1. Pharmacokinetics

The most common approach for modeling pharmacokinetics is based on compartmental models. The concept is to consider the body as one or several compartments and describe the drug concentration in each compartment [Bailey2005]. The drug is assumed to be homogeneously distributed in each compartment, and the transport rate between compartments is assumed to be proportional to the drug concentration. Although this is a crude simplification of reality, it is a powerful tool for the prediction of the drug concentration in the body. Moreover, the freedom of the number of compartments allows to reach a good compromise between accuracy and complexity depending on available data.

Three-compartment model for propofol and remifentanil.

1.1. Propofol and remifenantil

For propofol and remifentanil, a three-compartment model is commonly used, see the figure above. The first compartment is the central compartment, which represents blood and highly perfused organs such as the heart, brain, kidney and liver. Then the two peripherical compartments respectively represent the highly perfused tissues such as muscles, and the poorly perfused tissues such as fat. The model results in a dynamical system described by the following state-space representation:

\[\begin{split}\begin{pmatrix} \dot{x}_1(t) \\ \dot{x}_2 (t)\\ \dot{x}_3(t) \end{pmatrix} = \begin{pmatrix} -(k_{10} + k_{12} + k_{13}) & k_{12} & k_{13} \\ k_{21} & -k_{21} & 0\\ k_{31} & 0 & -k_{31} \end{pmatrix} \begin{pmatrix} x_1 (t)\\ x_2 (t)\\ x_3(t) \end{pmatrix} + \begin{pmatrix} \frac{1}{V_1} \\ 0\\ 0 \end{pmatrix} u(t)\end{split}\]

where \(x_1(t), x_2(t)\) and \(x_3(t)\) respectively represent drug concentration in blood, muscle and fat in \(\mathrm{mg/ml}\) for propofol and \(\mathrm{µg/ml}\) for remifentanil. The coefficients can be obtained from the equations below, all in \(\mathrm{1/s}\):

\[k_{10} = \frac{Cl_1}{V_1},\quad k_{12} = \frac{Cl_2}{V_1},\quad k_{13} = \frac{Cl_3}{V_1},\quad k_{21} = \frac{Cl_2}{V_2},\quad k_{31} = \frac{Cl_3}{V_3}\]

where \(V_i\) and \(Cl_i\) for \(i \in \{1,2,3\}\) are, respectively, the volumes and the clearance rates of each compartment. The input \(u(t)\) is the drug infusion rate in \(\mathrm{mg/s}\) for propofol and \(\mathrm{µg/s}\) for remifentanil.

Multiple studies have been conducted to estimate the parameters of the PK model for propofol and remifentanil depending on the patient’s characteristics (age, height, weight, and sex). The most commonly used in clinical practice are the models of [Scnnider1999] and [Marsh1991] for propofol, and the model of [Minto1997] for remifentanil. However, two more recent studies made by Eleveld et al. ([Eleveld2018], [Eleveld2017]) respectively for propofol and remifentanil, gather much more data and could be considered as the state of the art.

Simulated propofol concentration for a 3-compartment model.

Remark: The 3-compartment model of propofol and remifentanil is often presented with state representing drug mass rather than concentration. In this case the model is given by

\[\begin{split}\begin{pmatrix} \dot{m}_1(t) \\ \dot{m}_2 (t)\\ \dot{m}_3(t) \end{pmatrix} = \begin{pmatrix} -(k_{10} + k_{12} + k_{13}) & \textcolor{blue}{k_{21}} & \textcolor{blue}{k_{31}} \\ \textcolor{blue}{k_{12}} & -k_{21} & 0\\ \textcolor{blue}{k_{13}} & 0 & -k_{31} \end{pmatrix} \begin{pmatrix} m_1 (t)\\ m_2 (t)\\ m_3(t) \end{pmatrix} + \begin{pmatrix} 1 \\ 0\\ 0 \end{pmatrix} u(t)\end{split}\]

where \(m_i\) for \(i \in \{1,2,3\}\) are the mass of drug in each compartment and blood concentration is given by \(\frac{m_1(t)}{V_1}\). Note that both models are equivalent.

