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Endocrinology Department of the Portuguese Cancer Institute (J.F.R., L.G.S.), 1099-023 Lisboa, Portugal; Abel Salazar’s Institute of Biomedical Sciences, University of Oporto (H.G.F.), 4099-003 Oporto, Portugal; and Department of Chemistry (H.G.F.), Faculty of Science and Technology, New University of Lisbon, 2829-516 Caparica, Portugal
Address all correspondence and requests for reprints to: Dr. J. F. Raposo, Endocrinology Department, Portuguese Cancer Institute, R. Prof. Lima Basto, 1099-023 Lisboa, Portugal. E-mail: . filipe.raposo{at}mail.telepac.pt
Abstract
A mathematical model of calcium homeostasis is presented in which the controlling factors are the plasma concentrations of calcium, PTH, and calcitriol, and the effector organs are the parathyroids, bone, kidney, and intestine. Other factors can be added as the need arises. The model is aimed at simulating what happens in a single individual, but its parameters and variables were adjusted to the corresponding published average values.
Simulations of published observations in humans undergoing the infusion of calcium or its chelators are presented. With a single exception, these simulations provided a good fit to the data. The response of the system to extrinsic perturbations was characterized by simulating chronic infusions of calcium, PTH, and calcitriol. Finally, the steady state response to perturbations in some of its parameters (the secretory mass of the parathyroids and the affinity and/or sensitivity of the calcium, PTH, and calcitriol receptors) and to renal failure were also investigated in an attempt to analyze the pathogenesis of clinical hypo- or hypercalcemias.
In its present form the model cannot be used to base clinical decisions in individual cases. However, it requires modest computational resources, and clinicians with a modest mathematical background can manipulate it. It is a useful tool for the analysis of general mechanisms of the diseases of calcium metabolism and for the design of clinical experiments aimed at characterizing these diseases. The model can also be the core of future autoadaptive extensions to be used in individual patients.
IN HUMANS, THE extracellular pool of dissolved calcium (in ionic and bound forms) is less than 35 mmol, and the rates of exchange with the environment and bone are approximately 5 and 88 mmol/d, respectively (1). Despite these rapid exchanges, the fluctuations in the concentration of free (1–1.4 mM) or total (2.2–2.6 mM) calcium in healthy individuals (2) are very small. This implies that the rates of exchange between the extracellular fluid compartment (ECC) on one side and the environment and bone on the other are tightly controlled.
A number of controlling factors have been identified (1), but the adjustment of the calcium exchanges is mainly achieved by regulating the ECC concentrations of two main regulators, PTH and calcitriol. Free ECC calcium is also a feedback regulator. The target organs are bone, kidney, and intestine.
ECC PTH stimulates the rate of production of calcitriol by promoting 1
-hydroxylation of 25-hydroxyvitamin D in the kidney (3), stimulates the efflux of calcium from the bone (4), and regulates the threshold of calcium excretion by the kidney (5). ECC calcitriol inhibits its own rate of synthesis, regulates a fraction of the calcium absorbed in the intestine (6), and inhibits calcium excretion by the kidney. ECC calcium (7) and calcitriol (8) inhibit the rate of production of PTH by the parathyroid glands. Calcitriol also promotes the calcium exchange between bone and ECC, but only in the presence of PTH (6). Finally, the rate at which calcium is eliminated by the kidney (9) or deposited in the bone (1) is also a function of its ECC concentration.
The calcium ECC concentration is thus controlled by a nonlinear, multiloop system. Its behavior cannot be predicted by simple intuition. Explicit predictions imply the construction of mathematical models, as is now common practice in the area of blood glucose regulation (10, 11).
The phosphate concentration in the ECC is also reasonably steady (0.97–1.45 mM) (2) despite its involvement in a large number of metabolic processes (12). As calcium phosphates are also the main mineral components of the bone, the exchanges of calcium and phosphate between bone and the ECC are tightly coupled (1). The external balance of phosphate is finely regulated by a still unknown mechanism (13). Its urinary excretion is also partially under the control of PTH (13).
The characterization of calcium homeostasis can now be made with some detail in health and disease (14), because calcium, phosphate, PTH, and calcitriol in solution can be measured routinely with considerable accuracy. Published data thus obtained in humans enable model adjustments.
