Optimizing Combination Therapy under a Bivariate Weibull Distribution, with Application to Toxicity and Efficacy Responses

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1 Optimizing Combination Therapy under a Bivariate Weibull Distribution, with Application to Toxicity and Efficacy Responses Kim, Sungwook (Peter) University of Missouri Issac Newton Institute for Mathematical Sciences Nov,

2 1 Context Two Dependent Binary Outcomes at dose x Probabilities 2 Motivation Motivation for Bivariate Weibull Distribution Plots 3 Model Development Methods for Constructing Bivariate Distributions Bivariate Exponential Distribution Bivariate Weibull Distribution 4 Information matrix Likelihood Information matrix 5 Rest of This Project Rest of This Project

3 Two Dependent Binary Outcomes at dose x Two Dependent Binary Outcomes at dose x Consider two dependent binary outcomes,* U for efficacy and V for toxicity. Outcome probabilities given the dose level x are defined as follow. p uv = Pr(U = u, V = v X = x), u, v = 0, 1 Toxicity 1 0 Efficacy 1 p 11 p 10 p 1. 0 p 01 p 00 p 0. p.1 p.0 1 *Motivated by Dragalin, Fedorov and Wu (2006).

4 Probabilities Outcome Probabilities as function of dose Outcome probabilities are p 11 = y 0 p 10 = y 0 z z p 01 = z y p 00 = y z 0 f (y, z)dzdy f (y, z)dzdy 0 f (y, z)dzdy f (y, z)dydz where f (y, z) is the bivariate Weibull density.

5 Motivation for Bivariate Weibull Distribution Why the Bivariate Weibull Distribution? 1 Natural Dose Range 0 < Dose <

6 Motivation for Bivariate Weibull Distribution Why the Bivariate Weibull Distribution? 1 Natural Dose Range 0 < Dose < 2 Flexibility 6 parameters to regress on one drug; can extend to multiple drugs and covariates. **Other possible bivariate distributions considered -Bivariate Gamma distribution -Bivariate Beta distribution -Bivariate F-distribution

7 Plots Plots Bivariate Exponential Dist. σ = 1 Bivariate Weibull Dist. σ = 1.2

8 Plots Plots Bivariate Weibull Dist. σ = 1.4 Bivariate Weibull Dist. σ = 1.6

9 Plots Plots Bivariate Weibull Dist. σ = 1.8 Bivariate Weibull Dist. σ = 2

10 Plots Plots Bivariate Weibull Dist. σ = 2.5 Bivariate Weibull Dist. σ = 3

11 Methods for Constructing Bivariate Distributions Methods for Constructing Bivariate Distribution 1 Combine a Conditional Distribution with a Marginal Distribution h(x, y) = f (x)g(y x)

12 Methods for Constructing Bivariate Distributions Methods for Constructing Bivariate Distribution 1 Combine a Conditional Distribution with a Marginal Distribution h(x, y) = f (x)g(y x) 2 Copula Methods - The Inversion Method - Geometric Methods - Algebraic Methods - Rüschendorf s Method - Distortion Function

13 Bivariate Exponential Distribution Start with Independent Exponential Distribution, Freund (1961) Let T 1 and T 2 have distributions f (t 1 ) = β 1 exp( β 1 t 1 ) for t 1 > 0 f (t 2 ) = β 2 exp( β 2 t 2 ) for t 2 > 0. Then the probability that T 1 fails (occurs) at t 1 and T 2 has not yet failed is t P(T 1 < t, T 2 > t 1 ) t=t1 = β 1 e β 1t 1 e β 2t 1 = β 1 e (β 1+β 2 )t 1.

14 Bivariate Exponential Distribution Freund (1961) uses Conditional Method Introduce to Dependency. 1 The probability that T 1 fails (occurs) at t 1 and T 2 has not yet failed is t P(T 1 < t, T 2 > t 1 ) t=t1 = β 1 e β 1t 1 e β 2t 1 = β 1 e (β 1+β 2 )t 1.

