Renin-angiotensin system model

In this study, we use the RAS model with the female-specific parameter set from [19]. The model variables consist of the systemic concentrations of AGT, Ang I, Ang IV, Ang (1–7), AT1R-bound Ang II, and AT2R-bound Ang II. AT1R-bound Ang II inhibits renin secretion through a feedback loop. For a detailed description of the underlying RAS model used in this study, see [18,19].

Impact of estrogen decline in aging women on the RAS.

Estrogen levels impact the RAS by activating and inhibiting various key processes. Specifically, estrogen activates AGT production and AT2R binding, while inhibiting renin secretion, ACE activity, and AT1R binding [8]. These processes are illustrated in Fig 1.

After the onset of menopause, estrogen levels decrease dramatically. To model this decline in estrogen levels, we use the estrogen model for post-menopausal women from [20]. The normalized level of estrogen, denoted by E, is modeled through explicit time dependence so that

(1)

where t is the time in years (i.e., age), years is the age of onset of estrogen decline, and years is a parameter fit to normalized estrogen data from [21] that determines the slope of estrogen decline in relation to women’s final menstrual period (see [20] for details). Note that before , E = 1 so that estrogen is assumed to remain at baseline level pre-menopause. Indeed, estrogen levels vary during the menstrual cycle [22]. In this study, we do not consider variations in estrogen over the menstrual cycle in pre-menopausal women and consider baseline estrogen level pre-menopause as the average level through the full cycle.

To include the impact of estrogen on the RAS, we use functions of the form:

(2)(3)

to model activation (g⁺) or inhibition (g^–) with

where i denotes the target mechanism, is a parameter that is determined for each relevant target mechanism, and E is given by Eq. 1. These are Hill-type activation and inhibition functions, normalized such that when E = 1, for any value of . Thus, at baseline estrogen levels, there are no changes.

The enzyme renin is secreted by the juxtaglomerular cells of the kidneys to convert AGT to Ang I. Estrogen inhibits renin levels [8,23,24]. We let be the impact of estrogen on renin secretion. The parameter value for was chosen based on experimental results from [24] which compared plasma estrogen to plasma renin levels under baseline conditions and estrogen injection. Renin secretion is modeled by

(4)

where is baseline renin secretion and is the feedback of [AT1R-bound Ang II] on renin secretion from [18]. Plasma renin concentration (PRC) is modeled by

(5)

where denotes the half life of renin as in [18].

AGT is synthesized in the liver and is stimulated by estrogen [8,23,25]. In post-menopausal women, AGT levels decline from pre-menopause levels [23]. To model the impact of estrogen on AGT, we multiply the production of AGT, denoted by , by . The value of was based on rat experimental data from [25] which quantified the increase in hepatic AGT mRNA levels under estrogen injection [25]. AGT concentration is modeled by

(6)

where

(7)

is plasma renin activity (PRA) with parameter and is the half-life for AGT as in [18]. The equation for PRA is a slight modification from the equation used in [18,19] to include the impact of AGT on PRA as well. The parameter was fitted to predict the same model output in the original blood pressure model from [19].

ACE converts Ang I into Ang II. Researchers have reported that both ACE expression and activity are inhibited by estrogen in rodent models [26–29]. We let represent the inhibition of estrogen levels on ACE activity. The parameter is based on differences in ACE activity under estrogen injection in ovariectomized rats, as reported in [27]. Ang I can be converted independently of ACE into Ang II by chymases (chym) [30] or into Ang (1–7) by neprilysin (nep) [31]. Ang II can be converted to Ang (1–7) by angiotensin-converting enzyme 2 (ACE2), or into Ang IV by aminopeptidases (II, IV) [32]. The concentration of Ang I and Ang II are modeled by

(8)(9)

where denotes the corresponding enzyme activities and and are the half lives of Ang I and Ang II, respectively, as in [18]. is an input used for Ang II infusion (see Section Parameter fitting to PTH and Ang II infusion experiments) and is set to 0 in simulations with no Ang II infusion.

Ang II is reported to remain around the same in aging humans [33]. Ovariectomized rats showed a small decrease in plasma Ang II levels, which was reversed by estrogen replacement at pre-ovariectomy levels [34,35]. We note that Ang II levels are often reported in post-menopausal women on estrogen replacement therapy, but researchers have found significant differences in the impact of estrogen replacement therapy on the RAS when taking via oral or transdermal therapy [36]. This is likely due to the high exposure to estrogen in the liver when ingesting oral estrogen.

Ang II binds to AT1R and AT2R, triggering different physiological responses. Estrogen inhibits Ang II binding to AT1R, but activates Ang II binding to AT2R [8,29,37]. AT1R density is increased over 2-fold in ovariectomized rats [38]. Conversely, AT2R receptor density measured in the left ventricle is significantly reduced in ovariectomized rats, but returns to baseline with estrogen replacement [37]. We based our parameter value for and on these changes in receptor densities. AT1R-bound Ang II ([AT1R-AngII]) and AT2R-bound Ang II ([AT2R-AngII]) concentrations are modeled by

(10)(11)

with and denoting the half-lives of AT1R-bound Ang II and AT2R-bound Ang II from [18]. and denote the binding rates of Ang II to AT1R and AT2R, respectively, as in [18]. and represent the impact of estrogen on the respective binding rate.

