## Abstract

A model of Sr metabolism was developed by using plasma and urinary Sr kinetic data obtained in groups of postmenopausal women who received four different oral doses of Sr and collected during the Sr administration period (25 days) and for 28 days after cessation of treatment. A nonlinear compartmental formalism that is appropriate for study of non-steady-state kinetics and allows dissociation of variables pertaining to Sr metabolism (*system 1*) from those indirectly operating on it (*system 2*) was used. At each stage of model development, the dose-dependent model response was fitted to the four sets of data considered simultaneously (1 set per dose). A seven-compartment model with internal Sr distribution and intestinal, urinary, and bone metabolic pathways was selected. It includes two kinds of nonlinearities: those accounting for saturable intestinal and bone processes, which behave as intrinsic nonlinearities because they are directly dependent on Sr, and extrinsic nonlinearities (dependent on *system 2*), which suggest the cooperative involvement of plasma Sr changes in modulating some intestinal and bone mineral metabolic pathways. With the set of identified parameter values, the initial steady-state model predictions are relevant to known physiology, and some peculiarities of model behavior for long-term Sr administration were simulated.

- mathematical model
- bone mineral metabolism
- mineral intestinal absorption
- postmenopausal women

despite the numerous nonlinearities involved in metabolic processes per se and in their regulation (e.g., hormonal), most compartmental and/or noncompartmental models of in vivo mineral metabolism are linear. On the basis of tracer kinetic data and with the assumption that the system is maintained in a constant steady state, at least over the experimental duration, these models have been widely used to obtain quantitative estimates of processes such as intestinal absorption and urinary excretion. These models have also been used to estimate internal distribution pools (IDP) and other transfer pathways of many minerals; they may be macronutrients, such as Ca (1) and Mg (2), or trace elements, such as Zn (11) and Se (28). In some respects, compartmental and noncompartmental models of in vivo radioactive Sr kinetics (9,20, 30) differ from other investigations, in that the trace element Sr has been mainly used as a “tracer” for Ca in comparative metabolic studies.

Over the past few years, interest has increased in the mechanisms underlying the regulation of trace element metabolism in healthy subjects or animals given high, but nontoxic, doses of the mineral (18, 39, 44, 45) or elements that interact with it (23, 31, 35). Linear compartmental analysis has been used to identify the sites of regulation by comparing the parameters of a model built from initial steady-state tracer kinetic data with those obtained under other steady state(s) after changes in the mineral intake, i.e., long after the suspected regulatory mechanisms have been in full nonlinear operation.

Mathematical modeling may help describe the physiological characteristics of regulatory and/or adaptive processes of mineral metabolism, provided the study is carried out in a non-steady state.^{1} Under these conditions, the nonlinear expressions result in variations that are observed under physiological conditions (37) or experimentally induced by displacing the metabolism or its related controlling system from the initial (steady) state. In any case, the compartmental formalism, with or without tracer data (8), must explicitly incorporate the nonlinearities postulated to be appropriate. Typical recent applications of such nonlinear models include the study of endocrine-metabolic systems in non-steady state (5, 24).

It is difficult to obtain information about the nonlinear processes that help regulate Ca metabolism in vivo under physiological conditions because of the powerful self-regulatory system governing extracellular Ca homeostasis (27, 38). The situation is quite different for Sr, a mineral trace element very similar to Ca in its physicochemical properties. A fairly high oral intake of Sr is well tolerated and results in a large increase in extracellular Sr concentration, without any apparent hormonal response.

The purpose of our study is to use compartmental modeling to analyze the kinetics of Sr metabolism that is greatly shifted from its physiological state. We used experimental plasma and urinary data from four groups of ∼10 postmenopausal women given four oral doses of Sr (S-12911, molecule containing 2 atoms of nonradioactive Sr; Institut de Recherches Internationales Servier). The kinetic data cover the increase in plasma Sr concentration over the 25-day period for which Sr was given [administration period (AdP)] and its decrease over the 28 days after Sr administration [postadministration period (PAdP)]. This report describes the development of a nonlinear seven-compartment model of human Sr metabolism and the underlying assumptions. The building strategy led to the introduction of two kinds of nonlinearity. The first accounts for the saturable processes [Michaelis-Menten (M-M) and Langmuir-type equations] that behave as time-implicit nonlinear functions and are intrinsic to Sr metabolism, because they form part of the system of differential equations for Sr metabolism. The second is extrinsic to Sr metabolism and requires a specific time-explicit differential formulation that defines the kinetic behavior of biological component(s) other than Sr. Surprisingly, these last model properties may reflect self-regulatory processes specifically operating on Sr metabolism. Because there has been no evidence that Sr has any physiological role, an attractive alternative is that our modeling procedure has revealed some difficult-to-measure regulatory processes linked to Ca metabolism because of the analogies between Sr and Ca. At the stage of the present study, the building procedure is reported with the primary goal to show how pertinent information relative to nonlinear expressions can be extracted from the data. Therefore, only a minimal interpretation of the model with regard to Sr metabolism per se is made.

## PROTOCOL AND DATA

Total Sr plasma and urine concentrations were measured by inductively coupled plasma emission spectrophotometry. All data were obtained during a study of the acceptability of repeated oral doses of S-12911. Four groups, each composed of 9–10 healthy postmenopausal women, were followed for 54 days. All subjects gave informed consent. The women were 59.5 ± 4.9 yr old and of normal height (158 ± 5 cm) and weight (62 ± 10 kg). They consumed a normal diet (actual daily dietary Ca and Sr intake data are not available). They were given 0.5, 1, 2, or 4 g of S-12911 for 25 days (AdP) in two daily doses: the first was given 1 h after breakfast and the other 1 h before dinner, i.e., with intervals of 10 and 14 h between doses, respectively. Plasma samples were collected just before administration (*time 0*) and on *days 1, 2, 7, 14, 21, 25*, and *26–54*, with detailed kinetics after the morning daily administration on *days 1, 14*, and*25* and with sampling frequency of 1–3 days during the 28 days after withdrawal of repeated administration (PAdP; Fig.1). The number of measurement times over the experimental duration was 38 (26 and 12 during AdP and PAdP, respectively) per individual, with a total of ∼350 data points, for each dose.

Twelve-hour urine collections were obtained after the end of treatment, on *days 25, 28–32, 35, 46*, and *53,* and their volumes were estimated. Other data on total and ionized plasma Ca and urinary Ca and creatinine were obtained on *days 1, 15, 27*, and *54*.

## FORMALISM AND NUMERICAL METHODS

### Mathematical Formalism

The model was built to describe not only Sr metabolism, but also other biological variables that may be sensitive to changes in Sr concentration and act on metabolic processes associated with Sr. We used a modeling approach based on the nonlinear compartmental formalism (7). Two different, but interdependent, systems of ordinary differential equations were considered simultaneously
Equation 1
Equation 2
where *system 1* is related to Sr metabolism, with*y _{i}
* denoting the amount (i.e., concentration) of Sr in

*compartment i*, and

*system 2*refers to biological variable(s) (

*z*) other than Sr.

*F*(

*t*) and

*G*are the flow rates of

*y*and

*z*to

*compartment i*from a source apart from the material included in a particular differential system.

*Systems 1*and

*2*are nonlinear, because any G,

*K,*and

*H*are a priori dependent on some

*y*and/or

_{i}*z*and

_{i}. K*H*are then called the fractional transfer functions (FTF) ascribed to

*y*and

*z,*respectively.

