Mathematical modeling of postmenopausal osteoporosis and its treatment by the anti-catabolic drug denosumab

Abstract

Denosumab, a fully human monoclonal antibody, has been approved for the treatment of postmenopausal osteoporosis. The therapeutic effect of denosumab rests on its ability to inhibit osteoclast differentiation. Here, we present a computational approach on the basis of coupling a pharmacokinetics model of denosumab with a pharmacodynamics model for quantifying the effect of denosumab on bone remodeling. The pharmacodynamics model comprises an integrated systems biology-continuum micromechanics approach, including a bone cell population model, considering the governing biochemical factors of bone remodeling (including the action of denosumab), and a multiscale micromechanics-based bone mechanics model, for implementing the mechanobiology of bone remodeling in our model. Numerical studies of postmenopausal osteoporosis show that denosumab suppresses osteoclast differentiation, thus strongly curtailing bone resorption. Simulation results also suggest that denosumab may trigger a short-term bone volume gain, which is, however, followed by constant or decreasing bone volume. This evolution is accompanied by a dramatic decrease of the bone turnover rate by more than one order of magnitude. The latter proposes dominant occurrence of secondary mineralization (which is not anymore impeded through cellular activity), leading to higher mineral concentration per bone volume. This explains the overall higher bone mineral density observed in denosumab-related clinical studies.

Keywords: bone remodeling; micromechanics; pharmacokinetics; systems biology.

Figure 1

Pharmacokinetics model representation of denosumab: compartment I represents the subcutaneous tissue with denosumab being present at mass concentration m_den,sub, whereas compartment II represents the blood serum with denosumb being present at mass concentration m_den,ser; the compartment concentrations are governed by absorption rate k_abs, degradation rates k_org and k_enz, and the administered dose D_den (which is related to Δm_den,sub).

Figure 2

Calibration of the denosumab pharmacokinetics model against the clinical data of Bekker et al. : (a) comparison between temporal evolutions of model-predicted and measured serum concentrations, and (b) model-predicted versus measured serum concentrations;  mg/kg,  mg/kg, and  mg/kg.

Figure 3

Graphical sketch, adapted from Pivonka et al. , showing all mechanisms considered in the model presented in this paper, with novel contributions colored green and red: guidance of cell developments (OB_u → OB_p → OB_a and OC_p → OC_a) occurs biochemically (governed by TGF-β and the RANK-RANKL-OPG pathway, including the action of PTH) via related activator and repressor functions, π_act and π_rep, and biomechanically (the macroscopic loading Σ relates to microscopic loading σ_bm, causing microscopic deformations represented by the microscopic strain energy density Ψ_bm, the latter entering the bone cell population model); evolutions of the different cell developmental stages follows cell differentiation (considered through maximum differentiation rates , , and ), cell proliferation (considered through maximum proliferation rate ), and cell apoptosis (considered through maximum apoptosis rates and ); the mechanisms based on which the effects of PMO and denosumab administration, respectively, are depicted in the red-colored box (dotted thick lines) and in the green-colored box (dashed thick lines), respectively; see Sections 3.2–3.4 for details on related model extensions.

Figure 4

Simulation of postmenopausal osteoporosis (PMO) with disease initiation at t_PMO,ini = 0: (a) prescribed temporal evolutions of the reduction factor of the disease-related RANKL production rate, , and of the mechanoresponsiveness reduction factor, , (b) model-predicted evolutions of the concentrations of active osteoclasts, , and active osteoblasts, , normalized with respect to the concentration before onset of PMO, and , (c) the increase of the vascular porosity of osteoporotic cortical bone over time, Δf_vas(t), simulated by our model, compared with corresponding experimental results, and (d) the phase diagram comparing bone resorption versus bone formation responses associated to the simulated porosity increase of osteoporotic cortical bone; the arrows indicate path directions.

Figure 5

Simulation results obtained for administration regimes involving one administration of denosumab: (a)–(d) phase diagrams (in logarithmic scales) for the simulated doses, (a) D_den = 0, (b) mg/kg, (c) mg/kg, and (d) mg/kg, with the temporal progress indicated by respective markers (days, ⋄…5 days, ○ …182.5 days, ▽ …185 days, △ …190 days, days, ▹…700 days, and days after onset of postmenopausal osteoporosis), and (e) the corresponding temporal evolutions of the volume fractions of the extravascular bone matrix, f_bm, as postmenopausal osteoporosis progresses.

Figure 6

Simulation results obtained for administration regimes involving multiple administrations of denosumab: temporal evolutions of volume fractions of the extravascular bone matrix, f_bm, during progression of PMO (a) for different administered doses ( mg/kg,  mg/kg, and  mg/kg) and (c) different administration intervals ( month,  months,  months, and  months), as well as the corresponding phase diagrams (b) and (d), where each loop corresponds to one of the “hills” in (a) and (c), representing one administration interval; the paths depicted in (b) and (d) are directed counterclockwise.

Figure 7

Study of the sensitivity of the model-predicted bone matrix volume fraction f_bm after 10 years of postmenopausal osteoporosis progress with respect to a varying anabolic strength parameter λ, computed for an administration regime involving multiple administrations of denosumab, with a constant administration interval Δt_den = 9 months and administered doses D_den = 0, mg/kg,  mg/kg, and  mg/kg.

Figure 8

Two scenarios investigated for denosumab administration regimes involving multiple injections: in scenario (i), m_den,ser starts from a level below in each administration interval ς, while in scenario (ii), m_den,ser remains higher than at all times; within one administration interval scenario (i) comprises three domains ①, ②, and ③ based on whether m_den,ser is lower or higher than (the transition between the domains is indicated by the white-faced circle-shaped marker, representing and black-faced circle-shaped marker, representing ), following Equations (A.5)–(A.7), whereas scenario (ii) follows Equation (A.8).