We chose parameter values for based on reported changes in the respective RAS levels in the literature as described above. The values for for each RAS component impacted by estrogen levels are listed in Table 1. Model simulations of the estrogen-RAS model in isolation are shown in S2 Appendix.

Download:

• PNG
  larger image
• TIFF
  original image

Table 1. Values of the parameters that determine the impact of estrogen on renin-angiotensin system components. The parameters are unitless.

https://doi.org/10.1371/journal.pcbi.1013101.t001

RAS inhibitor simulations.

To simulate the RAS inhibitors, we use the same approach as in [18]. Specifically, to simulate treatment with an angiotensin receptor blocker (ARB), we let

(12)

where is the baseline value for and is the percent inhibition of AT1R binding by the ARB. Similarly, to simulate treatment with an ACE inhibitor (ACEi), we let

(13)

where is the baseline value for and is the percent inhibition of ACE by the ACEi.

Calcium homeostasis and bone biology model

The mathematical model developed by Peterson and Riggs [13] was chosen as the basis for the calcium homeostasis and bone remodeling components of the composite model because it includes detailed bone, renal, and blood compartments, and predicts levels of PTH, calcitriol, bone cell activity, calcium, and phosphate. We further incorporated specific additions from published model extensions [14,16]:

• Gaweda et al. [16] extended the original model to include additional regulatory mechanisms governing calcium and phosphate homeostasis. In this study, we included the impact of phosphate levels on parathyroid gland capacity as added by Gaweda et al. [16] (equation for H_5,11 in [16]). Note that Gaweda et al. [16] used the Peterson and Riggs [13] model as a basis for their extensions so all other parameters and biological components are the same as in [13].
• To evaluate long-term structural changes within the bone compartment, we incorporated the bone mineral density (BMD) compartment introduced by Riggs et al.[14].
• To investigate estrogen-driven changes in calcium homeostasis and bone remodeling, we integrated the estrogen effects added in Riggs et al.[14]. The authors included the impact of estrogen on latent and active TGF-, osteoblast signaling, and renal calcium handling. TGF- is a key cytokine in bone remodeling that regulates osteoblast and osteoclast proliferation and activity. Here, TGF- refers to canonical transforming growth factor- signaling as represented in the underlying Peterson–Riggs framework [13], rather than the broader TGF- superfamily including bone morphogenetic proteins. Estrogen inhibits latent TGF- formation, while stimulating its activation. TGF- impacts bone remodeling activity through stimulating the expression of RANK on osteoclasts, stimulating osteoclast apoptosis, stimulating pre-osteoblast formation, and inhibiting osteoblast proliferation. Estrogen directly inhibits both the formation of pre-osteoblasts as well as the apoptosis rate of osteoblasts. Riggs et al. [14] also included the impact of estrogen on renal calcium reabsorption by decreasing renal calcium reabsorption with declining estrogen levels post-menopause. All of these estrogen effects are included in the model used in this study.

A model schematic is shown in Fig 1. The model equations are listed in S1 Appendix.

Composite calcium homeostasis, bone remodeling, and RAS model

To formulate a composite model, we connected the estrogen-RAS model (Section Renin-angiotensin system model) with the calcium homeostasis and bone biology model (Section Calcium homeostasis and bone biology model) through known regulatory mechanisms. Specifically, the composite model includes the impact of AT1R-bound Ang II and calcium on PTH secretion (Section AT1R-bound Ang II and calcium impact on PTH secretion), the impacts of extracellular calcium levels and the calciotropic hormones on renin secretion (Section Calciotropic hormones, calcium, and renin secretion), and the impact of AT1R-bound Ang II on bone remodeling (Section AT1R-bound Ang II activation of RANKL and OPG).

To model those interactions, we follow the approach in [13] and characterize the changes in responses influenced by a given stimulant using hyperbolic functions:

(14)

(15)

where x denotes the stimulus variable, is the maximum anticipated response, is the steepness of the response (note when the functions are sigmoidal), is the value of x that produces the half-maximal response, and is the minimum anticipated response. The minus or plus superscript in H denotes whether the response is a decrease or increase from the baseline steady state, respectively. Note that when , the response is a classical response often used in modeling in pharmacology (see [39]). These hyperbolic response functions were chosen because they provide a simple phenomenological representation of saturating physiological regulation with bounded responses. The parameters , , , and allow flexible control of maximal/minimal response, sensitivity, and response steepness while maintaining smooth nonlinear behavior. This formulation is also consistent with the approach used in the underlying Peterson and Riggs framework [13]. Parameter values for the equations added in this study are listed in Table 2. The newly introduced parameters in Tables 1, 2 represent the principal coupling (‘crosslinking’) parameters connecting the estrogen-RAS subsystem with the calcium homeostasis and bone remodeling subsystems. These parameters govern the strength and sensitivity of the inter-system regulatory interactions introduced in this study. Additional parameter values can be found in [13,14,16] or using our publicly available model code.

Download:

• PNG
  larger image
• TIFF
  original image

Table 2. Parameter values for coupling the RAS components to the calcium homeostasis and bone remodeling model. Additional parameters can be found in [13,14,16].

https://doi.org/10.1371/journal.pcbi.1013101.t002

AT1R-bound Ang II and calcium impact on PTH secretion.

