| 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: | \( \begin{align*} C_1(t_0) &= C_{1residual} \\ C_2(t_0) &= C_{2residual} \\ C_3(t_0) &= C_{3residual} \\ C_4(t_0) &= C_{4residual} + \frac{F D}{V_1} \end{align*}\) |