Utility-Based Optimization of Combination Therapy Using Ordinal Toxicity and Efficacy in Phase I/II Clinical Trials

Transcription

1 Utility-Based Optimization of Combination Therapy Using Ordinal Toxicity and Efficacy in Phase I/II Clinical Trials Peter F. Thall Department of Biostatistics University of Texas, M.D. Anderson Cancer Center Memorial Sloan Kettering Cancer Center October 2, 2009

2 UTILITY-BASED DOSE-FINDING 1 Collaborators Hoang Nguyen Biostatistics Dept, M.D. Anderson, Texas Computer Programming, Prior Calibration, Model Refinements, Graphics

3 UTILITY-BASED DOSE-FINDING 1 Collaborators Hoang Nguyen Biostatistics Dept, M.D. Anderson, Texas Computer Programming, Prior Calibration, Model Refinements, Graphics Nadine Houede Institut Bergonie, Bordeaux, France Medical Application, Utility Elicitation and Restaurant Selection

4 UTILITY-BASED DOSE-FINDING 1 Collaborators Hoang Nguyen Biostatistics Dept, M.D. Anderson, Texas Computer Programming, Prior Calibration, Model Refinements, Graphics Nadine Houede Institut Bergonie, Bordeaux, France Medical Application, Utility Elicitation and Restaurant Selection Xavier Paoletti Institut Curie, Paris, France Gaussian Copula Formulas and Wine Selection

5 UTILITY-BASED DOSE-FINDING 1 Collaborators Hoang Nguyen Biostatistics Dept, M.D. Anderson, Texas Computer Programming, Prior Calibration, Model Refinements, Graphics Nadine Houede Institut Bergonie, Bordeaux, France Medical Application, Utility Elicitation and Restaurant Selection Xavier Paoletti Institut Curie, Paris, France Gaussian Copula Formulas and Wine Selection Andrew Kramar Parc Euromedecine, Montpellier, France Detailed Editing and La Joie de Vivre

6 UTILITY-BASED DOSE-FINDING 2 OUTLINE

7 UTILITY-BASED DOSE-FINDING 2 OUTLINE 1. Combination Bio-Chemotherapy for Bladder Cancer

8 UTILITY-BASED DOSE-FINDING 2 OUTLINE 1. Combination Bio-Chemotherapy for Bladder Cancer 2. Data Structure and Probability Models

9 UTILITY-BASED DOSE-FINDING 2 OUTLINE 1. Combination Bio-Chemotherapy for Bladder Cancer 2. Data Structure and Probability Models Bivariate Almost Ordinal (Toxicity, Efficacy)

10 UTILITY-BASED DOSE-FINDING 2 OUTLINE 1. Combination Bio-Chemotherapy for Bladder Cancer 2. Data Structure and Probability Models Bivariate Almost Ordinal (Toxicity, Efficacy) A Flexible Parametric Model

11 UTILITY-BASED DOSE-FINDING 2 OUTLINE 1. Combination Bio-Chemotherapy for Bladder Cancer 2. Data Structure and Probability Models Bivariate Almost Ordinal (Toxicity, Efficacy) A Flexible Parametric Model Establishing Priors

12 UTILITY-BASED DOSE-FINDING 2 OUTLINE 1. Combination Bio-Chemotherapy for Bladder Cancer 2. Data Structure and Probability Models Bivariate Almost Ordinal (Toxicity, Efficacy) A Flexible Parametric Model Establishing Priors 3. Trial Design and Conduct

13 UTILITY-BASED DOSE-FINDING 2 OUTLINE 1. Combination Bio-Chemotherapy for Bladder Cancer 2. Data Structure and Probability Models Bivariate Almost Ordinal (Toxicity, Efficacy) A Flexible Parametric Model Establishing Priors 3. Trial Design and Conduct Utilities

14 UTILITY-BASED DOSE-FINDING 2 OUTLINE 1. Combination Bio-Chemotherapy for Bladder Cancer 2. Data Structure and Probability Models Bivariate Almost Ordinal (Toxicity, Efficacy) A Flexible Parametric Model Establishing Priors 3. Trial Design and Conduct Utilities Safety Rules

