The change in the NMR chemical shift of (23)Na(+) induced by the shift reagent TmDOTP was examined under various experimental conditions typical of cells, including changed Na(+), K(+), PO(4)(3-), and Ca(2+) concentrations, pH and temperature. A mathematical model was developed relating these factors to the observed chemical shift change relative to a capillary-sphere reference. This enabled cation concentrations to be deduced quantitatively from experimental chemical shifts, including those observed during biological time courses with cell suspensions containing TmDOTP DOTP, 1,4,7,10-tetraazacyclododecane-1,4,7,10-tetrakis (methylenephosphonate) [corrected]. The model was applied to a (23)Na NMR time course in which monensin, a sodium ionophore, was introduced to human erythrocytes, changing the concentration of cations which may bind TmDOTP, and also resulting in cell volume changes. Using the model with experimentally determined conditions, the chemical shift was predicted and closely followed the experimental values over time. In addition to the model, parameter fitting was achieved by calculating the likelihood distribution of parameters, and seeking the maximum likelihood with a Bayesian type of analysis.