*System 1*is intrinsically nonlinear when at least one FTF

*K*depends on

*y*. Similarly,

*system 2*becomes intrinsically nonlinear as soon as FTF

*H*or input

*G*depends on

*z*. Because the kinetics of the two systems are not independent,

*systems 1*and

*2*are extrinsically nonlinear: At least one

*K*is a function of

*z*, and, reciprocally, one

*H*and/or

*G*is dependent on

*y*. Thus the kinetics of Sr (

*y*,

*system 1*) influence the kinetics of

*z*(

*system 2*), which, in turn, affect

*system 1*.

The *y _{i}
* associated with plasma is

*y*

_{1}. The initial conditions,

*y*(0) =

_{i}*y*(

_{i}*t*= 0) and

*z*(0) =

_{i}*z*(

_{i}*t*= 0), are calculated with the assumption that

*systems 1*and

*2*are in steady state (d

*y*d

_{i}/*t*= 0 and d

*z*d

_{i}/*t*= 0) at

*time 0*with

*F*(

*t*) =

*F*(

*t*= 0), i.e., before oral Sr has any effect, displacing Sr metabolism from its initial physiological state.

#### Remark 1.

All the experimental plasma data are concentrations. Hence, except for*y*
_{1} (*compartment 1*), any*y _{i}
* and parameters having a concentration dimension must be used with care. Biological interpretation of

*system 1*requires an estimate of the apparent distribution volume (V

_{app}, assumed to be constant over the experimental period). This enables concentration (

*y*

_{1}) and apparent concentration (C

_{app}for other

*y*) to be transformed to a mass dimension. This conversion was done by using urinary excretion data with their 12-h integral values calculated from the Sr concentration and volume (see

_{i}*Eq. 3*). It is also possible that V

_{app}does not apply to

*system 2*. Under these conditions and in the absence of direct experimental data for

*system 2*, variations in

*z*have no meaning as absolute values; only

*z*kinetic behavior is of interest.

#### Remark 2.

When *K*, *H*, or input *G* is dependent on*y* and/or *z* variables, their dependence on*system 1* and/or *system 2* must be specified; e.g.,*K _{ij}
* =

*f*(

*k*

*,y,z*) with

*l*= 1, 2, 3, … and

*k*having a constant value ≥0. Otherwise, they are strictly linear, have constant values ≥0 (e.g.,

*K*=

_{ij}*k*), and are referred to as the input rate (

_{ij}*g*) and the fractional transfer coefficient (FTC) called

*k*or

*h*. For any FTF in

*system 1*, the term intrinsic nonlinearity (INL) or extrinsic nonlinearity (ENL) has been used to specify its dependence on its own (intrinsic) variables (

*y*) or on the (extrinsic) variables associated with

*system 2*(

*z*).

#### Remark 3.

The general form (*systems 1* and *2*) defined above was considered only if the kinetics of some *z* variables do not satisfy the assumption of a pseudo-steady-state assumption: indeed, if the kinetics of a given *z* are very rapid, at each time, d*z/*d*t *≈ 0, and *z* approximates its asymptotic value, *
* [e.g., as assumed for formulation of the M-M equation (32)]. Then the dependence of*system 1* on *z* [through some *K = f*(*z*)] can be directly formulated as a time-implicit nonlinear function of *y*, i.e., equivalent to an INL, as opposed to its dependence on the time-explicit differential form of *z*, with its corresponding ENL. Otherwise, when the kinetics of *y* and *z* are not very different, d*z/*d*t ≠* 0 must be considered to take into account the effects of *z* changes on *y*kinetics, and the general form of both differential systems must be used.

#### Remark 4.

For a given set of parameter values together with the value of the initial Sr plasma concentration, *y*
_{1}(*t*= 0) ≠ 0, an estimate of the other initial conditions*y _{i}
*(0) and the initial values of input functions

*F*

_{i}_{0}(

*t*= 0) are required. If Sr metabolism is approximated to be physiologically in a constant steady state, an analytic calculation is possible for some simple situations (a single positive and real solution exists for d

*y*d

_{i}/*t*= d

*z*d

_{i}/*t*= 0). When multiple input functions operate within a given model structure, the relations specifying the interdependence between the inputs and the irreversible outputs must be defined a priori.

#### Remark 5.

The compartmental formalism presented here is similar to that successfully used to model endocrine-metabolic systems such as glucose metabolism and its hormonal control (4). It also has certain similarities to the conceptual framework developed for modeling non-steady-state pharmacokinetic/pharmacodynamic data (33).

### Numerical Methods

The nonlinear differential systems in a non-steady state generally have no analytic solution, even when they are specified with defined nonlinearities and a given set of parameter values. Thus *systems 1* and *2* must be simultaneously solved using numerical integration (17).

The fit of the model response to the experimental data was carried out by nonlinear parameter estimation with *y*
_{1}(*compartment 1*) compared with plasma Sr kinetics. The method of Levenberg-Marquardt (16, 22) was used to minimize the weighted residual sum of squares (WRSS) as criterion function (8). WRSS was computed using the mean plasma Sr concentration and the reciprocal of the measured error variance at each sampling time and for all the doses studied, except in the preliminary study of *stage 1* of model building, during which each dose of Sr was considered separately. Because the optimization method is described as a local one, the initial set of parameter values was chosen manually for its ability to describe the overall behavior fairly satisfactorily. The minimization procedure was repeated using a randomly noised initial set of parameter values to avoid obtaining a local and/or false minimum WRSS. This was done until the values of WRSS and the identified parameter values did not differ between several successive iterations.

To examine the goodness of fit to experimental data, we used, in addition to visual inspection, the run test to check the independence of weighted residuals. We also detected misfitting by observing the values of the partial WRSS computed for each set of data, because the model predictions are usually related to four sets of data (1 per Sr dose). As an indication of validity for increasing complexity in the model structure, the significance of the improvement of fit was checked with an *F* test.

The accuracy of the identified parameter values (practical or a posteriori identifiability) was estimated from the computed variance-covariance matrix, and precision was expressed as coefficient of variation (CV) in terms of percent fractional standard deviation (7). Theoretical (a priori) identifiability was not analyzed because of the high degree of complexity reached by the nonlinear model structures. Thus the unique estimate of the set of parameter values cannot be completely ensured, despite use of various initial sets of parameter values for optimization. Moreover, when the tested model structure was nonidentifiable, some parameter values diverged during optimization and/or there was a very large (much more than 100%) computed CV. Here, a model is ruled out only if one or several of its parameter values has a CV ≥ 100%. A correct accuracy is assumed if the calculated CV is such that the parameter differs significantly from zero for *P* < 0.05.

## PHYSIOLOGICAL BACKGROUND

It is generally assumed that Ca and Sr share the same main metabolic pathways, involving the gastrointestinal (GI) tract, kidneys, and bone, that are organized around an IDP (Fig.2). However, this scheme is an oversimplification. For instance, the central IDP is, a priori, a multicompartmental substructure made up of the extracellular fluids (plasma included), soft tissues, intracellular sectors, and the so-called exchangeable mineral at the bone surface. It will also be useful to take into account a measure of complexity in the gut during model building and interpretation, because intestinal mineral absorption and secretion vary along the GI tract (6). The same is true for bone, because bone mineral metabolism involves, on the one hand, bone accretion and resorption, mainly linked to bone remodeling in adults via local osteoblast-osteoclast activities (26), and, on the other hand, the physicochemical processes by which the bone mineral solid phase is formed (nucleation and crystal growth) from solute ions, matures (changes in size, shape, and chemical composition), and dissolves (12). These processes also depend on the dynamic equilibrium-nonequilibrium (25) at the bone surface that governs its relations with ions in the adjacent extracellular fluid. All these processes can give rise to a complexity not shown in Fig. 2. On the contrary, this scheme may be simplified. For example, renal glomerular filtration and tubular reabsorption are so rapid that only the net urinary mineral excretion may be obtained from our kinetic data.

