A mathematical model of calcium and phosphorus metabolism in two forms of hyperparathyroidism

Parathyroid hormone (PTH) plays a critical role in calcium and phosphorus metabolism. Interestingly, in two forms of hyperparathyroidism (excessive amount of PTH in the serum), the metabolic disturbances in patients with chronic kidney disease (CKD) significantly differ from those with primary hyperparathyroidism (PHP). Since an intuitive understanding of these PTH-linked regulatory mechanisms are hardly possible, we developed a mathematical model using clinical data (1586 CKD and 40 PHP patients). The model was composed of a set of ordinary differential equations, in which the regulatory mechanism of PTH together with other key factors such as 1,25-Dihydroxyvitamin D (1,25(OH)₂D) and calcium was described in the tissues including bone, the kidney, the serum, and the parathyroid glands. In this model, an increase in PTH was induced by its autonomous production in PHP, while PTH in CKD was elevated by a decrease in feedback inhibition of 1,25(OH)₂D in the serum, as well as an increase in stimulation by phosphorus in the serum. The model-based analysis revealed characteristic differences in the outcomes of hyperparathyroidism in CKD and PHP. The calcium exchange in bone, for instance, was predicted significantly higher in PHP than CKD. Furthermore, we evaluated the observed and predicted responses to the administration of calcimimetics, a recently developed synthetic drug that modulated efficacy of calcium-sensing receptors. The results herein support the notion that the described model would enable us to pose testable hypotheses about the actions of PTH, providing a quantitative analytical tool for evaluating treatment strategies of PHP and CKD.