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David J Jörg

Doris H Fuertinger

Alhaji Cherif

David A Bushinsky

Ariella Mermelstein

Jochen G Raimann

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
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Figures
Tables
Additional files

5 figures, 4 tables and 2 additional files

Figures

Figure 1

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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 …

Figure 2 with 1 supplement

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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 …

Figure 2—figure supplement 1

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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, …

Figure 3

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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 …

Appendix 1—figure 1

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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 τ_e (Appendi…

Appendix 3—figure 1

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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. …

Tables

Appendix 3—table 1

Values of fit weights $W^{β}$ used in Equation 29.

Weight	Value
$W^{(BMD)}$	300
$W^{(CTX)}$	1
$W^{(P1NP)}$	1
$W^{(BSAP)}$	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.

Publication	Medication(s)	Dosings	Figure(s)				Table(s)
Publication	Medication(s)	Dosings	BMD	CTX	P1NP	BSAP	BMD
Black et al., 2006	Alendronate	5–10 mg Q1D	2	3	3	—	—
Bone et al., 2011	Denosumab	60 mg Q6M	3b	4b	4a	—	—
Cosman et al., 2016	Romosozumab	210 mg Q1M	3b	3e	3d	—	—
	Denosumab	60 mg Q6M	3b	3e	3d	—	—
Leder et al., 2014	Teriparatide	20 mcg Q1D	2d	4c,f	4b,e	—	—
Leder et al., 2015	Teriparatide	20 mcg Q1D	3	4	—	—	—
	Denosumab	60 mg Q6M	3	4	—	—	—
Lewiecki et al., 2019	Romosozumab	210 mg Q1M	3b	—	—	—	—
	Denosumab	60 mg Q6M	3b	—	—	—	—
Looker et al., 1998	[Age-dependent BMD]	—	—	—	—	—	7
McClung et al., 2006	Denosumab	6 mg Q3M, 14 mg Q6M, 210 mg Q6M	2b	2e	—	2f	—
McClung et al., 2017	Denosumab	6–14 mg Q3M, 14–210 mg Q6M	2b	—	—	—	—
McClung et al., 2018	Romosozumab	140 mg Q1M, 210 mg Q1M	3c	4b	4a	—	—
	Denosumab	60 mg Q6M	3c,d	4b,d	4a,c	—	—
	Alendronate	70 mg Q1W	3d	4d	4c	—	—
Recknor et al., 2015	Blosozumab	180 mg Q4W, 180 mg Q2W, 270 mg Q2W	3b	4d	4a	—	—
Saag et al., 2017	Alendronate	70 mg Q1W	3b	3d	3c	—	—

