- Author:
- Hanne <Hanne@hanne-nielsens-macbook.local>
- Date:
- 2010-06-11 13:08:47+12:00
- Desc:
- Added images in ai svg and png format
- Permanent Source URI:
- https://models.physiomeproject.org/workspace/pivonka_zimak_smith_gardiner_dunstan_sims_martin_mundy_2008/rawfile/5b5911153fb79230c7d470da81ce891f1c04f8e8/pivonka_2008_matlab.cellml
<?xml version='1.0' encoding='utf-8'?>
<model xmlns="http://www.cellml.org/cellml/1.0#" xmlns:cmeta="http://www.cellml.org/metadata/1.0#" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:bqs="http://www.cellml.org/bqs/1.0#" xmlns:cellml="http://www.cellml.org/cellml/1.0#" xmlns:dcterms="http://purl.org/dc/terms/" xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#" cmeta:id="pivonka_2008" name="pivonka_2008">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
<articleinfo>
<title>Model structure and control of bone remodeling: a theoretical study</title>
<author>
<firstname>Catherine</firstname>
<surname>Lloyd</surname>
<affiliation>
<shortaffil>Auckland Bioengineering Institute, The University of Auckland</shortaffil>
</affiliation>
</author>
</articleinfo>
<section id="sec_status">
<title>Model Status</title>
<para>
This CellML model runs in both OpenCell and COR to reproduce the published results (figure 5). The units have been checked and they are consistent. This particular version of the model has an increased rate of osteoclast differentiation (D_OC_p) for the first 100 days, and the values for "model 1" have been taken from table 2 of the published paper. This CellML model translation was based on the author's original Matlab code and we would like to thank them for sharing this code with us.
</para>
</section>
<sect1 id="sec_structure">
<title>Model Structure</title>
<para>
ABSTRACT: It is generally accepted that RANKL is highly expressed in osteoblast precursor cells while OPG is highly expressed in mature osteoblasts, but to date no functional utility to the BMU has been proposed for this particular ligand-decoy-receptor expression profile. As discovered in the mid 90s, the RANK-RANKL-OPG signaling cascade is a major signaling pathway regulating bone remodeling. In this paper we study theoretically the functional implications of particular RANKL/OPG expression profiles on bone volume. For this purpose we formulate an extended bone-cell dynamics model describing functional behaviour of basic multicellular units (BMUs) responsible for bone resorption and formation. This model incorporates the RANK-RANKL-OPG signaling together with the regulating action of TGF-beta on bone cells. The bone-cell population model employed here builds on the work of Lemaire et al. (2004) [1], but incorporates the following significant modifications: (i) addition of a rate equation describing changes in bone volume with time as the key 'output function' tracking functional behaviour of BMUs, (ii) a rate equation describing release of TGF-beta from the bone matrix, (iii) expression of OPG and RANKL on both osteoblastic cell lines, and (iv) modified activator/repressor functions. Using bone volume as a functional selection criterion, we find that there is a preferred arrangement for ligand expression on particular cell types, and further, that this arrangement coincides with biological observations. We then investigate the model parameter space combinatorially, searching for preferred 'groupings' of changes in differentiation rates of various cell types. Again, a criterion of bone volume change is employed to identify possible ways of optimally controlling BMU responses. While some combinations of changes in differentiation rates are clearly unrealistic, other combinations of changes in differentiation rates are potentially functionally significant. Most importantly, the combination of parameter changes representing the signaling pathway for TGF-beta gives a unique result that appears to have a clear biological rationale. The methodological approach for the investigation of model structure described here offers a theoretical explanation as to why TGF-beta has its particular suite of biological effects on bone-cell differentiation rates.
