Phase I clinical trial designs: range and trend of expected toxicity level in standard A+B designs and an extended isotonic design treating toxicity as a quasi-continuous variable

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Content PHASE I CLINICAL TRIAL DESIGNS: RANGE AND TREND OF EXPECTED TOXICITY LEVEL IN STANDARD A+B DESIGNS AND AN EXTENDED ISOTONIC DESIGN TREATING TOXICITY AS A QUASI-CONTINUOUS VARIABLE by Zhengjia Chen A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (BIOSTATISTICS) May 2009 Copyright 2009 Zhengjia Chen ii Dedication To my mother, Shuyi Luo, and father, Hualong Chen To my brother, Xinjia Chen To my wife, Xiaomei Zhuang To my sons, Jianda Chen, Jianze Chen, and Jianwei Chen iii Acknowledgements First of all, I would like to express my deepest gratitude to my two mentors: Dr. Stanley Azen and Dr. Mark Krailo. This dissertation would not have been done without their outstanding knowledge, experienced guidance, and insightful comments. I am greatly indebted to Dr. Krailo for his tremendous help on my academic career, my immigration to the United States, and the life of my family in the past 5 years. I am also extremely indebted to Dr. Azen for bringing me to the field of Biostatistics and leading me through the Ph.D training program in the past 10 years and helping on my immigration to the United States. I really appreciated my other committee members, Dr. Rand Wilcox, Dr. Daniel Stram, and Dr. Anny Xiang for their insightful comments, valuable suggestion, and indispensable support very much. I would also like to thank Dr. Jeff Sun for his valuable suggestion and encouragement. Thanks also go to many USC faculty and staffs, particularly, Kim Sigmund, Amy Fan, Dongyun Yang, Huiyan Ma, Chaoyan Li, and Weicheng Wu for their various helps in academic, social, and personal aspects of my life. Finally, I would like to thank Children’s Oncology Group for its excellent research environment and invaluable database which are essential for my Ph.D dissertation. iv Table of Contents Dedication ..................................................................................................................ii Acknowledgements ....................................................................................................iii List of Tables .............................................................................................................vii List of Figures ............................................................................................................x Abbreviations .............................................................................................................xi Abstract ......................................................................................................................xiii Chapter I: Introduction ...............................................................................................1 Chapter II: Issues of Phase I Clinical Trial ................................................................3 2.1 Purpose of Phase I Clinical Trial .....................................................................3 2.2 Toxicity ............................................................................................................4 2.3 Tested Dose Levels Selection. .........................................................................5 2.4 Relationship between Drug Dose and Toxicity. ..............................................6 2.4.1 Power Function Model. .............................................................................7 2.4.2 Logistic Model. .........................................................................................9 2.4.3 Hyperbolic Tangent Model. ......................................................................10 Chapter III: Designs of Phase I clinical trials ............................................................13 3.1 Classification of Phase I Designs. ....................................................................13 3.2 Rule Based Designs .........................................................................................14 3.2.1 Standard Design. .......................................................................................14 3.2.2 Biased Coin Up and Down Design (BCD) ...............................................16 3.2.3 Isotonic Design. ........................................................................................17 3.2.4 Other Rule Based Designs ........................................................................18 3.3 Model Based Methods .....................................................................................19 3.3.1 Continual Reassessment Method ..............................................................19 3.3.2 Escalation with Overdose Control. ...........................................................23 3.4 Two-Stage Designs ..........................................................................................24 3.4.1 Two Stage Rule Based Designs. ...............................................................25 3.4.2 Two Stage Model Based Designs. ............................................................27 3.4.3 Mixed Two-Stage Designs. .......................................................................28 3.5 Disadvantages and Advantage of Different Designs. ......................................30 v Chapter IV: The Limit and Trend of Expected Toxicity Level at MTD of Standard Phase I Designs ...........................................................................................31 4.1 Introduction ......................................................................................................32 4.2 Standard 3+3 Design without Dose De-escalation ..........................................34 4.2.1 Scheme of the Design ...............................................................................34 4.2.2 Simulation Using the Formula Developed by Lin and Shih. ....................35 4.2.3 Simulation Results. ...................................................................................37 4.3 Standard 3+3 Design with Dose De-escalation ...............................................40 4.3.1 Scheme of the Design ...............................................................................40 4.3.2 Simulation Using the Formula Developed by Lin and Shih. ....................42 4.3.3 Simulation Results. ...................................................................................43 4.4 Other Standard Algorithm-based A+B Designs. .............................................44 4.5 Discussion. .......................................................................................................50 Chapter V: An Extended Isotonic Design for Phase I Trials with Differentiation of Graded Toxicity ............................................................................54 5.1 Introduction ......................................................................................................54 5.1.1 Motivation and Purpose of the Extended Isotonic Design. .......................55 5.1.2 Current Major Phase I Designs. ................................................................56 5.1.3 Why Isotonic Design Is Chosen as A Framework? ..................................57 5.1.4 Basic Idea of Our Extended Isotonic Design ............................................58 5.2 Toxicity Response Is Treated as A Continuous Variable. ...............................59 5.2.1 Composite Equivalent Toxicity Score System. ........................................60 5.2.2 The Chosen Values of Parameter α and β. ................................................64 5.2.3 Normalized Equivalent Toxicity Score .....................................................72 5.3 Design of Extended Isotonic Design. ...............................................................73 5.3.1 Pool Adjacent Violator Algorithm and Isotonic Regression. ...................73 5.3.2 Determination of Target Normalized Equivalent Toxicity Score. ............74 5.3.3 Summary of Extended Isotonic Design ....................................................81 5.4 An Example of Extended Isotonic Design. ......................................................82 Chapter VI: Evaluation of Extended Isotonic Design ................................................87 6.1 Simulation Study Methods ...............................................................................87 6.1.1 Simulation Scenarios. ...............................................................................87 6.1.2 Number of Monte Carlos Simulation. .......................................................91 6.2 The Different Designs. .....................................................................................95 6.2.1 Extended Isotonic Design Treating Toxicity Response as A Quasi-Continuous Variable. ...............................................................................95 6.2.2 Isotonic Design Treating Toxicity Response as A Binary Variable. .............................................................................................................96 6.2.3 Standard 3+3 Design with Dose De-escalation. .......................................96 6.2.4 Continuous Reassessment Method. ..........................................................97 6.2.5 Continuous Reassessment Method with Normalized Equivalent Toxicity Score (CRM-NETS). ...........................................................................99 vi 6.3 Result of Simulation ........................................................................................100 6.3.1 Comparison of Accuracy of MTD Estimation. .........................................100 6.3.2 Comparison of Patient Distribution, Sample Size, and Study Length. ...............................................................................................................104 6.4 Summary of Simulation ...................................................................................106 Chapter VII: Application of Extended Isotonic Design in Real Phase I Trials of Children’s Oncology Group ........................................................................110 7.1 Pseudo-Trial with EID Using Data of ADVL0311 ..........................................110 7.2 Pseudo-Trial with EID Using Data of A09712 ................................................116 7.3 Sensitivity Analysis .........................................................................................123 7.3.1 Results Using Data of ADVL0311. ..........................................................124 7.3.2 Results Using Data of A09712. ................................................................127 7.4 Summary of Application of EID ......................................................................130 Chapter VIII: Pseudo-Trails with Extended Isotonic Design Using Real Data with Bootstrap Method ......................................................................................132 8.1 Simulations of EID Using Data of ADVL0311 with Bootstrap Method. ..................................................................................................................134 8.2 Simulations of EID Using Data of A09712 with Bootstrap Method. ..............141 8.3 Summary of Application of EID with Bootstrap Method ................................147 Chapter IX: Discussion of Extended Isotonic Design ...............................................149 Chapter X: Future research ........................................................................................153 10.1. Within-patient Escalation. .............................................................................153 10.2. Designs for Combinations of Agents. ...........................................................154 10.3. Designs for Delayed or Cumulative Toxicity. ..............................................155 10.4. Designs Accounting for Covariates and Patient-Specific MTD. ..................156 References ..................................................................................................................158 Appendices .................................................................................................................164 Appendix A: Programme of EID for Simulation Studies of Chapter VI ...............164 Appendix B: Programme of EID for Bootstrap Replication of ADVL0311 ............................................................................................................169 vii List of Tables Table 4.1: METL at MTD from Standard 3+3 Design. .............................................38 Table 4.2: METL at MTD from Standard 2+2 Design. .............................................46 Table 4.3: METL at MTD from Standard 4+4 Design. .............................................47 Table 4.4: METL at MTD from Standard 5+5 Design. .............................................48 Table 5.1: Mapping of Adjusted Grade and Original Toxicity ..................................62 Table 5.2: Summary of ETS Estimated with Different α and β on Different Extreme Toxicity Profiles ..........................................................................................67 Table 5.3: Summary of NETS Estimated with Different α and β on Different Extreme Toxicity Profiles ..........................................................................69 Table 5.4: Composite Toxicity Equivalent Score Estimation System .......................72 Table 5.5: TNETS Based on TTP with 33% DLT and Ratios (1:1:1:1 and 1:1). ............................................................................................................................77 Table 5.6: TNETS Based on TTP with 33% DLT and Ratios (1:2:3:4 and 1:2). ............................................................................................................................77 Table 5.7: TNETS Based on TTP with 33% DLT and Ratios (4:3:2:1 and 2:1). ............................................................................................................................78 Table 5.8: TNETS Based on TTP with 20% DLT and Ratios (1:1:1:1 and 1:1). ............................................................................................................................78 Table 5.9: TNETS Based on TTP with 20% DLT and Ratios (1:2:3:4 and 1:2). ............................................................................................................................78 Table 5.10: TNETS Based on TTP with 33% DLT and Ratios (4:3:2:1 and 2:1). ............................................................................................................................79 Table 5.11: TNETS Based on TTP with 50% DLT and Ratios (1:1:1:1 and 1:1). ............................................................................................................................79 Table 5.12: TNETS Based on TTP with 50% DLT and Ratios (1:2:3:4 and 1:2). ............................................................................................................................80 viii Table 5.13: TNETS Based on TTP with 50% DLT and Ratios (4:3:2:1 and 2:1). ............................................................................................................................80 Table 5.14: An Example of Simulation Study of the EID .........................................86 Table 6.1: True Toxicity Profiles of Different Scenarios ..........................................89 Table 6.2: Minimum Number of Simulations Required under Different α and ε with 6 Dose Levels ...........................................................................................94 Table 6.3: Percentage for Each Dose Recommended as MTD under Different Scenarios. ...................................................................................................103 Table 6.4: Comparisons of Patients Distribution, Sample Size, and Number of Cohorts by Different Designs. .................................................................107 Table 7.1: Dose Level and Detail Treatment of ADVL0311 .....................................111 Table 7.2: Summary of Patient’s Enroll Order, Dose Level, Toxicity, ETS, and NETS of ADVL0311 ..........................................................................................112 Table 7.3: A Pseudo-trial with EID Using the Patients of ADVL0311 with Parameter α=-2 and β=0.5. .........................................................................................114 Table 7.4: Dose Level and Detail Treatment of A09712 ...........................................117 Table 7.5: Summary of Patient’s Enroll Order, Dose Level, Toxicity, ETS, and NETS of A09712.................................................................................................112 Table 7.6: A Pseudo-trial with EID Using the Patients of A09712 with Parameter α=-2 and β=0.5. .........................................................................................121 Table 7.7: A Pseudo-trial with EID Using the Patients of ADVL0311 with Parameter α=-2 and β=2 .............................................................................................124 Table 7.8: A Pseudo-trial with EID Using the Patients of ADVL0311 with Parameter α=-2 and β=1 .............................................................................................124 Table 7.9: A Pseudo-trial with EID Using the Patients of ADVL0311 with Parameter α=-2 and β=0.5 ..........................................................................................125 Table 7.10: A Pseudo-trial with EID Using the Patients of ADVL0311 with Parameter α=-2 and β=0.25 ...............................................................................125 ix Table 7.11: A Pseudo-trial with EID Using the Patients of ADVL0311 with Parameter α=-2 and β=0.1 .................................................................................126 Table 7.12: A Pseudo-trial with EID Using the Patients of A09712 with Parameter α=-2 and β=2. ............................................................................................127 Table 7.13: A Pseudo-trial with EID Using the Patients of A09712 with Parameter α=-2 and β=1. ............................................................................................127 Table 7.14: A Pseudo-trial with EID Using the Patients of A09712 with Parameter α=-2 and β=0.5 ..........................................................................................128 Table 7.15: A Pseudo-trial with EID Using the Patients of A09712 with Parameter α=-2 and β=0.25 ........................................................................................128 Table 7.16: A Pseudo-trial with EID Using the Patients of A09712 with Parameter α=-2 and β=0.1 ..........................................................................................129 Table 8.1: Dose levels, Toxicities, ETS, and NETS with Different Values of α and β of Patients in ADLV0311 .........................................................................135 Table 8.2: Simulation Results with Data of ADVL0311 ...........................................139 Table 8.3: Summary of dose level, toxicity, ETS, and NETS with different values of α and β for patients in A09712 ...................................................................142 Table 8.4: Simulation Results with Data of A09712. ................................................146 x List of Figures Figure 2.1: The Power Function Model of Relation between Dose and Toxicity ......................................................................................................................8 Figure 2.2: The Logistic Model of Relation between Dose and Toxicity .................10 Figure 2.3: The Hyperbolic Tangent Model of Relation between Dose and Toxicity ......................................................................................................................12 Figure 4.1: Escalation Schema for Standard A+B Design without Dose De-escalation ..............................................................................................................35 Figure 4.2: ETL (%) at MTD from the Standard 3+3 Design by Number of Tested Dose Levels. ...................................................................................................39 Figure 4.3: METL (%) at MTD from the Standard 3+3 Designs by Log 10 Transformation of Number of Dose Levels. ..............................................................40 Figure 4.4: Escalation Schema for Standard A+B Design with Dose De- escalation....................................................................................................................42 Figure 4.5: Ratio of ETLs from Different Standard Designs without Dose De-escalation by Log 10 Transformation of Number of Tested Dose Levels. ............49 Figure 4.6: Ratio of ETLs From Different Standard Designs with Dose De-escalation by Log 10 Transformation of Number of Tested Doses. ......................50 Figure 5.1: Additional ETS Score Contributed by Other Toxicities besides the Most Severe Toxicity with Different Parameter α and β. ....................................71 xi Abbreviations Abbreviation Explanation ANETS Average Normalized Equivalent Toxicity Score BCD Biased Coin Up and Down Design COG Children’s Oncology Group CRM Continuous Reassessment Method CRM-NETS Continuous Reassessment Method with Normalized Equivalent Toxicity Score CTEP Clinical Trial Evaluation Program DLT Dose Limiting Toxicity EID Extended Isotonic Design ETL Expected Toxicity Level ETS Equivalent Toxicity Score EWOC Escalation With Overdose Control FDA Food and Drug Administration ID Isotonic Design IND Investigational New Drug Application IR Isotonic Regression METL Mean Expected Toxicity Level MLE Maximum Likelihood Estimation MTD Maximum Tolerated Dosage NCI National Cancer Institute xii NDA New Drug Application NETS Normalized Equivalent Toxicity Score PANETS Pooled Average Normalized Equivalent Toxicity Score PAVA Pool Adjacent Violators Algorithm PBP Product of Beta Prior QMLEs Quasi Maximum Likelihood Estimates RWRs Random Walk Rules SD Standard Deviation TITE-CRM Time to Event Continuous Reassessment Method TNETS Target Normalized Equivalent Toxicity Score TTB Total Toxicity Burden TTP Target Toxicity Profile TTL Target Toxicity Level WBC White Blood Count xiii Abstract The current designs of Phase I trials are comprehensively reviewed and classified by algorithm (assumption and dose-toxicity relationship) as rule based designs vs model based designs or by number of stages as one stage designs vs two stage designs. Standard 3+3 designs are most widely used for their practical simplicity. Through simulation study, the expected toxicity levels (ETL) at maximum tolerated dosage (MTD) are originally found to decrease monotonically from about 30% to 0% as the number of dose levels increase from 3 to infinity, which solves the previously unexamined issue of the number of dose levels that are planned in a study. We conclude that the number of specified dose levels is an important factor affecting substantially the ETL at MTD and recommend that fewer than 20 dose levels be designated. In Standard 3+3 designs, target toxicity level (TTL) can not be pre-specified and toxicity response is treated as a binary indicator of dose limiting toxicity (DLT), discarding lots of valuable toxicity information. Therefore, a novel toxicity score system is proposed to measure quantitatively the overall severity of multiple toxicities of each patient, and then coupled with Isotonic Regression (IR) to create an extended isotonic design (EID) which allow pre-specification of TTL, treat toxicity response as a quasi-continuous variable, and fully utilize all toxicity information. Simulation studies and applications of EID to two real Phase I trials demonstrate that EID can always estimate an more accurate MTD with less sample size according to the exact toxicity profile while designs treating toxicity response as a binary variable can’t accomplish that. Our EID is practical, objective, model free, simple to use, xiv and more accurate in MTD estimation so that it is of great practical value and will help to begin a new era in which toxicity response is really treated as a quasi-continuous variable. 1 Chapter I: Introduction Medicine is very important for the health care of human beings. It is a long, difficult, and expensive process to develop safe and effective new medicines. It takes 12 years on average for an experimental drug to travel from lab to medicine. Only five in 5,000 compounds that enter preclinical testing make it to human testing. One of these five tested in people is approved. New medicine development procedure includes, Preclinical Testing, Investigational New Drug Application (IND), Phase I Clinical Trials, Phase II Clinical Trials, Phase III Clinical Trials, New Drug Application (NDA), and Approval. The company must continue to submit periodic reports of new medicine to Food and Drug Administration (FDA), including any cases of adverse reactions and appropriate quality-control records. For some medicines, FDA requires additional studies (Phase IV) to evaluate long-term effects. A Phase I trial is one of the most important steps in the drug’s development and the first clinical trial in human subjects after the laboratory and animal study for a therapeutic agent showing potential cure effect of disease. Since initial experience with a new agent may substantially influence its fate, a careful and thoughtful approach to the design of Phase I trials is essential. Clinical research involving humans poses serious ethical concerns and clinical trials involving oncology patients and cytotoxic drugs have been among the most problematic of all. A cancer Phase I trial has therapeutic intent. Typically, patients enrolled in a Phase I trial in the field of oncology usually are very sick 2 patients who have already tried and failed on the existing standard treatments and are willing to help development of new treatment. Thus, from a therapeutic perspective, one should design cancer Phase I trials to minimize both the number of patients treated at low, non-therapeutic doses as well as the number given severely toxic overdoses. The total number of subjects included in Phase I studies varies with the drug, but is generally in the range of 20 to 80. 3 Chapter II: Issues of Phase I Clinical Trial 2.1 Purpose of Phase I Clinical Trial The main purpose of a Phase I trial is to understand how well the drug can be tolerated in a small number of human beings and its safety profile, including the safe dosage range. It is a widely accepted assumption that the therapeutic effect of drug depends on its toxicity and increase monotonically with its dosage level. Higher doses are correlated with both severe toxicity and better therapeutic effect. Therefore, a balance is to be achieved between toxicity level and therapeutic benefit. To achieve the best benefit of therapeutic agent, patient should be treated the maximum dosage of drug under the condition patients can tolerate its associated toxicity with closely monitoring. In the precise words, main goals of a Phase I trial are to determine the dose-toxicity relationship of the new therapeutic agent and estimate the MTD of the agent given the specified highest tolerable toxicity level and under safe administration (Rosenberger et al., 2002; Potter, 2006). Phase I trial also determines how a drug is absorbed, distributed, metabolized and excreted, and the duration of its action. In order to estimate its pharmacokinetics and pharmacodynamics and understand how it is processed in the human’s body, blood and other fluid samples can be collected at various time points and then analyzed. Sufficient information about the drug's pharmacokinetics and pharmacological effects collected during Phase I trials can help to design a well-controlled scientifically valid Phase II studies. Other purposes of Phase I studies include collecting side effects associated with 4 increasing doses; evaluating drug metabolism, structure-activity relationships, and the mechanism of action in humans; exploring biological phenomena and processes of diseases; and obtaining early evidence on effectiveness if possible. 2.2 Toxicity In the National Cancer Institute (NCI) Common Toxicity Criteria (NCI 2003), according to their severity and types, toxicities are classified into 5 grades as below: Grade 0: no toxicity; Grade 1: mild toxicity; Grade 2: moderate toxicity; Grade 3: severe toxicity; Grade 4: life-threatening toxicity; Grade 5: Death. The DLT is usually defined as a group of grade 3 or higher non-hematologic toxicities and grade 4 hematologic non-transient toxicities. Studies of continuously administered agents often classify some grade 2 toxicities as DLT. The highest acceptable DLT level is usually defined as a TTL. It can be said that TTL determines the MTD of the new therapeutic agent. The potential therapeutic effect of drug varies with type and stage of disease as well as target patient population. TTL is usually specified during the design stage of a Phase I trial. If the toxicity associated with the drug is transient or correctable or non-fatal condition and moderate, TTL can be set higher, such as more than 33%. On the other hand, TTL will be defined low and monitored strictly if the toxicities can cause 5 permanent severe harms or are lethal to human beings. If deaths occur and are related to the testing drug, the trial will be suspended and the escalation scheme must be re- evaluated. 2.3 Tested Dose Levels Selection. Tested range of doses of the therapeutic agent must be defined in the Phase I trial. Before a Phase I trial starts, the animal studies of the target therapeutic drug have been performed. The tested doses usually are a series of small fractions of the dosage which causes harm in animal studies. For example, one-tenth of the mice lethal dose, or one- third toxic low dose in dogs, or one-tenth of other animal dose (LD10) which will kill 10% of the animals will be selected as starting dose. For children, an even lower dose, such as the 5% of LD10, may be chosen instead (Potter D. 2006). If several drugs are combined, all information for each individual drug from previous trials needs to be considered. Since dosage is continuous, a certain increment of dosage can be used between two consecutive trials. For example, dose increment pattern may adapt some variation of Fibonacci Series, a sequence of numbers first created by Leonardo Fibonacci (fi-bo-na-chee) in 1202. In a Fibonacci series, each number equals the sum of the two preceding numbers. An example is (1,1,2,3,5,8,13,21,...). A modified Fibonacci series has been widely used to define the subsequent dose steps in the practical Phase I trials. Usually, the increments of dose for succeeding levels are 100%, 67%, 50%, and 40%, followed by 33% for all subsequent levels so that increments decrease with the increasing dose levels (Potter D. 2006). 6 2.4 Relationship between Drug Dose and Toxicity. A well accepted assumption is that toxicity probability increases monotonically with increasing drug dose even though downturn of toxicity probability at high dose levels could happen in some special cases which are not common and not considered here. There are non-parametric and parametric ways to describe toxicity-dose relation. In non-parametric way, with the only assumption that toxicity is non-decreasing with dose, the IR is most widely used to define the increased toxicity risk as dose increases (Leung et al., 2001). In the parametric way, a distribution with some parameters is adapted to model the toxicity-dose curve. In biological point of view, human body has his/her stabilization and self-salvage system to protect himself/herself from mild toxicity when drug dose is at a low level below a certain threshold level, but toxicity probability increases at an accelerated speed once the stabilization and self-salvage system have been broken, and reaches rapidly the worst condition, death, and then levels off. Therefore a model whose distribution is of sigmoid shape is appropriate to describe the relationship between dose and toxicity probability. In Phase I clinical trial, the dose levels usually have been defined before the initiation of the trial based on the previous related trials or animal studies. Let d i (i = 1, …, K) with {d 1 ≤d 2 ≤d 3 ≤… ≤d K } be the predefined dose levels approved by the regulatory agency. We assume that N patients have enrolled in the trial and received one of these K dose levels. Let Y 1 , Y 2 , …, Y n be the binary toxicity outcomes (Y j =1 if toxicity; 0 if no toxicity, j=1 to n) of the N enrolled patients, respectively. Assuming a homogeneity among patients, the probability of toxicity for patient i is defined as p i with p i =P{Y j = 1 | 7 d i }, i = 1, …, K. In general, we can use a distribution function with two unknown parameters, α and β, to describe the relationship between toxicity probability and dose level. ) , , ( β α i i d F p = Let Γ be the toxicity probability associated with the MTD, denoted as TTL, and Let μ be the MTD of the tested drug corresponding to Γ, and hence μ is a quantile of distribution F. Then their relation can be further defined as below: ) , , ( 1 β α μ Γ = − F In the literature, three families of monotonic distributions, power function, hyperbolic tangent function, and logistic function have been widely investigated and used to describe the toxicity-dose curve. Each of these 3 distributions is further described in detail. 2.4.1 Power Function Model. With the same notation as defined before, the model of toxicity-dose relation using power function is given by β α ) ( d p = Where dosage is divided by α to make sure that α / d ranges from 0 to 1 given all possible dosages. The parameter β defines the speed of increased toxicity risk with 8 increasing dosage. When the TTL has been pre-selected as Г, the corresponding MTD, μ, can be estimated using the following equation: ) ) ln( exp( β α Γ = u There are three kinds of shapes, convex, linear, and concave, in the power function distribution depending on the selected value of β. The follow figure gives some sample relation distribution between dose and toxicity assuming power function model covering these 3 shapes. Figure 2.1: The Power Function Model of Relation between Dose and Toxicity 9 2.4.2 Logistic Model. When logistic model is used, the dose-toxicity relationship is assumed to be of logistic form as following: d p it * ) ( log β α + = Where α defines the starting toxicity probability at the lowest testing dose level and β defines the speed of increased toxicity risk with increasing dose level. In another form, the relation can also be expressed as following: ) * exp( 1 ) * exp( d d p β α β α + + + = When the TTL has been pre-selected as Г, the corresponding MTD, μ, can be estimated using the following equation: β α μ − Γ − Γ = ) 1 ln( Assuming the logistic model, we investigate different shapes of curves reflecting the slow, median, and fast increasing toxicity risk using different values of β. The value of α is set to make the start toxicity risk close to 0. These curves are plotted below: 10 Figure 2.2: The Logistic Model of Relation between Dose and Toxicity 2.4.3 Hyperbolic Tangent Model. The hyperbolic tangent function is assumed for the dose-toxicity relationship as the following: β α } 2 1 ) tanh( { + + = d p Where α defines the starting toxicity probability at the lowest testing dose level and β defines the speed of increased toxicity risk with increasing dose level. 11 The definition of hyperbolic tangent function is defined as below: ... 5 3 1 1 ) 2 exp( 1 ) 2 exp( ) exp( ) exp( ) exp( ) exp( ) ( ) sinh( ) tanh( 2 2 + + + = + − = − + − − = = χ χ χ χ χ χ χ χ χ χ χ χ conh Using exponential form, it can also be expressed as: β α α } 2 1 ) 1 )) ( 2 (exp( 2 1 )) ( 2 exp( { + + + − + = d d p When the TTL has been pre-selected as Г, the corresponding MTD, μ, can be estimated using the following equation: α β μ − − Γ = − ) 1 ) ) ln( exp( 2 ( tanh 1 When the tangent function is expressed using exponential form, it can also be expressed as: α β μ − ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − Γ − = 1 ) ) ln( exp( 1 1 ln 2 1 With the hyperbolic tangent model, we investigate different shapes of curves reflecting the slow, medium, and fast increasing toxicity risk using different values of β. The value of α is set to make the start toxicity risk close to 0. The following figure gives some sample distribution between dose and toxicity assuming hyperbolic model. 12 Figure 2.3: The Hyperbolic Tangent Model of Relation between Dose and Toxicity Both α and β in the three models can be estimated from the dose and toxicity data using parameter estimation methods, such as, maximum likelihood method, etc. after the trial finishes. Among the three families of distribution, the power function is the least used one because it is not sigmoid shape and this range can go below 0 and exceed 1. The logistic model is one of the most popular models because of its sigmoid shape distribution and its value range of 0 to 1 exclusively. The hyperbolic tangent is also commonly employed because of this sigmoid shape distribution even though it is more complicated than logistic model. 13 Chapter III: Designs of Phase I clinical trials The main goal of Phase I clinical trial of a new agent is to find a MTD dose with the highest acceptable toxicity probability level. During the estimation of MTD, if too many patients are treated at an overdosed dose level, it will put patients in an unacceptable toxicity risk which it is deemed unethical (Potter, 2006). On the other hand, it is also unethical to treat too many patients at an under-dosed dose level which is obviously lack of sufficient toxicity for the therapeutic effect (Potter, 2006). Design of Phase I clinical trial is a scheme for estimating MTD correctly, ethically, and efficiently. Currently, a lot of designs for Phase I clinical trials have been proposed in the literature. Some have been demonstrated successful and widely implemented in the practical Phase I trials, such as standard3+3 design etc. However, many have only been shown to be feasible with simulation studies, but have never been used in the real trials (Potter D. 2006). 3.1 Classification of Phase I Designs. The designs for Phase I trials can be classified by their algorithms or the number of stages in the design. By algorithm, the designs can be classified into two categories: rule based and model based. In rule based designs, non-decreasing dose toxicity relationship is the only well-accepted assumption required. Therefore rule based designs are well-suited for first in human clinical trials in which dose toxicity relationship is not well understood. The common rule based designs include standard design, ID, biased 14 coin up and down design, accelerated titration design, Storer’s up and down design, etc. In model based designs, three parametric dose-toxicity functions, logistic model, hyperbolic model, and power function, have been usually employed to depict the relationship between dose and toxicity. Model based designs may fail to find an MTD in the first in human studies that is based on observed DLTs (Potter, 2006). The common model based designs are continuous reassessment method (CRM), escalation with overdose control (EWOC), etc. By the number of stages in the design, the designs of Phase I clinical trials can also be classified into two categories: one stage design or two stage design. One stage design can be either rule based or model based. In two stage designs, rule based or model based designs can be implemented in each stage separately. 