15 UTILITY-BASED DOSE-FINDING 2 OUTLINE 1. Combination Bio-Chemotherapy for Bladder Cancer 2. Data Structure and Probability Models Bivariate Almost Ordinal (Toxicity, Efficacy) A Flexible Parametric Model Establishing Priors 3. Trial Design and Conduct Utilities Safety Rules 4. Computer Simulations

16 UTILITY-BASED DOSE-FINDING 2 OUTLINE 1. Combination Bio-Chemotherapy for Bladder Cancer 2. Data Structure and Probability Models Bivariate Almost Ordinal (Toxicity, Efficacy) A Flexible Parametric Model Establishing Priors 3. Trial Design and Conduct Utilities Safety Rules 4. Computer Simulations

17 UTILITY-BASED DOSE-FINDING 3 Dose-Finding for Combination Therapy of Bladder Cancer

18 UTILITY-BASED DOSE-FINDING 3 Dose-Finding for Combination Therapy of Bladder Cancer Goals: Find an optimal dose pair of combination therapy = chemotherapeutic (chemo) agents gemcitabine + cisplatin + biological agent for untreated advanced bladder cancer, based on Toxicity and Efficacy (a phase I/II trial)

19 UTILITY-BASED DOSE-FINDING 3 Dose-Finding for Combination Therapy of Bladder Cancer Goals: Find an optimal dose pair of combination therapy = chemotherapeutic (chemo) agents gemcitabine + cisplatin + biological agent for untreated advanced bladder cancer, based on Toxicity and Efficacy (a phase I/II trial) Treatment Regime: In each 28-day cycle, the patient receives 1. biological agent orally each day at dose levels d 1 = 1, 2, 3 or 4 2. gemcitabine on days (1, 8, 15) at dose levels d 2 = 1, 2 or 3 (750, 1000 or 1250 mg/m 2 /day) 3. a fixed dose of 70 mg/m 2 cisplatinum on day 2 = d = (d 1, d 2 ) {1, 2, 3, 4} {1, 2, 3}

20 UTILITY-BASED DOSE-FINDING 4 The Dose Pair Domain D

21 UTILITY-BASED DOSE-FINDING 4 The Dose Pair Domain D (1, 3) (2, 3) (3, 3) (4, 3) (1, 2) (2, 2) (3, 2) (4, 2) d 2 (1, 1) (2, 1) (3, 1) (4, 1) d 1 d 1 = dose of biological agent d 2 = dose of chemo agent gemcitabine

22 UTILITY-BASED DOSE-FINDING 5 Clinical Outcomes (evaluated over two 28-day cycles )

23 UTILITY-BASED DOSE-FINDING 5 Clinical Outcomes (evaluated over two 28-day cycles ) Toxicity includes AEs (fatigue, diarrhea, mucositis) related to the biological agent and chemo-related AEs (renal tox, neurotoxicity) Y 1 = 0 if no grade 3,4 (severe) TOX Y 1 = 1 if grade 3,4 TOX occurs, but resolved within 2 weeks Y 1 = 2 if grade 3,4 TOX occurs, & not resolved within 2 weeks

24 UTILITY-BASED DOSE-FINDING 5 Clinical Outcomes (evaluated over two 28-day cycles ) Toxicity includes AEs (fatigue, diarrhea, mucositis) related to the biological agent and chemo-related AEs (renal tox, neurotoxicity) Y 1 = 0 if no grade 3,4 (severe) TOX Y 1 = 1 if grade 3,4 TOX occurs, but resolved within 2 weeks Y 1 = 2 if grade 3,4 TOX occurs, & not resolved within 2 weeks Efficacy is evaluated by the end of two cycles (day 56) Y 2 = 0 if progressive disease (PD) at any time in the first 2 cycles Y 2 = 1 if stable disease (SD) at day 56 Y 2 = 2 if complete or partial remission (CR/PR) at day 56