AT1R-bound Ang II concentration, denoted by [AT1R-AngII], increases PTH secretion from the parathyroid gland [6]. To model this effect, we let denote the impact of AT1R-bound Ang II on PTH secretion:

(16)

where the parameters correspond to the function form of type Eq. 15.

Conversely, PTH secretion is inhibited by extracellular calcium levels ([Ca]) [44]. To model this effect, we let be of the form of Eq. 14:

(17)

PTH concentration ([PTH]) in the blood is modeled by

(18)

where is the secretion of PTH from the parathyroid gland and is degradation as in [13]. The parameters for Eqs. 16 and 17 were fit using the Ang II only experiment data from Grant et al. [40] (Section Parameter fitting to PTH and Ang II infusion experiments). The parameter values are listed in Table 2.

Calciotropic hormones, calcium, and renin secretion.

Renin secretion, denoted by , is impacted by [PTH], calcitriol ([Ctriol]), and [Ca]. Specifically, increased [Ca] and [Ctriol] inhibit renin secretion, while [PTH] stimulates renin secretion [6].

Extracellular calcium levels are detected by the calcium sensing receptor (CaSR) in the juxtaglomerular cells [45]. To model the impact of CaSR on , we let:

(19)

Ortiz-Capisano et al. [41,42] found that renin release was 40% lower when using a calcimimetic or at elevated calcium levels in the juxtaglomerular cells of rat kidneys. We used this result to estimate parameters so that at high calcium levels, renin release is decreased by 40%.

To model stimulation of by [PTH], we let:

(20)

The parameters (listed in Table 2) were fit using the PTH only infusion experiment from Grant et al. [40] (Section Parameter fitting to PTH and Ang II infusion experiments).

To model inhibition of by calcitriol, we let:

(21)

The parameters were selected to produce a physiologically consistent inhibitory relationship between calcitriol and renin secretion guided by the trends reported in Tomaschitz et al. in human participants [43]. Taken together, we incorporate the combined impact of calcium, PTH, and calcitriol on renin secretion by modifying Eq. 4 to

(22)

AT1R-bound Ang II activation of RANKL and OPG.

Shimizu et al. [7] reported that in ovariectomized rats, infusion of Ang II increased RANKL expression and led to a significant decrease in bone volume, but this impact was abolished by treatment with an AT1R blocker and not an AT2R blocker. Their results showed that RANKL production is activated by AT1R-bound Ang II levels. Additionally, Shimizu et al. [7] found that OPG expression is also increased in osteoblast cells due to infusion of Ang II, but to a much lesser extent than RANKL.

The impact of AT1R-bound Ang II on RANKL and OPG expression is modeled using the form in Eq. 15 for activation so that:

(23)(24)

These are included in the equations for the production of RANKL and OPG, respectively (Eqs. F.6 and F.8 in S1 Appendix). The values for and represent the highest possible amount of activation of RANKL and OPG by AT1R binding, respectively. These values were set to the fold-change in RANKL and OPG expression after Ang II infusion in Shimizu et al. [7]. The parameter values are listed in Table 2.

Sobol sensitivity analysis

Mathematical models of complex biological systems, like the one used in this study, involve a large number of parameters with unknown or uncertain values. To investigate the impact of variations in parameter values on model output, we can use methods such as sensitivity analysis [46,47]. The Sobol analysis was also used to assess the robustness of model outputs to uncertainty and variability in parameter values. Global sensitivity analysis methods evaluate the model at many points in the parameter space, providing an assessment of how individual parameters contribute to variations in model output across the parameter space, including interactions with other parameter changes [46–48].

The Sobol method is a variance-based global sensitivity analysis technique that decomposes the variance in the model output into contributions from each individual parameter as well as its interactions with other parameters [46,49]. The results of a Sobol sensitivity analysis are called “Sobol sensitivity indices” (or “Sobol indices” for short) and are computed for each parameter included in the analysis. First-order Sobol indices (denoted by ) are the fraction of the variance in the model output the individual parameter contributes to. Total order Sobol indices (denoted by ) quantify the fraction of output variance the parameter contributes to, including interactions with other parameter changes. By evaluating Sobol indices, we can identify which parameters are the most influential on model output. For more information about Sobol sensitivity analysis and its use in applications in biology, we refer readers to [46–49].

We conducted a Sobol sensitivity analysis using simulations of the full composite model (described in Section Composite calcium homeostasis, bone remodeling, and RAS model). The model outputs we considered were bone mineral density (Eq. F.13 in S1 Appendix), AT1R-bound Ang II (Eq. 10), PTH (Eq. 18), calcitriol (Eq. D.3 in S1 Appendix), extracellular calcium (Eq. B.1 in S1 Appendix), and renin concentration (Eq. 5) over a simulation time from 20 to 80 years with age-related estrogen decline. We computed Sobol indices at every 10 years of simulation time.

The full composite model contains 150 parameters, making a Sobol sensitivity analysis over the full parameter set computationally expensive and prone to numerical instability. To make the analysis computationally tractable, we used a two-step procedure.

