A model of calcium homeostasis in the rat

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1 Am J Physiol Renal Physiol 3: F47 F6, 6. First ublished June 9, 6; doi:.5/ajrenal.3.6. A model of calcium homeostasis in the rat avid Granjon,, Olivier Bonny, and Aurélie Edwards Sorbonne Universités, UPMC Univ Paris 6, Université Paris escartes, Sorbonne Paris Cité, INSERM UMRS 38, CNRS ERL 88, Centre de Recherche des Cordeliers, Paris, France; and eartment of Pharmacology and Toxicology, University of Lausanne, and Service of Nehrology, Lausanne University Hosital, Lausanne, Switzerland Submitted 3 Aril 6; acceted in final form 7 June 6 Granjon, Bonny O, Edwards A. A model of calcium homeostasis in the rat. Am J Physiol Renal Physiol 3: F47 F6, 6. First ublished June 9, 6; doi:.5/ajrenal.3.6. We develoed a model of calcium homeostasis in the rat to better understand the imact of dysfunctions such as rimary hyerarathyroidism and vitamin deficiency on calcium balance. The model accounts for the regulation of calcium intestinal utake, bone resortion, and renal reabsortion by arathyroid hormone (), vitamin 3, and Ca itself. It is the first such model to incororate recent findings regarding the role of the calcium-sensing recetor (CaSR) in the kidney, the resence of a raidly exchangeable ool in bone, and the delayed resonse of vitamin 3 synthesis. Accounting for two (fast and slow) calcium storage comartments in bone allows the model to roerly redict the effects of bishohonates on the lasma levels of Ca ([Ca ] ),, and vitamin 3. Our model also suggests that Ca exchange rates between lasma and the fast ool vary with both sex and age, allowing [Ca ] to remain constant in site of sex- and age-based hormonal and other differences. Our results suggest that the inconstant hyercalciuria that is observed in rimary hyerarathyroidism can be attributed in art to counterbalancing effects of and CaSR in the kidney. Our model also correctly redicts that calcimimetic agents such as cinacalcet bring down [Ca ] to within its normal range in rimary hyerarathyroidism. In addition, the model rovides a simulation of CYP4A inactivation that leads to a situation reminiscent of infantile hyercalcemia. In summary, our model of calcium handling can be used to deciher the comlex regulation of calcium homeostasis. calcium; ; vitamin 3 ; homeostasis; mathematical model CALCIUM PLAYS AN ESSENTIAL role in organisms as it is involved in many biological rocesses such as cardiac activity regulation, muscle contraction, blood clotting, and bone formation. More than 99% of total body calcium is found in bones, only % of which can be readily exchanged with the extracellular fluid. The latter contains.% of total body calcium, 6% of which is ionized; the remainder is bound to albumin or other ions such as hoshate, bicarbonate, and citrate (4, 9). In rats, the lasma concentration of total calcium is normally comrised between. and.6 mm and that of ionized calcium between. and.3 mm (74, 9). Given the key biological roles of calcium, its levels in lasma need to be tightly regulated. The main organs involved in calcium homeostasis are the intestine, bones, and kidneys, all of which are linked via the lasma comartment. The intestinal eithelium absorbs 6% of calcium intake (, 35). The bones constitute a reservoir that can either accumulate calcium (by accretion) or release it (by resortion), as required. The kidneys freely filter Ca and subsequently reabsorb Address for rerint requests and other corresondence: A. Edwards, Centre de Recherche des Cordeliers, ERL 88, UMRS 38, 5 Rue de l école de Médecine, 756 Paris, France ( aurelie.edwards@crc.jussieu.fr). 98% of the filtered load; thus fractional Ca urinary excretion is %. Calcium fluxes between these comartments are regulated by several hormones, rincially arathyroid hormone () and calcitriol, i.e., the active form of vitamin 3 [,5(OH) -vitamin ]. In resonse to a decrease in Ca lasma levels, is secreted by the arathyroid glands; it then binds to its recetor on bone and kidney cells, where it, resectively, acts to increase bone resortion and renal calcium reabsortion (6, 74, 9). In bone, acts on osteoblasts, thereby stimulating the synthesis of recetor activator of nuclear factor-b ligand (RANK-L), which is the rimary mediator of osteoclast differentiation and activation (). also inhibits the roduction of osteorotegerin; the latter acts as a decoy recetor that binds to RANK-L and revents it from interacting with its cellular recetor RANK. In the kidney, elicits an increase in the aracellular ermeability of the thick ascending limb (TAL) to Ca ; it also enhances the exression of calcium transort roteins in the distal convoluted and connecting tubules, such as the transient recetor otential vanilloid tye 5 (TRPV5) and the sodiumcalcium exchanger (NCX) (7). also triggers the conversion of the inactive form of vitamin 3 (i.e., 5-hydroxyvitamin ) to its active form, calcitriol, in the roximal tubule (47). Calcitriol binds to the vitamin recetor (VR) in the intestine, bones, kidneys, and arathyroid glands. In the intestine, calcitriol (thereafter denoted vitamin 3 ) acts to increase intestinal calcium absortion by increasing the ermeability of tight junctions as well as uregulating the activity and exression of calcium transort roteins such as the transient recetor otential vanilloid tye 6 (TRPV6), calbindin 9k, and lasma membrane Ca ums (PMCA) (37). Vitamin 3 also enhances Ca reabsortion in the kidney by uregulating the exression of TRPV5, NCX, and calbindin 8k (9). Lastly, in bone, under a low calcium diet, vitamin 3 stes u the roduction of RANK-L by osteoblasts to accelerate the