Steady-state volume of distribution of two-compartment models with simultaneous linear and saturated elimination

Abstract

The model-independent estimation of physiological steady-state volume of distribution ($V_{dss,p}$), often referred to non-compartmental analysis (NCA), is historically based on the linear compartment model structure with central elimination. However the NCA-based steady-state volume of distribution ($V_{dss,nca}$) cannot be generalized to more complex models. In the current paper, two-compartment models with simultaneous first-order and Michaelis–Menten elimination are considered. In particular, two indistinguishable models $\mathrm{M}_1$ and $\mathrm{M}_2$, both having central Michaelis–Menten elimination, while first-order elimination exclusively either from central or peripheral compartment, are studied. The model-based expressions of the steady-state volumes of distribution $V_{dss,\mathrm{M}_i}\,\,(i=1,2)$ and their relationships to NCA-based $V_{dss,nca}$ are derived. The impact of non-linearity and peripheral elimination is explicitly delineated in the formulas. Being concerned with model identifiability and indistinguishability issues, an interval estimate of $V_{dss,p}$ is suggested.

Appendices

Appendix 1: Derivation Eq. 4 based on a two-compartment linear model with central elimination

For a two-compartment linear model with central elimination, the drug disposition of drug amount at central ($A_1(t)$) and peripheral compartments ($A_2(t)$) after a constant intravenous infusion rate $R_0$ is

$$\begin{aligned} \left\{ \begin{array}{ll} &A'_1(t)= R_0+ k_{21}A_2(t)-(k_{12}+k_{el})A_1(t),\\ &A'_2(t)= k_{12}A_1(t)-k_{21}A_2(t), \end{array} \right. \end{aligned}$$

where $k_{12}$ and $k_{21}$ are distribution rate constants from the central to the peripheral compartment and vice versa, respectively; $k_{el}$ is the linear elimination rate constant from the central compartment. Thus the concentration at the central compartment is $C_p(t)=A_1(t)/V_1$, where $V_1$ is the volume for the central compartment.

Following the physiological definition (Eq. 1), the steady-state drug amount in each compartment ($A^{ss}_{1}$ and $A^{ss}_{2}$) satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} &R_0+k_{21}A^{ss}_{2}-(k_{12}+k_{el})A^{ss}_{1}=0,\\ &k_{12}A^{ss}_{1}-k_{21}A^{ss}_{2}=0. \end{array} \right. \end{aligned}$$

(35)

Solving the system (35), we obtain

$$\begin{aligned} A_1^{ss}=\frac{R_0}{k_{el}},\quad A_2^{ss}=\frac{k_{12}R_0}{k_{21}k_{el}},\quad \text {and}\quad C_p^{ss}=\frac{A_1^{ss}}{V_1}=\frac{R_0}{k_{el}V_1}. \end{aligned}$$

(36)

Accordingly, the steady-state volume of distribution using compartment analysis is

$$\begin{aligned} V_{dss,\mathrm{M}}=\frac{A_{1}^{ss}+A_{2}^{ss}}{C_p^{ss}}=V_1\left[ 1+\frac{k_{12}}{k_{21}}\right] . \end{aligned}$$

On the other hand, the drug disposition of the two-compartment linear model after a single intravenous bolus administration is

$$\begin{aligned} \left\{ \begin{array}{ll} &A'_1(t)= k_{21}A_2(t)-(k_{12}+k_{el})A_1(t),\\ &A'_2(t)= k_{12}A_1(t)-k_{21}A_2(t), \end{array} \right. \end{aligned}$$

(37)

where initial condition $A_{1}(0)=Dose$ and $A_{2}(0)=0$. Since system (37) is linear, the time course of drug plasma concentration can be written as

$$\begin{aligned} C_p(t)=A_1(t)/V_1=C_1e^{-\lambda _1 t}+C_2e^{-\lambda _2 t}, \end{aligned}$$

where the exponents ($\lambda _1, \lambda _2$) and coefficients ($C_1, C_2$) are determined as

$$\begin{aligned} \lambda _1+\lambda _2=k_{12}+k_{21}+k_{el}, \quad \lambda _1\lambda _2=k_{21}k_{el} \end{aligned}$$

and

$$\begin{aligned} C_1=\frac{Dose}{V_1} \frac{k_{21}-\lambda _1}{\lambda _2-\lambda _1},\quad C_2=\frac{Dose}{V_1} \frac{k_{21}-\lambda _2}{\lambda _1-\lambda _2}. \end{aligned}$$

After some straightforward calculations we obtain

$$\begin{aligned} AUC=\int _0^{\infty }C_p(t)\,dt=\frac{C_1}{\lambda _1}+\frac{C_2}{\lambda _2}=\frac{Dose}{V_1}\frac{1}{k_{el}} \end{aligned}$$

