A physiologically based mathematical model of integrated calcium homeostasis and bone remodeling

Bone 46 (2010) 49–63

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Bone j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / b o n e

A physiologically based mathematical model of integrated calcium homeostasis and bone remodeling Mark C. Peterson a,⁎,1, Matthew M. Riggs b a b

Amgen, Inc., One Amgen Center Drive, MS 28-3-B, Thousand Oaks, CA 91320, USA Metrum Research Group LLC, 2 Tunxis Road, Suite 112, Tariffville, CT 06081, USA

a r t i c l e

i n f o

Article history: Received 25 February 2009 Revised 24 August 2009 Accepted 26 August 2009 Available online 2 September 2009 Edited by: T. Jack Martin Keywords: Systems biology RANK–RANKL–OPG Parathyroid hormone Endocrine regulation of calcium and bone Bone remodeling

a b s t r a c t Bone biology is physiologically complex and intimately linked to calcium homeostasis. The literature provides a wealth of qualitative and/or quantitative descriptions of cellular mechanisms, bone dynamics, associated organ dynamics, related disease sequela, and results of therapeutic interventions. We present a physiologically based mathematical model of integrated calcium homeostasis and bone biology constructed from literature data. The model includes relevant cellular aspects with major controlling mechanisms for bone remodeling and calcium homeostasis and appropriately describes a broad range of clinical and therapeutic conditions. These include changes in plasma parathyroid hormone (PTH), calcitriol, calcium and phosphate (PO4), and bone-remodeling markers as manifested by hypoparathyroidism and hyperparathyroidism, renal insufficiency, daily PTH 1–34 administration, and receptor activator of NF-κB ligand (RANKL) inhibition. This model highlights the utility of systems approaches to physiologic modeling in the bone field. The presented bone and calcium homeostasis model provides an integrated mathematical construct to conduct hypothesis testing of influential system aspects, to visualize elements of this complex endocrine system, and to continue to build upon iteratively with the results of ongoing scientific research. © 2009 Elsevier Inc. All rights reserved.

Introduction Bone biology is physiologically complex and has been investigated at multiple levels, from the subcellular level, to individual endocrine relationships, as well as at the clinical outcomes level after therapeutic intervention or disease state progression. Works conducted to date on the mechanisms and interdependencies of this system have been well described in the literature. In a piecewise manner, a multitude of publications spanning the last century are present, with quantification of healthy bone dynamics, correlated organ dynamics, related disease sequelae from endocrine imbalances, secondary physiologic changes due to disease progression, and results of therapeutic interventions. However, none of these works assemble a single integrated model that joins the current knowledge of calcium homeostasis with the present understanding of bone biology. In reviewing the literature related to bone physiology, multiple discrete elements of physiology are described. In the healthy individual, Abbreviations: 1-α-OH, 1-alpha hydroxylase; BMD, bone mineral density; BMU, bone morphogenic unit; Ca, calcium; ECF, extracellular fluid; GFR, glomerular filtration rate; NF-κB, nuclear factor-kappa B; OPG, osteoprotegerin; PTH, parathyroid hormone; PO4, phosphate; QD, once daily; RANK, receptor activator of NF-κB; RANKL, receptor activator of NF-κB ligand; RK4, fourth-order Runge–Kutta algorithm; ROB, responding osteoblast; TGF-β, transforming growth factor beta. ⁎ Corresponding author. Fax: +1 858 795 9274. E-mail address: [email protected] (M.C. Peterson). 1 Current address: Biogen Idec, 5200 Research Place, San Diego, CA 92122, USA. 8756-3282/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.bone.2009.08.053

these work in concert to produce stable bones, a homeostatic state of bone turnover, and calcium balance. These elements, and others, work together intricately to maintain bone integrity via paracrine, endocrine, and intracellular mechanisms. Important components include the parathyroid gland and hormone, the kidney, the gut, the osteoclast/ osteoblast cellular team, often referred to as bone morphogenic units (BMUs), as well as proven and purported intracellular controlling mechanisms. Physiologic perturbations of these components, through therapeutic interventions and diseases, are known to impact bone health, both positively and negatively. As these mechanisms of bone regulation become clearer through reports in the literature, such as the more recent work on WNT signaling, a next logical step is to unify the available knowledge into a predictive model of the overall physiologic system. The literature provides several models that quantify aspects of bone biology and related physiology. Raposo et al. published an intricate general model to describe systemic calcium homeostasis [1]. This model included the major organ and system attributes that influence calcium regulation. While the authors demonstrated the model's ability to describe several disease state effects, there were simplifying assumptions in the mathematical construct, including linear assumptions about calcium exchange to and from bone. Subsequently, Lemaire et al. published a quantitative cellular model describing the linking of osteoclasts and osteoblasts [2]. This model unified several of the controlling and linking mechanisms involved in BMU concerted activities building and degrading bone. However, this model did not facilitate changes elicited through calcium homeostatic mechanisms,

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nor did it account for the anabolic nature of once-daily PTH administration. Thirdly, Bellido et al. proposed an intracellular control mechanism that describes differential responses to PTH [3]. This model qualitatively describes the relative activities of the intracellular signaling pathway involving Runx2, Bcl-2, and CREB transcription factors (Runx2–Bcl-2–CREB system) which was purported to influence the differential effects of continuous and intermittently administered PTH on catabolic and anabolic bone changes, respectively. To date, these models have existed independently. The system of bone biology and calcium homeostasis, with the relative wealth of reported physiologic understanding, together with various component models, offers a unique opportunity to develop a physiologically based mathematical model. Similar in concept to the intricately compiled model of the vascular system by Guyton et al. [4], yet simpler in construct, the current model (Fig. 1) incorporates aspects from the subcellular level through clinical outcomes measures of bone dynamics. It joins together three models previously described and provides a kinetic (quantitative) construct of the Runx2–Bcl-2– CREB system. The presented model provides a first iteration of a tool designed to allow exploration of disease states and therapeutic interventions, with the ability to be further refined as the wealth of knowledge in this area expands. Materials and methods Model development paradigm Development of the presented physiologic system model was motivated by a desire to produce a model that was generalized as

much as possible, structured based upon available literature reported observations of the system and hypothesized mechanistic relationships, and able to be extended in the future with new data and understandings. The objective was to construct a model that could predict short- and long-term changes in bone biology and related homeostatic conditions when the system was perturbed by either therapeutic intervention or disease state progressions. A critical consideration when constructing the model was the requirement for physiologic calcium homeostasis maintenance as required to sustain life while predicting changes in bone. Three previously published models [1-3] emerged as underpinning works for constructing the broader mathematical aspects of the presented model. Deficiencies in these previously published works due to available knowledge, applied assumptions, and attributes that were modified based on current literature were the following: (1) addition of a generalized link integrating the effects of bone remodeling on Ca exchange between bone and plasma, (2) integration of the effects of Ca homeostatic mechanisms on bone remodeling, and (3) development of the purported mechanism and qualitative model of Bellido et al. [3] into a mathematical kinetic model to describe the hypothetical differences in intracellular osteoblast signaling, thus describing the differential effects on bone remodeling of chronically versus intermittently elevated PTH. Overall, the presented model represents a first published iteration of a unifying generalized model of bone physiology and calcium homeostasis, and provides a construct for further development as the body of literature expands. An essential component to defining the model was the requirement for the model to simultaneously describe all utilized data under a single set of parameters within a single set of mathematical

Fig. 1. Schematic of physiologic system model to describe calcium homeostasis and bone remodeling.

