MichealisMenten 3compartments, bolus  

Parameters  
\(V_m\)  Maximum rate of elimination. Maximum rate of elimination of the drug from the circulation through a saturable pathway, [mg/h]  
\(K_m\)  MichaelisMenten 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]  
\(Q_2\)  Intercompartmental clearance 12. Ratio of the drug’s distribution rate between compartments 1 and 2 over its circulating concentration, [L/h]  
\(Q_3\)  Intercompartmental clearance 13. 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 3compartments, infusion  

Parameters  
\(V_m\)  Maximum rate of elimination. Maximum rate of elimination of the drug from the circulation through a saturable pathway, [mg/h] 
\(K_m\)  MichaelisMenten 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] 
\(Q_2\)  Intercompartmental clearance 12. Ratio of the drug’s distribution rate between compartments 1 and 2 over its circulating concentration, [L/h] 
\(Q_3\)  Intercompartmental clearance 13. 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 3compartments, extravascular  

Parameters  
\(V_m\)  Maximum rate of elimination. Maximum rate of elimination of the drug from the circulation through a saturable pathway, [mg/h] 
\(K_m\)  MichaelisMenten 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] 
\(Q_2\)  Intercompartmental clearance 12. Ratio of the drug’s distribution rate between compartments 1 and 2 over its circulating concentration, [L/h] 
\(Q_3\)  Intercompartmental clearance 13. 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:  \( \begin{align*} C_1(t_0) &= C_{1residual} \\ C_2(t_0) &= C_{2residual} \\ C_3(t_0) &= C_{3residual} \\ C_4(t_0) &= C_{4residual} + \frac{F D}{V_1} \end{align*}\) 