(38)

and

$$\begin{aligned} AUMC=\int _0^{\infty }tC_p(t)\,dt=\frac{C_1}{\lambda ^2_1}+\frac{C_2}{\lambda ^2_2}=\frac{Dose}{V_1}\frac{1}{k^2_{el}}\left[ 1+\frac{k_{12}}{k_{21}}\right] . \end{aligned}$$

Therefore, the steady-state volume of distribution using non-compartmental analysis is

$$\begin{aligned} V_{dss,nca}=\frac{Dose}{AUC}\frac{AUMC}{AUC} =Dose\frac{\frac{Dose}{V_1}\frac{1}{k^2_{el}}\left[ 1+\frac{k_{12}}{k_{21}}\right] }{(\frac{Dose}{V_1}\frac{1}{k_{el}})^2} =V_1\left[ 1+\frac{k_{12}}{k_{21}}\right] =V_{dss,\mathrm{M}}. \end{aligned}$$

Appendix 2: Derivation of indistinguishability of $\mathrm{M}_1$ and $\mathrm{M}_2$ (Eqs. 9)

We will claim that the relationships (Eqs. 9) in the case of intravenous bolus administration, the similar procedure can be used to obtain Eqs. 9 for the case of intravenous infusion. On the one hand, we assume that $C_{1,\mathrm{M}_1}(t)=C_{1,\mathrm{M}_2}(t)$ for all positive time $t\ge 0$. The equation $C_{1,\mathrm{M}_1}(0^+) = C_{1,\mathrm{M}_2}(0^+)$ directly implies $Dose/V_1=Dose/{\tilde{V}}_1$ which means

$$\begin{aligned} V_{1}={\tilde{V}}_{1}. \end{aligned}$$

Since the functions of $A_{1,\mathrm{M}_1}(t)$ and $A_{1,\mathrm{M}_2}(t)$ are infinite differentiable at the right hand side of time 0, then we have $A'_{1,\mathrm{M}_1}(0^+)=A'_{1,\mathrm{M}_2}(0^+)$ which is equivalent to

$$\begin{aligned} (k_{12}+k_{el})Dose+\frac{V_{max}Dose}{V_1K_m+Dose}=\tilde{k}_{12}Dose+\frac{{\tilde{V}}_{max}Dose}{{\tilde{V}}_1\tilde{K}_m+Dose} \end{aligned}$$

(39)

Rearrangement Eq. 39 leads to the following quadratic function with respect to Dose

$$\begin{aligned} a\cdot Dose^2 + b\cdot Dose + c = 0, \end{aligned}$$

(40)

where

$$\begin{aligned} a = \tilde{k}_{12}-(k_{12}+k_{el}),\quad b= (K_m+\tilde{K}_m)a+{\tilde{V}}_{max}-V_{max},\quad c= V_1^2K_m\tilde{K}_ma+V_1(K_m{\tilde{V}}_{max}-\tilde{K}_mV_{max}). \end{aligned}$$

Since Eq. 40 is valid for any Dose which implies that all coefficients a, b and c have to be zeroes, thence we have

$$\begin{aligned} \tilde{k}_{12} = k_{12}\, +\, k_{el},\quad {\tilde{V}}_{max} = V_{max},\quad \tilde{K}_m = K_m. \end{aligned}$$

(41)

Next we need to identify the relation of $k_{21}$ and $k_{el}$. Taking derivatives of $A'_{1,\mathrm{M}_1}(t)$ and $A'_{1,\mathrm{M}_2}(t)$ with respect to time t give rise to

$$\begin{aligned} A''_{1,\mathrm{M}_1}(t) &= \,k_{21}A'_{2,\mathrm{M}_1}(t)-(k_{12}+k_{el})A'_{1,\mathrm{M}_1}(t)-\frac{V_1V_{max}K_mA'_{1,\mathrm{M}_1}(t)}{(V_1K_m+A_{1,\mathrm{M}_1}(t))^2}\nonumber \\& = \,k_{21}k_{12}A_{1,\mathrm{M}_1}(t)-k^2_{21}A_{2,\mathrm{M}_1}-(k_{12}+k_{el})A'_{1,\mathrm{M}_1}(t) -\frac{V_1V_{max}K_mA'_{1,\mathrm{M}_1}(t)}{(V_1K_m+A_{1,\mathrm{M}_1}(t))^2} \end{aligned}$$

