The `tci`

package implements target-controlled infusion (TCI) algorithms for intravenous drugs with pharmacokinetics (PK) described by compartmental ordinary differential equation (ODE) based models. TCI systems are used in open- or closed-loop settings to partially or fully automate the delivery of medication to a patient, respectively. For a given patient, a pharmacokinetic or pharmacokinetic-pharmacodynamic (PK-PD) model is identified that describes the patient’s predicted response to the medication. This model is inverted by a TCI algorithm to supply the user (i.e. clinician) with infusion rates and durations that are expected to achieve a desired concentration or effect in the patient.

Within the `tci`

package, users can select the Jacobs (1990) algorithm for plasma-targeting, the Shafer and Gregg (1992) algorithm for effect-site targeting, or define a custom TCI algorithm. Calculations are made using closed-form solutions to 1-, 2-, and 3-compartment PK models, as well as 3-compartment models with a fourth effect-site compartment. PK model code is based on solutions and code published by Abuhelwa, Foster, and Upton (2015). PK models based on other packages, such as mrgsolve, PKPDsim, and RxODE can be adapted for use within `tci`

and are illustrated in a separate vignette. When an invertible PD model is specified, the algorithms in `tci`

can be extended to PD outcomes.

Finally, `tci`

provides a number of functions for simulating responses from patients receiving medications under open- or closed-loop TCI system administration. Data can be simulated assuming normal or log-normally distributed errors that are additive or multiplicative with respect to the response model. Closed-loop control is implemented using Bayesian updates to a patient-specific prior model. Use of these functions is demonstrated in a separate vignette. S3 plotting methods are additionally provided for visualizing dosing regimes and patient responses.

```
library(tci)
library(knitr)
library(gridExtra)
```

For illustration of `tci`

functions, we use a three compartment model with a fourth effect-site compartment. In reality, this is a four-compartment model; however, the fourth compartment is assumed to have negligible volume, such that its presence doesn’t interfere with the three-compartmental dynamics. This is achieved by setting the volume of the fourth compartment to be a minuscule fraction of the central compartment volume and setting \(k_{1e} = k_{e0}\).(Shafer and Gregg 1992) The system of equations describing the three-compartment, effect-site model are below.

\[\begin{align} \dot{C}(t) = A*C(t) + B*k_R(t) \end{align}\]

\[\begin{align} A = \begin{pmatrix} -(k_{10}+k_{12}+k_{13}) & \frac{V_2}{V_1}k_{21} & \frac{V_3}{V_1}k_{31} & 0\\ \frac{V_1}{V_2}k_{12} & -k_{21} & 0 & 0 \\ \frac{V_1}{V_3}k_{13} & 0 & -k_{31} & 0 \\ k_{e0}/10^5 & 0 & 0 & -k_{e0} \end{pmatrix} \end{align}\]

\[\begin{align} B = [1,0,0,0]^T \end{align}\]

Infusion schedules must be passed alongside PK functions and are either created directly by the user or as the output of other `tci`

functions described later. Infusion schedules must contain column names “infrt,” “begin,” and “end” designating to infusion rates with beginning and end times. The `create_intvl`

function can be used to expand doses in the correct format by specifying infusion rates and end times with a start time that defaults to zero.

```
# e.g. infusion rates of 100 ug/min for 30 sec intervals at 0,4 minutes
<- create_intvl(
dose as.matrix(cbind(time = c(0.5,4,4.5,10),
infrt = c(100,0,100,0))),
inittm = 0
)
dose#> infrt begin end
#> [1,] 100 0.0 0.5
#> [2,] 0 0.5 4.0
#> [3,] 100 4.0 4.5
#> [4,] 0 4.5 10.0
```

