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A Semi-Empirical Cell Dynamics Model for Bone Turnover Under External Stimulus

[+] Author and Article Information
E. Owen Carew1

Department of Mathematical Sciences,  Kent State University at Salem, Salem, OH, 44460ecarew@kent.edu

1

Correspondence author.

J Biomech Eng 134(2), 024503 (Mar 19, 2012) (9 pages) doi:10.1115/1.4005761 History: Received February 23, 2011; Revised January 09, 2012; Posted January 24, 2012; Published March 14, 2012; Online March 19, 2012

The normal periodic turnover of bone is referred to as remodeling. In remodeling, old or damaged bone is removed during a ‘resorption’ phase and new bone is formed in its place during a ‘formation’ phase in a sequence of events known as coupling. Resorption is preceded by an ‘activation’ phase in which the signal to remodel is initiated and transmitted. Remodeling is known to involve the interaction of external stimuli, bone cells, calcium and phosphate ions, and several proteins, hormones, molecules, and factors. In this study, a semi-empirical cell dynamics model for bone remodeling under external stimulus that accounts for the interaction between bone mass, bone fluid calcium, bone calcium, and all three major bone cell types, is presented. The model is formulated to mimic biological coupling by solving separately and sequentially systems of ODEs for the activation, resorption, and formation phases of bone remodeling. The charateristic time for resorption (20 days) and the amount of resorption (∼0.5%) are fixed for all simulations, but the formation time at turnover is an output of the model. The model was used to investigate the effects of different types of strain stimuli on bone turnover under bone fluid calcium balance and imbalance conditions. For bone fluid calcium balance, the model predicts complete turnover after 130 days of formation under constant 1000 microstrain stimulus; after 47 days of formation under constant 2000 microstrain stimulus; after 173 days of formation under strain-free conditions, and after 80 days of formation under monotonic increasing strain stimulus from 1000 to 2000 microstrain. For bone fluid calcium imbalance, the model predicts that complete turnover occurs after 261 days of formation under constant 1000 microstrain stimulus and that turnover never occurs under strain-free conditions. These predictions were not impacted by mean dynamic input strain stimuli of 1000 and 2000 microstrain at 1 Hz and 1000 microstrain amplitude. The formation phase of remodeling lasts longer than the resorption phase, increased strain stimulus accelerates bone turnover, while absence of strain significantly delays or prevents it, and formation time for turnover under monotonic increasing strain conditions is intermediate to those for constant strain stimuli at the minimum and maximum monotonic strain levels. These results are consistent with the biology, and with Frost’s mechanostat theory.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Imposed strain stimuli as defined by Eq. 3 plotted over 1 s: (a) S1 – constant 1000μɛ (k1  = 0, k2  = 0, ɛ = 0.001), (b) S2 – constant 2000μɛ (k1=0,k2=0,ɛ0=0.002), (c) S3 – zero strain (k1=0,k2=0,ɛ0=0), (d) S4 – monotonic increasing strain from 1000μɛ at t=0 to 2000μɛ at t=τA (k1=0,k2=1,ɛ0=0.001,ɛmax=0.002), (e) S5 – 1 Hz dynamic strain, 1000μɛ amplitude, 1000μɛ mean (k1=1,k2=0,ω=1,ɛ0=0.001,ɛmax=0.002), and (f) S6 – 1 Hz dynamic strain, 1000μɛ amplitude, 2000μɛ mean (k1=1,k2=0,ω=1,ɛ0=0.002,ɛmax=0.003)

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Figure 2

Simulation of the formation phase of bone remodeling under constant strain stimulus S1 (Fig. 1) and bone fluid calcium (BFC) balance conditions. Predicted transients for: (a) bone fluid calcium (BFC), (b) bone calcium, (c) osteoblast concentration, and (d) bone mass. Model parameters are: τA=1 min (∼7×10-4 days), rAC5=0.3mMh-1 (BFC balance), α1=10-15%(Ca2+)-1day-1, α2=5×10-11%(Ca+)-1day-1, α3=10-9mM-1, α4=3×106day-1, and τR=20 days. BFC and bone calcium concentrations which are related by Eq. 1 remain at steady state. Osteoblast concentration increases nonlinearly to steady state and bone mass increases linearly, during formation. Osteocytes, responding osteoblasts, and osteoclasts are not updated during formation. Bone turnover is predicted after 150 days (τF=130 days, ΣΔρ≈0.0015%).

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Figure 4

Simulation of the formation phase of bone remodeling under strain-free conditions S3 (Fig. 1) and BFC balance conditions. Osteoblast concentration decreases nonlinearly to steady state in the early stages of the formation phase leading to a delay in bone turnover. Model parameters and comments about calcium, other bone cells, and bone mass transients given in Fig. 2 apply. Bone turnover is predicted after 193 days (τF=173 days, ΣΔρ≈0.001%).

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Figure 5

Simulation of the formation phase of bone remodeling under monotonic increasing strain stimulus S4 (Fig. 1) and BFC balance conditions. Model parameters and comments about calcium, bone cells, and bone mass transients given for Fig. 2 apply. Bone turnover is predicted after 100 days (τF=80 days, ΣΔρ=0.0019%) and formation time is intermediate to those for constant 1000μɛ and 2000μɛ stimuli.

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Figure 3

Simulation of the formation phase of bone remodeling under constant strain stimulus S2 (Fig. 1) and BFC balance conditions. Model parameters and comments about calcium, bone cells, and bone mass transients given for Fig. 2 apply. Bone turnover is predicted after 67 days (τF=47 days, ΣΔρ≈0.0038%).

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Figure 6

Simulation of the formation phase of bone remodeling under constant strain stimulus S1 (Fig. 1) and BFC imbalance conditions. Here, rAC5=0.295mMh-1 (BFC imbalance) - other parameters are as given in Fig. 2 BFC decreases while bone calcium increases, nonlinearly, to steady state. Bone mass increases nonlinearly, and then linearly. Comments about osteoblast and other bone cell transients given in Fig. 2 apply. Bone turnover is predicted after 281 days (τF=261 days, ΣΔρ≈0.0002%).

Grahic Jump Location
Figure 7

Simulation of the formation phase of bone remodeling under strain-free conditions S3 (Fig. 1) and BFC imbalance conditions. Model parameters are identical to those for Fig. 6. Osteoblast concentration decreases nonlinearly to steady state as in Fig. 4 Bone mass increases nonlinearly during formation, peaks at t≈68 days (τF=48 days, ΣΔρ≈0.23%) and starts to decrease linearly, indicating that bone never turns over and there is loss of bone mass. Comments about calcium transients given in Fig. 6 and other bone cell transients given in Fig. 2, apply.

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