# Michaelis-Menten 1-compartment

Michaelis-Menten 1-compartment, bolus
Parameters
$$V$$ Volume of distribution. Ratio of the drug’s amount present in the body over its circulating concentration, [L]
$$V_m$$ 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]
Equations
Equation: $$\frac{dC}{dt} = -\frac{\frac{V_m}{V}\times C}{K_m + C}$$
Initial conditions: $$C(t_0) = C_{residual} + \frac{D}{V}$$
Michaelis-Menten 1-compartment, infusion
Parameters
$$V$$ Volume of distribution. Ratio of the drug’s amount present in the body over its circulating concentration, [L]
$$V_m$$ 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]
$$T_{inf}$$ Infusion time, [h]
Equations
Equation: \begin{align*} \frac{dC}{dt} &= \begin{cases} \frac{k_0}{V} – \frac{\frac{V_m}{V}\times C}{K_m + C} , &\text{for t\leq t_0 + T_{inf}}\\ -\frac{\frac{V_m}{V}\times C}{K_m + C}, &\text{for t> t_0 + T_{inf}} \end{cases} \end{align*}
Initial conditions: $$C(t_0) = C_{residual}$$
Michaelis-Menten 1-compartment, extravascular
Parameters
$$V$$ Volume of distribution. Ratio of the drug’s amount present in the body over its circulating concentration, [L]
$$V_m$$ 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]
$$k_a$$ Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [h⁻¹]
Equations
Equation: \begin{align*} \frac{dC_1}{dt} &= k_a C_2 -\frac{\frac{V_m}{V}\times C_1}{K_m + C_1} \\ \frac{dC_2}{dt} &= – k_a C_2 \end{align*}
Initial conditions: \begin{align*} C_1(t_0) &= C_{1residual} \\ C_2(t_0) &= C_{2residual} + \frac{F D}{V} \\ \end{align*}