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equations. Therefore, clinical data were used to identify and estimate parameters sensitive to the investigated system interventions. The utilized literature included data from an anabolic therapy [5], an antiresorptive bone therapeutic [6,7], and data from patients with varying degrees of renal insufficiency [8]. These data included both pretreatment data and information during and after discontinuation of treatment. Multiple dose levels of therapeutics were available to define hyperbolic response boundaries and steepness. Various levels of renal insufficiency provided critical information on the sensitivities of the system to kidney function. Within each dataset, multiple bioanalytical response variables were available, and again, fit such that the model was able to describe all variables and conditions simultaneously. Specifically, total serum calcium [6,7], parathyroid hormone [6,7], bone-specific alkaline phosphatase [6,7], C-telopeptide [6,7], 1,25-dihydroxyvitamin D [8], phosphate [8], and urine Ntelopeptide [5] were included from each of these reported investigations. If not already reported as such, values were fitted as the relative percent change from baseline. These clinical data were used to fit parameter estimates using Berkeley Madonna (version 8.0.1, University of California at Berkeley). Ordinary differential equations were used to describe the rates of change for each component and were solved using the fixed step size integration algorithm available in Berkeley Madonna (Fourth-order Runge–Kutta algorithm (RK4)). Parameter estimation used the simplex search algorithm with least squares minimization employed by the Curve Fit option within Berkeley Madonna. General mathematical considerations Due to the substantial number of model parameters relative to the available data, parameter identifiability was a concern. To minimize identifiability problems for estimated rate constants, a steady-state assumption was used to initiate differential equations. This approach assumes equal production and elimination rates, such that the evaluation of a differential equation (rate of change) for some moiety of the system, d/dt(moiety), is assumed to be equal to zero. Therefore, d/dt(moiety) = kin − kout⁎moiety is solved under this steady-state assumption and at the initial condition for the moiety as kin = kout⁎moietyinitial condition. In the interest of model parsimony, expressions with hyperbolic or sigmoidal Emax terms were often solved for their initial conditions. For example, if the hyperbolic expression Emax⁎moiety/(EC50 + moiety) = 1 under the initial conditions, then Emax could be expressed as (EC50 + moietyinitial condition)/moietyinitial condition. These hyperbolic components within the differential equations that characterize change responses are denoted in three ways, Hx, H−x, or H+x, and parameterized as follows: Hx = α4xγ = δγ + xγ




Hx− = α − ðα − ρÞ4xγ = δγ + xγ γ γ γ Hþ x = ρ + ðα − ρÞ4x = δ + x





In these equations, H represents the hyperbolic term and x indicates the stimulus variable. No superscript, a minus, or a plus indicates whether the response is a classical Emax, a decrease from steady-state, or an increase from steady-state, respectively. γ is a sigmoidicity term influencing the steepness of response, α is the maximum anticipated response, ρ is the minimum anticipated response, and δ is the value of x that produces the half-maximal response. Parameters included in the model but without values provided in Tables 1 or 2 were solved for explicitly under the initial conditions, as described above. If not reported for a given H, γ is assumed to be equal to unity (one). Of note, this Emax expression was used to replace many of the tangential expressions provided in [1] in an effort to provide physiologically interpretable parameter estimates. These parameterizations also

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Table 1 Initial conditions. Site

Compartment

Gut

Ca Calcitriol-dependent Ca absorption PO4 Ca PO4 Calcitriol PTH PO4 1-alpha-OH PT pool PT max capacity Ca immediately exchangeable (IC) Ca non-IC PO4 IC PO4 non-IC Responding Osteoblast Osteoblast Osteoclast Latent TGF-β Active TGF-β RANK RANKL OPG Runx2 CREB Bcl-2

Vasculature

Intracellular Kidney PT gland Bone

Osteoblast intracellular

Initial value

Units

1.3 0.5

mmol Unitless

3 4 5 6 7 8 9 10 11 12

0.839 32.9 16.8 1260 53.9 3226 126 0.5 1 100

mmol mmol mmol mmol mmol mmol mmol/h Unitless Unitless mmol

13 14 15 16

24900 Equimolar to Ca Equimolar to Ca 0.00104122

mmol mmol mmol Unitless

0.00501324 0.001154 228.1 0.2281 10 0.4 4 10 10 100

Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless

Differential equation number 1 2

17a 18 19 20 21a 22a 23a 26 27 28

a Initial conditions (differential equation number) for osteoblast fast (17a) and slow (17b) components, as well as RANK–RANKL (24) and OPG–RANKL (25) complexes, were solved from initial conditions.

allowed for varying maximum effects and initial set points. Additionally, the Emax expression is consistent with, and may be derived from, enzyme kinetics and receptor theory [9], and thus, was considered most appropriate for this biologic model.

Table 2 Estimates for non-hyperbolic parameters. Parameter k1–4 k3–5 k4–12 k5–8 k8–5 k6D k7D k9D k9S k11 k15-14 k17D k27D k18D k19-20 k21D k21-24 k24-21 k22D k23D k26S k28D δ2,1 α10 bT6'4 δT6’4

Estimate 0.0495 0.365 3.667 51.8 0.01927 0.1 7.143 0.05 6.3 0.0001604 2.444E-05 0.0006055 0.002795 0.02917 2.98E-05 0.003237 0.00000624 0.1120 0.002933 15.89 6.93 0.693 0.75 0.01 0.03 90

Units −1

h h−1 mmol/h h−1 h−1 h−1 h−1 h−1 h−1 h−1 h−1 h−1 h−1 h−1 h−1 h−1 h−1 h−1 h−1 h−1 h−1 h−1 — — — mM

Parameter

Estimate

Units

φ12–4 φ4–12 φ5-u φ17a φk17D φ28,k17D α20,21 α7,22 α17,22 δ7,23 γ18,12–4 γ24,12–4 α17,19S γ18,19–20 γ19–20 πc0 D(1) D(3) F3 GFR Vvasc Vic Vbone

0.1078 0.1144 1.142 0.7976 0.3132 1.01 0.1518 1.307 0.1738 3.85 1.697 0.6038 0.01113 0.5939 0.9191 0.2281 1.0 0.4 0.7 6.0 14 32.3 46.4

— — — — — — — — — mM — — — — — — mmol mmol — L/h L L L

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Disease state and therapeutics evaluations The following therapeutic scenarios were used to explore parameter sensitivities and aid in the identification of the mathematical representations of the system. As mentioned previously, and of notable importance, was the requirement of the model to describe all of the aforementioned datasets within the single structure and with a single set of parameter estimates. Endogenous parathyroid hormone (PTH) is produced by the parathyroid (PT) gland in response to changing levels of Ca in the extravascular compartment. Changes in the production of PTH can be affected short term by the excretion rate of PTH or long-term changes in the mass of the PT gland, which affects the limits of PTH production. In this way, states such as primary or secondary hyperparathyroidism can be simulated and produce appropriate short-term and long-term changes in PTH levels. An example of secondary hyperparathyroidism manifested through renal impairment was explored and compared to clinical data observed at various stages of pre-dialysis renal impairment [8]. This was implemented within the model as a progressive decrease in glomerular filtration rate (GFR) from 100 mL/min at baseline to approximately 16 mL/min 12 months later, resulting in decreased PO4 excretion, increased plasma PO4, and a subsequent increase in PT gland capacity. Primary hyperparathyroidism was mimicked through a longitudinal increase in endogenous PTH production that resulted in a 3-fold elevation of plasma PTH by the end of the 12-month evaluation. This effect of constantly elevated PTH was contrasted with the effect of intermittent PTH elevations, as produced by once-daily administration. The former case is known to lead to bone loss because of markedly elevated osteoclast function relative to osteoblast function [10,11], whereas the latter case has been associated with anabolic bone formation [5,12]. Observations of concentrations and effects of teriparatide (PTH 1–34) were used to assess intermittent PTH administration. Teriparatide is a recombinantly produced, truncated form of naturally occurring parathyroid hormone that includes the first 34 amino acids of the N-terminal portion of the protein [13]. Teriparatide is administered subcutaneously at a clinical dose of 20 μg daily to produce anabolic increases in bone mineral density [5]. The pharmacokinetics of teriparatide are well described and are consistent with PTH, where the absorption half-life following subcutaneous administration is approximately 1.5 h and the elimination half-life is approximately 8.5 min [5]. These values were used to simulate plasma levels of teriparatide following 20 μg daily administration for 12 months. Teriparatide was assumed to manifest equivalent actions to endogenous PTH, and cellular sensitivities and responses were set equivalent to PTH. A final scenario considered for the model was inhibition of RANK– RANKL interaction through competitive RANKL binding. In vivo, this process is accomplished by the endogenously produced soluble RANKL decoy receptor OPG. Recently, an exogenously administered therapeutic, denosumab, has allowed for further evaluation of the effects of RANKL inhibition. Denosumab is a fully human monoclonal IgG2 antibody that binds with high affinity and specificity to RANKL, inhibiting its binding to the RANK receptor on osteoclasts [6,7]. In the absence of RANKL, osteoclasts are not formed (i.e., no differentiation) and enter apoptosis. Denosumab also has been shown not to cross react with other tumor necrosis factor (TNF) family members, TRAIL, or CD-40 ligand [14]. Thus, it represents a discrete intercession point in the biological link between osteoclasts and osteoblasts. Since the actions of denosumab impact only this single critical element of the osteoblast/osteoclast units (a.k.a. bone morphogenic units), all levels of markers, electrolytes, and cytokines can be considered to be downstream results of RANKL inhibition. The pharmacokinetics of denosumab have been well described [15], and were used to simulate

