--- title: "Drawing PK curves with pmxTools" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Drawing PK curves with pmxTools} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ```{r, include = FALSE} knitr::opts_chunk$set( collapse = TRUE, comment = "#>" ) ``` ```{r setup, warning=F, message=F} library(pmxTools) library(patchwork) ``` `pmxTools` includes functions for constructing pharmacokinetic (PK) concentration-time curves for a range of common compartmental systems using PK parameters and microconstants. The underlying solutions are based on Julie Bertrand and France MentrĂ©'s [Mathematical Expressions of the Pharmacokinetic and Pharmacodynamic Models implemented in the Monolix software](https://www.facm.ucl.ac.be/cooperation/Vietnam/WBI-Vietnam-October-2011/Modelling/Monolix32_PKPD_library.pdf), published in 2008. The options available are by necessity confined to one-, two- and three-compartmental linear models with intravenous bolus, infusion or oral dosing (the latter with zero- or first-order absorption). If more complex systems are needed, you are probably better off using an ODE-based simulation package such as [`rxode2`](https://nlmixr2.github.io/rxode2/) or [`mrgSolve`](https://mrgsolve.org/). ## Specifying model parameters Models take dose, time and PK parameters as inputs. Names for some PK parameters, especially volumes and intercompartmental clearances, vary by model type. Parameter | Description | Used for ----------|----------------------------------------------------|------------------------------------------------ `t` | Time | Single-dose `tad` | Time after last dose | Steady state `dose` | Dose | All `CL` | Clearance | All `V`, `V1` | Central volume of distribution | All `V2` | First peripheral volume of distribution | Two- and three-compartment models `V3` | Second peripheral volume of distribution | Three-compartment models `Q`, `Q1`, `Q2` | Intercompartmental clearance (between V1 and V2) | Two- and three-compartment models `Q3` | Intercompartmental clearance (between V1 and V3) | Three-compartment models `tlag` | Lag time | Models with lag time `tinf` | Duration of infusion | Models with infusion `dur` | Duration of zero-order absorption | Models with zero-order absorption `ka` | First-order absorption rate | Models with first-order absorption ## One-compartment, single dose Available functions include: Model | Variant | Call ------------------------------|--------------------------------------------|---------------------------------------- 1-compartment | Single-dose, bolus | `calc_sd_1cmt_linear_bolus(t, dose, CL, V)`   | Single-dose, infusion | `calc_sd_1cmt_linear_infusion(t, dose, CL, V, tinf)`   | Single-dose, zero-order oral absorption | `calc_sd_1cmt_linear_oral_0(t, dose, CL, V, dur)`   | Single-dose, zero-order oral absorption with lag time | `calc_sd_1cmt_linear_oral_0_lag(t, dose, CL, V, dur, tlag)`   | Single-dose, first-order oral absorption | `calc_sd_1cmt_linear_oral_1(t, dose, CL, V, ka)`   | Single-dose, first-order oral absorption with lag time | `calc_sd_1cmt_linear_oral_1_lag(t, dose, CL, V, ka, tlag)` For example - ```{r 1cptdemo, fig.height=10, fig.width=7,message=FALSE} library(ggplot2) t <- seq(0, 24, by=0.1) df1 <- data.frame(t = t, Cp1 = calc_sd_1cmt_linear_bolus(t = t, dose = 600, CL = 5, V = 20), Cp2 = calc_sd_1cmt_linear_infusion(t = t, dose = 600, CL = 5, V = 20, tinf=2), Cp3 = calc_sd_1cmt_linear_oral_0(t = t, dose = 600, CL = 5, V = 20, dur=2), Cp4 = calc_sd_1cmt_linear_oral_0_lag(t = t, dose = 600, CL = 5, V = 20, dur=2, tlag=1), Cp5 = calc_sd_1cmt_linear_oral_1(t = t, dose = 600, CL = 5, V = 20, ka=1.5), Cp6 = calc_sd_1cmt_linear_oral_1_lag(t = t, dose = 600, CL = 5, V = 20, ka=1.5, tlag=1)) p1.1 <- ggplot(df1, aes(t, Cp1)) + geom_line() + labs(title="Bolus dose") p1.2 <- ggplot(df1, aes(t, Cp2)) + geom_line() + labs(title="Infusion") p1.3 <- ggplot(df1, aes(t, Cp3)) + geom_line() + labs(title="Zero-order oral") p1.4 <- ggplot(df1, aes(t, Cp4)) + geom_line() + labs(title="Zero-order oral with lag time") p1.5 <- ggplot(df1, aes(t, Cp5)) + geom_line() + labs(title="First-order oral") p1.6 <- ggplot(df1, aes(t, Cp6)) + geom_line() + labs(title="First-order oral with lag time") p1.1 + p1.2 + p1.3 + p1.4 + p1.5 + p1.6 + plot_layout(nrow=3) ``` ## One-compartment, steady state Available functions include: Model | Variant | Call ------------------------------|--------------------------------------------|---------------------------------------- 1-compartment | Steady state, bolus | `calc_ss_1cmt_linear_bolus(tad, tau, dose, CL, V)`   | Steady state, infusion | `calc_ss_1cmt_linear_infusion(tad, tau, dose, CL, V, tinf)`   | Steady state, zero-order oral absorption | `calc_ss_1cmt_linear_oral_0(tad, tau, dose, CL, V, dur)`   | Steady state, zero-order oral absorption with lag time | `calc_ss_1cmt_linear_oral_0_lag(tad, tau, dose, CL, V, dur, tlag)`   | Steady state, first-order oral absorption | `calc_ss_1cmt_linear_oral_1(tad, tau, dose, CL, V, ka)`   | Steady state, first-order oral absorption with lag time | `calc_ss_1cmt_linear_oral_1_lag(tad, tau, dose, CL, V, ka, tlag)` For example - ```{r 1cptssdemo, fig.height=10, fig.width=7,message=FALSE} t <- seq(0, 24, by=0.1) df1ss <- data.frame(t = t, Cp1 = calc_ss_1cmt_linear_bolus(t = t, tau = 24, dose = 600, CL = 1, V = 20), Cp2 = calc_ss_1cmt_linear_infusion(t = t, tau = 24, dose = 600, CL = 1, V = 20, tinf=2), Cp3 = calc_ss_1cmt_linear_oral_0(t = t, tau = 24, dose = 600, CL = 1, V = 20, dur=2), Cp4 = calc_ss_1cmt_linear_oral_0_lag(t = t, tau = 24, dose = 600, CL = 1, V = 20, dur=2, tlag=1.5), Cp5 = calc_ss_1cmt_linear_oral_1(t = t, tau = 24, dose = 600, CL = 1, V = 20, ka=1.5), Cp6 = calc_ss_1cmt_linear_oral_1_lag(t = t, tau = 24, dose = 600, CL = 1, V = 20, ka=1.5, tlag=1.5)) p1.1ss <- ggplot(df1ss, aes(t, Cp1)) + geom_line() + labs(title="Bolus dose") p1.2ss <- ggplot(df1ss, aes(t, Cp2)) + geom_line() + labs(title="Infusion") p1.3ss <- ggplot(df1ss, aes(t, Cp3)) + geom_line() + labs(title="Zero-order oral") p1.4ss <- ggplot(df1ss, aes(t, Cp4)) + geom_line() + labs(title="Zero-order oral with lag time") p1.5ss <- ggplot(df1ss, aes(t, Cp5)) + geom_line() + labs(title="First-order oral") p1.6ss <- ggplot(df1ss, aes(t, Cp6)) + geom_line() + labs(title="First-order oral with lag time") p1.1ss + p1.2ss + p1.3ss + p1.4ss + p1.5ss + p1.6ss + plot_layout(nrow=3) ``` ## Two-compartment, single dose Available functions include: Model | Variant | Call ------------------------------|--------------------------------------------|---------------------------------------- 2-compartment | Single-dose, bolus | `calc_sd_2cmt_linear_bolus(t, dose, CL, V1, V2, Q)`   | Single-dose, infusion | `calc_sd_2cmt_linear_infusion(t, dose, CL, V1, V2, Q, tinf)`   | Single-dose, zero-order oral absorption | `calc_sd_2cmt_linear_oral_0(t, dose, CL, V1, V2, Q, dur)`   | Single-dose, zero-order oral absorption with lag time | `calc_sd_2cmt_linear_oral_0_lag(t, dose, CL, V1, V2, Q, dur, tlag)`   | Single-dose, first-order oral absorption | `calc_sd_2cmt_linear_oral_1(t, dose, CL, V1, V2, Q, ka)`   | Single-dose, first-order oral absorption with lag time | `calc_sd_2cmt_linear_oral_1_lag(t, dose, CL, V1, V2, Q, ka, tlag)` For example - ```{r 2cptdemo, fig.height=10, fig.width=7, message=FALSE} t <- seq(0, 24, by=0.1) df2 <- data.frame(t = t, Cp1 = calc_sd_2cmt_linear_bolus(t = t, dose = 600, CL = 5, V1 = 20, V2 = 80, Q = 1), Cp2 = calc_sd_2cmt_linear_infusion(t = t, dose = 600, CL = 5, V1 = 20, V2 = 80, Q = 1, tinf=2), Cp3 = calc_sd_2cmt_linear_oral_0(t = t, dose = 600, CL = 5, V1 = 20, V2 = 80, Q = 1, dur=2), Cp4 = calc_sd_2cmt_linear_oral_0_lag(t = t, dose = 600, CL = 5, V1 = 20, V2 = 80, Q = 1, dur=2, tlag=1), Cp5 = calc_sd_2cmt_linear_oral_1(t = t, dose = 600, CL = 5, V1 = 20, V2 = 80, Q = 1, ka=1.5), Cp6 = calc_sd_2cmt_linear_oral_1_lag(t = t, dose = 600, CL = 5, V1 = 20, V2 = 80, Q = 1, ka=1.5, tlag=1)) p2.1 <- ggplot(df2, aes(t, Cp1)) + geom_line() + labs(title="Bolus dose") p2.2 <- ggplot(df2, aes(t, Cp2)) + geom_line() + labs(title="Infusion") p2.3 <- ggplot(df2, aes(t, Cp3)) + geom_line() + labs(title="Zero-order oral") p2.4 <- ggplot(df2, aes(t, Cp4)) + geom_line() + labs(title="Zero-order oral with lag time") p2.5 <- ggplot(df2, aes(t, Cp5)) + geom_line() + labs(title="First-order oral") p2.6 <- ggplot(df2, aes(t, Cp6)) + geom_line() + labs(title="First-order oral with lag time") p2.1 + p2.2 + p2.3 + p2.4 + p2.5 + p2.6 + plot_layout(nrow=3) ``` ## Two-compartment, steady state Available functions include: Model | Variant | Call ------------------------------|--------------------------------------------|---------------------------------------- 2-compartment | Steady state, bolus | `calc_ss_2cmt_linear_bolus(tad, tau, dose, CL, V1, V2, Q)`   | Steady state, infusion | `calc_ss_2cmt_linear_infusion(tad, tau, dose, CL, V1, V2, Q, tinf)`   | Steady state, zero-order oral absorption | `calc_ss_2cmt_linear_oral_0(tad, tau, dose, CL, V1, V2, Q, dur)`   | Steady state, zero-order oral absorption with lag time | `calc_ss_2cmt_linear_oral_0_lag(tad, tau, dose, CL, V1, V2, Q, dur, tlag)`   | Steady state, first-order oral absorption | `calc_ss_2cmt_linear_oral_1(tad, tau, dose, CL, V1, V2, Q, ka)`   | Steady state, first-order oral absorption with lag time | `calc_ss_2cmt_linear_oral_1_lag(tad, tau, dose, CL, V1, V2, Q, ka, tlag)` For example - ```{r 2cptssdemo, fig.height=10, fig.width=7,message=FALSE} t <- seq(0, 24, by=0.1) df2ss <- data.frame(t = t, Cp1 = calc_ss_2cmt_linear_bolus(tad = t, tau = 24, dose = 600, CL = 1, V1 = 20, V2 = 80, Q = 0.25), Cp2 = calc_ss_2cmt_linear_infusion(tad = t, tau = 24, dose = 600, CL = 1, V1 = 