Finally, the plasma Sr concentration in the normal adult is 0.5–1 μM, with >95% of the total body Sr (∼400 mg) in the bone, as for Ca, and a normal diet provides a daily Sr intake of ∼2 mg (10).

## MODEL BUILDING

Visual inspection of the mean plasma Sr values at each sampling time indicates a dose-dependent rise in plasma Sr concentration during AdP with, as expected, a slowing of the rate with treatment duration. It is followed by a regular decrease during PAdP (Fig. 1). However, there are indications for nonlinear dose dependence(s) of the kinetic behavior during each of these periods. The concentrations reached after each dose during the last part of AdP (mean value computed with data from *day 14*) show a departure from strict linear dose dependence, and the curve tends to flatten for the highest doses. During PAdP, the influence of the dose on the fall in plasma Sr may be estimated from the concentrations normalized to the mean value during the last part of AdP. The decrease in plasma Sr concentration is significantly slower for the smallest dose than for the other three doses (see Fig. 4
*B*). These changes indicate that nonlinear dose-dependent processes are involved in plasma Sr kinetics. One of our objectives is to develop a model capable of justifying and describing these nonlinearities.

Our model of in vivo Sr metabolism was therefore built in three stages. The two preliminary stages used partial kinetic data to obtain the information required to construct the final model(s) on the basis of all available data. The general idea underlying *stages 1* and*2* was to identify fairly qualitative (structural) features, such as the number of compartments, the nature of the involved nonlinearity (or nonlinearities), and its (their) optimal location(s) within the model that helped fit these partial kinetic data, including the dose dependence. We started with a minimal model structure and increased its complexity, inasmuch as each model was required to have a sound physiological meaning.

### Stage 1: IDP Compartmental Substructure

*Stage 1* considered only the PAdP plasma and urinary data collected from 612 h (25.5 days) to 1,272 h (53 days; Fig.1
*A*). It was thus possible to formulate some attractive, but not necessarily definitive, simplifications and approximations (seeappendix
) and use them as the conceptual basis for this first modeling stage.

Each dose was first considered separately, and the ability of linear or nonlinear compartmental models to fit each of the four sets of PAdP plasma data was tested. At least three compartments were required for strictly linear models, except for one dose, for which only two compartments were needed. Not only the Sr input function [*f*
(*t*) in *Eq.EA4
*], but also all the other identified FTC, varied with the dose under these conditions. When one extrinsic *z* nonlinearity was introduced within the model, we chose to use *system 2* in its generic form (see *Assumptions 1* and *2* inappendix
). With inclusion of the time-explicit (*Eq.EA1, a* and *b*) or time-implicit (*Eq. EA2
*) version of *z*, in no case, did the goodness of fit obtained with this kind of nonlinear model significantly improve compared with that obtained by the best linear model. The parameter values defining*system 2* and the corresponding *system 1* FTF also showed large uncertainties (CV > 100%). However, the optimal structure for the nonlinear models was a two-compartment model at all the doses examined. These results seem to indicate that an extrinsic*z* nonlinearity is not detrimental to reproducing, separately, the rates of Sr loss during PAdP, although such nonlinearity is not entirely justified under this condition.

When we used another strategy, we obtained very different results. We examined the capacity of a given model structure plus a single set of parameter values [except for the input function*f*
(*t*), which obviously differed with the dose] to fit the set of the four dose-related PAdP plasma data. Including one nonlinearity within the model significantly improved the goodness of fit over that obtained using a strictly linear model under these conditions, whatever the form (time implicit or time explicit) used for the generic *z*. The location of the nonlinearity within the structure was also important, because the criterion differs according to the kind of FTF that*z*
_{(n+1)} modulates. For instance, the fit was less good when the irreversible (*K*
_{01}) exit depends on*z*
_{(n+1)}, rather than the reversible (*K*
_{21}) exit. Interestingly, we obtained the best fit for both peculiar model structures and when the time-explicit, rather than the time-implicit, form of*z*
_{(n+1)} was employed.

The main properties and fitting characteristics of these structurally different models (number of compartments and presence, nature, and form of *z* nonlinearity) are shown in Table1. These models had the same physiologically relevant feature (basic 3-compartment model; Fig.3). These include two irreversible exits: one from *compartment 1* and the other from *compartment 2*, which may reflect urinary excretion and internal uptake (probably by bone). The optimal criterion value was obtained with the nonlinear three-compartment structure using *system 2* under the logistic form (*Eq. EA1, c* and *d*) or the generic form (*Eq. EA1, a* and *b*), provided it was implemented in its time-explicit form. The goodness of fit differed significantly from the other linear or nonlinear versions of the two- or three-compartment model. In other respects, the time-explicit form of *z*
_{(n+1)} gives an index of cooperativity (*p*; see *Assumption 2* inappendix
) that is much greater than 1, i.e., a positive cooperativity, whereas a negative cooperativity (*p* < 1) is associated with the time-implicit form.

As expected, when linear models with the three-compartment structure (Fig. 3) were used, *k*
_{01} and*k*
_{02} values were very inaccurate (CV largely >100%). This is due to a structural indetermination: only the sum of these irreversible exits could be identified. This indetermination seemed to be removed by using a nonlinear FTF. For instance, the set of parameter values defining *system 1* was accurate enough (significantly different from 0) when *K*
_{21} was modulated by the time-explicit form of the logistic*z*
_{(n+1)}. However, this was not so for the accuracy of parameters estimated for *system 2*.

Different criterion values were produced (Fig.4
*A*) when optimization was carried out from the basic structure (Fig. 3), with*K*
_{21} as a logistic time-explicit ENL and various, but fixed, *p* values. However, these did not differ significantly from the optimal one for sufficiently high* p*values (*p *≥ 2). A similar minimal criterion was maintained, despite different *p* values because of compensatory changes in the values of other parameters defining *system 1* and/or*system 2*. The entire set of parameter values (*systems 1* and *2*) was well defined for fixed values of*p* ≥ 3. Otherwise, some parameters varied greatly with CV > 100% for *system 2*. Figure 4
*B*illustrates the goodness of fit to plasma Sr data obtained with this model for *p* = 3. For each of the four doses, the data and the model response (*compartment 1*) were normalized to the theoretical *y*
_{1} values at 612 h to better appreciate the dose dependence of the kinetic behaviors. The statistical tests used to estimate the goodness of fit were satisfactory.

