INTRODUCTION

TOP ABSTRACT INTRODUCTION NONLINEAR COMPARTMENTAL MODEL PHYSIOLOGICAL INTERPRETATION... DISCUSSION APPENDIX A APPENDIX B APPENDIX C REFERENCES

IN THE ABSENCE OF EVIDENCE for a physiological role for Sr, the biological interest in this mineral and its metabolism was focused on analogies and discrepancies with Ca²⁺ as its closest divalent cation in the periodic table and the essential mineral in numerous key extracellular and intracellular processes. Sr and Ca are members of the alkaline earth series, with comparable features of cation chemistry (ionic radius, charge-to-size ratio, high coordination number, and H₂O exchange) (46). Thus they show physicochemical similarities in their interactions with organic or inorganic components. These similarities are such that the major metabolic pathways of both cations, intestinal absorption, deposition in and removal from bone, and excretion in urine and feces, are believed to involve identical processes. However, in vivo and in vitro kinetic studies, often using a tracer radionuclide of Sr²⁺ and Ca²⁺ (27, 36), have revealed, at least quantitatively, behavioral differences at gastrointestinal (GI), renal, and, possibly, bone levels (10, 42). They have been interpreted in terms of discrimination between Sr and Ca (44). Globally, retention of Ca relative to Sr is favored at the organism level.

Physicochemical analytic studies have shown that synthetic hydroxyapatite (HA), with partial substitution of Ca²⁺ by Sr²⁺ into the crystal lattice, can be produced from calcium phosphate-supersaturated aqueous solutions containing Sr²⁺, the level of discrimination against Sr²⁺ relative to Ca²⁺ being directly dependent on the rate of crystal growth and the associated crystal perfection (24, 30). Recently, Sr²⁺ in solution was described as easily adsorbed at the crystal surface and as inhibiting the rates of HA crystal growth and dissolution (9). In vivo data indicate that Sr²⁺ can be incorporated into biological apatite (5, 12), with possible limitation in the number of substitutions.

For situations in which cations interact with organic components, the behavioral differences between Ca and Sr may be analyzed in terms of the physicochemical, stereochemical, and structural characteristics of the organic species, i.e., from nearly no difference between ion reactivities to a nearly complete specificity for Ca as a result of the small difference between Ca and Sr chemistry.

Among the main factors involved in discrimination between Sr and Ca are their relative abundance and availability in the natural environment (46). Sr is a trace element, whereas Ca is a macronutrient. Their concentrations in biological fluids also differ greatly. There are extracellular fluids with millimolar concentrations of Ca and micromolar concentrations of Sr. The very high extracellular Ca-to-Sr molar ratio means that Sr cannot significantly compete with Ca under physiological conditions; therefore, there has been little need for selection, during evolution, of mechanisms specific for Ca, rather than Sr. In vitro experiments confirm this. Sr²⁺ can replace Ca²⁺ in activating the parathyroid Ca²⁺-sensing receptor (PCaR) (25) at a concentration that is twice that needed for Ca²⁺. Similarly, the apical Ca²⁺ channels isolated from intestine (31) and kidney (41) are also permeable to Sr²⁺, with higher apparent permeability for Ca²⁺ than for Sr²⁺. However, the intracellular fluid contains 0.1 µM free Ca²⁺ (Ca_i) and Ca_i is an important second messenger for numerous cellular processes (e.g., proliferation, differentiation, apoptosis, and activity). Inasmuch as Sr is not as effectively regulated as Ca_i, their intracellular concentrations might be similar, hence, the requirement for high specificity for Ca by most intracellular proteins influencing resting Ca_i and Ca_i signaling and regulatory mechanisms. For instance, Sr²⁺ is 600-fold less potent than Ca²⁺ in causing calmodulin-induced inhibition of liver inositol trisphosphate-induced Ca²⁺ release, probably because Sr²⁺ binds 30 times less well than Ca²⁺ to calmodulin (29).

This high degree of discrimination against Sr relative to Ca for vital intracellular functions is likely the reason for hypocalcemia and/or hypocalcified bone and the decrease in 1,25-dihydroxyvitamin D synthesis, which are the first manifestations of Sr administration, only if there is a drastic increase in plasma Sr concentration. Nevertheless, plasma Sr concentration can be dose dependently increased 100-fold over a large range of Sr salt administration, while a molar Ca-to-Sr ratio higher than unity is always maintained. This has no apparent toxic effect and causes no significant change in plasma Ca concentration or in calcitropic hormones (17, 26).