(42)

and

$$\begin{aligned} A''_{1,\mathrm{M}_2}(t) &= \tilde{k}_{21}A'_{2,\mathrm{M}_2}(t)-\tilde{k}_{12}A'_{1,\mathrm{M}_2}(t)-\frac{{\tilde{V}}_1{\tilde{V}}_{max}\tilde{K}_mA'_{1,\mathrm{M}_2}(t)}{({\tilde{V}}_1\tilde{K}_m+A_{1,\mathrm{M}_2}(t))^2}\nonumber \\&= \tilde{k}_{21}\tilde{k}_{12}A_{1,\mathrm{M}_2}(t)-\tilde{k}_{21}(\tilde{k_{21}}+\tilde{k}_{el})A_{2,\mathrm{M}_2}(t)-\tilde{k}_{12}A'_{1,\mathrm{M}_2}(t)-\frac{{\tilde{V}}_1{\tilde{V}}_{max}\tilde{K}_mA'_{1,\mathrm{M}_2}(t)}{({\tilde{V}}_1\tilde{K}_m+A_{1,\mathrm{M}_2}(t))^2}. \end{aligned}$$

(43)

Substituting $t=0^+$ into Eq. 42 and Eq. 43 and using Eq. 41, the identity $A''_{1,\mathrm{M}_1}(0^+) = A''_{1,\mathrm{M}_2}(0^+)$ yields

$$\begin{aligned} k_{12}k_{21} = \tilde{k}_{12}\tilde{k}_{21}, \end{aligned}$$

(44)

which is equivalent to

$$\begin{aligned} \tilde{k}_{21} = \frac{k_{21}k_{12}}{k_{12}+k_{el}}. \end{aligned}$$

Taking derivatives of $A''_{1,\mathrm{M}_1}(t)$ and $A''_{1,\mathrm{M}_2}(t)$ again leads to

$$\begin{aligned} A'''_{1,\mathrm{M}_1}(t) = k^3_{21}A_{2,\mathrm{M}_1}(t)-k_{12}k^2_{21}A_{1,\mathrm{M}_1}(t)-(k_{12}+k_{el})A''_{1,\mathrm{M}_1}(t) + \left( k_{12}k_{21}+\frac{2V_1V_{max}K_mA'_{1,\mathrm{M}_1}(t)}{(V_1K_m+A_{1,\mathrm{M}_1}(t))^3}\right) A'_{1,\mathrm{M}_1}(t) -\frac{V_1V_{max}K_mA''_{1,\mathrm{M}_1}(t)}{V_1K_m+A_{1,\mathrm{M}_1}(t)} \end{aligned}$$

and

$$\begin{aligned} A'''_{1,\mathrm{M}_2}(t) = \tilde{k}_{21}\left( \tilde{k}_{21}+\tilde{k}_{el}\right) ^2A_{2,\mathrm{M}_2}(t)-\tilde{k}_{12}\tilde{k}_{21}\left( \tilde{k}_{21}+\tilde{k}_{el}\right) A_{1,\mathrm{M}_2}(t)-\tilde{k}_{12}A''_{1,\mathrm{M}_2}(t) +\left( \tilde{k}_{12}\tilde{k}_{21}+\frac{2{\tilde{V}}_1{\tilde{V}}_{max}\tilde{K}_mA'_{1,\mathrm{M}_2}(t)}{({\tilde{V}}_1\tilde{K}_m+A_{1,\mathrm{M}_2}(t))^3}\right) A'_{1,\mathrm{M}_2}(t)-\frac{{\tilde{V}}_1{\tilde{V}}_{max}\tilde{K}_mA''_{1,\mathrm{M}_2}(t)}{{\tilde{V}}_1\tilde{K}_m+A_{1,\mathrm{M}_2}(t)}. \end{aligned}$$

Therefore, $A'''_{1,\mathrm{M}_1}(0^+) = A'''_{1,\mathrm{M}_2}(0^+)$ yields

$$\begin{aligned} k_{21} = \tilde{k}_{21}+\tilde{k}_{el}, \end{aligned}$$

which implies

$$\begin{aligned} \tilde{k}_{el} = k_{21}-\tilde{k}_{21} = \frac{k_{21}k_{el}}{k_{12}+k_{el}}. \end{aligned}$$

(45)

On the other hand, we will claim that $C_{1,\mathrm{M}_1}(t) = C_{1,\mathrm{M}_2}(t)$ for all $t\ge 0$ if Eq. 9 are valid. In fact, by the method of variation of constants we have

$$\begin{aligned} A_{2,\mathrm{M}_1}(t) = \int _{0}^t e^{-k_{21}(t-s)}k_{12}A_{1,\mathrm{M}_1}(s)\,ds, \end{aligned}$$