serum levels of denosumab following 60 mg administration once every 6 months. Results As described earlier, the framework of this physiologic systems model is based on three previously published models describing: (1) calcium homeostasis [1]; (2) bone resorption and formation kinetics mediated by PTH, the RANK–RANKL–OPG axis, and TGF-β [2], and (3) osteoblastic intracellular signaling [3]. The calcium homeostasis components of the model, which describe the kinetics of Ca, PO4, and relevant endocrine factors (PTH, calcitriol, 1-alpha hydroxylase and PT gland capacity), are structured similarly to previously reported work [1]. The bone-remodeling components are also either structured as, or restructured from, previously reported models [2,3] and initialized from literature-supported values. The second and third components collectively describe the genesis, differentiation, and removal rates of responding osteoblasts, active osteoblasts (also described throughout as osteoblasts), and osteoclasts. The interdependence and kinetics of these cells are mediated through two controlling mechanisms: (1) the RANK–RANKL–OPG ligand–receptor interactions and (2) a net outcome of partial rate differences of osteoblast intracellular enzymatic activities. Both of these mechanisms are influenced by PTH and TGF-β. The presented model is composed of 28 differential equations that describe the integrated components governing bone and calcium homeostasis, and is graphically presented in Fig. 1. Physiologic spaces and organ systems that are discretely defined by the differential equations and, in most cases, influenced by other cytokines and electrolyte levels are listed in Table 1. Initial conditions and set point values for each are also provided with applied units. Of note, those components related directly to bone (osteoblast, osteoclast, TGF-β, RANK–RANKL–OPG, Runx2, Bcl-2, CREB) and the PT gland (PT pool and PT max capacity) were considered in relative terms, and therefore, reported as unitless. The characterized physiologic spaces and organ systems included in the model are the following: (1) gut—describes oral absorption of Ca and PO4, influenced by plasma calcitriol and PO4; (2) vasculature— assumed to be in equilibrium with extracellular fluid (ECF) for Ca, PO4, PTH, and calcitriol; (3) intracellular PO4 levels; (4) kidney— describes 1-α-hydroxylase and calcitriol production, renal excretion Table 3 Estimates for hyperbolic function parameters. Function

Parameter

Estimate

Function

Parameter

Estimate

H+2,1 H+2,1 H+2,1 H1–4 H1–4 H±6,2 H±6,2 H±6,2 H6,4 H6,4 H4-u H7,4-u H-4,10–7 H-4,10–7 H-4,10-7 H7,9 H7,9 H-5,9 H-5,9 H-5,9 H-6,11 H-6,11 H-6,11

ρ γ δ α δ ρ α δ α δ δ α ρ α γ δ γ α δ γ α ρ γ

0.25 4 0.75 0.9 1 0.003 0.037 90 2 90 1.573 1.0615 6249 96.25 11.74 1.549 0.1112 1.525 1.302 8.252 4.103 0.9 12.50

H+20,16 H+20,16 H+20,16 H+20,17 H+20,17 H+20,17 H-20,17D H-20,17D H-20,17D H28,17D H28,17D H+24,18S H+24,18S H+24,18S H+20,18D H+20,18D H+20,18D H-22,18D H-22,18D H-22,18D H+7,26D H+7,26D H+7,27S H+7,27S

γ α ρ γ α ρ γ ρ α α α α γ ρ γ α ρ α ρ γ α ρ α ρ

1.810 4.176 0.2016 0.1223 0.2517 5.585E-05 2.924 0.1743 0.7114 0.001802 3.816 3.545 8.531 0.3883 1.017 2.175 0.2004 3.803 0.4698 3.092 5.125 0.125 3.897 0.5

M.C. Peterson, M.M. Riggs / Bone 46 (2010) 49–63

of Ca and PO4 influenced by plasma Ca, PO4, calcitriol, PTH, and GFR; 5) PT gland—describes PTH production influenced by plasma Ca and calcitriol; 6) bone—describes Ca and PO4 levels via bidirectional diffusion and osteoclast- and osteoblast-promoted exchange with plasma; and (7) osteoblast intracellular component—describes the differential influence of PTH on bone metabolism purported to be controlled by the intracellular Runx2–Bcl-2–CREB system. Within the differential equations, parameters can be identified as either being within hyperbolic functions, or not. To aid identification, rate constants, composite rate constants, and physiologic parameters are listed in Table 2 as non-hyperbolic function parameters. Those embedded in hyperbolic terms are listed in Table 3 as hyperbolic function parameters. The complete model system of differential equations is provided below and descriptions of each component are in the section which follows.

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Parathyroid gland d = dt Að10Þ = ð1 − Að10ÞÞ4α10 4ð0:854T6−V4 + 0:15Þ
− Að10Þ4α10 4 0:854Tþ 6 V4 + 0:15
γ4;10 F T6 V4 = 1 F ðEXPðbT6 V4 4ðAð6Þ = Vvasc − δT6 V4 4 Að4Þ0 = Að4Þ Þ
γ4;10 − EXPð− bT6 V4 4ðAð6Þ = Vvasc − δT6 V4 4 Að4Þ0 = Að4Þ Þ
γ4;10 = ðEXPðbT6 V4 4ðAð6Þ = Vvasc − δT6 V4 4 Að4Þ0 = Að4Þ Þ
γ4;10 Þ + EXPð−bT6 V4 4ðAð6Þ = Vvasc − δT6 V4 4 Að4Þ0 =Að4Þ −

d = dt Að11Þ = k11 4H6;11 − k11 4Að11Þ Bone

Differential equations

d = dt Að12Þ = r4 − 12 − r12 − 4 + k13 − 12 4Að13Þ − k12 − 13 4Að12Þ

Gut

d = dt Að13Þ = − k13 − 12 4Að13Þ + k12 − 13 4Að12Þ

d = dt Að1Þ =

þ Dð1Þ4H2;1

− r1 − 4

d = dt Að14Þ = r5 − 14 − r14 − 5 + k15 − 14 4Að15Þ − k14 − 15 4Að14Þ

r1 − 4 = ðH1 − 4 4ðAð2Þ = 0:5Þ = ðAð1Þ + δ2;1 Þ + k1 − 4 Þ4Að1Þ d = dt Að2Þ =

þ H6;2 4ð1 −

Að2ÞÞ −

− H6;2 4Að2Þ

d = dt Að15Þ = k14 − 15 4Að14Þ − k15 − 14 4Að15Þ c
þ c d = dt Að16Þ = k17D 4Að17Þ0 = π0 4H20;16 − k17D 4Að17Þ0 4π0

d = dt Að3Þ = Dð3Þ4F3 − k3 − 5 4Að3Þ Vasculature

þ

= ðAð16Þ0 4H20;17 Þ4Að16Þ Að17Þ = Að17aÞ + Að17bÞ

d = dt Að4Þ = r12 − 4 − r4 − 12 − r4 − u + r1 − 4 r12 − 4 = k4 − 12 4ð1 − u12 − 4 Þ + k4 − 12 4u12 − 4 4H18;12 − 4   4 Að24Þ4Að18Þ0 =ðAð24Þ0 4Að18ÞÞ γ24;12 − 4

d = dt Að17aÞ = k17D 4Að17Þ0 4πc0 = ðAð16Þ0 4Hþ 20;17 Þ4Að16Þ4u17a 0
4 k17aD = k17D − k17aD 4Að17aÞ k17aD = ðk17D 4Að17Þ0 + k17D0 4uk17D 4Að17aÞ0 4u17a

r4 − 12 = k4 − 12 4ðAð4Þ = Að4Þ0 Þ4ðð1 − u4 − 12 Þ + u4 − 12 4ðAð17Þ = Að17Þ0 ÞÞ r4 − u = ð2 − H6;4 Þ4ð0:34GFR4Að4Þ − H4 − u 4H7;4 − u Þ

− k17D0 4uk17D 4Að17Þ0 Þ = Að17aÞ0 0

−

c

k17D = u28;k17D 4k17D 4H20;17D = π0 − H28;17D þ

c

d = dt Að17bÞ = k17D 4Að17Þ0 4π0 = ðAð16Þ0 4H20;17 Þ4Að16Þ 0

d = dt Að5Þ = r5 − 14 − r14 − 5 − r5 − u + r3 − 5 − r5 − 8 + r8 − 5

4ð1 − u17a Þ 4 uk17D − k17D 4 uk17D 4 Að17bÞ

r5 − 14 = 0:4644r4 − 12

d = dt Að18Þ = k18D 4π0 4Að18Þ0 4H24;18S − k18D 4H20;18D 4H22;18D 4Að18Þ

r14 − 5 = 0:4644r12 − 4


γ17;19S d = dt Að19Þ = k19 − 20 4Að19Þ0 4 Að17Þ= Að17Þ0
γ19 − 20 − k19 − 20 4 Að19Þ=Að19Þ0 4
γ18;19 − 20 Að18Þ=Að18Þ0 4Að19Þ