Of the many factors that may influence calcium and phosphate metabolism (1), those mentioned above are not only the best studied, but also probably the most important from a clinical point of view. In this paper we describe a minimal mathematical model of ECC calcium concentration regulation that encodes only those mechanisms (through kinetic relationships) required to simulate the selected published observations.
Materials and Methods
The model is depicted in Fig. 1
. The three hormonal targets in this system are the bone with a time response of minutes to hours (1), the kidney with a time response to perturbations on the order of hours (9), and the intestine that responds to changes in the ECC concentration of calcitriol with a delay of days (6). A fraction of the calcium excreted in the urine, the diffusional movements of calcium across the gut wall, and the deposition of calcium in the bone seem to be determined by simple, hormone-independent, physical-chemical factors such as the concentrations of ionic calcium in ECC and bone, local nucleation mechanisms.
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Eleven compartments are encoded. Eight of them do not require a special description: the total amounts of calcium, phosphate, PTH, and calcitriol in the ECC; the calcium and phosphate exchangeable pools in bone; the total amount of intracellular phosphate; and the total amount of kidney 1-hydroxylase. This last quantity is equated to the rate of synthesis of calcitriol. Three other pools were introduced as a way of encoding for the slower response times of the calcium exchanges to calcitriol and of the renal phosphate excretion to its external balance. The parathyroid gland pool represents the fraction of the secretory mass of gland that is active at a given instant. The calcium transport pool of the intestine is the calcitriol-dependent fraction of the intestine calcium transporters. The kidney phosphate transporter pool is the external balance-dependent fraction of the phosphate transporters of the nephron. All of these three last pools are dimensionless. The movements of calcium and phosphate in and out of bone obey the relation of Ca/P in hydroxyapatite.
The following definitions apply to the model quantities: parameters (rates of breakdown, maximum rates of synthesis or transport, permeabilities, hormone affinities, etc.), which are constant unless subjected to an external perturbation; intrinsic variables (ECC concentration of calcium, PTH, or calcitriol), which are computed by the model and cannot be manipulated directly in a real situation; and extrinsic variables (calcium and phosphate intakes), which can be manipulated externally.
Certain clinical and experimental simulations imply the perturbation of parameters (for example, the secretory mass of the parathyroid gland in surgical removals) as if they were extrinsic variables. To avoid numeric discontinuities the parameters or extrinsic variables were relaxed using an exponentially smoothed out time step function, with a time constant 2 orders of magnitude smaller than the smallest time constant of the response of the system.
The time response of the model depends strongly on the choice of initial, steady state values, and as shown in Table 1 there is a large number of variables and parameters that have to be adjusted. To restrict the arbitrariness of these adjustments the following constraints were adopted.
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Calcium absorption in the intestine was modeled according to data reported by Sheikh et al. (15). Passive absorption was set at 10% of the total absorption. The rate constants of degradation of PTH, calcitriol, and 1-hydroxylase were taken from the reports by Parfitt (1) and Fraser (16), respectively. The turnover rates were computed directly as the rate constants times the corresponding concentrations.
The model was solved with Berkeley Madonna software (www.berkeleymadonna.com), kindly provided by Prof. Robert Macey.
The basal (control or normal) values and dimensions of the parameters and variables are given in Table 1
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Results
Of the many possible simulations we present those of published experiments in humans; of chronic infusions of calcium, PTH, and calcitriol; and of clinical malfunctions of regulation of the plasma concentration of calcium.
Fitting observations to acute experiments
A plot of the average values of transient hypocalcemic episodes in 95 patients for 4 d (17) together with our simulations are given in Fig. 2A. There was no combination of parameters that provided a good fit of the model to these data. Figure 1, B and C, report good fits to published PTH and calcium concentrations in hypercalcemic and hypocalcemic clamps in seven healthy volunteers (18). Satisfactory simulations (Fig. 1, D, E, F, and H) were also obtained of experiments in which infusions of calcium gluconate were performed in humans (9, 19, 20).