First, we conducted a Sobol sensitivity analysis varying the 121 parameters belonging to the calcium homeostasis and bone remodeling subsystem by ±10%. These parameters included those governing bone remodeling, PTH secretion and degradation, calcitriol synthesis and degradation, renal calcium and phosphate handling, intestinal calcium and phosphate absorption, and extracellular calcium and phosphate levels. Parameters with first-order Sobol indices for at least one output variable at any simulation time point were retained. This screening step identified 30 sensitive calcium/bone subsystem parameters.

Second, we combined these 30 retained parameters with all newly introduced parameters from Tables 1 and 2, including all estrogen-RAS and RAS-calcium-bone coupling parameters, yielding 59 parameters in the final Sobol sensitivity analysis. Thus, no newly introduced coupling parameters were excluded during the screening step. The final Sobol analysis varied parameters by ±25% of baseline values. The results of the Sobol sensitivity analysis are described in the Sensitivity analysis section.

PTH and Ang II infusion experiments simulations

We conducted simulations of infusion of (i) Ang II, (ii) PTH, and (iii) PTH and Ang II together (described in the Parameter fitting to PTH and Ang II infusion experiments section). Model results, together with the experimental data reported for the clinical studies done by Grant et al. [40], are shown in Fig 2.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 2. Simulation results for infusion of Ang II (“Ang II only”), PTH (“PTH only”), and Ang II and PTH together (“Ang II + PTH”).

Shown are trajectories for (A) plasma renin activity, (B) angiotensin II plasma concentration, (C), intact parathyroid hormone plasma concentration, and (D) angiotensin receptor type 1-bound Ang II during and following the 1 hour infusion period. Lines indicate simulations, while the dots and error bars in panels (A–C) are from the relevant experimental data presented in Grant et al. [40]. Ang II: angiotensin II, AT1R: angiotensin receptor type 1, iPTH: intact parathyroid hormone, PRA: plasma renin activity.

https://doi.org/10.1371/journal.pcbi.1013101.g002

When Ang II is infused in isolation, Ang II levels rise, resulting in an increase in AT1R-bound Ang II levels as receptor occupancy increases in response to the higher circulating ligand concentration (Fig 2(B), 2(D)). This stimulates PTH secretion so that endogenous PTH (intact PTH, denoted by [iPTH]) increases (Fig 2(C)). Grant et al. [40] reported that during an Ang II infusion, [iPTH] increased, while PRA decreased slightly likely due to the negative feedback on renin secretion by AT1R-bound Ang II (Fig 2 “Ang II only” data). Model simulations capture both the rise in [iPTH] (Fig 2(C) “Ang II only”) and slight decrease in PRA (Fig 2(A) “Ang II only”) during infusion.

When infusing PTH in healthy volunteers, PRA nearly doubled on average (Fig 2(A) “PTH only” data) [40]. Model simulations predicted a 69% increase in PRA (Fig 2(A) “PTH only”). This difference may reflects the absence of FGF-23 in the model. PTH increases circulating FGF-23 levels [50], which subsequently suppresses calcitriol, thereby increasing renin secretion [51]. Including FGF-23 effects could help strengthen and sustain the PTH induced renin increase. While total PTH levels increased substantially during the PTH infusion experiment, endogenous PTH production steadily decreased. This effect is captured by the model (Fig 2(C) “PTH only” data).

When infusing both Ang II and PTH together in healthy volunteers, the authors reported that Ang II levels were similar to those reported in the Ang II only experiment, likely due to PTH having only a small impact on Ang II levels [40]. In our model simulations, Ang II levels were slightly higher in the Ang II and PTH infusion experiment than the Ang II only experiment, but were still within the range reported by Grant et al. [40] at the end of the 1 hour infusion time (Fig 2(B) “Ang II + PTH”). Grant et al. [40] reported that PRA levels remained near baseline for the Ang II and PTH infusion experiment. In our model simulations, the “Ang II + PTH” simulations resulted in a transient increase in PRA but declines to near the levels prior to infusion by the end of the simulation time (Fig 2(A)). This discrepancy may stem from unmodeled PTH receptor dynamics. In vivo, Ang II suppresses renin release by binding to juxtaglomerular AT1R, while PTH stimulates renin release by binding to juxtaglomerular parathyroid hormone receptor 1 (PTH1R) [52]. However, the current model lacks PTH receptors and instead assumes that PTH directly stimulates renin secretion, whereas Ang II must first undergo receptor binding. Consequently, the stimulation by PTH is faster acting, which may resulting in the observed transient increase in renin secretion before it is counteracted by the AT1R-bound Ang II mediated inhibition.

Impact of estrogen decline and RAS inhibitors

How do declining estrogen levels in post-menopausal women impact the RAS? How do changes in the RAS and declining estrogen levels contribute to altered bone remodeling? Does bone mineral density improve with RAS inhibition? To investigate these questions, we conducted simulations of the composite calcium homeostasis, bone remodeling, and estrogen-RAS model with estrogen decline observed in post-menopausal women with and without drug interventions.

In each simulation, estrogen decline starts at 50 years, which is about the average age of menopause onset in women. We conducted simulations with key RAS inhibitors (ACEi and ARB) starting at age 60 years, unless otherwise indicated. For ARB simulations, we let (Eq. 12) vary from 88–98.3% as reported for different types and typical dose levels of ARBs by Hallow et al. [18]. For ACEi, we conducted simulations with (Eq. 13) varying from 94–97.2% as reported for common types of ACEi and doses by Hallow et al. [18]. The results are shown in Figs 3 and 4.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 3. Impact of post-menopausal estrogen decline and renin-angiotensin system inhibitors on the renin-angiotensin system.