Thus a nonlinear three-compartment model, including an ENL showing positive cooperativity in its dependence on *compartment 1*, is able to account for the observed PAdP plasma Sr kinetics, including dose dependence, from a single set of parameter values. Obviously, it is too early for a physiological interpretation of the model. Figure5
*A* illustrates the predicted early kinetic behavior of *z*
_{(n+1)}(*z*
_{4} for the optimal model using the logistic nonlinearity). The four doses studied resulted in very different behaviors. For the three higher doses, the expected*z*
_{4} maximum value (1.0) is reached at different times after initiation of oral Sr administration, but all were reached long before the beginning of PAdP. In contrast, the lowest dose resulted in a very late increase, with *z*
_{4}< 1 at the end of the AdP. Among the set of parameters, only*f*
(*t*), i.e., the part of the entry relative to Sr administration, differs according to the dose: the input functions predicted from the nonlinear model are not strictly proportional to the dose, and the dose dependence presents a smooth, regular, S-shaped curve, at variance with the curve predicted from a linear three-compartment model (Fig. 5
*B*). This result indicates that other nonlinear processes, probably related to the GI system, should be considered at a further stage of model building. This is all the more likely, given that, when searching to identify an optimal nonlinear model with*f*
(*t*) compelled to a strict linear dose dependence, the minimal criterion reached is significantly higher than the optimal value obtained in the absence of this constraint.

Finally, at this stage, it is possible, using the urinary data, to transform C_{app} to mass of Sr. With a model structure (Fig.3) that *1*) dissociates the irreversible exit from*compartment 1* (*k*
_{01}), including at least urinary excretion from another (internal) exit (*k*
_{02}), and *2*) optimally fits the PAdP plasma kinetic data, it is possible to estimate the relation of V_{app} to *k*
_{01}. Because no significant variation in urinary creatinine (experimental data not shown) appears during the PAdP, Sr urinary clearance (0.286 l/h or 4.77 ml/min) can be estimated from linear regression between each experimental 12-h urinary Sr PAdP value and the predicted time-corresponding integral of *y*
_{1}. The following relation is then retained
Equation 3which gives rise to V_{app} > 60 liters with the above model.

### Stage 2: GI Compartmental Substructure

The earliest data for the time immediately following the start of oral Sr (from *time 0* to 10 h, FAdP) are particularly useful for identifying the structural and nonlinear properties of GI metabolic pathways. Of course, this may be detrimental to the recognition of other aspects of Sr metabolism. Our objective was to find, using the new assumptions in appendix
, the minimal nonlinear compartmental structure that reproduces the four dose-FAdP data when they are analyzed simultaneously using a single set of parameter values, except the input function*F _{i}
*

_{0}(

*t*) into the GI compartment, which is obviously dose dependent.

The problem of the GI structure is not trivial, because structural indetermination becomes evident once the system becomes even moderately complex, when linear processes alone were assumed. Fortunately, the goodness of fit obtained with a model that includes an irreversible exit from *compartment 1*,* k*
, in addition to the urinary excretion, *k*
(see*Assumption 8* in appendix
) and a one- or two-compartment linear structure for the IDP, is significantly improved when the FTF from the GI compartment to *compartment 1*becomes an INL via an M-M equation (*Eq. EB3
*) compared with the corresponding strictly linear model (Table2). The increased number of GI compartments also improves the fit, in contrast to the IDP, for which one compartment is sufficient. Furthermore, the fit was not significantly improved (2.50 vs. 2.76) when this three-compartment minimal model (Fig. 6
*A*) was identified without the*f*
(*t*) dose-dependence constraint (*Assumption 4* in appendix
) and the estimated values of*f*
(*t*) remained linearly correlated with the Sr intake.

The parameter values of the strictly linear model are very imprecise, whereas the presence of a saturable transfer function alone (*Eq.EB3,* with *k*
= 0) results in satisfactory accuracy (the parameters significantly differ from 0) for an optimal nonlinear structure with one IDP compartment (*compartment 1*) and two GI compartments (*compartments 4* and *5*). *Compartment 4* does not exchange directly with *compartment 1* (Fig. 6
*A*). Accepting the input *F*
_{40}(*t*), i.e., physiological and supplementary Sr entry, it serves as a source for the rest of the system, especially for the other GI compartment (*compartment 5*), from which Sr is nonlinearly absorbed into *compartment 1* or excreted into the environment (feces).

The irreversible transfer from the first to the second GI compartment*(k*
_{54}
*,* in Fig. 6
*A*) introduces a lag time [mean residence time (τ = 1/*k _{ij}
*) in

*compartment 4*of ∼1 h] between the moment of oral Sr intake and its absorption, which is linked to the observed time of the plasma concentration peak (∼4 h). An illustration of the fit with the FAdP experimental data is given in Fig. 6

*B*. The relation of the GI compartment to

*compartment 1*is unidirectional

*(k*

_{51}= 0) and restricted to the saturable part of

*Eq. EB3*(

*k*= 0). Indeed, the consideration of an intestinal absorption that is partly linear does not significantly improve the criterion value and makes the value of this parameter very inaccurate. This, together with the poor characterization of the IDP substructure and the lack of significance of the additional irreversible exit,

*k*, is probably due to the short time (10 h) over which plasma kinetics were examined (FAdP). Therefore, we have not confined the use of these findings to the minimal structure (Fig. 6

*A*), but we believe that a more complex relation between

*compartment 1*and the GI tract could be identified from the overall kinetics data (

*stage 3*).

### Stage 3: Overall Compartmental Structure

*Stages 1* and *2* provided useful information about model structure and the nature of the nonlinearities required to adequately fit the PAdP or FAdP data. *Stage 3* used all the available experimental data (AdP + PAdP). The partial criterion values corresponding to PAdP and FAdP were used to assess the overall goodness of fit. Indeed, it may be expected that when a model fits optimally to all plasma kinetic data, each partial criterion should not differ from its corresponding optimal value previously obtained (Tables1 and 2). The aim of *stage 3* was to achieve a model that*1*) includes the structure reflecting the GI system (*stage 2*), *2*) shows the internal distribution of Sr, including bone metabolism, and* 3*) takes into account nonlinearities of any origin, intrinsic (INL) or extrinsic (ENL). The general procedure used so far, based on the capacity of models to fit the experimental data, was used to examine the requirement for the two very different nonlinearities studied in *stages 1* and*2* and find an efficient combination of these nonlinearities. Because this may make the structure more complex, we used any relation that avoids obvious structural indetermination or substantially diminishes the number of parameter values to be estimated. Most of the assumptions made in *stage 2* were kept in *stage 3*, except for the new hypotheses described in appendix
.

Figure 7
*A* illustrates the minimal structure used to start this model building. It combines the two substructures from the preceding stages and includes five compartments for *system 1*: two (*compartments 4*and *5*) describe the GI tract, and the others (*compartments 1–3*) refer to IDP. We used only two nonlinear FTF:* K*
_{15}, associated with an INL involving saturable and nonsaturable processes according to *Eq.EB3
*, and *K _{21}
*, associated with a logistic ENL (

*Eq. EA1, c*and

*d*) illustrated by the (

*n*+1)th compartment.

The structural complexity was increased to obtain an adequate fit of the model response to all the data when the four Sr intakes were analyzed simultaneously (Table 3). Only one extra compartment was added to *system 1* (or 2 extra compartments, if *Eq. EC3
* was applied with the*pseudocompartment 6** changed to *compartment 6*). This additional compartment was needed only when the overall criterion (23.81) approached the best value obtained (17.98). It is an IDP compartment (*compartment 7*) in a reversible linear relation to *compartment 1*, having a higher turnover than other IDP processes. A second logistic ENL [*z*
_{(n+2)}] was included, with parameters different from those of the initial one,*z*
_{(n+1)}, because a sigmoidal influence on the relation of the GI compartment to *compartment 1* appeared to be necessary. Finally, a simplified version of the Langmuir-type INL was introduced (*Eq. EC2b
*). Only the inhibition term of the complete *Eq. EC2
* was used, not the saturable process. The Langmuir-type INL operates at the level of *K*
_{21}, on an FTF that is also modulated by*z*
_{(n+1)}.