25 UTILITY-BASED DOSE-FINDING 6 Interim Within-Patient Treatment Modifications

26 UTILITY-BASED DOSE-FINDING 6 Interim Within-Patient Treatment Modifications 1. If grade 1 or 2 non-haematologic TOX in cycle 1, the dose of the biological agent is reduced 2. If the patient does not recover from a grade 3, 4 non-haematologoc TOX in two weeks, the bio agent is stopped, but the patient may continue to receive the chemo agents (physician decision) 3. If Y 1 = 2 (unresolved gr. 3,4 TOX) or Y 2 = 2 (PD) then treatment is stopped

27 UTILITY-BASED DOSE-FINDING 6 Interim Within-Patient Treatment Modifications 1. If grade 1 or 2 non-haematologic TOX in cycle 1, the dose of the biological agent is reduced 2. If the patient does not recover from a grade 3, 4 non-haematologoc TOX in two weeks, the bio agent is stopped, but the patient may continue to receive the chemo agents (physician decision) 3. If Y 1 = 2 (unresolved gr. 3,4 TOX) or Y 2 = 2 (PD) then treatment is stopped 4. Y 1 = 2 and no PD before day 56 = Y 2 is inevaluable

28 UTILITY-BASED DOSE-FINDING 7 The Outcome Domain Y 2 0 = PD 1 = SD 2 = CR/PR 0 (0, 0) (0, 1) (0, 2) Y 1 1 (1, 0) (1, 1) (1, 2) 2 (2, 0) (2, 1) (2, 2) (2, Ineval) 10 elementary outcomes, including {Y 1 = 2 and Y 2 inevaluable}

29 UTILITY-BASED DOSE-FINDING 8 A General Probability Model

30 UTILITY-BASED DOSE-FINDING 8 A General Probability Model Outcomes Y = (Y 1, Y 2 ) where ordinal Y j {1,..., m j } where m j = # levels of outcome j = 1, 2

31 UTILITY-BASED DOSE-FINDING 8 A General Probability Model Outcomes Y = (Y 1, Y 2 ) where ordinal Y j {1,..., m j } Doses d = (d 1, d 2 ) where m j = # levels of outcome j = 1, 2

32 UTILITY-BASED DOSE-FINDING 8 A General Probability Model Outcomes Y = (Y 1, Y 2 ) where ordinal Y j {1,..., m j } where m j = # levels of outcome j = 1, 2 Doses d = (d 1, d 2 ) Marginals π k,y (d, θ) = Pr(Y k = y d, θ)

33 UTILITY-BASED DOSE-FINDING 8 A General Probability Model Outcomes Y = (Y 1, Y 2 ) where ordinal Y j {1,..., m j } where m j = # levels of outcome j = 1, 2 Doses d = (d 1, d 2 ) Marginals π k,y (d, θ) = Pr(Y k = y d, θ) Joint pmf π(y d, θ) = Pr(Y = y Z = 1, d, θ) where y = (y 1, y 2 ) is observed if Z = 1

34 UTILITY-BASED DOSE-FINDING 9 Likelihood Z = I(Y 2 is evaluable), ζ = Pr(Z = 1) δ(y) = I(Y = y, Z = 1) δ 1 (y 1 ) = I(Y 1 = y 1 ) L(Y, Z d, θ) = 1 [ ζ y { } ] δ( y) Z [ π(y d, θ) (1 ζ) 2 { } ] δ1 (y 1 ) 1 Z π 1,y1 (d, θ) Z=0 y 1 =0

35 UTILITY-BASED DOSE-FINDING 10

36 UTILITY-BASED DOSE-FINDING 11 The Aranda-Ordaz (AO) Model (1983)

37 UTILITY-BASED DOSE-FINDING 11 The Aranda-Ordaz (AO) Model (1983) Given linear term η(d, α) the AO model is Pr(Y = 1 d, α) = ξ{η(d, α), λ} = 1 (1 + λe η(d, α) ) 1/λ, λ > 0 1. For η(d, α) = α 0 + α 1 log(d), d = 0 = ξ = 0 = The outcome is impossible if no treatment is given

