Modeling osteoporosis to design and optimize pharmacological therapies comprising multiple drug types

  1. David J Jörg  Is a corresponding author
  2. Doris H Fuertinger
  3. Alhaji Cherif
  4. David A Bushinsky
  5. Ariella Mermelstein
  6. Jochen G Raimann
  7. Peter Kotanko
  1. Biomedical Modeling and Simulation Group, Global Research and Development, Fresenius Medical Care Germany, Germany
  2. Renal Research Institute, United States
  3. Department of Medicine, University of Rochester School of Medicine and Dentistry, United States
  4. Icahn School of Medicine at Mount Sinai, United States
5 figures, 4 tables and 2 additional files

Figures

Schematic of the osteoporosis model describing the cell dynamics and signaling pathways within a ‘representative bone remodeling unit (BRU)’.

Regulatory interactions between different model components are indicated by colored boxes (see legend). TGFβ, transforming growth factor beta; BMP, bone morphogenetic protein; PDGF, platelet-derived growth factor; IGF, insulin-like growth factor; FGF, fibroblast growth factor.

Figure 2 with 1 supplement
With a single set of parameters, the calibrated model can quantitatively predict the effects of various drugs in different dosing regimens, alone and in combination.

(A) Comparison of simulated total hip bone mineral density (BMD, black curves) and clinical data (dots), including aging behavior (green dots) and treatment behavior (black dots) of various sequential drug treatments, including denosumab, romosozumab, alendronate, and teriparatide. Hybrid aging/treatment datasets were created combining data from Looker et al., 1998 (aging dataset, green dots in panel A; in total N=3251 subjects 20 years and older), as well as Recknor et al., 2015 (blosozumab 180 mg Q2W: N=25), McClung et al., 2018 (placebo/deno.: N=18, alendro./romo./deno.: N=21), and Leder et al., 2015 (deno./teri.: N=27, teri. + deno./deno.: N=23) (treatment datasets, black dots in panels A and B) as indicated, see ‘Methods.’ (B) Zoom into the treatment regions shown in panel (A) including BMD (black) and baseline changes of the bone resorption marker C-terminal telopeptide (CTX, red) and the bone formation marker procollagen type 1 amino-terminal propeptide (P1NP, blue). Colored bars above the plots indicate the medication scheme (see legend). Data points show population averages; average types and error bar types as reported in the respective original publication. In both panels, BMD is displayed as a fraction of its value at t0=25 years.

Figure 2—figure supplement 1
Continuation of Figure 2 comparing model predictions and clinical data from various studies, all conventions identical.

See Appendix 3—table 2 for a list of data sources and Appendix 3—table 3 for goodness-of-fit measures. Dosing: mg, milligrams; mcg, micrograms; Q x M, dose administered every x months; Q x W, every x weeks; Q x D, every x days; R, romosozumab; A, alendronate; D, denosumab; T, teriparatide.

The model predicts differential outcomes for different sequences of the same drugs at constant total medication load.

(A) Simulated progression of bone mineral density (BMD) and C-terminal telopeptide (CTX) and procollagen type 1 amino-terminal propeptide (P1NP) concentrations for different sequences (columns) of the three drugs denosumab (D), alendronate (A), and romosozumab (R) as indicated. Simulated treatment starts at age 67. The total amount of drug administered is identical among columns. Clinical results on the sequence ARD (column 5) were reported in McClung et al., 2018, see also Figure 2. (B) Maximum simulated BMD (relative to baseline at treatment start) achieved during the course of treatment for different drug sequences. (C) Simulated BMD 10 years after treatment end (relative to baseline at treatment start) for different drug sequences.

Appendix 1—figure 1
Parameterization of the aging behavior and creation of hybrid aging/treatment datasets for model calibration and validation.

(A) Age dependence of estradiol serum levels. Clinical data (dots) modified from Sowers et al., 2008. The curve shows a fit of the function given by Equation 18 to determine the parameter τe (Appendix 3—table 4). (B) Bone mineral density (BMD) age dependence for different ethnic groups as indicated. Data modified from Looker et al., 1998; reported age bin averages have been used to represent the center of the age bin. (C) BMD age dependence shown in panel (B), where all curves have been normalized to their earliest value (t0 = 25y). (D) Schematic of how hybrid aging/treatment datasets were generated by merging the same aging dataset with different treatment datasets; for details, see ‘Methods.’