We have reported the development of a nonlinear compartmental model for human Sr metabolism (39). In contrast to most mathematical analyses of Sr kinetics, our model focuses on Sr metabolism per se, postponing its comparison with Ca metabolism. Because it is based on non-steady-state kinetic data, it includes several nonlinearities that must be interpreted to validate our model. The primary concern of this work is to study, within the context of the Sr-Ca similarity-dissimilarity outlined above, the relevance of the model to Ca metabolism and to yield and illustrate, using model refinements, suitable explanations about the identified nonlinearities and their role in the homeostatic and/or adaptive regulation of this metabolism.

	NONLINEAR COMPARTMENTAL MODEL

TOP ABSTRACT INTRODUCTION NONLINEAR COMPARTMENTAL MODEL PHYSIOLOGICAL INTERPRETATION... DISCUSSION APPENDIX A APPENDIX B APPENDIX C REFERENCES

Briefly, the compartmental model described previously (39) was based mainly on plasma Sr concentration kinetic data collected in postmenopausal women given twice-daily oral doses of Sr (S-12911, Institut de Recherches Internationales Servier). These non-steady-state data were obtained for four doses of Sr (1.95, 3.89, 7.78, and 15.57 mmol/day) and included the increase in plasma Sr concentration during Sr administration (AdP, the first 25 days) and its decrease after cessation of treatment [postadministration period (PAdP), the other consecutive 27 days]. The plasma Sr kinetics and the fit to these experimental data obtained at the end of model building have been described previously (Fig. 1 in Ref. 39).

The model consists of two distinct, but interdependent, compartmental systems (Fig. 1).

View larger version (34K): [in this window] [in a new window]   Fig. 1.   Nonlinear compartmental model of Sr kinetics developed in Ref. 39. Solid lines, overall Sr metabolism (system 1); dashed lines, biological variables other than Sr (z variables of system 2 interacting with system 1). Circles numbered 1 to 7 are kinetically distinct entities associated with Sr metabolism in bone (compartments 2, 6, and 3), gastrointestinal (GI) tract (compartments 4 and 5), and the internal distribution pool (compartment 1, including plasma, and compartment 7). Circles numbered (n + 1) and (n + 2) are compartments representing z(n+1) and z(n+2) (system 2). k and K denote fractional transfer coefficients (kij) and fractional transfer functions (Kij); F(t) represents input pathways associated with dietary Sr and oral Sr doses. Bold arrows denote nonlinearities that are intrinsic [Michaelis-Menten (M-M) for K15 and Langmuir-type for K21] or extrinsic due to reciprocal interactions between system 1 and system 2 (curved arrows). For further explanation, see APPENDIX A.

System 1 is directly concerned with Sr metabolism and includes seven compartments describing whole body Sr and its relations to the surrounding milieu. These compartments are organized into three subsystems. 1) The GI tract consists of two compartments (compartments 4 and 5): one (compartment 4) accepts the Sr from food (and the oral Sr administration), and the other (compartment 5) is associated with the irreversible excretion of Sr in the feces and the bidirectional transfer of Sr to (intestinal absorption) or from (endogenous intestinal secretion) the internal distribution pool (IDP) via compartment 1. 2) The bone subsystem has three compartments connected to compartment 1 through bidirectional (compartments 2 and 3) or unidirectional (compartment 6) relations. Compartment 6 is directly concerned with irreversible Sr movements associated with bone mineral accretion and removal. Bone formation operates from one compartment (compartment 2) lying between compartment 1 and compartment 6. 3) The IDP consists of two compartments. From compartment 1, Sr is excreted via the kidneys; compartment 1 includes plasma and other extracellular and, probably, intracellular sectors in rapid equilibrium with the plasma [apparent distribution volume (V_app) ~40 liters]. Compartment 7 is a rapidly exchangeable pool with no precise physiological identity. This system 1 also contains two intrinsic nonlinearities (INL) that involve nonlinear relations to a given system 1 compartment. These INL account for a saturable Michaelis-Menten (M-M)-type process (Eq. A1) involving intestinal absorption and for a Langmuir-type reaction (Eq. A3) involving mineral transfer from the extracellular fluids to the bone. This second nonlinearity does not behave as a simple saturable process, because it includes the inhibitory effect of one compartment that is different from the source compartment 1, compartment 6, or compartment 3, according to whether one refers to one or the other of both optimal models retained in the companion article (39). With reference to this inhibition variable, the optimal model here is model L6 or model L3.

