Density functional theory (DFT) calculations have been performed using the uB3LYP/6-31++G(d,p) model to calculate the solution phase one-electron reduction potentials (E(calc)) and absolute pKa values of a series of copper bis(thiosemicarbazonato) complexes. The effects of solvation in water and dimethylsulfoxide (DMSO) are incorporated as a self-consistent reaction field (SCRF) using the integral equation formalism polarisable continuum model (IEFPCM) and are found to be essential for quantitative agreement with an average error in E(calc) of -0.02 V compared to experiment. The bonding and spin densities are examined through the use of Natural Bond Order analysis and the results used to rationalise the calculated and observed reduction potentials. Calculated estimates of pKa values of several copper(II) species are presented and their implications for the mechanisms of transport and trapping within hypoxic cells are considered. Reduction is found to be a prerequisite for protonation of the complexes which suggests their transport in the blood stream as neutral species, and the mechanistic sequence is identified as a sequential electrochemical-chemical (EC) process. The complex equilibria of protonation, reoxidation and dissociation are discussed and the copper(I) diprotonated, cationic complex of diacetyl bis(4-methyl-3-thiosemicarbazonato)copper(II), Cu(I)ATSMH2(+), is identified as a possible candidate for the initial species trapped in hypoxic cells.