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 …

Medication(s)		MAPE				WMAPE				Shown in
Medication(s)	Data ref.	BMD (%)	CTX (%)	P1NP (%)	BSAP (%)	BMD (%)	CTX (%)	P1NP (%)	BSAP (%)	Shown in
Calibration datasets
Alendronate 5–10 mg Q1D	Black et al., 2006	0.9	7.0	21.1	—	0.6	4.7	13.5	—	Appendix 3—figure 1
Alendronate 70 mg Q1W	Saag et al., 2017	0.5	34.4	13.7	—	0.2	20.3	10.2	—	Appendix 3—figure 1
Blosozumab 180 mg Q4W	Recknor et al., 2015	0.7	16.5	23.8	—	0.4	10.8	18.7	—	Appendix 3—figure 1
Blosozumab 270 mg Q2W	Recknor et al., 2015	0.3	26.8	13.2	—	0.2	15.8	4.8	—	Appendix 3—figure 1
Denosumab 14 mg Q3M → denosumab 60 mg Q6M	McClung et al., 2017	0.3	—	—	—	0.1	—	—	—	Appendix 3—figure 1
Denosumab 14 mg Q6M	McClung et al., 2006	0.2	73.9	—	17.0	0.0	46.3	—	0.2	Appendix 3—figure 1
Placebo	Recknor et al., 2015	0.4	7.3	15.4	—	0.3	7.2	15.3	—	Appendix 3—figure 1
Teriparatide 20 mcg Q1D	Leder et al., 2014	0.4	17.1	8.5	—	0.2	9.1	1.6	—	Appendix 3—figure 1
Teriparatide 20 mcg Q1D→ denosumab 60 mg Q6M	Leder et al., 2015	0.4	65.5	—	—	0.2	8.6	—	—	Appendix 3—figure 1
Validation datasets
Alendronate 70 mg Q1W → romosozumab 140 mg Q1M → denosumab 60 mg Q6M	McClung et al., 2018	0.5	25.4	20.0	—	0.3	17.1	3.6	—	Figure 2
Blosozumab 180 mg Q2W	Recknor et al., 2015	0.3	21.3	20.3	—	0.1	13.2	12.6	—	Figure 2
Denosumab 60 mg Q6M → teriparatide 20 mcg Q1D	Leder et al., 2015	0.4	103.1	—	—	0.3	44.6	—	—	Figure 2
Placebo	McClung et al., 2018	0.6	6.2	10.5	—	0.5	6.2	10.5	—	Figure 2
Placebo→ denosumab 60 mg Q6M	McClung et al., 2018	0.6	13.2	10.1	—	0.4	5.1	7.4	—	Figure 2
Teriparatide 20 mcg Q1D + denosumab 60 mg Q6M → denosumab 60 mg Q6M	Leder et al., 2015	0.7	183.7	—	—	0.4	66.4	—	—	Figure 2
Alendronate 70 mg Q1W → romosozumab 140 mg Q1M → placebo	McClung et al., 2018	0.5	22.9	16.8	—	0.4	15.1	4.2	—	Figure 2—figure supplement 1
Placebo→ denosumab 60 mg Q6M	Cosman et al., 2016	0.5	81.9	9.6	—	0.0	44.4	4.0	—	Figure 2—figure supplement 1
Placebo→ denosumab 60 mg Q6M	Lewiecki et al., 2019	0.4	—	—	—	0.1	—	—	—	Figure 2—figure supplement 1
Placebo→ denosumab 60 mg Q6M	McClung et al., 2017	0.6	—	—	—	0.3	—	—	—	Figure 2—figure supplement 1
Romosozumab 210 mg Q1M → alendronate 70 mg Q1W	Saag et al., 2017	1.0	30.0	37.1	—	0.5	19.4	11.7	—	Figure 2—figure supplement 1
Romosozumab 210 mg Q1M → denosumab 60 mg Q6M	Cosman et al., 2016	1.2	92.5	22.4	—	0.6	50.7	6.9	—	Figure 2—figure supplement 1
Romosozumab 210 mg Q1M → denosumab 60 mg Q6M	Lewiecki et al., 2019	1.0	—	—	—	0.5	—	—	—	Figure 2—figure supplement 1
Romosozumab 210 mg Q1M → denosumab 60 mg Q6M	McClung et al., 2018	0.6	14.2	57.8	—	0.3	6.5	15.9	—	Figure 2—figure supplement 1
Romosozumab 210 mg Q1M → placebo	McClung et al., 2018	0.7	15.0	53.3	—	0.5	10.2	18.5	—	Figure 2—figure supplement 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.