3.2 Rule Based Designs 3.2.1 Standard Design. The “standard designs” are rule based up-and-down methods used in the clinical trial evaluation program (CTEP) Phase I protocol templates (CTEP website). While ‘conventional’ or ‘standard’ methods have become standard practice among many Phase I clinical trialists, they are not designed with the intention of producing accurate estimates of a target quantile. Rather they are designed to screen drugs quickly and identify a dose level that does not exhibit too much toxicity on a very small group of patients. These standard designs have two kinds of categories: without and with dose de-escalation. In the standard design without dose de-escalation, three patients are assigned to the first dose level. If no DLT is observed, the trial proceeds to the next dose level and another 15 cohort of three patients is enrolled. If at least two out of the three patients experience at least one DLT, then the previous dose level is considered as the MTD; otherwise, if only one patient experiences the DLT, then three additional patients are enrolled at the same dose level. If at least one of the three additional patients experience the DLT, then the previous dose is considered as the MTD; otherwise, the dose will be escalated. The standard design with dose de-escalation allows three new patients to be treated at a previous dose level if only three patients are treated at that level previously. Dose reduction continues until a dose level is reached at which six patients are treated and at most one DLT is observed in the six patients. The MTD is estimated as the highest dose level at which at most one of six patients experience DLT, and the immediate higher dose level has at least two patients who experience DLTs. If the first dose is not tolerable, then the MTD cannot be established within the confines of the study. Storer (1989) is probably the first to examine characteristics of the standard design from the standpoint of the statistician. The operating characteristics of the standard design were discussed in Reiner et al. (1999) and Lin and Shih (2001). Note that any design with sampling that is asymmetric about the MTD will yield a biased result; thus the standard design, and all other designs that approach the MTD from below, will tend to yield a low estimate of the MTD. In the next chapter, we will discuss the relationship between expected toxicity level and number of dose level in the simulation study. The standard designs are simple and can usually determine a reasonable MTD so that they are the most widely used methods for Phase I clinical trial. But they also have 16 many shortcomings, for example: the methods are not designed around a quantile of interest; not all toxicity data are used to determine the MTD; and the MTD is not a dose with any particularly specified probability of toxicity. These cons of standard designs lead us to explore an EID for Phase I clinical trial. 3.2.2 Biased Coin Up and Down Design (BCD) BCD is a specific up-and-down design using random walk rules (RWRs) which can allow one to center dose level assignments unimodally around any target quantile of interest by providing a unifying theory. Durham and Flournoy (1994) for the first time introduced the BCD in which the trial starts from the lowest dose level and increases the dose level one by one until the observation of first DLT, thereafter the random walk rule is employed to find the dose with toxicity level Г. If Г ≤ 0.5, the BCD deescalates a dose if the last patient treated has DLT, and randomizes with probability b = Г/(1- Г) to the next higher dose and 1 - b to the same dose otherwise. If Г > 0.5, the BCD escalates a dose if the last patient treated has DLT and randomizes with probability b = (1- Г)/ Г to the next lower dose and 1-b to the same dose otherwise. Given the MTD has P DLT = b/(1 + b), the frequency distribution, f d , of assigned doses is asymptotically unimodal and centered at MTD. Any selected Г = b/(1 + b) can be achieved by adjusting the b. There are five estimators of the target dose which are derived using mode, maximum likelihood, weighted least squares, sample averages, and IR, respectively. The estimator from a linearly interpolated IR is better than others in mean square error and required number of subjects for convergence. BCD is non-parametric, simple, and easy to implement. It has a 17 workable finite distribution theory and operates on a finite lattice of dosages. Specific designs can be chosen according to any quantile of interests. But BCD also has some disadvantages: 1) does not consider the response of some previous patients; 2) can not control the variance of the estimators of quantile; 3) does not have an appropriate stopping rule because it is non-parametric in merit; 4) hard to convince others that the dose allocation should be determined by tossing a coin; 5) can also dictate assignment to a dose where serious toxicity has already occurred. 3.2.3 Isotonic Design. Leung and Wang (2001), for the first time, introduced an isotonic design (ID) in which only a non-decreasing dose toxicity relationship is the required assumption. In their ID, M patients are recruited in each cohort treated at the same dose level, and then IR is employed to estimate the risk, Q, of patients experiencing DLT at each dose via the pool-adjacent-violators algorithm (PAVA) (Robertson et al., 1988) after the response of toxicity from newly treated cohort has been obtained. The dose allocation rationale is to treat new cohort at a dose level closer to the pre-specified target acceptable toxicity level, Г, than the current dose level, so that dose escalates or de-escalates one level when Q at the new level is closer to Г than the current value of Q and stays at the same level otherwise. The trial stops when the same dose has been tested consecutively for four cohorts or a maximum number of patients have been treated. The recommended dose level for the next cohort is the MTD. Through simulation studies, the ID was compared with the Standard Design, Storer’s Up-and-Down Designs (Storer, 1989), the modified 18 CRM (Goodman, 1995), and EWOC (Babb et al., 1998). In order to obtain meaningful comparison, a fixed sample size of 24 with a cohort size of 3 was used in the ID without a stopping rule and no dose level could be skipped during escalation and de-escalation in all methods. The ID performs substantially better than the Standard Design and compares favorably with Storer’s Up-and-Down Designs, the modified CRM (Goodman, 1995), and EWOC (Babb et al., 1998). Moreover, the ID is model free and especially appropriate in the cases where the parametric dose-toxicity relationship is not well understood. 3.2.4 Other Rule Based Designs There are many other rule based designs. For example, Narayana rule and “k-in-a- row” rule (Ivanova et al., 2003) are two up-and-down methods designed to target dose with probability of toxicity, Г, of the form, Г = 1 – (0.5) 1/k , (k > 0). In the Narayana rule, R and T are the fraction of patients with DLTs and the number of DLTs among the last k patients treated at the current dose level, respectively. Then the next patient is treated at the next lower level if R > Г and T > 0; or at the next higher level if R < Г and T = 0; or at the same level otherwise. Similarly in the k-in-a-row rule, the next patient is treated at the next lower dose level if the last patient experiences DLT; or at the next higher level if the last k patients are all treated at the same level and have no DLTs; or at the same level otherwise. In both designs the chances of patients assigned to the most toxic doses are reduced by using more response information. All previous responses are used to calculate a local estimate of the probability of toxicity in the Narayana rule and up to k most recent 19 responses are used in the “k-in-a-row” rule. Through simulation studies, Ivanova et al. (2003) showed that the Narayana rule is asymptotically superior to the k-in-a-row rule and both designs estimated MTD more precisely than the CRM in most of the cases. CRM is also more likely to treat patients with overdose because of its fast dose escalation. All rule-based designs are intrinsically robust because they do not employ any dose toxicity model and can estimate a reasonable MTD using a stopping rule based either on observed DLTs or on convergence criteria. Ad hoc additional dose levels can also be added when needed without any impact on their robustness. Most of rule-based designs are practically simple and easy to implement. At present, standard designs are still the most popular in the Phase I clinical trials. 3.3 Model Based Methods 3.3.1 Continual Reassessment Method O’Quigley et al. (1990) originally introduced the CRM, a Bayesian approach to fully and efficiently use all data and prior information available in a Phase I study. Like in rule based designs, a TTL, Г, is pre-specified and the goal is to estimate the dose associated with the TTL, Г. A parametric model depicting the dose toxicity relationship and a prior distribution for each unknown parameter of the model are required to implement CRM. The posterior mean of each parameter is computed using the prior for the parameter and all available toxicity data for the probability of toxicity, P DLT , of each dose level. The computation is conducted and P DLT of each dose level is updated with composite toxicity data available when a new patient is recruited. The main idea of CRM 20 is to treat each patient at the dose level with P DLT closest to Г. The MTD is defined as the dose level of the last patient treated in the trial. In the originally proposed CRM, a one parameter model of dose toxicity function and single patient cohort are used. Furthermore, the first patient is proposed to be treated at a dose level determined purely by a guess prior in the original CRM, this makes the method impractical. Korn et al. (1994) proposed a modified CRM in which the trial starts at the lowest dose level, no dose level is allowed to be skipped during the dose escalation, and the trial stops when the same dose has been recommended for new patient consecutively for a fixed number of times. But patients still may be treated at excessively toxic doses in the modified CRM because of its single patient per cohort and the length of study is still very long because of the restriction that the toxicity of all treated patients must be obtained to calculate the new dose level for the next patient. In addition to the modification of Korn at al. (1994), Faries (1994), in his modified CRM, added another rule that no dose escalation is allowed for the next patient when the last patient has DLT. This rule can avoid treating patients at overly toxic doses compared with the standard 3+3 design. The number of patients per cohort is a critical factor of the CRM design. Goodman et al. (1995) proposed a modified CRM similar to that of Korn et al. (1994), but using two- and three-patient cohorts. Through simulation studies, they found that the modified CRM with two- and three-patient cohorts has the advantages of less toxicity and shorter study duration; and its accuracy of estimated MTD is comparable to that of the single-patient CRM; and the exponential prior for β does not perform as well as the uniform prior in general. The MTD of standard designs is not estimated with specific 21 quantile so that there is no uniform standard to compare it with modified CRM. The modified CRM with two- and three-patient cohorts still treats more patients at over-toxic doses in general, but it has less chance to underestimate the MTD. This modified CRM became practical and were implemented in some Phase I clinical trials. Shen and O’Quigley (1996) investigated their asymptotic characteristics. O’Quigley and Chevret (1996) and Chevret (1993) further conducted a lot of simulation studies on the CRM. Application of a stopping rule other than a fixed sample size in CRM design is also a significant modification. Goodman et al. (1995) originally proposed a stopping rule using the width of the posterior distribution for β as a criterion in order to reduce sample size with little loss of precision. Heyd and Carlin (1999) further investigated the performance of several stopping rules based on the width of the Bayesian posterior probability interval of β in the logistic model logit(P DLT (x i )) = α + βx i . These stopping rules generated bigger biases in the estimated MTD than stopping at a fixed sample size while reducing the study sample size substantially. Zohar and Chevret (2001) studied several other stopping rules and found that rules using Bayesian gain functions work most consistently in different dose toxicity scenarios. But these stopping rules were neither compared with stopping at a fixed sample size nor with the stopping rule used by Goodman et al. (1995). Some other stopping rules for the CRM have also been introduced and discussed (O’Quigley and Reiner (1998) and O’Quigley (2002a)). Gasparini and Eisele (2000) introduced a curve-free CRM using a curve-free prior, the product-of-beta prior (PBP). No specific dose-toxicity relationship curve specification except a monotonicity constraint is required for the statistical modeling of 22 the probabilities of DLT. A guess for the prior P0 i is still needed. The probabilities of DLT are modeled as Θi = (1 - P0 i+1 )/(1 - P0 i ), where Θi is distributed as a beta variate and the multivariate distribution of the Θi is updated after the toxicity response of each new patient has been obtained. The expected P DLT (x i ) is estimated from the distribution. The order restriction, P DLT (x i+1 ) > P DLT (x i ), is retained when P DLT (x i+1 )) is decoupled from P DLT (x i ). They found that the curve-free CRM has the practical advantage of treating fewer patients at excessively toxic doses than the CRM in the simulation studies. Cheung (2002) pointed out that the dose escalation scheme of the curve free CRM may be stuck at the lowest level if a vague prior is used. O’Quigley (2002a) states that the preference for curve-free CRM over model-based CRM is the result of a more fortunate choice of arbitrary specification parameters and the two designs are operationally equivalent in terms of sequential dose allocation and final recommendation. The practical value of the curve free CRM still needs to be further evaluated. Many other contributions have also been made to the CRM. Although it is argued that the data will eventually swamp the prior, the specification of prior is still very important in the Phase I clinical trial where the sample size is usually small. The medians, modes, and expectations of the priors are found to frequently far from the P0 i (Gasparini and Eisele, 2000). The P DLT ( ξ i ) estimated from the posterior mean of β is different from the posterior mean of P DLT ( ξ i ) and the dose escalation is more aggressive when using P DLT ( ξ i ) estimated from the posterior mean of β, especially at the early stage of a trial with fewer patients (Ishizuka and Ohashi, 2001). The CRM has been demonstrated to greatly increase the probability of estimating the correct MTD than the 23 standard algorithm based designs in a lot of simulation studies in the literature (O’Quigley, et al. 1990; O’Quigley and Chevret, 1991; Chevret, 1993; Goodman et al. 1995; Heyd and Carlin, 1999; Møller, 1995; Zacks et al. 1998). However, CRM still has not been widely used due to its model based, intensive computation, requirement of prior distribution, and complicated interpretation (Rosenberger et al. 2002; Potter, 2006; O’Quigley and Chevret, 1991). 3.3.2 Escalation with Overdose Control. In order to address the ethical requirement that the probability of patient being treated at overdose is under a pre-specified value, Babb et al. (1998) introduced an adaptive dose escalation scheme called EWOC. The constraint on overdosing of EWOC is a superior feature over the CRM. The theoretical foundation of EWOC was elaborated by Zacks et al. (1998). A two-parameter model, logit(P DLT (x i )) = α + β x i , is first used to depict the dose, x i , and DLT relationship and then the joint posterior for α and β is transformed to a joint posterior for the MTD and the probability of DLT at the lowest dose level, ρ 0 . EWOC is also designed to approach rapidly the MTD in addition to the overdose constraint so that it starts from the lowest dose level and single patient per cohort is used. After the toxicity response of last enrolled patient has been obtained, the joint posterior for the MTD and ρ 0 is updated using all the available information and the next coming patient is treated at the 25 th percentile of the marginal posterior for the MTD. The trial stops after a fixed number of patients have been treated and then the MTD is computed as its posterior mean or estimated by minimizing the posterior 24 expected loss in a loss function. In order to be safe and shorten the length of the trial, no dose level can be skipped during the dose escalation procedure and multiple patient cohorts can be used instead in EWOC. Through simulation studies, EWOC has been shown to be effective in overdose control and have comparable accuracy of estimated MTD as CRM. Fewer patients are treated at non-optimal dose levels, resulted in less DLT, and the estimated MTD has smaller average bias and mean squared error in EOWC than in some other non-parametric designs, such as four up-and-down designs and two stochastic approximation methods. It seems the EWOC is an alternative promising design for Phase I clinical trial, especially when the ethnic requirement of overdose control is a big concern. Model based methods, such as the CRM and the EWOC, are complicated to be explained to non-statisticians and computationally challenging to implement. The key to their usefulness lies in the packaging of these methods in user-friendly software that runs quickly and is well-documented. 3.4 Two-Stage Designs A Phase I clinical trial can have one stage or two stages. The design in each stage of the two stage designs can be rule based or model based separately. Usually there are three combinations. The first one is a two stage rule based design in which the designs in both stages are rule based; the second one is a two stage model based design in which the design of both stages are model based; and the third one is a mixed two stage design in 25 which the design of the first stage is rule based and the design of the second one is model based. So far, to our knowledge, no two stage design in which the first stage is model based design and the second stage is rule based design has been proposed and discussed. 3.4.1 Two Stage Rule Based Designs. Several two stage rule base designs have been introduced (Storer, 1989; Korn et al., 1994; Simon et al., 1997). The purpose of speeding up the dose escalation from very low dose level to the vicinity of MTD initiated the two stage rule based designs. 3.4.1.1 Storer’s Two-stage Designs Storer (1989) developed a two-stage design in which the first stage uses an accelerated method and the second stage uses a standard 3+3 design. In the first stage, only 1 patient is treated in each dose level and dose escalates if the patient experiences no DLT. The second stage using the standard 3+3 designs will not start until first DLT is observed. The sample size is fixed as 24 patients in the second stage. The MTD is determined using maximum likelihood estimation (MLE) and logistic model for dose toxicity relationship. With an accelerated first stage, fewer patients are treated at low doses. But Korn et al. (1994) found Storer’s first stage too aggressive and suggested that two-patient cohorts in the first stage be used and grade 2 toxicity in the stage 2 be considered. The sample size of single-patient designs is also too small to conduct pharmacokinetic measurements (Korn et al., 1994). 26 3.4.1.2 Accelerated Titration Methods. Accelerated titration methods are two stage designs in which the first stage is a rule based design using 1 patient per dose level until the observation of first DLT and the second stage could be rule based or model based design. Simon et al. (1997) further proposed several “accelerated titration designs”. All designs have 40% dose-step increments except that the dose-step increment is 2 times in the stage 1 of design 4. Design 1 is the one stage standard 3+3 design without intra-patient dose escalation. But intra-patient dose escalation is allowed in the both stages of design 2 to 4 if the worst toxicity is grade 0-1 in the previous course for that patient. The first stage of designs 2 to 4 uses 1 patient per dose level until the first DLT or two patients who experienced grade 2 toxic effects are observed. The second stage using the standard design begins at the lower dose level. In design 2, only the toxicities occurred during the first cycle of therapy are considered in the first stage, but these toxicities in any cycle count in designs 3 and 4. After the trial has completed, all patients’ dose and response information will be used to fit the model: y ij = log(d ij + κD ij )+ η i + ε ij , where y ij is a latent variable indicating the toxicity experienced by the i th patient during the j th cycle, d ij is the dose, D ij is the cumulative dose up to the j th cycle, η i is a random patient effect, ε ij is the error term, and κ is a parameter to be estimated. The interpretation of y ij is as follows: if K1 < y ij < K2, the toxicity is grade 2; if K2 < y ij < K3, the toxicity is grade 3; if y ij > K3, the toxicity is greater than or equal to grade 4. The values of K1, K2, and K3 are estimated during the fitting procedure. MTD is determined by the fitting of data instead of using the same way as the one in the standard design. Accelerated titration design are robust, can shorten the 27 duration of trial, reduce the number of patients treated at low dose, allow intra-patient dose escalation, estimate the MTD as a quantile, and extract maximal information about toxicity, including cumulative effects (Simon et al., 1997). They also have some cons, such as: it is risky to escalate rapidly with single patients and difficult to observe the cumulative effects of treatment when allowing intra-patient dose escalation, and no differentiation of different toxicities of the same grade is implemented when modeling toxicity response. 3.4.2 Two Stage Model Based Designs. Moller et al. (1995) modified the original CRM into a two stage model based design in which single patient cohorts are used for the lowest four dose levels and two patient cohorts are treated at the higher dose levels during the first stage, and then original single patient CRM is used in the second stage. They demonstrated with simulation studies that their two stage design had fewer toxicities and converged faster than the original CRM while the accuracies of MTDs estimated from both designs are comparable. Zohar and Chevret (2003) proposed a different two stage CRM design in which the first stage starts from the lowest dose level and multiple patient cohort can be used; the second stage will not start until a MTD has been estimated in the first stage and then the subsequent patients are only treated at that MTD till the accuracy of the MTD is within a bound. In all Bayesian based CRM design regardless of number of stages, the validity of the underlying assumptions and the accuracy of the prior information have significant 28 impact on their performance. The lack of prior information about true MTD is common in some first in human studies. For example, Simon et al. (1997) shown that it was often that more than 20 dose levels is required to be tested before reaching the MTD which is 1000 times as higher as the starting dose and ad hoc additional dose levels are often required. Model based designs are not robust so that they are inappropriate for the situation. 3.4.3 Mixed Two-Stage Designs. In the literature, several mixed two stage designs have been proposed in which first stage is rule-based and ends upon observation of the first DLT and the second stage is a CRM. Two parameter logistic models and MLE are used in all these mixed two stage designs. A requisite assumption of these designs is that the true MTD is in the vicinity of the dose with the first observation of DLT. To avoid Bayesian method and a guess of priori, O’Quigley and Shen (1996) modified the original CRM into a two stage design in which the first stage is the standard design till the observation of DLT and the second stage is the CRM using the MLE to estimate the parameters of dose toxicity model and determine dose allocation. Through simulation studies, they found that the two stage design performed similarly as the one stage CRM in selecting correct MTD when the true MTD is not very high and but would have fewer toxic events. When the true MTD is high the two stage design is prone to underestimate the MTD. Wang and Faries (2000) introduced a similar mixed two stage design in which two patient cohorts are used in the first stage till observation of 1 or 2 DLTs in the same 29 cohort and in the second stage 20 single patient cohorts are treated sequentially at the dose level closest to the MTD estimated with MLE fitting a logistic model, logit(P DLT (x i )) = α + βx i (where x i is the true dose) to the available data. To guarantee converge of the logistic regression and existence of parameter estimates, one observation of a pseudo- patient with DLT at the highest dose level is added. Storer (2001) proposed a mixed two stage design in which the first stage is the Storer’s first stage and in the second stage three patient cohorts are accrued and treated at the level closest to the MLE of the MTD using a logistic model logit(P DLT (x i )) = α + βx i , in which β is fixed at 0.75 and x i is the log of the dose. There is an exception that the first cohort of the second stage is treated at the dose level immediately below the last dose level of the first stage. Potter (2002) introduced a similar mixed two stage method as the Storer’s two stage (2001) with the exceptions that in the first stage dosage increases 50% in each escalation and in the second stage both α and β in the model logit(P DLT (x i )) = α + βx i are estimated. To make sure the monotonicity of logistic regression, 20 observation of pseudo-patients without DLT at a very low dose and 20 with DLT at a very high dose are added. The first stage of mixed two stage design is a rule based design is and MLE instead of Bayesian method is used in its second stage so that initialization information is used as minimally as possible. Therefore, the mixed two stage design outperforms the two stage model based design 30 3.5 Disadvantages and Advantage of Different Designs. All ruled based designs are robust and simple to implement and usually give a reasonable MTD under some certain rules. Applying some sort of models, such as IR, to data can improve the accuracy of the MTD. Model based designs required a parametric model of dose toxicity relationship and may greatly improve the probability of estimating the correct MTD compared with rule based designs when certain assumptions are satisfied. However, model based designs are not robust and should not be used unless their underlying assumptions can be met with confidence. The accuracy of estimated MTD depended substantially on the number of the observed DLTs and sample size is also an important factor. Overall, different designs, no matter rule based or model based, usually perform similarly when they are similar in sample size and aggressiveness. Thus, simple designs, especially standard designs, are still very popular in the Phase I clinical trial practices. 31 Chapter IV: The Limit and Trend of Expected Toxicity Level at MTD of Standard Phase I Designs Although many other types of designs have been proposed in recent years, the standard algorithm-based 3+3 designs are still widely used for their practical simplicity. At early stage, a common belief was that the ETL at the MTD should be 33% (Storer, 1989; Storer and Willson, 1992; Mick, 1996). Recently, simulation studies by Kang and Ahn (Kang and Ahn, 1996; Kang and Ahn, 2002) indicated that ETL is between 17% and 21% if the dose-toxicity curve is S-shaped where the toxicity rate is small in low dose levels. He et al (2006) further confirmed that the standard 3+3 designs produce the ETL in the range of 19 % to 24%. However they only investigated designs where the number of dose levels was at most 20. They conjectured, but did not verify, that the limiting value of ETL lay in the interval (0%, 16%). In this chapter, we perform simulation studies using the approach of He et al. (2006) to generate arbitrary increasing dose toxicity relationships and applying the formula for ETL developed by Lin and Shih (2001) to investigate the relationship between the ETL and the number of dose levels tested in standard algorithm-based k+k designs for k=2, 3, 4 and 5. We find that the ETL decreases monotonically as the number of the dose levels specified for the design increases and will approach 0 when the number of dose levels approaches infinity. We conclude that the number of specified dose levels substantially affects the ETL at MTD and is an important factor in the design. We recommend that, for escalation plans where 32 the number of dose levels is determined prior to the start of the study, fewer than 20 levels be designated. 4.1 Introduction The main purpose of a phase I cancer clinical trial is to determine the MTD of a drug for a specific mode of administration. Under the assumption that the benefit of new cytotoxic agent increases with dose, but that increases in dose are associated with increases in the frequency of side effects, the MTD is sought as the highest dosage of a new drug at which the patients not to experience serious side effects with high frequency. These identified serious side effects are designated as DLTs in the subsequent discussion (Rosenberger and Haines, 2002). The TTL is an investigator-specified probability of DLT, p * , the highest acceptable probability of DLT. Many factors are considered in establishing the TTL, such as the expected benefit in terms of disease control and the risk of death or disease progression if ineffective therapy is employed. The ETL is the probability of DLT at MTD defined. For designs that are based on algorithms and do not provide estimation of dose associated with the TTL (He et al., 2006), the TTL will likely differ from the ETL. Although many other types of designs have been proposed in recent years (O’Quigley et al., 1990; Shen and O’Quigley, 1996; Durham and Flournoy, 1994; Durham and Flournoy, 1997; Whitehead and Brunier, 1995; Whitehead, 1997; Babb et al. 1998; Leung and Wang, 2001), the standard algorithm-based designs, especially the 3+3 designs, are still widely used for their practical simplicity (Kang and Ahn, 1996; Kang 33 and Ahn, 2002). With these designs, a trial begins with the selection of a starting dose based on animal studies, usually one-tenth of the lethal dose in mice or based on the information from previous trials. The dose administered to patients is escalated in cohorts, where the series of dose levels tested are defined as part of the study design. If many doses are selected and the doses are closely spaced in the scale of the probability of DLT, large percent of patients will be treated at dosages with a small probability of DLT. Such doses are less likely to provide benefit to the population. On the other hand, if too few doses are selected and the doses are coarsely spaced, the estimated MTD likely will not have the probability of DLT in vicinity of the TTL which could result in a high percent of patients being treated at dosages with a substantial potential for DLT (Potter, 2006). In the standard 3+3 design, the MTD is selected algorithmically; the trial is not designed to select a dose with a specified TTL. Indeed, in these algorithmically based designs, a particular TTL can be substantially different from the ETL. ETL was initially anticipated to be 33% because a batch of 3 patients are treated at a time and the dose level stops increasing when 33% (1/3 or 2/6) patients experience DLT (Storer, 1989; Storer and Willson, 1992; Mick, 1996). Later, through simulation studies, Kang and Ahn (Kang and Ahn, 1996; Kang and Ahn, 2002) found the ETL at the estimated MTD in the simulation studies varied from 17% to 21% when the dose-toxicity curve was S-shaped and the toxicity rate was small in low dose levels. Recently, He et al.(2006), using the formulae developed by Lin and Shih (2001), further confirmed that the standard 3+3 designs produce the ETL in the range of 19 % to 24%. But He et al. (2006) only 34 investigated designs where the number of dose levels was at most 20. They conjectured, but did not verify, that the limiting value of ETL lay in the interval (0%, 16%). In this chapter, we simulate the true dose toxicity relationship and use the formulae of Lin and Shih (2001) to calculate the ETL. With this, we investigate the limiting behavior of the ETL in relation to the number of dose levels as the number of dose levels increases infinitely in standard algorithm-based 3+3 designs as well as 2+2, 4+4, and 5+5 designs. 4.2 Standard 3+3 Design without Dose De-escalation 4.2.1 Scheme of the Design The general standard A+B design has two kinds of different implementations: one without dose de-escalation and the other one with dose de-escalation. A diagram for the general A+B design without dose de-escalation adapted from Lin and Shih (2001) is presented in Figure 4.1. The standard 3+3 design without dose de-escalation is a special case in which A=B=3 and C=D=E=1. With this design, three patients are assigned to the first dose level. If no DLT is observed, the trial proceeds to the next dose level and another cohort of three patients is enrolled. If at least two out of the three patients experience at least one DLT, then the previous dose level is considered as the MTD; otherwise, if only one patient experiences the DLT, then three additional patients are enrolled at the same dose level. If at least one of the three additional patients experience the DLT, then the previous dose is considered as the MTD; otherwise, the dose will be escalated. 35 Figure 4.1: Escalation Scheme for Traditional A+B Design without Dose De-escalation 4.2.2 Simulation Using the Formula Developed by Lin and Shih. In the literature, a well accepted assumption is that toxicity increases monotonically with dose level. Three parametric dose-toxicity functions, logistic model, hyperbolic model and power function, have been usually employed to depict the relationship between dose and toxicity. We, however, employ the method proposed by He et al. (2006). We generate N =5,000 sets of probabilities of DLT for dose levels d 1 ,…,d k as θ 1 ,…,θ k from the uniform distribution (0; 1) with θ 1 ≤ θ 2 ≤…≤ θ k through 36 simulation. The thetas ( θ 1 ,…,θ k ) are order statistics from the U(0,1) distribution. For each of the N sets of the θ 1 ,…,θ k , ETL is obtained by the exact computation using the formulae developed by Lin and Shih (2001). For the standard 3+3 design without dose de-escalation in which A=B=3 and C=D=E=1, in each simulated trial, we define θ j as the probability of a patient having DLT at dose level j (1 ≤ j < n) and j P 0 = P(<C/A when treated at dose j), and j Q 0 = P( ( ≥C/A but ≤ D/A) + ≤ E/(A+B) when treated at dose j). Then: x A j x j C x j x A P − − = − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∑ ) 1 ( 1 0 0 θ θ m B j m j D C x x E m x A j x j j m B x A Q − = − = − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∑∑ ) 1 ( ) 1 ( 0 0 θ θ θ θ Let P i be defined as the probability of dose i chosen as the MTD, and () 1 0 1 0 1 0 0 1 ) ( ) 1 ( + + = − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = < ≤ ∏ i i i j j j i Q P Q P K i P 1 0 1 0 1 ) 1 ( Q P i P i − − = < ∏ = + = ≥ K j j j i Q P K i P 1 0 0 )) ( ) ( For each simulated dose-toxicity relationship, ETL can be derived as: ETL= ∑ − = 1 1 K i P(DLT at MTD | MTD = Dose i) P(MTD = Dose i | dose 1 ≤ MTD < dose K) ∑ ∑ − = − = = 1 1 1 1 K i i K i i i P P ETL θ 37 The results of the simulations are combined to evaluate the parameter estimates with the following summary statistics: where l L T E ˆ represents the expected toxicity level at the l th (l = 1 to N) simulation. The mean ETL (METL) is defined as N L T E L T ME N l l ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ∑ =1 ˆ ˆ The standard deviation (SD) of ETL is calculated as: 1 ) ˆ ˆ ( ) ˆ ( 1 2 − − = ∑ = N L T ME L T E L T E SD N l l These represent estimates of the true mean and standard deviation over the space of all monotonically increasing dose-toxicity relationships. 4.2.3 Simulation Results. The METLs in the Standard 3+3 Phase I Design without dose de-escalation by the number of dose levels are summarized in Table 4.1. When the number of dose levels is as low as 3, the highest METL is 28.8%. The METL decreases monotonically as the number of dose levels increases. The METLs range from 24% to 17% when the number of dose level varies from 5 to 25. The METL decreases to below 1/6 (16.7%) when the number of dose level is larger than 30. As the number of dose levels increases to 1000, the METLs decreases substantially to as low as 5.8% (SD = 0.2%). The METLs further decrease to 1.2% (SD = 0.01%) when the number of dose level is 100,000. We expect that the ETL approaches 0 when the number of dose level increases infinitely (Table 4.1 and Figure 4.2 and 4.3). 38 Table 4.1: METL at MTD from Standard 3+3 Design. Total number of dose levels Standard 3+3 Phase I Design Mean (95% CI) Without Dose De-escalation With Dose De-escalation 3 28.8 (28.3, 29.3) 28.0 (27.5, 28.5) 4 26.2 (25.8, 26.6) 25.0 (24.6, 25.4) 5 24.5 (24.2, 24.8) 23.2 (22.9, 23.5) 6 23.3 (23.1, 23.5) 22.1 (21.9, 22.3) 7 22.5 (22.3, 22.7) 21.3 (21.1, 21.5) 8 21.9 (21.7, 22.1) 20.8 (20.6, 21.0) 9 21.5 (21.4, 21.6) 20.4 (20.2, 20.6) 10 21.1 (21.0, 21.2) 20.0 (19.9, 20.1) 11 20.7 (20.6, 20.8) 19.7 (19.6, 19.8) 12 20.3 (20.2, 20.4) 19.5 (19.4, 19.6) 13 20.0 (19.9, 20.1) 19.2 (19.1, 19.3) 14 19.7 (19.6, 19.8) 18.9 (18.8, 19.0) 15 19.5 (19.4, 19.6) 18.7 (18.6, 18.8) 16 19.2 (19.1, 19.3) 18.5 (18.4, 18.6) 17 19.0 (18.9, 19.1) 18.4 (18.3, 18.5) 18 18.8 (18.7, 18.9) 18.2 (18.1, 18.3) 19 18.6 (18.5, 18.7) 18.0 (17.9, 18.1) 20 18.4 (18.35, 18.45) 17.9 (18.85, 17.95) 21 18.2 (18.15, 18.25) 17.7 (18.65, 17.75) 22 18.0 (17.96, 18.04) 17.6 (17.56, 17.64) 23 17.8 (17.76, 17.84) 17.4 (17.36, 17.44) 24 17.7 (17.66, 17.74) 17.3 (17.26, 17.34) 25 17.5 (17.46, 17.54) 17.1 (17.06, 17.14) 26 17.4 (17.36, 17.44) 17.0 (16.96, 17.04) 27 17.2 (17.16, 17.24) 16.9 (16.86, 16.94) 28 17.1 (17.06, 17.14) 16.8 (16.76, 16.84) 29 17.0 (16.96, 17.04) 16.6 (16.57, 16.63) 30 16.8 (16.76, 16.84) 16.5 (16.47, 16.53) 40 15.7 (15.67, 15.73) 15.5 (15.47, 15.53) 50 14.8 (14.77, 14.83) 14.7 (14.67, 14.73) 80 13.0 (12.98, 13.02) 12.9 (12.88, 12.92) 100 12.1 (12.08, 12.12) 12.1 (12.08, 12.12) 1000 5.8 (5.79, 5.81) 5.8 (5.79, 5.81) 10000 2.6 (2.60, 2.60) 2.6 (2.60, 2.60) 100000 1.2 (1.20, 1.20) 1.2 (1.20, 1.20) 39 Figure 4.2: ETL (%) at MTD from the Standard 3+3 Design by Number of Tested Dose Levels. 10 12 14 16 18 20 22 24 26 28 30 0 20406080 100 Number of Doses Expected Toxicity Level (%) Without Deescalation With Deescalation 40 Figure 4.3: METL (%) at MTD from the Standard 3+3 Designs by Log 10 Transformation of Number of Dose Levels. 0 5 10 15 20 25 30 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Log 10 (Number of Doses) Expected Toxicity Level (%) Without Deescalation With Deescalation 4.3 Standard 3+3 Design with Dose De-escalation 4.3.1 Scheme of the Design The diagram for the general A+B design with dose de-escalation adapted from Lin and Shih (2001) is presented in Figure 4.4. The standard 3+3 design with dose de- escalation is a special case in which A=B=3 and C=D=E=1. This design is essentially 41 similar to the 3+3 design without dose de-escalation, but provides for re-examination of lower doses when DLTs are observed with excessive frequency at higher levels. The algorithm allows three new patients to be treated at a previous dose level if only three patients are treated at that level previously. Dose reduction continues until a dose level is reached at which six patients are treated and at most one DLT is observed in the six patients. The MTD is defined as the highest dose level at which at most one of six patients experience DLT, and the immediate higher dose level has at least two patients who experience DLTs. If the first dose is not tolerable, then the MTD cannot be established within the confines of the study. 42 Figure 4.4: Escalation Schema for Standard A+B Design with Dose De-escalation 4.3.2 Simulation Using the Formula Developed by Lin and Shih. For the standard 3+3 design with dose de-escalation in which A=B=3 and C=D=E=1, let θ j , j P 0 , and j Q 0 be defined as in section 4.2.2. Let j P 1 = P( ≥C/A but ≤ D/A when treated at dose j), and j Q 1 = P(<C/A + ≤ E/(A+B) when treated at dose j), and j Q 2 = P(<C/A + > E/(A+B) when treated at dose j). 43 Then we got: x A j x j D C x j x A P − = − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∑ ) 1 ( 1 θ θ m B j m j C x x E m x A j x j j m B x A Q − − = − = − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∑∑ ) 1 ( ) 1 ( 1 00 1 θ θ θ θ m B j m j C x B x E m x A j x j j m B x A Q − − =− + = − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∑∑ ) 1 ( ) 1 ( 1 01 2 θ θ θ θ The probability that a particular dose level is chosen as MTD is given by: () ( ) ∑ ∏ ∏ + = − + = − = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = < ≤ K i k k k k i j j i i i j j j i Q P Q Q Q Q P K i P 1 0 0 1 1 2 1 0 1 1 0 0 1 ) ( ) 1 ( () ∑ ∏ = − = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = < K k k k k j j i Q P Q i P 1 0 0 1 1 2 1 ) 1 ( ∏ = + = ≥ K k k k i Q P K i P 1 0 0 )) ( ) ( The ETL is provided by the formula given above. The formulae to calculate the METL and standard deviation of ETL of the 5000 simulations are the same as those for the design without dose de-escalation in Section 4.2.2. 