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- Clinical Pharmacology—The Quarterback of Drug Development

Can the dose be too low so that the drug will not work?

Can the dose be too high that it may cause some potential problem? How do people learn about dosing information? This book answers some of these questions. Dosing information on the drug label is based on discussion and agreement between the pharmaceutical manufacturer and the drug regulatory agency.

A drug label is a high level summary of results obtained from many scientific experiments. Scientists with biological, chemical, medical, or statistical background working in the pharmaceutical industry designed and executed these experiments to obtain information to help understand the dosing information. This book introduces the drug development process, the design and analysis of clinical trials.

Many of the discussions are based on applications of statistical methods in the design and analysis of dose response studies. The potential readers include pharmacokienticists, clinical scientists, clinical pharmacologists, pharmacists, project managers, pharmaceutical scientists, clinicians, programmers, data managers, regulatory specialists, and study report writers.

This book is also a good reference for professionals working in a drug regulatory environment, for example, the FDA. In addition, statistical and medical professionals in academia may find this book helpful in understanding the drug development process and practical concerns in selecting doses for a new drug. Naitee Ting received his Ph.

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Ting is currently an Associate Director of Biostatistics in Pfizer Global Research and Development, supporting clinical development of new drugs. He has over 18 years of experiences in designing and analyzing late phase clinical trials. During his tenure at Pfizer, Dr.

Ting has published more than 20 statistical papers in peer-reviewed journals and book chapters. However, continuous outcomes are also often observed in clinical trials.

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For example, shrinkage in a solid tumor, changes in blood pressure, or changes of glomerular filtration rate. In , Aiyang Tao [14] studied bivariate continuous outcomes by bivariate normal distribution, and use AMCP-MOD method which is a fixed design with pre-determined randomization strategy to find optimal dose for phase III. In our article, we propose Archimedean Copula joint model to evaluate efficacy and toxicity simultaneously. We mainly focus on two cases: bivariate continuous outcome and a mixture of continuous and categorical outcomes.

Compared with other joint models, Archimedean Copula joint model has four main advantages: 1 The Copula joint model has no restrictions on probability distributions of efficacy and toxicity. Bivariate mixed outcome is one of most analytically difficult cases, because they do not follow an obvious multivariate distribution. Based on this nice property of Archimedean Copula, we can easily structure the joint distribution via marginal distribution of continuous outcome and the conditional distribution of discrete outcome. In addition, Archimedean Copula is not restricted to radial symmetry.

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Take three popular Archimedean Copulas as examples. The Clayton Copula is an asymmetric Archimedean Copula, exhibiting greater dependence in the negative tail than in the positive.

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The Frank Copula is a symmetric Archimedean Copula. The Gumbel Copula is an asymmetric Archimedean Copula, exhibiting greater dependence in the positive tail than in the negative. According to different situations, we can choose corresponding Archimedean Copula. In fact, Archimedean Copula has another merit that they allow modeling dependence in arbitrary high dimensions with only one parameter.

However, we mainly focus on bivariate efficacy —toxicity outcome in this article, so this merit is not obvious. Continuous reassessment method CRM [15] is a popular algorithm in Phase I cancer clinical trials which is proposed to select maximum tolerated dose MTD [16] , [17] , [18].

It is also an adaptive design which means design is guided by examination of the accumulated data at an interim point in the clinical trial. It aims to 1 keep to a minimum number of patients treated at unacceptable high toxic dose levels. According to its good character, we adopt CRM approach and extend it to bivariate trials to select the optimal dose for phase III in our article.

The remainder of the paper is organized as follows. ACE inhibitors inhibitors of angiotensin-converting enzyme are used primarily in treatment of hypertension and heart failure. However, renal impairment is a significant adverse effect of all ACE inhibitors. The reason for this is still unknown.

Some suggest that it is associated with their effect on angiotensin II-mediatedd homeostatic functions such as renal blood flow. Renal blood flow may be affected by angiotensin II because it vasoconstricts the efferent arterioles of the glomeruli of the kidney, thereby increasing glomerular filtration rate GFR. In one clinical trial an ACE inhibitor is used to treat hypertension. The efficacy endpoint is the change of sitting blood pressure from baseline. Anticoagulants are pivotal agents for the prevention and treatment of thromboembolic disorders.

The efficacy of Anticoagulant is to lower the VTE venous thromboembolism incidence rate. Unfortunately, such an effect can be accompanied by an increase in major bleeding, especially postoperative, during the treatment. Therefore, when choosing the optimal dose, we should consider the VTE incidence rate and bleeding event simultaneously.

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Copula is recently popular strategy for joint modeling in statistical applications. The history of Copula may begin with Frechet [19] , who studied the problem in low-dimension. In , Sklar researched in-depth in this respect, by introducing the notion, and the name, of a Copula, and proving the theorem that now bears his name [20]. In our study, Let and respectively denote the efficacy and toxicity outcomes obtained from subject within fixed dose group. For each dose d i , we assume , and. The marginal distributions can be arbitrary continuous distributions.

## Clinical Pharmacology—The Quarterback of Drug Development

To specify the marginal models, let and be suitable link models chosen from common candidate models like Emax, Exponential, logit, etc. For Archimedean Copula, the Copula function is defined as: where is the so called generator, , , and is the association parameter measuring dependence between efficacy and toxicity.

Some popular Archimedean Copulas, their corresponding generators and association parameters are listed in Table 1. Note that all marginal parameters are inherited from the with meaning, and the parameters for association are brought into the Copula parameter. For Frank Copula and Clayton Copula, only if , the outcomes are independent, and when or , outcomes have positive and negative associations; For Gumbel Copula, when , the outcomes are independent, and when , outcomes have positive associations.

When efficacy and toxicity are both continuous outcomes.

go The joint distribution determined by Archimedean Copula takes the form in Table 1 , and the density function of Archimedean Copula is given by: where and represent corresponding marginal density function. When the margins appear to be mixed outcomes, we adopt another strategy to build joint model. Assume efficacy is continuous and toxicity is categorical. Similarly, let and , but is discrete distribution here. Let where is the left-hand limit of. Then, the joint density is given by. Take Frank Copula for example, the joint distribution determined by Frank Copula takes the form.

Copula is popular in statistical applications as it allows one not only to conveniently build joint regression model, but also to easily estimate the parameters in joint regression model by using Copula density function. Once we get the Copula joint density function, we can use maximum likelihood estimation, which has come to be quite mature and be integrated in many softwares.