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). Their results are plotted together with those presented by Peacock (5) and with the corresponding simulations. Transient and steady state characterization of the system
After a constant infusion of 2.5 mmol/h calcium (Fig. 3, A–F), the rise in ECC calcium starts immediately (Fig. 3A) as does its urinary excretion as a result of the increase in filtered load. The fall in PTH caused by the calcium feedback inhibition follows the ECC concentration of this ion. After 6–7 h ECC calcium begins to fall because PTH is falling despite the fact that a constant calcium infusion is still on. Meanwhile, ECC calcitriol is falling because PTH is falling, but with a delay reflecting its smaller turnover rate. By the end of 1 wk (Fig. 3D), the system is approaching a steady state, which is clearly established after 5–6 months (Fig. 3E). This steady state is characterized by higher than normal ECC calcium and urinary excretion and lower than normal ECC PTH and calcitriol.
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, F–H, report simulations of prolonged infusions of calcium, PTH, and calcitriol ranging from negative (exemplified by the infusion of a calcium chelator) to positive values. When increasing amounts of calcium (-0.5 to 2.5 mmol/h) were infused until a steady state was reached (Fig. 3F), ECC calcium rose, although at a higher rate for negative infusions, driving down ECC PTH and, through it, ECC calcitriol. The calcium urinary output rose not only because its filtered load increased, but also because tubular uptake decreased as a result of the fall in ECC PTH.
In PTH infusions (Fig. 3G) there is an increase in the ECC concentration of the four variables. For calcium, calcitriol and urinary output saturation is reached once the rate of infusion is equal to the basal secretory rate. The range of the renal response to PTH resulted from the fitting of the model to Peacock’s data (5). As more data become available, further adjustments will be made. Figure 3H reports an infusion of calcitriol. The response of PTH is the direct result of the calcitriol inhibition, whereas the biphasic response of calcium urinary output is the result of the monotonical increase in ECC calcium and a steep fall in ECC PTH.
Pathophysiology of disorders of calcium homeostasis (Figs. 4 and 5)
We assumed that in clinical situations of hyper- or hypocalcemia the ECC concentration of calcium are still regulated, but with a higher or a lower than normal set-point. Figure 4A shows the effect of an increase in the parathyroid secretory mass. There is an increase in ECC PTH that induces an increased rate of calcitriol synthesis, a positive balance of calcium, and a consequent increase in ECC calcium. The resulting increase in filtered load explains the increase in calcium urinary output.
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Simulations F, G, and H illustrate the dissociation between calcium and PTH curves. In familial hypocalciuric hypercalcemia (Fig. 4F), the affinity of the calcium receptor in parathyroid gland and kidney tubule is diminished. The anomaly in the gland would make PTH secretion less sensitive to the inhibition by calcium, producing as a result a rise in the ECC concentrations of both calcium and PTH and a decrease in urinary calcium excretion. The anomaly in the kidney lowers urinary calcium excretion. Because of the dramatic fall in calcium excretion, the rise in the concentration of this ion in the ECC more than compensates for the lower affinity of the calcium receptor. An analysis of the initial transient would show a peak in ECC PTH, followed by an attenuated peak of calcitriol. After their corresponding peaks there is a slow fall while ECC calcium rises, so that at steady state there is hypercalcemia, and concentrations of PTH and calcitriol are slightly below the initial values, but still in the normal range. In situations A–F calcitriol follows PTH until the stimulation of calcitriol synthesis reaches saturation (A, C, and D).
In Jansen’s disease (22), the primary alteration is an increase in the affinity of the PTH receptor in bone and kidney. In calcitriol intoxication the increased ECC concentration of this hormone inhibits PTH secretion and stimulates calcium absorption in the intestine. The secretion of PTH is inhibited in both cases by the high concentration of calcitriol. The calcium threshold in the kidney is lowered because of the low plasma concentration of PTH. The increased calcium excretion reflects an increase in the filtered load resulting from a mild increase in ECC calcium.
Figure 5A simulates the surgical removal of parathyroid tissue. The primary effect is a fall in PTH secretion and consequently a fall in the ECC concentrations of calcium and calcitriol. The fall in calcium excretion is the combined effect of the fall in filtered load with a moderate fall in the renal threshold for calcium excretion.
