Modeling of bone formation and resorption mediated by parathyroid hormone: response to estrogen/PTH therapy
Catherine
Lloyd
Auckland Bioengineering Institute, The University of Auckland
Model Status
This CellML model represents the second model from the published paper (eqs 38-40). This model builds on the core model (which describes changes in the concentration of PTH, the number of osteoclasts and the number of osteoblasts) by considering the effects of estrogen administration (for a period of 12 days in an interval of 28 days). The model runs in OpenCell to recreate the published results (figure 7a). The units are all dimensionless (in accordance with the published paper) and are therefore consistent. However a dimensionless time does mean that the model will not run in COR.
Model Structure
ABSTRACT: Bone, a major reservoir of body calcium, is under the hormonal control of the parathyroid hormone (PTH). Several aspects of its growth, turnover, and mechanism, occur in the absence of gonadal hormones. Sex steroids such as estrogen, nonetheless, play an important role in bone physiology, and are extremely essential to maintain bone balance in adults. In order to provide a basis for understanding the underlying mechanisms of bone remodeling as it is mediated by PTH, we propose here a mathematical model of the process. The nonlinear system model is then utilized to study the temporal effect of PTH as well as the action of estrogen replacement therapy on bone turnover. Analysis of the model is done on the assumption, supported by reported clinical evidence, that the process is characterized by highly diversified dynamics, which warrants the use of singular perturbation arguments. The model is shown to exhibit limit cycle behavior, which can develop into chaotic dynamics for certain ranges of the system's parametric values. Effects of estrogen and PTH administrations are then investigated by extending on the core model. Analysis of the model seems to indicate that the paradoxical observation that intermittent PTH administration causes net bone deposition while continuous administration causes net bone loss, and certain other reported phenomena may be attributed to the highly diversified dynamics which characterizes this nonlinear remodeling process.
model diagram
Schematic diagram depicting the Rattanakul et al 2003 model - effects of PTH on osteoclast differentiation by osteoblasts. Exogenous application of PTH, and the effects of oestrogen, are also depicted.
The original paper reference is cited below:
Modeling of bone formation and resorption mediated by parathyroid hormone: response to estrogen/PTH therapy, Chontita Rattanakul, Yongwimon Lenbury, Nateetip Krishnamara, and David J. Wollkind, 2003,BioSystems, 70, 55-72. PubMed ID: 12753937
$\frac{d x}{d \mathrm{time}}=\frac{\mathrm{a1}}{\mathrm{k1}+y}-\mathrm{b1}x$
$\frac{d y}{d \mathrm{time}}=\begin{cases}\mathrm{epsilon}(\frac{(\mathrm{a2}+\mathrm{a3}x)yz}{\mathrm{k2}+x^{2.0}}-\mathrm{b2}y) & \text{if $(\mathrm{time}\ge 0.0)\land (\mathrm{time}< 10000.0)$}\\ \mathrm{epsilon}(\frac{(\mathrm{a2}+\mathrm{a3}x)yz}{\mathrm{k2}+x^{2.0}}-\mathrm{b2}y+\mathrm{kc}y) & \text{if $(\mathrm{time}\ge 10000.0)\land (\mathrm{time}< 27280.0)$}\\ \mathrm{epsilon}(\frac{(\mathrm{a2}+\mathrm{a3}x)yz}{\mathrm{k2}+x^{2.0}}-\mathrm{b2}y) & \text{if $(\mathrm{time}\ge 27280.0)\land (\mathrm{time}< 50320.0)$}\\ \mathrm{epsilon}(\frac{(\mathrm{a2}+\mathrm{a3}x)yz}{\mathrm{k2}+x^{2.0}}-\mathrm{b2}y+\mathrm{kc}y) & \text{if $(\mathrm{time}\ge 50320.0)\land (\mathrm{time}< 67600.0)$}\\ \mathrm{epsilon}(\frac{(\mathrm{a2}+\mathrm{a3}x)yz}{\mathrm{k2}+x^{2.0}}-\mathrm{b2}y) & \text{if $(\mathrm{time}\ge 67600.0)\land (\mathrm{time}< 90640.0)$}\\ \mathrm{epsilon}(\frac{(\mathrm{a2}+\mathrm{a3}x)yz}{\mathrm{k2}+x^{2.0}}-\mathrm{b2}y+\mathrm{kc}y) & \text{if $(\mathrm{time}\ge 90640.0)\land (\mathrm{time}< 107920.0)$}\end{cases}$
$\frac{d z}{d \mathrm{time}}=\mathrm{epsilon}\mathrm{delta}(\mathrm{a4}x-\frac{\mathrm{a5}xz}{\mathrm{k3}+x}+\mathrm{b3}z)$
Rattanakul et al's 2003 mathematical model of bone formation and resorption mediated by parathyroid hormone: response to estrogen/PTH therapy.
number of osteoblastic cellsznumber of osteoclastic cellsyPTHconcentration of parathyroid hormonexDavidWollkindJc.lloyd@auckland.ac.nzThe Bioengineering InstituteThe University of Auckland0.1100000250BioSystemsChontitaRattananakul2007-07-24T00:00:00+00:00CatherineLloydMayosteoblastparathyroid hormoneosteoporosisosteoclastModeling of bone formation and resorption mediated by parathyroid hormone: response to estrogen/PTH therapy557072Modeling of bone formation and resorption mediated by parathyroid hormone: response to estrogen/PTH therapy (core model + exogenous estrogen administration)The University of Auckland, Bioengineering InstituteYongwimonLenburyNateetipKrishnamaraThis is a CellML description of Rattanakul et al's 2003 mathematical model of bone formation and resorption mediated by parathyroid hormone: response to estrogen/PTH therapy.12753937keywordCatherine Lloyd2003-06-00 00:00