Equilibration rate constant, k_e0, to determine effect-site concentration for the Masui remimazolam population pharmacokinetic model in general anesthesia patients

Abstract

Effect-site concentration is widely used to determine drug dosage in anesthesia practice. To obtain effect-site concentration, a pharmacokinetic model with a corresponding equilibration rate constant between plasma and effect-site, k_e0, is necessary. Remimazolam, a novel short-acting benzodiazepine, has been approved as anesthetic/sedative. Recently, a remimazolam pharmacokinetic model has been published using a large dataset including wide range of subject characteristics (416 males and 246 females, age 18–93 years, total body weight 34–149 kg, height 133–204 cm, body mass index 14–61 kg m⁻², ASA physical status: I–IV, and Asian, White, American African, and 2 other races). This Masui model can be applicable to various patients, but a pharmacodynamic model including k_e0 was not developed simultaneously. A previous article has indicated that the time to peak effect of drug after its bolus should be used to determine k_e0 for a pharmacokinetic model without simultaneous development of corresponding pharmacodynamic model. The k_e0 value can be calculated using numerical analysis but not algebraic solution. We provide the detail method of the numerical analysis and a tool to have k_e0 value easily for the Masui remimazolam PK model. Additionally, we provide a multiple regression model to have k_e0 value for the PK model.

Appendices

Appendix 1

A PK parameter set of three compartment model using distribution volumes and clearances includes the distribution volume of i th compartment (V_i), elimination clearance (CL), intercompartmental clearance between central and i th compartment (Q_i). Using these parameters, three compartment model can be described as the simultaneous differential Equation 5:

$$\begin{array}{l}\left\{\begin{array}{l}\frac{\mathrm{d}{A}_{1}}{\mathrm{d}t}=-\left({k}_{10}+{k}_{12}+{k}_{13}\right){A}_{1}+{k}_{21}{A}_{2}+{k}_{31}{A}_{3}\\ \frac{\mathrm{d}{A}_{2}}{\mathrm{d}t}={k}_{12}{A}_{1}-{k}_{21}{A}_{2}\\ \frac{\mathrm{d}{A}_{3}}{\mathrm{d}t}={k}_{13}{A}_{1}-{k}_{31}{A}_{3}\end{array}\right.\end{array}$$

(5)

where A_i is the drug amount in the i th compartment, t is the time, k₁₀ is the equilibration rate constant, k_ij is the equilibration rate constant from i th to j th compartment, and rate constants are calculated as follows: k₁₀ = CL/V₁, k₁₂ = Q₂/V₁, k₁₃ = Q₃/V₁, k₂₁ = Q₂/V₂, and k₃₁ = Q₃/V₃. These PK parameters of A, B, C, α, β, and γ in Eq. 2 can be described using Eqs. 6 and 7:

$$\left\{\begin{array}{l}\alpha +\beta +\gamma ={k}_{10}+{k}_{12}+{k}_{13}+{k}_{21}+{k}_{31}\\ \alpha \beta +\beta \gamma +\gamma \alpha =\left({k}_{10}+{k}_{13}\right){k}_{21}+\left({k}_{10}+{k}_{12}\right){k}_{31}+{k}_{21}{k}_{31}\\ \alpha \beta \gamma ={{k}_{10}k}_{21}{k}_{31}\end{array}\right.$$

(6)

$$\left\{\begin{array}{c}A=\frac{\left({k}_{21}-\alpha \right)\left({k}_{31}-\alpha \right)}{\left(\beta -\alpha \right)\left(\gamma -\alpha \right)}\\ B=\frac{\left({k}_{21}-\beta \right)\left({k}_{31}-\beta \right)}{\left(\alpha -\beta \right)\left(\gamma -\beta \right)} \\ C=\frac{\left({k}_{21}-\gamma \right)\left({k}_{31}-\gamma \right)}{\left(\alpha -\gamma \right)\left(\beta -\gamma \right)}\end{array}\right.$$

(7)

The values of −α, −β, and −γ are the solutions of the cubic Equation 8:

$${x}^{3}+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}=0$$

(8)

Cardano’s method [11] offers the algebraic solution of Eq. 8. Equation 8 can be converted to Eq. 10 using Eq. 9:

$$x=y-\frac{{a}_{2}}{3}$$

(9)

$$\begin{array}{c}{y}^{3}+px+q=0\end{array}$$

(10)

where $p={a}_{1}-\frac{{{a}_{2}}^{2}}{3}$ and $q={a}_{0}-\frac{1}{3}{a}_{1}{a}_{2}+\frac{2}{27}{{a}_{2}}^{3}$.

The solutions of y are calculated as y_i in Eq. 11:

$$\begin{array}{c}{y}_{i}=2a\mathrm{cos}\left(\frac{1}{3}\mathrm{arccos}\frac{b}{2a}+\frac{2i\pi }{3}\right)\end{array}$$

(11)

where i = 0, 1, or 2, $a=\sqrt{-\frac{p}{3}}$, and $b=-\frac{q}{{a}^{2}}$.

Equations 9 and 11 lead Eq. 12:

$$\begin{array}{c}{x}_{i}=2a\mathrm{cos}\left(\frac{1}{3}\mathrm{arccos}\frac{\mathrm{b}}{2\mathrm{a}}+\frac{2\mathrm{i\pi }}{3}\right)-\frac{{\mathrm{a}}_{2}}{3}\end{array}$$

(12)

where i = 0, 1, or 2.

