### Non-Compartment Model to One-Compartment Model Transformation

When a drug follows a one-compartment model of oral dose (1), the following non-compartment model PK parameters,

**w** = (

*AUC, T*
_{
max,
}
*T*
_{
1/2
}), are necessary to recover the one-compartment model parameters, β = (

*k*
_{
a,
}
*k*
_{
e,
}
*V*).

where,

*F* is an assumed known bioavailability, and dose denotes the oral dose. If however, only oral clearance,

*CL*
_{
po
} is reported, instead of

*AUC*, then

*CL*
_{
po
}
*= V × K*
_{
e
}. On the other hand, when dosing is through IV, only

**w** = (

*AUC, T*
_{
1/2
}), are necessary to recover the one compartment (3), with

**β** = (

*k*
_{
e,
}
*V*). The transformation formulas are defined in (4).

Similarly, if *CL*
_{
iv
} is reported, instead of *AUC*, then *CL*
_{
IV
}
*= V × k*
_{
e
}. These one-compartment-model and non-compartment model parameters and transformation were defined and discussed in great detail by [16].

### Non-Compartment Model to Two-Compartment Model Transformation

If a drug’s pharmacokinetics follows a two-compartment model with oral dose (5), the following non-compartment model PK parameters,

**w** = (

*Vd*,

*AUC, T*
_{
max,
}
*CL*
_{
iv,
}
*T*
_{
1/2,slow,
}
*T*
_{
1/2,fast
}), are necessary to recover the two-compartment model parameters,

**β** = (

*k*
_{
a,
}
*k*
_{
e,
}
*V*
_{
1,
}
*k*
_{
12,
}
*k*
_{
21
}). Their transformations are defined in (6).

If a drug’s pharmacokinetics follows a two-compartment model with IV dose (7), the following non-compartment model PK parameters,

**w** = (

*Vd*,

*AUC, CL*
_{
iv,
}
*T*
_{
1/2, slow,
}
*T*
_{
1/2, fast
}), are necessary to recover the two-compartment model parameters,

**β** = (

*k*
_{
e,
}
*V*
_{
1,
}
*k*
_{
12,
}
*k*
_{
21
}). Their transformations are defined in (8).

### A Multivariate Nonlinear Mixed Effect Model (Model Specification)

Based on the multiple transformation equations between non-compartment model PK parameters and one or two compartment models, a multivariate nonlinear mixed effect model is established to estimate the population level PK parameters and their between study variances. Denote

*w*
_{
jk
} as the observed

*j* th non-compartment PK parameter (

*j=1*,…,

*J*
_{
k
}) from study

*k* (

*k*=

*1,..,K*). Please note that not every study published all of the non-compartment parameters, hence

*J*
_{
k
} varies from study to study.

**β**
_{
k
} is the study level compartment-model PK parameter vector, and

*g*
_{
j
} (

**β**
_{
k
}) represents the transformation function. Because non-compartment model PK parameter,

*w*
_{
jk
}, is usually published in the form of a sample mean, model (9) shows that its variance is

, where

is the within study variance (assumed to be homogeneous across studies), and

*n*
_{
k
} is study

*k* sample size

Model (9) also shows that the observed non-compartment model parameters,
, are independent. This is a multivariate nonlinear regression model.

Study level compartment model parameter

**β**
_{
k
} is assumed to follow multivariate normal distribution (10), in which

**µ** is the population PK parameter vector and

**Ω**
_{
k
} is its general covariance matrix.

.

The joint likelihood of population/subject parameters and their covariance is shown in equation (

11).

where
is a *J×1* (
) observed non-compartment model PK parameter vector;
is a *J×p* indicator matrix, and **X**
_{
k
} is a *J*
_{
k
}
*×p* matrix indicating the corresponding transformation function; **g**(.) is a *p×1* transformation function vector;
is a study level compartment-model PK parameter vector;
is a diagonal *J×J* covariance matrix for **W**, *a*nd
;
is a *Kp×p* design matrix relating study-specific parameter **β** to population parameter **µ**, and **I**
_{
k
} is an identity matrix; and
is a *Kp×Kp* covariance matrix for study-specific parameter **β**.

This multivariate nonlinear mixed model (11) is different from the conventional univariate nonlinear mixed model [17] structurally in the additional design matrix **X** in front of the nonlinear function ( i.e. transformation function **g**(.)). Model (11) is a meta-analysis approach, in which sample mean non-compartment model PK parameters are formulated. Among the existing nonlinear mixed model meta-analysis literatures, some dealt with the subject-level data from multiple studies [18, 19]; the others dealt with sample mean drug concentration data [20, 21]; and none of them discussed the meta-analysis on summarized PK parameters through the non-compartment model.

### A Multivariate Nonlinear Mixed Effect Model (Estimation and Inference)

As a conditional first order linearization approach provides the least biased estimate in estimating the PK parameter with comparable efficiency [22, 23]), it is chosen as the estimation approach for this multivariate nonlinear mixed model. This conditional first order linearization approach was firstly introduced by Lindstrom and Bates [24]. We revise their derivation based on our special meta-analysis multivariate nonlinear mixed model (11). This two-step estimation scheme is described as following.

Step 1: given the current estimate of variance component

and

, minimize the following objective function,

*L*
_{
1
}, with respect to (

**β, µ**).

.

Computationally, minimizing

*L*
_{
1
} on (

**β, µ**) is an iterative process. Within each iteration, a linearization is applied to

**Xg(β)** with respect to

**β**, and a linear mixed model (13) is fitted [

24].

Parameters (

**μ, b, β**)’s estimates and their covariance are

Step 2: given the current estimate,

, minimize the following objective function,

*L*
_{
2
} , with respect to

**θ**, which is the variance component parameter vector in (

**Ω, Σ**), and it is of dimension

*q*.

This

*L*
_{
2
} likelihood function is the restricted maximum likelihood for variance component estimates. The scores and the elements of information matrix for

**θ** are defined in (17).

Hence, **θ** can be estimated through an iterative Fisher algorithm. An alternative derivation of this two-step first order linearization is through a second order Laplace’s approximation [25–27].