Dose-Finding Experiments in Clinical Trials. Up and Down and

Transcription

1 Workshop on the Design Analysis of Clinical Trials Institute of Mathematical Sciences National University of Singapore 27 October 2011 Dose-Finding Experiments in Clinical Trials Department of Statistics University of Missouri, Columbia, MO

2 Basic Response Function Shape Typical dose-response curves Efficacy, p 1. Toxicity, p. 1 Efficacy without toxicity, p 10 curtesy of V. Fedorov

3 Outline for Toxicity, e.g., 4 (compared with Adaptive D-Optimal ) 5 Some Remarks About Tradeoffs

4 Outline

5 Popular 3+3 Rule (Korn et al., 1994 ) X(m) = dose allocated to mth cohort of 3 subjects. 1 Begin at a safe dose: X(1) = x 1. 2 If X(m) = x j, x j+1 if 0/3 positive responses; go to 2; X(m + 1) = x j if 1/3 positive responses; go to 3; stop if 2/3 positive responses. 3 Evaluate an additional cohort of 3 patients at x j : { x j+1 if 1/6 respond positively; go to 2 X(m + 2) = stop if 2/6 respond positively. When the trial stops, say at x k, go forward with x k 1. The risk of choosing an incorrect level is large Reiner, Paoletti O Quigley (1999).

6 Outline

7 is a dose escalation procedure, period. There is no going down. up down von Bétsky, 1947; Dixon & Mood, 1948

8 P(Toxicity) Suppose the dose is decreased if a toxicity is observed

9 P(Toxicity) P(No Toxicity) Cluster allocations around LD50 1 decrease dose if toxicity is observed 2 increase dose if no toxicity is observed Derman, 1957, Durham &, 1994

10 P(Up) = (.3/.7)P(No Toxicity) clusters allocations around the 30% toxicity rate Derman, 1957; Durham &, 1994; Durham &, 1995; Durham, Haghighi, 1995; Giovagnoli & Pintacuda, 1998, Bortot, Giovagnoli, 2005; Baldi Antognini, Bortot, & Giovagnoli, 2008

11 Treatment Allocations Targeting.3 Change P(UP) to cluster allocations around LD30 1 go down if toxicity 2 go up if no toxicity heads wp.3/.7 (1) Durham &, Stylianou (2002) recommend estimating target by isotonic regression.

12 Treatment Allocations Targeting.3 Change P(UP) to cluster allocations around LD30 1 go up if 0 toxicity in group of size 2 2 go down if 1 2 toxicities 3 P(up) = P(down) ( 2 0 ) (1 p) 0 = ( 2 1 ) ( 2 p ) p 2 (1) Wetherill, 1963 (2) Tsutatkawa, (3) Gezmu &, 2006; Stylianou (2002) recommend estimating target by isotonic regression.

13 Expected Allocations Targeting LD10, Many Doses

14 Expected Allocations Targeting LD10, Many Doses 45

15 P(Toxicity) P(No Efficacy) Sounds good, but wrong target 1 decrease dose if toxicity is observed 2 increase dose if no efficacy is observed Derman, 1957; Durham &, 1994

16 Up if no Efficacy Down if Toxicity Sounds good, but wrong target 1 Go down if toxicity 2 Go up if no efficacy Allocations cluster around dose where which is not the dose P( no toxicity ) = P( efficacy ) arg max P( no toxicity efficacy ). x X (Trival corollary of Durham &, 1994). simulated in Gooley et al, 1994; discussed in Kpamegan &, 2001; see also Ivanova, 2003

17 : Discretized Kiefer-Wolfowitz Allocations cluster around dose arg max x X P{ no toxicity efficacy } Right target; but clinically unappealing If the nth pair of subjects has been treated at X(n) 2 X(n) + 2, the midpoint of the (n + 1)th pair is V(n) = X(n + 1) = X(n) + V (n), 1 if treatment at X(n) 2 results in success treatment at X(n) + 2 in failure. 0 if the nth pair of treatments result in 2 successes or 2 failures. 1 if the treatment at X(n) 2 results in failure the treatment X(n)+ 2 in success. Kpamegan &, 2001, 2008

18 Outline

19 Transition Probabilities From dose x i, the transition probability in one trial in n 0 trials, respectively, are p ik = P{X(n + 1) = x k X(n) = x i }; p ik (n) = P{X(n + 1) = x k X(1) = x i }; (i, k) = 1,..., K. Of course, p ik (0) = δ ik, where δ ik is Kronecker s delta function. Note: because the transitions are assumed to follow a rom walk, p ij = 0 for i j > 1.