(46)

and

$$\begin{aligned} A_{2,\mathrm{M}_2}(t) = \int _{0}^t e^{-(\tilde{k}_{21}+\tilde{k}_{el})(t-s)}\tilde{k}_{12}A_{1,\mathrm{M}_2}(s)\,ds. \end{aligned}$$

(47)

Replacing $A_{2,\mathrm{M}_1}(t)$ and $A_{2,\mathrm{M}_2}(t)$ by Eq. 46 and Eq. 47 in the first equation of system $\mathrm{M}_1$ (Eq. 5), respectively, these yield an initial value problem of a system of an integral-differential equation with respect to the time course of observed drug concentration, that is,

$$\begin{aligned} \left\{ \begin{array}{ll} & C'_{1,\mathrm{M}_1}(t)=k_{12}k_{21}\int _{0}^te^{-k_{21}(t-s)}C_{1,\mathrm{M}_1}(s)\,ds-(k_{12}+k_{el})C_{1,\mathrm{M}_1}(t)-\frac{V_{max}C_{1,\mathrm{M}_1}(t)}{V_1(K_m+C_{1,\mathrm{M}_1}(t))},\\& C_{1,\mathrm{M}_1}(0^+) = Dose/V_1, \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{array}{ll} &C'_{1,\mathrm{M}_2}(t) = \tilde{k}_{12}\tilde{k}_{21}\int _{0}^t e^{-(\tilde{k}_{21}+\tilde{k}_{el})(t-s)}C_{1,\mathrm{M}_2}(s)\,ds-\tilde{k}_{12}C_{1,\mathrm{M}_2}(t)-\frac{{\tilde{V}}_{max}C_{1,\mathrm{M}_2}(t)}{{\tilde{V}}_1(\tilde{K}_m+C_{1,\mathrm{M}_2}(t))},\\ & C_{1,\mathrm{M}_2}(0^+)=Dose/{\tilde{V}}_1. \end{array} \right. \end{aligned}$$

According to the Eq. 9, $C_{1,\mathrm{M}_1}(t)$ and $C_{1,\mathrm{M}_2}(t)$ have the same dynamical behavior. Therefore, the qualitative theory of differential equation implies $C_{1,\mathrm{M}_1}(t)\equiv C_{1,\mathrm{M}_2}(t)$ under the conditions Eq. 9 [28].

Appendix 3: Indistinguishability of $\mathrm{M}_{e}$ from $\mathrm{M}_1$ and $\mathrm{M}_2$

The differential equations of model $\mathrm{M}_e$ are

$$\begin{aligned} \left\{ \!\!\!\! \begin{array}{lll} &\displaystyle \frac{d}{dt}A_{1,\mathrm{M}_e}(t) = f(t)+\hat{k}_{21}A_{2,\mathrm{M}_e}(t)\quad-(\hat{k}_{12}+\hat{k}_{el,1})A_{1,\mathrm{M}_e}(t)-\frac{\hat{V}_{max}A_{1,\mathrm{M}_e}(t)}{\hat{K}_m\times \hat{V}_1+A_{1,\mathrm{M}_e}(t)},\\ & \displaystyle \frac{d}{dt}A_{2,\mathrm{M}_e}(t) = \hat{k}_{12}A_{1,\mathrm{M}_e}(t)-(\hat{k}_{21}+\hat{k}_{el,2})A_{2,\mathrm{M}_e}(t),\\ & A_{1,\mathrm{M}_e}(0)=0, \quad A_{2,\mathrm{M}_e}(0)=0, \end{array} \right. \end{aligned}$$

(48)

where f(t) is drug input. Similar to the derivation of Eq. 7, the dynamic behavior of plasma concentration with time of system $\mathrm{M}_e$ (Eq. 48) can be expressed as

$$\begin{aligned} \displaystyle \frac{d}{dt}C_{1,\mathrm{M}_2}(t) = \frac{f(t)}{\hat{V}_1}+\hat{k}_{12}\hat{k}_{21}\int _{0}^te^{-(\hat{k}_{21}+\hat{k}_{el,2})(t-s)}C_{1,\mathrm{M}_1}(s)\,ds-(\hat{k}_{12}+\hat{k}_{el,1})C_{1,\mathrm{M}_2}(t)\nonumber -\frac{\hat{V}_{max}C_{1,\mathrm{M}_e}(t)}{\hat{V}_1(\hat{K}_m+C_{1,\mathrm{M}_e}(t))}. \end{aligned}$$

(49)

Comparing the equivalence of Eqs. 7, 8 and Eq. 49, and the qualitative theory of integro-differential equations [28], it follows that $\mathrm{M}_e$ is indistinguishable from models $\mathrm{M}_1$ and $\mathrm{M}_2$ under the conditions Eqs. 20.