maturation of osteoclasts, thereby favoring bone resortion (87, 96). In turn, the synthesis and secretion of are regulated by the calcium-sensing recetor (CaSR). The binding of Ca to arathyroid gland CaSR triggers an intracellular signaling cascade that results in inhibition of secretion and that ultimately cleaves intravesicular into inactive forms (8, 76). CaSR is also resent in other tissues such as bone, brain, and intestine (6). In the kidney, CaSR has been shown to enhance urinary Ca excretion via a -indeendent athway (48, 6). As summarized in Fig., the mechanisms that regulate lasma Ca levels are highly couled and can be hard to deciher. The major objective of this study was to develo a mathematical model of calcium homeostasis in the rat so as to htt:// X/6 Coyright 6 the American Physiological Society ownloaded from ( ) on Aril 4, 9. F47

2 F48 Fig.. Schematic diagram of calcium exchanges between lasma, bone, intestine, and kidney. Calcium fluxes between comartments are deicted with black arrows. Regulation by arathyroid hormone (shown in dashed arrows) acts to enhance bone resortion and renal calcium reabsortion and to stimulate the conversion of the inactive form of vitamin 3 into its active form. Regulation by vitamin 3 (dotted arrows) acts to increase bone resortion, intestinal calcium absortion, and renal calcium reabsortion; vitamin 3 also reresses its own synthesis. In addition, lasma calcium levels regulate arathyroid hormone () secretion via the calcium-sensing recetor (CaSR) and inhibit the synthesis of vitamin 3. CaSR-mediated effects on renal calcium reabsortion are not shown. Calcium intake intestine Fecal excretion - intestinal absortion resortion bone lasma Ca + accretion 3 bone: raidly exchangeable ool + arathyroid gland + reabsortion filtration - + kidney Urinary excretion better understand the imact of dysfunctions (such as rimary hyerarathyroidism) on calcium balance and concentration. Several models in human have been develoed (4, 57, 69, 74); however, they do not incororate several findings that the resent study seeks to account for: the role of the renal CaSR, the delayed resonse of vitamin 3 synthesis to changes in levels, and the raidly exchangeable ool in bone [with the excetion, for the latter, of the model of Peterson and Riggs (69)]. MATHEMATICAL MOEL Our model of calcium homeostasis is based on conservation equations that yield the lasma concentration of Ca,, and vitamin 3 in rats, as well as Ca fluxes between lasma, bone, intestine, and kidneys. Model arameters aly to a rat weighing 3 g (i.e., mo old). Synthesis and Secretion is the main hormone involved in the short-term regulation of lasma calcium levels. The synthesis of in arathyroid glands is downregulated by vitamin 3 (8, 8), and its secretion is regulated by the calcium-sensing recetor (CaSR), which can detect minute variations in [Ca ], on the order of. mm (i.e., 4% of its equilibrium value). The half-life of is 4 6 min, and its lasma concentration ranges between and 8 M (, 39). We resectively denote the concentration of within arathyroid cells and in lasma by [] g and []. With the assumtion of a constant cell volume, changes in [] g with time are given by: d g g k rod 3 k deg rod 3 g FCa g () The first term on the right-hand-side reresents the synthesis of, at a basal rate (k g rod ) that can be modulated by vitamin 3. The reression factor 3 rod is adjusted so that at equilibrium the synthesis of is reduced by %. The second term on the left hand side characterizes elimination from the cell, either by degradation (with a first-order rate constant k g deg ) or by exocytosis. The exocytosis function is defined as: FCa g exo g exo Ca nca Ca nca K nca Ca () This function accounts for the inhibition of secretion by CaSR (73): g exo is the imal rate of secretion from the gland, g exo characterizes the degree to which CaSR can imally inhibit secretion, K Ca is the binding affinity of Ca to CaSR, and n(ca) is a steeness factor. As discussed by Shresta et al. (8), to roerly account for the effects of both hyercalcemia and hyocalcemia on secretion, it is necessary to use an adative n, which can be calculated as: n exo nca e exorca n n exo and n exo, resectively, characterize the sensitivity of the recetor under hyocalcemia and hyercalcemia, R is the threshold between these two states, and exo is an amlification term. When [Ca ] is well below R, n(ca) tends towards high values (), as commonly used to fit hyocalcemia rofiles (, 73, 8). Conversely, when [Ca ] rises above R, n(ca) takes on a lower value. Assuming a constant lasma volume (V ), changes in [] with time are given by: d g exo g exo Ca nca Ca nca K Ca nca V c exo V g k deg (4) where V c is the arathyroid gland volume. The degradation rate constant in lasma (k deg ) is chosen so that the half-life of in lasma fits the exerimental data of Lewin et al. (6). Vitamin 3 The biosynthesis athway of vitamin 3 is comlex (8). Its recursor 5-hydroxyvitamin (denoted inact 3 below) is converted to vitamin 3 [that is,,5(oh) 3, or calcitriol] in the roximal tubule by the enzyme CYP7B, also known as -(OH)-ase. For simlicity, we assume that the level of inact 3 remains constant. The equilibrium lasma concentration of vitamin 3 is m in a - to 5-g rat (43, 89) and 8 M in older rats (3 7 g) (39, 64). In arathyroidectomized (PTX) rats, the level of vitamin 3 falls sharly, by a factor of at least (89). The level of inact 3 also aears to decrease with age; its baseline value is taken as 5 nm (39). The dynamic evolution of the lasma concentration of vitamin 3 (denoted [ 3] ) is determined as: (3) AJP-Renal Physiol doi:.5/ajrenal ownloaded from ( ) on Aril 4, 9.