20 Probabilities of a Rom Walk Rule Given sample space X = {x 1 < x 2 < < x K } the initial dose X(1), the rom walk rule is determined by 1 p j,j 1 =P(decreasing the dose from x j to x j 1 ) 2 p j,j+1 =P(increasing the dose from x j to x j+1 ) 3 p jj =P(performing another trial at x j ); 4 p j,j 1 + p jj + p j,j+1 = 1 for all k.

21 Example: A Biased Coin Rule 1 go down if toxicity 2 go up if no toxicity coin flip yields heads 3 use same dose if no toxicity coin flip yields tails Then p j,j 1 = P{toxicity X(n) = x j }; p j,j+1 = P{no toxicity X(n) = x j heads}, p jj = P{no toxicity X(n) = x j tails}.

22 Example: Group Rule Treat in cohorts of size s. Let Y (m) be the number of toxicities in mth cohort. Let c L c U be two integers such that 0 c L < c U s. Given the mth cohort is treated at X(m) = x j on the interior of X, the m + 1st cohort of s subjects is assigned to x j 1 if Y (m) c U ; X(m + 1) = x j+1 if Y (m) c L ; x j if c L < Y (m) < c U. p j,j 1 = P{Y (m) c U X(m) = x j }; p j,j+1 = P{Y (m) c L X(m) = x j }

23 Example: Opposing Failure Functions 1 go down if treatment is toxic 2 go up treatment is not effective (no efficacy) Then p j,j 1 = P{toxicity X(n) = x j }; p j,j+1 = P{no efficacy X(n) = x j }

24 Transition Probability Matrix p 11 p p 21 p 22 p P = pk 1,K 2 p K 1,K 1 p K 1,K 0 0 p K,K 1 p KK. (1) The (i, k) element of P n is p ik (n). {p ij : x i x j 1} depend on the definition of a particular up down rule on the probability of response at the current dose x i.

25 Simple Regularity Conditions If the allocation rule allows each dose to be reached, eventually, from every other dose, the matrix P, hence the up down rule, is called regular. When {p i,i 1 } {p i,i+1 } are non-degenerate functions of an increasing response function F(x), bounded away from 0 1, if at least one {p ii } is nonzero so the allocation process will not be not periodic, then the allocation rule will be regular.

26 Asymptotic Properties of Regular Rules For any initial treatment distribution p, the asymptotic probability of allocating at x k is lim P p{x(n) = x k } = w k, k = 1,..., K n exist are unique. Empirical proportion of allocations to x k is W k (n) w k as n by LLNs for regular Markov chains. n ((W1 (n),..., W K (n)) w) ( ( )) n N 0, diag σ1 2 ( P),..., σk 2 ( P). Convergence is exponentially fast!!!

27 Deriving the Asymptotic Allocation Distribution For rom walks, asymptotic allocation probabilities {w k } are solutions to the balance equations: w k = w k 1 p k 1,k + w k p kk + w k+1 p k+1,k, k = 1,..., K, yield the explicit solution: w k = k λ j, j=1 λ j = p j 1,j p j,j 1, k = 2,..., K ; λ 1 1 = 1+ K k=2 j=2 k λ j. The formula for λ 1 stardizes all the asymptotic allocation probabilities so they sum to one.

28 Unimodality of the Allocation Distribution If {λ j } decreases with j, then w will be unimodal with w j < w j+1 { so long as λj > 1, or equivalently, if p j,j 1 < p j 1,j ; w j = w j 1 { if λj = 1; or equivalently, if p j,j 1 = p j 1,j ; w j < w j+1 { when λj < 1 or equivalently, if p j,j 1 > p j 1,j. If {λ j } decreases with j λ 1 1 λ K, the mode of the asymptotic allocations will be max{x j X : p j,j 1 p j 1,j }

29 Example: D&F Biased Coin Rule Let b = P{heads}. The mode of the asymptotic allocations will be max x j X : p j,j 1 p j 1,j P{toxicity X(n) = x j } P{no toxicity heads X(n) = x j 1 } P{toxicity X(n) = x j } b (1 P{toxicity X(n) = x j 1 }) Set b equal to b P{toxicity X(n) = x j} 1 P{toxicity X(n) = x j } P{ toxicity x = target} 1 P{toxicity x = target}

30 D&F BCR with Logistic Responses P{toxicity at x}= F(x) = (1 + exp(α + βx)) 1 with X = {x 1 < x 1 +, x < < x K }. F (x Γ ) = Γ. The asymptotic allocation proportions are a mixture of two discrete distributions: W w j = lim k (n) n n ( ) xj (x Γ 0.5 ) (1 Γ)φ σ ( ) xj (x Γ ) + Γφ, σ where σ = /β; mixing parameter is Γ = P{target toxicity}.