The final structural arrangement that includes these modifications is shown in Fig. 7
*B*. It was developed using iterative optimization, testing various changes required to give a satisfactory behavior for a reasonable degree of complexity. The framework in which this was done is summarized in Table 3. Each of the nonlinearities (the M-M INL or the logistic ENL) can greatly improve the global criterion (*models 2* and *3* in Table 3) over a strictly linear structure (*model 1*). However, this does not ensure a satisfactory fit to the PAdP data. The logistic ENL has parameter values very different from those identified in *stage 1* in this situation: *p* is close to 1, and*g*
and*h*
_{0(n+1)} (*Eq. EA1, c* and*d*) have much higher values than previously observed. The fit tended to be improved when both nonlinearities were simultaneously included in the model. However, elementary combination of the two kinds of nonlinearities (*model 4*) or simplistic interaction between them (*models 5* and *6*) did not give consistent results. Indeed, the sets of identified parameter values were erratic, with very large and unrealistic estimates of V_{app} (V_{app} > 200 liters), even though the saturable part of* K*
_{15} was, according to*Eq. EC1
*, converted to a logistic ENL through an extrinsic*z* variable that was identical to [*z*
_{(n+1)} in *model 5*] or different from [*z*
_{(n+2)} in*model 6*] that modulating *K*
_{21}. The parameter values were also very inaccurately detected for *systems 1* and *2* (CV > 100%). The three criterion values decreased substantially when the logistic*z*
_{(n+1)} influenced*K*
_{15} and the reverse process*K*
_{51} [with* K*
_{51} =*k*
_{51}
*z*
_{(n+1)},*Eq. EA3
*], whereas the computed CV of the set of parameter values became <100% for *system 1*, despite the addition of an extra compartment (*model 7*). On the contrary, some parameters in *system 2* remained very inaccurate, with CV > 100%. Therefore, the Langmuir-type INL (*Eq. EC2
*) that regulates the FTF *K*
_{21} was added. This slightly morphed the overall criterion, but the parameter values were very inaccurate. This was resolved by using the simplified normalized version *Eq. EC2b,* with the integral of the irreversible exit from *compartment 2* toward the*pseudocompartment 6** as inhibition variable (*model 8*). The Langmuir-type INL was further kept, because a posteriori examination showed that the *system 1* parameter values were accurate, including the inhibition constant*k*
in *Eq. EC2b
*. Explicit consideration of two logistic* z *had interesting consequences for the model-fitting properties. In fact, the way INL and ENL were combined had a great influence on the accuracy of parameter values, whereas the total number of parameters did not vary (*models 9–11*). The fit was moderately improved, but an optimal criterion associated with parameter values significantly different from 0 for *system 1* and with CV < 100% for *system 2* was obtained only when the *K*
_{15} and*K*
_{51} FTF involved in the relation of the GI compartment to *compartment 1* depended on distinct extrinsic variables [*z*
_{(n+1)} and*z*
_{(n+2)} and conversely]. The values and accuracy of parameters defining *system 1* (IDP and GI) and *system 2* of a model in which *K*
_{21}and* K*
_{51} depended on*z*
_{(n+1)}, even though*K*
_{15} was modulated by the other extrinsic variable,* z*
_{(n+2)}, are reported in Table 4. Under such conditions, the*system 2* parameters [mainly,*g*
and*g*
] vary greatly, a problem reminiscent of the indetermination observed during *stage 1* of model building (Fig. 4
*A*) and linked to the fact that the parameters* p*
_{(n+1)} and*p*
_{(n+2)} were not fixed at a given value. Furthermore, comparing *model 12* with *model 11* (Table 3) reveals that an explicit bone compartment (*compartment 6*), instead of a simple irreversible exit from*compartment 2*, does not significantly alter the fit. The calculated supplementary FTC, *k*
_{16}, is very small, and changing the inhibition variable (*compartment 6*for *model 12* and *compartment 3* for *model 13*) used for the Langmuir-type INL (*Eq. EC2b
*) has very little effect on criterion values, provided the inhibitory compartment has a low enough turnover.

The compartmental structure shown in Fig. 7
*B* summarizes the main characteristics of the retained optimal *models 12* and*13* (Table 3). These models give essentially identical theoretical responses using global criterion values. However, the values of partial criteria corresponding to FAdP and PAdP only approach the optimal values obtained at *stages 1* and *2* of model building. The predicted kinetic behavior of *compartment 1* compared with various mean values of plasma Sr throughout the experiment (AdP + PAdP) is shown in Fig. 1
*A* for the four doses studied. Figure 1
*B* shows a comparison of the theoretically predicted and experimental data for two of the doses used and at different times of AdP for the period immediately following the morning oral Sr dose.

Finally, we used another strategy to check the optimality of these structures. We progressively simplified the structure of *model 12* (Table 3) by reducing the number of variables in *systems 1* and *2* or the transfer processes that are “targets” for INL and ENL. Only one of the feasible simplifications (*model 14* in Table 3) had a relatively good value of the global criterion, although it was significantly higher than that obtained with *model 12* or *13*. This model has the same number of variables and parameters as the optimal model. It differs in that* K*
_{51} is not modulated by*z*
_{(n+1)}. Under this condition,*k*
_{51} became zero, making the relation of the GI compartment to *compartment 1* irreversible.*K*
_{15} thus defined the net intestinal absorption of Sr, including a nonsaturable process that is linearly dependent on the GI Sr concentration and a saturable process that is modulated by an extrinsic* z* variable. Also, the FAdP and PAdP criteria are far from their minimal value obtained at *stages 1* and*2,* and the *system 2* parameter values were very inaccurate, with CV > 100% (*model 14* in Table 3).

In summary, the final model (Fig. 7
*B*) that fits all the experimental data well and gives satisfactorily accurate parameter estimates (statistically, at least, for *system 1*) requires several intricate nonlinear processes. The partial nonlinear structures identified during the preliminary *stages 1* and *2*(Fig. 3 and 6
*A*) can be included in a single model, provided the behavior of IDP and the GI compartment are influenced by ENL, i.e., by *z *functions kinetically expressed and showing high cooperativity in their dependence on plasma Sr concentration.

