$$ \newcommand{\esp}[1]{\mathbb{E}\left(#1\right)} \newcommand{\var}[1]{\mbox{Var}\left(#1\right)} \newcommand{\deriv}[1]{\dot{#1}(t)} \newcommand{\prob}[1]{ \mathbb{P}\!(#1)} \newcommand{\eqdef}{\mathop{=}\limits^{\mathrm{def}}} \newcommand{\by}{\boldsymbol{y}} \newcommand{\bc}{\boldsymbol{c}} \newcommand{\bpsi}{\boldsymbol{\psi}} \def\pmacro{\texttt{p}} \def\like{{\cal L}} \def\llike{{\cal LL}} \def\logit{{\rm logit}} \def\probit{{\rm probit}} \def\one{{\rm 1\!I}} \def\iid{\mathop{\sim}_{\rm i.i.d.}} \def\simh0{\mathop{\sim}_{H_0}} \def\df{\texttt{df}} \def\res{e} \def\xomega{x} \newcommand{\argmin}[1]{{\rm arg}\min_{#1}} \newcommand{\argmax}[1]{{\rm arg}\max_{#1}} \newcommand{\Rset}{\mbox{$\mathbb{R}$}} \def\param{\theta} \def\setparam{\Theta} \def\xnew{x_{\rm new}} \def\fnew{f_{\rm new}} \def\ynew{y_{\rm new}} \def\nnew{n_{\rm new}} \def\enew{e_{\rm new}} \def\Xnew{X_{\rm new}} \def\hfnew{\widehat{\fnew}} \def\degree{m} \def\nbeta{d} \newcommand{\limite}[1]{\mathop{\longrightarrow}\limits_{#1}} \def\ka{k{\scriptstyle a}} \def\ska{k{\scriptscriptstyle a}} \def\kel{k{\scriptstyle e}} \def\skel{k{\scriptscriptstyle e}} \def\cl{C{\small l}} \def\Tlag{T\hspace{-0.1em}{\scriptstyle lag}} \def\sTlag{T\hspace{-0.07em}{\scriptscriptstyle lag}} \def\Tk{T\hspace{-0.1em}{\scriptstyle k0}} \def\sTk{T\hspace{-0.07em}{\scriptscriptstyle k0}} \def\thalf{t{\scriptstyle 1/2}} \newcommand{\Dphi}[1]{\partial_\pphi #1} \def\asigma{a} \def\pphi{\psi} \newcommand{\stheta}{{\theta^\star}} \newcommand{\htheta}{{\widehat{\theta}}}$$

Introduction

We aim to implement several of the models used in this excellent tutorial on TMDD models:

Dua P, Hawkins E, van der Graaf P. A Tutorial on Target-Mediated Drug Disposition (TMDD) Models. CPT: Pharmacometrics & Systems Pharmacology. 2015;4(6):324-337. link

Following Dua et al., ``Target-mediated drug disposition (TMDD) is the phenomenon in which a drug binds with high affinity to its pharmacological target site (such as a receptor) to such an extent that this affects its pharmacokinetic characteristics.’’

For more biological and mathematical details about the models used here, readers are referred to the paper of Dua et al. and the many references therein.

All the R codes and the interactive Shiny applications (generated using shinymlx) used in this webpage are available here


A one compartment model

The model

The one-compartment model was first proposed by Mager and Jusko in 2001:

Mager, D. E., & Jusko, W. J. (2001). General pharmacokinetic model for drugs exhibiting target-mediated drug disposition. Journal of pharmacokinetics and pharmacodynamics, 28(6), 507-532. [PudMed]

Dua et al. propose a mathematical description of this model and a Berkeley Madonna code for this base TMDD model.

\[ \begin{alignat}{2} \frac{dL}{dt} &= - k_{eL}L - k_{on} LR + k_{off}P && \quad ; \quad L(0)= 0 \\ \frac{dR}{dt} &= k_{in} - k_{out}R - k_{on} LR + k_{off}P && \quad ; \quad R(0)=R_0 =k_{in}/k_{out} \\ \frac{dP}{dt} &= k_{on} LR - k_{off}P - k_{eP}P && \quad ; \quad P(0)=0 \end{alignat} \] This model was now implemented in Mlxtran:

[LONGITUDINAL]
input = {kel, kep, kon, kout, koff, Vc, R0}

PK:
; Dosemg: Dose injected into central compartment (mg/kg)
; MWlig=150000; molecular weight of ligand (Da) 
; (Dosemg*1e-3/MWlig)*1e9 ; Dose conversion (nmol/kg)

depot(target=L, p=100/15/Vc)  ; L is a concentration (nM)

EQUATION:
kin = kout*R0 ; Synthesis of receptor (nM/day)

;Initial Conditions
L_0 = 0
R_0 = R0
P_0 = 0

;Ordinary Differential Equations
ddt_L = -kel*L-kon*L*R+koff*P 
ddt_R = kin-kout*R-kon*L*R+koff*P
ddt_P = kon*L*R-koff*P-kep*P

Comparison of 2 dosing regimens

We perform the same numerical experiment in order to show the nonlinear (dose- and time-dependent) behavior of the pharmacokinetics of the ligand.

Hide the R code
library(gridExtra)

p <- c(
kel  = 0.024, # Elimination of ligand (1/day)
kep  = 0.201, # Elimination of complex (1/day)
kout = 0.823, # Elimination of receptor (1/day)
koff = 0.9,   # Dissociation rate (1/day)
kon  = 0.592, # Binding rate (1/(nMday))
R0   = 2.688,  # inital concentration of receptor  in central compartment (nM)
Vc   = 0.04  # volume of central compartment (L/kg)
)

# Dosing Regimen
Dosemg <- c(0.05, 2)
adm1   <- list(amount=Dosemg[1], time=0)
adm2   <- list(amount=Dosemg[2], time=0)
g1     <- list(treatment = adm1)
g2     <- list(treatment = adm2)

# Observations
tobs <- seq(0,100,by=1)
out  <- list(name = c('L', 'R', 'P'), time = tobs)

# Run Simulx
res <- simulx(model    = "model/tmdd1.txt",
               parameter= p,
               output   = out,
               group    = list(g1, g2))

# Plot results
pl1 <- ggplot(data=res$L) + geom_line(aes(x=time, y=L, colour=group), size=0.5) + 
  theme(legend.position="none") + scale_y_log10() 
pl2 <- ggplot(data=res$R) + geom_line(aes(x=time, y=R, colour=group), size=0.5) + 
  scale_colour_discrete(name="Dose\n(mg/kg)", breaks=c("1","2"),labels=Dosemg) +
   theme(legend.title=element_text(size=10), legend.position=c(0.8,0.2))  + scale_y_log10() 
pl3 <- ggplot(data=res$P) + geom_line(aes(x=time, y=P, colour=group), size=0.5) + 
   theme(legend.position="none") + scale_y_log10() 

grid.arrange(pl1,pl2,pl3, nrow=1)


A Shiny application

Hide the R code
adm <- list(
  tfd    = list(widget="slider", value=0,  min=0,  max=24, step=2),
  nd     = list(widget="slider", value=1,  min=1,  max=10, step=1),
  ii     = list(widget="slider", value=12, min=4,  max=24, step=2),
  amount = list(widget="slider", value=1,  min=.1, max=2,  step=.1)
)

out1 <- list(name = 'L', time = tobs)
out2 <- list(name = 'R', time = tobs)
out3 <- list(name = 'P', time = tobs)

shiny.app <- shinymlx(model     = "model/tmdd1.txt",
                      parameter = list(p[1:4],p[5:7]),
                      output    = list(out1, out2, out3),
                      treatment = adm,
                       style     = "navbar2",
                       title     = "TMDD - Model 1")

shiny::runApp(shiny.app)



A two compartments model

This model, first described by Mager and Jusko in 2001, assumes that only unbound drug can distribute into a peripheral tissue compartment, while other ligands remain in the central compartment. See Dua et al. for more details



Quasi Equilibrium (QE) model

Quasi-equilibrium pharmacokinetic model for drugs exhibiting target-mediated drug disposition. Mager DE, Krzyzanski W, Pharm Res. 2005 Oct; 22(10):1589-96. [PudMed]

The QE model proposed by Mager and Krzyzanski assumes that equilibrium between the binding and dissociation of the complex has been achieved. See Dua et al. for more details



Quasi-steady-state (QSS) model

Parameter values proposed by Dua et al.

Approximations of the target-mediated drug disposition model and identifiability of model parameters. Gibiansky L, Gibiansky E, Kakkar T, Ma P, J Pharmacokinet Pharmacodyn. 2008 Oct; 35(5):573-91. [PudMed]

The QSS model assumes that the binding rate is balanced by the sum of the dissociation and internalization rates. See Dua et al. for more details


Parameter values for denosumab

Population pharmacokinetic analysis of denosumab in patients with bone metastases from solid tumours. Gibiansky L, Sutjandra L, Doshi S, Zheng J, Sohn W, Peterson MC, Jang GR, Chow AT, P?rez-Ruixo JJ Clin Pharmacokinet. 2012 Apr 1; 51(4):247-60. [PudMed]


Michaelis-Menten (MM) model

The MM model is derived from the Michaelis-Menten equation for enzyme kinetics, which relates reaction rate to concentration. See Dua et al. for more details


Multiple targets TMDD model

Target-mediated drug disposition model for drugs that bind to more than one target. Gibiansky L, Gibiansky E, J Pharmacokinet Pharmacodyn. 2010 Aug; 37(4):323-46. [PudMed]

Some drugs have the ability to bind to multiple targets in one cell; for example, they could bind to both cell membrane (M) and soluble (S) targets. See Dua et al. for more details



FcRn mediated recycling

Pharmacokinetics of anti-hepcidin monoclonal antibody Ab 12B9m and hepcidin in cynomolgus monkeys. Xiao JJ, Krzyzanski W, Wang YM, Li H, Rose MJ, Ma M, Wu Y, Hinkle B, Perez-Ruixo JJ, AAPS J. 2010 Dec; 12(4):646-57. [PudMed]

Xiao et al. extended the QE two-compartment model with a depot to include the process of FcRn-mediated endosomal recycling. See Dua et al. for more details