4.3.3 Simulation Results. The METLs in the Standard 3+3 Phase I Design with dose de-escalation by the number of dose levels are summarized in Table 4.1. When the number of dose levels is as low as 3, the highest METL is 28.0%. The METL decreases monotonically with the increased number of dose levels. The METLs range from 23% to 17% when the number 44 of dose level varies from 5 to 25. The METL decreases to below 1/6 (16.7%) when the number of dose levels is larger than 30. It further decreases to 1.2% (SD = 0.01%) when the number of dose level reaches 100,000, the largest value considered in our simulation. As with the design that does not incorporate dose de-escalation, we expect that the METL approaches 0 as the number of dose level increases to infinity (Table 4.1 and Figure 4.2 and 4.3). The METLs from Standard 3+3 Phase I Design with dose de-escalation are noticeably lower than those from the design without dose de-escalation when the number of dose levels is less than 100 (Table 4.1 and Figure 4.2 and 4.3). The difference decreases with the increasing number of tested dose levels (Table 4.1 and Figure 4.2 and 4.3). From Table 4.1, we see that the METLs are approximately 18% in designs that examined 20 dose levels. Thus 20 dose levels or less in a Phase I trial is desirable in order to select an appropriate MTD which can produce an amount of toxicity required for the therapeutic effect of a cytotoxic agent. 4.4 Other Standard Algorithm-based A+B Designs. We also conduct similar simulation studies to investigate the METLs in the other standard A+B designs, such as 2+2 design (A=B=2 and C=D=E=1), 4+4 design (A=B=4 and C=D=E=1), and 5+5 design (A=B=5 and C=D=E=1). These designs are not comparable with respect to their cutoffs. The analysis shows that decreasing the number of patients in each cohort still leads to the same conclusion, but the setup does not permit direct comparison of the cohort sizes. The results are also summarized in the Table 4.2, 45 Table 4.3, and Table 4.4. We find that, similar to the standard 3+3 design, METLs decrease monotonically as number of dose levels increases and approached 0 when number of dose levels increases infinitely. The designs with dose de-escalation give lower METLs than the corresponding designs without dose de-escalation in standard 2+2, 4+4, and 5+5 designs as in the 3+3 designs. However, the difference diminishes as the number of dose levels increases. The ratios of METLs from standard 2+2, 4+4, and 5+5 designs compared with the corresponding standard 3+3 designs versus the log 10 (number of dose levels) are plotted in Figure 4.5 (without dose de-escalation) and Figure 4.6 (with dose de-escalation). As expected, for a fixed number of dose levels, the METLs increase as the number of patients enrolled in a cohort decreases. The ratios of their METLs compared with corresponding standard 3+3 designs appear to approach limiting values of 1.33, 0.83, and 0.75, for 2+2, 4+4, and 5+5 designs, respectively as the number of dose levels increases. These results demonstrate that the trends of ETLs by increasing number of dose levels found in 3+3 designs are similarly existed in the other standard A+B designs. 46 Table 4.2: METL at MTD from Standard 2+2 Design. Total number of dose levels Standard 2+2 Phase I Design Mean (95% CI) Without Dose De-escalation With Dose De-escalation 3 31.8 (31.4, 32.2) 30.9 (30.4, 31.4) 4 30.9 (30.6, 31.2) 29.7 (29.4, 30.0) 5 30.3 (30.1, 30.5) 29.0 (28.7, 29.3) 6 29.7 (29.5, 29.9) 28.5 (28.3, 28.7) 7 29.2 (29.0, 29.4) 28.1 (27.9, 28.3) 8 28.8 (28.7, 28.9) 27.8 (27.7, 27.9) 9 28.4 (28.3, 28.5) 27.5 (27.4, 27.6) 10 28.0 (27.9, 28.1) 27.1 (27.0, 27.2) 11 27.6 (27.5, 27.7) 26.8 (26.7, 26.9) 12 27.2 (27.1, 27.3) 26.5 (26.4, 26.6) 13 26.9 (26.8, 27.0) 26.2 (26.1, 26.3) 14 26.5 (26.4, 26.6) 25.9 (25.8, 26.0) 15 26.2 (26.1, 26.3) 25.6 (25.5, 25.7) 16 25.9 (25.8, 26.0) 25.4 (25.3, 25.5) 17 25.6 (25.55, 25.65) 25.1 (25.05, 25.15) 18 25.3 (25.25, 25.35) 24.8 (24.75, 24.85) 19 25.0 (24.95, 25.04) 24.6 (24.56, 24.64) 20 24.7 (24.65, 24.75) 24.4 (24.36, 24.44) 21 24.5 (24.46, 24.54) 24.1 (24.06, 24.14) 22 24.2 (24.16, 24.24) 23.9 (23.86, 23.94) 23 24.0 (23.96, 24.04) 23.7 (23.66, 23.74) 24 23.8 (23.76, 23.84) 23.5 (23.46, 23.54) 25 23.5 (23.46, 23.54) 23.3 (23.26, 23.34) 26 23.3 (23.26, 23.34) 23.1 (23.06, 23.14) 27 23.1 (23.06, 23.14) 22.9 (22.86, 22.94) 28 22.9 (22.86, 22.94) 22.7 (22.66, 22.74) 29 22.7 (22.66, 22.74) 22.5 (22.46, 22.54) 30 22.6 (22.56, 22.64) 22.4 (22.36, 22.44) 40 21.0 (20.96, 21.04) 20.8 (20.76, 20.84) 50 19.7 (19.66, 19.74) 19.6 (19.57, 19.63) 80 17.2 (17.17, 17.23) 17.2 (17.17, 17.23) 100 16.1 (16.08, 16.12) 16.1 (16.08, 16.12) 1000 7.6 (7.59, 7.61) 7.6 (7.59, 7.61) 10000 3.5 (3.50, 3.50) 3.5 (3.50, 3.50) 100000 1.6 (1.60, 1.60) 1.6 (1.60, 1.60) 47 Table 4.3: METL at MTD from Standard 4+4 Design. Total number of dose levels Standard 4+4 Phase I Design Mean (95% CI) Without Dose De-escalation With Dose De-escalation 3 27.4 (26.9, 28.9) 26.7 (26.2, 27.2) 4 23.8 (23.4, 24.2) 22.9 (22.5, 23.3) 5 21.6 (21.3, 21.9) 20.5 (20.2, 20.8) 6 20.1 (19.8, 20.4) 19.0 (18.7, 19.3) 7 19.1 (18.9, 19.3) 18.0 (17.8, 18.2) 8 18.4 (18.2, 18.6) 17.3 (17.1, 17.5) 9 17.9 (17.7, 18.1) 16.8 (16.6, 17.0) 10 17.4 (17.3, 17.5) 16.4 (16.2, 16.6) 11 17.0 (16.9, 17.1) 16.0 (15.9, 16.1) 12 16.7 (16.6, 16.8) 15.8 (15.7, 15.9) 13 16.4 (16.3, 16.5) 15.5 (15.4, 15.6) 14 16.1 (16.0, 16.2) 15.3 (15.2, 15.4) 15 15.8 (15.7, 15.9) 15.1 (15.0, 15.2) 16 15.6 (15.5, 15.7) 14.9 (14.8, 15.0) 17 15.4 (15.3, 15.5) 14.7 (14.6, 14.8) 18 15.2 (15.1, 15.3) 14.6 (14.5, 14.7) 19 15.1 (15.0, 15.2) 14.4 (14.3, 14.5) 20 14.9 (14.8, 15.0) 14.3 (14.2, 14.4) 21 14.7 (14.6, 14.8) 14.1 (14.04, 14.16) 22 14.6 (14.55, 14.65) 14.1 (14.05, 14,15) 23 14.4 (14.35, 14.45) 13.9 (13.85, 13.95) 24 14.3 (14.26, 14.34) 13.8 (13.76, 13.84) 25 14.2 (14.16, 14.24) 13.7 (13.66, 13.74) 26 14.1 (14.06, 14.14) 13.6 (13.56, 13.64) 27 14.0 (13.96, 14.04) 13.6 (13.56, 13.64) 28 13.9 (13.86, 13.94) 13.5 (13.46, 13.54) 29 13.8 (13.76, 13.84) 13.4 (13.37, 13.43) 30 13.6 (13.56, 13.64) 13.3 (13.27, 13.33) 40 12.7 (12.67, 12.73) 12.5 (12.47, 12.53) 50 12.1 (12.07, 12.13) 11.9 (11.88, 11.92) 80 10.6 (10.58, 10.62) 10.5 (10.48, 10.52) 100 9.9 (9.88, 9.92) 9.9 (9.88, 9.92) 1000 4.7 (4.69, 4.71) 4.7 (4.69, 4.71) 10000 2.2 (2.20, 2.20) 2.2 (2.20, 2.20) 100000 1.0 (1.00, 1.00) 1.0 (1.00, 1.00) 48 Table 4.4: METL at MTD from Standard 5+5 Design. Total number of dose levels Standard 5+5 Phase I Design Mean (95% CI) Without Dose De-escalation With Dose De-escalation 3 26.5 (26.0, 27.0) 26.0 (25.49, 26.5) 4 22.5 (22.1, 22.9) 21.8 (21.4, 22.2) 5 20.0 (19.7, 20.3) 19.1 (18.8, 19.4) 6 18.3 (18.0, 18.6) 17.4 (17.1, 17.7) 7 17.1 (16.9, 17.3) 16.2 (15.9,16.5) 8 16.3 (16.1, 16.5) 15.3 (15.1, 15.5) 9 15.7 (15.5, 15.9) 14.8 (14.6, 15.0) 10 15.2 (15.0, 15.4) 14.3 (14.1, 14.5) 11 14.8 (14.7, 14.9) 13.9 (13.7, 14.1) 12 14.4 (14.3, 14.5) 13.6 (13.5, 13.7) 13 14.1 (14.0, 14.2) 13.3 (13.2, 13.4) 14 13.8 (13.7, 13.9) 13.0 (12.9, 13.1) 15 13.6 (13.5, 13.7) 12.8 (12.7, 12.9) 16 13.3 (13.2, 13.4) 12.6 (12.5, 12.7) 17 13.2 (13.1, 13.3) 12.4 (12.3, 12.5) 18 13.0 (12.9, 13.1) 12.3 (12.2, 12.4) 19 12.8 (12.7, 12.9) 12.2 (12.1, 12.3) 20 12.7 (12.6, 12,8) 12.1 (12.0, 12.2) 21 12.5 (12.4, 12.6) 11.9 (11.8, 12.0) 22 12.4 (12.3, 12.5) 11.8 (11.7, 11.9) 23 12.3 (12.2, 12.4) 11.7 (11.6, 11.8) 24 12.2 (12.15, 12.25) 11.6 (11.55, 11.65) 25 12.1 (12.05, 12.15) 11.6 (11.55, 11.65) 26 12.0 (11.96, 12.04) 11.5 (11.46, 11.54) 27 11.9 (11.86, 11.94) 11.4 (11.36, 11.44) 28 11.8 (11.76, 11.84) 11.3 (11.26, 11.34) 29 11.7 (11.66, 11.74) 11.2 (11.16, 11.24) 30 11.6 (11.56, 11.64) 11.2 (11.16, 11.24) 40 10.8 (10.77, 10.83) 10.5 (10.48, 10.52) 50 10.3 (10.28, 10.32) 10.1 (10.08, 10.12) 80 9.1 (9.08, 9.12) 9.0 (8.98, 9.02) 100 8.5 (8.48, 8.52) 8.4 (8.38, 8.42) 1000 4.1 (4.09, 4.11) 4.1 (4.09, 4.11) 10000 1.9 (1.90, 1.90) 1.9 (1.90, 1.90) 100000 0.9 (0.90, 0.90) 0.9 (0.90, 0.90) 49 Figure 4.5: Ratio of ETLs from Different Standard Designs without Dose De-escalation by Log 10 Transformation of Number of Tested Dose Levels. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0123 456 Log10 (Number of Doses) Ratio Design 2+2 vs 3+3 Design 4+4 vs 3+3 Design 5+5 vs 3+3 50 Figure 4.6: Ratio of ETLs From Different Standard Designs with Dose De-escalation by Log 10 Transformation of Number of Tested Doses. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0123456 Log10 (Number of Doses) Ratio Design 2+2 vs 3+3 Design 4+4 vs 3+3 Design 5+5 vs 3+3 4.5 Discussion. In our study, we investigate the relationship between METL and number of dose levels employed in both standard 3+3 designs with or without dose de-escalation mechanism as well as some other standard A+B designs. Based on our simulation results, the design without dose de-escalation gives higher METLs than the design with dose de- 51 escalation for all the number of dose levels examined, particularly noticeable when the number of dose levels is less than 30. This result is consistent with the data from He et al. (2006). This can be explained by the fact that MTD is usually chosen at higher dose level in design without dose de-escalation because this design does not allow re-examination of a dose level once it is considered to have an acceptable toxicity profile. The desired TTL combined with the number of patients available for a phase I trial should inform the choice of whether a dose de-escalation design is preferred to a design that does not incorporate dose de-escalation. We find that the standard 3+3 designs produce the METLs in the range of 17% to 28% when the number of dose levels used in the design is less than 25. These results are consistent with those from previous published studies (Kang and Ahn, 1996; Kang and Ahn, 2002; He et al., 2006; Smith et al., 1996; Korn et al., 1994). Smith et al.(Smith et al., 1996) found that ETL varied from 23% to 28% with a revised standard design, in which if two or more patients experienced DLTs at a dose level then the dose level below was defined as the MTD. Korn et al. (Korn et al., 1994) reported the ETL in the range from 17% to 21% in their revised standard design in which the dose level may deescalate several times. From our simulations, the estimates of the ETLs and their associated standard deviations decrease monotonically as the number of dose levels considered in the design increases. This is consistent with the reports from previous study (He et al., 2006). This phenomenon can be explained heuristically by noted that, as the number of dose levels increases, the trial requires to test more and more dose levels with probability of DLTs 52 below the TTL. Consequently there are more chances that 2 out the 6 tested patients will show DLTs and the trial stops before reaching the dose with the TTL. From our results, the ETL decreases from about 28% to 1.2%. The number of dose level tested can substantially affect the ETL and the number of dose levels is an important factor to be considered when the phase I trial is designed. In our study, we assume only an increasing relationship between dose and toxicity. Table 4.1, 4.2, 4.3, and 4.4 provide a reference for the association between ETL and number of dose levels considered in a design when the exact shape of the dose-toxicity relationship is not well understood. We, for the first time, find that the ETL in general approach 0 when number of dose levels goes to the infinity. Our simulation is predicated on a model where the probability of DLT at the lowest dose level and the difference in the probability of DLT between two consecutive dose levels tend to 0 as the number of dose levels tends to infinity so that a trial keeps testing the dose level of DLT probability close to 0 until trial stops because of getting 2 or more patients with DLTs out of 6 tested patients by chance. We admit that the limit of ETL is bound by the probability of DLT at the lowest dose level. From the expression for the ETL, we see that the ETL will not tend to 0 if 0, i kiN k ε θ ∃> ∋∀ ≥ . Such a structure, however, implies that 0 dose of the agent under study would be associated with a non-zero probability of DLT. This is not a plausible model for most agents used in drug studies. Therefore, our conclusion is valid in general. In summary, our study indicates that the ETL decreases monotonically as the number of the dose level tested increases and will in general approach 0 when the number 53 goes to infinity. We conclude that the number of dose levels incorporated into a study design will substantially affect the ETL and is an important design parameter to consider. We recommend that a maximum of 20 dose levels be incorporated into the design if an ETL between 28% and 18% is desired. 54 Chapter V: An Extended Isotonic Design for Phase I Trials with Differentiation of Graded Toxicity 5.1 Introduction From our study in the previous chapter, we can see that ETL of the standard 3+3 designs has a varying range (28%-0%), substantially differs from TTL (33%). Moreover the standard 3+3 design has many other weaknesses event though it is used in the CTEP Phase I protocol templates (available at the CTEP website) and has been most widely used. For example, standard 3+3 designs can not predefine an exact TTL and estimate MTD accordingly. The MTD is not a dose with any particular probability of toxicity and the MTD estimate has large variability. Not all toxicity data at every dose level are used to determine the MTD. Many patients are likely to be treated at low doses and at least two patients will be treated at dose levels above the MTD. Dose-escalation does not stop upon the observation of first DLT at a dose level. There is a difficulty in dose levels selection: if the levels are closely spaced, many patients will be required to complete study, and many of these will be treated at low doses; however, if the levels are coarsely spaced, the accuracy of the MTD will be poor, and patients will be more likely to be treated at excessively toxic doses. These obvious weaknesses trigger the invention of many other Phase I designs (Potter, 2006). 55 5.1.1 Motivation and Purpose of the Extended Isotonic Design. After a comprehensive review of the current Phase I designs proposed in the literature and several years of experiences in Phase I clinical trials, we find that toxicity response is reduced to be a binary indicator as 1 for DLT, 0 for no DLT in most of the current Phase I designs, including the standard 3+3 designs. In the NCI Common Toxicity Criteria (National Cancer Institute, 2003), the DLT is defined as a group of grade 3 or 4 non-hematologic and grade 4 hematologic toxicities as well as death (grade 5). This kind of too coarse classification of DLT can not satisfy several practical requirements for a lot of clinical trials. First, patients often experience practically multiple toxicities of different grades varying from 0 (no toxicity of that type) to 4 (life threatening) instead of single toxicity. Second, the toxicities are not equally severe. For example, a grade 4 renal toxicity of severe and possibly irreversible effects is much more dangerous than a grade 3 reversible toxicity of other type, such as a grade 3 neutropenia. Third, some different toxicities usually occur together, such as: fatigue and nausea/vomiting, myelosuppression and fever. Fourth, some, but not all, low-grade non DLT toxicity occurred at a lower dose may be likely followed by occurrence of same toxicity of a higher grade at a higher dose. It will provide a more reliable basis for dose escalation to distinguish grades 0, 1, and 2 rather than treat them equally as “non toxicity”. Although convenient to reduce the multiple and ordinal toxicities to only an indictor of the occurrence of DLT, it discards useful information of toxicity which should be used in the dose allocation algorithm (Yuan et al., 2006; Bekele and Thall, 2004; Wang et al., 2000). 56 The accuracy of the MTD depends more on the observed toxicity response, especially the DLTs if most patients do not have a DLT, than on the sample size (Potter D. 2006). Therefore all toxicity information of all patients should be fully utilized in the trial in order to maximize the efficiency. In this chapter, we are going to develop a new design which can fulfill our 3 goals, first to fully utilize the severity information of all toxicities during the dose allocation procedure and estimation of the MTD by treating toxicity response as a Quasi-Continuous variable instead of a binary outcome; second to estimate the MTD exactly according to pre-specified acceptable toxicity level; third to keep the robustness and practical simplicity of ruled based design. 5.1.2 Current Major Phase I Designs. At present, lots of designs for Phase I trials have been proposed in the literature (Rosenberger et al., 2002; Potter, 2006). The standard 3+3 designs (Simon et al., 1997) and CRM (O’Quigley et al., 1990) are the two major designs. In standard 3+3 designs, the TTL can neither be defined before the trial, nor be estimated exactly after the trial (Potter, 2006). The Bayesian CRM allows exact definition of targeted toxicity level and estimation of the corresponding MTD (O’Quigley et al., 1990). A lot of simulation studies in the literature showed that the CRM were superior over the standard method by greatly improving the probability of estimating the correct MTD (O’Quigley et al., 1990; O’Quigley and Chevret, 1991; O’Quigley and Shen, 1996; Goodman et al., 1995; Chevret, 1993; Heyd and Carlin, 1999; Møller, 1995; Zacks et al., 1998). However, CRM still has not been widely used due to its model based, intensive computation, requirement 57 of prior distribution, and complicated interpretation (Rosenberger et al., 2002; Potter, 2006; O’Quigley et al., 1990; O’Quigley and Chevret, 1991). The standard 3+3 designs are still the most popular designs because of their practical simplicity and no modeling required beyond a well-accepted assumption of non-decreasing relationship between dose and toxicity (Leung and Wang, 2001; Kang and Ahn, 2001; Kang and Ahn, 2002). In 2001, Leung and Wang proposed an ID for Phase I trials (Leung and Wang, 2001). Their ID is a simple model free approach which only requires the same well- accepted assumption of non-decreasing relationship between dose and toxicity as standard designs. Therefore it is as simple as the popular standard design and can be easily implemented in practice (Leung and Wang, 2001). But different from the standard designs, the ID employs IR (Bartholomew, 1983) to estimate the toxicity probability of each dose level and adjust the estimates to enforce a non-decreasing order starting from the lowest dose level. The escalation and de-escalation algorithm depends on the distance between the newly estimated toxicity probability of each dose level and the pre-defined TTL. The next dose level is the one which is the closest to the TTL. Through simulation study, Leung and Wang demonstrated that their design performed substantially better than the standard Phase I design and compared favorably with CRM, Storer’s Up-and- Down Design, and EWOC design (Leung and Wang, 2001). 5.1.3 Why Isotonic Design Is Chosen as A Framework? ID is chosen as a framework for our new design to incorporating all toxicity information because compared with other major Phase I designs, such as standard designs and model based designs, it has some attractive advantages. 58 During the dose allocation procedure, the standard 3+3 designs only use toxicity observed at the current dose and disregard toxicity at other doses. ID takes into account the toxicity information at all dose levels by using IR. In standard 3+3 designs, the TTL can not be exactly predefined and MTD can not be estimated exactly according to a TTL. In ID, TTL is predefined and MTD is estimated according to TTL using an IR. In standard 3+3 design, the assumption of a non-decreasing dose toxicity relationship is not used parametrically or non-parametrically. ID uses semi-parametrically the assumption of a non-decreasing dose toxicity relationship by using IR. Model based designs require a reliable dose toxicity relationship model and a prior distribution of parameter. But the model is likely very different from true dose- toxicity model and strongly informative priors may not be accurate for it. But ID only required a widely accepted assumption of a non-decreasing dose toxicity relationship. Model based designs require intensive computation and substantial effort to implement. But ID is practically simple. Model based designs are not applicable when no sufficient knowledge about the relationship between dose and toxicity is available, especially for a brand new chemical agent. But ID can handle these situations because of its model free characteristics and robustness. 5.1.4 Basic Idea of Our Extended Isotonic Design The original ID does not accommodate the differentiation of graded toxicity. In order to address them in design, we propose an EID. Our EID is an extended design of ID with an original introduction of NETS for each toxicity that patient experienced in order 59 to fully utilize all toxicity information. In our EID, toxicity is first assigned an adjusted toxicity grade that reflects its severity. A composite equivalent toxicity score (ETS) measuring all toxicities experienced by each patient is estimated and normalized (0, 1). The normalized ETS (NETS) of each patient is incorporated into the IR to determine dose allocation instead of the probability of dichotomized DLT. MTD is estimated according to the pre-defined acceptable toxicity level in terms of NETS using IR. 5.2 Toxicity Response Is Treated as A Continuous Variable. The key point of our EID is to treat toxicity response as a Quasi-Continuous variable and multiple toxicities of each patient experienced are all taken into account. Although several recent studies try to differentiate toxicity response beyond binary, none of them is shown to be really applicable in dealing with multiple toxicities a patient experiences and measuring them quantitatively and comprehensively (Yuan et al., 2006; Bekele and Thall, 2004; Wang et al., 2000). As early as 2000, Wang et al. first used a weight to reflect the differentiation in the severity of grade 3 and 4 toxicities in dose decision algorithm (Wang et al., 2000). In their extended CRM, the original TTL, p0, is used until the occurrence a grade 4 toxicity, after which the weighted TTL, p1= p0/w, will be used instead. By taking the impact of grade 4 toxicity into account, it can reduce the chance of selecting the higher dose level as MTD (Wang et al., 2000). Then, in 2004, Bekele and Thall applied the total toxicity burden (TTB) to measure the severity of toxicities (Bekele and Thall, 2004). The dose allocation procedure (increasing, staying at the same dose, or decreasing) is based on the 60 comparison between the observed TTB and the average TTB value, TTBc, of the same outcome in a hypothetical collection of cohorts of variant possible outcomes. The estimated MTD is the average TTBc value of the “staying at the same dose” cohorts. This method gave substantially improved results, but required the extensive involvement of oncologists and was practically complicated and hard to interpret (Bekele and Thall, 2004). Recently, Yuan et al. proposed a method called the Quasi-CRM in which numeric ETS was employed to incorporate the impacts of toxicity grade on the dose escalation decisions of the standard CRM by using the quasi-Bernoulli likelihood (Yuan et al., 2006). The Quasi-CRM was shown to be superior over the standard CRM and comparable to a univariate version of the Bekele and Thall method (Bekele and Thall, 2004) in some simulation studies. But no application of the Quasi-CRM in the real Phase I clinical trial was provided. 5.2.1 Composite Equivalent Toxicity Score System. In the literature, so far no comprehensive and practical score system or function has been proposed for assessing quantitatively the composite severity of multiple observed toxicities per patient in current Phase I trial designs. Without such a score system, there is no way to treat toxicity response as a Quasi-Continuous variable and fully utilize all toxicity information. Therefore we originally propose such a comprehensive score system for estimation of the composite ETS of each patient who experiences multiple toxicities during the procedures of dose allocation and MTD 61 estimation in the design of Phase I trials. In our score system, a composite ETS, ranging from 0 to 6, is estimated for patients mainly according to the toxicity grade (0 to 4) and whether toxicity is DLT, both of which have intuitive clinical interpretation and have been widely accepted (National Cancer Institute, 2003). When drug-related death (grade 5) happens, the trial needs to be suspended and re-evaluated so that it is not considered in our system. But if necessary, the highest composite ETS score, such as 7, can be assigned to death with corresponding modification. As composite ETS of each patient will be normalized (0, 1) later, therefore, initial composite ETS estimation has lots of flexibility. Suppose that there are n 1 , n 2 , . . . , n K patients who received the treatment at dose level d 1 , d 2 , . . . , d K , (d 1 ≤ d 2 ≤ . . . ≤ d K ) respectively. Let T j,k,i be the i th (1 ≤ i ≤ I) toxicity of j th (1 ≤ j ≤ n k ) patient among the n k patients who received the dose level d k (1 ≤ k ≤ K) and its corresponding toxicity grade be G j,k,i . DLTs are usually pre-defined specifically in each trial as a group of grade 4 hematological toxicity (grade 4 neutropenia or thrombocytopenia of any duration) or grade 3 or 4 nonhematological toxicity (excluding nausea, vomiting, and alopecia) and febrile neutropenia (grade 3 or 4). We further differentiate grade 3 toxicities into grade 3 non DLT and grade 3 DLT, and grade 4 toxicities into grade 4 non DLT and grade 4 DLT. It is assumed that low grade non DLT is less severe than high grade non LDT, non DLT is less severe than DLT, and grade 3 DLT is less severe than grade 4 DLT. Therefore we create an adjusted grade, ' , , i k j G , for each toxicity as 0 for grade 0 toxicity or no toxicity, 1 for grade 1 toxicity, 2 for grade 2 toxicity, 3 for grade 3 non DLT, 4 for grade 4 non DLT, 5 for grade 3 DLT, and 6 for grade 4 DLT (Table 5.1). 62 Table 5.1: Mapping of Adjusted Grade and Original Toxicity Original Grade / whether DLT Grade 0 Grade 1 Grade 2 Grade 3 Non DLT Grade 4 non DLT Grade 3 DLT Grade 4 DLT Adjusted Grade 0 1 2 3 4 5 6 Let the maximum adjusted grade, max , ,k j G , among the total I toxicities that patient j at dose level d k experienced be defined as: ) ( max ' , , 1 max , , i k j I i k j G G ≤ ≤ = Let the composite ETS for patient j at dose level k be defined as S j,k . A summary of the range of composite ETS is presented in the Table 5.4. The composite ETS score for a patient with no toxicity or only grade 0 toxicity is 0. The composite ETS score for patient who has only 1 toxicity (adjusted grade = 1) is arbitrarily assigned as 0.1. The composite ETS score for a patient who has only 1 toxicity (adjusted grade ≥ 2) is estimated as below: 1 max , , , − = k j k j G S The composite ETS score for the patient who had 2 or more toxicities (adjusted grade ≥ 1) is estimated as below: ) ) 1 ( * exp( 1 ) ) 1 ( * exp( 1 1 max , , ' , , 1 max , , ' , , max , , , ∑ ∑ = = − + + − + + − = I i k j i k j I i k j i k j k j k j G G G G G S β α β α 63 The chosen values of parameters, α and β, in the above equations are flexible so that chemical agent and study specificities can be taken into account in the score system. The value of α mainly reflects the differences in the impacts between the most severe toxicity and other toxicities each patient experiences. The value of parameter β represents the increasing “speed” of composite severity or ETS by additional toxicity. From the rules described above, we can see that the maximum adjusted toxicity grade minus 1 is the integer part of the final composite ETS score and the maximum adjusted toxicity grade is the highest possible composite ETS score of each patient. The composite ETS preserves the relative order of the highest toxicity grade of each patient. A large number of toxicities of the same adjusted grade will generate a composite ETS score just slightly less than that generated by a single toxicity of the next higher adjusted grade. All toxicity grades are taken into account although lower grades will tend to contribute less to the final composite ETS score. The composite ETS, 4.0, is the cut point separating patient with or without DLT. In this way, the composite ETS retains the information provided by the binary indicator of DLT while also permitting two patients who both do or do not express DLT to be distinguishable if they have different overall toxicity profiles. There is also some flexibility left for investigators in the composite ETS estimation system. The impact of different types of toxicities with same grade can be differentiated by assigning a higher adjusted grade to a more severe toxicity and a lower one for the less severe toxicity before computation of the composite ETS. For example, we can give a relative higher adjusted grade to non reversible toxicity than reversible 64 toxicity, or to long lasting toxicity than transient toxicity, or to non-hematological toxicity than hematological toxicity. Vice versus, different toxicities of different grades can be assigned same adjusted grade if they are close in overall severity. Based on the adjusted grade of toxicity, a scoring function may be obtained by interpolating between them and continuous adjusted grade can be assigned to some toxicity of continuous measurements, such as neutropenia. The exact composite ETS score can be changed according to practical needs, but the relative order of each case should be retained during the composite ETS estimation. 5.2.2 The Chosen Values of Parameter α and β. In Section 5.2.1, the values of α and β in the equation for calculation of ETS for patient with multiple toxicities can be chosen subjectively. Beta has to be greater than or equal to 0 ( β ≥ 0) in order to enforce the constraint that ETS is a non-decreasing function of the non-maximum-grade toxicities. According to our score system, for 2 patients with same maximum adjusted grade, the first patient with only 1 toxicity and the other one with 2 toxicities with same adjusted grades, the difference of ETS, Δ, for the 2 patients is as below: 21 exp( ) exp( ) ,0 1 exp( ) 1 exp( ) SS αβα β αβ α + Δ= − = ≥ ∀ ≥ ++ + Because the ETS for a patient with no toxicity or only grade 0 toxicity is arbitrarily assigned as 0 and the ETS for patient who has only 1 toxicity (adjusted grade = 1) is arbitrarily assigned as 0.1, it is consistent for Δ to be close to 0.1 too. 65 exp( ) exp( 2) 0.12, 2 1exp( ) 1exp( 2) α α α − Δ≥ ≥ ≥ ∀ ≥ − ++− Therefore α = -2 is an approximately appropriate choice for the consistency of our score system. When β is equal to 0, our method reverts to the method counting only the maximum grade for each patient. Therefore we will only investigate situations ( β > 0) that are different from the standard method of escalation. In order to explore the upper limit of β and lower limit of α, ETSs and NETSs are estimated with some extreme values of α and β based on some extreme toxicity profiles and summarized on the Table 5.2 and Table 5.3, respectively. From these 2 tables, we can see that when the absolute value of α is bigger, there is longer “distance” for faster “speed” with bigger value of β. Whether the chosen values of α and β are appropriate depends on the actual toxicity profiles of patients. In order to differentiate profiles with wider range multiple toxicities and reflect on the estimated ETS and NETS, the values of α and β should be smaller. On the other hand, the absolute values of α and β can be relatively bigger when different profiles have narrow range of multiple toxicities. In the Table 5.3, we see that once β is too large, such as 10 or 100, with α =0, the NETS of a patient with 1 grade 2 toxicity and 1 grade 1 toxicity immediately reaches the upper endpoint of the interval, leaving no chance for being near the midpoint of interval in Table 5.4. On the other hand, if β is too small, even the NETS of the patient with the most severe toxicity profile is still very close to the lower endpoint of the interval, leaving no chance of increasing to the midpoint of interval in Table 5.4. Therefore, the value of β should be chosen appropriately in order to differentiate all different toxicity 66 profiles of any patients enrolled in the population and reflect all the differences in the final estimated NETSs in the real trial. In addition, the chosen value of β should guarantee the assumption that the average NETS that will be observed is near the midpoint of each interval in Table 5.4. The results in Table 5.2 and Table 5.3 indicate that larger values of β make the interval midpoint of the NETS scores unachievable for most patient-specific toxicity profiles. But we can explore, verify, and recommend some appropriate and applicable values of α and β for practice of EID in the real Phase I trial, using some real trial data. 67 Table 5.2: Summary of ETS Estimated with Different α and β on Different Extreme Toxicity Profiles Toxicities Summary α = -10 α = -5 α = 0 β = 0.1 β = 1 β = 10 Β = 100 β = 0.1 β = 1 β = 10 β = 100 β = 0.1 β = 1 β = 10 β = 100 1 grade 1 toxicity 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 10 grade 1 toxicities 0.0001 0.27 1 1 0.02 0.98 1 1 0.71 1 1 1 50 grade 1 toxicities 0.006 1 1 1 0.48 1 1 1 0.99 1 1 1 1 grade 2 toxicity 1 grade 1 toxicity 1.00005 1.00008 1.007 2 1.007 1.01 1.5 2 1.51 1.62 2 2 5 grade 2 toxicities 5 grade 1 toxicities 1.00009 1.03 2 2 1.01 1.82 2 2 1.66 2 2 2 1 grade 3 DLT 1 grade 2 toxicity 1 grade 1 toxicity 4.00005 4.00008 4.02 5 4.007 4.01 4.73 5 4.51 4.65 5 5 1 grade 3 DLT 5 grade 2 toxicities 5 grade 1 toxicities 4.00006 4.0009 5 5 4.009 4.12 5 5 4.57 4.95 5 5 5 grade 3 DLT 1 grade 2 toxicity 1 grade 1 toxicity 4.00007 4.005 5 5 4.01 4.40 5 5 4.61 4.99 5 5 5 grade 3 DLT 5 grade 2 toxicities 5 grade 1 toxicities 4.00009 4.05 5 5 4.01 4.88 5 5 4.67 5 5 5 1 grade 4 DLT 1 grade 2 toxicity 1 grade 1 toxicity 5.00005 5.00008 5.007 6 5.007 5.01 5.5 6 5.51 5.62 6 6 (Table continued on next page) 68 Table 5.2 (cont.) Toxicities Summary α = -10 α = -5 α = 0 β = 0.1 β = 1 β = 10 Β = 100 β = 0.1 β = 1 β = 10 β = 100 β = 0.1 β = 1 β = 10 β = 100 1 grade 4 DLT 5 grade 2 toxicities 5 grade 1 toxicities 5.00006 5.0006 6 6 5.009 5.08 6 6 5.56 5.92 6 6 5 grade 4 DLT 1 grade 2 toxicity 1 grade 1 toxicity 5.00007 5.004 6 6 5.01 5.38 6 6 5.61 5.99 6 6 5 grade 4 DLT 5 grade 2 toxicities 5 grade 1 toxicities 5.00009 5.03 6 6 5.01 5.82 6 6 5.66 6 6 6 1 grade 4 DLT 1 grade 3 DLT 1 grade 2 toxicity 1 grade 1 toxicity 5.00005 5.0002 5.97 6 5.008 5.02 6 6 5.53 5.79 6 6 1 grade 4 DLT 1 grade 3 DLT 5 grade 2 toxicities 5 grade 1 toxicities 5.00006 5.001 6 6 5.009 5.16 6 6 5.58 5.97 6 6 5 grade 4 DLT 5 grade 3 DLT 1 grade 2 toxicity 1 grade 1 toxicity 5.0001 5.21 6 6 5.02 5.98 6 6 5.70 6 6 6 5 grade 4 DLT 5 grade 3 DLT 5 grade 2 toxicity 5 grade 1 toxicity 5.0001 5.66 6 6 5.02 6 6 6 5.74 6 6 6 69 Table 5.3: Summary of NETS Estimated with Different α and β on Different Extreme Toxicity Profiles Toxicities Summary α = -10 α = -5 α = 0 β = 0.1 β = 1 β = 10 β = 100 β = 0.1 β = 1 β = 10 β = 100 β = 0.1 β = 1 β = 10 β = 100 1 grade 1 toxicity 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.017 10 grade 1 toxicities 0.00002 0.05 0.17 0.17 0.003 0.16 0.17 0.17 0.12 0.17 0.17 0.17 50 grade 1 toxicities 0.001 0.17 0.17 0.17 0.08 0.17 0.17 0.17 0.17 0.17 0.17 0.17 1 grade 2 toxicity 1 grade 1 toxicity 0.17 0.17 0.17 0.33 0.17 0.17 0.25 0.33 0.25 0.27 0.33 0.33 5 grade 2 toxicities 5 grade 1 toxicities 0.17 0.17 0.33 0.33 0.17 0.3 0.33 0.33 0.28 0.33 0.33 0.33 1 grade 3 DLT 1 grade 2 toxicity 1 grade 1 toxicity 0.67 0.67 0.67 0.83 0.67 0.67 0.79 0.83 0.75 0.77 0.83 0.83 1 grade 3 DLT 5 grade 2 toxicities 5 grade 1 toxicities 0.67 0.67 0.83 0.83 0.67 0.69 0.83 0.83 0.76 0.83 0.83 0.83 5 grade 3 DLT 1 grade 2 toxicity 1 grade 1 toxicity 0.67 0.67 0.83 0.83 0.67 0.73 0.83 0.83 0.77 0.83 0.83 0.83 5 grade 3 DLT 5 grade 2 toxicities 5 grade 1 toxicities 0.67 0.67 0.83 0.83 0.67 0.81 0.83 0.83 0.78 0.83 0.83 0.83 1 grade 4 DLT 1 grade 2 toxicity 1 grade 1 toxicity 0.83 0.83 0.83 1 0.84 0.84 0.92 1 0.92 0.94 1 1 (Table continued on next page) 70 Table 5.3 (cont.) Toxicities Summary α = -10 α = -5 α = 0 β = 0.1 β = 1 β = 10 β = 100 β = 0.1 β = 1 β = 10 β = 100 β = 0.1 β = 1 β = 10 β = 100 1 grade 4 DLT 5 grade 2 toxicities 5 grade 1 toxicities 0.83 0.83 1 1 0.84 0.85 1 1 0.93 0.99 1 1 5 grade 4 DLT 1 grade 2 toxicity 1 grade 1 toxicity 0.83 0.83 1 1 0.84 0.90 1 1 0.94 1 1 1 5 grade 4 DLT 5 grade 2 toxicities 5 grade 1 toxicities 0.83 0.84 1 1 0.84 0.97 1 1 0.94 1 1 1 1 grade 4 DLT 1 grade 3 DLT 1 grade 2 toxicity 1 grade 1 toxicity 0.83 0.83 0.99 1 0.84 0.84 1 1 0.92 0.97 1 1 1 grade 4 DLT 1 grade 3 DLT 5 grade 2 toxicities 5 grade 1 toxicities 0.83 0.83 1 1 0.84 0.86 1 1 0.93 0.99 1 1 5 grade 4 DLT 5 grade 3 DLT 1 grade 2 toxicity 1 grade 1 toxicity 0.83 0.87 1 1 0.84 1 1 1 0.95 1 1 1 5 grade 4 DLT 5 grade 3 DLT 5 grade 2 toxicity 5 grade 1 toxicity 0.83 0.94 1 1 0.84 1 1 1 0.96 1 1 1 71 The composite ETS score is consisted of the contributions by the maximum adjusted grade of toxicities and additional toxicities besides the most severe toxicity of each patient. The values of α and β have substantial influences on the final ETS of each patient. Figure 5.1 shows some examples on the contribution of ETS by additional toxicities besides the most severe toxicity for a wide range of values of β when α is fixed as -2. We recommend α as -2 and β as 0.1, 0.25, or 0.5, respectively in the practical Phase I trial. These values are further demonstrated appropriate and applicable by real trials in Chapter VII and VIII. Figure 5.1: Additional ETS Score Contributed by Other Toxicities besides the Most Severe Toxicity with Different Parameter α and β. 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 Additional Contribution of ETS 0 2 4 6 8 10 12 14 16 18 20 Number of additional maximun grade toxicities Alpha=-2,Beta=2 Alpha=-2,Beta=1 Alpha=-2,Beta=0.5 Alpha=-2,Beta=0.25 Alpha=-2,Beta=0.1 72 5.2.3 Normalized Equivalent Toxicity Score After the composite ETS of each patient has been obtained, it is further normalized. The NETS of patient j in the dose level d k is defined as max , * , / S S S k j k j = where S max < ∞ is the composite ETS for patient with the most severe combined toxicities among all patients (S max = 6 in our system). Thus * ,k j S є [0, 1], k = 1, 2,. . . , K. The NETS, * ,k j S , is discrete and not exactly continuous because of the finite number of possible toxicities (National Cancer Institute, 2003). Therefore, NETS is called quasi- continuous and can be viewed as fractional events. The average NETS (ANETS) of n k patients in the dose level d k , S k *, is defined as [] k n j k j k n S S k / 1 * , * ∑ = = where k = 1, 2, . . . , K, Table 5.4: Composite Toxicity Equivalent Score Estimation System Most severe toxicity Maximum adjusted grade Range of composite ETS* Range of composite NETS* Mid- range NETS Grade 0 0 0 0 0 Grade 1 1 [0.1 - 1) [1/60 – 1/6) 0.092 Grade 2 2 [1 - 2) [1/6 – 1/3) 0.25 Grade 3 non DLT 3 [2 - 3) [1/3 – ½) 0.417 Grade 4 non DLT 4 [3 – 4) [1/2 – 2/3) 0.583 Grade 3 DLT 5 [4 – 5) [2/3 – 5/6) 0.75 Grade 4 DLT 6 [5 – 6) [5/6 – 1) 0.917 * The symbol “[“ means that the following number is included and the symbol “)“ means that the number before is excluded from the range. 73 5.3 Design of Extended Isotonic Design. Same as in the original ID by Leung and Wang (Leung and Wang, 2001), the only assumption for the EID is that toxicity is non-decreasing with dose. The new idea of our EID design is to fully utilize the severity information of toxicity by applying the ANETS, S k *, in the IR instead of probability of binary DLT as in the original ID during the dose allocation procedure and estimation of the MTD for the new agent. 5.3.1 Pool Adjacent Violator Algorithm and Isotonic Regression. The ANETS at d k , the dose level k (1 ≤ k ≤ K), must satisfy the non-decreasing dose toxicity risk relationship. When the non-decreasing relationship is violated among the range from dose d r to dose d h with d r below d k (r ≤ k) and d h above d k (h ≥ k), the pooled estimate of ANETS at dose d k , h k r q , , ˆ , is estimated with the PAVA as below: ∑ ∑ = = = h r i i h r i i i h k r n n S q ) ( ) ( ˆ * , , The final pooled ANETS (PANETS) at dose d k as k q ˆ with satisfaction of non- decreasing dose toxicity risk relationship among all dose levels is estimated using all toxicity response with IR as below: )) ˆ ( ( ˆ , , 1 max min h k r k r K h k k q q ≤ ≤ ≤ ≤ = 74 The k q ˆ estimated from the equation above must be at least as large as any of or the maximum of h k q , , 1 ˆ , h k q , , 2 ˆ , h k q , , 3 ˆ , . . . , h k k q , , ˆ for any h (h ≥ k) and be smaller than any of or the minimum of k k r q , , ˆ , 1 , , ˆ + k k r q , 2 , , ˆ + k k r q , . . . , h k r q , , ˆ for any r (r ≤ k) when toxicity is monotonic so that it can guarantee the non-decreasing assumption. Let q* be Target Normalized Equivalent Toxicity Score (TNETS) which is an analog as TTL and find a MTD with a PANETS, MTD q ˆ , such that | MTD q ˆ - q*| ≤ | i q ˆ - q*| where i = 1 to K . 