</para>
<informalfigure float="0" id="fig_reaction_diagram">
<mediaobject>
<imageobject>
<objectinfo>
<title>model diagram</title>
</objectinfo>
<imagedata fileref="pivonka_2008.png"/>
</imageobject>
</mediaobject>
<caption>Schematic illustration of bone-cell population model: (a) osteoblastic cell line and (b) osteoclastic cell lines taken into account together with RANK-RANKL-OPG pathway (DOB, DOC...differentiation rates of osteoblasts and osteoclasts; EOB, EOC...elimination/apoptosis rates of osteoblasts and osteoclasts; RANKL binds to RANK and promotes differentiation of OCp; OPG binds to RANKL and so inhibits OCp differentiation).</caption>
</informalfigure>
<para>
The original paper reference is cited below:
</para>
<para>
Model structure and control of bone remodeling: a theoretical study, Pivonka P, Zimak J, Smith DW, Gardiner BS, Dunstan CR, Sims NA, Martin TJ, Mundy GR, 2008, <emphasis>Bone</emphasis>, 43, 249-263. <ulink url="http://www.ncbi.nlm.nih.gov/pubmed/18514606">PubMed ID: 18514606</ulink>
</para>
</sect1>
</article>
</documentation>
<units name="day">
<unit units="second" multiplier="86400.0"/>
</units>
<units name="pM">
<unit units="mole" prefix="pico"/>
<unit units="litre" exponent="-1"/>
</units>
<units name="per_pM">
<unit units="pM" exponent="-1"/>
</units>
<units name="flux">
<unit units="pM"/>
<unit units="day" exponent="-1"/>
</units>
<units name="percent" base_units="yes"/>
<units name="first_order_rate_constant">
<unit units="day" exponent="-1"/>
</units>
<units name="second_order_rate_constant">
<unit units="pM" exponent="-1"/>
<unit units="day" exponent="-1"/>
</units>
<component name="environment">
<variable units="day" public_interface="out" name="time"/>
</component>
<component cmeta:id="OB_p" name="OB_p">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#OB_p">
<dc:title>OB_p</dc:title>
<dcterms:alternative>responding osteoblast population</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="dimensionless" public_interface="out" name="OB_p" initial_value="6.194e-4"/>
<variable units="first_order_rate_constant" name="D_OB_u" initial_value="7.000e-4"/>
<variable units="first_order_rate_constant" name="pd_OB_p" initial_value="0.0"/>
<variable units="dimensionless" public_interface="in" name="pi_TGF_beta_act"/>
<variable units="dimensionless" public_interface="in" name="pi_TGF_beta_rep"/>
<variable units="first_order_rate_constant" public_interface="in" name="D_OB_p"/>
<variable units="day" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> OB_p </ci>
</apply>
<apply>
<plus/>
<apply>
<minus/>
<apply>
<times/>
<ci> D_OB_u </ci>
<ci> pi_TGF_beta_act </ci>
</apply>
<apply>
<times/>
<ci> D_OB_p </ci>
<ci> pi_TGF_beta_rep</ci>
<ci> OB_p </ci>
</apply>
</apply>
<ci> pd_OB_p </ci>
</apply>
</apply>
</math>
</component>
<component cmeta:id="OB_a" name="OB_a">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#OB_a">
<dc:title>OB_a</dc:title>
<dcterms:alternative>active osteoblast population</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="dimensionless" public_interface="out" name="OB_a" initial_value="5.584e-4"/>
<variable units="first_order_rate_constant" name="A_OB_a" initial_value="1.890e-1"/>
<variable units="first_order_rate_constant" name="pd_OB_a" initial_value="0.0"/>
<variable units="dimensionless" public_interface="in" name="pi_TGF_beta_rep"/>
<variable units="first_order_rate_constant" public_interface="in" name="D_OB_p"/>
<variable units="dimensionless" public_interface="in" name="OB_p"/>
<variable units="day" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> OB_a </ci>
</apply>
<apply>
<plus/>
<apply>
<minus/>
<apply>
<times/>
<ci> D_OB_p </ci>
<ci> pi_TGF_beta_rep </ci>
<ci> OB_p </ci>
</apply>
<apply>
<times/>
<ci> A_OB_a </ci>
<ci> OB_a </ci>
</apply>
</apply>
<ci> pd_OB_a </ci>
</apply>
</apply>
</math>
</component>
<component cmeta:id="OC_a" name="OC_a">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#OC_a">
<dc:title>OC_a</dc:title>
<dcterms:alternative>active osteoclast population</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="dimensionless" public_interface="out" name="OC_a" initial_value="8.070e-4"/>
<variable units="first_order_rate_constant" name="D_OC_p"/>