In autosomal dominant hypocalcemia (23) (Fig. 5B), the affinity of the calcium receptor in the kidney [XR(Ca(k)/Ca)] and parathyroid [XR(PTH/Ca)] was increased. To obtain lower than the average normal levels of serum PTH in our simulation, the increase in the affinity of the parathyroid receptor was larger than that in the kidney to compensate for the fall in serum calcium. The overall effect is the end result of the interplay between the inhibition of PTH secretion by calcium and calcitriol and also between the opposite effects of PTH (which raises the renal calcium threshold) and calcium and calcitriol (which promote renal excretion).
Plots C and D report two situations in which there is an increase in the effectiveness of the calcium negative feedback loop resulting from an increase in affinity [XR(PTH/Ca)] or in sensitivity [b(PTH/Ca)] of the parathyroid calcium receptor. The second mechanism is clearly more effective, but only a more extensive analysis will show the importance of the set-point chosen for the basal state.
Figure 5, E and F, correspond to simulations of pseudohypoparathyroidism (24) and illustrate the effect of changing the affinity or the sensitivity of the PTH receptor. Studies in vitro with transfected cells have shown that the production of intracellular cAMP as a function of PTH concentration follows an S-shaped curve as in our model and is decreased in Gs protein mutant cells. As no clear saturation is shown, it is not possible to determine whether there is a diminished affinity, a diminished sensitivity, or both (25).
A better insight of the last panel is provided by plots of the first 50 h of the response (Fig. 5, G and H). After a control period of 2 h, the sensitivity of the PTH receptor is reduced in 60%. Figure 5G illustrates the dramatic effect of this perturbation on the threshold for urinary calcium excretion (UCaTH). The urinary excretion increases immediately and is followed by a delay in the fall in ECC calcium and a consequent increase in ECC PTH. Figure 5H describes the time course of the activation of 1-hydroxylase and the plasma concentrations of PTH and calcitriol, three closely associated variables.
Parathyroid growth and secretion are regulated by calcium (26). In hypocalcemic patients secondary hyperparathyroidism may appear, reflecting an increase in the secretory mass of the parathyroid (27). In renal failure several factors may induce secondary hyperparathyroidism: low concentrations of calcium and calcitriol and high concentrations of phosphates in serum. In our simulations of the data reported by Pitts et al. (28), a simple reduction in the calcium tubular load and in the synthesis of calcitriol was not enough to reproduce published clinical observations. The simulations presented in Fig. 6 compare these observations with the simulations for different values of creatinine clearance, assuming that the parathyroid secretory mass was 4.8, 2.2, and 1.7 times the control for fractional creatinine clearances of 0.093, 0.26, and 0.60 of the control. The calcium concentration in ECC reflects the opposite effects of a reduction in urinary excretion as a result of the renal failure and a reduction in calcium absorption in the intestine reflecting the fall in calcitriol concentration.
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Published models of calcium metabolism deal with spontaneous oscillations (29, 30, 31), with the relationship between bone calcium and calcium homeostasis (32, 33, 34, 35, 36), or with the simulation of experiments with radioactive calcium (37, 38, 39). Particularly useful in the context of the model presented here were analytic descriptions of the response of the parathyroid gland to the ECC calcium concentration (7, 40). We constructed our model along the lines of those developed for the chick by Hurwitz and co-workers (41, 42, 43).
Our model was built stepwise from the bottom up, starting with the kinetic descriptions of relevant processes (actions of controlling variables) that are available in the literature or for which plausible descriptions can be postulated. Feedforward or feedback branches then linked these components. The addition of new components was determined either by the need to simulate the biochemical patterns of published experiments performed in humans or by the ability to simulate accepted biochemical patterns of diseases of calcium homeostasis. There was no need to include other factors, mentioned in the literature, such as sex hormones (androgens and estradiol) and magnesium.
In its present form the model is not totally constrained, in the sense that there is no a unique set of parameters that can be used to simulate all of the published clinical or experimental data collected here. This shortcoming is to be expected of models of this complexity, in particular when almost all published data were gathered from different individuals or groups of individuals. A factor complicating further the simulations is the fact that most pathological situations are first seen after a prolonged and insidious period of development during which changes other than those resulting directly from the initial perturbation play a role in determining the clinical picture, whereas simulations are equivalent to perturbations applied to otherwise normal subjects.
We were able to simulate a number of published acute experiments performed in humans, with the exception of those reported by Bourrrel et al. (17). This failure is not surprising because their average published values of calcemia remained unchanged after a marked reduction in plasma PTH.