38 UTILITY-BASED DOSE-FINDING 11 The Aranda-Ordaz (AO) Model (1983) Given linear term η(d, α) the AO model is Pr(Y = 1 d, α) = ξ{η(d, α), λ} = 1 (1 + λe η(d, α) ) 1/λ, λ > 0 1. For η(d, α) = α 0 + α 1 log(d), d = 0 = ξ = 0 = The outcome is impossible if no treatment is given 2. For η(d, α) = α 0 + α 1 d, d = 0 = ξ = 1 (1 + λe α 0) 1/λ = baseline Pr(Y=1) without treatment

39 UTILITY-BASED DOSE-FINDING 11 The Aranda-Ordaz (AO) Model (1983) Given linear term η(d, α) the AO model is Pr(Y = 1 d, α) = ξ{η(d, α), λ} = 1 (1 + λe η(d, α) ) 1/λ, λ > 0 1. For η(d, α) = α 0 + α 1 log(d), d = 0 = ξ = 0 = The outcome is impossible if no treatment is given 2. For η(d, α) = α 0 + α 1 d, d = 0 = ξ = 1 (1 + λe α 0) 1/λ = baseline Pr(Y=1) without treatment 3. λ=1 = ξ{η(d, α), 1} = eη(d,α) 1+e η(d,α) (logistic)

40 UTILITY-BASED DOSE-FINDING 11 The Aranda-Ordaz (AO) Model (1983) Given linear term η(d, α) the AO model is Pr(Y = 1 d, α) = ξ{η(d, α), λ} = 1 (1 + λe η(d, α) ) 1/λ, λ > 0 1. For η(d, α) = α 0 + α 1 log(d), d = 0 = ξ = 0 = The outcome is impossible if no treatment is given 2. For η(d, α) = α 0 + α 1 d, d = 0 = ξ = 1 (1 + λe α 0) 1/λ = baseline Pr(Y=1) without treatment 3. λ=1 = ξ{η(d, α), 1} = eη(d,α) 1+e η(d,α) (logistic) 4. lim λ 0 ξ(η(d, α), λ) = 1 exp{ e η(d,α) } (compl. log-log)

41 UTILITY-BASED DOSE-FINDING 12

42 UTILITY-BASED DOSE-FINDING 13 Prob(Response) Prob(Response) Dose Dose Prob(Response) Prob(Response) Dose Dose Prob(Response) Prob(Response) Dose Dose

43 UTILITY-BASED DOSE-FINDING 14 A Generalized Aranda-Ordaz (GAO) Model

44 UTILITY-BASED DOSE-FINDING 14 A Generalized Aranda-Ordaz (GAO) Model To Accommodate Two Linear Terms: η 1 η 1 (d 1, α 1 ) for d 1 and η 2 η 1 (d 2, α 2 ) for d 2

45 UTILITY-BASED DOSE-FINDING 14 A Generalized Aranda-Ordaz (GAO) Model To Accommodate Two Linear Terms: η 1 η 1 (d 1, α 1 ) for d 1 and η 2 η 1 (d 2, α 2 ) for d 2 ξ {η 1, η 2, λ, γ} = 1 {1 + λ(e η 1 + e η 2 + γe η 1+η 2 )} 1/λ γ accounts for interaction between the two agents γ = 0 = Additive effects e η 1 and e η 2 in the GAO model

46 UTILITY-BASED DOSE-FINDING 15 Linear terms determining the marginal of Y k

47 UTILITY-BASED DOSE-FINDING 15 Linear terms determining the marginal of Y k For j = dose, k = outcome, y = value of Y k η (j) k,y (d j, α (j) k ) = α(j) k,y,0 + α(j) k,y,1 (d j d j )

48 UTILITY-BASED DOSE-FINDING 15 Linear terms determining the marginal of Y k For j = dose, k = outcome, y = value of Y k 1. α (1) k,y,0 and α (2) k,y,0 η (j) k,y (d j, α (j) k ) = α(j) k,y,0 + α(j) k,y,1 (d j d j ) are intercepts