Appendix 3—figure 1
Calibration datasets comparing model predictions and clinical data from various studies.

All conventions identical to Figure 2. Drug administrations are provided in the bottom row. See Appendix 3—table 2 for a list of data sources and Appendix 3—table 3 for goodness-of-fit measures. Dosing: mg, milligrams; mcg, micrograms; Q x M, dose administered every x months; Q x W, every x weeks; Q x D, every x days; B, blosozumab; A, alendronate; D, denosumab; T, teriparatide.

Tables

Appendix 3—table 1
Values of fit weights Wβ used in Equation 29.
WeightValue
W(BMD)300
W(CTX)1
W(P1NP)1
W(BSAP)1
  1. BMD, bone mineral density; CTX, C-terminal telopeptide; P1NP, procollagen type 1 amino-terminal propeptide; BSAP, bone-specific alkaline phosphatase

Appendix 3—table 2
Data sources used to calibrate and validate the model.

Columns titled ‘Figure(s)’ indicate the plot panels in the respective publication that were digitized. BMD always refers to total hip bone mineral density.

PublicationMedication(s)DosingsFigure(s)Table(s)
BMDCTXP1NPBSAPBMD
Black et al., 2006Alendronate5–10 mg Q1D233——
Bone et al., 2011Denosumab60 mg Q6M3b4b4a——
Cosman et al., 2016Romosozumab210 mg Q1M3b3e3d——
Denosumab60 mg Q6M3b3e3d——
Leder et al., 2014Teriparatide20 mcg Q1D2d4c,f4b,e——
Leder et al., 2015Teriparatide20 mcg Q1D34———
Denosumab60 mg Q6M34———
Lewiecki et al., 2019Romosozumab210 mg Q1M3b————
Denosumab60 mg Q6M3b————
Looker et al., 1998[Age-dependent BMD]—————7
McClung et al., 2006Denosumab6 mg Q3M, 14 mg Q6M, 210 mg Q6M2b2e—2f—
McClung et al., 2017Denosumab6–14 mg Q3M, 14–210 mg Q6M2b————
McClung et al., 2018Romosozumab140 mg Q1M, 210 mg Q1M3c4b4a——
Denosumab60 mg Q6M3c,d4b,d4a,c——
Alendronate70 mg Q1W3d4d4c——
Recknor et al., 2015Blosozumab180 mg Q4W, 180 mg Q2W, 270 mg Q2W3b4d4a——
Saag et al., 2017Alendronate70 mg Q1W3b3d3c——
  1. Q, every; M, month; D, day; W, week; CTX, C-terminal telopeptide; P1NP, procollagen type 1 amino-terminal propeptide; BSAP, bone-specific alkaline phosphatase.

Appendix 3—table 3
Goodness-of-fit measures for calibration and validation datasets.

Mean absolute percentage error (MAPE) and windowed minimal absolute percentage error (WMAPE) as defined in Equation 30 and Equation 31, respectively. The column ‘Shown in’ indicates the figure in this article that shows the respective simulation and data plot.

Medication(s)MAPEWMAPEShown in
Data ref.BMD (%)CTX (%)P1NP (%)BSAP (%)BMD (%)CTX (%)P1NP (%)BSAP (%)
Calibration datasets
Alendronate 5–10 mg Q1DBlack et al., 20060.97.021.1—0.64.713.5—Appendix 3—figure 1
Alendronate 70 mg Q1WSaag et al., 20170.534.413.7—0.220.310.2—Appendix 3—figure 1
Blosozumab 180 mg Q4WRecknor et al., 20150.716.523.8—0.410.818.7—Appendix 3—figure 1
Blosozumab 270 mg Q2WRecknor et al., 20150.326.813.2—0.215.84.8—Appendix 3—figure 1
Denosumab 14 mg Q3M
→

