README

tci

The tci package implements target-controlled infusion (TCI) algorithms to calculate infusion schedules for compartmental pharmacokinetic (PK) and pharmacokinetic-pharmacodynamic (PK-PD) models. Using these infusion schedules, functions in tci can be used to simulate PK or PK-PD responses with or without model misspecification. Generated responses can be used to simulate closed-loop control by implementing Bayesian updates to the PK or PK-PD model based on the “observed” (i.e., simulated) data.

Closed-form solutions are provided for one, two, or three compartment mammillary models (i.e., all peripheral compartments are joined to a central compartment), as well as a three-compartment model with an adjoining effect-site. Alternative models, potentially based on ordinary differential equations (ODE), can be specified using other R packages and adapted for use with tci functions (see the “Custom PK models and algorithms” vignette).

tci implements both plasma-and effect-site targeting TCI algorithms based on work by Jacobs (1990) and Shafer and Gregg (1992), respectively. Users can implement alternative user-defined TCI algorithms, however. Again, see the “Custom PK models and algorithms” vignette for further details.

Several population PK models commonly used for TCI are implemented in tci. These include the Marsh, Schnider, and Eleveld models for propofol, and the Minto, Kim, and Eleveld models for remifentanil. Use of these models is illustrated in the “Population PK models” vignette.

Installation

The tci package can be installed from CRAN using the command.

install.packages("tci")

The most recent version can be downloaded from GitHub using the devtools package and loaded as follows.

devtools::install_github("jarretrt/tci")

library(tci)
library(ggplot2) # for plotting

Examples

Equations implementing 1-, 2-, 3-compartment and 3-compartment-effect structural PK models are included in the tci package. The function pkmod will automatically infer the correct structure based on the parameter names.

# 3-compartment model with effect site
(mod3ecpt <- pkmod(pars_pk = c(cl = 10, q2 = 2, q3 =20, v = 15, v2 = 30, v3 = 50, ke0 = 1.2)))

## tci pkmod object
## See ?update.pkmod to modify or add elements
## 
## PK model 
##  4-compartment PK model 
##  PK parameters: cl = 10, q2 = 2, q3 = 20, v = 15, v2 = 30, v3 = 50, ke0 = 1.2 
##  Initial concentrations: (0,0,0,0) 
##  Plasma compartment: 1 
##  Effect compartment: 4 
## 
## Simulation
##  Additive error SD: 0 
##  Multiplicative error SD: 0 
##  Logged response: FALSE

Acceptable parameter names can be viewed by calling list_parnms(). Less-commonly used parameters, such as clearance from a peripheral compartment, are also permissible.

# acceptable parameter names
list_parnms()

## Acceptable names for 'pars_pk' vector (case-insensitive) 
## 
## First compartment options
##  Central volume: 'v','v1' 
##  Elimination: 'cl','cl1','k10','ke' 
## 
## Second compartment options
##  Peripheral volume: 'v2' 
##  Transfer: 'q','q2','k12','k21' 
##  Elimination: 'cl2','k20' 
## 
## Third compartment options
##  Second peripheral volume: 'v3' 
##  Transfer: 'q3','k13','k31' 
##  Elimination: 'cl3','k30' 
## 
## Effect-site
##  Elimination: 'ke0'

Elements of pkmod objects can be modified through an update.pkmod method. Perhaps most usefully, this allows for partial modifications to PK-PD parameters. For example, the effect-site equilibrium constant can be easily updated.

update(mod3ecpt, pars_pk = c(ke0 = 0.9), init = c(1,0.2,0.3,1))

## tci pkmod object
## See ?update.pkmod to modify or add elements
## 
## PK model 
##  4-compartment PK model 
##  PK parameters: cl = 10, q2 = 2, q3 = 20, v = 15, v2 = 30, v3 = 50, ke0 = 0.9 
##  Initial concentrations: (1,0.2,0.3,1) 
##  Plasma compartment: 1 
##  Effect compartment: 4 
## 
## Simulation
##  Additive error SD: 0 
##  Multiplicative error SD: 0 
##  Logged response: FALSE

Most functions in the tci package pass additional arguments to update.pkmod allowing for easy modification of pkmod objects as needed.

TCI Infusion schedules

TCI algorithms are implemented using the function tci_inf (manual infusions are implemented by inf_manual). The user supplies a set of targets, times at which the target is set, and a pkmod object. The TCI algorithm (defaults to type = "plasma") is iteratively applied to calculate infusion rates required to reach each target in turn. By default, infusion rates are updated in increments of 1/6, corresponding to every 10-second intervals if infusions rate units are in amount per minute. Infusion rates themselves must have the same units as the PK elimination parameters. If elimination rates are in different units, such as hours, then the TCI update frequency should be modified by the argument dtm.