49 UTILITY-BASED DOSE-FINDING 15 Linear terms determining the marginal of Y k For j = dose, k = outcome, y = value of Y k η (j) k,y (d j, α (j) k ) = α(j) k,y,0 + α(j) k,y,1 (d j d j ) 1. α (1) k,y,0 and α (2) k,y,0 are intercepts 2. α (1) k,y,1 and α (2) k,y,1 are dose effects

50 UTILITY-BASED DOSE-FINDING 15 Linear terms determining the marginal of Y k For j = dose, k = outcome, y = value of Y k η (j) k,y (d j, α (j) k ) = α(j) k,y,0 + α(j) k,y,1 (d j d j ) 1. α (1) k,y,0 and α (2) k,y,0 2. α (1) k,y,1 and α (2) k,y,1 are intercepts are dose effects 3. α (j) k = (α (j) k,1,0, α(j) k,1,1, α(j) k,2,0, α(j) k,2,1 )

51 UTILITY-BASED DOSE-FINDING 15 Linear terms determining the marginal of Y k For j = dose, k = outcome, y = value of Y k η (j) k,y (d j, α (j) k ) = α(j) k,y,0 + α(j) k,y,1 (d j d j ) 1. α (1) k,y,0 and α (2) k,y,0 2. α (1) k,y,1 and α (2) k,y,1 are intercepts are dose effects 3. α (j) k = (α (j) k,1,0, α(j) k,1,1, α(j) k,2,0, α(j) k,2,1 ) 4. θ k = (α (1) k, α (2) k, λ k, γ k )

52 UTILITY-BASED DOSE-FINDING 16 Marginal Distributions

53 UTILITY-BASED DOSE-FINDING 16 Marginal Distributions For each outcome k = 1, 2 and levels y = 1,..., m k, Pr(Y k y Y k y 1, d, θ k ) = ξ {η (1) k,y (d 1, α (1) k ξ k,y(d, θ k ) = ), η(2) k,y (d 2, α (2) k ), λ k, γ k }

54 UTILITY-BASED DOSE-FINDING 16 Marginal Distributions For each outcome k = 1, 2 and levels y = 1,..., m k, Pr(Y k y Y k y 1, d, θ k ) = ξ {η (1) k,y (d 1, α (1) k ξ k,y(d, θ k ) = ), η(2) k,y (d 2, α (2) k ), λ k, γ k } π k,0 (d, θ k ) = 1 ξ k,1 ( d, θ k ) π k,y (d, θ k ) = {1 ξ k,y+1 ( d, θ k )} y j=1 ξ k,j ( d, θ k ), 1 y m k 1 π k,mk (d, θ k ) = m k j=1 ξ k,j ( d, θ k )

55 UTILITY-BASED DOSE-FINDING 17 In the three-level ordinal outcome case, the unconditional marginal probabilities are π k,0 (d, θ k ) = 1 ξ k,1 ( d, θ k ) π k,1 (d, θ k ) = ξ k,1 ( d, θ k ) ξ k,1 ( d, θ k )ξ k,2 ( d, θ k ) π k,2 (d, θ k ) = ξ k,1 ( d, θ k ) ξ k,2 ( d, θ k )

56 UTILITY-BASED DOSE-FINDING 18 Bivariate Distribution of (Y 1, Y 2 )

57 UTILITY-BASED DOSE-FINDING 18 Bivariate Distribution of (Y 1, Y 2 ) Use a Gaussian copula, given by C ρ (u, v) = Φ ρ {Φ 1 (u), Φ 1 (v)} for 0 u, v 1. Φ ρ = cdf of a bivariate std normal with correlation ρ Φ = usual N(0,1) cdf

58 UTILITY-BASED DOSE-FINDING 19 The cdf of each Y k is 0 ify < 0 1 π F k (y d, θ k ) = k,1 (d, θ k ) π k,2 (d, θ k ) if0 y < 1 1 π k,2 (d, θ k ) if1 y < 2 1 ify 2