For the angiotensin receptor blocker simulations (labeled “ARB” in the legend), (Eq. 12) was varied between 88–98.3% based on the ranges reported in [18]. For the angiotensin-converting enzyme inhibitor simulations (labeled “ACEi” in the legend), (Eq. 13) was varied between 94–97.2% as reported in [18]. Lines represent the mean values over the varied values for the respective treatment with shaded area representing the range. AGT: angiotensinogen, Ang I: angiotensin I, Ang II: angiotensin II, AT1R: angiotensin type 1 receptor, AT2R: angiotensin type 2 receptor, RAS: renin angiotensin system, RASi: RAS inhibitor, ARB: angiotensin receptor blocker, ACEi: angiotensin-converting enzyme inhibitor.

https://doi.org/10.1371/journal.pcbi.1013101.g003

Download:

• PNG
  larger image
• TIFF
  original image

Fig 4. Impact of post-menopausal estrogen decline and RAS inhibitors on (A) bone mineral density, (B) osteoclast levels, (C) RANK-RANKL levels, and (D) RANK-OPG levels.

The values are normalized to their baseline levels and shown as a percent of the baseline value. For the angiotensin receptor blocker simulations (labeled “ARB” in the legend), (Eq. 12) was varied between 88–98.3% based on the ranges reported in [18]. For the angiotensin-converting enzyme inhibitor simulations (labeled “ACEi” in the legend), (Eq. 13) was varied between 94–97.2% as reported in [18]. Dashed lines represent the mean values over the varied values for the respective treatment, with shaded areas representing the corresponding ranges. Data in panel (A) are from the female data in Looker et al. [53]. Both simulations and data are for femoral neck bone mineral density. BMD: bone mineral density, RANK: receptor activator of NF-kappaB, RANKL: RANK ligand, OPG: osteoprotegerin, RASi: RAS inhibitor, ARB: angiotensin receptor blocker, ACEi: angiotensin-converting enzyme inhibitor.

https://doi.org/10.1371/journal.pcbi.1013101.g004

ACEi and ARB inhibit ACE activity and AT1R binding, respectively. While these are different components of the RAS, both drugs result in reduced levels of AT1R-bound Ang II (Fig 3(E)). The impact of these drugs on the various components of the RAS are drug and dose-dependent. The varied range in values of had a negligible effect on our model simulation results (Fig 3, “ACEi”). In the ARB simulations, the impact on the RAS has a wider variation based on the value of (Fig 3, “ARB”). Note that we simulated a wider range for than .

We first consider the impact of ACEi and ARB on the RAS components. ARB directly inhibits the binding of Ang II to AT1R. As a result, AT1R-bound Ang II levels decreased by about 66–89%, depending on efficacy (Fig 3(E), “ARB” vs. “no RASi”). Recall that AT1R-bound Ang II inhibits renin secretion (Fig 1). Hence, decreased AT1R-bound Ang II levels result in increased renin levels (Fig 3(B)) as well as increased downstream Ang I (Fig 3(C)) and Ang II levels (Fig 3(D)). The lower AT1R binding along with increased Ang II levels results in more Ang II binding to AT2R, so that AT2R-bound Ang II levels increase by about 4-fold on average (Fig 3(F) “ARB” vs. “no RASi”’). ACEi impact the RAS by inhibiting the activity of ACE, which is the enzyme that converts Ang I into Ang II (Fig 1). Ang II levels decreased by an average of 52% in the ACEi simulations despite increased Ang I levels (Fig 3(D), 3(C) “ACEi” vs. “no RASi”). As a result, both AT2R-bound Ang II and AT1R-bound Ang II decreased by about 52% from the no RAS inhibitor case (Fig 3(E), 3(F), “ACEi” vs. “no RASi”).

Next, we analyze the age-related estrogen decline effect (S2 Appendix) on bone mineral density and its response to ACEi and ARB. Estrogen maintains bone density by regulating factors involved in bone remodeling that promote the differentiation and activation of osteoblasts, while inhibiting osteoclast activity (Fig 1). When estrogen levels decline, osteoclast activity increases, which increases bone resorption, ultimately resulting in a decrease in bone mineral density (Fig 4(A), 4(B) “no RASi”). The bone mineral density simulations results with the decline in estrogen levels is consistent with the femoral neck BMD data reported by Looker et al. for age-stratified women, as measured by dual-energy X-ray absorptiometry [53] (Fig 4(A) “no RASi” data). This decrease is a result of a combination of the direct impacts of estrogen on bone remodeling and increased AT1R-bound Ang II levels, which stimulates RANKL production to bind with RANK (Fig 4(C); Fig 3(E) “no RASi”). RANK-RANKL stimulates osteoclast differentiation and activation. Throughout the estrogen decline simulations, the calciotropic hormones, PTH and calcitriol, as well as key electrolytes, calcium and phosphate, remained within normal range (Fig B in S3 Appendix).