15 Bivariate Exponential Distribution Freund (1961) uses Conditional Method Introduce to Dependency. 1 The probability that T 1 fails (occurs) at t 1 and T 2 has not yet failed is t P(T 1 < t, T 2 > t 1 ) t=t1 = β 1 e β 1t 1 e β 2t 1 = β 1 e (β 1+β 2 )t 1. 2 Specify, furthermore, that the probability density of T 2 given that T 1 fails first at t 1 is t P(T 2 < t T 2 > t 1 ) t=t2 >t 1 = β 2e β 2 (t 2 t 1 ) for 0 < t 1 < t 2.

16 Bivariate Exponential Distribution Freund (1961) uses Conditional Method Introduce to Dependency. 1 The probability that T 1 fails (occurs) at t 1 and T 2 has not yet failed is t P(T 1 < t, T 2 > t 1 ) t=t1 = β 1 e β 1t 1 e β 2t 1 = β 1 e (β 1+β 2 )t 1. 2 Specify, furthermore, that the probability density of T 2 given that T 1 fails first at t 1 is t P(T 2 < t T 2 > t 1 ) t=t2 >t 1 = β 2e β 2 (t 2 t 1 ) for 0 < t 1 < t 2. 3 f (T 1 = t 1, T 2 = t 2 ) = β 1 e (β1+β2)t1 β 2e β 2 (t2 t1) for 0 < t 1 < t 2 <

17 Bivariate Exponential Distribution Freund (1961) Analogously 1 The probability that T 2 fails at t 2 and T 1 has not yet failed is t P(T 2 < t, T 1 > t 2 ) t=t2 = β 2 e β 2t 2 e β 1t 2 = β 2 e (β 1+β 2 )t 2.

18 Bivariate Exponential Distribution Freund (1961) Analogously 1 The probability that T 2 fails at t 2 and T 1 has not yet failed is t P(T 2 < t, T 1 > t 2 ) t=t2 = β 2 e β 2t 2 e β 1t 2 = β 2 e (β 1+β 2 )t 2. 2 The probability density of T 1 given that T 2 fails first at t 2 is t P(T 1 < t T 1 > t 2 ) t=t1 >t 2 = β 1e β 1 (t 1 t 2 ) for 0 < t 2 < t 1.

19 Bivariate Exponential Distribution Freund (1961) Analogously 1 The probability that T 2 fails at t 2 and T 1 has not yet failed is t P(T 2 < t, T 1 > t 2 ) t=t2 = β 2 e β 2t 2 e β 1t 2 = β 2 e (β 1+β 2 )t 2. 2 The probability density of T 1 given that T 2 fails first at t 2 is t P(T 1 < t T 1 > t 2 ) t=t1 >t 2 = β 1e β 1 (t 1 t 2 ) for 0 < t 2 < t 1. 3 f (T 1 = t 1, T 2 = t 2 ) = β 2 e (β1+β2)t2 β 1e β 1 (t1 t2) for 0 < t 2 < t 1 <

20 Bivariate Exponential Distribution Freund (1961) In Summary Bivariate Exponential Distribution f (t 1, t 2 ) = { β 1 e (β1+β2)t1 β 2 e β 2 (t2 t1) β 2 e (β1+β2)t2 β 1 e β 1 (t1 t2) for 0 < t 1 < t 2 < for 0 < t 2 < t 1 < = { β 1 (β 2 )e (β 2 )t2 (β1+β2 β 2 )t1 β 2 (β 1 )e (β 1 )t1 (β1+β2 β 1 )t2 for 0 < t 1 < t 2 < for 0 < t 2 < t 1 <

21 Bivariate Exponential Distribution Marshall-Olkin (1967) Introduced Dependency between Exponential Random Variables using Copula Methods F M (t 1, t 2 ) = P[T 1 > t 1, T 2 > t 2 ] = exp[ β 1 t 1 β 2 t 2 β 3 Max(t 1, t 2 )], β 1, β 2, β 3 > 0. Proschan-Sullo (1974) repeat steps taken by Freund, starting with marginal distribution T 1 and T 2 obtained from Marshall-Olkin to get bivariate exponential distribution with three parameters.