59 UTILITY-BASED DOSE-FINDING 19 The cdf of each Y k is 0 ify < 0 1 π F k (y d, θ k ) = k,1 (d, θ k ) π k,2 (d, θ k ) if0 y < 1 1 π k,2 (d, θ k ) if1 y < 2 1 ify 2 The joint pmf of Y is π(y d, θ) = 2 a=1 2 ( 1) a+b C ρ (u a, v b ) b=1 u 1 = F 1 (y 1 d, θ), v 1 = F 2 (y 2 d, θ) u 2 = F 1 (y 1 1 d, θ), v 2 = F 2 (y 2 1 d, θ)

60 UTILITY-BASED DOSE-FINDING 20 π(y) = Φ ρ {Φ 1 F 1 (y 1 ), Φ 1 F 2 (y 2 )} Φ ρ {Φ 1 F 1 (y 1 1), Φ 1 F 2 (y 2 )} Φ ρ {Φ 1 F 1 (y 1 ), Φ 1 F 2 (y 2 1)} + Φ ρ {Φ 1 F 1 (y 1 1), Φ 1 F 2 (y 2 1)} θ = (θ 1, θ 2, ρ) and dim(θ) = 21

61 UTILITY-BASED DOSE-FINDING 21 Establishing Priors

62 UTILITY-BASED DOSE-FINDING 21 Establishing Priors 1. Elicit prior means of π k,1 (d) and π k,2 (d) for each d and outcome k = 1, 2

63 UTILITY-BASED DOSE-FINDING 21 Establishing Priors 1. Elicit prior means of π k,1 (d) and π k,2 (d) for each d and outcome k = 1, 2 2. Use an extension of the least squares method of Thall and Cook (2004) to solve for prior means of the 21 model parameters

64 UTILITY-BASED DOSE-FINDING 21 Establishing Priors 1. Elicit prior means of π k,1 (d) and π k,2 (d) for each d and outcome k = 1, 2 2. Use an extension of the least squares method of Thall and Cook (2004) to solve for prior means of the 21 model parameters 3. Calibrate prior variances to obtain reasonably small prior effective sample sizes of the priors on π k,1 (d) and π k,2 (d) = ESS =.20 to.70

65 UTILITY-BASED DOSE-FINDING 21 Establishing Priors 1. Elicit prior means of π k,1 (d) and π k,2 (d) for each d and outcome k = 1, 2 2. Use an extension of the least squares method of Thall and Cook (2004) to solve for prior means of the 21 model parameters 3. Calibrate prior variances to obtain reasonably small prior effective sample sizes of the priors on π k,1 (d) and π k,2 (d) = ESS =.20 to.70

66 UTILITY-BASED DOSE-FINDING 22 Establishing Utilities

67 UTILITY-BASED DOSE-FINDING 22 Establishing Utilities The Delphi method: A tool for establishing consensus among experts (Dalkey, 1969; Brook, Chassin, Fink, et al., 1986)

68 UTILITY-BASED DOSE-FINDING 22 Establishing Utilities The Delphi method: A tool for establishing consensus among experts (Dalkey, 1969; Brook, Chassin, Fink, et al., 1986) 1. Elicit utilites from each of several individuals

69 UTILITY-BASED DOSE-FINDING 22 Establishing Utilities The Delphi method: A tool for establishing consensus among experts (Dalkey, 1969; Brook, Chassin, Fink, et al., 1986) 1. Elicit utilites from each of several individuals 2. Show all elicited values (anonymously) to all individuals, and allow them to adjust their utilties

70 UTILITY-BASED DOSE-FINDING 22 Establishing Utilities The Delphi method: A tool for establishing consensus among experts (Dalkey, 1969; Brook, Chassin, Fink, et al., 1986) 1. Elicit utilites from each of several individuals 2. Show all elicited values (anonymously) to all individuals, and allow them to adjust their utilties 3. Repeat 2 or 3 times, and compute the mean across experts of the final values

