Richard Ernest Bellman

Some mathematical aspects of chemotherapy—II: The distribution of a drug in the body

In a previous paper (Bellman, Jacquez, and Kalaba,Bull. Math. Biophysics,22: 181–198, 1960) a model of the processes occurring in the exchange of a drug between capillary plasma, extracellular space and intracellular space was developed. This included the possibility of a reaction between the drug and a component of the intracellular space. The equations developed thus describe the events within a capillary bed. In the present paper, a simplified model of the body is set up. Each organ is treated as a single capillary bed and is linked to other organs via the circulation, in the parallel and/or series arrangements found in the body. Mixing in the circulation is included at the simplest possible level. The concentration of drug entering any one capillary bed is determined by the concentrations leaving all other capillary beds, the time lags, and mixing involved in the circulation. The equations describing these processes in conjunction with the equations of the processes occurring within each capillary bed lead to a large set of differential-difference equations.