The results presented by Attie et al. (21) were reasonably fitted, but as the JCa(PU)/[Ca(P)] curves depend very strongly on time, which cannot be obtained from their paper, the fitting may be spurious. The fitting of Peacock’s results is not surprising, because to build the model we fitted the kinetics of the urinary excretion of calcium to his data.
The experiments with chronic infusions cannot be performed in practice, but are conceptually useful, because they may provide clinical insights. If we exclude the surgically induced situations, most of the biochemical patterns that the clinician observes in anomalies of calcium homeostasis have at least three characteristics. 1) They are in a quasi steady state. 2) They reflect the effect of the lack or excess of a regulating factor (PTH or calcitriol). 3) They reflect an anomaly in the gain of one or more of the feedback loops due to an increase or decrease in the number or affinity of some receptor (to calcium, or PTH, or calcitriol). The simulations of prolonged steady infusions cover only the first two characteristics.
Until molecular probes are created it is unlikely that a precise functional diagnosis of anomalies of the calcium homeostasis can be established. It is unlikely that clinical hypercalcemia is due to a lack of regulation, because in such a case one should expect the ECC calcium concentration to oscillate widely with calcium intake. More likely it results from a shift upward of the set-point of its regulation as a result either of a change in the gain of one or more feedback loops or of a change in the affinity of the receptors for its regulating agents (calcium, PTH, or calcitriol). The simulations reported in Fig. 4 illustrate this philosophy. Quantitative results should be interpreted with caution and should be considered only as probable indicators of what might happen.
An obvious shortcoming of the model is the description of what happens to the bone. Of the bone calcium, only the so-called exchangeable pool was considered, but we know that over the long term the amount of calcium that can be mobilized is much larger, in particular through activation of the osteoblast-osteoclast pathway (44). An encoding of this mechanism may follow the strategy used to describe the effect of calcitriol on intestinal calcium absorption or parathyroid secretion. The only inhibitory effects on the parathyroid considered were either directly on the rate of secretion (calcium) or on the fraction of the secretory mass that is active (calcitriol). We did not include a feedback loop reflecting the effect of the ECC concentrations of calcium, phosphate, and calcitriol (6) on cell proliferation in the parathyroid gland, because we lack sufficient data to support an analytical description of these actions. As shown in Fig. 6, without this mechanism it is not possible to simulate the hyperparathyroidism of renal failure.
Fitting the model to data from a particular patient should be undertaken with prudence, because it was constructed from averaged data collected by different researchers on different populations. However, it can be the core of future autoadaptive extensions applicable to individual patients.
The model is in itself a product of evidence-based thought, because it is a synthesis from a minimal set of analytical descriptions of quantifiable published processes and data. It provides a powerful tool for the analysis of general mechanisms of the diseases of calcium metabolism and for the design of clinical experiments aimed at characterizing these diseases.
The Mathematical Model
The core of the model is a set of 11 time-dependent nonlinear differential equations that are simple statements of the conservation principle applied to the pools of calcium [QCa(p)], phosphate [QP(p)], PTH [QPTH(p)], and calcitriol [QD(p)] in the ECC, exchangeable pool of calcium and phosphate in bone [QCa(b), QP(b)], intracellular phosphate [QP(c)], 1-hydroxylase [QE(K)] in the kidney, the calcitriol-dependent intestinal calcium transport pool [QTCa(i)], the external balance-dependent phosphate kidney transport pool [QTP(K)], and the secretory mass of the parathyroid gland (QCPT).
The concentrations of calcium, PTH, and calcitriol are assumed to be the same in plasma and ECC. Calcium concentrations refer to total calcium, because most data in the literature are so expressed.