3 d 3 k conv min n conv t conv n t conv n K conv conv Ca 3 conv 3 Ca conv inact k 3 3 deg 3 (5) deg The first term on the right-hand-side reresents the conversion of inact 3 to vitamin 3, which is catalyzed by -(OH)-ase. This conversion occurs at a minimum rate k min conv, and we assume that it is activated by with a delay of 4 h and a imum gain conv (5, Ca 63). The factors conv and 3 conv, resectively, account for the reression of -(OH)-ase by calcium and vitamin 3 (). The second term reresents the degradation rate of vitamin 3 in lasma, which is catalyzed by CYP4A and regulated by calcitriol and (among other factors). The constant k 3 deg is adjusted so that the half life of vitamin 3 in lasma is 5 8 h, and deg accounts for -induced reression of CYP4A. Calcium Exchanges Between Organs The intestinal comartment. Calcium crosses the intestinal barrier via both aracellular and transcellular routes (53). The aracellular athway, across which transort is driven by the luminal-to-lasma calcium concentration gradient, is not saturable and revails under a high calcium diet. In contrast, the transcellular athway redominates when calcium intake is low. Ca entry into the cell is mediated by TRPV6 (note that the concentration of Ca is several orders of magnitude lower in the cytosol than in the lumen). Within the cytosol, Ca binds to calbindin 9k, which facilitates its transfer to the basolateral side, where Ca is extruded mostly (9%) by PMCA, with a small contribution from Na /Ca (NCX) exchangers. There may also be other Ca transorters, yet to be identified, in the gut (3, 7). Vitamin 3 regulates intestinal Ca transort via both the transcellular and aracellular routes (53). Across the aracellular athway, it is thought to modulate the exression of claudins and (4). We assume that there is no net accumulation of calcium in the intestinal comartment over time, so that intestinal calcium excretion (E Ca) is the difference between calcium intake (I Ca) and net intestinal absortion [ abs( 3)]: E Ca I Ca abs 3 (6) Exerimental findings suggest that at most 7% of ingested calcium is absorbed into the bloodstream () and that two-thirds of intestinal calcium transort are modulated by vitamin 3 (6). The rate of net intestinal calcium absortion is thus determined as: abs 3 I Ca (7) K 3 abs The first term within arentheses on the right-hand side reresents the fraction of absorbed calcium that is vitamin 3 indeendent, and the second term reresents the fraction that is regulated by vitamin 3. The bone comartment. The bone comartment can be divided into a raidly exchangeable ool and a slowly exchangeable ool; the amount of Ca they, resectively, contain is denoted as N Caf and N Cas. The raidly exchangeable (or fast ) ool is thought to mediate the short-term regulation of calcium homeostasis, but it remains oorly characterized (7, 7). We assume that the amount of Ca in the fast ool is given by: dn Caf Ca k f Ca Ca V k f N Caf ac N Caf (8) The first two terms on the right-hand-side reresent the flow of Ca from the lasma to the raidly exchangeable ool and vice versa, which we assume obey first-order kinetics. Since it remains unclear whether and vitamin 3 act directly on the raidly exchangeable ool, we assume no such regulation. The last term accounts for Ca accretion from the fast to the slow ool, at a rate ac. The amount of Ca in the slow bone ool varies with time as: dn Cas ac N Caf res, 3 (9) where the Ca resortion rate, res(, 3), is determined as: res, 3 min res res. K res K 3 res () We assume that resortion is activated by (saturating levels of) and 3 (4, 54), and that resortion increases sixfold when stimulation by and vitamin 3 is imal; that is, min res res 6 min res. The kidney comartment. The fraction of calcium that is not bound to lasma roteins is filtered nonselectively in roortion to the glomerular filtration rate (GFR). Most of the filtered load (98%) is then reabsorbed: roughly two-thirds across the tight junctions of the roximal tubule, 5% via the aracellular route in the thick ascending limb (TAL), and % in distal convoluted (CT) and connecting (CNT) tubules via transcellular athways (4). The contribution and mechanisms of Ca transort in the collecting duct remain oorly understood and are not exlicitly considered here. As in the intestine, we assume that there is no accumulation of calcium in the kidney over time, so that the fractions of the filtered load that are reabsorbed ( reab) and excreted ( u) sum to unity. In other words: reab u () Ca reabsortion in the kidney (relative to its filtered load) is given by the sum of fractional Ca reabsortion in the roximal tubule ( PT), the thick ascending limb ( TAL), and the CT-CNT ( CT): reab PT TAL CT () inhibits Ca reabsortion in the roximal tubule indirectly: it inhibits the activity of the sodium/roton exchanger NHE3, thereby reducing the transmembrane electric otential difference that drives the aracellular reabsortion of Ca (4, 67). We assume that PT is given by: where PT PT PT PT n. (3) ref PT is taken as.6, PT as.5, ref as 5 M, and n PT as 5. In addition, TAL is modulated by Ca (via CaSR) and, and CT by and vitamin 3. In the TAL, we assume that: where TAL TAL TALCa TAL (4) TALCa CASR Ca C ref AJP-Renal Physiol doi:.5/ajrenal ownloaded from ( ) on Aril 4, 9. F49 n (5) TAL

4 F5 TAL (6) K TAL The imum value of TAL is set to.5; TAL is taken as.5, CASR as.75, and as.75. We assume that C ref equals.33 mm and n TAL equals 4, so that the regulation by CaSR is very sensitive to small changes in Ca levels. Similarly, fractional Ca reabsortion in the CT-CNT is calculated as: CT CT CT, 3 (7) where CT, 3 CT.8 K CT. 3 3 K 3 CT (8) The imum value of CT equals., with CT set to.8 and CT to.. Equation 8 assumes that the contribution of is greater than that of vitamin 3 (9). The lasma comartment. Net changes in the total amount of calcium in lasma result from the combined effect of intestinal calcium absortion, Ca resortion from the slow bone ool, Ca exchanges with the fast ool, and urinary Ca excretion, such that: V dca tot abs 3 res, 3 k Ca f N Caf Ca k f Ca V u GFRCa (9) where [Ca tot ] is the lasma concentration of total calcium. As described above, a fraction b of calcium in lasma is bound to roteins such as albumin. We assume that b remains fixed (and equal to.4) such that: dca tot dca dcabound, () dca b where [Ca bound ] is the lasma concentration of bound calcium. Combining the revious two equations, we obtain: dca b V abs 3 res, 3 k Ca f N Caf Ca k f Ca V u GFRCa () Parameters related to the metabolism of, vitamin 3, and calcium are, resectively, listed in Tables,, and 3. Numerical Methods The differential equations were integrated using the MATLAB solver dde3. Simulations were erformed on a ersonal comuter with an Intel-based rocessor. EXPERIMENTAL MEASUREMENTS All exeriments were conducted in accordance with the regulations of the Veterinarian Office of the Canton de Vaud. Five male and five female mice aged 55 days and weighing g were killed, and their skeleton was reared following the general aroach of Pramod and coworkers (7). Their skin was removed, as well as much of the viscera and muscles. The uncleaned skeletons were washed with water and immersed in a 95% ethanol solution