PK models are defined to have S3 class “pkmod” and can be evaluated through the `predict.pkmod`

method in combination with a dosing schedule. Evaluation of a model additionally requires a set of PK model parameters. Starting concentrations, if unspecified, will be set to zero in each compartment. The units of measurement used for infusion rates and concentrations are not inherent to any structural PK model and may change depending on the drug and application. For the examples below, we will consider the intravenous anesthetic propofol, which is commonly described by a three-compartment model and used in TCI applications. Population models of propofol describe volumes in terms of liters (L), clearance parameters in terms of milliliters per minute (mL/min), and amounts of drug in terms of milligrams (mg). The infusion rate describes the quantity of propofol administered per unit of time on the same scale as the rest of the parameters in the PK model, and is therefore given in terms of mg/min.

```
# model parameters
<- c(k10=1.5,k12=0.15,k21=0.09,k13=0.8,
pars_3cpt k31=0.8,v1=10,v2=15,v3=100,ke0=1)
# predict concentrations of a three-compartment model with effect-site at
# times 1, 2, 8 minutes
predict(pkmod3cptm,
pars = pars_3cpt,
inf = dose,
tms = c(1,2,8))
#> time c1 c2 c3 c4
#> [1,] 1 1.0812467 0.1708101 0.09872216 1.1317615
#> [2,] 2 0.3558635 0.2129614 0.07501638 0.7619403
#> [3,] 8 0.1675395 0.3914383 0.03653374 0.3157391
```

The same arguments can similarly be passed to a `plot.pkmod`

method to display the patient responses over time.

```
# plot concentrations
plot(pkmod3cptm, inf = dose, pars = pars_3cpt,
title = "Concentrations for a 3 compartment model with an effect site")
```

The `tci_plasma`

and `tci_effect`

functions implement plasma and effect-site targeting, respectively. Each takes as arguments, a target concentration, `Cpt`

, an infusion duration, `dt`

, a `pkmod`

class function, and PK model parameters. The output of each function is a single infusion rate that, if administered for `dt`

, will result in the target concentrations at time `dt`

for a patient with PK described by the specified model. Initial concentrations can optionally be passed as well through the argument “init.” Letting `dt`

be expressed in terms of minutes, we can calculate the infusion rate (mg/min) required to reach a target concentration of 2 mg/L for a one-minute infusion.

```
# calculate infusion rates
<- tci_plasma(Cpt = 2, # target plasma concentration
kR_plasma dt = 1, # duration of infusion
pkmod = pkmod3cptm, # pk model
pars = pars_3cpt) # pk model parameters
kR_plasma#> kR.v1
#> 49.58785
<- tci_effect(Cet = 2, dt = 1, pkmod = pkmod3cptm, pars = pars_3cpt)
kR_effect
# set up dosing schedules
<- create_intvl(
inf_2min_plasma data.frame(time = c(1, 20),
infrt = c(kR_plasma,0))
)
<- create_intvl(
inf_2min_effect data.frame(time = c(1, 20),
infrt = c(kR_effect,0))
)
# plot results
<- plot(pkmod3cptm,
plt_plasma pars = pars_3cpt,
inf = inf_2min_plasma,
title = "Plasma targeting a concentration of 2 mg/L at 1 minute")
<- plot(pkmod3cptm,
plt_effect pars = pars_3cpt,
inf = inf_2min_effect,
title = "Effect-site targeting a concentration of 2 mg/L at 1 minute")
grid.arrange(plt_plasma, plt_effect)
```

The Shafer-Gregg effect-site algorithm identifies the infusion rate that will achieve the target concentration in the effect-site as quickly as possible, without subsequent overshoot. Due to the hysteresis between the central and effect-site compartments, this requires administering a larger dose and attaining a higher peak plasma concentration than with plasma-targeting. As the duration of the infusion increases, the ratio of peak plasma concentrations is reduced. By default, the infusion duration is set to 10 seconds, which is consistent with many commercial TCI devices.(Morton 2009)

TCI algorithms are iteratively applied through the function ‘tci’ to calculate infusion rates for a series of targets, using the same arguments. Effect-site targeting is used by default for models with more than one compartment and plasma targeting for one compartment models; however, a TCI algorithm can be specified through the `tci_alg`

or `tci_custom`

arguments. In the example below, a concentration of 2 mg/L is targeted for the first 5 minutes, followed by 3 mg/L for the following five minutes.

```
# target concentrations of 2 for 0-5 minutes and 3 for 5-10 minutes.
<- c(0,5,10)
tci_times <- c(2,3,3)
tci_targets
# infusions for effect-site targeting
<- tci(Ct = tci_targets,
inf_3cpt_effect tms = tci_times,
pkmod = pkmod3cptm,
pars = pars_3cpt)
plot(inf_3cpt_effect, title = "Effect-site targeting for three-compartment model")
```