22 Bivariate Exponential Distribution Proschan-Sullo (1974) Bivariate Exponential Distribution Proschan-Sullo proposed BVE which is the combination of both Freund (1961) and Marshall-Olkin (1967). β 1 (β 2 + β 3)e (β 2 +β3)t2 (β1+β2 β 2 )t1 for 0 < t 1 < t 2 < f (t 1, t 2 ) = β 2 (β 1 + β 3)e (β 1 +β3)t1 (β1+β2 β 1 )t2 for 0 < t 2 < t 1 < β 3 e (β1+β2+β3)t1 for 0 < t 1 = t 2 <

23 Bivariate Exponential Distribution Recall: Transformation from Exponential to Weibull Using the transformation Y = T 1 σ 1 and Z = T 1 σ 2 If T 1 is distributed β 1 e β 1t 1, then Y is distributed β 1 σy σ 1 e β 1y σ. If T 2 is distributed β 2 e β 2t 2, then Z is distributed β 2 σz σ 1 e β 2z σ.

24 Bivariate Weibull Distribution Bivariate Weibulll Distribution Bivariate Weibulll Distribution David D. Hanagal (2005) proposed a bivariate Weibull model by means of the transformations T 1 = Y σ and T 2 = Z σ, σ > 0 from the bivariate exponential distribution proposed by Proschan and Sullo. (1974). β 1(β 2 + β 3)σ 2 (yz) σ 1 e (β 2 +β 3)z σ (β 1 +β 2 β 2 )y σ for 0 < y < z < f (y, z) = β 2(β 1 + β 3)σ 2 (yz) σ 1 e (β 1 +β 3)y σ (β 1 +β 2 β 1 )zσ for 0 < z < y < β 3σy σ 1 e (β 1+β 2 +β 3 )y σ for 0 < y = z < Bivariate Regression θ 1 = (θ 11, θ 12), θ 2 = (θ 21, θ 22), x=(1, x) T 1 = Y σ e σθ 1x, T 2 = Z σ e σθ 2x

25 Bivariate Weibull Distribution Outcome Probabilities as function of dose Outcome probabilities are p 11 = y 0 p 10 = y 0 z z p 01 = z y p 00 = y z 0 f (y, z)dzdy f (y, z)dzdy 0 f (y, z)dzdy f (y, z)dydz where f (y, z) is the bivariate Weibull density.

26 Bivariate Weibull Distribution Range of Integration for p 10 : Case 1 (y z ) and Case 2 (y < z )

27 Bivariate Weibull Distribution p 10 = P(U = 1, V = 0 X = x) where U= I(Efficacy), and V= I(Toxicity) p 10 = P(U = 1, V = 0 X = x) = y z 0 f (y, z)dydz = [ z y f (y, z)i (y < z)dzdy + f (y, z)i (y < z)dzdy 0 z z y y y ] y + f (y, z)i (z = y)dy + f (y, z)i (z < y)dzdy I (y z ) z z z [ y ] + f (y, z)i (y < z)dzdy I (y < z ). 0 z

28 Bivariate Weibull Distribution p 1. = P(U = 1 X = x) where U= I(Efficacy) p 1. = P(U = 1 X = x) = y 0 0 f (y, z)dydz = y y 0 0 f (y, z)i (z < y)dzdy+ = β 2(β 1 + β ( 3) 1 β 1 + β 2 β 1 (β 1 + β 3) β ( 1 + β 1 + β 2 + β 3 y 0 y (1 e (β 1 +β 3)y σ ) 1 e (β 1+β 2 +β 3 )y σ ) + f (y, z)i (y < z)dzdy+ y 0 f (y, z)i (y = z)dy 1 ( )) 1 e (β σ 1+β 2 +β 3 )y (β 1 + β 2 + β 3 ) β ( ) 1 1 e (β 1+β 2 +β 3 )y σ. β 1 + β 2 + β 3