## DISCUSSION

In contrast to earlier kinetic studies, most recent compartmental modeling studies of in vivo mineral metabolism have used multiple tracers (stable or radioactive isotopes) given orally and/or intravenously and different sites of experimental measurements (21, 31, 39). These protocols are designed to develop complete models that represent all the metabolic pathways within some high-order, linear compartmental structures. These must be theoretically and practically identifiable (7) while remaining consistent with known physiology. The present modeling study has used quite different experimental data, because they concern the variations in plasma and urinary concentration of the mineral (Sr) itself after various oral doses of the mineral. Nevertheless, the complexity of the identified structure (Fig. 7
*B*) is comparable with that of models for other minerals [Zn (19) and Mo (39)], even though the theoretical identifiability of the resulting model(s) has not been formally stated. The Sr metabolism described by *system 1* has 7 compartments and 14 FTF. Thus, insofar as the present modeling procedure is confined to the characterization of a system structure plus the estimate of parameters and variables of physiological interest, this study seems satisfactory. This is mainly due to the qualitative and quantitative aspects of kinetic and dynamic information provided by the plasma and urinary data. We should recall that the data describe the rise in plasma Sr concentration during oral ingestion of Sr and its decrease after termination of oral Sr administration; we had a total of 38 samples per individual covering the experimental period. The short-term effect of oral Sr was also monitored, and data were obtained for four different doses of Sr. Finally, a total of 38 individuals (∼10 per Sr dose) were examined. A major feature of our modeling procedure was that we looked for a single model, with just one set of parameter values defining a given compartmental structure, that could describe the entire set of data, including dose dependence. As asserted by Wagner (41), nonlinearities can be recognized in such a situation, but the corollary is that these nonlinearities significantly influence the identified kinetic behavior and the a priori identifiability. The nonlinearities of intrinsic or extrinsic origin become part of the model structure and are thus important for the model kinetic response. Conversely, a compartmental model built using each set of data separately will probably have a structure that is different from the nonlinear one described here. Use of separate data sets disregards one major expression of the nonlinearities, and the linear characteristics of the model should be strengthened with, as a consequence, inherent difficulties in the choice of model structure linked to structural indetermination or a priori unidentifiability. For instance, an unidentifiable linear model may become identifiable once it contains a nonlinearity, as shown by Godfrey (13).

We broke down the overall problem of model building by using two preliminary stages. This allowed us to look for structural and nonlinear characteristics of two subsystems before constructing a single integrated model. These subsystems deal with the multicompartmental substructure that includes the internal distribution of Sr, obtained from the later PAdP data, and the GI substructure, from the earlier FAdP data.

Each of these preliminary stages revealed interesting nonlinear properties of human Sr metabolism in a non-steady state. For instance, accounting for the PAdP data not only requires a peculiar nonlinearity, but we went further into the use of the nonlinear compartmental formalism (*Eqs. 1
* and *
2
*). Indeed, this nonlinearity must be time explicit, which means that the extrinsic variable responsible for this nonlinearity does not follow the classical rapid-equilibrium (pseudo-steady-state) assumption but requires its time-explicit expression via the differential *system 2* (see *Remark 3*). This is particularly important, because incorrect application of the asymptotic (time implicit) formulation of *system 2*, with the consecutive substantially improved fit compared with that obtained with linear models (Table 1), could result in an erroneous characterization of this nonlinearity: the time-implicit form shows a negative cooperativity in its dependence on*compartment 1* Sr concentration, instead of the highly positive one found with the time-explicit version.

The second preliminary stage involved the kinetic analysis of the short-term effects of Sr immediately after the first oral dose. It revealed a direct link between our study and the present use of Sr as a marker of intestinal Ca absorption (42). Vitamin D increases the intestinal absorption of Ca and Sr (3), providing indirect evidence that Sr and Ca use the same transcellular pathway. Our analysis of the data collected during FAdP for all four Sr doses does ascertain that a saturable M-M-like process (*Eq.EB3
*) is involved in the human intestinal absorption of Sr. Our companion article (36) also shows that the parameters defining this process are quantitatively in agreement with Sr directly using some of the transcellular processes involved in the physiological regulation of Ca absorption.

With the overall nonlinear compartmental structure describing all the experimental data (Fig. 7
*B*), it is possible to verify its biological significance, because the parameter values of the optimal models (*model 12* or *13* in Table 3) have satisfactory CV values (CV always <30%; Table 4 for *system 1* in *model 12*). The present study, as the first part of a more detailed study (36), checks the model validity without directly addressing the physiological interpretation of the INL and ENL. This may be done by examining the predictions of the model for human Sr metabolism under physiological conditions (steady state) and by simulating the effects of long-term oral Sr.

### Initial Steady-State Predictions

The initial values of the IDP and GI compartments have no physiological meaning when expressed as C_{app}. We therefore used the estimated V_{app} *(*see *Remark 1*and *Eq. 3
*) to compute the masses of Sr in *compartments 1–7* and the daily transfer rates within the model or with the outside, with the assumption that overall human Sr metabolism is in steady state under physiological conditions. Table5 shows the mass and rate values predicted using parameters identified for *model 12* (there are minor differences between *models 12* and *13*). This model gives the mass of Sr within the IDP compartments (∼14 mg in *compartments 1* + *2* +*3* + *7*; Fig. 7
*B*) as only 3–4% of the total body Sr (∼400 mg), whereas *compartment 6*, equivalent to the bone solid phase in healthy adults, contains >96% of the total mass. The large value of V_{app} (∼50 liters) is more than three times the extracellular fluid volume and suggests that it represents the rapidly exchangeable extracellular and/or intracellular Sr in *compartment 1. Compartments 3* and *7*, in linear bidirectional exchange with*compartment 1*, must be concerned with the relatively slow exchange of intracellular or bone Sr. *Compartment 2* appears to be an intermediary entity between the free Sr in extracellular fluid and Sr within the bone mineral solid phase (*compartment 6*). The irreversible transfer from *compartment 2* to*compartment 6* that is assumed to match that related to the return of Sr from *compartment 6* to *compartment 1*may be compared with Sr movements in bone mineral accretion (apposition + augmentation) and removal (resorption + diminution). The bone Sr turnover is estimated at 0.256 mg/day, slightly lower than the predicted urinary Sr excretion (0.32 mg/day). It is associated with a long mean residence time of Sr in*compartment 6*, ∼4 yr. This value was estimated by assuming knowledge of the total body Sr mass (*Eq. EC3
*). GI*compartments 4* and *5* each contain ∼0.1 mg of Sr, which is negligible compared with the Sr content of*compartment 1*. According to our steady-state hypothesis, the daily urinary excretion also represents the net balance between the GI compartment and *compartment 1*. Because the predicted mean daily amount of Sr ingested is 1.54 mg/day, the net intestinal absorption of Sr appears to be ∼20% of the ingested Sr, with ∼80% excreted in the feces. All these values are in good agreement with the known physiological parameters of Sr metabolism in healthy adults (10, 34). Thus each of the two optimal models is physiologically relevant for human Sr status, at least using*system 1*, although the biological nature of some compartments (*compartments 3* and *7*) remains unclear. They differ only in the inhibition variable involved in the Langmuir-type function (*Eq. EC2b
*): the integrative*compartment 6* for *model 12* vs. the reversible *compartment 3* for *model 13*.

### Long-Term Predictions

The ability of a model to correctly predict situations not directly connected to the experimental protocol used to build it is generally considered to be an indication of its validity. This is particularly true when the model is nonlinear. Using the same twice-daily oral administration, we have predicted the Sr mass in bone (mainly *compartment 6*) and simulated the responses of the optimal *models 12* and *13* by prolonging the four oral doses of Sr. These predictions have a pharmacological, rather than physiological, meaning. *Compartment 6* increases with a nonlinear dose dependence for both models, so that the relative quantity of Sr in the bone mineral solid phase is reduced as the dose of Sr increases (Fig. 8). However, increasing curves predicted by *models 12* and *13*differ, and the difference increases with the simulation time. This shows a weakness in our modeling procedure. It also indicates that a choice between the two candidate models (i.e., satisfactory models giving analogous goodness of fit) could be made on the basis of long-term experimental data. Unfortunately, the few available data on the Sr content of bone after long-term Sr administration to humans give only the relative amounts of Sr and Ca in trabecular bone biopsies. This precludes any direct comparison between these experimental data and the model predictions that concern only Sr, unless the effect(s) of Sr administration on bone Ca metabolism can also be predicted (36). Nevertheless, the percent molar ratio (Sr/Ca) computed from *compartment 6* and from other IDP compartments outside *compartment 1* (*compartments 2 *+ *3*+* 6 *+ *7*) is ∼0.25 for the smallest dose and ∼1 for the highest dose, if we restrict our predictions to 1 yr for *model 13* and assume that the bone Ca mass of the women remains at ∼800 g throughout the Sr administration. These values account fairly well for the available experimental measurements on bone biopsies (data not shown).