System 2 is relative to biological variables other than Sr [compartments (n + 1) and (n + 2); Fig. 1]. Each extrinsic variable is associated with a one-compartment structure with an entry flow as an S-shaped growth function known as the logistic equation, affected by the Sr concentration in compartment 1 with a high (~3 or more) cooperativity order (see Eqs. A5 and A6 for formulation). It acts on system 1 through feedback modulation of some Sr transfer fluxes. The action of system 2 on system 1 introduces additional nonlinear functions in system 1, called extrinsic nonlinearities (ENL). The model includes two parameter-distinct time-explicit variables: z_(n+1), acting on the intestinal endogenous secretion rate and on the Langmuir-type process of Sr transfer from compartment 1 to bone compartment 2, and z_(n+2), acting on the saturable intestinal Sr absorption rate.

	PHYSIOLOGICAL INTERPRETATION AND MODEL REFINEMENTS

TOP ABSTRACT INTRODUCTION NONLINEAR COMPARTMENTAL MODEL PHYSIOLOGICAL INTERPRETATION... DISCUSSION APPENDIX A APPENDIX B APPENDIX C REFERENCES

We have examined the meaning of the structure (Fig. 1) by testing the physiological reliability of the optimal models (models L6 and L3) and their relevance to Ca metabolism at four levels: 1) the initial steady state of mineral metabolism itself (for Sr and Ca), 2) the physiological significance of intestinal and bone INL and their applicability to Ca metabolism, 3) the kinetic and dynamic properties of the z logistic variables and their possible extension from dependence on Sr to dependence on Ca, and 4) the nature of the complex interactions between INL and ENL in system 1 and their potential meaning for the regulation of some Ca metabolic pathways and for the effects of Sr on Ca metabolism.

The relevance of the Sr model to Ca metabolism was checked with the assumption that compartment 1 has a constant concentration of 2.5 or 1.25 mM. The experimental data (not reported) indicate that Sr has no effect on total and/or ionized plasma Ca concentration.

Model refinements were proposed to theoretically illustrate some physiological interpretations. If required, parameter estimation and a posteriori identifiability with precision of parameter values expressed as coefficient of variation (CV, in %) were carried out as described previously (39).

Initial Steady State: From Sr to Ca Metabolism

The results presented in Table 1 show that applying model L6 to Ca metabolism gives rise to a number of predicted theoretical values that agree with the known characteristics of Ca metabolism. With application of the same model parameters for Ca and Sr, except for the urinary excretion, the relative mass distribution of Ca within the body (IDP and bone) is similar to that of Sr. The total Ca mass is ~900 g, with >800 g in compartment 6 and only ~30 g (3.4% of the total mass) in the other compartments. The fluxes into and out of compartment 6 (bone mineral solid phase) give a Ca bone turnover of ~600 mg/day, 1.6 times the Ca urinary excretion (340 mg/day), with a mean residence time in compartment 6 of ~4 yr. On the contrary, applying the GI parameter to Ca metabolism reveals a considerable discrepancy between Sr and Ca. The model predicts an unrealistic value of >9 g/day for mean Ca ingestion, whereas the net intestinal balance (340 mg/day) represents <4% of the predicted Ca daily ingestion (~96% is excreted in feces), although its absolute value is not inconsistent. This meaningless model prediction can be made realistic by reducing K₅₁, the value of the FTF linked to the return of mineral from compartment 1 toward compartment 5 (Fig. 2). Dividing k₅₁ by 8-10 gives an expected daily Ca ingestion of ~1 g, with a net intestinal absorption >30%. This causes no change in the other properties that fit the expected processes involved in Ca metabolism of these human subjects. Thus, in addition to urinary excretion, our results suggest that some of the mechanisms influencing the bidirectional relation of the GI compartment to compartment 1 differ quantitatively between Ca and Sr metabolism (see below).

View larger version (10K): [in this window] [in a new window]   Fig. 2.   Initial steady-state prediction for daily dietary Ca intake required when model L3 is applied to Ca metabolism as a function of decreasing intestinal endogenous mineral secretion (K51 divided by an increasing divisor number). Dashed lines delimit the expected physiological range of daily dietary Ca intake.

GI and Bone INL

Intestinal M-M-type process: Sr-Ca similarity-dissimilarity. The transfer function K₁₅ from GI compartment 5 to compartment 1 in our model (Fig. 1) is governed by an M-M equation in addition to a simple linear process (Eq. A1). This kind of representation has been used to account for in vitro data on Ca intestinal absorption (45), demonstrating that the transfer of mineral from the GI compartment to extracellular fluids is the sum of two distinct processes: one is saturable and mediated by species required for the intracellular mineral transfer but present in limited amounts; the other is unsaturable and remains proportional to mineral concentration in the lumen.

View larger version (10K): [in this window] [in a new window]   Fig. 3.   Predicted dependence of linear (solid line) and nonlinear (dashed line) Sr intestinal absorption rates on compartment 5 concentration. Values are expressed as apparent concentration.

