9. BASIC PBPK MODEL EQUATIONS

A Physiologically Based PharmacoKinetic (PBPK) model for humans descibes the body as a set of interconnected compartments, or continuous stirred tank reactors (CSTRs). Each compartment can describe either an organ or a tissue. The set of differential equations of the PBPK model are derived through the mass balance across various compartmens (see Figure 5.1, on Page ), as follows.

A mass balance around the equilibrium lung compartment results in:

$$c_{\mbox{arterial}}= \frac{Q_{\rm cardiac}c_{\mbox{venous}}+ ... ...rdiac}+ Q_{\rm alveolar}/ P_{\mbox{\scriptsize blood/air}}}\ ,$$ (9.1)

where,

$$c_{\mbox{venous}}= \frac{1}{Q_{\rm cardiac}} \sum_{j=1}^{n} Q_j c_j \;$$ (9.2)

and $Q_j$ and $c_j$ are volumetric blood flow and concentration with respect to compartment $j$.

The mass balance on all compartments $j$ in the PBPK model, other than the viable skin and stratum corneum compartments, is given by:

$$V_j \frac{d c_j}{dt} = Q_j\left (c_{\mbox{arterial}}- \frac{c_j}{P_{j{\rm /blood}}}\right ) - R_j.$$ (9.3)

The rate of metabolism, $R_j$, in any given compartment is given by:

$$R_j = \frac{V\!{\rm max}_{j}c_{v,j}}{K\!\mbox{\small m}_j+ c_{v,j}} \;$$ (9.4)

where $V\!{\rm max}_{j}$ and $K\!\mbox{\small m}_j$ are Michaelis-Menten constants.

Roy [176] presents a thorough description of PBPK models for humans.