Thus the quality of the model predictions for Sr metabolism alone makes it possible to examine the physiological interpretation of the model nonlinear features. The physiological nature of the INL and ENL in the model, together with their interactions, must be identified to make our model significant. As for the regulation of many biological systems, these nonlinearities could involve more-or-less direct self-regulatory processes. However, this could be physiologically questionable for Sr for at least three reasons: *1*) we know nothing of the physiological role of Sr; *2*) there is no evidence for Sr homeostasis; and *3*) the daily intakes of oral Sr used to reveal Sr-dependent nonlinearities were up to 1,000 times the normal intake. Thus we need to know whether the processes are Sr specific or directly related to the physiological regulation of Ca metabolism and its homeostasis, inasmuch as Sr and Ca are physicochemically similar.

## Acknowledgments

The authors thank Dr. G. Milhaud for helpful discussions.

## Appendix

### Assumptions Used in Stage 1 of Model Building

In view of the nature of the PAdP data treated at this stage of the modeling procedure (regular decrease in plasma Sr concentration after cessation of relatively long-term oral Sr administration, with a suspected peculiar nonlinear dose dependence), a set of assumptions and approximations relative to formal and biological considerations is given as follows.

#### Assumption 1.

*System 1* is assumed to be intrinsically linear, so that any nonlinear behavior of the data originates from peculiarities of*system 2*, whether it be expressed under its time-explicit or time-implicit form, and from its interactions with *system 1*.

#### Assumption 2.

Well-defined *system 2* specifications were chosen for their generic analytic properties. The chosen *system 2* is a monocompartmental structure. It is extrinsically nonlinear via*y*
_{1}, i.e., via the variable including plasma Sr. Consequently, it depends only indirectly on the Sr dose. In its time-explicit form, *system 2* is shown for *i*= *n* + 1 as follows
Equation A1with
Equation A1a
Equation A1band *p* is a constant value >0.

Because we were interested only in the kinetic behavior,*z*
_{(n+1)} was normalized by its minimum asymptotic value obtained with *y*
_{1} = 0. Then, in absolute value, *g*
= *h*
.

Under the pseudo-steady-state assumption (see* Remark 3*), the generic *z*
_{(n+1)} is approximated to its asymptotic form and becomes a time-implicit function of*y*
_{1} following an equation of the Hill-type
Equation A2with *k*
^{1} =*k*
^{2},* p* > 0, and, for increasing values of *y _{1}
*, generic

_{(n+1)}, from 1, when

*y*

_{1}= 0, toward the value of

*k*

^{3}

*/k*

^{4}. Thus, depending on whether

*k*

^{3}

*/k*

^{4}is >1 or <1, generic

_{(n+1)}is an increasing or a decreasing function of

*y*

_{1}. Its deviation from a simple hyperbola, obtained for

*p*= 1 (M-M equation), appears if

*p*> 1 (positive cooperativity) or

*p*< 1 (negative cooperativity).

#### Assumption 3.

Another specific formulation was used for *system 2*. It is based on consideration of a nonlinear growth function, the logistic equation (40), that is activated by*y*
_{1}. *Equation EA1, a* and *b*, is replaced by
Equation A1c
Equation A1dwith* p* > 0.

Normalizing* z*
_{(n+1)} by its maximum value, *L*=1, for*h*
_{0(n+1)}≤*g*
, the asymptotic value of*z*
_{(n+1)} is ≥0, whatever the value of*y*
_{1}, and* z*
_{(n+1)}always grows toward 1 with a sigmoidal curve when*y*
_{1} increases. In contrast to the generic form (*Eq. EA1, a* and *b*), this *system 2* is now intrinsically nonlinear, because*G*
_{(n+1)0} is a function of*z*
_{(n+1)}.

#### Assumption 4.

*z*
_{(n+1)} acts on one or several transfer processes in *system 1,* so that the resulting FTF obeys the following equation
Equation A3

#### Assumption 5.

The initial *y*
_{1} value,*y*
_{1}(0), is the mean experimental plasma Sr concentration computed at *time 0*. It is used to estimate, under the steady-state hypothesis and for any set of parameter values, the other initial conditions and initial value of the input function assumed to take place directly on *compartment 1*, *F*
_{10}(*t *= 0). This input function is then
Equation A4with* f*
=*F*
_{10}(*t *=* 0*) and*f*
(*t*) the entry linked to one of the four doses of Sr administered. At 612 h, 12 h after the last oral Sr administration, the influence of Sr entry from the gut was assumed to be associated only with the normal nutritional intake of Sr, *f*
. Thus*f*
(*t*) = 0 for 0 ≥ *t* ≥ 612 h or > 0 for 0 <*t* < 612 h. In the latter case,*f*
(*t*) is approximated to a constant value (we did not take into account the fact that the oral Sr was given according to a precise protocol), and its dose dependence was directly in proportion to the Sr dose.*f*
Is assumed to be constant from 0 to 1,272 h and to balance the sum of the irreversible exits, Σ*k*
_{0i}
*y _{i},* at

*time*0: it includes the physiological net input flux from the diet and other irreversible Sr influxes (possibly from bone).

## Appendix

### Assumptions Used in Stage 2 of Model Building

We used a new set of assumptions, mainly dictated by GI mineral metabolism and the nature of the FAdP data.

#### Assumption 1.

The compartmental representation of the GI tract is inferred to be a series of unidirectional compartments: from the proximal compartment, which receives dietary and supplementary exogenous Sr, to the most distal compartment, from which the fecal Sr is excreted. Some of these compartments can transfer Sr with the plasma (*compartment 1*) due to intestinal absorption (from lumen to plasma) and/or intestinal secretion (from *compartment 1* to lumen).

#### Assumption 2.

The input of exogenous Sr was defined as the physiological Sr entry (from the normal diet) plus the oral Sr given twice daily
Equation B1with *i* (*i* ≠ 1), the GI compartment receiving dietary and supplementary Sr.*f*
Is approximated to a constant rate, and*f*
(*t*) takes the form of a time-explicit Dirac-type function, such as
where χ_{i} denotes the impulsive increase in the present value of *compartment i* after each oral intake. It is applied only once, at *time 0*, because only FAdP data are used.

#### Assumption 3.

χ_{i} Has a (apparent) concentration dimension, and the oral Sr dose is known as a mass [the smallest oral dose of S-12911 contained 972.5 μmol of Sr (D_{1})]. The compartmental formalism requires that the relation of χ_{i} to the overall V_{app} must be verified as
Equation B2where D_{α} is the mass of Sr ingested at each oral dose, with α = 1, 2, 3, and 4 in reference to each of the four doses.