View larger version (11K): [in this window] [in a new window]   Fig. 4.   Refined nonlinear compartmental substructure for GI Sr metabolism. Compartments are as shown in Fig. 1, except for intermediary compartment 1', which is associated with intestinal intracellular mineral. Rate-limiting (M-M-type) process of transcellular mineral transport from GI compartment 5 to compartment 1 is the apical entry into epithelial cells indicated by K1'5 that, as for K51', is modulated by a z extrinsic variable. k15 denotes paracellular nonsaturable (linear) intestinal absorption. See Fig. 1 legend for explanation of symbols.

Bone Langmuir-type function, an Sr-specific process. The second INL is the so-called Langmuir-type function. It operates on K₂₁, the FTF related to the transfer of mineral from compartment 1 to the major bone compartment 6 via compartment 2 (Fig. 1). The interpretation of the present nonlinearity differs from that of intestinal absorption, because it concerns the bone, where not only the cellular and organic components may interact with mineral, but also the various mineral physicochemical reactions may be directly responsible for the nonlinearity. For instance, the adsorption of mineral species at the surface of the bone solid phase or other deeper sites, such as ion integration or substitution inside the crystal lattice, may be saturable processes, because they depend on free sites, the number (concentration) of which may be restricted. Similarly, the reactivity of these sites may change with the composition of the liquid or the solid phase. Impurities (foreign ions) may act as solutes or as constituents of the crystal lattice and so inhibit some of the numerous steps involved in bone mineralization (1-3, 9).

View larger version (17K): [in this window] [in a new window]   Fig. 5.   Refined nonlinear compartmental substructure for bone Sr metabolism. Explicit representation of mineral adsorption at the bone surface is shown as the first reversible step of Sr (and Ca) incorporation into the bone solid phase. System 2 compartment (n + 3) characterizes free adsorption sites. Compartment 1 includes plasma. Intermediary compartments 1' and 3 are associated with mineral bound to adsorption sites and compartments 2 and 6 with mineral incorporated into the first and deep layers of bone solid phase. See Fig. 1 legend for explanation of symbols. , Fractional transfer associated with recovery of free sites. Compartment 3 acts as a noncompetitive inhibitor. For further explanation, see APPENDIX B.

3

1

ENL Acting on G1 and/or Bone

Kinetic and dynamic behavior of the logistic z variables. As reported for z_(n+1) in model L3 (Fig. 6A), the kinetic behavior of this variable reveals important differences between the various Sr doses. There is almost no change over time with the smallest dose (D₁), whereas z_(n+1) increases acutely during oral Sr administration (AdP) with the highest one (D₄), to reach its maximum after ~40 h, with only slight changes. For the other two doses (D₂ and D₃), z_(n+1) produced a more moderate change, intermediate between D₁ and D₄. During PAdP, z_(n+1) tended to decrease at a rate that depended on the value reached at the end of AdP. The kinetics of z_(n+2) (not shown) are rather similar to the kinetics of z_(n+1), with some differences linked to distinctive parameter values (Table 2): the slopes of the sharp rise during AdP and the fall during PAdP are more pronounced for z_(n+2). From a dynamic point of view, the expected sigmoidal curves (Fig. 6B) obtained when the asymptotic z values (computed from the identified parameter values given in Table 2) are plotted against y₁ increasing values also differ according to whether z_(n+1) or z_(n+2) is considered, with concentrations for the half-maximal activating effect (HMC) slightly above 30 or 50 µM. These curves clearly indicate that the modulation of Sr metabolism (system 1) by the logistic z functions through their effects on the target FTF (K₁₅, K₅₁, and K₂₁) is not due to kinetic features alone but is also dose dependent. The asymptotic values of y₁ for the different Sr doses, i.e., the compartment 1 values obtained after prolonged simulation of model L3 with continued exogenous Sr, only gave the maximal increase (the z unit value) with the two highest doses for z_(n+1) and z_(n+2) (Fig. 6B). D₁ shows a smaller increment of z_(n+2) than z_(n+1).

View larger version (9K): [in this window] [in a new window]   Fig. 6.   Kinetic and dynamic behavior of system 2 logistic variables. A: predictions, from model L3, of z(n + 1) variations over experimental duration [during administration period (AdP) and after cessation of treatment (PAdP)] for the 4 oral Sr doses: D1 (solid line), D2 (dashed line), D3 (dashed-dotted line), and D4 (dashed-dotted-dotted line). B: computed dependence of the asymptotic value of z(n+1) (solid line) and z(n+2) (dashed line ) on compartment 1 Sr concentration. Vertical arrows, asymptotic compartment 1 Sr concentration reached for each Sr dose. Each z variable is expressed in normalized concentration (defined in APPENDIX A).

