Efficacy (0.2, 0.4, 0.4, 0.4, 0.4)



• Results for MLESc %MTD E % Selected Avg Tx Avg E % no B % Gd B

1 5, 44, 42, 9, 0 L 0, 1, 5, 1, 0 20 11 92 0

U 0, 1, 5, 2, 0

2 0, 0, 3, 18, 79 L 23, 38, 22, 3, 0 13 17 14 50

U 1, 23, 38, 24, 0

3 0, 2, 26, 70, 2 L 62, 25, 2, 0, 0 12 25 11 64

U 7, 57, 24, 1, 0

4 0, 2, 25, 71, 2 L 5, 50, 18, 9, 0 15 26 18 76

U 0, 22, 24, 35, 1

5 6, 37, 38, 12, 0 L 18, 62, 8, 1, 0 20 34 12 61

U 2, 37, 38, 11, 0

2. CBDDTake home messages:

• CBDD– monotonic dose-toxicity relationship

– easy explanation and implementation

– allocations of MTD faster

– extension to OBD with a reasonable small number of patients

– biased estimation of toxicity for low and high doses (this method is not used for estimation)


3. DRUG COMBINATIONS


• Richly et al. (Annals of Oncology 2006; 17:866-73)

Sorafenib Doxorubicin

50mg/m^2 60mg/m^2 70mg/m^2

200mg bid 0.08 0.12 0.17

400mg bid 0.17 0.25 0.36

600mg bid 0.33 0.46 0.60

3. DRUG COMBINATIONS• Statistical challenges for combination phase I trials

– small number of patients in a phase I trial– much larger searching space– partial order of dose levels

1,1 ≺ 1,2 , 2,1 ≺ 1,3 , 2,3 , 3,1 ≺ 3,2 , 2,3 ≺ 3,3– possibility of more than one MTD combinations

• There is a need to find efficient and easy algorithms– utilize semi-order and monotonic dose response

relationship– utilize prior knowledge– utilize information from all dose groups– utilize reasonable model assumptions– easy to specify model parameters and simple computation


3. DRUG COMBINATIONS• One dimensional approaches:

– search a pre-specified path (Richly, et al, 2006)


Agent 1 Agent 2

Dose 1 Dose 2 Dose 3

Dose 1 (1,1) p11 (1,2) p12 (1,3) p13

Dose 2 (2,1) p21 (2,2) p22 (2,3) p23

Dose 3 (3,1) p31 (3,2) p32 (3,3) p33



Working model:

• Let (i, j) be the combination of agent A at dose i and agent B at dose level j– toxicity response is a binary variable, 1– toxicity rate ~ ,– mean toxicity rate ⁄

• Monotonic dose response: whenever , ≺ ,• Partial order on dose combinations

– strict ordering (SO): , ≺ , iff , ,– diagonal ordering (DO): , ≺ , iff



Working Model

• Selection of prior distribution parameters– When estimated toxicity rate is available

, for , ≺ ,where select a constant and use

and 1for the initial prior distribution ,

– When estimated toxicity rate is not available but the toxicity of single agent is known, estimate by

1 1 1– Alternatively specified by estimated inter-quartiles

38



• For patients being treated at a single dosecombination and had DLTs, we will updateprior distributions for all dose combinations asfollowing:

∗ ; ∗ ;

For , ≺ , : ∗ ; ∗ ;For , ≺ , : ∗ ; ∗ ;

• The partial order of posterior mean toxicity is preserved.



• Prior parameters before observing t22

Agent 1 Agent 2

Dose 1 Dose 2 Dose 3 Dose 4

Dose 1 a11, b11 a12, b12 a13, b13 a14, b14

Dose 2 a21, b21 a22, b22 a23, b23 a24, b24

Dose 3 a31, b31 a32, b32 a33, b33 a34, b34


• Prior parameters after observing t22=1 underSOAgent 1 Agent 2


Dose 1 a11, b11 a12, b12 a13, b13 a14, b14

Dose 2 a21, b21 a22+1, b22 a23+1, b23 a24+1, b24

Dose 3 a31, b31 a32+1, b32 a33+1, b33 a34+1, b34



• Prior parameters after observing t22=1 under DO

Agent 1 Agent 2


Dose 1 a11, b11 a12, b12 a13, b13 a14+1, b14

Dose 2 a21, b21 a22+1, b22 a23+1, b23 a24+1, b24

Dose 3 a31, b31 a32+1, b32 a33+1, b33 a34+1, b34



• Prior parameters after observing t22=0 under SO& DO

Agent 1 Agent 2


Dose 1 a11, b11+1 a12, b12+1 a13, b13 a14, b14

Dose 2 a21, b21+1 a22, b22+1 a23, b23 a24, b24

Dose 3 a31, b31 a32, b32 a33, b33 a34, b34




Starting cohort:

1. Start from lowest dose combination

2. Determine the size of dose cohort

3. Escalate the dose levels until the first toxicity is observed

4. Use the Bayesian utility function to determine the next dose selection based on posterior distributions derived using working model and data updates

3. DRUG COMBINATIONSDose Allocation:• Utility functions:

– , ∑ , – ,

• Expected utility:, ,

∗ , ,∗

• We can select dose maximize the utility or the higher one in adjacent combinations

.


3. DRUG COMBINATIONSSample size and Stopping rules• Range of sample size:

– don’t stop before n=nmin

– must stop at n=nmax

• Stopping rules:– Evidence that the lowest dose is too toxic

|– Current recommended dose is very likely a MTD

min, ≺ ,

|




• Simulation and comparisons can be found in (Lee, Fan, and Lu: Journal of Biopharmaceutical Statistics, 2017, 27:1,34-43).– most patients allocated near MTD levels

– smaller average sample size

– more likely to give recommendation doses when there are MTDs

– not perform well when MTD are at the lowest or highest corners

3. DRUG COMBINATIONSTake home messages:• Needs new designs for phase I combination trials

– Our CBDD is flexible and good for finding MTDs

– The CBDD algorithm searching for OBDs for single drug can be used here too

• Joint Phase I/II design searching for safety and efficacy simultaneously can be challenge in combinations


4. STOCHASTIC MODELS• Focus on descriptive not on possible biological

relationship of drugs, body, and diseases

• Early phase designs rarely used PK and PD information

• Current paradigm is difficult for more complicated combinations in drugs and strategies

• Integration of biological meaningful parameters in phase I/II designs

• Dose -> exposure -> events of interests



4. STOCHASTIC MODELS

Chiang and Comforti model assumptions• Time to tumor after exposure of toxic material at age

follows a survival distribution with an intensity function as

, ,• is the internal factor for tumor incidence,

which is a function of age at the time of exposure ( )• , , is the external factor that dominates the

intensity function; 0 is the absorbing parameter and (>0) is the discharge parameter

• For an exposure with dose at time , the remaining toxicity in the body is



4. STOCHASTIC MODELS, , is a function of current exposure.

• If the exposure started from time 0 and continues to t, the remaining toxic materials (exposure) in the body is

1 , , ,

• The cumulative time to cancer is

1 1

• For a simple survival model with =0,

111

• Rao’s extension (1990), for 0 < 1,

111




0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

weeks

Cum

ulat

ive

Dis

tribu

tion

Func

tion

53

Cumulative Probability Functions by Dose



Dose Response Curves by Time

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

dose

Pro

babi

lity

of T

umor

Occ

urre

nce

4. STOCHASTIC MODELSDiscontinuation of exposure

• If the exposure of toxic materials terminates at time s, the cumulative time to cancer is

11

• Thus

lim→

1 1

• The smaller , the smaller of lifetime chance for cancer.


4. STOCHASTIC MODELSWe extend their ideas to dose finding studies.• For a drug administrated between time interval , , the drug

remaining in body (or blood concentration) at time is

; , ⋀ ⋀

0

• Cumulative dose for the patient up to time is

; , ; , =

1 ∧ ∨


4. STOCHASTIC MODELSMost cancer treatments are organized in cycles.• Let be the length of the treatment and be

the length of cycle.• If a patient will receive a maximum of

treatment cycles from , , …, .

• Let the number of treatment cycles to time , and is the maximum number of cycles.


4. STOCHASTIC MODELS• For a patient who received drug at dose , his

or her blood/body concentration at time is

; , , ; ,

1 ⋀

11

1⋀



Toxicity dose response curves• Using the idea by Chiang and Conforti (89), the

intensity function for the 1st dose limiting toxicity (DLT) event is

; , , ; , ,• The cumulative distribution function is

• In reality, also, and are not both identifiable.


