Bayesian Joint Modeling of Bone Mineral Density And Repeated Time-To-Fracture Event For Multiscale Bone Systems Model Extension Elodie L. Plan*, Kyle T. Baron, Marc R. Gastonguay, Jonathan L. French, William R. Gillespie, and Matthew M. Riggs Background
Results
· Physiologically-based multiscale systems (PBMS) model describes cellular mechanisms and bone dynamics in bone-related diseases.[1,2] · Fracture rate considered as most meaningful endpoint affected by disease progression and drug [3] intervention.
Evaluation w.r.t. NHANES data
Objectives To develop a model simultaneously characterizing bone mineral density (BMD) and fracture risk based on time since final menstrual period (FMP).
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10 20 30 40 50 60
FMPage (yr) : (20,42]
BMD: Retrospective prediction, from examination time to FMP, through fracture time point(s).
FMPage (yr) : (42,64] ●
1.4
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BMD (g cm2)
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Structural parameter estimates: b = 0.84, str = −1.66, spo = −0.85, sfi = −0.34 (close to reported literature values[5] ).
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Random effect included as residual variability: σ = 0.131 (95% credible interval (CI) 0.127–0.136).
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Centered covariate effects added on all parameters.
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10 20 30 40 50 60 Time since FMP (yr)
Fig. 3: Posterior predictive distributions obtained from BMD fit
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Fracture hazard
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θh ×(1+θBM D ×(BMD(t)−BMD))
θh = −5.5 (95%CI −5.49 – −5.54), θBM D = 1.5 (95%CI 1.49–1.52) h(t) accumulates from t 0 = FMP. ∆DIC(Mconst ant −M var y ing ) = 30
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(BMD = 0.8 g/cm2 , α = 1)
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h(t) = e
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Fracture hazard
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Proportion of fracture−free women
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Proportion of fracture−free women
Fracture risk: Time-varying hazard reflects increase due to time-dependent BMD decline in final model.
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Time since FMP (yr)
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F M Page and time indirect covariates with dynamically predicted BMD vs. direct with observed BMD.
Time since FMP (yr)
Fig. 4: Simulation with constant hazard
Fig. 5: Simulation with final model
Simulation with PBMS model 20
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TGF β (%)
Fracture (count)
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BMD (g cm2) : (0.736,1.1]
BMD (g cm2) : (1.1,1.46]
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Age (yr)
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0.005 0.010 0.015 0.020 0.025 100
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Fracture hazard
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Age (yr)
Fig. 6: Estrogen, bone remodeling factors and markers, calcium, BMD and fracture risk time-courses for 100 ♀ with FMP at 50 yr (SD 8) 0
Piecewise BMD model predictions resembled those based on the mechanistic model[2] reflecting the estrogen loss effect on a series of bone markers.
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PBMS model extended to reflect changes in expected fracture-free time driven by bone markers.
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Developed model spans several magnitudes in time and space: slow changes in survival can be predicted from more rapid changes in bone markers.
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BMD (g cm2)
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Survival (probability)
0.0 0.2 0.4 0.6 0.8 1.0
BMD (g cm2) : (0.375,0.736]
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BMD (g cm2)
Value 0.0 0.2 0.4 0.6 0.8 1.0
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2.26 2.28 2.30 2.32 2.34
· 2005-2008 NHANES demographics, dual energy X-ray absorptiometry, body measures, osteoporosis, and reproductive health datasets. · 1605 postmenopausal ♀ of 63 (95% interpercentiles (IP) 27–85) yr mean age and 45 (95% IP 26–57) yr mean FMP age; 1 femoral neck BMD measure and 0–5 (204 total) fracture events each.
Calcium concentration (mmol)
60
Osteoblasts (%)
110
90
40
[4]
Value
20
Data
100 150 200 250 300 350
Methods
Osteoclasts (%)
40
TGF βactive (%) 140
100
60
95
80 100
Estrogen (%)
Fig. 1: Multiscale bone systems model extension to fracture risk
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50 60 Time since FMP (yr)
Fig. 7: BMD and probability to not experience fractures for 100 ♀ with BMD of 0.8 g/cm2 (SD 0.04) at FMP
Fig. 2: Number of fractures since FMP per observed BMD strata
Conclusions
Models
BMD: Piecewise linear model from literature[5] BMD(t) = b + str × t −1,2yr + spo × t 2,5yr + sfi × t 5,∞yr Included covariates: BMI, ethnicity, and F M Pa ge . FractureRrisk: Repeated time-to-event model −(
tj
h(u)du)α
[6,7]
S(t) = e Investigated covariates: observed BMD, BMD(t), F M Pa ge , and time (α6=1, Weibull distribution). t j−1
Software
WinBUGS, BlackBox[8] , R (deSolve, mrgSim).
· Simultaneous modeling of BMD time-course and repeated time-to-fracture events from publicly available data enabled the characterization of the fracture risk in > 1500 postmenopausal ♀. · Next steps include, among others, testing drug effects from previously explored therapeutics, performing external evaluation with estrogen therapy, and including uncertainty in deterministic model. · This model will be made available in the data and model library ™.
References [1] Peterson MC and Riggs MM. Bone, 2010. [2] Riggs MM et al. Measurement and Kinetics of In Vivo Drug Effects: Advances in Simultaneous PKPD Modeling. The Netherlands, 2010. [3] Food and Drug Administration (FDA). Background Document for Meeting of Advisory Committee for Reproductive Health Drugs and Drug Safety and Risk Management Advisory Committee, 2011. [4] Centers for Disease Control and Prevention (CDC). National Center for Health Statistics (NCHS). National Health and Nutrition Examination Survey Data (NHANES).
[5] Greendale GA et al. J Bone Miner Res, 2012. [6] Garnett C & Holford NHG. 5th International Symposium on Measurement and Kinetics of In Vivo Drug Effects. The Netherlands, 2006. [7] Cox EH et al. J Pharmacokinet Biopharm, 1999. [8] BUGSModelLibrary, PKPD Model Library for WinBUGS, Metrum Institute.
*
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