Age-related decreases in estrogen levels result in a 4.5-fold increase in osteoclast activity (Fig 4(B) “no RASi”). However, with ARB, AT1R-bound Ang II levels decrease (Fig 3(E)), resulting in reduced RANKL production, so that osteoclast levels return to about 40% above baseline levels (Fig 4(B), 4(C)). As a result, by the end of the simulation time (i.e., t = 80 years), bone mineral density is at about 90% of baseline for ARB users versus 75% before estrogen decline with no treatment (Fig 4(A) “ARB” vs. “no RASi”). ACEi also decreased AT1R-bound Ang II levels by a lesser amount (Fig 3(E)) so that osteoclast levels are lowered to about 2-fold baseline levels. This results in a slight improvement in bone mineral density to about 85% of the level before estrogen decline (Fig 4(A) “ACEi”). We observed that the range in the amount of AT1R-bound Ang II inhibition due to varied ARB efficacy did not have a large impact the change in bone levels on these drugs (Fig 3, Fig 4). Thus, the improvement to bone mineral density may not depend significantly on the dose of the RAS inhibitors within their typically prescribed ranges. This does not indicate a lack of dose dependence in the BMD response, but rather that the dependence on dose is not significant within the pharmacologically relevant RAS inhibition ranges.

We conducted simulations of RAS inhibitor treatment starting at different ages with age-related estrogen decline as in Eq. 1. In these simulations, we let and based on the average value for the respective drugs from [18]. We did this because varying did not have much impact on bone mineral density (Fig 4(A)). The results are shown in Fig 5. In all simulations, bone mineral density improved on both the ARB and ACEi drugs with a greater improvement with ARB treatment. When the administration of ARB or ACEi was started before the onset of estrogen decline (i.e., starting a RAS inhibitor before 50 years old), bone mineral density increased from baseline by about 5% (Fig 5(A), 5(C). When estrogen decline started at 50 years, the bone mineral density decreased, but stays above 95% for patients on ARB (Fig 5(A)). When the administration of ACEi began before estrogen decline, bone mineral density declined below 90% of baseline during age-related estrogen decline (Fig 5(C)). Our model simulations suggest that early RAS inhibition prevents bone loss driven by age-related estrogen decline.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 5. Model simulation results for varied starting age of angiotensin receptor blocker (A–B) and angiotensin-converting enzyme inhibitor (C–D). in all simulations for ARB and in all ACEi simulations.

The bone mineral density is femoral neck bone mineral density. AT1R: angiotensin type 1 receptor, ARB: angiotensin receptor blocker, ACEi: angiotensin-converting enzyme inhibitor, RASi: RAS inhibitor.

https://doi.org/10.1371/journal.pcbi.1013101.g005

In summary, the RAS, calcium, and bone remodeling model developed in this study represents the expected decrease in bone mineral density due to age-related estrogen decline. This is driven by both the direct impacts of estrogen on bone remodeling and calcium regulation, as well as RAS activation by estrogen decline. Model simulations suggest that ARBs offer stronger protection for bone health than ACEi, and that result is consistent when starting treatment at different ages.

Sensitivity analysis

We conducted a Sobol sensitivity analysis for the full composite model during age-related estrogen decline in women (see Sobol sensitivity analysis section for details). The parameters with the largest average values in Fig C(A) in S3 Appendix are the parameters that have the largest impact on variation in bone mineral density. The top five parameters, based on the average value, are all related to the regulation of RANK and RANKL (Fig C(A) in S3 Appendix). The RANK-RANKL complex directly impacts bone remodeling by stimulating osteoclasts (Fig 1). Specifically, the parameters and are baseline production rates of RANKL and RANK, respectively. The parameter relates to the impact of PTH on RANKL production. The parameters k₃ and k₄ are the binding and unbinding rate of the RANK-RANKL complex, respectively. The next parameter with the largest impact on bone mineral density is , which represents the minimum impact of AT1R on RANKL production (Eq. 23), thereby linking the RAS to bone remodeling. represents the sensitivity of renin secretion to calcium levels. This is the parameter with the largest impact on [AT1R-bound Ang II] (Fig C(B) in S3 Appendix).

The newly introduced parameters in Tables 1 and 2 represent the principal coupling (‘crosslinking’) parameters connecting the estrogen-RAS subsystem with the calcium homeostasis and bone remodeling subsystems. These parameters govern the strength and sensitivity of the inter-system regulatory interactions introduced in this study. In summary, the top parameters that impact bone mineral density all primarily relate to RANK, RANKL and RANK-RANKL binding.

AT1R-bound Ang II is the key biological component that connects the RAS with the calcium homeostasis and bone remodeling model (Fig 1). The parameters with the 5 largest average and values for the model output of [AT1R-bound Ang II] are , , , , and (Fig C(B) in S3 Appendix). represents half-maximal response of renin secretion to calcium levels, linking the calcium regulatory system to the RAS. represents the baseline fluxes of calcium from the plasma to the bone, impacting variation in calcium and renin levels (Fig D in S3 Appendix). Calcium level changes affect the RAS via the CaSR in the renin-secreting juxtaglomerular cells of the kidneys. , , and are all directly related to the RAS (Section Impact of estrogen decline in aging women on the RAS). Specifically, determines baseline renin secretion, is the binding rate for AT1R, and is baseline conversion rate from renin concentration to activity. Our sensitivity analysis shows that variations in [AT1R-bound Ang II] are driven by the CaSR impact on renin secretion and the RAS cascade.

