Draft: a Mathematical Model of Acute Response of Parathyroid Hormone to Changes in Plasma Ionized Calcium in Humans

A complex bio-mechanism, referred to as calcium homeostasis, regulates plasma ionized calcium (Ca++) concentration in the human body to within a narrow physiologic range which is crucial for maintaining normal physiology and metabolism. Various metabolic disorders and pathologic conditions origi∗Address all correspondence to this author. nate from acute and/or chronic disturbances/disorders in calcium homeostatic system. This system relies on numerous subsystems which operate in different time-scales ranging from minutes to weeks. In this thesis we focus on a particular sub-system that operates on the time-scale of minutes; the dynamics involves the response of the parathyroid glands to acute changes in plasma Ca++ concentration. We develop a two-pool, linear timevarying model describing the dynamics of the sub-system. We 1 Copyright c ⃝ 2008 by ASME show that this model can predict dynamics observed in clinical tests of induced hypoand hypercalcemia in normal humans. NOMENCLATURE PTH Parathyroid hormone PTG Parathyroid gland PTHP PTH in plasma pool PTHC PTH in PTC CaP Ionized calcium in plasma pool 25D 25 hydroxy Vitamin D 25DP 25D in plasma pool 1,25D 1,25 dihydroxy Vitamin D 1,25DP 1,25D in plasma pool DNA Deoxyribonucliec acid RNA Ribonucliec acid mRNA Messenger ribonucliec acid VDR Vitamin D receptor CaR Calcium receptor x1 Amount of parathyroid hormone in PTG pool x1,SS,N Amount of normal PTH in PTG pool x1,SS,Max Amount of maximal steady-state PTH in PTG pool x1,SS,Max Amount of maximal steady-state PTH in PTG pool x1,SS,Min Amount of minimal steady-state PTH in PTG pool x2 Amount of PTH in plasma pool x2,SS,N Amount of normal PTH in plasma pool x2,SS,Max Amount of maximal steady-state PTH in plasma pool x2,SS,Min Amount of minimal steady-state PTH in plasma pool Ca++ Ionized calcium Ca(t) Ionized calcium concentration in plasma pool CaSS,N Normal Ionized calcium concentration in plasma pool t Time τ1 Half-life of PTH in PTG pool τ2 Half-life of PTH in plasma pool λ1 Decay rate constant of PTH in PTG pool λ2 Clearance rate constant of PTH in plasma pool λCa(t) secretion rate function of PTH from PTG pool INTRODUCTION Calcium homeostasis refers to a complex bio-mechanism that regulates plasma ionized calcium (Ca++) concentration in the human body to within a narrow physiologic range which is crucial for maintaining normal physiology and metabolism. Plasma Ca++ plays a vital role in normal functioning of muscles, nerves, platelets, neutrophils, and coagulation factors, cell growth, cell division, secretion of hormones and other regulators [1], and mineralization of bones [2]. This complex biosystem comprises numerous sub-systems interconnected via positive and negative feedback pathways. Various metabolic disorders and pathologic conditions originate from acute and/or chronic disturbances/disorders in the calcium homeostasis. They can be caused by any one or a combination of the following factors: a) changes in plasma Ca++ levels and/or vitamin D levels by a physiologically significant amount, b) impaired synthesis and/or secretion of parathyroid hormone (PTH), and c) pathological conditions of parathyroid glands and kidneys. These disorders may affect normal bone remodeling process causing various metabolic bone diseases, such as osteoporosis, a major public health concern in the United States [3,4], besides causing many other abnormalities and disease conditions (eg., primary and secondary hyperparathyroidism). The present understanding of calcium homeostasis and its disorders is based on the traditional approach of biological and medical science which involves discovering various signaling pathways, identifying critical bio-markers, and employing statistical analysis. However, calcium homeostasis is a sophisticated bio-system where numerous sub-systems interact at different time-scales. A perturbation in one sub-system has corresponding effects in the interconnected sub-systems and these in turn have effects on the other sub-systems. These cascade-effects propagate in positive and negative feedback pathways in multiple directions. It becomes impractical, if not impossible, to keep track of all of these interactions in order to derive an understanding of the bio-system as a whole using the traditional approach. Systems biology, an emerging science, relies on the integration of mathematical models for individual sub-systems into a single model, enabling us to study the effects of disturbances in the various inputs, pools and processes of the overall bio-system. We took the systems biology approach in our overall calcium homeostasis modeling efforts. In this paper, we first developed a qualitative model of the overall bio-system in a normal human. We then focus on the sub-system that involves the acute dynamical interaction between plasma Ca++ and PTH (called Ca-PTH axis for ease of notation). This model can successfully predict dynamics observed in clinical tests of induced hypoand hypercalcemic clamp tests. QUALITATIVE MODEL OF CALCIUM HOMEOSTASIS A detailed qualitative model describing the interactions between various components and signalling pathways of calcium homeostasis is presented in this section. The intestine, bone, kidney, and plasma are the four major pools of Ca++ in the human body. The average total plasma calcium concentration in the human body is 2.1-2.6 mmol/L [5]. At normal physiological state, about 1.1-1.3 mmol/L is ionized calcium, 0.9-1.1 mmol/L is protein-bound calcium, 0.18 mmol/L is complexed calcium and 180 nmol/L is intracellular free calcium [5]. The ionized form of calcium, which is just 0.5% of the total body calcium, is metabolically active and tightly regulated by the homeostatic system. The condition when Ca++ level falls below the normal 2 Copyright c ⃝ 2008 by ASME range is called hypocalcemia and the condition when it climbs above the normal range is called hypercalcemia. An overall daily Ca++ balance in the plasma of a healthy adult is maintained by fluxes of calcium between the plasma and the intestine, bone and kidney (for details see [6]). The fluxes are actively regulated by two chief hormones: the metabolically active form of vitamin D and PTH. The metabolism and regulation of vitamin D and PTH are presented in the next sections, respectively. Vitamin D Metabolism and Regulation The general steps in the metabolism of vitamin D are shown in Fig. 1 (the presentation here follows the detailed exposition in [7]). There are two sources of vitamin D in the human body: daily diet and skin. Daily diet provides vitamin D2 (er7-dehydro cholesterol