31 BCR Component Distributions are Discrete Normal φ(z) = P{Z = z} where Z is a rom variable defined on a discrete real valued points X, φ(z j ) = exp( 1 2 z2 j ) K i=1 exp( 1 2 z2 j ), z k X.

32 The Distribution of Allocations The modes of the two discrete normal distributions are separated only by one stepsize. If Γ = 0.5, the component discrete normal distributions are equally weighted. Whereas if Γ is very small, the asymptotic distribution is dominated by a single discrete normal distribution. Because many common response function models are very similar, except in the extreme tails, this result will hold in general, approximately, it provides insight into the distributions of allocations to be expected when running the biased coin design.

33 In addition, the asymptotic allocation probabilities can be written as a mixture of allocation probabilities conditioned on the responses: w j = (1 Γ)P{X(n) = x Y (n) = 0}+ΓP{X(n) = x Y (n) = 1}. In general, the mode of {w j } is bounded within x Γ ±. But with logistic responses, the difference between the mode of {w j } x Γ can be bounded more tightly. Let κ = max{j λ j > 1}. Then x Γ is bounded as x κ < x Γ < x κ +.5, 0.0 Γ 0.5

34 Group Up-Down Rule: UD(s, c L, c U ) any combination of c L, c U s such that p j,j 1 = p jj +1 will cause the treatment mode to be approximately x q. Y (m) B(s, q), so p j,j 1 = s r=c U ( s r c L ( s p j,j+1 = r r=0 ) F(x j ) r ( 1 F(x j ) ) s r ) F(x j ) r ( 1 F(x j ) ) s r. G&F show {p j,j 1 } increases with {j} {p j,j+1 } decreases with {j} {λ j } decreases, so allocations are unimodal [Durham &, 1994]; treatment mode is arg max x X {p j,j 1 p jj +1}.

35 Example: UD(s, 0, 1) Go up if 0 of s toxicities are observed in the cohort; Go down if 1 toxicities is observed. The approximate criterion for the allocation mode is s r=1 ( s r p j,j 1 = p j,j+1 ) F(x j ) r ( 1 F(x j ) ) ( ) s r s = F(x 0 j ) 0 ( 1 F(x j ) ) s 0 1 ( 1 F(x j ) ) s = ( 1 F(xj ) ) s ( 1 F(xj ) ) s = 1/2 So choose s to give P{no toxicity at target dose}= s (1/2).

36 Example: Opposing Failure Functions 1 go down if treatment is toxic 2 go up treatment is not effective (no efficacy) Balance equation is p j,j 1 = p j,j+1 P{toxicity X(n) = x j } = P{no efficacy X(n) = x j } Since the allocation mode is where these two intensity functions are equal, allocation mode is not at the point of maximum P{efficacy no toxicity}

37 Estimating the Dose µ with Target Toxicity Γ 1 parametric - MLE With location-scale response model p i = F i = F ( xi α β ), the MLE of µ is ˆµ = ˆβF 1 (Γ) + ˆα ML estimates often do not exist in small studies. Because allocations are concentrated around a target dose, ML estimates of ˆβ are typically very poor, yielding poor estimates of µ. 2 non-parametric - isotonic regression Quality of estimate depends on sample sizes at doses neighboring the target. Estimates of toxicity at each dose converge as binomial rom variables.

38 Outline

39 Current work at Isaac Newton Institute with Val Fedorov Yuehui Wu at GSK.

40 1 Estimate the target dose 2 Treat next subject at that dose Bayesian frequentist best intention designs are proliferating: Recent examples include Li, Durham (1995), West (1997), Conaway (2004), Smith, et al.(2006), Bartroff Lai (2010), Cheung (2010) Thall (2010).