Ca as an inducer of the z function: evidence for mineral self-regulation. Evidently, the high level of cooperativity revealed by the identified value of p_(n+1) and p_(n+2) (~3 and 4, respectively; Table 2), defining the order at which y₁ activates the z logistic functions, accounts for the main peculiarities reported above. Another interesting characteristic is that, despite showing large inaccuracy (Table 2), the identified values for g and g, i.e., the constant parameters independent of y₁ in Eq. A6, have CV < 100%. Thus some organic and/or mineral species other than Sr and assumed to be constant throughout the experiment could influence the logistic functions. Inasmuch as Ca influences many cellular processes, we investigated whether the Sr dependence of extrinsic z functions could result from a direct interaction between Sr and Ca. We have merely conjectured that Sr mimics the inducing effects of Ca on some Ca²⁺-dependent physiological processes. We changed the nonlinear Eq. A6, as shown in APPENDIX C, by Eq. C1 to clarify the dependence of Ca and Sr on the z functions.

View larger version (10K): [in this window] [in a new window]   Fig. 7.   Theoretical dependence of z(n+1) (solid line) and z(n+2) (dashed line) asymptotic value on Sr (A) and Ca (B) compartment 1 concentration when Sr and Ca activate the z logistic functions. Arrows, half-maximal plasma mineral concentration for each computed curve. In A, thin and thick curves are obtained in the absence and presence, respectively, of a physiological plasma Ca concentration. In B, stippled vertical bar represents range of normal plasma free Ca concentration.

Interaction Between INL and ENL: How Sr Affects Sr and Ca Metabolism

Time-varying FTF. Although variations in K₅₁ only reflect variations of z_(n+1) (Fig. 6A) with, for D₃ and D₄, a maximal increase during AdP of about six times its initial value, K₂₁, although modulated by the same logistic function [z_(n+1)], changes differently with time, with a maximum for D₄ of less than three times the initial value (Fig. 8A). This difference is due to the greater influence of the Langmuir-type than the logistic function: the Langmuir-type function tends to progressively decrease K₂₁ when compartment 3 (the inhibitory variable in Eq. A3) rises. Conversely, z_(n+1) increases this FTF, but with its own y₁-dependent kinetics. K₂₁ continuously decreases during AdP at the lowest Sr dose (Fig. 8A) because of the very small z_(n+1) increment (Fig. 6A). In contrast, K₂₁ increases sharply at the two highest doses, further counterbalanced by the opposite effect of the Langmuir-type nonlinearity. The relative weights of the nonlinearities are such that K₂₁ for D₄ has a value that is lower than that for the intermediary D₂ and D₃ at the end of AdP. This could be interpreted as D₂ and D₃ being more efficient than D₄ for the transfer of mineral toward bone. This result could be important for the effect of Sr on Ca transfer to bone via bone formation/mineralization, because this metabolic process, characterized by K₂₁, is common to Ca and Sr, and the extracellular Ca concentration (Y₁) does not change significantly under our experimental conditions.

View larger version (15K): [in this window] [in a new window]   Fig. 8.   Time variations of nonlinear fractional transfer functions predicted from model L3 over experimental time (AdP and PAdP) for D1-D4 (see Fig. 1 legend for explanation of lines). A: complex kinetic behavior of mineral transfer from compartment 1 to bone mineral solid phase, resulting, at K21 level, from interaction of a Langmuir-type intrinsic nonlinearity (INL) with a logistic extrinsic nonlinearity (ENL; see Eq. A3). B and C: because of the interaction of an M-M-type INL with a logistic ENL (see Eq. A1), K15 also has peculiar kinetics with complex dose-dependence relationship. B: long-term variations in K15. C: short-term (24-h) kinetics for K15 compared with quasi-constant value of K51 (see Eq. A2) observed late in AdP for D3 and D4.

Dose-dependent effects of Sr on Ca metabolism. The above analysis has revealed an attractive property of the model related to the complex dose dependence of GI and bone FTF. If Sr and Ca share the same metabolic pathways, then the variations in FTF in response to exogenous Sr must also affect Ca metabolism. We have therefore departed from a strict physiological framework to examine the effects of exogenous Sr on the metabolism of Ca. The asymptotic behavior of the model variables was computed for Ca and Sr (see APPENDIX C) using different values for compartment 1 Sr concentration (y₁) and a constant compartment 1 Ca concentration (Y₁ = 2,500 µM) plus the set of parameter values previously identified from model L3.