5.3.2 Determination of Target Normalized Equivalent Toxicity Score. The determination of TNETS is a clinical decision based on the target toxicity profile (TTP) at the MTD. In order to define the TTP, we need to ask physicians or investigators of the Phase I trial some questions. The questions for them are listed as below: 1). If the treatment were to become standard, what proportion of patients would get a DLT and still be acceptable for all patients? A). 20% B). 33% C). 50% D). Others, specify _________. 75 2). Within a selected total probability of DLT for a patient who is treated at MTD level, what are the ratios for Grade 3 DLT and Grade 4DLT do you find acceptable? A). 1:1 B). 2:1 C). 1:2 D). Others, specify _________. 3). Among patients treated at MTD level, many will not have DLT, but must still have some side effects, what is the smallest percentage of patients who will have no side effect that would be acceptable? A). 0% B). 5% C). 10% D). Others, specify _________. 4). Within a selected total probability of non-DLT after deduction of the probability for no toxicity or grade 0 toxicity in question 3) for a patient who is treated at MTD level, what are the ratios for Grade 1 toxicity, Grade 2 toxicity, Grade 3 non-DLT, and Grade 4 non-DLT that would be acceptable? A). 1:1:1:1 B). 4:3:2:1 C). 1:2:3:4 D). Others, specify _________. 76 If the TTL is pre-specified as 33% in designs treating toxicity response as binary variable, then correspondingly, the highest acceptable probability of DLT in our EID is also defined as 33%. Further, we can assume that the probabilities of grade 3 and 4 DLT should be equal and the non DLT toxicity probabilities of each grade should also be equal in the detail TTP as well as a certain probability of no toxicities, such as 7%. Therefore, the corresponding TTP is consisting of 7%, 15%, 15%, 15%, 15%, 16.5% and 16.5% for grade 0 toxicity or no toxicity, grade 1 toxicity, grade 2 toxicity, grade 3 non DLT, grade 4 non DLT, grade 3 DLT, and grade 4 DLT being the most severe toxicity, respectively (Table 5.5). The range of NETS is from (g-1)/6 to g/6 for a patient whose toxicity maximum adjusted grade is g and the corresponding mid-range value of NETS is ((g-1)/6+g/6)/2 (Table 5.4). To be specific, the mid-range NETS values are 0, 0.092, 0.25, 0.417, 0.583, 0.75, and 0.917 for grade 0 toxicity, grade 1 toxicity, grade 2 toxicity, grade 3 non DLT, grade 4 non DLT, grade 3 DLT, and grade 4 DLT being the toxicity with maximum adjusted grade for a patient, respectively (Table 5.4). Therefore the TNETS, q*, for design in which toxicity response is treated as quasi-continuous variable is determined to be 47.6% (0.07*0 + 0.15*0.092 + 0.15*0.250 + 0.15*0.417 + 0.15*0.583 + 0.165*0.75 + 0.165*0.917 = 0.476) (Table 5.5). 77 Table 5.5: TNETS Based on TTP with 33% DLT and Ratios (1:1:1:1 and 1:1). Target Toxicity Profile 67% non DLT toxicity (1:1:1:1) 33% DLT (1:1) TTL =33% Grade 0 or None Grade 1 Grade 2 Grade 3 non DLT Grade 4 non DLT Grade 3 DLT Grade 4 DLT TNETS = 0.476 7% 15% 15% 15% 15% 16.5% 16.5% 100% (Sum) Mid-range NETS 0 0.092 0.25 0.417 0.583 0.75 0.917 NA Contribution 0 0.0138 0.0375 0.0626 0.0875 0.1238 0.1513 0.476 (Sum) If one has a desired TTP, it should hold for all studies. Otherwise the TTP and the corresponding TNETS can be determined with the answers to the 4 questions above. The following tables (Table 5.6 to Table 5.13) summarize the different TNETSs for some specific TTPs according to the different answers of the above questions from physicians and investigators of Phase I trial. Table 5.6: TNETS Based on TTP with 33% DLT and Ratios (1:2:3:4 and 1:2). Target Toxicity Profile 67% non DLT toxicity (1:2:3:4) 33% DLT (1:2) TTL =33% Grade 0 or None Grade 1 Grade 2 Grade 3 non DLT Grade 4 non DLT Grade 3 DLT Grade 4 DLT TNETS = 0.535 7% 6% 12% 18% 24% 11% 22% = 100% (Sum) Mid-range NETS 0 0.092 0.25 0.417 0.583 0.75 0.917 NA Contribution 0 0.0055 0.03 0.0751 0.1399 0.0825 0.2017 0.535 (Sum) 78 Table 5.7: TNETS Based on TTP with 33% DLT and Ratios (4:3:2:1 and 2:1). Target Toxicity Profile 67% non DLT toxicity (4:3:2:1) 33% DLT (2:1) TTL =33% Grade 0 or None Grade 1 Grade 2 Grade 3 non DLT Grade 4 non DLT Grade 3 DLT Grade 4 DLT TNETS =0.418 7% 24% 18% 12% 6% 22% 11% = 100% (Sum) Mid-range NETS 0 0.092 0.25 0.417 0.583 0.75 0.917 NA Contribution 0 0.0221 0.045 0.05 0.035 0.165 0.1009 0.418 (Sum) Table 5.8: TNETS Based on TTP with 20% DLT and Ratios (1:1:1:1 and 1:1). Target Toxicity Profile 80% non DLT toxicity (1:1:1:1) 20% DLT (1:1) TTL =20% Grade 0 or None Grade 1 Grade 2 Grade 3 non DLT Grade 4 non DLT Grade 3 DLT Grade 4 DLT TNETS = 0.415 6% 18.5% 18.5% 18.5% 18.5% 10% 10% 100% (Sum) Mid-range NETS 0 0.092 0.25 0.417 0.583 0.75 0.917 NA Contribution 0 0.017 0.0463 0.0771 0.1079 0.075 0.0917 0.415 (Sum) Table 5.9: TNETS Based on TTP with 20% DLT and Ratios (1:2:3:4 and 1:2). Target Toxicity Profile 80% non DLT toxicity (1:2:3:4) 20% DLT (1:2) TTL =20% Grade 0 or None Grade 1 Grade 2 Grade 3 non DLT Grade 4 non DLT Grade 3 DLT Grade 4 DLT TNETS =0.481 6% 7.4% 14.8% 22.2% 29.6% 7% 13% = 100% (Sum) Mid-range NETS 0 0.092 0.25 0.417 0.583 0.75 0.917 NA Contribution 0 0.0068 0.037 0.0926 0.1726 0.0525 0.1192 0.481 (Sum) 79 Table 5.10: TNETS Based on TTP with 33% DLT and Ratios (4:3:2:1 and 2:1). Target Toxicity Profile 80% non DLT toxicity (4:3:2:1) 20% DLT (2:1) TTL =20% Grade 0 or None Grade 1 Grade 2 Grade 3 non DLT Grade 4 non DLT Grade 3 DLT Grade 4 DLT TNETS = 0.34928 7% 29.6% 22.2% 14.8% 7.4% 13% 7% = 100% (Sum) Mid-range NETS 0 0.092 0.25 0.417 0.583 0.75 0.917 NA Contribution 0 0.0272 0.0555 0.0617 0.0431 0.0975 0.0642 0.34928 (Sum) Table 5.11: TNETS Based on TTP with 50% DLT and Ratios (1:1:1:1 and 1:1). Target Toxicity Profile 50% non DLT toxicity (1:1:1:1) 50% DLT (1:1) TTL =50% Grade 0 or None Grade 1 Grade 2 Grade 3 non DLT Grade 4 non DLT Grade 3 DLT Grade 4 DLT TNETS = 0.564 6% 11% 11% 11% 11% 25% 25% = 100% (Sum) Mid-range NETS 0 0.092 0.25 0.417 0.583 0.75 0.917 NA Contribution 0 0.0101 0.0275 0.0459 0.0641 0.1875 0.2293 0.564 (Sum) 80 Table 5.12: TNETS Based on TTP with 50% DLT and Ratios (1:2:3:4 and 1:2). Target Toxicity Profile 50% non DLT toxicity (1:2:3:4) 50% DLT (1:2) TTL =50% Grade 0 or None Grade 1 Grade 2 Grade 3 non DLT Grade 4 non DLT Grade 3 DLT Grade 4 DLT TNETS = 0.614 6% 4.4% 8.8% 13.2% 17.6% 17% 33% = 100% (Sum) Mid-range NETS 0 0.092 0.25 0.417 0.583 0.75 0.917 NA Contribution 0 0.00405 0.022 0.055 0.1026 0.1275 0.3026 0.614 (Sum) Table 5.13: TNETS Based on TTP with 50% DLT and Ratios (4:3:2:1 and 2:1). Target Toxicity Profile 80% non DLT toxicity (4:3:2:1) 20% DLT (2:1) TTL =50% Grade 0 or None Grade 1 Grade 2 Grade 3 non DLT Grade 4 non DLT Grade 3 DLT Grade 4 DLT TNETS = 0.515 7% 17.6% 13.2% 8.8% 4.4% 33% 17% = 100% (Sum) Mid-range NETS 0 0.092 0.25 0.417 0.583 0.75 0.917 NA Contribution 0 0.0162 0.033 0.0367 0.0257 0.2475 0.1559 0.515 (Sum) 81 5.3.3 Summary of Extended Isotonic Design Our EID is similar to the original ID proposed by Leung and Wang (Leung and Wang, 2001) and is summarized in detail as below: Start from the lowest dose level d k , k = 1: 1. Treat a cohort of M patients at d k . 2. Update the PANETS, i q ˆ , of each dose d i (i = 1 to K) with the toxicity response of current cohort using IR. The dose d k is the current dose used and a dose level for next cohort is decided according to following rule: If k q ˆ < q*, then Escalate if q* − k q ˆ ≥ 1 ˆ + k q − q*, Where k < K Continue at same dose, Otherwise If k q ˆ ≥ q*, then De-escalate if q* − 1 ˆ − k q ≥ k q ˆ − q*, Where k > 1 Continue at same dose, Otherwise 3. Iterate between steps 1 and 2 and stop when a fixed number of cohorts have been tested and the recommended MTD is not the highest tested dose level or 3 consecutively cohorts have been treated at the same dose level. 4. The recommended dose level for the next cohort is considered the MTD. 82 An IR and a simple comparison are performed in the iteration. Safety constraint is imposed so that no untried dose may be skipped when escalating, nor can the dose de- escalate too fast by skipping some dose levels. We use the same simple practical stopping rule as the one proposed by Leung and Wang (Leung and Wang, 2001) that the trial stops when the same dose had been tested consecutively for 3 cohorts or a fixed number of cohorts have been tested and the recommended MTD is not the highest tested dose level. 5.4 An Example of Extended Isotonic Design. An example of EID is described in the Table 5.14. The true NETSs at the dose levels 1 to 6 are 0.34, 0.43, 0.48, 0.54, 0.61, and 0.71, respectively. The TNETS, q*, is 0.476 so that the d 3 is the true MTD. There are 3 patients in each cohort. The simulation starts at d 1 , the lowest dose level and the total NETS for the firs cohort of 3 patients tested at this dose level is 0.73, resulted at an average NETS as 0.24. Then the estimated risks for the 6 dose levels using IR are all 0.24. Therefore the dose level increases to d 2 . The second cohort (3 patients) is tested at d 2 and has total NETS of 0.96, resulting average NETS of 0.32 at d 2 . After the second cohort, the estimated risks for the 6 dose levels using IR are 0.24, 0.32, 0.32, 0.32, 0.32, and 0.32 for dose levels 1 to 6, respectively. The updated risk of current tested dose level (dose level 2) is 0.32 which is less than 0.476, the TNETS, so that the recommended dose level for next cohort increases to d 3 . 83 The third cohort (3 patients) is tested at d 3 and has a total NETS as 1.36, resulting average NETS as 0.45 at d 3 . After the third cohort, the estimated risks for the 6 dose levels using IR are 0.24, 0.32, 0.45, 0.45, 0.45, and 0.45 for dose levels 1 to 6, respectively. The updated risk of current tested dose level (dose level 3) is 0.45 which is less than the TNETS (0.476) so that the recommended dose level for next cohort increases to d 4 . The fourth cohort (3 patients) is tested at d 4 and has a total NETS of 2.35, resulting an average NETS as 0.45 at d 4 . After the fourth cohort, the estimated risks for the 6 dose levels using IR are 0.24, 0.32, 0.45, 0.78, 0.78, and 0.78 for dose levels 1 to 6, respectively. The updated risk of current tested dose level (dose level 3) is 0.78 which is less than TNETS (0.476) so that the recommended dose level for next cohort decreases to d 3 . The fifth cohort (3 patients) is tested at d 3 and has a total NETS as 1.48 (2.84 - 1.36). The third cohort (3 patients) had been treated at d 3 before so that the average NETS at d 3 becomes 0.47 with the toxicity response of both third and fifth cohorts. After the fifth cohort, the estimated risks for the 6 dose levels using IR are 0.24, 0.32, 0.47, 0.78, 0.78, and 0.78 for dose levels 1 to 6, respectively. The updated risk of current tested dose level (dose level 3) of 0.47 is the closest to the TNETS (0.476) among the 6 dose levels so that the recommended dose level for next cohort remains as d 3 . The sixth cohort (3 patients) is tested at d 3 and has a total NETS as 1.53 (4.27 - 2.84). The third, fifth cohorts (6 patients) had been treated at d 3 before so that the average 84 NETS at d 3 becomes 0.47 with the toxicity responses of third, fifth, and sixth cohorts. After the sixth cohort, the estimated risks for the 6 dose levels using IR are 0.24, 0.32, 0.47, 0.78, 0.78, and 0.78 for dose levels 1 to 6, respectively. The updated risk of current tested dose level (dose level 3) of 0.47 is the closest to the TNETS (0.476) among the 6 dose levels so that the recommended dose level for next cohort remains as d 3 . The seventh cohort (3 patients) is tested at d 3 and has a total NETS of 1.52 (5.79 - 4.27). The third, fifth, and sixth cohorts (9 patients) had been treated at d 3 before so that the average NETS at d 3 becomes 0.476 with the toxicity responses of third, fifth, sixth, and seventh cohorts. After the seventh cohort, the estimated risks for the 6 dose levels using IR are 0.24, 0.32, 0.48, 0.78, 0.78, and 0.78 for dose levels 1 to 6, respectively. The updated risk of current tested dose level (dose level 3) of 0.48 is the closest to the TNETS (0.476) among the 6 dose levels so that the recommended dose level for next cohort remains as d 3 . The eighth cohort (3 patients) is tested at d 3 and has a total composite NETS as 0.60 (6.39 - 5.79). The third, fifth, sixth, and seventh cohorts (12 patients) had been treated at d 3 before so that the average NETS at d 3 becomes 0.43 with the toxicity responses of third, fifth, sixth, seventh, and eighth cohorts. After the seventh cohort, the estimated risks for the 6 dose levels using IR are 0.24, 0.32, 0.43, 0.78, 0.78, and 0.78 for dose levels 1 to 6, respectively. The updated risk of current tested dose level (dose level 3) of 0.43 is the closest to the TNETS (0.476) among the 6 dose levels so that the recommended dose level for next cohort remains as d 3 . After the eighth cohort, there are 85 4 consecutive cohorts (fifth, sixth, seventh, and eighth) which have been treated at the same dose level (dose level 3) so that the trial stops and recommends the dose level 3 (d 3 ) as the MTD. 86 Table 5.14: An Example of Simulation Study of the EID* Cohort Dose level (True NETS**) Current tested dose Recom- mended next dose d 1 (0.34) d 2 (0.43) d 3 (0.48) d 4 (0.54) d 5 (0.61) d 6 (0.71) 1 No. Patients Tested 3 0 0 0 0 0 1 2 Sum NETS 0.73 0 0 0 0 0 Estimated PANETS 0.24 0.24 0.24 0.24 0.24 0.24 2 No. Patients Tested 3 3 0 0 0 0 2 3 Sum NETS 0.73 0.96 0 0 0 0 Estimated PANETS 0.24 0.32 0.32 0.32 0.32 0.32 3 No. Patients Tested 3 3 3 0 0 0 3 4 Sum NETS 0.73 0.96 1.36 0 0 0 Estimated PANETS 0.24 0.32 0.45 0.45 0.45 0.45 4 No. Patients Tested 3 3 3 3 0 0 4 3 Sum NETS 0.73 0.96 1.36 2.35 0 0 Estimated PANETS 0.24 0.32 0.45 0.78 0.78 0.78 5 No. Patients Tested 3 3 6 3 0 0 3 3 Sum NETS 0.73 0.96 2.84 2.35 0 0 Estimated PANETS 0.24 0.32 0.47 0.78 0.78 0.78 6 No. Patients Tested 3 3 9 3 0 0 3 3 Sum NETS 0.73 0.96 4.27 2.35 0 0 Estimated PANETS 0.24 0.32 0.47 0.78 0.78 0.78 7 No. Patients Tested 3 3 12 3 0 0 3 3 Sum NETS 0.73 0.96 5.79 2.35 0 0 Estimated PANETS 0.24 0.32 0.48 0.78 0.78 0.78 8 No. Patients Tested 3 3 15 3 0 0 3 3*** Sum NETS 0.73 0.96 6.39 2.35 0 0 Estimated PANETS 0.24 0.32 0.43 0.78 0.78 0.78 * EID (Accrual of 3 patients per cohort and TNETS is 0.476). ** True NETS of the dose level. *** Recommended MTD from the trial is dose level 3. 87 Chapter VI: Evaluation of Extended Isotonic Design Normally about 20-80 patients are recruited in a Phase I trial. As its sample size is small, large sample properties are not appropriate for evaluating a phase I trial design and simulation studies under different dose-toxicity assumptions are the most convincing ways to evaluate different methods (Storer, 1989). Therefore, in this section, simulation studies will be performed to compare our EID with ID (Leung and Wang 2001), Standard 3+3 design (Lin and Shih 2001), and CRM (O’Quigley et al. 1990; O’Quigley and Shen 1996; Goodman et al. 1995; Heyd et al. 1999) in the aspects of estimation of MTD, sample size, robustness, simplicity, patient distribution by dose levels. A simulation study is also conducted to further compare our EID with CRM-NETS which is the Quasi-CRM by Yuan et al. (Yuan et al., 2006) with application of our NETS to treat toxicity response as a quasi-continuous variable. 6.1 Simulation Study Methods 6.1.1 Simulation Scenarios. Table 6.1 summarizes the true probability of toxicity of each kind of adjusted grade being the most severe toxicity with maximum adjusted grade for a patient at each of the 6 dose levels among 5 different scenarios. For example, the probability is 0.20 for a grade 3 DLT being the most severe toxicity with maximum adjusted grade among all toxicities that a patient treated at the dose level 1 of Target Scenario will have. The probabilities of DLTs are 8%, 24%, 33%, 44%, 56%, and 76% for the dose level 1, 2, 3, 88 4, 5, and 6, respectively in all 5 scenarios. But the detail toxicity profile is different in each of them. The calculation of NETS for each dose level is the same as that of TNETS so that the NETS for each dose level varies according to the exact toxicity profile in different scenarios. In Target Scenario, the probability being the most severe toxicity with maximum adjusted grade for each kind of non DLT toxicities (grade 1, 2, 3 non DLT, and 4 non DLT toxicity) is the same and that for each kind of DLTs (Grade 3 and 4 DLT) is the same. This pattern is the same as the targeted one in calculation of TNETS. Target Scenario can be viewed as an ideal and reference case in which the detail toxicity profile is the TTP given a certain DLT probability. In this scenario, DLT probability and NETS are matched pretty well in measuring the overall toxicity severity of each dose level so that the dose level 3 is the true MTD in designs treating toxicity response as binary (TTL=33%) or quasi-continuous variable (TNEST=0.476). Toxicity profile skews to low grade toxicity and is less toxic in each dose level in Medium-Under-Toxic-Scenario than in Target Scenario. Therefore, NETS of each dose level is smaller in Medium-Under-Toxic-Scenario than in Target Scenario because of the relatively higher probability of low grade toxicity being the most severe toxicity in each dose level. On the contrast, Medium-Over-Toxic-Scenario has more toxic toxicity profile than Target Scenario with higher probability of higher grade toxicity being the most severe toxicity in each dose level so that NETS of each dose level is higher in Medium- Over-Toxic-Scenario than in Target Scenario. 89 Extreme-Over-Toxic-Scenario is an extreme over toxic case of Medium-Over- Toxic-Scenario and Extreme-Under-Toxic-Scenario is an extreme under toxic case of scenario B. Therefore, the order of increasing toxicity burden is Extreme-Under-Toxic- Scenario, Medium-Under-Toxic-Scenario, Target- Scenario, Medium-Over-Toxic- Scenario, to Extreme-Over-Toxic-Scenario, given the same probability of DLT at each dose level. The designs in which toxicity response is treated as binary variable should have no difference in estimation of MTD between Target Scenario, Medium-Under- Toxic-Scenario, Medium-Over-Toxic-Scenario, Extreme-Over-Toxic-Scenario, and Extreme-Under-Toxic-Scenario. But in the design in which toxicity response is treated as quasi-continuous variable, MTD is estimated differently according to the exact toxicity profile in each scenario. Table 6.1: True Toxicity Profiles of Different Scenarios Scenario Most severe Toxicity Maximum Adjusted Grade Mid- range NETS Probability that the grade is the maximum adjusted grade patient will have at the dose level 1 2 3 4 5 6 Target- Scenario 0 0 0 0.11 0.09 0.07 0.05 0.03 0.01 1 1 0.092 0.2 0.16 0.15 0.12 0.1 0.05 2 2 0.25 0.2 0.17 0.15 0.13 0.1 0.06 3 non DLT 3 0.417 0.2 0.17 0.15 0.13 0.1 0.06 4 non DLT 4 0.583 0.21 0.17 0.15 0.13 0.11 0.06 3 DLT 5 0.75 0.04 0.12 0.165 0.22 0.28 0.38 4 DLT 6 0.917 0.04 0.12 0.165 0.22 0.28 0.38 Composite NETS for the dose 0.341 0.427 0.476 0.54 0.607 0.713 Probability of DLTs for the dose 0.08 0.24 0.33 0.44 0.56 0.76 (Table continued on next page) 90 Table 6.1 (cont.) Scenario Most severe Toxicity Maximum Adjusted Grade Mid- range NETS Probability that the grade is the maximum adjusted grade patient will have at the dose level 1 2 3 4 5 6 Medium- Under- Toxic- Scenario 0 0 0 0.11 0.09 0.07 0.05 0.03 0.01 1 1 0.092 0.324 0.268 0.24 0.204 0.164 0.092 2 2 0.25 0.243 0.201 0.18 0.153 0.123 0.069 3 non DLT 3 0.417 0.162 0.134 0.12 0.102 0.082 0.046 4 non DLT 4 0.583 0.081 0.067 0.06 0.051 0.041 0.023 3 DLT 5 0.75 0.06 0.16 0.22 0.3 0.37 0.51 4 DLT 6 0.917 0.02 0.08 0.11 0.14 0.19 0.25 Composite NETS for the dose 0.269 0.363 0.41 0.483 0.556 0.67 Probability of DLTs for the dose 0.08 0.24 0.33 0.44 0.56 0.76 Medium- Over- Toxic- Scenario 0 0 0 0.11 0.09 0.07 0.05 0.03 0.01 1 1 0.092 0.081 0.067 0.06 0.051 0.041 0.023 2 2 0.25 0.162 0.134 0.12 0.102 0.082 0.046 3 non DLT 3 0.417 0.243 0.201 0.18 0.153 0.123 0.069 4 non DLT 4 0.583 0.324 0.268 0.24 0.204 0.164 0.092 3 DLT 5 0.75 0.02 0.08 0.11 0.14 0.19 0.25 4 DLT 6 0.917 0.06 0.16 0.22 0.3 0.37 0.51 Composite NETS for the dose 0.408 0.486 0.526 0.593 0.653 0.751 Probability of DLTs for the dose 0.08 0.24 0.33 0.44 0.56 0.76 (Table continued on next page) 91 Table 6.1 (cont.) Scenario Most severe Toxicity Maximum Adjusted Grade Mid- range NETS Probability that the grade is the maximum adjusted grade patient will have at the dose level 1 2 3 4 5 6 Extreme- Over- Toxic- Scenario 0 0 0 0 0 0 0 0 0 1 1 0.092 0 0 0 0 0 0 2 2 0.25 0 0 0 0 0 0 3 non DLT 3 0.417 0 0 0 0 0 0 4 non DLT 4 0.583 0.92 0.76 0.68 0.56 0.44 0.24 3 DLT 5 0.75 0 0 0 0 0 0 4 DLT 6 0.917 0.08 0.24 0.32 0.44 0.56 0.76 Composite NETS for the dose 0.61 0.663 0.69 0.73 0.77 0.837 Probability of DLTs for the dose 0.08 0.24 0.33 0.44 0.56 0.76 Extreme- Under- Toxic- Scenario 0 0 0 0.46 0.38 0.34 0.28 0.22 0.12 1 1 0.092 0.46 0.38 0.34 0.28 0.22 0.12 2 2 0.25 0 0 0 0 0 0 3 non DLT 3 0.417 0 0 0 0 0 0 4 non DLT 4 0.583 0 0 0 0 0 0 3 DLT 5 0.75 0.08 0.24 0.33 0.44 0.56 0.76 4 DLT 6 0.917 0 0 0 0 0 0 Composite NETS for the dose 0.102 0.215 0.271 0.356 0.44 0.581 Probability of DLTs for the dose 0.08 0.24 0.33 0.44 0.56 0.76 6.1.2 Number of Monte Carlos Simulation. The next important question in our Monte Carlos Simulation is: How to estimate the probability of different dose levels chosen as MTD with pre-specified confidence level and absolute error? The answer is that we can achieve these goals by determining appropriate number of Monte Carlos Simulations. 92 Let S be the number of successes during N identical independent experiments of a Bernoulli random variable with a success probability p Є (0, 1). The Chernoff – Hoeffding bound (Chernoff, 1952, Hoeffding, 1963) asserts that, for any ε, α Є (0, 1), α ε − > ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ < − 1 Pr p N S if ) 2 ( ) 2 ln( 2 ε α > N Let p i denote the probability that toxicity level d i , i = 1, … , k is selected as the MTD in the Phase I trial. The goal of the simulation study is to estimate p i . Suppose we simulate a design N times under identical independent conditions and observe that there are S i times the level d i is selected as MTD. Then, it is natural to take N S p i i = ˆ , i = 1, … , k as an estimate for p i . A crucial problem is: How we should choose N? More specifically, for prescribed margin of absolute error ε Є (0, 1) and confidence parameter α Є (0, 1), how large N is sufficient to ensure that {} α ε − > = < − 1 ,..., 1 , ˆ Pr k i p p i i , According to Chernoff-Hoeffding bound and Bonferroni’s adjustment (Bonferroni, 1935), the answer is that the minimum number of simulation runs can be chosen as ) 2 ( ) 2 ln( 2 ε α k N > 93 We can show the above formula by using Chernoff-Hoeffding bound and Bonferroni’s inequality (Bonferroni, 1935). Applying Chernoff-Hoeffding bound, for ) 2 ( ) 2 ln( 2 ε α k N > , we have the following {} ) / ( 1 ˆ Pr k p p i i α ε − > < − for any i = 1 , … , k. Bonferroni’s inequality asserts that, for any two events A and B, {} { } { } 1 Pr Pr Pr − + ≥ ∩ B A B A which can be derived from the fact that {} { } { } { } B A B A B A ∩ − + = ∪ ≥ Pr Pr Pr Pr 1 . Now if we define { } ε < − = 1 1 ˆ p p A , { } ε < − = 2 2 ˆ p p B , then {} 2 , 1 , ˆ = < − = ∩ i p p B A i i ε . Applying Bonferroni’s inequality, we have {} k k k i p p i i α α α ε 2 1 1 1 1 2 , 1 , ˆ Pr − = − − + − ≥ = < − Similarly, we define {} 2 , 1 , ˆ = < − = i p p A i i ε , { } ε < − = 3 3 ˆ p p B , then {} 3 , 2 , 1 , ˆ = < − = ∩ i p p B A i i ε . Applying Bonferroni’s inequality again, we have {} k k k i p p i i α α α ε 3 1 1 1 2 1 3 , 2 , 1 , ˆ Pr − = − − + − ≥ = < − . 94 Repeatedly applying Bonferroni’s inequality and by induction, we have {} α ε − > = < − 1 ,..., 1 , ˆ Pr k i p p i i . For example, in our simulation, there are totally 6 dose levels so that k = 6. To achieve an overall confidence level of 99%, α is set as 0.01. The prescribed margin of absolute error ε is set to 0.01. Then, the number of simulations N is determined as: 35451 ) 01 . 0 2 ( ) 01 . 0 6 2 ln( 2 = × ÷ × > N Under these conditions, it can be interpreted as that we have 99% confidence that the percentage of each of the 6 dose levels chosen as MTD is estimated with an absolute error less than 0.01 when the number of simulations performed is at least 35451. Table 6.2: Minimum Number of Simulations Required under Different α and ε with 6 Dose Levels Alpha Level ( α) Minimum Number of Simulations Absolute Error Level ( ε) 0.001 0.01 0.02 0.03 0.05 0.1 0.001 4696331 46964 11741 5219 1879 470 0.01 3545039 35451 8863 3939 1419 355 0.05 2740320 27404 6851 3045 1097 275 0.1 2393746 23938 5985 2660 958 240 The Table 6.2 summarizes the minimum number of simulations required under different combinations of parameters α and ε when there are 6 dose levels tested. In our simulation, we perform 40,000 times of Monte Carlo simulations to achieve 99% 95 confidence level and an absolute error of less than 0.01 in the estimated percentage under each scenario. 6.2 The Different Designs. In the simulation study of all 5 designs, the standard comparison outcome variables are percentage of each dose level being chosen as MTD, percentage of patients treated at each dose level, average sample size, and number of cohorts being used. The percentage of each dose level being chosen as MTD is used to compare the probability of selecting correct MTD in each design. The percentage of patients treated at each dose level is an indicator of whether a design can avoid treating too many patients at overdosed or under dosed level. The average sample size is a measurement of the cost of a design and the average number of cohorts is a surrogate as the length of trial when patients are readily available for entry into trial. The comparison of these outcome variables can give a pretty comprehensive idea about how each design performs. 6.2.1 Extended Isotonic Design Treating Toxicity Response as A Quasi-Continuous Variable. The detail design of EID in the simulation is described as in Section 5. The TNETS level at the MTD is set to be 47.6% as in Section 5.2. In the simulated EID, 3 patients per cohort are used and trial stops according to two stopping criteria: 1) 4 consecutive cohorts have been tested at the same dose level; 2) the maximum 20 cohorts have been tested when the first stopping criteria have never been met. After a simulated 96 trial stops, the recommended dose level for next cohort is the MTD. In the case all tested dose levels are overdosed, the first dose level is determined as MTD. On the other hand, the highest dose level is chosen as MTD when all tested dose levels are under-dosed. 6.2.2 Isotonic Design Treating Toxicity Response as A Binary Variable. ID is set up similarly as the EID above with the only differences are that binary toxicity response and the TTL of 33% are used instead of quasi-continuous toxicity response and the TNETS of 47.6%. 6.2.3 Standard 3+3 Design with Dose De-escalation. We simulate the standard 3+3 design with dose de-escalation which is the representative of standard design and the most widely used design in the field. This design is similar to the 3+3 design without dose de-escalation, but allows re-examination of lower doses when DLTs are observed with excessive frequency at higher levels. With this design, 3 patients are assigned to the first dose level. If no DLT is observed, the trial proceeds to the next dose level and another cohort of 3 patients is enrolled. If at least 2 out of the 3 patients experience DLT, then the dose level decreases; otherwise, if only 1 patient experiences DLT, then 3 more patients are enrolled at the same dose level. If none of the 3 additional patients experience DLT, the dose will be escalated; otherwise, the dose level decreases. Dose reduction continues until a dose level is reached at which 6 patients have been treated and at most 1 DLT is observed. The MTD is defined as the highest dose level at which at most 1 of 6 patients experience 97 DLT, and the immediate higher dose level has at least 2 patients who experience DLTs [9]. In the case all tested dose levels are overdosed, the first dose level is selected as MTD. On the other hand, the highest dose level is chosen as MTD in case all tested dose levels are under-dosed. The standard design does not specify a fixed sample size of patients to be treated in advance. This strategy uses a maximum sample size of 6 times of predefined dose levels to find the MTD which is not estimated with specific quantile. 6.2.4 Continuous Reassessment Method. CRM is a representative of model based design. We simulate a Likelihood-based modified CRM (O’Quigley et al. 1990; O’Quigley and Shen 1996; Goodman et al. 1995; Heyd et al. 1999) in which toxicity response is treated as a binary outcome and a logistic model is employed to depict the dose toxicity relationship. The probability of DLT at a certain dose level, d i , is described as below: i d DLT P it * )) ( ( log β α + = Where α defines the starting probability of DLT at the lowest dose level and β defines the speed of increased probability of DLT with increasing dose. In another form, the probability of DLT can also be expressed as following: ) * exp( 1 ) * exp( ) ( i i d d DLT P β α β α + + + = 98 The parameters α and β can be updated with Baysian method or MLE method, both of which have been shown to perform similarly by simulation studies (O’Quigley and Shen, 1996). The MLE method is used in our simulated CRM design. The likelihood function is expressed as below: ( ) ∏ = − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + = N i y i i y i i i i d d d d L 1 1 ) * exp( 1 ) * exp( 1 ) * exp( 1 ) * exp( β α β α β α β α Where the toxicity response y i is treated as a binary outcome, 1 for DLT and 0 for non DLT. In order to have less toxicity and shorter study duration, 3 patients per cohort is used in our simulated CRM design. The trial starts treating the first cohort from the lowest dose level and keeps escalating one dose level at a time until a DLT occurs, and thereafter the MLE is employed to update the parameters α and β in the logistic relationship between dose and probability of DLT after each additional cohort. The probability of DLT for each dose level is updated with updated parameter α and β. The recommended dose level for next cohort is the one which has a probability of DLT closest to the TTL, 33%. The trial stops when the same dose has been recommended for 4 consecutive cohorts or a maximum 20 cohorts have been treated. MTD is the recommended dose level for the next coming cohort after the trial stops. 99 6.2.5 Continuous Reassessment Method with Normalized Equivalent Toxicity Score (CRM-NETS). We also simulate a CRM-NETS design (Yuan et al. 2006) which is same as the CRM described in Section 6.2.4 except that toxicity response is measured quantitatively with NETS, S * , and the quasi-Bernoulli likelihood is employed (Yuan et al. 2006) instead of the regular likelihood function. The dose-toxicity relationship is assumed to be of logistic form as following: i d S it * ) ( log * β α + = Where α defines the starting NTES at the lowest tested dose level and β defines the speed of increased NETS with increasing dose level. In another form, the relation can also be expressed as following: ) * exp( 1 ) * exp( * i i d d S β α β α + + + = The so-called quasi-Bernoulli likelihood function (Yuan et al. 2006) is constructed using a family of probability distributions that may not contain the true distribution and it is defined as below: ( ) ∏ = − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + = N i S i i S i i i i d d d d L 1 1 * * ) * exp( 1 ) * exp( 1 ) * exp( 1 ) * exp( β α β α β α β α Estimators obtained by maximizing the quasi-Bernoulli likelihood function are called quasi maximum likelihood estimates (QMLEs). Under some regularity conditions, 100 QMLEs are strongly consistent if the “quasi” distributions belong to linear exponential families such as the binomial family (Yuan et al. 2006; Gourieroux, Monfort, and Trognon, 1984; McCullagh and Nelder, 1989). The trial starts from the lowest dose level and keeps escalating one dose level at a time until a DLT occurs, and then the QMLE is employed to update the parameters α and β after each additional cohort. The recommended dose level for next coming cohort is the dose which has the NETS closest to the TNETS, 47.6%. 6.3 Result of Simulation 6.3.1 Comparison of Accuracy of MTD Estimation. The TTL is 33% for the ID and CRM in which toxicity response is treated as binary variable. Correspondingly, the TNETS is 47.6% for the EID and CRM-NETS in which toxicity response is treated as quasi-continuous variable. Comparisons of performances in correct MTD estimation between the 5 designs are summarized in the Table 6.3. The simulation results from all 5 scenarios are the same for the ID, Standard design, and CRM, respectively, because the probability of DLTs in each dose level is the same across different scenarios. However the overall toxicity severity of each dose level is different substantially and significantly across the different scenarios, especially between Extreme-Over-Toxic-Scenario and Extreme-Under-Toxic-Scenario. These results demonstrate that the designs treating toxicity response as binary variable can only utilize probability of DLTs for dose levels, but discard a lot of valuable toxicity 101 information and fail to differentiate more detail toxicity profile. In the contrast, the simulation results from the 5 scenarios are different for EID and CRM-NETS, convincing that treating toxicity response as quasi-continuous variable with the score system we propose can successfully differentiate the toxicity profile and fully utilize all toxicity information in MTD estimation. In Target Scenario, toxicity profile of each dose level is similar as the TTP used to calculate the TNETS and the true MTD is the dose level 3 for all 5 designs. The percentages of each dose level chosen as MTD are pretty similar between EID and ID or between CRM and CRM-NETS. These suggest that designs treating toxicity response as binary or quasi-continuous variable perform similarly in MTD estimation when toxicity profile in each dose level is close to the TTP. All designs except the Standard design perform similarly in the MTD estimation. Obviously, Standard design (17.3%) substantially underestimates the MTD compared with the other 4 designs (> 31%). In under toxic scenarios, such as Medium-Under-Toxic-Scenario and Extreme- Under-Toxic-Scenario, the non DLT toxicity profile skews to the low grade toxicities when compared with the TTP used to calculate the TNETS. There is higher probability for grade 3 DLT than the grade 4 DLT given that the total probability of DLT does not change. The true MTD is a dose level with a higher probability of DLT than 33%. In EID and CRM-NETS, MTD is estimated according to the exact toxicity profile instead of the coarse probability of DLT so that MTD is not underestimated for them while MTD is obviously underestimated in ID, Standard design, and CRM. On the other hand, in over 102 toxic scenarios, such as Medium-Over-Toxic-Scenario and Extreme-Over-Toxic- Scenario, the non DLT toxicity profile skews to the high grade toxicities and there is higher probability for grade 4 DLT than the grade 3 DLT given the same total probability of grade 3 and 4 DLTs. The true MTD is a dose level with a lower probability of DLT than 33%. In EID and CRM-NETS, MTD is estimated according to the exact toxicity profile so that the over toxic dose levels have less chance to be chosen as MTD while MTD is overestimated in the ID, Standard design, and CRM. These simulation results demonstrate that treating toxicity as quasi-continuous variable can reduce significantly the chance of underestimating MTD in case of under toxic scenarios and overestimating MTD in case of over toxic scenarios, especially in the extreme cases like Extreme-Over- Toxic-Scenario and Extreme-Under-Toxic-Scenario. Both EID and CRM-NETS can successfully differentiate toxicity profile and select correct MTD when toxicity profile deviates substantially from the TTP. They have similar probability of selecting the correct MTD in Target Scenario and Extreme-Over- Toxic-Scenario. CRM-NETS outperforms EID by 5.1% (40.7% - 35.6%) in Medium- Under-Toxic-Scenario, but EID outperforms CRM-NETS by about 3.8% (39.9% - 36.1%) and 9.9% (48.3% - 38.4%) in the Medium-Over-Toxic-Scenario and Extreme-Under- Toxic-Scenario, respectively. In overall, EID performs better in selecting correct MTD than CRM-NETS. 103 Table 6.3: Percentage for Each Dose Recommended as MTD under Different Scenarios. Scenario Dose Probability of DLTs for the dose Composite NETS for the dose Probability (%) the dose chosen as MTD EID** ID* StandardCRM* CRM- NETS** Target Scenario 1 0.08 0.34 12.2 16.0 45.1 10.0 29.4 2 0.24 0.43 33.0 34.0 33.2 37.8 29.2 3 0.33 0.48 34.5 33.8 17.3 31.6 35.4 4 0.44 0.54 17.1 14.1 4.0 18.8 5.9 5 0.56 0.61 3.1 2.0 0.4 1.9 0 6 0.76 0.71 0.1 0 0 0 0 Medium- Under- Toxic- Scenario 1 0.08 0.27 2.7 16.0 45.1 10.0 14.8 2 0.24 0.36 14.8 34.0 33.2 37.8 14.7 3 0.33 0.41 30.4 33.8 17.3 31.6 26.6 4 0.44 0.48 35.6 14.1 4.0 18.8 40.7 5 0.56 0.56 15.4 2.0 0.4 1.9 3.2 6 0.76 0.67 1.2 0 0 0 0.14 Medium- Over- Toxic- Scenario 1 0.08 0.41 35.6 16.0 45.1 10.0 52.5 2 0.24 0.49 39.9 34.0 33.2 37.8 36.1 3 0.33 0.53 19.9 33.8 17.3 31.6 11.4 4 0.44 0.59 4.3 14.1 4.0 18.8 0.02 5 0.56 0.65 0.3 2.0 0.4 1.9 0 6 0.76 0.75 0 0 0 0 0 Extreme- Over- Toxic- Scenario 1 0.08 0.61 100.0 16.0 45.1 10.0 100 2 0.24 0.66 0 34.0 33.2 37.8 0 3 0.33 0.69 0 33.8 17.3 31.6 0 4 0.44 0.73 0 14.1 4.0 18.8 0 5 0.56 0.77 0 2.0 0.4 1.9 0 6 0.76 0.84 0 0 0 0 0 (Table continued on next page) 104 Table 6.3 (cont.) Scenario Dose Probability of DLTs for the dose Composite NETS for the dose Probability (%) the dose chosen as MTD EID** ID* StandardCRM* CRM- NETS** Extreme- Under- Toxic- Scenario 1 0.08 0.10 0 16.0 45.1 10.0 7.2 2 0.24 0.21 1.1 34.0 33.2 37.8 5.2 3 0.33 0.27 5.7 33.8 17.3 31.6 7.5 4 0.44 0.36 20.4 14.1 4.0 18.8 15.7 5 0.56 0.44 48.3 2.0 0.4 1.9 38.4 6 0.76 0.58 24.5 0 0 0 25.9 * Designs treating toxicity response as binary variable is targeted with TTL = 0.33. ** Designs treating toxicity response as quasi-continuous variable is targeted with TNETS = 0.476. 