<variable units="first_order_rate_constant" name="A_OC_a" initial_value="7.000e-1"/>
<variable units="first_order_rate_constant" name="pd_OC_a" initial_value="0.0"/>
<variable units="dimensionless" public_interface="in" name="pi_RANKL_act"/>
<variable units="dimensionless" public_interface="in" name="pi_TGF_beta_act"/>
<variable units="day" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> OC_a </ci>
</apply>
<apply>
<plus/>
<apply>
<minus/>
<apply>
<times/>
<ci> D_OC_p </ci>
<ci> pi_RANKL_act </ci>
</apply>
<apply>
<times/>
<ci> A_OC_a </ci>
<ci> pi_TGF_beta_act </ci>
<ci> OC_a </ci>
</apply>
</apply>
<ci> pd_OC_a </ci>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> D_OC_p </ci>
<piecewise>
<piece>
<cn type="e-notation" cellml:units="first_order_rate_constant">2.100<sep/>-2</cn>
<apply>
<and/>
<apply>
<gt/>
<ci> time </ci>
<cn cellml:units="day"> 0.0 </cn>
</apply>
<apply>
<leq/>
<ci> time </ci>
<cn cellml:units="day"> 100.0 </cn>
</apply>
</apply>
</piece>
<otherwise>
<cn type="e-notation" cellml:units="first_order_rate_constant">2.100<sep/>-3</cn>
</otherwise>
</piecewise>
</apply>
</math>
</component>
<component cmeta:id="BV" name="BV">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#BV">
<dc:title>BV</dc:title>
<dcterms:alternative>bone volume</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="dimensionless" name="BV" initial_value="100.0"/>
<variable units="dimensionless" name="OC_a_initial" initial_value="1.0"/>
<variable units="dimensionless" name="OB_a_initial" initial_value="1.0"/>
<variable units="first_order_rate_constant" name="k_form" initial_value="1.571e0"/>
<variable units="first_order_rate_constant" initial_value="200.00" name="k_res"/>
<variable units="dimensionless" public_interface="in" name="OC_a"/>
<variable units="dimensionless" public_interface="in" name="OB_a"/>
<variable units="day" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> BV </ci>
</apply>
<apply>
<minus/>
<apply>
<times/>
<ci> k_form </ci>
<apply>
<minus/>
<ci> OB_a </ci>
<ci> OB_a_initial </ci>
</apply>
</apply>
<apply>
<times/>
<ci> k_res </ci>
<apply>
<minus/>
<ci> OC_a </ci>
<ci> OC_a_initial </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component cmeta:id="TGF_beta" name="TGF_beta">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#TGF_beta">
<dc:title>TGF_beta</dc:title>
<dcterms:alternative>TGF-beta concentration</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="dimensionless" public_interface="out" name="pi_TGF_beta_act"/>
<variable units="dimensionless" public_interface="out" name="pi_TGF_beta_rep"/>
<variable units="pM" initial_value="1.416e-3" name="KD_TGF_repress"/>
<variable units="pM" initial_value="4.545e-3" name="KD_TGF_activate"/>
<variable units="pM" name="alpha" initial_value="1.0"/>
<variable units="dimensionless" public_interface="in" name="OC_a"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> pi_TGF_beta_act </ci>
<apply>
<divide/>
<apply>
<times/>
<ci> alpha </ci>
<ci> OC_a </ci>
</apply>
<apply>
<plus/>
<ci> KD_TGF_activate </ci>
<apply>
<times/>
<ci> alpha </ci>
<ci> OC_a </ci>
</apply>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> pi_TGF_beta_rep </ci>
<apply>
<divide/>
<cn cellml:units="dimensionless"> 1 </cn>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1 </cn>
<apply>
<divide/>
<apply>
<times/>
<ci> alpha </ci>
<ci> OC_a </ci>
</apply>
<ci> KD_TGF_repress </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component cmeta:id="PTH" name="PTH">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#PTH">
<dc:title>PTH</dc:title>
<dcterms:alternative>PTH concentration</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="dimensionless" public_interface="out" name="pi_PTH_act"/>
<variable units="dimensionless" public_interface="out" name="pi_PTH_rep"/>
<variable units="flux" name="beta_PTH" initial_value="2.500e2"/>
<variable units="first_order_rate_constant" name="Deg_PTH" initial_value="8.600e1"/>
<variable units="flux" name="P_PTH_d" initial_value="0.0"/>
<variable units="pM" name="PTH_tot"/>
<variable units="pM" name="KD_PTH_repress" initial_value="1.5e2"/>
<variable units="pM" name="KD_PTH_activate" initial_value="2.226e-1"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> pi_PTH_act </ci>