during 4 days and then in a % KOH solution. The KOH bath was reeated during 3 extra days (to remove remaining arts of skin, gristles). The skeletons were then Table. arameters cleaned with hot water in a fine-mesh strainer to revent small bones from being lost and then weighed. Two samles of final KOH solutions from each mice were examined to ensure that no calcium was dissolved in the bath. Skeletons were dried into a stove during day at 37 C. Finally, they were reduced to ashes in ceramic cus in an oven at 8 C during 4 days (6 h heating cooling time). Ashes were collected, weighed, and ultimately dissolved in a 6 mol/l HCl solution (4 ml). Their calcium content was assessed by the NM- BAPTA method on a Cobas analyzer. RESULTS Parameter Symbol Value Reference g synthesis rate k g rod.8 mol min Estimated Inhibition of g synthesis by vitamin 3 3 rod 5 3 M Estimated g degradation rate k g deg.35 min () Maximal secretion rate of g g exo.59 min Fitted from (39) Maximal inhibition of secretion by Ca g exo.57 min Fitted from (39) Binding of Ca to CaSR K Ca.6 mm (39) exo n (8) exo n 5 (8) R. mm (8) exo 6 mm (8) degradation rate k deg. min () Volume of arathyroid glands V c. l (45) g and, resectively, denote arathyroid hormone in the arathyroid gland and in lasma; CaSR, calcium-sensing recetor. Exerimental Measurements Most of the variables used for the model were obtained from the literature. However, bone calcium content was measured directly in mice and extraolated to a 3 g rat. The calcium content of the whole mouse skeleton was mmol er animal (n ) with a mean weight of g or.4 g of calcium. To extraolate these data to a 3-g rat, we assumed that the ratio of bone calcium to body weight is the same in rats and mice. With this hyothesis, the total amount of calcium in the rat skeleton was estimated as 5 mmol. Based on this value and other model arameters, the amount of calcium in the raidly exchangeable ool was estimated as.6 mmol or.5% of the total amount in bone. Model Validation Predicted values of Ca,, and vitamin 3 lasma concentrations at steady-state, summarized in Table 4, all fall within the exerimental range. The lasma concentration of Ca (denoted [Ca ] ) is redicted as. mm. Intestinal Ca absortion is comuted as.6 mol/min, urinary excretion is.47 mol/min, and the balance (. mol/min) is the net flux of Ca into bone (i.e., the difference between accretion and resortion). We also validated our model by comaring the redicted and measured rofiles of [Ca ] as a function of time under different scenarios. Acutely induced hyocalcemia. Lewin et al. (6) infused the chelating agent EGTA to normal and arathyroidectomized rats during h. They observed a stee decrease in [Ca ] during the first min, followed by a two-hase return to AJP-Renal Physiol doi:.5/ajrenal ownloaded from ( ) on Aril 4, 9.

5 Table. Vitamin 3 arameters Parameter Symbol Value Reference Minimum roduction rate of vitamin 3 min k conv min (89) Maximal increase in vitamin 3 roduction rate min conv 6. 5 min (89) Plasma concentration inact of 3 [ inact 3 ] 5 nm (39) Activation of vitamin 3 roduction by K conv 3 M Estimated sensitivity coefficient n conv 6 estimated Time delay for inact effects on 3 conversion 4 min (55) Inhibition of 3 roduction by Ca Inhibition of vitamin 3 roduction by itself egradation rate of vitamin 3 Inhibition of CYP4A by Ca conv.3 mm Estimated 3 conv.8 M estimated k 3 deg.9 min (6, 9, 5) deg.5 M Estimated equilibrium over the next 3 h. We simulated this exeriment by accounting for the reaction between calcium and EGTA in the lasma comartment, as described in the APPENIX. As shown in Fig., the dynamic behavior of [Ca ] (normalized by its initial, reinjection value) that our model redicts closely matches the exerimental data in normal rats. We also comared model redictions of the lasma concentration of Table 3. Ca arameters (denoted [] ) with measured values reorted in another study. As observed by Fox (39), when EGTA was infused at a exonentially decreasing rate during h, [] exhibited a eak-and-lateau behavior: [] reached its eak value in min, and its lateau value was twice as high as its steady-state value. As illustrated in Fig., the redicted and exerimental time courses of [] are in good agreement. Acutely induced hyercalcemia. Finally, we simulated the exeriment erformed by Bronner and Stein (7), in which rats received an intravenous calcium gluconate bolus. As shown in Fig., redicted values of [Ca ] fit the exerimental data well in the initial hase, but the model redicts too raid a return to steady state. Part of this discreancy may stem from the fact that the rats in these exeriments weighed significantly less than our model rats (8 vs. 3 g), such that there must have existed imortant differences in lasma volume, [] and/or calcium bone content between the two oulations. Model Predictions We then used the model to better understand how the lasma concentration of Ca is regulated following various metabolic erturbations. Primary hyerarathyroidism. Primary hyerarathyroidism was simulated by increasing the base-case synthesis rate (k g rod ) by a factor ranging from to. Shown in Fig. 3 are the redicted steady-state values of [], [Ca ], and [ 3 ], as well as intestinal calcium absortion, urinary excretion, resortion, and accretion, as a function of k g rod. The focus of these simulations is not on transient (dynamic) rofiles, but on concentrations and fluxes once equilibrium is reached. Our model suggests that [] increases aroximately linearly with k g rod. Since stimulates bone Ca resor- Parameter Symbol Value Reference Calcium intake I Ca.3 mol/min (5, 6) Stimulation of absortion by 3 K 3 abs M Estimated Rate of Ca transfer from lasma to fast bone ool Ca k -f.53 min Estimated Rate of Ca transfer from fast bone ool to lasma Ca k f- 5 min Estimated Initial calcium content in bone N Cab 5 mmol Measured Initial calcium content in raidly exchangeable ool N Caf.6 mmol Measured Accretion rate ac.958 min (5) Minimal resortion rate min res.4 mol/min (5) Maximal resortion rate res.7 mol/min (5) Stimulation of resortion by 3 K 3 res M Estimated Stimulation of resortion by K res.75 M Estimated Glomerular filtration rate GFR ml/min (97) Minimal fractional reabsortion of Ca in the PT PT.6 (9, 68) Stimulation of Ca reabsortion in PT by PT.5 Estimated Sensitivity of Ca reabsortion in PT to ref 5 M Estimated Minimal fractional reabsortion of Ca in the TAL TAL.5 (9, 68) Stimulation of Ca reabsortion in TAL by.75 Estimated Sensitivity of Ca reabsortion in TAL to K TAL. M Estimated Stimulation of Ca reabsortion in TAL by CaSR CaSR.75 Estimated Sensitivity of Ca reabsortion in TAL to Ca C ref.33 mm Estimated Minimal fractional reabsortion of Ca in the CT-CNT CT.8 (9, 68) Stimulation of Ca reabsortion in the CT-CNT by and vitamin 3 CT. Estimated Sensitivity of Ca reabsortion in the CT-CNT to K CT.8 M Estimated Sensitivity of Ca reabsortion in the CT-CNT to vitamin 3 K 3 CT 8 M Estimated Plasma volume V ml (56) Fraction of bound calcium b.4 (88) Parameters aly to a 3 g ( mo-old) male rat. PT, roximal tubule; TAL, thick ascending limb; CT, distal convoluted tubule; CNT, connecting tubule. F5 AJP-Renal Physiol doi:.5/ajrenal ownloaded from ( ) on Aril 4, 9.