6.3.2 Comparison of Patient Distribution, Sample Size, and Study Length. Comparison of patient distribution, sample size, and number of cohorts by different designs are summarized in the Table 6.4. The pattern of probability of DLTs by dose levels are the same in Target Scenario, Medium-Under-Toxic-Scenario, Medium- Over-Toxic-Scenario, Medium-Over-Toxic-Scenario, and Extreme-Under-Toxic- Scenario so that the distribution of patients, sample size, and number of cohorts are the same across all 5 scenarios in the ID, Standard design, and CRM. But they are different across the 5 scenarios in the EID and CRM-NETS. The patient distribution is centered at the true MTD in all 5 designs and skews to low dose levels because the dose allocation is escalated one by one from the lowest level in all designs. Standard design needs the least total sample size and patient distribution skews most heavily to the low dose levels. The first reason is that in standard design, the true MTD is a lower dose level with an expected toxicity level of about 22% which is 105 much lower than 33% (He et al. 2006) so that it needs to test fewer dose levels from the lowest level to the true MTD. The second one is that each dose level can only be treated no more than 2 cohorts with 6 patients. The distribution of patients by dose level is much flatter in other 4 designs. In Target Scenario in which the true MTD is the same for all designs, the patient distributions in the ID and CRM are similar to those in the EID and CRM-NETS. In the other 4 scenarios, there is no way to compare the patient distribution between the designs treating toxicity response as binary variable and as quasi-continuous variable because the estimated MTDs are not the same for them and patients are approximately distributed around their estimated MTDs. The total sample size increases with the number of dose levels between the lowest dose level and the true MTD. In over toxic scenarios, such as Medium-Under-Toxic- Scenario and Extreme-Under-Toxic-Scenario, the EID and CRM-NETS need smaller total sample size than the ID and CRM because the true MTD is a lower dose level when estimated according to exact toxicity profile than to probability of DLT. The results are totally opposite in the under toxic scenarios, such as Medium-Over-Toxic-Scenario and Extreme-Over-Toxic-Scenario. The sample size and number of cohorts are very similar between EID and CRM- NETS in all 5 scenarios, suggesting similar cost and length of trial with them. In all 5 scenarios, the percentage of patients treated at the true MTD is higher in EID than in CRM-NETS and the patient distribution skews more heavily to the low dose levels in 106 CRM-NETS than in EID (Table 6.4). These results suggest that CRM-NETS are more likely to treat patients at under-dosed dose level than EID and the therapeutic effect for patient is better in EID than in CRM-NETS. 6.4 Summary of Simulation From simulation results, we can see that designs treating toxicity response as quasi-continuous variable with our novel score system (EID and CRM-NETS) can fully utilize all toxicity information and estimate MTD more accurately according to exact toxicity profile instead of coarse probability of DLT without additional cost and extended length of trial, especially in case of significantly more or less severe grade 4 toxicities among all DLTs. The therapeutic effect for patient is better in our EID than in CRM- NETS although both need similar sample size and length of trial. Moreover, our EID is model free, objective, robust, and much simpler to use. 107 Table 6.4: Comparisons of Patients Distribution, Sample Size, and Number of Cohorts by Different Designs. Scenario Dose PDLT NETS Percentage (%) and Standard Deviation (SD) of Patients Treated at the Dose Level EID ID Standard CRM CRM ETS Percent SD Percent SD Percent SD Percent SD Percent SD Target- Scenario 1 0.08 0.34 23.1 23.9 26.3 25.9 39.3 23.3 21.2 22.4 36.6 35.2 2 0.24 0.43 32.5 26.7 35.2 27.0 35.0 14.5 36.5 30.4 29.2 28.4 3 0.33 0.48 26.5 24.0 25.9 25.0 18.3 16.4 25.2 27.3 23.5 25.7 4 0.44 0.54 13.6 18.8 10.5 17.6 6.19 11.1 14.2 22.7 7.93 14.1 5 0.56 0.61 3.82 9.55 2.00 7.24 1.14 4.68 2.23 8.24 1.21 3.57 6 0.76 0.71 0.56 2.61 0.13 1.24 0.10 1.19 0.64 2.29 1.60 3.61 Average sample size 27.6 9.26 25.5 8.39 13.8 4.47 26.6 8.08 29.1 10.6 Average cohort number 9.20 3.09 8.48 2.80 4.60 1.49 8.87 2.69 9.71 3.54 Medium -Under- Toxic- Scenario 1 0.08 0.27 13.8 12.6 26.3 25.9 39.3 23.3 21.2 22.4 22.5 26.6 2 0.24 0.36 22.1 21.6 35.2 27.0 35.0 14.5 36.5 30.4 21.7 23.5 3 0.33 0.41 26.5 22.6 25.9 25.0 18.3 16.4 25.2 27.3 22.4 23.7 4 0.44 0.48 23.8 21.7 10.5 17.6 6.19 11.1 14.2 22.7 23.6 23.1 5 0.56 0.56 11.5 16.7 2.00 7.24 1.14 4.68 2.23 8.24 6.12 10.8 6 0.76 0.67 2.32 6.16 0.13 1.24 0.10 1.19 0.64 2.29 3.70 6.20 Average sample size 30.3 9.05 25.5 8.39 13.8 4.47 26.6 8.08 34.0 11.3 Average cohort number 10.1 3.02 8.48 2.80 4.60 1.49 8.87 2.69 11.3 3.77 (Table continued on next page) 108 Table 6.4 (cont.) Scenario Dose PDLT NETS Percentage (%) and Standard Deviation (SD) of Patients Treated at the Dose Level EID ID Standard CRM CRM ETS Percent SD Percent SD Percent SD Percent SD Percent SD Medium -Over- Toxic- Scenario 1 0.08 0.41 41.0 33.7 26.3 25.9 39.3 23.3 21.2 22.4 55.5 38.5 2 0.24 0.49 35.4 27.6 35.2 27.0 35.0 14.5 36.5 30.4 30.1 30.3 3 0.33 0.53 17.3 22.0 25.9 25.0 18.3 16.4 25.2 27.3 11.6 19.6 4 0.44 0.59 5.31 11.9 10.5 17.6 6.19 11.1 14.2 22.7 1.80 4.79 5 0.56 0.65 0.94 4.12 2.00 7.24 1.14 4.68 2.23 8.24 0.37 1.88 6 0.76 0.75 0.09 0.88 0.13 1.24 0.10 1.19 0.64 2.29 0.69 2.44 Average sample size 24.7 9.15 25.5 8.39 13.8 4.47 26.6 8.08 24.4 9.06 Average cohort number 8.23 3.05 8.48 2.80 4.60 1.49 8.87 2.69 8.14 3.02 Extreme -Over- Toxic- Scenario 1 0.08 0.61 100 0 26.3 25.9 39.3 23.3 21.2 22.4 100 0 2 0.24 0.66 0 0 35.2 27.0 35.0 14.5 36.5 30.4 0 0 3 0.33 0.69 0 0 25.9 25.0 18.3 16.4 25.2 27.3 0 0 4 0.44 0.73 0 0 10.5 17.6 6.19 11.1 14.2 22.7 0 0 5 0.56 0.77 0 0 2.00 7.24 1.14 4.68 2.23 8.24 0 0 6 0.76 0.84 0 0 0.13 1.24 0.10 1.19 0.64 2.29 0 0 Average sample size 12 0 25.5 8.39 13.8 4.47 26.6 8.08 15 0 Average cohort number 4 0 8.48 2.80 4.60 1.49 8.87 2.69 5 0 (Table continued on next page) 109 Table 6.4 (cont.) Scenario Dose PDLT NETS Percentage (%) and Standard Deviation (SD) of Patients Treated at the Dose Level EID ID Standard CRM CRM ETS Percent SD Percent SD Percent SD Percent SD Percent SD Extreme -Under- Toxic- Scenario 1 0.08 0.10 9.70 2.84 26.3 25.9 39.3 23.3 21.2 22.4 15.6 18.9 2 0.24 0.21 11.6 7.83 35.2 27.0 35.0 14.5 36.5 30.4 14.7 16.7 3 0.33 0.27 14.7 12.8 25.9 25.0 18.3 16.4 25.2 27.3 14.2 16.9 4 0.44 0.36 21.3 17.9 10.5 17.6 6.19 11.1 14.2 22.7 17.2 19.5 5 0.56 0.44 28.2 19.8 2.00 7.24 1.14 4.68 2.23 8.24 20.8 21.0 6 0.76 0.58 14.5 16.3 0.13 1.24 0.10 1.19 0.64 2.29 17.5 18.7 Average sample size 33.4 8.14 25.5 8.39 13.8 4.47 26.6 8.08 35.4 11.4 Average cohort number 11.1 2.71 8.48 2.80 4.60 1.49 8.87 2.69 11.8 3.80 110 Chapter VII: Application of Extended Isotonic Design in Real Phase I Trials of Children’s Oncology Group Ultimately, to assess adequately our EID’s utility, it is necessary to evaluate its performance in actual cancer Phase I clinical trials. We apply our EID to two real completed Phase I clinical trials, ADVL0311 and A09712, of the Children’s Oncology Group (COG). The original designs of these two studies were Standard 3+3 designs with dose de-escalation. We conduct two pseudo-trials with EID using the patients and their toxicity information from these two studies and investigate if the MTDs recommended by EID are the same as those defined by the original studies. 7.1 Pseudo-Trial with EID Using Data of ADVL0311 The title of the study ADVL0311 is a Phase I study of pemetrexed (LY231514, Alimta) (NSC # 698037, IND # 67952) in children and adolescents with recurrent solid tumors. This Phase I study is to estimate the MTD of pemetrexed administered as a 10 minute IV infusion every 21 days in children and adolescents with recurrent solid tumors. The starting dose is 400 mg/m 2 , and subsequent levels will be 30% higher than the previous (Table 7.1). DLTs includes any Grade III or Grade IV nonhematologic toxicity attributable possibly, probably or definitely to the pemetrexed with the specific exclusions (Grade III nausea and vomiting, Grade III transaminase (AST/ALT) elevation which returns to Grade ≤ I or baseline prior to the time for the next treatment course, 111 Grade III fever or infection, and Alopecia) and Grade 4 neutropenia or Grade 4 thrombocytopenia of > 7 days duration, which requires transfusion therapy on greater than 2 occasions in 7 days, or which causes a delay of ≥ 14 days beyond the planned interval between treatment courses. Table 7.1: Dose Level and Detail Treatment of ADVL0311 Dose Level Treatment 1 400 mg/m 2 over 10 minutes 2 520 mg/m 2 over 10 minutes 3 670 mg/m 2 over 10 minutes 4 870 mg/m 2 over 10 minutes 5 1130 mg/m 2 over 10 minutes 6 1470 mg/m 2 over 10 minutes 7 1910 mg/m 2 over 10 minutes 8 2480 mg/m 2 over 10 minutes Patients had experienced variant toxicities from no toxicity to many DTLs. The composite ETS ranging from 0 to 6.0 is calculated with α=-2 and β=0.5 for each patient according to his/her toxicities and the rules described in the Chapter V. Then the ETS of each patient is divided by the maximum ETS (6.0) to obtain the NETS with range 0 to 1 (Table 7.2). 112 Table 7.2: Summary of Patient’s Enroll Order, Dose Level, Toxicity, ETS, and NETS of ADVL0311 Enroll Order Patient ID Dose level Had DLT? Toxicities Summary ETS NETS 1 714061 1 No 1 grade 3 non DLT toxicity 3 grade 1 toxicity 2.182 0.364 2 739198 1 No 4 grade 1 toxicity 0.378 0.063 3 724155 1 No 1 grade 3 non DLT toxicity 1 grade 2 toxicity 2 grade 1 toxicity 2.209 0.368 4 740865 2 No 9 grade 1 toxicity 0.881 0.147 5 717893 2 No 2 grade 4 non DLT toxicity 1 grade 3 non DLT toxicity 3 grade 1 toxicity 3.321 0.553 6 741409 2 No 1 grade 3 non DLT toxicity 8 grade 2 toxicity 10 grade 1 toxicity 2.912 0.485 7 742323 3 No 1 grade 3 non DLT toxicity 3 grade 2 toxicity 4 grade 1 toxicity 2.417 0.403 8 742406 3 Inevaluable NA NA NA 9 742544 3 No 1 grade 2 toxicity 4 grade 1 toxicity 1.269 0.211 10 712496 3 No 1 grade 4 non DLT toxicity 1 grade 3 non DLT toxicity 1 grade 2 toxicity 4 grade 1 toxicity 3.294 0.549 11 708773 4 No 1 grade 4 non DLT toxicity 5 grade 2 toxicity 3.818 0.636 12 744553 4 No 1 grade 4 non DLT toxicity 2 grade 3 non DLT toxicity 1 grade 2 toxicity 3.269 0.545 13 728327 4 No 1 grade 2 toxicity 4 grade 1 toxicity 1.269 0.211 14 745379 5 No No toxicity 0 0 15 745924 5 Inevaluable NA NA NA 16 746285 5 No 2 grade 4 non DLT toxicity 1 grade 2 toxicity 6 grade 1 toxicity 3.378 0.563 17 741265 5 No 2 grade 2 toxicity 1.182 0.197 18 600812 6 Yes 1 grade 4 DLT toxicity 1 grade 3 non DLT toxicity 1 grade 2 toxicity 2 grade 1 toxicity 4.214 0.702 (Table continued on next page) 113 (Table 7.2 Cont.) Enroll Order Patient ID Dose level Had DLT? Toxicities Summary ETS NETS 19 703789 6 No 1 grade 3 non DLT toxicity 2 grade 2 toxicity 5 grade 1 toxicity 2.378 0.396 20 748842 6 No No toxicity 0 0 21 718998 6 No 2 grade 3 non DLT toxicity 3 grade 1 toxicity 2.269 0.378 22 748678 6 No 3 grade 1 toxicity 0.269 0.045 23 749850 6 No 1 grade 3 non DLT toxicity 4 grade 2 toxicity 16 grade 1 toxicity 2.881 0.480 24 750716 7 No 1 grade 3 non DLT toxicity 2 grade 2 toxicity 8 grade 1 toxicity 2.500 0.417 25 735053 7 No 2 grade 2 toxicity 4 grade 1 toxicity 1.378 0.230 26 741575 7 No 1 grade 4 non DLT toxicity 2 grade 3 non DLT toxicity 6 grade 2 toxicity 15 grade 1 toxicity 3.893 0.649 27 724127 8 No 1 grade 4 non DLT toxicity 1 grade 2 toxicity 5 grade 1 toxicity 3.245 0.541 28 69243 8 Yes 1 grade 4 DLT toxicity 5 grade 4 non DLT toxicity 9 grade 3 non DLT toxicity 7 grade 2 toxicity 4 grade 1 toxicity 5.968 0.995 29 739883 8 No 3 grade 3 non DLT toxicity 1 grade 2 toxicity 2 grade 1 toxicity 2.417 0.403 30 749014 8 Yes 1 grade 4 DLT toxicity 3 grade 4 non DLT toxicity 12 grade 3 non DLT toxicity 11 grade 2 toxicity 14 grade 1 toxicity 5.993 0.999 31 751395 7 No 2 grade 2 toxicity 2 grade 1 toxicity 1.269 0.211 32 737478 7 No 8 grade 4 non DLT toxicity 8 grade 3 non DLT toxicity 13 grade 2 toxicity 5 grade 1 toxicity 4.000 0.667 33 754963 7* No 3 grade 4 non DLT toxicity 1 grade 3 non DLT toxicity 2 grade 2 toxicity 5 grade 1 toxicity 3.622 0.604 * Recommended MTD by original standard 3+3 design with dose de-escalation. 114 We conduct a pseudo-trial with EID using some patients according to their enroll order and their NETS in the Table 7.2 after exclusion of the two inevaluable patients. The pseudo-trial with EID is summarized in Table 7.3. There are 3 patients in each cohort of the pseudo-trial. Table 7.3: A Pseudo-trial with EID Using the Patients of ADVL0311 with Parameter α=-2 and β=0.5. Cohort Patient ID PANEST of the dose level Dose Level 1 2 3 4 5 6 7 8 Current Next 1 1, 2, 3 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 1 2 2 4, 5, 6 0.26 0.40 0.40 0.40 0.40 0.40 0.40 0.40 2 3 3 7, 9,10 0.26 0.39 0.39 0.39 0.39 0.39 0.39 0.39 3 4 4 11,12,13 0.26 0.39 0.39 0.46 0.46 0.46 0.46 0.46 4 5 5 14,15,17 0.26 0.38 0.38 0.38 0.38 0.38 0.38 0.38 5 6 6 18,19,20 0.26 0.37 0.37 0.37 0.37 0.37 0.37 0.37 6 7 8 24,25,26 0.26 0.37 0.37 0.37 0.37 0.37 0.43 0.43 7 8 9 27,28,29 0.26 0.37 0.37 0.37 0.37 0.37 0.43 0.65 8 7 11 31,32,33 0.26 0.37 0.37 0.37 0.37 0.37 0.46 0.65 7 7* *MTD Recommended by EID. Both the original ADVL0311 trial and the pseudo-trial with EID recommend the same MTD, the dose level 7. The toxicity profile of patients in ADVL0311 is not different very much from the TTP employed to define the TNETS (47.6%) so that the final estimated MTDs from both method match. But the sample sizes used in the 2 trials are different. In the pseudo trial with EID, the trial stops with only 8 cohorts, resulting in a sample size of 24. Actually, if there were more patients who have been treated at the dose level 7 in the original trial, the pseudo-trial should have treated 2 more consecutive cohorts (6 patients) at the dose level 7 before it stops because 3 consecutive cohorts have 115 been treated at the same dose level 7. Therefore, the complete sample size of the pseudo- trial should be 30 (24+6) which is similar to the sample size (31) of the original trial using the Standard 3+3 design with dose de-escalation. Although both designs require similar sample sizes, during the dose allocation procedure, more patients are treated at the final recommended MTD and fewer patients are treated at under-dosed or over-dosed dose levels in the pseudo-trial with EID. For example, the 3 patients (21, 22, and 23) who were treated at the dose level 6 in the Standard design are not required in the pseudo trial using EID because, in the previous cohort (patient 18, 19 and 20) treated at dose level 6, although the patient 18 had a grade 3 DLT as well as other low grade toxicities, the patient 20 had no toxicity and thus reduced the PANETS of the dose level 6 to 0.39 which is below the TNETS (47.6%) so that the dose level 6 is considered under-dosed and dose level escalates to dose level 7 for the next cohort. Dose level 6 is also finally decided as under-dosed at the end of pseudo- trial. This demonstrates that our EID can assess the toxicity risk of each dose level comprehensively and more accurately according to the exact toxicity profile of all patients at a specific dose level by employing the average NETS. On the other hand, in the cohort (patient 27, 28, 29) treated at the dose level 8, all of these 3 patients had very severe multiple toxicities so that the PANETS of dose level 8 is 0.66 which is substantially higher than the TNETS (47.6%). Therefore the recommended dose level for the next cohort (patient 31, 32, 33) is de-escalated to dose level 7 so that the patient 30 is not required in the pseudo-trial and can avoid being treated at dose level 8 which is an 116 over toxic dose level. In the original trial using Standard design, the patient 30 was unnecessarily treated at dose level 8 and had extremely severe composite toxicities (1 grade 4 DLT, 3 grade 4 non DLT toxicities, 12 grade 3 non DLT toxicities, 11 grade 2 toxicities, and 14 grade 1 toxicities). The pseudo trial with data of ADVL0311 demonstrates that, using EID, investigators can fully consider composite effect of multiple toxicities of each patient and differentiate quantitatively the toxicities, especially life-threatening ones during dose allocation procedure. Moreover EID can successfully estimate correct MTD according to the exact toxicity profile, increase therapeutic effect of a trial by reducing the chance of treating unnecessary patients at under-dosed dose level, and protect patients from being exposed to over toxic dose level by estimating the toxicity risk of each dose level quantitatively and accurately with less number of treated patients. 7.2 Pseudo-Trial with EID Using Data of A09712 The title of study A09712 is a Phase I study of motexafin gadolinium (XCYTRIN, NSC #695238) and involved field radiation therapy for intrinsic pontine glioma of childhood. The main purpose of this study was to determine the MTD and schedule of motexafin gadolinium given prior to radiation therapy. The starting dose is 1.7 mg/kg/dose from Monday to Friday for 3 weeks. The detail dose at each dose level is summarized in Table 7.4. 117 Table 7.4: Dose Level and Detail Treatment of A09712 Dose level Treatment 1 1.7mg/kg/dose M-F*3w 2 1.7mg/kg/dose MWF*6w 3 1.7mg/kg/dose M-F*6w 4 1.9mg/kg/dose M-F*6w 5 3.4mg/kg/dose M-F*6w 6 4.4mg/kg/dose M-F*6w 7 5.5mg/kg/dose M-F*6w 8 7.1mg/kg/dose M-F*6w 9 9.2mg/kg/dose M-F*6w DLTs were defined as any Grade 4 hematologic toxicity that persists for more than 7 days or that requires platelet transfusions for a period of time exceeding 7 days during the assigned weeks of concurrent chemotherapy and radiation therapy and Grade 3 or 4 non-hematologic toxicity with the exception of Grade 3 nausea and/or vomiting which can be controlled within 7 days. Patients had varying toxicities from no toxicity to multiple DTLs. A composite ETS ranging from 0 to 6.0 is estimated with α=-2 and β=0.5 for each patient according to his/her toxicities and rules described in chapter V. The ETS of each patient is divided by the maximum ETS (6.0) to get the NETS with range 0 to 1 (Table 7.5). We conduct a pseudo-trial with EID using some patients according to their enroll order and their NETS except the 3 inevaluable patients (Table 7.6). There are 2, 3, or 4 patients in each cohort according to the original data of A09712. 118 Table 7.5: Summary of Patient’s Enroll Order, Dose Level, Toxicities, ETS, and NETS of A09712 Enroll Order Patient ID Dose Level Had DLT? Toxicity Summary ETS NETS 1 74823 1 No 2 grade 2 toxicity 3 grade 1 toxicity 1.321 0.220 2 79631 1 No No toxicity 0 0 3 81562 1 No 1 grade 2 toxicity 2 grade 1 toxicity 1.182 0.197 4 83068 1 No 1 grade 2 toxicity 4 grade 1 toxicity 1.269 0.211 5 86441 2 No 1 grade 1 toxicity 0.100 0.017 6 86625 2 No No toxicity 0 0 7 86861 2 No 2 grade 1 toxicity 0.182 0.030 8 87167 2 No 2 grade 1 toxicity 0.182 0.030 9 600668 3 No 3 grade 2 toxicity 5 grade 1 toxicity 1.562 0.260 10 600907 3 Inevaluable NA NA NA 11 601169 3 No 1 grade 2 toxicity 1 grade 1 toxicity 1.148 0.191 12 601307 3 No 1 grade 1 toxicity 0.100 0.017 13 700073 3 No No toxicity 0 0 14 702795 4 No No toxicity 0 0 15 703732 4 No 1 grade 1 toxicity 0.100 0.017 16 705476 4 Yes 1 grade 3 DLT toxicity 2 grade 3 non DLT toxicity 5 grade 2 toxicity 1 grade 1 toxicity 4.426 0.738 17 709539 4 No 1 grade 2 toxicity 2 grade 1 toxicity 1.182 0.197 18 711039 4 No 4 grade 1 toxicity 0.378 0.063 19 711051 4 No 1 grade 2 toxicity 3 grade 1 toxicity 1.223 0.204 20 714200 5 No 1 grade 2 toxicity 1 grade 1 toxicity 1.148 0.191 21 714201 5 No 2 grade 1 toxicity 0.182 0.030 22 714337 5 No 2 grade 1 toxicity 0.182 0.030 23 714655 5 No No toxicity 0 0 24 717897 6 No 1 grade 2 toxicity 1 grade 1 toxicity 1.148 0.191 25 717942 6 No 2 grade 2 toxicity 1 grade 1 toxicity 1.223 0.204 (Table continued on next page) 119 (Table 7.5 Cont.) Enroll Order Patient ID Dose Level Had DLT? Toxicity Summary ETS NETS 26 717959 6 No 2 grade 2 toxicity 2 grade 1 toxicity 1.269 0.211 27 725184 7 No 1 grade 2 toxicity 1.000 0.167 28 725313 7 No 1 grade 2 toxicity 1 grade 1 toxicity 1.148 0.191 29 726041 7 Inevaluable NA NA 30 728784 7 No 2 grade 2 toxicity 2 grade 1 toxicity 1.269 0.211 31 732123 8 No 2 grade 2 toxicity 1 grade 1 toxicity 1.182 0.197 32 732190 8 No 2 grade 2 toxicity 1 grade 1 toxicity 1.182 0.197 33 732228 8 No 2 grade 2 toxicity 2 grade 1 toxicity 1.269 0.211 34 734789 9 Yes 1 grade 3 DLT toxicity 2 grade 3 non DLT toxicity 4 grade 2 toxicity 2 grade 1 toxicity 4.401 0.734 35 735536 9 Yes 1 grade 3 DLT toxicity 1 grade 2 toxicity 1 grade 1 toxicity 4.154 0.692 36 735225 8 Inevaluable NA NA NA 37 736059 8 Yes 1 grade 3 DLT toxicity 1 grade 3 non DLT toxicity 2 grade 2 toxicity 3 grade 1 toxicity 4.269 0.711 38 736071 8 Yes 1 grade 3 DLT toxicity 1 grade 3 non DLT toxicity 4 grade 2 toxicity 5 grade 1 toxicity 4.401 0.734 39 739016 7 Yes 1 grade 3 DLT toxicity 1 grade 2 toxicity 2 grade 1 toxicity 4.168 0.695 40 739776 7 Yes 1 grade 3 DLT toxicity 1 grade 3 non DLT toxicity 2 grade 2 toxicity 1 grade 1 toxicity 4.231 0.705 41 741007 7 No 2 grade 2 toxicity 6 grade 1 toxicity 1.500 0.250 (Table continued on next page) 120 (Table 7.5 Cont.) Enroll Order Patient ID Dose Level Had DLT? Toxicity Summary ETS NETS 42 743243 6 Yes 1 grade 3 DLT toxicity 2 grade 3 non DLT toxicity 2 grade 2 toxicity 2 grade 1 toxicity 4.310 0.718 43 743979 6 No 2 grade 2 toxicity 2 grade 1 toxicity 1.269 0.211 44 746027 6* No 1 grade 3 non DLT toxicity 3 grade 2 toxicity 6 grade 1 toxicity 2.500 0.417 * Recommended MTD by original standard 3+3 design with dose de-escalation. 121 Table 7.6: A Pseudo-trial with EID Using the Patients of A09712 with Parameter α=-2 and β=0.5. Cohort Patients in the cohort PANETS of the dose level Dose Level 1 2 3 4 5 6 7 8 9 Current Recommended 1 1,2,3 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 1 2 2 5,6,7 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 2 3 3 9,11,12 0.08 0.08 0.16 0.16 0.16 0.16 0.16 0.16 0.16 3 4 4 14,15,16 0.08 0.08 0.16 0.25 0.25 0.25 0.25 0.25 0.25 4 5 5 20,21,22 0.08 0.08 0.16 0.17 0.17 0.17 0.17 0.17 0.17 5 6 6 24,25,26 0.08 0.08 0.16 0.17 0.17 0.20 0.20 0.20 0.20 6 7 7 27,28,30 0.08 0.08 0.16 0.17 0.17 0.20 0.20 0.20 0.20 7 8 8 31,32,33 0.08 0.08 0.16 0.17 0.17 0.20 0.20 0.20 0.20 8 9 9 34,35 0.08 0.08 0.16 0.17 0.17 0.20 0.20 0.20 0.71 9 8 10 37,38 0.08 0.08 0.16 0.17 0.17 0.20 0.20 0.41 0.71 8 8* *Recommend MTD by EID. **Recommended MTD by original standard 3+3 design with dose de-escalation. 122 From above table, the original A09712 trial recommended dose level 6 as MTD, but the pseudo-trial recommends the dose level 8 as MTD. The toxicity profile of the patients in A09712 is an extreme under toxic scenario which is very different from the TTP employed to define the TNETS (47.6%). There are no grade 4 non DLT toxicities and no grade 4 DLTs among all patients of A09712. But in the TTP, it assumes equal probabilities of grade 3 and grade 4 non LDT toxicities among all non DLT toxicities, and equal probabilities of grade 3 and 4 DLTs among a total 33% probability of DLTs. The overall toxicity profile of study A09712 is substantially under toxic and toxicity distribution skews to the low grade, less severe toxicities. The dose level 6 recommended in the original A09712 is a substantially underestimated MTD with far less toxic effect than the targeted. The pseudo-trial with EID identifies the dose level 8 as MTD according to the exact toxicity profile of patients with the TNEST as of 47.6%. We could further confirm that the dose level 6 and 7 are under-dosed and the dose level 8 is the true MTD with the estimated PANETS closest to the TNETS should we have included the additional two cohorts (patient 39, 40, 41, 42, 43, and 44) treated at dose level 7 and 6 which are not required in our pseudo-trial. Standard design usually underestimates MTD with less than the target toxicity level, 33%. The toxicity profile of A09712 is extremely under toxic and the use of EID will adjust for the deviation and estimate a correct MTD with enough toxicity level to ensure therapeutic effect of chemical agent. In the pseudo trial with EID, the trial stops with only 10 cohorts, resulting in a sample size of 28. Actually, if there were more patients who have been treated at the dose 123 level 8 in the original trial, the pseudo-trial should have treated 2 more consecutive cohorts (6 patients) at the dose level 8 before it stops because 3 consecutive cohorts have been treated at the same dose level 8. Therefore, the complete sample size of the pseudo- trial should be 34 (28+6) which is 7 patients less than that (41) of the original trial. Both of the 2 patients in the 2nd cohort (patient 37 and 38) treated at does level 8 had a grade 3 DLT, but the PANETS of dose level 8 (patient 31, 32, 33, 37, and 38) is still not above the TNETS (47.6%) so that the recommended dose level for the next cohort is still dose level 8. But in the original trial with Standard design, the dose level 8 was judged as over toxic based on that 2 out 5 treated patients had DLT regardless of the facts that both DLTs were grade 3, their additional non DLT toxicities were moderate, and moreover the toxicities of other 3 patients (31, 32, and 33) were very mild. These demonstrate the downside of coarsely categorizing composite toxicity response into binary outcome as DLT or non-DLT under cases in which the toxicity response profile is substantially deviated from the TTP. In EID, the toxicity response is treated as quasi-continuous variable and overall toxicity severity is measured quantitatively according to the overall exact toxicity profile without discarding any information so that it has more power to estimate a correct MTD with less sample size. 7.3 Sensitivity Analysis In the score system, the overall score is based mainly on the most severe toxicity with the maximum adjusted grade, but still leave some flexibility for investigator by 124 choosing different values of parameter α and β in the calculation of composite ETS according to specific purposes of the study and practical needs. Here we do sensitivity analyses to study how the real COG studies perform when different values of parameter α and β are used. We fix α as -2 and choose 5 different values for β as 2, 1, 0.5, 0.25, and 0.1 to represent the different increasing slopes of composite ETS by additional toxicities. 7.3.1 Results Using Data of ADVL0311. The sensitivity analysis results using the real data of study ADVL0311 are summarized in the following 5 tables. In each pseudo-trial, the estimated PANETS of each dose level changes with different values of α and β. The bigger the value of β, the higher the estimated PANETS of each dose level. The TNETS is 0.476 in all pseudo- trials. In all pseudo-trials, the final recommended MTD is always the dose level 7 under all the combinations of α and different values of β. Table 7.7: A Pseudo-trial with EID Using the Patients of ADVL0311 with Parameter α=-2 and β=2 PANETS at Dose Level Dose Level Next Dose 1 2 3 4 5 6 7 8 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 1 2 0.34 0.44 0.44 0.44 0.44 0.44 0.44 0.44 2 3 0.34 0.44 0.49 0.49 0.49 0.49 0.49 0.49 3 4 0.34 0.44 0.49 0.54 0.54 0.54 0.54 0.54 4 5 0.34 0.44 0.45 0.45 0.45 0.45 0.45 0.45 5 6 0.34 0.44 0.44 0.44 0.44 0.44 0.44 0.44 6 7 0.34 0.44 0.44 0.44 0.44 0.44 0.50 0.50 7 8 0.34 0.44 0.44 0.44 0.44 0.44 0.50 0.71 8 7 0.34 0.44 0.44 0.44 0.44 0.44 0.52 0.71 7 7* * Recommend MTD for TNETS as of 0.476. 125 Table 7.8: A Pseudo-trial with EID Using the Patients of ADVL0311 with Parameter α=-2 and β=1. PANETS at Dose Level Dose Level Next Dose 1 2 3 4 5 6 7 8 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 1 2 0.30 0.42 0.42 0.42 0.42 0.42 0.42 0.42 2 3 0.30 0.42 0.44 0.44 0.44 0.44 0.44 0.44 3 4 0.30 0.42 0.44 0.50 0.50 0.50 0.50 0.50 4 5 0.30 0.41 0.41 0.41 0.41 0.41 0.41 0.41 5 6 0.30 0.41 0.41 0.41 0.41 0.41 0.41 0.41 6 7 0.30 0.41 0.41 0.41 0.41 0.41 0.48 0.48 7 8 0.30 0.41 0.41 0.41 0.41 0.41 0.48 0.68 8 7 0.30 0.41 0.41 0.41 0.41 0.41 0.50 0.68 7 7* * Recommend MTD for TNETS as of 0.476. Table 7.9: A Pseudo-trial with EID Using the Patients of ADVL0311 with Parameter α=-2 and β=0.5 PANETS at Dose Level Dose Level Next Dose 1 2 3 4 5 6 7 8 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 1 2 0.26 0.40 0.40 0.40 0.40 0.40 0.40 0.40 2 3 0.26 0.39 0.39 0.39 0.39 0.39 0.39 0.39 3 4 0.26 0.39 0.39 0.46 0.46 0.46 0.46 0.46 4 5 0.26 0.38 0.38 0.38 0.38 0.38 0.38 0.38 5 6 0.26 0.37 0.37 0.37 0.37 0.37 0.37 0.37 6 7 0.26 0.37 0.37 0.37 0.37 0.37 0.43 0.43 7 8 0.26 0.37 0.37 0.37 0.37 0.37 0.43 0.65 8 7 0.26 0.37 0.37 0.37 0.37 0.37 0.46 0.65 7 7* * Recommend MTD for TNETS as of 0.476. 126 Table 7.10: A Pseudo-trial with EID Using the Patients of ADVL0311 with Parameter α=-2 and β=0.25 PANETS at Dose Level Dose Level Next Dose 1 2 3 4 5 6 7 8 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 1 2 0.25 0.35 0.35 0.35 0.35 0.35 0.35 0.35 2 3 0.25 0.35 0.37 0.37 0.37 0.37 0.37 0.37 3 4 0.25 0.35 0.37 0.43 0.43 0.43 0.43 0.43 4 5 0.25 0.35 0.35 0.35 0.35 0.35 0.35 0.35 5 6 0.25 0.35 0.35 0.35 0.35 0.35 0.35 0.35 6 7 0.25 0.35 0.35 0.35 0.35 0.35 0.39 0.39 7 8 0.25 0.35 0.35 0.35 0.35 0.35 0.39 0.62 8 7 0.25 0.35 0.35 0.35 0.35 0.35 0.43 0.62 7 7* * Recommend MTD for TNETS as of 0.476. Table 7.11: A Pseudo-trial with EID Using the Patients of ADVL0311 with Parameter α=-2 and β=0.1 PANETS at Dose Level Dose Level Next Dose 1 2 3 4 5 6 7 8 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 1 2 0.25 0.31 0.31 0.31 0.31 0.31 0.31 0.31 2 3 0.25 0.31 0.36 0.36 0.36 0.36 0.36 0.36 3 4 0.25 0.31 0.36 0.42 0.42 0.42 0.42 0.42 4 5 0.25 0.31 0.34 0.34 0.34 0.34 0.34 0.34 5 6 0.25 0.31 0.34 0.34 0.34 0.35 0.35 0.35 6 7 0.25 0.31 0.34 0.34 0.34 0.35 0.36 0.36 7 8 0.25 0.31 0.34 0.34 0.34 0.35 0.36 0.59 8 7 0.25 0.31 0.34 0.34 0.34 0.35 0.40 0.59 7 7* * Recommend MTD for TNETS as of 0.476. 127 7.3.2 Results Using Data of A09712. The sensitivity analysis results using the real data of study A09712 are summarized in the following 5 tables. The TNETS is 0.476 in all 5 pseudo-trials. In each pseudo-trial, the estimated PANETS of each dose level changes with different values of α and β. The bigger the value of β, the higher the estimated PANETS of each dose level. But in all pseudo-trials, the final recommended MTD is always the dose level 8 under all combinations of α and different values of β. Table 7.12: A Pseudo-trial with EID Using the Patients of A09712 with Parameter α=-2 and β=2. PANETS at Dose Level Dose Level 1 2 3 4 5 6 7 8 9 Current Recomme nded 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 1 2 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 2 3 0.11 0.11 0.19 0.19 0.19 0.19 0.19 0.19 0.19 3 4 0.11 0.11 0.19 0.28 0.28 0.28 0.28 0.28 0.28 4 5 0.11 0.11 0.19 0.20 0.20 0.20 0.20 0.20 0.20 5 6 0.11 0.11 0.19 0.20 0.20 0.27 0.27 0.27 0.27 6 7 0.11 0.11 0.19 0.20 0.20 0.25 0.25 0.25 0.25 7 8 0.11 0.11 0.19 0.20 0.20 0.25 0.25 0.27 0.27 8 9 0.11 0.11 0.19 0.20 0.20 0.25 0.25 0.27 0.77 9 8 0.11 0.11 0.19 0.20 0.20 0.25 0.25 0.49 0.77 8 8* *: Recommend MTD for TNETS as of 0.476. 128 Table 7.13: A Pseudo-trial with EID Using the Patients of A09712 with Parameter α=-2 and β=1 PANETS at Dose Level Dose Level 1 2 3 4 5 6 7 8 9 Current Recomme nded 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 1 2 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 2 3 0.09 0.09 0.18 0.18 0.18 0.18 0.18 0.18 0.18 3 4 0.09 0.09 0.18 0.27 0.27 0.27 0.27 0.27 0.27 4 5 0.09 0.09 0.18 0.18 0.18 0.18 0.18 0.18 0.18 5 6 0.09 0.09 0.18 0.18 0.18 0.23 0.23 0.23 0.23 6 7 0.09 0.09 0.18 0.18 0.18 0.22 0.22 0.22 0.22 7 8 0.09 0.09 0.18 0.18 0.18 0.22 0.22 0.22 0.22 8 9 0.09 0.09 0.18 0.18 0.18 0.22 0.22 0.22 0.75 9 8 0.09 0.09 0.18 0.18 0.18 0.22 0.22 0.44 0.75 8 8* * Recommend MTD for TNETS as of 0.476. Table 7.14: A Pseudo-trial with EID Using the Patients of A09712 with Parameter α=-2 and β=0.5 PANETS at Dose Level Dose Level 1 2 3 4 5 6 7 8 9 Current Recomme nded 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 1 2 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 2 3 0.08 0.08 0.16 0.16 0.16 0.16 0.16 0.16 0.16 3 4 0.08 0.08 0.16 0.25 0.25 0.25 0.25 0.25 0.25 4 5 0.08 0.08 0.16 0.17 0.17 0.17 0.17 0.17 0.17 5 6 0.08 0.08 0.16 0.17 0.17 0.20 0.20 0.20 0.20 6 7 0.08 0.08 0.16 0.17 0.17 0.20 0.20 0.20 0.20 7 8 0.08 0.08 0.16 0.17 0.17 0.20 0.20 0.20 0.20 8 9 0.08 0.08 0.16 0.17 0.17 0.20 0.20 0.20 0.71 9 8 0.08 0.08 0.16 0.17 0.17 0.20 0.20 0.41 0.71 8 8* * Recommend MTD for TNETS as of 0.476. 129 Table 7.15: A Pseudo-trial with EID Using the Patients of A09712 with Parameter α=-2 and β=0.25 PANETS at Dose Level Dose Level 1 2 3 4 5 6 7 8 9 Current Recomme nded 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 1 2 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 2 3 0.07 0.07 0.14 0.14 0.14 0.14 0.14 0.14 0.14 3 4 0.07 0.07 0.14 0.24 0.24 0.24 0.24 0.24 0.24 4 5 0.07 0.07 0.14 0.16 0.16 0.16 0.16 0.16 0.16 5 6 0.07 0.07 0.14 0.16 0.16 0.19 0.19 0.19 0.19 6 7 0.07 0.07 0.14 0.16 0.16 0.19 0.19 0.19 0.19 7 8 0.07 0.07 0.14 0.16 0.16 0.19 0.19 0.19 0.19 8 9 0.07 0.07 0.14 0.16 0.16 0.19 0.19 0.19 0.70 9 8 0.07 0.07 0.14 0.16 0.16 0.19 0.19 0.40 0.70 8 8* * Recommend MTD for TNETS as of 0.476. Table 7.16: A Pseudo-trial with EID Using the Patients of A09712 with Parameter α=-2 and β=0.1 PANETS at Dose Level Dose Level 1 2 3 4 5 6 7 8 9 Current Recom- mended 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 1 2 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 2 3 0.07 0.07 0.13 0.13 0.13 0.13 0.13 0.13 0.13 3 4 0.07 0.07 0.13 0.24 0.24 0.24 0.24 0.24 0.24 4 5 0.07 0.07 0.13 0.16 0.16 0.16 0.16 0.16 0.16 5 6 0.07 0.07 0.13 0.16 0.16 0.19 0.19 0.19 0.19 6 7 0.07 0.07 0.13 0.16 0.16 0.19 0.19 0.19 0.19 7 8 0.07 0.07 0.13 0.16 0.16 0.19 0.19 0.19 0.19 8 9 0.07 0.07 0.13 0.16 0.16 0.19 0.19 0.19 0.69 9 8 0.07 0.07 0.13 0.16 0.16 0.19 0.19 0.39 0.69 8 8* * Recommend MTD for TNETS as of 0.476. 130 The results of pseudo-trials with the real data from ADVL0311 and A09712 demonstrate that performance of EID is very robust. There are two reasons for the robustness of EID. The first is that the integer part of final ETS depends on the most severe toxicity with the maximum adjusted grade of each patient and values of α and β affect the increasing “speed” of decimal part of final ETS contributed by additional toxicities. The second is that an IR is employed in the EID so that dose allocation and MTD estimation depend on the relative order of PANETS of each dose level and not sensitive to its actual value. The chosen value of α and β will affect equally all dose levels in a trial so that they will not change substantially the relative order of toxicity risks (PANETS) of dose levels, especially when the number of patients treated at same dose level is small and percentage of patients with multiple toxicities is minimal. In addition, the gaps of toxicity risks (PANETS) between two consecutive dose levels are pretty large to tolerate the effects by different values of β. 7.4 Summary of Application of EID The application of our EID to the two real COG Phase I trials has demonstrated that our EID can avoid underestimating or overestimating MTD by treating toxicity response as quasi-continuous variable and measuring the toxicity severity quantitatively. Our EID is also found to have more power to estimate MTD with less sample size by fully utilizing all toxicity response information. Our EID can increase therapeutic effect of a trial by reducing the chance of treating unnecessary patients at under-dosed dose level and protect patients from being exposed to over toxic dose level by estimating the 131 toxicity risk (PANETS) of each dose level more accurately with fewer treated patients in the dose allocation procedure. Sensitivity analyses further show that our EID is very objective and robust. 132 Chapter VIII: Pseudo-Trails with Extended Isotonic Design Using Real Data with Bootstrap Method In last chapter, the application of EID is demonstrated by pseudo-trials using the actual patients according to their entry orders and received dose levels. As the sample sizes of the two real trials are small and patients are sampled without replacement according to their treated dose levels and entry orders so that these pseudo-trials can’t estimate the comprehensive performance in general real situation with infinite sample sizes and random entry order. Therefore, in this chapter, we further conduct pseudo-trials of EID by randomly sampling patients with bootstrap method (sampling with replacement) from patients of ADVL0311 and A09712, respectively. Bootstrap method is used to randomly sample a patient at a time from the patient pool at the right dose level. Three (3) patients per cohort are used. The toxicities and treated dose levels of sampled patients are used in the pseudo-trial with EID, but their entry orders in the original trial are ignored. If more than three patients were enrolled at a dose level, as was done in A09712 when three patients were enrolled during escalation and three during de-escalation, all six were always available during the bootstrapping. The bootstrap method is a sampling method with replacement so that sample spaces of any dose levels will not run out even though there are only 3 patients treated at some dose levels and more than 3 patients are needed to be sampled from those dose levels. Same patient may be sampled several times and some 133 patients may never be sampled in a pseudo-trial. No new dose levels can be added ad hoc during a pseudo-trial. The recommended new dose level for next coming cohort is still the highest existed dose level when the sampling performed at the highest existed dose level indicates that further escalation is warranted. On the other hand, the recommended new dose level for next coming cohort is still the lowest existed dose level when the sampling performed at the lowest existed dose level indicates that further de-escalation is warranted. Actually, the probabilities of these two cases are very small. For example, in all pseudo-trials with ADVL0311 or A09712, the probabilities are 0 for the case that MTD can’t be estimated because the dose-seeking process called for a dose to be investigated that was below the doses available from the actual trials data. The probabilities are 0 for the case that MTD can’t be estimated because the ANETSs of all existed dose levels are smaller than the TNETS (0.476) in all pseudo-trials with A09712. But in the pseudo-trials with ADVL0311, the probabilities that the escalation process could not determine an MTD because an escalation to a dose higher than any available from the database were indicated 0.1675%, 0.0425%, 0.0125%, 0%, and 0% with β=0.01, β=0.25, β=0.5, β=1, and β=2, respectively.. We also conduct different pseudo-trials with patients’ different NETSs estimated with different values of α and β to investigate the impacts on MTD estimation by the chosen values of α and β. The TNETS is 0.476 in all pseudo-trials with EID and bootstrap method is used in this chapter. A total of 40,000 simulations are conducted in each situation. The comparison outcomes are the percentage of each dose level chosen as MTD, average sample size, and average cohort number. 134 8.1 Simulations of EID Using Data of ADVL0311 with Bootstrap Method. The ETS and NETS of patients in ADVL0311 are calculated according to their toxicities and rules described in chapter V. Five (5) different values of β and 1 fixed value of α are used to estimate 5 sets of ETS and NETS of all patients and summarized in the Table 8.1. The estimated ETS and NETS increase with the increasing value of β (Table 8.1). But the influence of value of β on the final value of NETS decreases when the most severe toxicity of individual patient becomes more severe and NETS becomes bigger (Table 8.1). We conduct pseudo-trials with EID by randomly sampling patients of ADVL0311 and using their NETS in the Table 8.1. The simulation pseudo-trial results are summarized in Table 8.2. 