The calciotropic hormones are the primary regulators of calcium and bone remodeling. Additionally, these hormones have a bidirectional relationship with the RAS (Fig 1). In our Sobol sensitivity analysis, the parameters with the largest impact on PTH were , , , and (Fig E(A) in S3 Appendix). represents the maximal stimulation by PTH on 1-hydroxylase, the enzyme that synthesizes calcitriol. and characterize the maximal effect and half-maximal concentration for phosphate mediated calcitriol inhibition. PTH and calcitriol regulate one another, as calcitriol inhibits PTH secretion and PTH stimulates calcitriol production (Fig 1). Both PTH and calcitriol are sensitive to the parameter which is the baseline production rate of 1-hydroxylase, and to the parameters and , which denote the baseline rate of calcium exchange between the bone and plasma and vice versa (Fig E in S3 Appendix). The latter is likely due to the sensitivity of PTH to calcium levels, which are impacted by these two parameters as well (Fig D(A) in S3 Appendix).

While TGF is an important regulator of osteoblast and osteoclast activity in the model, parameters associated with TGF signaling did not rank among the most influential Sobol indices for the outputs considered here. This does not imply that TGF is physiologically unimportant; rather, within the parameter ranges and simulation conditions studied, variation in RANK/RANKL regulation and calcium homeostasis contributed more strongly to output variance.

The Sobol sensitivity analysis results shown here are the average (bar) and standard deviation (error bar) of and values over the full simulation time from age 20–80 years. We also analyzed the results over the simulation time from age 20–50, i.e., before estrogen decline. The parameters with the largest impact on model output were the same as those for the full simulation time. This suggests that the dominant sources of output variance arise from parameters governing the core calcium, RAS, and bone remodeling pathways, whereas the parameters mediating estrogen-dependent regulation have comparatively smaller contributions to global output variance within the parameter ranges considered.

This does not indicate that estrogen has little physiological influence in the model. Rather, it indicates that the model outputs are relatively less sensitive to the parameters mediating the estrogen impacts than to parameters governing the core RAS, bone and calcium pathways. During the menopause simulations used in the Sobol analysis, estrogen declines significantly over time, producing a substantial system-wide effect. The presence of estrogen decline itself is the dominant driver of the postmenopausal changes, while variation in the parameters controlling the precise shape of the estrogen impact produces comparatively smaller changes in model output at the global scale considered in the Sobol analysis.

In summary, our global sensitivity analysis shows that, in our model, bone remodeling is most sensitive to RANK-RANKL binding and that the RAS is sensitive to both calcium levels and the calciotropic hormones. These are the key biological components that impact RAS levels and bone remodeling.

Clinical perspectives

Although estrogen replacement therapy (or hormone replacement therapy) can effectively manage menopause symptoms and post-menopausal osteoporosis, its benefits must be weighed against the heightened risks of blood clots, stroke, and breast cancer, especially with long-term use [56,57]. Furthermore, side effects such as weight gain, diarrhea, and mood changes can affect patient well-being and adherence [57]. The balance of risks and benefits associated with hormone replacement therapy varies considerably based on individual physiology, time since menopause onset, and the specific type of hormone replacement therapy [56]. Other current osteoporosis treatments, including antiresorptives and osteoanabolic therapies, have limitations on their duration of use, typically ranging from one to five years [58]. Therefore, the development of alternative long-term osteoporosis treatments are needed to broaden the range of effective options for patients. Our mathematical modeling analysis presented in this study supports clinical evidence suggesting that ARBs may contribute to maintaining bone health during post-menopausal estrogen decline by reducing AT1R-bound Ang II levels. An important future extension would be to investigate hormone replacement therapy, including the impact of treatment timing, dose, and administration route on the coupled RAS–calcium–bone remodeling system.

Indeed, there are a number of clinical applications that this model could be extended to study. The RAS is a key hormone system in blood pressure regulation. The RAS model used in this study is part of a larger model that involves whole-body blood pressure regulation [18,19]. Using a similar approach to [59], the model presented in this study may be incorporated into the blood pressure model [19] through the RAS to investigate the interactions of hypertension and bone remodeling. Chronic kidney disease (CKD) is a disorder of the loss of kidney function that affects over 10% of the global population [60]. Many patients with CKD experience CKD-metabolic bone disorder, where their impaired renal function results in loss of bone mineral density. Previous mathematical modeling studies have used the underlying calcium homeostasis and bone remodeling model to study this disorder [15,16,61]. For example, in [15], the authors simulated a progressive loss of renal function using an exponential decline in glomerular filtration rate (GFR). However, none of these studies have considered the RAS. Many CKD patients take RAS inhibitors to protect their kidney function, prevent hypertension, and have been shown to improve bone health [62]. Furthermore, CKD impacts the RAS, leading to decreased conversion of Ang I to Ang1–7 via neprilysin and increased conversion of Ang I to Ang II via ACE [63]. The model presented in this study may be used to investigate how RAS inhibitors may impact bone density and electrolyte balance in patients with CKD, by incorporating CKD mechanisms from [15,16,61] with additional CKD effects on the RAS [63].

Limitations and future directions

In this study, we developed a mathematical model that describes the interactions between the RAS, calciotropic hormones, calcium regulation, and bone remodeling. One major limitation of this model is the large number of unknown parameters, resulting in uncertainty in the model outcomes. In this study, we analyzed the impacts of variations in parameter values on model output by conducting a global sensitivity analysis using the Sobol method. However, we do recognize that there may be parameters and mechanisms that are not included or could be changed in this model.