6 F5 Table 4. Steady-state values under normal conditions Equilibrium Range Source [Ca ],mm...3 (, 5, 74, 9) [],M (,, 39, 9) [ 3],M (, 3, 43) Intestinal Ca absortion, mol/min (38, 6) Fractional intestinal absortion, F intest, % () Ca accretion, mol/min.5 Ca resortion, mol/min.48 Net Ca flux into bone, mol/min. Urinary Ca excretion, mol/min (5, 6, 78, 97) Fractional excretion, u,%.95 (5, 6, 78, 97) tion and reabsortion, as well as the synthesis of vitamin 3 (which in turn enhances intestinal calcium absortion), [Ca ] and [ 3 ] rise in arallel with []. Given that the effects of on vitamin 3 roduction and calcium exchanges between comartments are modeled as saturable rocesses, the rate at which [Ca ] and [ 3 ] increase with synthesis diminishes as k g rod is elevated beyond times its baseline value. Note that the rise in [Ca ] leads to a artial, CaSRmediated inhibition of exocytosis. This, combined with the inhibitory effects of vitamin 3 on synthesis, exlains why the rate at which [] varies with k g rod is slow at first; the sloe increases once k g rod reaches times its baseline value (Fig. 3). Interestingly, our results suggest that fractional urinary calcium excretion and total urinary calcium excretion (U Ca ) may not vary monotonically with k g rod. Indeed, urinary Ca excretion is regulated by counterbalancing mechanisms: whereas an increase in vitamin 3 favors enhanced Ca reabsortion, increases in [Ca ], via the renal calcium-sensing recetor (CaSR), act instead to inhibit renal calcium reabsortion; moreover, inhibits and stimulates Ca transort, resectively, in the roximal tubule and in the distal nehron. Our model redicts that urinary calcium excretion first increases raidly with increasing k g rod, as CaSR-mediated effects revail. As k g rod is elevated above times its baseline value, [] increases at a much greater rate than [Ca ] and -mediated effects thus become dominant; given the counteracting effects of along the nehron, U Ca is redict to decrease slightly before rising steely (Fig. 3). It is equal to.4 times its base-case value when k g rod is multilied by, and 4.7 times at its lateau level (Fig. 4). A.6 C. Calcium-gluconate injection [Ca + ] (mm) EGTA injection during min [Ca + ] (mm) Fig.. Comarison of model results (solid curves) with exerimental data (circles). A: lasma levels of Ca ([Ca ] ) rofile obtained by Lewin et al. (6) following a -min infusion of EGTA. B: lasma concentration of measured by Fox (39) during a -h administration of EGTA. C: [Ca ] rofile obtained by Bronner and Stein (7) following the raid intravenous injection of calcium-gluconate. B EGTA injection during min Time (min) [] (M) Time (min) AJP-Renal Physiol doi:.5/ajrenal ownloaded from ( ) on Aril 4, 9.

7 A Normalized concentrations B Normalized calcium fluxes [Ca + ] [] [ 3 ] urinary excretion intestinal absortion bone resortion bone accretion Normalized synthesis F53 Fig. 3. Predicted effects of rimary hyerarathyroidism on concentrations and fluxes at steady state. A: normalized lasma concentration (relative to base-case value) of Ca (solid curve), (dashed curve), and vitamin 3 (dotted curve), as a function of synthesis (normalized by its base-case value). The rate of synthesis is increased by a factor ranging from to. B: urinary Ca excretion (dashed curve), intestinal calcium absortion (dotted curve), bone accretion (solid curve), and bone resortion (dot dashed curve), as a function of synthesis (normalized by its base-case value); fluxes are normalized by their base-case value. These results were obtained assuming a constant GFR. There is evidence that GFR decreases under hyercalcemic conditions; a large enough GFR reduction could abolish the CaSR-induced U Ca increase. According to our simulations, a 3% GFR decrease, as reorted in Ref. 59, lowers urinary Ca excretion below its base-case value if k g rod is increased by a factor 6 above its baseline value (and levels by a factor 4.5); above that, urinary Ca excretion is redicted to be higher than in the base case. The extent to which modulates Ca transort in the roximal tubule remains to be determined. To evaluate the imact of this uncertainty on model results, we varied the -deendent comonent of Ca reabsortion in the PT (i.e., the arameter PT in Eq. 3) between and.; its baseline value is.5. In these simulations, the total ercentage of Ca reabsorbed by the PT was ket fixed at 65%, and the synthesis rate (k g rod ) was increased by a factor ranging from to 3. In the absence of effects on roximal tubule Ca transort (i.e., PT ), urinary calcium excretion fluctuates less as k g rod is varied. As k g rod is increased above times its baseline value, -mediated stimulation of Ca reabsortion in the TAL and CT/CNT redominates and U Ca decreases (Fig. 4). Conversely, when the (inhibitory) effects of in the roximal tubule are enhanced (i.e., PT.), urinary calcium excretion varies much more significantly with k g rod. Assuming a constant GFR, absolute urinary calcium excretion (which equals.47 mol/min in the base case) is redicted to reach a imal value of.6 mol/min for PT,..4 Calcium excretion (µmol min - ) without effects on PT base case with increased effects Fig. 4. Predicted urinary Ca excretion as a function of the normalized synthesis rate, deending on the relative contribution of to Ca reabsortion in the roximal tubule. The rate of synthesis is increased by a factor ranging from to 3. The arameters PT and PT are, resectively, taken as.65 and (dashed curve; without effects on roximal tubule reabsortion),.6 and.5 (solid curve, base-case values), and.55 and. (dotdashed curve) Normalized synthesis AJP-Renal Physiol doi:.5/ajrenal ownloaded from ( ) on Aril 4, 9.