Functions implementing the Marsh, Schnider, and Eleveld population PK models for propofol (Marsh et al. 1991; Schnider et al. 1998; Eleveld et al. 2018) (PK-PD for Eleveld model) are currently provided and can be used to calculate PK and PK-PD parameters at patient covariates or to simulate parameters for new patients given inter-patient parameter variability.

```
<- data.frame(AGE = c(20,40,65),
patient_dat TBM = c(50,70,90),
HGT = c(150,170,200),
MALE = c(TRUE,FALSE,TRUE))
# evaluate at covariate values and return clearance parameters
<- schnider_poppk(patient_dat, rand = FALSE, rate = FALSE)
patient_pk
patient_pk#> AGE TBM HGT MALE LBM V1 V2 V3 CL Q2 Q3 KE0
#> 1 20 50 150 TRUE 40.77778 4.27 31.803 238 1.186933 0.498 0.836 0.456
#> 2 40 70 170 FALSE 49.80657 4.27 23.983 238 2.012072 0.978 0.836 0.456
#> 3 65 90 200 TRUE 73.08000 4.27 14.208 238 2.131152 1.578 0.836 0.456
# evaluate TCI for patient 1
<- tci(Ct = tci_targets,
tci_patient1 tms = tci_times,
pkmod = pkmod3cptm,
pars = patient_pk[1,],
tci_alg = "effect")
head(round(tci_patient1,3))
#> infrt begin end dtm Ct c1_start c2_start c3_start c4_start c1_end
#> [1,] 159.105 0.000 0.167 0.167 2 0.000 0.000 0.000 0.000 5.914
#> [2,] 0.000 0.167 0.333 0.167 2 5.914 0.008 0.002 0.223 5.361
#> [3,] 0.000 0.333 0.500 0.167 2 5.361 0.022 0.005 0.618 4.859
#> [4,] 0.000 0.500 0.667 0.167 2 4.859 0.036 0.008 0.947 4.405
#> [5,] 0.000 0.667 0.833 0.167 2 4.405 0.048 0.011 1.216 3.993
#> [6,] 0.000 0.833 1.000 0.167 2 3.993 0.059 0.013 1.434 3.621
#> c2_end c3_end c4_end
#> [1,] 0.008 0.002 0.223
#> [2,] 0.022 0.005 0.618
#> [3,] 0.036 0.008 0.947
#> [4,] 0.048 0.011 1.216
#> [5,] 0.059 0.013 1.434
#> [6,] 0.068 0.015 1.607
```

`tci`

functions are compatible with other user-specified population PK models provided that they return parameters with either elimination rate-constants named “k10,” “k12,” etc. (capitalization optional), or as clearance parameters “CL,” “Q2,” etc.

Abuhelwa, Ahmad Y., David J R Foster, and Richard N. Upton. 2015. “ADVAN-style analytical solutions for common pharmacokinetic models.” *Journal of Pharmacological and Toxicological Methods* 73: 42–48. https://doi.org/10.1016/j.vascn.2015.03.004.

Eleveld, D J, P Colin, A R Absalom, and M M R F Struys. 2018. “Pharmacokinetic-pharmacodynamic model for propofol for broad application in anaesthesia and sedation.” *British Journal of Anaesthesia* 120 (5): 942–59. https://doi.org/10.1016/j.bja.2018.01.018.

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.

Marsh, B., M. White, N. Morton, and G. N. C. Kenny. 1991. “Pharmacokinetic model driven infusion of propofol in children.” *British Journal of Anaesthesia* 67 (1): 41–48. https://doi.org/10.1093/bja/67.1.41.

Morton, Neil S. 2009. *Total Intravenous Anesthesia and Target-Controlled Infusion*. Fourth Edi. 4. Elsevier Inc. https://doi.org/10.1016/B978-0-323-42974-0.00008-2.

Schnider, Thomas W., Charles F. Minto, Pedro L. Gambus, Corina Andresen, David B. Goodale, Steven L. Shafer, and Elizabeth J. Youngs. 1998. “The Influence of Method of Administration and Covariates on the Pharmacokinetics of Propofol in Adult Volunteers.” *Anesthesiology* 88: 1170–82.

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.