Calcium and phosphate move across the kidney into urine [Jpu(Ca), Jpu(P)], across the intestine into ECC [Jip(Ca), Jip(P)], from ECC to bone [Jpb(Ca), Jpb(P)], and in the opposite direction [Jbp(Ca), Jbp(P)]. Although the intracellular pool of calcium can be neglected, the intracellular pool of phosphate cannot. Therefore, the movements of phosphate from the cells into the ECC Jcp(P) and in the opposite direction Jpc(P) were considered. PTH is produced (SPTH) and degraded at a rate kPTH. Kidney 1-hydroxylase [QE(k)] is produced at a rate (SE) under the control of PTH and degrades at a rate kE. Calcitriol is produced in the kidney at a rate that we can express as the amount of kidney 1-hydroxylase [QE(k)] and degrades at a rate KD. The fraction of active intestinal translocators [QTCa(i)] is inactivating at a rate k2,D', whereas the fraction of inactive translocators [1 - QTCa(i)] is activating at a rate k1,D'. The fraction of active parathyroid cells (QCPT) is inactivating at a rate k4,D', whereas the fraction of inactive cells [1 - (QCP)] is activating at a rate k3,D'. Both of these rates are under the control of ECC calcitriol.
The instantaneous sizes of the pools are the result of the interplay of all of these fluxes as follows, in which d/dt [Qx(y)] is the time derivative of pool Q of component x [calcium, PTH, calcitriol, calcium transporters (Tca), phosphate transporters (TP), active parathyroid cells (C)] in compartment y (p, plasma; c, cell; b, bone; k, kidney; i, intestine; PT, parathyroid glands).
d/dt [QCa(p)] = Jbp(Ca) - Jpb(Ca) -Jpu(Ca) + Jip(Ca)
d/dt [QP(p)] = Jbp(P) - Jpb(P) - Jpu(P) +Jip(P) - Jpc(P) + Jcp(P)
d/dt [QP(c)] = Jpc(P) - Jcp(P)
d/dt [QPTH(p)] = SPTH - kPTH x CPTH(P)
d/dt [QCa(b)] = -Jbp(Ca) + Jpb(Ca)
d/dt [Qp(b)] = -Jbp(P) + Jpb(P)
d/dt [QE(k)] = SE - kE x QE(k)
d/dt [QD(p)] = QE(k) - kD x QD(p)
d/dt [QTCa(i)] = (1 - QTCa(i)) x k1,D' - QTCa(i) x k2,D'
d/dt [QC(PT)] = (1 - QC(PT)) x k3,D' - QC(PT) x k4,D'
d/dt [QTP(k)] = [1 - QTP(k)] x k1,P' - QTp(P) x k2,P'
The functions relating substrate concentrations (x) to transport fluxes or reaction rates (Y) are of four types. 1) Y(x)1± = Y(x)1± (Max) x{a[1 ± TANH (b(x - xR)] + d}, where Y±1(Max) is the maximum rate, and a + d = 1 The ranges of function between [ ] are 0
2 or 20 according to whether the plus (activation) or minus (inhibition) sign is used (45). x is the substrate concentration, xR is the value of X for which Y = 1, a is the fraction of Y that depends on x, set initially at 0.85 for all cases in the steady state, whereas d (0.15) is the independent fraction. b is the slope of Y in relation to x (sensitivity) at xR. In the cases of QTCa(i), QC(PT), and QTP(K), a was set at 1 x k1,P' [Yk (Jext)1-] and k2,P' [Yk (Jext)1+, which are both functions of the external phosphate balance (Jext) defined as the instantaneous difference between phosphate intake [Jip(P)] and phosphate excretion by the kidney [Jpu(P)].
2) Y2+ = Y2+(Max) x x/(x + xR) or Y2- = Y(x)2 - (Max) x xR/(x + xR). These are simple Michaelis-Menten functions denoting positive or negative dependence, respectively, of Y on x. xR is the affinity, and Y2± (Max) is the maximum rate. Functions of types Y1 and Y2 were used to establish bounds to the dynamic range of the variables. Most simulations took place within the quasilinear part of these functions.
Kinetics of types 1 and 2 exhibit saturation because they describe processes in which there is activation or inactivation of receptors as the first step of a chain of events that is different from cell type to cell type. For example, PTH activates osteoblasts in the bone and resorption of calcium in the kidney tubule. At the present stage it is not possible to derive the overall kinetics from the molecular mechanisms of the different steps, which are mostly unknown. The functions used are the simplest approximations according to whether the process continues at a certain rate in the absence of the controlling factor (type 1, S shaped) or not (type 2, MM).