8 F54 mol/min for PT.5, and.37 mol/min for PT. (Fig. 4). In the remainder of the study, PT is fixed at.5. Exerimental evidence suggests that CaSR and act via indeendent athways in the TAL (6). However, their resective contribution to Ca handling in that segment also remains unclear. To robe the imact of this uncertainty, we conducted simulations in which we enhanced the contribution of CaSR relative to that of in the TAL (while keeing the total contribution fixed), and vice versa. More secifically, we first increased CaSR from.75 to., while decreasing from.75 to.5 (Eq. 4). Conversely, we then raised from.75 to. while reducing CaSR from.75 to.5. As exected, when the imal amlitude of CaSR-mediated inhibition is enhanced and that of -mediated stimulation is reduced as described above, the model redicts that total urinary calcium excretion is higher than with base-case arameters, by.6 mol/min at a given value of k g rod (Fig. 5). Conversely, when the imal amlitude of CaSRmediated inhibition is reduced and that of -mediated stimulation is augmented as described above, U Ca is.3 mol/min lower than with base-case arameters at a given value of k g rod (Fig. 5). Altogether, these results suggest that the extent to which rimary hyerarathyroidism affects urinary Ca excretion is strongly deendent on the balance between - and CaSRmediated effects along the nehron. The influence of other factors, such as the hydration status and ossible variations in vitamin levels, was not considered here. Primary hyoarathyroidism. We then simulated the oo- site scenario, hyoarathyroidism, by decreasing k g rod from its base-case value to zero. eicted in Fig. 6 are the steady-state values of [], [Ca ], [ 3 ], as well as Ca fluxes between comartments. As shown, the lasma concentration of decreases slowly for small reductions in k g rod and faster as k g rod nears zero; indeed, the comensatory effects exerted by vitamin 3 and Ca diminish as their own concentration decreases. In the absence of, [ 3 ] and [Ca ] are, resectively, 59 and 4% lower than in the base case: whereas vitamin 3 synthesis is not stimulated by any more, it is also less inhibited by calcium under these hyocalcemic conditions. In the absence of, calcium bone resortion is reduced by 55% (relative to the base case) as a result of the [] and [ 3 ] decrease, while calcium bone accretion is reduced by 4% owing to the [Ca ] decrease. Intestinal absortion is 33% lower than in the base case. Urinary Ca excretion was comuted assuming a constant GFR, as suggested by studies in hyocalcemic dogs (77). U Ca is determined by the balance between counteracting effects: whereas the reduced levels of and vitamin 3 act to lower Ca reabsortion, lower lasma Ca levels decrease the filtered load. As shown in Fig. 6, the latter effects redominate (i.e., U Ca decreases) until synthesis is almost fully abolished. Vitamin 3 deficiency. To investigate the effects of vitamin 3 deficiency, we decreased the concentration of its recursor ([ inact 3 ]) by a factor ranging from to. In these simulations too, we focused on steady-state rofiles. As shown in Fig. 7, diminishing [ inact 3 ] reduces the rate of vitamin 3 synthesis, which subsequently decreases the rate of intestinal Ca absortion and bone resortion, thereby lowering [Ca ].In arallel, diminishing [ inact 3 ] reduces the inhibitory effects of vitamin 3 on roduction, thereby elevating []. Together, the [Ca ] decrease and [] increase accelerate inact the conversion of 3 to vitamin 3 [i.e., by stimulating -(OH)-ase], therefore mitigating the effects of decreasing the concentration of its recursor. Owing to these feedback mechanisms, when [ inact 3 ] is halved, [ 3 ] is redicted to decrease by % only. Intestinal Ca absortion concomitantly decreases from.6 to. mol/min. Relative to its base-case value, [] is 8% higher when [ inact 3 ] is halved. The increase in levels favors enhanced Ca reabsortion along the nehron, but the [ 3 ] decrease acts in the oosite direction. The net result is a decrease in fractional Ca excretion (from.95 to.75%), i.e., an increase in fractional reabsortion. Note that since the filtered load of Ca is lower, absolute Ca reabsortion is nevertheless slightly reduced (from.37 to.8 mol/min). Similarly, the [] increase stimulates Ca resortion whereas the [ 3 ] decrease has oosite effects. There is near comensation; i.e., bone resortion diminishes slightly, from.48 (base case) to.47 mol/min when [ inact 3 ] is halved. The overall effect of the reduction in calcium absortion, resortion, and urinary excretion is a 4% decrease in [Ca ] (from. to.6 mm)..5 Fig. 5. Predicted urinary Ca excretion as a function of the normalized synthesis rate, deending on the relative contribution of and the renal CaSR to Ca reabsortion in the thick ascending limb. The rate of synthesis is increased by a factor ranging from to 3. The arameters CaSR and in Eqs. 5 and 6 are, resectively, taken as.75 and.75 (solid curve; base-case values),. and.5 (dotdashed curve), and.5 and. (dotted curve). Calcium excretion (µmol min - ) base case CaSR >> CaSR << Normalized synthesis AJP-Renal Physiol doi:.5/ajrenal ownloaded from ( ) on Aril 4, 9.

