Model of calcium homeotasis, PTH synthesis and secretion, and vitamin D3 effects 

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Transport mechanisms in organs involved in calcium homeostasis Bone

Bone can be divided in two compartments, namely a rapidly exchangeable pool and a slowly exchangeable pool (Figure 1.2). The rapid pool contains calcium that is immediately available for exchange, probably stored as brushite, a very unstable salt composed of calcium and phosphate (also known as dicalcium dihydrate, CaHPO4.2H2O). The existence of this pool was evinced by radiolabeled calcium studies, which indicated the presence of a fast calcium deposit in the haversian system of cortical long bones [54, 137, 158]. It contributes to the minute-to-hour regulation of calcemia, for example during the night when calcium intake is reduced. The size and localization of this pool remain to be refined, but it is thought to allow for calcium exchanges with plasma (about 5 to 6g of calcium per day) that are 10-fold higher than those between the deep pool and plasma. The deep pool is difficult to mobilize and is localized in the calcified matrix of bone. It is composed of hydroxyapatite, as previously described, and it exchanges 0.25 to 0.5 g of calcium per day with plasma.
The bone remodeling process is continuous and involves 2 types of cells:
• Osteoclasts bind to calcified bone matrix and release its components into the bloodstream. Their maturation is complex and highly regulated from monocytes in bone marrow [230].
• Osteoblasts constitute 4 to 6 % of bone cells and their lifespan is about 3 months. They crystallize the bone matrix, regulate osteoclasts maturation, and interact with osteocytes more deeply in the bone [151].
Bone remodeling begins with an activation phase during which pre-osteoclasts are recruited and merge together, leading to large multinucleated cells (Figure 1.3). This step is regulated by PTH, the insulin growth factor I (IGF-I), and the tumor necrosis factor α (TNF-α) as well as estrogens. The resorption phase by activated osteoclasts lasts 2 to 4 weeks. The mature osteoclast is a polarized cell that has an integrin crown delimiting a basolateral compartment next to the bone marrow and an apical compartment that has a large ruffled border in contact with the bone matrix, called Howship lacuna. It contains some proteases with an acid pH. The calcium concentration in the lacuna can reach 40 mM. The acidification process participates in dissolving hydroxyapatite crystals, releasing calcium and phosphate, which are then carried into plasma either by crossing osteoclasts or by transcytosis. After a transition period lasting about 9 days, the formation phase or accretion is led by osteoblasts. The building of a new bone matrix takes 4 to 6 months. First, entwined collagen fibers are laid down by osteoblasts and form a non calcified matrix, the osteoid. Then, vesicles containing calcium and phosphate are added to this matrix thereby mineralizing the bone. A quiescent phase ends the cycle: osteoblasts stay and can differentiate into osteocytes or vanish by apoptosis.
Osteocytes are the most abundant bone cells (90-95%), with a life time about 25 years [8, 166]. They are formed from osteoblast differentiation, but unlike osteoblasts, they are located in the bone matrix, also called the osteoplast. Cellular processes called canaliculi of 250-300 nm diameter constitute a network connecting osteocytes to each other as well as to other bone cells such as osteoblasts and osteoclasts. Osteocytes are the major producer of RANKL, responsible for the maturation of osteoclats (Figure 1.3). Conversely, they are capable of mineralizing the osteoid laid down by osteoblasts. Additionally, they are the major source of FGF23, which is involved in the regulation of phosphate homeostasis (Section 1.2.3) [8]. The role of osteocytes in calcium homeostasis was uncertain for a long time but is now established, through the concept of osteolytic osteolysis [8, 166]. The mecanism is thought to be similar to that of osteoclasts since proton pumps are expressed by osteocytes [8]. However, the relative contribution of osteoblasts and osteocytes remains to be elucidated. Moreover, comparing the quantity of osteocytes to that of osteoclasts, they could be a good candidate for the rapid regulation of calcium homeostasis [8].