Several physiological quantities shown as fixed values in Fig 1 were treated as prescribed inputs or baseline parameters rather than dynamically regulated state variables. This modeling choice was made to limit model complexity and parameter uncertainty in this first integrated framework. Additionally, the present study focused on a representative post-menopausal estrogen decline trajectory rather than investigating inter-individual variability in menopause timing or decline rate. Future extensions could incorporate alternative estrogen decline regimes and dynamic regulation of additional physiological compartments.

In addition, while calcium and phosphate are dynamically regulated within the present framework, we did not investigate dedicated calcium or phosphate perturbation protocols, which could be explored in future studies to further characterize mineral homeostasis within the integrated model.

A worthwhile future direction is to accurately model an aging population. In this study, we conducted simulations that represent the estrogen decline that is seen in post-menopausal women. However, aging populations, including post-menopausal women, have shown other physiological changes related to the systems in the model presented in this study. For example, aging populations have shown a blunted response to ACE inhibitors, have a lower GFR, decreased intestinal calcium absorption, lower vitamin D levels, and higher PTH levels [64–66]. These effects are not included in the model developed in this study at this time. Indeed, future work may consider the specific physiological impacts of aging for a more comprehensive description of post-menopausal physiology.

Given the importance of estrogen and using a female-specific RAS, the model presented in this study only represents a female population. However, the same RAS subsystem could be used with the male-specific parameter set from [19]. Sex differences have been found in the RAS [67] and renal handling of calcium [68,69]. Men also can form osteoporosis as they age, which is not driven by the declining estrogen levels found in post-menopausal women. Rianon et al. [70] found that there are sex- and race-specific effects of ACEi on the prevention of age-related bone loss. Specifically, the authors reported that ACEi were effective at protecting against bone loss in African-American elderly men [70]. This is different than reports that did not consider sex and race differences and only reported significant changes in fracture risk in patients using ARB. Notably the systemic study by Wu et al. [12] found no significant difference in fracture risk in male patients on ACEi. The model presented in this study may be extended to consider how underlying sex and racial differences may explain altered treatment outcomes [71].

In this study, we do not include the hormone aldosterone, which is connected with the RAS as a part of the renin angiotensin aldosterone system. Aldosterone is released from the adrenal cortex in response to stimulation by AT1R-bound Ang II [72]. Researchers have shown that aldosterone also has a bidirectional relationship with the calciotropic hormones, as it both stimulates and is stimulated by PTH [6]. Additionally, aldosterone increases renal calcium reabsorption [73]. Clinical reports have shown that the drug spironolactone (an aldosterone inhibitor), has a significant impact on the bone health of hypertensive patients [74]. Indeed, this hormone may be added to this model to study the impacts of hypertension, spironolactone, hyperaldosteronism, or other relevant conditions and medications on bone density and calcium regulation. Additionally, in this study, we assume that the RAS activity in the bone is the same as the systemic RAS, though this may not always be the case. Experimental evidence has shown that there is a bone-specific RAS [75]. Future work may involve modeling a bone-specific RAS to consider the impacts on bone remodeling similar to the renal-specific RAS models in [17,76].

FGF-23 is produced by osteoblasts and osteocytes and has numerous system wide effects that impact the RAS, bone remodeling, and calcium homeostasis [77]. FGF-23 acts through the fibroblast growth factor receptor/Klotho complex expressed in the kidneys and parathyroid glands. Effects include the suppression of calcitriol, PTH, and phosphate reabsorption. Furthermore, FGF-23 levels have been implicated as a clinically relevant biomarker for osteoporosis. High circulating FGF-23 levels are associated with an increased incidence of fragility fractures and decreased BMD [77]. Future model extensions could incorporate FGF-23 dynamics.

Additionally, the current model assumes that PTH directly stimulates renin secretion. However, experimental evidence has shown that PTH stimulates renin secretion through binding to juxtaglomerular PTH1R receptors [52]. Including explicit PTH1R-mediated signaling dynamics may better capture the complex endocrine feedback mechanisms governing calcium homeostasis and the renin-angiotensin system.

Estrogen levels vary during the different phases of the menstrual cycle in menstruating women [22]. Women who take oral contraceptives also have different estrogen levels [78]. The mathematical model presented here may be adapted to consider how changes in estrogen impacts the RAS, calcium homeostasis, and bone remodeling during the different phases of the menstrual cycle or on oral contraceptives [79]. During pregnancy, estrogen levels rise dramatically, while post-parturition, the estrogen levels drop below pre-pregnancy levels to support lactation [80]. Each of these changes results in major impacts on physiology, including the RAS, calcium homeostasis, and bone remodeling [80,81]. Previously, we developed mathematical models that represent calcium homeostasis in rats that consider sex differences and maternal adaptations of pregnancy and lactation [82], but these models do not include a detailed bone compartment or the impact of the RAS. The model presented in this study may be similarly adapted to consider the major changes that women undergo during pregnancy and lactation. Future work could further evaluate the model against longitudinal menopause-transition cohorts, such as SWAN, that include antihypertensive medication use and bone-related outcomes [83].