(309d) A Control Engineering Perspective to Modeling Calcium Regulation and Related Pathologies

Calcium (Ca) homeostasis, the maintenance of a stable plasma Ca concentration in the body, is essential to normal physiological function where the total plasma Ca concentration in a healthy human body must be maintained within a very narrow range (2.2-2.4mM) [1] in the face of significant variations in Ca ingestion and/or excretion. Meeting such stringent requirements is the task of an effective, high performance physiological regulatory system that employs parathyroid hormone (PTH) and calcitriol to regulate Ca flux between the plasma and kidneys, intestines and bones. Disorders in the regulatory organs cause abnormal hormonal secretion and activity and contribute to chronic imbalances in plasma Ca levels. Furthermore, these changes in hormonal activity often lead to long-term physiological problems: for example, osteoporosis (increased loss of bone mineral density) arises as a consequence of primary hyperparathyroidism (PHPT) – a condition characterized by hyper secretion of PTH [2].

Our primary objective is to develop a quantitative understanding of how normocalcemia (the maintenance of Ca concentration in the appropriate narrow range) is achieved, and to employ such understanding to elucidate the mechanisms by which associated pathological conditions such as PHPT, hypoparathyroidism, kidney failure and vitamin D deficiency affect Ca homeostasis.

In this presentation, we discuss our representation of the physiological process of Ca regulation in the form of an engineering control system block diagram with explicitly identified sensor, controller, actuators and process. Mathematical models are generated for each component sub-process by applying standard conservation of mass principles augmented with available mechanistic information. The result is a series of ordinary differential equations (ODE) that describe the dynamics of: (i) PTH in the controller; (ii) calcitriol from the kidneys in the actuator; (iii) bone cells (osteoblasts and osteoclasts) from the bone in the actuator; and (iv) plasma calcium (the “process”). The overall model is then validated with published clinical data. Pathological conditions are simulated as defects in appropriate control system components (sensor, controller and/or actuator) and the simulation results validated against published clinical data of the Ca-related disorders in question.

We demonstrate how our modeling approach provides information about important quantities that are typically unavailable by measurement and are hence not available in published data sets, and how such information can be used to understand Ca homeostasis and the onset and propagation of Ca-related disorders.

References

1. Raposo, J.F., L.G. Sobrinho, and H.G. Ferreira, A minimal mathematical model of calcium homeostasis. J Clin Endocrinol Metab, 2002. 87(9): p. 4330-40.

2. Fraser, W.D., Hyperparathyroidism. Lancet, 2009. 374(9684): p. 145-58.