Parameter	Description	Value	Unit	Origin	Model equation
Core model
$ω_{C^{*}}$	Reference pre-osteoclast to osteoclast differentiation rate	0.93	$d^{- 1}$	Calibration	Equation 10
$e_{C^{*}}$	Estrogen threshold for downregulation of pre-osteoclast to osteoclast differentiation	0.94	1	Calibration	Equation 10
$s_{C^{*}}$	Sclerostin threshold for upregulation of pre-osteoclast to osteoclast differentiation	$8.60 \times 10^{6}$	1	Calibration	Equation 10
$η_{C}$	Reference osteoclast apoptosis rate	0.02	$d^{- 1}$	Calibration	Equation 13
$e_{C}$	Estrogen threshold for upregulation of osteoclast apoptosis	0.99	1	Calibration	Equation 13
$r_{C}$	Resorption signal threshold for upregulation of osteoclast apoptosis	10.10	1	Calibration	Equation 13
$ν_{C}$	Max. rel. effect of regulatory factors on osteoclast apoptosis	$1.23 \times 10^{- 4}$	1	Calibration	Equation 13
$ω_{B^{*}}$	Reference pre-osteoblast to osteoblast differentiation rate	0.32	$d^{- 1}$	Calibration	Equation 10
$s_{B^{*}}$	Sclerostin threshold for downregulation of pre-osteoblast to osteoblast differentiation	$1.63 \times 10^{2}$	1	Calibration	Equation 10
$η_{B}$	Reference osteoblast apoptosis rate	$8.68 \times 10^{- 3}$	$d^{- 1}$	Calibration	Equation 13
$ω_{B}$	Reference osteoblast to osteocyte conversion rate	$6.24 \times 10^{- 4}$	$d^{- 1}$	Calibration	Equation 11
$η_{Y}$	osteocyte apoptosis rate	$1.10 \times 10^{- 4}$	$d^{- 1}$	Estimate	Equation 14
$κ_{s}$	Sclerostin degradation rate	0.05	$d^{- 1}$	Estimate; see Suen et al., 2015; Ominsky et al., 2015.	Equation 15
$e_{s}$	Estrogen threshold for downregulation of sclerostin secretion	9.60	1	Calibration	Equation 15
$λ_{C}$	Reference bone resorption rate per unit density osteoclast	$3.82 \times 10^{- 6}$	$d^{- 1}$	Calibration	Equation 16
$λ_{B}$	Reference bone formation rate per unit density osteoblast	$1.29 \times 10^{- 6}$	$d^{- 1}$	Calibration	Equation 16
$s_{Ω}$	Sclerostin threshold for downregulation of bone formation	$3.04 \times 10^{3}$	1	Calibration	Equation 16
$r_{Ω}$	Resorption signal threshold for upregulation of bone formation	$1.02 \times 10^{3}$	1	Calibration	Equation 16
$ν_{Ω}$	Max. rel. effect of the resorption signal on bone formation	$1.08 \times 10^{2}$	1	Calibration	Equation 16
$γ$	Equilibration rate of the bone mineral content	$6.65 \times 10^{- 3}$	$d^{- 1}$	Calibration	Equation 17
$c_{0}$	Reference bone mineral content	0.80	1	Estimate	Equation 17
$t_{e}$	Onset of estrogen decline	50.00	$y$	Estimate	Equation 18
$τ_{e}$	Time scale of estrogen decline	2.60	$y$	Indep. fit (Appendix 1—figure 1A)	Equation 18
Bone turnover markers
$q_{CTX}$	Exponent relating the bone resorption rate to CTX levels	1.16	1	Calibration	Equation 15
$q_{P1NP}$	Exponent relating the bone formation rate to P1NP levels	1.45	1	Calibration	Equation 15
$q_{BSAP}$	Exponent relating the bone formation rate to BSAP levels	0.92	1	Calibration	Equation 15
Medication extension: sclerostin antibodies
$E_{blosozumab}$	Efficacy: blosozumab	0.01	1	Calibration	Equation 21
$T_{blosozumab}$	Effective half-life: blosozumab	7.00	$d$	$T_{romosozumab}$	Equation 21
$E_{romosozumab}$	Efficacy: romosozumab	0.01	1	$E_{blosozumab}$	Equation 21
$T_{romosozumab}$	Effective half-life: romosozumab	7.00	$d$	Solling et al., 2018	Equation 21
$δ_{s}$	Sclerostin/antibody unbinding rate	0.05	$d^{- 1}$	$κ_{s}$	Equation 25
Medication extension: RANKL antibodies
$E_{denosumab}$	Efficacy: denosumab	$4.34 \times 10^{3}$	1	Calibration	Equation 21
$T_{denosumab}$	Effective half-life: denosumab	10.00	$d$	Bekker et al., 2004	Equation 21
$β_{C^{*}}^{rAb}$	Max. rel. effect of RANKL antibodies on pre-osteoclast to osteoclast differentiation	0.87	1	Calibration	Equation 24
$β_{b}^{rAb}$	Max. rel. effect of RANKL antibodies on mineralization	0.02	1	Calibration	Equation 24
Medication extension: bisphosphonates
$E_{alendronate}$	Efficacy: alendronate	$2.97 \times 10^{- 5}$	1	Calibration	Equation 22
$T_{alendronate}$	Effective half-life: alendronate	$1.53 \times 10^{2}$	$d$	Calibration	Equation 22
$η_{C}^{bp}$	Max. contribution of bisphosphonates to osteoclast apoptosis rate	1.00	$d^{- 1}$	Calibration	Equation 26
Medication extension: PTH analogs
$E_{teriparatide}$	Efficacy: teriparatide	0.27	1	Calibration	Equation 22
$T_{teriparatide}$	Effective half-life: teriparatide	0.04	$d$	Satterwhite et al., 2010	Equation 22
$β_{B}^{pth}$	Max. rel. effect of PTH analogs on osteoblast apoptosis	1.31	1	Calibration	Equation 27
$β_{C^{*}}^{pth}$	Max. rel. effect of PTH analogs on pre-osteoclast to osteoclast differentiation	4.28	1	Calibration	Equation 27

CTX, C-terminal telopeptide; P1NP, procollagen type 1 amino-terminal propeptide; BSAP, bone-specific alkaline phosphatase; PTH, parathyroid hormone.

Additional files

Transparent reporting form: https://cdn.elifesciences.org/articles/76228/elife-76228-transrepform1-v1.docx
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Source code 1 Simulation codes needed to reproduce the results in the paper.: https://cdn.elifesciences.org/articles/76228/elife-76228-code1-v1.zip
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Figures

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

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.

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

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

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

Calibration datasets comparing model predictions and clinical data from various studies.

Tables

Values of fit weights $W^{β}$ used in Equation 29.

Data sources used to calibrate and validate the model.

Goodness-of-fit measures for calibration and validation datasets.

Full list of parameters of the core model and the medication extensions.

Additional files

Transparent reporting form

Source code 1

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

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.

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

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

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

Calibration datasets comparing model predictions and clinical data from various studies.

Values of fit weights Wβ<math id="inf114"><msup><mi>W</mi><mi>β</mi></msup></math> used in Equation 29.

Data sources used to calibrate and validate the model.

Goodness-of-fit measures for calibration and validation datasets.

Full list of parameters of the core model and the medication extensions.

Transparent reporting form

Source code 1

Values of fit weights $W^{β}$ used in Equation 29.