Parathyroid hormone secretion by multiple distinct cell populations, a time dynamic mathematical model

Abstract

The acute response of parathyroid hormone to perturbations in serum ionized calcium ([Ca(2+)]) is physiologically complex, and poorly understood. The literature provides numerous observations of quantitative and qualitative descriptions of parathyroid hormone (PTH) dynamics. We present a physiologically based mathematical model of PTH secretion constructed from mechanisms suggested in the literature, and validated against complex [Ca(2+)] clamping protocols from human data. The model is based on two assumptions. The first is that secretion is a fraction of cellular reserves, with the fraction being determined by the kinetics of [Ca(2+)] with its receptor. The second is that there are multiple distinct populations of parathyroid cells, with different secretory parameters. The steady state and transient PTH secretion responses of the model are in agreement with human experimental PTH responses to different hypocalcemia and hypercalcemia stimuli. This mathematical model suggests that a population of secreting cells is responsible for the PTH secretory dynamics observed experimentally.

Keywords: Calcium; hysteresis; parathyroid hormone; simulation.

Figure 1.

Model schematic. Each of the 20 subpopulations (denoted by superscripts 1, … 20) of parathyroid cells is subdivided into a sensitive and insensitive population (denoted by subscripts 1 and 2). Each subpopulation type has its own intracellular pool (VPTH) and secretion fraction (curved arrow), but there is only one serum pool.

Figure 2.

Ideal calcium profiles in the training and validation protocols defined in Schwarz et al. (1998). (A) Curve used for training the responses; (B and C) curves used in validation. All calcium measurements are given in mmol/L.

Figure 3.

Model validation results: validation of the population model against the hypocalcemia‐normocalcemia‐hypocalcemia protocol as reported by Schwarz et al. (1998). The gray curves denote 25 model outputs, while the black boxes show the individual data redrawn from Schwarz. The calcium curve corresponding to this protocol is Figure 2B.

Figure 4.

Model validation results: validation of the population model against the hypocalcemia‐extreme hypocalcemia protocol as reported by Schwarz et al. (1998). The gray curves denote 25 model outputs, while the black boxes show the individual data redrawn from Schwarz. The calcium curve corresponding to this protocol is Figure 2C.

Figure 5.

Comparison of Brown's model (dotted line), a model consisting of a single pair of insensitive and sensitive cells (dashed line), and a model composed of 20 such pairs to the calibration protocol (solid line). The experimental output that generates this output is shown in black boxes with standard deviation bars shown. The calibration used a subset of this data, namely the responses at 0 min (baseline), 5 and 10 min (initial hysteretic response), 30–60 and 150–170 min (hypocalcemic steady state), and 90–120 min (hypercalcemic steady state). The calcium curve generating this output is given by Figure 2A.