Mathematical Modeling of Bone Remodeling in response to Osteoporosis Treatments
1Khamir Mehta, 2Antonio Cabal, & 3David Ross
1Applied Mathematics and Modeling, Informatics IT , Merck & Co. 2Modeling and Simulation, Merck & Co, 3 School of Mathematical Sciences, Rochester Institute of Technology
4.2 : Effect of PTH 1. Abstract
A : Effect of PTH pulse size
The use of physiologically based middle‐out models, where the level of complexity incorporated in the model comes from the specific demands of the decisions required, is growing fast in the pharmaceutical industry. In this work we present an innovative model we developed for investigating the effect on bone remodeling of osteoporosis treatments. An imbalance in the activity of osteoclasts (cells that resorb bone matrix) and osteoblasts (cells that form new bone) can cause osteoporosis, wherein there is a net and progressive loss of bone. A better understanding of the molecular pathways regulating their activity in bone remodeling can guide the development of novel osteoporosis treatments. We have developed a semi‐mechanistic model of bone remodeling that incorporates, integrates, and extends available physiological information on bone remodeling. The mechanistic basis of the model establishes a unified structure, wherein it can include multiple novel (e.g calcilytics), and existing (e.g bisphosponates, PTH) treatment options, administered sequentially or simultaneously. We show some of the results of the model simulations in response to the common treatment strategies and compare them with published clinical data. Our model agrees with the data and elucidates on a long‐standing puzzle : the key pathways governing the switch between the anabolic and catabolic action of the PTH. We think that our model is a first step in understanding the bone remodeling process, enabling the testing of new hypotheses and the development of treatment strategies including combination therapies for osteoporosis. B : Effect of PTH pulse shape
2. Background
2.1 Bone Remodeling & Osteoporosis Continuous formation and resporption of bone via osteoblasts and osteoclasts •
A balance between the activity of bone resorbing agents (osteoclasts) and bone formation activity (osteoblasts) determines bone health
– Bone health chiefly measured by measuring Bone Mineral Density (BMD)
– The balance between osteoblast and osteoclast activity is tightly regulated by hormones and molecules: e.g PTH, Ca, etc.
www.umich.edu/news/Releases/2005/Feb05/bone.html
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Pulsed-Finkelstein
Continuous
MK5442
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14
PTH in Plasma (pM)
2.2 Osteoporosis Treatment Strategies
C : Comparison with Clinical Data
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•
Several different treatment options available :
• Bisphosphonates (e.g. Alendronate) act on the osteoclasts ability to resorb bone cells
• Direct Parathyroid Hormone (PTH) (e.g. Forteo) act on preferentially increasing the activity of osteoblasts. • Receptor activator of nuclear factor kappa‐B ligand (RANKL) inhibitors (e.g. Denosumab) act on RANKL, controlling their role in bone resorbption
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40
60
Time (days)
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http://models.cellml.org/exposure/73aca737d34378a6195760
b5164a0dd4/lemaire_tobin_greller_cho_suva_2004_g.cellml/
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Proposed role of PTH/PTHR on osteoblast apoptosis (Bellido et al 2003)
http://stke.sciencemag.org/cgi/content‐nw/full/jbc;278/50/50259/FIG8
– Has transparent, modular structure, based on simple mechanisms of action and can include multiple treatment strategies and link drug interventions to common biomarkers of bone health
– Incorporates available literature information and experimental observations relevant to bone remodeling. Model parameters and initial conditions are taken/estimated from available literature and/or match data
– Model Structure : 28 ODEs ; ~ 46 parameters. Parameters/initial conditions are constrained to ensure steady state corresponding to homeostasis 4. Results
4.1 Effect of Alendronate Treatment
0
10
20
30
Time (months)
40
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The Receptor activator of nuclear factor
kappa-B ligand (RANKL) is found in
osteoblasts surface and plays a critical role in
bone resorbtion.via osteoblast activation.