Phosphate distribution in the organism

In the organism phosphate is the skeleton of DNA, and the second major element of bone after calcium. It plays a key role in metabolic pathways, such as ATP hydrolysis, and in transduction signals (MAPK,MAPKK). It is present in two forms in the body, as inorganic phosphate and as a component of organic compounds. Phosphate is distributed as follows: 85% is found in bone and teeth (as hydroxyapatite), 14% in tissues and cells (as phospholipids, phosphoproteins), and 1% in the extracellular compartment. In plasma, 50% of phosphate species is present as H2PO−4 /HPO24− , 40% forms complexes with other ions such as Ca2+, Na+ and Mg2+, and the last 10% is bound to proteins [131, 72]. Sodium-phosphate complexes account for 30% of total plasma phosphate. All but the protein-bound forms, that is, 90% of plasma phosphate, are freely filtered by the kidney.

Phosphate transport in organs

Phosphate crosses the intestinal barrier via paracellular and transcellular pathways [138, 85]. About 70% of the ingested phosphate is absorbed mainly in the duodenum and jejunum [203]. On a low phosphate diet, the transcellular route predominates, whereas on a high phosphate diet, paracellular transport prevails. There are several phosphate transporters in intestinal cells, namely Pit1, Pit2, NaPi-IIb (apical side); Pit transporters belong to the Type III sodium-phosphate co-transporters also known as SLC20, whereas NaPi-IIb is part of Type II sodium-phosphate co-transporters or SLC34. NaPi-IIb mediates 90 % of total absorption. The regulation of phosphate absorption by vitamin D is controversial; in the jejunum, the effects of vitamin D appear to be small [219, 45, 138, 172]. Moreover, it has been shown that aging could reduce the stimulatory effect of vitamin D3 together with NaPi-IIb expression [226].
The ionized and ion-bound forms of phosphate are freely filtered by the kidney, representing about 90% of the total plasma phosphate, but this fraction can decrease in case of hypercalcemia through calcium-phosphate fetuin complexes [162]. Approximately 70 to 80 % of the filtered load is reab-sorbed in the proximal tubule [203], mostly via the cotransporter NaPi-IIa (or SLC34A1), which accounts for 80% of phosphate reabsorption in that segment, but also by NaPi-IIc. About 10% is reabsorbed in the distal tubule but the nature of the transporter(s) involved has yet to be discov-ered [23]. In the proximal tubule, phosphate reabsorption is negatively regulated by PTH, FGF23 [184, 114], and high phosphate intake. However, the role of 1,25(OH)2D in the regulation of renal phosphate reabsorption remains unclear [85].