135 Table 8.1: Dose levels, Toxicities, ETS, and NETS with Different Values of α and β of Patients in ADLV0311 Dose Level Patient ID Toxicities α = -2 β = 2 α = -2 β = 1 α = -2 β = 0.5 α = -2 β = 0.25 α = -2 β = 0.1 ETS NETS ETS NETS ETS NETS ETS NETS ETS NETS 1 714061 1 grade 3 non DLT toxicity 3 grade 1 toxicity 2.500 0.417 2.269 0.378 2.182 0.364 2.148 0.358 2.130 0.355 1 739198 4 grade 1 toxicity 0.982 0.164 0.731 0.122 0.378 0.063 0.223 0.037 0.154 0.026 1 724155 1 grade 3 non DLT toxicity 1 grade 2 toxicity 2 grade 1 toxicity 2.661 0.443 2.339 0.390 2.209 0.368 2.159 0.360 2.134 0.356 2 740865 9 grade 1 toxicity 1.000 0.167 0.998 0.166 0.881 0.147 0.500 0.083 0.231 0.039 2 717893 2 grade 4 non DLT toxicity 1 grade 3 non DLT toxicity 3 grade 1 toxicity 3.953 0.659 3.622 0.604 3.321 0.553 3.202 0.534 3.148 0.525 2 741409 1 grade 3 non DLT toxicity 8 grade 2 toxicity 10 grade 1 toxicity 3.000 0.500 2.999 0.500 2.912 0.485 2.542 0.424 2.244 0.374 3 742323 1 grade 3 non DLT toxicity 3 grade 2 toxicity 4 grade 1 toxicity 2.991 0.498 2.791 0.465 2.417 0.403 2.237 0.373 2.159 0.360 3 742544 1 grade 2 toxicity 4 grade 1 toxicity 1.881 0.313 1.500 0.250 1.269 0.211 1.182 0.197 1.142 0.190 3 712496 1 grade 4 non DLT toxicity 1 grade 3 non DLT toxicity 1 grade 2 toxicity 4 grade 1 toxicity 3.924 0.654 3.562 0.594 3.294 0.549 3.192 0.532 3.145 0.524 (Table continued on next page) 136 Table 8.1 (cont.) Dose Level Patient ID Toxicities α = -2 β = 2 α = -2 β = 1 α = -2 β = 0.5 α = -2 β = 0.25 α = -2 β = 0.1 ETS NETS ETS NETS ETS NETS ETS NETS ETS NETS 4 708773 1 grade 4 non DLT toxicity 5 grade 2 toxicity 4.000 0.667 3.993 0.666 3.818 0.636 3.438 0.573 3.214 0.536 4 744553 1 grade 4 non DLT toxicity 2 grade 3 non DLT toxicity 1 grade 2 toxicity 3.881 0.647 3.500 0.583 3.269 0.545 3.182 0.530 3.142 0.524 4 728327 1 grade 2 toxicity 4 grade 1 toxicity 1.881 0.313 1.500 0.250 1.269 0.211 1.182 0.197 1.142 0.190 5 745379 No toxicity 0 0 0 0 0 0 0 0 0 0 5 746285 2 grade 4 non DLT toxicity 1 grade 2 toxicity 6 grade 1 toxicity 3.982 0.664 3.731 0.622 3.378 0.563 3.223 0.537 3.154 0.526 5 741265 2 grade 2 toxicity 1.500 0.250 1.269 0.211 1.182 0.197 1.148 0.191 1.130 0.188 6 600812 1 grade 4 DLT toxicity 1 grade 3 non DLT toxicity 1 grade 2 toxicity 2 grade 1 toxicity 4.690 0.782 4.354 0.726 4.214 0.702 4.161 0.694 4.135 0.689 6 703789 1 grade 3 non DLT toxicity 2 grade 2 toxicity 5 grade 1 toxicity 2.982 0.497 2.731 0.455 2.378 0.396 2.223 0.370 2.154 0.359 6 748842 No toxicity 0 0 0 0 0 0 0 0 0 0 6 718998 2 grade 3 non DLT toxicity 3 grade 1 toxicity 2.881 0.480 2.500 0.417 2.269 0.378 2.182 0.364 2.142 0.357 6 748678 3 grade 1 toxicity 0.881 0.147 0.500 0.083 0.269 0.045 0.182 0.030 0.142 0.024 6 749850 1 grade 3 non DLT toxicity 4 grade 2 toxicity 16 grade 1 toxicity 3.000 0.500 2.998 0.500 2.881 0.480 2.500 0.417 2.231 0.372 (Table continued on next page) 137 Table 8.1 (cont.) Dose Level Patient ID Toxicities α = -2 β = 2 α = -2 β = 1 α = -2 β = 0.5 α = -2 β = 0.25 α = -2 β = 0.1 ETS NETS ETS NETS ETS NETS ETS NETS ETS NETS 7 750716 1 grade 3 non DLT toxicity 2 grade 2 toxicity 8 grade 1 toxicity 2.998 0.500 2.881 0.480 2.500 0.417 2.269 0.378 2.168 0.361 7 735053 2 grade 2 toxicity 4 grade 1 toxicity 1.982 0.330 1.731 0.289 1.378 0.230 1.223 0.204 1.154 0.192 7 741575 1 grade 4 non DLT toxicity 2 grade 3 non DLT toxicity 6 grade 2 toxicity 15 grade 1 toxicity 4.000 0.667 3.998 0.666 3.893 0.649 3.516 0.586 3.236 0.539 8 724127 1 grade 4 non DLT toxicity 1 grade 2 toxicity 5 grade 1 toxicity 3.818 0.636 3.438 0.573 3.245 0.541 3.173 0.529 3.139 0.523 8 69243 1 grade 4 DLT toxicity 5 grade 4 non DLT toxicity 9 grade 3 non DLT toxicity 7 grade 2 toxicity 4 grade 1 toxicity 6.000 1.000 6.000 1.000 5.968 0.995 5.670 0.945 5.286 0.881 8 739883 3 grade 3 non DLT toxicity 1 grade 2 toxicity 2 grade 1 toxicity 2.991 0.498 2.791 0.465 2.417 0.403 2.237 0.373 2.159 0.360 8 749014 1 grade 4 DLT toxicity 3 grade 4 non DLT toxicity 12 grade 3 non DLT toxicity 11 grade 2 toxicity 14 grade 1 toxicity 6.000 1.000 6.000 1.000 5.993 0.999 5.818 0.970 5.354 0.892 (Table continued on next page) 138 Table 8.1 (cont.) Dose Level Patient ID Toxicities α = -2 β = 2 α = -2 β = 1 α = -2 β = 0.5 α = -2 β = 0.25 α = -2 β = 0.1 ETS NETS ETS NETS ETS NETS ETS NETS ETS NETS 7 751395 2 grade 2 toxicity 2 grade 1 toxicity 1.881 0.313 1.500 0.250 1.269 0.211 1.182 0.197 1.142 0.190 7 737478 8 grade 4 non DLT toxicity 8 grade 3 non DLT toxicity 13 grade 2 toxicity 5 grade 1 toxicity 4.000 0.667 4.000 0.667 4.000 0.667 3.960 0.660 3.519 0.586 7 754963 3 grade 4 non DLT toxicity 1 grade 3 non DLT toxicity 2 grade 2 toxicity 5 grade 1 toxicity 4.000 0.667 3.953 0.659 3.622 0.604 3.321 0.553 3.182 0.530 (Table continued on next page) 139 Table 8.2: Simulation Results with Data of ADVL0311 Dose Level Treatment Probability (%) the dose chosen as MTD α=-2 β=0.1 α=-2 β=0.25 α=-2 β=0.5 α=-2 β=1 α=-2 β=2 1 400 mg/m 2 over 10 minutes 0 0 0 0.13 4.89 2 520 mg/m 2 over 10 minutes 0.04 0.34 3.35 15.15 33.06 3 670 mg/m 2 over 10 minutes 0.41 3.11 18.3 37.84 47.69 4 870 mg/m 2 over 10 minutes 3.77 8.29 9.79 8.77 2.85 5 1130 mg/m 2 over 10 minutes 0.35 0.46 0.95 1.35 1.02 6 1470 mg/m 2 over 10 minutes 0.33 3.40 8.27 10.2 4.53 7** 1910 mg/m 2 over 10 minutes 89.61 82.58 59.00 26.52 5.97 8 2480 mg/m 2 over 10 minutes 5.50 (0.1675*) 1.83 (0.0425*) 0.35 (0.0125*) 0.05 0 Average Sample Size 39.8 (6.7) 39.4 (8.2) 36.2 (9.7) 32.1 (11.02) 26.7 (9.2) Average Number of Cohorts 13.3 (2.2) 13.1 (2.7) 12.1 (3.2) 10.7 (3.7) 8.9 (3.1) Original Sample Size 31 Original Cohort Number 11 *: All dose levels with ANETSs less than the TNETS (0.476). **: The MTD recommended by original trial with standard 3+3 design with dose de-escalation _: The most frequently recommended MTD by pseudo-trials with EID. The original ADVL0311 trial with standard 3+3 design with dose de-escalation recommended the dose level 7 as MTD. In the pseudo trials with EID, the dose level 7 is most frequently recommended as MTD when β is 0.1, 0.25, or 0.5 and α is -2. But when β increases to 1 or 2 and α is still -2, the dose level 3 becomes the most frequently recommended dose level as MTD. Among the 3 patients treated at the dose level 3, 1 patient (712496) has a lot of toxicities (1 grade 4 non DLT toxicity, 1 grade 3 non DLT 140 toxicity, 1 grade 2 toxicity, and 4 grade 1 toxicity), especially 1 grade 4 non DLT toxicity with the maximum adjusted grade as of 4 so that his ETS is greater than 3 and his NETS is more than 0.5, above the TNETS (0.476). Another patient (742323) has 1 grade 3 non DLT toxicity, 3 grade 2 toxicity, and 4 grade 1 toxicity so that his NETS is estimated to be more than the TNETS (0.476) when β increases to 1 or 2 and α is still -2. With bootstrap method, there are at least 3 2 3 2 3 2 × × chance that all 3 sample patients are only from these 2 patients. When all 3 sample patients are only from these 2 patients, the PANETS of the dose level 3 will be more than the TNTS (0.476) and dose escalation stops before it reaches the true MTD, the dose level 7, so that dose level 3 becomes the most frequently recommended MTD. These results suggest that too big value of β may lead to underestimate MTD and is not appropriate to use. The sample size and number of cohorts of the pseudo-trials with EID increase with the decreasing β values used because the sample size and number of cohorts usually increase with “distance” (in scale of dose levels) between the starting lowest dose level and final recommended MTD. The original trial with standard 3+3 design with dose de- escalation uses 31 eligible patients and 11 cohorts. The average sample size and number of cohorts of the pseudo-trials with EID are marginally bigger than that of the original trial except the case in which β is 2 and α is -2. This can be explained by the difference in the stop rule between 2 designs. EID stops if there have been 3 consecutive cohorts treated at the same dose level while standard 3+3 design with dose de-escalation stops with no more than 2 cohorts (6 patients) treated at any dose levels. 141 The probabilities are 0 for the case that MTD can’t be estimated because the ANETSs of all existed dose levels are bigger than the TNETS (0.476). But the probabilities that MTD can’t be estimated because the ANETSs of all existed dose levels are smaller than the TNETS (0.476) are 0.1675%, 0.0425%, 0.0125%, 0%, and 0% with β=0.01, β=0.25, β=0.5, β=1, and β=2, respectively. The percentage of dose level 7 being chosen as MTD decreases from 90% to 59% when β increases from 0.1 to 0.5. This suggests that the value of β can substantially affect the performance of EID and too big value of β increases the chance of stopping the trial before dose level escalates to the dose level with the smallest absolute value of difference between its ANETS and the TNETS (0.476). The pseudo trials with data of ADVL0311 suggest EID with β as 0.1, 0.25, or 0.5 and α as -2 can successfully estimate correct MTD with no significant additional cost of sample sizes and number of cohorts in general. 8.2 Simulations of EID Using Data of A09712 with Bootstrap Method. The ETS and NETS of patients in A09712 are calculated according to their toxicities and rules described in chapter V. Five (5) different values of β and 1 fixed value of α are used to estimate 5 sets of ETS and NETS of all patients and summarized in the Table 8.3. The estimated ETS and NETS increase with the increasing value of β (Table 8.3). But the influence of value of β on the final value of NETS decreases when the most severe toxicity of individual patient becomes more severe and NETS becomes bigger (Table 8.3). 142 Table 8.3: Summary of dose level, toxicity, ETS, and NETS with different values of α and β for patients in A09712 Dose Level Patient ID Toxicities α = -2 β = 2 α = -2 β = 1 α = -2 β = 0.5 α = -2 β = 0.25 α = -2 β = 0.1 ETS NETS ETS NETS ETS NETS ETS NETS ETS NETS 1 74823 2 grade 2 toxicity 3 grade 1 toxicity 1.953 0.325 1.622 0.270 1.321 0.220 1.202 0.200 1.148 0.191 1 79631 No toxicity 0 0 0 0 0 0 0 0 0 0 1 81562 1 grade 2 toxicity 2 grade 1 toxicity 1.500 0.250 1.269 0.211 1.182 0.197 1.148 0.191 1.130 0.188 1 83068 1 grade 2 toxicity 4 grade 1 toxicity 1.881 0.313 1.500 0.250 1.269 0.211 1.182 0.197 1.142 0.190 2 86441 1 grade 1 toxicity 0.100 0.017 0.100 0.017 0.100 0.017 0.100 0.017 0.100 0.017 2 86625 No toxicity 0 0 0 0 0 0 0 0 0 0 2 86861 2 grade 1 toxicity 0.500 0.083 0.269 0.045 0.182 0.030 0.148 0.025 0.130 0.022 2 87167 2 grade 1 toxicity 0.500 0.083 0.269 0.045 0.182 0.030 0.148 0.025 0.130 0.022 3 600668 3 grade 2 toxicity 5 grade 1 toxicity 1.999 0.333 1.924 0.321 1.562 0.260 1.294 0.216 1.175 0.196 3 601169 1 grade 2 toxicity 1 grade 1 toxicity 1.269 0.211 1.182 0.197 1.148 0.191 1.133 0.189 1.125 0.187 3 601307 1 grade 1 toxicity 0.100 0.017 0.100 0.017 0.100 0.017 0.100 0.017 0.100 0.017 3 700073 No toxicity 0 0 0 0 0 0 0 0 0 0 4 702795 No toxicity 0 0 0 0 0 0 0 0 0 0 4 703732 1 grade 1 toxicity 0.100 0.017 0.100 0.017 0.100 0.017 0.100 0.017 0.100 0.017 4 705476 1 grade 3 DLT toxicity 2 grade 3 non DLT toxicity 5 grade 2 toxicity 1 grade 1 toxicity 4.992 0.832 4.802 0.800 4.426 0.738 4.240 0.707 4.160 0.693 4 709539 1 grade 2 toxicity 2 grade 1 toxicity 1.500 0.250 1.269 0.211 1.182 0.197 1.148 0.191 1.130 0.188 (Table continued on next page) 143 Table 8.3 (cont.) (Table continued on next page) Dose Level Patient ID Toxicities α = -2 β = 2 α = -2 β = 1 α = -2 β = 0.5 α = -2 β = 0.25 α = -2 β = 0.1 ETS NETS ETS NETS ETS NETS ETS NETS ETS NETS 4 711039 4 grade 1 toxicity 0.982 0.164 0.731 0.122 0.378 0.063 0.223 0.037 0.154 0.026 4 711051 1 grade 2 toxicity 3 grade 1 toxicity 1.731 0.289 1.378 0.230 1.223 0.204 1.165 0.194 1.136 0.189 5 714200 1 grade 2 toxicity 1 grade 1 toxicity 1.269 0.211 1.182 0.197 1.148 0.191 1.133 0.189 1.125 0.187 5 714201 2 grade 1 toxicity 0.500 0.083 0.269 0.045 0.182 0.030 0.148 0.025 0.130 0.022 5 714337 2 grade 1 toxicity 0.500 0.083 0.269 0.045 0.182 0.030 0.148 0.025 0.130 0.022 5 714655 No toxicity 0 0 0 0 0 0 0 0 0 0 6 717897 1 grade 2 toxicity 1 grade 1 toxicity 1.269 0.211 1.182 0.197 1.148 0.191 1.133 0.189 1.125 0.187 6 717942 2 grade 2 toxicity 1 grade 1 toxicity 1.731 0.289 1.378 0.230 1.223 0.204 1.165 0.194 1.136 0.189 6 717959 2 grade 2 toxicity 2 grade 1 toxicity 1.881 0.313 1.500 0.250 1.269 0.211 1.182 0.197 1.142 0.190 7 725184 1 grade 2 toxicity 1.000 0.167 1.000 0.167 1.000 0.167 1.000 0.167 1.000 0.167 7 725313 1 grade 2 toxicity 1 grade 1 toxicity 1.269 0.211 1.182 0.197 1.148 0.191 1.133 0.189 1.125 0.187 7 728784 2 grade 2 toxicity 2 grade 1 toxicity 1.881 0.313 1.500 0.250 1.269 0.211 1.182 0.197 1.142 0.190 8 732123 2 grade 2 toxicity 1 grade 1 toxicity 1.500 0.250 1.269 0.211 1.182 0.197 1.148 0.191 1.130 0.188 8 732190 2 grade 2 toxicity 1 grade 1 toxicity 1.500 0.250 1.269 0.211 1.182 0.197 1.148 0.191 1.130 0.188 8 732228 2 grade 2 toxicity 2 grade 1 toxicity 1.881 0.313 1.500 0.250 1.269 0.211 1.182 0.197 1.142 0.190 144 Table 8.3 (cont.) Dose Level Patient ID Toxicities α = -2 β = 2 α = -2 β = 1 α = -2 β = 0.5 α = -2 β = 0.25 α = -2 β = 0.1 ETS NETS ETS NETS ETS NETS ETS NETS ETS NETS 9 734789 1 grade 3 DLT toxicity 2 grade 3 non DLT toxicity 4 grade 2 toxicity 2 grade 1 toxicity 4.988 0.831 4.769 0.795 4.401 0.734 4.231 0.705 4.157 0.693 9 735536 1 grade 3 DLT toxicity 1 grade 2 toxicity 1 grade 1 toxicity 4.310 0.718 4.198 0.700 4.154 0.692 4.136 0.689 4.126 0.688 8 736059 1 grade 3 DLT toxicity 1 grade 3 non DLT toxicity 2 grade 2 toxicity 3 grade 1 toxicity 4.881 0.813 4.500 0.750 4.269 0.711 4.182 0.697 4.142 0.690 8 736071 1 grade 3 DLT toxicity 1 grade 3 non DLT toxicity 4 grade 2 toxicity 5 grade 1 toxicity 4.988 0.831 4.769 0.795 4.401 0.734 4.231 0.705 4.157 0.693 7 739016 1 grade 3 DLT toxicity 1 grade 2 toxicity 2 grade 1 toxicity 4.401 0.734 4.231 0.705 4.168 0.695 4.142 0.690 4.128 0.688 7 739776 1 grade 3 DLT toxicity 1 grade 3 non DLT toxicity 2 grade 2 toxicity 1 grade 1 toxicity 4.769 0.795 4.401 0.734 4.231 0.705 4.168 0.695 4.137 0.690 7 741007 2 grade 2 toxicity 6 grade 1 toxicity 1.998 0.333 1.881 0.313 1.500 0.250 1.269 0.211 1.168 0.195 (Table continued on next page) 145 Table 8.3 (cont.) Dose Level Patient ID Toxicities α = -2 β = 2 α = -2 β = 1 α = -2 β = 0.5 α = -2 β = 0.25 α = -2 β = 0.1 ETS NETS ETS NETS ETS NETS ETS NETS ETS NETS 6 743243 1 grade 3 DLT toxicity 2 grade 3 non DLT toxicity 2 grade 2 toxicity 2 grade 1 toxicity 4.943 0.824 4.599 0.766 4.310 0.718 4.198 0.700 4.147 0.691 6 743979 2 grade 2 toxicity 2 grade 1 toxicity 1.881 0.313 1.500 0.250 1.269 0.211 1.182 0.197 1.142 0.190 6* 746027 1 grade 3 non DLT toxicity 3 grade 2 toxicity 6 grade 1 toxicity 2.998 0.500 2.881 0.480 2.500 0.417 2.269 0.378 2.168 0.361 *Recommended MTD by original standard 3+3 design with dose de-escalation. 146 We conduct pseudo-trials with EID by randomly sampling patients of A09712 and using their NETS in the Table 8.3. The simulation pseudo-trial results are summarized in Table 8.4. Table 8.4: Simulation Results with Data of A09712. Dose Level Treatment Probability (%) the dose chosen as MTD α=-2 β=0.1 α=-2 β=0.25 α=-2 β=0.5 α=-2 β=1 α=-2 β=2 1 1.7mg/kg/dose M- F*3w 0 0 0 0 0 2 1.7mg/kg/dose MWF*6w 0 0 0 0 0 3 1.7mg/kg/dose M- F*6w 0 0 0 0.05 0.27 4 1.9mg/kg/dose M- F*6w 0.01 0.02 0.02 0.07 0.19 5 3.4mg/kg/dose M- F*6w 0 0 0 0.02 0.23 6* 4.4mg/kg/dose M- F*6w 1.62 1.97 2.91 6.99 20.19 7 5.5mg/kg/dose M- F*6w 7.33 8.02 11.05 22.96 34.54 8** 7.1mg/kg/dose M- F*6w 83.50 83.74 82.97 69.89 44.58 9 9.2mg/kg/dose M- F*6w 7.54 6.26 3.05 0.03 0.01 Average Sample Size 41.0 (4.5) 41.1 (4.7) 41.1 (5.0) 41.1 (5.9) 40.0 (7.1) Average Number of Cohorts 13.7 (1.5) 13.7 (1.6) 13.7 (1.7) 13.8 (2.0) 13.3 (2.4) Original Sample Size 40 Original Cohort Number 13 *: The MTD recommended by original trial with standard 3+3 design with dose de-escalation. **: The most frequently recommended MTD by pseudo-trials with EID. From the Table 8.4, the original A09712 trial recommended the dose level 6 as MTD, but the pseudo-trials most frequently recommend the dose level 8 as MTD among all cases with 5 different sets of α and β. The toxicity profile of the patients in A09712 is 147 an extremely under toxic scenario with no grade 4 non DLT toxicities and no grade 4 DLTs. The dose level 6 recommended in the original A09712 is a substantially underestimated MTD with far less toxic effect. The pseudo-trials with EID identify the dose level 8 as MTD according to the exact toxicity profile of patients and the TNEST as of 0.476. These results demonstrated that the use of EID can adjust for the deviation of toxicity profile and always estimate a correct MTD associated with the target toxicity level to ensure therapeutic effect of chemical agent. The probabilities are 0 for the case that MTD can’t be estimated because the ANETSs of all existed dose levels are all either bigger or smaller than the TNETS (0.476). The frequency of correctly estimating dose level 8 as MTD decreases from 83.5% to 44.58% with the increasing values of β, also suggesting that the value of β can substantially affect the performance of EID. But the decreases are very small, less than 1%, when value of β increases from 0.1 to 0.5. The average sample sizes (about 41 patients) and number of cohorts (about 13.7 cohorts) of the pseudo-trials with EID are not only pretty stable regardless the different β values used, but also similar to those (40 patients and 13 cohorts) of the original trial with standard 3+3 design with dose de- escalation. 8.3 Summary of Application of EID with Bootstrap Method The application of our EID with bootstrap method to the two real COG Phase I trials has also demonstrated that our EID can generally estimate an correct MTD under any toxicity profiles, especially deviated profiles, by treating toxicity response as quasi- 148 continuous variable and measuring the toxicity severity quantitatively and estimating MTD according to exact toxicity profile. The pseudo trials with data of ADVL0311 or A09712 suggest that EID with β as 0.1, 0.25, or 0.5 and α as -2 can successfully estimate the dose level with NETS closest to the TNETS as of 0.476 with the no significant additional sample sizes and number of cohorts in general. Therefore, we recommend the use of EID with β as 0.1, 0.25, or 0.5 and α as -2 in the practical Phase I clinical trials. 149 Chapter IX: Discussion of Extended Isotonic Design In real trials, patient usually has multiple toxicities so that no designs can claim to be able to treat toxicity response as continuous if the case of multiple toxicities per patient is not considered and multiple toxicities are not measured comprehensively and quantitatively (Wang et al., 2000;Yuan et al., 2006; Bekele and Thall, 2004). We propose a novel score system to treat toxicity as quasi-continuous variable and estimate an overall toxicity severity for a patient quantitatively. To reduce arbitrariness and be objective, our score system is mainly based on the generally accepted toxicity grade and type defined by NCI as well as study specific definition of DLT. The significant cut point of patient with (ETS>4.0) or without (ETS<4.0) DLT used in traditional designs is retained in the score system. Moreover, IR (Bartholomew, 1983) is coupled with the composite NETS to estimate MTD. IR depends mainly on the relative order of composite toxicity severity, is not sensitive to the exact score so that IR can reduce the impact from the arbitrariness of the score system. On the other hand, there are still some “wiggle room” left for investigators by choosing different values for the parameter α and β. Moreover, the mapping between the original grade of toxicity and adjusted grade proposed in the Table 5.1 can be violated as long as investigator has convincing reasons to justify. Therefore, a balance between flexibility and objectiveness is achieved very well in our score system. 150 At present, lots of designs for Phase I trials have been proposed in the literature (Rosenberger, et al. 2002; Potter, 2006). Designs can be classified by their algorithm as two major groups: rule-based and model-based. The Standard 3+3 design (Simon et al., 1997) is the most popular rule-based design and CRM (O’Quigley, et al. 1990) is the typical model-based design. But the ID proposed by Leung and Wang (Leung and Wang, 2001) is chosen as a framework to couple with the score system to create our EID because of its attractive advantages over those two kinds of designs. Our EID keeps all advantages of ID and has additional advantages: 1) toxicity is treated as quasi-continuous variable with our novel score system; 2) toxicities of different grade and type can be differentiated in detail; 3) all toxicity information is fully used in the dose allocation; 4) a correct MTD is estimated under any scenarios according to the exact toxicity profile; 5) EID performs substantially better than the ID in selecting correct MTD. Our EID reduce the chance to choose the over toxic dose level in case the probability of grade 4 toxicity is relatively high and correct the underestimated MTD when the probability of grade 4 toxicity is relatively low. Our EID requires more effort to implement than the ID, such as more involvement of physicians in identifying toxicities, calculation of NETS, and predefining TNETS. But the efforts are warranted by the advantages of the EID over the ID in which a lot of toxicity information is ignored in the dose allocation and MTD estimation procedures. The performance of all one- and two-stage model-guided designs depends, to varying degrees, on the validity of the underlying assumptions and the accuracy of the 151 prior information. Although many authors have demonstrated that the designs perform quite well under model misspecification when the dose levels bracket the true MTD, performance of the designs can be unacceptable when that criterion is not met. Moreover, it is often that there is rather little prior information about the true MTD, particularly in first-in-human studies in which the dose levels initially selected may be far below the true MTD. In the examples of real Phase I studies described by Simon et al. (Simon et al., 1997), it was common for a study to require 20 dose levels and for the MTD to be 1000 times the starting dose. To accommodate this situation, additional dose levels are often added as needed in a Phase I study, and thus it may not be possible to elicit priors that have the requirements necessary for acceptable performance of the model-guided designs. Many model-guided designs are inappropriate for this situation because they lack the requisite robustness of the rule-based designs. A rule based design which minimizes the number of patients treated at low doses has the greatest value. Our EID has all advantages of ID over model-based designs. In addition, it has better therapeutic effect for patients than the model-based CRM-NETS. In our EID, we applied the idea of NETS to the original ID to incorporate fully the differentiation of toxicity. The only assumption in our design is the non-decreasing dose toxicity relationship which is a well accepted among Phase I clinical trial. Moreover, usually no sufficient knowledge about the relationship between dose and toxicity is available in phase I trials, especially for a brand new chemical agent. Our method can handle these situations because of its model free characteristic. The Quasi-CRM requires a specific 152 dose toxicity relationship model and a prior distribution of parameter (Yuan et al., 2006). But the model is likely very different from true dose-toxicity model and strongly informative priors may not be accurate for it (O’Quigley, 2002). Intensive computation and requiring substantial effort to implement in original CRM are also disadvantages of Quasi-CRM (O’Quigley and Shen, 1996). BT method is somewhat too study specific and depends too much on Physicians (Bekele and Thall, 2004). Moreover the method is a little bit too subjective without proposing any universal and objective rules. These concerns prevent BT method from being widely implemented in the Phase I trial practice (Bekele and Thall, 2004). In summary, we propose a novel score system to treat toxicity response as quasi- continuous variable and fully utilize all toxicity information. Simulation studies and practical applications demonstrate that our EID can estimate correct MTD according to exact toxicity profile instead of coarse probability of DLT without additional cost and extended length of trial. The therapeutic effect for patient in our EID is better than designs treating toxicity as binary variable and CRM-NETS. Moreover, our EID is model free, objective, robust, and simple to use. Therefore, the EID we propose would be of great practical value to the field of Phase I clinical trial and will help to begin a new era in which toxicity response is treated as quasi-continuous variable. 153 Chapter X: Future research Given the fact that most of the current designs can be expected to perform similarly when compared on a level playing field, there is probably little need for more design effort on the simple Phase I study. However, further work on designs for late toxicity and other specialized situations seems warranted, and more thought should be given to including within-patient escalation in all designs. 10.1. Within-patient Escalation. In the literature, most Phase I designs only considered single dose per subject. But in some situations, such as eligible patients are rare or difficult to obtain, or oncologists try to increase patients’ chances for therapeutic benefit for subjects, only intra-patient dose escalation is not enough. Some Phase I trial designs within patient dose escalation have been introduced in order to let patients have more therapeutic benefit and obtain data on inter-patient variability (Sheiner et al., 1991; Simon et al., 1997; Sheiner et al., 1991). There are some pre-requisites for the execution of these designs with within patient dose escalation. First, the toxicity event is transient and reversible, second the washout period of the residual (carry-over) effects from the doses must be short and third it is practically possible and useful to treat a subject at multiple doses. The designs with within patient dose escalation can improve the efficiency and reduce sample size substantially. Moreover, multiple doses per patient can find the dose at which the toxicity 154 occurs first for the patient, leading to the better understanding of the variations of drug effects on different patients and better treatments for patients. Therefore, designs allowing within patient dose escalation is a possible new direction for Phase I trials when multiple doses per patient are feasible. 10.2. Designs for Combinations of Agents. In the literature, there are many designs for the combination of agents, in which the MTD is a contour in the space determined by the doses of multiple agents (Thall et al., 2003; Ivanova and Wang, 2004; Conaway et al., 2004). The simple and common assumption about the relationship between different agents is “additivity” in which there are no interaction effects between agents. The assumption of “additivity” is not valid when agent interaction exists, or different agents have different functioning mechanisms, or no overlapping DLTs are selected for combination of agents (Greco et al., 1995). Like in the design for single agent, a series of combinational dose levels for multiple agents are required to be pre-designated in most designs for combination of agents. Three different algorithms for escalating agents are commonly used. In the first algorithm, the agents are escalated alternately; in the second one, the agents are escalated simultaneously; in the third one, for example with only two agents, one agent is increased to the MTD while keeping the other one low, then the first one is decreased from its MTD one or two levels while the second one is escalated to the MTD. More discuss about the guidance and problems of combination of agents can be found by Korn and Simon (Korn and Simon, 1993). 155 10.3. Designs for Delayed or Cumulative Toxicity. In most of the current Phase I designs, dose allocation for the next cohort can not determined until the toxicity responses from all treated patients have been obtained. These designs work very well when the evaluation period of toxicity is not long. But in the cases of late-onset effects of radiation or toxicities from chemo-preventive agents, the trial will become impractically long. In rule based design, the duration of trial may be reduced by increasing the number of patients per cohort or modifying the dose allocation algorithm to base on the available toxicity information in case of too long evaluation period or it can’t afford to wait that long to accrue new cohorts of patients. But to our knowledge, no such rule based design has been proposed. For model based design, there are several publications discussed the late onset toxicities (Cheung and Chappell, 2000; Hüsing et al., 2001; Braun et al., 2003; Braun et al., 2005; Legedza and Ibrahim, 2000). For example, Cheung and Chappell (2000) introduced a time-to-event CRM (TITE- CRM), in which each patient contributes a factor of P DLT (x i ) y (1 – w P DLT (x i )) (1-y) to a modified likelihood function, where the weight, w, is 1 for the patient who has shown DLT, otherwise the fraction of the evaluation period that has passed for the patient. The TITE-CRM method uses the maximum likelihood estimation in the dose allocation procedure and is valid under the assumption that the probability density distribution of DLT is uniform during the evaluation period. In the cases when toxicities occur more likely in the later part of evaluation period or is caused by cumulative agent administration, a greater weight should be given to the later times. Through simulation 156 studies, TITE-CRM is proved to shorten the duration of trial by 3 to 6 fold than CRM while retaining the accuracy of estimated MTD and administration safety. 10.4. Designs Accounting for Covariates and Patient-Specific MTD. Most of the time, the generalizability of the results from Phase I trial is assumed valid unless specified. Patient-specific or group specific MTD is seldom used in medical practice and most clinicians are prone not to use the agents whose toxicity and efficacy change substantially at the patients at extreme values of some covariates unless the patient-specific doses have been demonstrated to have both less toxicity and greater efficacy. In the cases where patient- or group-specific MTDs need to be estimated, patient covariates should be incorporated into the design. In the rule based design, patients are classified into ordered groups according to the level of covariates, such organ dysfunction. A monotone relationship between MTD and level of covariates is usually assumed and a modified dose allocation algorithm is employed to estimate MTD for each group. In the model based design, little or no changes are required in order to incorporate patients’ covariates (Babb and Rogatko, 2001; Legedza and Ibrahim, 2001; O’Quigley and Paoletti, 2003; Piantadosi and Liu, 1996). For example in a study by Mick and Ratain (1993), DLT of many agents is myelosuppression and the log of the white blood count (WBC) is modeled as a linear function of dose and log baseline WBC. MTD for a certain patient is the particular dose that patient should receive in order to have a specific level of WBC so that MTD varies over patients with different baseline WBC. Like the ordinary model based design, the modified designs depend a lot on the validity of the used model. 157 In addition, complete inclusion of all important covariates in the model is also critical to the validity of the design. Therefore, considerable amount information about an agent must have been accumulated before the patient specific MTD of the agent can be estimated. Currently, CTEP is sponsoring some studies in patients with renal or hepatic dysfunction and a protocol template is available on CTEP website. 158 References Ahn C. An evaluation of Phase I cancer clinical trial designs. 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Journal of Biopharmaceutical Statistics.2003;13(1):87–101. 164 Appendices Appendix A: Programme of EID for Simulation Studies of Chapter VI LIBNAME HOME "C:\USER\PDDISSERTATION\BIOSTATISTICS_PHD\ISOTONIC\DATA"; %MACRO SIM(WHICH_SCENARIO,MAX,BATCH,COH); DATA SCENARIO; SET HOME.SCENARIO_ALL; IF SCENARIO="&WHICH_SCENARIO"; IF ADJ_GRADE IN (0,1,2,3,4,5,6); IF GRADE = "0" THEN DO; NETS_LOW=0; NETS_HIGH=0; WIDE=NETS_HIGH-NETS_LOW; END; IF GRADE = "1" THEN DO; NETS_LOW=1/60; NETS_HIGH=1/6; WIDE=NETS_HIGH-NETS_LOW; END; IF GRADE = "2" THEN DO; NETS_LOW=1/6; NETS_HIGH=1/3; WIDE=NETS_HIGH-NETS_LOW; END; IF GRADE = "3 non DLT" THEN DO; NETS_LOW=1/3; NETS_HIGH=1/2; WIDE=NETS_HIGH-NETS_LOW; END; IF GRADE = "4 non DLT" THEN DO; NETS_LOW=1/2; NETS_HIGH=2/3; WIDE=NETS_HIGH-NETS_LOW; END; IF GRADE = "3 DLT" THEN DO; NETS_LOW=2/3; NETS_HIGH=5/6; WIDE=NETS_HIGH-NETS_LOW; END; IF GRADE = "4 DLT" THEN DO; NETS_LOW=5/6; NETS_HIGH=1; WIDE=NETS_HIGH-NETS_LOW; END; RUN; proc transpose data=SCENARIO out=DOSE_1WIDE prefix=DOSE_1; by SCENARIO; id ADJ_GRADE; var DOSE_1; run; proc transpose data=SCENARIO out=DOSE_2WIDE prefix=DOSE_2; by SCENARIO; id ADJ_GRADE; var DOSE_2; run; proc transpose data=SCENARIO out=DOSE_3WIDE prefix=DOSE_3; by SCENARIO; id ADJ_GRADE; var DOSE_3; run; 165 proc transpose data=SCENARIO out=DOSE_4WIDE prefix=DOSE_4; by SCENARIO; id ADJ_GRADE; var DOSE_4; run; proc transpose data=SCENARIO out=DOSE_5WIDE prefix=DOSE_5; by SCENARIO; id ADJ_GRADE; var DOSE_5; run; proc transpose data=SCENARIO out=DOSE_6WIDE prefix=DOSE_6; by SCENARIO; id ADJ_GRADE; var DOSE_6; run; proc transpose data=SCENARIO out=NETSWIDE prefix=NETS; by SCENARIO; id ADJ_GRADE; var NETS; run; proc transpose data=SCENARIO out=NETS_LOWWIDE prefix=NETS_LOW; by SCENARIO; id ADJ_GRADE; var NETS_LOW; run; proc transpose data=SCENARIO out=NETS_HIGHWIDE prefix=NETS_HIGH; by SCENARIO; id ADJ_GRADE; var NETS_HIGH; run; proc transpose data=SCENARIO out=WIDEWIDE prefix=WIDE; by SCENARIO; id ADJ_GRADE; var WIDE; run; DATA ALL; MERGE NETSWIDE NETS_LOWWIDE NETS_HIGHWIDE WIDEWIDE DOSE_1WIDE DOSE_2WIDE DOSE_3WIDE DOSE_4WIDE DOSE_5WIDE DOSE_6WIDE; BY SCENARIO; DROP _NAME_; RUN; /************************DESIGN WITH ET SCORE***************/ DATA ISOTONIC_ET; SET ALL; ARRAY DOSE_P[6,7] DOSE_10 DOSE_11 DOSE_12 DOSE_13 DOSE_14 DOSE_15 DOSE_16 DOSE_20 DOSE_21 DOSE_22 DOSE_23 DOSE_24 DOSE_25 DOSE_26 DOSE_30 DOSE_31 DOSE_32 DOSE_33 DOSE_34 DOSE_35 DOSE_36 DOSE_40 DOSE_41 DOSE_42 DOSE_43 DOSE_44 DOSE_45 DOSE_46 DOSE_50 DOSE_51 DOSE_52 DOSE_53 DOSE_54 DOSE_55 DOSE_56 DOSE_60 DOSE_61 DOSE_62 DOSE_63 DOSE_64 DOSE_65 DOSE_66; ARRAY WIDE[7] WIDE0 WIDE1 WIDE2 WIDE3 WIDE4 WIDE5 WIDE6; ARRAY NETS[7] NETS0 NETS1 NETS2 NETS3 NETS4 NETS5 NETS6; ARRAY NETS_LOW[7] NETS_LOW0 NETS_LOW1 NETS_LOW2 NETS_LOW3 NETS_LOW4 NETS_LOW5 NETS_LOW6; ARRAY NETS_HIGH[7] NETS_HIGH0 NETS_HIGH1 NETS_HIGH2 NETS_HIGH3 NETS_HIGH4 NETS_HIGH5 NETS_HIGH6; ARRAY TOXX[6] TOX1 TOX2 TOX3 TOX4 TOX5 TOX6; ARRAY PTT[6] PT1 PT2 PT3 PT4 PT5 PT6; ARRAY WEIGHT {6,6,6}; ARRAY RISKK[6] RISK1 RISK2 RISK3 RISK4 RISK5 RISK6; DO REP=1 TO &MAX; COUNT=0; DO COHORT=1 TO &COH; 166 IF COHORT=1 THEN DO; DO I=1 TO 6; TOXX[I]=0; PTT[I]=0; END; END; IF COHORT=1 THEN DOSELEVEL=1; ELSE DOSELEVEL=NEXT_DOSELEVEL; PTT[DOSELEVEL]=SUM(PTT[DOSELEVEL],&BATCH); DO J=1 TO &BATCH; PRAND=UNIFORM(123456789+COHORT+J+REP); ADJ_GRADE=0; IF PRAND LE DOSE_P(DOSELEVEL,1) THEN ADJ_GRADE=0; ELSE IF PRAND LE (DOSE_P(DOSELEVEL,1)+DOSE_P(DOSELEVEL,2)) THEN ADJ_GRADE=1; ELSE IF PRAND LE (DOSE_P(DOSELEVEL,1)+DOSE_P(DOSELEVEL,2)+DOSE_P(DOSELEVEL,3)) THEN ADJ_GRADE=2; ELSE IF PRAND LE (DOSE_P(DOSELEVEL,1)+DOSE_P(DOSELEVEL,2)+DOSE_P(DOSELEVEL,3) +DOSE_P(DOSELEVEL,4)) THEN ADJ_GRADE=3; ELSE IF PRAND LE (DOSE_P(DOSELEVEL,1)+DOSE_P(DOSELEVEL,2)+DOSE_P(DOSELEVEL,3) +DOSE_P(DOSELEVEL,4)+DOSE_P(DOSELEVEL,5)) THEN ADJ_GRADE=4; ELSE IF PRAND LE (DOSE_P(DOSELEVEL,1)+DOSE_P(DOSELEVEL,2)+DOSE_P(DOSELEVEL,3) +DOSE_P(DOSELEVEL,4)+DOSE_P(DOSELEVEL,5)+DOSE_P(DOSELEVEL,6)) THEN ADJ_GRADE=5; ELSE IF PRAND LE (DOSE_P(DOSELEVEL,1)+DOSE_P(DOSELEVEL,2)+DOSE_P(DOSELEVEL,3) +DOSE_P(DOSELEVEL,4)+DOSE_P(DOSELEVEL,5)+DOSE_P(DOSELEVEL,6) +DOSE_P(DOSELEVEL,7)) THEN ADJ_GRADE=6; NETSRAND=UNIFORM(1256789+COHORT+J+REP+ADJ_GRADE+1); NETS_ADJ=NETSRAND*WIDE(ADJ_GRADE+1)+NETS_LOW(ADJ_GRADE+1); TOXX[DOSELEVEL]=SUM(TOXX[DOSELEVEL],NETS_ADJ); END; DO I=1 TO 6; DO R=1 TO I; DO S=I TO 6; DO K=R TO S; IF K=R THEN TOTALTOX=0; IF K=R THEN TOTALPT=0; TOTALTOX=SUM(TOTALTOX,TOXX[K]); TOTALPT=SUM(TOTALPT,PTT[K]); END; IF TOTALPT GT 0 THEN WEIGHT[R,I,S]=TOTALTOX/TOTALPT; ELSE WEIGHT[R,I,S]=0; END; END; END; DO I=1 TO 6; TEMPMIN=1; DO S=I TO 6; TEMPMAX=0; DO K=1 TO I; TEMPMAX=MAX(TEMPMAX,WEIGHT[K,I,S]); END; TEMPMIN=MIN(TEMPMIN,TEMPMAX); END; RISKK[I]=TEMPMIN; END; 167 IF RISKK[DOSELEVEL] LT 0.476 THEN DO; IF (DOSELEVEL LT 6) THEN DO; IF ((0.476-RISKK[DOSELEVEL]) GE (RISKK[DOSELEVEL+1]-0.476)) THEN DO; NEXT_DOSELEVEL=SUM(DOSELEVEL,1); END; END; ELSE DO; NEXT_DOSELEVEL=DOSELEVEL; END; END; ELSE DO; IF (DOSELEVEL GT 1) THEN DO; IF (0.476-RISKK[DOSELEVEL-1]) LT (RISKK[DOSELEVEL]-0.476) THEN NEXT_DOSELEVEL=DOSELEVEL-1; ELSE DO; NEXT_DOSELEVEL=DOSELEVEL; END; END; ELSE DO; NEXT_DOSELEVEL=DOSELEVEL; END; END; IF NEXT_DOSELEVEL GE 6 THEN NEXT_DOSELEVEL=6; IF NEXT_DOSELEVEL=DOSELEVEL THEN COUNT=SUM(COUNT,1); ELSE COUNT=0; IF COUNT GT 3 THEN ERALYSTOP=1; OUTPUT; IF COUNT GT 3 THEN DO; GO TO R1; END; END; R1: ERALYSTOP=0; END; KEEP PT1 PT2 PT3 PT4 PT5 PT6 COUNT TOX1 TOX2 TOX3 TOX4 TOX5 TOX6 RISK1 RISK2 RISK3 RISK4 RISK5 RISK6 NEXT_DOSELEVEL DOSELEVEL COHORT REP ERALYSTOP NETSRAND NETS_ADJ PRAND ADJ_GRADE; RUN; DATA Isotonic_et1; SET Isotonic_et; BY REP; IF LAST.REP; RUN; PROC FREQ DATA=Isotonic_et1; TITLE 'Isotonic1'; TABLE NEXT_DOSELEVEL/OUT=TAB; RUN; DATA TABOUT; SET TAB; SCENARIO="&WHICH_SCENARIO"; KEEP SCENARIO COUNT NEXT_DOSELEVEL PERCENT; RUN; DATA FINALTAB; SET FINALTAB TABOUT; IF SCENARIO NE ""; RUN; 168 DATA Isotonic_et2; SET Isotonic_et1; TOTAL_PTS=SUM(PT1,PT2,PT3,PT4,PT5,PT6); PTC1=PT1/TOTAL_PTS; PTC2=PT2/TOTAL_PTS; PTC3=PT3/TOTAL_PTS; PTC4=PT4/TOTAL_PTS; PTC5=PT5/TOTAL_PTS; PTC6=PT6/TOTAL_PTS; RUN; PROC MEANS NOPRINT DATA=Isotonic_et2; VAR PTC1 PTC2 PTC3 PTC4 PTC5 PTC6 TOTAL_PTS COHORT; OUTPUT OUT=PAT_PERC; RUN; DATA PAT_PERC_OUT; SET PAT_PERC; SCENARIO="&WHICH_SCENARIO"; RUN; DATA FINALCOH; SET FINALCOH PAT_PERC_OUT; IF SCENARIO NE ""; RUN; %MEND; DATA FINALTAB; RUN; DATA FINALCOH; RUN; %SIM(A,40000,3,20); %SIM(B,40000,3,20); %SIM(C,40000,3,20); %SIM(D,40000,3,20); %SIM(E,40000,3,20); %SIM(F,40000,3,20); %SIM(G,40000,3,20); DATA HOME.FINALTAB_EID; SET FINALTAB; COUNT_EID = COUNT; PERCENT_EID = PERCENT; DOSELEVEL=NEXT_DOSELEVEL; KEEP SCENARIO DOSELEVEL COUNT_EID PERCENT_EID; RUN; DATA HOME.FINALCOH_EID; SET FINALCOH; STAT_EID=_STAT_; DROP _STAT_; RUN; 169 Appendix B: Programme of EID for Bootstrap Replication of ADVL0311 LIBNAME HOME "C:\USER\PDDISSERTATION\BIOSTATISTICS_PHD\ISOTONIC\DATA"; DATA WKADVL0311; SET HOME.Bootstrap_advl0311_beta2; IF ETS NE .; NETS=ROUND(100*ETS/6)/100; DOSE_LEVEL=TREATMENTID; PATIENT_ID=PTNT_ID; RUN; PROC SORT DATA=WKADVL0311; BY DOSE_LEVEL PATIENT_ID; RUN; DATA WKADVL0311N; RETAIN ORDER_ID; SET WKADVL0311; BY DOSE_LEVEL; IF FIRST.DOSE_LEVEL THEN ORDER_ID=1; ELSE ORDER_ID=ORDER_ID+1; RUN; DATA NUMBER_PTS; SET WKADVL0311N; BY DOSE_LEVEL ORDER_ID; IF LAST.DOSE_LEVEL; NN=1; KEEP DOSE_LEVEL ORDER_ID NN; RUN; DATA CM; DO I=1 TO 8; DO J=1 TO 6; DOSE_LEVEL=I; ORDER_ID=J; OUTPUT; END; END; KEEP DOSE_LEVEL ORDER_ID; RUN; PROC SORT DATA=CM; BY DOSE_LEVEL ORDER_ID; RUN; PROC SORT DATA=WKADVL0311N; BY DOSE_LEVEL ORDER_ID; RUN; DATA WKADVL0311N1; MERGE CM WKADVL0311N; BY DOSE_LEVEL ORDER_ID; RUN; DATA WKADVL0311N2; SET WKADVL0311N1; NN=1; KK=DOSE_LEVEL*10+ORDER_ID; RUN; PROC SORT DATA=WKADVL0311N2; BY NN KK; RUN; proc transpose data=WKADVL0311N2 out=WKADVL0311_WIDE prefix=NETS; BY NN; id kk; var NETS; run; PROC SORT DATA=NUMBER_PTS; BY NN DOSE_LEVEL; RUN; proc transpose data=NUMBER_PTS out=NUMBER_PTS_WIDE prefix=PTS_DOSE; BY NN; id DOSE_LEVEL; var ORDER_ID; run; DATA MALL; MERGE NUMBER_PTS_WIDE WKADVL0311_WIDE; BY NN; RUN; 170 DATA HOME.PATIENTS_POOL_ADVL0311; SET MALL; RUN; DATA ALL; SET HOME.PATIENTS_POOL_ADVL0311; RUN; %MACRO SIM(MAX,BATCH,COH); /************************DESIGN WITH ET SCORE***************/ DATA ISOTONIC_ET; SET ALL; ARRAY NETSS[8,6] NETS11 NETS12 NETS13 NETS14 NETS15 NETS16 NETS21 NETS22 NETS23 NETS24 NETS25 NETS26 NETS31 NETS32 NETS33 NETS34 NETS35 NETS36 NETS41 NETS42 NETS43 NETS44 NETS45 NETS46 NETS51 NETS52 NETS53 NETS54 NETS55 NETS56 NETS61 NETS62 NETS63 NETS64 NETS65 NETS66 NETS71 NETS72 NETS73 NETS74 NETS75 NETS76 NETS81 NETS82 NETS83 NETS84 NETS85 NETS86; ARRAY PTS_DOSE[8] PTS_DOSE1 PTS_DOSE2 PTS_DOSE3 PTS_DOSE4 PTS_DOSE5 PTS_DOSE6 PTS_DOSE7 PTS_DOSE8; ARRAY TOXX[8] TOX1 TOX2 TOX3 TOX4 TOX5 TOX6 TOX7 TOX8; ARRAY PTT[8] PT1 PT2 PT3 PT4 PT5 PT6 PT7 PT8; ARRAY WEIGHT {8,8,8}; ARRAY RISKK[8] RISK1 RISK2 RISK3 RISK4 RISK5 RISK6 RISK7 RISK8; DO REP=1 TO &MAX; COUNT=0; DOSELEVEL=. ; NEXT_DOSELEVEL = . ; DO COHORT=1 TO &COH; IF COHORT=1 THEN DO; DO I=1 TO 8; TOXX[I]=0; PTT[I]=0; END; END; IF COHORT=1 THEN DOSELEVEL=1; ELSE DOSELEVEL=NEXT_DOSELEVEL; DO J=1 TO &BATCH; PRAND=UNIFORM(123456789+COHORT+J+REP); ORDERID=FLOOR(PRAND*PTS_DOSE[DOSELEVEL])+1; NETS_SIM=NETSS[DOSELEVEL,ORDERID]; IF NETS_SIM NE . THEN PTT[DOSELEVEL]=SUM(PTT[DOSELEVEL],1); TOXX[DOSELEVEL]=SUM(TOXX[DOSELEVEL],NETS_SIM); PT_ID=REP*1000+DOSELEVEL*100+PTT[DOSELEVEL]; END; DO I=1 TO 8; DO R=1 TO I; DO S=I TO 8; DO K=R TO S; IF K=R THEN TOTALTOX=0; IF K=R THEN TOTALPT=0; TOTALTOX=SUM(TOTALTOX,TOXX[K]); TOTALPT=SUM(TOTALPT,PTT[K]); END; IF TOTALPT GT 0 THEN WEIGHT[R,I,S]=TOTALTOX/TOTALPT; ELSE WEIGHT[R,I,S]=0; END; END; END; 171 DO I=1 TO 8; TEMPMIN=1; DO S=I TO 8; TEMPMAX=0; DO K=1 TO I; TEMPMAX=MAX(TEMPMAX,WEIGHT[K,I,S]); END; TEMPMIN=MIN(TEMPMIN,TEMPMAX); END; RISKK[I]=TEMPMIN; END; IF RISKK[DOSELEVEL] LE 0.476 THEN DO; IF (DOSELEVEL LT 8) THEN DO; IF ((0.476-RISKK[DOSELEVEL]) GE (RISKK[DOSELEVEL+1]-0.476)) THEN DO; NEXT_DOSELEVEL=SUM(DOSELEVEL,1); END; END; ELSE DO; NEXT_DOSELEVEL=DOSELEVEL; END; END; ELSE DO; IF (DOSELEVEL GT 1) THEN DO; IF (0.476-RISKK[DOSELEVEL-1]) LT (RISKK[DOSELEVEL]-0.476) THEN NEXT_DOSELEVEL=DOSELEVEL-1; ELSE DO; NEXT_DOSELEVEL=DOSELEVEL; END; END; ELSE DO; NEXT_DOSELEVEL=DOSELEVEL; END; END; IF NEXT_DOSELEVEL GE 8 THEN NEXT_DOSELEVEL=8; IF NEXT_DOSELEVEL=DOSELEVEL THEN COUNT=SUM(COUNT,1); ELSE COUNT=0; IF COUNT GT 3 THEN ERALYSTOP=1; OUTPUT; IF COUNT GT 3 THEN DO; GO TO R1; END; END; R1: ERALYSTOP=0; END; KEEP PT1 PT2 PT3 PT4 PT5 PT6 PT7 PT8 COUNT TOX1 TOX2 TOX3 TOX4 TOX5 TOX6 TOX7 TOX8 RISK1 RISK2 RISK3 RISK4 RISK5 RISK6 RISK7 RISK8 NEXT_DOSELEVEL DOSELEVEL COHORT REP ERALYSTOP PRAND NETS_SIM PRAND ORDERID PT_ID; RUN; DATA Isotonic_et1; SET Isotonic_et; BY REP; IF LAST.REP; RUN; 172 PROC FREQ DATA=Isotonic_et1; TITLE 'Isotonic1'; TABLE NEXT_DOSELEVEL/OUT=TAB; RUN; DATA TABOUT; SET TAB; KEEP COUNT NEXT_DOSELEVEL PERCENT; RUN; DATA FINALTAB; SET FINALTAB TABOUT; RUN; DATA Isotonic_et2; SET Isotonic_et1; TOTAL_PTS=SUM(PT1,PT2,PT3,PT4,PT5,PT6,PT7,PT8); PTC1=PT1/TOTAL_PTS; PTC2=PT2/TOTAL_PTS; PTC3=PT3/TOTAL_PTS; PTC4=PT4/TOTAL_PTS; PTC5=PT5/TOTAL_PTS; PTC6=PT6/TOTAL_PTS; PTC7=PT7/TOTAL_PTS; PTC8=PT8/TOTAL_PTS; RUN; PROC MEANS NOPRINT DATA=Isotonic_et2; VAR PTC1 PTC2 PTC3 PTC4 PTC5 PTC6 PTC7 PTC8 TOTAL_PTS COHORT; OUTPUT OUT=PAT_PERC; RUN; DATA PAT_PERC_OUT; SET PAT_PERC; RUN; DATA FINALCOH; SET FINALCOH PAT_PERC_OUT; RUN; %MEND; DATA FINALTAB; RUN; DATA FINALCOH; RUN; %SIM(40000,3,20); DATA HOME.TAB_EID_BOOTSTRAP_0311_beta2; SET FINALTAB; COUNT_EID = COUNT; PERCENT_EID = PERCENT; DOSELEVEL=NEXT_DOSELEVEL; KEEP DOSELEVEL COUNT_EID PERCENT_EID; RUN; DATA HOME.COH_EID_BOOTSTRAP_0311_beta2; SET FINALCOH; STAT_EID=_STAT_; DROP _STAT_; RUN;