# effect-site targeting for three-compartment effect site model
inf_3ecpt <- inf_tci(target_vals = c(2,3,4,4), target_tms = c(0,2,3,10), 
                     pkmod = mod3ecpt, type = "effect")
head(inf_3ecpt)

##        begin     end inf_rate Ct c1_start   c2_start  c3_start  c4_start
## [1,] 0.00000 0.16667 643.6921  2 0.000000 0.00000000 0.0000000 0.0000000
## [2,] 0.16667 0.33333   0.0000  2 6.033672 0.03532348 0.2079682 0.5966263
## [3,] 0.33333 0.50000   0.0000  2 4.301795 0.09138590 0.5235366 1.4096671
## [4,] 0.50000 0.66667   0.0000  2 3.136188 0.13103476 0.7267367 1.8178340
## [5,] 0.66667 0.83333  19.3835  2 2.350071 0.15960595 0.8548449 1.9785480
## [6,] 0.83333 1.00000   0.0000  2 2.000000 0.18174014 0.9390928 2.0109479
##        c1_end     c2_end    c3_end    c4_end
## [1,] 6.033672 0.03532348 0.2079682 0.5966263
## [2,] 4.301795 0.09138590 0.5235366 1.4096671
## [3,] 3.136188 0.13103476 0.7267367 1.8178340
## [4,] 2.350071 0.15960595 0.8548449 1.9785480
## [5,] 2.000000 0.18174014 0.9390928 2.0109479
## [6,] 1.586605 0.19939667 0.9931813 1.9678507

Population PK models

Population PK models are implemented by the function poppkmod. The user must supply a data frame with the set of covariates (e.g., weight, age) required by the model. Several published population PK models are currently implemented in tci. For propofol, these include the Marsh, Schnider, and Eleveld models. For remifentanil, they include the Minto, Kim, and Eleveld models. See ?poppkmod or the population PK model vignette for details. list_pkmods() will list available population PK models and covariates required by each.

# data frame of patient covariates
data <- data.frame(ID = 1:5, AGE = seq(20,60,by=10), 
                   TBW = seq(60,80,by=5), HGT = seq(150,190,by=10), 
                   MALE = c(TRUE,TRUE,FALSE,FALSE,FALSE))
# Eleveld population PK model for propofol
pkpd_elvd <- poppkmod(data = data, drug = "ppf", model = "eleveld")

As with the pkmod class, poppkmod objects can be used by inf_tci and have predict and simulate methods to predict and simulate PK-PD responses, respectively.

PK-PD parameter values can be drawn at random from the inter-/intra-individual variability distribution, as described by the pkmod Omega matrix, by either 1) setting the argument sample = TRUE when calling poppkmod, or 2) by using the function sample_iiv.

set.seed(1)
pkpd_elvd_iiv <- sample_iiv(pkpd_elvd)

Simulations

Simulations are best implemented through the function simulate_tci, which allows for model misspecification as well as Bayesian updates to model parameters based on previously observed data (i.e., “closed-loop” control). simulate_tci can be used for both pkmod or poppkmod classes. Required arguments to simulate_tci are 1) a prior PK model (pkmod_prior) that is used to calculate infusion rates and may be updated throughout the simulation if update times are provided, 2) a true PK model (pkmod_true) that is used to simulate observations, 3) TCI target values, 4) TCI target times, and 5) times to simulate observations.

# TCI target values (PD response)
target_vals <- c(75,60,50,50)
# values are in terms of minutes. 1/6 = 10 seconds
# TCI target times
target_tms <- c(0,3,6,10)
# observation times 
obs_tms <- seq(1/6,10,1/6)

# simulate without updates ("open-loop")
sim_ol <- simulate_tci(pkmod_prior = pkpd_elvd, pkmod_true = pkpd_elvd_iiv, 
             target_vals, target_tms, obs_tms, type = "effect", seed = 1)

simulate_tci returns an object with class sim_tci that can be plotted using the ggplot2 library.

plot(sim_ol)

Modifications can be made to the plot to show a subset of responses, concentrations instead of PD response values, infusion rates, and simulated data.

plot(sim_ol, yvar = "c4", id = c(1,3,5), show_inf = TRUE, wrap_id = TRUE)

Closed-loop simulations can be implemented by specifying a set of update times. We illustrate this with updates each minute and a processing delay of 20 seconds.

sim_cl <- simulate_tci(pkmod_prior = pkpd_elvd, pkmod_true = pkpd_elvd_iiv, 
             target_vals, target_tms, obs_tms, update_tms = 1:10, delay = 1/3,
               type = "effect", seed = 1)

## [1] "Simulating ID=1"
## [1] "Simulating ID=2"
## [1] "Simulating ID=3"
## [1] "Simulating ID=4"
## [1] "Simulating ID=5"

Since plot.sim_tci returns a ggplot2 object, it is easy to modify aspects such as titles and axis labels using ggplot2 functions.

plot(sim_cl) + 
  xlab("Minutes") + 
  ylab("Bispectral Index") + 
  ggtitle("Closed-loop simulation of Eleveld propofol model", 
          subtitle = "Minute updates, processing delay of 20 seconds")

Jacobs, James R. 1990. “Algorithm for Optimal Linear Model-Based Control with Application to Pharmacokinetic Model-Driven Drug Delivery.” IEEE Transactions on Biomedical Engineering 37 (1): 107–9. https://doi.org/10.1109/10.43622.

Shafer, Steven L., and Keith M. Gregg. 1992. “Algorithms to rapidly achieve and maintain stable drug concentrations at the site of drug effect with a computer-controlled infusion pump.” Journal of Pharmacokinetics and Biopharmaceutics 20 (2): 147–69. https://doi.org/10.1007/BF01070999.