9 F55 A - % Change B % Change [Ca + ] [] [ 3 ] urinary excretion intestinal absortion bone resortion bone accretion Fig. 6. Predicted effects of rimary hyoarathyroidism on concentrations and fluxes at steady state. A: fractional change in the lasma concentration of Ca,, and vitamin 3, as a function of the ercentage of synthesis inhibition. B: fractional change in urinary Ca excretion, intestinal calcium absortion, bone accretion, and bone resortion, as a function of the ercentage of synthesis inhibition % Inhibition of synthesis As [ inact 3 ] is decreased by more than a factor of, vitamin 3 concentrations are redicted to fall more raidly: the counterbalancing effects of Ca and on its conversion by -(OH)-ase (see above) are less otent at very low [ inact 3 ] values. Concomitantly, intestinal calcium absortion, bone resortion, and urinary Ca excretion decrease more raidly, and variations in [Ca ] and [] with [ inact 3 ] become more significant (Fig. 7). Since bone accretion decreases at a faster rate than bone resortion, calcium retention diminishes, but it remains ositive even under vitamin 3 deletion. CYP4A inhibition. To mimic the effects of an inactivating CYP4A mutation, we set the degradation rate constant k 3 deg to zero. The model then redicts a 8-fold increase in [ 3 ],a 95% reduction in [] (since vitamin 3 inhibits synthesis), and a 44% increase in [Ca ]. Calcium intestinal absortion and calcium bone resortion resectively increase by 49 and 4%. The [] reduction, in combination with the increase in the filtered load of Ca, enhances total calcium excretion by a factor of.6. In comarison, in CYP4A knockout mice, [ 3 ] and [Ca ] were, resectively, increased by a factor of 9 and (86). Inhibition of resortion. Bishoshonates, which inhibit bone resortion, are used to treat several bone diseases, such as osteoorosis and Paget s disease. To investigate the effects of bishoshonates on calcium metabolism, we erformed simulations in which the imal resortion rate (the sum of res min and res in Eq. ) was rogressively reduced to zero. We first examined the dynamic effects of halving the imal rate of resortion on the dynamic evolution of lasma Ca,, and vitamin 3 levels. As deicted in Fig. 8, diminishing the resortion rate leads to a sudden decrease in [Ca ], which in turn triggers a raid increase in levels and arallel reductions in accretion and urinary Ca excretion. Since the effects of on vitamin 3 roduction occur with a time delay (taken as 4hinourmodel), [ 3 ] varies slowly over the first 4 h. When it starts to rise significantly, intestinal calcium absortion is augmented, which in turn elevates [Ca ] and conversely lowers []. ue to these feedback mechanisms, oscillations ensue, but they damen raidly. At equilibrium, the redicted value of [Ca ] is 5% lower than its basal value, which is similar to the values reorted by Fleisch (38). The equilibrium values of [] and [3] are, resectively, 7 and 3% higher than in the base case, which is also comarable to exerimental measurements (34). We then varied the resortion rate over a large range, from its base-case value to zero. Shown in Fig. 9 are redicted concentrations and fluxes at steady state (i.e., once equilibrium has been reached) as a function of the resortion rate. As described above, inhibiting resortion lowers the lasma level of Ca, raises that of and vitamin 3, enhances Ca absortion, decreases Ca accretion, and reduces both absolute and fractional urinary Ca excretion. Our results suggest that the (steady-state) lasma concentration of increases with decreasing resortion, before reaching saturation (Fig. 9): as [Ca ] decreases, CaSR-mediated inhibition of exocytosis is reduced. Since stimulates the synthesis of vitamin 3 and inhibits its degradation, [ 3 ] increases in arallel with (Fig. 9). In counterbalance, the increase in vitamin 3 levels lowers the rate of [] synthesis. Intestinal calcium absortion, which is stimulated by vitamin 3, increases as the resortion rate is lowered, whereas Ca accretion decreases (Fig. 9). The model redicts that the rate of accretion falls more slowly than that of resortion, so that Ca retention increases, as observed exerimentally (38, 75). Urinary Ca excretion decreases with the resortion rate, which indicates that the reabsortion-enhancing effects of and vitamin 3 (as a result of and vitamin 3 elevation) AJP-Renal Physiol doi:.5/ajrenal ownloaded from ( ) on Aril 4, 9.

10 F56 Fig. 7. Predicted effects of vitamin 3 deficiency on steady-state concentrations and fluxes. A: normalized lasma concentration of Ca,, and vitamin 3, as a function of [ inact 3 ] (normalized by its base-case value). B: normalized urinary Ca excretion, intestinal calcium absortion, bone accretion, and bone resortion, as a function of [ inact 3 ] (normalized by its base-case value). A Normalized concentrations B Normalized calcium fluxes [Ca + ] [] [ 3 ].9.8 urinary excretion intestinal absortion bone resortion bone accretion Normalized [ 3 inact ] are greater than the excretion-enhancing effects of Ca (via CaSR). Cinacalcet treatment. Cinacalcet, a calcimimetic agent acting on the calcium sensing recetor by increasing its sensitivity to calcium, is commonly used to treat secondary hyerarathyroidism in chronic renal failure or, more rarely, rimary hyerarathyroidism. Cinacalcet decreases exocytosis from arathyroid glands and decreases Ca reabsortion in the TAL (6). To assess the theoretical imact of Cinacalcet on rats with hyerarathyroidism, we erformed the following simulations. To mimic rimary hyerarathyroidism, we increased the rate of synthesis by a factor of 85, so that the lasma concentration of is -fold its basal value. To mimic the effects of Cinacalcet, we fixed the rate of exocytosis to its minimal value, and imized the inhibitory effects of CaSR on Ca reabsortion in the TAL. As illustrated in Table 5, Cinacalcet is redicted to return both [] and [ 3 ] in rats with hyerarathyroidism to their levels in control animals; the lasma calcium concentration is thus rescued, i.e., its redicted value (.4 mm) falls within the normal range. The rates of calcium exchanges between comartments are also very close to their base-case values (i.e., without rimary hyerarathyroidism). Effects of age and sex on calcium homeostasis. Maintenance of the calcium balance varies between the two sexes; it also evolves with age. In articular, the lasma concentration of vitamin 3 is 5% lower in female rats than in age-matched male rats (65). Vitamin 3 levels (and intestinal calcium absortion) also decrease with age (95). Conversely levels increase with age but do not vary significantly between age-matched female and male rats (39, 44, 65). Bone turnover is also widely affected by aging (3, 3). esite these differences, lasma calcium concentration varies very little as a function of either age or sex, under hysiological conditions. We used the model to examine how [Ca ] can remain stable given these documented variations in hormonal levels. We considered - and 8-mo-old, male and female rats, and accounted for differences in and vitamin 3 synthesis, as described above. We also accounted for sex- and age-based differences in lasma volume (V ), glomerular filtration rate (GFR), calcium intake (I Ca ), the sensitivity of intestinal absortion to vitamin 3 (K 3 abs ), and the rate of bone resortion (i.e., the arameters min res and res ), as summarized in Table 6. The model redicts that to maintain [Ca ]. mm in all four rat grous, it is also necessary to osit sex- and age-based differences in the rate of calcium exchanges between lasma and the raidly exchangeable bone ool (i.e., k Ca f- and k Ca -f ), as also suggested by one study (). However, other model arameters need not change (Table 6). Intestinal absortion is redicted to be halved in -mo-old females comared with young males, mainly because vitamin 3 levels are 5% lower in females. Bone resortion follows the same trend. Bone accretion is decreased in females because they have less calcium in the raidly exchangeable bone ool than males. This results in a lower bone turnover in females and a net calcium flux into the dee bone that is halved. Urinary calcium excretion is redicted to be reduced in females, as a result of the lower GFR, even though fractional calcium excretion is higher in females (owing to lower levels of vitamin 3 ). In 8-mo-old animals, the intestinal absortion and bone accretion of calcium are substantially reduced (aroximately 5% in both sexes). Given that levels increase with age, urinary calcium excre- AJP-Renal Physiol doi:.5/ajrenal ownloaded from ( ) on Aril 4, 9.