Hormonal control of phosphate homeostasis

FGF23 is a 32 kDa protein that belongs to the family of fibroblast growth factors; FGFs are involved in diverse functions such as development, repair, and metabolism [133]. FGF23 exists under 2 different shapes: 25(FGF23)251 which is biologically active, and 25(FGF23)179 which lacks the binding domain to klotho, resulting in an inactive pattern [139]. FGF23 is synthetized in bone by osteocytes and osteoblasts [28].
FGF23 is an important regulator of phosphate homeostasis. It inhibits phosphate reabsorption in the kidney by reducing the expression of NaPi-IIa [18]. It may also indirectly affect serum calcium by acting on vitamin D3, as further described below. FGF23-null mice have hyperphosphatemia and increased vitamin D levels. In addition, their bone turnover is unexpectedly decreased, for reasons that remain unclear [95].
Vitamin D3 is the most important systemic regulator of FGF23 production. Indeed, VDR-null mice have undetectable levels of FGF23 [139]. Several other regulators are known: DMP1 and PHEX inhibit the synthesis of FGF23 through mechanisms that remain to be elucidated [139]. Conversely, CYP27B1 (1-α-(OH)ase) and phosphate enhance the production of FGF23 [186]. It has also been shown that PTH activates the orphan receptor Nurr1 to induce FGF23 transcription [141]. Moreover, FGF23 synthesis is blunted in PTX rats and conversely enhanced in primary hyperparathyroidism. However, the effects of PTH could be indirect, that is, mediated by vitamin D3 [15, 139].
FGF23 is known to protect cells against the toxicity of vitamin D3 (Figure 1.12). The overex-pression of FGF23 triggers the suppression of vitamin D3 synthesis. On the other hand, the plasma levels of vitamin D3 are 3-fold higher in FGF23-null mice than in control mice [139, 48, 15]. FGF23 regulates the metabolism of vitamin D3 by inhibiting the expression of CYP27B1 and increasing that of CYP24A1 [48, 184].
Several studies suggest that FGF23 affects the synthesis and secretion of PTH, in part via the MAPK signaling pathways [15]. Intravenous injections of FGF23 decrease the gene expression and secretion of PTH in a dose-dependent manner [124]. On the other hand, other studies suggest a stimulatory effect of FGF23 on PTH secretion, as summarized in reference [139]. In parathyroid glands, FGF23 increases the expression of CaSR and VDR, and thereby plays a hypocalcemic role [42]. In case of hypocalcemia, PTH mRNA is doubled and if FGF23 is added, PTH mRNA is drastically reduced. Interestingly, in hypercalcemia, PTH mRNA is reduced but the addition of FGF23 does not change anything. Besides, the expression of VDR and CaSR is stimulated by FGF23 in case of low calcium concentration, whereas FGF23 has no effect in hypercalcemia even if the expression of VDR snd CaSR is doubled. Uremic patients may develop a resistance to FGF23 since PTH levels are not decreased despite the presence of FGF23. Note that recently,

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Table of contents :