þ

c

r5 − u = 0:884GFR4Að5Þ − 0:884GFR4u5 − u r3 − 5 = k3 − 5 4Að3Þ r5 − 8 = k5 − 8 4Að5Þ

− 10004k19 − 20 4Að20Þ

d = dt Að6Þ = Að9Þ − k6D 4Að6Þ −

d = dt Að7Þ = H4;10 − 7 4ðAð10Þ = 0:5Þ4Að11Þ − k7D 4Að7Þ

d = dt Að21Þ = ððk21D 4Að21Þ0 + k21 − 24 4Að21Þ0 4Að22Þ0 − k24 − 21 4 γ20;21

Að24Þ0 Þ = Að20Þ0

Intracellular phosphate

− k21D 4Að21Þ

d = dt Að22Þ = k22S − k22D 4Að22Þ − k21 − 24 4ðAð23Þ4Að22Þ

Kidney

− H5;9

γ20;21

Þ4Að20Þ

− k21 − 24 4Að21Þ4Að22Þ + k24 − 21 4Að24Þ

d = dtAð8Þ = r5 − 8 − r8 − 5

d = dt Að9Þ =

−


γ19 − 20 4 d = dt Að20Þ = k19 − 20 4 Að19Þ=Að19Þ0
γ18;19 − 20 4Að19Þ Að18Þ=Að18Þ0

r8 − 5 = k8 − 5 4Að8Þ

− k9S 4H7;9 4H5;9

þ

− k9D 4Að9Þ

= 1 for Að5ÞbAð5Þ0

k22S

+ Að21Þ4Að22ÞÞ + k24 − 21 4ðAð24Þ + Að25ÞÞ
= k22D 4Að22Þ0 4 Að17Þ=Að17Þ0 γ17;22 4α7;22 4ðAð7Þ = Vvasc Þ
= ðδ7;22 4 Að17Þ=Að17Þ0 γ17;22 + ðAð7Þ = Vvasc ÞÞ

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d = dt Að23Þ = k23D 4Að23Þ0 4 Að16Þ = Að16Þ0 4ððAð7Þ = Vvasc Þ + ðδ7;23 4Að16Þ = Að16Þ0 Þ = ð24ðAð7Þ = Vvasc ÞÞ − k21 − 24 4Að23Þ4Að22Þ+ k24 − 21 4Að25Þ− k23D 4Að23Þ d = dt Að24Þ = k21 − 24 4Að21Þ4Að22Þ − k24 − 21 4Að24Þ d = dt Að25Þ = k21 − 24 4Að23Þ4Að22Þ − k24 − 21 4Að25Þ Osteoblast intracellular components þ

d = dt Að26Þ = k26S − H7;26D 4Að26Þ þ

d = dt Að27Þ = k27S 4H7;27S − k27D 4Að27Þ d = dt Að28Þ = k28D 4Að26Þ4Að27Þ − k28D 4Að28Þ

Description of model components The following description of the model necessarily differs in both grouping and order from the differential equations listed above. This is due to the interdependencies of the endocrine, cytokine, and system components of the model. Thus, to enable the reader to link the following information about the model to the differential equations, the equation numbers (e.g., D(1)) and mass amounts (e.g., A(1)) have been provided at appropriate points. Gut The input function for oral Ca (A(1)) reflects the rate and extent (bioavailability) of an orally administered amount of Ca (D(1)) and the influence of calcitriol on each component. The resultant flux of Ca from the intestine is defined by multiplying two terms: one for the rate of Ca absorption, which is saturable and dependent on the amount of Ca ingested, and one for bioavailability, which is dependent on the number of intestinal transporters (A(2)). Parameter estimates for oral Ca absorption rate and extent, as well as those quantifying transporter kinetics, are fixed to literature reported values [16-20]. Lastly, passive absorption is reported to account for approximately 16% of total oral Ca absorption under the initial conditions [21,22]. Thus, a first-order term (k1–4) to account for non-saturable passive (paracellular) absorption is included. The daily dose of oral Ca (D(1)) has been set equal to the 24-h Ca excretion from the kidney under the initial steady-state conditions. Currently, the model does not account for Ca loss through feces, sweat, or desquamation, nor does it provide for the slow calcium leak of approximately 1 mmol/day from bone observed clinically [23]. These factors may need to be incorporated during further refinements of the model. Calcium Extracellular calcium is regulated through exchanges with bone, excretion through the kidney, and input through oral dietary consumption. Approximately 99% of the body's total Ca is stored in bone, where approximately 100 mmol of the 25,000–30,000 mmol of total skeletal Ca is considered to be available for immediate exchange with plasma Ca [23-25]. The immediately exchangeable fraction and the remaining non-immediately exchangeable bone Ca are represented as A(12) and A(13), respectively. Exchanges occur between the immediately exchangeable fraction, A(12) and the plasma Ca pool (A(4)), as well as the non-immediately exchangeable Ca pool, A(13). This two-compartmental representation of bone Ca is

similar to a previous model used to describe Ca tracer kinetics with appropriate modifications applied [26]. The total amount of Ca exchanged daily between bone and plasma (88 mmol/day) has been reported to result from both passive diffusion and active processes [23]. The model is parameterized consistent with this publication, such that approximately 90% of Ca exchanged under the steady-state initial condition occurs passively. Active components, accounting for the remaining 10% of daily Ca exchange, are controlled by osteoblasts and osteoclasts, which affect the accretion into and release from bone, respectively. The active release of Ca from bone is modulated by RANK–RANKL interaction, reflecting the in vitro observation that RANKL mediates acid-induced bone Ca efflux [27]. Finally, the addition of the non-immediately exchangeable bone Ca compartment to the current model provides an appropriate longer-term accounting of total bone Ca. Renal Ca excretion represents a relatively rapid (hours) control mechanism for regulating plasma Ca. Chiefly regulated by PTH, increased PTH levels lead to a decrease in Ca excretion (Ca sparing) and, conversely, decreased PTH levels result in an increase in Ca excretion (Ca loss). Approximately 10 g (250 mmol) of calcium are filtered by the glomeruli per day [28], whereas only a small fraction (∼1–10 mmol/day) of this filtered load is excreted [23]. Thus, tubular reabsorption plays a critical role in regulating the amount of Ca lost to the urine. To accurately represent the kidney, an expression that describes the ‘renal threshold’ for calcium, above which there is a linear increase in Ca excretion with increased plasma Ca concentration, is used [28-30]. The parameterization includes expressions for both the filtered load (fu⁎GFR; [fu = free fraction or fraction unbound]) and tubular reabsorption. This aspect of the model enables evaluations of Ca homeostatic responses and bone-remodeling changes arising from altered GFR conditions, such as progressive renal disease. Although the relationship between Ca excretion and plasma Ca concentration is linear when above a threshold plasma Ca concentration, Ca excretion becomes nonlinear near and below this threshold [28-30]. An explanation for this nonlinearity is the involvement of two basolateral efflux transporters: the sodium/calcium dependent exchange (NCX) transporter whose activity increases with increased PTH, and the high-affinity plasma membrane calcium–ATPase (PMCA) transporter. On the apical side, Ca enters the cell from the lumen through the epithelial transient receptor potential cation channel, subfamily V, members 5 and 6 (TRPV5 and TRPV6), whose expression is upregulated with increased calcitriol [31]. For the current model, it is assumed that 50% of filtered Ca is reabsorbed independent of PTH, consistent with the slope of 0.5 observed between Ca excretion and plasma Ca concentrations above the threshold [23,29,30]. This reabsorption component is intended to reflect the high-affinity PMCA transporter. All additional reabsorption is assumed to be dependent on PTH, as with the NCX transporters. For this reabsorption, a saturable relationship with a multiplicative hyperbolic effect for PTH is used. Consistent with the effect of calcitriol on the apical transport, a hyperbolic effect of calcitriol on renal excretion is used [1]. The fraction of Ca unbound in plasma is set to 0.6 [23,29] and the initial, assumed “normal” GFR is set to 100 mL/min. Parathyroid hormone PTH, produced by the parathyroid (PT) gland, is released into plasma for systemic circulation. Production is assumed to be tonic, and does not include pulsatile secretion as has been reported [32]. The production rate, which is highly responsive to changes in plasma Ca concentration, is described through a sigmoidal relationship [33]. Parameter estimates are based on literature reports of the responsiveness of PTH to plasma Ca concentration changes [33-36]. PTH production also is influenced by the overall capacity of the PT gland, which is inversely related to plasma calcitriol concentration. The