Abstract (if available)

Abstract The current designs of Phase I trials are comprehensively reviewed and classified by algorithm (assumption and dose-toxicity relationship) as rule based designs vs model based designs or by number of stages as one stage designs vs two stage designs. Standard 3+3 designs are most widely used for their practical simplicity. Through simulation study, the expected toxicity levels (ETL) at maximum tolerated dosage (MTD) are originally found to decrease monotonically from about 30% to 0% as the number of dose levels increase from 3 to infinity, which solves the previously unexamined issue of the number of dose levels that are planned in a study. We conclude that the number of specified dose levels is an important factor affecting substantially the ETL at MTD and recommend that fewer than 20 dose levels be designated. In Standard 3+3 designs, target toxicity level (TTL) can not be pre-specified and toxicity response is treated as a binary indicator of dose limiting toxicity (DLT), discarding lots of valuable toxicity information. Therefore, a novel toxicity score system is proposed to measure quantitatively the overall severity of multiple toxicities of each patient, and then coupled with Isotonic Regression (IR) to create an extended isotonic design (EID) which allow pre-specification of TTL, treat toxicity response as a quasi-continuous variable, and fully utilize all toxicity information. Simulation studies and applications of EID to two real Phase I trials demonstrate that EID can always estimate an more accurate MTD with less sample size according to the exact toxicity profile while designs treating toxicity response as a binary variable can’t accomplish that. Our EID is practical, objective, model free, simple to use, and more accurate in MTD estimation so that it is of great practical value and will help to begin a new era in which toxicity response is really treated as a quasi-continuous variable.