1.2. Atracurium

For atracurium, the two-compartment model proposed in [Ward1983] is considered, see the figure below.

Two-compartment model for atracurium.

The first compartment is the central compartment, which represents blood and highly perfused organs such as the heart, brain, kidney and liver. Then the peripherical compartment represents the highly perfused tissues such as muscles. The model results in a dynamical system described by the following state-space representation:

\[\begin{split}\begin{pmatrix} \dot{x}_1(t) \\ \dot{x}_2 (t) \end{pmatrix} = \begin{pmatrix} -\frac{(k_{10} + k_{12})}{60} & \frac{k_{21}}{60} \frac{V_2}{V_1} \\ \frac{k_{12}}{60} \frac{V_1}{V_2} & -\frac{k_{21}}{60} \end{pmatrix} \begin{pmatrix} x_1 (t)\\ x_2 (t) \end{pmatrix} + \begin{pmatrix} \frac{1}{V_1 * Weight} \\ 0 \end{pmatrix} u(t)\end{split}\]

where \(x_1(t)\) and \(x_2(t)\) respectively represent drug concentration in blood and muscles in \(\mathrm{\mu g/ml}\). The coefficients can be obtained from the equations below, all in \(\mathrm{1/s}\):

\[k_{10} = \frac{Cl_1}{V_1},\quad k_{21} = \frac{\alpha \beta}{k_{10}},\quad k_{12} = \alpha + \beta - k_{10} - k_{21}\]

where \(V_1\) and \(Cl_1\) are, respectively, the volume and the clearanc rate of the primary compartment. The coefficients \(\alpha\) and \(\beta\) are computed as:

\[\alpha = \frac{\log(2)}{t^{\alpha}_{1/2}}, \quad \beta = \frac{\log(2)}{t^{\beta}_{1/2}}\]

where \(t^{\alpha}_{1/2}\) and \(t^{\beta}_{1/2}\) are the half live times of the drug.

The input \(u(t)\) is the atracurium infusion rate in \(\mathrm{\mu g/s}\).

Remark: The 2-compartment model of atracurium is often presented with state representing drug mass divided by the patient’s weight rather than concentration. In this case the model is given by

\[\begin{split}\begin{pmatrix} \dot{m}_1(t) \\ \dot{m}_2 (t) \end{pmatrix} = \begin{pmatrix} -\frac{(k_{10} + k_{12})}{60} & \textcolor{blue}{\frac{k_{21}}{60}} \\ \textcolor{blue}{\frac{k_{12}}{60}} & -\frac{k_{21}}{60} \end{pmatrix} \begin{pmatrix} m_1 (t)\\ m_2 (t) \end{pmatrix} + \begin{pmatrix} \frac{1}{Weight} \\ 0 \end{pmatrix} u(t)\end{split}\]

where \(m_1\) and \(m_2\) are the mass of drug in each compartment and blood concentration is given by \(\frac{m_1(t)}{V_1}\). Note that both models are equivalent.

1.3. Norepinephrine

Norepinephrine is commonly used during anesthesia to higher blood pressure. It is an endogenous substance, which means that it is naturally produced by the body. It mainly acts on α1 and α2 adrenergic receptors, but also on β1 and β2 adrenergic receptors. It’s main effect, on α receptors, is vasoconstriction, which leads to an increase in blood pressure. It has a short half-life, between 1-6 minutes depending on the studies.

For this drug, there is less studies focused on modelling, and the model structure varies between the propositions.

In [Beloeil2005], the authors have proposed a single compartment model given by the following equation to model the PK of norepineprhine in shocked adult patients:

\[\dot{x}(t) = -\frac{Cl}{V} x(t) + \frac{1}{V} u(t)\]

where \(V\) and \(Cl\) are, respectively, the volume and the clearance rate of the single compartment. \(x(t)\) is directly the blood concentration of norepinephrine.

