Bayesian estimation of biokinetic parameters for a two compartment model describing Calcium-45 exchange

Studies of the metabolism of alkaline earths in adult man summarized by the International Committee on Radiological Protection have contributed to the calculation of risk estimates for the bone seeking elements radium, strontium, and calcium. The studies also clarified the roles that these elements assume in radiocarcinogenesis. In particular, the retention of radium and strontium on the surface and in the volume of bones has been found to increase the risk of bone cancers.

It has been observed that the elements radium, strontium and calcium have similar chemical and metabolic properties. Paired data consisting of the specific activity of Calcium-45 in the blood and the surface of bone for rabbits sacrificed at various times has been obtained through autoradiographic studies by Rowland. Based on this data a model consisting of a blood compartment and a bone surface compartment has been established to describe the metabolism of Calcium-45 after intravenous injection in rabbits. Previous work by Box et. al. demonstrated the application of Bayesian inference techniques for the estimation of model parameters. Bayesian inference techniques are statistical methods derived from Bayes' Theorem which allow for the estimation of parameters with description of the parameters' remaining uncertainty given the available data. This methodology has been applied to obtain posterior densities of the biokinetic parameters of a two compartment model of Calcium-45 exchange between blood and bone surface with excretion.

The mode of the posterior density for the blood to bone surface biokinetic parameter. estimates the rate of exchange to be 0.0204 min^-1. The posterior mode for the exchange rate from bone surface to blood is estimated to be 0.0124 min^-1, and the posterior mode of the rate of excretion is estimated to be 0.00599 min^-1 The mean of each parameter has been estimated to be 0.0171 ± 0.0090 min^-1 for bone surface to blood, 0.0249 ± 0.0112 min^-1 for blood to bone surface, and 0.00665 ± 0.00223 min^-1 for excretion.