designs of dose selection studies in phase i oncology...
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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.
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OUTLINE1. Introduction
2. Curve-free Bayesian decision-theoretic designs (CBDDs)
3. CBDD for drug combinations
4. Parametric stochastic model approaches
5. Discussion
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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)
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1. INTRODUCTION
• All treatments should not harm patients
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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
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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.
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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
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2. CBDD – Working Model
Working data and updated prior Beta(a,b) at diff doses– monotonic dose-toxicity relationship
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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
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2. CBDD – Working Model
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
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2. CBDD – STARTING COHORT
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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.
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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
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Dose Level (mg/m2)
25 30 35 40 45
Toxicity rate (pi)
0.001 0.143 0.364 0.439 0.600
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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
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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)
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2. CBDD - AN EXAMPLE
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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
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2. CBDD - AN EXAMPLE
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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
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2. CBDD - AN EXAMPLE
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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
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2. CBDD - AN EXAMPLE
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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
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2. CBDD - AN EXAMPLE
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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
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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
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2. CBDD - SIMULATIONS• The new design out performs CRM and CRML
in– dose allocation
– dose recommendation
– mean number of patients
– mean number of toxicity
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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
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2. CBDD - Extension to OBD
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• 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.
2. CBDD - Extension to OBD
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• 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
2. CBDD - Extension to OBD
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• 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,…,
⁄
2. CBDD - Extension to OBD
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• 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
,…, ⁄
2. CBDD - Extension to OBD
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• 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.
2. CBDD - Extension to OBD
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• 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.
2. CBDD - Extension to OBD
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• 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)
2. CBDD - Extension to OBD
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• 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)
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3. DRUG COMBINATIONS
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• 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
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3. DRUG COMBINATIONS• One dimensional approaches:
– search a pre-specified path (Richly, et al, 2006)
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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
3. DRUG COMBINATIONS
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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
3. DRUG COMBINATIONS
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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
3. DRUG COMBINATIONS
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• 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.
3. DRUG COMBINATIONS
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• 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
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• Prior parameters after observing t22=1 underSOAgent 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+1, b22 a23+1, b23 a24+1, b24
Dose 3 a31, b31 a32+1, b32 a33+1, b33 a34+1, b34
3. DRUG COMBINATIONS
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• Prior parameters after observing t22=1 under DO
Agent 1 Agent 2
Dose 1 Dose 2 Dose 3 Dose 4
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
3. DRUG COMBINATIONS
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• Prior parameters after observing t22=0 under SO& DO
Agent 1 Agent 2
Dose 1 Dose 2 Dose 3 Dose 4
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
3. DRUG COMBINATIONS
3. DRUG COMBINATIONS
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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
.
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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, ≺ ,
|
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3. DRUG COMBINATIONS
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• 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
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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
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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
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4. STOCHASTIC MODELS
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
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4. STOCHASTIC MODELS
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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
4. STOCHASTIC MODELS
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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.
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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 ∧ ∨
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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.
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4. STOCHASTIC MODELS• For a patient who received drug at dose , his
or her blood/body concentration at time is
; , , ; ,
1 ⋀
11
1⋀
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4. STOCHASTIC MODELS
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.
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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 ∨
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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.
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4. STOCHASTIC MODELS
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.
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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
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4. STOCHASTIC MODELS
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.
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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 .
*
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4. STOCHASTIC MODELS
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 _______
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4. STOCHASTIC MODELS
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
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4. STOCHASTIC MODELS
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.
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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
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5. DISCUSSIONS
• What if we can directly control dose exposure in phase I trial?
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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?
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THANK YOU
Questions?
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