**Cost minimization of a phase II clinical study**

Lee Kien Foo, Stephen Duffull

University of Otago, New Zealand

**Objectives:**

The purpose of phase II studies is to provide initial exploration of drug efficacy and safety in the target patient population. Population pharmacokinetics (PK) of the drug are often explored in phase I studies and point estimates of these parameters (fixed effects, variances of between subject variability (BSV) and residual error) with biomarker information can be used to assist in deciding future dose and dosing regimens. However, since PK parameter values for the patient population may not be the same as for healthy volunteers, a study designed solely on phase I PK may carry a significant risk of failure.

Optimal designs have concentrated mainly on improving the precision of parameter estimates by optimizing the dose and/or sampling schedule. For these designs, the upper boundary of the design space provides the most precise parameter estimates. More recently, designs that include a cost function have been investigated as either a fixed total cost [1] or via different cost functions [2]. In both cases, the cost is determined based on the use of resources and not linked to the success or failure of the study. A decision based on precision only would yield expensive but successful studies whereas a study that considered cost only may be prone to failure. In this project, we propose an approach to determine designs that minimize the expected cost of a clinical study. The expected cost accounts for the probability of success of the study.

Design variables considered here are the number of patients (Np), number of samples per patient (Ns) and their schedule (S), and the defined daily dose (DDD) of the drug. The objective of our study is to locate optimum designs which naturally balances the cost of a clinical study with the probability of study success without setting arbitrary constraints for the design variables.

**Methods:**

The method will be defined with a specific simulation example. A simple example is chosen for illustration but the proposed method can be generalised to other experiments.

In our example, all patients will receive the same dose, for 3 doses at 24 hours fixed dose interval. Based on prior biomarker data the clinical team has defined a therapeutic window for the 3rd dose to be in the range [0.3unitL^{-1}, 1.3unitL^{-1}]. A clinical study was defined as success if more than 60% of patients have a 3rd dose trough concentration within the therapeutic range.

*Expenditure of a clinical study*For a given design (Np, Ns, S and DDD), the expenditure (X) of the study is defined as

X = $[Np x (Cp + ds(Ns)(Cs) + dn(Cd)(DDD))].

Cp is the cost per patient, ds is days ta take samples, Cs is cost per sample, Cd is cost per dose and Nd is the number of DDD given. ds = dn = 3 in our example.

*Cost of a clinical study *The total cost T of a successful study was set to the expenditure of the study (X). The total cost T of a failed study was set to the sum of the expenditure of the study (X) plus expenditure to redo a successful study, with an empirical design, (X

^{empirical}) and the expenditure for the time penalty of having to repeat the study (X

^{time}).

In our example, the costs were arbitrarily set to: Cp = $10000, Cd =$10 and Cs will be investigated at $100, $500 and $1000. The empirical design is Np = 70, Ns = 8 and DDD = 1 unit. This design has at least an 80% chance of success if there were no uncertainty in the fixed effect parameters and variance of the random effect parameters, and is used to define X^{empirical }. X^{time} is assumed to be either $0 or half of X^{empirical}. The proposed method is used to locate a design that minimizes the expected cost, E[T].

*Uncertainty in population PK parameters*The uncertainty of a set of population PK parameters was incorporated with an additional level of hierarchy where the parameters are assumed to follow specific hyper-prior distributions. The hyper-parameters were calculated with formulas derived by [3].

*Cost optimization*The design variables (Np, Ns, S, DDD) were optimised using an exchange algorithm. At each iteration of the exchange algorithm, the hyperprior was updated to reflect the information content of the proposed study which was evaluated with POPT [4]. The trough concentration of the 3rd dose was calculated for each patient in the study and random error included. E[T] was determined by Monte-Carlo sampling of 1000 replications of each study design for population PK parameters generated from the hyperprior distribution, where on each realisation of a study the success or failure and total cost T of the study was evaluated.

*Simulation study*

The PK model was given by a one compartment first order input and output and combined error model. The fixed effects estimates were assumed to be multivariate log normal with nominal mean of CL = 0.03Lh^{-1}, V = 1L and Ka = 1h^{-1}. The variances of the BSV are assumed to be the same with value 0.1. The proportional and additive errors were assumed to be normal with mean 0 and variance 0.1 and 0.05, respectively.

The nominal means of CL, V and Ka were assumed to follow a normal hyper-prior distribution. The variances of the BSV were assumed to follow an inverse Wishart distribution. The variance of the proportional error was assumed to follow an inverse Gamma distribution.

**Results: **

In all cases the design that minimized the expected cost did not consist of any design variables being located at the boundary of the design space.

*Without time penalty *

The optimal design when Cs = $100 was Np = 33, Ns = 18 and DDD = 3unit. E[T] was $582,520 providing a power of 92%.

When Cs = $500, the optimal design was Np = 46, Ns = 8 and DDD = 3unit. E[T] was $1,185,771 with a power of 89%.

When Cs was $1000, the optimal design was Np = 58, Ns = 6 and DDD = 3unit. E[T] was $1,884,100 with a power of 89%.

*With time penalty*

The optimal design when Cs = $100 was Np = 38, Ns = 17 and DDD = 3unit. E[T] was $618,980 with a power of 97%.

When Cs = $500, the optimal design was Np = 53, Ns = 8 and DDD = 3unit. E[T] was $1,279,500 with a power of 95%.

When Cs = $1000, the optimal design is Np = 63, Ns = 6 and DDD = 3unit. E[T] was $2,012,600 with a power of 93%.

** Conclusions: **

An approach was developed to minimize the cost of a clinical study, where the cost considers both the cost of success and failure. The designs did not reach boundary values of the design space. These designs naturally balances the cost of study and probability of study success. Although as secondary outcome, it was interesting to see that (1) including an additional cost penalty for time delays changes the overall study cost and increases patient recruitment and (2) all designs tended to favour higher power (about 90% for all cases). Further work with real examples is warranted.

**References:**

[1] Retout S, Comets E, Bazzoli C and Mentre F. Design optimization in nonlinear mixed effects models using cost functions: application to a joint model of infliximab and methotrexate pharmacokinetics. Communication in Statistics - Theory and Methods, 2009, 8: 3351 - 3368.

[2] Bazzoli C , Retout S and Mentre F. Design evaluation and optimization in multi-response nonlinear mixed effects models with cost functions: application to the pharmacokinetics of zidovudine and its active metabolite. PAGE 19 (2010) http://www.page-meeting.org/?abstract=1710

[3] Dokoumetzidis A and Aarons L. Analytical expressions for combining population pharmacokinetic parameters from different studies. Journal of Biopharmaceutical Statistics. 2008, 18:662 - 676.

[4] http://www.winpopt.com/