Because *Eqs. EB2
* and *
3
* must be satisfied simultaneously, the value of χ_{i} can be deduced from any value of *k*
_{01} and vice versa.

#### Assumption 4.

Unless otherwise stated, the*f*
(*t*) dose dependence was usually constrained to be in direct proportion to the Sr ingested. Thus only one*f*
(*t*) has to be identified, and the others can be easily calculated from it.

#### Assumption 5.

By analogy with intestinal Ca absorption, Sr may be absorbed via a linear (nonsaturable, paracellular) and a carrier-mediated (saturable, transcellular) pathway, the carrier-mediated pathway being described by the M-M equation (43). Thus the overall FTF that includes a saturative relation on the GI *compartment j*, from which intestinal absorption takes place, has the following expression
Equation B3with *k*
defining the linear part of the process, *k*
the maximal velocity, and *k*
the concentration of *compartment j* for which the rate is half-maximal.

#### Assumption 6.

In contrast to *stage 1* of model building, only *system 1* may account for any nonlinearity in the FAdP data, its dependence on *system 2* being precluded. In other terms,*system 1* may be an intrinsically nonlinear differential system, at least one of its fractional transfer functions depending on some of its own variables, *K _{ij}
* =

*f*(

*k*

*,y*)

*,*i.e., being an INL.

#### Assumption 7.

The initial steady state of *system 1* at *time 0*involves two different relations between input and output functions:*1*) Sr excretion (mainly urinary and fecal) is balanced by the oral input of exogenous Sr,*f*
(see *Eq.EB1
*). The net external Sr balance is then approximately zero.*2*) If the model predicts other irreversible Sr outputs from IDP, they are counterbalanced by a distinct Sr input related to*compartment 1*,* F*
_{10}(*t*), such as
with *i* compartment(s) belonging to IDP.

#### Assumption 8.

The FTC associated with the urinary excretion of Sr is calculated by applying *Eq. 3
*, V_{app} being obtained from*Eq. EB2
*. If another output can take place from*compartment 1*,* k*
_{01} has to be split into* k*
and*k*
, the urinary and the other output transfer coefficients, respectively, with

## Appendix

### Assumptions Used in Stage 3 of Model Building

#### Assumption 1.

At this final stage of modeling, *system 1* includes some FTFs having the general form [*K _{ij}
* =

*f*(

*k*

*,y,z*)], which combines ENL and INL. Thus the influence of

*system 2*on

*system 1*is not restricted to the relation in

*Eq.EA3,*and

*z*can modulate transfer functions otherwise involving INL. For instance, if we choose to modulate the saturable part of

*Eq. EB3*, the FTF associated with intestinal absorption takes the following form Equation C1where the maximum velocity changes according to the

*z*kinetics (

*k*

*z*).

#### Assumption 2.

As observed during *stage 1* of model building, generic and logistic *z* give similar optimal fits to data (Table 1). We therefore use only the logistic *system 2 *(see *Eq. EA1, c* and *d*), but generalized for *i* = (*n* + 1)*, *(*n* + 2),*…,*(*n *+ *m*) when several parameter-distinct monocompartmental substructures are employed in*system 2*.

#### Assumption 3.

The impulsive Dirac function,*f*
(*t*) in*Eq. EB1
*, applied to the GI compartment receiving dietary and supplementary Sr (*Assumption 2* in appendix
), is effective at *time 0* and at other times during AdP, depending on the experimental protocol.

#### Assumption 4.

We used another INL applied to bone Sr metabolism that assumes that some bone FTF may be subjected to a process limiting the capacity of bone to accept large amounts of Sr. The following form was chosen
Equation C2where *b* denotes the *y _{b}
*belonging to an IDP or bone compartment other than the source

*compartment j, k*is the maximal transfer rate from

*compartment j*to

*compartment i, k*is the concentration of

*compartment j*for which

*K*is half-maximal in the absence of any inhibitory effect due to

_{ij}*y*, and

_{b}*k*, the constant of inhibition, is expressed as a C

_{app}. This FTF may or may not be modulated by

*z*. This type of function is analogous to an M-M equation accounting for inhibition by a species other than the substrate of the reaction. Because M-M and Langmuir-type equations are similar (32), this kind of INL may be directly applied to physicochemical processes involved in bone mineral metabolism, such as the adsorption of mineral ions to the surface of existing bone solid phase (15). In this context,

*Eq. EC2*, which also assumes that some Sr mineral species may be inhibitory, was denoted a Langmuir-type equation,

*y*being the inhibition variable.

_{b}Because the saturation process through *y _{j}
*may not have operated during the optimization procedure (

*k*very large compared with the value reached by

*y*), whereas the inhibition through

_{j}*y*was effective, the following form of

_{b}*Eq. EC2*was considered Equation C2awhere

*k*is the ratio of

*k*to

*k*chosen in

*Eq. EC2*and, thus, takes a

*t*

^{−1}dimension.

*k*becomes the inhibition constant (

*k*in

*Eq.EC2*).

Finally, we used a normalized version of *Eq. EC2a
* in this stage of model building to minimize any parameter indetermination. This version is such that* y _{b}
* is replaced by

*y*−

_{b}*y*(

_{b}*0*); i.e., at

*time 0*,

*K*=

_{ij}*k*Equation C2b

#### Assumption 5.

It has been emphasized (see *Remark 4*) that characterization of the irreversible exits from *system 1* requires that the input-output relations used to define the initial (*t* = 0) steady state be made clear. We postulated that any irreversible output from an IDP compartment other than 1 is related to the irreversible uptake of Sr by bone, balanced at *time 0* by a peculiar input flux into *compartment 1*,*f _{10}
* (initial net internal Sr balance = 0). Under these conditions, any irreversible output from

*compartment 1*in addition to urinary excretion (

*Assumption 8*inappendix ) is interpreted as one of the Sr excretions contributing to

*f*, the entry of exogenous physiological Sr into the GI compartment that ensures zero net external Sr balance at

*time 0*. Otherwise, the character of the additional exit remains ambiguous, because it may be due to Sr uptake by bone and/or Sr excretion other than urinary excretion.

#### Assumption 6.

It is also possible to carry out the Sr internal zero balance at*time 0* by considering that a bone compartment collects the “irreversible” bone Sr uptake and returns it directly to*compartment 1*. If it is assumed that the initial total body Sr mass (SrbM) is known, this new compartment at *time 0* can be estimated as follows
Equation C3with* j *≠* i* belonging to IDP. This relation, which invalidates *f*
_{10}, allows estimation of the associated* k*
_{1i}, which must be compared with a unidirectional bone mineral removal, including first-order bone resorption.

## Footnotes

This study was financially supported by the Centre National de la Recherche Scientifique and the Institut de Recherches Internationales Servier.

Address for reprint requests and other correspondence: J. F. Staub, UMR 7052 Centre National de la Recherche Scientifique, Laboratoire de Recherches Orthopédiques, Faculté de Médecine Lariboisière-St-Louis, 10 Ave. de Verdun, 75010 Paris, France (E-mail:staub{at}ccr.jussieu.fr).

↵1 As stated by Phair (29), “a strong case can be made that understanding non-steady states is the ultimate goal of kinetic analysis” and, in this context, reference must be made to the application to biology of nonlinear dynamic systems with their potentiality for more-or-less complex self-oscillatory behavior (14).

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

*advertisement*” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.First published November 7, 2002;10.1152/ajpregu.00227.2002

- Copyright © 2003 the American Physiological Society