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hyperbolic tangent expressions relating PT gland capacity (A(10)) with plasma calcitriol concentration and the parameter estimates are retained from the previous model [1]. During evaluation of secondary hyperparathyroidism data it was noted that the expression as parameterized in previous models failed to adequately provide for the observed extent of PT gland hyperplasia, and subsequent PTH hypersecretion [37]. Where clinical studies have reported approximately 7- to 15-fold greater plasma PTH concentrations in patients with secondary hyperparathyroidism compared to control subjects [8,34], the hyperbolic tangent functions only provided for a 2-fold increase in PTH production capacity. Additionally, observed PT gland hyperplasia is believed to result from transcriptional changes induced by hyperphosphatemia [38,39], in addition to the inverse relationship with calcitriol [40]. Therefore, an additional component is included in the model to improve the predictions of the magnitude and time-course for changes associated with PO4. This sigmoidal expression (A(11)) produces a multiplicative increase in the capacity of the PT gland to secrete PTH. Of note, this added expression also allows for the PT gland capacity to decrease over time if hyperphosphatemia is corrected. The reversibility of hyperplasia is questionable, and conflicting reports conclude that decreased PT gland size was [41], or was not [42], observed. If the latter is true, then the expressions for A(11) may warrant further development for evaluation of therapeutic intervention in secondary hyperparathyroidism. Regarding PTH degradation in vivo, PTH has a short plasma halflife of approximately 3–5 min [43]. The rate constant for the firstorder elimination of PTH is retained from the previous model [1], producing an approximate half-life of 5.8 min, consistent with the literature. Phosphate Like Ca, the majority of total body phosphorus (∼85%) is contained in bone, and the remaining 15% is more widely distributed than Ca [44,45], including an intracellular component that was determined to influence the kinetic description of extracellular PO4 [1]. The rate parameters for PO4 exchanges between plasma (A(5)) and the immediately exchangeable bone compartment (A(14)), as well as between plasma and the intracellular fluid compartment (A(8)), are retained from previous work [1] and described as first-order processes. The exchange fluxes of PO4 between plasma and bone are the same as the respective Ca fluxes multiplied by the stoichiometric factor of 0.464, the reported molar ratio of phosphorous to calcium in hydroxyapatite [1]. This is consistent with the reported molar ratio of calcium to phosphate Ca/PO4 of 40:25 [23]. This approach assumes that PO4 exchange with bone is directly proportional to, and under the same influences as, Ca exchange. Phosphate enters the model system through the gut compartment (A(3)), and dietary intake is defined as a constant infusion into the gut of approximately 1000 mg (10.5 mmol) of PO4 daily. The bioavailable fraction of dietary PO4 is assumed to be 0.7 [1,45]. Gut PO4 absorption rate is assumed to be first-order with the half-life in the intestine set to approximately 1.9 h. Consistent with Ca, PO4 renal excretion is defined in terms of the plasma PO4 fraction filtered by the glomeruli and the corresponding amount reabsorbed. This parameterization allows for the definition of a GFR-dependent renal resorption threshold consistent with literature reports [46-49]. The free fraction of PO4 in plasma was set at 88% [45,49,50]. 1-α-Hydroxylase and Calcitriol Calcitriol (A(6)) formation is dependent on 1-α-hydoxylase (A(9)), which is formed in the kidney. This production in the kidney is stimulated by increased PTH, vitamin D depletion, and/or hypo-

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phosphatemia. Conversely, opposite changes in these factors inhibit 1α-hydroxylase formation [23]. The baseline production rate and elimination rate constants were extracted from previous work [1], and consistent with that report, the current model construct does not yet include a vitamin D kinetics component or the effects of vitamin D as a precursor on calcitriol levels. For future model expansion, this aspect could be added with the calcitriol production rate described as a first-order process, dependent on vitamin D concentrations. The model does include a kinetic description of plasma PO4 concentrations (see previous section), however, and data on renal insufficiency [8] were used to quantify the effect increasing PO4 levels on calcitriol production. Calcitriol (1,25-dihydroxycholecalciferol) acts to increase total body Ca by promoting gut Ca absorption and by decreasing renal Ca excretion [23]. Its production rate is described as a direct effect of 1α- hydoxylase and its elimination is first-order. RANK/RANKL/OPG Differential equations and cytokine effects influencing the involvement of the receptor–ligand complexation of RANK–RANKL (A (24)) and OPG–RANKL (A(25)) in bone homeostasis were adopted from the literature [2] and revised. The three moieties each have production rates, affecting cytokine relationships, binding terms, and removal rates defined or solved for under initial conditions. RANK production is structured as inducible by active TGF-β (A(20)) [51], and degradation is described as a first-order process. The production rate of RANKL includes a hyperbolic function relating increased PTH concentration with increased RANKL production [52,53], and a term for the fractional change in osteoblasts, which is used to reflect the capacity for RANKL expression on osteoblast surfaces. As with RANK, RANKL degradation is described as a first-order process. Binding rate constants account for association and dissociation of RANK with RANKL (A(22)) and are assumed to be equal to the complexation constants for RANKL with OPG (A(23)). OPG production is linearly affected by responding osteoblasts (A(16)) and inversely related to PTH [52,53] levels through a hyperbolic equation. The initial condition for OPG is set empirically to 10 times that of RANKL and 0.4 times that of RANK, with the production rate solved under these initial conditions. The first-order removal rate for OPG is fixed at approximately twice the value previously reported [2] based on initial sensitivity testing. PTH-mediated osteoblast regulation mechanisms Directionally appropriate responses of osteoclasts and osteoblasts in conditions of chronically elevated PTH (e.g., hyperparathyroidism) and once-daily (QD) administration of PTH are accomplished using current theories on differential intracellular enzyme degradation rates [3,54-56]. It is believed that QD PTH administration is associated with an increase in Runx2-associated intracellular anti-apoptotic signaling. This effect is diminished following continuous PTH administration due to increased production of Smurf-2 and Runx2 degrading enzyme [3,54-56]. According to this theory, the increase in the anti-apoptotic signaling is kinetically faster than the increased production of degrading enzyme, and thus, the anti-apoptotic effect is observed with QD PTH administration. During continuous PTH elevations, however, the anti-apoptotic effect is abrogated by the increase in degradation enzyme. This theory was used to develop a series of differential equations controlling the Runx2-related intracellular osteoblast signaling. As this aspect of the model is based on a putative mechanism, data are not yet available in the literature. Therefore, parameters are set empirically and several assumptions about mathematical relationships are made as follows. The initial conditions for both Runx2 (A(26)) and CREB (A(27)) are set at a value of 10, and Bcl-2 (A(28)) is set as the product of these two (100). Bcl-2 is assumed to follow

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first-order elimination with a rate constant fixed to provide a half-life of 1 h. This assumption is based on a report that mRNA for Runx2 had a half-life of less than 2 h [57]. Therefore, although empirical, it is reasonable to assume that the entire intracellular system responds on a similar timescale of minutes to hours and relatively rapidly compared to other system components. Runx2 and CREB production rates are solved under these initial conditions, and both are assumed to follow first-order elimination. The elimination rate constants and the effects of PTH on Runx2 elimination and CREB production are estimated as separate hyperbolic expressions. Bcl-2 affects osteoblast survival by decreasing the elimination rate coefficient (k17D') for osteoblast apoptosis through a sigmoidal expression (H28,17D), which decreases with a Bcl-2 increase of greater than 5% from initial conditions.

Latent and active TGF-β TGF-β resides within bone as a latent form produced by osteoblasts [58] and is converted to its active form during bone resorption by osteoclasts [59,60]. The involvement of TGF- β is described by two differential equations, one for each form, latent (A (19)) and active (A(20)). The production rate of the latent form is a function of osteoblast numbers, and the removal rate is regulated by osteoclasts. The production rate of active TGF-β is defined as the removal rate of the latent form. The removal rate constant for active TGF-β is assumed to be 1000-fold higher than the removal rate constant of latent TGF-β, providing a 1000-fold excess of latent TGF-β in bone extracellular matrix under the initial conditions [59,61]. The mathematical description of TGF- β as two distinct moieties, latent and active, permits a time delay, rather than the direct concordance used by LeMaire et al. [2], between active TGF-β concentration from osteoclast numbers. In other words, an increase in osteoclast activity will lead to an increase in active TGF-β concentration, but not instantaneously. This allows for differential rates of change in these cell types and the concentrations of signaling cytokines known to be imbedded in bone matrix [59,62]. The differential effects of active TGF-β on responding osteoblast production versus maturation, as well as osteoclast and active osteoblast apoptosis, are described through sigmoidal expressions. The first three of these effects were initially described in a previous bone model [2]. The last two effects, for osteoblast apoptosis, are added to the current model to reflect the reported apoptotic effect [63].

