Population Pharmacokinetic Modeling in the Presence of Missing Time-Dependent Covariates: Impact of Body Weight on Pharmacokinetics of Paracetamol in Neonates

Abstract

Body weight is the primary covariate in pharmacokinetics of many drugs and dramatically changes during the first weeks of life of neonates. The objective of this study is to determine if missing body weights in preterm and term neonates affect estimates of model parameters and which methods can be used to improve performance of a population pharmacokinetic model of paracetamol. Data for our analysis were obtained from previously published studies on the pharmacokinetics of intravenous paracetamol in neonates. We adopted a population model of body weight change in neonates to implement three previously introduced methods of handling missing covariates based on data imputation, likelihood function modification, and full random effects modeling. All models were implemented in NONMEM 7.4, and population parameters were estimated using the FOCE method. Our major finding was that missing body weights minimally affect population estimates of pharmacokinetic parameters but do affect the covariate relationship parameters, particularly the one describing dependence of clearance on body weight. None of the tested methods changed estimates of between-subject variability nor impacted the predictive performance of the model. Our analysis shows that a modeling approach towards handling missing covariates allows borrowing information gathered in various studies as long as they target the same population. This approach is particularly useful for handling time-dependent missing covariates.

Appendices

Appendix 1. The likelihood function for IMPUT approach

Given the independence of random variables ε_Cij, η_CLj, η_Vj, the likelihood function of observing APAP plasma concentrations $\boldsymbol{C}={\left\{{C}_{ij}\right\}}_{1\le i\le N,1\le j\le {n}_i}$ at times $\boldsymbol{t}={\left\{{t}_{ij}\right\}}_{1\le i\le N,1\le j\le {n}_i}$ with imputed body weights ${\boldsymbol{BW}}_{\boldsymbol{m}}={\left\{{BW}_{ij}\right\}}_{1\le i\le N,1\le j\le {m}_i}$ is of the following form (28):

$$ p\left(\boldsymbol{C};{\boldsymbol{BW}}_{\boldsymbol{m}},\boldsymbol{\theta}, \boldsymbol{\gamma}, \boldsymbol{t}\right)=\prod \limits_{i=1}^N\underset{{\mathbb{R}}^2}{\int}\prod \limits_{j=1}^{n_i}\varphi \left({C}_{ij};\frac{A\left({t}_{ij}\right)}{V\left({t}_{ij}\right)},{\sigma}_C^2\right){\Phi}_2\left(\boldsymbol{\eta}; \mathbf{0},\boldsymbol{\Omega} \right)d\boldsymbol{\eta} $$

(14)

where θ = (CL₀, V₀, BW_CL, BW_V) and $\boldsymbol{\gamma} =\left({\sigma}_C^2,{\omega}_{CL}^2,{\omega}_V^2\right)$ are vectors of model parameters. The function φ describes the normal distribution

$$ \varphi \left(y;\mu, {\sigma}^2\right)=\frac{1}{\sqrt{2\pi {\sigma}^2}}\mathit{\exp}\left(-\frac{{\left(y-\mu \right)}^2}{2{\sigma}^2}\right) $$

(15)

and Φ is a probability density function for the multivariate normal distribution:

$$ {\Phi}_2\left(\boldsymbol{\eta}; \mathbf{0},\boldsymbol{\Omega} \right)={\left(\frac{1}{\sqrt{2\pi }}\right)}^2\frac{1}{\sqrt{\det \boldsymbol{\Omega}}}\mathit{\exp}\left(-\frac{1}{2}{\boldsymbol{\eta}}^{\prime }{\boldsymbol{\Omega}}^{-\mathbf{1}}\boldsymbol{\eta} \right) $$

(16)

and $\boldsymbol{\Omega} =\mathbf{\operatorname{diag}}\left({\omega}_{CL}^2,{\omega}_V^2\right)$.

Appendix 2. The likelihood function for LIKELIHOOD approach

Let for subject i^th 1 ≤ j ≤ m_i index times for which BWs are measured and m_i + 1 ≤ j ≤ n_i index times for missing BWs. Also, let 1 ≤ i ≤ N_V be reserved for neonates born by vaginal delivery, and N_V + 1 ≤ i ≤ N for neonates born by cesarean delivery. Then, the likelihood function of observing APAP plasma concentrations becomes (26):

$$ {\displaystyle \begin{array}{c}\left(\boldsymbol{C};\boldsymbol{BW},\boldsymbol{\theta}, \boldsymbol{\gamma}, \boldsymbol{t}\right)=\prod \limits_{i=1}^N\underset{{\mathbb{R}}^2}{\int}\prod \limits_{j=1}^{m_i}\varphi \left({C}_{ij};\frac{A\left({t}_{ij}\right)}{V\left({t}_{ij}\right)},{\sigma}_C^2\right){\Phi}_2\left(\boldsymbol{\eta}; \mathbf{0},\boldsymbol{\Omega} \right)d\boldsymbol{\eta} \cdot \\ {}\prod \limits_{i=1}^N\underset{{\mathbb{R}}^9}{\int}\prod \limits_{j={m}_i+1}^{n_i}\varphi \left({C}_{ij};\frac{A\left({t}_{ij}\right)}{V\left({t}_{ij}\right)},{\sigma}_C^2\right){\Phi}_9\left(\boldsymbol{\eta}; \mathbf{0},{\boldsymbol{\Omega}}_{\boldsymbol{C}}\right)d\boldsymbol{\eta} \end{array}} $$

(17)

Here, $\boldsymbol{BW}={\left\{{BW}_{ij}\right\}}_{1\le i\le N,1\le j\le {m}_i}$ denotes the observed BWs, and

$$ {\boldsymbol{\Omega}}_{\boldsymbol{C}}=\mathbf{\operatorname{diag}}\left({\omega}_{CL}^2,{\omega}_V^2,{\omega}_{kinbase}^2,{\omega}_{koutmax}^2,{\omega}_{koutbase}^2,{\omega}_{T50}^2,{\omega}_{BW0}^2,{\omega}_{TlagC}^2,{\omega}_{TlagV}^2\right) $$

Only model parameters θ and γ were estimated, while the fixed effect parameters for the BW model and the entries of Ω_C other than ${\omega}_{CL}^2,{\omega}_V^2$ were fixed.

Appendix 3. The Likelihood Function for FREM Approach

The joint likelihood function for observed APAP plasma concentrations $\boldsymbol{C}={\left\{{C}_{ij}\right\}}_{1\le i\le N,1\le j\le {n}_i}$and observed BWs $\boldsymbol{BW}={\left\{{BW}_{ij}\right\}}_{1\le i\le N,1\le j\le {m}_i}$becomes:

$$ p\left(\boldsymbol{C},\boldsymbol{BW};{\boldsymbol{\theta}}_{\boldsymbol{F}},{\boldsymbol{\gamma}}_{\boldsymbol{F}},\boldsymbol{t}\right)=\prod \limits_{i=1}^N\underset{{\mathbb{R}}^2}{\int }{\prod}_{j=1}^{m_i}\varphi \left({C}_{ij};\frac{A\left({t}_{ij}\right)}{V\left({t}_{ij}\right)},{\sigma}_C^2\right){\Phi}_2\left(\boldsymbol{\eta}; \mathbf{0},\boldsymbol{\Omega} \right)d\boldsymbol{\eta} \bullet \kern5.75em \prod \limits_{i=1}^N\underset{{\mathbb{R}}^7}{\int}\prod \limits_{j=1}^{m_i}\varphi \left({BW}_{ij}; BW\left({t}_{ij}\right),{\sigma}_{BW}^2 BW{\left({t}_{ij}\right)}^2\right){\Phi}_7\left(\boldsymbol{\eta}; \mathbf{0},{\boldsymbol{\Omega}}_{\boldsymbol{W}}\right)d\boldsymbol{\eta} $$

(18)

where

$$ {\displaystyle \begin{array}{c}{\boldsymbol{\theta}}_{\boldsymbol{F}}=\Big(C{L}_0,{V}_0,B{W}_{CL},B{W}_V,{k}_{inbase},{k}_{inPNA},{k}_{outmax},{k}_{outbase},{k}_{outPNA},\\ {}{k}_{inbase GA},{T}_{50},H,{W}_0,{T}_{lagV},{T}_{lagC},B{W}_{0 SEX},B{W}_{0 GA}\Big)\end{array}} $$

and

$$ {\boldsymbol{\gamma}}_{\boldsymbol{F}}=\left({\sigma}_C^2,{\sigma}_{BW}^2,{\omega}_{CL}^2,{\omega}_V^2,{\omega}_{kinbase}^2,{\omega}_{koutmax}^2,{\omega}_{koutbase}^2,{\omega}_{T50}^2,{\omega}_{BW0}^2,{\omega}_{TlagC}^2,{\omega}_{TlagV}^2\right) $$

are vectors of model parameters. Here

$$ {\boldsymbol{\Omega}}_{\boldsymbol{W}}=\mathbf{\operatorname{diag}}\left({\omega}_{kinbase}^2,{\omega}_{koutmax}^2,{\omega}_{koutbase}^2,{\omega}_{T50}^2,{\omega}_{BW0}^2,{\omega}_{TlagC}^2,{\omega}_{TlagV}^2\right) $$

Only CL₀, V₀, BW_CL, BW_V, W₀, BW_0SEX, BW_0GA, and ${\sigma}_C^2,{\sigma}_{BW}^2,{\omega}_{CL}^2,{\omega}_V^2,{\omega}_{BW0}^2$were estimated, while the remaining model parameters were fixed.