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$A phase I study of vincristine, escalating doses of irinotecan, temozolomide and bevacizumab (VIT-B) in pediatric and adolescent patients with recurrent or refractory solid tumors of non-hematopo...$

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A phase I study of vincristine, escalating doses of irinotecan, temozolomide and bevacizumab (VIT-B) in pediatric and adolescent patients with recurrent or refractory solid tumors of non-hematopo...

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Creator Chen, Zhengjia (author)

Core Title Phase I clinical trial designs: range and trend of expected toxicity level in standard A+B designs and an extended isotonic design treating toxicity as a quasi-continuous variable

School Keck School of Medicine

Degree Doctor of Philosophy

Degree Program Biostatistics

Publication Date 02/26/2009

Defense Date 12/08/2008

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag binary indicator,dose limiting toxicity,extended isotonic design,isotonic regression,maximum tolerated dosage,OAI-PMH Harvest,phase I clinical trial designs,quasi-continuous variable,toxicity response

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Azen, Stanley Paul (committee chair), Krailo, Mark D. (committee chair), Stram, Daniel O. (committee member), Wilcox, Rand R. (committee member), Xiang, Anny Hui (committee member)

Creator Email nelsonczj@yahoo.com,zhengjic@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m1987

Unique identifier UC1291204

Identifier etd-Chen-2612 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-151232 (legacy record id),usctheses-m1987 (legacy record id)

Legacy Identifier etd-Chen-2612.pdf

Dmrecord 151232

Document Type Dissertation

Rights Chen, Zhengjia

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu