PD models

Overview

For the PK, consider the the drug concentration defined as follows: $$ C(t)=\sum_{j=1}^J\mathit{1}{{t-t_j>T{inf_j}}}\frac{d_{k,j}}{T_{inf_j}}\frac{1}{kV}(1-e^{-kT_{inf_j}})+\mathit{1}{{t-t_j\leq T{inf_j}\&t-t_j\geq0}}\frac{d_{k,j}}{T_{inf_j}}\frac{1}{kV}(1-e^{-k(t-t_j)}) $$

where $T_{inf_j}$ is the duration of the infusion of the $j^{th}$ administration, $V$ is the distribution volume, $Cl$ is the clearance of elimination, and $k$ is the micro-constant defined as $k = Cl/V.$ We assumed that the delay between successive doses was greater than the infusion duration, meaning that $t_{j+1} − t_j > T_{inf_J}$ for $j \in {1, \cdots, J − 1}.$

 library(StatComp21019)
 data(data)
 attach(data)
 t<-seq(from=0,to=35,by=0.1)
 plot(t,Concentration(t=t,d=rep(25,7)),type = "l",lty=2,ylab="Concentration ")
 lines(t,Concentration(t=t,d=dose2),type = "l")

For the PD, the objective was to model cytokine mitigation when intrapatient dose-escalation was implemented. The cytokine production is stimulated by the drug concentration but inhibited by cytokine exposure through the AUC. The cytokine response is defined as follows: $$ \frac{\mathrm{d}E(t)}{\mathrm{d}t}=\frac{E_{max}C(t)^H}{EC^{50}+C(t)^H}{1-\frac{I_{max}AUC_E(t)}{\frac{IC_{50}}{K^{J-1}}+AUC_E(t)}}-k_{deg}E(t) $$

where $AUC_E(t)$ is the cumulative cytokine exposure.$E_{max}$ is Maximum cytokine release rate.$EC^{50}$ is Drug exposure for half-maximum release.$H$ is Hill coefficient for cytokine release.$I_{max}$ is Maximum inhibition of cytokine release.$IC_{50}$ is Cytokine exposure for half-maximum inhibition.$k_{deg}$ is Degradation rate for cytokine.$K$ is Priming factor for cytokine release.

t<-seq(from=0,to=35,by=0.02)
 plot(t,Cytokine(t=t,d=rep(25,7))$EE,type="l",lty=2,ylab="E(t)")
 abline(h=max(Cytokine(t=t,d=rep(25,7))$EE),lty=2)
 lines(t,Cytokine(t=t,d=c(1,5,10,rep(25,4)))$EE,type="l")
 abline(h=max(Cytokine(t=t,d=c(1,5,10,rep(25,4)))$EE),lty=1)