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Peterson & Riggs (2010). A physiologically based mathematical model of integrated calcium homeostasis and bone remodeling.

July 2016, model of the month by Vincent Knight-Schrijver
Original model: BIOMD0000000613


Introduction

In a landmark case for Quantitative Systems Pharmacology (QSP), this model, by Peterson and Riggs (2010)[1, BIOMD0000000613], was used in a U.S. Food and Drug Administration (FDA) advisory meeting for review of a Biological License Application (BLA) [BLA 125511].
The application under scrutiny was for use of a recombinant parathyroid hormone, Natpara, in treating hypoparathyroidism. The inclusion of this particular model for the review was important because elevated parathyroid hormone (PTH) promotes excessive calcium (Ca) excretion via the kidney. This hypercalciuria is an adverse effect of PTH administration and causes nephrolithiasis and kidney damage in patients.
Thus, to answer questions on correct dosing strategies, the committee adapted the model and predicted optimal dose regimens for the drug to adhere to the therapeutic window.

The original model was developed to simulate the processes which govern the homeostatic control of bone remodelling (the continual restructuring of the skeleton). Bone remodelling is largely mediated by two opposing reactions, resorption and formation. These processes are regulated, respectively, by osteoclasts (OCs) and osteoblasts (OBs), and the balance of these two processes heavily influences the flux of Ca through the body by means of storage and release of Ca into, and from, the skeleton. The activity of both OBs and OCs is controlled through a large signalling component including transforming growth factor beta (TGF-beta), PTH and RANK, RANKL and OPG. Further control is also governed by the availability of Ca and phosphate (PO4) ions and so the complexity of bone remodelling in vivo requires tight homeostatic regulation.

Model

Previous models independently addressed the multiple facets of bone remodelling homeostasis but the model presented here combined them to fully capture the dynamic nature of bone tissue [1, BIOMD0000000613]. This encompassed models of Ca homeostasis [2] and the key elements of cellular signalling [3],[4,BIOMD0000000278]. The model could then be used to simulate bone modelling homeostasis over time and predict the perturbative effects of diseases and therapy upon bone physiology and remodelling.

Figure 1

Figure 1. The model structure, highlighting the interactions between calcium homeostasis and bone remodelling. The model includes the intake of Ca and PO4 from the gut and the excretion of Ca and PO4, including reabsorption, through the kidneys. Net Ca is then lost or gained through the OC and OB activity, illustrating the balance between bone resorption or formation. Figure taken from [1].

Results

One key part of this model was to represent the activity of PTH. Endogenous PTH is normally secreted by the PTH gland but altered PTH secretion rates can result in hyperparathyroidism and hypoparathyroidism. The model was able to reproduce the effects of these disease states upon the plasma concentrations of Ca and PO4 [Figure 2, column 1].

Figure 2

Figure 2.. The effects of PTH disease states upon Ca, PO4, osteoblasts and osteoclasts. Simulations were made with the curated SBML version of the model. For hyperparathyroidism, the PTH concentration was simply assigned as the sum of PTH_0 + a Henri-Michaelis-Menten function with S = time, V = (PTH_0 x 3) and Km = 1000. For hypoparathyroidism, the PTH concentration was fixed to PTH = 0.5 x PTH_0.

Through simulated pulsatile PTH administration, the model demonstrates that the apoptosis rate of osteoblasts can be decreased over time [Figure 3]. This benefit of PTH administered in this manner is in stark contrast to the tonic PTH elevation as seen in hyperparathyroidism. In hyperparathyroidism, the PTH-mediated reduction in osteoblast apoptosis rates is counteracted by an even greater increase in osteoclast activity thereby promoting bone desctruction. The model hints at how this effect does not occur in pulsatile administration.

Figure 3

Figure 3. Simulations of once-daily teriparatide administration for 30 days. Osteoblast apoptosis is predicted to decrease and as a result the number of osteoblasts increase. Osteoclast numbers also increase. However the osteoclast pool is significantly lower than with tonic PTH elevation. Simulations were carried out with the curated SBML model; the apoptosis parameter is "kbprime".


Conclusion

The presented model integrates previous knowledge and theoretical models. The outcome is a larger and potentially more useful kinetic model with systems pharmacology utility for predicting drug dosing, mechanisms and long-term adverse effects in patients with hypoparathyroidism. The model, in a landmark case, has been reused in official drug review proceedings for the final stages of drug development and illustrates the added value that QSP brings to biology and drug discovery. The results suggested the use of the dose regimens seen in the latest dose comparison clinical study: NCT02781844.

References

  1. Peterson, MC, Riggs, MM (2010). A physiologically based mathematical model of integrated calcium homeostasis and bone remodeling. Bone, 46, 1:49-63.
  2. Raposo, JF, Sobrinho, LG, Ferreira, HG (2002). A minimal mathematical model of calcium homeostasis. J. Clin. Endocrinol. Metab., 87, 9:4330-40.
  3. Bellido, T, Ali, AA, Plotkin, LI, Fu, Q, Gubrij, I, Roberson, PK, Weinstein, RS, O'Brien, CA, Manolagas, SC, Jilka, RL (2003). Proteasomal degradation of Runx2 shortens parathyroid hormone-induced anti-apoptotic signaling in osteoblasts. A putative explanation for why intermittent administration is needed for bone anabolism. J. Biol. Chem., 278, 50:50259-72.
  4. Lemaire, V, Tobin, FL, Greller, LD, Cho, CR, Suva, LJ (2004). Modeling the interactions between osteoblast and osteoclast activities in bone remodeling. J. Theor. Biol., 229, 3:293-309.
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