20, V2 = 80, Q = 0.25, tinf=2), Cp3 = calc_ss_2cmt_linear_oral_0(tad = t, tau = 24, dose = 600, CL = 1, V1 = 20, V2 = 80, Q = 0.25, dur=2), Cp4 = calc_ss_2cmt_linear_oral_0_lag(tad = t, tau = 24, dose = 600, CL = 1, V1 = 20, V2 = 80, Q = 0.25, dur=2, tlag=1.5), Cp5 = calc_ss_2cmt_linear_oral_1(tad = t, tau = 24, dose = 600, CL = 1, V1 = 20, V2 = 80, Q = 0.25, ka=1.5), Cp6 = calc_ss_2cmt_linear_oral_1_lag(tad = t, tau = 24, dose = 600, CL = 1, V1 = 20, V2 = 80, Q = 0.25, ka=1.5, tlag=1.5)) p2.1ss <- ggplot(df2ss, aes(t, Cp1)) + geom_line() + labs(title="Bolus dose") p2.2ss <- ggplot(df2ss, aes(t, Cp2)) + geom_line() + labs(title="Infusion") p2.3ss <- ggplot(df2ss, aes(t, Cp3)) + geom_line() + labs(title="Zero-order oral") p2.4ss <- ggplot(df2ss, aes(t, Cp4)) + geom_line() + labs(title="Zero-order oral with lag time") p2.5ss <- ggplot(df2ss, aes(t, Cp5)) + geom_line() + labs(title="First-order oral") p2.6ss <- ggplot(df2ss, aes(t, Cp6)) + geom_line() + labs(title="First-order oral with lag time") p2.1ss + p2.2ss + p2.3ss + p2.4ss + p2.5ss + p2.6ss + plot_layout(nrow=3) ``` ## Three-compartment, single dose Available functions include: Model | Variant | Call ------------------------------|--------------------------------------------|---------------------------------------- 3-compartment | Single-dose, bolus | `calc_sd_3cmt_linear_bolus(t, dose, CL, V1, V2, V3, Q2, Q3)`   | Single-dose, infusion | `calc_sd_3cmt_linear_infusion(t, dose, CL, V1, V2, V3, Q2, Q3, tinf)`   | Single-dose, zero-order oral absorption | `calc_sd_3cmt_linear_oral_0(t, dose, CL, V1, V2, V3, Q2, Q3, dur)`   | Single-dose, zero-order oral absorption with lag time | `calc_sd_3cmt_linear_oral_0_lag(t, dose, CL, V1, V2, V3, Q2, Q3, dur, tlag)`   | Single-dose, first-order oral absorption | `calc_sd_3cmt_linear_oral_1(t, dose, CL, V1, V2, V3, Q2, Q3, ka)`   | Single-dose, first-order oral absorption with lag time | `calc_sd_3cmt_linear_oral_1_lag(t, dose, CL, V1, V2, V3, Q2, Q3, ka, tlag)` For example - ```{r 3cptdemo, fig.height=10, fig.width=7, message=FALSE} t <- seq(0, 24, by=0.1) df3 <- data.frame(t = t, Cp1 = calc_sd_3cmt_linear_bolus(t = t, CL = 3.5, V1 = 20, V2 = 500, V3 = 200, Q2 = 0.5, Q3 = 0.05, dose = 600), Cp2 = calc_sd_3cmt_linear_infusion(t = t, CL = 3.5, V1 = 20, V2 = 500, V3 = 200, Q2 = 0.5, Q3 = 0.05, dose = 600, tinf=2), Cp3 = calc_sd_3cmt_linear_oral_0(t = t, CL = 3.5, V1 = 20, V2 = 500, V3 = 200, Q2 = 0.5, Q3 = 0.05, dose = 600, dur=2), Cp4 = calc_sd_3cmt_linear_oral_0_lag(t = t, CL = 3.5, V1 = 20, V2 = 500, V3 = 200, Q2 = 0.5, Q3 = 0.05, dose = 600, dur=2, tlag=1), Cp5 = calc_sd_3cmt_linear_oral_1(t = t, CL = 3.5, V1 = 20, V2 = 500, V3 = 200, Q2 = 0.5, Q3 = 0.05, dose = 600, ka=1.5), Cp6 = calc_sd_3cmt_linear_oral_1_lag(t = t, CL = 3.5, V1 = 20, V2 = 500, V3 = 200, Q2 = 0.5, Q3 = 0.05, dose = 600, ka=1.5, tlag=1)) p3.1 <- ggplot(df3, aes(t, Cp1)) + geom_line() + labs(title="Bolus dose") p3.2 <- ggplot(df3, aes(t, Cp2)) + geom_line() + labs(title="Infusion") p3.3 <- ggplot(df3, aes(t, Cp3)) + geom_line() + labs(title="Zero-order oral") p3.4 <- ggplot(df3, aes(t, Cp4)) + geom_line() + labs(title="Zero-order oral with lag time") p3.5 <- ggplot(df3, aes(t, Cp5)) + geom_line() + labs(title="First-order