41 Dose finding versus Estimation

42 Selecting the MTD Proportion of Correct Selection of MTD by by U&D with groups of size 2 after 8 cohorts Senario MTD U&D Uniform Gamma Normal Lognormal Weibull Logistic From Oron & Hoff (2011) Oron & Hoff give similar results for 16 cohorts; similar estimation performance also found by Durham, Rosenberger (1997)

43 Number of Cohorts Allocated to MTD: left; U&D right; Oron, et al., 2011 Number of Runs , Normal Scenario Number of Runs U&D, Normal Scenario Number of Cohorts Allocated to MTD Number of Cohorts Allocated to MTD Number of Runs Number of Runs , Gamma Scenario Number of Cohorts Allocated to MTD, Lognormal Scenario Number of Runs Number of Runs U&D, Gamma Scenario Number of Cohorts Allocated to MTD U&D, Lognormal Scenario Number of Cohorts Allocated to MTD Number of Cohorts Allocated to MTD Figure 5: Between-run between-scenario variability. The histograms depict the distribution of the number of cohorts (excluding the first one) that have been allocated the true MTD during a single specific run. The ensemble size is 1000 runs. Scenarios are Normal (top), Gamma (middle) Lognormal (bottom); designs are one-parameter power (left) GU&D (right), both with cohort size 2.

44 Impossibility Theorem: d j =MTD x n F n 1, y n F n 1 Bernoulli (f (x n )) {x n } n=1 is a design { } Sequence of estimators MTDn n=1 consistent with respect to a given design if MTD n d j as for all increasing functions f, or equivalently, by discreteness, P( N such that for all n N, MTD n = d j ) = 1 is said to be strongly Under this framework, there exists no design that satisfies for all increasing functions f, P( N such that for all n N, x n = d j ) = 1) = 1, or equivalently, that P(x n d j i.o.) = 0 Azriel, Mel & Rinott, 2011

45 Azriel (2011) proves Cheung & Chappell (2002) conjecture for consistency Oron et al. (2011) sampled dose-response curves found conditions met 26% of the time. Azriel sampled curves that satisfy the commonly used power function prior found conditions satisfied 43% of the time. How to use this information? Cheung (2011) provides instructions for constructing priors that will satisfy his conditions. Is constructing priors to make an algorithm work consistent with Bayesian philosophy? What about historical experience expert opinion?

46 that are consistent for the MTD from Azriel, et al., 2011 Using MTD based on isotonic estimators, consistent designs have been proposed by 1 Ivanova, et al. (2003??) 2 Ivanova Kim (2009) 3 Azriel, et al. (2011) Robbins Munro procedures with suitably bounded gain /or restricted estimators.

47 Tradeoffs For increasing response functions, the dose that minimizes the variance of the estimated target is the target itself, except if target quantile is extreme. This suggests a good design will converge quickly to the target dose. Converging quickly to the target dose costs in terms of information about slope of toxicity function at the target. Converging quickly can cause convergence to the wrong dose; it may converge before reaching the target. (Chang & Ying, 2009)

48 Tradeoffs For long memory designs, first dose is very influential on converge rate. This has been long known for stochastic approximation is periodically rediscovered, as by Resche-Rigon, Zohar Chevret (2008), concerning the. Long memory designs, based on maximum likelihood or Bayesian estimates, may not converge or converge to the wrong dose with substantial probability. Up & down designs with Markovian rules for changing doses are short memory designs, with well-known analytical finite operating characteristics, that converge exponentially fast to a distribution clustered around the target.

49 with Utility ζ(x, θ) Type I best dose: x (θ) = arg min { ζ(x, θ) U }, x X where U is a predefined constant, e.g., ζ(x, θ) = E(y x, θ) = θ 1 + θ 2 x y Normal(E(y x, θ), 1); ζ(x, θ) = P(y = 1 x, θ) = F(θ 1 + θ 2 x); y is binary; F is a cdf. Type II best dose: x (θ) = arg max [ζ(x, θ)], x X e.g., ζ(x, θ) = E(y x, θ) = θ 1 + θ 2 x + θ 3 x 2 y Normal(E(y x, θ), 1); ζ(x, θ) = P(y = 1 x, θ) = F(θ 1 + θ 2 x + θ 3 x 2 ); y is binary; F is a cdf.