11 A.3 F57 [Ca + ] (mm).. B [] (M) C [ 3 ] (M) Calcium excretion (µmol min - ) E Fig. 8. Predicted effects of inhibiting bone resortion on the evolution of concentrations and fluxes with time. In this simulation, the imal bone resortion rate is divided by starting at t h.a, B, and C: resectively, deict the lasma concentration of Ca,, and vitamin 3, as a function of time. : urinary Ca excretion. E: intestinal calcium absortion, accretion, and resortion. Calcium Flux (µmol min - ) Time (hours) intestinal absortion bone resortion bone accretion tion is redicted to be lower (5 and 7%, resectively, in 8-mo-old male and female rats). ISCUSSION Scoe of Model We have develoed a mathematical model that describes Ca exchanges among the intestine, lasma, slowly and raidly exchangeable bone ools, and the kidneys, as well as their regulation by and vitamin 3. To our knowledge, this is the first such model of Ca homeostasis that alies to the rat, accounts for the raidly exchangeable ool in bone, and considers the imact of the renal CaSR. As shown in Fig., the model adequately reroduces exerimental findings in different scenarios. Model Limitations We should nevertheless acknowledge a number of limitations. Model arameters aly to 3-g rats, a weight at which rats are tyically used for hysiological exeriments but for which exerimental calcium measurements are limited. We thus extraolated measured values of calcium bone content in mice. The model also assumes fixed values for lasma volume AJP-Renal Physiol doi:.5/ajrenal ownloaded from ( ) on Aril 4, 9.

12 F58 A % Change [Ca + ] [] [ 3 ] Fig. 9. Predicted effects of inhibiting bone resortion on steady-state concentrations and fluxes. A: fractional change in the lasma concentration of Ca,, and vitamin 3, as a function of the ercentage of inhibition. B: fractional change in urinary Ca excretion, intestinal calcium absortion, bone accretion, and bone resortion, as a function of the ercentage of inhibition. B % Change urinary excretion intestinal absortion bone accretion % Inhibition of resortion Table 5. Effects of Cinacalcet administration to rats with rimary hyerarathyroidism Control PHPT PHPT with Cinacalcet [Ca ],mm [],M [ 3],M Intestinal Ca absortion, mol/min Ca accretion, mol/min Ca resortion, mol/min Net Ca flux into bone, mol/min Urinary Ca excretion, mol/min Primary hyerarathyroidism (PHPT) is simulated by increasing the rate of synthesis (k g rod ) by a factor of 85 (so that () is multilied by ). In the resence of Cinacalcet, the rate of exocytosis is set to its minimum value (.7 min vs..99 in the base case), and the inhibitory effects of calcium-sensing recetor (CaSR) on Ca reabsortion in the thick ascending limb (TAL) are imized [ TAL(Ca) from Eq. 5 is set to.75 vs.. in the base case]. and GFR, whereas these may vary in tandem with Ca and/or under certain conditions. Taking into consideration the numerous factors that regulate lasma volume and GFR is beyond the scoe of this model. Instead, we accounted for observed changes in GFR in secific simulations. In addition, our model does not consider the interactions between calcium and hoshate, which are articularly relevant in bone. Accounting for this couling is challenging because the regulation of hoshate homeostasis has not been fully characterized. Fibroblast growth factor 3 (FGF3), a bone-derived hormone that regulates systemic hoshate homeostasis, has recently been recognized as an imortant comonent of the bone-arathyroid-kidney axis; in articular, FGF3 inhibits the synthesis of vitamin 3 and the secretion of (8). Of note, FGF3 may have also a direct effect on calcium transort in the kidney (5). A better quantitative understanding of hoshate metabolism is needed before it can be added to the model. Since we do not reresent the metabolism of comounds other than Ca,, and vitamin 3, we cannot account for the allosteric factors that modulate the binding affinity of CaSR to Ca, such as sodium and ionic strength, nor can we account for the effects of lasma volume, sodium reabsortion, and thiazides on the fraction of Ca that is reabsorbed in the roximal tubule ( PT ). Similarly, the model does not consider other known regulators of calcium homeostasis, including klotho, calcitonin, and steroids. Sex hormones are also known to regulate calcium homeostasis by modifying calcium absortion in the intestine, calcium bone metabolism, and renal calcium reabsortion. The model does not exlicitly reresent the mechanisms underlying these regulations and takes into account only some asects of sex-based differences (body size, calcium intake, as well as and vitamin 3 levels) (3, 49, 66). Finally, the model is hamered by its descrition of organs as black boxes, meaning that transort and regulation mechanisms at the molecular and cellular scales are described instead by simlified relationshis, such as first-order kinetics and Michaelis-Menten equations. There exists detailed, cellularbased models of calcium homeostasis in kidney (3), and intestine (33, 46, 83); these could be integrated in the resent mathematical model in the future. Similarly, models of bone AJP-Renal Physiol doi:.5/ajrenal ownloaded from ( ) on Aril 4, 9.