denosumab 60 mg Q6M
McClung et al., 20170.3———0.1———Appendix 3—figure 1
Denosumab 14 mg Q6MMcClung et al., 20060.273.9—17.00.046.3—0.2Appendix 3—figure 1
PlaceboRecknor et al., 20150.47.315.4—0.37.215.3—Appendix 3—figure 1
Teriparatide 20 mcg Q1DLeder et al., 20140.417.18.5—0.29.11.6—Appendix 3—figure 1
Teriparatide 20 mcg Q1D→
denosumab 60 mg Q6M
Leder et al., 20150.465.5——0.28.6——Appendix 3—figure 1
Validation datasets
Alendronate 70 mg Q1W
→
romosozumab 140 mg Q1M
→
denosumab 60 mg Q6M
McClung et al., 20180.525.420.0—0.317.13.6—Figure 2
Blosozumab 180 mg Q2WRecknor et al., 20150.321.320.3—0.113.212.6—Figure 2
Denosumab 60 mg Q6M
→
teriparatide 20 mcg Q1D
Leder et al., 20150.4103.1——0.344.6——Figure 2
PlaceboMcClung et al., 20180.66.210.5—0.56.210.5—Figure 2
Placebo→
denosumab 60 mg Q6M
McClung et al., 20180.613.210.1—0.45.17.4—Figure 2
Teriparatide 20 mcg Q1D + denosumab 60 mg Q6M
→
denosumab 60 mg Q6M
Leder et al., 20150.7183.7——0.466.4——Figure 2
Alendronate 70 mg Q1W
→
romosozumab 140 mg Q1M
→
placebo
McClung et al., 20180.522.916.8—0.415.14.2—Figure 2—figure supplement 1
Placebo→
denosumab 60 mg Q6M
Cosman et al., 20160.581.99.6—0.044.44.0—Figure 2—figure supplement 1
Placebo→
denosumab 60 mg Q6M
Lewiecki et al., 20190.4———0.1———Figure 2—figure supplement 1
Placebo→
denosumab 60 mg Q6M
McClung et al., 20170.6———0.3———Figure 2—figure supplement 1
Romosozumab 210 mg Q1M
→
alendronate 70 mg Q1W
Saag et al., 20171.030.037.1—0.519.411.7—Figure 2—figure supplement 1
Romosozumab 210 mg Q1M
→
denosumab 60 mg Q6M
Cosman et al., 20161.292.522.4—0.650.76.9—Figure 2—figure supplement 1
Romosozumab 210 mg Q1M
→
denosumab 60 mg Q6M
Lewiecki et al., 20191.0———0.5———Figure 2—figure supplement 1
Romosozumab 210 mg Q1M
→
denosumab 60 mg Q6M
McClung et al., 20180.614.257.8—0.36.515.9—Figure 2—figure supplement 1
Romosozumab 210 mg Q1M
→
placebo
McClung et al., 20180.715.053.3—0.510.218.5—Figure 2—figure supplement 1
  1. Q, every; M, month; D, day; W, week; BMD, bone mineral density; CTX, C-terminal telopeptide; P1NP, procollagen type 1 amino-terminal propeptide; BSAP, bone-specific alkaline phosphatase.