71 UTILITY-BASED DOSE-FINDING 22 Establishing Utilities The Delphi method: A tool for establishing consensus among experts (Dalkey, 1969; Brook, Chassin, Fink, et al., 1986) 1. Elicit utilites from each of several individuals 2. Show all elicited values (anonymously) to all individuals, and allow them to adjust their utilties 3. Repeat 2 or 3 times, and compute the mean across experts of the final values Nadine Houede did this via questionnaire with 8 French medical oncologists who treat bladder cancer patients, using a utility scale of 0 to 100, and finished after 2 rounds

72 UTILITY-BASED DOSE-FINDING 24 French Utilities Elicited Using the Delphi Method

73 UTILITY-BASED DOSE-FINDING 24 French Utilities Elicited Using the Delphi Method Y 2 = 0 Y 2 = 1 Y 2 = 2 Y 2 PD SD CR/PR Inevaluable Y 1 = Y 1 = Y 1 =

74 UTILITY-BASED DOSE-FINDING 25 Using Utilities to Select Dose Pairs

75 UTILITY-BASED DOSE-FINDING 25 Using Utilities to Select Dose Pairs For U(y) = the numerical utility of outcome y, the mean utility for a patient treated with d is u(d, θ) = E Y {U(Y) d, θ} = y U(y) π(y d, θ)

76 UTILITY-BASED DOSE-FINDING 25 Using Utilities to Select Dose Pairs For U(y) = the numerical utility of outcome y, the mean utility for a patient treated with d is u(d, θ) = E Y {U(Y) d, θ} = y U(y) π(y d, θ) Given current data n = {(Y 1, d 1 ),, (Y n, d n )}, select d opt (data n ) = argmax d D θ u( d, θ)p(θ data n )dθ = argmax d U(y) D y θ π (y d, θ)p(θ data n )dθ

77 UTILITY-BASED DOSE-FINDING 26 Additional Safety Rules

78 UTILITY-BASED DOSE-FINDING 26 Additional Safety Rules 1. Do Not Skip Untried Dose Pairs: If (d 1, d 2 ) is the current dose pair, then escalation is allowed to as yet untried pairs (d 1 + 1, d 2 ), (d 1, d 2 + 1), or (d 1 + 1, d 2 + 1).

79 UTILITY-BASED DOSE-FINDING 26 Additional Safety Rules 1. Do Not Skip Untried Dose Pairs: If (d 1, d 2 ) is the current dose pair, then escalation is allowed to as yet untried pairs (d 1 + 1, d 2 ), (d 1, d 2 + 1), or (d 1 + 1, d 2 + 1). 2. Stop The Trial if All Dose Pairs are Too Toxic: For π max 1,2 = fixed upper limit on Pr(Y 2 = 2), and p U = a fixed upper probability cut-off (e.g..80 to.90), terminate accrual if min d Pr{π 1,2(d, θ) > π max 1,2 data} > p U

80 UTILITY-BASED DOSE-FINDING 27 Simulations

81 UTILITY-BASED DOSE-FINDING 27 Simulations N max = 48, choose d for cohorts of 3 patients, start at d = (2,2), safety stopping rule applied with π max 1,2 =.33 and p U =.80.

82 UTILITY-BASED DOSE-FINDING 27 Simulations N max = 48, choose d for cohorts of 3 patients, start at d = (2,2), safety stopping rule applied with π max 1,2 =.33 and p U =.80. A simulation scenario is a vector π true (d) of fixed probability values for the 12 d pairs, with ρ true = u true (d) = y U( y) π true (y d)

83 UTILITY-BASED DOSE-FINDING 27 Simulations N max = 48, choose d for cohorts of 3 patients, start at d = (2,2), safety stopping rule applied with π max 1,2 =.33 and p U =.80. A simulation scenario is a vector π true (d) of fixed probability values for the 12 d pairs, with ρ true = u true (d) = y U( y) π true (y d) 1000 runs per scenario

84 UTILITY-BASED DOSE-FINDING 28

85 UTILITY-BASED DOSE-FINDING 29 Scenario 1 Chemo Agent Dose True Utility Prob Select Num Patients Chemo Agent Dose Chemo Agent Dose Biol Agent Dose 1 Biol Agent Dose 2 Biol Agent Dose 3 Biol Agent Dose 4

