Path lengths and initial derivatives in arbitrary and Hessenberg compartmental systems

Abstract

It is shown that if the matrix of an n-compartment system is upper Hessenberg (zero entries below the subdiagonal) and nonzero initial conditions exist in the first compartment only, then the solution for the nth compartment is β e 1∗ e 2∗…∗ e n where β is a constant, e i ( t)=exp(λ it ), λ i is the ith root of the matrix and ∗ denotes convolution. Conversely, the solution is a compartment can take this form only if the matrix of the system is Hessenberg (up to a permutational similarity). These results depend upon the relations which obtain among the distribution of zero entries in sequential powers of the matrix, the initial derivatives in the compartments, and the path length between compartments in the directed diagram of the system. These relations are developed and employed in an assessment of the product precursor criterion as proposed by Rescigno and Segre.