Simulated Clinical Trials: Principle, Good Practices, and Focus on Virtual Patients Generation

Abstract

It is a well-known fact that clinical trials is a challenging process essentially for financial, ethical, and scientific concern. For twenty years, simulated clinical trials (SCT for short) has been introduced in the drug development. It becomes more and more popular mainly due to pharmaceutical companies which aim to optimize their clinical trials (duration and expenses) and the regulatory agencies which consider simulations as an alternative tool to reduce safety issues. The whole simulation plan is based on virtual patients generation. The natural idea to do so is to perform Monte Carlo simulations from the joined distribution of the covariates. This method is named Discrete Method. This is trivial when the parameters of the distribution are known, but, in practice, data available come from historical databases. A preliminary estimation step is necessary. For Discrete Method that step may be not effective, especially when there are a lot of covariates mixing continuous and categorical ones. In this chapter, simulation studies illustrate that the so-called Continuous Method may be a good alternative to the discrete one, especially when marginal distributions are moderately bi-modal.

Appendices

Appendix 1

For $i=1,2$, $\mu ^Y_i$ and $\sigma ^Y_i$ (resp. $\mu ^Z_i$ and $\sigma ^Z_i$) denote the expectation and the standard deviation of $(Y|X=i)$ (resp. $(Z|X=i)$) distribution, $\rho _i$ denotes the coefficient of correlation between $(Y|X=i)$ and $(Z|X=i)$, and $\pi$ denotes the probability that $X=1$. The relationships between conditional and marginal expectations are ${\mathbf{{E}} \left[ Y \right] } = \pi . \mu ^Y_1 + (1- \pi ) . \mu ^Y_2$ and ${\mathbf{{E}} \left[ Z \right] } = \pi . \mu ^Z_1 + (1- \pi ) . \mu ^Z_2$. The relationships between conditional and marginal variances are:

$$\begin{aligned} {\mathbf{{V}} \left[ Y \right] }&= {\mathbf{{E}} \left[ {\mathbf{{V}} \left[ Y|X \right] } \right] } + {\mathbf{{V}} \left[ {\mathbf{{E}} \left[ Y|X \right] } \right] },\\ {\mathbf{{V}} \left[ Y \right] }&= \pi (\sigma ^Y_1)^2 + (1- \pi ) (\sigma ^Y_2)^2 + \left( \pi (\mu ^Y_1)^2 + (1- \pi ) (\mu ^Y_2)^2 - (\pi \mu ^Y_1 + (1- \pi ) \mu ^Y_2)^2 \right) ,\\ {\mathbf{{V}} \left[ Z \right] }&= \pi (\sigma ^Z_1)^2 + (1- \pi ) (\sigma ^Z_2)^2 + \left( \pi (\mu ^Z_1)^2 + (1- \pi ) (\mu ^Z_2)^2 - (\pi \mu ^Z_1 + (1- \pi ) \mu ^Z_2)^2 \right) . \end{aligned}$$

The relationships between conditional covariances and covariance are:

$$\begin{aligned} {{\text {cov}}\left[ Y,Z \right] }&= \pi \sigma ^Y_1 \sigma ^Z_1 \rho _1 + (1- \pi ) \sigma ^Y_2 \sigma ^Z_2 \rho _2 + \left( \pi \mu ^Y_1 \mu ^Z_1 + (1- \pi ) \mu ^Y_2\mu ^Z_2 - {\mathbf{{E}} \left[ Y \right] } {\mathbf{{E}} \left[ Z \right] } \right) . \end{aligned}$$

Appendix 2

Consider Y a random variable log-normally-distributed with expectation (resp. standard deviation) denoted $\mu _Y$ (resp. $\sigma _Y$), Z a random variable $\mathscr {N}{(\mu _Z,\sigma _Z)}$ distributed, and $\rho _{Y,Z}$ the coefficient of correlation. In order to generate the random vector (Y, Z), consider $Y=\exp (U)$ and (U, Z) a Gaussian vector where U is $\mathscr {N}{(\mu _U,\sigma _U)}$ distributed and $\rho _{U,Z}$ denotes the coefficient of correlation. The parameters of (U, Z) and these of (Y, Z) are linked by the relationships:

$$\begin{aligned} \mu _Y&= {\mathbf{{E}} \left[ Y \right] } = {\mathbf{{E}} \left[ \exp (U) \right] } = \exp \left( \mu _U + \frac{\sigma _U^2}{2}\right) \\ \sigma _Y^2&= {\mathbf{{V}} \left[ Y \right] } = {\mathbf{{E}} \left[ \exp (2U) \right] } - ({\mathbf{{E}} \left[ \exp (U) \right] })^2 = \exp (\sigma _U^2-1) \exp \left( 2 \mu ^U + \sigma _U^2 \right) \end{aligned}$$

An application of Stein’s Lemma with $g(x) = \exp (x)$ we have:

$$\begin{aligned} \rho _{Y,Z}&= {{\text {cov}}\left[ Y,Z \right] } = {{\text {cov}}\left[ e^U,Z \right] } = {\mathbf{{E}} \left[ \exp (U) \right] } {{\text {cov}}\left[ U,Z \right] } =\mu _Y \rho _{U,Z} \end{aligned}$$

After some algebra, this yields to the following parametrization:

$$\begin{aligned} \mu _U = \ln ( \mu _Y ) -\frac{1}{2}\ln \left( 1+\frac{\sigma _Y^2}{(\mu _Y)^2}\right) , \quad \sigma _U^2 = \ln \left( 1+\frac{\sigma _Y^2}{(\mu _Y)^2} \right) \quad \text {and} \quad \rho _{U,Z} = \frac{\rho _{Y,Z}}{\mu _Y}. \end{aligned}$$