View larger version (9K): [in this window] [in a new window]   Fig. 9.   Complex dose-dependent effect of Sr on GI and bone Ca metabolism according to model L3. A: asymptotic Ca mass in compartment 6 (bulk bone solid phase) as a function of compartment 1 Sr concentration. B: asymptotic oral Ca intake required to maintain compartment 1 at physiological plasma Ca concentration (2.5 mM) as a function of compartment 1 Sr concentration.

An integrative mechanism for intestinal secretion of endogenous mineral. Finally, we examined the bidirectionality of the relation of GI compartment 5 to compartment 1. The satisfactory initial models (Fig. 1), as well as the refined GI compartmental structure with the intermediary intestinal cellular compartment explicitly considered (compartment 1' in Fig. 4), require reversibility between lumen and plasma. This kinetic reversibility could be due to peculiar features of some process involved in the facilitated membrane transport of mineral, as suggested for intestinal Ca absorption. Two mechanisms of Ca entry from the gut lumen into the enterocyte via the apical membrane have been proposed: the first mechanism may be essentially irreversible, with a Ca²⁺ channel similar to that recently detected in the proximal small intestine [mainly in the duodenum (19)]; the second mechanism, associated with a lipid-soluble mobile carrier, could be reversible to some extent (47). Each of these processes, channels (19), channel-like transporters (31), or mobile carriers (47), is also saturable, with an apparent M-M constant in the same range as that estimated by our model. Finally, multiple signaling pathways may regulate them, a property consistent with their modulation, in our model, by extrinsic z functions.

View larger version (11K): [in this window] [in a new window]   Fig. 10.   Computed S-shaped curves of y1 dependence for 3 logistic variables: z(n+1) (solid line), which modulates bone fractional transfer function, K21; z(n+2) (dashed line), which modulates a first saturable irreversible process of intestinal Sr absorption; and z(n+3) (dashed-dotted line), which acts on another intestinal absorption process that is reversible because of modulation of intestinal endogenous mineral secretion by this same extrinsic z(n+3) variable.

	DISCUSSION

TOP ABSTRACT INTRODUCTION NONLINEAR COMPARTMENTAL MODEL PHYSIOLOGICAL INTERPRETATION... DISCUSSION APPENDIX A APPENDIX B APPENDIX C REFERENCES

The present study was prompted by the need to analyze the relevance of a nonlinear compartmental model, developed in the companion paper (39) to describe human Sr metabolism, for Ca metabolism. Given the model structure (Fig. 1) and the new quantitative and mainly qualitative information it offers, our purpose was to test the feasibility of physiological mechanisms that could underlie the nonlinear processes included in this model, with their intrinsic or extrinsic nature. Using model refinements, we were able to check a set of assumptions related to the origin of these nonlinearities and compare refined model predictions with the few available data concerning Ca metabolism under Sr supplementation. (Only data relative to total and free plasma Ca concentration and daily urinary Ca excretion were assessed over the experimental duration.) Of course, considering the theoretical and practical identifiability requirements, the refined structures were often overmodeled. Thus any refined model is only an illustration of a mechanistic hypothesis, rather than a fully justified step of modeling. Nevertheless, this approach has several advantages and seems to be of importance for a better understanding of Ca metabolism, its regulation, and the Ca-Sr discrimination (see below).

First, and as a necessary condition for further discussion, the relevance of the Sr model to Ca metabolism is supported by the overall agreement of the model predictions at the initial state, including discrimination of Ca over Sr. As it was for Sr metabolism (39) and under the assumption that the whole Sr or Ca metabolism is in an asymptotic steady state (zero net mineral bone balance), the quantitative characteristics of Ca metabolism predicted at time 0 (under physiological conditions) are consistent with experimental results for Ca mass distribution and the rates of the main metabolic pathways when, in addition to urinary excretion, some GI processes involved in the net mineral intestinal balance discriminate in favor of Ca. The estimated absorption of Ca is then ~1.5 times that of Sr, close to that reported in the literature (44). Thus these observations corroborate several published in vivo investigations.

Second, we take advantage of the peculiarities of our model and use its constitutive intrinsic intestinal and bone nonlinearities to examine the underlying mechanisms and the processes that may distinguish between Sr and Ca in the GI tract or bone. This kind of study was not possible for kidney mineral metabolism, because the rates of glomerular filtration/renal secretion and tubular reabsorption are too fast for our model. Consequently, the Sr-to-Ca ratio of urinary clearance, estimated to be ~2, in agreement with other evaluations (21), was used to account for the Sr-Ca discrimination during tubular reabsorption.