29 Bivariate Weibull Distribution p.1 = P(V = 1 X = x) where V= I(Toxicity) p.1 = P(V = 1 X = x) = z 0 0 f (y, z)dzdy = z z 0 0 f (y, z)i (y < z)dydz+ = β 1(β 2 + β ( 3) 1 β 1 + β 2 β 2 (β 2 + β 3) β ( 2 + β 1 + β 2 + β 3 z 0 z (1 e (β 2 +β 3)z σ ) 1 e (β 1+β 2 +β 3 )z σ ) + f (y, z)i (z < y)dydz+ z 0 f (y, z)i (y = z)dz 1 ( )) 1 e (β σ 1+β 2 +β 3 )z (β 1 + β 2 + β 3 ) β ( ) 3 1 e (β 1+β 2 +β 3 )z σ. β 1 + β 2 + β 3

30 Likelihood Likelihood L(Θ u, v; x) = p11 uv pu(1 v) 10 p (1 u)v 01 p (1 u)(1 v) 00 l(θ u, v; x) = (u)(v)ln(p 11 )+(u)(1 v)ln(p 10 )+(1 u)(v)ln(p 01 )+(1 u)(1 v)ln(p 00 ) (**p 11 = 1 p 10 p 01 p 00 )

31 Information matrix Partial derivative (Tentative) l(θ u,v;x) Θ = ABC; where Θ = {θ 11, θ 12, θ 21, θ 22, β 1, β 2, β 3, β 1, β 2, σ} A = diag( θ 1 θ 11, θ 1 θ 12, θ 2 θ 21, θ 2 θ 22, 1, 1, 1, 1, 1, 1)

32 Information matrix Partial derivative (Tentative) BC = p 10 θ 1 p 10 p 01 θ 1 p 01 p 00 θ 1 p 00 θ 1 p 10 θ 1 p 01 θ 1 p 00 θ 2 p 10 θ 2 p 01 θ 2 p 00 θ 2 p 10 θ 2 p 01 θ 2 p 00 β 1 β 1 β 1 p 10 β 2 p 01 β 2 p 00 β 2 p 10 β 3 p 01 β 3 p 00 β 3 p 10 p 01 p 00 β 1 β 1 β 1 p 10 p 01 p 00 β 2 β 2 β 2 p 10 p 01 p 00 σ σ σ u(1 v) uv p 10 1 p 10 p 01 p 00 (1 u)(v) uv p 01 1 p 10 p 01 p 00 (1 u)(1 v) uv p 00 1 p 10 p 01 p 00

33 Information matrix Information matrix (Tentative) Since, l(θ u,v;x) Θ = ABC; [ ( l(θ u, ) ( ) ] T v; x) l(θ u, v; x) I (Θ) = E = E ( (ABC)(ABC) T ) Θ Θ = E ( ABCC T B T A T ) = AB [ E ( CC T )] (AB) T.

34 Information matrix Information matrix (Tentative) ( ) E CC T = p 10 1 p 10 p 01 p 00 1 p 10 p 01 p 00 1 p 10 p 01 p p 10 p 01 p 00 p 01 1 p 10 p 01 p 00 1 p 10 p 01 p p 10 p 01 p 00 1 p 10 p 01 p 00 p 00 1 p 10 p 01 p 00 } {{ } D p 10 ( ) 1 = p p p 01 p p 00 }{{} D (Because U is distributed Bernoulli(p 1. ), and V is distributed Bernoulli(p.1 ) and U and V are dependent.)

35 Rest of This Project Rest of This Project 1 Canonical Form

36 Rest of This Project Rest of This Project 1 Canonical Form 2 Compute Optimal Design

37 Rest of This Project Rest of This Project 1 Canonical Form 2 Compute Optimal Design 3 Penalty Function

38 Rest of This Project Fin Thank you for your attention!