4. STOCHASTIC MODELS• Combining and into one parameter , we have the

cumulative distribution for time of 1st DLT event as the following:

; , , 1 ; , ,

1 ACU ; 1 , 1

1

1 11

1∧ 1 ∨


4. STOCHASTIC MODELSSome properties• The current toxicity model is an exponential dose response

model.- In order to get non-exponential dose response model, one can

make toxicity intensity function as a non-linear function, such as a quadratic function of the blood/body concentration at time (allow for quadratic dose response).

• Another option is to use ; , , to replace the dose in conventional models (such as logistic model, thus becoming time-adjusted).

• The unknown parameters are and . However, may be able to estimated from PK/PD studies. The model will be much simpler in this case.

• As shown in Chiang and Conforti (89), the conditional survival probability for time to a DLT can be derived.



Advantage 1: Late on-set and/or long term DLTs• If we are interested in toxicity until T, MTD can be

found by

• Note:- T can be longer than the length of a phase I trial.- T is different from the time t0 in TITE-CRM.

• Based on estimated model parameters, we can predict toxicity rate at any time.


4. STOCHASTIC MODELSAdvantage 2: Simultaneously model toxicity and efficacy endpoints:• hazard rate for the first toxicity event at time t: a function of dose

concentration:; , , ; , , = ∑ ; , ,

• probability of response: a logistic function of cumulative exposure (AUC) by the end of experiment duration T

| , , = ; , , ; , ,

where ; , , =∑ ; , ,• likelihood function based on joint observation of , , ,• application of GLR test (Bartroff, Lai, and Narasiham, SIM 2014) for

inference and dose selection in a combined phase I/II study



Advantage 3: Optimization the treatment plan

• Statistical optimization of treatment doses using TITE-CRM.

• In the new model, the probability of toxicity is a function of length of cycles , it can also be used to optimize the length of cycle to reduce the toxicity rate.


4. STOCHASTIC MODELSAdvantage 4: dose for drug combinations• Parametric and non-parametric models have been

developed for combination doses for combination therapies, including TITE-CRM (Wages, et al., SIM, 2011).

• In our model, we can link the toxicity hazard rate function as a function of blood/organ concentration of both drugs and their interactions at time of .

*



Advantage 4: dose for drug combinations

• Note that two treatment can have different cycles and length of treatment, the model is flexible enough for concurrent use or sequential use of two drugs. - Simultaneous treatments (maintain partial ordering)

Treatment 1 __g________ __g________ __g________ __g________ __g________

Treatment 2 __g________ __g________ __g________ __g________ __g________

- Sequential treatments (no ordering)Option 1 __g1_______ __g2________ __ g1_______ __ g2________ __ g1________

Option 2 __ g2_______ __ g1________ __ g2 _______ __ g1________ __ g2 _______



Advantages:• biological relevant model parameters• flexibility in addressing delayed toxicity• easy to evaluate different drug administration schedules• easy extension to drug combinations • phase I/II (Bartroff, Lai, and Narasimhan, 2014 Stat Med)

Disadvantages:• model justifications – need biological experimental data

to support• need detailed data phase I data to verify the model



Take home messages:• Beyond non-parametric approach

- Integration of PK/PD into dose finding studies- Development of models based on parameters with

biological interpretations- Use of biological based lower dimensional models to

reduce the complexity of phase I combination trials.• Beyond fixed time point

- Chiang (1985) suggested biostatisticians use stochastic process approaches to model complex biological processes and experiments

- Dose finding study is an excellent area to use his suggestion because most PK/PD analyses are based on dynamic models.


5. DISCUSSIONS

1. Curve-free Bayesian Decision-theoretical Designs (CBDD)

– Single agent MTD and OBD– Drug combinations

2. Beyond empirical modeling to biologically based stochastic modeling

– Parametric approaches are needed– Chiang and Comforti model is one possible

option


5. DISCUSSIONS

• What if we can directly control dose exposure in phase I trial?


natbiomedeng. 2017;1,0070

Eliminating population pharmacological variations

5. DISCUSSIONS

• What if we can monitor treatment effect real time?– Liquid biopsy?

• How should we design phase I/II trials using these new technologies?


THANK YOU

Questions?


Documents

Designs of Dose Selection Studies in Phase I Oncology Trialsims.nus.edu.sg/events/2017/quan/files/ying.pdf · dose-finding studies Zohar et al (SIM 2011). – Looking for the MTD