In [Oualha2014], the focused was one shock child patient and the endogenous production is also considered in the model:

\[\dot{x}(t) = -\frac{Cl}{V} x(t) + \frac{1}{V} (u_{endo} + u(t))\]

In [Li2024], the authors studied healthy patient and considered a two compartments model with endogenous production and a delayed input:

\[\begin{split}\begin{align} \dot{x_1}(t) & = \frac{1}{V_1}\left( - (Cl_1 + Cl_2)x_1(t) + Cl_2 x_2(t) + u_{endo} + u(t - T_{lag}) \right) \\ \dot{x_2}(t) & = \frac{Cl_2}{V_2}\left(x_1(t) - x_2(t)\right) \end{align}\end{split}\]

A comparison of those three model is given in the figure below.

Blood concentration of norepinephrine for a constant injection of 0.1 µg/s

1.4. References

[Bailey2005]

J. M. Bailey and W. M. Haddad, “Drug dosing control in clinical pharmacology,” IEEE Control Systems Magazine, vol. 25, no. 2, pp. 35–51, Apr. 2005, doi: https://doi.org/10.1109/MCS.2005.1411383.

[Scnnider1999]

T. W. Schnider et al., “The Influence of Age on Propofol Pharmacodynamics,” Anesthesiology, vol. 90, no. 6, pp. 1502-1516., Jun. 1999, doi: https://doi.org/10.1097/00000542-199906000-00003.

[Marsh1991]

B. Marsh, M. White, N. morton, and G. N. C. Kenny, “Pharmacokinetic model Driven Infusion of Propofol in Children,” BJA: British Journal of Anaesthesia, vol. 67, no. 1, pp. 41–48, Jul. 1991, doi: https://doi.org/10.1093/bja/67.1.41.

[Minto1997]

C. F. Minto et al., “Influence of Age and Gender on the Pharmacokinetics and Pharmacodynamics of Remifentanil: I. Model Development,” Anesthesiology, vol. 86, no. 1, pp. 10–23, Jan. 1997, doi: https://doi.org/10.1097/00000542-199701000-00004.

[Eleveld2018]

D. J. Eleveld, P. Colin, A. R. Absalom, and M. M. R. F. Struys, “Pharmacokinetic–pharmacodynamic model for propofol for broad application in anaesthesia and sedation” British Journal of Anaesthesia, vol. 120, no. 5, pp. 942–959, mai 2018, doi: https://doi.org/10.1016/j.bja.2018.01.018.

[Eleveld2017]

D. J. Eleveld et al., “An Allometric Model of Remifentanil Pharmacokinetics and Pharmacodynamics,” Anesthesiology, vol. 126, no. 6, pp. 1005–1018, juin 2017, doi: https://doi.org/10.1097/ALN.0000000000001634.

[Beloeil2005]

H. Beloeil, J.-X. Mazoit, D. Benhamou, and J. Duranteau, “Norepinephrine kinetics and dynamics in septic shock and trauma patients,” BJA: British Journal of Anaesthesia, vol. 95, no. 6, pp. 782–788, Dec. 2005, doi: https://doi.org/10.Beloeil20051093/bja/aei259.

[Oualha2014]

M. Oualha et al., “Population pharmacokinetics and haemodynamic effects of norepinephrine in hypotensive critically ill children,” British Journal of Clinical Pharmacology, vol. 78, no. 4, pp. 886–897, 2014, doi: https://10.1111/bcp.12412.

[Li2024]

Y. Li et al., “Population Pharmacokinetic Modelling of Norepinephrine in Healthy Volunteers Prior to and During General Anesthesia,” Clin Pharmacokinet, vol. 63, no. 11, pp. 1597–1608, Nov. 2024, doi: https://10.1007/s40262-024-01430-y.

[Ward1983]

S. Ward et al., “Pharmacokinetics of Atracurium Besylate in Healty Patients (after a single i.v. bolus dose),” British Journal of Anesthesia, vol. 55, no. 2, pp. 113-116, Feb. 1983, doi: 10.1093/bja/55.2.113.