# Michaelis-Menten 3-compartments

Michealis-Menten 3-compartments, bolus
Parameters
$$V_m$$ Maximum rate of elimination. Maximum rate of elimination of the drug from the circulation through a saturable pathway, [mg/h]
$$K_m$$ Michaelis-Menten constant. Substrate concentration at which a saturable elimination rate reaches half of its maximum value, [mg/L]
$$V_1$$ Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
$$V_2$$ Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
$$V_3$$ Peripheral volume of distribution, third compartment. Volume into which a drug is considered to distribute thirdly with further retardation, from and back to the central compartment, [L]
$$Q_2$$ Inter-compartmental clearance 1-2. Ratio of the drug’s distribution rate between compartments 1 and 2 over its circulating concentration, [L/h]
$$Q_3$$ Inter-compartmental clearance 1-3. Ratio of the drug’s distribution rate between compartment 1 and 3 over its circulating concentration, [L/h]
$$k_{12}$$ $$=\frac{Q_2}{V_1}$$
$$k_{21}$$ $$=\frac{Q_2}{V_2}$$
$$k_{13}$$ $$=\frac{Q_3}{V_1}$$
$$k_{31}$$ $$=\frac{Q_3}{V_3}$$
Equations
Equation: \begin{align*} \frac{dC_1}{dt} &= k_{21} C_2 – k_{12} C_1 + k_{31} C_3 – k_{13} C_1 -\frac{\frac{V_m}{V_1}\times C_1}{K_m + C_1} \\ \frac{dC_2}{dt} &= k_{12} C_1 -k_{21} C_2 \\ \frac{dC_3}{dt} &= k_{13} C_1 -k_{31} C_3 \end{align*}
Initial conditions: \begin{align*} C_1(t_0) &= C_{1residual} + \frac{D}{V_1} \\ C_2(t_0) &= C_{2{residual}} \\ C_3(t_0) &= C_{3{residual}} \end{align*}
Linear 3-compartments, infusion
Parameters
$$V_m$$ Maximum rate of elimination. Maximum rate of elimination of the drug from the circulation through a saturable pathway, [mg/h]
$$K_m$$ Michaelis-Menten constant. Substrate concentration at which a saturable elimination rate reaches half of its maximum value, [mg/L]
$$V_1$$ Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
$$V_2$$ Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
$$V_3$$ Peripheral volume of distribution, third compartment. Volume into which a drug is considered to distribute thirdly with further retardation, from and back to the central compartment, [L]
$$Q_2$$ Inter-compartmental clearance 1-2. Ratio of the drug’s distribution rate between compartments 1 and 2 over its circulating concentration, [L/h]
$$Q_3$$ Inter-compartmental clearance 1-3. Ratio of the drug’s distribution rate between compartment 1 and 3 over its circulating concentration, [L/h]
$$T_{inf}$$ Infusion time, [h]
$$k_{12}$$ $$=\frac{Q_2}{V_1}$$
$$k_{21}$$ $$=\frac{Q_2}{V_2}$$
$$k_{13}$$ $$=\frac{Q_3}{V_1}$$
$$k_{31}$$ $$=\frac{Q_3}{V_3}$$
Equations
Equation: \begin{align*} \frac{dC_1}{dt} &= \begin{cases} k_{21} C_2 – k_{12} C_1 -\frac{\frac{V_m}{V_1}\times C_1}{K_m + C_1} + \frac{k_0}{V_1}, &\text{for t\leq t_0 + T_{inf}}\\ k_{21} C_2 – k_{12} C_1 -\frac{\frac{V_m}{V_1}\times C_1}{K_m + C_1}, &\text{for t> t_0 + T_{inf}} \end{cases} \\ \frac{dC_2}{dt} &= k_{12} C_1 -k_{21} C_2 \end{align*}
Initial conditions: \begin{align*} C_1(t_0) &= C_{1residual} \\ C_2(t_0) &= C_{2{residual}} \end{align*}
Linear 3-compartments, extravascular
Parameters
$$V_m$$ Maximum rate of elimination. Maximum rate of elimination of the drug from the circulation through a saturable pathway, [mg/h]
$$K_m$$ Michaelis-Menten constant. Substrate concentration at which a saturable elimination rate reaches half of its maximum value, [mg/L]
$$V_1$$ Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
$$V_2$$ Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
$$V_3$$ Peripheral volume of distribution, third compartment. Volume into which a drug is considered to distribute thirdly with further retardation, from and back to the central compartment, [L]
$$Q_2$$ Inter-compartmental clearance 1-2. Ratio of the drug’s distribution rate between compartments 1 and 2 over its circulating concentration, [L/h]
$$Q_3$$ Inter-compartmental clearance 1-3. Ratio of the drug’s distribution rate between compartment 1 and 3 over its circulating concentration, [L/h]
$$k_a$$ Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [h⁻¹]
$$F$$ Bioavailability. Fraction of the drug’s administered dose that reaches unchanged the systemic circulation, [%]
$$k_{12}$$ $$=\frac{Q_2}{V_1}$$
$$k_{21}$$ $$=\frac{Q_2}{V_2}$$
$$k_{13}$$ $$=\frac{Q_3}{V_1}$$
$$k_{31}$$ $$=\frac{Q_3}{V_3}$$
Equations
Equation: \begin{align*} \frac{dC_1}{dt} &= k_{21} C_2 – k_{12} C_1 + k_{31} C_3 – k_{13} C_1 + k_a C_4 -\frac{\frac{V_m}{V_1}\times C_1}{K_m + C_1} \\ \frac{dC_2}{dt} &= k_{12} C_1 -k_{21} C_2 \\ \frac{dC_3}{dt} &= k_{13} C_1 -k_{31} C_3 \\ \frac{dC_4}{dt} &= -k_{a} C_4 \end{align*}
Initial conditions: \begin{align*} C_1(t_0) &= C_{1residual} \\ C_2(t_0) &= C_{2residual} \\ C_3(t_0) &= C_{3residual} \\ C_4(t_0) &= C_{4residual} + \frac{F D}{V_1} \end{align*}