86 UTILITY-BASED DOSE-FINDING 30

87 UTILITY-BASED DOSE-FINDING 31 Scenario 2 Chemo Agent Dose True Utility Prob Select Num Patients Chemo Agent Dose Chemo Agent Dose Biol Agent Dose 1 Biol Agent Dose 2 Biol Agent Dose 3 Biol Agent Dose 4

88 UTILITY-BASED DOSE-FINDING 32

89 UTILITY-BASED DOSE-FINDING 33 Scenario 3 Chemo Agent Dose True Utility Prob Select Num Patients Chemo Agent Dose Chemo Agent Dose Biol Agent Dose 1 Biol Agent Dose 2 Biol Agent Dose 3 Biol Agent Dose 4

90 UTILITY-BASED DOSE-FINDING 34

91 UTILITY-BASED DOSE-FINDING 35 Scenario 4 Chemo Agent Dose True Utility Prob Select Num Patients 2.9 Chemo Agent Dose Chemo Agent Dose Biol Agent Dose 1 Biol Agent Dose 2 Biol Agent Dose 3 Biol Agent Dose 4

92 UTILITY-BASED DOSE-FINDING 36 Additional Simulation Results

93 UTILITY-BASED DOSE-FINDING 36 Additional Simulation Results 1. Safe : π max 1,2 =.33 and p U =.80 = Large PSTOP in toxic scenarios

94 UTILITY-BASED DOSE-FINDING 36 Additional Simulation Results 1. Safe : π max 1,2 =.33 and p U =.80 = Large PSTOP in toxic scenarios 2. Insensitive to Cohort Size : c = 1 versus c = 3 give the virtually same results under Scenarios 1 4, but slightly larger PSTOP for c = 1 (93% versus 90%) under the toxic Scenario 5

95 UTILITY-BASED DOSE-FINDING 36 Additional Simulation Results 1. Safe : π max 1,2 =.33 and p U =.80 = Large PSTOP in toxic scenarios 2. Insensitive to Cohort Size : c = 1 versus c = 3 give the virtually same results under Scenarios 1 4, but slightly larger PSTOP for c = 1 (93% versus 90%) under the toxic Scenario 5 3. Consistent : Pr{Select d having largest u true (d)} with N max

96 UTILITY-BASED DOSE-FINDING 36 Additional Simulation Results 1. Safe : π max 1,2 =.33 and p U =.80 = Large PSTOP in toxic scenarios 2. Insensitive to Cohort Size : c = 1 versus c = 3 give the virtually same results under Scenarios 1 4, but slightly larger PSTOP for c = 1 (93% versus 90%) under the toxic Scenario 5 3. Consistent : Pr{Select d having largest u true (d)} with N max 4. Stupid Priors Give Stupid Designs : A naive uninformative prior with E(θ) = 0 and all prior standard deviations = 1000 gives terrible results

97 UTILITY-BASED DOSE-FINDING 37 Conclusions

98 UTILITY-BASED DOSE-FINDING 37 Conclusions 1. Utilities including Future Patient Benefit, Cost, Expected Profit, etc. may lead to better designs and more publications.

99 UTILITY-BASED DOSE-FINDING 37 Conclusions 1. Utilities including Future Patient Benefit, Cost, Expected Profit, etc. may lead to better designs and more publications. 2. French utilities are not necessarily the same as American utilities = Think globally but elicit utilities locally.

100 UTILITY-BASED DOSE-FINDING 37 Conclusions 1. Utilities including Future Patient Benefit, Cost, Expected Profit, etc. may lead to better designs and more publications. 2. French utilities are not necessarily the same as American utilities = Think globally but elicit utilities locally. 3. Many extensions are possible, and this also may lead to better designs and more publications.

101 UTILITY-BASED DOSE-FINDING 37 Conclusions 1. Utilities including Future Patient Benefit, Cost, Expected Profit, etc. may lead to better designs and more publications. 2. French utilities are not necessarily the same as American utilities = Think globally but elicit utilities locally. 3. Many extensions are possible, and this also may lead to better designs and more publications.