The GI compartmental substructure has two distinctive properties. 1) Nonsaturable (linear, paracellular) and saturable (M-M type, transcellular) transport processes are clearly indicated and quantitatively identified from in vivo kinetic and dynamic (dose-dependent Sr kinetics) data. The parameter values for the Sr-saturable process (maximum velocity 21.6 mmol/day and M-M constant ~0.8 mM) are on the same order as those expected for physiological intestinal Ca transport, consistent with the idea that Sr and Ca intestinal absorption use similar transcellular paths (4, 42). Thus our results are at variance with the existence of Sr-specific processes for intestinal absorption or the suggestion that Sr absorption is entirely passive. Furthermore, this saturable process is comparable to the first step in transepithelial mineral transport, as illustrated in the refined model with an intermediary intracellular compartment between the intestinal lumen and extracellular fluids. This may be the rate-limiting apical mineral uptake by enterocytes that is mediated by a Ca channel (19) and/or a channel-like transporter (31) with a half-saturation concentration of 0.4-1 mM (14, 31), which is quite close to that estimated here. Thus, according to our model, the discrimination between Ca and Sr in the GI tract is not mainly due to this process. 2) The GI compartmental substructure includes a bidirectional relation between the gut and plasma. This bidirectionality anticipates the intestinal secretion of endogenous Ca or Sr to be reabsorbed. This reabsorption is required to fit the Sr experimental data and is important, because, together with the absorption process per se, it determines the net intestinal balance. The secretion of endogenous mineral is predicted to be nonsaturable, at least for the experimental Sr concentrations, but can be regulated, because it depends on one of the z extrinsic functions. It is thus comparable with the transcellular process that may operate mainly in the distal intestine (43). There may be an intestinal carrier with the properties of a mobile carrier that could transport mineral reversibly from both sides of the apical membrane, as experimentally seen using Sr as an analog of Ca and showing that both minerals can be transported and apparently countertransported (47). Such a reversible carrier might be important for the intestinal secretion of endogenous Ca and Sr and the regulation of the net intestinal mineral balance (45). The refined GI model that explores this eventuality shows that a part of mineral entry into enterocytes could be reversible at a mechanistic level. In this context, the extrusion of Ca or Sr across the apical intestinal membrane as a source for the intestinal secretion might account for discrimination between Ca and Sr, if it is more specific for Sr than for Ca or if the concentration of free Sr²⁺ in the enterocyte is higher than the concentration of free Ca²⁺. It is known that two major enterocyte proteins, the cytosolic Ca binding protein and the basolateral plasma ATP-dependent Ca pump, have a much greater affinity for Ca than for Sr (20, 32). So, as proposed by Wasserman (44) and supported by our refined model, the higher efficiency of Ca_i than of intracellular Sr homeostasis in intestinal epithelium could indirectly contribute to the preferential net intestinal absorption of Ca over Sr.

At the initial steady state, our Sr model can be applied to Ca metabolism without any discrimination in favor of Ca over Sr in the bone. This agrees with most of the in vivo studies using Sr and/or Ca radioactive tracers, under steady-state conditions for Ca and Sr metabolism, that show little skeletal discrimination (10). Nevertheless, this behavior seems to contradict in vitro physicochemical evidence indicating that the substitution/incorporation of Sr into the apatite lattice depends on the growth rate, perfection (24), and age (30) of the crystal or that Sr inhibits HA growth and dissolution (9). The behavior predicted from the refined model that attempts to represent the Langmuir-type nonlinearity as a process of mineral adsorption at the bone surface plus noncompetitive inhibition is significant. First, it illustrates the physicochemical mechanisms involved in the first steps of mineral incorporation into bone solid phase and thus helps clarify the dynamic properties of the bone liquid-solid interface. Second, it reveals the preferential movement of Ca from the bone surface toward the bulk bone solid phase. According to our study, the residence time in the bone surface is ~28 days for Sr and 10 times less for Ca. Such a discrepancy between Ca and Sr at the bone surface is predominantly revealed in the non-steady-state situation (kinetic discrimination). In physicochemical terms, the schema is based on the adsorption of Ca²⁺ (and Sr²⁺) at the bone surface followed by the formation of unstable primary solid entities [similar to the surface nuclei as building units for crystal growth (35)], which may be the source of ions for extracellular fluids or used to increase the mineral bulk bone solid phase. Third, this refined model suggests that there are few free adsorption sites under physiological conditions, most of them being occupied by Ca²⁺, in accordance with the report of Groer and Marshall (16). Sr probably enters the bone solid phase as a foreign component of apatite via adsorption and subsequent solid solution formation (13) and may, in this way, participate in the overall process of bone solid phase formation. Fourth, the model predicts that Sr²⁺ not only competes with Ca²⁺ at the adsorption sites but also inhibits the first step of mineralization by stabilizing the binding of mineral species containing Sr to adsorption sites, so forming a small mineral pool rich in Sr at the bone surface. This effect appears to be specific to Sr, in that Ca does not behave in the same way. There has been little discussion of the physiological significance of a small bone mineral pool rich in Sr at the liquid-solid bone interface, except for the suggestion by Lengemann (22) that the "selection against Sr in bone appears to occur at some point after the initial entry of Sr²⁺ into the bone and is associated with Sr²⁺ incorporation into a less labile form of bone mineral." Finally, our model predicts that, at least for some particular Sr extracellular concentration range, this inhibition of bone solid phase formation by Sr is favorably counterbalanced in vivo by the action of an extrinsic z function that probably activates osteoblastic lineage cells and bone formation (see just below).