Appendix 3—table 4
Full list of parameters of the core model and the medication extensions.
ParameterDescriptionValueUnitOriginModel equation
Core model
ωC*Reference pre-osteoclast to osteoclast differentiation rate0.93d-1CalibrationEquation 10
eC*Estrogen threshold for downregulation of pre-osteoclast to osteoclast differentiation0.941CalibrationEquation 10
sC*Sclerostin threshold for upregulation of pre-osteoclast to osteoclast differentiation8.60×1061CalibrationEquation 10
ηCReference osteoclast apoptosis rate0.02d-1CalibrationEquation 13
eCEstrogen threshold for upregulation of osteoclast apoptosis0.991CalibrationEquation 13
rCResorption signal threshold for upregulation of osteoclast apoptosis10.101CalibrationEquation 13
νCMax. rel. effect of regulatory factors on osteoclast apoptosis1.23×10-41CalibrationEquation 13
ωB*Reference pre-osteoblast to osteoblast differentiation rate0.32d-1CalibrationEquation 10
sB*Sclerostin threshold for downregulation of pre-osteoblast to osteoblast differentiation1.63×1021CalibrationEquation 10
ηBReference osteoblast apoptosis rate8.68×10-3d-1CalibrationEquation 13
ωBReference osteoblast to osteocyte conversion rate6.24×10-4d-1CalibrationEquation 11
ηYosteocyte apoptosis rate1.10×10-4d-1EstimateEquation 14
κsSclerostin degradation rate0.05d-1Estimate; see Suen et al., 2015; Ominsky et al., 2015.Equation 15
esEstrogen threshold for downregulation of sclerostin secretion9.601CalibrationEquation 15
λCReference bone resorption rate per unit density osteoclast3.82×10-6d-1CalibrationEquation 16
λBReference bone formation rate per unit density osteoblast1.29×10-6d-1CalibrationEquation 16
sΩSclerostin threshold for downregulation of bone formation3.04×1031CalibrationEquation 16
rΩResorption signal threshold for upregulation of bone formation1.02×1031CalibrationEquation 16
νΩMax. rel. effect of the resorption signal on bone formation1.08×1021CalibrationEquation 16
γEquilibration rate of the bone mineral content6.65×10-3d-1CalibrationEquation 17
c0Reference bone mineral content0.801EstimateEquation 17
teOnset of estrogen decline50.00yEstimateEquation 18
τeTime scale of estrogen decline2.60yIndep. fit (Appendix 1—figure 1A)Equation 18
Bone turnover markers
qCTXExponent relating the bone resorption rate to CTX levels1.161CalibrationEquation 15
qP1NPExponent relating the bone formation rate to P1NP levels1.451CalibrationEquation 15
qBSAPExponent relating the bone formation rate to BSAP levels0.921CalibrationEquation 15
Medication extension: sclerostin antibodies
EblosozumabEfficacy: blosozumab0.011CalibrationEquation 21
TblosozumabEffective half-life: blosozumab7.00dTromosozumabEquation 21
EromosozumabEfficacy: romosozumab0.011EblosozumabEquation 21
TromosozumabEffective half-life: romosozumab7.00dSolling et al., 2018Equation 21
δsSclerostin/antibody unbinding rate0.05d-1κsEquation 25
Medication extension: RANKL antibodies
EdenosumabEfficacy: denosumab4.34×1031CalibrationEquation 21
TdenosumabEffective half-life: denosumab10.00dBekker et al., 2004Equation 21
βC*rAbMax. rel. effect of RANKL antibodies on pre-osteoclast to osteoclast differentiation0.871CalibrationEquation 24
βbrAbMax. rel. effect of RANKL antibodies on mineralization0.021CalibrationEquation 24
Medication extension: bisphosphonates
EalendronateEfficacy: alendronate2.97×10-51CalibrationEquation 22
TalendronateEffective half-life: alendronate1.53×102dCalibrationEquation 22
ηCbpMax. contribution of bisphosphonates to osteoclast apoptosis rate1.00d-1CalibrationEquation 26
Medication extension: PTH analogs
EteriparatideEfficacy: teriparatide0.271CalibrationEquation 22
TteriparatideEffective half-life: teriparatide0.04dSatterwhite et al., 2010Equation 22
βBpthMax. rel. effect of PTH analogs on osteoblast apoptosis1.311CalibrationEquation 27
βC*pthMax. rel. effect of PTH analogs on pre-osteoclast to osteoclast differentiation4.281CalibrationEquation 27
  1. CTX, C-terminal telopeptide; P1NP, procollagen type 1 amino-terminal propeptide; BSAP, bone-specific alkaline phosphatase; PTH, parathyroid hormone.

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  1. David J Jörg
  2. Doris H Fuertinger
  3. Alhaji Cherif
  4. David A Bushinsky
  5. Ariella Mermelstein
  6. Jochen G Raimann
  7. Peter Kotanko
(2022)
Modeling osteoporosis to design and optimize pharmacological therapies comprising multiple drug types
eLife 11:e76228.
https://doi.org/10.7554/eLife.76228