Matrix proof of flow,volume and mean transit time theorems for regional and compartmental systems

The relations (inflow) = (dose)/(area under indicator curve), and (volume of distribution) = (throughflow) × (mean transit time) are derived by a matrix method for a system of interconnected subsystems, within which spatial indicator activity gradients may exist, and for compartments, within which the indicator activity is spatially uniform. The inflow theorem, is different from the outflow theorem. Equivalent labeling of multi-input systems reduces them formally to single input systems. Foreign indicator flow-volume kinetics are more general than, and include as a special case, tracer flux-mass (metabolic) kinetics. Volume of distribution in the indicator steady state may be different from the equilibrium volume of distribution.