Drawing PK curves with pmxTools

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 Mentré’s Mathematical Expressions of the Pharmacokinetic and Pharmacodynamic Models implemented in the Monolix software, 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 RxODE or mrgSolve.

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 -


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 -


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 -


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 -


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 -


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 -


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.


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()