Inhibition of RANKL via interventions can
hence result in net increase in bone mineral
density. Here we present the effect of change
in RANKL concentration and how it affects
the concentration of bone cells (osteoblasts,
and osteoclasts), and ultimately the BMD. We
also show the observed biomarkers
corresponding to osteoblasts and osteoclasts
(P1NP and CTX)The simulations were
performed with 3 interventions in the RANKL
concentrations
at
every
180
days
(corresponding to biannual drug interventions,
as in typical Denosumab treatments [6]). The
results shown here are run for 2 years. The
red curve shown in the figure shows a control
with no change, and the blue curve is the
negative control, in which the RANKL was
increased, which expectedly leads to decrease
in bone mineral density
Lemaire model
• Model Features
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4.3 : Effect of RANKL Model Schematic
Peterson & Riggs model
http://wires.wiley.com/WileyCDA/WiresArticle/wisId‐WSBM115.html
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-10
-30
0
0
3. Model Development
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20
-20
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Deal and Gideon, Clv Cln J of Medicine (2003)
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30
% change in BMD
•
It is known that PTH has a catabolic effect
when it is administered continuously and an
anabolic effect when it is administered in
pulses of appropriate shape and frequency.
Figures on the left shows the results of
simulations that we performed with our model
of the effects on BMD of continuous PTH
administration and pulsed PTH administration
over the course 1 year of treatment. The
curves correspond to our simulations of the
daily dosing regimen.
A)Effect of PTH pulse size. Simulations
corresponding to increasing dose (increased
peak amplitude, seen in the figure on the left)
were performed and the corresponding
changes in the bone cell concentrations were
plotted on the right. The outer right panel
indicates the observed changes in the
biomarkers for osteoblasts (P!NP), osteoclasts
(CTX) and Bone Mineral Density (BMD). An
increasing dose of PTH does not necessarily
translate into greater increase in BMD.
B)Effect of PTH pulse shape. Studies, for
example that of Cosman and co-workers [5],
have shown that the shapes of plasma PTH
pulses affect the potency of PTH doses in
improving bone density. Here we show the
effect of pulse shape; we simulate scenarios
wherein patients were dosed similar amount
of PTH, albeit with different time profile as
seen in the figure on the left. The black curves
correspond to our simulation of the same 1year dosing regimen with a continuous dose of
PTH that again yields similar net daily increase
in PTH..
C)Comparison with Data. We compare our
results with clinical results as published by
Fiinkelstein [4]. In the study , patients were
given two years of daily PTH treatment
followed by a year of no treatment and another
year of daily treatment. Simulated patients were
given daily doses of PTH with pulse shape
consistent with 37 microgram sub-cutaneous
injection, as per the study. The pulse shape
can be seen on the left plot, while the BMD is
plotted on the right along with the clinical
observations. We have plotted the continuous
PTH treatment (in cyan) as a negative control.
5. Conclusions
 We have developed a mathematical model of dynamics of bone remodeling based on available physiological observations/data. The model, in form of ODEs, quantifies the relationships between the key molecular pathways governing bone remodeling, and links, via reasonable assumptions, the cell and molecular concentrations to the biomarkers measured in the laboratory.  The results presented here show the utility of the unified model that we have developed, in understanding the effect of various interventions for osteoporosis mitigation. Model results are consistent with the known effects of PTH, Bisphosphonates and anti‐RANKL on the bone remodeling process, and also agrees with available clinical data on BMD.  The model allows the comparison of osteoporosis therapies already on the market and new, innovative therapies in different stages of development. The model is a platform for evaluating potential new therapies under various administration protocols to characterize their efficacy and ease of implementation by comparing them with alternatives. The model also enables the generation of testable hypotheses and predictions of the possible outcomes of clinical trials. 6. Acknowledgements/Key References
As part of a study of various treatments, Miller and co-workers [6] treated postmenopausal women with low lumbar spine T-scores (scores between
-4 and -1.8) with Alendronate. The patients were given once-weekly alendronate (70 mg/Kg) treatments for two years. The treatments were given
for 2 years, and then stopped. The yellow dots in this graph represent the data acquired by Miller and co-workers. They’re graphed, here, along
with the error bars that appeared in Miller’s analysis; see Figure 4 of the paper [6]. The solid curve in A represents the CTX profile in our simulation
of Miller’s Alendronate trial; the solid curve in B represents the BMD results from our simulation. The model computations indicate that the
upswing in CTX accompanies a comparable upswing in osteoclast population.
Acknowledgements : Wendy Comisar; Rajiv Shrestha; Jeff Saltzman; Craig Fancourt ;Drew Denker ; Teun Post ; Rik deGreef ; Junghoon Lee. All from Merck Research
Laboratories for useful discussions and inputs.
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