Responding and active osteoblasts The rate of differentiation of mesenchymal progenitors into responding osteoblasts is set equal to a rate constant times the initial conditions solution to the system of equations for osteoblasts (responding and active). This production rate is affected by active TGF-β as described previously, and the first-order elimination rate constant for responding osteoblasts is defined under the initial conditions. The mathematical expressions for active osteoblasts are similar in structure to those for responding osteoblasts, with the input rate defined as the output rate from the responding osteoblast compartment and the elimination defined as a first-order removal, with the first-order rate constant influenced by intracellular signaling pathways, endogenous PTH levels, and PTH administration patterns as previously described. Active osteoblasts are further described as being removed by either a “slow” or “fast” removal rate. This is consistent with estimates of between 50% and 70% of osteoblasts undergoing apoptosis [64] and the rest differentiating into osteocytes. It is plausible that cells undergoing apoptosis are removed from the system at a different rate than those undergoing differentiation. Thus, the two distinguished terminal subpopulations may represent separate, kinetically distinct pathways of active osteoblast elimination [65]. The first-order removal rate for active osteoblasts is estimated using teriparatide and denosumab clinical data [6,7,66] and the relative starting ratio of osteoclasts to osteoblasts is set to ∼0.23, or approximately 4–5 osteoblasts per osteoclast, consistent with reported values [67]. Osteoclasts The osteoclast production rate is solved using initial conditions of osteoclast numbers. The influence of RANK, RANKL, and OPG (see previous section for details) enters the model within the system of equations for osteoclasts, where RANKL is required for both production [62] and survival of osteoclasts [68]. The effect of RANKL on osteoclast production is described through a hyperbolic relationship that is dependent on the relative level of RANK in complex with RANKL (A(24)). The removal rate for osteoclasts is described using a first-order rate constant, and is inversely related to active TGF-β. Osteoclast survival is also dependent on RANKL, accommodating the dependence of osteoclasts on RANKL as an anti-apoptotic survival factor [68].

Fig. 2. Percent of baseline (%) following RANKL inhibition (60 mg denosumab every 6-month administration) for (A) plasma calcium (dashed line) and phosphate (solid line), (B) plasma PTH (dashed line) and calcitriol (solid line), and (C) bone-related osteoclast (dashed line) and osteoblast (solid line) function. A horizontal reference (dotted line) is included on each figure at the baseline value of 100%. Circles (O) represent observed plasma calcium (A), PTH (B) and serum c-telopeptide (C). Triangles (Δ) represent observed bone-specific alkaline phosphatase (C). Observed values were reproduced from [6].

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Fig. 3. Percent of baseline (%) following once-daily PTH 1–34 (20 μg teriparatide) administration for (A) plasma calcium (solid line) and phosphate (dot dash line), (B) plasma PTH (solid gray line) and calcitriol (solid line), and (C) bone-related osteoclast (dashed line) and osteoblast (solid line). The solid bands in panels (A) and (B) represent the peak-totrough fluctuations in plasma calcium (A) and PTH (B), respectively. Fig. 4 provides a focused view of these within-day steady-state changes. A horizontal reference (dotted line) is included on each figure at the baseline value of 100%. An additional horizontal reference (dashed line) is provided on (A) at 103% to represent the approximate mean observed increase in plasma calcium during once-daily teriparatide (20 μg) administration. Plasma calcium of 112% (10.5 mg/dL) was considered to be the upper limit of normal [13]. Circles (O) and triangles (Δ) represent observed urine N-telopeptide and bone-specific alkaline phosphatase (C), respectively. Observed values were reproduced from [70].

Therapeutic and disease state evaluations Five diverse system perturbation cases are provided to demonstrate model robustness, recalling that the integrated model is required to adequately describe all sets of observed data under a single set of parameters and a single structural model. The first scenario demonstrates the ability of the model to describe the effects of RANKL inhibition. The next four scenarios represent differing effects of fluctuations in PTH resulting from separate sequelae: (1) repeated daily administration of PTH 1–34, (2) model predicted changes resulting from primary hyperparathyroidism, and (3) secondary hyperparathyroidism resulting from progressive renal insufficiency. The fourth PTH scenario represents model predicted changes resulting from primary hypoparathyroidism. In addition to these five scenarios, the model is used to generate predictions of cytokine levels and intracellular transcription factor levels for the purported causal Runx2–Bcl-2–CREB system following continuous or pulsatile PTH administration. Predictions of the magnitude and time-course of changes for each moiety within the model can be obtained and presented as predicted amounts (i.e., concentrations, masses, and cell counts). However, for ease of interpretation of the

induced changes, predictions are presented as the change relative to baseline. RANKL inhibition provided by the RANKL inhibiting antibody denosumab, administered every 6 months, provided an opportunity to evaluate the model under bone catabolic conditions. Fig. 2 presents the observed central tendency data and model predictions following denosumab administered every 6 months at a dose of 60 mg. RANKL inhibition is expected to decrease both osteoclastogenesis and osteoclast survival, leading to a marked decrease in osteoclasts (Fig. 2C). This decrease leads to a decrease in TGF-β activation and a subsequent decrease in osteoblasts (Fig. 2C). The relative magnitude of bone resorption changes affect a decrease in Ca exchange from bone to plasma leading to a transient decrease in plasma Ca (Fig. 2A) and small changes in phosphate (Fig. 2A). The regulatory mechanism of Ca-sensing in the PT gland then increases production of PTH, increasing plasma PTH (Fig. 2B) and plasma calcitriol (Fig. 2B). These latter two responses serve to correct the effect of decreased bone Ca accretion to plasma Ca. The model appropriately describes the direction and magnitude of responses for each of these markers resulting from RANKL inhibition compared to available clinical observations over the time-course [6].

Fig. 4. Daily steady-state changes in plasma calcium (dashed line, left axis) and PTH (solid line, right axis) predicted by the model after 1 month of once-daily PTH 1–34 administration (20 μg teriparatide). Predictions were consistent with clinical observations [13].

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Fig. 5. Primary hyperparathyroidism instituted in the model as a progressive increase in endogenous PT gland production of PTH to affect an approximate 3-fold increase in plasma PTH leading to percent of baseline (%) values for (A) plasma calcium (dashed line) and phosphate (solid line), (B) plasma PTH (dashed line) and calcitriol (solid line), and (C) bonerelated osteoclast (dashed line) and osteoblast (solid line) function. A horizontal reference (dotted line) is included on each figure at the baseline value of 100%. Predictions were consistent with clinical observations [72].

Next are presented cases of PTH treatment and simulations of PTH disease states. Intermittently elevated PTH achieved with once-daily administration of PTH is presented in Figs. 3 and 4. Daily PTH administration causes overall mild increases in calcium due to the calcium sparing effect (Figs. 3A and 4) and slight increases in phosphate (Fig. 3A). The administration of PTH at 20 μg transiently increases the levels of circulating PTH (endogenous + PTH 1–34) to approximately 650% of endogenous levels, and is represented on a daily scale in Fig. 4. Over a period of 12 months, daily PTH administration results in a pronounced increase in osteoblasts and osteoclasts [69,70]. The model accounts for their differential effects through kinetic differences in osteoblast intracellular signaling (Fig. 1) that leads to an anti-apoptotic effect and prolonged osteoblast survival and cell numbers (Fig. 3C). The effects of PTH on RANKL and OPG also continue, leading to osteoclast increases (Fig. 3C). Each of these model predicted effects, again, are consistent with clinical findings [71]. Primary hyperparathyroidism, induced in the model as a progressive increase in endogenous PT gland production of PTH, is affected within the model as an approximate 3-fold increase in plasma PTH by the end of the 12-month evaluation period (Fig. 5B). Plasma calcitriol production

