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# PK libraries / One compartment models

Lixoft provides PK model libraries. The purpose of this library is to provide the user an extensive set of classical PK model to help him during the modelization. These model can be split between the number of compartments, their type of absorption and their type of elimination.

### Parameters

• V: volume of distribution,
• k: elimination rate constant,
• Cl: clearance of elimination,
• Vm: maximum elimination rate (in amount per time unit),
• Km: Michaelis-Menten constant (in concentration unit),
• ka: absorption rate constant,
• Tlag: lag time,
• Tk0: absorption duration for zero order absorption

### Parameterisation

There are two parameterizations for one compartment models, (V and k) or (V and Cl). The equations are given for the first parameterization (V; k). The equations for the second parameterizations (V; Cl) are derived using $k = \frac{Cl}{V}$.

### Hypothesis and initial conditions

We assume that $C_c(t)=C_e(t)=0$ for all time before the first dose.

### Linear elimination model

#### IV bolus

Additional hypothesis : At each dose, the concentration in the central compartment $C_c$ has an increasing step of $\frac{D_i}{V}$

Single dose $\begin{array}{l}\left\{\begin{array}{ccl}C_c(t)&=&\frac{D}{V}e^{-k(t-t_{D})}\\ C_e(t)&=&\frac{D}{V}\frac{k_{e0}}{k_{e0}-k}(e^{-k(t-t_{D})}- e^{-k_{e0}(t-t_{D})})\end{array} \right. \end{array}$

Multiple doses : $\begin{array}{l}\left\{\begin{array}{ccl}C_c(t)&=&\sum^{n}_{i=1}\frac{D_{i}}{V}e^{-k(t-t_{D_{i}})}\\ C_e(t)&=&\sum^{n}_{i=1}\frac{D_i}{V}\frac{k_{e0}}{k_{e0}-k}(e^{-k(t-t_{D_i})}- e^{-k_{e0}(t-t_{D_i})})\end{array} \right. \end{array}$

Steady state $\begin{array}{l}\left\{\begin{array}{ccl}C_c(t)&=&\frac{D}{V}\frac{e^{-k(t-t_D)}}{1-e^{-k\tau}}\\C_e(t)&=&\frac{D}{V}\frac{k_{e0}}{k_{e0}-k}(\frac{e^{-k(t-t_D)}}{1-e^{-k\tau}}-\frac{e^{-k_{e0}(t-t_D)}}{1-e^{-k_{e0}\tau}}) \end{array}\right. \end{array}$

### IV infusion

Dose type Time is less or equal to the last dose timing plus the infusion timing, i.e. $t \leq t_D+T_{inf}$ Time is greater than the last dose timing plus the infusion timing, i.e. $t > t_D+T_{inf}$
Single dose $\begin{array}{l}\left\{\begin{array}{ccl}C_c(t)&=&\frac{D}{T_{inf}}\frac{1}{kV}(1-e^{-k(t-t_{D})})\\C_e(t)&=&\frac{D}{T_{inf}}\frac{1}{kV(k_{e0}-k)}[k_{e0}(1-e^{k(t-t_D)})\\&&-k(1-e^{-k_{e0}(t-t_D)})] \end{array} \right. \end{array}$ $\begin{array}{l}\left\{\begin{array}{ccl}C_c(t)&=&\frac{D}{T_{inf}}\frac{1}{kV}(1-e^{-kT_{inf}})e^{-k(t-t_{D}-T_{inf})}\\C_e(t)&=&\frac{D}{T_{inf}}\frac{1}{kV(k_{e0}-k)}[k_{e0}(1-e^{-kT_{inf}})e^{-k(t-t_D-T_{inf})}\\&&-k(1-e^{-k_{e0}T_{inf}})e^{-k_{e0}(t-t_D-T_{inf})}]\end{array} \right. \end{array}$
Multiple doses $\begin{array}{l}\left\{\begin{array}{ccl}C_c(t)&=&\sum^{n-1}_{i=1}\frac{D_{i}}{T_{inf,i}}\frac{1}{kV}(1-e^{-kT_{inf,i}})e^{-k(t-t_{D_{i}}-T_{inf,i})}\\&&+\frac{D_{n}}{T_{inf,n}}\frac{1}{kV}(1-e^{-k(t-t_{D_{n}}})\\C_e(t)&=&\sum^{n-1}_{i=1}\frac{D_i}{T_{inf,i}}\frac{1}{kV(k_{e0}-k)}[k_{e0}(1-e^{-kT_{inf,i}})e^{-k(t-t_{D_{i}}-T_{inf,i})}\\&&-k(1-e^{-k_{e0}T_{inf,i}})e^{-k_{e0}(t-t_{D_{i}}-T_{inf,i})}]\\&&+\frac{D_n}{T_{inf,n}}\frac{1}{kV(k_{e0}-k)}[k_{e0}(1-e^{-k(t-t_{D_{n}})})\\&&-k(1-e^{-k_{e0}(t-t_{D_{n}})})] \end{array} \right. \end{array}$ $\begin{array}{l}\left\{\begin{array}{ccl}C_c(t)&=&\sum^{n}_{i=1}\frac{D_{i}}{T_{inf,i}}\frac{1}{kV}(1-e^{-kT_{inf,i}})e^{-k(t-t_{D_{i}}-T_{inf,i})}\\C_e(t)&=&\sum^{n}_{i=1}\frac{D_i}{T_{inf,i}}\frac{1}{kV(k_{e0}-k)}[k_{e0}(1-e^{-kT_{inf,i}})e^{-k(t-t_{D_{i}}-T_{inf,i})}\\&&-k(1-e^{-k_{e0}T_{inf,i}})e^{-k_{e0}(t-t_{D_{i}}-T{inf,i})}] \end{array} \right. \end{array}$