Basic Model

These are often referred to as ADME, and taken together describe the drug concentration in the body when medicine is prescribed. These ADME processes are typically described by zeroth-order or first-order rate reactions modelling the dynamics of the quantity of drug q , with a given rate parameter k, for example:

\[\frac{dq}{dt} = -k^{*}\]

\[\frac{dq}{dt} = -kq\]

The body itself is modelled as one or more compartments, each of which is defined as a kinetically homogeneous unit (these compartments do not relate to specific organs in the body, unlike Physiologically based pharmacokinetic, PBPK, modeling). There is typically a main central compartment into which the drug is administered and from which the drug is excreted from the body, combined with zero or more peripheral compartments to which the drug can be distributed to/from the central compartment (See Fig 2). Each peripheral compartment is only connected to the central compartment.

Two Compartment Model

The following example PK model describes the two-compartment model shown diagrammatically in Fig 2. The time-dependent variables to be solved are the drug quantity in the central and peripheral compartments, q_c and q_p1 (units: [ng]) respectively.

\[\frac{dq}{dt} = Dose(t) - \frac{q_{c}}{V_{c}}CL - Q_{p1} (\frac{q_{c}}{V_{c}} - \frac{q_{p1}}{V_{p1}}),\]

\[\frac{dq_{p1}}{dt} = Q_{p1} (\frac{q_{c}}{V_{c}} - \frac{q_{p1}}{V_{p1}}) .\]

This model describes an intravenous bolus dosing protocol, with a linear clearance from the central compartment (non-linear clearance processes are also possible, but not considered here). The input parameters to the model are:

• The dose function Dose(t), which could consist of instantaneous doses of X ng of the drug at one or more time points, or a steady application of X ng per hour over a given time period, or some combination.

• V_c [mL], the volume of the central compartment

• V_p1 [mL], the volume of the first peripheral compartment

• CL [mL/h], the clearance/elimination rate from the central compartment

• Q_p1 [mL/h], the transition rate between central compartment and peripheral compartment 1

Subcutaneous Model

Another example model we will show uses subcutaneous dosing, and adds an additional compartment from which the drug is absorbed to the central compartment

\[\frac{dq_{0}}{dt} = Dose(t) - k_{\alpha}q_{0},\]

\[\frac{dq_{c}}{dt} = k_{\alpha}q_{0} - \frac{q_{c}}{V_{c}}CL - Q_{p1} (\frac{q_{c}}{V_{c}} - \frac{q_{p1}}{V_{p1}}),\]

\[\frac{dq_{p1}}{dt} = Q_{p1} - (\frac{q_{c}}{V_{c}} - \frac{q_{p1}}{V_{p1}}).\]

where .. math:: k_α [/h] is the “absorption” rate for the s.c dosing.

N Compartment Model

When dealing with more complex systems the following equation is used, which builds on the previous equations to enable multi-compartment modeling.

\[\frac{dq_{0}}{dt} = Dose(t) - k_{\alpha}q_{0},\]

\[\frac{dq_{c}}{dt} = k_{\alpha}q_{0} - \frac{q_{c}}{V_{c}}CL - \sum_{i=1}^{N}Q_{pi} (\frac{q_{c}}{V_{c}} - \frac{q_{pi}}{V_{pi}}) ,\]

\[\frac{dq_{pi}}{dt} = Q_{pi}(\frac{q_{c}}{V_{c}} - \frac{q_{pi}}{V_{pi}}).\]