On a Multicompartment Migration Model with Chronic Feeding

In uptake and retention studies, the biosystem can often be thought of as comprising K distinct compartments with constant rates of passage of particles from any compartment to a different compartment. Before the theoretical uptake and retention curve for any compartment can be plotted and compared with the experimentally determined curve for the same compartment, it must be possible to estimate these rates from the data of the experiment. A mathematical model of such a system is deduced which is sufficiently general to comprehend both the chronic and acute feeding situations. A method is presented for estimating the migration rate constants. If λiidenotes the theoretical constant rate of migration of particles from region i to region j, and$\tilde\lambda_{ii}$denotes an estimate of λiicalculated from the experimental data, then it is shown that \begin{equation*}\tag{1.1}\tilde\lambda_{ij} = \frac{1}{m_i}\sum^K_{k=1} w_{ik} \sum^{n-1}_{\nu=1} \frac{u^*_{k\nu} [m_j(u^*_{j, \nu+1} - u^*_{i, \nu-1}) - \delta_{1j}(f_{\nu+1} - f_{\nu-1})]}{t_{\nu+1} - t_{\nu-1}},\\ i = 1, \cdots, K, i \neq j,\end{equation*} where the meaning of the symbols occurring above is as follows: tris the time, measured from the beginning of the experiment, at which the vth observation of the system is made, ν = 1, ⋯, n - 1 (t0= 0); miis the size, measured in appropriate units, of the ith compartment or region (for example, total volume in milliliters of blood, or total mass in grams of bone), and is here assumed to be constant in a first approximation; fv= f(tv), where f(t) (the feeding function) measures the total amount of radioactivity administered to the system up to time t; u* kνis the concentration of radioactivity in region k at time tν; wikis the element in row i and column k in the inverse of the matrix [∑n-1 ν=1u* iνu* kν], and δij= 1, if j = 1, δij= 0, if j ≠ 1.