The mechanistic interpretation of the intrinsic M-M and Langmuir-type nonlinearities discussed above must be considered in the more general context of the overall nonlinear behaviors of our model (Fig. 1). Indeed, the more interesting result of our modeling procedure is the characterization of time-explicitly formulated z variables operating at GI and bone levels and identified in the absence of any direct observation on the physical entities with which they could be compared. These distinctive z functions are mainly related to the dynamic information in our data, a single set of model parameter values being identified to fit the experimental kinetics due to four oral doses of Sr, considered simultaneously. As discussed below, these z nonlinear functions can be compared with controlling variables that are sensitive to extracellular Sr and/or Ca concentrations and act in feedback on GI and bone mineral metabolism.

First, we need to know what these z extrinsic variables may represent. With their own nonlinear characteristics (see APPENDIX A), they are involved in the expression of an inherently saturable process with a typical sigmoidal response that is well adapted to cellular functions as diverse as hormone secretion and cell proliferation and differentiation. They probably implicate the activities of particular cellular system(s) influenced by changes in Sr and/or Ca concentration. The effects may be local or systemic and act on cells involved in mineral metabolism, such as osteoblastic lineage cells for bone and enterocytes for the GI tract.

Second, we need to know what makes these functions sensitive to Sr and Ca. The peculiar dependence of the z variables on Sr, and probably on Ca, strongly suggests the involvement of Ca²⁺ (polyvalent cation)-sensing receptor(s) (CaR), with their known high degree of cooperativity for Ca and/or other cations (6, 37). These CaR proteins respond to polyvalent cations other than Ca, and the concentration producing a half-maximal effect may be higher or lower than that of Ca (mM). In some respects, under cover of the main hypothesis that Sr and Ca interact in an additive manner, our results indicate that Sr is 25 times more potent than Ca on the suggestive CaR. This finding agrees with experimental studies indicating that some mineral trace elements, e.g., Pb²⁺, Cd²⁺, and Co²⁺, additively activate CaR and are more potent CaR agonists than Ca²⁺ (18, 37). However, this sensitivity is not sufficient for the physiological concentration of Sr to significantly modify the predicted response to Ca. Our findings also suggest that the CaR is functionally similar to, but distinct from, the PCaR, because Sr²⁺ is a less potent agonist than Ca²⁺ for this receptor (25). This is consistent with the finding that oral Sr has no effect on hormone secretion or renal function, which depends on PCaR activation (6).

Third, we also need to know in what context the regulatory role of these functions is operating. We believe that the extrinsic nonlinearities represent the expression of peculiar physiological Ca self-regulatory processes revealed via Sr²⁺ interactions with CaR. Kinetically, the variables assumed to be effective in controlling GI and bone metabolic paths have slow turnover rates, with a mean residence time of 10 days, and then may be operating in adaptive, rather than homeostatic, processes for Ca metabolism. Hence, direct (non-hormone-dependent) self-regulatory CaR-dependent processes could occur in the GI or bone cellular system, as proposed by Brown and Pollack (6). CaR have been identified in intestinal epithelial cells (8, 15) and in osteoblastic lineage cells (34, 48). These osteoblastic cells respond to high concentrations of extracellular Ca²⁺ (34) and Sr²⁺ (7) and possess an Al³⁺-sensitive CaR different from PCaR (33).

Finally, the physiological interpretation of our model could help us understand the pharmacological effects of oral Sr and the potential of Sr in the treatment of osteoporosis (28). There seems to be a plasma Sr concentration (40-100 µM) at which the beneficial action of Sr on bone-forming cells surpasses its inhibitory physicochemical action. The predicted long-term effect of the maintenance of plasma Sr concentration at ~60 µM on bone mineral mass (Fig. 9) could be significant in managing the therapeutic effects of an oral Sr dose. A normal Ca net balance in the GI tract can be obtained with a rather high, but physiological, dietary Ca intake in conjunction with this optimal plasma Sr concentration. Of course, these estimations must be used with care, because they are asymptotic predictions that are justified only if the parameter values do not vary over the longer-term Sr administration.