# The sojourn time and its prospective use in pharmacology

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## Abstract

Sojourn time in a given compartment i when the material has been injected in compartment j (S_{ji}) corresponds to the average time spent by the particles of the material in i before their definitive exit from that compartment. Sojourn time is different from the mean residence time\((\bar t_{ji} )\), which is the average age of the particles leaving the system. If\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{K} \) denotes the transfer matrix of the compartmental system, then\( - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{K} ^{ - 1} \) provides the sojourn times in each compartment given initial arrival in each of the other compartments. It can be shown that\(S_{ji} = x_i (s)/x_{j,0} \left| {_{s = 0} } \right.\) that is the transfer function between compartment j and i, where x_{i}(s) is the Laplace transform of the function x_{i}(t) and x_{j,0} indicates the dose introduced in compartment j at time 0. It can be observed that x_{i}¦_{s=0} corresponds to the value of AUC in compartment i. Since AUC_{i}/x_{j,0}=F_{ji} · AUC/x_{i,0} (F_{ji}=fraction of the dose in j reaching i), one has S_{ji}=F_{ji}S_{ii} · AUC_{i} corresponds to a rectangle of height equal to x_{j,i} and base equal to S_{ii}. Therefore in a compartment i a drug acts on the average for a time equal to S_{ii} and the number of molecules in it depends on the dose and on F_{ji}. In compartments which are not sampled the value of AUC can be calculated by a simulated curve or by\( - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{K} ^{ - 1} \) From the height of the rectangle whose area is equal to AUC one should subtract the threshold θ for a given effect; the resulting should indicate the intrinsic efficacy of the drug. These considerations could be applied in pharmacology, toxicology, and chemotherapy.

## Key words

sojourn time mean residence time compartments## Preview

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