50 Samples of Type I & II Naive BI Allocation Sequences 3+3

51 In Contrast, Adaptive D-optimal Design Goal: allocate doses to gain maximum information in terms of a selected criterion (cf. Fedorov Hackel (1997) Atkinsen, Donev Tobias (2007). Often criteria are defined in terms of covariance matrix of parameters estimates, which is inverse to the information matrix (exactly in linear models, asymptotically in general). D optimality criterion allocations are selected to maximize the determinant of the information matrix (minimize the variance covariance matrix). Dose allocation may be constrained by bounding a penalty function, e.g., to reduce the likelihood of doses with small P(efficacy) /or high P(toxicity). Cost of observations can be incorporated. Dose allocations for each new cohort are based on updated estimates of the covariance matrix.

52 Predicted Dose from naive BI Adaptive D-Optimal design with cost C = 2 under linear model.

53 Parameter estimates (n = 100) from naive BI Adaptive D-Optimal design with cost C = 2 under linear model.

54 Predicted best dose (n = 100) from naive BI Adaptive D-Optimal design with cost C = 2 under quadratic model.

55 Parameter estimates (n = 100) from naive BI Adaptive D-Optimal design with cost C = 2 under quadratic model.

56 It is Important to Underst Interactions between Objective Criteria In typical phase one dose-finding studies, inference patient gain objectives compliment each other. In typical phase two dose-finding studies, inference patient gain objectives contradict each other.

57 Conflict Between Objectives with Unimodal Success Probability Functions With s maximizing patient gain, goal is a One Point Design at argmax x {P(Success x)} that minimizes the variance of MLE of argmax{p(success x)} typically are at least two point designs This has been proven for several parametric models of toxicity efficacy believed to hold very generally when marginal P{toxicity dose} P{efficacy dose} are common two parameter models. Fan Chaloner (2003), Rabie (2004), Dragalin, Fedorov Wu (2006).

58 Ways to Deal with the Conflict between Criteria Choose between the Goals of Estimation Patient Gain Use Compound Criteria Penalize Undesirable Outcomes Dragalin, Fedorov Wu (2006) Pronzato (2010) suggests starting with estimation objective then converting to patient gain objective after enough information about the response function has been obtained to give reliable estimates.

59 Thank you!

60 Outline

61 1 Azriel, D. (2010) Dissertation Technical Report. Israel 2 Aziel, D. (2011) A note on the robustness of the continual reassessment method. Preprint. 3 Azriel, D., Mel, M., Rinott, Y., The treatment versus experimentation dilemma in dose-finding studies. J. Statist. Plann. Inference 141 (8), Baldi Antognini, Alessro Paola Bortot Alessra GiovagnoliAISM (2008) Romized group up down experiments AISM 60:4559 DOI /s Romized group up down experiments 5 Bortot, P., Giovagnoli, A. (2005) Up down experiments of first second order. Journal of Statistical Planning Inference 134(1),

62 continued 1 Chang, H-H., Ying, Z. (2009) Nonliner sequential designs for logistic item response theory models with applications to comperterized adaptive tests. Annals of Statistics Cheung, Y., Letter to the editor. Biometrics 58 (1), Cheung, Y., Chappell, R., A simple technique to evaluate model sensitivity in the continual reassessment method. Biometrics 58 (3), Derman, C., Non-parametric up down experimentation. Ann. Math. Statist. 28, Dixon, W. J., Mood, A. M., A method for obtaining analyzing sensitivity data. J. Am. Statist. Assoc. 43,

63 continued 1 Dragalin, V., Fedorov, V., Wu, Y. (2006) Adaptive designs for selecting drug combinations based on efficacy-toxicity response. JSPI 2 Durham, SD,, N (1995). Up down designs I: Stationary treatment distributions. Adaptive. Lecture-Notes Monograph Series , N, Rosenberger, WF (eds.). Hayward, CA: Institute of Mathematical Statistics. 3 Durham, SD,, N, Haghighi, AA (1995). Up down designs II: Exact treatment moments. Adaptive. Lecture-Notes Monograph Series , N, Rosenberger, WF (eds.). Hayward, CA: Institute of Mathematical Statistics. 4 Durham, SD,, N, Rosenberger, WF (1997). A rom walk rule for phase I clinical trials. Biometrics