1 Introduction 
1.1 General introduction on calcium homeostasis
1.1.1 Overview of calcium homeostasis
1.1.2 Transport mechanisms in organs involved in calcium homeostasis
Bone
The intestine
The kidney
1.1.3 Regulation of calcium homeostasis
Parathyroid hormone
The calcium sensing receptor (CaSR)
Vitamin D
Calcitonin
1.2 Overview of phosphate homeostasis
1.2.1 Phosphate distribution in the organism
1.2.2 Phosphate transport in organs
Intestinal absorption of phosphate
Renal phosphate handling
1.2.3 Hormonal control of phosphate homeostasis
FGF23, a regulator of phosphate homeostasis
Phosphate, a regulator of PTH and vitamin D
1.2.4 Precipitation of calcium-phosphate
Acid base equilibria between phosphate components
Binding of calcium and phosphate
Mechanisms of calcium-phosphate precipitation
Regulation of bone mineralization by fetuin-A
Precipitation of calcium and phosphate in the urine
1.3 Historical perspectives on Mathematical Models of Calcium Homeostasis
1.3.1 The first calcium kinetic studies
1.3.2 A global model of calcium Homeostasis
1.3.3 Dividing the bone compartment into two exchangeable pools
1.3.4 Toward more elaborate models
The model of Hurwitz
The latest models of calcium homeostasis
1.3.5 PTH modeling review
A sigmoidal relationship between PTH and calcium
The first model of PTH synthesis and secretion
A more elaborate model of PTH synthesis and secretion
An asymmetric exocytosis function for PTH secretion
A subpopulation of PT cells to model PTH dynamics
1.3.6 About Vitamin D and FGF23
Modeling the vitamin D3 synthesis pathway
Modeling the effects of FGF23
1.4 Aim of this thesis
1.4.1 Building of model of calcium homeostasis
2 Experimental measurements of some bone parameters 
2.1 Aim of these experiments
2.2 Method
2.2.1 Determination of the total calcium and phosphate bone content in mice
Kinetic study of calcium exchanges between plasma and the bone compartment
2.3 Results
2.3.1 Calcium and phosphate content in the bone
2.3.2 Kinetic part
2.4 Discussion
2.4.1 About the total pool of calcium and phosphate in bone
2.4.2 About the kinetic experiments
2.4.3 Conclusion
3 Model of calcium homeotasis, PTH synthesis and secretion, and vitamin D3 effects 
3.1 Mathematical model of PTH synthesis and secretion
3.1.1 Modeling PTH synthesis
3.1.2 PTH exocytosis from the cell
3.1.3 PTH dynamics in plasma
3.1.4 Parameters
3.1.5 Mathematical analysis of the PTH model
Non-dimensionalization
Existence and unicity
Explicit solutions
Expected values at steady state
Building of the plane phase
Simulation results
3.1.6 Possible improvements to the PTH model
3.2 Mathematical model of calcium homeostasis with PTH
3.2.1 Calcium balance in the intestine, bone, kidney and plasma.
3.2.2 Mathematical analysis of the PTH-calcium model
Non-dimensionalization of the model
Identification of steady state
Numerical stability of the steady state
3.2.3 Parameters of the PTH-calcium model
Determination of unknown parameters
Parameters of the calcium/PTH model
3.3 Mathematical model of calcium homeostasis with PTH and vitamin D3
3.3.1 Hormonal part of the model
Balance equation for vitamin D3
Effect of vitamin D3 on PTH production
3.3.2 Effect of vitamin D3 on calcium metabolism
Regulation of intestinal calcium absorption
Regulation of bone remodeling
Regulation of kidney reabsorption
3.3.3 Equations of the model
Equations of the hormonal system
Equations of calcium homeostasis regulated by PTH and D3
Parameters of the calcium homeostasis model
3.4 Summary of the mathematical properties of the model
3.5 Discussion and improvements to the model
4 A Mathematical Model of Calcium Homeostasis in the Rat 
4.1 Introduction
4.2 Mathematical Model
4.2.1 PTH synthesis and secretion
4.2.2 Vitamin D3
4.2.3 Calcium exchanges between organs
The intestinal compartment
The bone compartment
The kidney compartment
The plasma compartment
4.2.4 Determination of unknown parameters
4.2.5 Numerical methods
4.3 Results
4.3.1 Experimental Measurements
4.3.2 Model Validation
Acutely induced hypocalcemia
Acutely induced hypercalcemia
4.3.3 Model Predictions
4.4 Discussion
5 A Model of Calcium and Phosphate Homeostasis in the Rat 
5.1 Mathematical model
5.1.1 Hormone conservation equations
FGF23 dynamics
Conservation of Vitamin D3
5.1.2 Conservation equation for PTH
5.1.3 Modeling calcium-phosphate binding in plasma and bone
Formation of CaHPO4 and CaH2PO+4 salts in plasma
Regulation of bone mineralization by fetuin-A
Calcium and phosphate in the rapid bone pool
5.1.4 Equations for phosphate
Intestinal absorption of phosphate
Intracellular phosphate
Hormonal control of phosphate reabsorption in the kidney
Phosphate in the bone compartment
Phosphate binding to Na+
Phosphate in plasma
5.1.5 Determination of unknown parameters
5.1.6 Improvement to the calcium homeostasis model
Calcium binding to proteins
Conservation equation for plasma calcium
5.2 Summary of the main equations
5.3 Results
5.3.1 Model validation
Calcium and Phosphate in Primary Hyperparathyroidim
FGF23 deficiency
Intravenous injection of Phosphate
Phosphate gavage
5.3.2 Model Predictions
Primary hyperparathyroidism
Primary hypoparathyroidism
Vitamin D3 deficiency
5.4 Discussion
5.4.1 Scope of the model
5.4.2 Model limitations
5.4.3 Calcium and phosphate metabolism
5.4.4 Calcium and phosphate metabolism dysfunctions
6 Discussion and General Conclusion 
6.1 What does our model bring to the understanding of calcium homeostasis?
6.2 What is still missing?
6.3 Further extensions?

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