(conversion from vitamin D) and levels increase slightly (Fig. 5B) as a result of increased production of 1-α-hydroxylase (not shown). Plasma Ca increases remarkably (Fig. 5A), as expected through the multifaceted Ca sparing effects of PTH via increased renal tubular reabsorption, increased oral absorption, and increased release of Ca from bone. This latter effect is illustrated through a notably greater increase in osteoclasts (Fig. 5C) relative to osteoblasts (Fig. 5C), resulting in net bone resorption. This net bone resorption also results in increased release and plasma levels of PO4 (Fig. 5A). Each of these predicted effects is consistent in direction and magnitude with actual clinical changes observed in primary hyperparathyroidism [72]. Interestingly, and consistent with clinical observations [8], distinctly different effects for plasma calcitriol, PO4, and Ca follow the development of secondary hyperparathyroidism due to progressive renal failure as compared to primary hyperparathyroidism. These differences arise from a decrease in renal PO4 excretion due to declining renal function, and a subsequent increase in plasma PO4 (Fig. 6B). The resulting hyperphosphatemia inhibits the production of 1-α-hydroxylase and diminishes calcitriol production (Fig. 6C). Decreased calcitriol levels then lead to decreased oral Ca absorption

Fig. 6. Progressive renal insufficiency instituted as (A) a progressive decrease in GFR from 100 mL/min to approximately 16 mL/min 10 years later leading to changes in percent of baseline (%) for (B) plasma calcium (dashed line) and phosphate (solid line), (C) plasma PTH (dashed line) and calcitriol (solid line), and (D) bone-related osteoclast (dashed line) and osteoblast (solid line) function. A horizontal reference (dotted line) is included in panels (B) to (D) at the baseline value of 100%. Triangles (Δ) represent observed plasma phosphate (B) and calcitriol (C). Circles (O) represent observed GFR (A) and plasma PTH (C). Observed data were estimated from [8] by adjusting to a reference of 100 mL/min. The predicted increase in net bone resorption (marked osteoclast function increase without considerable osteoblast change) is consistent with clinical observations of bone mineral density loss in chronic renal failure patients [8].

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Fig. 7. Primary hypoparathyroidism instituted in the model as an immediate 50% lowering of endogenous PT gland production of PTH leading to changes in percent of baseline (%) for (A) plasma calcium (dashed line) and phosphate (solid line), (B) plasma PTH (dashed line) and calcitriol (solid line), and (C) bone-related osteoclast (dashed line) and osteoblast (solid line) function. A horizontal reference (dotted line) is included on each figure at the baseline value of 100%. Predictions were consistent with clinical observations [73].

and plasma levels (Fig. 6B). These decreases in plasma calcitriol and Ca elicit an increase in PT gland production of PTH leading to secondary hyperparathyroidism (Fig. 6C). The chronically elevated PTH markedly increases the number of osteoclasts (Fig. 6D) through stimulation of RANKL and inhibition of OPG production. The increased numbers of osteoclasts activate proportionally more TGF-β, that in turn increases production of responding osteoblasts that mature into active osteoblasts (Fig. 6D). The magnitude of increase in osteoblasts is considerably lower than that for osteoclasts leading to a net increase in bone resorption, consistent with clinical observations of bone mineral density loss in patients with hyperparathyroidism [72]. By convention, hypoparathyroidism should lead to the opposite of the Ca sparing effects of directly elevated PTH. The model appropriately accounts for these expected effects, where a 50% lowering of PTH (Fig. 7B) decreases plasma calcitriol (Fig. 7B) through diminished 1-αhydroxylase production, and decreases osteoclasts (Fig. 7C) through decreased RANKL and OPG. This latter change, again, leads to diminishing TGF-β activation and related decreases in responding and active osteoblasts (Fig. 7C). The greater magnitude decreases in osteoclasts relative to osteoblasts leads to a net decrease in Ca release from bone. This, in addition to a decrease in gut Ca absorption secondary to decreased calcitriol, and decreased renal Ca reabsorption, leads to a remarkable decrease in plasma Ca (Fig. 7A). No remarkable changes in plasma PO4 are predicted (Fig. 7A). As with the other cases, these predictions are again consistent with clinical observations [73].

The differential bone-remodeling effects under conditions of continuously elevated PTH (catabolic, Fig. 5) versus once-daily administered PTH (anabolic, Fig. 3) are evaluated in the model as resulting from proposed kinetic differences in the intracellular signaling factors Runx2 and CREB (Figs. 8 and 9). In the model, the osteoblast apoptosis rate coefficient (k17D') is decreased with increased Bcl-2. Bcl-2 is derived as the product of Runx2 and CREB. During primary hyperparathyroidism (Fig. 8), the model predicts an increase in CREB, but also a decrease in Runx2, leading to a null effect (no change) in the product (Bcl-2). Therefore, the relative rate of osteoblast apoptosis under conditions of continuously elevated PTH is unaffected by this anti-apoptotic intracellular mechanism in the model. The net result of chronically increased PTH is a marked increase in osteoclasts, elicited through direct actions of PTH on the RANK/RANKL/OPG system (Fig. 1), with a slight increase in osteoblasts (Fig. 5C). The osteoblast change is predicted to result to a downstream effect from increased levels of active TGF-β resulting from the increase in osteoclasts. This downstream TGF-β effect leads to only a moderate and delayed reduction in the osteoblast apoptosis (k17D', Fig. 8), affected in the model through the hyperbolic expression H−20,17D. H−20,17D decreases with increased active TGF-β. Following once-daily administration of PTH (Fig. 9), the relative rate of osteoblast apoptosis is affected by this anti-apoptotic intracellular mechanism because Runx2 levels diminish only intermittently and CREB levels are constantly elevated due to kinetic differences in these

Fig. 8. Predicted relative effect (% of baseline) of primary hyperparathyroidism, instituted in the model as an approximate 3-fold increase in plasma PTH (light dashed line) after 12 months, on the intracellular signaling factors Runx2 (solid line) and CREB (dot dash line) and the osteoblast apoptosis rate coefficient (k17D′) (bold dashed line). A horizontal reference (dotted line) is included at the baseline value of 100%.

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Fig. 9. Predicted relative effect (% of baseline changes) of once-daily PTH 1–34 (20 μg teriparatide) administration on intracellular signaling factors Runx2 (light solid line) and CREB (dot dash line), the osteoblast apoptosis rate coefficient (k17D′) (bold solid line) and total (endogenous + PTH 1–34) plasma PTH (light dashed line). The effect of constantly elevated PTH on k17D′ is provided for reference (bold dashed line). A horizontal reference (dotted line) is also included at the baseline value of 100%. These predictions, in concert with Fig. 8, provide a kinetic description of the hypothetical mechanism described by Bellido et al. [3].

pathways. Therefore, in addition to a constant reduction in osteoblast apoptosis due to the downstream effect on TGF-β, there is an added rhythmic reduction in osteoblast apoptosis from this intracellular mechanism. This leads to the markedly greater elevation in osteoblasts relative to constantly elevated PTH (Figs. 3C and 5C, respectively). The effect of constantly elevated PTH on the osteoblast apoptosis is provided as reference (Fig. 9). These predictions (Figs. 8 and 9) provide a kinetic description of the hypothetical mechanism described by Bellido et al. [3]. Discussion In this paper we present a first iteration of a physiologically based mathematical model of integrated calcium homeostasis and bone remodeling. The model presented was developed based upon an underpinning physiologic tenant that the body must maintain stringent calcium homeostasis, and that changes in bone and overall physiology are related to this need. The present model joins and builds upon two earlier published quantitative models: a model of systemic calcium homeostasis [1] and a cellular model of the BMU relationship [2]. While each of these previous models demonstrated significant advances in quantitative systems modeling, they had limitations in their abilities to predict observations following simulated administrations of therapeutics or disease state progressions. These limitations resulted chiefly from the difference between the objectives of those modeling efforts and the objectives of the present work. For example, bone in the calcium homeostasis model of Raposo et al. [1] was represented as a linear firstorder compartment with an obvious absence of endocrine effecting components or feedback mechanisms controlling the responses to cytokines and TNF family members. This model simplification was reasonable in the context of that work, as it did not detract from the overall ability of the model to describe calcium fluctuations in the majority of the disease scenarios demonstrated. However, in the present model, where description of longer-term bone remodeling was an objective, a more detailed description of calcium flow resulting from BMU activity was necessary. Similarly, the excellent quantitative cellular model of LeMaire et al. describing the linking of osteoclasts and osteoblasts [2] unified several of the controlling and linking mechanisms. However, that model assumed that TGF-β was in direct equilibrium with osteoclasts and that the RANK–RANKL–OPG system was maintained at a steady-state condition. Additionally, it did not facilitate changes elicited through calcium homeostatic mechanisms, nor did it account for the anabolic nature of once-daily PTH administration. These limiting assumptions were addressed, at least in

