Rate constants and primary PK parameters in NONMEM
Reviewer #3 questions the use of primary PK parameters (CL, Vss, ADVAN4 TRANS3) instead of rate constants. An explanation on the background of the NONMEM translator (TRANS) subroutine needs to be provided here.
The TRANSsubroutine in NM-TRAN performs the translation between user-chosen pharmacokinetic parameters computed in $PK and the set of parameters used internally in the kinetic equations computed in the ADVAN subroutine. For ADVANswhich implement a linear kinetic model, such as ADVAN4, these internal parameters are in fact the rate constants of the model (Beal (1992). The TRANS routine requested by the Co-Rapporteur, TRANS3, is a translator routine that performs a reparameterization of the primary PK parametersused in the NM-TRAN control stream (CL, Vss)to the rate constantsused by ADVAN4 (K, K23and K32) to compute the kinetic equations for a 2-compartment kinetic model. The translator routine used by the Applicant, TRANS1, is a "dummy" translator that does not perform any reparameterization of the basic PK parameters and is the default in NM-TRAN(e.g. Beal, Boeckmann and Sheiner 1992 p.6). It is therefore important to note that, whether the NONMEM control stream is specified using rate constants or primary PK parameters, the associated NM-TRAN translator routines (TRANS1 and TRANS3, respectively) ensure that the internal model fitting procedures compute the kinetic equations in terms of rate constants(Beal (1992), Boeckmann(1992), Beal (1998).
Hence, for every PK model developed with the use of primary PK parameters under ADVAN4 TRANS3in NM-TRAN, there exists an equivalent model specified in rate constants which is passed on to the PREDPP estimation step such that the following relationships hold(Beal (1992):
(1)
Typically, random effects are estimated on both CLand V, as well as covariates such as body weight, age and estimates of renal function. The presence of random effectson CLand Vcomplicates the translation to the internally used rate constants considerably, and necessitates a more mathematically involved analysis of the relationships between CL, V and K, the rate constant used the kinetic equations computed inside NONMEM. For a model specified in primary PK parameters in the NMTRAN control stream under ADVAN4 TRANS3, the random effects onV and CL together with any associated covariate effects are typically parameterized as follows:
, where
, where
Here TVV and TVCL represent the products of and with any covariates estimated on V and CL, respectively, while the random effects on V and CL, and , by definition have the following properties:
A model specified in rate constants such as used internally in the kinetic equations solved by PREDPP can be similarly parameterized:
, where
, where
with
From (1) it follows that these two models are equivalent in CL, V and K when It can be shown that is the case when
(2)
and
(3)
In terms of the covariance matrix these conditions imply
(4)
Equations 2, 3 and 4 demonstrate that specifying the model in terms of primary PK parameters under TRANS3, introduces an additional, hidden layer of complexity between the model thus specified and the model that is actually computed inside NONMEM. For example, it follows from equation 2 that any covariates evaluated on Vwill internally also be evaluated (inversely) on K, although that is not apparent from the TRANS3 control stream. This allows for computational dependencies between the covariate effects estimated on Vand those estimated on CL, which remain hidden for the unsuspecting end-user.
Moreover, the random effects structure specified under TRANS3 may appear simpler than the one evaluated internally by PREDPP. It is good practice in model development to start out with a simple model and to progress to more complex models. Therefore, when random effects are identified on both CL and V, we usually start with a model that assumes a simple diagonal variance-covariance matrix between and(i.e. . However, it follows from equation 4 that the variance-covariance matrix of the corresponding model that is evaluated internally by NONMEM assuming is considerably more complex, with :
In conclusion, when deciding which TRANS routine to use in our population PK model development, we are faced with a trade-off between pharmacological interpretability (specifying the control stream in terms of primary PK parameters which are transformed by TRANS3 to the rate constantsused internally in the kinetic equations) and statistical transparency (specifying the control stream directly using the internally used rate constants which are passed on unchanged by the dummy translator TRANS1). For the submitted PPK model for canagliflozin, the Authors opted for maximal statistical transparency during model development using TRANS1 and subsequently performed the transformation of K to the more pharmacologically meaningful CL.
Redeveloping the submitted population PK model using primary PK parameters
By reversing the transformations performed by TRANS3, any model developed with the use of rate constants under ADVAN4 TRANS1 can be redeveloped into an equivalent model written in terms of primary PK parameters that can be run under ADVAN4 TRANS3, such that the kinetic equations evaluated in PREDPP are identical for both models. Hence, the final PPK model submitted here (in the following referred to as Model A) was redeveloped into an equivalent, internally identical model expressed in primary PK parameters (Model B) by reversing Eqs. 1 to 4 as below.
From (1) it follows that model A can be redeveloped into an equivalent, internally identical model expressed in primary PK parameters (Model B) when the following relationships hold:
(5)
As previously, using the same notation, it can be shown that this is the case when
(6)
and
(7)
In terms of the covariance matrix these conditions imply
(8)
Because in the final model submitted by the Applicant , the variance-covariance matrix of the internally identical model as coded in terms of primary PK parameters can be simplified to
Table 1 shows that when Model B was run under ADVAN4 TRANS3, it yielded an identical objective function value (OFV) as Model A, thus confirming that the internal kinetic equations underlying both versions of the model are identical.
Table 1:Modeling resultsModel / Description / n1 / OFV / ΔOFV2
A / Final PPK model for canagliflozin submitted by the Applicant run under ADVAN4 TRANS1 / -8516.552
B / Model A redeveloped using primary PK parameters run under ADVAN4 TRANS3 / 0 / -8516.552 / 0.000
1Number of additional parameters relative to parent model
2 Difference in OFV relative to parent model, * indicates statistical significance at p< 0.001
The shrinkage on CL/Fin Model B was calculated from the output of model B using Eqs. 7 and 8 as follows (Savic 2009):
It was found that the shrinkage on CL/F in Model B amounted to 21%, which is acceptable (not greater than 20-30%, Savic 2009), thus upholding the evaluation of the clinical relevance of the identified covariates as reported. This indicates that the shrinkage on CL/Fas estimated under TRANS3 is determined primarily by the shrinkage on the internally estimated rate parameter K (21% in the final submitted PPK model) and not by the shrinkage on V/F (>55% in the final submitted PPK model).
From Eq. (6) it follows that the covariate structure on CL/Fin Model B is identical to the product of the covariates identified on K and V/Fin Model A (including their estimated exponents). Hence, CL/Fin Model B not only is affected by significant covariate effects of eGFR and Dose, but also Body Weight, Sex and Age.
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24-Dec-2012