In this formulation the analytical functions do not describe the kinetics of the activation or inactivation of the receptor itself as is observed in vitro in transfected cells. It is conceivable that different reaction sequences initiated by the same receptor, but in different organs, obey different kinetics, and that the receptors in vivo behave differently from in vitro. In our model we used the same kinetics for each receptor independently of the target organ, as there is no experimental evidence to force us to complicate the model further.
Consequently, the expressions of affinity or sensitivity of a receptor are used as an abbreviation for affinity or sensitivity of the whole process. Uncoupling at any point along the chain is equivalent to a large decrease in the affinity or sensitivity of the receptor, so that the rate of the process becomes negligible.
3) Y3(x) = k x x. This is a first order nonsaturating kinetics. k is a reaction rate, and x may be a concentration or a pool. Equations of this type were used to compute the degradations of PTH, calcitriol, and 1-hydroxylase. Similar equations were used to compute the intestinal fluxes of calcium [kCa(i)] and phosphate [kP(i)], where k is an effective permeability.
4) Y4(x) = x x f(xT). Y4(x) is the calciuria (x = Ca) or phosphaturia (x = P), and xT is the ratio between the renal threshold for calcium (Cathr) and the plasma calcium concentration (Ca(p)) or the ratio between the renal threshold for phosphate (Pthr) and the phosphate concentration in plasma (P(p)). Function f(xT) was originally proposed by Walton and Bijvoet (46) as a normalized description of the phosphaturia. f(xT) is a second degree polynomial with coefficients AZ, BZ, and CZ, where Z is either calcium (Ca) or phosphate (P).
The general computational strategy follows that reported by Lew et al. (47). The system is first solved for a steady state (control) situation. This is also a way of testing for its stability. To compute the steady state, all of the time derivatives are assumed to be 0. The resulting 11 simultaneous equations are then solved after the initial values of the variables, and the values of an appropriate number of parameters are substituted in the corresponding functions. The fluxes [Jab(x)] of component x from compartments a to b and the secretory rate (SPTH rate of PTH secretion) are described as follows:
Jbp(Ca) = Y(D(p))2+ x Y(QCa(b))2+ x Y(PTH(p))1+
Jpb(Ca) = kCa(b) x QCa(p)
Jbp(P) = Stoic(Ca/P) x Jbp(Ca)
Jpb(P) = Stoic(Ca/P) x Jpb(Ca)
Jpu(Ca) = Y(D(p))2+ x Y(Ca(p))2- x Xf(CaT)
Jpu(P) = Xf(PT)
Jip(Ca) = Y[Ca(i)]2+ x (1 - QTCa(i)) x k1,D' - QTCa(i) x k2,D' + kCa(i) x [CCa(i) - CCa(p)]
Jip(P) = JP(ing)
SPTH = Y(Ca(p))1- x [k3,D' x [1 - QC(PT)] - k4,D' xQC(PT)]
Jpc(P) = k4,p x CP(p)
Jcp(P) = k3,p x QP(c)
The absorption of phosphate across the intestine [Jip(P)] was assumed to be identical to the phosphate intake [JP(ing)]. This is equivalent to setting JP(ing) at 0.7 of its control value, in agreement with the observation that 70% of the ingested phosphate is absorbed over a range of 0.81–3.5 mmol/h (13). The movements of phosphate in [Jpb(P)] and out [Jbp(P)] of the bone are tightly coupled to those of calcium [Jpb(Ca) and Jbp(Ca), respectively] by a stoichiometry [Stoic(Ca/P)] of 0.464, which is the ratio P/Ca in hydroxyapatite.
The following auxiliary equations were used to compute the variables and parameters defined above:
Cathr = 1.95 + Y(PTH(p))2+
CaT = Cathr/CCa(p)
Pthr = Y(PTH(p))2- x QTP(k)
PT = Pthr/CP(p)
k1,D' = Yi(D(p))1+
k2,D' = Yi(D(p))1-
k3,D' = YPT(D(p))1+
k4,D' = YPT(D(p))1-
k1,p' = Yk (Jext)1+
Acknowledgments
Footnotes
This work was supported by Association for Endocrine Oncology.
Abbreviations: ECC, Extracellular fluid compartment; UCaTH, threshold for urinary calcium excretion.
Received November 21, 2001.
Accepted June 13, 2002.
References
s. J Biol Chem 276:165–171
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