part, in the present model to facilitate predictions of short- and longterm changes affected through bone therapeutics, and observed as changes in bone resorption and formation markers. Overall, significant modifications and additions were made to the previous models, all in the interest of forming a single, revised, composite model of calcium and bone homeostasis. While predictions of bone mineral density (BMD) have not been generated, the ability of the model to reproduce observations of bone resorption markers, formation markers, PTH levels, and other related system observations supports its further development and use for this purpose in the future. Importantly, following the fusion and described modifications of these two quantitative models of calcium homeostasis and BMU function the unified model still failed to describe a critical component of the clinical literature and physiology, differential responses to PTH. Further modification and model complexity was required for the model to correctly predict clinical observations of the differential effects of PTH on bone metabolism following daily treatment with 20 μg/day teriparatide as compared to sustained hyperparathyroidism. The model had to be able to predict the differential responses to PTH in order for it to be considered reasonable under the current state of scientific knowledge and documented observations. To enable predictions of the differing responses to PTH resulting from exposure modulation, a mathematical construct was developed that quantifies the intracellular control mechanism for responses to PTH that had been qualitatively proposed by Bellido et al. [3]. Several theoretical models have been published previously that describe the differential effects of PTH when administered continuously versus intermittently [74-78]. However, current literature proposes that differential rates of intracellular enzyme activities may be responsible [3]. The qualitative literature description of the conceptual model based on in vitro experiments describes the relative activities of an intracellular signaling pathway, governed by Runx2, Bcl-2, and CREB. The pathway is purported to influence the differential effects of continuous and intermittently administered PTH on catabolic and anabolic bone changes, respectively. To test this proposed mechanism's plausibility, a mathematical component was developed and incorporated into the existing model, representing a possible controlling mechanism for osteoblast apoptosis. The model was then evaluated with and without this aspect included (results described, figures not shown). With the additional component, the model was able to differentially describe changes in bone and calcium homeostasis arising from intermittent or continuous changes in PTH levels reported for teriparatide treatment and conditions of hyperparathyroidism. It

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should be noted that the mathematical description of this mechanism represents one of many plausible mechanism(s) of PTH action, where these mechanisms are an active focus of many research programs. As our knowledge of the controlling mechanisms becomes better defined, model refinement of this component should be considered. As demonstrated in Figs. 3 and 5, this component allowed the model to predict the literature data. Further, to ensure that the results were not an artifact of total administered amounts of PTH, additional simulations (not shown) were performed using infusions of PTH that produced continuous concentrations at levels consistent with the maximum concentrations observed during daily administrations. As expected from clinical data, the continuously elevated PTH affected a markedly greater increase in osteoclast vs. osteoblast activity, and while osteoclast activity increased under both conditions, the osteoblast activity increased more following intermittent administration (Fig. 3), consistent with clinical observations and the proposed mechanism by Bellido et al. Of note, components related directly to bone (osteoblast, osteoclast, TGF-β, RANK–RANKL–OPG, Runx2, Bcl-2, and CREB) and the PT gland (PT pool and PT max capacity) were considered in relative terms and, therefore, reported as unitless. For components that were initially drawn from the Lemaire et al. model [2] (osteoblasts, osteoclasts, and RANK–RANKL–OPG), the previously reported values were re-calibrated based on initial data fits and known physiology. For example, Lemaire et al. reported reference active osteoblasts and osteoclasts as 0.0007282 and 0.009127 pM, respectively. Instead, our model provides initial osteoblast and osteoclast conditions that were calibrated such that the relative starting ratio of osteoclasts to osteoblasts is set to ∼0.23, or approximately 4–5 osteoblasts per osteoclast, consistent with reported values [67]. The RANK–RANKL– OPG initial values were then calibrated based on these new osteoblast and osteoclast values. The assignment of these values as unitless acknowledges that, although consistent in relative magnitude, they are not estimated from actual experimental data of detailed cell counts per unit area. Therefore, in the future these numerical values maybe refined and compared to such data. To the authors' knowledge, this work represents the first time a mathematical description of the mechanism proposed by Bellido et al. has been constructed and published. It also represents the first known report of a plausible integrated mathematical description of the differential effects of PTH embedded into a physiologic model of bone metabolism, incorporating bone cell components, relevant organ systems integral to systemic calcium homeostasis, and used to predict clinical observations of relevant drug therapies. Of note, an alternative mechanism for the differential effects of PTH, based on the concept of receptor desensitization, has been derived mathematically by Potter et al. [74] and was published recently. This construct was considered for inclusion in the current model. However, the publication for this model provided only conceptual predictions, rather than comparisons of the predictions to clinical data. Our testing of this mechanistic concept failed to provide predictions consistent with the observed clinical literature (results not shown). Therefore, the intracellular mechanism described above was pursued with more successful results. Uses for this model in understanding bone physiology and associated endocrine-related phenomena may be significant. Through an iterative process of estimation, simulation, and modification, the overall model has the ability to enable further understanding of the biology and biological changes affected during disease progression and therapeutic modifications of disease states. For example, the model has been useful for understanding plasma Ca changes observed during denosumab or teriparatide therapy as the expected effects of changes in Ca exchange between plasma and bone, as well as homeostatic changes in gut absorption and renal excretion, and allows the quantification of these effects. The model also may lend insight into affected changes in cytokines (e.g., RANKL), which are difficult to measure with confidence or at the site of action. It has the

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ability to allow a researcher to make in silico hypothesis assessments of these pathways and guide confirmatory biological experimentation to further our understanding of these mechanisms. The potential applications of this sort of quantitative systems physiologic model extend from basic science, to numerous areas of clinical research, and drug development. The evolution of this quantitative systems physiologic model and others like it is by no means complete. There are elements of the model that are known areas for expansion and revision in accordance with additional development and with new discoveries. Natural next steps for expansion of this model are to develop a predictive link to changes in BMD, to explore differential effects between therapeutic agents on different BMD sites (e.g., hip vs. spine), to incorporate descriptions for additional disease states or biological changes (e.g., progressive effects following estrogen loss during menopause and vitamin D deficiency), and to include/explore emerging signaling pathways (e.g., Wnt). For example, extension of the model is ongoing in an effort to describe longitudinal BMD changes affected through changes in the bone markers. Exploration of differential site effects will require more extensive consideration of site-specific BMU changes and data in the literature to support region specific model components, including differential effects for trabecular and cortical bone. As the mathematical components are added (and/or updated), they can be tested against the accumulated data, and then the revised model used to move the collective understanding of the field forward, pursuing an accelerated rate of research and discovery. In conclusion, the integrated physiologically based model that has been developed, which simultaneously describes Ca homeostasis and bone remodeling with respect to both the timescale (longitudinal effects) and magnitude (e.g., % of baseline) of changes due to disease progression and/or therapeutic intervention, marks a notable achievement toward model-based research. The authors believe it highlights an area of opportunity in the field of bone biology to fuse systems modeling approaches (i.e., mathematical descriptions of cellular mechanisms and multiple endocrine feedbacks) with clinical outcomes modeling, which are often described with heuristic models, in an effort to produce physiologically appropriate changes over time. The generality of this model, as displayed by its utility to describe a range of therapeutics and disease states, from progressive renal impairment and hypoparathyroidism and hyperparathyroidism to several therapeutic intervention mechanisms for the treatment of osteoporosis (PTH, RANKL inhibition), lends credence to the development of these types of models, and to this model's use as a platform for continued evaluation, expansion, and use in the research of bone and Ca homeostasis therapeutics and diseases. Acknowledgments Funding provided by Amgen, Inc. through a collaborative scientific partnership between Amgen, Inc. and Metrum Research Group LLC. The authors wish to thank Heidi Costa and Joe Hebert for their assistance in preparing this manuscript. References [1] Raposo JF, Sobrinho LG, Ferreira HG. A minimal mathematical model of calcium homeostasis. J Clin Endocrinol Metab 2002;87:4330–40. [2] Lemaire V, Tobin FL, Greller LD, Cho CR, Suva LJ. Modeling the interactions between osteoblast and osteoclast activities in bone remodeling. J Theor Biol 2004;229:293–309. [3] Bellido T, Ali AA, Plotkin LI, Fu Q, Gubrij I, Roberson PK, et al. Proteasomal degradation of Runx2 shortens parathyroid hormone-induced anti-apoptotic signaling in osteoblasts. A putative explanation for why intermittent administration is needed for bone anabolism. J Biol Chem 2003;278:50259–72. [4] Guyton AC, Coleman TG, Cowley Jr AW, Liard JF, Norman Jr RA, Manning Jr RD. Systems analysis of arterial pressure regulation and hypertension. Ann Biomed Eng 1972;1:254–81. [5] Forteo® teriparitide (rDNA origin) injection package insert. Indianapolis: Eli Lilly and Company; 2004. p. 1–22.

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