<apply>
<divide/>
<ci> PTH_tot </ci>
<apply>
<plus/>
<ci> PTH_tot </ci>
<ci> KD_PTH_activate </ci>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> pi_PTH_rep </ci>
<apply>
<divide/>
<cn cellml:units="dimensionless"> 1 </cn>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1 </cn>
<apply>
<divide/>
<ci> PTH_tot </ci>
<ci> KD_PTH_repress </ci>
</apply>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> PTH_tot </ci>
<apply>
<divide/>
<apply>
<plus/>
<ci> beta_PTH </ci>
<ci> P_PTH_d </ci>
</apply>
<ci> Deg_PTH </ci>
</apply>
</apply>
</math>
</component>
<component cmeta:id="OPG" name="OPG">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#OPG">
<dc:title>OPG</dc:title>
<dcterms:alternative>OPG concentration</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="pM" public_interface="out" name="OPG"/>
<variable units="flux" name="beta_OPG" initial_value="1.625e8"/>
<variable units="dimensionless" name="beta1_OB_p_OPG" initial_value="0.0"/>
<variable units="dimensionless" name="beta2_OB_a_OPG" initial_value="1.0"/>
<variable units="pM" name="OPG_max" initial_value="2.000e8"/>
<variable units="flux" name="OPG_eff"/>
<variable units="flux" name="P_OPG_d" initial_value="0.0"/>
<variable units="first_order_rate_constant" name="Deg_OPG" initial_value="3.500e-1"/>
<variable units="flux" name="Degr_OPG"/>
<variable units="flux" name="prod_OPG"/>
<variable units="dimensionless" public_interface="in" name="pi_PTH_rep"/>
<variable units="dimensionless" public_interface="in" name="OB_p"/>
<variable units="dimensionless" public_interface="in" name="OB_a"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> OPG </ci>
<apply>
<divide/>
<apply>
<plus/>
<ci> P_OPG_d </ci>
<ci> OPG_eff </ci>
</apply>
<apply>
<plus/>
<apply>
<divide/>
<ci> OPG_eff </ci>
<ci> OPG_max </ci>
</apply>
<ci> Deg_OPG </ci>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> OPG_eff </ci>
<apply>
<times/>
<apply>
<plus/>
<apply>
<times/>
<ci> beta1_OB_p_OPG </ci>
<ci> beta_OPG </ci>
<ci> OB_p </ci>
</apply>
<apply>
<times/>
<ci> beta2_OB_a_OPG </ci>
<ci> beta_OPG </ci>
<ci> OB_a </ci>
</apply>
</apply>
<ci> pi_PTH_rep </ci>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> prod_OPG </ci>
<apply>
<plus/>
<apply>
<times/>
<ci> OPG_eff </ci>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 1 </cn>
<apply>
<divide/>
<ci> OPG </ci>
<ci> OPG_max </ci>
</apply>
</apply>
</apply>
<ci> P_OPG_d </ci>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> Degr_OPG </ci>
<apply>
<times/>
<ci> Deg_OPG </ci>
<ci> OPG </ci>
</apply>
</apply>
</math>
</component>
<component cmeta:id="RANKL" name="RANKL">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#RANKL">
<dc:title>RANKL</dc:title>
<dcterms:alternative>RANKL concentration</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="dimensionless" public_interface="out" name="pi_RANKL_act"/>
<variable units="dimensionless" name="RANKL"/>
<variable units="dimensionless" name="K_A2_RANKL" initial_value="3.412e-2"/>
<variable units="dimensionless" name="K_A1_RANKL" initial_value="1.000e-3"/>
<variable units="pM" name="RANK" initial_value="1.000e1"/>
<variable units="pM" name="RANKL_eff"/>
<variable units="pM" name="RANKL_tot"/>
<variable units="pM" name="K_RANKL_OB_p"/>
<variable units="pM" name="K_RANKL_OB_a"/>
<variable units="pM" name="OPG_RANKL"/>
<variable units="dimensionless" name="R_RANKL" initial_value="2.703e6"/>
<variable units="pM" name="RANKL_RANK"/>
<variable units="flux" name="pd_RANKL"/>
<variable units="flux" name="P_RANKL_d" initial_value="0.0" />
<variable units="pM" name="R1_OB_p_RANKL" initial_value="1.0"/>
<variable units="pM" name="R2_OB_a_RANKL" initial_value="0.0"/>
<variable units="flux" name="beta_RANKL" initial_value="1.684e4"/>
<variable units="first_order_rate_constant" name="Deg_RANKL" initial_value="1.013e1"/>
<variable units="pM" name="KD_RANKL_activate" initial_value="4.457e0"/>
<variable units="dimensionless" public_interface="in" name="pi_PTH_act"/>
<variable units="dimensionless" public_interface="in" name="OB_p"/>
<variable units="dimensionless" public_interface="in" name="OB_a"/>
<variable units="pM" public_interface="in" name="OPG"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> RANKL </ci>
<apply>