64 continued 1 Durham, S. D.,, N., Rom walks for quantile estimation. In: Statistical decision theory related topics V (West Lafayette, IN, 1992). Springer, New York, pp Fan, SK Chaloner, K (2003) Optimal designs limiting optimal designs for a trinomial response. JSPI 3 Gezmu, M.,, N., Group up down designs for dose- nding. J. Statist. Plann. Inference 136 (6), Giovagnoli, A., Pintacuda, N., Properties of frequency distributions induced by general up down methods for estimating quantiles. J. Statist. Plann. Inference 74 (1),

65 continued 1 Gooley, T. A., Martin, P. J., Lloyd, D. F., Pettinger, M. Simulation as a design tool for Phase I/II clinical trials: An example from bone marrow transplantation. Controlled Clinical Trials 15, Goodman, S., Zahurak, M., Piantadosi, S., Some practical improvements in the continual reassessment method for phase I studies. Stat. Med. 14, Hardwick, J., Meyer, M.C. Stout, Q.F. (2003) Directed walk designs for dose response problems with competing failure modes. Biometrics 4 Kiefer, J. Wolfowitz, J. (1952). Stochastic approximation of the maximum of a regression function. Annals of Mathematical Statistics 25,

66 continued 1 Kpamegan, EE,, N (2001). An optimizing up down design. Optimum Design Atkinson, A.C., Bogacka, B., Zhigljavsky, A. (eds.). The Netherls: Kluwer Academic Publishers. 2 Kpamegan, EE,, N (2008). Up down designs for selecting the dose with maximum success probability. Sequential Analysis Ivanova, A.,, N., Chung, Y., Cumulative cohort design for dose finding. J. Statist. Plann. Inference 137,

67 continued 1 Ivanova, A., Kim, S.H., 2009.Dosefinding for continuous ordinal outcomes with a monotone objective function: a unified approach.biometrics 65, Ivanova, A., Haghighi, A., Mohanty, S., Durham,S.,2003.Improvedup down designs for phasei trials. Statistics in Medicine 22, Oron, A. P., Small-sample behavior of long-memory Phase I cancer designs. preprint In: DAE 2009, Columbia, MO. Invited Conference Talk. 4 Oron, A.P., Hoff, P.D. (2011). Small sample behavior of novel phase I designs. preprint. 5 Oron, AP, Azriel, D, Hoff, PD (2011) Dose-finding designs: The role of convergence properties. International Journal of Biostatistics. In Press.

68 continued 1 O Quigley, J, Shen, LZ. (1996) Continual reassessment method: a likelihood approach. Biometrics Pronzato, L (2010). Penalized optimal designs for dose-finding. J. Statist. Plann. Inference Rabie, H.,, N. (2004). Optimal designs for contingent response models. MoDa 7 Advances in Model-Oriented Design Analysis A.D. Bucchianico, H. Lauter H.P. Wynn, editors. Heidelberg: Physica-Verlag. 4 Resche-Rigon, M., Zohar, S., Cheveret, S. (2008) Adaptive designs or dose-finding in non-cancer phase II trials: influence of early unexpected outcomes. 5 Stylianou, M,, N (2002). Dose finding using isotonic regression estimates in an up down biased coin design. Biometrics

69 Other (Treatment above Everything) Adaptive R M (ARM) Plug in estimated parameters search for the needed dose, made the next observation at this dose G. Wetherill (1963) Sequential Estimation of Quantal Response Curves, JRSS (B), 25, 1-48 T. Lai, H. Robbins (1982), Adaptive Design in Regression Control, Proc. Natl. Acad. Sci. USA, 75, C.F.J. Wu (1985), Efficient Sequential with Binary Data, JASA, 80, Continual Reassessment Method () ~ ARM + Discrete doses + Dose escalation + Bayesian blending J. O Quigley, M. Pepe, L. Fisher (1990), Continual Reassessment Method, Biometrics, 46, J. O Quigley, L. Shen (1996), Continual Reassessment method: Likelihood Approach, Biometrics, 52, 673 Adaptive Response (Utility) Optimization Plug in estimated parameters search for the optimal dose, made an observation at this dose Zh. Li, S. Durham, N. (1995), An Adaptive Design for Maximization of a Contingent Binary Response, In Adaptive design, IMS lecture Notes, Volume 25. Desirability Maximization i i Plug in estimated parameters search for the dose that maximizes a desirability function + Dose escalation + Bayesian blending P. Thall, J. Cook (2004), Dose-Finding Based on Efficacy-Toxicity Trade-offs, Biometrics, 60, curtesy of V. Fedorov