Finally, α, β, and γ are calculated using Eq. 13:

$$\begin{array}{c}\left\{\begin{array}{c}\alpha =maximum\left\{{x}_{i}\right\}\\ \beta =median\left\{{x}_{i}\right\}\\ \gamma =minimum\left\{{x}_{i}\right\}\end{array}\right.\end{array}$$

(13)

where i = 0, 1, or 2.

Appendix 2

Equation 14 provides approximately k_e0 if the numerical analysis (Appendix 1) is impossible to be applied. Before obtaining the following equation, numerical analysis clarified that Eq. 3 had only one k_e0 value between 0.15 and 0.26 min⁻¹ (this is the range of k_e0 for the Masui PK model for a patient with characteristics within the original study) for each pharmacokinetic parameter set.

$$\begin{array}{l}\begin{array}{l}{k}_{\mathrm{e}0}=-9.06+\mathrm{F}\left(\mathrm{age}\right)+F\left(\mathrm{TBW}\right)+F\left(\mathrm{height}\right)+0.999\cdot F\left(\mathrm{sex}\right)+F\left(\mathrm{ASAPS}\right)\\ -4.50\cdot \mathrm{F}2\left(\mathrm{age}\right)\cdot F2\left(\mathrm{TBW}\right)-4.51\cdot F2\left(\mathrm{age}\right)\cdot F2\left(\mathrm{height}\right)\\ +2.46\cdot \mathrm{F}2\left(\mathrm{age}\right)\cdot F2\left(\mathrm{sex}\right)+3.35\cdot F2\left(\mathrm{age}\right)\cdot F2\left(\mathrm{ASAPS}\right)\\ -12.6\cdot F2\left(\mathrm{TBW}\right)\cdot F2\left(\mathrm{height}\right)+0.394\cdot F2\left(\mathrm{TBW}\right)\cdot F2\left(\mathrm{sex}\right)\\ +2.06\cdot F2\left(\mathrm{TBW}\right)\cdot F2\left(\mathrm{ASAPS}\right)+0.390\cdot F2\left(\mathrm{height}\right)\cdot F2\left(\mathrm{sex}\right)\\ +2.07\cdot F2\left(\mathrm{height}\right)\cdot F2\left(\mathrm{ASAPS}\right)+5.03\cdot F2\left(\mathrm{sex}\right)\cdot F2\left(\mathrm{ASAPS}\right)\\ +99.8\cdot F2\left(\mathrm{age}\right)\cdot F2\left(\mathrm{TBW}\right)\cdot F2\left(\mathrm{height}\right)+5.11\cdot F2\left(\mathrm{TBW}\right)\cdot F2\left(\mathrm{height}\right)\cdot F2\left(\mathrm{sex}\right)\\ -39.4\cdot F2\left(\mathrm{TBW}\right)\cdot F2\left(\mathrm{height}\right)\cdot F2\left(\mathrm{ASAPS}\right)-5.00\cdot F2\left(\mathrm{TBW}\right)\cdot F2\left(\mathrm{sex}\right)\cdot F2\left(\mathrm{ASAPS}\right)\\ -5.04\cdot F2\left(\mathrm{height}\right)\cdot F2\left(\mathrm{sex}\right)\cdot F2\left(\mathrm{ASAPS}\right)\end{array}\end{array}$$

(14)

where

$$\left\{\begin{array}{l}F\left(\mathrm{age}\right)=0.228-2.72\cdot {10}^{-5}\cdot age+2.96\cdot {10}^{-7}\cdot {\left(\mathrm{age}-55\right)}^{2}\\ -4.34\cdot {10}^{-9}\cdot {\left(\mathrm{age}-55\right)}^{3}+5.05\cdot {10}^{-11}\cdot {\left(\mathrm{age}-55\right)}^{4}\\ F\left(\mathrm{TBW}\right)=0.196+3.53\cdot {10}^{-4}\cdot TBW-7.91\cdot {10}^{-7}\cdot {\left(\mathrm{TBW}-90\right)}^{2}\\ F\left(\mathrm{height}\right)=0.148+4.73\cdot {10}^{-4}\cdot height-1.43\cdot {10}^{-6}\cdot {\left(\mathrm{Height}-167.5\right)}^{2}\\ F\left(\mathrm{sex}\right)=0.237-2.16\cdot {10}^{-2}\cdot sex\\ F\left(\mathrm{ASAPS}\right)=0.214+2.41\cdot {10}^{-2}\cdot ASAPS\\ F2\left(\mathrm{age}\right)=F\left(\mathrm{age}\right)-0.227\\ F2\left(\mathrm{TBW}\right)=F\left(\mathrm{TBW}\right)-0.227\\ F2\left(\mathrm{height}\right)=F\left(\mathrm{height}\right)-0.226\\ F2\left(\mathrm{sex}\right)=F\left(\mathrm{sex}\right)-0.226\\ F2\left(\mathrm{ASAPS}\right)=F\left(\mathrm{ASAPS}\right)-0.226\end{array}\right.$$

age (years), TBW total body weight (kg), height (cm), sex: 0 for male and 1 for female, and ASAPS: 0 for ASA-PS I/II and 1 for ASA-PS III/IV.