<divide/>
<apply>
<plus/>
<ci> beta_RANKL </ci>
<ci> P_RANKL_d </ci>
</apply>
<apply>
<times/>
<ci> RANKL_tot </ci>
<apply>
<plus/>
<apply>
<divide/>
<ci> beta_RANKL </ci>
<ci> RANKL_eff </ci>
</apply>
<ci> Deg_RANKL </ci>
</apply>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> RANKL_eff </ci>
<apply>
<times/>
<apply>
<plus/>
<apply>
<times/>
<ci> K_RANKL_OB_p </ci>
<ci> OB_p </ci>
</apply>
<apply>
<times/>
<ci> K_RANKL_OB_a </ci>
<ci> OB_a </ci>
</apply>
</apply>
<ci> pi_PTH_act </ci>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> K_RANKL_OB_p </ci>
<apply>
<times/>
<ci> R1_OB_p_RANKL </ci>
<ci> R_RANKL </ci>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> K_RANKL_OB_a </ci>
<apply>
<times/>
<ci> R2_OB_a_RANKL </ci>
<ci> R_RANKL </ci>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> RANKL_tot </ci>
<apply>
<plus/>
<cn cellml:units="pM"> 1 </cn>
<apply>
<times/>
<ci> K_A2_RANKL </ci>
<ci> RANK </ci>
</apply>
<apply>
<times/>
<ci> K_A1_RANKL </ci>
<ci> OPG </ci>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> pd_RANKL </ci>
<apply>
<plus/>
<apply>
<minus/>
<apply>
<times/>
<ci> beta_RANKL </ci>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 1 </cn>
<apply>
<times/>
<ci> RANKL </ci>
<apply>
<divide/>
<ci> RANKL_tot </ci>
<ci> RANKL_eff </ci>
</apply>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci> Deg_RANKL </ci>
<ci> RANKL </ci>
<ci> RANKL_tot </ci>
</apply>
</apply>
<ci> P_RANKL_d </ci>
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</math>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> pi_RANKL_act </ci>
<apply>
<divide/>
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<apply>
<plus/>
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</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> OPG_RANKL </ci>
<apply>
<times/>
<ci> K_A1_RANKL </ci>
<ci> OPG </ci>
<ci> RANKL </ci>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> RANKL_RANK </ci>
<apply>
<times/>
<ci> K_A2_RANKL </ci>
<ci> RANKL </ci>
<ci> RANK </ci>
</apply>
</apply>
</math>
</component>
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<map_components component_2="environment" component_1="BV"/>
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<map_variables variable_2="pi_TGF_beta_act" variable_1="pi_TGF_beta_act"/>
<map_variables variable_2="OC_a" variable_1="OC_a"/>
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<map_components component_2="OC_a" component_1="RANKL"/>
<map_variables variable_2="pi_RANKL_act" variable_1="pi_RANKL_act"/>
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<connection>
<map_components component_2="BV" component_1="OC_a"/>
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<connection>
<map_components component_2="BV" component_1="OB_a"/>
<map_variables variable_2="OB_a" variable_1="OB_a"/>
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<connection>
<map_components component_2="OPG" component_1="OB_p"/>
<map_variables variable_2="OB_p" variable_1="OB_p"/>
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<connection>
<map_components component_2="OPG" component_1="OB_a"/>
<map_variables variable_2="OB_a" variable_1="OB_a"/>
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<connection>
<map_components component_2="OPG" component_1="PTH"/>
<map_variables variable_2="pi_PTH_rep" variable_1="pi_PTH_rep"/>
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<connection>
<map_components component_2="RANKL" component_1="OPG"/>
<map_variables variable_2="OPG" variable_1="OPG"/>
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<connection>
<map_components component_2="RANKL" component_1="OB_p"/>
<map_variables variable_2="OB_p" variable_1="OB_p"/>
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<connection>
<map_components component_2="RANKL" component_1="OB_a"/>
<map_variables variable_2="OB_a" variable_1="OB_a"/>
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<connection>
<map_components component_2="RANKL" component_1="PTH"/>
<map_variables variable_2="pi_PTH_act" variable_1="pi_PTH_act"/>
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Model structure and control of bone remodeling: a theoretical study
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