oral") p3.6 <- ggplot(df3, aes(t, Cp6)) + geom_line() + labs(title="First-order oral with lag time") p3.1 + p3.2 + p3.3 + p3.4 + p3.5 + p3.6 + plot_layout(nrow=3) ``` ## Three-compartment, steady state Available functions include: Model | Variant | Call ------------------------------|--------------------------------------------|---------------------------------------- 3-compartment | Single-dose, bolus | `calc_ss_3cmt_linear_bolus(tad, tau, dose, CL, V1, V2, V3, Q2, Q3)`   | Single-dose, infusion | `calc_ss_3cmt_linear_infusion(tad, tau, dose, CL, V1, V2, V3, Q2, Q3, tinf)`   | Single-dose, zero-order oral absorption | `calc_ss_3cmt_linear_oral_0(tad, tau, dose, CL, V1, V2, V3, Q2, Q3, dur)`   | Single-dose, zero-order oral absorption with lag time | `calc_ss_3cmt_linear_oral_0_lag(tad, tau, dose, CL, V1, V2, V3, Q2, Q3, dur, tlag)`   | Single-dose, first-order oral absorption | `calc_ss_3cmt_linear_oral_1(tad, tau, dose, CL, V1, V2, V3, Q2, Q3, ka)`   | Single-dose, first-order oral absorption with lag time | `calc_ss_3cmt_linear_oral_1_lag(tad, tau, dose, CL, V1, V2, V3, Q2, Q3, ka, tlag)` For example - ```{r 3cptssdemo, fig.height=10, fig.width=7,message=FALSE} t <- seq(0, 24, by=0.1) df3ss <- data.frame(t = t, Cp1 = calc_ss_3cmt_linear_bolus(tad = t, tau = 24, dose = 600, CL = 1, V1 = 20, V2 = 500, V3 = 200, Q2 = 0.5, Q3 = 0.05), Cp2 = calc_ss_3cmt_linear_infusion(tad = t, tau = 24, dose = 600, CL = 1, V1 = 20, V2 = 500, V3 = 200, Q2 = 0.5, Q3 = 0.05, tinf=2), Cp3 = calc_ss_3cmt_linear_oral_0(tad = t, tau = 24, dose = 600, CL = 1, V1 = 20, V2 = 500, V3 = 200, Q2 = 0.5, Q3 = 0.05, dur=2), Cp4 = calc_ss_3cmt_linear_oral_0_lag(tad = t, tau = 24, dose = 600, CL = 1, V1 = 20, V2 = 500, V3 = 200, Q2 = 0.5, Q3 = 0.05, dur=2, tlag=1.5), Cp5 = calc_ss_3cmt_linear_oral_1(tad = t, tau = 24, dose = 600, CL = 1, V1 = 20, V2 = 500, V3 = 200, Q2 = 0.5, Q3 = 0.05, ka=1.5), Cp6 = calc_ss_3cmt_linear_oral_1_lag(tad = t, tau = 24, dose = 600, CL = 1, V1 = 20, V2 = 500, V3 = 200, Q2 = 0.5, Q3 = 0.05, ka=1.5, tlag=1.5)) p3.1ss <- ggplot(df3ss, aes(t, Cp1)) + geom_line() + labs(title="Bolus dose") p3.2ss <- ggplot(df3ss, aes(t, Cp2)) + geom_line() + labs(title="Infusion") p3.3ss <- ggplot(df3ss, aes(t, Cp3)) + geom_line() + labs(title="Zero-order oral") p3.4ss <- ggplot(df3ss, aes(t, Cp4)) + geom_line() + labs(title="Zero-order oral with lag time") p3.5ss <- ggplot(df3ss, aes(t, Cp5)) + geom_line() + labs(title="First-order oral") p3.6ss <- ggplot(df3ss, aes(t, Cp6)) + geom_line() + labs(title="First-order oral with lag time") p3.1ss + p3.2ss + p3.3ss + p3.4ss + p3.5ss + p3.6ss + plot_layout(nrow=3) ``` ## Multiple dosing Being able to display single-dose and steady state curves for various commonly-seen systems is all very well but it is often more useful to show a concentration-time curve which demonstrates accumulation and time to steady state. The `pk_curve()` function makes this straightforward by using the principle of superposition. ```{r pkcurve1, fig.height=4, fig.width=7,message=FALSE} dfcurve1 <- pk_curve(t=seq(0,168,by=0.1), model="3cmt_oral", ii=24, addl=12, params=list(CL=1.5, V1=25, V2=2, V3=5, Q2=0.5, Q3=0.25, ka=1)) ggplot(dfcurve1, aes(t, cp)) + geom_line() ```